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Undergraduate Industrial Organization Solution to Practice Questions Keigo Makino Yuta Toyama Last Updated: January 5, 2020 1 Monopoly 1.1 Monopoly with CES Demand Function Consider a monopolist in the market whose marginal cost is constant and denoted by c. The market demand is given by Q = AP -γ , where γ> 0 (a) Calculate the price elasticity. (Answer) The definition of the price elasticity is ε dQ dP × P Q . With CES demand function, ε = -γAP -(γ+1) × P AP -γ = -γ. (b) Solve the monopolist’s problem. What is the optimal price? [Hint: Consider the two cases when γ> 1 and γ (0, 1] separately. (Answer) The profit of the monopolist is π(P )= P · AP -γ - cAP -γ = AP -γ+1 - cAP -γ . 1. Suppose γ> 1. The FOC is dp =(-γ + 1)AP -γ + cγAP -γ-1 =0. The optimal price is P * = γ-1 . 2. Suppose γ 1. Then dp = AP -γ (-γ +1+ P ) is always positive where P> 0. To maximize π(P ), P should be infinitely large, which is not realistic. Remember that a monopolist always operates in a price region where |ε|≥ 1. See the lecture slide for details. 1.2 Monopoly with Carbon Tax Consider a steel market in which there exists a monopolist named Nippon Steel Company (hereafter NSC). The market demand for a steel is given by Q = 100 - 2P , where Q is the quantity of steel production and P is the market price of steel. The NSC produces steel with constant marginal cost mc = 20. Note that price and cost is measured in 1 USD and the production quantity is measured in a ton. 1

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Undergraduate Industrial Organization

Solution to Practice Questions

Keigo Makino Yuta Toyama

Last Updated: January 5, 2020

1 Monopoly

1.1 Monopoly with CES Demand Function

Consider a monopolist in the market whose marginal cost is constant and denoted by c. The market

demand is given by Q = AP−γ , where γ > 0

(a) Calculate the price elasticity.

(Answer) The definition of the price elasticity is ε ≡ dQdP ×

PQ . With CES demand function,

ε = −γAP−(γ+1) × P

AP−γ= −γ.

(b) Solve the monopolist’s problem. What is the optimal price? [Hint: Consider the two cases

when γ > 1 and γ ∈ (0, 1] separately.

(Answer) The profit of the monopolist is π(P ) = P ·AP−γ − cAP−γ = AP−γ+1 − cAP−γ.

1. Suppose γ > 1. The FOC is dπdp = (−γ + 1)AP−γ + cγAP−γ−1 = 0. The optimal price is

P ∗ = cγγ−1 .

2. Suppose γ ≤ 1. Then dπdp = AP−γ(−γ + 1 + cγ

P ) is always positive where P > 0. To maximize

π(P ), P should be infinitely large, which is not realistic.

Remember that a monopolist always operates in a price region where |ε| ≥ 1. See the lecture slide

for details.

1.2 Monopoly with Carbon Tax

Consider a steel market in which there exists a monopolist named Nippon Steel Company (hereafter

NSC). The market demand for a steel is given by Q = 100 − 2P , where Q is the quantity of steel

production and P is the market price of steel. The NSC produces steel with constant marginal cost

mc = 20. Note that price and cost is measured in 1 USD and the production quantity is measured

in a ton.

1

(a) Solve the profit maximization problem for the NSC.

(Answer) The maximization problem for the NSC is

maxP,Q

PQ−mc ·Q = P (100− 2P )− 20(100− 2P ).

The FOC is 100− 4P + 40 = 0. The optimal price and quantity are P ∗ = 35 and Q∗ = 30.

Steel production is associated with emissions of CO2, which has a detrimental effect on envi-

ronment. Denote the total external cost from producing CO2 as E. Assume that one unit of steel

production leads to an environmental damage equivalent to 2 USD.

(b) Calculate consumer surplus CS, producer surplus PS, and the total external cost E.

(Answer) PS = 35 · 30− 20 · 30 = 450. E = 2 · 30 = 60. CS = 30 · (50− 35) · 12 = 225.

(c) The government now introduces a carbon tax to reduce CO2 emissions. Denote the carbon

tax by τ . Under carbon tax regime, the NSC has to pay τ for each unit of production. Let

τ = 2, so that the carbon tax is exactly same as the social cost of steel production. Solve the

profit maximization problem in this case.

