Monetary Theory and PolicyProblem Solutions by pengxuebo

VIEWS: 2 PAGES: 109

									           Monetary Theory and Policy:
               Problem Solutions∗
                               Carl E. Walsh
                    University of California, Santa Cruz
                                 March 16, 1999



Contents
1 Chapter 2: Money in a General Equilibrium Framework                                2

2 Chapter 3: Money and Transactions                                                  8

3 Chapter 4: Money and Public Finance                                               20

4 Chapter 5: Money and Output in the Short Run                                      27

5 Chapter 6: Money and the Open Economy                                             43

6 Chapter 8: Discretionary Policy and Time Inconsistency                            59

7 Chapter 9: Monetary-Policy Operating Procedures                                   81

8 Chapter 10: Interest Rates and Monetary Policy                                    97

9 Typos                                                                           109
   ∗ Solutions c 1998 by Carl E. Walsh. Comments and corrections should be sent to wal-

shc@cats.ucsc.edu.




                                          1
1      Chapter 2: Money in a General Equilibrium
       Framework
     1. Calvo and Leiderman (1992): A commonly used specification of the de-
        mand for money, originally due to Cagen (1956), assumes m = Ae−αit
        where A and α are parameters and i is the nominal rate of interest. In
        the Sidrauski (1967) model, assume that utility is separable in consump-
        tion and real money balances: u(ct , mt ) = w(ct ) + v(mt ), and further
        assume that v(mt ) = mt (B − D ln mt ) where B and D are positive pa-
        rameters. Show that the demand for money is given by mt = Ae−αt it
        where A = e( D −1) and αt = w (ct )/D.
                      B




    The basic condition from which one can derive the demand for money in
Sidrauski’s money-in-the-utility function model is given by equation (2.23) on
page 57. This equation states that the ratio of the marginal utility of money to
the marginal utility of consumption depends on the nominal rate of interest:
                             um (ct , mt )     it
                                           =        ≈ it
                             uc (ct , mt )   1 + it
Notice that the expression has been simplified by employing the approximation
x/(1 + x) ≈ x for small x. Sidrauski developed his model in continuous time,
in which case the first order condition takes the exact form
                                  um (ct , mt )
                                                = it
                                  uc (ct , mt )
Using the proposed utility function, um = B − D − D ln mt and uc = w (ct ), so
this condition becomes
                 um   B − D − D ln mt   B/D − 1 − ln mt
                    =                 =                 = it
                 uc       w (ct )         w (ct )/D
Rearranging yields
                                       B        w (ct )it
                           ln mt = (     − 1) −
                                       D          D
or
                                                 w (ct )
                              mt = e( D −1) e−
                                        B
                                                   D     it




     2. Suppose u(ct , mt ) = ∞ β i [ln ct + mt e−γmt ], γ > 0 and β = 0.95. As-
                               i=0
        sume the production function is f (k) = k0.5 and δ = 0.02. What rate of
        inflation maximizes steady-state welfare? How do real money balances at
        the welfare maximizing inflation rate depend on γ?


                                            2
    The steady-state welfare maximizing nominal rate of interest is iss = 0 (see
section 2.3.1.2, pages 61-64) at which point um = 0. If R is the gross real rate
of interest (one plus the real rate of interest), 1 + i = R(1 + π) and the rate of
inflation that yields a zero nominal rate of inflation is
                                           1
                                  πss =      −1
                                           R
In the steady-state, R is equal to 1/β, or 1/R = β = 0.95 (see equation 2.19,
page 54). Hence, the optimal rate of inflation is 0.95 − 1 = −0.5 or a 5% rate
of deflation.
    To determine how money demand depends on the parameter γ, use the rep-
resentative agent’s first order condition (see equation 2.23, page 57), evaluated
at the steady-state nominal rate of interest:
                                  um     iss
                                     =
                                  uc   1 + iss
Given the form of the utility function, this becomes
                       um                           iss
                          = ct (1 − γmt ) e−γmt =
                       uc                         1 + iss
At the welfare maximizing inflation rate, iss = 0, which requires 1 − γmss = 0
or
                                                       1
                                    mss =
                                                       γ
Thus, real money demand is decreasing in γ.


  3. Assume that mt = Ae−αit where A and α are constants. Calculate the
     welfare cost of inflation in terms of A and α, expressed as a percentage of
     steady-state consumption (normalized to equal 1). Does the cost increase
     or decrease with α? Explain why.

    A traditional method for determining the welfare cost of inflation involves
calculating the area under the money demand function. That is, the loss in
consumer surplus when the interest rate is equal to i > 0 is given by
                                                   i
                            l(i, A, α) ≡               Ae−αx dx
                                               0

Evaluating this integral yields
                                           A
                            l(i, A, α) =     1 − e−αi                         (1)
                                           α



                                           3
as the welfare cost of an inflation rate of π = i − r if r is the real rate of return.
The effect of α on this cost is
  ∂l(i, A, α)   αA(ie−αi ) − A(1 − e−αi )          A
              =                           =               e−αi (1 + αi) − 1 ≤ 0
      ∂α                   α2                      α2
The sign depends on e−αi (1 + αi) − 1, but this is always negative ( e−x (1 + x)
is maximized when x = 0 at which point e−x (1 + x) = 1; it then declines with
x).
    From the specification of the money demand equation, the interest elasticity
of money demand is −αi, so money demand is more sensitive to the interest
rate the larger is α. As the nominal interest rate rises with an increase in infla-
tion, households respond by reducing their demand for money, thereby helping
to reduce the distortion generated by the inflation tax. The greater the interest
sensitive of money demand, the lower will be the welfare cost of the inflation
tax.

  4. Suppose a nominal interest rate of im is paid on money balances. These
     payments are financed by a combination of lump-sum taxes and printing
     money. Let a be the fraction financed by lump-sum taxes. The govern-
     ment’s budget identity is τ t + vt = im mt , with τ t = aim mt and vt = θmt .
     Using Sidrauski’s model,
       (a) Show that the ratio of the marginal utility of money to the marginal
           utility of consumption will equal r + π − im = i − im . Explain why.
       (b) Show how i−im is affected by the method used to finance the interest
           payments on money. Explain the economics behind your result.

   (a) The budget constraint in the basic Sidrauski model must be modified to
take into account the interest payments on money and that net transfers ( τ in
equation (2.12), page 52) consists of two components, the first being the lump-
sum transfer v and the second being the lump-sum tax τ . Thus, the budget
constraint becomes
                                  1 + im
         f(kt−1 ) + (1 − δ)kt−1 +        mt−1 − τ t + vt = ct + kt + mt
                                   1+π
where population growth has been ignored for simplicity. The value function for
the problem is still given by (2.13), but the first order conditions change because
of the change in the budget constraint. In particular, equation (2.15) becomes
                                                 β(1 + im )
             um − β [fk + 1 − δ] Vω (ω t+1 ) +              Vω (ω t+1 ) = 0          (2)
                                                   1+π
The first order condition for consumption (see 2.14), together with the envelope
condition (see 2.17) implies
     uc (ct , mt ) = β [fk + 1 − δ] Vω (ω t+1 ) = β [fk + 1 − δ] uc (ct+1 , mt+1 )
                   = βRuc (ct+1 , mt+1 )

                                          4
Using this result, equation (2) can be rearranged, resulting in
      um          β(1 + im ) uc (ct+1 , mt+1 )       β(1 + im )          1
            = 1−                               =1−
      uc            1+π         uc (ct , mt )           1+π             βR
                 β(1 + im )   R(1 + π) − (1 + im )    i − im
            = 1−            =                      =
                 R(1 + π)           R(1 + π)           1+i
The ratio of the marginal utility of money to consumption is set equal to the
opportunity cost of money. Since money now pays a nominal rate of interest
im , this opportunity cost is i − im , the difference between the nominal return on
capital and the nominal return on money.
    (b) From the government’s budget constraint, interest payments not financed
through lump-sum taxes must be financed by printing more money. Hence, v =
θm = (1 − a)im m, or the rate of money growth will equal θ = (1 − a)im . In
the steady-state, π = θ. This means that π = (1 − a)im . Hence, the opportunity
cost of money is given by

              i − im ≈ r + π − im = r + (1 − a)im − im = r − aim

where r = R − 1. Paying interest on money affects the opportunity cost of
money only if a > 0. Printing money to finance interest payments on money
only results in inflation; this raises the nominal interest rate i, thereby offsetting
the effect of paying interest.


  5. Assume u(c, m) = −c−a 1 + (m∗ − m)2 , a > 0. Normalize so that the
     steady-state value of consumption is equal to 1 (css = 1). Using equation
     (2.23) of the text, show that there exist two steady state equilibrium values
     for real money balances if aI ss < 1. (Recall that I = i/(1 + i) where i is
     the nominal rate of interest.)

   From equation (2.23),
                                um (css , mss )
                                                = I ss
                                uc (css , mss )
Using the utility function specified in the question,

                            um (c, m) = 2c−a (m∗ − m)

and

                      uc (c, m) = ac−(1+a) 1 + (m∗ − m)2

Therefore
               um (css , mss )         2(css )−a (m∗ − mss )
                               =                                   =I           (3)
               uc (css , mss )   a(css )−(1+a) [1 + (m∗ − mss )2 ]



                                          5
We now need to solve this equation for mss . Let x ≡ (m∗ −mss ). Then equation
(3) becomes

                       2(css )−a x = aI(css )−(1+a) 1 + x2

If css is normalized to equal 1, this becomes

                                   2x = aI 1 + x2

which can be rewritten more explicitly as a quadratic in x:
                                            2
                                   x2 −       x+1=0
                                           aI
From the quadratic formula,
                          2          4
                         aI   ±    (aI)2   −4         1         1
                   x=                            =      ±           −1
                                   2                 aI       (aI)2

There will be two real solutions if and only if
                                         1
                                             −1>0
                                       (aI)2
which holds for

                                           aI < 1

If this condition is satisfied, both solutions for x are positive (so that mss < m∗ ).
This can be verified by noting that aI < 1 implies

                                   1         1
                                     −           −1>0
                                  aI       (aI)2



  6. In Sidrauski’s money-in-the-utility-function model augmented to include
     variable labor supply, money is superneutral if the representative agent’s
     preferences are given by

                        β i u(ct+i , mt+i , lt+i ) =        β i (ct+i mt+i )b lt+i
                                                                               d


      but not if they are given by

                     β i u(ct+i , mt+i , lt+i ) =      β i (ct+i + kmt+i )b lt+i
                                                                             d


      Discuss. (Assume output depends on capital and labor and the aggregate
      production function is Cobb-Douglas.)


                                             6
   The steady-state values of css , kss , lss , yss must satisfy the following four
equations (see pages 65-67):
                               ul
                                  = −fl (kss , 1 − lss )                           (4)
                               uc

                                                    1
                            fk (kss , 1 − lss ) =     −1+δ                         (5)
                                                    β


                            css = f(kss , 1 − lss ) − δkss                         (6)


                                y ss = f(kss , 1 − lss )                           (7)

If the single period utility function is of the form (ct+i mt+i )b lt+i , then
                                                                    d


                               ul  dcb mb ld−1 dc
                                  = b−1 b d =
                               uc  bc m l      bl
is independent of m and equations (4) - (7) involve only the four unknowns css ,
kss , lss , y ss . These can be solved for css , kss , lss , and yss independently of m
or inflation. Superneutrality holds.
    If the utility function is (ct+i + kmt+i )b lt+i , then
                                                   d


                        ul   d(c + km)b ld−1   d(c + km)
                           =                 =
                        uc   b(c + km)b−1 ld       al

which is not independent of m. Thus, equations (4) - (7) will involve 5 un-
knowns ( css , kss , lss , y ss , and mss ) and cannot be solved independently of the
money demand condition and inflation. Superneutrality does not hold.


   7. Suppose the representative agent does not treat τ t as a lump sum transfer,
      but instead assumes her transfer will be proportional to her own holdings
      of money (since in equilibrium, τ is proportional to m). Solve for the
      agent’s demand for money. What is the welfare cost of inflation?

    This question is basically the same as Question 4. If the transfer is viewed by
the individual as proportional to her own money holdings, then this is equivalent
to the individual viewing money as paying a nominal rate of interest. If this is
financed via lump-sum taxes, changes in inflation do not change the opportu-
nity cost of holding money — a rise in inflation that depreciates the individual’s
money holdings is offset by the increase in the transfer the individual anticipates
receiving.




                                            7
2     Chapter 3: Money and Transactions
    1. Suppose the production function for shopping takes the form ψ = c =
       ex (ns )a mb , where a and b are both positive but less than 1 and x is a
                                                                        1−Φ
                                                                            l1−η
                                                                      1−Φ + 1−η
       productivity factor. The agent’s utility is given by v(c, l) = c
       where l = 1 − n − ns and n is time spend in market employment.

       (a) Derive the transaction time function g(c, m) = ns .
       (b) Derive the money in the utility function specification implied by
           the shopping production function. How does the marginal utility
           of money depend on the parameters a and b? How does it depend on
           x?
        (c) Is the marginal utility of consumption increasing or decreasing in m?

    (a) From the shopping production function,

                                                 c      1/a
                           g(c, m) = ns =                                       (8)
                                               ex mb
    (b) From the definition of the agent’s utility,

                        c1−Φ   (1 − n − ns )1−η
           v(c, l) =         +
                        1−Φ         1−η
                                                       1/a 1−η
                        c1−Φ   1 − n − exc b
                                         m
                   =         +                                   ≡ u(c, n, m)
                        1−Φ           1−η
The marginal utility of money is

                  ∂u(c, n, m)       ∂v(c, l)    ∂l
                                =
                     ∂m               ∂l        ∂m
                                                           −η    bns
                                                     1/a
                                =    1 − n − (ns )                              (9)
                                                                 am

The time spend shopping can be written as c1/a e−x/a m−b/a . The marginal pro-
ductivity of money in reducing shopping time is given by (b/a)(ns /m), so an
increase in b/a increases the effect additional money holdings have in reducing
the time needed for shopping. Additional money holdings result in more leisure
(and more utility) when b/a is large, thus acting to increase the marginal utility
                                       ∂l
of money. In terms of equation (9), ∂m rises with b/a. But the marginal utility
of leisure a decreasing function of total leisure, so ∂v(c,l) declines.
                                                        ∂l
    The effect of x on the marginal utility of money, for given c and n, operates
through ns and represents a productivity shift; an increase in x reduces the time
needed for shopping for given values of c and m. This affects the productivity
of m in the shopping time production function. The marginal product of money
in reducing shopping time is (b/a)c1/a e−x/a m−(1+b/a) . This is decreasing in x;

                                        8
a higher x decreases the marginal effect of m in reducing shopping time, so
money is less “productive.”
    (c) An increase in consumption affects utility in two ways. First, consump-
tion directly yields utility; vc > 0. This represents the effect of consumption
on utility, holding leisure constant. Since leisure is being held constant, vc is
independent of m. Second, higher consumption increases the time devoted to
shopping, as this reduces the time available for leisure. This effect will depend
on the level of money holding. From (8), consumption and money are comple-
ments in producing shopping time, and higher money holdings reduce the effect
of higher c on ns . This means that with higher money holdings, an increase in
c has less of an effect in reducing leisure time and will therefore lead a rise in
consumption to have a larger overall positive effect on utility.


  2. Define superneutrality. Carefully explain whether the Cooley-Hansen cash-
     in-advance model exhibits superneutrality. What role does the cash-in-
     advance constraint play in determining whether superneutrality holds?

    A model exhibits the property of superneutrality if the real equilibrium (out-
put, consumption, capital, etc.) is independent of the rate of nominal money
growth. Superneutrality normally is interpreted to refer to the steady-state equi-
librium of a model. As demonstrated in Chapter 2, the Sidrauski model dis-
plays superneutrality with respect to the steady-state, but changes in the in-
flation rate will generally affect the short-run equilibrium. If the ratio of the
marginal utilities of leisure and consumption is independent of money holdings,
then Sidrauski’s model is superneutral in the short-run also (see pages 65-67).
    The Cooley-Hansen model does not display superneutrality. Different rates
of inflation affect the opportunity cost of holding money. Through the cash-
in-advance constraint, inflation affects the marginal cost of consumption since
consumption is treated as a cash good. Higher inflation induces a substitution
away from cash goods and towards credit goods. In Cooley and Hansen’s model,
leisure is a credit good; cash is not needed to purchase leisure. As a result,
changes in the steady-state rate of inflation alter the demand for leisure and the
supply of labor. This was shown in equation (3.29) on page 110, where Θ was
equal in the steady-state to one plus the inflation rate.


  3. Is the steady-state equilibrium in the Cooley-Hansen cash-in-advance model
     affected by any of the following modifications? Explain.

      (a) Labor is supplied inelastically (normalize so that n = 1, where n is
          the supply of labor).
      (b) Purchases of capital are also subject to the cash-in-advance constraint
          (i.e. one needs money to purchase both consumption and investment
          goods);


                                        9
       (c) The growth rate of money follows the process ut = γut−1 + ϕt where
           0 < γ < 1 and ϕ is a mean zero i.i.d. process.

    (a) Following on the previous problem, one important modification when la-
bor is supplied inelastically is that Cooley and Hansen’s model will now display
superneutrality. Without a labor-leisure choice, the model becomes essentially
the model of section 3.3.1.
    (b) Referring to the model of section 3.6.1, the cash-in-advance constraint
would become

                   Pt ct + Pt [kt − (1 − δ)kt−1 ] ≤ Mt−1 + Tt−1

where kt − (1 − δ)kt−1 is equal to net purchases of capital. Dividing by Pt , this
becomes
                                     mt−1
                           ct + it ≤       + τ t ≡ at                        (10)
                                       Πt
where it is net investment ( kt − (1 − δ)kt−1 ).
   The value function for this problem is

              V (at , kt−1 ) = max {u(ct , 1 − nt ) + βEt V (at+1 , kt )}

where at+1 = Πt+1 + τ t+1 , kt = f(kt−1 , nt ) + (1 − δ)kt−1 + at − ct − mt and
                mt

the maximization is subject to the cash in advance constraint (10). Let λ be
the Lagrangian multiplier associated with the budget constraint and let      be
the Lagrangian multiplier associated with the cash-in-advance constraint. If we
assume a standard Cobb-Douglas production function ( yt = ezt kt−1 n1−α ), then
                                                                  α
                                                                      t
the budget constraint is

                 ezt kt−1 n1−α + (1 − δ)kt−1 + at ≥ ct + kt + mt
                      α
                           t

and the first order conditions for ct , kt , mt , and nt , together with the envelope
conditions, are

                            uc (ct , 1 − nt ) − λt −    t   =0                 (11)

                          βEt Vk (at+1 , kt ) − λt −        t   =0             (12)

                                 1
                        βEt            Va (at+1 , kt ) − λt = 0                (13)
                               Πt+1

                                               yt
              −un (ct , 1 − nt ) + (1 − α)          βEt Vk (at+1 , kt ) = 0    (14)
                                               nt

                               Va (at , kt−1 ) = λt +       t                  (15)


                                          10
                                              yt
                  Vk (at , kt−1 ) = λt α          +1−δ +                t (1   − δ)         (16)
                                             kt−1
The Lagrangian       appears in this last condition because higher capital at the
start of the period reduces the cash needed to achieve a given value of kt ; only
net purchases ( kt − (1 − δ)kt−1 ) are subject to the cash-in-advance constraint.
    Since (15) implies Et Va (at+1 , kt ) = Et λt+1 + t+1 , equations (11) - (16)
can be used to derive the following conditions, which should be compared to
equations (3.51), (3.52), and the two equations following (3.52) on page 126:

                                   uc (ct , 1 − nt ) = λt +        t                        (17)

                                          λt+1 + t+1
                                  βEt                           = λt                        (18)
                                             Πt+1

                                                                       yt
                  −un (ct , 1 − nt ) + (λt +           t ) (1   − α)           =0           (19)
                                                                       nt

           λt +     t   = βEt Vk (at+1 , kt ) = βEt Rt λt+1 +               t+1 (1   − δ)   (20)

where Rt = α yt+1 + 1 − δ. The first two equations are identical to (3.51) and
                kt
(3.52). The next two differ. According to (??), the marginal utility of leisure is
set equal to the utility value of the marginal product of labor, but now account
must be taken of the fact that any additional income requires cash to be spent.
That is why the marginal product of labor is multiplied by λt + t and not just
λt . According the (20), the value of an additional purchase of capital (which
costs λt + t ) is the additional future return (the Rt λt+1 term) and the value of
relaxing the future cash-in-advance constraint that comes from reducing future
net purchases (the t+1 (1 − δ) term).
    Turning to an analysis of the steady-state, (18) implies that
                                                   Πss
                                     ss
                                          = λss        −1
                                                    β
which implies Πss ≥ β will be requires for the existence of a steady-state since
  must be nonnegative. Now eliminate from the steady-state version of (20):
                        Πss                                     Πss
              λss              = β Rss λss + λss                    − 1 (1 − δ)
                         β                                       β
                                                                                             ss
or, recalling that Rss −1+δ is the steady-state marginal product of capital α kss ,
                                                                              y


                                  yss       Πss        1
                              α       =                  −1+δ
                                  kss        β         β
which depends on the rate of inflation. Thus, superneutrality does not hold when
capital purchases are also subject to the cash-in-advance constraint. Notice that

                                                  11
this conclusion would hold even if labor is supplied inelastically as in part (a)
of this question (see Problem 5 below). By imposing a tax on capital purchases,
inflation affects the steady-state capital stock and kss is decreasing in Πss .
    For a complete discussion of the implications of making the cash-in-advance
constraint apply to both consumption and capital or only to consumption, see
Abel (1985).
    (c) The steady-state depends on the average rate of money growth since that
pins down average inflation. It does not depend on the transitory dynamics of
the monetary supply process, although the short-run dynamics will.

  4. Money-in-the-utility-function and cash-in-advance constraints are alterna-
     tive means for constructing models in which money has positive value in
     equilibrium.
      (a) What strengths and weaknesses do you see with each of these ap-
          proaches?
      (b) Suppose you wanted to study the effects of the growth of credit cards
          on money demand. Which approach would you adopt? Why?

    Both the money-in-the-utility function approach and the cash-in-advance ap-
proach are best viewed as convenient short-cuts for generating a role for money.
If we believe that the major role money plays is to facilitate transactions, then
in some ways the CIA approach has an advantage in making this transactions
role more explicit. It forces one to think more about the exact nature of the
transactions technology and the timing of payments (e.g., can current period in-
come be used to purchase current period consumption?). On the other hand, the
rather rigid restrictions the CIA typically places on transactions are certainly
unattractive. In modern economies we normally have multiple means that can be
used to facilitate the transactions we undertake. Also, the generally exogenous
distinction between cash and credit goods is troublesome, since most things are
a bit of both.
    The MIU approach can be viewed as being based on some specification of
a shopping time model, and the notion of a production function for shopping
time allows for less rigid substitution between different means of carrying out
transactions. The example in the text emphasized the use of time or money
for transactions, but one could allow a variety of means of payment to enter
the production function as imperfect substitutes. Of course that treats the de-
gree of substitution as exogenous, which is also unsatisfactory. We would really
like a model that accounts for why certain means of payment are used in some
circumstances and others in different circumstances.
    By emphasizing the link between transactions and money demand, the CIA
approach probably provides the more natural starting point for an analysis of
credit card usage. For an interesting recent analysis, see D.L. Brito and P.R.
Hartley, “Consumer Rationality and Credit Cards,” Journal of Political Economy,
103 (2), April 1995, 400-433.



                                       12
  5. Consider the model of Section 3.3.1. Suppose that money is required to
     purchase both consumption and investment goods. The cash-in-advance
     constraint then becomes ct + xt ≤ mt−1 /Πt + τ t where x is investment.
     Assume the aggregate production function takes the form yt = ezt kt n1−α .
                                                                        α
                                                                          t
     Show that the steady-state capital-labor ratio is affected by the rate of
     inflation. Does a rise in inflation raise or lower the steady-state capital-
     labor ratio. Explain.

    Most of this problem is already worked out as part of the solution to Problem
3.b. The model of Section 3.3.1 assumed utility depended only on consumption,
so there was no labor-leisure choice. Otherwise, the set-up is similar to Problem
3.b, so the equations defining the steady-state are, from (17) - (20),

                                   uc (css ) = λss +       ss
                                                                             (21)


                                         λss +    ss
                                  β                     = λss                (22)
                                            Πss


                        λss +      ss
                                        = β [Rss λss +     ss
                                                                (1 − δ)]     (23)

Equation (19) has been dropped since there is no labor supply decision, and
utility in (21) depends only on consumption. From (22),

                                                  Πss
                                    ss
                                         = λss        −1
                                                   β

so (23) becomes

                        Πss                           Πss
                                 = β Rss +                − 1 (1 − δ)
                         β                             β

For convenience, normalize n to 1. Then Rss + 1 − δ ≡ αy ss /kss + 1 − δ =
α(kss )α−1 + 1 − δ, and equation (23) implies

                                              Πss        1
                             α(kss )α−1 =                  −1+δ
                                               β         β

Hence, the steady-state capital-labor ratio is
                                                                       −1
                                   1        Πss        1              1−α
                    k   ss
                             =                           −1+δ
                                   α         β         β

which is decreasing in the inflation rate ( Πss ). Higher inflation implies a higher
tax on capital purchases and this lowers the steady-state stock of capital.



                                                 13
  6. Consider the following model:
                                          ∞
                       Preferences: Et         β i [ln ct+i + b ln dt+i ]
                                         i=0


                                                                     mt−1
     Budget constraint: ct + dt + mt + kt = Akt−1 + τ t +
                                              a
                                                                            + (1 − δ)kt−1
                                                                     1 + πt
                                                                                    (24)

                                                                     mt−1
                    Cash-in-advance constraint: ct ≤ τ t +                          (25)
                                                                     1 + πt
     where m denotes real money balances and πt is the inflation rate from
     period t − 1 to period t. The two consumption goods, c and d, represent
     cash (c) and credit (d) goods. The net transfer τ is viewed as a lump-sum
     payment (or tax) by the household.

      (a) Does this model exhibit superneutrality? Explain.
      (b) What is the rate of inflation that maximizes steady-state utility?

