Advanced Macroeconomics 1st Edition David Romer - Solutions

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Explain consumption, investment, inflation and monetary policy
10.17 The efficiency of the decentralized equilibrium in a search economy. Con- sider the model of Section 10.8. Let the interest rate, r, approach zero, and assume that the firms are owned by the households; thus welfare can be measured as the sum of utility and profits per unit time, which equals
10.16 Consider the model of Section 10.8. Suppose the economy is initially in equi- librium, and that A then falls permanently. Suppose, however, that entry and exit are ruled out; thus the total number of jobs, F + V, remains constant. How do unemployment and vacancies behave over time in response
10.15 Suppose we replace the assumption in equation (10.76) that the worker and the firm divide the surplus from their relationship equally with the assump- tion that fraction f of the surplus goes to the worker and fraction 1-f goes to the firm: (1-fVE - Vu) = f(V - Vy). (a) How does this change
10.14 Describe how each of the following affects equilibrium employment in the model of Section 10.8: (a) An increase in the job breakup rate,b. (b) An increase in the interest rate, r. (c) An increase in the effectiveness of matching, K.
10.13 In the setup described in Problem 10.12, suppose that w is distributed uni- formly on [ a,+ a], and that C < u (a) Find V in terms of p,a, and C. (b) How does an increase in a affect V? Explain intuitively.
10.12 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 cumulative distribution function. Each time the worker samples a job from this distribution, he or she
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 in. All workers who choose the secondary sector obtain a job. But there is a fixed number, N,, of
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, 1981.) Consider a union and a firm with the objective functions assumed in part (a) of Problem 10.9.
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 union with the ob- jective function [U(w)-K]L+U(wa)(N-L), U'() > 0, where N is the num- ber of union
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 are only two possible values of A, Ag and AG (AB < AG), each occurring with probability one-half. 1
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 employed when A = AG and un- employed when A = Ag receive an unemployment insurance benefit of B > 0
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 C," the consumption of un- employed workers The firm's profits in state are therefore A, F(L,)-[CEL, +
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 unemployed the longest are hired first. (a) Consider a steady state where there is no shirking.
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 rate,b. (c) A positive multiplicative shock to the production function (that is, sup- pose the production
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 almost any plausible bargaining framework, this makes it easier for workers to extract conces-
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 unions that are able to obtain a wage that exceeds the nonunion wage by proportion . Thus, w = M. (1+)wn.
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 inflation. Thus equations (9.39) and (9.40) are replaced by m(t) = Ce-b) and (t) = B[(t)(t)], 0 < S*.
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 rate gy. What rate of inflation leads to the highest path of seignorage?
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 multiplier for that instrument. Ek and u are independent, mean-zero disturbances that are unknown when 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 independent, mean-zero shocks with variances of and M, and wherea, h,and k are positive.
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 delegation analyzed in Section 9.5, suppose that the poli- cymaker's preferences are believed to be
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 that the social wel- fare function is yy - an/2, where y is a random variable with mean y 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 function iscan/2]. The policymaker is chosen randomly from a pool of possible policymakers with
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 backward from the last period.) 9.10. Consider the situation analyzed in Problem 9.8, but assume that there
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) One-period punishment. equals if -1 = 1 and equals b/a other- wise; = each period. = (b) Severe punishment.
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 the Lucas supply curve, (9.8), each period. Expected inflation is determined as follows. If has
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 equation (9.9). Is society made better or worse off by this change? Explain intuitively.
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. Specifically, assume that = Ame and that is constant at zero. Suppose that the money supply is given by
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 that monetary policy is deter- mined by m, m-1+a+ &, where is a white-noise disturbance. Assume that
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 hold each period. Suppose, however, that the money supply follows a random walk: m = m-1+ut, where
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 = car and m - p = b+ hy-ki, where y, m, and p are the logs of output, the money supply, and the price
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 of the money stock and the price level, and where we are implicitly assuming that output and the
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 its investment using a mix of equity and risk-free debt. Specifically, consider the financing of
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 off, in period 1 and 2 in period 2. is certain, but 2 is uncertain. The firm maximizes expected profits
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 uncertain is future values ofa. This problem asks you to show that it is an equilibrium for q(t) and
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 2.6.)
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 depreciates at rate &; thus k(t) = I(t)- Sk(t). Consider the representative firm's maximization problem.
= 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 increasing in PH, so that I = I(PH), with I() > 0, and that HI-SH Assume also that the rent is a decreasing
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 to fraction f of the value of their capital holdings after time 7 Assume the industry is initially in
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 over time In each case, assume that K and q are initially at their long run equilibrium values (a) A
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 to k(t) = f(k(t)) c(t)- (n+g)k(t). elc(t)-*/(1-9)] dt subject (a) What is the current value
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 of its product, P, and the quantity, Y, as given. Finally, input markets are competitive; thus the
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 individual's consumption is 1. In the bad state, fraction A of the population consumes 1-(6/A) and the
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; thus C Y, where Y, is the ex- ogenously determined output per person and C, is consumption per person.
