Question

Provide step by step solutions and explanations:

63a) Write out the Solow Equation for the growth rate of capital per worker as a function of y/k, s, n, and depreciation.

63b) Given the production function: Y = A*K ^.5 * L ^.5 . Derive the equation for y as a function of just A and k.

63c) Given the production function: Y = A*K ^.5 * L ^.5 . Assume Technology is 1. Savings rate = 5%, depreciation rate = 10% and population growth rate = 2%.

I: What is the steady state amount of capital per worker? II: What is the steady state output per worker ? III : If the depreciation rate is increased during a prolong war to 30% - what is the new steady state of capital ? IV: What is the steady state output per worker ? V: By what percent does output per worker fall.

Answer 1

63(a)

Major components to calculate growth rate of capital per worker are New Investments per worker and depreciation per worker

New Investments = sy Where s = saving rate and y = output per worker = f(k) and this new investment will increase capital stock per worker

(n + d)k = depreciation, where k = capital per worker and it will decrease capital stock.

Hence, growth rate of capital per worker is given by:

k = sy - (n + d)k

63(b)

Y = AK ^.5L ^.5

Dividing both sides by L we get

y = Y/L = Y = A*K ^.5 * L ^.5 /L = A(K/L)^0.5 = Ak^0.5

=> y = Ak^0.5

63(c)

(I) Given: Savings rate(s) = 5% = 0.05, depreciation rate(d) = 10% = 0.1 and population growth rate (n) = 2% = 0.02 and technology(A) = 1

Steady state occurs when: k = sy - (n + d)k

=> 0.05*k^0.5 - (0.1 + 0.02)k = 0

=> k = 0.17------------Steady state level

(II)Steady state level of output per worker = Ak^0.5 = k^0.5 = 0.42

(III) Now d = 0.3

Hence,

Steady state occurs when: k = sy - (n + d)k

=> 0.05*k^0.5 - (0.3 + 0.02)k = 0.024

=> k = 0.17------------Steady state level

(IV) New Steady state level of output per worker = Ak^0.5 = k^0.5 = 0.16

(V)

Change in output per worker = ((0.16 - 0.42)/0.42)*100 = (-)61.9

Hence, Output per worker will fall by 61.9%