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foundations macroeconomics

Questions and Answers of Foundations Macroeconomics

11.21. Growth and seignorage, and an alternative explanation of the inflationgrowth relationship. (Friedman, 1971.) Suppose that money demand is given by ln (M/P) = a − bi + ln Y, and that Y is
11.20. Rational political business cycles. (Alesina and Sachs, 1988.) Suppose the relationship between output and inflation is given by yt = y +b(πt − Et −1πt), where b > 0 and where Et −1
11.18. Consider the situation analyzed in Problem 11.16, but assume that there is only some finite number of periods rather than an infinite number. What is the unique equilibrium? (Hint: Reason
11.17. Other equilibria in the Barro–Gordon model. Consider the situation described in Problem 11.16. Find the parameter values (if any) for which each of the following is an equilibrium:(a)
11.16. Solving the dynamic-inconsistency problem through punishment. (Barro and Gordon, 1983.) Consider a policymaker whose objective function is∞t=0 βt(yt − aπ2 t /2), where a > 0 and 0 < β <
11.15. In the model of delegation analyzed in Section 11.7, suppose that the policymaker’s preferences are believed to be described by (11.59), with a > a, when πe is determined. Is social
11.14. The tradeoff between low average inflation and flexibility in response to shocks with delegation of control over monetary policy. (Rogoff, 1985.)Suppose that output is given by y = y n + b(π
11.13. A model of reputation and monetary policy. (This follows Backus and Driffill, 1985, and Barro, 1986.) Suppose a policymaker is in office for two periods. Output is given by (11.53) each
11.12. (Fischer and Summers, 1989.) Suppose inflation is determined as in Section 11.7. Suppose the government is able to reduce the costs of inflation;that is, suppose it reduces the parameter a in
11.11. The importance of using rather than saving your ammunition in the presence of the zero lower bound. Suppose inflation is described by the accelerationist Phillips curve, ˙π(t) = λy(t),λ >
11.10. Uncertainty and policy. (Brainard, 1967.) Suppose output is given by y =x + (k + εk)z + u, where z is some policy instrument controlled by the government and k is the expected value of the
11.9. Money versus interest-rate targeting. (Poole, 1970.) Suppose the economy is described by linear IS and money-market equilibrium equations that are subject to disturbances: y = c− ai + ε1, m
11.8. Consider the system given by (11.41).(a) What does the system simplify to when φπ = 1? What are the eigenvalues of the system in this case? Suppose we look for self-fulfilling movements in
11.7. Consider the model of Section 11.4. Suppose, however, the aggregate supply equation, (11.16), is πt = πt−1 +α(yt−1 − y n t−1)+επt , where επ is a white-noise shock that is
11.6. Regime changes and the term structure of interest rates. (See Mankiw and Miron, 1986.) Consider an economy where money is neutral. Specifically, assume that πt = mt and that r is constant at
11.5. Policy rules, rational expectations, and regime changes. (See Lucas, 1976, and Sargent, 1983.) Suppose that aggregate supply is given by the Lucas supply curve, yt = y + b(πt − πe t ), b >
11.4. Suppose you want to test the hypothesis that the real interest rate is constant, so that all changes in the nominal interest rate reflect changes in expected inflation. Thus your hypothesis is
11.3. Assume, as in Problem 11.2, that prices are completely unresponsive to unanticipated monetary shocks for one period and completely flexible thereafter.Assume also that y = c − ar and m − p
11.2. Consider a discrete-time model where prices are completely unresponsive to unanticipated monetary shocks for one period and completely flexible thereafter. Suppose the IS equation is y = c −
11.1. Consider a discrete-time version of the analysis of money growth, inflation, and real balances in Section 11.1. Suppose that money demand is given by mt − pt = c − b(Et pt +1 − pt), where
10.17. The efficiency of the decentralized equilibrium in a search economy. Consider the steady state of the model of Section 10.6. Let the discount rate, r, approach zero, and assume that the firms
10.16. Consider the static search and matching model analyzed in equations(10.71)–(10.75). Suppose, however, that the matching function, M(•), is not assumed to be Cobb–Douglas or to have
10.15. Consider the model of Section 10.6.(a) Use equations (10.65) and (10.69), together with the fact that VV = 0 in equilibrium, to find an expression for E as a function of the wage and exogenous
10.14. Consider the model of Section 10.6. Suppose the economy is initially in equilibrium, and that y then falls permanently. Suppose, however, that entry and exit are ruled out; thus the total
10.13. Consider the steady state of the Mortensen-Pissarides model of Section 10.6.(a) Suppose that φ = 0. What is the wage? What does the equilibrium condition (10.70) simplify to?
