Answer to Question #284320 in Microeconomics for beki

Question

In the short-run production period, given the total cost of production as:    

TC = wL + rK, where K is fixed

Prove that w/AP and SMC=w/MP

Answer 1

The relation between average product and average cost:

Let y be the output produced by z1 units of input 1:

y = TP(z1).

Then we have

AVC(y) = VC(y)/y = w1z1/TP(z1),

or

AVC(y) = w1/[TP(z1)/z1].

But

TP(z1)/z1 = AP(z1),

so that

AVC(y) = w1/AP(z1).

The relation between marginal product and marginal cost:

Take some amount z1 of input 1 and increase it a bit. The resulting increase in cost is

c = w1z1.

The change in the output is

y = MP(z1)z1.

Now, short run marginal cost is the rate of change of variable cost: c/y. So substituting we have

SMC(y) = w1/MP(z1).