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foundations macroeconomics

Questions and Answers of Foundations Macroeconomics

3.10. (This follows Krugman, 1979; see also Grossman and Helpman, 1991b.) Suppose the world consists of two regions, the "North" and the "South." Output and capital accumulation in region (i = N,S)
3.9. (This follows Rebelo, 1991.) Assume that there are two factors of production, capital and land. Capital is used in both sectors, whereas land is used only in producing consumption goods.
3.8. Suppose that output at firm i is given by Y, KLIK L1, where K, and L, are the amounts of capital and labor used by the firm, K and I are the aggregate amounts of capital and labor, and a > 0,
3.7. Learning-by-doing. Suppose that output is given by equation (3.22), Y(r) = KitA()L)], that L is constant and equal to 1, that K(t) = SY(t), and that knowledge accumulation occurs as a side
3.6. Consider the model of Section 3.3 with 3 + 8 = 1 and n- 0. (a) Using (3.14) and (3.15), find the value that A/K must have for gk and 94 to be equal. (b) Using your result in part (a), find the
3.5. Consider the economy described in Section 3.3, and assume 3 + 0 < 1 and n > 0. Suppose the economy is initially on its balanced growth path, and that there is a permanent increase in s. (a) How,
3.4. Consider the economy analyzed in Section 3.3. Assume that + 0, and that the economy is on its balanced growth path. Describe how each of the following changes affects the 94 0 and 9x 0 lines
3.3. Lags in a model of growth with an explicit knowledge-production sector. Assume that final-goods production is given by Y(t) = A(1)(1 - a)L, where a is the fraction of the population engaged in
3.2. Consider two economies (indexed by i 1, 2) described by Y,(t) K, (t)" and K(t)=s, Y,(t), where > 1. Suppose that the two economies have the same initial value of K, but that s > s. Show that Y/Y
3.1. Consider the model of Section 3.2 with # < 1. (a) On the balanced growth path, AgA(t), where g is the balanced- growth-path value of ga. Use this fact and equation (3.7) to derive an expression
2.20. Explosive paths in the Samuelson overlapping-generations model (See Black, 1974; Brock, 1975; and Calvo, 1978a.) Consider the setup described in Problem 2.18. Assume that x is zero, and assume
2.19. The source of dynamic inefficiency. There are two ways in which the Dia- mond and Samuelson models differ from textbook models. First, markets are incomplete: because individuals cannot trade
2.18. Stationary monetary equilibria in the Samuelson overlapping-generations model. (Again this follows Samuelson, 1958.) Consider the setup described in Problem 2.17. Assume that x < 1+n. Suppose
2.17. The basic overlapping-generations model. (This follows Samuelson, 1958, and Allais, 1947.) Suppose, as in the Diamond model, that N, 2-period-lived individuals are born in period t and that N,
2.16. Social security in the Diamond model. Consider a Diamond economy where g is zero, production is Cobb-Douglas, and utility is logarithmic. (a) Pay-as-you-go social security. Suppose the
2.15. Depreciation in the Diamond model and microeconomic foundations for the Solow model. Suppose that in the Diamond model capital depreciates at rate , so that r = f'(k) - 8. (a) How, if at all,
2.14. A discrete-time version of the Solow model. Suppose Y = F(K, AL), with F() having constant returns to scale and the intensive form of the production function satisfying the Inada conditions.
2.13. Consider the Diamond model with logarithmic utility and Cobb-Douglas pro- duction. Describe how each of the following affects k+1 as a function of kr. (a) A rise in n. (b) A downward shift of
2.12. Precautionary saving, non-lump-sum taxation, and Ricardian equivalence. (This follows Leland, 1968, and Barsky, Mankiw, and Zeldes, 1986.) Consider an individual who lives for two periods. The
2.11. The analysis of government policies in the Ramsey-Cass-Koopmans model in the text assumes that government purchases do not affect utility from private consumption. The opposite extreme is that
2.10. Using the phase diagram to analyze the impact of unanticipated and antic- ipated temporary changes. Analyze the following two variations on Problem 2.9: (a) At time 0, the government announces
2.9. Using the phase diagram to analyze the impact of an anticipated change. Consider the policy described in Problem 2.8, but suppose that instead of announcing and implementing the tax at time 0,
2.8. Capital taxation in the Ramsey-Cass-Koopmans model. Consider a Ramsey- Cass-Koopmans economy that is on its balanced growth path. Suppose that at some time, which we will call time 0, the