   (a) The model exhibits superneutrality if the real variables k, c, and d are
independent of π in the steady-state. If we define at ≡ τ t + mt−1 , the value
                                                                1+π t
function can be defined as

V (at , kt−1 ) = max {ln ct+i + b ln dt+i
                                       mt
                 +βEt V τ t+1 +             , Akt−1 + (1 − δ)kt−1 + at − ct − dt − mt
                                                a
                                   1 + πt+1
where the maximization is subject to

                                      ct ≤ at

Let denote the Lagrangian multiplier associated with this cash-in-advance con-
straint. From the first order conditions for the agent’s decision problem,
                         1
                            − βEt Vk (at+1 , kt ) −     t   =0                      (26)
                         ct

                            b
                               − βEt Vk (at+1 , kt ) = 0                            (27)
                            dt

                   1
                         βEt Va (at+1 , kt ) − βEt Vk (at+1 , kt ) = 0              (28)
                1 + πt+1

                      Va (at , kt−1 ) = βEt Vk (at+1 , kt ) +    t                  (29)


                                         14
                Vk (at , kt−1 ) = β aAkt−1 + 1 − δ Et Vk (at+1 , kt )
                                       a−1
                                                                                      (30)
plus the two constraints (24) and (25). Equation (30) implies that, in the steady-
state,
                                                                                 −1
                                                           1           1        1−α
    1=β      aA(k )
                  ss a−1
                           +1−δ        ⇒k   ss
                                                 =                       −1+δ         (31)
                                                          αA           β
This means that the steady-state capital stock is independent of the inflation
rate.
    Let λt ≡ βEt Vk (at+1 , kt ). From (26) and (27),
                                  dt
                                     = 1+ t               b                           (32)
                                  ct     λt
Equations (28) and (29) imply
                                      λt+1 + t+1
                             βEt                 = λt
                                       1 + πt+1
In the steady-state, this implies
                       λss +     ss               ss
                                                                  1 + π ss
                                      = 1+                =
                          λss                    λss                 β
and combining this with (32),
                                 dss        1 + π ss
                                     =                        b                       (33)
                                 css           β
so the relative consumption of c and d depends on the rate of inflation. The
real equilibrium does not display superneutrality.
    (b) From the steady-state marginal product of capital condition (31), we have
the standard result that the real rate of return will equal 1/β. Letting R ≡ 1/β,
equation (33) can be written as
                                 dss = (1 + iss )bcss
where i = R(1 + π). Letting Z = A(kss )a − δkss , in the steady-state we have
from the budget constraint (24) Z = css + dss = css + (1 + iss )bcss or css = Z
where    = [1 + (1 + iss )b]−1 . Hence, steady-state utility of the representative
agent can be expressed as
    1                                  1
       [ln css + b ln dss ] =               [ln css + b ln(1 + iss )bcss ]
   1−β                                1−β
                                       1
                             =              [ln Z + b ln(1 + iss )b Z]
                                      1−β
                                       1
                             =              [(1 + b) ln Z + b ln(1 + iss )b]
                                      1−β
                                      1+b                1+b         b
                             =              ln       +       ln Z +     ln(1 + iss )b
                                      1−β                1−β        1−β

                                            15
Now maximize this with respect to the nominal rate of interest i. Since Z was
shown earlier to be independent of the inflation rate, the first order condition
is
                     1+b          b                     1      b
                 −                                +                 =0
                     1−β    1 + (1 + iss )b           1 − β 1 + iss
or
                                 1          1+b
                                      =
                              1 + iss   1 + (1 + iss )b
which implies
                               1 + (1 + iss )b
                                               =1+b
                                  1 + iss
This holds if and only if

                                        iss = 0

So the optimal rate of inflation will be the rate that yields a zero nominal rate
of interest.


     7. Consider the following model:
                                              ∞
                        Preferences: Et           β i (ln ct+i + ln dt+i )
                                          i=0


                                                                       mt−1
       Budget constraint: ct + dt + mt + kt = Akt−1 + τ t +
                                                a
                                                                              + (1 − δ)kt−1
                                                                       1 + πt


       where m denotes real money balances and πt is the inflation rate from
       period t − 1 to period t. Utility depends on the consumption of two types
       of good; c must be purchased with cash, while d can be purchased using
       either cash or credit. The net transfer τ is viewed as a lump-sum payment
       (or tax) by the household. If a fraction q of d is purchased using cash,
       then the household also faces a cash-in-advance constraint that takes the
       form
                                             mt−1
                                 ct + qdt ≤          + τt
                                             1 + πt
       What is the relationship between the nominal rate of interest and whether
       the cash-in-advance constraint is binding? Explain. Will the household
       ever use cash to purchase d (i.e. will the optimal q ever be greater than
       zero)?


                                          16
    The basic model is similar to the one studied in Problem 6, differing only in
the utility function and the cash-in-advance constraint. The value function is
V (at , kt−1 ) = max {ln ct+i + ln dt+i
                                      mt
                 +βEt V τ t+1 +             , Akt−1 + (1 − δ)kt−1 + at − ct − dt − mt
                                                a
                                   1 + πt+1
where at = τ t + mt−1 and ct + qdt ≤ at . and we require that 0 ≤ q ≤ 1
                    1+π t
since q is the fraction of the d good purchased with cash. Actually, the relevant
consideration is whether q is positive or not. Let θ denote the Lagrangian on
the constraint q ≥ 0. The first order conditions for the household’s decision
problem for the current are simply stated here as, modifying them to reflect the
different utility function and cash-in-advance constraint:
                         1
                            − βEt Vk (at+1 , kt ) −     t   =0
                         ct

                         1
                            − βEt Vk (at+1 , kt ) − q   t   =0
                         dt

                   1
                         βEt Va (at+1 , kt ) − βEt Vk (at+1 , kt ) = 0
                1 + πt+1

                      Va (at , kt−1 ) = βEt Vk (at+1 , kt ) +    t


               Vk (at , kt−1 ) = β aAkt−1 + 1 − δ Et Vk (at+1 , kt )
                                      a−1


In addition, we need the first order condition for the optimal choice of q. This
takes the form
                            − t dt + θ t ≤ 0 qt θ t = 0
where the condition qt θt = 0 is the complementary slackness condition associated
with the inequality constraint on q. Since q cannot be reduced below zero, the
optimum can have − t dt +θt < 0 at q = 0; utility could be increased by reducing
q even further, but the non-negativity constraint binds. As long as the nominal
rate of interest is positive, > 0, and d > 0. this implies that θ > 0 from
which the condition qθ = 0 implies that q = 0. So, as long as the nominal rate
of interest is positive, the household will never use cash to purchase d.

  8. Trejos and Wright (1993) find that if no search is allowed while bargaining
     takes place, output tends to be too low (the marginal utility of output
     exceeds the marginal production costs). Show that output is also too low
     in a basic cash-in-advance model. (For simplicity, assume only labor is
     needed to produce output according to the production function y = n.)
     Does the same hold true in a money-in-the-utility-function model?


                                         17
    In a basic cash-in-advance model, inflation taxes cash goods. Suppose the
nominal rate of interest is positive; relative to the case of a zero nominal interest
rate, households will be consuming fewer cash goods (which bear the inflation
tax) and more credit goods. Since leisure is a credit good, inflation will tend to
lower output by increasing the demand for leisure and reducing labor supply. For
example, equation (3.29) on page 110 shows how inflation reduces labor supply
relative to the case of a zero nominal rate of interest.
    If we modify the model of Section 3.3.2.1 by ignoring capital, and assume the
production function is y = n, then the value function for the decision problem
of the household becomes (see section 3.6 of the Chapter Appendix):

                                                      nt + at − ct
           V (at ) = max u(ct , 1 − nt ) + βV                      + τ t+1
                                                         Πt+1

where at = τ t + mt−1 /Πt , and the maximization is subjective to the cash-in-
advance constraint ct ≤ at . If t is the Lagrangian multiplier associated with
the cash-in-advance constraint, then the first order necessary conditions are

                                        βV (nt + at − ct − mt )
                  uc (ct , 1 − nt ) −                           −             =0
                                                Πt+1                      t



                                           βV (nt + at − ct − mt )
                    −ul (ct , 1 − nt ) +                           =0
                                                   Πt+1


                                     βV (nt + at − ct − mt )
                        V (at ) =                            +
                                             Πt+1                     t


           βV (nt +at −ct −mt )
Let λt ≡          Πt+1          .   Then these first order conditions imply

                                                                 −1
                     ul (ct , 1 − nt )      λt
                                       =             = 1+    t
                                                                      ≤1
                     uc (ct , 1 − nt )   λt +    t          λt

As long as the cash-in-advance constraint is binding, > 0 and ul /uc is greater
than it would be in the case in which = 0. Since ul /uc is increasing in labor
supply, labor supply and output is reduced relative to the     = 0 case. In this
framework, the marginal cost of output is ul since this is the utility cost of
supplying additional labor. The marginal utility of the output that is produced
is uc . Since ul < uc when the cash-in-advance constraint binds, the marginal
utility of output exceeds the marginal cost of production.
    In a basic money-in-the-utility-function model, the relevant condition was
given by equation (2.34) on page 66. The marginal utility cost of supplying
more labor ul is just equal to the marginal utility of consumption times that
additional output produced fn uc . So the marginal cost of production and the
marginal utility of output are equal. This doesn’t mean money and inflation
don’t affect output. A positive nominal interest rate reduces real money holdings

                                              18
relative to the social optimum. How that affects labor supply (and output) will
depend on how a decrease in m affects ul /uc and the effect could go either way.
For the utility function used in the linear version of the money-in-the-utility-
function model of Chapter 2, equation (2.45) shows that a lower value of m will,
for given c and y, act to increases labor supply for Φ < 1 and decrease labor
supply for Φ > 1. Thus, if, for example, Φ < 1, consumption and money are
complements; an increase in m increases the marginal utility of consumption.
Higher inflation that reduces m also leads to a fall in the marginal utility of
consumption. Households will shift towards consuming more leisure and fewer
consumption goods. The decline in labor supply as more leisure is consumed will
lower output.




                                      19
3     Chapter 4: Money and Public Finance
    1. Consider the version of the Sidrauski (1967) model studied in Problem
       1 of Chapter 2. Utility was given by u(ct , mt ) = w(ct ) + v(mt ), with
       w(ct ) = ln ct and v(mt ) = mt (B − D ln mt ) where B and D are positive
       parameters. Steady-state revenue from seigniorage is given by θm, where
       θ is the growth rate of the money supply.

       (a) Is there a “Laffer Curve” for seigniorage (i.e. are revenues increasing
           in θ for all θ ≤ θ∗ and decreasing in θ for all θ > θ ∗ for some θ∗ ?
       (b) What rate of money growth maximizes steady-state revenues from
           seigniorage?
       (c) Assume now that the economy’s rate of population growth is n and
           reinterpret m as real money balances per capita. What rate of infla-
           tion maximizes seigniorage? How does it depend on n?

    (a) From Problem Set 2, we know that the demand for money in this model
is given by mt = Ae−i/ct D where ln A = D − 1. Hence, seigniorage in the
                                           B

steady-state is equal to

                            sss = πAe−(r         +π)/css D
                                            ss




Taking the derivative with respect to π,
∂sss                    π                                    π
     = Ae−(r +θ)/c D − ss Ae−(r +π)/c D = Ae−(r +π)/c D 1 − ss
            ss    ss           ss    ss        ss    ss

 ∂π                   c D                                  c D


This is positive (i.e. seigniorage is increasing in inflation) for π < css D, and
negative for π > css D. Hence, there is a Laffer curve.
    (b) Steady-state seigniorage is maximized for π = θ = css D.
    (c) With population growth at the rate γ, the growth rate of per capital money
balances is given by

                                    θ−π−γ

Hence, in the steady-state, π = θ − γ. Steady-state seigniorage will be still be
maximized at an inflation rate of css D, but this now corresponds to a rate of
money growth of css D + γ.


    2. Suppose that government faces the following budget identity:

                             bt = Rbt−1 + gt − τ t yt − st

      where the terms are one period debt, gross interest payments, govern-
      ment purchases, income tax receipts and seigniorage. Assume seigniorage

                                       20
      is given by f (πt ) where π is the rate of inflation. The interest factor
                                                        ∞
      R is constant and the expenditure process {gt+i }i=0 is exogenous. The
      government sets time paths for the income tax rate and for inflation to
      minimize
                                     ∞
                                Et         β i [h(τ t+i ) + k(πt+i )]
                                     i=0

      where the functions h and k represent the distortionary costs of the two
      tax sources. Assume the functions h and k imply positive and increasing
      marginal costs of both revenue sources.

       (a) What is the intratemporal optimality condition linking the choices of
           τ and π at each point in time?
       (b) What is the intratemporal optimality condition linking the choice π
           at different points in time?
       (c) Suppose y = 1, f (π) = aπ, h(τ ) = bτ 2 and k(π) = cπ2 . Evaluate the
           inter- and intratemporal conditions. Find the optimal settings for τ t
           and π t in terms of bt−1 and    R−i gt+i .
       (d) Using your results from part (c), when will optimal financing imply
           constant planned tax rates and inflation over time?

   (a) Solving the budget constraint forward, the government’s decision problem
can be written as min Et ∞ β i (h(τ t+i ) + k(πt+i )) subject to
                            i=0

          Rbt−1 + Et      R−i gt+i − Et             R−i [τ t+i yt+i + f (π t+i )] = 0

Let λ be the Lagrangian multiplier associated with this constraint. The first
order conditions are

                          Et β i h (τ t+i ) − λR−i yt+i = 0

and

                        Et β i k (πt+i ) − λR−i f (π t+i ) = 0

Hence, the condition linking taxes and seigniorage at each date t + i take the
form
                        Et h (τ t+i )     λ     Et k (πt+i )
                                      =       =
                         Et yt+i        (βR)i   Et f (πt+i )
    (b) The first order conditions for seigniorage at dates t + i and t + j take
the form β i Et k (st+i ) = λR−i Et f (πt+i ) and β j Et k (st+j ) = λR−j Et f (πt+j ),
or
                               k (πt+i )                 k (πt+j )
                    Et (βR)i              = λ = Et (βR)j
                               f (π t+i )                f (πt+j )


                                               21
   (c) Given the assumed functional forms, the first order conditions become

                                          λ      c
                        2bEt τ t+i =          =2   Et πt+i
                                        (βR)i    a

which implies that Et τ t+i =    c
                                 ab    Et π t+i for all i. The intertemporal condition
becomes
      c                   c
  2     Et πt+i (βR)i = 2   Et π t+j (βR)j or for j = 0, Et πt+i = (βR)−i πt
      a                   a


These results imply Et τ t+i = ab (βR)−i πt .
                               c

   Now we can evaluate the government’s budget constraint recalling at y = 1):

       Rbt−1 + Et       R−i gt+i      = Et      R−i (τ t+i yt+i + f (πt+i ))
                                                   c
                                      =       R−i ( (βR)−i πt + a(βR)−i πt )
                                                   ab
                                           c
                                      =       + a πt       (βR2 )−i
                                           ab
This implies that
                         c         −1
                 πt =       +a          B Rbt−1 + Et       R−i gt+i
                         ab
where B = βR2 /(βR2 − 1).
   Finally, the optimal tax rate is given by
               c       c              c       −1
        τt =      πt =                   +a        B Rbt−1 + Et       R−i gt+i
               ab      ab             ab
   (d) Et πt+i = (βR)−i πt = πt if and only if βR = 1 (i.e. R = 1/β).


  3. Mankiw (1987) suggested that the nominal interest rate should evolve as a
     random walk under an optimal tax policy. Suppose the real rate of interest
     is constant and that the equilibrium price level is given by equation (4.24).
     Suppose the nominal money supply is given by mt = mp + vt where mp is
                                                               t              t
     the central bank’s planned money supply and vt is a white noise control
     error. Let θ be the optimal rate of inflation. There are different processes
     for mp that lead to the same average inflation rate but different time series
     behavior of the nominal interest rate. For each of the processes for mp     t
     given in a and b, show that average inflation is θ; also show whether the
     nominal interest rate is a random walk.

       (a) mp = θ(1 − γ)t + γmt−1 ;
            t
      (b) mp = mt−1 + θ.
           t




                                           22
   Equation (4.24) states that
                                       mt   αEt pt+1
                               pt =       +                                 (34)
                                      1+α    1+α
For the money process in part (a), this becomes
                           θ(1 − γ)t + γmt−1 + vt αEt pt+1
                    pt =                         +                          (35)
                                    1+α            1+α
and the no-bubbles solution is of the form

                            pt = p0 + at + bmt−1 + cvt

where a, b, and c are coefficients to be determined. This solution implies

  Et pt+1 = p0 + a(t + 1) + bmt = p0 + a(t + 1) + b [θ(1 − γ)t + γmt−1 + vt ]


Using this and the trial solution in equation (35) yields
                             θ(1 − γ)t + γmt−1 + vt
 p0 + at + bmt−1 + cvt     =
                                      1+α
                               α [p0 + a(t + 1) + b (θ(1 − γ)t + γmt−1 + vt )]
                             +
                                                    1+α
                               θ(1 − γ) + αa + αbθ(1 − γ)         α(p0 + a)
                           =                                  t+
                                           1+α                      1+α
                                      γ(1 + αb)              1 + αb
                                  +               mt−1 +             vt
                                        1+α                   1+α
This will hold for all realizations of vt and mt−1 if
                                   α(p0 + a)
                            p0 =             ⇒ p0 = αa
                                     1+α


                θ(1 − γ) + αa + αbθ(1 − γ)
          a=                                     ⇒ a = θ(1 − γ)(1 + αb)
                           1+α

                            γ(1 + αb)          γ
                      b=              ⇒b=
                              1+α         1 + α(1 − γ)

                                             1
                                   c=
                                        1 + α(1 − γ)
Substituting for b in the expression for a,
                                  αγ                           1+α
         a = θ(1 − γ) 1 +                      = θ(1 − γ)
                             1 + α(1 − γ)                   1 + α(1 − γ)


                                          23
   Hence, the equilibrium price level evolves according to
                          1+α                           1
  pt = θ(1 − γ)                        (α + t) +                (γmt−1 + vt )
                       1 + α(1 − γ)                1 + α(1 − γ)
Average inflation will equal
                                       1+α                      γ mt−1
                  pt     = θ(1 − γ)                         +
                                    1 + α(1 − γ)              1 + α(1 − γ)
                                       1+α                         γθ
                         = θ(1 − γ)                         +
                                    1 + α(1 − γ)              1 + α(1 − γ)
                         = θ
   Expected inflation is equal to
                              1+α                              1
Et pt+1 − pt    = θ(1 − γ)                         +                     (γmt − γmt−1 − vt )
                           1 + α(1 − γ)                   1 + α(1 − γ)
                              1+α                              1
                = θ(1 − γ)                         +                     (γ mt − vt )
                           1 + α(1 − γ)                   1 + α(1 − γ)
With a constant real rate of interest, as was assumed in deriving equation (4.24),
the nominal rate of interest will equal
                                   1+α                      1
    it   = r0 + θ(1 − γ)                           +                     (γ mt − vt )
                                1 + α(1 − γ)           1 + α(1 − γ)
                            1
         = i0 +                     (γ mt − vt )
                       1 + α(1 − γ)
Since mt = θ(1−γ)+γ mt−1 +vt , mt is a first order autoregressive process
and (1 − γL) mt = θ(1 − γ) + vt . So the nominal interest rate is of the form
                                      it = i0 + zt − vt
where zt AR(1) and vt is white noise. Quasi-first differencing,
               (1 − γL)it    = (1 − γ)i0 + (1 − γL)(zt − vt )
                             = (1 − γ)i0 + θ(1 − γ) + vt − (1 − γL)vt
                             = (1 − γ)i0 + θ(1 − γ) + γvt−1
 Thus, the nominal interest rate, so the nominal interest rate will follow the first
order autoregressive process
                                it = i0 + γit−1 + γvt−1


   With the money supply process in (b), equation (4.24) becomes
                                   θ + mt−1 + vt αEt pt+1
                            pt =                +                                       (36)
                                       1+α        1+α


                                             24
and the equilibrium solution for pt is of the form pt = p0 + bmt−1 + cvt . Using
this in (36),

                           θ + mt−1 + vt α [p0 + b(θ + mt−1 )]
 p0 + bmt−1 + cvt     =                 +
                               1+α               1+α
                             1 + αb      αp0     1 + αb                   1
                      =             θ+       +           mt−1 +              vt
                             1+α        1+α       1+α                    1+α

or p0 = (1 + α)θ, b = 1, and c = 1/(1 + α). With
                                                      1
                          pt = (1 + α)θ + mt−1 +         vt
                                                     1+α
average inflation is just θ. Expected inflation is

                                              1                  1
            Et pt+1 − pt = mt − mt−1 +           vt = θ +           vt
                                             1+α                1+α

    With a constant real rate of interest, as was assumed in deriving equation
(4.24), the nominal rate of interest will equal

                                                1
                             it = r0 + θ +         vt
                                               1+α

so that it is equal to a constant plus a white noise error; it is not a random
walk.


  4. Suppose the Correia-Teles model of Section 4.5.3 is modified so that output
     is equal to f(n) where f is a standard neoclassical production function
     exhibiting positive but diminishing marginal productivity of n. If f (n) =
     na for 0 < a < 1, does the optimality condition given by (4.44) continues
     to hold?

   Start with the budget constraint when f(n) = na . Equation (4.42) becomes
               ∞
    dt−1   =         Di ct+i − (1 − τ t+i )f (1 − lt+i − ns ) + Rt It+i mt−1+i
                                                          t+i
               i=0
                ∞
           =         Di ct+i − (1 − τ t+i )(1 − lt+i − ns )a + Rt It+i mt−1+i
                                                        t+i
               i=0

while the first order condition for labor supply, equation (4.39), is modified to
become
                              ul
                                      = aλt (1 − lt − ns )a−1                    (37)
                           (1 − τ t )                  t




                                          25
   Following the same steps outlines on page 170, we use the fact that dt−1 = 0
and Di = β i D0 λt+i /λt to obtain
       ∞
 D0
             β i λt+i ct+i − λt+i (1 − τ t+i )(1 − lt+i − ns )a − λt+i It+1+i mt+i = 0
                                                           t+i
 λt    i=0



From (37), λt+i (1−τ t+i ) = ul (1−lt −ns )1−a /a. Making this substitution (along
                                         t
with the others discussed in the text) yields
                ∞
           D0                                       ul (1 − lt − ns )1−a
0 =                   β i uc ct+i − ul ηgns −
                                          t+i
                                                                  t
                                                                           (1 − lt+i − ns )a
                                                                                        t+i
           λt   i=0
                                                              a
                 ∞
           D0                                       ul
   =                  β i uc ct+i − ul ηgns −
                                          t+i          (1 − lt − ns )
                                                                  t
           λt   i=0
                                                    a

This implies
                 ∞
                                       ul                    1
                       β i uc ct+i −      (1 − lt ) − ul       − η ns
                                                                    t+i = 0
                 i=0
                                       a                     a

which corresponds to equation (4.43) on page 171. Notice that the only modifi-
                                              1
cation is that ns is multiplied by the factor a − η; in the example of the text,
a = 1 and this became 1 − η.
    The first-order condition for the optimal choice of m in the social welfare
problem is

                                          1
                                 β i ψul ( − η) −      t+i   g =0
                                          a
                                            1
which replaces (4.44). As long as β i ψul ( a − η) − t+i must be nonzero, the
optimum still involves g = 0, or a zero nominal interest rate.




                                               26
4     Chapter 5: Money and Output in the Short
      Run
                                                                                 b 1−Φ
                                                                         c( M )               1−η
    1. Assume household preferences are given by U =              + Ψ (1−N)
                                                                            P
                                                           1−Φ           1−η
       and aggregate output is given by Y = K α N 1−α . Linearize around the
       steady-state the labor market equilibrium condition equation (5.26) from
       the monopolistic competition model. How does the result depend on q?
       Explain.

    Equation (5.26) states that
                                           Ul
                                              = qM P L
                                           Uc
For the functional forms specified in the question, this becomes
                                   (1 − N)−η              Y
                                               = q(1 − α)                                     (38)
                                  c−Φ (mb )1−Φ            N
where m = M/P .
    Using the methods employed in the text, we can define X ss as the steady-
                        ˆ
state value of X and x as the percent deviation of x around its steady-state
                       ˆ
value, so x = xss (1 + x). Then, (38) can be written as
                                           −η
                             Lss (1 + ˆ
                                      l)                                                ˆ
                                                                              Y ss (1 + y )
                                                         1−Φ
                                                               = q(1 − α)                     (39)
                         −Φ                          b                                  ˆ
                                                                              N ss (1 + n)
           [css (1     ˆ
                     + c)]       (mss (1     ˆ
                                           + m))

where L = 1 − N. Since
                                    (Lss )−η                           Y ss
                                                    1−Φ
                                                          = q(1 − α)
                                                                       N ss
                         (css )−Φ (mss )b

equation (39) becomes

                                      (1 + ˆ −η
                                           l)                           ˆ
                                                                   (1 + y )
                                                         1−Φ
                                                               =                              (40)
                             (1 +   c)−Φ [(1
                                    ˆ           +   m)b ]
                                                    ˆ                   ˆ
                                                                   (1 + n)
Notice that q has dropped out; the labor market deviations around the steady-
state will not be affected by q. Now take logs of (40) and employ the approxi-
mation that ln(1 + x) ≈ x for small x, to obtain

                               −ηˆ + Φˆ − b(1 − Φ)m = y − n
                                 l    c           ˆ   ˆ ˆ

Since Lss (1 + ˆ = 1 − N ss (1 + n), this implies ˆ = − Nss n. Using this we have,
                                                         ss
               l)                ˆ                l     L ˆ

                                 N ss
                      −η −            ˆ                ˆ   ˆ ˆ
                                      n + Φˆ − b(1 − Φ)m = y − n
                                           c
                                 Lss


                                                    27
or
                                  1
                       ˆ
                       n=                                            ˆ
                                                   (ˆ − Φˆ + b(1 − Φ)m)
                                                    y    c
                              1 + η Nss
                                      ss

                                    L




     2. The Chari, Kehoe, and McGratten (1996) model of price adjustment led
        to equation (5.30). Using equation (5.29), show that the parameter a in
                          √         √
        (5.30) equals (1 − γ)/(1 + γ).

     Equation (5.29), page 200, states that
                   1   1−γ                                  γ
            pt =               [pt−1 + Et pt+1 ] +                   [mt + Et mt+1 ]   (41)
                   2   1+γ                                 1+γ
If m follows a random walk as was assumed in deriving (5.30), Et mt+1 = mt
and (5.29) becomes
                          1   1−γ                                   2γ
                   pt =              [pt−1 + Et pt+1 ] +                   mt
                          2   1+γ                                  1+γ
                                           1       1−γ
It will be convenient to define b ≡         2       1+γ   and rewrite this as

                          pt = b [pt−1 + Et pt+1 ] + (1 − 2b) mt                       (42)

Let the proposed solution be

                                  pt = a1 pt−1 + a2 mt

From (5.30), Et pt+1 = a1 pt + a2 mt = a1 (a1 pt−1 + a2 mt ) + a2 mt , so equation
(42) becomes
            pt   = b [pt−1 + a1 (a1 pt−1 + a2 mt ) + a2 mt ] + (1 − 2b) mt
                 = b 1 + a2 pt−1 + [1 − 2b + ba2 (1 + a1 )] mt
                           1

For this to equal the proposed solution for all realizations of pt−1 and mt requires
that

                                    a1 = b 1 + a2
                                                1

and
                                                                    1 − 2b
                   a2 = 1 − 2b + ba2 (1 + a1 ) ⇒ a2 =
                                                                 1 − b(1 + a1 )
The first of these conditions requires that a1 be the solution to
                                               1
                                 a2 −
                                  1                 a1 + 1 = 0
                                               b


                                                   28
or
                                                           √
                                                b−1 ±       b−2 − 4
                                     a1 =
                                                           2
                                      1    1−γ
Recalling that b was equal to         2    1+γ     , this becomes

                                                   2
                         1+γ               1+γ
                     2   1−γ    ±    4     1−γ         −4
       a1   =
                                     2
                                                       2
                         1+γ               1+γ
            =                    ±                         −1
                         1−γ               1−γ
                         1+γ           1
            =                    ±                     1 + 2γ + γ 2 − (1 − 2γ + γ 2 )
                         1−γ          1−γ
                                              √ 2             √ 2
                  1+γ        1     √      1− γ           1+ γ
            =          ±2            γ=             and
                  1−γ      1−γ              1−γ            1−γ
                   √     √
Since 1 − γ = (1 + γ)(1 − γ), we can write these two potential solutions as
                                        √          2        √
                                      1− γ                1− γ
                                                        =   √
                                       1−γ                1+ γ
and
                                        √          2             √
                                      1+ γ                     1+ γ
                                                        =        √
                                       1−γ                     1− γ
Both only the first of these is less than 1 in absolute value, so the stable solution
has
                                             √
                                         1− γ
                                   a1 =      √
                                         1+ γ
    Returning to the condition for a2 , and using the value for a1 just found,
together with the definition of b,
                                                                         1−γ
                                  1 − 2b                           1−    1+γ
                a2       =                    =                                   √
                               1 − b(1 + a1 )   1−             1   1−γ
                                                                          1+
                                                                                1− γ
                                                                                  √
                                                               2   1+γ          1+ γ
                                           2γ
                                          1+γ                            2γ
                         =                                     =
                                     1−γ           1                           1−γ
                               1−    1+γ
                                                   √
                                                 1+ γ              1+γ−         √
                                                                              1+ γ
                                                         √
                                2γ                     1− γ
                         =       √ =1−                   √
                               γ+ γ                    1+ γ
which verifies that a2 = 1 − a1 .



