5 Assume an economy that is perfectly competitive with a fix

(5) Assume an economy that is perfectly competitive with a fixed exogenous workforce, ???, of unknown size. The economy is characterized by the following production function: ?? = ??(??, ??) = ????? 1/2?? 1/2 , where labor is (L), capital is (K), output (Y) and total factor productivity is ???. What is the equilibrium rental price for capital (r) if Y=100, K=1, and the magnitudes of ??? and ???are unknown?

Solution

If Y=100 and K=1

100=A(1)^(1/2)L^(1/2)

10000/A^2=L

Now in competetive enviornment we have Price=MC

MPK=0.5A(L/K)^0.5 and Marginal Cost=r

MPK=r

0.5A(L)^0.5=r

r=0.5A(10000/A^2)^0.5

r=0.5A(100/A)=50