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货币银行书后的答案solution
Monetary Theory and Policy (2nd ed.), Problems and Solutions?Carl E. Walsh January 2003Contents1 Introduction 2 Chapter 2: Money-in-the-Utility Function 3 Chapter 3: Money and Transactions 4 Chapter 4: Money and Public Finance 5 Chapter 5: Money, In?ation, and Output in the Short-Run 6 Chapter 6: Money and the Open Economy 7 Chapter 7: The Credit Channel of Monetary Policy 8 Chapter 8: Discretionary Policy and Time Inconsistency 9 Chapter 9: Monetary-Policy Operating Procedures 10 Chapter 10: Interest Rates and Monetary Policy 11 Chapter 11: Policy Analysis in New Keynesian Models 1 2 16 30 39 57 74 75 98 113 1261IntroductionThis manual contains the solutions to all the end-of-chapter problems in Monetary Theory and Policy, 2nd edition (Cambridge, MA: MIT Press, 2003). Please report any errors or suggested corrections to walshc@ucsc.edu. Corrections will be posted at http://econ.ucsc.edu/~walshc/mtp_2e. I would like to thank Kevin Salyer and Federico Ravenna for suggesting problems.?° cThe MIT Press, 2003.12Chapter 2: Money-in-the-Utility Function1. The MIU model of section 2.2 implied that the marginal rate of substitution between money and consumption was set equal to it /(1 + it ) (see 2.12). That model assumed that agents entered period t with resources ω t and used those to purchase capital, consumption, nominal bonds, and money. The real value of these money holdings yielded utility in period t. Assume instead that money holdings chosen in period t do not yield P i utility until period t + 1. Utility is β U (ct+i , Mt+i /Pt+i ) as before, but the budget constraint takes the form ω t = ct + Mt+1 + bt + kt Ptand the household chooses ct , kt , bt , and Mt+1 in period t. The household’s real wealth ω t is given by ω t = f (kt?1 ) + (1 ? δ)kt?1 + (1 + rt?1 )bt?1 + mt . Derive the ?rst order condition for the household’s choice of Mt+1 and show that Um (ct+1 , mt+1 ) = it . Uc (ct+1 , mt+1 ) (Suggested by Kevin Salyer.) Let the value function be V (ω t , mt ) = max {U (ct , mt ) + βV (ω t+1 , mt+1 )} subject to ω t+1 = f (kt ) + (1 ? δ)kt + (1 + rt )bt + mt+1 and ω t ? ct ? mt+1 (1 + π t+1 ) ? bt ? kt = 0. In this setup, rt is the real return on bonds. Let λt denote the Lagrangian on this last constraint. First order conditions for ct , mt+1 , bt , and kt plus the envelope theorem give Uc (ct , mt ) = λt βVm (ω t+1 , mt+1 ) + βVω (ω t+1 , mt+1 ) = λt (1 + π t+1 ) β(1 + rt )Vω (ω t+1 , mt+1 ) = λt Vω (ω t , mt ) = λt Vm (ω t , mt ) = Um (ct , mt ). Now combing the second and third of these to obtain βVm (ω t+1 , mt+1 ) + βVω (ω t+1 , mt+1 ) = β(1 + π t+1 )(1 + rt )Vω (ω t+1 , mt+1 )2or βVm (ω t+1 , mt+1 ) = [(1 + π t+1 )(1 + rt ) ? 1] βVω (ω t+1 , mt+1 ) = it βVω (ω t+1 , mt+1 ). Hence, Vm (ω t+1 , mt+1 ) = it . Vω (ω t+1 , mt+1 ) But the last of the ?rst order conditions implies Vm (ω t+1 , mt+1 ) = Um (ct+1 , mt+1 ), while the ?rst and fourth yield Vω (ω t+1 , mt+1 ) = λt+1 = Uc (ct+1 , mt+1 ). Thus, Um (ct+1 , mt+1 ) = it . Uc (ct+1 , mt+1 ) 2. (Carlstrom and Fuerst 2001): Assume that the representative household’s utility depends on consumption and the level of real money balances available for spending on consumption. Let At /Pt be the real stock of money that enters the utility function. If capital is ignored, the household’s P i objective is to maximize β U (ct+i , At+i /Pt+i ) subject to the budget constraint Yt + Mt?1 (1 + it?1 )Bt?1 Mt Bt + τt + = Ct + + , Pt Pt Pt Ptwhere income Yt is treated as an exogenous process. Assume that the stock of money that yields utility is the real value of money holdings after bonds have been purchased but before income has been received or consumption goods have been purchased: At Mt?1 (1 + it?1 )Bt?1 Bt = + τt + ? . Pt Pt Pt Pt (a) Derive the ?rst order conditions for Bt and for At . Let zt = Mt?1 /Pt + τ t + (1 + it?1 )Bt?1 /Pt . Note that At /Pt ≡ at = zt ? bt . Let the value function be V (zt ) = max {U (ct , zt ? bt ) + βV (zt+1 } , where zt+1 Mt (1 + it )Bt + τ t+1 + Pt+1 Pt+1 ? ? ? ? Pt Pt mt + τ t+1 + (1 + it ) bt , = Pt+1 Pt+1 = Yt + zt ? ct ? mt ? bt = 0. 3andAt time t, the household chooses ct , mt , bt . Let λt denote the Lagrangian on this last constraint. First order conditions plus the envelope theorem give Uc (ct , zt ? bt ) = λt ? ? Pt β Vz (zt+1 ) = λt Pt+1 ? ? Pt ?Ua (ct , at ) + β(1 + it ) Vz (zt+1 ) = λt Pt+1 Vz (zt ) = Ua (ct , zt ? bt ) + λt Now combining the second and third of these to eliminate the value function, one can obtain ?Ua (ct , zt ? bt ) + (1 + it )λt = λt , or Ua (ct , at ) = it λt . Using this in the fourth equation, Vz (zt ) = Ua (ct , at )+λt = λt (1+it ). Now updating this one period and using it to eliminate Vz (zt+1 ) in the second F OC, we are left with Uc (ct , at ) = λt ? ? ? Pt 1 + it+1 λt+1 = β λt+1 . Pt+1 1 + πt+1 (1) (2)λt = β(1 + it+1 )?(b) How do these conditions di?er from those obtained in the text? In the basic MIU model of section 2.2, equation (2.10) and (1) are identical. Using (2.10) and (2.13), equation (2.6) can be written (after setting n = 0) as ? ? 1 + it λt = β λt+1 , 1 + π t+1 which di?ers from (2) in that it rather than it+1 appears. This di?erence in timing is related to the dependence of utility on Mt /Pt in the model of the text and on At /Pt = Mt?1 /Pt +τ t +[(1 + it?1 )Bt?1 ? Bt ] /Pt in the model of this problem. In the basic MIU model of section 2.2, the household can reduce current consumption at a cost of Uc (ct , mt ) = λt and purchase bonds instead, yielding a real payo? of (1 + it )/(1 + πt+1 ) in period t + 1. This gross interest income could be spend on consumption, but it did not serve to augment the period t + 1 holdings of money that enter the utility function. In the model of this problem, the same option to lower current consumption to purchase bonds is open, but purchasing bonds has an addition cost since purchasing additional bonds reduces At /Pt which reduces current utility by Ua (ct , zt ? bt ) = it λt in period t. Thus, the utility cost of purchasing a bond is Uc (ct , zt ? bt ) + Ua (ct , zt ? bt ) = (1 + it )λt . The bonds pays o? 1 + it )/(1 + πt+1 ) in period t + 1 and this return, 4since it is paid out before the goods market opens (i.e., it adds to ) yields the marginal utility of At+1 /Pt+1 (equal to it+1 λt+1 ) plus the marginal utility of consumption ( λt+1 ). That is, the gross interest income from the bond augments the real stock of money that enters time t utility and can be spend on period t consumption. Hence, ? ? ? ? 1 + it 1 + it+1 (1+it+1 )λt+1 ? λt = β λt+1 . (1+it )λt = β 1 + π t+1 1 + π t+1 3. (Calvo and Leiderman 1992): A commonly used speci?cation of the demand for money, originally due to Cagen (1956), assumes that 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 consumption 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 parameters. Show that the demand for money is given by mt = Ae?