                                                   29
   3. Equation (5.28) was obtained from equation (5.27) by assuming R = 1.
      Show that in general,
                         Rss                         1                       Rss                1
           pt =                            pt +         Et pt+1 +                      vt +        Et vt+1
                       1 + Rss                      Rss                    1 + Rss             Rss




   Equation (5.27) of the text states that

                                          Et Ptθ Vt Yt + R−1 Pt+1 Vt+1 Yt+1
                                                          t+1
                                                              θ
                                Pt =                      1                 1
                                                                                                             (43)
                                                           −1
                                           qEt Pt1−q Yt + Rt+1 Pt+1 Yt+1
                                                                 1−q




If we evaluate this at the steady state, recalling that θ = (2 − q)/(1 − q),

                                      (P ss )θ V ss Y ss 1 + (Rss )−1             1
                       P   ss
                                =               1
                                                                                 = P ss V ss                 (44)
                                      q (P ss ) 1−q   Y   ss   1+    (Rss )−1     q

   Let lower case letters denote percentage deviations from the steady-state, so
1+ x = Xt /X ss . Then the left side of equation (5.27) can be written P ss (1+ pt )
while the right side becomes
                                                        −1
(P ss ) V ss Y ss (1 + pt )θ (1 + vt )(1 + yt ) + (Rss ) Et (1 + rt+1 )−1 (1 + pt+1 )θ (1 + vt+1 )(1 + yt+1 )
       θ

                  1                               1                                                             1
         (P ss ) 1−q qY ss (1 + pt ) 1−q (1 + yt ) + (Rss )−1 Et (1 + rt+1 )−1 (1 + pt+1 ) 1−q (1 + yt+1 )


Now using the approximations (1 + x)s ≈ 1 + sx and (1 + x)(1 + z) ≈ 1 + x + z,
this becomes

  P ss V ss (1 + θpt + vt + yt ) + (Rss )−1 Et (1 + θpt+1 + vt+1 + yt+1 − rt+1 )

     q     1+          1
                      1−q       pt + yt + (Rss )−1 Et 1 +                   1
                                                                           1−q    pt+1 + yt+1 − rt+1


After some cancellation, equation (43) can be written
                                                                −1
                  (1 + θpt + vt + yt ) + (Rss )                      Et (1 + θpt+1 + vt+1 + yt+1 − rt+1 )
(1 + pt ) =
                  1+             1
                                1−q     pt + yt + (Rss )−1 Et 1 +                 1
                                                                                 1−q   pt+1 + yt+1 − rt+1
                                                                                                             (45)
This expression is of the form
                                                      1 + z + R−1 (1 + d)
                                          1+x=                                                               (46)
                                                      1 + s + R−1 (1 + c)


                                                               30
which can be written as

                  (1 + x) 1 + s + R−1 (1 + c) = 1 + z + R−1 (1 + d)

Multiplying out the left side,

1 + x + s + sx + R−1 (1 + x) + R−1 (x + xc) ≈ 1 + x + s + R−1 (1 + c) + R−1 x
                                            = 1 + R−1 x + 1 + s + R−1 (1 + c)

yielding for equation (46),

                1 + R−1 x + 1 + s + R−1 (1 + c) ≈ 1 + z + R−1 (1 + d)

or
          z + R−1 (1 + d) − s − R−1 (1 + c)         R                    1
     x≈                                     =                    z−s+         (d − c)
                     (1 + R−1 )                    1+R                   R


                                                                              1
Returning to equation (45), we have x = pt , z = θpt + vt + yt , s =         1−q   pt + y t ,
                                                            1
d = Et (θpt+1 + vt+1 + yt+1 − rt+1 ), and d = Et           1−q   pt+1 + yt+1 − rt+1 , so

                          Rss                     1
                 pt =               pt + vt +           Et (pt+1 + vt+1 )
                        1 + Rss                  Rss

                1
since θ −      1−q   = 1. This is our desired result.


     4. Using the equilibrium condition (5.42) for the price level, show that equi-
        librium output is independent of any policy response to εt−1 or vt−1 .

     Equation (5.42) states that

               d2 mt + a(1 + d2 )Et−1 pt + Et pt+1 − d2 vt + ut − (1 + d2 )εt
        pt =                                                                           (47)
                                     (1 + a)(1 + d2 )

Suppose that monetary policy does respond to εt−1 and vt−1 by setting the nom-
inal supply of money according to

                                  mt = b1 εt−1 + b2 vt−1

Substituting this expressing into (47),

        d2 (b1 εt−1 + b2 vt−1 ) + a(1 + d2 )Et−1 pt + Et pt+1 − d2 vt + ut − (1 + d2 )εt
pt =
                                        (1 + a)(1 + d2 )
                                                                                    (48)


                                            31
    Assuming all the disturbance terms are serially uncorrelated, we can use the
method of undetermined coefficients to find the solution for the equilibrium price
level. Inspection of (48) suggests the following guess:
                   pt = γ 0 + γ 1 εt−1 + γ 2 vt−1 + γ 3 vt + γ 4 ut + γ 5 εt          (49)
where xt ≡ −d2 vt + ut − (1 + d2 )εt . Using (49),
                             Et−1 pt = γ 0 + γ 1 εt−1 + γ 2 vt−1                      (50)
and
                               Et pt+1 = γ 0 + γ 1 εt + γ 2 vt                        (51)
   Substituting equations (49) - (51) into (48), yields, with some rearranging,
(1 + a)(1 + d2 )pt     = d2 (b1 εt−1 + b2 vt−1 ) + a(1 + d2 ) (γ 0 + γ 1 εt−1 + γ 2 vt−1 )
                         +γ 0 + γ 1 εt + γ 2 vt − d2 vt + ut − (1 + d2 )εt
                       = γ 0 [1 + a(1 + d2 )] + [d2 b1 + aγ 1 (1 + d2 )] εt−1
                         + [d2 b2 + γ 2 a(1 + d2 )] vt−1
                         + [γ 1 − (1 + d2 )] εt + [γ 2 − d2 ] vt + ut                 (52)
We could now replace pt on the left side of this equation with the proposed
solution given in equation (49) and equate coefficients to solve for the values of
the γ i coefficients. However, to determine the effect of the policy reaction on
output, we do not need to do this. From equation (5.34) on page 205, output
depends on the price surprise term pt − Et−1 pt . We can obtain this by taking
expectations of (52) based on t−1 information and subtract the result from (52).
So first taking expectations,
(1 + a)(1 + d2 )Et−1 pt     = γ 0 [1 + a(1 + d2 )]
                              + [d2 b1 + aγ 1 (1 + d2 )] εt−1 + [d2 b2 + γ 2 a(1 + d2 )] vt−1
Subtracting this from (52),
      (1 + a)(1 + d2 ) (pt − Et−1 pt ) = [γ 1 − (1 + d2 )] εt + [γ 2 − d2 ] vt + ut
This is independent of the policy response coefficients b1 and b2 .
    Because any systematic policy response to εt−1 or vt−1 is fully incorporated
into the public’s expectations at the start of period t, it cannot generate any price
surprise; pt (and Et−1 pt ) adjust fully to anticipated or predictable movements
in the period t nominal supply of money.

  5. Assume nominal wages are set for one period but that they can be indexed
     to the price level:
                                         0
                                   wt = wt + b(pt − Et−1 pt )
                                    c


       where w0 is a base wage and b is the indexation parameter (0 ≤ b ≤ 1).

                                              32
       (a) How does this change modify the aggregate supply equation given by
           (5.18)?
      (b) Assume the indexation parameter is set to minimize Et−1 (nt − n)2 .
          Using your modified aggregate supply equation, together with (5.35)
          - (5.37) and a money supply process mt = ω t , show that the opti-
          mal degree of wage indexation is increasing in the variance of ω and
          decreasing in the variance of ε (Gray 1978).

  (a) From equation (5.16) and the new specification for the contract wage,
employment is given by
                                    0
                         nt = yt − wt + b(pt − Et−1 pt ) + pt
                                              0
If the base wage is set according to (5.15), wt = Et−1 (yt + pt − nt ), and

                 nt − Et−1 nt = yt − Et−1 yt + (1 − b) (pt − Et−1 pt )         (53)

Notice that if b = 0 (no indexation), we obtain the expression in the test (equa-
tion 5.16). At the other extreme, if b = 1, nominal wages are completely indexed
and adjust fully to unexpected changes in the price level. As a result, the real
wage and employment are insulated from price level movements.
    Since the model underlying equations (5.34) - (5.37) was based on the as-
sumption that labor supply was inelastic, we can set Et−1 nt = 0 since all vari-
ables should be interpreted as deviations around a steady-state.
    Substituting (53) into the production function (5.8),

  yt − Et−1 yt    = (1 − α) [yt − Et−1 yt + (1 − b) (pt − Et−1 pt )] + zt − Et−1 zt
                  = a(1 − b) (pt − Et−1 pt ) + εt                               (54)

where, as in the text, a = (1 − α)/α and εt = (zt − Et−1 zt ) /α. Equation (54)
shows that, relative to (5.18), the effect of a price surprise on output is now
a(1 − b) < a.
    (b) To solve for the variance of employment, use (5.35) - (5.37) to solve for
yt and pt , using the assumption that mt = ω t . From (5.36) and (5.37),

                    rt   = it + pt − Et−1 pt
                         = d2 (yt + pt + vt − ω t ) + pt − Et−1 pt

Substituting this into the aggregate spending equation (5.35),

          yt     = Et yt+1 − [d2 (yt + pt + vt − ω t ) + pt − Et−1 pt ] + ut   (55)
                   Et yt+1 − [(1 + d2 )pt − Et−1 pt ] + ut + d2 (ωt − vt )
                 =
                                          1 + d2
If we assume the productivity shock z is serially uncorrelated, then Et yt+1 =
Et zt+1 = 0 and (55) implies

                         yt − Et−1 yt = − (pt − Et−1 pt ) + st                 (56)


                                           33
where st ≡ [ut + d2 (ωt − vt )] /(1 + d2 ). Solving (54) and (56) for yt − Et−1 yt
and pt − Et−1 pt ,

                                         a(1 − b)             1
                  yt − Et−1 y =                     st +              εt
                                       1 + a(1 − b)      1 + a(1 − b)

                                                   st − εt
                              pt − Et−1 pt =
                                                1 + a(1 − b)

Using these results in (53),

            nt − Et−1 nt      = yt − Et−1 yt + (1 − b) (pt − Et−1 pt )
                                  (1 − b)(1 + a)                b
                              =                    st +                 εt                (57)
                                   1 + a(1 − b)            1 + a(1 − b)
so
                                                                                          2
                                       (1 − b)(1 + a)                  b
     Et−1 (nt − n)2    = Et−1                              st +                εt − n
                                        1 + a(1 − b)              1 + a(1 − b)
                                                2                           2
                              (1 − b)(1 + a)                      b
                       =                            σ2 +                        σ2 + n2
                               1 + a(1 − b)          s
                                                             1 + a(1 − b)        ε


The value of the indexation parameter is picked to minimize this expression.
The first order condition for this problem is

                  −(1 − b)(1 + a)2                         (1 + a)b
              2                     3     σ2 + 2
                                           s                          3   σ2 = 0
                                                                           ε
                      [1 + a(1 − b)]                 [1 + a(1 − b)]

which implies

                               −(1 − b)(1 + a)σ 2 + bσ2 = 0
                                                s     ε

or the optimal degree of indexation is

                                 (1 + a)σ2              σ2
                       b∗ =               s
                                       2 + σ2
                                              =1−        ε
                              (1 + a)σs     ε     (1 + a)σ 2 + σ 2
                                                           s     ε

Hence,

                                          0 ≤ b∗ ≤ 1

with the inequalities strict if both σ2 and σ 2 are positive. Define γ ≡ σ2 /σ2 as
                                      ε       s                          ε   s
the relative variance of productivity shocks to demand side shocks (arising from
money supply shocks ω, money demand shocks v, and aggregate spending shocks
u). Then
                                                 γ
                                    b∗ = 1 −
                                               1+a+γ


                                               34
which is decreasing in γ. As aggregate demand shocks become more important
(and γ falls), it is optimal to have a higher degree of nominal wage indexation to
insulate real wages and employment from fluctuating. Real productivity shocks
do call for real wage adjustments, so if ε shocks are important, then the optimal
degree of indexation is lower in order to allow for some real wage movements
as the price level changes.


     6. The basic Taylor model of price level adjustment was derived under the
        assumption that the nominal wage set in period-t remained unchanged for
        periods t and t + 1. Suppose instead each period t contract specifies a
        nominal wage x1 for period t and x2 for period t + 1. Assume these are
                          t                    t
        given by x1 = pt + κyt and x2 = Et pt+1 + κEt yt+1 . The aggregate price
                    t                   t
        level at time t is equal to pt = 1 (x1 + x2 ). If aggregate demand is given
                                          2  t    t−1
        by yt = mt − pt and mt = m0 + ω t , what is the effect of a money shock
        ω t on pt and yt ? Explain why output shows no persistence after a money
        shock.

   From the definition of the aggregate price level and the contract nominal
wages,
                              1
                       pt   =   [pt + κyt + Et−1 pt + κEt−1 yt ]
                              2
                            = κyt + Et−1 pt + κEt−1 yt                        (58)

which can be compared to the equation at the bottom of page 216. Notice that
pt−1 does not appear in (58), since x2 is now set based on Et−1 pt rather than
                                     t−1
on pt−1 and Et−1 pt as in the specification given by equation (5.44).
   Substituting the assumed specification for aggregate demand into (58),

                  pt   = κ (mt − pt ) + Et−1 pt + κEt−1 (mt − pt )
                         κmt + (1 − κ)Et−1 pt + κm0
                       =                                                      (59)
                                      1+κ
where use has been made of the fact that Et−1 mt = m0 under the assumed
money supply process. We can write the solution to this as

                                     pt = γ 0 + γ 1 ω t

for γ 0 and γ 1 such that

                                    κ (m0 + ω t ) + (1 − κ)γ 0 + κm0
                   γ 0 + γ 1 ωt =
                                                  1+κ
or
                                               κ
                                       γ1 =
                                              1+κ


                                            35
and
                                        γ 0 = m0
Aggregate output is then given by
                                                              1
               yt = mt − pt = m0 + (1 − γ 1 )ω t − γ 0 =               ωt
                                                             1+κ
Since output is equal to the white noise error ω, it displays no persistence.
   In the standard formulation, some nominal wages in effect during period t
were set in earlier periods on the basis of the price level in those earlier periods.
This imparts sluggishness to the adjustment of the price level; pt can no longer
jump to fully offset any change in the period t nominal supply of money. With
the alternative specification used in this problem, nominal wages in effect in
period t depend only on period t prices and previous expectations about pt . Thus,
output in period t is only affected by movements in mt that were unpredictable
when the oldest contract still in effect was set.


  7. The p-bar model led to the following two equations for pt and yt :
                                  d2 mt + Et pt+1 − d2 vt + ut
                           pt =                                − yt
                                            1 + d2

              d2 (mt − Et−1 mt ) + Et pt+1 − Et−1 pt+1 − d2 vt + ut
       yt =                                                         + (1 − γ)yt−1
                                    1 + d2


      Assume γ = 1 and
                                      mt = mt−1 + aut
      Show that the variance of yt depends on the parameter a. What value of
      a would minimize the impact of IS shocks (u) on output?

    If γ = 1, lagged output drops out of the model, and from the assumed process
for money, mt − Et−1 mt = mt−1 + aut − mt−1 = aut . Thus, the output equation
can be written as
                         Et pt+1 − Et−1 pt+1 − d2 vt + (1 + ad2 ) ut
                  yt =                                                          (60)
                                           1 + d2
Notice that as long as neither Et pt+1 nor Et−1 pt+1 are affected by an IS shock,
the impact of u on output would be neutralized if a were set equal to −1/d2 . To
check whether price expectations are affected, use the price and output equations,
together with the money supply process to obtain
          (1 + d2 )pt    = d2 (mt−1 + aut ) + Et pt+1 − d2 vt + ut
                           − [Et pt+1 − Et−1 pt+1 − d2 vt + (1 + ad2 ) ut ]     (61)

                                           36
or

                   (1 + d2 )pt = d2 (mt−1 + aut ) + Et−1 pt+1 − ad2 ut              (62)
Consider the following proposed solution for pt :

                                  pt = δ 0 + δ 1 mt−1 + δ 2 ut

Based on this solution,

                       pt+1     = δ 0 + δ 1 mt + δ 2 ut+1
                                = δ 0 + δ 1 (mt−1 + aut ) + δ 2 ut+1

So

                              Et pt+1 = δ 0 + δ 1 (mt−1 + aut )

and

                                  Et−1 pt+1 = δ 0 + δ 1 mt−1

Substituting these expressions into (62),

               (1 + d2 )pt    = d2 (mt−1 + aut ) + Et−1 pt+1 − ad2 ut
                              = d2 (mt−1 + aut ) + δ 0 + δ 1 mt−1 − ad2 ut
                              = δ 0 + (δ 1 + d2 ) mt−1

Using the proposed solution to eliminate pt ,

                 (1 + d2 ) [δ 0 + δ 1 mt−1 + δ 2 ut ] = δ 0 + (δ 1 + d2 ) mt−1

equating coefficients implies

                                 (1 + d2 )δ 0 = δ 0 ⇒ δ 0 = 0

                              (1 + d2 )δ 1 = (δ 1 + d2 ) ⇒ δ 1 = 1

                                            δ2 = 0

so pt = mt−1 .
   We can now collect these results to evaluate equation (60) for output:
            Et pt+1 − Et−1 pt+1 − d2 vt + (1 + ad2 ) ut   [1 + a(1 + d2 )] ut − d2 vt
     yt =                                               =
                              1 + d2                               1 + d2


and the variance of output is
                                                 2                   2
                                1 + a(1 + d2 )                d2
                      σ2 =                           σ2 +                σ2
                       y
                                   1 + d2             u
                                                            1 + d2        v




                                              37
Thus, to insulate output from demand shock, a should be set equal to
                                            1
                                 a∗ = −          <0
                                          1 + d2
Notice that this is a smaller response than found earlier when the possible effects
of u on expected future prices were ignored (see equation 60). A positive ut that
results in a fall in mt causes private agents to revise downward their forecast
of the future price level: Et pt+1 − Et−1 pt+1 = aut if a is negative. But from
(60), this acts to reduce yt . Thus, to stabilize yt , the reduction in mt needs to
be smaller than would be the case if price expectations did not matter.


  8. Derive the equilibrium expression for pt and yt corresponding to equa-
     tions (5.39) and (5.41) for the case in which the aggregate productivity
     disturbance is given by zt = ρzt−1 + et , −1 < ρ < 1.

   The equations of the model that led to equations (5.39) and (5.40) were given
by (5.34) - (5.37) and are repeated here:

            Aggregate supply:        yt = Et−1 yt + a(pt − Et−1 pt ) + εt     (63)


                  Aggregate demand:         yt = Et yt+1 − rt + ut            (64)


                  Money demand:         mt − pt = yt − d−1 it + vt
                                                        2                     (65)


            Fisher equation:       it = rt + Et pt+1 − pt = rt + Et πt+1      (66)

As discussed in the text (page 208), equations (64) - (66) can be combined to
yield equation (5.38) of the text:

                       d2 (mt − pt ) + Et π t+1 + Et yt+1 − d2 vt + ut
                yt =                                                          (67)
                                            1 + d2
   As discussed on page 204 (and in footnote 43),

                                  Et yt+1 = Et zt+1

Using the assumed process for z specified in the question,

                                    Et yt+1 = ρzt

Equation (67) then becomes

                         d2 (mt − pt ) + Et πt+1 + ρzt − d2 vt + ut
                  yt =
                                           1 + d2


                                          38
Equating this expression for yt with the expression for yt given by the aggregate
supply relationship (63) and solving for pt (and using the fact that ρzt−1 + et =
zt ) results in

       d2 mt + a(1 + d2 )Et−1 pt + Et π t+1 + ρzt − d2 vt + ut − (1 + d2 ) (ρzt−1 + εt )
pt =
                                       d2 + a(1 + d2 )


which corresponds to (5.39).
   Taking expectations of this expression as of time t − 1 and subtracting the
result from pt yields

                  d2 (mt − Et−1 mt ) + Et πt+1 − Et−1 πt+1 + ρet − d2 vt + ut − (1 + d2 )εt
pt − Et−1 pt =
                                               d2 + a(1 + d2 )


so that real output, from (63) is

yt    = Et−1 yt + εt
              d2 (mt − Et−1 mt ) + Et πt+1 − Et−1 πt+1 + ρet − d2 vt + ut − (1 + d2 )εt
        +a
                                           d2 + a(1 + d2 )
                         d2
      = ρzt−1 +                      εt
                   d2 + a(1 + d2 )
          a [d2 (mt − Et−1 mt ) + Et πt+1 − Et−1 πt+1 + ρet − d2 vt + ut ]
        +
                                   d2 + a(1 + d2 )

where the only difference from equation (5.41) is the presence of ρet . This is the
innovation in zt that persists into period t + 1. This affects aggregate demand
in period t under the assumption that agents are forward looking.


     9. Suppose that the nominal money supply evolves according to mt = +
        γmt−1 + ωt for 0 < γ < 1 and ωt a white noise control error. If the rest
        of the economy is characterized by equations (5.34) - (5.37), solve for the
        equilibrium expressions for the price level, output, and the nominal rate
        of interest. What is the effect of a positive money shock (ωt > 0) on the
        nominal rate? How does this result compare to the γ = 1 case discussed
        in the text? Explain.

    To answer this problem, make use of the results in Section 5.7.3 of the Ap-
pendix to Chapter 5. Solving the basic model leads to equation (5.85) for the
price level:

               d2 mt + a(1 + d2 )Et−1 pt + Et pt+1 − d2 vt + ut − (1 + d2 )εt
        pt =                                                                       (68)
                                     (1 + a)(1 + d2 )



                                            39
Given the assumed process for mt , guess that the equilibrium price level is given
by

                     pt = b0 + b1 mt−1 + b2 ut + b3 εt + b4 vt + b5 ωt

Based on this guess,

                                 Et−1 pt = b0 + b1 mt−1

and

                          Et pt+1 = b0 + b1 ( + γmt−1 + ω t )

Substituting these into (68), we have that the following condition must hold for
all realizations of mt−1 and the random disturbances:

(1 + a)(1 + d2 )pt     = d2 ( + γmt−1 + ω t ) + a(1 + d2 ) [b0 + b1 mt−1 ]
                         + [b0 + b1 ( + γmt−1 + ω t )] − d2 vt + ut − (1 + d2 )εt
                       = d2 + a(1 + d2 )b0 + b0 + b1 + [d2 γ + a(1 + d2 )b1 + b1 γ] mt−1
                         +ut − (1 + d2 )εt − d2 vt + (d2 + b1 ) ωt

Using the proposed solution for pt , this requires that

                  (1 + a)(1 + d2 )b0 = d2 + a(1 + d2 )b0 + b0 + b1


                     (1 + a)(1 + d2 )b1 = d2 γ + a(1 + d2 )b1 + b1 γ


                                 (1 + a)(1 + d2 )b2 = 1


                             (1 + a)(1 + d2 )b3 = −(1 + d2 )


                                (1 + a)(1 + d2 )b4 = −d2


                              (1 + a)(1 + d2 )b5 = d2 + b1

Solving these yields the following solution for pt :

                     pt = b0 + b1 mt−1 + b2 ut + b3 εt + b4 vt + b5 ωt


                       (1 + d2 )         γd2
         pt   =                  +                mt−1
                     1 − γ + d2       1 − γ + d2
                       ut − (1 + d2 )εt − d2 vt           d2
                     +                          +                      ωt
                          (1 + a)(1 + d2 )        (1 + a)(1 − γ + d2 )

                                            40
Using this result for pt , the equilibrium expressions for output can be obtained
from (5.81) as
                  ut − (1 + d2 )εt − d2 vt             d2
       yt     = a                          +                        ω t + εt
                     (1 + a)(1 + d2 )          (1 + a)(1 − γ + d2 )
                     ut − d2 vt                  d2                     1
              = a                  +                          ωt +           εt
                  (1 + a)(1 + d2 )      (1 + a)(1 − γ + d2 )          1+a
while from (5.83) and (5.84),
  it        = d2 (yt − mt + pt + vt )
                   γd2          d2                        1−γ
            =             +              (ut + vt ) −                  d2 (γmt−1 + ω t )
              1 − γ + d2      1 + d2                    1 − γ + d2
    From these results, we can see that a positive realization of ω increases the
current price level and output and lowers the nominal rate of interest. Because
output rises, the real rate of interest rt must fall to ensure aggregate demand
and supply are equal at the temporarily higher level of output. When the money
supply follows a random walk ( γ = 1), a money supply shock has no effect on
the nominal interest rate, leading to a fall in the real rate but an equal rise in
expected inflation. When γ < 1, expected inflation rises less since the money
stock, after an initial increase, regresses back to its initial value.
    To focus on the effect of ω on expected inflation, set u, v, and ε equal to
zero, and use the equilibrium expression for pt to obtain
                                     γd2                 γd2
             Et pt+1 − pt   =                    +                   (γ − 1) mt−1
                                  1 − γ + d2         1 − γ + d2
                                        (1 + a)γ − 1
                                +                         d2 ω t
                                    (1 + a)(1 − γ + d2 )
while the impact on the real rate of interest is
                                                  a
                        rt = −yt = −                          d2 ω t
                                         (1 + a)(1 − γ + d2 )
A rise in γ implies that ω has a larger impact on the expected future price level
since it has a larger impact on the future money supply. This means the current
price level rises more in response to a positive realizations of ω. In turn, the
larger unanticipated rise in pt generates a larger output increase and a larger
corresponding fall in the real rate of interest. Since pt rises more when γ is
large, expected future inflation is less affected since the rise in the level of the
money supply and the price level is more persistent.

 10. An increase in average inflation lowers the real demand for money. Demon-
     strate this by using the model given by equations (5.34) - (5.37) and as-
     suming the nominal money supply grows at a constant trend rate so
     that mt = t to show that real money balance mt − pt are decreasing in
       .


