Homework 9 Answer Sheet

1. Suppose all firms in a monopolistically competitive industry were merged into one large firm. Would that new firm produce as many different brands? Would it produce only a single brand? Explain.

Monopolistic competition is defined by product differentiation. Each firm earns economic profit by distinguishing its brand from all other brands. This distinction can arise from underlying differences in the product or from differences in advertising. If these competitors merge into a single firm, the resulting monopolist would not produce as many brands, since much brand competition is internecine. However, it is unlikely that only one brand would be produced after the merger. Producing several brands with different prices and characteristics is one method of splitting the market into sets of customers with different price elasticities, which may also stimulate overall demand.

2. Consider the following duopoly. Demand is given by P = 10 - Q, where Q = Q1 + Q2. The firmsí cost functions are C1(Q1) = 4 + 2Q1 and C2(Q2) = 3 + 3Q2.

a. Suppose both firms have entered the industry. What is the joint profit-maximizing level of output? How much will each firm produce? How would your answer change if the firms have not yet entered the industry?

If both firms enter the market, and they collude, they will face a marginal revenue curve with twice the slope of the demand curve:

MR = 10 - 2Q.

Setting marginal revenue equal to marginal cost (the marginal cost of Firm 1, since it is lower than that of Firm 2) to determine the profit-maximizing quantity, Q:

10 - 2Q = 2, or Q = 4.

Substituting Q = 4 into the demand function to determine price:

P = 10 - 4 = $6.

The profit for Firm 1 will be:

p 1 = (6)(4) - (4 + (2)(4)) = $12.

The profit for Firm 2 will be:

p 2 = (6)(0) - (3 + (3)(0)) = -$3.

Total industry profit will be:

p T = p 1 + p 2 = 12 - 3 = $9.

If Firm 1 were the only entrant, its profits would be $12 and Firm 2ís would be 0.

If Firm 2 were the only entrant, then it would equate marginal revenue with its marginal cost to determine its profit-maximizing quantity:

10 - 2Q2 = 3, or Q2 = 3.5.

Substituting Q2 into the demand equation to determine price:

P = 10 - 3.5 = $6.5.

The profits for Firm 2 will be:

p 2 = (6.5)(3.5) - (3 + (3)(3.5)) = $9.25

b. What is each firmís equilibrium output and profit if they behave noncooperatively? Use the Cournot model. Draw the firmsí reaction curves, and show the equilibrium.

In the Cournot model, Firm 1 takes Firm 2ís output as given and maximizes profits. The profit function derived in 2.a becomes

p 1 = (10 - Q1 - Q2 )Q1 - (4 + 2Q1 ), or

Setting the derivative of the profit function with respect to Q1 to zero, we find Firm 1ís reaction function:

Similarly, Firm 2ís reaction function is

To find the Cournot equilibrium, we substitute Firm 2ís reaction function into Firm 1ís reaction function:

Substituting this value for Q1 into the reaction function for Firm 2, we find Q2 = 2.

Substituting the values for Q1 and Q2 into the demand function to determine the equilibrium price:

P = 10 - 3 - 2 = $5.

The profits for Firms 1 and 2 are equal to

p 1 = (5)(3) - (4 + (2)(3)) = 5 and

p 2 = (5)(2) - (3 + (3)(2)) = 1.

Figure 12.2.b

c. How much should Firm 1 be willing to pay to purchase Firm 2 if collusion is illegal, but the takeover is not?

In order to determine how much Firm 1 will be willing to pay to purchase Firm 2, we must compare Firm 1ís profits in the monopoly situation versus those in an oligopoly. The difference between the two will be what Firm 1 is willing to pay for Firm 2.

Substitute the profit-maximizing quantity from part a to determine the price:

P = 10 - 4 = $6.

The profits for the firm are determined by subtracting total costs from total revenue:

p 1 = (6)(4) - (4 + (2)(4)), or

p 1 = $12.

We know from part b that the profits for Firm 1 in the oligopoly situation will be $5; therefore, Firm 1 should be willing to pay up to $7, which is the difference between its monopoly profits ($12) and its oligopoly profits ($5). (Note that any other firm would pay only the value of Firm 2ís profit, i.e., $1.)

Note, Firm 1 might be able to accomplish its goal of maximizing profit by acting as a Stackelberg leader. If Firm 1 is aware of Firm 2ís reaction function, it can determine its profit-maximizing quantity by substituting for Q2 in its profit function and maximizing with respect to Q1:

, or , or

Therefore

Substituting Q1 and Q2 into the demand equation to determine the price:

P = 10 - 4.5 - 1.25 = $4.25.