2.7. Derive an expression analogous to (2.37) for the case of a positive depreciation rate.
2.6. Describe how each of the following affect the = 0 and k = 0 curves in Figure 2.5, and thus how they affect the balanced-growth-path values of c and k: (a) A rise in 0. (b) A downward shift of
2.5. The productivity slowdown and saving. Consider a Ramsey-Cass-Koopmans economy that is on its balanced growth path, and suppose there is a permanent fall in g. (a) How, if at all, does this
2.4. Consider a household with utility given by (2.1)-(2.2). Assume that the real interest rate is constant, and let W denote the household's initial wealth plus the present value of its lifetime
2.3. Assume that the instantaneous utility function u(C) in equation (2.1) is In C. Consider the problem of a household maximizing (2.1) subject to (2.5). Find an expression for C at each time as a
2.2. The elasticity of substitution with constant-relative-risk-aversion utility. Consider an individual who lives for two periods and whose utility is given by equation (2.46). Let P and P2 denote
2.1. Consider N firms each with the constant returns to scale production function Y = F(K, AL), or (using the intensive form) Y = ALf(k). Assume f'() > 0, f" () < 0. Assume that all firms can hire
13. (a) In the model of convergence and measurement error in equations (1.33)- (1.34), suppose the true value of b is -1. Does a regression of In(Y/N)1979 In(Y/N)1870 on a constant and In(Y/N)1870
1.12. Consider a Solow economy on its balanced growth path. Suppose the growth- accounting techniques described in Section 1.7 are applied to this economy. (a) What fraction of growth in output per
1.11. Embodied technological progress. (This follows Solow, 1960, and Sato, 1966.) One view of technological progress is that the productivity of capital goods built att depends on the state of
1.10. Natural resources in the Solow model. At least since Malthus, some have argued that the fact that some factors of production (notably land and natural resources) are available in finite supply
1.9. The Harrod-Domar model. (See Harrod, 1939, and Domar, 1946.) Suppose the production function is Leontief, Y(t) = min[ck K (t), CLet L(t)], where cx, CL, and g are all positive. As in the Solow
1.8. Suppose that, as in Problem 1.7, capital and labor are paid their marginal products. In addition, suppose that all capital income is saved and all labor income is consumed. Thus K = [F (K,
1.7. Factor payments in the Solow model. Assume that both labor and capital are paid their marginal products. Let w denote aF (K, AL)/L and r denote OF (K, AL)/OK. (a) Show that the marginal product
1.1.6. Suppose that, despite the political obstacles, the United States permanently reduces its budget deficit from 3% of GDP to zero. Suppose that initially s = 0.15 and that investment rises by the
1.5. Find the elasticity of output per unit of effective labor on the balanced growth path, y, with respect to the rate of population growth, n. If ak (k"). 9. 2%, and 6 = 3%, by about how much does
1.3. Consider the constant elasticity of substitution (CES) production function, Y = K-10+(AL-1/jet-1), where 0 < < and 1. (or is the elasticity of substitution between capital and effective labor.
1.2. Suppose that the production function is Cobb-Douglas. (a) Find expressions for k*, y*, and c* as functions of the parameters of the model, s, n, 8, 9, anda. (b) What is the golden-rule value of
1.1. Consider a Solow economy that is on its balanced growth path. Assume for simplicity that there is no technological progress. Now suppose that the rate of population growth falls. (a) What
Consider the model of crises in Section 13.9, and suppose T is distributed uniformly on some interval [μ − X,μ + X], where X > 0 and μ − X ≥ 0. Describe how, if at all, each of the following
Debt as a means of mitigating the common-pool problem. (Chari and Cole, 1993.) Consider the same setup as in Problem 13.14.Suppose, however, that there is an initial level of debt, D. The government
The common-pool problem in government spending. (Weingast, Shepsle, and Johnsen, 1981.) Suppose the economy consists of M > 1 congressional districts. The utility of the representative person living
Conditionality and reform. Consider the model in Section 13.6.Suppose an international agency offers to give the workers and capitalists each an amount F > 0 if they agree to reform. Use analysis
Crises and reform. Consider the model in Section 13.7.Suppose, however, that if there is no reform, workers and capitalists both receive payoffs of −C rather than 0, where C ≥ 0.(a) Find
Consider the Alesina Drazen model. Describe how, if at all, each of the following developments affects workers’ proposal and the probability of reform:(a) A fall in T.(b) A rise in B.(c) An equal
The Persson--Svensson model. (Persson and Svensson, 1989.) Suppose there are two periods. Government policy will be controlled by different policymakers in the two periods. The objective function of
Consider the Tabellini Alesina model in the case where α can only take on the values 0 and 1. Suppose, however, that there are 3 periods. The period-1 median voter sets policy in periods 1 and 2,
Consider the Tabellini Alesina model in the case where α can only take on the values 0 and 1. Suppose that the amount of debt to be issued, D, is determined before the preferences of the period-1
Consider the Tabellini Alesina model in the case where α can only take on the values 0 and 1. Suppose that there is some initial level of debt, D0. How, if at all, does D0 affect the deficit in
The Condorcet paradox. Suppose there are three voters, 1, 2, and 3, and three possible policies, A, B, and C. Voter 1’s preference ordering is A, B, C; voter 2’s is B, C, A; and voter 3’s is C,
If the tax rate follows a random walk (and if the variance of its innovations is bounded from below by a strictly positive number), then with probability 1 it will eventually exceed 100 percent or be
Consider the Barro tax-smoothing model. Suppose there are two possible values of G(t) GH and GL with GH > GL . Transitions between the two values follow Poisson processes (see Sections 7.4 and 11.2).
