Question

3 Growth Model Suppose that output (Y) in an economy is given by the following aggregate production function: Y = K + NE where Kt is capital and Nt is the population. Furthermore, assume that capital depreciates at rate 8 and that savings is a constant proportion s of income. You may assume that 8 > S. 1. Suppose that the population remains constant. Solve for the steady-state level of capital per worker. 2. Now suppose that the population grows at rate n. Solve for the steady-state level of capital per worker. 3. Based on your answer to part 2) above, solve for the steady-state growth rates in terms of n) of the following: (a) capital per worker (b) output per worker (c) capital (d) output

Answer 1

up YE = kt & NE Deprecates at rate s saving a s et assume 80s Suppose population remain constant. so Yt = kt & Nt for steady state level of capital per florker change in 'per worker' term. t = kt & NE Tyta ki te At steady state it + NE OK = 0 sy = sk syt - Sk* Akt +NE = Sk*. O k* = SkrtNET

skat + SNE = ske [ 8-8] kk + ENE = 0 [s-8] ke s salt capital per ke s s No FEEGSCH iss worker S-8 Now suppose population grows at rate n. that means population is not constant so YE I kt tnt Cap per worker' term - YE - KE & NE Et E 나 나 Tlet NEP LE - Yt = kt + Pt at steady state - DK - Sflf) = [stn] kk sket + PE] = (sto] Lt =0 na pop growity a rate

Skatept = [sth] kA SPE = [sth] kant - She 8pt = [sto-8] ki Capital per worker k = 8 h [sth-s] sto-& sto-8 t 't stn - 8 L sto-sl y = I Stn Stn-B It I oweput per worker

(C) Capitag (ке) Where LL Labour ЕЕ ЕКЕ - Kt = ktolt capital - KE = A .LE бир 0- - Kt NE У – Г 5 S+) - барч9- at steady state,