Question

Consider the following production function, ? = ?1⁄4?1⁄4?1⁄4 , where y = output, K = the

amount of capital, L = the number of employment, m = quantity of variable materials hired. Let r

(unit price of capital) = $5, w (wage per employment) = $3, pm (unit price of variable material) =

$12; Suppose that the firm is minimizing its cost of production in the short run,

(a) Suppose in short run, the amount of capital is fixed at ?= 16. What is this firm’s short run

total cost function ?(?)?

[6 marks]

(b) Draw this firm’s short run supply curve.

[3 marks]

(c) Suppose the market price of the output is $120. What is this firm’s short run profit?

[3 marks]

(d) Show this firm’s producer’s surplus is the sum of firm’s profit and its fixed cost.

[3 marks]

Answer 1

a.) The attached image derives the cost function in terms of m.

The cost function is in terms of m. i.e.

C = 80+24m

Substituting value of m in equation of production we have,

m = y2 / 8

So cost function in terms of y ( output ) is -

C = 80+24 ( y2 / 8 )

or, C= 80 + 3 y2

b. The short run supply curve is given by the Marginal cost curve -

MC = 6y

See the image below -

c. Given Revenue = 120y

Therefore profit function is given by -

At maximum, y =20 units.

and maximum profit in short run = $1120

d. From the image attached below , we can see that producer surplus is the area of the triangle colored in orange.

Area of Triangle = 1/2 x base x height

or 1/2 x 20 x 120 = 1200

also Profit + fixed cost = 1120+80= $1200

Hence Producer surplus is = Profit + fixed cost.