αt it , B where A = e( D ?1) and αt = w0 (ct )/D. The basic condition from which one can derive the demand for money in Sidrauski’s money-in-the-utility function model is given by (2.12). 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 simpli?ed by employing the approximation x/(1 + x) ≈ x for small x. Sidrauski developed his model in continuous time, in which case the ?rst 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 = w0 (ct ), so this condition becomes B ? D ? D ln mt um B/D ? 1 ? ln mt = = = it 0 (c ) uc w t w0 (ct )/D Rearranging yields ln mt = ( or mt = e( D ?1) e?BB w0 (ct )it ? 1) ? D Dw0 (ct ) it D4. Assume that mt = Ae?αit , where A and α are constants. Calculate the welfare cost of in?ation in terms of A and α, expressed as a percentage of 5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 in?ation 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 Z i l(i, A, α) ≡ Ae?αx dx0Evaluating this integral yields l(i, A, α) = ? A? 1 ? e?αi α (3)The sign depends on e?αi (1 + αi) ? 1, but this is always nonpositive ( e?x (1 + x) is maximized when x = 0 at which point e?x (1 + x) = 1; it then declines with x). From the speci?cation 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 in?ation, households respond by reducing their demand for money, thereby helping to reduce the distortion generated by the in?ation tax. The greater the interest sensitive of money demand, the lower will be the welfare cost of the in?ation tax. P t 5. Suppose W = β (ln ct + mt e?γmt ), γ & 0, and β = 0.95. Assume .5 that the production function is f (kt ) = kt and δ = 0.02. What rate of in?ation maximizes steady-state welfare? How do real money balances at the welfare-maximizing rate of in?ation depend on γ? The steady-state welfare maximizing nominal rate of interest is iss = 0 (see section 2.3) 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 in?ation that yields a zero nominal rate of in?ation is πss = 1 ?1 Ras the welfare cost of an in?ation rate of π = i ? r if r is the real rate of return. The e?ect of α on this cost is ? ? ¤ A ? ?αi αA(ie?αi ) ? A(1 ? e?αi ) ?l(i, A, α) = e (1 + αi) ? 1 ≤ 0 = 2 2 ?α α αIn the steady-state, R is equal to 1/β, or 1/R = β = 0.95 (see 2.18, page 52). Hence, the optimal rate of in?ation is 0.95 ? 1 = ?0.5 or a 5% rate of de?ation. To determine how money demand depends on the parameter γ, use the representative agent’s ?rst order condition (see 2.12), evaluated at the steady-state nominal rate of interest: um iss = uc 1 + iss 6Given the form of the utility function, this becomes um iss = ct (1 ? γmt ) e?γmt = uc 1 + iss At the welfare maximizing in?ation rate, iss = 0, which requires 1 ? γmss = 0 or 1 mss = γ Thus, real money demand at iss = 0 is decreasing in γ. 6. Suppose that the utility function (2.38) is replaced by u (ct , mt , lt ) = ? 1 1?Φ ?1?Φ ?? ¤ 1 ? 1?b 1?b 1?b 1?η lt . act + (1 ? a)mt(a) Derive the ?rst order conditions for the household’s optimal money holdings. The basic condition for optimal money holdings in the MIU model is that the marginal rate of substitution between money and consumption equal the opportunity cost of holding money. With the speci?cation of the utility function given in this problem, this condition is um (1 ? a)m?b it t = = ?b uc 1 + it act which is the same as obtained with (2.38). Because utility can be written as v(c, m)φ(l), the marginal rate of substitution between money and consumption is independent of leisure. (b) Show how (2.43) and (2.44) are altered with this speci?cation of the utility function. With nonseparable utility, the marginal utility of consumption now depends on both real money holdings and leisure: ? ? ? 1?b ¤ 1 ?Φ 1?b 1?b uc (ct , mt , lt ) = act + (1 ? a)mt ×When linearized, this becomes¤ b ? 1?η ?1?Φ ?b ? 1?b act act + (1 ? a)m1?b 1?b lt t ?? ? ?1?Φ ? 1?b ¤ b?Φ 1?η lt = act + (1 ? a)m1?b 1?b ac?b . t t ? ? lt ?1 ct ? ?2 mt + (1 ? Φ)(1 ? η)? .Since ? = ?n? t /(1 ? n), (2.43) is modi?ed to become l n Et [?1 (?t+1 ? ct ) ? ?2 (mt+1 ? mt ) ? ?3 (? t+1 ? nt )] ? rt = 0 (4) c ? ? ? n ? ? 7where ?3 ≡ (1?Φ)(1?η)n/(1?n). For the labor market equilibrium, the marginal rate of substitution between leisure and consumption must equal the marginal product of labor. This condition takes the form ? ? ?? ¤ 1 1?η 1?Φ ? 1?b 1 act + (1 ? a)m1?b 1?b lt t 1?Φ ? ?? ??Φ ? 1?b ¤ 1?Φ 1?η ?η (1 ? η)lt act + (1 ? a)m1?b 1?b lt t Yt ? ? =α , ? 1?b ¤ b?Φ ? 1?η ?1?Φ ?b Nt act act + (1 ? a)m1?b 1?b lt t or Xcb Yt t =α . alt Ntwhere X = ac1?b + (1 ? a)m1?b . Linearizing this expression yields t t ? ? n xt + b?t + ? c ? ? nt = yt ? nt . ? 1?n From the de?nition of X, xt = (1 ? b)γ?t + (1 ? b)(1 ? γ)mt ? c ? ? ss 1?b ¤ where γ = a(css )1?b / a(c ) + (1 ? a)(mss )1?b . Hence ? ? n (1 ? b)γ?t + (1 ? b)(1 ? γ)mt + b?t + c ? c ? ? nt = yt ? nt , ? 1?n or ?? ? ? nss ? ? ? nt = yt ? [b + (1 ? b)γ] ct ? (1 ? b)(1 ? γ)mt , (5) ? 1+ 1 ? nss which can be compared to (2.44). Notice that (5) is independent of η. 7. Suppose the utility function (2.38) is replaced by ? ?h i1?Φ 1 u (ct , mt , lt ) = . ac c1?b + am m1?γ + al l1?η t t 1?Φ(a) Derive the ?rst order conditions for the household’s optimal money holdings. The basic condition for optimal money holdings in the MIU model is that the marginal rate of substitution between money and consumption equal the opportunity cost of holding money. With the speci?cation of the utility function given in this problem, this condition is um (1 ? γ)am m?γ it t = = ?b uc 1 + it (1 ? b)ac ct The marginal rate of substitution between money and consumption is independent of leisure. 8(b) Show how (2.43) and (2.44) are altered with this speci?cation of the utility function. With nonseparable utility, the marginal utility of consumption now depends on both real money holdings and leisure: h i?Φ 1?η uc (ct , mt , lt ) = ac c1?b + am m1?γ + al lt (1 ? b)ac c?b . t t t ? ? ? Ψ1 ct + Ψ2 mt + Ψ3 nt , where Ψ1 = ? [b + Φ(1 ? b)xc ] , Ψ2 = ?Φ(1 ? γ)xm , ? ? n Ψ3 = Φ(1 ? η)(1 ? xc ? xm ) , 1?n xc = ac c1?b ac c1?b + am m1?γ + al l1?ηWhen linearized, this becomesand variables without time subscripts are steady-state values. Use has also been made of the fact that ? = ?n? t /(1 ? n). Equation (2.43) is l n modi?ed to become Et [Ψ1 (?t+1 ? ct ) + Ψ2 (mt+1 ? mt ) + Ψ3 (? t+1 ? nt )] ? rt = 0. c ? ? ? n ? ? For the labor market equilibrium, the marginal rate of substitution between leisure and consumption must equal the marginal product of labor. This condition takes the form ? ?h i1?Φ 1 u (ct , mt , lt ) = . ac c1?b + am m1?γ + al l1?η t t 1?Φ h i?Φ 1?η (1 ? b)ac c?b . uc (ct , mt , lt ) = ac c1?b + am m1?γ + al lt t t t i?Φ h 1?η ?η (1 ? η)al lt ac c1?b + am m1?γ + al lt t t Yt =α , h i?Φ Nt 1?γ 1?η (1 ? b)ac c?b ac c1?b + am mt + al lt t t?η Yt (1 ? η)al lt =α . Nt (1 ? b)ac c?b torLinearizing this expression yields ? ? n b?t + η c ? ? nt = yt ? nt . ? 1?n9Hencewhich can be compared to (2.44). Notice that (6) is independent of mt . ? 