Textbook Exercises:

4. A firm faces the following average revenue (demand) curve: P = 120 - 0.02Q where Q is weekly production and P is price, measured in cents per unit. The firm’s cost function is given by C = 60Q + 25,000. Assume that the firm maximizes profits.

a. What is the level of production, price, and total profit per week?

The profit-maximizing output is found by setting marginal revenue equal to marginal cost. Given a linear demand curve in inverse form, P = 120 - 0.02Q, we know that the marginal revenue curve will have twice the slope of the demand curve. Thus, the marginal revenue curve for the firm is MR = 120 - 0.04Q. Marginal cost is simply the slope of the total cost curve. The slope of TC = 60Q + 25,000 is 60, so MC equals 60. Setting MR = MC to determine the profit-maximizing quantity:

120 - 0.04Q = 60, or

Q = 1,500.

Substituting the profit-maximizing quantity into the inverse demand function to determine the price:

P = 120 - (0.02)(1,500) = 90 cents.

Profit equals total revenue minus total cost:

= (90)(1,500) - (25,000 + (60)(1,500)), or

= $200 per week.

b. If the government decides to levy a tax of 14 cents per unit on this product, what will be the new level of production, price, and profit?

Suppose initially that the consumers must pay the tax to the government. Since the total price (including the tax) consumers would be willing to pay remains unchanged, we know that the demand function is

P* + T = 120 - 0.02Q, or P* = 120 - 0.02Q - T,

where P* is the price received by the suppliers. Because the tax increases the price of each unit, total revenue for the

monopolist decreases by TQ, and marginal revenue, the revenue on each additional unit, decreases by T:

MR = 120 - 0.04Q - T

where T = 14 cents. To determine the profit-maximizing level of output with the tax, equate marginal revenue with marginal cost:

120 - 0.04Q - 14 = 60, or

Q = 1,150 units.

Substituting Q into the demand function to determine price:

P* = 120 - (0.02)(1,150) - 14 = 83 cents.

Profit is total revenue minus total cost:

83 1,150 60 1,150 25,000 1450cents, or

$14.50 per week.

Note: The price facing the consumer after the imposition of the tax is 97 cents. The monopolist receives 83 cents. Therefore, the consumer and the monopolist each pay 7 cents of the tax.

If the monopolist had to pay the tax instead of the consumer, we would arrive at the same result. The monopolist’s cost function would then be

TC = 60Q + 25,000 + TQ = (60 + T)Q + 25,000.

The slope of the cost function is (60 + T), so MC = 60 + T. We set this MC to the marginal revenue function from part (a):

120 - 0.04Q = 60 + 14, or

Q = 1,150.

Thus, it does not matter who sends the tax payment to the government. The burden of the tax is reflected in the price of the good.

5. The following table shows the demand curve facing a monopolist who produces at a constant marginal cost of $10.

Price Quantity

18 016 414 812 1210 168 20 6 24 4 28 2 32 0 36

a. Calculate the firm’s marginal revenue curve.

To find the marginal revenue curve, we first derive the inverse demand curve. The intercept of the inverse demand curve on the price axis is 18. The slope of the inverse demand curve is the change in price divided by the change in quantity. For example, a decrease in price from 18 to 16 yields an

increase in quantity from 0 to 4. Therefore, the slope is 12

and the demand curve is P 18 0.5Q.

The marginal revenue curve corresponding to a linear demand curve is a line with the same intercept as the inverse demand curve and a slope that is twice as steep. Therefore, the marginal revenue curve is

MR = 18 - Q.

b. What are the firm’s profit-maximizing output and price? What is its profit?

The monopolist’s maximizing output occurs where marginal revenue equals marginal cost. Marginal cost is a constant $10. Setting MR equal to MC to determine the profit-maximizing quantity:

18 - Q = 10, or Q = 8.

To find the profit-maximizing price, substitute this quantity into the demand equation: P 18 0.5 8 $14.

Total revenue is price times quantity: TR 14 8 $112.

The profit of the firm is total revenue minus total cost, and total cost is equal to average cost times the level of output produced. Since marginal cost is constant, average variable cost is equal to marginal cost. Ignoring any fixed costs, total cost is 10Q or 80, and profit is

112 80 $32.

c. What would the equilibrium price and quantity be in a competitive industry?

For a competitive industry, price would equal marginal cost at equilibrium. Setting the expression for price equal to a marginal cost of 10: 18 0.5Q 10 Q 16 P 10.

Note the increase in the equilibrium quantity compared to the monopoly solution.

d. What would the social gain be if this monopolist were forced to produce and price at the competitive equilibrium? Who would gain and lose as a result?

The social gain arises from the elimination of deadweight loss. Deadweight loss in this case is equal to the triangle above the constant marginal cost curve, below the demand curve, and between the quantities 8 and 16, or numerically (14-10)(16-8)(.5)=$16. Consumers gain this deadweight loss plus the monopolist’s profit of $32. The monopolist’s profits are reduced to zero, and the consumer surplus increases by $48.

6. Suppose that an industry is characterized as follows: C 100 2Q2 Firm total cost functionMC 4Q Firm marginal cost functionP 90 2Q Industry demand curveMR 90 4Q Industry marginal revenue curve.

a. If there is only one firm in the industry, find the monopoly price, quantity, and level of profit.

If there is only one firm in the industry, then the firm will act like a monopolist and produce at the point where marginal revenue is equal to marginal cost:

MC=4Q=90-4Q=MR Q=11.25.

For a quantity of 11.25, the firm will charge a price P=90-2*11.25=$67.50. The level of profit is $67.50*11.25-100-2*11.25*11.25=$406.25.

b. Find the price, quantity, and level of profit if the industry is competitive.