Profits for Firm 1 are:

p 1 = (4.25)(4.5) - (4 + (2)(1.25)) = $6.125,

and profits for Firm 2 are:

p 2 = (4.25)(1.25) - (3 + (3)(1.25)) = -$1.4375.

Although Firm 2 covers average variable costs in the short run, it will go out of business in the long run. therefore, Firm 1 should drive Firm 2 out of business instead of buying it. If this is illegal, Firm 1 would have to resort to purchasing Firm 2, as discussed above.

3. A monopolist can produce at a constant average (and marginal) cost of AC = MC = 5. The firm faces a market demand curve given by Q = 53 - P.

a. Calculate the profit-maximizing price and quantity for this monopolist. Also calculate the monopolistís profits.

The monopolist wants to choose quantity to maximize its profits:

max p = PQ - C(Q),

p = (53 - Q)(Q) - 5Q, or p = 48Q - Q2.

To determine the profit-maximizing quantity, set the change in p with respect to the change in Q equal to zero and solve for Q:

Substitute the profit-maximizing quantity, Q = 24, into the demand function to find price:

24 = 53 - P, or P = $29.

Profits are equal to

p = TR - TC = (29)(24) - (5)(24) = $576.

b. Suppose a second firm enters the market. Let Q1 be the output of the first firm and Q2 be the output of the second. Market demand is now given by

Q1 + Q2 = 53 - P.

Assuming that this second firm has the same costs as the first, write the profits of each firm as functions of Q1 and Q2.

When the second firm enters, price can be written as a function of the output of two firms: P = 53 - Q1 - Q2. We may write the profit functions for the two firms:

or

and

or

c. Suppose (as in the Cournot model) each firm chooses its profit-maximizing level of output under the assumption that its competitorís output is fixed. Find each firmís "reaction curve" (i.e., the rule that gives its desired output in terms of its competitorís output).

Under the Cournot assumption, Firm 1 treats the output of Firm 2 as a constant in its maximization of profits. Therefore, Firm 1 chooses Q1 to maximize p 1 in b with Q2 being treated as a constant. The change in p 1 with respect to a change in Q1 is

This equation is the reaction function for Firm 1, which generates the profit- maximizing level of output, given the constant output of Firm 2. Because the problem is symmetric, the reaction function for Firm 2 is

d. Calculate the Cournot equilibrium (i.e., the values of Q1 and Q2 for which both firms are doing as well as they can given their competitorsí output). What are the resulting market price and profits of each firm?

To find the level of output for each firm that would result in a stationary equilibrium, we solve for the values of Q1 and Q2 that satisfy both reaction functions by substituting the reaction function for Firm 2 into the one for Firm 1:

By symmetry, Q2 = 16.

To determine the price, substitute Q1 and Q2 into the demand equation:

P = 53 - 16 - 16 = $21.

Profits are given by

p i = PQi - C(Qi) = p i = (21)(16) - (5)(16) = $256.

Total profits in the industry are p 1 + p 2 = $256 +$256 = $512.

*e. Suppose there are N firms in the industry, all with the same constant marginal cost, MC = 5. Find the Cournot equilibrium. How much will each firm produce, what will be the market price, and how much profit will each firm earn? Also, show that as N becomes large the market price approaches the price that would prevail under perfect competition.

If there are N identical firms, then the price in the market will be

.

Profits for the iíth firm are given by

,

Differentiating to obtain the necessary first-order condition for profit maximization,

.

Solving for Qi,

.

If all firms face the same costs, they will all produce the same level of output, i.e.,
Qi = Q*. Therefore,

We may substitute for Q = NQ*, total output, in the demand function:

Total profits are

p T = PQ - C(Q) = P(NQ*) - 5(NQ*)

or

p T

or

p T

or

p T

Notice that with N firms

and that, as N increases (N ® )

Q = 48.

Similarly, with

as N ® ,

P = 53 - 48 = 5.

With P = 5, Q = 53 - 5 = 48.

Finally,

so as N ® ,

p T = $0.

In perfect competition, we know that profits are zero and price equals marginal cost. Here, p T = $0 and P = MC = 5. Thus, when N approaches infinity, this market approaches a perfectly competitive one.