8. Suppose a nominal interest rate of im is paid on money balances. These payments are ?nanced by a combination of lump-sum taxes and printing money. Let a be the fraction ?nanced by lump-sum taxes. The government’s budget identity is τ t + vt = im mt , with τ t = aim mt and v = θmt . Using Sidrauski’s model, do the following: (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. The budget constraint in the basic Sidrauski model must be modi?ed to take into account the interest payments on money and that net transfers ( τ in 2.4) consists of two components, the ?rst being the lump-sum transfer v and the second being the lump-sum tax τ . Thus, the budget constraint becomes f (kt?1 ) + (1 ? δ)kt?1 + 1 + im mt?1 ? τ t + vt = ct + kt + mt 1+π? ? ?? n 1+η ? c nt = yt ? b?t , ? 1?n(6)where population growth has been ignored for simplicity. The value function for the problem is still given by (2.5), but the ?rst order conditions change because of the change in the budget constraint. In particular, (2.15) becomes um ? β [fk + 1 ? δ] Vω (ω t+1 ) + β(1 + im ) Vω (ω t+1 ) = 0 1+π (2.4.1)The ?rst 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 ) Using this result, (2.4.1) can be rearranged, resulting in ? ? ? ?? ? β(1 + im ) uc (ct+1 , mt+1 ) um β(1 + im ) 1 = 1? =1? uc 1+π uc (ct , mt ) 1+π βR R(1 + π) ? (1 + im ) i ? im β(1 + 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 di?erence between the nominal return on capital and the nominal return on money.10(b) Show how i?im is a?ected by the method used to ?nance the interest payments on money. Explain the economics behind your result. From the government’s budget constraint, interest payments not ?nanced through lump-sum taxes must be ?nanced 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 a?ects the opportunity cost of money only if a & 0. Printing money to ?nance interest payments on money only results in in? this raises the nominal interest rate i, thereby o?setting the e?ect of paying interest. 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 ?nanced via lump-sum taxes, changes in in?ation do not change the opportunity cost of holding money ― a rise in in?ation that depreciates the individual’s money holdings is o?set by the increase in the transfer the individual anticipates receiving. 9. Suppose money is a productive input into production so that the aggregate production function becomes y = f (k, m). Incorporate this modi?cation into the model of section 2.2. Is money still superneutral? Explain. In the steady state, the household’s Euler condition implies that the gross real rate of return will equal 1/β. Hence, 1 ? δ + fk = 1 , βwhere fk is the marginal produce of capital. When real money balances enters directly into the production function, the marginal product of capital will depend on m. Thus, in the steady state, 1 ? δ + fk (kss , mss ) = 1 , βand the steady-state capital stock will depend on the steady-state level of real money balances. Money will a change in the money growth rate will a?ect in?ation and the equilibrium level of money holdings. The level of capital in the steady-state will need to adjust to ensure fk (kss , mss ) = β ?1 ? 1 + δ. 10. In Sidrauski’s MIU model augmented to include a variable labor supply, money is superneutral if the representative agent’s preferences are given by X X d β i u(ct+i , mt+i , lt+i ) = β i (ct+i mt+i )b lt+i 11but not if they are given by X i X i d β u(ct+i , mt+i , lt+i ) = β (ct+i + kmt+i )b lt+i .Discuss. (Assume output depends on capital and labor and the aggregate production function is Cobb-Douglas.) The steady-state values of css , kss , lss , y ss must satisfy the following four equations: ul = fl (k ss , 1 ? lss ) uc fk (kss , 1 ? lss ) = 1 ?1+δ β (7) (8) (9) (10)d (ct+i mt+i )b lt+i ,css = f (k ss , 1 ? lss ) ? δkss y ss = f (k ss , 1 ? lss ) If the single period utility function is of the form ul dcb mb ld?1 dc = b?1 b d = uc bc m l blthenis independent of m and (7) - (10) involve only the four unknowns css , kss , lss , y ss . These can be solved for css , k ss , lss , and y ss independently of m or in?ation. Superneutrality holds. If the utility function is (ct+i + d kmt+i )b lt+i , then 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, (7) - (10) will involve 5 unknowns ( css , k ss , lss , y ss , and mss ) and cannot be solved independently of the money demand condition and in?ation. Superneutrality does not hold. 11. 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 = θm). Solve for the agent’s demand for money. What is the welfare cost of in?ation? 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 ?nanced via lump-sum taxes, changes in in?ation do not change the opportunity cost of holding money ― a rise in in?ation that depreciates the individual’s money holdings is o?set by the increase in the transfer the individual anticipates receiving. (See also problem 8.) 12. By simulating the model of section 2.5.2, investigate the impact of a money growth shock on employment as η varies from 1 to 10. Provide an intuitive explanation for your ?ndings. Assume ut = γut?1 + φt , where φt is 12x 10 0-4-0 . 5-1-1 . 5-2η η η η η-2 . 5= = = = =0 1 2 4 16 35-3 0 5 10 15 20 25 30Figure 1: Ch. 2, Problem 12: E?ect of η on the employment response to a money growth rate shock (γ = 0.5) white noise and γ = .5. How do your results change if γ = .8? Money growth innovations a?ect the real equilibrium in the MIU model by altering expected in?ation and the marginal utility of consumption which, in turn, alters labor supply. For the baseline parameter values, faster money growth reduces labor supply, employment, and output. The more sensitive labor supply is to changes in the real wage (the larger is η), the greater will be the decline in employment in response to faster money growth. The more serially correlated money growth is, the greater will be the impact of φt on future money growth and in?ation. Therefore, we should expect that the impact of φt on employment increases as γ increases and as η increase. This is con?rmed by the simulation results shown in ?gure 1 for γ = 0.5 and ?gure 2 for γ = 0.8 (note the scale in the two ?gures).13x 10 0-3-0 . 2-0 . 4-0 . 6-0 . 8-1η η η η η= = = = =0 1 2 4 16-1 . 2 0 5 10 15 20 25 30 35Figure 2: Ch. 2, Problem 12: E?ect of η on the employment response to a money growth rate shock (γ = 0.8) 13. By simulating the model of section 2.5.2, investigate the impact of a money growth shock on employment as b varies from .1 to 20. Assume ut = γut?1 + φt , where φt is white noise and γ = .5. Recall that a money growth innovation reduces employment when b & Φ and increases it when b & Φ. So for b = .1, a positive innovation in φt will increase employment, since Φ = 2 in the benchmark calibration, with the e?ect decreasing as b increases, becoming a negative e?ect that grows in absolute value as b becomes larger and larger. Figure 3 shows the e?ects for b = 0.1, 0.2, and 0.5. Figure 4 shows the responses for b = 1, 4, and 20.140.45 0.4 0.35 0.3 0.25 b= 0.1 0.2 0.15 0.1 0.05 0 -0 . 