(Answer)The maximization problem for the NSC is

maxP,Q

PQ− (mc+ τ) ·Q = P (100− 2P )− (20 + 2)(100− 2P ).

The FOC is 100− 4P + 44 = 0. The optimal price and quantity are P ∗ = 36 and Q∗ = 28.

(d) Calculate consumer surplus CS, producer surplus PS, the total external cost E, and the tax

revenue T = τq∗ where q∗ is the production quantity.

(Answer) PS = 36 ·28−22 ·28 = 392. E = 2 ·28 = 56. CS = 28 ·(50−36) · 12 = 196. T = 2 ·28 = 56.

(e) Define the total welfare by CS+TS+T −E. Can we achieve higher welfare under the carbon

tax? If not, discuss why.

(Answer) No. The welfare in (b) is PS+CS−E = 615. The welfare in (d) is PS+CS+T−E = 588,

which is lower than the welfare in (b). Since the monopolist already produces less than the socially-

optimal level, the loss of consumer surplus and producer surplus might overwhelm the gains from

pollution mitigation. See Buchanan (1969) for theoretical argument and Fowlie, Reguant, and Ryan

(2014) for empirical studies.1

1James Buchanan (1969) “External Diseconomies, Corrective Taxes, and Market Structure,“ American EconomicReview. Fowlie, Meredith, Mar Reguant, and Stephen P. Ryan. "Market-based emissions regulation and industrydynamics." Journal of Political Economy 124.1 (2016): 249-302.

2

2 Price Discrimination

2.1 Price Discrimination in General

(a) Write down an example of 2nd (self-selection) and 3rd (market segmentation) degree price

discrimination. Explain why these examples are 2nd or 3rd degree price discrimination.

(Answer) Skipped. See the lecture slide.

(b) To make the price discrimination successful, firms need to prohibit resale of products. Using

the example, discuss why firms have to prohibit (or hinder) resale.

(Answer) Skipped. See the lecture slide.

2.2 2nd Degree Price Discrimination

As a marketing manager of a software company, you have to set the price of a new product. The

market is divided into two equally sized segments: Professional users willing to pay $500 and non-

professional users willing to pay $200 for the full version of the software. The production cost of

the soft ware is zero.

(Notice) I did not make explicit this above, but the most important assumption in this question

is that the seller cannot distinguish who is professional or non-professional. If the seller can dis-

tinguish them, the seller can offer a separate price menu for each consumer type, which is the case

of the third-degree price discrimination (discrimination by indicator). Here, we consider the case

of the second-degree price discrimination (discrimination by self-selection).

(a) Suppose there is a scaled down version of the product, which is worth $100 to non-professionals

but worthless to professionals. Assume the production cost of this version is also zero. What

is the optimal price of each version?

(Answer)

# of consumers Full (product 1) Scale-down (product 2) Intermediate (product 3)

Professionals 1 wp1 = 500 wP2 = 0 wp3 = 250

Non-professional 1 wn1 = 200 wn2 = 100 wn3 = 150

Let pjbe the price of product j. The consumer chooses product j that gives the highest net-utility

given by wi,j − pj for i = p and n. As explained in the lecture, p1 = 500 and p2 = 100. The profit

is 500 + 100 = 600.

(b) The company also has an intermediate version of the product, for which professionals are

willing to pay $250 and non-professional $150. Again, assume that the production cost is 0.

Which versions of the product should the firm sell to maximize profits?

(Answer) Consider the case when a firm sells both product 1 and 3. In this case the price is

p1 = 400 and p3 = 150. The seller needs to lower the price for product 1 so that professionals

3

prefer to buy product 1 rather than product 3. The keywords are incentive compatibility constraint

and participation constraint. See the lecture slide 4 for the details. The profit is 450 + 100 = 550.

Therefore, the firm prefers to sell product 1 and 2 rather than product 1 and 3.

2.3 3rd Degree Price Discrimination

A market consists of two population segments, A and B. An individual in segment A has demand

for your product q = 50−p. An individual in segment B has demand for your product q = 120−2p.

Segment A has 1000 people in it. Segment B has 1200 people in it. Total cost of producing q units

is C = 20q.