4. This exercise is a continuation of Exercise 3. We return to two firms with the same constant average and marginal cost, AC = MC = 5, facing the market demand curve
Q1 + Q2 = 53 - P. Now we will use the Stackelberg model to analyze what will happen if one of the firms makes its output decision ahead of the other one.

a. Suppose Firm 1 is the Stackelberg leader (i.e., makes its output decisions ahead of Firm 2). Find the reaction curves that tell each firm how much to produce in terms of the output of its competitor.

Firm 1, the Stackelberg leader, will choose its output, Q1, to maximize its profits, subject to the reaction function of Firm 2:

max p 1 = PQ1 - C(Q1),

subject to

Substitute for Q2 in the demand function and, after solving for P, substitute for P in the profit function:

To determine the profit-maximizing quantity, we find the change in the profit function with respect to a change in Q1:

Set this expression equal to 0 to determine the profit-maximizing quantity:

53 - 2Q1 - 24 + Q1 - 5 = 0, or Q1 = 24.

Substituting Q1 = 24 into Firm 2ís reaction function gives Q2:

Substitute Q1 and Q2 into the demand equation to find the price:

P = 53 - 24 - 12 = $17.

Profits for each firm are equal to total revenue minus total costs, or

p 1 = (17)(24) - (5)(24) = $288 and

p 2 = (17)(12) - (5)(12) = $144.

Total industry profit, p T = p 1 + p 2 = $288 + $144 = $432.

Compared to the Cournot equilibrium, total output has increased from 32 to 36, price has fallen from $21 to $17, and total profits have fallen from $512 to $432. Profits for Firm 1 have risen from $256 to $288, while the profits of Firm 2 have declined sharply from $256 to $144.

b. How much will each firm produce, and what will its profit be?

If each firm believes that it is the Stackelberg leader, while the other firm is the Cournot follower, they both will initially produce 24 units, so total output will be 48 units. The market price will be driven to $5, equal to marginal cost. It is impossible to specify exactly where the new equilibrium point will be, because no point is stable when both firms are trying to be the Stackelberg leader.

5. Two firms compete in selling identical widgets. They choose their output levels Q1 and Q2 simultaneously and face the demand curve

P = 30 - Q,

where Q = Q1 + Q2. Until recently, both firms had zero marginal costs. Recent environmental regulations have increased Firm 2ís marginal cost to $15. Firm 1ís marginal cost remains constant at zero. True or false: as a result, the market price will rise to the monopoly level.

True.

If only one firm were in this market, it would charge a price of $15 a unit. Marginal revenue for this monopolist would be

MR = 30 - 2Q,

Profit maximization implies MR = MC, or

30 - 2Q = 0, Q = 15, (using the demand curve) P = 15.

The current situation is a Cournot game where firm 1's marginal costs are zero and firm 2's marginal costs are 15. We need to find the best response functions:

Firm 1ís revenue is

Pq1 = (30 - q1 -q2) q1 = 30q1 - - q1q2

and its marginal revenue is given by:

MR1 = 30 - 2q1 - q2

Profit maximization implies MR1 = MC1 or

30 - 2q1 - q2 = 0 Þ q1 = 15 - (1/2) q2

which is firm 1ís best response function.

Firm 2ís revenue is

P q2 = (30 - q1 - q2) q2

= 30q2 - q1q2 - ,

and its marginal revenue is given by:

MR2 = 30 - q1 - 2q2

Profit maximization implies MR2 = MC2, or

30 - q1 - 2q2 = 15 Þ q2 = (15/2) - (1/2) q1

which is firm 2ís best response function.

Cournot equilibrium occurs at the intersection of best response functions. Substituting for q1 in the response function for firm 2 yields:

q2 = (15/2) - (1/2) [ 15 - (1/2) q2 ],

Thus

q2 = 0, q1 = 15, and

P = 30 - q1 + q2 = 15, which is the monopoly price.

6. Suppose that two identical firms produce widgets and that they are the only firms in the market. Their costs are given by C1 = 30Q1 and C2 = 30Q2, where Q1 is the output of Firm 1 and Q2 is the output of Firm 2. Price is determined by the following demand curve:

P = 150 - Q

where Q = Q1 + Q2.

a. Find the Cournot-Nash equilibrium. Calculate the profit of each firm at this equilibrium.

To determine the Cournot-Nash equilibrium, we first calculate the reaction function for each firm, then solve for price, quantity, and profit. Profit for Firm 1, TR1 - TC1, is equal to

Therefore,

Setting this equal to zero and solving for Q1 in terms of Q2:

Q1 = 60 - 0.5Q2.

