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foundations macroeconomics

Questions and Answers of Foundations Macroeconomics

10.11 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
10.10 Does worker influence on the wage and employment after shocks to labor demand are realized affect the cyclical characteristics of the labor market? (a) (This is based on McDonald and Solow,
10.9 Does worker influence on the wage after shocks to labor demand are real- ized affect the cyclical characteristics of the labor market? (a) (This follows McDonald and Solow, 1981.) Consider a
10.8 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
10.7 Unemployment insurance. (This follows Feldstein, 1976.) Consider a firm with revenues AF(L). A has two possible values, AB and AG (AB < Ac), each of which occurs half the time. Workers who are
10.6 Implicit contracts without variable hours. Suppose that each worker must either work a fixed number of hours or be unemployed. Let C denote the consumption of employed workers in state 1, and
10.5 The fair wage-effort hypothesis. (Akerlof and Yellen, 1990.) Suppose there is a large number of firms, N, each with profits given by F(eL) - wL, F' () > 0,F")
1.10.4 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, p.(b) An increase in the job breakup
10.2. Efficiency wages and bargaining. Summers (1988, p. 386) states "In an efficiency wage environment, firms that are forced to pay their workers premium wages suffer only second-order losses. In
10.1. Union wage premia and efficiency wages. (Summers, 1988.) Consider the efficiency-wage model analyzed in equations (10.12)-(10.19). Suppose, how- ever, that fraction f of workers belong to
9.17. (Cagan, 1956.) Suppose that instead of adjusting their real money holdings gradually toward the desired level, individuals adjust their expectation of in- flation gradually toward actual
9.16. Growth and seignorage, and an alternative explanation of the inflation- growth relationship. (Friedman, 1971.) Suppose that money demand is given by In(M/P) abi +In Y, and that Y is growing at
9.15. Uncertainty and policy. (Brainard, 1967.) Suppose output is given by y = x+(k+ Ex)zu, where z is some policy instrument controlled by the gov- ernment and k is the expected value of the
9.14. Money versus interest-rate targeting. (Poole, 1970.) Suppose the econ- omy is described by linear IS and LM curves that are subject to distur bances: y cai is, mp hy ki +EIM, where IS and L are
9.13. (a) In the model of reputation analyzed in Section 9.5, is social welfare higher when the policymaker turns out to be a Type 1, or when he or she turns out to be a Type 2? (b) In the model of
9.12. 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 + b), and
9.11. More on solving the dynamic-inconsistency problem through reputation. (This is based on Cukierman and Meltzer, 1986.) Consider a policymaker who is in office for two periods and whose objective
9.10. Consider the situation analyzed in Problem 9.8, but assume that there is only some finite number of periods rather than an infinite number. What is the unique equilibrium? (Hint: reason
9.9. Other equilibria in the Barro-Gordon model. Consider the situation described in Problem 9.8. Find the parameter values (if any) for which each of the follow- ing is an equilibrium: (a)
9.8. Solving the dynamic-inconsistency problem through punishment. (Barro and Gordon, 1983b.) Consider a policymaker whose objection function is By a12), where a > 0 and 0 < < 1. y, is determined by
9.7. (Fischer and Summers, 1989.) Suppose inflation is determined as in Section 9.4. Suppose the government is able to reduce the costs of inflation; that is, suppose it reduces the parameter a in
9.6. Regime changes and the term structure of interest rates. (See Blanchard, 1984; Mankiw and Miron, 1986; and Mankaw, Miron, and Weil, 1987.) Con- sider an economy where money is neutral.