                                             41
    Answering this problem involves repeating the steps outlined in Section 5.7.3
of the Chapter 5 Appendix, replacing the money supply process given in (5.86)
with the one in the problem. Since the focus is on the effects of the deterministic
trend     on real money balances, it will simplify the problem if all stochastic
disturbances terms are ignored. In this case, the expression for the equilibrium
price level given in equation (5.85) becomes

                                  d2 mt + a(1 + d2 )Et−1 pt + Et pt+1
                    pt    =
                                           (1 + a)(1 + d2 )
                                  d2 t + a(1 + d2 )Et−1 pt + Et pt+1
                          =                                                                  (69)
                                           (1 + a)(1 + d2 )

For our guess for the solution, equation (5.87) is replaced by

                                        pt = p0 +   pt

so that Et−1 pt = p0 +       pt   = pt and Et pt+1 = p0 +          p (t + 1).   Substituting these
into (69),

                  d2 t + a(1 + d2 ) p0 + p t + p0 + p (t + 1)
         pt   =
                                    (1 + a)(1 + d2 )
                  [1 + a(1 + d2 )] p0 + p      d2 + a(1 + d2 ) p +                   p
              =                             +                                            t
                      (1 + a)(1 + d2 )               (1 + a)(1 + d2 )

which equals p0 +    pt   if and only if

                              d2 + a(1 + d2 ) p +          p
                         =                                         ⇒        =
                     p
                                 (1 + a)(1 + d2 )                       p


and
                                   [1 + a(1 + d2 )] p0 +       p
                          p0 =                                     =
                                       (1 + a)(1 + d2 )                d2

   With this expression for the equilibrium price level, real money balances will
equal

                          mt − pt = t −             + t        =−
                                               d2                      d2

which is decreasing in the growth rate of nominal money balances . From the
solution for the price level, the rate of inflation is equal to . Higher values
of , and therefore higher rates of inflation, increase the opportunity cost of
holding money. This reduces the real demand for money, and, in equilibrium,
real money balances are lower at higher rates of money growth.




                                              42
5      Chapter 6: Money and the Open Economy
    1. Suppose mt = m0 + γmt−1 and m∗ = m∗ + γ ∗ m∗ . Use equation (6.24)
                                         t      0       t−1
       to show how the behavior of the nominal exchange rate under flexible
       prices depends on the degree of serial correlation exhibited by the home
       and foreign money supplies.

   Equation (6.24) on page 249 gives the following expression for the nominal
exchange rate:
                              ∞              i
                        1            δ
                st =                              mt+i − m∗             ∗
                                                          t+i − ct+i − ct+i
                       1+δ    i=0
                                    1+δ

Using the result that ct+i − c∗ = ct − c∗ , this becomes
                              t+i       t

                                                 ∞              i
                                         1                δ
                 st = − (ct − c∗ ) +                                mt+i − m∗                  (70)
                               t
                                        1+δ      i=0
                                                         1+δ                t+i



which is equation (6.25) of the text. We now have to use the specified processes
                                                                       i
for the nominal money supplies to evaluate the                    δ
                                                                 1+δ       mt+i − m∗
                                                                                   t+i terms.
    Since mt = m0 + γmt−1 ,

                        mt+1 = (m0 + γmt ) = (1 + γ)m0 + γ 2 mt−1

and

                              mt+2 = m0 (1 + γ + γ 2 ) + γ 3 mt−1

Similarly,1
                                                  i
                             mt+i     = m0             γ j + γ i+1 mt−1
                                                 j=0

                                                 1 − γ i+1
                                      = m0                 + γ i+1 mt−1
                                                   1−γ
For the foreign money supply,

                                         1 − (γ ∗ )i+1
                           m∗ = m∗
                                 0                     + (γ ∗ )i+1 m∗
                            t+i
                                           1 − (γ ∗ )               t−1

    1 This   uses the fact that 1 + a + a2 + ... + ak can be written as

    1 + a + a2 + ... − ak+1 + ak+2 + ...          =      1 + a + a2 + ... − ak+1 1 + a + a2 + ...

                                                         1    ak+1
                                                  =         −
                                                        1−a   1−a
for −1 < a < 1.


                                                   43
                                                                                             ∞
     Now, for notational easy, let b =               δ
                                                    1+δ     . We need to evaluate            i=0 b
                                                                                                  i
                                                                                                      mt+i − m∗
                                                                                                              t+i :

        ∞                                       ∞
                                                              1 − γ i+1      1 − (γ ∗ )i+1
             bi    mt+i − m∗              =          bi m0              − m∗
                                                                           0
       i=0
                           t+i
                                                i=0
                                                                1−γ            1 − (γ ∗ )
                                                   ∞
                                                +         bi γ i+1 mt−1 − (γ ∗ )i+1 m∗
                                                                                     t−1
                                                    i=0

Taking each term individually,
                  ∞                                              ∞
                                1 − γ i+1                m0
                        bi m0                   =                    bi 1 − γ i+1
                  i=0
                                  1−γ                   1−γ    i=0
                                                                                  ∞
                                                         m0         1
                                                =                      −γ     bi γ i
                                                        1−γ        1−b    i=0
                                                         m0          1     γ
                                                =                       −
                                                        1−γ        1 − b 1 − bγ

                  ∞
                                1 − (γ ∗ )i+1             m∗
                                                           0           1     γ∗
                        bi m∗
                            0                       =                     −
                  i=0
                                  1 − (γ ∗ )            1 − γ∗       1 − b 1 − bγ ∗

                          ∞                                  ∞
                                                                              γmt−1
                                i i+1
                               bγ       mt−1 = γmt−1               bi γ i =
                         i=0                                i=0
                                                                              1 − bγ

                   ∞                                         ∞
                                                                                  γ ∗ m∗
                         bi (γ ∗ )i+1 m∗ = γ ∗ m∗                  bi (γ ∗ )i =        t−1

                   i=0
                                       t−1      t−1
                                                             i=0
                                                                                  1 − bγ ∗

     Now collecting all these results, equation (70) becomes
                                1      m0       1          γ                        m∗
                                                                                     0         1     γ∗
st   = − (ct − c∗ ) +                               −                         −                   −
                t
                               1+δ    1 − γ 1 − b 1 − bγ                          1 − γ∗     1 − b 1 − bγ ∗
                                       γmt−1 γ ∗ m∗   t−1
                                    +          −
                                        1 − bγ    1 − bγ ∗
Since b = δ/(1 − δ), 1/(1 − b) is equal to 1 + δ, and an expression like 1/(1 − bγ)
is equal to (1 + δ)/(1 + δ(1 − γ)). So we can write the nominal exchange rate as
                        m0             γ              m∗
                                                       0               γ∗
st   = − (ct − c∗ ) +         1−                 −          1−
                t
                      1−γ         1 + δ(1 − γ)      1 − γ∗        1 + δ(1 − γ ∗ )
                                          ∗ ∗
                         γmt−1          γ mt−1
                 +                 −
                      1 + δ(1 − γ) 1 + δ(1 − γ ∗ )
                               1+δ                    1+δ
     = − (ct − c∗ ) + m0                  − m∗ 0
                t
                           1 + δ(1 − γ)           1 + δ(1 − γ ∗ )
                         γmt−1          γ ∗ m∗
                                             t−1
                 +                 −
                      1 + δ(1 − γ) 1 + δ(1 − γ ∗ )

                                                     44
    To gain some insights from this expression, suppose that the money supplies
in both countries display the same autoregressive coefficient; γ = γ ∗ and m0 =
m∗ . In this case,
  0

                                          γ
               st = − (ct − c∗ ) +                   mt−1 − m∗
                             t
                                     1 + δ(1 − γ)            t−1


 The nominal exchange rate depends on the initial difference in the money sup-
plies. If γ = 1, this difference is permanent (the money supplies follow random
walks with drift), and st = − (ct − c∗ ) + mt−1 − m∗ , reflecting the perma-
                                      t               t−1
nent difference associated with the difference in price levels when mt−1 = m∗ .t−1
If γ < 1 but positive, then both mt and m∗ follow stable processes that converge
                                          t
to m0 /(1 − γ). Any difference mt−1 − m∗ is now transitory and so has a
                                            t−1
smaller impact on the current nominal exchange rate.
    When γ = γ ∗ , the comparison is more complicated, since the money supplies
in the two countries regress towards their steady-state values at different rates.
The nominal exchange rate depends on the discounted value of the differences
in the paths followed by m and m∗ .


  2. In the model of Section 6.3 used to study policy coordination, aggregate-
     demand shocks were set equal to zero in order to focus on a common
     aggregate-supply shock. Suppose instead that the aggregate supply shocks
     are zero, and the demand shocks are given by u ≡ x + φ and u∗ ≡ x + φ∗
     so that x represents a common demand shock and φ and φ∗ are uncor-
     related country-specific demand shocks. Derive policy outcomes under
     coordinated and (Nash) noncoordinated policy setting. Is there a role for
     policy coordination in the face of demand shocks? Explain.

    Problems 2 - 5 use the model of section 6.3, so it will be convenient to develop
some general expressions for output and inflation first, and then apply them to
the special cases considered in each of the problems. The basic model is given
in equations (6.35) - (6.39) on page 261 of the text. Equilibrium expressions
for output in the two countries are given by equations (6.43) and (6.44) on page
264.
    Under a coordinated policy, the objective is to minimize
                        1     2           ∗
                          E λyt + π2 + λ(yt )2 + (π∗ )2
                        2          t


and the first order conditions for π and π∗ are (see page 265),
                                                   ∗
                          λb2 A1 yt + πt + λb2 A2 yt = 0                       (71)

                                  ∗
                          λb2 A1 yt + π ∗ + λb2 A2 yt = 0
                                        t                                      (72)



                                        45
It will prove convenient to use these to get expressions for πt −π ∗ , the difference
                                                                   t
in inflation between the two countries, and π t + π∗ , the sum of inflation in the
                                                     t
two countries. First subtracting and then adding (71) and (72),
                                                          ∗
                        π t − π ∗ = −λb2 (A1 − A2 )(yt − yt )
                                t


                                                    ∗
                              πt + π∗ = −λb2 (yt + yt )
                                    t

where we have used the fact that A1 + A2 = 1 (see page 264). Since the term
A1 − A2 will appear frequently, let H ≡ A1 − A2 .
                                                ∗             ∗
    Now using (6.43) and (6.44) to find yt − yt and yt + yt , and substituting
                                           ∗           ∗
the results into the expressions for πt − πt and πt + πt , we obtain

              πt − π∗
                    t     = −λb2 H [b2 H(πt − π∗ )
                                                t
                                +H(et − e∗ ) + 2A3 (ut − u∗ )]
                                           t               t
                            −λb2 H 2 (et − e∗ ) + 2HA3 (ut − u∗ )
                                             t                 t
                          =                                                    (73)
                                         1 + λb2 H 2
                                                2



                    πt + π∗
                          t    = −λb2 [b2 (πt + π∗ ) + et + e∗ ]
                                                 t           t
                                 −λb2 (et + e∗ )
                               =              t
                                                                               (74)
                                    1 + λb2 2

For later reference, we can also derive
                ∗
          yt − yt     = b2 H(π t − π ∗ ) + H(et − e∗ ) + 2A3 (ut − u∗ )
                                     t             t                t
                                  λb2 H 2 (et − e∗ ) + 2HA3 (ut − u∗ )
                                                 t                 t
                      = −b2 H
                                               1 + λb2 H 2
                                                      2

                        +H(et − e∗ ) + 2A3 (ut − u∗ )
                                  t               t
                              1
                      =               [H(et − e∗ ) + 2A3 (ut − u∗ )]           (75)
                         1 + λb2 H 2
                                2
                                                t               t


and

               ∗          λb2 (et + e∗ )                1
         yt + yt = −b2             2
                                     t
                                         + et + e∗ =         (et + e∗ )        (76)
                            1 + λb2              t
                                                     1 + λb2
                                                           2
                                                                    t


    Note from these expressions that average inflation in the two countries re-
sponds only to region-wide supply shocks (see equation 74), while inflation will
respond differently in the two countries to the extent that there are different
supply shocks or different demand shocks (see equation 73).
    Adding together (73) and (74) yields, in the cooperative equilibrium,

                      λb2 H [H(et − e∗ ) + 2A3 (ut − u∗ )] λb2 (et + e∗ )
          2πc,t = −                  t                t
                                                          −           t
                                  1 + λb2 H 2
                                         2                   1 + λb22



                                          46
or2
                  1    H 2 (et − e∗ )  et + e∗   λb2 HA3 (ut − u∗ )
       π c,t   = − λb2         2 H 2 + 1 + λb2 −
                                  t          t                  t
                  2    1 + λb2               2     1 + λb2 (H)2
                                                          2
                             1 + λb2 H 2 et − 2A1 A2 (et − e∗ ) HA3 (ut − u∗ )
                                    2                       t
               = −λb2                                          +            t
                                                                                      (77)
                                  [1 + λb2 H 2 ] [1 + λb2 ]
                                         2              2        1 + λb2 H 2
                                                                       2

Then, from (74),
                             et + e∗
       π∗      = −πc,t +           t
        c,t
                             1 + λb2
                                   2
                             1 + λb2 H 2 e∗ + 2A2 A2 (et − e∗ ) HA3 (ut − u∗ )
                                    2      t                t
               = −λb2                                          −            t
                                                                                      (78)
                                  (1 + λb2 H 2 ) (1 + λb2 )
                                         2              2        1 + λb2 H 2
                                                                       2

From (75) and (76),
                      1         1
      yc,t     =                             [H(et − e∗ ) + 2A3 (ut − u∗ )]
                      2    1 + λb2 H 2
                                  2
                                                      t                t

                                         1        1
                                 +                     (et + e∗ )
                                         2     1 + λb2
                                                     2
                                                              t

                      A1 1 + λb2 H 2 et + A2 1 − λb2 H 2 e∗
                               2                       2  t              A3 (ut − u∗ )
               =                                            +                      t
                                                                                       (79)
                             (1 + λb2 H 2 ) (1 + λb2 )
                                     2             2                     1 + λb2 H 2
                                                                                2

and

       ∗           A1 1 + λb2 H 2 e∗ + A2 1 − λb2 H 2 et
                            2       t               2                A3 (ut − u∗ )
      yc,t =                                             −                     t
                                                                                      (80)
                          (1 + λb2 H 2 ) (1 + λb2 )
                                  2             2                    1 + λb2 H 2
                                                                            2

    These results all pertain to the case of a coordinated policy in the two coun-
tries. Under policy without coordination, each country takes the inflation rate
in the other as given in a Nash equilibrium. The home country’s policy maker
                                  2
sets inflation to minimize E λyt + π2 while the foreign country policy maker
                                        t
                                    ∗
sets inflation to minimize E λ(yt )2 + (π∗ )2 . The first order conditions take
the form

                                       λb2 A1 yt + πt = 0

and
                                               ∗
                                       λb2 A1 yt + π∗ = 0
                                                    t

It follows that
                                                          ∗
                               π t − π ∗ = −λb2 A1 (yt − yt )
                                       t
   2 This uses the fact that (A − A )2 = (1 − A − A )2 = (1 − 2A )2 . Expanding the square
                               1    2          2   2            2
yields 1 − 4A2 (1 − A2 ) = 1 − 4A1 A2 .


                                                 47
and
                                                        ∗
                             π t + π ∗ = −λb2 A1 (yt + yt )
                                     t

Using equations (6.43) and (6.44), these become

        πt − π∗
              t   = −λb2 A1 [b2 H(πt − π∗ ) + H(et − e∗ ) + 2A3 (ut − u∗ )]
                                          t            t               t
                      λb2 A1 [H(et − e∗ ) + 2A3 (ut − u∗ )]
                  = −                 t                t
                                                                                        (81)
                                 1 + λb2 A1 H
                                        2

and

                  πt + π∗
                        t    = −λb2 A1 [−b2 (πt + π∗ ) + et + e∗ ]
                                                    t          t
                                     λb2 A1            ∗
                             = −                (et + et )                              (82)
                                  1 + λb2 A1
                                          2

Adding these together,

                  λb2 A1 [H(et − e∗ ) + 2A3 (ut − u∗ )]           λb2 A1
 2π t     = −                     t                t
                                                        +                      (et + e∗ )
                             1 + λb2 A1 H
                                    2                           1 + λb2 A1
                                                                       2
                                                                                      t

                         2A1 1 + λb2 H et + 2A2 e∗
                                      2               t   2λb2 A1 A3 (ut − u∗ )
          = −λb2 A1                                     −                   t
                          [1 + λb2 A1 H] [1 + λb2 A1 ]
                                 2              2            1 + λb2 A1 H
                                                                    2

or

                           A1 1 + λb2 H et + A2 e∗
                                      2            t           A3 (ut − u∗ )
        πN,t = −λb2 A1                                 +                  t
                                                                                        (83)
                          (1 + λb2 A1 H) (1 + λb2 A1 )
                                 2              2              1 + λb2 A1 H
                                                                      2


From (82),

                                             λb2 A1
                  π∗      = −πN,t −                       (et + e∗ )
                   N,t
                                           1 + λb2 A1
                                                  2
                                                                 t

                                           A1 1 + λb2 H e∗ + A2 et
                                                      2     t
                          = −λb2 A1
                                          (1 + λb2 A1 H) (1 + λb2 A1 )
                                                 2              2

                                              A3 (ut − u∗ )
                                         −               t
                                                                                        (84)
                                              1 + λb2 A1 H
                                                     2

The sum and differences of output in the noncooperative equilibrium are

                               ∗     H(et − e∗ ) + 2A3 (ut − u∗ )
                         yt − yt =           t                t
                                                                                        (85)
                                           1 + λb2 A1 H
                                                   2

and

                                ∗           1
                          yt + yt =                     (et + e∗ )                      (86)
                                        1 + λb2 A1
                                              2
                                                               t




                                             48
Hence,

                       A1 1 + λb2 H et + A2 e∗
                                  2            t     A3 (ut − u∗ )
             yN,t =                                +            t
                                                                           (87)
                      (1 + λb2 A1 H) (1 + λb2 A1 )
                             2              2        1 + λb2 A1 H
                                                            2

and

             ∗         A1 1 + λb2 H e∗ + A2 et
                                  2     t                  A3 (ut − u∗ )
            yN,t =                                     −              t
                                                                           (88)
                      (1 + λb2 A1 H) (1 + λb2 A1 )
                             2              2              1 + λb2 A1 H
                                                                  2

   We can now address the specific questions posed in Problem 2. For this
problem, u ≡ x+φ and u∗ ≡ x+φ∗ so that x represents a common demand shock
and φ and φ∗ are uncorrelated country-specific demand shocks, and et ≡ e∗ ≡ 0.
                                                                          t
thus, under a cooperative policy, equations (77), (78), (79), and (80) become

                                    λb2 HA3
                       πc,t = −                    (φt − φ∗ )
                                   1 + λb2 H 2
                                         2
                                                          t



                                    λb2 A3
                        π∗ =                      (φt − φ∗ )
                         c,t
                                  1 + λb2 H 2
                                        2
                                                         t




                                      A3
                        yc,t =                    (φt − φ∗ )
                                  1 + λb2 H 2
                                        2
                                                         t


and
                                       A3
                        ∗
                       yc,t = −                    (φt − φ∗ )
                                   1 + λb2 H 2
                                         2
                                                          t


   From (83), (84), (87), and (88), equilibrium inflation rates and outputs
without cooperation are
                                     λb2 A1 A3
                      πN,t = −                      (φt − φ∗ )
                                   1 + λb2 A1 H
                                          2
                                                           t



                                    λb2 A1 A3
                      π∗ =                         (φt − φ∗ )
                       N,t
                                  1 + λb2 A1 H
                                         2
                                                          t



                                       A3
                       yN,t =                      (φt − φ∗ )
                                  1 + λb2 A1 H
                                        2
                                                          t


and
                                        A3
                       ∗
                      yN,t = −                      (φt − φ∗ )
                                   1 + λb2 A1 H
                                         2
                                                           t




                                        49
Note that inflation response less to the relative demand shocks φt − φ∗ under
                                                                       t
the coordinated policy than under the noncooperative policy as can be seen, for
example, by comparing the coefficients in the equilibrium expressions for πc,t
and πN,t :
                                  λb2 HA3        λb2 A1 A3
                                       2 H 2 < 1 + λb2 A H
                                 1 + λb2              2 1

(To see that the inequality follows, rewrite the comparison as

                   λb2 HA3 1 + λb2 A1 H < λb2 A1 A3 1 + λb2 H 2
                                 2                        2

Dividing both sides by λb2 A3 ,

                             H 1 + λb2 A1 H < A1 1 + λb2 H 2
                                     2                 2

which becomes H < A1 ; recalling that H = A1 − A2 , and both A1 and A2 are
positive, this inequality always holds.) This contrasts with the case considered
in the text; with only a common supply shock, inflation responds more under a
coordinated policy (see page 267).
    To investigate the potential role for policy coordination, we can evaluate the
home country’s loss function under the two policies. With coordination,
                                             2                        2
                        1          A3                  λb2 HA3
          Lc       =      λ                      +                            σ2 + σ2 ∗
                        2       1 + λb2 H
                                      2               1 + λb2 H 2
                                                            2
                                                                               φ    φ


                        1       λA2
                                  3
                   =                        σ2 + σ2 ∗                                         (89)
                        2    1 + λb2 H
                                   2
                                             φ    φ


In the noncooperative equilibrium,
                                              2                           2
                       1           A3                       λb2 A1 A3
       LN      =         λ                        +                               σ2 + σ2 ∗
                       2      1 + λb2 A1 H
                                    2                     1 + λb2 A1 H
                                                                 2
                                                                                   φ    φ


                       1 λA2 1 + λb2 A2
                           3         2 1
               =                                  σ2 + σ2 ∗                                   (90)
                       2 [1 + λb2 A1 H]2
                                2
                                                   φ    φ


Comparing (89) and (90), coordination yields a gain if and only if

                                            LN > Lc

which occurs when
                                  1 + λb2 A2
                                        2 1                   1
                                                2     >
                                [1 +   λb2 A1 H]
                                         2
                                                          1 + λb2 H
                                                                2

This comparison reduces to
                                                                              2
                         1 + λb2 H
                               2       1 + λb2 A2 > 1 + λb2 A1 H
                                             2 1          2



                                                 50
Multiplying out both sides, this becomes

           1 + λb2 H + λb2 A2 + λ2 b4 HA2 > 1 + 2λb2 A1 H + λ2 b4 A2 H 2
                 2       2 1        2   1          2            2 1

or

                     H(1 − 2A1 ) + A2 + λb2 H(1 − H)A2 > 0
                                    1     2          1                        (91)

But 1 − 2A1 = (A1 + A2 ) − 2A1 = −(A1 − A2 ) = −H, so (91) becomes

                        −H 2 + A2 + λb2 H(1 − H)A2 > 0
                                1     2          1

Since A2 − H 2 = (A1 − H)(A1 + H) > 0 and 1 − H > 0, the inequality holds.
        1
    Consider what happens in the face of a positive demand shock to the home
country ( φ > 0). The home country will deflate ( π < 0) to partially offset
the impact of the demand shock on domestic output. Because the home policy
authority takes foreign inflation as given in a Nash equilibrium, it expects this
deflation to produce a real appreciation (see equation 6.42; ρ falls when π falls),
reducing the impact of inflation on domestic output. More inflation volatility is
needed to maintain output stability. Thus, offseting demand shocks is perceived
to be more costly. With a coordinated policy, π is reduced while π ∗ is increased,
thus serving to stabilize output will smaller fluctuations in inflation.


     3. Continuing with the same model as in the previous question, how are real
        interest rates affected by a common aggregate-demand shock?

    In the notation of Problem 2, x was a common aggregate demand shock.
As was shown as part of the solution to Problem 2, neither home nor foreign
inflation is affected by a common demand shock (equations 77, 78, 79, 80, 83,
84, 87, and 88 all depended only on u − u∗ from which a common demand shock
cancels out). With output and inflation independent of x, it must be that a
common demand shock alters real interest rates to offset the demand shock and
maintain aggregate demand constant (since output remains constant). Thus, a
positive value of x should raise real interest rates in both countries; a negative
x should lower real rates.
    To verify this, first note from equation (6.42) on page 263 that a common
demand shock will leave the real exchange rate unchanged; the right side of (6.42)
depends only on terms like π − π∗ , e − e∗ , and u − u∗ , none of which are affected
by a common demand shock (see equations (73) and 81) With ρ unaffected, the
interest parity condition (6.39) implies that r∗ − r must remain unchanged, so
both interest rates change by the same amount. Adding together the aggregate
demand equations (6.37) and (6.38), we can obtain

                         ∗     1
                   rt + rt =                                   ∗
                                    [ut + u∗ − (1 − a3 )(yt + yt )]
                                           t
                               a2



                                         51
If the x disturbance is the only shock, ut + u∗ = 2x and, from either (76) or
                                              t
            ∗
(86), yt + yt is unaffected, so

                                    ∗       1
                              rt + rt = 2        x
                                            a2
Since we have seen that with only an x shock, r − r∗ = 0, it follows that

                                     ∗      1
                               rt = rt =         x
                                            a2



  4. Policy coordination with asymmetric supply shocks: Continuing with the
     same model as in the previous two questions, assume there are no demand
     shocks but that the supply shocks e and e∗ are uncorrelated. Derive policy
     outcomes under coordinated and uncoordinated policy setting. Does co-
     ordination or noncoordination lead to greater or smaller inflation response
     to supply shocks? Explain.