0 5 0 5 10 15 20 25 30 35 b= 0.5 b= 0.2Figure 3: Ch. 2, Problem 13: Employment response for b = 0.1, 0.2, and 0.5.x 10 10-4864b= 120 b= 4 -2 b= 20 -4-6 0 5 10 15 20 25 30Figure 4: Ch. 3, Problem 13: Employment response for b = 1, 4, and 20153Chapter 3: Money and Transactions1. 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 productivity factor. The agent’s utility is given by v(c, l) = c1?Φ /(1 ? Φ) + l1?η /(1 ? η), where l = 1 ? n ? ns and n is time spend in market employment. (a) Derive the transaction time function g(c, m) = ns . From the shopping production function, ? c ?1/a g(c, m) = ns = x b (11) e m(b) Derive the money in the utility function speci?cation 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? From the de?nition of the agent’s utility, v(c, l) = c1?Φ (1 ? n ? ns ) + 1?Φ 1?η ? ?1/a ?1?η ? 1 ? n ? exc b m c1?Φ + ≡ u(c, n, m) 1?Φ 1?η ?v(c, l) ?l ? ?l ?m ?1?η=The marginal utility of money is ?u(c, n, m) ?m == (1 ? n ? ns )?η?bns am?(12)The time spend shopping can be written as c1/a e?x/a m?b/a . The marginal productivity of money in reducing shopping time is given by (b/a)(ns /m), so an increase in b/a increases the e?ect 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 of money. ?l In terms of (12), ?m rises with b/a. But the marginal utility of leisure is a decreasing function of total leisure, so ?v(c,l) declines. ?l The e?ect of x on the marginal utility of money, for given c and n, operates through ns and represents
an increase in x reduces the time needed for shopping for given values of c and m. This a?ects 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) . Thi a higher x decreases the marginal e?ect of m in reducing shopping time, so money is less “productive.” 16(c) Is the marginal utility of consumption increasing or decreasing in m? An increase in consumption a?ects utility in two ways. First, consumption dir vc & 0. This represents the e?ect 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 e?ect will depend on the level of money holding. From (11), consumption and money are complements 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 e?ect in reducing leisure time and will therefore cause a rise in consumption to have a larger overall positive e?ect on utility. 2. De?ne superneutrality. Carefully explain whether the Cooley-Hansen CIA model exhibits superneutrality. What role does the CIA constraint play in determining whether superneutrality holds? A model exhibits the property of superneutrality if the real equilibrium (output, consumption, capital, etc.) is independent of the rate of nominal money growth. Superneutrality normally is interpreted to refer to the steady-state equilibrium of a model. As demonstrated in Chapter 2, the Sidrauski model displays superneutrality with respect to the steady-state, but changes in the in?ation rate will generally a?ect 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. The CooleyHansen model does not display superneutrality. Di?erent rates of in?ation a?ect the opportunity cost of holding money. Through the cash-in-advance constraint, in?ation a?ects the marginal cost of consumption since consumption is treated as a cash good. Higher in?ation induces a substitution away from cash goods and towards credit goods. In Cooley and Hansen’s model, leis cash is not needed to purchase leisure. As a result, changes in the steady-state rate of in?ation alter the demand for leisure and the supply of labor. This was shown in (3.34) on page 114, where Θ was equal in the steady-state to one plus the in?ation rate. 3. Is the steady-state equilibrium in the Cooley-Hansen CIA model a?ected by any of the following modi?cations? Explain. (a) Labor is supplied inelasticly (normalize so that n = 1, where n is the supply of labor). Following on the previous problem, one important modi?cation when labor is supplied inelasticly is that Cooley and Hansen’s model will now display superneutrality. Without a labor-leisure choice, the model becomes essentially the model of section 3.3.1.17(b) Purchases of capital are also subject to the CIA constraint (i.e., one needs money to purchase both consumption and investment goods). 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 ct + it ≤ mt?1 + τ t ≡ at Πt (13)where it = kt ? (1 ? δ)kt?1 is net investment. The value function for this problem is V (at , kt?1 ) = max {u(ct , 1 ? nt ) + βEt V (at+1 , kt )}mt where at+1 = Πt+1 +τ t+1 , kt = f (kt?1 , nt )+(1?δ)kt?1 +at ?ct ?mt and the maximization is subject to the cash in advance constraint (13). 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 α production function ( yt = ezt kt?1 n1?α ), then the budget constraint t is α ezt kt?1 n1?α + (1 ? δ)kt?1 + at ≥ ct + kt + mt tand the ?rst order conditions for ct , kt , mt , and nt , together with the envelope conditions, are uc (ct , 1 ? nt ) ? λt ? ?t = 0 βEt Vk (at+1 , kt ) ? λt ? ?t = 0 ? ? 1 βEt Va (at+1 , kt ) ? λt = 0 Πt+1 ? ? yt ?un (ct , 1 ? nt ) + (1 ? α) βEt Vk (at+1 , kt ) = 0 nt Va (at , kt?1 ) = λt + ?t ? ? yt Vk (at , kt?1 ) = λt α + 1 ? δ + ?t (1 ? δ) kt?1 (14) (15) (16) (17) (18) (19)The Lagrangian ? appears in this last condition because higher capital at the start of the period reduces the cash needed to achieve
only net purchases ( kt ? (1 ? δ)kt?1 ) are subject to the?cash-in-advance constraint. Since (18) implies Et Va (at+1 , kt ) = ? Et λt+1 + ?t+1 , (14) - (19) can be used to derive the following 18where Rt = α yt+1 + 1 ? δ. The ?rst two equations are identical to kt (3.59) and (3.60). The next two di?er. According to (20), 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 (23), 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, (21) implies that ? ss ? Π ?ss = λss ?1 β which implies Πss ≥ β will be requires for the existence of a steadystate since ? must be nonnegative. Now eliminate ? from the steadystate version of (23): ? ss ? ? ss ? ? ? Π Π λss = β Rss λss + λss ? 1 (1 ? δ) β β or, recalling that Rss ? 1 + δ is the steady-state marginal product of y ss capital α kss , ? ss ? ? ? y ss Π 1 α ss = ?1+δ k β β which depends on the rate of in?ation. Thus, superneutrality does not hold when capital purchases are also subject to the cash-in-advance constraint. Notice that this conclusion would hold even if labor is supplied inelasticly as in part (a) of this question (see Problem 6 below). By imposing a tax on capital purchases, in?ation a?ects the steadystate capital stock and k ss 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).conditions, which should be compared to (3.59), (3.60), (3.61) and (3.62): uc (ct , 1 ? nt ) = λt + ?t (20) ? ? λt+1 + ?t+1 βEt (21) = λt Πt+1 ? ? yt ?un (ct , 1 ? nt ) + (λt + ?t ) (1 ? α) =0 (22) nt ¤ ? λt + ?t = βEt Vk (at+1 , kt ) = βEt Rt λt+1 + ?t+1 (1 ? δ) (23)19(c) The growth rate of money follows the process ut = γut?1 + ?t , where 0 & γ & 1 and ? is a mean zero, independently, and identically distributed process. The steady-state depends on the average rate of money growth since that pins down average in?ation. It does not depend on the transitory dynamics of the monetary supply process, although the short-run dynamics will. 4. Use (3.60), (3.62), and (3.63) to show that the nominal interest rate is positive as long as the CIA constraint is expected to bind in the future. The three equations are ? ? λt+1 + ?t+1 λt = βEt (24) Πt+1 λt = βEt Rt λt+1 ? ? 