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foundations macroeconomics

Advanced Macroeconomics 5th Edition David Romer - Solutions

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 only two possible values of A, AB and AG (AB < AG ), each occurring with probability 1 2 .We can think
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 consumption of unemployed workers. The firm’s profits in state i are therefore Ai F (Li ) −
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 workers the firm hires, w is the wage it pays, and e is workers’ effort. Effort is given by e =
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 unemployed the longest are hired first.(a) Consider a steady state where there is no shirking. Derive an
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, b.(c) A positive multiplicative shock to the production function (that is, suppose the production
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 only second-order losses. In almost any plausible bargaining framework, this makes it easier for
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 are able to obtain a wage that exceeds the nonunion wage by proportion μ. Thus, wu = (1 + μ)wn ,
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 be able to increase the total consumption of the nonwithdrawers in the two periods to cb∗2 ?
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 the contract described in the text: anyone who deposits one unit in period 0 can withdraw ca∗1 in
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 quantity they demand of the risky asset rather than to their expectations of the price of 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 the model in Problem 10.7.Suppose, however, that the representative hedge-fund manager,
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 this affect your answer in part (b)(iii) of the problem for how the noise traders affect the price in
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 about consumption in period 2. Like the sophisticated investors, they have utility U(C) =
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 assets. The first is a safe asset in perfectly elastic supply. Its rate of return is normalized to zero:
(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 in the text, which uses the properties of lognormal distributions, is nonetheless correct.
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 verifying output, but the entrepreneur can hide fraction 1 − f of the project’s output from the
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. Describe how each of the following affects D:(a) A small increase in W .(b) A small increase in
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 may have heterogeneous preferences about consumption in different states.(a) What are the equilibrium
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 only one investment project. It pays RG in state G and RB in state B, with RG > RB > 0. We will refer
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 investment using a mix of equity and risk-free debt. Specifically, consider the financing of the
(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 in period 1 and π2 in period 2. π1 is certain, but π2 is uncertain. The firm maximizes expected
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 change, and on their behavior over time.In each case, assume q and K are initially at Point E + in
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 what is uncertain is future values of a.This problem asks you to show that it is an equilibrium for
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 Section 2.6.)
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 depreciates at rate δ; thus κ(t) = I (t) − δκ(t). Consider the representative firm’s maximization
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 = I(pH ), with I(•) > 0, and that H = I − δH. Assume also that the rent is a decreasing
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 fraction f of the value of their capital holdings at some time in the future, time T. Assume the
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 over time. In each case, assume that K and q are initially at their long-run equilibrium values.(a) A
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. Set up the problem of choosing the path of LA(t) to maximize the lifetime utility of the
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 ∞t=0 e−βt[c(t)1−θ/(1 − θ)]dt subject to k(t) = f (k(t)) − c(t) − (n + g)k(t).(a) What
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 garbage the individual creates at time t; for simplicity, we allow for the possibility that G can be
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 after-tax real interest rate relevant to home ownership is r − τ i , where r is the pretax real
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 that buys a new capital good at time t can deduct fraction D(s) of the purchase price from its
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 equally spaced.)
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(•) functions. Discuss how the change in assumptions changes the results, and explain the intuition.
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 assumptions about the household’s preferences and the distribution of Y as in Problem 8.16, plot, as a
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 probability 1/(N +1). Likewise, assume that C can only take on the values 0,e, 2e, 3e, ... , and find
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. Specifically, it asks you to approximate the value function by value-function iteration, along the lines of
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 > 0. The interest rate is zero and the individual has no initial wealth, so the individual’s
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 > 0, θ > 0, where Zit is the ‘‘reference” level of consumption. Assume the interest rate is
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, that goods are durable; specifically, Ct = (1 − δ)Ct−1 + Xt, where Xt is purchases in period t
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 individual’s consumption is 1. In the bad state, fraction λ of the population consumes 1 − (φ/λ)
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 = Yt, where Yt is the exogenously determined output per person and Ct is consumption per person.
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 (b) of Problem 8.8 plus (1 + r)t b , b > 0.(i) Is consumers’ first-order condition 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 + r)t ].(a) Show that equilibrium requires Pt = Et[(Dt +1 + Pt +1)/(1 + r)] (assume that if the stock
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 − τ )r] ≤ Y1 + Y2/[1 + (1 − τ )r], whereτ is the tax rate. The government’s revenue is 0
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 that Et [Ct +1] = Ct (and thus that consumption follows a random walk) and that ∞s=0 Et [Ct
