Question

1. Consider a monopolist with the cost function C(q) = 6q, facing the market demand function D(p) = 20 − 2p.

   1. (a) Find the monopoly quantity and price, the monopolist’s profit and the con- sumer surplus.

   2. (b) Now suppose that the government gives to the monopolist a subsidy of $2 per unit sold. Find the monopoly quantity and price, the monopolist’s profit, the consumer surplus, and the cost of the subsidy.

   3. (c) How does this subsidy affect total surplus (taking into account its cost)?

Answer 1

MC = dC/dq = 6

Q = D(p) = 20 - 2p, so

p = (20 - Q)/2 = 10 - 0.5Q

(a)

TR = p x Q = 10Q - 0.5Q²

MR = dTR/dQ = 10 - Q

Setting MR = MC,

10 - Q = 6

Q = 4

P = 10 - 0.5 x 4 = 10 - 2 = 8

TR = PQ = 8 x 4 = 32

TC = 6 x 4 = 24

Profit = TR - TC = 32 - 24 = 8

When Q = 0, p = 10

Consumer surplus (CS) = (1/2) x (10 - 8) x 4 = 2 x 2 = 4

(b)

The subsidy will decrease MC by 2 and new MC = 6 - 2 = 4

Setting MR = New MC,

10 - Q = 4

Q = 6

p = 10 - 0.5 x 6 = 10 - 3 = 7

TR = 7 x 6 = 42

TC = 6 x 6 = 36

Profit = 42 - 36 = 6

CS = (1/2) x (10 - 7) x 6 = 3 x 3 = 9

Cost of subsidy = Unit subsidy x After-subsidy quantity = 2 x 6 = 12

(c)

The subsidy decreases total surplus (TS = CS + Profit).

TS before subsidy = 4 + 8 = 12

TS after subsidy = CS + PS - Cost of subsidy = 6 + 9 - 12 = 3

Decrease in total surplus = 12 - 3 = 9