1 + it λt = βEt λt+1 , 1 + π t+1 Equation (24) can be manipulated to yield ? ? ? ? ?t+1 λt+1 = βEt . λt ? βEt Πt+1 Πt+1 Multiplying and dividing the left side by (1 + it ) and using (25), the left side becomes ? ? ? ? ? ? ?? 1 (1 + it )λt+1 1 Et = λt 1 ? λt ? β 1 + it Πt+1 1 + it ? ? it = λt . 1 + it Hence, λt ? it 1 + it ? = βEt ? ?t+1 Πt+1 ? . (25) (26)In an equilibrium with a ?nite price level, the right side will be positive (and so the nominal interest rate will be positive) as long as the CIA constraint binds in some states of nature so that the expected value of ?t+1 is positive. 5. MIU and CIA models are alternative approaches to constructing models in which money has positive value in equilibrium. (a) What strengths and weaknesses do you see in each of these approaches? Both the money-in-the-utility function approach and the cash-in-advance approach 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 20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 income 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 speci?cation of a shopping time model, and the notion of a production function for shopping time allows for less rigid substitution between di?erent 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 degree 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 di?erent circumstances. (b) Suppose you wanted to study the e?ects of the growth of credit cards on money demand. Which approach would you adopt? Why? 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 3. 6. Consider the model of section 3.3.1. Suppose that money is required to purchase both consumption and investment goods. The CIA constraint then becomes ct +xt ≤ mt?1 /Πt + τ t , where x is investment. Assume that α 1?α the aggregate production function takes the form yt = ezt kt nt . Show that the steady-state capital-labor ratio is a?ected by the rate of in?ation. Does a rise in in?ation 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 setup is similar to Problem 3.b, so the equations de?ning the steady-state are, from (20) - (23), uc (css ) = λss + ?ss (27) ? ss ? λ + ?ss β = λss (28) Πss λss + ?ss = β [Rss λss + ?ss (1 ? δ)] (29) Equation (22) has been dropped since there is no labor supply decision,21and utility in (27) depends only on consumption. From (28), ? ss ? Π ss ss ? =λ ?1 β so (29) becomes ? Πss β ? ? ss ? ? ? Π ss =β R + ? 1 (1 ? δ) βFor convenience, normalize n to 1. Then Rss + 1 ? δ ≡ αy ss /kss + 1 ? δ = α(k ss )α?1 + 1 ? δ, and (29) implies ? ss ? ? ? Π 1 ss α?1 = α(k ) ?1+δ β β Hence, the steady-state capital-labor ratio is k ss =?1 ?? ? ? ss ? ? ?? 1?α 1 Π 1 ?1+δ α β βwhich is decreasing in the in?ation rate ( Πss ). Higher in?ation implies a higher tax on capital purchases and this lowers the steady-state stock of capital. 7. Consider the following model: Preferences: Et∞ X i=0β i [ln ct+i + θ ln dt+i ] mt?1 1 + πt (30)a Budget constraint: ct + dt + mt + kt = Akt?1 + τ t +mt?1 , (31) 1 + πt where m denotes real money balances and π t is the in?ation 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. CIA Constraint: ct ≤ τ t + (a) Does this model exhibit superneutrality? Explain. A model exhibits superneutrality if the real variables k, c, and d are independent of π in the steady-state. If we de?ne at ≡ τ t + mt?1 , the value function 1+π t can be de?ned as V (at , kt?1 ) = max {ln ct+i + b ln dt+i ? mt a +βEt V τ t+1 + , Akt?1 + (1 ? δ)kt?1 1 + πt+1 +at ? ct ? dt ? mt )} 22where the maximization is subject to ct ≤ at Let ? denote the Lagrangian multiplier associated with this cash-inadvance constraint. From the ?rst order conditions for the agent’s decision problem, 1 ? βEt Vk (at+1 , kt ) ? ?t = 0 ct b ? βEt Vk (at+1 , kt ) = 0 dt 1 βEt Va (at+1 , kt ) ? βEt Vk (at+1 , kt ) = 0 1 + π t+1 Va (at , kt?1 ) = βEt Vk (at+1 , kt ) + ?t ? ? a?1 Vk (at , kt?1 ) = β aAkt?1 + 1 ? δ Et Vk (at+1 , kt ) ?? ?? (32) (33) (34) (35) (36)plus the two constraints (30) and (31). Equation (36) implies that, in the steady-state, ?¤ ?? 1 = β aA(k ss )a?1 + 1 ? δ ? kss = 1 αA 1 ?1+δ β?1 ?? 1?α(37) This means that the steady-state capital stock is independent of the in?ation rate. Let λt ≡ βEt Vk (at+1 , kt ). From (32) and (33), ? ? ?t dt = 1+ b (38) ct λt Equations (34) and (35) imply ? ? λt+1 + ?t+1 = λt βEt 1 + π t+1 In the steady-state, this implies ? ? λss + ?ss ?ss 1 + πss = 1 + ss = ss λ λ β and combining this with (38), dss = css ? 1 + π ss β ? b (39)so the relative consumption of c and d depends on the rate of in?ation. The real equilibrium does not display superneutrality.23(b) What is the rate of in?ation that maximizes steady-state utility? From the steady-state marginal product of capital condition (37), we have the standard result that the real rate of return will equal 1/β. Letting R ≡ 1/β, (39) can be written as dss = (1 + iss )bcss where i = R(1 + π). Letting Z = A(k ss )a ? δkss , in the steady-state we have from the budget constraint (30) Z = css + dss = css + (1 + iss )bcss or css = ?Z where ? = [1 + (1 + iss )b]?1 . Hence, steadystate utility of the representative agent can be expressed as 1 [ln css + b ln dss ] = 1?β = = = 1 1?β 1 1?β 1 1?β 1+b 1?β [ln css + b ln(1 + iss )bcss ] [ln ?Z + b ln(1 + iss )b?Z] [(1 + b) ln ?Z + b ln(1 + iss )b] ln ? + 1+b b ln Z + ln(1 + iss )b 1?β 1?βNow maximize this with respect to the nominal rate of interest i. Since Z was shown earlier to be independent of the in?ation rate, the ?rst order condition is ? ? ? ? 1+b b 1 b ? =0 + 1 ? β 1 + (1 + iss )b 1 ? β 1 + iss or 1+b 1 = ss 1+i 1 + (1 + iss )b1 + (1 + iss )b =1+b 1 + iss This holds if and only if iss = 0 So the optimal rate of in?ation will be the rate that yields a zero nominal rate of interest. 8. Consider the following model: Preferences: Et∞ X i=0which impliesβ i [ln ct+i + ln dt+i ] mt?1 + (1 ? δ)kt?1 , 1 + πta Budget constraint: ct + dt + mt + kt = Akt?1 + τ t +24where m denotes real money balances and π t is the in?ation rate from period t ? 1 to period t. Utility depends on the consumption