10.12. Describe how each of the following affects steady-state employment in the Mortensen–Pissarides model of Section 10.6:(a) An increase in the job breakup rate, λ.(b) An increase in the
10.11. In the setup described in Problem 10.10, suppose that w is distributed uniformly on [μ − a,μ + a] and that C < μ.(a) Find V in terms of μ,a, and C.(b) How does an increase in a affect V
10.10. Partial-equilibrium search. Consider a worker searching for a job. Wages, w, have a probability density function across jobs, f (w), that is known to the worker; let F (w) be the associated
10.9. The Harris–Todaro model. (Harris and Todaro, 1970.) Suppose there are two sectors. Jobs in the primary sector pay wp; jobs in the secondary sector pay ws. Each worker decides which sector to
10.8. An insider-outsider model. Consider the following variant of the model in equations (10.39)–(10.42). The firm’s profits are π = A F (LI + LO) − wI LI −wO LO, where LI and LO are the
10.7. Implicit contracts under asymmetric information. (Azariadis and Stiglitz, 1983.) Consider the model of Section 10.5. Suppose, however, that only the firm observes A. In addition, suppose there
10.6. Implicit contracts without variable hours. Suppose that each worker must either work a fixed number of hours or be unemployed. Let C E i denote the consumption of employed workers in state i
10.5. The fair wage-effort hypothesis. (Akerlof and Yellen, 1990.) Suppose there are a large number of firms, N, each with profits given by F (eL)−wL, F(•) > 0, F (•) < 0. L is the number of
10.4. Suppose that in the Shapiro–Stiglitz model, unemployed workers are hired according to how long they have been unemployed rather than at random;specifically, suppose that workers who have been
10.3. Describe how each of the following affect equilibrium employment and the wage in the Shapiro–Stiglitz model:(a) An increase in workers’ discount rate, ρ.(b) An increase in the job breakup
10.2. Efficiency wages and bargaining. (Garino and Martin, 2000.) Summers (1988, p. 386) states, “In an efficiency wage environment, firms that are forced to pay their workers premium wages suffer
10.1. Union wage premiums and efficiency wages. (Summers, 1988.) Consider the efficiency-wage model analyzed in equations (10.12)–(10.17). Suppose, however, that fraction f of workers belong to
9.14. The Modigliani–Miller theorem. (Modigliani and Miller, 1958.) Consider the analysis of the effects of uncertainty about discount factors in Section 9.7.Suppose, however, that the firm
9.13. (This follows Bernanke, 1983a, and Dixit and Pindyck, 1994.) Consider a firm that is contemplating undertaking an investment with a cost of I. There are two periods. The investment will pay off
9.12. Consider the model of investment with kinked adjustment costs in Section 9.8.Describe the effect of each of the following on the q˙ = 0 locus, on the area where K˙ = 0, on q and K at the time
9.11. Consider the model of investment under uncertainty with a constant interest rate in Section 9.7. Suppose that, as in Problem 9.10, π (K) = a − bK and that C (I ) = αI 2/2. In addition,
9.10. Suppose that π (K) = a − bK and C (I ) = αI 2/2.(a) What is the q˙= 0 locus? What is the long-run equilibrium value of K?(b) What is the slope of the saddle path? (Hint: Use the approach
9.9. Suppose that the costs of adjustment exhibit constant returns in κ˙ and κ.Specifically, suppose they are given by C (κ˙/κ)κ, where C (0) = 0, C(0) = 0, C (•) > 0. In addition, suppose
9.8. A model of the housing market. (Poterba, 1984.) Let H denote the stock of housing, I the rate of investment, pH the real price of housing, and R the rent.Assume that I is increasing in pH, so
9.7. Consider the model of investment in Sections 9.2–9.5. Suppose it becomes known at some date that there will be a one-time capital levy. Specifically, capital holders will be taxed an amount
9.6. Consider the model of investment in Sections 9.2–9.5. Describe the effects of each of the following changes on the K˙ = 0 and q˙ = 0 loci, on K and q at the time of the change, and on their
9.5. Using the calculus of variations to find the socially optimal allocation in the Romer model. Consider the Romer model of Section 3.5. For simplicity, neglect the constraint that LA cannot be
9.4. Using the calculus of variations to solve the social planner’s problem in the Ramsey model. Consider the social planner’s problem that we analyzed in Section 2.4: the planner wants to
9.3. The major feature of the tax code that affects the user cost of capital in the case of owner-occupied housing in the United States is that nominal interest payments are tax-deductible. Thus the
