Question

mm LCIUTOT 9.U6 PM 3. The land of Grim can be described by the following production fune! tion Y = K L Moreover, there is no population growth nor is there technological progress. (a) Find the steady-state capital stock per worker, output per worker, and consumption per worker as a function of the saving rate and the depreciation rate. (b) Research shows that the depreciation rate in the Land of Grim is 10% per year. Make a table showing steady-state capital per worker, output per worker, and consumption per worker for sav. ing rates of 0%, 10%, 20%, 30% ... up to 100%. What saving rate maximizes output per worker? What saving rate maximizes consumption per worker? (c) Derive the formula for the marginal product of capital. Calcu- late the marginal product of capital for each value of the saving rate from part (b). What do these calculations tell you about the relationship between MP and steady-sate consumption per worker?

Answer 1

Answer Page Nol-1 Given data Y / L = (k/2)/2 Y = klz [y = output per wooker and K= capital per workeo] If s be savings rate and to d be depre- ciation rate, then in steady-state, 5/2=kly s/d = x (x2) s/d = k/ K = (5/) k'kll k'-1½ kh C = (1-8) xy = (1- s)x (s) d) = (5-5)/d

Page No - 2 Ky lo 1 3 21 0.1 0.1 1 0.1 0.2 4 0.1 0.3 9 0.1 0.4 16 0.1 0.5 0.1 0.6 36 oil 0.7 49 0.1 0.8 10.1 0.9 8 25 5 61 7 90.9 0.1 1 100 10 11E-15 output per worker in maximized when rate = 100% (= 1) saving cosumption per worker is maximized when savings rate = 50% (=0.5)

Page No - 3 e mpk = aylak = (1/2) x (2/4/2 = (4x)/(6) V2 MPK = (1/2) / (K/2) dsk k^(1/2) MPK 0. 0 0 0.1 0.1 1 10.50 0.1 0.2 4 2 0.25 0.1 0.3 9 3 0.17 0.1 0.4 16 4 0.13 0.1 0.5 25 5 0.10 0.1 0.6 36 10.1 0.7 49 7 0.07 0.1 0.8 64 8 . 0.06 10.1 0.9 81 9 0.06 0.1 1 100 100 .05 10.08 while Mpk is steadily decreasing, stead-state Consumption per worker first increases, reaches a maximum