Question

Consider a firm’s cost function c(y) = 4y^2 + 80, where the $80 is a quasi-fixed cost in the long run but a fixed cost in the short run. The marginal cost associated with this cost function is MC = 8y. Assume this firm operates in perfectly competitive markets.

(a) If the price of the firm’s output is $48, how many units will this firm choose to produce? What will be this firm’s profit?

(b) If the price of the firm’s output is $24, how many units will this firm choose to produce in the short run? How many units will this firm choose to produce in the long run? What are this firm’s short-run and long-run profits?

(c) Graph this firm’s long-run supply and short-run supply curves. How do these curves differ and why?

Answer 1

a)

A perfectly competitive increases its output as long as MC is lower tha or equal to Price to maximize profit.

So,

Set MC=Price .

8y=48

y=6

Optimal output is 6 units.

Total Revenue of firm=TR=P*y=48*6=$288

Total Cost of firm=TC=4y^2+80=4*6^2+80=$224

Profit=TR-TC=288-224=$64

b)

Set MC=Price

8y=24

y=3

Optimal output is 3 units.

Total Revenue of firm=TR=P*y=24*3=$72

Total Cost of firm=TC=4y^2+80=4*3^2+80=$116

Short run Profit =TR-TC=72-116=-$44

Given

TC=4y^2+80

ATC=TC/y=4y+(80/y)

Set ATC=MC to estimate the output where ATC is minimized..

4y+(80/y)=8y

4y=(80/y)

y^2=20

y*=4.47

ATC=4y+(80/y)=4*4.47+(80/4.47)=$35.78

In case of competitve market, long run price is equal to minimum ATC. So,

Price in long run=$36.78

Output of a firm in long run=4.47 units

Profit in long run=(P-ATC)*y=(35.78-35.78)*4.47=0

c)

In short run, a firm produces a positive quantity only if price>minimum AVC

In this case,

TVC=4y^2

AVC=TVC/y=4y

We observe that AVC is minimized at y=0

Minimum AVC=4y=4*0=0

In case of competitive firm, supply curve is given by

MC=P

8y=P

or

P=8y (function is deifined for P greater than or equal to zero)

or y=P/8 for

In long run, a firm produces a positive quantity only if price is higher than or equal to minimum ATC

We have calculated in earlier parts that minimum ATC is $35.78

So,

Long run supply curve for a firm is given by

or y=P/8 for

y=0 for P<35.78

We can conclude that

Given firm will produce for any positive value of price in short run

But it will produce a positive quantity in long run only if price is higher than or equal to $35.78

Long run and short run supply curve is same for a price higher than or equal to $35.78 i.e. short run supply curve and long run supply curve will overalap each other for a price of $35.78 or higher.