Suppose that we have (MPK), that each unit of capital is paid R, and that each unit of output is sold for price 𝑃. Show how to derive the equation for the real rental price of capital 𝑀𝑃𝐾 = 𝑅 /P

Answer 1

Assume you have a production function Q = f(L,K), where L = labor, K = capital.

Profit = total revenue - total costs

Total revenue = P * Q

Total Cost = R*K+ W*L where R is the rental rate of capital, W is the wage rate

Therefore profit = P*f(L,K) - (R*K * W*L)

To maximize profit with respect to K, take the partial derivative of profit with respect to K and set = 0. Note that the partial derivative of f(L,K) with respect to K = MPK.

0 = P*MPK - R

R = MPK*P

or

MPK= R/P

Answer 2

Technical Efficiency requires MRTS_KL= w/r. The Isoquant is tangent to the Isocost line.

MRTS_KL = MP(L)/MP(K) =w/r or w/MP(L)=r/MP(K) Marginal Cost = w/MP(L)=r/MP(K) you should be able to explain to me intuitively why this is the case.

In a competitive market a firm is a price taker and produces up to where MC=P so

MC=r/MP(K)=P

or r=P*MP(K)

you should be able to explain the intuition why hiring occurs up to the point where the value of the marginal product of the factor = the cost of using that factor

if r = P*MP(K) then dividing both sides by P yields r/p = MP(K).

If you need any help with this, please reach out. I love this stuff!

Lenny