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					                   MSc Finance: International Finance: Lecture 5
                   Monetary Policy with Forward-Looking Agents
                                          Anne Sibert

                                          Autumn 2008

                                     1. Introduction
Consider the following problem: A sophisticated scud missile passes overhead. The ob-
jective is to launch a rocket to destroy the scud missile as quickly as possible, to avoid the
missile taking evasive action. What you control is the angle at which you shoot o¤ the
rocket. This is a typical problem that a scientist might face. An important feature of the
problem is that the state (here, the location of the rocket) at a given time is a function
of past actions (here, the choice of the angle). With monetary policy, the current state is
in‡uenced by the future. Here is an example of how this can matter

               2. Monetary Policy with Forward-Looking Agents
The model described here is based on Kydland and Prescott [2] and Barro and Gordon
[1].
     I assume that social welfare is given by
                                          2              2
                                   =            (n    no ) ;   > 0:                              (1)
where is in‡    ation, n is employment, and no is socially optimal employment.1 All three
of these variables are in logarithm form. The interpretation of this is as follows. Optimal
in‡ ation is assumed to be zero. This is not important and it saves on notation. Society
dislikes deviations from this socially optimal rate and welfare falls at an increasing rate as
in‡ ation rises above or falls below zero. Society also dislikes deviations from the socially
optimal level of emplyment and welfare falls at an increasing rate as employment rises
above or falls below this optimal level. The parameter          tells us how society weights
losses due to in‡ ation deviations versus losses due to output deviations. The higher is ,
the greater the weight society places on employment deviations.
    We now assume a standard Keynesian model of employment determination. We sup-
pose that …rms and workers enter into wage contracts. These contracts specify a …xed
nominal wage W . After the wage is set, the price level is realised and …rms decide how
much labour to hire. Firms maximise their pro…ts by setting the marginal product of
labour equal to the real wage, W=P . Thus, the …rms have a downward sloping labour
demand curve.
    Let N denote workers’ and …rms’ most preferred level of employment, called the
natural rate. Let P e be the workers’ and …rms’ expectation of the price level. This is
where our model deviates from the Keynesian model. In deciding what nominal wage to
choose, the private sector forecasts the price level. We will initially assume that there
is no uncertainty in the model and that the private sector has perfect foresight. Thus,
it will turn out that this expectation will equal the actual price level. We assume that
   1 Suppose that the country is a small open economy and that PPP holds. Then the price level equals

the exchange rate, as in the monetary approach model of lectures 3 and 4. Thus, in‡ation equals the
depreciation of the exchange rate.



                                                 1
MSc Finance: International Finance: Lecture 5Monetary Policy with Forward-Looking Agents2


the workers and …rms choose the contractual nominal wage W such that if the price level
turns out to be P e , then the …rms choose employment of N . Suppose that workers and
…rms are wrong and the price level turns out to be higher than expected, say P 1 > P e .
Then W=P 1 < W=P e and the …rms choose employment of N 1 > N . If he price level
turns out to be lower than expected, say P 2 < P e , then W=P 2 > W=P e and the …rms
choose employment of N 2 < N . In general, it is clear that employment is increasing in
P=P e , with employment equal to N when P=P e = 1: We can write this idea in logarithm
form as

                               n=n +                (p      pe ) ;          > 0:               (2)
   To simplify the notation, we let        be one. We can rewrite equation (2) as

