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

Solution:

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

$AVC(y)=\frac{VC(y)}{y}=\frac{w_1z_1}{TP(z_1)}$ ,

or

$AVC(y)=\frac{w_1}{\frac{TP(z_1)}{z_1}}$ .

But

$\frac{TP(z_1)}{z_1}=AP(z_1)$ ,

so that

$AVC(y)=\frac{w_1}{AP(z_1)}$.

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: $\frac{c}{y}$. So substituting we have

$SMC(y)=\frac{w_1}{MP(z_1)}$ .