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foundations macroeconomics

Questions and Answers of Foundations Macroeconomics

5.8. 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
5.7. (a) Use an argument analogous to that used to derive equation (5.23) to show that household optimization requires b/(1 − t) = e−ρEt [wt(1 + rt +1)b/wt +1(1 − t +1)].(b) Show that this
5.6. 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.
5.5. 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
5.4. 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
5.3. 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)
5.2. 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
5.1. Redo the calculations reported in Table 5.1, 5.2, or 5.3 for any country other than the United States.
4.12. 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
4.11. 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
4.10. (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
4.9. 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 < α < 1, 0 < β < 1,
4.8. Consider the following model with physical and human capital:Y(t) = [(1 − aK )K(t)]α[(1 − aH)H(t)]1−α, 0 < α < 1, 0 < aK < 1, 0 < aH < 1, K˙(t) = sY(t) − δK K(t), H˙(t) = B[aK
4.7. (This follows Jones, 2002a.) 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,
4.6. 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
4.5. (This follows Mankiw, D. Romer, and Weil, 1992.) Suppose output is given by Y(t) = K(t)αH(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
4.4. Suppose the production function is Y = Kα(eφE L)1−α, 0 < α < 1. E is the amount of education workers receive; the rest of the notation is standard.Assume that there is perfect capital
4.3. Suppose output in country i is given by Yi = Ai QieφEi Li . Here Ei is each worker’s years of education, Qi is the quality of education, and the rest of the notation is standard. Higher
4.2. Endogenizing the choice of E. (This follows Bils and Klenow, 2000.) Suppose that the wage of a worker with education E at time t is begteφE . Consider a worker born at time 0 who will be in
4.1. The golden-rule level of education. Consider the model of Section 4.1 with the assumption that G(E) takes the form G(E) = eφE .(a) Find an expression that characterizes the value of E that
3.14. 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 YN (t) = A N (t)(1−aL )L N and
3.13. (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
3.12. (This follows Rebelo, 1991.) Assume that there are two sectors, one producing consumption goods and one producing capital goods, and two factors of production: capital and land. Capital is used
3.11. Learning-by-doing with microeconomic foundations. Consider a variant of the model in equations (3.22)–(3.25). Suppose firm i’s output is Yi(t) =Ki(t)α[A(t)Li(t)]1−α, and that A(t) =
3.10. (a) Show that (3.48) follows from (3.47).(b) Derive (3.49).
3.9. Suppose that policymakers, realizing that monopoly power creates distortions, put controls on the prices that patent-holders in the Romer model can charge for the inputs embodying their ideas.
3.8. Consider the model of Section 3.5. Suppose, however, that households have constant-relative-risk-aversion utility with a coefficient of relative risk aversion of θ. Find the equilibrium level
3.7. Learning-by-doing. Suppose that output is given by equation (3.22), Y(t) =K(t)α[A(t)L(t)]1−α; that L is constant and equal to 1; that K˙(t) = sY(t); and that knowledge accumulation occurs
3.6. Consider the model of Section 3.3 with β + θ > 1 and n > 0.(a) Draw the phase diagram for this case.(b) Show that regardless of the economy’s initial conditions, eventually the growth rates
3.5. Consider the model of Section 3.3 with β + θ = 1 and n = 0.(a) Using (3.14) and (3.16), find the value that A/K must have for gK and gA to be equal.(b) Using your result in part (a), find the
3.4. Consider the economy described in Section 3.3, and assume β + θ < 1 and n > 0. Suppose the economy is initially on its balanced growth path, and that there is a permanent increase in s.(a)
3.3. Consider the economy analyzed in Section 3.3. Assume that θ + β < 1 and n > 0, and that the economy is on its balanced growth path. Describe how each of the following changes affects the g˙A
3.2. Consider two economies (indexed by i = 1,2) described by Yi(t) = Ki(t)θ and K˙i(t) = siYi(t), where θ > 1. Suppose that the two economies have the same initial value of K, but that s1 > s2.
