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foundations macroeconomics

Questions and Answers of Foundations Macroeconomics

Consider deposit insurance in the Diamond Dybvig model of Section 10.6.(a) If fraction φ>θ of depositors withdraw in period 1, how large a tax must the government levy on each agent in period 1 to
Consider the Diamond Dybvig model described in Section 10.6, but suppose that ρR < 1.(a) In this case, what are ca∗1 and cb∗1 ? Is cb∗1 still larger than ca∗1 ?(b) Suppose the bank offers
Prices versus quantities in the DeLong Shleifer Summers Waldmann model.22 Consider modeling the noise traders in the model of equations(10.15) (10.23) of Section 10.4 in terms of shocks to the
This problem asks you to show that with some natural variants on the approach to modeling agency risk in Problem 10.7, consumption is not linear in the shocks, which renders the model intractable.(a)
Consider Problem 10.6.Suppose, however, that the demand of the period-0 noise traders is not fully persistent, so that noise traders’ demand in period 1 isρN0 + N1, ρ < 1. How, if at all, does
A simple model of agency risk. Consider the previous problem. For simplicity, assume A0 = 0. Now, however, there is a third type of agent: hedge-fund managers. They are born in period 0 and care only
Fundamental risk and noise-trader risk. Consider the following variant on the model of noise-trader risk in equations (10.15) (10.23). There are three periods, denoted 0, 1, and 2. There are two
(a) Show that in the model analyzed in equations (10.15) (10.23) of Section 10.4, the unconditional distributions of Ca 2t and Cn 2t are not normal.(b) Explain in a sentence or two why the analysis
A simpler approach to agency costs: limited pledgeability. (Lacker and Weinberg, 1989; Holmström and Tirole, 1998.) Consider the model of Section 10.2 with a different friction: there is no cost of
Consider the model of investment under asymmetric information in Section 10.2.Suppose that initially the entrepreneur is undertaking the project, and that(1 + r)(1 − W) is strictly less than RMAX.
Consider the model of Section 10.1.Suppose, however, that there are M households, and that household j ’s utility is Vj = U(C1) + βs jU(C2), where βs j > 0 for all j and s. That is, households
Consider the model of Section 10.1.Assume that utility is logarithmic, that β = 1, and that there are only two states, each of which occurs with probability one-half.In addition, assume there is
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 finances its
(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 π1
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 of the
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, suppose that
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 in
Suppose 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 capital
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 that I
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 equal to
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 behavior
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 negative.
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
Building intuition concerning the transversality condition. Consider an individual choosing the path of G to maximize ∞t=0 e−ρt− a 2 G(t)2dt, a > 0, ρ > 0.Here G(t) is the amount of
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
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 corporation
Consider a firm that produces output using a Cobb Douglas combination of capital and labor: Y = Kα L 1−α, 0
Problem 8.16 had you use a very primitive way of tackling the problem numerically. How might one do better? (Some candidates might involve interpolation or extrapolation, or not making the points you
Consider Problem 8.16.Change something about the model (the natural candidates are the utility function, the value of β, the value of r, and the distribution of Y) and find the new V(•) and C(•)
This problem asks you to use your analysis in Problem 8.16 to see how a onetime income shock affects the path of consumption starting from different situations. Specifically, under the same
Consider the following seemingly small variation on part (b) of Problem 8.16.Choose an N, and define e ≡ 200/N. Now, assume that Y can take on only the values 0,e, 2e, 3e, ... , 200, each with
Consider the dynamic programming problem that leads to Figure 8.4.This problem asks you to solve the problem numerically with one change: preferences are logarithmic, so that u(C) = ln C.
Time-inconsistent preferences. Consider an individual who lives for three periods. In period 1, his or her objective function is ln c 1 + δ ln c2 + δ ln c3, where 0
Precautionary saving with constant-absolute-risk-aversion utility. Consider an individual who lives for two periods and has constant-absoluterisk-aversion utility, U = − e−γC1 − e−γC2 , γ
Habit formation and serial correlation in consumption growth. Sup-pose that 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 >
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, however,
The equity premium and the concentration of aggregate shocks.(Mankiw, 1986.) Consider an economy with two possible states, each of which occurs with probability one-half. In the good state, each
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; thus Ct
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 part
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 Ct/(1 +
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 + C2/[1 + (1
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 imply
(This follows Hansen and Singleton, 1983.) Suppose instantaneous utility is of the constant-relative-risk-aversion form, u(Ct) = C1−θt /(1−θ), θ > 0. Assume that the real interest rate, r, is
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?
The time-averaging problem. (Working, 1960.) Actual data do not give consumption at a point in time, but average consumption over an extended period, such as a quarter. This problem asks you to
The average income of farmers is less than the average income of non-farmers, but fluctuates more from year to year. Given this, how does the permanent-income hypothesis predict that estimated
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
Consider the model of Section 7.8.Suppose, however, that monetary policy responds to current inflation and output: rt = φπ πt + φy yt + uMP t .(a) For the case of white-noise disturbances, find
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 set at
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
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 −
