Consider a monopolist in a market with linear inverse demand p(q) = 4 − q/2. The monopolist's cost function is c(q) = 2q.

(1) Write down the monopolist's profit function. Compute the profit-maximizing quantity and the corresponding price.

(2) Assume that a 2% tax is levied on the monopolist's profits. Does this have any effect on its choices of output level and output price?

(3) Consider now a quantity tax of $1 per output unit sold. Compute the optimal output level and the corresponding output price. How does this tax affect the monopolist's choices of output and price, and its profits? 

(Hint: Note that a quantity tax of of $1 per output unit sold is equivalent to raising the marginal cost by $1. Why?)

(4) We say that the monopolist passes on the tax to the consumer if it raises the price by more than the tax ($1 here). Is this the case with the quantity tax in (2)? 

 

1) Profit = 2Q -Q²/2

P = $3, Q = 2

2)

This tax has no impact on P and Q.

Only profits are reduced.

3)

P = $3.50

Q = 1

Profit = $0.50

4)

Only 50% of the tax is passed on to the consumer.

Step-by-step explanation

1)

Profit = Revenue -Cost = QP -Cost = Q(4-Q/2) -2Q = 2Q-Q²/2

Profit is maximum when dProfit/dQ = 0.

dProfit/dQ = 2-2Q/2 = 0 or

Q = 2.

P = 4-Q/2) =

$3

Profit = Revenue -Cost = PQ -C = ($3)(2) -2(2) = $6-$4

$2.

2)

If a 2% tax is imposed on profit, the

net profit function becomes (1-0.02)(Prtofit) =

1.96Q -0.98Q²/2.

Setting the derivative of this (profit) function equal to 0,

1.96-1.96Q/2 = 0,

Q = 2

P = $3.

Such a profits tax has NO impact on P and Q.

3)

If a $1 tax per unit is paid by the firm,

the cost function becomes C = 2Q + 1(Q) = 3Q

Marginal cost MC= dC/dQ = 3

Marginal revenue from the demand function P = 4-Q/2 is

MR = 4 -Q.

For maximum profit, MR = MC or

4-Q = 3

Q = 1

P =4-1/2 =

$3.50

Profit = Revenue - Cost = PQ -C = ($3.50)(1) -3(1) = $3.50-3 =

$0.50

4)

The consumer pays $0.50 more than before. That is, 50% of the tax is the consumer's burden.

Hence, 50% of the tax is passed on to the consumer.