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 = Q_{1} + Q_{2}. The firms’ cost functions are C_{1}(Q_{1}) = 4 + 2Q_{1} and C_{2}(Q_{2}) = 3 + 3Q_{2}.
a. Suppose both firms have entered the industry. What is the joint profitmaximizing 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 profitmaximizing 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 2’s would be 0.
If Firm 2 were the only entrant, then it would equate marginal revenue with its marginal cost to determine its profitmaximizing quantity:
10  2Q_{2} = 3, or Q_{2} = 3.5.
Substituting Q_{2} 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 firm’s equilibrium output and profit if they behave noncooperatively? Use the Cournot model. Draw the firms’ reaction curves, and show the equilibrium.
In the Cournot model, Firm 1 takes Firm 2’s output as given and maximizes profits. The profit function derived in 2.a becomes
p _{1} = (10  Q_{1}  Q_{2 })Q_{1}  (4 + 2Q_{1 }), or
Setting the derivative of the profit function with respect to Q_{1} to zero, we find Firm 1’s reaction function:
Similarly, Firm 2’s reaction function is
To find the Cournot equilibrium, we substitute Firm 2’s reaction function into Firm 1’s reaction function:
Substituting this value for Q_{1} into the reaction function for Firm 2, we find Q_{2} = 2.
Substituting the values for Q_{1} and Q_{2} 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 1’s 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 profitmaximizing 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 2’s 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 2’s reaction function, it can determine its profitmaximizing quantity by substituting for Q_{2} in its profit function and maximizing with respect to Q_{1}:
, or , or
Therefore
Substituting Q_{1} and Q_{2} 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 profitmaximizing price and quantity for this monopolist. Also calculate the monopolist’s profits.
The monopolist wants to choose quantity to maximize its profits:
max p = PQ  C(Q),
p = (53  Q)(Q)  5Q, or p = 48Q  Q^{2}.
To determine the profitmaximizing quantity, set the change in p with respect to the change in Q equal to zero and solve for Q:
Substitute the profitmaximizing 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 Q_{1} be the output of the first firm and Q_{2} be the output of the second. Market demand is now given by
Q_{1} + Q_{2} = 53  P.
Assuming that this second firm has the same costs as the first, write the profits of each firm as functions of Q_{1} and Q_{2}.
When the second firm enters, price can be written as a function of the output of two firms: P = 53  Q_{1}  Q_{2}. We may write the profit functions for the two firms:
or
and
or
c. Suppose (as in the Cournot model) each firm chooses its profitmaximizing level of output under the assumption that its competitor’s output is fixed. Find each firm’s "reaction curve" (i.e., the rule that gives its desired output in terms of its competitor’s 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 Q_{1} to maximize p _{1} in b with Q_{2} being treated as a constant. The change in p _{1} with respect to a change in Q_{1} 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 Q_{1} and Q_{2} for which both firms are doing as well as they can given their competitors’ 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 Q_{1} and Q_{2} that satisfy both reaction functions by substituting the reaction function for Firm 2 into the one for Firm 1:
By symmetry, Q_{2} = 16.
To determine the price, substitute Q_{1} and Q_{2} into the demand equation:
P = 53  16  16 = $21.
Profits are given by
p _{i} = PQ_{i}  C(Q_{i}) = 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 i’th firm are given by
,
Differentiating to obtain the necessary firstorder condition for profit maximization,
.
Solving for Q_{i},
.
If all firms face the same costs, they will all produce the same level of output, i.e.,
Q_{i} = 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
Q_{1} + Q_{2} = 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, Q_{1}, to maximize its profits, subject to the reaction function of Firm 2:
max p _{1} = PQ_{1}  C(Q_{1}),
subject to
Substitute for Q_{2} in the demand function and, after solving for P, substitute for P in the profit function:
To determine the profitmaximizing quantity, we find the change in the profit function with respect to a change in Q_{1}:
Set this expression equal to 0 to determine the profitmaximizing quantity:
53  2Q_{1}  24 + Q_{1}  5 = 0, or Q_{1} = 24.
Substituting Q_{1} = 24 into Firm 2’s reaction function gives Q_{2}:
Substitute Q_{1} and Q_{2} 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 Q_{1} and Q_{2} simultaneously and face the demand curve
P = 30  Q,
where Q = Q_{1} + Q_{2}. Until recently, both firms had zero marginal costs. Recent environmental regulations have increased Firm 2’s marginal cost to $15. Firm 1’s 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 1’s revenue is
Pq_{1} = (30  q_{1} q_{2}) q_{1} = 30q_{1}   q_{1}q_{2}
and its marginal revenue is given by:
MR_{1} = 30  2q_{1}  q_{2}
Profit maximization implies MR_{1} = MC_{1} or
30  2q_{1}  q_{2} = 0 Þ q_{1} = 15  (1/2) q_{2}
which is firm 1’s best response function.
