Question

4) Suppose that medieval England was a single, large, price-taking firm that produced one type of output with a constant-returns-to-scale Cobb-Douglas production function, q = 10.5K0.5. Labor and capital have inelastic supplies (everyone works and all capital is used). The Black Death killed 60% workers. If p is normalized to 1, K = L = 100, calculate the change in factor prices (wage and cost of capital).

Answer 1

Let the wage rate be w and cost of capital be r.

In the initial case, L=K=100 and p=1

q=L^0.5K^0.5

Marginal Product of labor=MPL=dq/dL=0.5K^0.5L^-0.5

Marginal revenue of product of labor=MRPL=p*MPL=1*0.5K^0.5L^-0.5=0.5K^0.5L^-0.5

Set MRPL=w for profit maximization

0.5K^0.5L^-0.5=w -------(1)

Put L=K=100

0.5*100^0.5*100^-0.5=w

w=0.5

Marginal Product of capital=MPK=dq/dK=0.5K^-0.5L^0.5

Marginal revenue of product of capital=MRPK=p*MPK=1*0.5K^-0.5L^0.5=0.5K^-0.5L^0.5

Set MRPK=r for profit maximization

0.5K^-0.5L^0.5=r ---------(2)

Put L=K=100

0.5*100^-0.5*100^0.5=r

r=0.5

Now the Black Death kills 60% of labor

So, Revised L=100*(1-60%)=40

In revised state, equation 1 gives

0.5K^0.5L^-0.5=w

0.5*100^0.5*40^-0.5=w

w'=0.790569

In revised state, equation 2 gives

0.5K^-0.5L^0.5=r

0.5*100^-0.5*40^0.5=r

r'=0.316228

Change in wage rate=(0.790569-0.5)/0.5=58.11%

Change in cost of capital=(0.316228-0.5)/0.5=-36.75%