3.1. Consider the model of Section 3.2 with θ < 1.(a) On the balanced growth path, A˙ = g∗A A(t), where g∗A is the balancedgrowth-path value of gA . Use this fact and equation (3.6) to derive
2.21. Explosive paths in the Samuelson overlapping-generations model. (Black, 1974; Brock, 1975; Calvo, 1978a.) Consider the setup described in Problem 2.19. Assume that x is zero, and assume that
2.20. The source of dynamic inefficiency. (Shell, 1971.) There are two ways in which the Diamond and Samuelson models differ from textbook models.First, markets are incomplete: because individuals
2.19. Stationary monetary equilibria in the Samuelson overlapping-generations model. (Again this follows Samuelson, 1958.) Consider the setup described in Problem 2.18. Assume that x < 1 + n. Suppose
2.18. The basic overlapping-generations model. (This follows Samuelson, 1958, and Allais, 1947.) Suppose, as in the Diamond model, that Lt two-period-lived individuals are born in period t and that
2.17. 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
2.16. Depreciation in the Diamond model and microeconomic foundations for the Solow model. Suppose that in the Diamond model capital depreciates at rate δ, so that rt = f(kt) − δ.(a) How, if at
2.15. A discrete-time version of the Solow model. Suppose Yt = F (Kt,AtLt), with F (•) having constant returns to scale and the intensive form of the production function satisfying the Inada
2.14. Consider the Diamond model with logarithmic utility and Cobb–Douglas production. Describe how each of the following affects kt +1 as a function of kt:(a) A rise in n.(b) A downward shift of
2.13. 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
2.12. Using the phase diagram to analyze the impact of unanticipated and anticipated temporary changes. Analyze the following two variations on Problem 2.11:(a) At time 0, the government announces
2.11. Using the phase diagram to analyze the impact of an anticipated change.Consider the policy described in Problem 2.10, but suppose that instead of announcing and implementing the tax at time 0,
2.10. 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,
2.9. A closed-form solution of the Ramsey model. (This follows Smith, 2006.)Consider the Ramsey model with Cobb–Douglas production, y(t) = k (t)α, and with the coefficient of relative risk
2.8. Derive an expression analogous to (2.39) for the case of a positive depreciation rate.
2.7. Describe how each of the following affects the c˙ = 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 θ.(b) A downward shift
2.6. 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
2.5. 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
2.4. Assume that the instantaneous utility function u(C ) in equation (2.1) is ln C. Consider the problem of a household maximizing (2.1) subject to (2.6).Find an expression for C at each time as a
2.3. (a) Suppose it is known in advance that at some time t0 the government will confiscate half of whatever wealth each household holds at that time. Does consumption change discontinuously at time
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 (2.43). Let P1 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
1.15. Derive equation (1.50). (Hint: Follow steps analogous to those in equations[1.47] and [1.48].)
1.14. (a) In the model of convergence and measurement error in equations (1.38)and (1.39), suppose the true value of b is −1. Does a regression of ln(Y/N )1979 − ln(Y/N )1870 on a constant and
1.13. Consider a Solow economy on its balanced growth path. Suppose the growthaccounting techniques described in Section 1.7 are applied to this economy.(a) What fraction of growth in output per
1.12. Embodied technological progress. (This follows Solow, 1960, and Sato, 1966.)One view of technological progress is that the productivity of capital goods built at t depends on the state of
1.11. Go through steps analogous to those in equations (1.28)–(1.31) to find how quickly y converges to y∗ in the vicinity of the balanced growth path. (Hint:Since y = f (k), we can write k =
1.10. Suppose that, as in Problem 1.9, 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
1.9. Factor payments in the Solow model. Assume that both labor and capital are paid their marginal products. Let w denote ∂F (K, AL)/∂L and r denote[∂F (K,AL)/∂K] − δ.(a) Show that the
1.8. Suppose that investment as a fraction of output in the United States rises permanently from 0.15 to 0.18. Assume that capital’s share is 1 3 .(a) By about how much does output eventually rise
1.7. 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 αK (k∗) = 1 3 , g = 2%, and δ = 3%, by about
1.6. 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 happens
1.5. 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, δ, g, and α.(b) What is the golden-rule
1.4. Consider an economy with technological progress but without population growth that is on its balanced growth path. Now suppose there is a one-time jump in the number of workers.(a) At the time
1.3. Describe how, if at all, each of the following developments affects the breakeven and actual investment lines in our basic diagram for the Solow model:(a) The rate of depreciation falls.(b) The
1.2. Suppose that the growth rate of some variable, X, is constant and equal to a > 0 from time 0 to time t1; drops to 0 at time t1; rises gradually from 0 to a from time t1 to time t2; and is
1.1. Basic properties of growth rates. Use the fact that the growth rate of a variable equals the time derivative of its log to show:(a) The growth rate of the product of two variables equals the sum