Firm 2’s revenue is
P q_{2} = (30  q_{1}  q_{2}) q_{2}
= 30q_{2}  q_{1}q_{2}  ,
and its marginal revenue is given by:
MR_{2} = 30  q_{1}  2q_{2}
Profit maximization implies MR_{2} = MC_{2}, or
30  q_{1}  2q_{2} = 15 Þ q_{2} = (15/2)  (1/2) q_{1}
which is firm 2’s best response function.
Cournot equilibrium occurs at the intersection of best response functions. Substituting for q_{1} in the response function for firm 2 yields:
q_{2} = (15/2)  (1/2) [ 15  (1/2) q_{2} ],
Thus
q_{2} = 0, q_{1} = 15, and
P = 30  q_{1} + q_{2} = 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 C_{1} = 30Q_{1} and C_{2} = 30Q_{2}, where Q_{1} is the output of Firm 1 and Q_{2} is the output of Firm 2. Price is determined by the following demand curve:
P = 150  Q
where Q = Q_{1} + Q_{2}.
a. Find the CournotNash equilibrium. Calculate the profit of each firm at this equilibrium.
To determine the CournotNash equilibrium, we first calculate the reaction function for each firm, then solve for price, quantity, and profit. Profit for Firm 1, TR_{1}  TC_{1}, is equal to
Therefore,
Setting this equal to zero and solving for Q_{1} in terms of Q_{2}:
Q_{1} = 60  0.5Q_{2}.
This is Firm 1’s reaction function. Because Firm 2 has the same cost structure, Firm 2’s reaction function is
Q_{2} = 60  0.5Q_{1} .
Substituting for Q_{2} in the reaction function for Firm 1, and solving for Q_{1}, we find
Q_{1} = 60  (0.5)(60  0.5Q_{1}), or Q_{1} = 40.
By symmetry, Q_{2} = 40.
Substituting Q_{1} and Q_{2} 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 CournotNash equilibrium.
b. Suppose the two firms form a cartel to maximize joint profits. How many widgets will be produced? Calculate each firm’s profit.
Because marginal cost is the same for both firms and is constant for all output, we may determine the joint profitmaximizing output by considering only one firm, i.e., let
Q_{1} = Q and Q_{2} = 0.
Profit is
p = 150Q  Q^{2}  30Q.
Therefore,
Solving for the profitmaximizing 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 Q_{1} = Q_{2} = 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 1’s profit differ from that found in part (b) above?
If Firm 1 were the only firm, it would solve the profitmaximization problem as in 6.b, i.e., Q_{1} = 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 firm’s profits?
Assuming their agreement is to split the market equally, Firm 1 produces 30 widgets. Firm 2 cheats by producing its profitmaximizing level, given Q_{1} = 30.Substituting Q_{1} = 30 into Firm 2’s reaction function:
Total industry output, Q_{T}, is equal to Q_{1} plus Q_{2}:
Q_{T} = 30 + 45 = 75.
Substituting Q_{T} into the demand equation to determine price:
P = 150  75 = $75.
Substituting Q_{1}, Q_{2}, 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 (CournotNash) equilibrium for each firm, assuming that each chooses the output level that maximizes its profits taking its rival’s output as given. What are the profits of each firm?
To determine the CournotNash 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  Q_{1}  Q_{2})Q_{1}  40Q_{1}, or
The change in p _{1} with respect to Q_{1} is
Setting the derivative to zero and solving for Q_{1} in terms of Q_{2} will give Texas Air’s reaction function:
Q_{1} = 30  0.5Q_{2}.
Because American has the same cost structure, American’s reaction function is
Q_{2} = 30  0.5Q_{1}.
Substituting for Q_{2} in the reaction function for Texas Air,
Q_{1} = 30  0.5(30  0.5Q_{1}) = 20.
By symmetry, Q_{2} = 20. Industry output, Q_{T}, is Q_{1} plus Q_{2}, or
Q_{T} = 20 + 20 = 40.
Substituting industry output into the demand equation, we find P = 60. Substituting Q_{1}, Q_{2}, and P into the profit function, we find
p _{1} = p _{2} = (60(20) 20^{2}  (20)(20) = $400
for both firms in CournotNash 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 Q_{1} is
Set the derivative to zero, and solving for Q_{1} in terms of Q_{2},
Q_{1} = 37.5  0.5Q_{2}.
This is Texas Air’s reaction function. Since American has the same cost structure as in 6.a., American’s reaction function is the same as before:
Q_{2} = 30  0.5Q_{1}.