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 plus (1 + ryb, b> 0. (i) Is consumers' first-order condition derived in part (a) of Problem 7.7
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 that equilibrium requires P, ED+1+P+1)/(1+r)] (assume that if the stock is sold, this happens after
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, however, that goods are durable; specifically. C = (1-8)C-1+ E, where E, is purchases in period 1 and 0
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 C+Cz/11+(1-7)r] +Y/1+(1-1)], where is the tax rate. The government's revenue is zero in period 1 and rr(Yi -
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- sumption follows a random walk) and that ElCl/(1+r) = A, - EY/(1+r). (b) Suppose that AY, AY-1+,
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 but not necessarily equal to the dis- count rate, p. (a) Find the Euler equation relating C, to
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) or carrying a coconut and looking for other people with coconuts (state C). If a person without a
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 it obtains with p = p* are K(p, p*),K>0. As usual, p* = p + by and y = m - p. Each firm faces a
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, r,:, (y,r), where 22 < (subscripts denote partial derivatives). Let r*(y) denote the profit-max mizing
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 representative rigid-price firm. Flexible-price firms set their prices after m is known; rigid-price firms
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 can either rise or fall, and that firms therefore follow a two-sided Ss policy: if p, p, (t) reaches
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 the price set by individuals = who set their prices at time 7. Assume that 1, so that p, (t) = m(t);
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 coefficients (Hint in the equation analogous to (6 63), is it still reasonable to impose A+ - 1?)
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 for simplicity that m is a random walk (so m, m + ur, where u is white noise and has a constant
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, fraction f set their prices in odd periods and fraction 1 f set their prices in even periods Thus
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 are set for t and t + 1, in t + 1, no prices are set, in t +2, prices are set for 1+2 and +3, and so
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, s+al,. Prices are flexible; thus (setting the constant term to zero for simplicity), p, w, + (1 -
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 addition, assume that money-market equilib- rium requires that total spending in the economy equal M.
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 unpredictable and unobservable. Thus the expected component of m, is c'Z-1, and the unexpected component
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 for good j. Suppose the individual has amount Y to spend on goods. Thus the budget constraint is o
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 the first-order condition for Li, and rearrange it to obtain an expres- sion for L; in terms of
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, and a
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 of aggregate supply in Section 5.47
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 and supply; this is known as the short-side rule. (a) Draw diagrams showing the situation in the
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 assumptions, assume NXP P 2 Exp*/P NX, 0, NXY 0, and Ey - NXy < 1. (a) Derive an expression for the slope
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, the price level, and the exchange rate, respectively; i is the nominal interest rate, and p is
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 selling bonds for dollars to keep the money stock unchanged. With this intervention, NX and CF must sum
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 exchange rate and perfect capital mobility, and (3) a floating exchange rate and imperfect capital
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 way to write the LM equation is M/P Li, Y T). (a) With this change to the IS-LM-AS model, can one
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 stock (or inventory stock) is proportional to the previous period's output: K* = CY-1. (3) Investment
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 concerning the division of their wealth between money and other assets, and that they are therefore
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) How do equal increases in G and T affect the position of the IS curve? Specifically, what is the
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) With this policy, what is the slope of the "LM curve" (that is, the set of com- binations of i and Y
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. Normalize the ini- tial value of Iny, In yo, to zero for simplicity. This problem asks you to con-
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, and G, fixed) is (1 + Ft+1)(K/K+1). (ii) Show that this implies that a In K+1/8 In K, evaluated at
(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 - a)A,, where Q = alns +(1a)A+In+N] - a(n+g).) (b) Defining Y, as the difference between In Y, and
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 the first-order condition analogous to equation (4.26) that relates current leisure and consumption
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 expected present value from the current period forward of lifetime utility of the representative
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 growth path; w* the value of w/A; * the value of L/N; and r* the value of r. (a) Use equations
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 (the e's) are absent, and that the instantaneous utility function is u(C)-C-0(C++). The v's are
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 infinitely-lived individuals. The representative individual maximizes the expected value of o u(C)/(1+p)',
(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 (4.23) and (4.26). (Note that [4.26] must hold in every period.) =0
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 consumption is thus (1+r)Y C); r, the rate of return, is potentially random. (i) Find the first-order
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 of amount Z > 0. (i) Does (4.23) continue to hold? Why or why not? (ii) Does the result in (a)
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 of the fact that the expectations of the 's are zero, what are the expectations of In A, In A2, and
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 deficit as a share of GDP. (d) The Standard and Poor 500 composite stock price index. (e) The
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 d[In y(t)]/dt. (b) Take a first-order Taylor approximation of the expression in part (a) in Ink and
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 growth rate of H? (b) Does the economy converge to a balanced growth path? If so, what are the growth

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