This is Firm 1ís reaction function. Because Firm 2 has the same cost structure, Firm 2ís reaction function is

Q2 = 60 - 0.5Q1 .

Substituting for Q2 in the reaction function for Firm 1, and solving for Q1, we find

Q1 = 60 - (0.5)(60 - 0.5Q1), or Q1 = 40.

By symmetry, Q2 = 40.

Substituting Q1 and Q2 into the demand equation to determine the price at profit maximization:

P = 150 - 40 - 40 = $70.

Substituting the values for price and quantity into the profit function,

p 1 = (70)(40) - (30)(40) = $1,600 and

p 2 = (70)(40) - (30)(40) = $1,600.

Therefore, profit is $1,600 for both firms in Cournot-Nash equilibrium.

b. Suppose the two firms form a cartel to maximize joint profits. How many widgets will be produced? Calculate each firmís profit.

Because marginal cost is the same for both firms and is constant for all output, we may determine the joint profit-maximizing output by considering only one firm, i.e., let

Q1 = Q and Q2 = 0.

Profit is

p = 150Q - Q2 - 30Q.

Therefore,

Solving for the profit-maximizing level of output,

120 - 2Q = 0, or Q = 60.

Substituting Q = 60 into the demand function to determine price:

P = 150 - 60 = $90.

Substituting P and Q into the profit function:

p = (90)(60) - (30)(60) = $3,600.

Because MC is constant, the firms may split quantities and profits. If they split quantity equally, then Q1 = Q2 = 30 and profits are $1,800 for each firm.

c. Suppose Firm 1 were the only firm in the industry. How would the market output and Firm 1ís profit differ from that found in part (b) above?

If Firm 1 were the only firm, it would solve the profit-maximization problem as in 6.b, i.e., Q1 = 60 and p 1 = $3,600.

d. Returning to the duopoly of part (b), suppose Firm 1 abides by the agreement, but Firm 2 cheats by increasing production. How many widgets will Firm 2 produce? What will be each firmís profits?

Assuming their agreement is to split the market equally, Firm 1 produces 30 widgets. Firm 2 cheats by producing its profit-maximizing level, given Q1 = 30.Substituting Q1 = 30 into Firm 2ís reaction function:

Total industry output, QT, is equal to Q1 plus Q2:

QT = 30 + 45 = 75.

Substituting QT into the demand equation to determine price:

P = 150 - 75 = $75.

Substituting Q1, Q2, and P into the profit function: p 1 = (75)(30) - (30)(30) = $1,350 and

p 2 = (75)(45) - (30)(30) = $2,475.

Firm 2 has increased its profits at the expense of Firm 1 by cheating on the agreement.

7. Suppose the airline industry consisted of only two firms: American and Texas Air Corp. Let the two firms have identical cost functions, C(q) = 40q. Assume the demand curve for the industry is given by P = 100 - Q, and that each firm expects the other to behave as a Cournot competitor.

a. Calculate the (Cournot-Nash) equilibrium for each firm, assuming that each chooses the output level that maximizes its profits taking its rivalís output as given. What are the profits of each firm?

To determine the Cournot-Nash equilibrium, we first calculate the reaction function for each firm, then solve for price, quantity, and profit. Profit for Texas Air, p 1, is equal to total revenue minus total cost:

p 1 = (100 - Q1 - Q2)Q1 - 40Q1, or

The change in p 1 with respect to Q1 is

Setting the derivative to zero and solving for Q1 in terms of Q2 will give Texas Airís reaction function:

Q1 = 30 - 0.5Q2.

Because American has the same cost structure, Americanís reaction function is

Q2 = 30 - 0.5Q1.

Substituting for Q2 in the reaction function for Texas Air,

Q1 = 30 - 0.5(30 - 0.5Q1) = 20.

By symmetry, Q2 = 20. Industry output, QT, is Q1 plus Q2, or

QT = 20 + 20 = 40.

Substituting industry output into the demand equation, we find P = 60. Substituting Q1, Q2, and P into the profit function, we find

p 1 = p 2 = (60(20) -202 - (20)(20) = $400

for both firms in Cournot-Nash equilibrium.

b. What would be the equilibrium quantity if Texas Air had constant marginal and average costs of 25, and American had constant marginal and average costs of 40?

By solving for the reaction functions under this new cost structure, we find that profit for Texas Air is equal to

The change in profit with respect to Q1 is

Set the derivative to zero, and solving for Q1 in terms of Q2,

Q1 = 37.5 - 0.5Q2.

