Question

1. lounchPad LounchPad . Country Country A and country B both have the production function Y = F(K, L) = K1/312/3 Does this production function have constant returns to scale? Explain. b. What is the per-worker production function, y = f(k)? c. Assume that neither country experiences population growth or technological progress and that 20 percent of capital depreciates each year. Assume further that country A saves 10 percent of output each year and country B saves 30 percent of output each year. Using your answer from part (b) and the steady-state condition that investment equals depreciation, find the steady-state level of capital per work- er for each country. Then find the steady-state levels of income per worker and consumption per worker. d. Suppose that both countries start off with a capital stock per worker of 1. What are the levels of income per worker and consumption per worker? e. Remembering that the change in the capital stock is investment less depreciation, use a calculator (or, better yet, a computer spread- sheet) to show how the capital stock per Worker will evolve over time in both coun- tries. For each year, calculate income per Worker and consumption per worker. How many years will it be before the consumption in country B is higher than the consumption in country A? roman and Japanese post-

Answer 1

Solution: (① Given y = F(k, L) = k113 2213 e production function have constant returns to scale or not? mathematically a productim function has constant return to scale if ny = F (ak, 1L) for any positive number d. and labor If we multiply both the amount of capital output is multiplied by same amount is, then amount of byd. · F Cak, / 4) = (ak) 113 (8L)213 =>y. y = k 13, 2/3 displays constant Jefurn to scale. ② per worker, y = f(K): Production function using results on constant Return to seale Ċ CRs) set y = YIL and obtain = y = YL = k1l3 (2/3/L

= k1l3 (2/3-1 = Kil3,163 Defining k=k/L we can rewrite the above expression as = kila countries A and B are: © The facts about f = depreciation rate 20.20 SA = Savinge rate in country A = 0.10 SP = Saving & rate in country B = 0.30. y = kl3 is the per worker production fonction per worker, Ak, The growth rate of the capital stock is given by investment per worker sf (k) minus the per worker Sk. depereciation :: Ak = 56(k) – Sk.

In steady state, the capital stock does not grow, so cue an write this as sh(K) = $k. I · Steady State k = k* such that = 5(k*)'3 = 8** ::** - CS($1312 Country A: K*A = (SA (8) 3/2 = (0110/0.20) 3/2 =0:35 Country B : k* B = CSB (8) 4/2 = 60.3070.2013/2 =(1.84] output per worker is given by yt = (1/3 Steady state yH A = (0.35)!2 and yte = (1.84912 €122]

per workers is given by: . Steady state consumption c* = (1-5) y* Country A; CA = (1-0:1) *(0.71) =0.64] p = (1-0.3)*(1.224) Country B; = (0.86] d) Both countries Start off with capital stock per worker perworker and consumption of 1' then levels of income er worker ase: at time zero (t =o) both countries have =) suppose l units of Capital per worker: koa = koß = 1 Sa =0.1, 86=0.3, 8€0.2, ko = 1 y = k13, C=C1-Dy ا ا ي = 1. Income (Y) per worker of Country A = y = k13

. Consumption (c) per worker of country a c =(1-Saly = (1-0.1960 = 0.90] • Income (4) per worker of country B: y = k13 = 0,113 Consumption (c) per worker of Country B: c=(1-56)y = (1-0,3)(0) =10.70 o -