If the industry is competitive then price is equal to marginal cost, so that 90-2Q=4Q, or Q=15. At a quantity of 15 price is equal to 60. The level of profit is therefore 60*15-100-2*15*15=$350.

c. Graphically illustrate the demand curve, marginal revenue curve, marginal cost curve, and average cost curve. Identify the difference between the profit level of the monopoly and the profit level of the competitive industry in two different ways. Verify that the two are numerically equivalent.

The graph below illustrates the demand curve, marginal revenue curve, and marginal cost curve. The average cost curve hits the marginal cost curve at a quantity of approximately 7, and is increasing thereafter (this is not shown in the graph below). The profit that is lost by having the firm produce at the competitive solution as compared to the monopoly solution is given by the difference of the two profit levels as calculated in parts a and b above, or $406.25-$350=$56.25. On the graph below, this difference is represented by the lost profit area, which is the triangle below the marginal cost curve and above the marginal revenue curve, between the quantities of 11.25 and 15. This is lost profit because for each of these 3.75 units extra revenue earned was less than extra cost incurred. This area can be calculated as

0.5*(60-45)*3.75+0.5*(45-30)*3.75=$56.25. The second method of graphically illustrating the difference in the two profit levels is to draw in the average cost curve and identify the two profit boxes. The profit box is the difference between the total revenue box (price times quantity) and the total cost box (average cost times quantity). The monopolist will gain two areas and lose one area as compared to the competitive firm, and these areas will sum to $56.25.

MC

MR

Demand

11.25 15

lost profit

Q

P

11. A monopolist faces the demand curve P = 11 - Q, where P is measured in dollars per unit and Q in thousands of units. The monopolist has a constant average cost of $6 per unit.

a. Draw the average and marginal revenue curves and the average and marginal cost curves. What are the monopolist’s profit-maximizing price and quantity? What is the resulting profit? Calculate the firm’s degree of monopoly power using the Lerner index.

Because demand (average revenue) may be described as P = 11 - Q, we know that the marginal revenue function is MR = 11 - 2Q. We also know that if average cost is constant, then marginal cost is constant and equal to average cost: MC = 6.

To find the profit-maximizing level of output, set marginal revenue equal to marginal cost:

11 - 2Q = 6, or Q = 2.5.

That is, the profit-maximizing quantity equals 2,500 units. Substitute the profit-maximizing quantity into the demand equation to determine the price:

P = 11 - 2.5 = $8.50.

Profits are equal to total revenue minus total cost,

= TR - TC = (AR)(Q) - (AC)(Q), or

= (8.5)(2.5) - (6)(2.5) = 6.25, or $6,250.

The degree of monopoly power is given by the Lerner Index:

P MCP

8 5 68 5

0 294.

.. .

Price

Q

2

4

6

8

12

4 6 102 12

10

8

AC = MC

MR D = AR

Profits

Figure 10.11.a

b. A government regulatory agency sets a price ceiling of $7 per unit. What quantity will be produced, and what will the firm’s profit be? What happens to the degree of monopoly power?

To determine the effect of the price ceiling on the quantity produced, substitute the ceiling price into the demand equation.

7 = 11 - Q, or

Q = 4,000.

The monopolist will pick the price of $7 because it is the highest price that it can charge, and this price is still greater than the constant marginal cost of $6, resulting in positive monopoly profit.

Profits are equal to total revenue minus total cost:

= (7)(4,000) - (6)(4,000) = $4,000.

The degree of monopoly power is:

P MCP

7 67

0143. .

c. What price ceiling yields the largest level of output? What is that level of output? What is the firm’s degree of monopoly power at this price?

If the regulatory authority sets a price below $6, the monopolist would prefer to go out of business instead of produce because it cannot cover its average costs. At any price above $6, the monopolist would produce less than the 5,000 units that would be produced in a competitive industry. Therefore, the regulatory agency should set a price ceiling of $6, thus making the monopolist face a horizontal effective demand curve up to Q = 5,000. To ensure a positive output (so that the monopolist is not indifferent between producing 5,000 units and shutting down), the price ceiling should be set at $6 + , where is small.

Thus, 5,000 is the maximum output that the regulatory agency can extract from the monopolist by using a price ceiling. The degree of monopoly power is

P MCP

6 66 6

0 as 0.

Exercise: 1. If a firm is pricing to maximize profit and has a price-marginal

cost margin of 45%, what is the elasticity of demand at that price? The rule of thumb is that the price-cost margin equals minus the inverse elasticity of demand, so a margin of 45 percent means that 0.45= -1/ or

2. A monopolist has the total cost function C=3Q2 . It faces the

demand curve P=1200-Q. a. What is the profit maximizing price and output? What is

totally profit?

b. What will the monopolist do if it faces a lump sum take of 50,000? Of $100,000?

c. What price ceiling would maximize total surplus? What is the deadweight loss associated with the monopolist (over the competitive outcome)?

a) Since the demand curve is P=1200-Q, the marginal revenue curve is

MR=1200-2Q. Solving for the quantity which equates MR and MC, we get 1200-2Q=6Q or =150 and =1200-150=$1050. Profit is

b) With lump-sum tax of $50000, the monopolist’s profit after taxes is $40000, so it will stay in business. Since the lump-sum tax is a fixed cost, the firm will still sell 150 units. With a lump-sum tax of $100000, profit after tax will be -$10000, so the firm will be better off to shut down in the long run.

c) The best price ceiling is that at which the quantity sold is maximized. This is the same as the competitive price. The quantity sold under a price ceiling is the minimum of the quantity demanded at that price and the quantity at which marginal cost equals price. Equating price and marginal cost to find the intersection of MC and the demand curve, we get 1200-Q=6Q or