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 θ of d is purchased using cash, then the household also faces a CIA constraint that takes the form ct + θdt ≤ mt?1 + τ t. 1 + πtWhat is the relationship between the nominal rate of interest and whether the CIA constraint is binding? Explain. Will the household ever use cash to purchase d (i.e., will the optimal θ ever be greater than zero)? The basic model is similar to the one studied in Problem 6, di?ering 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 a +βEt V τ t+1 + , Akt?1 + (1 ? δ)kt?1 1 + π t+1 +at ? ct ? dt ? mt )} where at = τ t + mt?1 and ct + θdt ≤ at . and we require that 0 ≤ θ ≤ 1 1+π t since θ is the fraction of the d good purchased with cash. Actually, the relevant consideration is whether θ is positive or not. Let ? denote the Lagrangian on the cash-in-advance constraint and let φ be the Lagrangian on the constraint θ ≥ 0. The ?rst order conditions for the household’s decision problem for the current are simply stated here as, modifying them to re?ect the di?erent utility function and cash-in-advance constraint: 1 ? βEt Vk (at+1 , kt ) ? ?t = 0 ct 1 ? βEt Vk (at+1 , kt ) ? θ?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 ? ? a?1 Vk (at , kt?1 ) = β aAkt?1 + 1 ? δ Et Vk (at+1 , kt ) ??t dt + φt ≤ 0 θt φt = 0 where the condition θt φt = 0 is the complementary slackness condition associated with the inequality constraint on θ. Since θ cannot be reduced 25In addition, we need the ?rst order condition for the optimal choice of θ. This takes the formbelow zero, the optimum can have ??t dt + φt & 0 at θ = 0; utility could be increased by reducing θ 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 φ 6= 0 from which the condition θt φt = 0 implies that θ = 0. So, as long as the nominal rate of interest is positive, the household will never use cash to purchase d. 9. Suppose the representative household enters period t with nominal money balances Mt?1 and receives a lump-sum transfer Tt . During period t, the bond market opens ?rst, and the household receives interest payments and purchases nominal bonds in the amount Bt . With its remaining money (Mt?1 + Tt + It?1 Bt?1 ? Bt ), the household enters the goods market and purchases consumption goods subject to Pt ct ≤ Mt?1 + Tt + It?1 Bt?1 ? Bt . The household receives income at the end of the period and ends period t with nominal money holdings Mt , given by ? ¤ α Mt = Pt ezt Kt?1 Nt1?α + (1 ? δ)Kt?1 ? Kt +Mt?1 +Tt +It?1 Bt?1 ?Bt ?Pt ct . If the household’s objective is to maximize & # ∞ ∞ X X c1?Φ (1 ? Nt+i )1?η t+i i i E0 β u(ct+i , 1 ? Nt+i ) = E0 β +Ψ , 1?Φ 1?η i=0 i=0 do the equilibrium conditions di?er from (3.59)-(3.63)? De?ne At ≡ Mt?1 + Tt + It?1 Bt?1 . The household’s decision problem can be expressed in terms of the value function as ( ) c1?Φ (1 ? Nt )1?η t V (At , Kt?1 ) = max +Ψ + βEt V (At+1 , Kt ) , 1?Φ 1?η where the maximization is over ct , Nt , Kt , Bt , and Mt , and is subject to Yt + (1 ? δ)Kt?1 + At Bt Mt ? ? ? ct ? Kt ≥ 0 Pt Pt Pt (40) (41)At Bt ? ? ct ≥ 0 Pt Pt and At+1 = Mt + Tt+1 + It Bt ,26α where Yt = ezt Kt?1 Nt1?α . Let λt and ?t denote the Lagrangian multipliers on the budget constraint (40) and the cash-in-advance constraint (41). The ?rst order conditions includec?Φ = λt + ?t t Ψ(1 ? Nt )?η = (1 ? α) ? Yt Nt ? λt(42) (43) (44) (45) (46) (47) (48)βEt VK (At+1 , Kt ) = λt βEt It VA (At+1 , Kt ) = λt + ?t Pt λt PtβEt VA (At+1 , Kt ) = VA (At , Kt?1 ) =λt + ?t Pt ? ? ? ? Yt + 1 ? δ λt . VK (At , Kt?1 ) = α Kt?1The ?rst two of these equations are identical to equations (3.59) and (3.61). Updating (47) one period and using (46) yields ? ? λt+1 + ?t+1 λt = βEt , (49) Pt Pt+1 which, when multiplied through by Pt is identical to (3.60). Updating (48) one period and using (44) yields ? ? ? ? Yt+1 λt = βEt α + 1 ? δ λt+1 Kt which is identical to (3.62). From (45) and (46), It (λt /Pt ) = (λt +?t )/Pt , or, since It = 1 + it , it = ?t /λt . Using (49), this implies ? ? λt+1 + ?t+1 λt + ?t = βEt It Pt Pt+1 or λt + ?t ? ? λt+1 + ?t+1 = (1 + it )λt = βEt ? ? 1 + it = βEt (1 + it+1 ) λt+1 , 1 + πt+1 1 + it 1 + π t+1 λt = βEt ? 1 + it+1 1 + π t+1 ? λt+1 (50) ? ?or27which di?ers from (3.63) in that it+1 appears rather than it . To understand the di?erence, recall that the model leading to equation (3.63) assumed the goods market open ?rst. Thus, cash needed for time t consumption needed to be accumulated during period t?1, and the opportunity cost of these funds was measured by it?1 . In the model speci?ed in this problem, the asset market opens ?rst. To increase ct , the agent must reduce period t purchases of bonds (Bt ), sacri?cing interest income of it . This means that the opportunity cost of the cash needed to purchase ct is it . The agent could reduce current consumption slightly, at a cost of the time t marginal utility of consumption uc (t), invest in bonds and increase t + 1 consumption by (1 + it )/(1 + π t+1 ), yielding utility (1 + it )/(1 + π t+1 )uc (t + 1). Since the marginal utility of consumption at t is equal to (1 + it )λt (from 42), (50) follows. 10. Trejos and Wright (1993) ?nd 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 CIA model. (For simplicity, assume that only labor is needed to produce output according to the production function y = n.) Does the same hold true in an MIU model? In a basic cash-in-advance model, in?ation taxes cash goods. Suppose the nominal rate of
relative to the case of a zero nominal interest rate, households will be consuming fewer cash goods (which bear the in?ation tax) and more credit goods. Since leisure is a credit good, in?ation will tend to lower output by increasing the demand for leisure and reducing labor supply. For example, (3.34) shows how in?ation reduces labor supply relative to the case of a zero nominal rate of interest. If we modify the model of Section 3.3.2 by ignoring capital, and assume the production function is y = n, then the value function for the decision problem of the household becomes (see 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 cashin-advance constraint ct ≤ at . If ?t is the Lagrangian multiplier associated with the cash-in-advance constraint, then the ?rst order necessary conditions are uc (ct , 1 ? nt ) ? βV 0 (nt + at ? ct ? mt ) ? ?t = 0 Πt+1 βV 0 (nt + at ? ct ? mt ) =0 Πt+1?ul (ct , 1 ? nt ) + V 0 (at ) =βV 0 (nt + at ? ct ? mt ) + ?t Πt+1 28Let λt ≡βV 0 (nt +at ?ct ?mt ) . Πt+1Then these ?rst order conditions imply ? ??1 ?t 1+ ≤1 λtul (ct , 1 ? nt ) λt = = uc (ct , 1 ? nt ) λt + ?tAs 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 (2.34) on page 65. 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 in?ation don’t a?ect output. A positive nominal interest rate reduces real money holdings relative to the social optimum. How that a?ects labor supply (and output) will depend on how a decrease in m a?ects ul /uc and the e?ect could go either way. For the utility function used in the linear version of the money-in-the-utility-function model of Chapter 2, (2.44) 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 m an increase in m increases the marginal utility of consumption. Higher in?ation 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.294Chapter 4: Money and Public Finance1. Suppose real income grows at the rate ?t & 0. How are (4.6) and (4.7) a?ected? Does seigniorage depend on ?? Explain. With real income growth, the ?nal term in (4.5) becomes (see (4.4)) ? ? 1 ht ? ht?1 . (1 + πt )(1 + ?t ) Hence, (4.6) becomes st ? 1 = ht ? ht?1 (1 + πt )(1 + ?t ) ? ? 