Question

4. Suppose that you estimate the following cost function for your company, which is a firm operating in a monopolistically competitive market:

TC = 180Q - 8Q² + 0.5Q³

You also estimate the following demand curve for the output you are producing.

P = 360 - 8Q

a. Derive the equations for average cost and marginal cost curves.

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?

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

Answer 1

a.

Average cost = Total cost / Q

= ( 180Q - 8Q² + 0.5Q³ ) /Q

Average cost = 180 - 8Q + 0.5Q²

Marginal cost = d(Total Cost) / dQ

= d( 180Q - 8Q² + 0.5Q³ ) / dQ

Marginal cost = 180 - 16Q + 1.5Q²

b.

Economies of scale exist till the point where the average cost is minimum

Average cost = 180 - 8Q + 0.5Q²

At minimum average cost quantity, d( average cost ) / dQ = 0

=> d( 180 - 8Q + 0.5Q² ) / dQ = 0

=> -8 + Q = 0

=> Q = 8

Therefore, the average cost is minimum till Q = 8

Therefore, the economies of scale will exist in the range 0 < Q <= 8 ( till Q = 8 )

The diseconomies of scale will exist in the range Q > 8 ( from Q = 8 )

c.

P = 360 - 8Q

Revenue = P x Q ( price x quantity )

= ( 360 - 8Q ) x Q

Revenue = 360Q - 8Q²

Marginal revenue = d( Revenue ) / dQ

MR = 360 - 16Q

Marginal cost = d(Total Cost) / dQ

= d( 180Q - 8Q² + 0.5Q³ ) / dQ

MC = 180 - 16Q + 1.5Q²

For profit maximization, the condition is MR = MC

=>  360 - 16Q = 180 - 16Q + 1.5Q²

=> 180 = 1.5Q²

=> Q = 10.95 ( Approx )

Therefore, we check at outputs 10 and 11

At Q = 10, P = 280

Revenue = 10*20 = 2800

TC = 1500 ( 180*10 - 8*(10)² + 0.5*(10)³ )

Profit = revenue - cost = 2800 - 1500 = 1300

At Q = 11, P = 272

Revenue = 11*272 = 2992

TC = 1677.5 ( 180*11 - 8*(11)² + 0.5*(11)³ )

Profit = 2992 - 1677.5 = 1314.5

As the profit is maximum at Q = 11, the producer would produce Q = 11 at a price P = 272

d.

At Q = 11, P = 272

Revenue = 11*272 = 2992

TC = 1677.5 ( 180*11 - 8*(11)² + 0.5*(11)³ )

Profit = 2992 - 1677.5 = 1314.5

The company is making a profit of 1314.5