   From equations (77), (78), (79), (80), (83), (84), (87), and (88) that were
derived in the process of solving Problem 2, the outcomes under coordinated and
noncoordinated policies when u ≡ u∗ ≡ 0 are


                              1 + λb2 H 2 et − 2A1 A2 (et − e∗ )
                                     2                       t
               πc,t = −λb2
                                   [1 + λb2 H 2 ] [1 + λb2 ]
                                          2              2



                              1 + λb2 H 2 e∗ + 2A1 A2 (et − e∗ )
                                    2
               π ∗ = −λb2                  t                 t
                 c,t
                                   [1 + λb2 H 2 ] [1 + λb2 ]
                                          2              2



                        A1 1 + λb2 H 2 et + A2 1 − λb2 H 2 e∗
                                 2                       2  t
               yc,t =
                               (1 + λb2 H 2 ) (1 + λb2 )
                                       2             2



                ∗       A1 1 + λb2 H 2 e∗ + A2 1 − λb2 H 2 et
                                 2       t               2
               yc,t =
                               (1 + λb2 H 2 ) (1 + λb2 )
                                       2             2



                                   A1 1 + λb2 H et + A2 e∗
                                              2            t
                 πN,t = −λb2 A1
                                  (1 + λb2 A1 H) (1 + λb2 A1 )
                                         2              2



                                   A1 1 + λb2 H e∗ + A2 et
                                              2
                 π∗ = −λb2 A1                       t
                  N,t
                                  (1 + λb2 A1 H) (1 + λb2 A1 )
                                         2              2



                                      52
                                  A1 1 + λb2 H et + A2 e∗
                                             2            t
                       yN,t =
                                 (1 + λb2 A1 H) (1 + λb2 A1 )
                                        2              2

and

                        ∗         A1 1 + λb2 H e∗ + A2 et
                                             2     t
                       yN,t =
                                 (1 + λb2 A1 H) (1 + λb2 A1 )
                                        2              2

   Under a coordinated policy, the response to a domestic supply shock home
inflation is

                                       1 + λb2 H 2 − 2A1 A2
                                             2
                      π c,t = −λb2                              et
                                      (1 + λb2 H 2 ) (1 + λb2 )
                                             2              2

while under noncoordinated policy it is

                                           A2 1 + λb2 H
                                            1         2
                   πN,t = −λb2                                    et
                                     (1 + λb2 A1 H) (1 + λb2 A1 )
                                            2              2

These expressions are messy, so an easier way of comparing outcomes is to
return to equation (73) and (74) for the case of cooperation and equations (81)
and (82) for the noncooperation case. For the sum of inflation rates, (74) and
(82) imply that π t +π∗ responds more to et under cooperation that in the absence
                         t
of cooperation (the coefficient on e in 82 is increasing in A1 — which is less than
1 — and under cooperation, the response in 74 is obtained by setting A1 = 1).
So (π t + π ∗ )c responds more than (πt + π∗ )N . On the other hand, (π t − π ∗ )c
              t                                 t                                t
responds less than (πt − π∗ )N as can be seen by comparing (73) and (81).3
                                t
The results that (πt + π∗ )c > (π t + π ∗ )N and (π t − π ∗ )c < (πt − π∗ )N allows
                             t             t                t              t
us to conclude that foreign inflation responds more (add the two inequalities
together), but it does not resolve whether domestic inflation responds more. We
saw in the text that cooperation leads to a larger inflation response to a common
supply shock than occurred without cooperation. Under cooperation in the face
of a home country supply shock, more of the adjustment is made by foreign
inflation than would occur without cooperation, and this acts to allow domestic
inflation to respond less. So inflation rate in the two countries diverge less
( (π t − π ∗ )c < (πt − π∗ )N ). But cooperation tends to lead to a stronger overall
           t               t
response ( (π t + π ∗ )c > (πt + π∗ )N ), so the net effect on π t is not clear.
                     t             t



   5. Assume the home-country policy maker acts as a Stackelberg leader and
      recognizes that foreign inflation will be given by equation (6.47). How
      does this change in the nature of the strategic interaction affect the home
      country’s response to disturbances?

   3 This uses the fact that A H > H 2 since A > A (see the definitions of the A s on page
                              1               1   2                            i
264).



                                           53
   Referring to section 6.3.3 of the text, the home policy authority picks in-
                       2
flation to minimize λyt + π2 , but now, rather than taking foreign inflation π∗
                              t                                                 t
as given, the home policy authority recognizes that π ∗ will be set according to
                                                        t
equation (6.47). Equation (6.47) was derived for the case of a single common
supply shock; et = e∗ = εt . The first order condition for the home country will
                     t
reflect this dependence of π∗ on π t . Thus, the first order condition for the home
                            t
country’s choice of inflation will be
                                       λb2 A1 A2
                                          2
                   λ b2 A1 − b2 A2                      yt + π t = 0             (92)
                                       1 + λb2 A2
                                             2 1

This should be compared to the expression immediately above equation (6.46) on
                                      λb2 A A
page 266. the extra term −λb2 A2 1+λb1 A2 yt is the effect of that the home
                                        2
                                           2 2
                                           2 1
country’s inflation rate has on home country output by causing foreign inflation
to adjust based on the reaction function given by (6.47).
    To solve (92), first note that the coefficient on yt can be rewritten as
                                  λb2 A1 A2
                                    2                     1 + λb2 H
                                                                2
                b2 A1 − b2 A2                  = b2 A1
                                 1 + λb2 A2
                                        2 1               1 + λb2 A2
                                                                2 1

since (A2 − A2 ) = H(A1 + A2 ) and A1 + A2 = 1. If we define
        1    2

                                      1 + λb2 H
                                            2
                                 S≡              <1
                                      1 + λb2 A2
                                            2 1
then the first order condition for the home country becomes
                                  λb2 A1 Syt + πt = 0
which contrasts with the first order conditon in the Nash case ( λb2 Ayt +πt = 0).
One why to interpret this is that because S < 1, the marginal output efects of a
rise in home inflation are now smaller, since the home policy maker recognizes
that higher π will induce the foreign country to reduce its inflation rate, causing
a depreciation for the home country ( ρ rises; see 6.42) that acts to offset the
expansionary impact of the rise in domestic inflation (see 6.35). It will be
optimal for π to response less. .
    Also, using the foreign country’s reaction function, home country output
(from 6.43) can be written
                                  λb2 A1 A2
                                    2                   λb2 A1
      yt    = b2 A1 πt − b2 A2          2 A2 π t +                     εt + εt
                                 1 + λb2 1            1 + λb2 A2
                                                             2 1
                            2                  2
                      1 + λb2 H          1 + λb2 A1 H
            = b2 A1              πt +                   εt
                      1 + λb2 A2
                            2 1           1 + λb2 A2
                                                 2 1

Now use the expression for home country output (equation 6.43 of the text) to
write the first order condition (92) as
           1 + λb2 A1 H
                 2                 1 + λb2 H
                                         2         1 + λb2 A1 H
                                                         2
 λb2 A1                   b2 A1               πt +              εt        + πt = 0
            1 + λb2 A2
                   2 1             1 + λb2 A2
                                         2 1        1 + λb2 A2
                                                           2 1




                                          54
or
                                                                       2
                                      λb2 A1 1 + λb2 A1 H
                                                   2
                        πt = −                                             2   εt
                                 (1 + λb2 A2 S s ) (1 + λb2 A2 )
                                        2 1               2 1

where
                                      1 + λb2 A1 H
                                            2                1 + λb2 H
                                                                   2
                              Ss ≡                              2
                                              (1 + λb2 A2 )
                                                     2 1

From (83), the response to a common supply shock in a Nash equilibrium is
                                                  λb2 A1
                                 πt = −                      εt
                                               (1 + λb2 A1 )
                                                       2




     6. In a small open economy with perfectly flexible nominal wages, the text
        showed that the real exchange rate and domestic CPI were given by
                                     ∞                 ∗
                                                   a2 rt+i + et+i − ut+i
                              ρt =         di Et
                                     i=0
                                                        a1 + a2 + b1

        and
                                      ∞                 i
                               1              1
                       pt =                                 Et [mt+i − zt+i − vt+i ]
                              1+c    i=0
                                             1+c

        where zt+i ≡ yt+i + (1 − h)ρt+i − crt+i . Assume r∗ = 0 for all t and
        that e, u, and z + v all follow first order autoregressive processes (e.g.
        et = ρe et−1 + xet for xe white noise). Let the nominal money supply be
        given by
                           mt = g1 et−1 + g2 ut−1 + g3 (zt−1 + vt−1 )
        Find equilibrium expressions for the real exchange rate, the nominal ex-
        change rate, and the consumer price index. What values of the parameters
        g1 , g2 , and g3 minimize fluctuations in st ? in qt ? in ρt ? Are there any
        conflicts between stabilizing the exchange rate (real or nominal) and sta-
        bilizing the consumer price index?

    The first thing to note is that under the assumptions of the problem, the real
exchange rate ρt is independent of the money supply process, depending only on
the exogenous behavior of r∗ , et , and ut . Thus, in this example, the behavior of
ρt is unaffected by the choice of the gi parameters. We can use the assumptions
of the problem to write the real exchange rate as
          ∞
                    ρi et − ρi ut               1                      1            1
  ρt =         di    e       u
                                  =                                         et −         ut   (93)
         i=0
                    a1 + a2 + b1           a1 + a2 + b1             1 − dρe      1 − dρu


                                                   55
    To evaluate the expression for the equilibrium price pt , we do need to use
the money supply process:
                 ∞           i
          1             1
pt   =                           Et [g1 et−1+i + g2 ut−1+i + g3 ω t−1+i − ωt+i ]
         1+c     i=0
                       1+c
                  ∞          i
          1             1
     =                            g1 ρi et−1 + g2 ρi ut−1 + g3 ρi ω t−1 − ρi (ρω ω t−1 + xωt )
         1+c     i=0
                       1+c            e            u            ω          ω


where ω t ≡ zt − vt . Collecting terms,
                 ∞           i
          1             1
pt   =                            g1 ρi et−1 + g2 ρi ut−1 + (g3 − ρω ) ρi ωt−1 − ρi xωt
         1+c     i=0
                       1+c            e            u                    ω         ω


              g1 et−1              g2 ut−1
     =                     +
           1 + c(1 − ρe )      1 + c(1 − ρu )
                      (g3 − ρω ) ωt−1             xωt
                   +                     −                                            (94)
                        1 + c(1 − ρω         1 + c(1 − ρω )

So pt is stabilized if g1 = g2 = 0 and g3 = ρω . Since neither e nor u affect p in
this setup, any response by m to these disturbances would simply add additional
variance to the price level. By setting g3 = ρω , policy is able to insulate pt from
the forecastable movements in ωt .
    The nominal exchange rate st is given by st = ρt − p∗ + pt where p∗ is the
foreign price level. For simplicity, set p∗ = 0. Then st = ρt + pt . Combining
(93) and (94), the nominal exchange rate is

                1              1                          1
st   =                              (ρ et−1 + xet ) −          (ρ ut−1 + xut )
           a1 + a2 + b1     1 − dρe e                 1 − dρu u
                g1 et−1              g2 ut−1          (g3 − ρω ) ωt−1            xωt
         +                    +                   +                    −
             1 + c(1 − ρe )       1 + c(1 − ρu )        1 + c(1 − ρω        1 + c(1 − ρω )

The effects of ω shocks on the nominal exchange rate are minimized if g3 = ρω .
The impact of et−1 is eliminated if

                       1                ρe                 g1
                                                 +                      =0
                  a1 + a2 + b1       1 − dρe          1 + c(1 − ρe )
or
                                    1 + c(1 − ρe )        ρe
                       g1 = −
                                    a1 + a2 + b1       1 − dρe

The effects ut−1 on st can be eliminated if

                        1                ρu                  g2
             −                                    +                     =0
                   a1 + a2 + b1       1 − dρu          1 + c(1 − ρu )


                                            56
or
                                  1 + c(1 − ρu )           ρu
                        g2 =
                                  a1 + a2 + b1          1 − dρu
     In terms of consumer prices qt , from equation (6.52) on page 270,

                          qt   = hpt + (1 − h)(st + p∗ )
                                                     t
                               = pt + (1 − h)ρt                                (95)

Combining this with (93) and (94),

              g1 et−1             g2 ut−1           (g3 − ρω ) ωt−1            xωt
qt   =                    +                    +                     −
           1 + c(1 − ρe )     1 + c(1 − ρu )          1 + c(1 − ρω       1 + c(1 − ρω )
                 1−h            1                           1
         +                            (ρ et−1 + xet ) −          (ρ ut−1 + xut )
             a1 + a2 + b1   1 − dρe e                    1 − dρu u
The effects of ω shocks on the consumer price index are minimized if g3 = ρω .
The impact of et−1 is eliminated if
                      g1                    1−h                ρe
                                   +                                  =0
                 1 + c(1 − ρe )          a1 + a2 + b1       1 − dρe
or
                                        1 + c(1 − ρe )         ρe
                   g1 = −(1 − h)
                                        a1 + a2 + b1        1 − dρe
The effects ut−1 on qt can be eliminated if

                       g2                   1−h                ρu
                                   −                                  =0
                 1 + c(1 − ρu )          a1 + a2 + b1       1 − dρu
or
                                       1 + c(1 − ρu )         ρu
                    g2 = (1 − h)
                                       a1 + a2 + b1        1 − dρu
    There are no conflicts (at least in this example) between stabilizing the nomi-
nal exchange rate and stabilizing the consumer price level in the face of ω shocks
( z − v). Since ω has no effect on the real exchange rate, stabilizing domestic
prices would also stabilize the nominal exchange rate and the consumer price
level. For e and u disturbances, the appropriate responses of m to stabilize q
are proportional to the optimal responses to stabilize s, but the responses are
smaller in absolute value by a factor 1 − h if the objective is to stabilize q. Both
e and u affect the real exchange rate. Since qt = pt + (1 − h)ρt , policy can
stabilize q by making p move to offset any movement in (1 − h)ρ. The nominal
exchange rate, however, is equal to ρt + pt , so it is stabilized it p fully adjusts
to offset movements in ρ.



                                            57
  7. Equation (6.42) for the equilibrium real exchange rate in the two-country
     model of section 6.3.1 takes the form ρt = AEt ρt+1 + vt . Suppose vt =
     γvt−1 + ψt , where ψt is a mean-zero, white-noise process. Suppose the
     solution for ρt is of the form ρt = bvt . Find the value of b. How does it
     depend on γ?

    First note that we can think of v as having a direct impact on ρ, holding
the expected future real exchange rate constant, and an indirect effect if v alters
Et ρt+1 . Under the proposed solution, Et ρt+1 = bEt vt+1 = bγvt . Substituting
this into the equilibrium condition for the real exchange rate,

                         ρt = Abγvt + vt = (1 + Abγ) vt

This can equal bvt for all realizations of vt if only if

                                  b = (1 + Abγ)

from which it follows that
                                           1
                                   b=
                                        1 − Aγ
    The expression for b shows that a rise in γ increases b and leads an inno-
vation in v to have a larger impact on the real exchange rate. This can be seen
more clearly by writing the solution for the real exchange rate as

                                ρt = bγvt−1 + bψt

If γ is large (i.e., close to 1 say), then innovations ψ to the v process are very
persistent. Therefore, in addition to the direct impact of v on ρ, there will be a
larger impact on Et ρt+1 if γ is large. Thus, the total impact of an innovation
ψt on the current real exchange rate ρt will be larger (i.e., b is larger).




                                        58
6     Chapter 8: Discretionary Policy and Time In-
      consistency
    1. Assume firms maximize profits in competitive factor markets with labor
       the only variable factor of production. Output is produced according to
       the production function Y = ALα , 0 < α < 1. Labor is supplied inelas-
       ticly. Nominal wages are set at the start of the period at a level consistent
       with market clearing, given expectations of the price level. Actual employ-
       ment is determined by firms once the actual price level is observed. Show
                                                                           1
       that, in log terms, output is given by y = αl∗ + 1−α (p − pe ) + 1−α ln αA,
                                                          α
               ∗
       where l is the log labor supply. (Note: the text contains a typo: al∗
       appears instead of αl∗ .)

   Given the assumed form of the production function, the demand for labor
can be obtained from the condition that firms set the marginal product of labor,
αALα−1 , equal to the real wage W/P :
                                                  −1
                                                 1−α
                                           W
                                Ld =
                                          αAP
If L∗ is the fixed supply of labor, the equilibrium real wage that equates labor
demand and labor supply is
                                     ∗
                                W
                                         = αA (L∗ )α−1
                                P
or, in log terms,
                            (w − p)∗ = ln αA − (1 − α)l∗
If workers and firms set the nominal wage at the start of the period, prior to
observing actual prices, then the contract wage consistent with labor market
clearing is
                            wc = Ep + ln αA − (1 − α)l∗
Actual (log) employment, given by the demand for labor, will be
                                   1
                        l = −         (wc − p − ln αA)
                                 1−α
                                   1
                            = −        Ep − p − (1 − α)l
                                 1−α
                                     1
                            = l∗ +       (p − Ep)
                                    1−α
and the log of output is equal to
                        y   = ln A + αl
                                          α
                            = ln A +         (p − Ep) + αl∗
                                         1−α

                                          59
  2. Suppose an economy is characterized by the following three equations:

                                   π = πe + ay + e

                                     y = −br + u

                                  m − π = −di + y + v

     where the first equation is an aggregate supply function written in the
     form of an expectations-augmented Phillips Curve, the second is an “IS” or
     aggregate-demand relationship, and the third is a money demand equation
     where m denotes the growth rate of the nominal money supply. The real
     interest rate is denoted by r and the nominal rate by i, with i = r + πe .
     Let the monetary authority implement policy by setting i to minimize the
     expected value of 1 λ(y − y ∗ )2 + π2 where y∗ > 0. Assume the policy
                         2
     authority has forecasts ef , uf , and v f of the shocks, but the public forms
     its expectations prior to the setting of i and without any information on
     the shocks.

      (a) Assume the monetary authority can commit to a policy of the form
          i = c0 + c1 ef + c2 uf + c3 vf prior to knowing any of the realizations of
          the shocks. Derive the optimal commitment policy (i.e., the optimal
          values of c0 , c1 , c2 , and c3 ).
      (b) Derive the time-consistent equilibrium under discretion. How does
          the nominal interest rate compare to the case under commitment?
          What is the average inflation rate?

   (a) From the IS relationship, y = −b(i − π e ) + u, so if we now use the
aggregate supply relationship, π = πe − ab(i − πe ) + au + e, or

                         π = (1 + ab)πe − abi + au + e                         (96)

Taking expectations of both sides, conditional on the public’s information, π e =
ie . Hence, inflation is

                         π = (1 + ab)ie − abi + au + e

and the objective function, expressed in terms of the policy instrument i becomes
        1
     L ≡ E λ (−b(i − ie ) + u − y∗ )2 + [(1 + ab)ie − abi + au + e]2           (97)
        2
Under the commitment policy, the central bank follows a policy of the form

                          i = c0 + c1 ef + c2 uf + c3 vf                       (98)


                                        60
where xf denotes the central bank’s forecast of x. With this policy rule, ie = c0 .
Substituting this result and the policy rule into (97) gives
                      1                                          2
              L ≡       Eλ −b(c1 ef + c2 uf + c3 v f ) + u − y ∗
                      2
                        1                                            2
                      + E c0 − ab c1 ef + c2 uf + c3 vf + au + e              (99)
                        2
  The objective is to minimizing this function by the choice of c0 , c1 , c2 and
c3 where the choices are made prior to actually observing any of the shocks or
forecasts. For c0 , the first order condition is

                E c0 − ab c1 ef + c2 uf + c3 vf + au + e = c0 = 0

For c1 , the first order condition (using c0 = 0) is

             0 = E λ −b(c1 ef + c2 uf + c3 vf ) + u − y ∗ (−bef )
                     +E −ab c1 ef + c2 uf + c3 vf + au + e (−abef )

If the shocks are mutually uncorrelated, this becomes

                            c1 (λ + a2 )b2 σf − abσf = 0
                                            e      e

where σf is the variance of the forecast of e and the result from rational expec-
        e
tations that E(eef ) = σf has been used Hence,
                        e
                                             a
                                  c1 =
                                         (λ + a2 )b
     For c2 , one obtains

             0 = E λ −b(c1 ef + c2 uf + c3 vf ) + u − y∗ (−buf )
                    +E −ab c1 ef + c2 uf + c3 vf + au + e (−abuf )

or

                        c2 (λ + a2 )b2 σ f − (λ + a2 )bσf = 0
                                         u              u

yielding
                                              1
                                      c2 =
                                              b
The first order condition for c3 is

            0 = Eλ −b(c1 ef + c2 uf + c3 v f ) + u − y ∗ (−bvf )
                    +E −ab c1 ef + c2 uf + c3 vf + au + e (−abvf ))

or

                                       c3 = 0


                                         61
     Hence, the optimal commitment policy is

                                    1             a
                             ic =       uf +           ef
                                    b           λ + a2

which does not depend on the money demand function at all.
    (b) Under discretion, the central bank treats expectations as given in choosing
i to minimize its expected loss function, based on its forecasts of the underlying
shocks. In this case, the expectations operator in the loss function (97) is con-
ditional on ef , uf , and vf . The first order condition for the choice of i under
discretion is

     (−b)λ −b(i − ie ) + uf − y ∗ − ab (1 + ab)ie − abi + auf + ef = 0

or

       λ b2 (i − ie ) − buf + by∗ − ab (1 + ab)ie − abi + auf + ef = 0

Solving for i,

                      (λb + a(1 + ab)) ie − λy∗ + (λ + a2 )uf + aef
                 i=                                                          (100)
                                        b(λ + a2 )

     Taking expectations based on the public’s information,
                                               λy∗
                                        ie =
                                                a
Hence, (100) becomes

                               λy ∗ 1                  a
                          i=       +       uf +             ef               (101)
                                a    b               λ + a2

Notice that policy under discretion responds to the stochastic shocks the same
way as policy would if commitment were possible. However, the nominal rate
under discretion is systematically higher than under commitment. Since expected
inflation is equal to ie ,
                                           λy ∗
                                    πe =        >0
                                            a



     3. Verify that the optimal commitment rule that minimizes the unconditional
        expected value of the loss function given by (8.10) is mc = − 1+a2 λ e.
                                                                       aλ




                                           62
   The loss function (8.10) is
                    1                       2 1               2
               V c = λ [a(b1 e + v) + e − k] + [b0 + b1 e + v]
                    2                         2
Under the commitment policy, the central bank chooses b0 and b1 to minimize
the unconditional expectation of V c . If the shocks are uncorrelated, we can write
this expectation as
             1                                1 2
       EV c = λ (1 + ab1 )2 σ2 + a2 σ2 + k2 +   b + b2 σ 2 + σ2
             2               e       v
                                              2 0    1 e      v


where σ2 is the variance of x. If we minimize this with respect to b0 and b1 ,
        x
the first order condition for b0 is

                                       b0 = 0

while that for b1 is

                             aλ(1 + ab1 )σ 2 + b1 σ 2 = 0
                                           e        e

or b1 = −aλ/(1 + a2 λ). Hence, the optimal commitment policy is

                                             aλ
                                  mc = −            e
                                           1 + a2 λ
as claimed.


  4. Suppose the central bank acts under discretion to minimize the expected
     value of equation (8.2). The central bank can observe e prior to setting
        m, but v is observed only after policy is set. Assume, however, that e
     and v are correlated, and that the expected value of v, conditional on e,
     is E [v|e] = qe where q = σ v,e /σ2 and σv,e is the covariance between e and
                                       e
     v.

       (a) Find the optimal policy under discretion. Explain how policy de-
           pends on q.
      (b) What is the equilibrium rate of inflation? Does it depend on q?

    (a) The loss function (8.2) is quadratic in inflation and the deviation of
output from a target level. Taking the model to consist of equations (8.2) -
(8.4), the central bank’s objective function can be written as
                       1                           1
         EV    =         Eλ [a(π − πe ) + e − k]2 + E [π]2
                       2                           2
                       1                                   1
               =         Eλ [a( m + v − m ) + e − k]2 + E [ m + v]2
                                              e
                       2                                   2


                                         63
If the central bank takes private sector expectations as given and can observe e
prior to setting policy, then the first order condition for the optimal choice of
  m is

             aλ [a( m + E [v|e] −   me ) + e − k] +   m + E [v|e] = 0

or

                 aλ [a( m + qe −    me ) + e − k] +   m + qe = 0

Hence,

                           a2 λ me + aλk − 1 + q(1 + a2 λ e
                     m=
                                       1 + a2 λ
Taking expectations, conditional on the public’s information set,

                                    me = aλk

so
                                        1 + q(1 + a2 λ e
                           m = aλk −                                      (102)
                                            1 + a2 λ
    The optimal respond to e will depend on q. If the central bank could observe
v, it would adjust m to offset the impact of v on inflation. If e provides
some information on which the central bank can base a forecast of v, then it
will adjust m to offset the forecasted impact of v on inflation. To see this,
note that (102) can be written as
                                        1
                         m = aλk −            e − qe
                                     1 + a2 λ
                                        1
                             = aλk −          e − E [v|e]
                                     1 + a2 λ

                                                              (1+q(1+a2 λ)e
    (b) The equilibrium rate of inflation is m + v or aλk −       1+a2 λ     + v.
this depends on q since the central bank’s choice of money growth is a function
of q. The average rate of inflation, or the inflation bias, is, however, equal to
aλk, and this is independent of q.


     5. Since the tax distortions of inflation are related to expected inflation,
        suppose the loss function (8.2) is replaced by
                                                        2
                              L = λ(y − yn − k)2 + (πe )

       where y = yn + a(π − πe ). How is Figure 8.2 modified by this change in
       the central bank’s loss function? Is there an equilibrium inflation rate?
       Explain.


                                       64
    The inflation loss is now assoicated with expected inflation. Since the central
bank is assumed, under discretion, to take expected inflation as given, it will now
view the costs of inflation as given, so increasing inflation a bit is perceived by
the central bank to yield benefit in terms of higher output but no cost.
    To see this, substitute the aggregate supply relationship into the loss function
to obtain
                                                 2        2
                           L = λ [a(π − πe ) − k] + (π e )

If this is minimized with respect to π, the first order condition is

                               2aλ [a(π − πe ) − k] = 0

or
                                                k
                                   π = πe +                                   (103)
                                               a2 λ
There is no expected rate of inflation such that π e = πe + k/a2 λ. Figure 8.2,
giving the central bank’s optimal choice of inflation, as a function of the public’s
expected rate of inflation, shows that the central bank’s reaction function given
by (103) does not cross the 45◦ line.




     6. Based on Jonsson (1995) and Svensson (1997b). Suppose equation (8.3)
        is modified to incorporate persistence in the output process:

              yt = (1 − θ)yn + θyt−1 + a(π t − π e ) + et ;
                                                 t            0<θ<1

       Suppose the policy maker has a two-period horizon with objective function
       given by

                                  L = min E [Lt + δLt+1 ]
                    1
       where Li =   2   λ(yi − yn − k)2 + π2 .
                                           i

        (a) Derive the optimal commitment policy.
        (b) Derive the optimal policy under discretion without commitment.
        (c) How does the presence of persistence (θ > 0) affect the inflation bias?

