Question

Suppose that we 1000 competitive firms that face a linear demand curve p(Y ) = a - Y and have constant marginal costs, c, for each firm. The maximum output of these identical competitive firms is a/1000.

a) How much of the good will be produced by each firm?

b) How much will be produced in total?

c) What will the price of the good be?

d) What will economic profits of the industry be?

e) What economic profits will the individual firm receive?

f) What is the socially optimal quantity of the good to produce and consume?

g) By how much will actual industry production deviate from socially optimal production?

Answer 1

(a)

For each firm, MR = MC.

TR = p x Y = aY - Y2

MR = dTR/dY = a - 2Y

a - 2Y = c

Y = (a - c)/2 = 0.5 x (a - c)

(b)

Total output (Q) = 1000 x Y = 1000 x 0.5 x (a - c) = 500 x (a - c)

(c)

Price = a - Y = a - 0.5 x (a - c) = a - 0.5a + 0.5c = 0.5 x (a + c)

(d)

For the industry,

Total revenue (TR) = p x Q = 0.5 x (a + c) x 500 x (a - c) = 250 x (a2 - c2)

Total cost (TC) = 1000 x c

Industry profit = TR - TC = 250 x (a2 - c2) - 1000c

(e)

Firm profit = Industry profit / 1000 = [250 x (a2 - c2) - 1000c] / 1000 = 0.25 x (a2 - c2) - c

(f)

In social optimal, p = MC.

a - Y = c

Y = a - c

Q = 1000 x (a - c)

(g)

Actual production will be lower than socially optimal production.

Deviation (decrease) = 1000 x (a - c) - 500 x (a - c) = 500 x (a - c)