                     n = n + (p        p   1)       (pe         p   1)      =n +       e
                                                                                           ;   (3)
where p 1 is last period’ price level (in logarithm form), = p p 1 is in‡
                         s                                                ation and e =
  e
p                              s
     p 1 is the private sector’ expectation of in‡ ation.
        Equation (3) is an expectations-augmented Phillips Curve. It shows that there
is a positive relationship between employment and unexpected in‡  ation. The di¤erence
between this Phillips Curve and the conventional Philips curve is a result of the private
sector anticipating the future actions of the government when it sets the contractual
nominal wage.
    Substituting equation (3) into equation (1) yields
                                       2                                e          2
                               =                (n +                         no ) :            (4)
                           o
                                                              s
     We assume that n < n : This means that private sector’ desired level of employment
is less than the socially optimal rate. A reason for this might be distortionary taxes (such
as income taxes) in labour markets. Let d no n > 0 denote the deviation between
the optimal and natural rates of employment. Use this notation to rewrite equation (4):
                                           2                        e         2
                                   =                    (                   d) :               (5)
   We see from equation (5) that because of the distortion in the labour market, the
government wants to increase employment. It can do this, but at the cost of higher
in‡ation.
   Recall that the timing is that the private sector forms its expectation of in‡      ation
and incorporates this expectation into its nominal wage contract. Then the central bank
chooses monetary policy. This implies that when the central bank picks its policy, it takes
expected in‡ation as given –that is, it treats it as a constant in its optimisation problem.
   We assume that the central bank can perfectly control in‡      ation. Thus, it chooses
to maximise equation (5). To …nd the …rst-order condition of the maximisation problem
di¤erentiate with respect to and set the result equal to zero
                                                                        e
                          d =d =           2        2 (                      d) = 0:           (6)
   To verify that the solution to this is indeed a maximum. We …nd …nd the second-order
condition; di¤erentiate (6) and verify the result is strictly negative.

                                   d2 =d        2
                                                    =       2       2 < 0:                     (7)
MSc Finance: International Finance: Lecture 5Monetary Policy with Forward-Looking Agents3


   Solving equation (6) for in‡ation yields

                                                 ( e + d)
                                         =                :                                 (8)
                                                  1+
    Thus, we have that in‡    ation is increasing in expected in‡   ation. This is because the
higher is expected in‡  ation, the higher actual in‡ation has to be to create a given amount
of unexpected in‡   ation. In‡  ation is increasing in d because the greater the deviation
between the optimal and the natural rates of employment, the more incentive the policy
maker has to create unexpected in‡    ation. In‡ation is also rising in the parameter ; this is
because the greater the weight that the policy maker puts employment deviations relative
to in‡ ation deviations, the more he is willing to in‡  ate.
    The public knows the preferences of the policy maker; hence, the public can solve
for in‡                                              s
        ation. Thus, we assume that the public’ expectation of in‡          ation equals actual
in‡ ation. Substituting e = into equation (8) yields

                                         ( + d)
                                    =           ,         = d:                              (9)
                                         1+
    Note from equation (3) that workers and …rms get their most preferred level of em-
ployment, n . The outcome also has strictly positive in‡ation. Everyone would be better
o¤ if the policy maker could commit himself to in‡  ation of zero. If zero in‡ ation were
expected, then workers and …rms would still have employment of n : This is the optimal
solution. But, the policy maker cannot attain the optimal solution. If zero in‡ation were
expected, then by equation (8), the central bank would want to reneg and set
                                                   d
                                             =       :                                     (10)
                                                 1+
    Anticipating this, the public will not expect zero in‡ation. Thus, we say that the opti-
mal solution of zero in‡ ation is not time consistent. This problem –that the government
will choose not to follow its optimal policy is called the time inconsistency problem.
    It appears that a solution is to appoint a conservative central banker – that is, a
central banker who puts no weight on employment deviations relative to output. Such a
central banker will always choose zero in‡   ation.

