7/31/2019 14weeksol

1/8

ECON 301 - WEE K 14 APRIL 22, 2011

1 Cournot Model with 2 firms (Duopoly)

As is well-known, the mobile industry in US is governed by the two major companies - Verizon wireless(v,

henceforth) and AT&T(a). Assume that it costs $10 to provide a person with wireless service for eachcompany. More precisely, their identical total cost function is given by

TC(yv) = 10yv and TC(ya) = 10ya

For simple analysis, we assume that people cannot distinguish the quality of their service. The inversedemand function for wireless service is given by

p(y) = 50y.

(a) Find analytically the best response functions for Verizon and AT&T, and plot them on the graph.

ANSWER Let y = yv + ya denote the number of total consumers in the industry. From Verizonsstance, in order to maximize its profits(denoted by v(yv), it must hold that its marginal revenuemust be equal to the marginal cost; that is, MRv = MCv. From the total cost function, MCv = 10 is

immediate. To figure out its marginal revenue function, we begin with its total revenue function;

TRv(yv) = p yv = (50yv ya) yv = 50yv y2v yayv

Differentiating with respect to yv gives us the marginal revenue:

TRv(yv)

yv= 50 2yv ya

According to the equilibrium condition(MRv = MCv), set up the equation as follows:

50 2yv ya = 10

Solving for yv gives us Verizons best response to ya:

yv = 20 12ya. (1)

As AT&T faces the same environment, its best response to yv must be

ya = 201

2yv. (2)

Refer to Figure 1 for their graphs; the blue line depicts the Verizons best response ( 1) and the red linethe AT&Ts (2). And their intersection determines Cournot-Nash equilibrium as we will see in theensuing part.

(b) Derive Cournot-Nash equilibrium. And find the profit margins of each individual company at theequilibrium.

ANSWER The Cournot-Nash equilibrium is determined by the intersection of the two best responsefunctions we obtain in part (a). Substituting (2) into (1) gives us

yv = 201

2

20

1

2yv

Solving for yv brings us the equilibrium quantity(number of Verizon users), yv =403 . By symmetry,

ya =403 . Therefore, the Cournot-Nash equilibrium is represented by (ya,yv) =

40

3,

40

3

To compute their accounting profits in the equilibrium,

1

7/31/2019 14weeksol

2/8

0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

30

35

40

45

ya

(AT&T)

yv

(Ver

izon)

CournotNash Equilibrium(40/3,40/3)

Cartel Equilibrium (10,10)

Figure 1: Cournot-Nash Equilibrium

1 Compute the corresponding market price.

When ya = yv = 403 , the total users in the market amount to 803 . Hence its price comes from themarket demand,

p(y) = 50y = 5080

3=

70

3

2 Set up the profit function.

Since the profit is measured by the difference between the firms total revenue and total cost,Verizons profit can be written by

v(yv) = p yv TC(yv)

=70

3

40

3 10

40

3

= 28009 400

3 =1600

9

The AT&Ts profits must be the same due to the symmetry.

(c) What is the deadweight loss(DWL)?

ANSWER To see the associated deadweight loss, refer to the diagram in Figure 2. Since we know thex,y-coordinate of each vertex of the deadweight loss triangle, it is not hard to see that

DWL =40

3

40

3

1

2=

800

9

(d) Suppose that the two firms form a cartel. What is the aggregatey = yv +ya and profit? Does collusion

benefit the two companies? Find the deadweight loss, and compare it to the one in part (c).

ANSWER A cartel is an explicit contract between two firms. It is a formal organization of the firmsthat agree to fix production in this context. Let y denote the total quantities of the two firms aspromised by the cartel. Then the marginal revenue generated from y becomes

MRc = 50 2y.

Since it should equal the marginal cost,

50 2y = 10.

2

7/31/2019 14weeksol

3/8

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

55

Equilibrium Price, Equilibrium Quantity(70/3,80/3)

DeadweightLoss

Figure 2: Deadweight Loss in Cournot-Nash

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

55

Equilibrium Price, Equilibrium Quantity(30,20)

DeadweightLoss

Figure 3: Deadweight Loss in Cartel

Hence y = 20. The corresponding equilibrium price is p = 30. Their aggregate profit is, therefore,

(y) = 30 20 20 10 = 400.