To determine Q_{1}, substitute for Q_{2} in the reaction function for Texas Air and solve for Q_{1}:
Q_{1} = 37.5  (0.5)(30  0.5Q_{1}) = 30.
Texas Air finds it profitable to increase output in response to a decline in its cost structure.
To determine Q_{2}, substitute for Q_{1} in the reaction function for American:
Q_{2} = 30  (0.5)(37.5  0.5Q_{2}) = 15.
American has cut back slightly in its output in response to the increase in output by Texas Air.
Total quantity, Q_{T}, is Q_{1} + Q_{2}, or
Q_{T} = 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 American’s actions?
Recall that profits for both firms were $400 under the original cost structure. With constant average and marginal costs of 25, Texas Air’s 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 Air’s average cost is 25. First, without investment, American’s profits would be:
(55)(15)  (40)(15) = $225.
Second, with investment by both firms, the reaction functions would be:
Q_{1} = 37.5  0.5Q_{2 }and Q_{2} = 37.5  0.5Q_{1}.
To determine Q_{1}, substitute for Q_{2} in the first reaction function and solve for Q_{1}:
Q_{1} = 37.5  (0.5)(37.5  0.5Q_{1}) = 25.
Substituting for Q_{1} in the second reaction function to find Q_{2}:
Q_{2} = 37.5  0.5(37.5  0.5Q_{2}) = 25.
Substituting industry output into the demand equation to determine price:
P = 100  50 = $50.
Therefore, American’s profits if Q_{1} = 30 and Q_{2} = 15 are
p _{2} = (100  30  15)(15)  (40)(15) = $225.
American’s profits if Q_{1} = Q_{2} = 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 costsaving 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 = Q_{E} + Q_{D}.
a. Unable to recognize the potential for collusion, the two firms act as shortrun perfect competitors. What are the equilibrium values of Q_{E}, Q_{D}, and P? What are each firm’s 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 Q_{E}, Q_{D}, and P? What are each firm’s profits?
To determine the CournotNash equilibrium, we first calculate the reaction function for each firm, then solve for price, quantity, and profit. Profits for Everglow are equal to TR_{E}  TC_{E}, or
The change in profit with respect to Q_{E} is
To determine Everglow’s reaction function, set the change in profits with respect to Q_{E} equal to 0 and solve for Q_{E}:
90  3Q_{E}  Q_{D} = 0, or
Because Dimlit has the same cost structure, Dimlit’s reaction function is
Substituting for Q_{D} in the reaction function for Everglow, and solving for Q_{E}:
By symmetry, Q_{D} = 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)(30^{2 })) = $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 Q_{E}, Q_{D}, and P? What are each firm’s profits?
Recall Everglow’s profit function:
If Everglow sets its quantity first, knowing Dimlit’s reaction function , we may determine Everglow’s reaction function by substituting for Q_{D} in its profit function. We find
.
To determine the profitmaximizing quantity, differentiate profit with respect to Q_{E}, set the derivative to zero and solve for Q_{E}:
Substituting this into Dimlit’s 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 Q_{E}, Q_{D}, and P? What are each firm’s profits?
If the firms split the market equally, total cost in the industry is ; therefore, . Total revenue is 100Q  Q^{2}; therefore, MR = 100  2Q. To determine the profitmaximizing quantity, set MR = MC and solve for Q_{T}:
This means Q_{E} = Q_{D} = 15.
Substituting Q_{T} 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 + q^{2}
The market demand for these seat covers is represented by the inverse demand equation:
P = 200  2Q,
where Q = q_{1} + q_{2} , total output.
a. If each firm acts to maximize its profits, taking its rival’s 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 firm’s cost function C(q) = 20q + q^{2} and the market demand function P = 200  2Q where total output Q is the sum of each firm’s output q_{1} and q_{2.}
We find the best response functions for both firms:
WW’s revenue is R_{1} = P q_{1} = (200  2(q_{1} + q_{2})) q_{1} = 200q_{1}  2q_{1}^{2}  2q_{1}q_{2}.
Its marginal revenue and cost functions are:
MR_{1} = 200  4q_{1}  2q_{2}
MC_{1} = 20 + 2q_{1}
Profit maximization implies:
MR_{1} = MC_{1} or 200  4q_{1}  2q_{2} = 20 + 2q_{1} which yields the best response function:
q_{1} = 30  (1/3)q_{2}.
By symmetry, BBBS’s best response function will be:
q_{2} = 30  (1/3)q_{1}.
Cournot equilibrium occurs at the intersection of these two best response functions, given by:
q_{1} = q_{2} = 22.5.
Thus,
Q = q_{1} + q_{2} = 45
P = 200  2(45) = $110.