9.2. Corporations in the United States are allowed to subtract depreciation allowances from their taxable income. The depreciation allowances are based on the purchase price of the capital; a
9.1. Consider a firm that produces output using a Cobb–Douglas combination of capital and labor: Y = KαL1−α, 0 < α < 1. Suppose that the firm’s price is fixed in the short run; thus it takes
8.15. Time-inconsistent preferences. Consider an individual who lives for three periods. In period 1, his or her objective function is ln c1 + δln c2 + δln c3, where 0 < δ < 1. In period 2, it is
8.14. Precautionary saving with constant-absolute-risk-aversion utility. Consider an individual who lives for two periods and has constant-absoluterisk-aversion utility, U = −e−γ C1 − e−γ
8.13. Habit formation and serial correlation in consumption growth. Supposethat the utility of the representative consumer, individual i, is given by T t=1[1/(1 + ρ)t](Cit/Zit)1−θ/(1 − θ), ρ
8.12. Consumption of durable goods. (Mankiw, 1982.) Suppose that, as in Section 8.2, the instantaneous utility function is quadratic and the interest rate and the discount rate are zero. Suppose,
8.11. The equity premium and the concentration of aggregate shocks. (Mankiw, 1986.) Consider an economy with two possible states, each of which occurs with probability 1 2 . In the good state, each
8.10. The Lucas asset-pricing model. (Lucas, 1978.) Suppose the only assets in the economy are infinitely lived trees. Output equals the fruit of the trees, which is exogenous and cannot be stored;
8.9. Bubbles. Consider the setup of the previous problem without the assumption that lims→∞ Et [Pt +s/(1 + r)s ] = 0.(a) Deterministic bubbles. Suppose that Pt equals the expression derived in
8.8. Consider a stock that pays dividends of Dt in period t and whose price in period t is Pt. Assume that consumers are risk-neutral and have a discount rate of r ;thus they maximize E [∞t =0
8.7. Consider the two-period setup analyzed in Section 8.4. Suppose that the government initially raises revenue only by taxing interest income. Thus the individual’s budget constraint is C1 +
8.6. A framework for investigating excess smoothness. Suppose that Ct equals[r/(1 + r)]{At + ∞s =0 Et [Yt +s]/(1 + r)s}, and that At +1 = (1 + r )(At + Yt − Ct).(a) Show that these assumptions
8.5. (This follows Hansen and Singleton, 1983.) Suppose instantaneous utility is of the constant-relative-risk-aversion form, u(Ct) = C1−θ t /(1 − θ), θ > 0. Assume that the real interest
8.4. In the model of Section 8.2, uncertainty about future income does not affect consumption. Does this mean that the uncertainty does not affect expected lifetime utility?
8.3. The time-averaging problem. (Working, 1960.) Actual data give not consumption at a point in time, but average consumption over an extended period, such as a quarter. This problem asks you to
8.2. The average income of farmers is less than the average income of nonfarmers, but fluctuates more from year to year. Given this, how does the permanent-income hypothesis predict that estimated
8.1. Life-cycle saving. (Modigliani and Brumberg, 1954.) Consider an individual who lives from 0 to T, and whose lifetime utility is given by U = T t =0 u(C(t))dt, where u(•) > 0, u(•) < 0. The
7.12. Consider the model of Section 7.8. Suppose, however, that monetary policy responds to current inflation and output: rt = φππt + φy yt + uMP t .(a) For the case of white-noise disturbances,
7.11. Consider a continuous-time version of the Mankiw–Reis model. Opportunities to review pricing policies follow a Poisson process with arrival rate α > 0.Thus the probability that a price path
7.10. The new Keynesian Phillips curve with partial indexation. Consider the analysis of the new Keynesian Phillips curve with indexation in Section 7.7.Suppose, however, that the indexation is only
7.9. Consider the new Keynesian Phillips curve with indexation, equation (7.76), under the assumptions of perfect foresight and β = 1, together with our usual aggregate demand equation, yt = mt −
7.8. (This follows Ball, 1994a.) Consider a continuous-time version of the Taylor model, so that p (t) = (1/T)Tτ=0 x (t − τ)dτ, where T is the interval between each individual’s price changes
7.7. State-dependent pricing with both positive and negative inflation. (Caplin and Leahy, 1991.) Consider an economy like that of the Caplin–Spulber model.Suppose, however, that m can either rise
7.6. Consider the experiment described at the beginning of Section 7.4. Specifically, a Calvo economy is initially in long-run equilibrium with all prices equal to m, which we normalize to zero. In