9.5. Policy rules, rational expectations, and regime changes. (See Lucas, 1976, and Sargent, 1983.) Suppose that aggregate supply is given by the Lucas supply curve, y = y + b(-), b> 0, and suppose
9.3. Assume, as in Problem 9.2, that prices are completely unresponsive to unan- ticipated monetary shocks for one period and completely flexible thereafter. Assume also that y=c-ar and m-p= b+hy-ki
9.2. Consider a discrete-time model where prices are completely unresponsive to unanticipated monetary shocks for one period and completely flexible there- after. Suppose the IS and LM curves are y =
9.1. Consider a discrete-time version of the analysis of money growth, inflation, and real balances in Section 9.2. Suppose that money demand is given by m, - P = c-b(EPP), where m and p are the logs
8.12. The Modigliani-Miller theorem. (Modigliani and Miller, 1958.) Consider the analysis of the effects of uncertainty about discount factors in Section 8.6. Suppose, however, that the firm finances
8.11. (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
8.10. Consider the model of investment under uncertainty with a constant interest rate in Section 8.6. Suppose that, as in Problem 8.9, (K) abk and that C(I) al/2; in addition, suppose that what is
8.9. Suppose that (K) = a - bK and C(I) = al/2. (a) What is the = 0 locus? What is the long-run equilibrium value of K? (b) What is the slope of the saddle path? (Hint: use the approach in Section
8.8. Suppose that the costs of adjustment exhibit constant returns in and K. Specifically, suppose they are given by C(k/k)k, where C(0) = 0, C'(0) = 0, C" () > 0. In addition, suppose capital
= 8.7. A model of the housing market. (This follows 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
8.6. Consider the model of investment in Sections 82-85 Suppose it becomes known at some date that there will be a one time capital levy, specifically, capital holders will be taxed an amount equal
8.5. Consider the model of investment in Sections 82-85 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 behavior
8.4. 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 maximize
8.1. Consider a firm that produces output using a Cobb-Douglas combination of capital and labor: Y=KL,0 < a < 1. Suppose that the firm's price is fixed in the short run; thus it takes both the price
7.10. The equity premium and the concentration of aggregate shocks. (Mankiw, 1986b.) Consider an economy with two possible states, each of which occurs with probability. In the good state, each
7.9. 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;
7.8. Bubbles. Consider the setup of the previous problem without the assumption that lim EP/(1+r)*1-0. (a) Deterministic bubbles. Suppose that P, equals the expression derived in part (b) of Problem
7.7. Consider a stock that pays dividends of D, in period and whose price in period t is Pr. Assume that consumers are risk-neutral and have a discount rate of r; thus they maximize ElC/(1+r (a) Show
7.6. Consumption of durable goods. (Mankiw, 1982.) Suppose that, as in Section 7.2, the instantaneous utility function is quadratic and the interest rate and the discount rate are zero. Suppose,
7.5. Consider the two-period setup analyzed in Section 7.4. Suppose that the gov ernment initially raises revenue only by taxing interest income. Thus the indi- vidual's budget constraint is
7.4. A framework for investigating excess smoothness. Suppose that C, equals [r/(1+][+ ElYs/(1+r], and that A+1-(1+r)A, + Y - Cr). (a) Show that these assumptions imply that EICC, (and thus that con-
7.3. (This follows Hansen and Singleton, 1983.) Suppose instantaneous utility is of the constant-relative-risk-aversion form, u(C) C(1-8), 8 > 0. Assume that the real interest rate, r, is constant
6.16. (This follows Diamond, 1982.)41 Consider an island consisting of N people and many palm trees. Each person is in one of two states, not carrying a coconut and looking for palm trees (state P)
6.15. Multiple equilibria with menu costs. (This follows Ball and D. Romer, 1991.) Consider an economy consisting of many imperfectly competitive firms. The profits that a firm loses relative to what
6.14. Consider an economy consisting of many imperfectly competitive, price- setting firms. The profits of the representative firm, firm i, depend o aggregate output, y, and the firm's real price,
6.13. Consider an economy consisting of some firms with flexible prices and some with rigid prices. Let p denote the price set by a representative flexible-price firm and p' the price set by a
6.12. State-dependent pricing with both positive and negative inflation. (This fol- lows Caplin and Leahy, 1991.) Consider an economy like that of the Caplin- Spulber model. Suppose, however, that m
6.11. (This follows Ball, 1994a) Consider a continuous time version of the Taylor model, so that p(t) = (1/1)fox(tr)dr, where T is the interval between each individual's price changes and x(tr) is