(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 constant but not necessarily equal to the discount rate, ρ.(a) Find the Euler equation relating Ct
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 examine the effects of this fact.Suppose that consumption follows a random walk: Ct = Ct−1 + et, where
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 consumption functions for farmers and nonfarmers differ?
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 individual’s income is Y0 + gt for 0 ≤ t < R, and 0 for R ≤ t ≤ T. The retirement age, R,
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 expressions analogous to (7.92)(7.94). What are the effects of an unfavorable inflation shock in this
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 time t is still being followed at time t + i is e−αi. The other assumptions of the model are the
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 partial. That is, if a firm does not have an opportunity to review its price in period t, its price in t
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 − pt.(a) Express pt+1 in terms of its lagged values and mt.(b) Consider an anticipated, permanent, one-time
(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 x(t−τ ) is the price set by firms that set their prices at time t−τ .Assume that φ = 1, so that
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 fall, and that firms therefore follow a simple two-sided Ss policy: if pi − p∗t reaches either S 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 1, there is a one-time, permanent increase in m to m1.Let us conjecture that the behavior of the
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 coefficients. (Hint: In the equation analogous to (7.33), is it still reasonable to impose λ + ν = 1?)
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 and t + 1; in t + 1, no prices are set; in t + 2, prices are set for t + 2 and t + 3; and so on. As
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 constant variance). Assume the profits a firm loses over two periods relative to always having pt =
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 their prices in odd periods and fraction 1− f set their prices in even periods. Thus the price level
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 representative rigid-price firm. Flexible-price firms set their prices after m is known; rigid-price firms set
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 unpredictable and unobservable. Thus the expected component of mt is c zt−1, and the unexpected
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 the first-order condition for Yi , and rearrange it to obtain an expression for Yi in terms of E [Pi
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 coconut and looking for palm trees(state P ) or carrying a coconut and looking for other people with
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; thus (setting the constant term to 0 for simplicity), pi = wi + (1 − α)i − s. Aggregating the
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 : πi = π(y,ri ), where π22 < 0 (subscripts denote partial derivatives). Let r∗(y) denote the
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 = p∗ are K(pi − p∗)2, K > 0. As usual, p∗ = p + φy and y = m − p. Each firm faces a
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 that A must take for the equations of the model to hold. (Hint: The fact that yt = AuIS t for all t
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 < 1 and eMP is white noise), and yt = Et yt+1 − (rt/θ). Find the expression analogous to (6.37).
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 traditional form, y(t) = −[i(t) − π(t)]/θ, θ > 0.The central bank sets the interest rate according
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 form, Yt = −rt/θ. The second is the money-market equilibrium condition, which we can write as m
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. Suppose that initially g is constant and equal to g L and that unemployment is at the level that causes
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 supply; this is known as the short-side rule. Draw diagrams showing the situation in the labor market
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 capital stock (or inventory stock) is proportional to the previous period’s output: K∗t = cYt−1. (3)
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 convert bonds, which pay a constant interest rate of i , into money, which pays no interest. If the
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 modify the utility function, (6.2), to be t βt[U(Ct) + B(Mt/Pt) −V(Lt)], B > 0, and B falls.
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 rather than four.
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 +1/∂ ln Kt (holding At, Lt, Ct, and Gt fixed) equals(1 + rt +1)(Kt/Kt +1).(ii) Show that this implies
(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 − (n + g )(t − 1)] + (1 − α) ~At, where Q ≡ α ln ˆs +(1 − α)(A + ln ˆ + N ) − α (n + g
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 5.4).(a) Find the first-order condition analogous to equation (5.26) that relates current leisure
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 present value from the current period forward of lifetime utility of the representative individual as
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 the balanced growth path; w∗ the value of w/A; ∗ the value of L/N; and r∗ the value of r.(a)
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 (the e’s) are absent and that the instantaneous utility function is u(Ct) = Ct − θ(Ct + νt)2.
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 lived individuals. The representative individual maximizes the expected value of ∞t=0 u(Ct)/(1 +
(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 condition is implied by (5.23) and (5.26). (Note that [5.26] must hold in every period.)
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 consumption is thus (1 + r)(Y1 − C1); r, the rate of return, is potentially random.(i) Find the
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 wealth of amount Z > 0.(i) Does (5.23) continue to hold? Why or why not?(ii) Does the result in (a )
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 (5.12)(5.15). How, if at all, does labor supply depend on the wage?(b) Consider the two-period problem
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 of the fact that the expectations of the εA’s are zero, what are the expectations of ln A1, ln
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 deficit as a share of GDP.(d) The Standard and Poor’s 500 composite stock price index.(e) The
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 that yi evolves according to dyi (t)/dt = −λ[yi (t) − y∗].(i) What is yi (t) as a function

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