This is Texas Airís reaction function. Since American has the same cost structure as in 6.a., Americanís reaction function is the same as before:

Q2 = 30 - 0.5Q1.

To determine Q1, substitute for Q2 in the reaction function for Texas Air and solve for Q1:

Q1 = 37.5 - (0.5)(30 - 0.5Q1) = 30.

Texas Air finds it profitable to increase output in response to a decline in its cost structure.

To determine Q2, substitute for Q1 in the reaction function for American:

Q2 = 30 - (0.5)(37.5 - 0.5Q2) = 15.

American has cut back slightly in its output in response to the increase in output by Texas Air.

Total quantity, QT, is Q1 + Q2, or

QT = 30 + 15 = 45.

Compared to 7a, the equilibrium quantity has risen slightly.

c. Assuming that both firms have the original cost function, C(q) = 40q, how much should Texas Air be willing to invest to lower its marginal cost from 40 to 25, assuming that American will not follow suit? How much should American be willing to spend to reduce its marginal cost to 25, assuming that Texas Air will have marginal costs of 25 regardless of Americanís actions?

Recall that profits for both firms were $400 under the original cost structure. With constant average and marginal costs of 25, Texas Airís profits will be

(55)(30) - (25)(30) = $900.

The difference in profit is $500. Therefore, Texas Air should be willing to invest up to $500 to lower costs from 40 to 25 per unit (assuming American does not follow suit).

To determine how much American would be willing to spend to reduce its average costs, we must calculate the difference in profits, assuming Texas Airís average cost is 25. First, without investment, Americanís profits would be:

(55)(15) - (40)(15) = $225.

Second, with investment by both firms, the reaction functions would be:

Q1 = 37.5 - 0.5Q2 and Q2 = 37.5 - 0.5Q1.

To determine Q1, substitute for Q2 in the first reaction function and solve for Q1:

Q1 = 37.5 - (0.5)(37.5 - 0.5Q1) = 25.

Substituting for Q1 in the second reaction function to find Q2:

Q2 = 37.5 - 0.5(37.5 - 0.5Q2) = 25.

Substituting industry output into the demand equation to determine price:

P = 100 - 50 = $50.

Therefore, Americanís profits if Q1 = 30 and Q2 = 15 are

p 2 = (100 - 30 - 15)(15) - (40)(15) = $225.

Americanís profits if Q1 = Q2 = 25 (when both firms have MC = AC = 25) are

p 2 = (100 - 25 - 25)(25) - (25)(25) = $625.

Therefore, the difference in profit with and without the cost-saving investment for American is $400. American should be willing to invest up to $400 to reduce its marginal cost to 25 if Texas Air also has marginal costs of 25.

*8. Demand for light bulbs can be characterized by Q = 100 - P, where Q is in millions of lights sold, and P is the price per box. There are two producers of lights: Everglow and Dimlit. They have identical cost functions:

Q = QE + QD.

a. Unable to recognize the potential for collusion, the two firms act as short-run perfect competitors. What are the equilibrium values of QE, QD, and P? What are each firmís profits?

Given that the total cost function is , the marginal cost curve for each firm is .

In the short run, perfectly competitive firms determine the optimal level of output by equating price and marginal cost.

Substitute the output for each of the firms in to the demand curve to determine the market price:

90 = 100 - P, or P = $10.

Determine the level of profits for each firm by subtracting total costs from total revenue:

.

Both firms are losing money by operating as perfect competitors in the short run.

b. Top management in both firms is replaced. Each new manager independently recognizes the oligopolistic nature of the light bulb industry and plays Cournot. What are the equilibrium values of QE, QD, and P? What are each firmís profits?

To determine the Cournot-Nash equilibrium, we first calculate the reaction function for each firm, then solve for price, quantity, and profit. Profits for Everglow are equal to TRE - TCE, or

The change in profit with respect to QE is

To determine Everglowís reaction function, set the change in profits with respect to QE equal to 0 and solve for QE:

90 - 3QE - QD = 0, or

Because Dimlit has the same cost structure, Dimlitís reaction function is

Substituting for QD in the reaction function for Everglow, and solving for QE:

By symmetry, QD = 30. Total industry output is 60.

Substituting industry output into the demand equation gives P:

60 = 100 - P, or P = $40.

Substituting total industry output and P into the profit function:

p I = (100 - 30 - 30)(30) - ((10)(30) + (0.5)(302 )) = $450.

c. Suppose the Everglow manager guesses correctly that Dimlit has a Cournot conjectural variation, so Everglow plays Stackelberg. What are the equilibrium values of QE, QD, and P? What are each firmís profits?