7.5. Repeat Problem 7.4 using lag operators.
7.4. Consider the Taylor model with the money stock white noise rather than a random walk; that is, mt = εt, where εt is serially uncorrelated. Solve the model using the method of undetermined
7.3. Synchronized price-setting. Consider the Taylor model. Suppose, however, that every other period all the firms set their prices for that period and the next. That is, in period t prices are set
7.2. The instability of staggered price-setting. Suppose the economy is described as in Problem 7.1, and assume for simplicity that m is a random walk(so mt = mt−1+ut, where u is white noise and
7.1. The Fischer model with unbalanced price-setting. Suppose the economy is described by the model of Section 7.2, except that instead of half of firms setting their prices each period, fraction f
6.16. Consider an economy consisting of some firms with flexible prices and some with rigid prices. Let pf denote the price set by a representative flexible-price firm and pr the price set by a
6.15. Observational equivalence. (Sargent, 1976.) Suppose that the money supply is determined by mt = c zt −1 + et, where c and z are vectors and et is an i.i.d.disturbance uncorrelated with zt
6.13. Thick-market effects and coordination failure. (This follows Diamond, 1982.)33 Consider an island consisting of N people and many palm trees.Each person is in one of two states, not carrying a
6.12. Indexation. (This problem follows Ball, 1988.) Suppose production at firm i is given by Yi = SLαi , where S is a supply shock and 0 < α ≤ 1. Thus in logs, yi = s + αi . Prices are
6.11. Consider an economy consisting of many imperfectly competitive, pricesetting firms. The profits of the representative firm, firm i, depend on aggregate output, y, and the firm’s real price,
6.10. Multiple equilibria with menu costs. (Ball and D. Romer, 1991.) Consider an economy consisting of many imperfectly competitive firms. The profits that a firm loses relative to what it obtains
6.9. (a) Consider the model in equations (6.27)–(6.30). Solve the model using the method of undetermined coefficients. That is, conjecture that the solution takes the form yt = AuIS t , and find
6.8. Consider the model in equations (6.27)–(6.30). Suppose, however, that there are shocks to the MP equation but not to the IS equation. Thus rt = byt +uMP t , uMP t = ρMPuMP t + eMP t (where
6.7. The liquidity trap. Consider the following model. The dynamics of inflation are given by the continuous-time version of (6.22)–(6.23): π˙(t) = λ[y(t)−y (t)],λ > 0. The IS curve takes the
6.6. The central bank’s ability to control the real interest rate. Suppose the economy is described by two equations. The first is the IS equation, which for simplicity we assume takes the
6.5. Productivity growth, the Phillips curve, and the natural rate. (Braun, 1984, and Ball and Moffitt, 2001.) Let gt be growth of output per worker in period t, πt inflation, and π W t wage
6.4. The analysis of Case 1 in Section 6.2 assumes that employment is determined by labor demand. Under perfect competition, however, employment at a given real wage will equal the minimum of demand
6.3. The multiplier-accelerator. (Samuelson, 1939.) Consider the following model of income determination. (1) Consumption depends on the previous period’s
6.2. The Baumol-Tobin model. (Baumol, 1952; Tobin, 1956.) Consider a consumer with a steady flow of real purchases of amount αY, 0 < α ≤ 1, that are made with money. The consumer chooses how
6.1. Describe how, if at all, each of the following developments affect the curves in Figure 6.1:(a) The coefficient of relative risk aversion, θ, rises.(b) The curvature of (•), ν falls.(c) We
5.16. Redo the regression reported in equation (5.55):(a) Incorporating more recent data.(b) Incorporating more recent data, and using M1 rather than M2.(c) Including eight lags of the change in log
5.15. The derivation of the log-linearized equation of motion for capital. Consider the equation of motion for capital, Kt +1 = Kt + Kαt (AtLt)1−α − Ct −Gt − δKt.(a) (i) Show that ∂ ln
5.14. (a) If the A˜t’s are uniformly 0 and if ln Yt evolves according to (5.39), what path does ln Yt settle down to? (Hint: Note that we can rewrite [5.39] as ln Yt − (n + g)t = Q + α[ln Yt
5.13. Consider the model of Section 5.5. Suppose, however, that the instantaneous utility function, ut, is given by ut = ln ct + b(1 − t )1−γ/(1 − γ ), b > 0, γ > 0, rather than by (5.7)
5.12. Suppose technology follows some process other than (5.8)–(5.9). Do st = sˆand t = ˆ for all t continue to solve the model of Section 5.5? Why or why not?
5.11. Solving a real-business-cycle model by finding the social optimum.34 Consider the model of Section 5.5. Assume for simplicity that n = g = A = N =0. Let V(Kt,At), the value function, be the
5.9. A simplified real-business-cycle model with taste shocks. (This follows Blanchard and Fischer, 1989, p. 361.) Consider the setup in Problem 5.8. Assume, however, that the technological

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