Profit for both firms will be equal and given by:
R  C = (110) (22.5)  (20(22.5) + 22.5^{2}) = $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 profitmaximizing 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 (2002Q)Q  2(20(Q/2) + (Q/2)^{2}) = 180Q  2.5Q^{2} and will be maximized at Q = 36.
Thus, we will have q_{1} = q_{2} = 36 / 2 = 18 and P = 200  2(36) = $128
Profit for each firm will be 18(128)  (20(18) + 18^{2}) = $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 = q_{1} + q_{2} = 22.5 + 18 = 40.5
P = 200 2(40.5) = $119.
Profit for WW = 22.5(119)  (20(22.5) + 22.5^{2}) = $1721.25.
Profit for BBBS = 18(119)  (20(18) + 18^{2}) = $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 
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 q_{2} which will be its best response to q_{1 }or:
q_{2} = 30  (1/3)q_{1}
WW profits will be:
p _{1} = Pq_{1}  C_{1} = (200  2q_{1}  2q_{2})q_{1}  20q_{1}  = (200 2q_{1}  2(30  (1/3)q_{1}))q_{1}  20q_{1}  q_{1}^{2 }= 21.4
= 120q_{1}  (7/3)
Profit maximization implies:
dp _{1}/ dq_{1} = 120  (14/3)q_{1} = 0 or q_{1} = 25.7 and q_{2} = 30  (1/3)(25.7) = 21.4
The equilibrium price and profits will then be:
P = 200  2(q_{1} + q_{2}) = 200  2(25.7 + 21.4) = $105.80
p _{1} = (105.80) (25.7)  (20) (25.7)  25.7^{2} = $1544.57
p _{2} = (105.80) (21.4)  (20) (21.4)  21.4^{2} = $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 1’s profits.
*10. Two firms compete by choosing price. Their demand functions are Q_{1} = 20  P_{1} + P_{2} and Q_{2} = 20 + P_{1}  P_{2}
where P_{1} and P_{2} are the prices charged by each firm respectively and Q_{1} and Q_{2} 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):
MR_{1} = 20  2P_{1} + P_{2}.
At the profitmaximizing price, MR_{1} = 0. Therefore,
This is Firm 1’s reaction function. Because Firm 2 is symmetric to Firm 1, its reaction function is
Substituting Firm 2’s reaction function into that of Firm 1:
By symmetry, P_{2} = $20.
To determine the quantity produced by each firm, substitute P_{1} and P_{2} into the demand functions:
Q_{1} = 20  20 + 20 = 20 and Q_{2} = 20 + 20  20 = 20.
Profits for Firm 1 are P_{1}Q_{1} = $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 2’s reaction function into account. Firm 1’s profit function is:
To determine the profitmaximizing price, find the change in profit with respect to a change in price:
Set this expression equal to zero to find the profitmaximizing price:
30  P_{1} = 0, or P_{1} = $30.
Substitute P_{1} in Firm 2’s reaction function to find P_{2}:
At these prices,
Q_{1} = 20  30 + 25 = 15 and Q_{2} = 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 noncartel 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)P^{1/2}.
Note that OPEC’s net demand is D = W  S.
a. Sketch the world demand curve W, the nonOPEC supply curve S, OPEC’s net demand curve D, and OPEC’s marginal revenue curve. For purposes of approximation, assume OPEC’s production cost is zero. Indicate OPEC’s optimal price, OPEC’s optimal production, and nonOPEC production on the diagram. Now, show on the diagram how the various curves will shift, and how OPEC’s optimal price will change if nonOPEC supply becomes more expensive because reserves of oil start running out.
OPEC’s net demand curve, D, is:
OPEC’s marginal revenue curve starts from the same point on the vertical axis as its net demand curve and is twice as steep. OPEC’s optimal production occurs where MR = 0 (since production cost is assumed to be zero), and OPEC’s optimal price in Figure 12.11.a.i is found from the net demand curve at Q_{OPEC}. NonOPEC production can be read off of the nonOPEC supply curve at a price of P*.
Figure 12.11.a.i
Next, suppose nonOPEC oil becomes more expensive. Then the supply curve S shifts to S*. This changes OPEC’s 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, nonOPEC production is _{.}. Notice that the curves must be drawn accurately to give this result.
Figure 12.11.a.ii
b. Calculate OPEC’s optimal (profitmaximizing) price. (Hint: Because OPEC’s 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 profitmaximizing 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 oilconsuming countries were to unite and form a "buyers’ cartel" to gain monopsony power. What can we say, and what can’t we say, about the impact this would have on price?
If the oilconsuming countries unite to form a buyers’ cartel, then we have a monopoly (OPEC) facing a monopsony (the buyers’ cartel). As a result, there is no welldefined 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 lemongrowing 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