6.10 Repeat Problem 6 9 using lag operators
6.9. Consider the Iaylor model with the money stock white noise rather than a random walk, that is, m = &t, where , is serially uncorrelated Solve the model using the method of undetermined
6.8. The instability of staggered price-setting. (See Fethke and Policano, 1986, Ball and Cecchetti, 1988, and Ball and D Romer, 1989) Suppose the economy is described as in Problem 6 7, and assume
6.7. The Fischer model with unbalanced price setting. Suppose the economy is as described by the model of Section 67, except that instead of half of the individuals setting their prices each period,
6.6. Synchronized price setting. Consider the Taylor model Suppose, however, that every other period all of the individuals set their prices for that period and the next That is, in period t prices
6.5. Indexation. (See Gray, 1976, 1978, and Fischer, 1977b. This problem follows Ball, 1988.) Suppose production at firm i is given by Y, SL, where S is a supply shock and 0 < a < 1. Thus in logs, y,
6.4. Suppose the economy is described by the model of Section 6.6. Assume, how- ever, that P is the price index described in part (c) of Problem 6.2 (with all the Z's equal to 1 for simplicity). In
6.3. Observational equivalence. (Sargent, 1976.) Suppose that the money supply is determined by m = c'z-1 +e, where c and z are vectors ande, is an i.i.d. disturbance uncorrelated with Z-1.e, is
6.2. (This follows Dixit and Stiglitz, 1977.) Suppose that the consumption index C in equation (6.2) is C = - Zdj](n-1), where C is the individ- ual's consumption of good j and Z; is the taste shock
6.1. Consider the problem facing an individual in the Lucas model when P/P is unknown. The individual chooses L, to maximize the expectation of U; U continues to be given by equation (6.3). (a) Find
5.16. Redo the regression reported in equation (5.40): (a) Incorporating more recent data. (b) Incorporating more recent data, and using M2 rather than M1.
5.15. Destabilizing price flexibility. (De Long and Summers, 1986b.) Consider the following closed-economy variant of the model in Problem 5.10: y = a(1-p), m-p-ki, poy. Assume a > 0, k > 0, 0 > 0,
5.14. Suppose that the production function is Y = AF(I) (where F'() > 0, F"() < 0, and A > 0), and that A falls. How does this negative technology shock affect the AS curve under each of the models
5.13. Consider the model of aggregate supply in Case 2 of Section 5.4. Suppose that aggregate demand at P equals YMAX, Show the resulting situation in the labor market.
5.12. The analysis of Case 1 in Section 5.4 assumes that employment is determined by labor demand. A more realistic assumption may be that employment at a given real wage equals the minimum of demand
5.11. Consider the model of aggregate demand in an open economy with imperfect capital mobility in Section 5.3, without the simplification assumed in equa- tion (5.22). In addition to our usual
5.10. The algebra of exchange-rate overshooting. Consider a simplified open- economy model: m-phy - ki, y = b( - p) - a(i - p), i = , p = 0y. The variables y, m, p, and are the logs of output, money,
5.9. Exchange-market intervention. Suppose that the central bank intervenes in the foreign exchange market by purchasing foreign currency for dollars, and that it sterilizes this intervention by
5.8. Describe how each of the following changes affect income, the exchange rate, and net exports at a given price level under: (1) a floating exchange rate and perfect capital mobility, (2) a fixed
5.7. (This follows Mankiw and Summers, 1986.) Suppose that the demand for real money balances depends on the interest rate, i, and on disposable income Y-T; in other words, suppose that the correct
5.6. The multiplier-accelerator. (Samuelson, 1939.) Consider the following model of income determination. (1) Consumption depends on the previous period's income: Ca+bY-1. (2) The desired capital
5.5. The Mundell effect. (Mundell, 1963.) In the IS-LM model, how does a fall in expected inflation,e, affect i, Y, and i - me?
5.4. The liquidity trap and the Pigou effect. Assume that the nominal interest rate is so low that the opportunity cost of holding money is negligible. Suppose that as a result people are indifferent
5.3. The government budget in the standard Keynesian model. (a) The balanced budget multiplier. (See Haavelmo, 1945.) Suppose that planned expenditure is given by (5.5), E = C(YT)+I(i - ) + G. (i)
5.2. The derivation of the LM curve assumes that M is exogenous. But suppose instead that the Federal Reserve has some target interest rate i and that it adjusts M to keep i always equal to i. (a)
5.1. Consider the IS-LM model presented in Section 5.2. In this model, what are di/dM and dy dM for a given value of P?
A Monte Carlo experiment, and the source of bias in OLS estimates of equation (4.56). Suppose output growth is described simply by Alny, = t, where the 's are independent, mean-zero disturbances.