Consider the Barro tax-smoothing model. Suppose that output, Y, and the real interest rate, r, are constant, and that the level of government debt outstanding at time 0 is zero. Suppose that there
Precautionary saving, non-lump-sum taxation, and Ricardian equivalence. (Leland, 1968, and Barsky, Mankiw, and Zeldes, 1986.) Consider an individual who lives for two periods. The individual has no
The stability of fiscal policy. (Blinder and Solow, 1973.) By definition, the budget deficit equals the rate of change of the amount of debt outstanding:δ (t) ≡ D(t). Define d(t) to be the ratio
(Cagan, 1956.) Suppose that instead of adjusting their real money holdings gradually toward the desired level, individuals adjust their expectation of inflation gradually toward actual inflation.
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 growing at
Rational political business cycles. (Alesina and Sachs, 1988.) Suppose the relationship between output and inflation is given by yt = yn + b(πt − Et−1πt), where b > 0 and where Et−1 denotes
The political business cycle. (Nordhaus, 1975.) Suppose the relationship between unemployment and inflation is described by πt = πt−1 − α(ut − un ) +εS t , α > 0, where the εS t ’s are
Consider the situation analyzed in Problem 12.19, but assume that there is only some finite number of periods rather than an infinite number. What is the unique equilibrium?
Other equilibria in the Barro Gordon model. Consider the situation described in Problem 12.19.Find the parameter values (if any) for which each of the following is an equilibrium:(a) One-period
Solving the dynamic-inconsistency problem through punishment.(Barro and Gordon, 1983.) Consider a policymaker whose objective function is ∞t=0 βt(yt − aπt 2/2), where a > 0 and 0
In the model of delegation analyzed in Section 12.8, suppose that the policymaker’s preferences are believed to be described by (12.69), with a > a , whenπe is determined. Is social welfare
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 = yn + b(π −
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 (12.63) each period. There
(Fischer and Summers, 1989.) Suppose inflation is determined as in Section 12.8.Suppose the government is able to reduce the costs of inflation; that is, suppose it reduces the parameter a in
Consider the Krugman model of Section 12.7.Assume the economy is in a steady state starting in period 3 and that i1 = 0.(a) Suppose i2 = 0.(i) How, if at all, does an increase in M2, holding M1 and
Consider the Krugman model of Section 12.7.Assume that i1 = 0 and that the economy is in steady state starting in period 2. Suppose, however, that y1(the value of y in period 1) need not equal y∗
Consider the Krugman model of Section 12.7.Assume the economy is in a steady state of the type described in that section starting in period 2. Suppose, however, that prices are completely sticky in
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 > 0, and
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
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 − p
Consider the system given by (12.41).(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~y and π
Consider the model of Section 12.4.Suppose, however, the aggregate supply equation, (12.16), is πt = πt−1 + α(yt−1 − yn t−1) + επt , where επ is a white-noise shock that is independent
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 zero.
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 = yn+b(πt−πe t ), b > 0, and
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 it = r
Assume, as in Problem 12.2, 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 = b +
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 − ar and
Consider a discrete-time version of the analysis of money growth, inflation, and real balances in Section 12.1.Suppose that money demand is given by mt − pt =c − b(Et pt+1 − pt), where m and p
The efficiency of the decentralized equilibrium in a search economy.Consider the steady state of the model of Section 11.4.Let the discount rate, r, approach zero, and assume that the firms are owned
Consider the static search and matching model analyzed in equations (11.71)(11.75). Suppose, however, that the matching function, M(•), is not assumed to be Cobb Douglas or to have constant
Consider the model of Section 11.4.(a) Use equations (11.65) and (11.69), together with the fact that VV = 0 in equilibrium, to find an expression for E as a function of the wage and exogenous
Consider the model of Section 11.4.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 number of
Consider the steady state of the Diamond Mortensen Pissarides model of Section 11.4.(a) Suppose that φ = 0. What is the wage? What does the equilibrium condition (11.70) simplify to?(b) Suppose that
Describe how each of the following affects steady-state employment in the Diamond Mortensen Pissarides model of Section 11.4:(a) An increase in the job breakup rate, λ.(b) An increase in the
In the setup described in Problem 11.10, 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 ?
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
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 be in.
An insider-outsider model. Consider the following variant of the model in equations (11.39) (11.42). The firm’s profits are π = AF (L I +LO )−wI L I − wO LO , where LI and LO are the numbers
Implicit contracts under asymmetric information. (Azariadis and Stiglitz, 1983.) Consider the model of Section 11.3.Suppose, however, that only the firm observes A. In addition, suppose there are
Implicit contracts without variable hours. Suppose that each worker must either work a fixed number of hours or be unemployed. Let CE i denote the consumption of employed workers in state i and CU i
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
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
Describe how each of the following affects 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 rate,
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
Union wage premiums and efficiency wages. (Summers, 1988.) Consider the efficiency-wage model analyzed in equations (11.12) (11.17). Suppose, however, that fraction f of workers belong to unions that

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