2. Using quarterly data since 1948, graph the Federal deficit as a percentage of GDP. Draw lines on the figure corresponding to business cycle peaks and troughs. What is the cyclical behavior of the
1. Using quarterly data since 1959, graph Federal government expenditures and receipts as a percentage of GDP. Separately, graph state and local government expenditures and receipts as a percentage
4. A constitutional amendment has been proposed that would force Congress to balance the budget each year (that is, outlays must equal revenues in each year). Discuss some advantages and
3.a. Use the fact that the nominal deficit equals the nominal primary deficit plus nominal interest payments on government debt to rewrite Equation (15.4) showing the change in the debt–GDP ratio
2. Both transfer programs and taxes affect incentives. Consider a program designed to help the poor that promises each aid recipient a minimum income of $10,000. That is, if the recipient earns less
1. Why is some state and local spending paid for by grants in aid from the Federal government instead of entirely through taxes levied by states and localities on residents? What are the advantages
9. Consider an economy in which the money supply consists of both currency and deposits. The growth rate of the monetary base, the growth rate of the money supply, inflation, and expected inflation
8. Real money demand in an all-currency economy with fixed real output and real interest rate, and constant inflation rate and money growth rate is L = 0.2Y - 500i, where Y is real income and i is
7. In this problem you are asked to analyze the question: By issuing new bonds and using the proceeds to pay the interest on its old bonds, can government avoid ever repaying its debts?a. Suppose
6. Find the largest nominal deficit that the government can run without raising the debt–GDP ratio, under each of the following sets of assumptions:a. Nominal GDP growth is 10% and outstanding
5. Suppose that all workers value their leisure at 90 goods per day. The production function relating output per day, Y, to the number of people working per day, N, is Y = 250N - 0.5N2 .
4. A constitutional amendment has been proposed that would force Congress to balance the budget each year (that is, outlays must equal revenues in each year). Discuss some advantages and
3.a. Use the fact that the nominal deficit equals the nominal primary deficit plus nominal interest payments on government debt to rewrite Equation (15.4) showing the change in the debt–GDP ratio
2. Both transfer programs and taxes affect incentives. Consider a program designed to help the poor that promises each aid recipient a minimum income of $10,000. That is, if the recipient earns less
1. Why is some state and local spending paid for by grants in aid from the Federal government instead of entirely through taxes levied by states and localities on residents? What are the advantages
9. Consider an economy in which the money supply consists of both currency and deposits. The growth rate of the monetary base, the growth rate of the money supply, inflation, and expected inflation
8. Real money demand in an all-currency economy with fixed real output and real interest rate, and constant inflation rate and money growth rate is L = 0.2Y - 500i, where Y is real income and i is
7. In this problem you are asked to analyze the question: By issuing new bonds and using the proceeds to pay the interest on its old bonds, can government avoid ever repaying its debts?a. Suppose
6. Find the largest nominal deficit that the government can run without raising the debt–GDP ratio, under each of the following sets of assumptions:a. Nominal GDP growth is 10% and outstanding
5. Suppose that all workers value their leisure at 90 goods per day. The production function relating output per day, Y, to the number of people working per day, N, is Y = 250N - 0.5N2 .
4. Suppose that the income tax law exempts income of less than $8000 from the tax, taxes income between $8000 and $20,000 at a 25% rate, and taxes income greater than $20,000 at a 30% rate.a. Find
3. Because of automatic stabilizers, various components of the government’s budget depend on the level of output, Y. The following are the main components of that budget: Tax revenues 1000 + 0.1Y
2. Congress votes a special one-time $1 billion transfer to bail out the buggy whip industry. Tax collections don’t change, and no change is planned for at least several years. By how much will
1. The following budget data are for a country having both a central government and provincial governments: Central purchases of goods and services 200 Provincial purchases of goods and services 150
8. In what ways is the government debt a potential burden on future generations? What is the relationship between Ricardian equivalence and the idea that government debt is a burden?
6. Give a numerical example that shows the difference between the average tax rate and the marginal tax rate on a person’s income. For a constant before-tax real wage, which type of tax rate most
4. What are the three main ways that fiscal policy affects the macroeconomy? Explain briefly how each channel of policy works.
3. How is government debt related to the government deficit? What factors contribute to a large change in the debt–GDP ratio?
2. Explain the difference between the overall government budget deficit, the current deficit, and the primary current deficit. Why are three deficit concepts needed?
1. What are the major components of government outlays? What are the major sources of government revenues? How does the composition of the Federal government’s outlays and revenues differ from

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