Question

Suppose that you estimate the following cost function for your company, which is a monopolistically competitive firm:

TC = 80Q - 4Q² + 0.25Q³, and the following demand curve for your product: P = 80Q - 2Q

a. Compute and plot the average cost and marginal cost.

b. Over what range of output does economies of scale exist? Diseconomies of scale?

c. How many units of output will you produce and what price will you charge for each unit? (Hint: A monopolistically competitive firm produces where MR=MC. First find the level of output by setting MR=MC, then substitute the value of Q into the price equation above to find the value of P.)

d. Is the company making a profit or loss at the suggested output level? How much?

Answer 1

(a)

TC = 80Q - 4Q² + 0.25Q³

AC = TC / Q

=> AC = (80Q - 4Q² + 0.25Q³) / Q

=> AC = 80 - 4Q + 0.25Q²

and

MC = dTC / dQ

=> MC = 80 - 4(2) Q + 0.25(3) Q²

=> MC = 80 - 8Q + 0.75Q²

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(b) Economies of scale exist when Average cost (AC) decreases as quantity increases,

Diseconomies of scale exist when Average cost (AC) increases as quantity increases.

=> AC = 80 - 4Q + 0.25Q².

Differntiate AC with respect to Q.

dAC / dQ = -4 + 0.50Q

Set dAC / dQ < 0

=> -4 + 0.50Q < 0

=> 0.50Q < 4

=> Q < (4 / 0.5)

=> Q < 8

AC decreases as Quantity increase for the quantity less than 8

It means economies of scale will occurs between 0 units of quantity to 8 units of quantity.

i.e., Economies of scale occurs when 0 < Q<8

and

AC increases as Quantity increases for quantity more than 8 units.

It means diseconomies of scale will occurs for quantity level above 8 units.

i.e., Diseconomies of scale occurs when Q > 8.

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(c) P = 80 - 2Q

=> TR = PQ

=> TR = (80 -2Q)Q

=> TR = 80Q - 2Q2

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MR = dTR / dQ

=> MR = 80 - 4Q

and

MC = 80 - 8Q + 0.75Q² .

Monopolistically competitive firm produces at MR = MC

=> 80 - 4Q = 80 - 8Q + 0.75Q²

=> 0.75Q² -8Q + 80 = 80 - 4Q

=> 0.75Q² - 8Q + 80 - 80 + 4Q =0

=> 0.75Q² - 4Q = 0

=> 0.75Q² = 4Q

=> (Q² / Q )= (4 / 0.75)

=> Q = 5.33

Output produced by monopolistically competitive firm is 5.33 units.

and

P = 80 - 2Q

=> P = 80 - 2(5.33)

=> P = 80 - 10.66

=> P = 69.34

Price charge by monopolistically competitive firm is 69.34

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(d)

TR = PQ

=> TR = 69.34 * 5.33

=> TR = 369.58

and

TC = 80Q - 4Q² + 0.25Q³

=> TC = 80(5.33) - 4(5.33)² + 0.25(5.33)³

=> TC = 426.4 - 113.64 + 37.85

=> TC = 350.61

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Profit = TR - TC

=> Profit = 369.58 - 350.61

=> Profit = 18.97

Company is making profit of $18.97