Question

Suppose that the production function Y = 0.5K0.5N0.5 1. Derive the steady state equations for output per worker, capital per worker and consump- tion per worker. 2. Assume that the depreciation rate is d=0.05. Compute steady state output per worker and consumption per worker for the following 11 values of savings rate: s = {0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1}. Plot the values of output per worker and consumption per worker on the same graph. Measure the saving rate on the horizontal axis and output per worker and consumption per worker on the vertical axis. If you want you can use Excel or any similar program. 3. Does the graph show that there is a value of s that maximizes output per worker? Does the graph show that there is a value of s that maximizes consumption per worker?

Answer 1

Production fonction est ? T. y = o.sKOS NOIS 0.5 K the pu-wookes Production function is - Now thes, ř - 0.5( Thus, y = y and ka k y = 0,5 vk Now we know, . Change in capital Stock Investment - Depreciation Here depreciation - Sk S=depreciation rate=0.05 2 and investment & flk) e - where s is the savings rate. Thus, Ak= sf () - Sk Now, at steady state ak o i - the flk) = 0.5VK Hence K* - OISTKA 0.05 => 0.05 Vt = 0.58 - 0.0025 k#= 0.2582 7 kit - 10082 scened with Calm Scanner

y = out but pu worker a sus and Steady state Consmption puworker CX = f(**) – Sk* -) luna 0.5 V kk - Ska ma oot | Now, we can derive Capital sou worka Marginal product of Impk y = disk? - 0.25 KTO'S 0.25 cmPk | Now, Steady state value of k that manimisis Consumption pu work will be at that level where mpk-8 =0 Now, we can compute out put puworker, Consmption pu worken in the graph following table . Scanned with CamScanner

Hue, y=0.5ut S=0.05, k²1 (* Ost Sk* mpk = 0.25 2 k. 8 k 1 y * Sk* e* MPK | МРk- a o To.o 10.0 10.0 10.0 0- 11.00. s o .os 0.45 0.25 0.2 10.075 0.2 4 1 0.2 0.8 Ot25 q 11.05 10.033 0. 39 T0:45 10.083 11 0:4 16: 0 2 10.8 11.2 10.625 10.575 0525 2.5 | 1:25 | 1:25 10:05 TO | 06360 | 31.84,2 10.04161-0-0084 10.7 49.0 3,5 2:45 1.05 0.0357-00143 10.8 64.0 4 3 . 2 018 10.031 -oola 10.9 18141514:05 10:45 10:027 -0.023 110001 5. 5"l! 00 0.025 0.025 Scanned_jacith CamScanner:

Now, we can graph the saving rate on the horizontal anis and output pu worku and hos con smption pu worker on the untical anis. Tere

CN 34t 33커 322 3) 4 29 4 2- 84 카카 | 261 25t 24 - 2.3 29 2t |19f 112 164 41 094 6 OS 0.5 배 10-21 1 + 어 여액S%여야 Swing rate Scanned with CamScanner

132 from the graph we can see that at the swings rate of 1 the out but pn worker is manimised a equal to s This is evident from the table also. I Also, this greeph shows that at swings rate of Lo.5, consumption pu workn es manimised equal to 1:25. At sarings rate o.S, MPK-s=0. which also in plin that consumption pu workes es manimised at s=0.5 . Scanned with CamScanner