Question

2. Assume the production function is y = 505-30, and the price of ris w = 1. (a) Derive the firm's total cost curve C(y), average cost curve AC(y), and marginal cost curve MC(y). (b) Assume that p > min AC(y), derive the firm's supply curve y(w,p)?

Answer 1

Solution:

Production function: y = 5x^1/3 - 30 and price of input x is w = 1

a) Rewriting the above production function in modified form: x = ((y + 30)/5)³

Total cost = price of input unit*total units of input employed

Total cost = w*x

TC(y) = 1*((y + 30)/5)³ = (y + 30)³/125

Average cost = total cost/total number of units

AC(y) = ((y + 30)³/125)/y = (y + 30)³/125y

Marginal cost = $\partial TC(y)/\partial y$ = 3*(y + 30)^3-1*1/125

MC(y) = 3(y + 30)²/125

b) Since, price p is greater than the minimum average cost (p > min AC(y)), it means the firm's shut down point is not incorporated. Beyond this point where price goes above the minimum of average cost, the firm's supply curve is same as the marginal cost curve for short run.

So, p(y) = MC(y) = 3(y + 30)²/125

As a function of w, MC = w*3(y + 30)²/125 (carrying on form TC(w, y), which can be derived by not substituting value for w in part (1))

Then, p = 3w(y + 30)²/125

125p/3w = (y + 30)²

y*(w, p) = (125p/3w)^1/2 - 30