                             3. The Time-Inconsistency Problem
Suppose that a policy maker must pick a series of actions between time 0 and time T .
We denote the time-t action by xt . The action might be a choice of in‡        ation or a tax or
some other policy. Suppose that at time zero, the policy maker found the optimal plan:
x0 ; x1 ; :::xT . If there is uncertainty, one might think of the optimal action at time t as a
vector specifying what to do in each possible state of the world. There are no surprises;
the policy maker knows what states of the world can occur. Thus, it seems that at time
t; 0 < t       T , the policy maker would …nd it optimal follow xt ; xt+1 ; :::xT . If, however,
there is a t where he would not want to do this, then the optimal plan is said to be time
inconsistent.
     In our previous example, think of the plan as covering periods zero and one. In period
zero, the private sector forms its expectations, picking expected in‡       ation equal to the
in‡  ation rate the government will choose in period one. There is no government policy.
In period one, the government chooses in‡       ation. If the government were able to commit
                                                                                   s
to a policy in period zero, it would take into account that the private sector’ expectation
MSc Finance: International Finance: Lecture 5Monetary Policy with Forward-Looking Agents4


of in‡ation will equal the in‡ation rate it chooses in period one. Thus, it would attempt to
in‡ uence this expectation. It would optimise by choosing zero in‡  ation. If the government
were then given the opportunity to change its mind in period one, it would want to do so.
This is because in period one, expectations have already been formed so the government
would not try to in‡  uence them. They are now treated as a constant.

                           4. Credibility vs. Flexibility
It appears that a solution to the time-inconsistency problem is to appoint a conservative
central banker –that is, a central banker who puts no weight on employment deviations
relative to output. Such a central banker will always choose zero in‡ ation.Now suppose
that the …rms’production functions are subject to a shock that shifts the labour demand
curve. Assume that the timing of events is as follows:

  1. Workers and …rms enter into nominal wage contracts specifying W .
  2. A stochastic mean-zero shock         shifts the labour demand curve.
  3. The central bank chooses in‡ation.

    Consider Figure 1 again. Suppose that the workers and …rms set W = P e and subse-
quently a shock shifts the labour demand curve in from N d to N1 .  d

    If the government sets in‡  ation equal to what the private sector expected before it
learned the realiszation of the shock, then employment will be N1 < N . It is to the
bene…t of the private sector if the central bank in‡     ates more than the private sector
expected so that P = P1 and N = N . Because the private sector forms its expectation
of in‡ ation and chooses W before the shock is realised and because the central bank
chooses in‡ ation after the shock is realised, the central bank has a stabilisation role.
    We can analyse this algebraically by rewriting equation (5) as
                                      2                      e           2
                             W =                 (               d      ) :            (11)
     where is a shock that shifts the labour demand function in if it is positive and out
if it is negative. Assume that has mean zero and variance 2 .
     The central bank chooses in‡ation after the shock has occurred, so it treats the shock
as a constant and maximises equation (11). The …rst-order condition is
                                                         e
                               2     2 (                     d       ) = 0;            (12)
   Solving yields
                                                     e
                                             (        +d+ )
                                     =                      :                          (13)
                                                     1+
    We assume that the private sector has rational expectations so that its expectation of
in‡ ation is the statistical expectation: e = E . Then taking expectations of both sides
of equation (13) yields

                                    (E + d + )                       (E + d)
                        E =E                   =                             :         (14)
                                      1+                              1+
   Solving yields
MSc Finance: International Finance: Lecture 5Monetary Policy with Forward-Looking Agents5




                                                   E = d:                                                                       (15)
   Substituting equation (15) into equation (13) yields

                                           ( d+d+ )
                                      =             = d+    :                                                                   (16)
                                             1+          1+
   In‡  ation is now equal to the in‡ation bias term d that we found last time plus a
stabilisation term = (1 + ).
   Substituting equations (15) and (16) into equation (11) gives social welfare

                         2                                      2                                    2                          2
   =       d+                                      d                =               d+                                  +d          :
                1+                    1+                                                 1+                       1+
                                                                                                                                (17)
   Taking the expected value yields
                                                   2                2                                         2
                              2                        +
                E   =             +       d2                    2       =           ( + 1) d2                     :             (18)
                                                   (1 + )                                                1+
   Suppose that instead the government appointed a conservative central banker or
appointed an independent monetary policy committee and ordered it to follow a zero-
in‡ation rule. Then = e and welfare is
                                                                        2
                                                   =        (d + ) :                                                            (19)
   Taking the expected value yields