    (a) Under commitment, the central bank follows a rule that specifies how
inflation will be set as a function of the state of the economy. The rule is chosen
before the central bank knows the current state of the economy, and the public
uses the rule to form their expectations about policy. They can do so since, by
assumption, the central bank is committed to following the rule. Since the state


                                          65
                                                      Figure 1: Dashed line is the central bank’s reaction function.
                                           0.35




                                            0.3
   Central Bank's Planned Inflation Rate




                                           0.25




                                            0.2




                                           0.15




                                            0.1




                                           0.05




                                             0
                                                  0            0.05        0.1            0.15          0.2   0.25     0.3
                                                                                   Expected Inflation




                                                                                 Figure 2:

at time t is characterized by yt−1 and et , one can specify the commitment policy
as
                                                                         πt = b0 + b1 xt−1 + b2 et                     (104)
for period t, and
                                                                        πt+1 = b0 + b1 xt + b2 et+1                    (105)
for period t + 1 where
                                                                            xt−1 ≡ yt−1 − yn
will be used to denote the output gap. Because the policy choice at time t may,
through xt , affect output at time t + 1, while any future effect of policy in t + 1
doesn’t matter (since the loss function only involves Lt and Lt+1 ), the optimal
response to et may differ from the optimal response to et+1 . For this reason,
the coefficients in (104) and (105) are allowed to differ.
    We need to find the optimal values of the coefficients in the policy rules that
minimize the unconditional value of the loss function L. Here, we can think of
the central bank deciding on the parameters of the policy rule prior to having
any information about the state of the economy. Thus, it evaluates things from
the perspective of the unconditional expectation of et (equal to zero) and xt−1
(also equal to zero)

                                                                                     66
   The public knows the value of xt−1 when forming expectations of time t, so
using the policy rule (104),

                                  πe = b0 + b1 xt−1
                                   t

and

                                    πt − πe = b2 et
                                          t

Using the aggregate supply relationship,

                              xt = θxt−1 + (1 + ab2 )et

Similarly, π t+1 − πe = b2 et+1 and
                    t+1

                 xt+1    = θxt + (1 + ab2 )et+1
                         = θ2 xt−1 + θ(1 + ab2 )et + (1 + ab2 )et+1

Substituting these expressions into the loss function yields
                        1
E [Lt + δLt+1 ] =         E λ [θxt−1 + (1 + ab2 )et − k]2 + (b0 + b1 xt−1 + b2 et )2
                        2
                          1                                                  2
                        + δEλ θ 2 xt−1 + θ(1 + ab2 )et + (1 + ab2 )et+1 − k
                          2
                          1                                              2
                        + δE [b0 + b1 (θxt−1 + (1 + ab2 )et ) + b2 et+1 ]
                          2
Now minimize this with respect to the parameters in the policy rule. Doing so
yields the following first order conditions:
    For b0 :

                              E [b0 + b1 xt−1 + b2 et ] = 0

or

                                        b0 = 0

     For b1 :

                          E [(b0 + b1 xt−1 + b2 et ) xt−1 ] = 0

or

                                        b1 = 0

     For b2 :

        0 = Eλ [(θxt−1 + (1 + ab2 )et − k) aet ] + (b0 + b1 xt−1 + b2 et ) et
            +λδE θ2 xt−1 + θ(1 + ab2 )et + (1 + ab2 )et+1 − k θaet
            +λδE [(b0 + b1 (θxt−1 + (1 + ab2 )et ) + b2 et+1 ) b1 aet ]

                                           67
                                                                2
or aλ(1 + ab2 )σ2 + b2 σ 2 + λδθ2 a(1 + ab2 )σ2 + λδa (b1 ) (1 + ab2 )σ2 = 0. Solving
                e        e                    e                        e
for b2 ,

                                     aλ + aλδθ2 + aλδ (b1 )2
                         b2 = −
                                  1 + a2 λ + a2 λδθ2 + a2 λδ (b1 )2

which depends on b1 . So turning to the first order condition for b1 ,

     δE [(b0 + b1 (θxt−1 + (1 + ab2 )et ) + b2 et+1 ) (θxt−1 + (1 + ab2 )et )] = 0

or δb1 E (θxt−1 + (1 + ab2 )et )2 = 0 which implies

                                           b1 = 0

Substituting this into the expression for b2 ,

                                           aλ 1 + δθ 2
                              b2 = −
                                         1 + a2 λ 1 + δθ 2

The first order condition for the optimal b2 is

          0 = δE λ θ 2 xt−1 + θ(1 + ab2 )et + (1 + ab2 )et+1 − k aet+1
              +δE [(b0 + b1 (θxt−1 + (1 + ab2 )et ) + b2 et+1 ) et+1 ]

or

                               δaλ(1 + ab2 )σ 2 + δb2 σ 2 = 0
                                              e         e

which yields

                                                aλ
                                    b2 = −
                                              1 + a2 λ

Finally, it is straightforward to show that b0 = 0. Thus, the optimal commitment
policy takes the form

                                          aλ 1 + δθ2
                            πc = −                              et                   (106)
                                        1 + a2 λ 1 + δθ2
                             t


and
                                               aλ
                                πc = −                   et+1                        (107)
                                 t+1
                                             1 + a2 λ

    Notice that average inflation is zero in each period. Also, the response to
et+1 given by b2 is exactly the same as obtained in, for example, equation (8.11)
of the text. Since in period t + 1 we have a standard one period problem, this is
not surprising. In period t, however, the optimal response to et differs from the

                                             68
standard optimal response to a supply shock in a one-period model if δθ is non-
zero. Assuming δ > 0 (the central bank cares about both period), the optimal
commitment policy in period t reduces to the standard result if θ = 0. If θ = 0,
then the optimal response to a period t supply shock as affected. Suppose θ > 0.
Then it becomes optimal to offset more of the impact of et on period t output
by responding more strongly to et ;

                               aλ 1 + δθ2           aλ
                                              >
                           1 + a2 λ 1 + δθ2       1 + a2 λ

Since xt will affect xt+1 , period t + 1 output can be made more stable by insu-
lating xt more from et .
    (b) Under discretion, the central bank picks π in each period, taking expec-
tations and the previous period’s output as given. Solving backwards, the central
bank will choose inflation in period t + 1 to minimize
                1                                           2    1
                  λ θxt + a(πt+1 − πe ) + et+1 − k              + π t+1
                2                   t+1
                                                                 2
taking xt and π e as given. The first order condition is
                t+1

                aλ θxt + a(π t+1 − π e ) + et+1 − k + πt+1 = 0
                                     t+1

or
                               a2 λπe − aλ (θxt + et+1 − k)
                                    t+1
                      πt+1 =
                                        1 + a2 λ
Hence,

                                πe = aλk − aλθxt
                                 t+1

Expected inflation is decreasing in last period’s output if output displays positive
persistence ( θ > 0). A positive value of xt means that output in period t + 1
will be closer to the desired level yn + k. Thus, the central bank’s incentive to
inflate is reduced. Anticipating this, the public expects lower inflation.
    Actual inflation in period t + 1 under discretion will equal

                                                   aλ
                      πd = aλk − aλθxt −                        et+1         (108)
                       t+1
                                                 1 + a2 λ

and output will be

                                                     1
                     yt+1 = yn + θ(yt − yn ) +                   et+1
                                                  1 + a2 λ

Notice that the response to et+1 is the same as under the optimal committment
policy given by equation (107).


                                        69
     The central bank’s loss in period t + 1 is
                                          2                                          2
           1           1                          1                 aλ
     Lt+1 = λ θxt +          et+1 − k         +     aλk − aλθxt −          et+1
           2        1 + a2 λ                      2               1 + a2 λ


    Now use these results to evaluate the central bank’s decision problem in period
t. The two period loss function is
                                                                              2
                         2       1             1
          L = λE [xt − k] + Eπ2 + δλE θxt +          et+1 − k
                              t
                                 2          1 + a2 λ
                                                            2
                    1                   aλ
                   + δE aλk − aλθxt −          et+1
                    2                 1 + a2 λ

Since xt = θxt−1 + a(πt − πe ) + et , the first order condition for the optimal
                               t
inflation rate in period t, taking xt−1 and expectations as given, is

                                                            1
            0 = aλ [xt − k] + πt + aθδλE θxt +                    et+1 − k
                                                         1 + a2 λ
                                                     aλ
                    −a2 λθδE aλk − aλθxt −                  et+1
                                                   1 + a2 λ
or

               0 = aλ [θxt−1 + a(πt − πe ) + et − k] + πt
                                       t
                   +aθδλ [θ(θxt−1 + a(πt − πe ) + et ) − k]
                                              t
                   −a2 λθδ [aλk − aλθ (θxt−1 + a(πt − πe ) + et )]
                                                          t

Solving for πt ,

              a2 λ 1 + δθ2 + aλθ2 δ 2                    aλ 1 + δθ + a2 λθδ
πt    =                                       πe +                                       k
            1 + a2 λ 1 + δθ2 + a2 λθ2 δ               1 + a2 λ 1 + δθ 2 + a2 λθ2 δ
                                               t


                        1 + δθ3 + a2 λθ 3 δ                             1 + δθ2 + a2 λθ2 δ
          −aλθ                                       xt−1 − aλ                                   et
                   1 + a2 λ 1 + δθ2 + a2 λθ2 δ                     1 + a2 λ 1 + δθ2 + a2 λθ2 δ

It follows that

            π e = aλ 1 + δθ + a2 λθδ k − aλθ 1 + δθ3 + a2 λθ3 δ xt−1
              t

and inflation under discretion is equal to

          πd
           t   = aλ 1 + δθ(1 + a2 λ) k − aλθ 1 + δθ3 + a2 λθ3 δ xt−1
                                1 + δθ 2 (1 + a2 λ)
                    −aλ                                     et                       (109)
                           1 + a2 λ 1 + δθ2 (1 + a2 λ)



                                          70
    (c) Optimal discretionary policy is characterized by equations (109) for period
t and (108) for period t+1. In period t, the fact that output displays persistence
does affect the inflation bias. Creating an output expansion in period t will tend
to raise output in period t + 1, so this increases the incentive to expand output
in period t. Anticipating this, the public expects higher inflation in period t, and
average inflation equals aλ 1 + δθ(1 + a2 λ) k which exceeds the one-period bias
aλk. The inflation bias is increasing in θ since a larger θ implies that any output
expansion in t will lead to a larger expansion in t + 1. In addition, a low value
of yt−1 (so xt−1 is negative) acts to move yt further from the central bank’s
desired level yn −k; this increases the incentive to inflate and raises the expected
and actual rate of inflation. Comparing (109) and (106) shows that the period t
response to et is also distorted under discretion. this is in contrast to the result
we have usually found. When θ = 0 , however, the response to et is larger, in
absolute value, under discretion


  7. Show that πd given by (8.18) is equal to inflation under discretion when the
     weight on inflation in (8.2) becomes 1+δ and the economy is characterized
     by (8.3) and (8.4).

      The central bank’s decision problem under discretion can be written as
                            1                           1
                    min       λ [a(π − π e ) + e − k]2 + (1 + δ) π 2
                            2                           2
      where π = m + v and the central bank observes e but not v before
      determining policy. The first order condition is

                        aλ [a( m −    me ) + e − k] + (1 + δ) m = 0

      or
                                      a2 λ me − aλe + aλk
                                m=
                                           1 + δ + a2 λ
      It follows that     me = aλk/(1 + δ). Hence

                                       a2 λ     aλk
                                                1+δ   − aλe + aλk
                              m =
                                            1 + δ + a2 λ
                                         aλk          aλe
                                  =            −
                                         1+δ      1 + δ + a2 λ
      and inflation is
                                     aλk              aλe
                             π=               −                +v
                                     1+δ          1 + δ + a2 λ
      as claimed in equation (8.18).



                                           71
  8. Suppose that the private sector forms expectations according to

                                  π e = π∗ if π t−1 = π e
                                    t                   t−1


                                    πe = aλk otherwise.
                                     t

      If the central bank’s objective function is given by (8.2) and its discount
      rate is β, what is the minimum value of π∗ that can be sustained in
      equilibrium?

   The central bank’s single period loss function, from (8.2), is
                                 1
                          Lt =     λ(yt − yn − k)2 + π2
                                 2                    t


Assume output is given by equation (8.3) of the text:

                                 yt = yn + a(π t − π e )
                                                     t

where the aggregate supply shock has been set to zero to parallel the analysis of
reputation in section 8.3.1.1. To determine the incentive the central bank has
to deviate from maintaining an inflation rate of π∗ , we need to consider what
happens if the central bank deviates in period t, incurs the punishment in period
t + 1 (the punishment being that the public expects an inflation rate of aλk) and
then returns to an inflation rate of π∗ in period t + 2. Let Lnd be the loss from
                                                               t
no-deviation in period t and Ld the loss from deviating. The central bank will
                                t
wish to deviate if

                           Ld + βLd < Lnd + βLnd
                            t     t+1  t      t+1

Following the discussion in the text, we can say that the central bank will have
an incentive to deviate if the gain exceeds the cost, or

                 G(π ∗ ) ≡ Lnd − Ld > β Ld − Lnd ≡ C(π∗ )
                            t     t      t+1  t+1                              (110)

   To evaluate these terms, start with the loss under the no-deviation case.
Not deviating means the central bank sets the rate of inflation equal to π∗ each
period. This is expected by the public, so output is equal to yn . The loss each
period is then
                                           1
                         Lnd = Lnd =         λk2 + (π∗ )2
                          t     t+1
                                           2
    If the central bank deviates in period t, then the public will expect an inflation
rate equal to the one-shot discretionary rate aλk in period t + 1. The best the
central bank can do is to inflation at this rate, ensuring yt+1 = yn and
                           1               1
                   Ld =      λk2 + (aλk)2 = λ 1 + a2 λ k2
                    t+1
                           2               2

                                           72
     Now we can determine Ld . Since the public is expecting π∗ , the central bank,
                                t
if it deviates, will pick π t to minimize
                                  1
                         Ld =       λ(a(πt − π∗ ) − k)2 + π 2
                          t
                                  2                         t


The first order condition is

                            (1 + a2 λ)πt − a2 λπ∗ − aλk = 0

or
                                          a2 λπ∗ + aλk
                                   πt =
                                             1 + a2 λ
Output in period t will equal

                             yt    = yn + a(πt − π∗ )
                                             aλk − π ∗
                                   = yn + a
                                              1 + a2 λ

and
                                                        2                      2
                      1           aλk − π∗                      a2 λπ∗ + aλk
           Ld    =      λ a                        −k       +
            t
                      2            1 + a2 λ                        1 + a2 λ
                                          2                         2
                      1       k + aπ ∗             a2 λπ∗ + aλk
                 =      λ                     +
                      2       1 + a2 λ                1 + a2 λ
                      1 (k + aπ ∗ )2
                 =     λ
                      2 1 + a2 λ
     We can now evaluate the gains and cost of deviation:

                     G(π∗ ) = Lnd − Ld
                               t     t
                                   1                1 (k + aπ ∗ )2
                             =       λk2 + (π ∗ )2 − λ
                                   2                2 1 + a2 λ
and

                C(π∗ ) = β Ld − Lnd
                            t+1  t+1
                                   1                 1
                        = β          λ 1 + a2 λ k2 −   λk2 + (π∗ )2
                                   2                 2
                              1
                        =       β a2 λ2 k2 − (π∗ )2
                              2
The inflation rate π ∗ can be sustained as an equilibrium if

                                     G(π∗ ) < C(π∗ )


                                              73
or

             1               1 (k + aπ∗ )2  1
               λk2 + (π∗ )2 − λ      2λ
                                           < β a2 λ2 k2 − (π ∗ )2
             2               2 1+a          2
Simplifying, this condition becomes

        1 + β(1 + a2 λ) (π∗ )2 − 2aλkπ∗ + 1 − β 1 + a2 λ       a2 λ2 k2 < 0   (111)

   The questions asks for the minimum value of π∗ that can be sustained as
an equilibrium, given the trigger strategy followed by the public in forming their
expectations. So we want to find the minimum value of π∗ that satisfies equation
(111).
   Solving the quadratic equation (111) for π∗ ,

                      2aλk ±    (2aλk)2 − 4 1 − β 2 (1 + a2 λ)2 a2 λ2 k2
               π∗ =
                                       2 (1 + β(1 + a2 λ))

Simplifying,

                                      aλk 1 ± β 1 + a2 λ
                               π∗ =
                                         1 + β(1 + a2 λ)

so the minumum π∗ is given by

                                      aλk 1 − β 1 + a2 λ
                           π∗ =
                            min
                                         1 + β(1 + a2 λ)

    Note that if 1 − β 1 + a2 λ < 0, π ∗ < 0 and π ∗ = 0 is a feasible equi-
                                        min
librium. This is the case shown in Figure 2 which plots G(π∗ ) − C(π∗ ) as a
function of π∗ for a = 1, λ = .5, β = .9 and k = .1.



     9. Assume that nominal wages are set at the start of each period but that
        wages are partially indexed against inflation. If wc is the contract base
        nominal wage, the actual nominal wage is w = wc + κ(pt − pt−1 ) where κ
        is the indexation parameter. Show how indexation affects the equilibrium
        rate of inflation under pure discretion. What is the effect on average
        inflation of an increase in κ? Explain why.

  Referring to question 1, consider the same production set up so that employ-
ment is again given (in log terms) by the marginal product condition:
                                       1
                               l=−        (w − p − ln αA)
                                      1−α


                                            74
                      Figure 3: Gain minus Cost of Deviating
G-C




      -0.04   -0.02        0       0.02        0.04   0.06     0.08   0.1   0.12




                                                π∗




                                    Figure 4:




                                          75
Substituting in the new expression for the nominal wage,
                           1
                    l=−       (wc − (1 − κ)p − κp−1 − ln αA)
                          1−α
 If we assume the base contract wage wc is set equal to the expected market
clearing level, wc = Ep + ln αA − (1 − α)l∗ where l∗ is the fixed supply of labor.
Actual employment is
                1
      l = −        (Ep + ln αA − (1 − α)l∗ − (1 − κ)p − κp−1 − ln αA)
              1−α
                 1−κ               κ
         = l∗ +      (p − Ep) −        (Ep − p−1 )
                 1−α            1−α
while output is

                           α(1 − κ)               ακ
            y     = αl∗ +            (p − Ep) −       (Ep − p−1 ) + e
                             1−α                1−α
                  = y ∗ + a(1 − κ) (p − Ep) − aκ (Ep − p−1 ) + e

where a = α/(1 − α). Under discretion, Ep and p−1 are taken as givens when
the central bank decides on its current price level. Thus, the important difference
introduced by indexation is that the effect of a price surprise is decreasing in the
indexation parameter κ.
    If we now derive the optimal discretionary policy outcome, using for example
the loss function (8.2) and repeating the analysis that led to equation (8.7), the
parameter a of the text is replaced everywhere with a(1 − κ) and the inflation
bias will be equal to

                                  aλ(1 − κ)k ≤ aλk

Notice that indexation ( 0 < κ < 1) reduces the average inflation bias. By
reducing the impact of a price surprise on real output, indexation reduces the
incentive to induce an output expansion.


 10. Suppose the central bank’s loss function is given by
                                  1
                         V cb =     λ y − yn − k)2 + (1 + δ)π 2
                                  2
     If y = yn + a(π − πe ) + e and π = m + v, verify that the inflation rate
     under discretion is given by equation (8.18).

   This problem actually is the same as number 7.




                                          76
 11. Beetsma and Jensen (1998): Suppose the social loss function is equal to
                               1
                          V s = E λ y − yn − k)2 + π2
                               2
     and the central bank’s loss function is given by
               1                                              2
         V cb = E (λ − θ) y − yn − k)2 + (1 + θ) π − πT           + tπ
               2
     where θ is a mean zero stochastic shock to the central bank’s preferences,
     π T is an inflation target assigned by the government, and tπ is a linear
     inflation contract with t a parameter chosen by the government. Assume
     that the private sector forms expectations before observing θ. Let y =
     yn + (π − πe ) + e and π = m + v. Finally, assume θ and the supply shock
     e are uncorrelated.

      (a) Suppose the government only assigns an inflation target (so t = 0).
          What is the optimal value for πT ?
      (b) Now suppose the government only assigns a linear inflation contract
          (so πT = 0). What is the optimal value for t?
      (c) Is the expected social loss lower under the inflation target arrange-
          ment or the inflation contract arrangement?

    (Notes: This statement of the problem corrects two typos in the text. Also,
in the text, Beetsma and Jensen is listed as forthcoming. It has now appeared
in the Journal of Money, Credit, and Banking, 30 (3), part 1, August 1998,
384-403.)
    (a) It will simplify to treat inflation as the central bank’s choice variable.
From the link between money growth and inflation ( π = m + v), one can
easily obtain the rate of money growth needed to achieve the desired expected
inflation rate.
    With only an inflation target, the central bank’s loss function is
                 1                                                 2
           V cb = E (λ − θ)(π − πe + e − k)2 + (1 + θ) π − πT
                 2
where the relationship between output and surprise inflation has been used to
eliminate y − yn . The first order condition for the optimal inflation setting,
taking private expectations as given, is

                (λ − θ)(π − πe + e − k) + (1 + θ) π − πT = 0               (112)

Taking expectations conditional on the public’s information set (which does not
include e or θ), −λk + π e − πT = 0 or

                                 π e = πT + λk



                                       77
Substituting this back into the central bank’s first order condition (112),

                                                  1+θ
                           π = πT + (λ − θ)k −        e                        (113)
                                                  1+λ
    To find the optimal value for the inflation target, use (113) to evaluate social
                                   1+θ
loss, noting that π − π e = −θk − 1+λ e:

                                        2                                2
          1                λ−θ                                   1+θ
     V s = E λ −(1 + θ)k +     e            + π T + (λ − θ)k −       e
          2                1+λ                                   1+λ

Minimizing this respect to the target inflation rate yields the first order condition

                                               1+θ
                         E π T + (λ − θ)k −        e =0
                                               1+λ
or

                                    πT = −λk

This is Svensson’s (1997) result: to offset the inflation bias, the inflation target
must be set below the socially optimal inflation rate (equal to zero in this case). If
the inflation term in the social loss function had allowed for a non-zero optimal
inflation rate, by having (π −π∗ )2 rather than simply π2 in V s , then the optimal
target would have been π ∗ − λk.
    (b) With a linear inflation contract but no inflation target, the central bank’s
loss function is
                   1
             V cb = E (λ − θ)(π − πe + e − k)2 + (1 + θ)π2 + tπ
                   2
and first order condition will be

                   (λ − θ)(π − πe + e − k) + (1 + θ)π + t = 0

Solving for inflation,

                                (λ − θ)(πe − e + k) − t
                           π=
                                         1+λ
The public will expect an inflation rate of λk − t. Hence, the central bank will
deliver an inflation rate of

                                       (1 + λ − θ)    1+θ
                      π = (λ − θ)k −               t−     e                    (114)
                                          1+λ         1+λ
Notice that the response to the supply shock is the same under either the target
(see equation 113) or the contract (see 114).



                                         78
   To find the optimal value of t from the government’s perspective, evaluate
                                                             1       1+θ
V , first noting that equation (114) implies π − πe = −θ k − 1+λ t − 1+λ e:
 s


                                                                         2
                          1            1      λ+θ
                Vs    =     Eλ −θ k −     t +     e−k
                          2           1+λ     1+λ
                                                                             2
                           1             (1 + λ − θ)    1+θ
                          + E (λ − θ)k −             t−     e
                           2                1+λ         1+λ

The first order condition for the optimal t is

                      θ              1        λ+θ
          0 = Eλ            −θ k −       t +       e−k
                    1+λ            1+λ        1+λ
                    (1 + λ − θ)            (1 + λ − θ)    1+θ
                 −E             (λ − θ)k −             t−     e
                        1+λ                   1+λ         1+λ

Evaluating the expectations, and making use of the assumption that E(θe) = 0,

           σ2         1                         σ2               1
      λ     θ
                         t − k − λk + t −        θ
                                                           k−       t =0
          1+λ        1+λ                       1+λ              1+λ

Combining terms, this can be written as

                                  σ2
                           1+      θ
                                          t − λ + σ2 k = 0
                                 1+λ               θ


or
                                     (1 + λ) λ + σ 2
                                t=                 θ
                                                     k                           (115)
                                        1 + λ + σ2
                                                 θ

   (c) To evaluate the social loss function under the alternative policies, it is
useful to start with the expressions for y − yn − k and π and their variances for
each type of policy. Let subscripts T and c denote the targeting regime and the
contract regime. Then, for the targeting regime,

                                λ−θ                    λ2 + σ2 2
                πT = −θk −          e ⇒ Eπ2 = σ 2 k2 +        θ
                                                                σe
                                1+λ                    (1 + λ)2
                                                θ


and
                                                        1+θ
                          yT − yn − k = −(1 + θ)k +         e
                                                        1+λ
which implies

                                                         1 + σ2
                     E (yT − yn − k)2 = (1 + σ2 )k2 +         θ
                                                                    σ2
                                                         (1 + λ)2
                                              θ                      e




                                          79
   For the contract regime, substitute (115) into (114) to obtain

                         (1 + λ − θ) (1 + λ) λ + σ2      1+θ
 πc   = (λ − θ)k −                               2
                                                   θ
                                                     k −     e
                            1+λ         1 + λ + σθ       1+λ
                  θ + σ2       1+θ            σ2 (1 + σ 2 ) 2 λ2 + σ2 2
      =     −          θ
                           k −     e ⇒ Eπ2 =    θ       θ
                                                           2k +
                                                                       θ
                                                                       2 σe
                1 + λ + σ2
                         θ     1+λ           (1 + λ + σ 2 )     (1 + λ)
                                                        θ



                                                  θ                  1+θ
                   yc − yn − k = − 1 +                          k+       e
                                             1 + λ + σ2
                                                      θ              1+λ

which implies

                                                  σ2                   1 + σ2
            E (yc − yn − k)2 = 1 +                 θ
                                                           2    k2 +        θ
                                                                                  σ2
                                                                       (1 + λ)2
                                                                                   e
                                          (1 + λ +     σ2 )
                                                        θ

    We can now evaluate social loss under the two policy regimes. For inflation
targeting,

                   1                  1 + σ2 2                 1 2 2 λ2 + σ2 2
       s
      VT   =         λ (1 + σ2 )k2 +       θ
                                              σe           +     σ k +        θ
                                                                                σe
                   2                 (1 + λ)2                  2 θ     (1 + λ)2
                             θ


                   1                        1 λ + σ2
           =         λ(1 + σ 2 ) + σ 2 k2 +         θ
                                                      σ2
                   2         θ       θ
                                            2 (1 + λ) e

while for the inflation contract regime,

                         1                  σ2                     1 + σ2
             Vcs     =     λ      1+         θ
                                                       2   k2 +         θ
                                                                             σ2
                         2                                        (1 + λ)2
                                                                              e
                                       (1 + λ + σ2 )
                                                 θ

                             1  σ2 (1 + σ2 ) 2 λ2 + σ 2 2
                         +        θ      θ
                                               k +        θ
                                                            σe
                             2 (1 + λ + σ 2 )2
                                          θ        (1 + λ)2
                         1        σ2          1 λ + σ2
                     =     λ+      θ
                                         k2 +         θ
                                                        σ2
                         2    1 + λ + σ2
                                       θ      2 (1 + λ) e
                      s
Subtracting Vcs from VT yields

                               1 2                             σ2
           VT − Vcs
            s
                         =       k   λ + (1 + λ)σ2 − λ +        θ
                               2                 θ
                                                           1 + λ + σ2
                                                                    θ
                               1 2 2                  1
                         =       k σθ (1 + λ) −
                               2                 1 + λ + σ2
                                                          θ

which is always positive if σ 2 > 0, i.e., if there is any uncertainty about the
                              θ
central bank’s preferences. Thus, in the face of preference uncertainty, the linear
inflation contract performs better than a simple inflation target.


                                             80
7      Chapter 9: Monetary-Policy Operating Pro-
       cedures
     1. Suppose equations (9.1) and (9.2) are modified as follows:

                                      yt = −αit + ut

                                    mt = −cit + yt + vt

       where ut = ρu ut−1 + ϕt , vt = ρv vt−1 + ψt and ϕ and ψ are white noise pro-
       cesses (assume all shocks can be observed with a one period lag). Assume
       the central bank’s loss function is E[y]2 .