We assume that the total profit, 400, is divided equally between the two companies, that is, 200 forVerizon and 200 for AT&T. Observe that each profit is greater than what we obtained in part (b).Therefore, the collusion is beneficial.

(e) Is the cartel sustainable if they compete for a short period?

ANSWER The cartel above could easily break down in a short term relationship. Observe that thecartel equilibrium, marked by the purple point in Figure 1, is neither on Verizons nor AT&Ts bestresponse. It implies that if AT&T produces 10 as they promised Verizon is willing to deviate asdepicted by the green arrow. Initially, they promised that each firm produces 10 units. From Verizonspoint of view, provided that AT&T produces 10 units, he wants to increase their production to 15units, which is the best response. So if they play one-shot game and never see each other in the

future, the cartel we described in part (d) cannot be supported as an equilibrium. However, if theyplay a repeated game(compete each other for a long time), the cartel can constitute an equilibrium ifthe valuation of their sustained relationship is sufficiently large. (It is beyond our class level.)

2 Cournot Model with Nfirms

Consider an industry with Nfirms that face the inverse demand, p(y) = 10 y, and suppose that eachfirms total cost function is TC(y) = 2y. (No Fixed Cost)

(a) Find the individual and aggregate production and the price in Cournot-Nash equilibrium.

ANSWER The Cournot-Nash equilibrium with Nfirms can be obtained in the following two ways:

1 Let me illustrative its formal way. Let yi denote firm is output level. Since there exist Nfirms inthe industry, we shall write their total outputs as

N

i=1

yi using the summation notation. Note that

the price associated with the total output yi is p = 10yi.We begin by firm is total revenue function. The underlying idea is the same as before: theoptimal quantity for firm i is determined at which its marginal revenue is equal to the marginalcost, that is 2. The total revenue function is

TR i(yi) = p yi =

10

N

i=1

yi

yi

3

7/31/2019 14weeksol

4/8

We differentiate it with respect to yi to obtain the marginal revenue:

TR i(yi)

yi= 10yi

N

i=1

yi,

which must be the marginal cost. Therefore, we come up with

10yi N

i=1

yi = 2 i.e. 8yi N

i=1

yi = 0.

This is firm is best response function. He wants to produce yi that solves out the equation above.Notice that the optimal choice for yi depends on the other firms decisions. The way to solve it isto use the symmetric argument just as we did in problem 1. By symmetry, firm 1s best responseequation is going to be

8y1 N

i=1

yi = 0.

Firm 2s is

8y2 N

i=1

yi = 0. (3)

Following this logic and recalling that there exist Nfirms, we encounter Nbest response equa-tions as follows:

8y1 N

i=1

yi = 0

8y2 N

i=1

yi = 0

8y3 N

i=1

yi = 0

...

8yNN

i=1

yi = 0.

For simple exposition, let Y = Ni=1 yi. Then summing these N equations delivers one simplefollowing equation

8N Y NY = 0.

Solving for Y gives us Y = 8NN+1

. Since each firm will produce the same output, it is straightfor-ward that each firms production is

yi =8

N+ 1

for all i. Again the equilibrium price follows from the market demand curve by substituting intothe aggregate quantity, i.e. Y;

p = 108N

N+ 1=

2N+ 10

N+ 1

4

7/31/2019 14weeksol

5/8

2 The easier(but informal) way is to apply the symmetry at the initial stage. Let y = y1 = y2 = = yN denote the symmetric Cournot-Nash equilibrium production for every firm. Supposethat the other firms produce y and firm i produces yi. Then the corresponding equilibrium pricecan be written by

p = 10 (N 1)yyi

Hence firm is total revenue is

TR i(yi) = p yi = {10 (N 1)yyi}yi

Differentiating with respect to yi gives us its marginal revenue

MRi(yi) = 10 (N 1)y 2yi

Again it should be equal to the marginal cost. Therefore, we have

10 (N 1)y 2yi = 2

Since we know that firm i also produces y, the equation boils down to

10 (N 1)y 2y = 2

Solving for y gives us the desired answer; y = 8N+1 , that is, in the Cournot-Nash equilibrium,

each firm would produce 8N+1 .

(b) In the graph with Nfor the horizontal axis and p for the vertical axis, plot the equilibrium price. Whatcan you say about the price as Ngoes to infinity?