(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 firm’s price changes and
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 or
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 period
Repeat Problem 7.4 using lag operators.
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
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 for t
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 has a
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 set
Consider an economy consisting of some firms with flexible prices and some with rigid prices. Let p f denote the price set by a representative flexible-price firm and pr the price set by a
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−1. et is
Consider the problem facing an individual in the Lucas model when Pi /P is unknown. The individual chooses Li to maximize the expectation of Ui ; Ui continues to be given by equation (6.74).(a) Find
Thick-market effects and coordination failure. (This follows Diamond, 1982.)31 Consider an island consisting of N people and many palm trees. Each person is in one of two states, not carrying a
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 flexible;
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, ri :
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 with pi
Consider the model in equations (6.29) (6.32). Suppose, however, that the Et[yt+1]term in (6.31) is multiplied by a coefficient ω, 0
(a) Consider the model in equations (6.29) (6.32). Solve the model using the method of undetermined coefficients. That is, conjecture that the solution takes the form yt = AuIS t , and find the value
Consider the model in equations (6.29) (6.32). Suppose, however, there are shocks to the MP equation but not the IS equation. Thus rt = byt +u MP t , u MP t = ρMPuMP t−1+e MP t (where −1 < ρMP
The liquidity trap. Consider the following model. The dynamics of inflation are given by the continuous-time version of (6.23) (6.24): π(t) = λ[y(t) − y(t)],λ > 0. The IS curve takes the
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 traditional
Productivity growth, the Phillips curve, and the natural rate. (Braun, 1984;Ball and Moffitt, 2001.) Let gt be growth of output per worker in period t, πt inflation, and πW t wage inflation.
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 and
The multiplier-accelerator. (Samuelson, 1939.) Consider the following model of income determination. (1) Consumption depends on the previous period’s income:Ct = a + bYt−1. (2) The desired
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 often to
Describe how, if at all, each of the following developments affects the curves in Figure 6.1:(a) The coefficient of relative risk aversion, θ, rises.(b) The curvature of (•), χ, falls.(c) We
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 money
The derivation of the log-linearized equation of motion for capital. Consider the equation of motion for capital, Kt +1 = Kt +Kαt (At Lt)1−α −Ct − Gt − δKt.(a) (i) Show that ∂ ln Kt
(a) If the ~At’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 −1 −
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) (see Problem
Suppose technology follows some process other than (5.8) (5.9). Do st = sˆ andt = ˆ for all t continue to solve the model of Section 5.5? Why or why not?
Solving a real-business-cycle model by finding the social optimum.33 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 expected
The balanced growth path of the model of Section 5.3.Consider the model of Section 5.3 without any shocks. Let y∗, k∗, c∗, and G∗ denote the values of Y/(AL), K/(AL), C/(AL), and G/(AL) on
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 disturbances
A simplified real-business-cycle model with additive technology shocks.(This follows Blanchard and Fischer, 1989, pp. 329 331.) Consider an economy consisting of a constant population of infinitely
(a) Use an argument analogous to that used to derive equation (5.23) to show that household optimization requires b/(1 − t) = e−ρ Et [wt(1 + rt +1)b/wt +1(1 − t +1)].(b) Show that this
Suppose an individual lives for two periods and has utility ln C1 + ln C2.(a) Suppose the individual has labor income of Y1 in the first period of life and zero in the second period. Second-period
Consider the problem investigated in (5.16) (5.21).(a) Show that an increase in both w1 and w2 that leaves w1/w2 unchanged does not affect 1 or 2.(b) Now assume that the household has initial
Suppose the period-t utility function, ut, is ut = ln ct + b (1 − t )1−γ/(1 − γ ), b > 0,γ > 0, rather than (5.7).(a) Consider the one-period problem analogous to that investigated in
Let A0 denote the value of A in period 0, and let the behavior of ln A be given by equations (5.8) (5.9).(a) Express ln A1, ln A2, and ln A3 in terms of ln A0, εA1, εA2, εA3, A, and g.(b) In light
Redo the calculations reported in Table 5.3 for the following:(a) Employees’ compensation as a share of national income.(b) The labor force participation rate.(c) The federal government budget
Redo the calculations reported in Table 5.1, 5.2, or 5.3 for any country other than the United States.
Convergence regressions.(a) Convergence. Let yi denote log output per worker in country i . Suppose all countries have the same balanced-growth-path level of log income per worker, y∗. Suppose also
Briefly explain whether each of the following statements concerning a crosscountry regression of income per person on a measure of social infrastructure is true or false:(a) ‘‘If the regression
(A different form of measurement error.) Suppose the true relationship between social infrastructure (SI ) and log income per person (y) is yi = a + bSIi + ei . There are two components of social
Increasing returns in a model with human capital. (This follows Lucas, 1988.) Suppose that Y(t) =K(t)α[(1 − aH )H(t)]β , H (t) =BaH H(t), and K(t) = sY(t).Assume 0
Consider the following model with physical and human capital:Y(t) = [(1 − aK )K(t)]α[(1 − aH )H(t)]1−α, 0 0, φ > 0, γ + φ < 1, L(t) = nL(t), A(t) = gA(t), where aK and aH are the
(This follows Jones, 2002.) Consider the model of Section 4.1 with the assumption that G(E) = eφE . Suppose, however, that E, rather than being constant, is increasing steadily: E(t) = m, where m >
Consider the model in Problem 4.5.(a) What are the balanced-growth-path values of k and h in terms of sk, sh , and the other parameters of the model?(b) Suppose α = 1 3 and β = 1 2 . Consider two
(This follows Mankiw, D. Romer, and Weil, 1992.) Suppose output is given by Y(t) =K(t)α H(t)β [A(t)L(t)]1−α−β , α > 0, β > 0, α + β < 1. Here L is the number of workers and H is their

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