                                               E   =            d2 +        2
                                                                                :                                               (20)
   The conservative conservative central bank or rule is better than than the central
banker who maximises welfare (this case is often called discretion) if

                                                            2                                                               2
         d2 +   2
                     >            ( + 1) d2                         ,       d2 +         2
                                                                                             <       ( + 1) d2 +                (21)
                                                       1+                                                              1+
                                                                                    2
                     ,       (1 + ) d2 + (1 + )                 2
                                                                    < ( + 1) d2 +                2
                                                                                                     ,    2
                                                                                                              < (1 + ) d2 :

    This is sensible. A central banker who maximises social welfare tends to be bad
because he produces an in‡    ation bias and tends to be good because he stabilises. The
conservative central banker does not produce an in‡      ation bias or stabilise. Thus, if the
variance of the labour-demand shock is su¢ ciently large, the stabilisation e¤ect dominates
and discretion is better than a rule. But, if the variance of the shock is su¢ ciently small,
then stabilisation is not that important and the rule is better than discrestion. Society
faces a tradeo¤: If it appoints a conservative central banker or if it legislates a zero-
in‡ation rule, it gains credibility (for low in‡ation), but it loses ‡exibility (in responding
to shocks). A challenge is to come up with monetary institutions that confer credibility
without sacri…cing too much ‡    exibility.

                                5. Introduction
We consider our simple monetary approach model in continuous time.
MSc Finance: International Finance: Lecture 5Monetary Policy with Forward-Looking Agents6


                         6. A Continuous-Time Model
In continuous time our model becomes

                                         mt        pt     =          y    it                     (22)
                                                   pt     =          et                          (23)
                                                   it     =          i + ee
                                                                         _t                      (24)
                                                   ee
                                                   _      =          _
                                                                     e                           (25)

    Equation (22) is the LM curve, equation (23) is purchasing power parity, equation (24)
is uncovered interest parity and equation (25) is perfect foresight. A dot over a variable
                                                                  _
is the conventional way of denoting a time derivative; that is, xt = dxt =dt. Uncovered
interest parity and perfect foresight together imply that the rate of depreciation of the
exchange rate equals the interest rate di¤erential. Since the interest rate di¤erential is
…nite, the derivative of the exchange rate with respect to time is …nite as well. Thus, the
exchange rate cannot "jump". Substituting equations (23) - (25) into equation (22) yields

                         mt        et = y               _
                                                        et ; where y = y                   i :   (26)

   Equation (26) is a …rst-order linear di¤ erential equation with a constant coe¢ cient.
The generic form for this is

                                             _
                                             xt = a + bxt + cyt :                                (27)
    Recall that there is a trick to solving this (it is the continuous-time analogue to our
lag-operator approach). Rewrite equation (27) as

                                             _
                                             xs     bxs = a + cys :                              (28)
   Notice that I have changed the dummy variable from t to s. This is harmless and will
be convenient because I want to solve for xt . Now multiply both sides of equation (28)
by e bs (where e is the exponential function, not the exchange rate!):
                                   bs                                 bs
                              e          _
                                        (xs        bxs ) = e               (a + cys ) :          (29)
                                                             bs
   The left-hand side is the derivative of e                      xs . Thus, we have

                             d           bs                          bs
                                e             xs       = e                (a + cys ) )           (30)
                             ds
                                         bs                          bs
                              d e             xs       = e                (a + cys ) ds:

    Now integrate both sides. What should the limits of integration be? What if we tried
integrating from s = 0 to s = t:

                        Rt          bs
                                                         Rt           bs
                         0
                             d e         xs        =         0
                                                                 e         (a + cys ) ds )       (31)
                                   bs         t          R
                                                         t
                                                                     bs
                              e         xs    0
                                                   =         e            (a + cys ) ds:
                                                         0