        (a) Under a money supply operating procedure, derive the value of mt
            that minimizes E[y]2 .
        (b) Under an interest rate operating procedure, derive the value of it
            that minimizes E[y]2 .
        (c) Explain why your answers in (a) and (b) depend on ρu and ρv .
        (d) Does the choice between a money supply procedure and an interest
            rate procedure depend on the ρi s? Explain.
        (e) Suppose the central bank sets its instrument for two periods (for
            example, mt = mt+1 = m∗ ) to minimize E[yt ]2 + βE[yt+1 ]2 where
            0 < β < 1. How is the instrument choice problem affected by the
            ρi s?

   a) Under a money supply procedure, the money demand relationship implies
that the interest rate is it = c−1 (yt + vt − mt ). Output is then equal to
                                      α
                         yt    = −      (yt + vt − mt ) + ut                  (116)
                                      c
                                    α (mt − vt ) + cut
                               =
                                          c+α
The objective is to pick mt to minimize E[y]2 . The first order condition is

                          2α   α (mt − vt ) + cut
                             E                    =0
                         c+α         c+α
Since the shocks are assumed to be observed with a one period lag, E (vt ) =
ρv vt−1 and E (ut ) = ρu ut−1, so this first order condition requires that mt satisfy

                          αmt − αρv vt−1 + cρu ut−1 = 0

or
                                               c
                              mt = ρv vt−1 −     ρ ut−1
                                               α u

                                          81
The money supply is adjusted to offset the forecasted effects of vt and ut on
output:
                                        cϕt − αψt
                                 yt =                                        (117)
                                          c+α


     b) Under an interest rate procedure, output is equal to

                                  yt = −αit + ut                             (118)

and it is chosen to minimize E[y]2 = E[−αit + ut ]2 . The first order condition
is

                              −2αE[−αit + ut ] = 0

or
                                          ρu ut−1
                                   it =
                                             α
The interest rate is adjusted to offset the predicted aggregate demand shock, while
money demand shocks do not affect output and so do not require any adjustment
in the interest rate instrument.
    c) As noted under parts (a) and (b), the optimal policy will involve trying to
insulate output from the two shocks ut and vt . If these could be observed before
policy is set, the optimal policies would be mt = vt − α ut under a money
                                                           c
                       1
procedure and it = α ut under an interest rate procedure. Under certainty
equivalence (which holds in the linear model with a quadratic objective function),
the optimal policy simply replaces ut and vt with the best forecast of the shocks,
ρu ut−1 and ρv vt−1 .
     d) The loss function under the m procedure is
                                                                         2
                              α ρv vt−1 −      c
                                                    ρu ut−1 − vt + cut
              L(m) = E                         α
                                                   c+α
                                                                    2
                             c(ut − ρu ut−1 ) − α(vt − ρv vt−1 )
                      = E
                                           c+α
                                           2
                             cϕt − αψt
                      = E
                               c+α
which is independent of both ρu and ρv .
   Under the interest rate procedure, the loss is

                        L(i) = E [ut − ρu ut−1 ]2 = E [ϕt ]2

which is also independent of ρu and ρv . Consequently, the comparison between
a money procedure and an interest rate procedure will not depend on either ρu

                                          82
or ρv . Since the predictable component of the shocks (the component that does
depend on ρu and ρv ) is offset under both policies, the comparison will only
depend on how well the different policies insulate output from the unforecastable
shocks ( ϕt and ψt ).
   e) If the objective is to set m or i at time t for two period to minimize
E[yt ]2 + βE[yt+1 ]2 , the analysis becomes more complicated and the comparisons
between the money supply and the interest rate policies will depend on the serial
correlation properties of the shocks. Starting with the money supply procedure,
we can use (116) for output to write the loss function as
                                         2                                   2
                  α (m∗ − vt ) + cut                α (m∗ − vt+1 ) + cut+1
              E                              + βE
                        c+α                                c+α
since, by assumption, m is fixed for two periods. The first order condition is
       2α         α (m∗ − vt ) + cut      α (m∗ − vt+1 ) + cut+1
              E                      + βE                                        =0
      c+α               c+α                      c+α
From the process followed by the disturbances, E(ut ) = ρu ut−1 and E(ut+1 ) =
ρ2 ut−1 , while E(vt ) = ρv vt−1 and E(vt+1 ) = ρ2 vt−1 . Using these in the first
 u                                               v
order condition, the optimal m∗ must satisfy
      α (m∗ − ρv vt−1 ) + cρu ut−1 + β α m∗ − ρ2 vt−1 + cρ2 ut−1 = 0
                                               v          u

or
                            α(1 + βρv )ρv vt−1 − c(1 + βρu )ρu ut−1
                     m∗ =
                                           α(1 + β)
Since m is fixed for two periods, it adjusts to offset what amounts to the average
discounted expected shocks over the two periods. As a consequence, output will
not be perfectly insulated from the forecasted components of ut , ut+1 , vt , or
vt+1 :
                      1+βρu                            1+βρv
            c ut −     1+β      ρu ut−1 − α vt −        1+β    ρv vt−1
 yt    =
                                     c+α
            c [(1 + β) ϕt + β (1 − ρu ) ρu ut−1 ] − α [(1 + β) ψt + β (1 − ρv ) ρv vt−1 ]
       =
                                         (1 + β) (c + α)
            cϕt − αψt          β       c(1 − ρu )ρu ut−1 − α(1 − ρv )ρv vt−1
       =                +                                                           (119)
               c+α          1+β                         c+α
(which should be compared with equation 117) and
                                 1+βρu                             1+βρv
                     c ut+1 −     1+β        ρu ut−1 − α vt+1 −     1+β      ρv vt−1
       yt+1   =
                                         c+α
                c (1 + β) ϕt+1 + ρu ϕt − (1 − ρu ) ρu ut−1
              =
                             (1 + β) (c + α)
                  α (1 + β) ψt+1 + ρv ψt − (1 − ρv ) ρv vt−1
                −
                               (1 + β) (c + α)

                                               83
or
         c ϕt+1 + ρu ϕt − α ψt+1 + ρv ψt           c (1 − ρu ) ρu ut−1 − α (1 − ρv ) ρv vt−1
yt+1 =                                   −
                       c+α                                      (1 + β) (c + α)


Forecast errors made in period t, and therefore not fully offset, continue to affect
output in period t + 1 if the disturbances are serially correlated ( ρu ϕt and ρv ψt
show up in the expressions for yt+1 ).
   Under an interest rate policy, the objective is to pick i∗ to minimize
                                       2                     2
                         E [−αi∗ + ut ] + βE [−αi∗ + ut+1 ]

so the first order condition is

                   −2α {E [−αi∗ + ut ] + βE [−αi∗ + ut+1 ]} = 0


                        −αi∗ + ρu ut−1 − αβi∗ + βρ2 ut+1 = 0
                                                  u

or
                                      (1 + βρu )ρu ut−1
                               i∗ =
                                          α(1 + β)

and output in the two periods will equal

                            1 + βρu                    β(1 − ρu )
             yt = ut −                ρu ut−1 = ϕt +              ρu ut−1
                             1+β                         1+β

and
                          (1 + βρu )ρu ut−1
         yt+1    = ut+1 −
                               (1 + β)
                             2
                   (1 + β) ρu ut−1 + ρu ϕt + ϕt+1 − (1 + βρu )ρu ut−1
                 =
                                         (1 + β)
                                  (1 − ρu )
                 = ϕt+1 + ρu ϕt −           ρ ut−1
                                   (1 + β) u

   Since the variance of output under the two policies will now depend on ρu
and ρu , the comparison of the loss functions under the two policies will no
longer be independent of the serial correlation properties of ut and vt . For
example, suppose ρu = 0 but ρv = 0. Under an interest rate policy, output does
not depend on the v disturbances, so yt = ϕt and yt+1 = ϕt+1 , but under the
money supply rule, equation (119) becomes

                       cϕt − αψt       β         α
                yt =             −                  (1 − ρv )ρv vt−1
                         c+α          1+β       c+α


                                           84
and
                    cϕt+1 − αψt+1               α          α (1 − ρv ) ρv vt−1
       yt+1 =                            −         ρv ψt +
                        c+α                    c+α          (1 + β) (c + α)
which still depends on ρv . Since σ2 = σ 2 /(1− ρ2 ), the variance of output under
                                   v     ψ       v
a money supply procedure is
                                  2                   2
                        c                      α                       (1 − ρv )2 ρ2
      E[yt ]2   =                     σ2 +                1 + ρ2 +                 v
                                                                                           σ2
                       c+α                    c+α                    (1 + β)2 (1 − ρ2 )
            m                          ϕ                       v                            ψ
                                                                                    v
                                  2                   2
                        c                      α                        (1 − ρv ) ρ2
                =                     σ2 +                1 + ρ2 +                 v
                                                                                           σ2
                       c+α                    c+α                    (1 + β)2 (1 + ρv )
                                       ϕ                       v                            ψ


while under an interest rate procedure,

                                             E[yt ]2 = σ2
                                                   i    ϕ




     2. Solve for the δ i s appearing in (9.11) and show that the optimal rule for
        the base is the same as that implies by the value of ∗ given in (9.10).

    The δ i s that appear in equation (9.11) are obtained by calculating the least
squares forecast of each shock based on the observed value of the interest rate.
For the model of section 9.3.2, the equilibrium expression for the interest rate
is given by equation (9.9):
                                              v−ω+u
                                        i=
                                             α+c+ +h
We can use this to calculate the forecasts of v, ω, and u, conditional on ob-
                                                    ˆ ˆ        ˆ
serving i. In the text, these forecasts are denoted v, ω , and u (see page 394).
From the least squares formula, the forecast of a variable y, conditional on x,
is y = σ2 x where σx,y is the covariance between x and y and σ 2 is the
         σx,y
   ˆ                                                                    x
           x
variance of x. (This assumes both y and x have zero means.)
    Applying this formula, we have
                      σ v,i
           ˆ
           v =                i
                      σ2
                      i                
                            σ2v
                        α+c+ +h                                           σ2
                =                       i = (α + c +      + h)            v
                                                                                       i
                        σ2 +σ 2 +σ 2
                         v    ω    u                                 σ2 + σ 2 + σ2
                                                                      v     ω    u
                       (α+c+ +h)2

so
                                             α+c+ +h
                                  δv =                        σ2
                                             σ2 + σ2 + σ2
                                              v    ω    u
                                                               v




                                                 85
Similarly,
                                          α+c+ +h
                               δω = −                      σ2
                                          σ2 + σ 2 + σ2
                                           v     ω    u
                                                            ω


and
                                         α+c+ +h
                                 δu =                     σ2
                                         σ2 + σ2 + σ2
                                          v    ω    u
                                                           u


   The next step is to substitute these expressions into the policy rule (9.11):
                             c+h
              b =          −       δu + δv − δω i
                               α
                            α+c+ +h            c+h 2
                   =                         −    σ + σ 2 + σ2 i
                            σ2 + σ 2 + σ2
                             v     ω    u       α u     v    ω


or, since b = i,
                           α+c+ +h                 c+h 2
                   =                           −      σ + σ2 + σ2
                           σ 2 + σ2 + σ2
                             v    ω    u            α u    v    ω


Solving this for   yields
                                                    σ 2 + ασ2
                               = − (c + h) + α        v     ω
                                                        σ2
                                                         u

which proves that the policy rule (9.11) yields the same response to the interest
rate as was found in equation (9.9), and the optimal policy rule is

                                               σ2 + ασ2              ∗
                   b = − (c + h) + α            v     ω
                                                               i=        i
                                                  σ2
                                                   u

  3. Suppose the money demand relationship is given by m = −c1 i + c2 y +v.
     Show how the choice of an interest rate versus a money supply operating
     procedure depends on c2 . Explain why the choice depends on c2 .

    Using the basic model given by equation (9.1) with the money demand equa-
tion specified in the problem, interest rates and output under a money supply
procedure are given by
                                        1
                                 i=          (c2 y − m + v)
                                        c1
and
                                         α
                       y     = −            (c2 y − m + v) + u
                                        c1
                                          1
                             =                   [a(m − v) + c1 u]
                                      c1 + αc2

                                             86
and the loss function E (y)2 is minimized when

         α           α(m − v) + c1 u       α                         αm
  2              E                   =2                     E              =0
      c1 + αc2          c1 + αc2        c1 + αc2                  c1 + αc2

or m = 0. The loss function is then equal to
                                             2
                                    1
                      L(m) =                     α2 σ2 + c2 σ 2
                                                          1 u
                                c1 + αc2             v


   In contrast, under an interest rate procedure, y = −αi + u, the variance of
output is minimized if i = 0, and the loss function then takes on the value

                                    L(i) = σ 2
                                             u

   An interest rate rule is preferred if L(i) < L(m), or if

                                           2c1
                             σ 2 > c2 +
                               v                   c2 σ 2
                                                        u
                                            α

which should be compared with equation (9.6) on page 390 of the text.
    A large value of c2 (a large income elasticity of money demand) makes
it more likely that a money supply procedure will be preferred. Consider the
impact on output of a positive u shock. If c2 is large, the resulting rise in
output has a large impact on money demand. This in turn causes interest rates
to rise, offsetting the original rise in output. Thus, u shocks have a smaller
impact on output under an m procedure when c2 is large. Similarly, a positive
v that increases money demand and raises interest rates under an m procedure,
will lower output but the decline in y has, when c2 is large, a strong impact in
lowering money demand. As a result, interest rates need to rise less to maintain
money market equilibrium after the positive v shock, and, with a smaller rise in
i, y falls less.
    These results can be illustrated by Figure 1. The negatively sloped solid line
is the IS equation y = −αi when u takes on its expected value of 0. The
positively sloped lines AA and BB give money market equilibrium when m = 0
and v takes on its expected value of 0 (that is, these lines show i = c2 y). c1
The line BB is drawn for a larger value of c2 . The dotted negatively sloped line
shows the results of a positive u shock; output rises less when c2 is large; output
rises from he the level associated with point C to D if c2 is small, but it rises
only to E when c2 is larger. A positive v shock shifts AA and BB to A A and
B B (since i = c1 (c2 y + v), the vertical shift is the same for both). Again,
                     1
output is less affected when c2 is large, falling only from C to F when c2 is
large, rather than C to G.




                                        87
Figure 5: Chapter 9, Problem 3 — The impact of c2 under a money supply
operating procedure



                                                                             B'

                                                                             B


                                                                             A'
  Interest Rate




                                                                             A

                                         G
                                                     E
                                             F
                                                         D
                      A'                         C



                      A
                           B'




                           B


                                                     Output




  4. Prices and aggregate supply shocks can be added to Poole’s analysis by
     using the following model:

                                       yt = yn + a(π t − Et−1 πt ) + et                 (120)


                                       yt = yn − α (it − Et πt+1 ) + ut                 (121)


                                        mt − pt = β 0 − βit + yt + vt                   (122)

                  Assume the central bank’s objective is to minimize E λy 2 + π 2 , and that
                  are disturbances are mean zero, white noise processes. Both Et−1 πt and
                  the policy instrument must be set prior to observing the current values of
                  the disturbances.

                  (a) Calculate the expected loss function if it is used as the policy instru-
                      ment. (Hint: Give the objective function, the instrument will always
                      be set to ensure expected inflation is equal to zero.)
                  (b) Calculate the expected loss function if mt is used as the policy in-
                      strument.
                   (c) How does the instrument choice comparison depend on


                                                     88
            i. the relative variances of the aggregate supply, demand, and money
               demand disturbances?
           ii. the weight on stabilizing output fluctuations λ?

   Note: There is a typo in this problem; the loss function should be
                                           2
                              E λ (y − yn ) + π2

(or one can simply assume yn = 0 as a normalization).
    a) Under an interest rate policy, the money demand equation given by (122)
is not needed. Using the “hint” and setting Et−1 πt = Et πt+1 = 0, equation
(140) implies yt − yn = −αit + ut . Using this in (120), inflation will equal
      1
πt = a (−αit + ut − et ). This means we can write the policy problem in terms
of the policy instrument it as
                                                                    2
                                  2           1
              min E λ (−αit + ut ) +            (−αit + ut − et )
                it                            a

The first order condition is
                                          1
               −2αE λ (−αit + ut ) +        (−αit + ut − et )       =0   (123)
                                          a
The problem specified that the policy instrument must be set before observing
the disturbances, so in evaluating the first order condition, E (ut ) = E(ut ) =
                                            1
E (et ) = 0. Thus, (100) becomes −αλit − α a it = 0 or

                                       it = 0

   With this setting for the nominal interest rate,

                           yt − yn = −αit + ut = ut                      (124)

and
                              1                     u t − et
                      πt =      (−αit + ut − et ) =
                              a                         a
Notice that under a policy that sets i = 0, inflation is a mean zero, serially
uncorrelated process, so Et−1 πt = Et πt+1 = 0 as was assumed.
   Using these results, the expected loss under an interest rate policy,

                                    (1 + a2 λ)σ 2 + σ 2
                           L(i) =               u     e
                                                                         (125)
                                            a2
   b) Under a money rule, we need to use equation (122), solving it for the
nominal interest rate and then use this result to eliminate it from equations
(120) and (121). Since equations (120) and (140) are expressed in terms of

                                         89
the rate of inflation while (122) involves the price level, we can either express
inflation as pt − pt−1 , and then solve for the price level, or, since pt−1 is known
when policy is set, we could replace mt − pt in (122) with mt + pt−1 − π t and
solve for the rate of inflation. Since the loss function is expressed in terms of
inflation, this latter approach is more convenient.
    From (122), the nominal rate of interest is
                                    c0 + πt − mt + pt−1 + yt + vt
                           it =                                                             (126)
                                                  c
Substituting this into (140), and using the earlier hint about expected inflation,
we have
                                     c0 + π t − mt + pt−1 + yt + vt
            yt − y n      = −α                                        + ut
                                                    c
                                α (mt − c0 − πt − pt−1 − yn − vt ) + cut
                          =
                                                 c+α
which can be solved jointly with (120) to yield
                              aα (mt − c0 − pt−1 − yn − vt ) + acut + αet
             yt − yn =                                                                      (127)
                                            a (c + α) + α

                        α (mt − c0 − pt−1 − yn − vt ) + cut − (c + α) et
              πt =                                                                          (128)
                                        a (c + α) + α
Now substitute these two solutions into the loss function. The policy problem is
then
                                               2                                            2
             aα (       − vt ) + acut + αet            α(       − vt ) + cut − (c + α) et
 min E λ            t
                                                   +        t
   t                    a (c + α) + α                             a (c + α) + α


where, for convenience, t has been defined as mt − c0 − pt−1 − yn and can be
viewed as the policy instrument. The first order condition for the choice of t
is
              2α                          aα t            α t
                               aλ                  +                           =0
        a (c + α) + α                a (c + α) + α   a (c + α) + α
where we have used the fact that at the time policy is chosen, Eut = Eet =
Evt = 0. The first order condition is satisfied for t = 0, or

                                      mt = c0 + pt−1 + yn

Using (127) and (128), output and inflation under a money instrument are
                                            acut − aαvt + αet
                                yt − yn =
                                              a (c + α) + α


                                               90
and
                                         cut − αvt − (c + α) et
                                  πt =
                                             a (c + α) + α
and the loss function is

                          1 + a2 λ       c2 σ2 + α2 σ2 + α2 λ + (c + α)2 σ2
                                             u       v                    e
                 L(m) =                                     2                       (129)
                                             [a (c + α) + α]
   c) The instrument choice hinges on a comparison of the loss L(i) given
in (125) and the loss L(m) given in (129), with an interest rate instrument
preferred if
                                          L(i) < L(m)
or if

   (1 + a2 λ)σ2 + σ 2   1 + a2 λ            c2 σ 2 + α2 σ 2 + α2 λ + (c + α)2 σ 2
                                                 u        v                     e
              u     e
                      <                                                             (130)
           a2                                    [a (c + α) + α]2
This can be rewritten with some rearranging as implying L(i) < L(m) if
                                2
                 (a (c + α) + α) − a2 c2             2a (c + α) + α(1 − a2 λ) 2
        σ2   >                               σ2 +                            σe
         v
                          a2 α2               u
                                                            (1 + a2 λ)
This shows that the comparison depends on the different variance terms. A
money oriented operating procedure is less likely to be desirable if money demand
shocks are large (i.e., σ2 is large). The coefficient on σ 2 is positive, so an
                          v                                   u
interest rate rule is more likely to be preferred if aggregate demand shocks are
important (i.e., σ2 is large), while it is also likely to be preferred if supply
                    u
disturbances are large (i.e., σ2 is large). Notice that if σ2 is zero (no supply
                                e                             e
shocks), the comparison is independent of the preference weight λ.
    The weight on stabilizing output fluctuations, λ, affects the comparison only
if σ2 > 0. In the absence of aggregate supply shocks, there is no conflict in
     e
this model between stabilizing output and stabilizing inflation, so the operating
procedure comparison will be independent of λ. When aggregate supply shocks
are present ( σ 2 > 0), then there can be conflicts between stabilizing output and
                e
stabilizing inflation. If output objectives are very important ( λ large), then it
is more likely an interest rate procedure will be preferred. To understand why,
consider what happens in the face of a positive aggregate supply shock. Under an
interest rate procedure, aggregate demand remains constant (see equation 124),
so output is stabilized and inflation must fall. Such a policy will be preferred
if λ is large. Under a money supply procedure, in comparison, output will rise
and inflation will fall. Since both adjust, inflation falls by less than under the i
policy. A policy maker who cares more about inflation stabilization (i.e., has a
lower λ) will prefer money supply operating procedure.



                                                91
  5. Using the intermediate target model of section (9.3.3) and the loss function
     (9.15), rank the policies that set it equal to ˆt , iT , and ˆt + ∗ xt .
                                                    ı t           ι

   The basic model of section 9.3.3 consists of the following equations:
                                 yt = a(πt − Et−1 πt ) + zt

                              yt = −α (it − Et πt+1 ) + ut

                             mt − pt−1 − π t = yt − cit + vt
These appeared as equations (9.12) - (9.14) on page 397. The loss function
(9.15) is
                                                             2
                                      V = E (π − π∗ )
   The first policy to evaluate sets
                                             1
                        it = ˆt = π∗ +
                             ı                 (ρu ut−1 − ρz zt−1 )
                                             α
(see equation 9.17, page 397). The value of the loss function under this policy
was given at the bottom of page 397:
                                                   2
                                              1
                                 V (ˆt ) =
                                    ı                   σ2 + σ2
                                                         ϕ    e                            (131)
                                              a
   Under the second policy,
                                             (1 + a)ϕt − et + aψt
                         it = iT = ˆt +
                                   ı
                               t
                                                 ac + α(1 + a)
(see equation 9.21 on page 399). The inflation rate is (see page 399)
                                             cϕt − (α + c)et − αψt
                        π t iT = π ∗ +
                             t
                                                 ac + α(1 + a)
and the value of the loss function under this policy was given in the middle of
page 400:
                                              2
                                1
            V (iT ) =                              c2 σ 2 + (α + c)2 σ2 + α2 σ2            (132)
                t
                          ac + α(1 + a)                 ϕ             e       ψ


   Comparing (131) and (132),
                             2                                         2
                         1                                   1
V (ˆt ) − V (iT ) =
   ı                              σ2 + σ2 −                                c2 σ 2 + (α + c)2 σ2 + α2 σ 2
              t
                         a         ϕ    e
                                                       ac + α(1 + a)            ϕ             e        ψ

                          2acα(1 + a) + α2 (1 + a)2 σ2 + 2αa2 (c + α)σ 2
                                                     ϕ                 e
                 =
                                             a2 (ac + α(1 + a))2
                                                         2
                                         1
                             −                               α2 σ2
                                   ac + α(1 + a)                 ψ



                                                  92
The first term is positive, indicating that the intermediate targeting rule leads to
a smaller loss ( V (ˆt ) > V (iT )) if the only disturbances are demand and supply
                    ı          t
shocks ( ϕ and e). As discussed in the text, however, the intermediate targeting
procedure can do worse if money demand shocks are important ( V (ˆt ) < V (iT )
                                                                       ı        t
if σ2 is large).
    ψ
    The final policy sets it equal to ˆt + ∗ xt , where xt is defined below equation
                                        ι
(9.22) on page 401 as
                                              1                1
                                  xt = 1 +        ϕt −              et + ψt
                                              a                a
      ∗
and       was defined, also on page 401, as

                              ∗        1        a(1 + a)σ 2 + aσ2
                                                           ϕ      e
                                  =
                                       α     (1 + a)2 σ 2 + σ2 + a2 σ 2
                                                        ϕ    e        ψ

Using the definition of ˆt , the rate of interest under this policy is
                       ι
            ∗                      1                                            1              1
it = ˆt +
     ι          xt = π∗ +            (ρu ut−1 − ρz zt−1 ) +          ∗
                                                                           1+       ϕt −         et + ψ t
                                   α                                            a              a


    To evaluate the loss function under this policy, use equation (9.16) of the
text to find the equilibrium rate of inflation when it = ˆt + ∗ xt :
                                                       ι
                       ∗            (a + α)π ∗ − α(ˆt + ∗ xt ) + ut − zt
                                                   ι
          π(ˆt +
            ι              xt ) =
                                                     a
                                                                                           ∗
                                     ∗   (ut − ρu ut−1 ) − (zt − ρz zt−1 ) − α                 xt
                                  = π +
                                                             a
                                     ∗     1
                                  = π +        (ϕt − et − α ∗ xt )
                                           a
Problem 2 showed that the value of ∗ was related to the best forecasts of ϕ and
e, conditional on observing x. We can write inflation under this policy as
                  ∗                     1
      π(ˆt +
        ι             xt ) = π ∗ +          {[ϕt − E(ϕt | x)] − [et − E(et | x)]}
                                        a
                                                                     1
Comparing this to the policy that lead to V (ˆt ), in which π − π∗ = a (ϕt − et ),
                                             ı
                                                        ∗
it is clear that the variance of inflation around π will be smaller with the
ˆt + ∗ xt policy since the variance of [ϕt − E(ϕt − | x)] is less than or equal
ι
to the variance of ϕ, and similarly for the comparison of the variances of
[et − E(et | x)] and et . Therefore,
                                      V ∗ ≡ V (ˆt +
                                               ι      ∗
                                                          xt ) ≤ V (ˆt )
                                                                    ı
   To compare the loss under the intermediate target policy iT and the policy
that optimal uses information (ˆt + ∗ xt ), use the equation near the bottom of
                               ι
page 400 to write
                                            iT = ˆt +
                                             t   ι         T
                                                               xt


                                                  93
Using equation (9.16). inflstion can be written as
                          (a + α)π ∗ − α ˆt + T xt + ut − zt
                                         ι
               π t (iT ) =
                     t
                                           a
                           ∗   ϕt − et − α T xt
                        = π +
                                       a
                                                                     1
so the loss function V (iT ) is equal to the variance of             a   ϕt − et − α   T
                                                                                           xt .
Inflation around π∗ under the ˆt + ∗ xt policy is
                                ι

                               ∗                  ϕt − et − α   T
                                                                    xt
                    πt (ˆt +
                        ι          xt ) = π ∗ +
                                                          a
                                             1
and the loss function V ∗ is the variance of a (ϕt − et − α ∗ xt ). But since                ∗

was chosen to minimize this variance, it must be that V ∗ ≤ V (iT ).

  6. Show that if the nominal interest rate is set according to (9.17), the ex-
                                                            ˆ
     pected value of the nominal money supply is equal to m given in (9.19).

   According to equation (9.17) on page 397, the interest rate takes the value
                                       1
                      it = π ∗ +           (ρu ut−1 − ρz zt−1 )
                                       α
With inflation given by equation (9.18) and output by (9;12), the money demand
equation (9.14) can be written as
mt − pt−1   = πt + a (πt − π∗ ) + zt − cit + vt
                             ϕt − et                 c
            = π∗ + (1 + a)              + zt − cπ∗ −   (ρu ut−1 − ρz zt−1 ) + vt
                                 a                   α
since E t−1 π t = π∗ . Taking expectations as of time t − 1 of this equation and
solving for E t−1 mt ,
                                      c               c
  Et−1 mt = (1 − c)π ∗ + pt−1 −         ρu ut−1 + 1 +   ρ zt−1 + ρv vt−1
                                      α               α z
                                        ˆ
which is the same as the expression for m in equation (9.19).

