Assume a perfectly competitive market and consider the following short-run cost

Question

Assume a perfectly competitive market and consider the following short-run cost function: c(y) = y^3 - 8y^2 + 30y + 5 (a) Derive the marginal cost function MC(y). (b) What is the variable cost function c_v(y)? (c) Derive the average variable cost function AVC(y). (d) Derive the average variable cost function AC(y). (e) Derive the average fixed cost function AFC(y). (f) Determine all the values of output such that marginal cost equals average variable costs. Which of these values will have the highest profit? (g) Determine at what price a firm with these cost functions would supply y = 6? (h) Determine the profit this firm would make with that price and y = 6.

Explanation / Answer

Cost Function = y3 - 8y2 + 30y + 5

a) MC = Differentiation of Cost function with respect to y

MC = 3y2 - 16y+ 30

b) Total Cost is sum of fixed and variable cost. Fixed cost is independent of quantity of prodution or y while variable cost depends on y.

TC =  y3 - 8y2 + 30y + 5

TVC = y3 - 8y2 + 30y

TFC = 5

c) AC = C / y => y2- 8y+ 30 + 5/y

d) AVC = VC / y => y3 - 8y2 + 30y/y = y2 - 8+ 30

e) AFC = FC / y => 5/y

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