Recall Everglowís profit function:

If Everglow sets its quantity first, knowing Dimlitís reaction function , we may determine Everglowís reaction function by substituting for QD in its profit function. We find

.

To determine the profit-maximizing quantity, differentiate profit with respect to QE, set the derivative to zero and solve for QE:

Substituting this into Dimlitís reaction function, we find Total industry output is 47.1 and P = $52.90. Profit for Everglow is $772.29. Profit for Dimlit is $689.08.

d. If the managers of the two companies collude, what are the equilibrium values of QE, QD, and P? What are each firmís profits?

If the firms split the market equally, total cost in the industry is ; therefore, . Total revenue is 100Q - Q2; therefore, MR = 100 - 2Q. To determine the profit-maximizing quantity, set MR = MC and solve for QT:

This means QE = QD = 15.

Substituting QT into the demand equation to determine price:

P = 100 - 30 = $70.

The profit for each firm is equal to total revenue minus total cost:

9. Two firms produce luxury sheepskin auto seat covers, Western Where (WW) and B.B.B. Sheep (BBBS). Each firm has a cost function given by:

C (q) = 20q + q2

The market demand for these seat covers is represented by the inverse demand equation:

P = 200 - 2Q,

where Q = q1 + q2 , total output.

a. If each firm acts to maximize its profits, taking its rivalís output as given (i.e., the firms behave as Cournot oligopolists), what will be the equilibrium quantities selected by each firm? What is total output, and what is the market price? What are the profits for each firm?

We are given each firmís cost function C(q) = 20q + q2 and the market demand function P = 200 - 2Q where total output Q is the sum of each firmís output q1 and q2.

We find the best response functions for both firms:

WWís revenue is R1 = P q1 = (200 - 2(q1 + q2)) q1 = 200q1 - 2q12 - 2q1q2.

Its marginal revenue and cost functions are:

MR1 = 200 - 4q1 - 2q2

MC1 = 20 + 2q1

Profit maximization implies:

MR1 = MC1 or 200 - 4q1 - 2q2 = 20 + 2q1 which yields the best response function:

q1 = 30 - (1/3)q2.

By symmetry, BBBSís best response function will be:

q2 = 30 - (1/3)q1.

Cournot equilibrium occurs at the intersection of these two best response functions, given by:

q1 = q2 = 22.5.

Thus,

Q = q1 + q2 = 45

P = 200 - 2(45) = $110.

Profit for both firms will be equal and given by:

R - C = (110) (22.5) - (20(22.5) + 22.52) = $1518.75

b. It occurs to the managers of WW and BBBS that they could do a lot better by colluding. If the two firms collude, what would be the profit-maximizing choice of output? What is the industry price? What is the output and the profit for each firm in this case?

If firms can collude, they should each produce half the quantity that maximizes total industry profits (i.e. half the monopoly profits).

Joint Profits will be (200-2Q)Q - 2(20(Q/2) + (Q/2)2) = 180Q - 2.5Q2 and will be maximized at Q = 36.

Thus, we will have q1 = q2 = 36 / 2 = 18 and P = 200 - 2(36) = $128

Profit for each firm will be 18(128) - (20(18) + 182) = $1,620

c. The managers of these firms realize that explicit agreements to collude are illegal. Each firm must decide on its own whether to produce the Cournot quantity or the cartel quantity. to aid in making the decision, the manager of WW constructs a payoff matrix like the real one below. Fill in each box with the (profit of WW, profit of BBBS). Given this payoff matrix, what output strategy is each firm likely to pursue?

If WW produces the Cournot level of output (22.5) and BBBS produces the collusive level (18), then:

Q = q1 + q2 = 22.5 + 18 = 40.5

P = 200 -2(40.5) = $119.

Profit for WW = 22.5(119) - (20(22.5) + 22.52) = $1721.25.

Profit for BBBS = 18(119) - (20(18) + 182) = $1458.

Both firms producing at the Cournot output levels will be the only Nash Equilibrium in this industry, given the following payoff matrix. (note: not only is this a Nash Equilibrium, but it is an equilibrium in dominant strategies.)

Profit Payoff Matrix

BB

BS

(WW profit, BBBS
profit)

Produce

Cournot q

Produce

Cartel q

WW

Produce

Cournot q

1518, 1518

1721, 1458

 

Produce

Cartel q

1458, 1721

1620, 1620

d. Suppose WW can set its output level before BBBS does. How much will WW choose to produce in this case? How much will BBBS produce? What is the market price, and what is the profit for each firm? Is WW better off by choosing its output first? Explain why or why not.