1 = ht ? ht?1 + 1 ? ht?1 (1 + πt )(1 + ?t ) ? ? (1 + πt )(1 + ?t ) ? 1 = ht ? ht?1 + ht?1 . (1 + πt )(1 + ?t ) ?Since πt and ?t are both growth rates, their product should be small (i.e., if in?ation is 2 percent and real income growth is 3 percent, π t ?t = 0.02 ? 0.03 = 0.0006), we can approximate seigniorage as ? ? π t + ?t st ≈ ht ? ht?1 + ht?1 1 + π t + ?t which is increasing in ?t . If households wish to maintain their real money holdings relative to income, faster income growth implies that the demand for money raises faster. Since the government is the monopoly supply of money, the increase in the growth rate of money demand increases the seigniorage revenues to the government. In a steady-state, constant real money balances relative to income implies that θ?π?? = 0, where θ is the growth rate of the nominal money supply. With h constant, seigniorage becomes ? ? ? ? π+? θ h= h. 1+π+? 1+θ Notice that the right side is the same as in (4.7). Seigniorage depends on the growth rate of nominal income ( π + ? = θ) since this is the rate at which nominal money demand rises. For a given θ, faster real growth just reduces the rate of in?ation, leaving seigniorage unchanged. 2. Consider the version of the Sidrauski (1967) model studied in problem 3 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. Approximate steady-state revenues from seigniorage are given by θm, where θ is the growth rate of the money supply.30(a) Is there a “La?er 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? ? From Problem Set 2, we know that the demand for money in this model is B given by mt = Ae?i/ct D where ln A = D ? 1. Hence, seigniorage in the steady-state is equal to sss = πAe?(rss+π)/css DTaking the derivative with respect to π, ?sss ?π = Ae?(rss ss π ? ss Ae?(r +π)/c D c D ? ? ss ss π = Ae?(r +π)/c D 1 ? ss c D ss+θ)/css DThis is positive (i.e. seigniorage is increasing in in?ation) for π & css D, and negative for π & css D. Hence, there is a La?er curve. (b) What rate of money growth maximizes steady-state revenues from seigniorage? Steady-state seigniorage is maximized for π = θ = css D. (c) Assume now that the economy’s rate of population growth is λ and reinterpret m as real money balances per capita. What rate of in?ation maximizes seigniorage? How does it depend on λ? 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 in?ation rate of css D, but this now corresponds to a rate of money growth of css D + γ. 3. 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, government purchases, income tax receipts, and seigniorage. Assume seigniorage is given by f (π t ), where π is the rate of in?ation. 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 in?ation to minimize ∞ X i Et β [h(τ t+i ) + k(πt+i )] ,i=0where the functions h and k represent the distortionary costs of the two tax sources. Assume that the functions h and k imply positive and increasing marginal costs of both revenue sources.31(a) What is the intratemporal optimality condition linking the choices of τ and π at each point in time? Solving the budget constraint forward, the government’s decision problem can be written as min Et subject to Rbt?1 + Et X∞ X i=0β i [h(τ t+i ) + k(π t+i )] XR?i gt+i ? EtR?i [τ t+i yt+i + f (πt+i )] = 0Let λ be the Lagrangian multiplier associated with this constraint. The ?rst order conditions are ? ¤ Et β i h0 (τ t+i ) ? λR?i yt+i = 0 and Hence, the condition linking taxes and seigniorage at each date t + i take the form λ Et k 0 (πt+i ) Et h0 (τ t+i ) = = i Et yt+i (βR) Et f 0 (π t+i ) (b) What is the intertemporal optimality condition linking the choice π at di?erent points in time? The ?rst order conditions for seigniorage at dates t+i and t+j take the form β i Et k0 (st+i ) = λR?i Et f 0 (π t+i ) and β j Et k0 (st+j ) = λR?j Et f 0 (πt+j ), or Et (βR)i k 0 (π t+i ) k 0 (πt+j ) = λ = Et (βR)j 0 0 (π f t+i ) f (πt+j ) ¤ ? Et β i k0 (πt+i ) ? λR?i f 0 (πt+i ) = 0(c) Suppose y = 1, f (π) = aπ, h(τ ) = bτ 2 , and k(π) = cπ 2 . Evaluate the inter- and intratemporal conditions. Find the optimal settings P ?i for τ t and πt in terms of bt?1 and R gt+i . Given the assumed functional forms, the ?rst order conditions become ?c? λ 2bEt τ t+i = =2 Et π t+i (βR)i a ?c? which implies that Et τ t+i = ab Et πt+i for all i. The intertemporal condition becomes ?c? ?c? 2 Et π t+i (βR)i = 2 Et π t+j (βR)j or for j = 0, Et πt+i = (βR)?i π t a a c These results imply Et τ t+i = ab (βR)?i π t . Now we can evaluate the government’s budget constraint recalling at y = 1): X X Rbt?1 + Et R?i gt+i = Et R?i (τ t+i yt+i + f (π t+i )) X c = R?i ( (βR)?i π t + a(βR)?i π t ) ab ?c ? X = (βR2 )?i + a πt ab 32This implies that πt = ?c i ??1 h X R?i gt+i B Rbt?1 + Et +a abwhere B = βR2 /(βR2 ? 1). Finally, the optimal tax rate is given by τt = ?c? ? c ?? c i ??1 h X R?i gt+i B Rbt?1 + Et πt = +a ab ab ab(d) Using your results from part c, when will optimal ?nancing imply constant planned tax rates and in?ation over time? Et π t+i = (βR)?i π t = πt if and only if βR = 1 (i.e. R = 1/β). 4. Suppose utility is given by U = c1?σ /(1?σ)+m1?θ /(1?θ). Find the function φ(P ) de?ned in (4.32) and verify that it has the shape shown in ?gure 4.3. Solve for the stationary equilibrium price level P ? such that P ? = ? ??θ ? φ(P ? ). The equilibrium condition becomes M /Pt = (R ? 1) c?σ /R. ? ??θ ? In the steady-state, this becomes M /Pt = (Pt+1 ? βPt ) c?σ /Pt+1 , or ? Pt+1 = 1 ? cσ ? Letting x ≡ cσ M ?θ , note that βPt ? ??θ ≡ φ(Pt ).? M Ptφ(0) = 0 and ¤ ? β 1 ? (1 ? θ)xPtθ φ (Pt ) = ? ?2 1 ? xPtθ0so In addition,0 & φ0 (0) = β & 1Pt →∞lim φ(Pt ) = limβPt = ∞. θ Pt →∞ 1 ? xPt5. Consider (4.37) implied by the ?scal theory of the price level. Seigniorage st was de?ned as it mt /(1 + it ). Assume that the utility function of the ? representative agent takes the form u(c, m) = ln c + b ln m. Show that st = bct and that the price level is independent of the nominal supply ? of money as long as τ t ? gt + bct is independent of Mt . With the logseparable utility function, the ?rst order condition for the representative agent’s money holdings requires that um bct it bct (1 + it ) = = ? mt = . uc mt 1 + it it 33Hence, we can write seigniorage as ? ? ? ?? ? it it bct (1 + it ) st = ? mt = = bct . 1 + it 1 + it it Substituting this into (4.37), Dt X = λt,t+i [τ t+i + bct+i ? gt+i ] . Pt i=0∞P∞ With Dt predetermined, Pt depends on i=0 λt,t+i [τ t+i + bct+i ? gt+i ]. If τ + bc ? g is independent of the nominal supply of money, then so is Pt . 