The derivation of the log-linearized equation of motion for capital, (4.52). Consider the equation of motion for capital, K+1 K+K (AL)- - C- Gr-8Kt. (a) (i) Show that In K+1/ln K, (holding A, L, C,
(a) If the A,'s are uniformly zero and if In Y, evolves according to (4.39), what path does In Y, settle down to? (Hint: note that we can rewrite (4.39) as In Y, (n+g)tQ+a[ln Y-1 - (n + g)(t-1)] + (1
Consider the model of Section 4.5. Suppose, however, that the instantaneous utility function, ut, is given by u = Inc + b(1)/(1 - y), b > 0, y > 0, rather than by (4.7) (see Problem 4.4). (a) Find
Suppose the behavior of technology is described by some process other than (4.8)-(4.9). Do st = and l, = l for all t continue to solve the model of Section 4.5? Why or why not?
Solving a real-business-cycle model by finding the social optimum.45 Con- sider the model of Section 4.5. Assume for simplicity that n = g = A = N = 0. Let V(K+,A,), the value function, be the
The balanced growth path of the model of Section 4.3. Consider the model of Section 4.3 without any shocks. Let y*, k*, c*, and G* denote the values of Y/AL, K/AL, C/AL, and G/AL on the balanced
A simplified real-business-cycle model with taste shocks. (This follows Blan- chard and Fischer, 1989, p. 361.) Consider the setup in Problem 4.8. Assume, however, that the technological disturbances
A simplified real-business-cycle model with additive technology shocks. (This follows Blanchard and Fischer, 1989, pp. 329-331.) Consider an econ- omy consisting of a constant population of
(a) Use an argument analogous to that used to derive equation (4.23) to show that household optimization requires b/(1 - l)e E[W+(1+r+1)b/ [W++1(1+1)]]. (b) Show that this condition is implied by
Suppose an individual lives for two periods and has utility In C + In C. (a) Suppose the individual has labor income of Y in the first period of life and zero in the second period. Second-period
Consider the problem investigated in (4.16)-(4.21). (a) Show that an increase in both w and w that leaves w/w unchanged does not affect l or l2. (b) Now assume that the household has initial wealth
Let In An denote the value of A in period 0, and let the behavior of In A be given by equations (4.8)-(4.9). (a) Express In A, In A2, and In A3 in terms of In A0, EAL, EA2, A3, A, and g. (b) In light
Redo the calculations reported in Table 4.3 for the following:44 (a) Employees' compensation as a share of national income. (b) The labor force participation rate. (c) The federal government budget
3.19. The speed of convergence in the model of Part B of this chapter. (a) Use the production function, (3.48), and the equations of motion for k and h, (3.49) and (3.50), to find an expression for
3.18. Increasing returns in a model with human capital. (This follows Lucas, 1988.) Suppose that Y(t) = K(t) [(1 - a)H(t), H(t) = Bay H(t), and K (t) = SY(t), and assume a + > 1.36 (a) What is the
3.17. Constant returns to physical and human capital together. Suppose the pro- duction function is Y(t) = K(t)H(t) (0 < a < 1), and that K and H evolve according to K(t) = SKY(t), H(t) = SHY(t). (a)
3.16. Use the production function, (3.43), and equations (3.55) and (3.56) to derive the expressions in equations (3.61) and (3.62) for the marginal products of physical and human capital on the
3.15. Consider the following variant of our model with physical and human capital:where a and a are the fractions of the stocks of physical and human capital used in the education sector. This model
3.14. Suppose that, despite the political obstacles, the United States permanently reduces its budget deficit from 3 percent of GDP to zero. Suppose that the economy is described by the model of Part
3.13. Consider the model of Part B of the chapter. (a) What is consumption per unit of effective labor on the balanced growth path? (b) What values of s and SH maximize this value?
3.12. Consider an economy described by the model of Part B of this chapter that is on its balanced growth path. Suppose there is a permanent increase in the rate of population growth. How does this
3.11. Delays in the transmission of knowledge to poor countries. (a) Assume that the world consists of two regions, the North and the South. The North is described by Yy(t) = Ax (t)(1-a)Ly and Ay(t)

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