WW is now able to set quantity first. WW knows that BBBS will choose a quantity q2 which will be its best response to q1 or:

q2 = 30 - (1/3)q1

WW profits will be:

p 1 = Pq1 - C1 = (200 - 2q1 - 2q2)q1 - 20q1 - = (200 -2q1 - 2(30 - (1/3)q1))q1 - 20q1 - q12 = 21.4

= 120q1 - (7/3)

Profit maximization implies:

dp 1/ dq1 = 120 - (14/3)q1 = 0 or q1 = 25.7 and q2 = 30 - (1/3)(25.7) = 21.4

The equilibrium price and profits will then be:

P = 200 - 2(q1 + q2) = 200 - 2(25.7 + 21.4) = $105.80

p 1 = (105.80) (25.7) - (20) (25.7) - 25.72 = $1544.57

p 2 = (105.80) (21.4) - (20) (21.4) - 21.42 = $1378.16.

WW is able to benefit from its first mover advantage by committing to a high level of output. Since firm 2 moves after firm 1 has selected its output, firm 2 can only react to the output decision of firm 1..If firm 1 produces its Cournot output as a leader, firm 2 produces its Cournot output as a follower. Hence, firm 1 cannot do worse as a leader than it does in the Cournot game. When firm 1 produces more, firm 2 produces less, raising firm 1ís profits.

*10. Two firms compete by choosing price. Their demand functions are Q1 = 20 - P1 + P2 and Q2 = 20 + P1 - P2

where P1 and P2 are the prices charged by each firm respectively and Q1 and Q2 are the resulting demands. (Note that the demand for each good depends only on the difference in prices; if the two firms colluded and set the same price, they could make that price as high as they want, and earn infinite profits.) Marginal costs are zero.

a. Suppose the two firms set their prices at the same time. Find the resulting Nash equilibrium. What price will each firm charge, how much will it sell, and what will its profit be? (Hint: Maximize the profit of each firm with respect to its price.)

To determine the Nash equilibrium, we first calculate the reaction function for each firm, then solve for price. With zero marginal cost, profit for Firm 1 is:

The marginal revenue is the slope of the total revenue function (here it is the slope of the profit function because total cost is equal to zero):

MR1 = 20 - 2P1 + P2.

At the profit-maximizing price, MR1 = 0. Therefore,

This is Firm 1ís reaction function. Because Firm 2 is symmetric to Firm 1, its reaction function is

Substituting Firm 2ís reaction function into that of Firm 1:

By symmetry, P2 = $20.

To determine the quantity produced by each firm, substitute P1 and P2 into the demand functions:

Q1 = 20 - 20 + 20 = 20 and Q2 = 20 + 20 - 20 = 20.

Profits for Firm 1 are P1Q1 = $400, and, by symmetry, profits for Firm 2 are also $400.

b. Suppose Firm 1 sets its price first, and then Firm 2 sets its price. What price will each firm charge, how much will it sell, and what will its profit be?

If Firm 1 sets its price first, it takes Firm 2ís reaction function into account. Firm 1ís profit function is:

To determine the profit-maximizing price, find the change in profit with respect to a change in price:

Set this expression equal to zero to find the profit-maximizing price:

30 - P1 = 0, or P1 = $30.

Substitute P1 in Firm 2ís reaction function to find P2:

At these prices,

Q1 = 20 - 30 + 25 = 15 and Q2 = 20 + 30 - 25 = 25.

Profits are

p 1 = (30)(15) = $450 and

p 2 = (25)(25) = $625.

If Firm 1 must set its price first, Firm 2 is able to undercut Firm 1 and gain a larger market share.

c. Suppose you are one of these firms, and there are three ways you could play the game: (i) Both firms set price at the same time. (ii) You set price first. (iii) Your competitor sets price first. If you could choose among these, which would you prefer? Explain why.

Your first choice should be (iii), and your second choice should be (ii). (Compare the Nash profits in part 10.a, $400, with profits in part 10.b., $450 and $625.) From the reaction functions, we know that the price leader provokes a price increase in the follower. By being able to move second, however, the follower increases price by less than the leader, and hence undercuts the leader. Both firms enjoy increased profits, but the follower does best.

