SIMON FRASER UNIVERSITY. Economics 483 Advanced Topics in Macroeconomics Spring 2014 Assignment 3 with answers

Transcription

1 BURNABY SIMON FRASER UNIVERSITY BRITISH COLUMBIA Paul Klein Office: WMC 3635 Phone: (778) paul klein URL: Economics 483 Advanced Topics in Macroeconomics Spring 2014 Assignment 3 with answers 1. Exercise 8.4 in McCandless and Wallace. Answer: See Answers to McCandless and Wallace. 2. Exercise 8.5 in McCandless and Wallace. Answer: See Answers to McCandless and Wallace. 3. Consider an overlapping generations environment where each member of each generation has preferences represented by The population growth rate is n, i.e. u(c h t (t), c h t (t + 1)) = ln c h t (t) + ln c h t (t + 1). N(t + 1) = n N(t). Each young person is endowed with one unit of labour as young and with nothing as old. Production takes place in each period according to Y t = K θ t L 1 θ t where K t is the aggregate savings of generation t 1 and L t is the aggregate labour supply in period t. Depreciation is complete, so that feasible allocations have to satisfy C(t) + K(t + 1) Y (t). Please turn over! 1

2 (a) Find that constant value of capital per worker that maximizes consumption per worker. (This is called the golden rule capital stock. See Phelps (1961).) What is the interest rate at this level of capital per worker? Answer: The resource constraint (when thought of as an equation rather than an inequality) is C(t) + K(t + 1) = K θ t L 1 θ t where apparently L(t) = N(t). Define consumption per worker via and capital per worker via c(t) = C(t) N(t) k(t) = K(t) N(t). With these definitions, the resource constraint becomes N(t)c(t) + N(t + 1)k(t + 1) = N(t)k θ (t). Dividing by N(t) and recalling that N(t + 1) = n N(t), we get c(t) + nk(t + 1) = k θ (t). The question asks for a constant level of capital per worker. Imposing constancy and solving for consumption per worker c, we get c = k θ nk. Maximizing this with respect to k, we set the derivative equal to zero. implying k gr = θk θ 1 = n ( ) 1 θ 1 θ. n At this level of capital, the interest rate (marginal product of capital) is equal to n. Notice that MP K = θk θ 1 N 1 θ = θk θ 1. 2

3 (b) Show that any allocation with a constant capital stock per worker that exceeds the value in (a) is not Pareto optimal. Answer: Suppose we are in an allocation where k(t) > k gr for all t. For any t, we can make everyone better off by setting k(s) = k gr for all s = t + 1. Clearly this means that there is more available for consumption in period t. Let s give all of that to the old. Then they are strictly better off. Meanwhile, by definition, consumption per person is higher in every period s = t + 1, t + 2,... Remark: Consumption per worker and consumption per person differ by a multiplicative constant. Therefore if one goes up, so does the other. (c) Characterize the set of parameter values such that the stationary competitive equilibrium capital stock per worker exceeds the golden rule capital stock per worker. Answer: K(t + 1) = N(t) 1 + w(t) = 1 + (1 θ)kθ (t)n 1 θ (t). In a stationary equilibrium, capital per worker is constant. Apparently or N(t + 1)k(t + 1) = k(t + 1) = 1 + (1 θ) N(t)kθ (t) θ n kθ (t) so that the stationary competitive equilibrium capital stock per worker satisfies so that k ce = k ce = θ n (kce ) θ ( θ ) 1 1 θ. n Evidently this is greater than the golden rule level just in case (1 θ) 1 + > θ. There are various equivalent formulations of this condition. One is 1 + > θ 1 θ. 3

4 Another is > θ 1 2θ. So people need to be sufficiently patient and the capital share needs to be sufficiently small. Indeed as θ approaches 1/2 from below, the condition becomes impossible to satisfy and the stationary competitive equilibrium is guaranteed to be Pareto optimal. By the way, this last formulation suggest that if = 1 and θ = 2/3 then the condition is satisfied because the right hand side is negative. Well, not really. Notice that, in this case, 1 + = 1 2 < θ 1 θ = 2 so the condition is not satisfied. What went wrong in going from one formulation to the next? Hint: Evidently 2 > 1. Now multiply both sides by 1. Then what happens? As it happens, a better formulation is (1 2θ) > θ. Clearly this cannot be satisfied if θ > 1/2. 4. Consider an overlapping generations environment without storage or capital. Suppose everyone s preferences are represented by u h t = ln c h t (t) + ln c h t (t + 1) and endowments are given by ω h t = [3, 1]. The population grows at rate n, i.e. N(t + 1) = n N(t). Let there be money, and denote the money stock in period t by M(t). Suppose the money stock grows at rate µ, i.e. M(t + 1) = µ M(t). Freshly printed money is handed out lump sum to the old. 4

5 (a) Define a stationary monetary competitive equilibrium. In particular, decide exactly what it is that is constant over time. Answer: My suggestion (not necessarily the only plausible one) is that savings per person should be constant over time. (b) Find the stationary monetary competitive equilibrium for an arbitrary µ. Answer: By definition of a stationary monetary equilibrium, we have p m (t + 1)M(t + 1)/N(t + 1) = p m (t)m(t)/n(t). Using our assumptions about population and monetary growth, we have By a no-arbitrage argument, we have p m (t + 1) p m (t) = n µ. r(t) = pm (t + 1) p m (t) Thus equilibrium saving per (young) person is s = µ n. Thus the consumption profile must be = n µ. c t = [3 s, s] = [3/2 + µ/(2n), 5/2 µ/(2n)]. All this of course only makes sense if savings is not negative so that µ 3n. (c) We know from Assignment 1 that any allocation that satisfies u h t / c h t (t) u h t / c h t (t + 1) < n for all h and all t 0 is not Pareto optimal. What values for µ are consistent with avoiding this undesirable outcome? Answer: The left hand side, in a competitive equilibrium, is the interest rate, which we just concluded is n/µ. Thus we want to avoid n µ < n i.e. we hope that or 1 µ 1 µ 1. 5

6 Deadline: Monday, February 24. References Phelps, E. S. (1961). The golden rule of accumulation: A fable for growthmen. American Economic Review 51 (4),