                                                                                                 b
        Recall that the de…nition of the "line" notation in the above equation is F (s)ja =
F (b)   F (a); b > a. Thus, equation (31) becomes
MSc Finance: International Finance: Lecture 5Monetary Policy with Forward-Looking Agents7




                             bt
                                                       R
                                                       t
                                                                   bs
                     e            xt    x0        =            e        (a + cys ) ds )                                   (32)
                                                           0

                                       bt
                                                                   R
                                                                   t
                                                                                bs
                                   e        xt    = x0 +                e            (a + cys ) ds =)
                                                                    0
                                                                         R
                                                                         t
                                            xt    = x0 ebt +                    eb(t      s)
                                                                                               (a + cys ) ds:
                                                                            0

    This gives x in terms of its past history and the initial value of x. We can see that x
will go o¤ to plus or minus in…nity if b > 0. When we solve our exchange rate model, this
approach will not work because the analogue of b will be strictly positive and the model
will be unstable. We will thus follow the same procedure we did before: we will solve
forward, rather than backwards. In terms of the above equation, we will integrate forward
from t to in…nity, rather than backward from 0 to t. Our boundary condition will not be
an initial value of x, but rather that x does not go to plus or minus in…nity when this is
not warranted by the fundamentals.
    Write equation (26) in the format of equation (27):


                                             mt       et       = y                    _
                                                                                      et )                                (33)
                                              _
                                             et       et       = y                   mt =)
                                                  1              y                   ms
                                            _
                                            es        es       =                        :

    I have a small notational problem. I have let e denote the exchange rate, so I will
call the exponential function "exp ". Then, multiplying both sides of equation (33) by
exp( s= ) and integrating forward yields


                                  1                                             y         ms
                    _
                    es                es exp( s= )                 =                            exp( s= ) )               (34)

                                                                                y         ms
                                  d [es exp( s= )]                 =                            exp( s= )ds )
                     Z       1                                                       Z    1
                                                                                1
                                  d [es exp( s= )]                 =                           (y    ms ) exp( s= )ds )
                         t
                                                                                     Zt 1
                                                                                1
       lim [es exp( s= )]              et exp( t= )                =                           (y    ms ) exp( s= )ds:
       s!1                                                                            t

    The …rst term on the left-hand side of equation (34) is only positive when there are
bubbles. This is because exp( s= ) goes to zero as s goes to in…nity. Thus, the …rst term
is zero unless the exchange rate is going to plus or minus in…nity faster than exp( s= )
is going to zero. Assuming that this term is zero, we have
MSc Finance: International Finance: Lecture 5Monetary Policy with Forward-Looking Agents8




                           Z     1
                       1
    et exp( t= )   =                 (y
                                  ms ) exp( s= )ds                            (35)
                             t
                          Z 1                   Z
                       y                      1 1
                   =          exp( s= )ds           ms exp( s= )ds
                           t
                                             Z 1t
                                     1     1
                   =     y exp( s= )jt            ms exp( s= )ds
                                              t
                                                             Z
                                                            1 1
                   =     y lim exp( s= ) + y exp( t= )           ms exp( s= )ds
                           s!1                                t
                                        Z 1
                                     1
                   =   y exp( t= )           ms exp( s= )ds:
                                              t

   This implies

                                             Z
                                           1 1
              et exp( t= ) = y exp( t= )         ms exp( s= )ds )                  (36)
                                              t
                               Z
                             1 1
              et exp( t= ) =       ms exp( s= )ds y exp( t= ) )
                                t
                               Z
                             1 1           s t
                        et =       ms exp(      )ds y :
                                          t

   The outcome is similar to the the discrete-time case. The exchange rate depends on
the current and all future values of the money supply.

                                     References
[1] Barro, R. and D. Gordon, "A Positive Theory of Monetary Policy in a Natural Rate
    Model," Journal of Political Economy 91, 589-610.
[2] Kydland, F. and E. Prescott (1977), "Rules Rather than Discretion: The Inconsistency
    of Optimal Plans," Journal of Political Economy 85, 473-91.

				
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