ANSWER Observe that the price, 2N+10N+1 , goes to 2 in the limit as N . It is worthwhile to notethat 2 is the marginal cost in this example as well as the market price in perfectly competitive market.Intuitively, as more firms enter the industry, the market price would drop to the marginal cost level,and thus the market becomes more efficient. See Figure 4.

0 5 10 15 200

1

2

3

4

5

6

7

8

9

10

Number of Firms(N)

EquilibriumP

rice(P)

Marginal Cost = 2

Figure 4: Equilibrium price converges to marginal cost as N .

(c) In the graph with Nfor the horizontal axis and y for the vertical axis, plot the equilibrium aggregateproduction level. What can you say about the aggregate production level as Ngoes to infinity?

ANSWER The aggregate production, 8NN+1 , is gradually increasing over Nand it converges to 8 in thelimit. See Figure 5.

5

7/31/2019 14weeksol

6/8

0 5 10 15 200

1

2

3

4

5

6

7

8

9

10

Number of Firms(N)

EquilibriumQ

uantities(y)

Figure 5: Equilibrium quantity converges to 8 as N .

(d) In the graph with Nfor the horizontal axis and DWL for the vertical axis, plot the equilibrium DWL.

What can you say about the DWL as Ngoes to infinity?

ANSWER Drawing a diagram is always helpful in figuring out deadweight loss. See Figure 6. Theblue line is the market demand curve, p(y) = 10 y, and the red (dotted) line is the firms constant

marginal cost curve. And the equilibrium quantity and price,(

8NN+1 ,

2N+1N+1

)is labeled point A. The

deadweight triangle is captured by ABC in the figure. Hence the area ofABC is

DWL =8

N+ 1

8

N+ 1

1

2=

32

(N+ 1)2

As Ngoes to infinity, the deadweight loss goes to 0. Therefore the market is Pareto efficient in thelimit since the deadweight loss goes away.

0 10

0

2

10

Quantity (y)

EquilibriumP

rice(P)

CB

A

Deadweight Loss

Figure 6: Deadweight Loss Triangle

0 5 10 15 20

0

1

2

3

4

5

6

7

8

9

10

Number of Firms(N)

DeadweightLoss

Figure 7: Deadweight Loss converges to 0

6

7/31/2019 14weeksol

7/8

3 Production Externalities

Firm S produces steel(s), and also produces a certain amount of pollution(x) which it dumps into a river.Firm F, a fishery, is located downstream and is adversely affected by Ss pollution. Suppose that the steelfirms cost function is given by

TCs(s, x) =1

4 s2 2x + x2,

and the fisherys cost function is

TCF(f, x) =1

2f2 + x f.

The price of steel and fish is ps = 1 and pf = 2, respectively.

(a) Find the amount of steel s and the amount of pollution x that maximizes the firms profit.

ANSWER x = 1 is immediate from the cost function. At x = 1, the steel firms cost would beminimized. To find the optimal s, observe that the marginal revenue is 1 but the marginal cost is 12 s.The optimal s will be 2 because the marginal revenue is equal to the marginal cost at s = 2.

(b) Given the amount of pollution x, find the optimal level of production of fish f, and find the fisherysprofit.

ANSWER Given x, the marginal cost function boils down to

TCFf

= f+ x

Setting it to be equal to marginal revenue 2,

f+ x = 2 f = 2 x.

Observe that the amount of fish fdecreases in x. Therefore, the pollution x has negative external effecton production of the fishery.

(c) Express the joint profit function.

ANSWER The profit function of the steel firm is

s = pss TCs = s1

4s2 + 2x x2 (-3)

And the profit function of the fishery is

f = pf f TCf = 2f1

2f2 x f (-2)

The joint profit function is obtained by simply adding the two profit functions above:

= s + f =s 14 s

2 + 2x x2

+

2f 12 f2 x f

(-1)

(d) Find the Pareto efficient level of production (s, x, f).

ANSWER The Pareto efficient production indicates (s, x, f) that maximizes the joint profit function inthe sense that it is the most socially desirable. To figure it out, we (partially) differentiate we got inpart (c) with respect to s, x, and f. It gives us

s= 1

1

2s = 0.

7

7/31/2019 14weeksol

8/8

x= 2 2x f = 0.

f= 2 f x = 0.

The Pareto efficient s immediately comes from the first equation; s = 2. And the Pareto efficient x and

f come from the last two equations: f = 2 and x = 0.

8