6. Mankiw (1987) suggested that the nominal interest rate should evolve as a random walk under an optimal tax policy. Suppose that the real rate of interest is constant and that the equilibrium price level is given by (4.29). Suppose that the nominal money supply is given by mt = mp + vt , where t mp is the central bank’s planned money supply and vt is a white noise t control error. Let θ be the optimal rate of in?ation. There are di?erent processes for mp that lead to the same average in?ation rate but di?erent time-series behavior of the nominal interest rate. For each of the processes for mp given below, demonstrate that average in?ation is θ. Is the nominal t interest rate is a random walk? (a) mp = θ(1 ? γ)t + γmt?1 . Equation (4.24) states that t pt = mt αEt pt+1 + 1+α 1+α (51)For the money process in part (a), this becomes pt = θ(1 ? γ)t + γmt?1 + vt αEt pt+1 + 1+α 1+α (52)and the no-bubbles solution is of the form pt = p0 + at + bmt?1 + cvt where a, b, and c are coe?cients 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 ]34Using this and the trial solution in (52) yields p0 + at + bmt?1 + cvt θ(1 ? γ)t + γmt?1 + vt 1+α α [p0 + a(t + 1) + b [θ(1 ? γ)t + γmt?1 + vt ]] + 1+α ? ? α(p0 + a) θ(1 ? γ) + αa + αbθ(1 ? γ) t+ = 1+α 1+α ? ? ? ? γ(1 + αb) 1 + αb + mt?1 + vt 1+α 1+α = α(p0 + a) ? p0 = αa 1+αThis will hold for all realizations of vt and mt?1 if p0 = ?a=? θ(1 ? γ) + αa + αbθ(1 ? γ) ? a = θ(1 ? γ)(1 + αb) 1+α b= γ γ(1 + αb) ?b= 1+α 1 + α(1 ? γ) c= 1 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) 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 in?ation will equal ? 1+α γ?m ?p = 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 in?ation 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) 35 ?With a constant real rate of interest, as was assumed in deriving (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 ?rst 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 is AR(1) and vt is white noise. Quasi-?rst di?erencing, (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?1So the nominal interest rate will follow the ?rst order autoregressive process it = i0 + γit?1 + γvt?1 0 (b) mp = mt?1 + θ. t becomes With the money supply process in (b), (4.24) pt =θ + mt?1 + vt αEt pt+1 + (53) 1+α 1+α and the equilibrium solution for pt is of the form pt = p0 + bmt?1 + cvt . Using this in (53), = θ + mt?1 + vt α [p0 + b(θ + mt?1 )] + 1+α 1+α ? ? ? ? ? ? 1 αp0 1 + αb 1 + αb θ+ + mt?1 + vt = 1+α 1+α 1+α 1+α 1 vt 1+α ? ?p0 + bmt?1 + cvtor p0 = (1 + α)θ, b = 1, and c = 1/(1 + α). With pt = (1 + α)θ + mt?1 +average in?ation is just θ. Expected in?ation is Et pt+1 ? pt = mt ? mt?1 ? 1 vt = θ ? 1+α 1 1+α vtWith a constant real rate of interest, as was assumed in deriving (4.24), the nominal rate of interest will equal ? ? 1 it = r0 + θ ? vt 1+α 36so that it is equal to a constant plus it is not a random walk. 7. (Correia and Teles 1999) Consider the optimal tax problem of section 4.4.3. The government wishes to maximize u(c, m, l) = v(c, m) + φ(l) subject to the economy’s resource constraint: f (1 ? l) ≥ c + g. (a) Derive the implementability constraint by using the ?rst order conditions (4.48)-(4.50) to eliminate the tax rates from the representative agent’s budget constraint (4.47). Using (4.48)-(4.50), the budget constraint (4.47) becomes ?v ? ?v ? c m (1 + τ )c + τ m m ? f (1 ? l) = c+ m ? f (1 ? l) = 0, λ λ or vc c + vm m + λf (1 ? l) = 0. (b) Set up the government’s optimization problem and derive the ?rst order conditions. The government’s problem is max v(c, m) + φ(l) + χ [vc c + vm m + λf (1 ? l)] + ψ [f (1 ? l) ? c ? g] ,c,m,lwhere χ is the Lagrangian multiplier on the implementability constraint and ψ is the Lagrangian multiplier on the resource constraint. The ?rst order conditions are vc + χ [vc + vcc c + vmc m] ? ψ = 0 vm + χ [vcm c + vm + vmm m] = 0 φ0 (l) ? χλf 0 ? ψf 0 = 0 vc c + vm m + λf (1 ? l) = 0 f (1 ? l) ? c ? g. (c) Show that the ?rst order condition for m is satis?ed if vm = vmc = vmm = 0. Argue that these conditions are met if the satiation level m? is equal to ∞. The ?rst order condition for m is vm + χ [vcm c + vm + vmm m] = 0. If vm = vmc = vmm = 0, this condition is satis?ed for all c and m. At the satiation level m? , vm = 0. Hence, the ?rst order condition will be satis?ed if vcm c + vmm m = 0 at m? . But if vm = 0 at m? = ∞, then we must also have vmm and vmc equal to zero.378. Suppose the Correia-Teles model of section 4.4.3 is modi?ed so that output is equal to f (n), where f is a standard neoclassical production function exhibiting positive but diminishing marginal productivity of n. Show that if f (n) = na for a & 0, the optimality condition given by (4.70) continues to hold. Start with the budget constraint when f (n) = na . Equation (4.68) becomes Rt?1 dt?1 = =∞ X i=0 ∞ X i=0? ¤ Di ct+i ? (1 ? τ t+i )f (1 ? lt+i ? ns ) + Rt?1+i It?1+i mt?1+i t+i ? ¤ Di ct+i ? (1 ? τ t+i )(1 ? lt+i ? ns )a + Rt?1+i It?1+i mt?1+i t+iFollowing the same steps outlines on page 190, we use the fact that dt?1 = 0 and Di = β i D0 λt+i /λt to obtain∞ ¤ D0 X i ? β λt+i ct+i ? λt+i (1 ? τ t+i )(1 ? lt+i ? ns )a ? λt+i It+i mt+i = 0 t+i λt i=0while the ?rst order condition for labor supply, (4.65), is modi?ed to become ul (54) = aλt (1 ? lt ? ns )a?1 t (1 ? τ t )From (54), λt+i (1 ? τ t+i ) = ul (1 ? lt ? ns )1?a /a. Making this substitution t (along with the others discussed in the text) yields 0 = = ? ? ? ? ∞ D0 X i ul (1 ? lt ? ns )1?a t β uc ct+i ? ul ηgns ? (1 ? lt+i ? ns )a t+i t+i λt i=0 a∞ ?u ? i D0 X i h l β uc ct+i ? ul ηgns ? (1 ? lt ? ns ) t+i t λt i=0 aThis implies∞ X? ? ? ? ?u ? 1 l β i uc ct+i ? (1 ? lt ) ? ul ? η ns t+i = 0 a a i=0which corresponds to (4.69) on page 191. Notice that the only modi?cation 1 is that ns is multiplied by the factor a ? η; in the example of the text, a = 1 and this became 1 ? η. The ?rst-order condition for the optimal choice of m in the social welfare problem is ? ? 1 β i ψul ( ? η) ? ?t+i g 0 = 0 a1 which replaces (4.70). As long as β i ψul ( a ? η) ? ?t+i must be nonzero, 0 the optimum still involves g = 0, or a zero nominal interest rate.385Chapter 5: Money, In?ation, and Output in the Short-Run1. Assume that nominal wages are set for one period but that they can be indexed to the price level:c 0 wt = wt + b(pt ? Et?1 pt ),where w0 is a base wage and b is the indexation parameter (0 ≤ b ≤ 1). (a) How does this change modify the aggregate supply equation given by (5.17)? From (5.16) and the new speci?cation for the contract wage, employment is given by ? 0 ¤ nt = yt ? wt + b(pt ? Et?1 pt ) + pt0 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 ) (55)Notice that if b = 0 (no indexation), we obtain the expression in the text (5.17). 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. If we assume labor supply is inelastic, we can set Et?1 nt = 0 since all variables should be interpreted as deviations around a steady-state. Substituting (55) into the production function (5.7), yt ? Et?1 yt = (1 ? α) [yt ? Et?1 yt + (1 ? b) (pt ? Et?1 pt )] +et ? Et?1 et = a(1 ? b) (pt ? Et?1 pt ) + (1 + a)εt (56)(b) Suppose the demand side of the economy is represented by a simple quantity equation, mt ? pt = yt and assume mt = vt where vt , is a mean zero shock. Assume the indexation parameter is set to minimize Et?1 (nt ? Et?1 n? )2 and show that the optimal degree of t wage indexation is increasing in the variance of v and decreasing in the varian