*11. The dominant firm model can help us understand the behavior of some cartels. Let us apply this model to the OPEC oil cartel. We shall use isoelastic curves to describe world demand W and noncartel (competitive) supply S. Reasonable numbers for the price elasticities of world demand and non-cartel supply are -1/2 and 1/2, respectively. Then, expressing W and S in millions of barrels per day (mb/d), we could write

W = 160P -1/2 and S = 3(1/3)P1/2.

Note that OPECís net demand is D = W - S.

a. Sketch the world demand curve W, the non-OPEC supply curve S, OPECís net demand curve D, and OPECís marginal revenue curve. For purposes of approximation, assume OPECís production cost is zero. Indicate OPECís optimal price, OPECís optimal production, and non-OPEC production on the diagram. Now, show on the diagram how the various curves will shift, and how OPECís optimal price will change if non-OPEC supply becomes more expensive because reserves of oil start running out.

OPECís net demand curve, D, is:

OPECís marginal revenue curve starts from the same point on the vertical axis as its net demand curve and is twice as steep. OPECís optimal production occurs where MR = 0 (since production cost is assumed to be zero), and OPECís optimal price in Figure 12.11.a.i is found from the net demand curve at QOPEC. Non-OPEC production can be read off of the non-OPEC supply curve at a price of P*.

Figure 12.11.a.i

Next, suppose non-OPEC oil becomes more expensive. Then the supply curve S shifts to S*. This changes OPECís net demand curve from D to D*, which in turn creates a new marginal revenue curve, MR*, and a new optimal OPEC production level of , yielding a new higher price of P*. At this new price, non-OPEC production is .. Notice that the curves must be drawn accurately to give this result.

Figure 12.11.a.ii

b. Calculate OPECís optimal (profit-maximizing) price. (Hint: Because OPECís cost is zero, just write the expression for OPEC revenue and find the price that maximizes it.)

Since costs are zero, OPEC will choose a price that maximizes total revenue:

Max p = PQ = P(W - S)

To determine the profit-maximizing price, we find the change in the profit function with respect to a change in price and set it equal to zero:

Solving for P,

c. Suppose the oil-consuming countries were to unite and form a "buyersí cartel" to gain monopsony power. What can we say, and what canít we say, about the impact this would have on price?

If the oil-consuming countries unite to form a buyersí cartel, then we have a monopoly (OPEC) facing a monopsony (the buyersí cartel). As a result, there is no well-defined demand or supply curve. We expect that the price will fall below the monopoly price when the buyers also collude, because monopsony power offsets monopoly power. However, economic theory cannot determine the exact price that results from this bilateral monopoly because the price depends on the bargaining skills of the two parties, as well as on other factors, such as the elasticities of supply and demand.

12. A lemon-growing cartel consists of four orchards. Their total cost functions are:

(TC is in hundreds of dollars, Q is in cartons per month picked and shipped.)

a. Tabulate total, average, and marginal costs for each firm for output levels between 1 and 5 cartons per month (i.e., for 1, 2, 3, 4, and 5 cartons).

The following tables give total, average, and marginal costs for each firm.

 

Firm 1

Firm 2

Units

TC

AC

MC

TC

AC

MC

0

20

__

__

25

__

__

1

25

25

5

28

28

3

2

40

20

15

37

18

9

3

65

22

25

52

17

15

4

100

25

35

73

18

21

5

145

29

45

100

20

27

 

Firm 3

Firm 4

Units

TC

AC

MC

TC

AC

MC

0

15

__

__

20

__

__

1

19

19

4

26

26

6

2

31

16

12

44

22

18

3

51

17

20

74

25

30

4

79

20

29

116

29

42

5

115

23

36

170

34

54

b. If the cartel decided to ship 10 cartons per month and set a price of 25 per carton, how should output be allocated among the firms?

The cartel should assign production such that the lowest marginal cost is achieved for each unit, i.e.,

Cartel

Unit Assigned

Firm

Assigned

1

2

2

3

3

1

4

4

5

2

6

3

7

1

8

2

9

4

10

3

Therefore, Firms 1 and 4 produce 2 units each and Firms 2 and 3 produce 3 units each.

c. At this shipping level, which firm has the most incentive to cheat? Does any firm not have an incentive to cheat?

At this level of output, Firm 2 has the lowest marginal cost for the producing one more unit beyond its allocation, i.e., MC = 21 for the fourth unit for Firm 2. In addition,
MC = 21 is less than the price of $25. For all other firms, the next unit has a marginal cost equal to or greater than $25. Firm 2 has the most incentive to cheat, while Firms 3 and 4 have no incentive to cheat