  7. Suppose the central bank is concerned with minimizing the expected value
     of a loss function of the form
                                    L = E[T R]2 + χE[if ]2
     which depends on the variances of innovations to total reserves and the
     funds rate (χ is a positive parameter). Using the reserve market model of
     section 9.4.2, find the values of φd and φb that minimize this loss function.
     Are there conditions under which a pure nonborrowed reserves or a pure
     borrowed reserves operating procedures would be optimal?


                                           94
    The reserves market model from section 9.4.2 consists of a total reserves
demand equation, a borrowed reserves demand equation, a supply of nonborrowed
reserves equation, and an equilibrium condition. These are specified as

                                     T R = −αif + vd


                               BR = b(if − id ) + v b


                             NBR = φd vd + φb v b + v s

and the equilibrium condition that

                                 T R = BR + N BR


               = αif + b(if − id ) + (φd − 1)vd + (1 + φb )v b + v s

    Substituting these first three equations into the equilibrium condition and
solving for the funds rate if yield
                        1
              if =              bid − (1 + φb )vb + (1 − φd )vd − vs
                       a+b

   If the objective is to pick φd and φb to minimize L = E[T R]2 + χE[if ]2 , the
first order conditions will be

                                          ∂if                     ∂if
                   −2αE[−αif + vd ]                  + 2χE[if ]           =0            (133)
                                          ∂φd                     ∂φd
and
                                          ∂if                     ∂if
                   −2αE[−αif + vd ]                  + 2χE[if ]           =0            (134)
                                          ∂φb                     ∂φb
                                     f
                                                                         ∂if
                              ∂i
To evaluate these, note that ∂φd          = −v d /(a + b), while         ∂φb
                                                                               = −vb /(a + b).
Hence, equation (133) implies

                                     −vd                        −vd
         0 = αE[αif − v d ]                     + 2χE[if ]
                                     a+b                        a+b
                           1 − φd                     α                  1 − φd
             = −α2                       σ2 +            σ2 − χ                    σ2
                          (a + b)2        d
                                                     a+b  d
                                                                        (a + b)2    d



Solving for φd ,
                                                     α(a + b)
                                (1 − φd ) =
                                                      α2 + χ


                                                95
                                             α(a + b)
                                φd = 1 −                                     (135)
                                              α2 + χ

     Equation (134) yields

             (1 + φb )vb     −v b              −(1 + φb )v b      −vb
     −αE α                            + χE                              =0
                a+b          a+b                  a+b             a+b

or

                            1 + φb               1 + φb
                   −α2                σ2 + χ                   σ2 = 0
                           (a + b)2    b
                                                (a + b)2        b



which can be solved for the optimal φb , yielding

                                      φb = −1                                (136)

    To understand these results, start first with the φb = −1 finding. A shock
to borrowed reserve demand should generate an equal but opposite movement
in nonborrowed reserves. This keeps total reserves unchanged. Since neither
total reserve supply or demand have changed, the funds rate is left unchanged.
Thus, setting φb = −1 and accommodating shifts in borrowed reserve demand
completely insulates both T R and if from vb shocks.
    From (135), the optimal value of φd is equal to 1 − α(a+b) is less than 1 and
                                                         α2 +χ
depends on the preference parameter χ. In response to a shock to total reserve
demand, reserve supply adjusts to fully accommodate the shift if φd = 1; this
would succeed in insulating the funds rate from the shock, but it would lead total
reserves to move one-for-one with vd . By setting φd < 1, reserve supply less
than fully accommodates the shift in reserve demand. This means the funds rate
rises (falls) if vd > 0 ( < 0). This means the funds rate moves more, but total
reserves move less, and this will be optimal since the policy maker cares about
both E[T R]2 and E[if ]2




                                        96
8     Chapter 10: Interest Rates and Monetary Pol-
      icy
    1. Suppose (10.1) is replaced by a Taylor sticky price adjustment model of the
       type studied in Chapter 5. Is the price level still indeterminate under the
       policy rule (10.5)? What if prices adjust according to the Fuhrer-Moore
       sticky inflation model?

    Equation (10.1) relates the level of real output to the price surprise term
pt − Et−1 pt ; the actual level of prices doesn’t really matter. With a Taylor-type
price adjustment model of the type discussed in section 5.5.1, the price level at
time t will depend on prices in previous periods. Employing a simple version
in which prices are a markup over wages, and nominal wages are set for two
periods with half of all wages set each period, the aggregate price level in period
t will be equal to
                                        1
                                 pt =     (xt + xt−1 )                         (137)
                                        2
where xt is the contract wage set in period t. If wage setting depends on the
expected price level over the two periods the wage is set and on the current state
of economic activity,
                                    1
                             xt =     (pt + Et pt+1 ) + kyt                    (138)
                                    2
as in equation (5.44) on page 216. Substituting (138) into (137),
                    1                                   1
             pt =     (pt + Et pt+1 + pt−1 + Et−1 pt ) + k (yt + yt−1 )
                    4                                   2
Multiplying both sides by 4 and rearranging, the Taylor adjustment model im-
plies
                       1                              2
                pt =     (pt−1 + Et pt+1 + Et−1 pt ) + k (yt + yt−1 )
                       3                              3
If we combine this with the IS equation (10.7) and Fisher equation (10.8), then
under an interest rate peg, equilibrium is obtained as the solution to
                       1                              2
                pt =     (pt−1 + Et pt+1 + Et−1 pt ) + k (yt + yt−1 )          (139)
                       3                              3

                                 yt = α0 − α1 rt + ut                          (140)


                               iT = rt + (Et pt+1 − pt )                       (141)

To see if the price level is determinate, consider what would happen if, at the end
of period t−1, the public expected the price level in all future periods to be higher

                                           97
by κ%. Since the model is specified in log form, we need to check whether adding
κ to pt , Et−1 pt and Et pt+1 would affect the equilibrium. Clearly, equations
(140) and (141) would be unaffected, (140) because it does not involve the price
level, and (141) because expected inflation is also unaffected if the price level
jumps by κ% and remains at this new higher level: (Et pt+1 + κ − (pt + κ)) =
Et pt+1 − pt . But equation (139) is affected; pt−1 is predetermined — it can’t
jump when expectations change. So when pt , Et pt+1 and Et−1 pt increase by
κ, the left side of 139) goes up be κ, while the right side only goes up by 2 κ
                                                                              3
because pt−1 can’t increase by κ; is no longer satisfied if pt jumps to pt + κ.
Equilibrium is determined by the historical price level.
    If a Fuhrer-Moore model of inflation adjustment is used, then equations (140)
and (141) remain unchanged, but the inflation process is different. From equa-
tion (5.61), the change in the contract wage is given by
                                     1
                              xt =     (π t + Et πt+1 ) + 2kyt
                                     2
and inflation is equal to
                       1
           πt   =        ( xt + xt−1 )
                       2
                       1
                =        (πt + Et π t+1 + π t−1 + Et−1 πt ) + k (yt + yt−1 )
                       4
Rearranging,
                       1                               4
                πt =     [π t−1 + Et πt+1 + Et−1 πt ] + k (yt + yt−1 )
                       3                               3
In terms of pt , this can be written as
                          1                               4
           pt = pt−1 +      [πt−1 + Et πt+1 + Et−1 π t ] + k (yt + yt−1 )      (142)
                          3                               3
Again. history pins down the price level. A jump in pt and all future expected
price levels leaves Et πt+1 and Et−1 πt unaffected. Lagged inflation πt−1 is also
unaffected since it is predetermined as of time t. But pt also depends on the level
of past prices pt−1 so a κ% jump would not leave the equilibrium unaffected.
    Both the Taylor and the Fuhrer-Moore models imply that the current price
level depends, in part, on the previous price level. Thus, the price level is deter-
minate if we can take the historical value of pt−1 as given.


  2. Derive the values of the unknown coefficients in (10.10) and (10.11) if the
     money supply process is given by (10.15).

    Equations (10.10) and (10.11) were used to derive the equilibrium processes
followed by the price level and the nominal interest rate when the nominal money


                                            98
supply was set according to (10.9). Suppose instead that mt is determined by
(10.15), repeated here as

                                mt =        +   0t   +   it − iT

The rest of model is given by equations (10.1) - (10.4). Using this new process
for mt , the proposed solutions for pt and it will need to be modified to allow
for the possibility that either the price level or the nominal rate of interest, or
both, may be affected by the deterministic trend that appears in the money supply
process. Thus, we need to consider the solutions

                    pt = b10 + b11 mt−1 + b12 et + b13 ut + b14 vt + b15 t                (143)

                    it = b20 + b21 mt−1 + b22 et + b23 ut + b24 vt + b25 t                (144)

    Combining (10.1), (10.2), and (10.4) yields (10.12) for the nominal rate of
interest:
                 α0 − yc   1
         it =            −    [a (pt − Et−1 pt ) + ut − et ] + Et pt+1 − pt               (145)
                   α1      α1
while (10.1), (10.3), and the new policy rule (10.15) yield

        pt   =       +   0t +     it − iT + cit − y c − a (pt − Et−1 pt ) − et − vt
             =       − i − yc +
                          T
                                       0t   + ( + c)it − a (pt − Et−1 pt ) − et − vt (146)

which can be compared to (10.13), obtained using the policy (10.9) in which the
money supply depended on lagged money.
   We need to solve for the unknown coefficients in (143) and (144) such that
equations (145) and (146) are satisfied for all realizations of the e, u, and v
and all t.
   Using the proposed solutions,

                           pt − Et−1 pt = b12 et + b13 ut + b14 vt

and

      Et pt+1    = b10 + b11 mt + b15 (t + 1)
                 = b10 + b11 + b15 − iT + (b11              0   + b15 ) t + it + b15 (t + 1)

Using these expressions for the expectational terms, together with the proposed
solutions, we can determine the unknown coefficients. For example, consider
the coefficients b11 and b21 on mt−1 From (145), these must satisfy

                                            b21 = −b11

while from (146), they must satisfy

                                       b11 = ( + c)b21


                                                99
or

                                   b11 = b21 = 0

    Proceeding in a similar manner, the coefficients b12 and b22 on et must
satisfy
                                       a              1
                             b22 = −      b12 − b12 −
                                       α1             α1
and

                             b12 = ( + c)b22 − ab12 − 1

or
                                         α1 + + c
                   b12 = −
                                α1 (1 + a) + ( + c)(α1 + a)

                                          α1 − 1
                    b22 =
                               α1 (1 + a) + ( + c)(α1 + a)
For the coefficients on ut :
                                       a              1
                             b23 = −      b13 − b13 +
                                       α1             α1
and

                              b13 = ( + c)b23 − ab13

or
                                             +c
                    b13 =
                               α1 (1 + a) + ( + c)(α1 + a)

                                           1+a
                    b23 =
                               α1 (1 + a) + ( + c)(α1 + a)
For the coefficients on vt :
                                          a
                                b24 = −      b14 − b14
                                          α1
and

                             b14 = ( + c)b24 − ab14 − 1

or
                                             α1
                   b14 = −
                                α1 (1 + a) + ( + c)(α1 + a)


                                          100
                                             α1 + a
                      b24 = −
                                  α1 (1 + a) + ( + c)(α1 + a)

For the trend,

                                  b25 = b15 − b15 = 0

and

                              b15 =     0   + ( + c)b25 =     0

Finally, for the constants

                                      α0 − y c
                      b20    =                      + b10 + b15 − b15
                                        α1
                                      α0 − y c
                             =                      + b10
                                        α1


                                      ( + c) (α0 − y c )
              b10 =     + iT +                           + ( + c)       0   − yc
                                            α1
    Collecting these results, the equilibrium processes for the nominal interest
rate and the price level are (ignoring the constant terms),

                              (α1 − 1) et + (1 + a) ut + (a + α1 )vt
                 it = b20 +
                                  α1 (1 + a) + ( + c)(α1 + a)


                                      ( + c) ut − (α1 + + c) et − α1 vt
              pt = b10 +     0t   +
                                        α1 (1 + a) + ( + c)(α1 + a)

These can be compared to the solution coefficients reported on page 437 and the
interest rate solution given in equation (10.14).


  3. Suppose the money supply process in section 10.4.2 is replaced with

                                  mt = γmt−1 + φqt−1 + ξ t

      so that the policy maker is assumed to response with a lag to the real
      rate shock, with the parameter γ viewed as a policy choice. Thus, policy
      involves a choice of γ and φ, with the parameter φ capturing the systematic
      response of policy to real interest rate shocks. Show how the effect of qt
      on the one and two period nominal interest rates depends on φ. Explain
      why the absolute value of the impact of qt on the spread between the long
      and short rates increase with φ.



                                              101
    This problem, and the following one, both use the model of section 10.4.2;
they differ in terms of the process followed by the nominal stock of money. It
will be convenient to solve the model for a more general specification of mt ,
allowing one to then obtain the solutions for Problems 3 and 4 (as well as the
results in the text) as special cases.
    The model consists of the following four equations (see equations 10.29 -
10.31 of the text):
                                         Rt = qt                            (147)

                              1
                       Rt =     [it − Et πt+1 + Et it+1 − Et πt+2 ]         (148)
                              2

                                   mt − pt = −ait + vt                      (149)

and
                         mt = γmt−1 + φqt−1 + ξ t − ρξ t−1                  (150)

where Rt is the two-real period interest rate, qt is an exogenous real rate shock,
it is the one period nominal rate, π is the inflation rate, the third equation is a
money demand relationship, and the money supply process used in section 10.4.2
(equation 10.32) has been replaced by the one specified in the question. Notice
that the case considered in the text had φ = ρ = 0, Problem 3 considers the case
with ρ = 0, and Problem 4 sets γ = 1, φ = 0 (and notice that to distinguish
between the coefficient on lagged money and that on the lagged shock ( ξ t−1 ) I
have renamed the latter ρ for the purposes of deriving the general solution; in
Problem 4, mt has a coefficient of 1 and the coefficient on ξ t−1 is called γ).
    To answer this question, we need to solve for the one and two-period nominal
rates, together with the interest rate spread,
                                                1
                              st ≡ I t − it =     (Et it+1 − it )
                                                2
    Using (149) to eliminate the one-period nominal rate from equation (148),
the price level prices must satisfy
                       1 pt − mt + vt
            qt     =                   − (Et pt+1 − pt )
                       2      a
                            pt+1 − mt+1 + vt+1
                       +Et                         − Et (pt+2 − pt+1 )
                                      a
or
            2aqt = pt − mt + vt + apt + Et pt+1 − Et mt+1 − aEt pt+2

Solving for pt ,
            (1 + a)pt = 2aqt + mt − vt − Et pt+1 + Et mt+1 + aEt pt+2


                                            102
Now use (150) to obtain
                                      Et mt+1 = γmt + φqt − ρξ t
so that
       (1 + a)pt = 2aqt + (1 + γ)mt − vt + φqt − ρξ t − Et pt+1 + aEt pt+2                   (151)
from which we can guess that the solution for pt is of the form
                                   pt = b1 mt + b2 qt + b3 vt + b4 ξ t
Using this to evaluate (151),
(1 + a) (b1 mt + b2 qt + b3 vt + b4 ξ t ) = 2aqt + (1 + γ)mt − b1 (γmt + φqt + ρξ t )
                                            +ab1 γ (γmt + φqt − ρξ t ) − vt + φqt − ρξ t
since Et pt+1 = b1 Et mt+1 = b1 (γmt + φqt − ρξ t ) and Et pt+2 = b1 Et mt+2 =
γb1 Et mt+1 = γb1 (γmt + φqt − ρξ t ). This equilibrium expression holds for all
realizations of mt and the random disturbances if4
                                                      1
                                          b1 =
                                                 1 + a(1 − γ)

                                         1                     aφ
                               b2 =                2a +
                                        1+a               1 + a(1 − γ)

                                                        1
                                              b3 = −
                                                       1+a
and
                                             a                ρ
                                 b4 = −
                                            1+a          1 + a(1 − γ)
or
                  1             1                             aφ                             aρ
pt =                      mt +                   2a +                 qt − vt −                      ξ
             1 + a(1 − γ)      1+a                       1 + a(1 − γ)                   1 + a(1 − γ) t
                                                                                               (152)
which can be compared to equation (10.34) of the text. The case in the text is
obtained by setting φ = ρ = 0.
    Now that we have the solution for the price level, the one-period nominal
interest rate is, from (149),
                    1
             it   =   (pt − mt + vt )
                    a
                            1−γ               1                               φ
                  = −                 mt +                          2+                qt
                        1 + a(1 − γ)         1+a                         1 + a(1 − γ)
                          1           ρ                             1
                    −                         ξ +                           vt               (153)
                        1+a      1 + a(1 − γ) t                    1+a
     4 In   deriving the solution for b1 , the fact that 1 − γ 2 = (1 + γ)(1 − γ) is used.


                                                   103
This implies that
                                      1−γ
               Et it+1     = −                     Et mt+1
                                   1 + a(1 − γ)
                                      1−γ
                           = −                     (γmt + φqt − ρξ t )
                                   1 + a(1 − γ)
The two period rate is
                        1
              It      =   [it + Et it+1 ]
                        2
                        1            1 − γ2                 1
                      =     −                     mt +           vt
                        2        1 + a(1 − γ)             1+a
                          1        1            φ (γ − a(1 − γ))
                        +                   2+                    qt
                          2     1+a               1 + a(1 − γ)
                          1     1         ρ (γ − a(1 − γ))
                        −                                   ξt                (154)
                          2 1+a             1 + a(1 − γ)
while the spread is
         1
St   =     (Et it+1 − it )
         2
         1         (1 − γ)2              1             1         2 − γ + a(1 − γ)
     =                         mt −            vt +                               ρξ t
         2      1 + a(1 − γ)           1+a           1+a           1 + a(1 − γ)
                 1            (2 − γ + a(1 − γ)) φ
         −                 2+                       qt                           (155)
             1+a                  1 + a(1 − γ)
    For the specific money process assumed for Problem 3, ρ = 0. Thus, the
terms involving ξ t all become equal to 0. Interest rates depend on φ because
when φ differs from zero, agents will adjust their forecast of the future money
supply once they observe qt . Suppose φ > 0; the money supply is increased
in response to a positive shock to the real rate of interest. Then a positive q
realization causes an upward revision in the future money supply and the future
price level. This raises expected future inflation and so the one-period rate it
rises more than in the φ = 0 case (see equation 153). With 0 < γ < 1,
the expected future money supply returns only gradually to its baseline after a
positive q shock. From (155), a positive q shock lowers the spread, but the
                                             ˙
absolute value of the impact increases with φ. To understand why, consider the
case in which γ = 1 so that the effect of q on m is permanent. Observing q > 0
raises Et mt+1 and Et πt+1 , contributing to the rise in it . But since the money
supply is now expected to remain at this higher level, Et πt+2 is unchanged. The
long rate rises less than the short rate and the difference between the two is
larger if qt has a large impact on mt+1 (i.e., if φ is large). When | γ |< 1, the
initial rise in mt is expected to gradually be reversed, so Et π t+1 will actually
fall, increasing it relative to Et it+1



                                          104
  4. Suppose the money supply process in section 10.4.2 is replaced with
                                     mt = mt−1 + ξ t − γξ t−1
     Does it depend on γ? Does It ? Explain.

    We can used equations (153), (154), and (155) from Problem 3 to answer
this question, simply modifying the parameters to reflect the new money supply
process. In particular, the coefficient on lagged money is now equal to 1 (this
coefficient was γ in Problem 3) and the coefficient on the lagged real rate shock
is now zero ( φ = 0). Finally, the coefficient on ξ t−1 ,which was called in ρ
Problem 3, is renamed γ in the current Problem.
    With these changes, the solutions for the one-period nominal rate and the
two-period nominal rate become
                                      1
                          it =               (2qt + vt − γξ t )                        (156)
                                     1+a

                                 1      1
                          It =                [2qt + vt − γξ t ]                       (157)
                                 2     1+a

    The one-period nominal rate does depend on γ; a positive realization of ξ t
increases the period t money supply. If this increase were permanent, the ex-
pected rate of inflation would not be affected as the current and expected future
price levels would rise in proportion to the increase in mt . With γ nonzero, how-
ever, some of the change in mt is offset in period t + 1 ( Et mt+1 = mt − γξ t ).
If 0 < ξ < 1, for example, the money stock is expected to be lower in t + 1 than
in period t. This reduces expected inflation and the nominal rate falls.
    The two-period nominal rate is
                                      1                  1
                            It =        (it + Et it+1 ) = it
                                      2                  2
since Et it+1 = 0. The future one-period rate is unaffected (since the money
supply remains constant at its t + 1 value — Et mt+2 = mt − γξ t ), so the two-
period rate moves half as much as the one-period rate.

                                                                           i
                                                     1         ∞
  5. Show that equation (10.43) implies rt =        1+D        i=0
                                                                      D
                                                                     1+D       Et if − πt+1+i .
                                                                                   t+i


   Equation (10.43) is repeated here:

                       rt − D [Et rt+1 − rt ] = if − Et πt+1
                                                 t

which can be written as
                       1                                  D
               rt =                   if − Et π t+1 +                Et rt+1
                      1+D              t
                                                         1+D


                                             105
Updating this one period and using the result to eliminate Et rt+1 yields
                 1
   rt     =          if − Et πt+1
                1+D   t

                   D       1                                          D
              +                   Et if − Et π t+2 +                       Et rt+2
                 1+D     1+D          t+1
                                                                     1+D
Continuing to recursively substitute forward results in
                                  ∞               i
                            1            D
                    rt =                              Et if − πt+1+i
                           1+D    i=0
                                        1+D               t+i


                                                  i
under the assumption that limi→∞            D
                                           1+D        Et rt+i = 0.


  6. Ball (1997) uses the following two equation model:

                                  yt+1 = a1 yt − a2 rt + ut+1

                                    πt+1 = πt + γyt + ηt+1

        The disturbances ut and η t are taken to be serially uncorrelated. At time t,
        the policy maker chooses rt , and the state variable at time t is πt +γyt ≡ κt .
        Assume the policy maker’s loss function is given by equation (10.54). The
        optimal policy rule takes the form θt = Aκt where θ t ≡ a1 yt −a2 rt . Derive
        the optimal value of A.

   Note: In the text, the coefficient on rt in the definition of θt is incorrectly
labelled as a3 .
   First rewrite the model in terms of κt and θ t :

                                      yt+1 = θt + ut+1

                                    πt+1 = κt + ηt+1

Though its choice of rt , the policy maker can determine θ t , so it simplifies the
problem to simply treat θt as the policy instrument.
   The loss function (10.54) can now be written as
                              ∞
                      1                                                2
                   L = Et   β i λ (θt + ut+1 )2 + κt + η t+1
                      2 i=1

The objective is to minimize this subject to the constraint that

                           κt+1   = πt+1 + γyt+1
                                  = κt + γθt + ηt+1 + γut+1

                                            106
     Define the value function

                1                    1                             2
V (κt ) = min     λEt (θt + ut+1 )2 + Et κt + ηt+1                     + βEt V κt + γθt + ηt+1 + γut+1
           θt   2                    2


Then the first order conditions include

                  λθt + γβEt V     κt + γθt + ηt+1 + γut+1 = 0                            (158)


                 V (κt ) = κt + βEt V      κt + γθ t + η t+1 + γut+1                      (159)

Multiplying the second of these by γ and adding it to the first yields

                                λθt + γV (κt ) = γκt

or
                                                      λ
                               V (κt ) = κt −              θt
                                                      γ
This implies

                                                           λ
                    Et V (κt+1 ) = Et κt+1 −                       Et θt+1
                                                           γ
                                                               λ
                                   = κt + γθt −                        Et θt+1
                                                               γ

Substituting this back into (158),

                                                  λ
                     λθ t + γβ κt + γθt −                 Et θt+1 = 0
                                                  γ
or
                                γβ                   βλ
                   θt = −                 κt +                          Et θt+1
                              λ + γ 2β             λ + γ2 β
When policy is set at time t, κt summaries the state, so optimal policy, given
the linear-quadratic structure, will be of the form θt = Aκt . Using this proposed
policy θ t = Aκt , and recalling that Et θ t+1 = AEt κt+1 = A(1 + γA)κt , this
becomes
                              γβ                   βλ
                Aκt = −                  κt +                          A(1 + γA)κt
                            λ + γ 2β             λ + γ 2β
which yields the following quadratic equation for A:

                      γβλA2 − λ − βλ + γ 2 β A + γβ = 0


                                           107
the solutions of which are

                     λ − βλ + γ 2 β +           (λ − βλ + γ 2 β)2 + 4γ 2 β 2 λ
             A1 =
                                                2γβλ
and

                     λ − βλ + γ 2 β −           (λ − βλ + γ 2 β)2 + 4γ 2 β 2 λ
             A2 =
                                                2γβλ
   To determine which of these solutions we want, note that

                            κt+1 = κt + γθt = (1 + γA) κt

so that κt will be stable only if A < 0 so that the coefficient 1 + γA is less than
1. Now consider the product of the two solutions A1 and A2 :
                                           2                          2
                          λ − βλ + γ 2 β       −    λ − βλ + γ 2 β        + 4γ 2 β 2 λ
         A1 A2   =
                                                   (2γβλ)2
                      −4γ 2 β 2 λ   1
                 =       2 β 2 λ2
                                  =− <0
                      4γ            λ

so one solution must be positive, the other negative. We are looking for the
negative solution, which is A2 , so our optimal policy rule is
                                                                  
                   λ − βλ + γ 2 β − (λ − βλ + γ 2 β)2 + 4γ 2 β 2 λ
           θt =                                                    κt
                                       2γβλ

In terms of the interest rate actually set by the policy maker, we can use the
definition of θt as a1 yt − a2 rt and κt as πt + γyt to obtain
                                  a1 yt − θt
                     rt    =
                                      a2
                                    a1             1
                           =             yt −             A2 (πt + γyt )
                                    a2             a2

which is in the form of a Taylor rule:

                                   a1 − γA2                 A2
                           rt =                    yt −          πt
                                      a2                    a2




                                           108
9     Typos
The most up-to-date list of known typos can be found through my web page at
http://econ.ucsc.edu/~walshc/ or by going directly to http://econ.ucsc.edu/~walshc/typos.html.
    Chapter 8

    1. Page 382, Problem 1: The coefficient on l∗ should be α, not a.
    2. Page 384, Problem 8: The central bank’s loss function should be the
       discounted sum of the single period loss function given by equation (8.2).
    3. Page 384, Problem 11: The weights on output and inflation in the loss
       function should be (λ − θ) and (1 + θ) so that they sum to 1 regardless of
       the realization of θ. Also set a = 1.

    Chapter 9

    1. Page 429, Problem 4: The term involving output in the loss function
       should be the output gap, y − yn , rather than simply y. Alternatively, one
       could just set yn = 0 as a normalization.

    Chapter 10

    1. Page 452, unnumbered equation at bottom of page: The expression for the
       impact of a money supply shock on the long term nominal interest rate
       should be multiplied by 1/2 (see the coefficient on m in equation 10.36).
    2. Page 476, Problem 6. In the last line of the problem, the coefficient on r
       in the definition of θ should be a2 , not a3 as appears.




                                       109

								
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