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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