(a) What is total market demand for your product?

(Answer) Total market demand is

D(p) =

1000 · (50− p) + 1200 · (120− 2p) = 194000− 3400p if p < 50

1200 · (120− 2p) if p ∈ [50, 60]

0 if p > 60

(b) Assume that you must charge the same price to both segments. What is the profit-maximizing

price? What are your profits?

(Answer) If p < 50, the profit is π(p) = (194000− 3400p)(p− 20). From the FOC is −400(−655 +

17p) = 0. The optimal price under p < 50 is 65517 ≈ 38.5 and the profit is 19845000

17 ≈ 1167352.94.

If p ∈ [50, 60], the profit is π(p) = 1200 · (120 − 2p)(p − 20). The optimal price under p ∈ [50, 60]

is p = 50 and the profit is 720000. If p > 60, the profit is always zero. Thus the optimal price is

p∗ = 65517 and the maximized profit is π(p∗)1984500017 ≈ 1167352.94.

(c) Imagine now that members of segment A all wear a scarlet “A” on their shirts or blouses and

that you can legally charge different prices to these people. What price do you charge to the

scarlet “A” people? What price do you charge to those without the scarlet “A”?

(Answer) Let the former price be pA and the latter one be pB. The profit maximization problem is

maxpA,pB

1000 · (50− pA)(pA − 20) + 1200 · (120− 2pB)(pB − 20).

The FOCs are 70− 2pA = 0 and 160− 4pB = 0. The optimal pricing is p∗A = 35 and p∗B = 40. The

profit is 1185000. The profit becomes higher if you can use the third degree price discrimination.

3 Oligopoly

4

3.1 Bertrand Competition

Two firms produce a specialized microchip (Trium X406) used in certain home appliances. Demand

is given by Q = 100 − 2p (Q in thousands of units, p in $). Whichever firm sets the lowest price

gets all of the demand, and there are no capacity constraints. Both producers have a marginal cost

of $30. A cost-reducing innovation would allow Firm 1 to decrease its cost down to $20. The cost

of the innovation is $520 per period.

(a) Should Firm 1 go ahead and acquire the innovation?

(Answer) No.

I implicitly assume that these firms divide the demand equally if two firms set the same price.

If firm 1 go ahead and acquire the innovation, the pricing of firm 1 is slightly below 302 and the

profit is close to (100 − 2 · 30)(30 − 20) − 520 = −120. If firm 1 does not acquire, the profit is12 · (100− 2 · 30)(30− 30) = 0.

(b) Macroeconomic conditions indicate that demand is expected to increase by up to 40% in the

future? Would this change your answer?

(Answer) Yes. Firm 1 should acquire the innovation.

Assume firms maximize their expected profit. If firm 1 go ahead and acquire the innovation, the

expected profit is close to 1.4 · (100 − 2 · 30)(30 − 20) − 520 = 40. If firm 1 does not acquire, the

expected profit is 12 · 1.4 · (100− 2 · 30)(30− 30) = 0.

3.2 Applied question in Cournot Competition

Suppose there are two producers of silver in the world: Mexico and Peru. In 2014, total production

was 280 (millions of ounces), whereas price was $18 per ounce. Marginal cost is constant at the

level of $10 per ounce (Mexico) and $12 per ounce (Peru). Finally, suppose that firms compete

according to the Cournot model (output levels are determined simultaneously and the equilibrium

price is such that total demand equals total supply.

(a) Assuming that inverse demand curve is linear, i.e., P = a− bQ. Determine the coefficients of

the demand curve (a, b) that are consistent with the observed data.

(Answer) In the lecture slide, I proved

q∗1 =a− 2c1 + c2

3band q∗2 =

a− 2c2 + c13b

Q∗ =2a− c1 − c2

3band P ∗ =

a+ c1 + c23

2See Problem 4.2 (a) for details.

5

From the observed data, we know that c1 = 10, c2 = 12, Q∗ = 280 and P ∗ = 18. Then, the system

of equation is

840b =2a− 22,

54 =a+ 22.

We get (a, b) = (32, 120). Note that q∗1, q

∗2 > 0.

(b) Suppose that a third country, China, discovers silver mines with a marginal cost of production

equal to Peru. What impact do you expect this will have on the price of silver? Explain it

by words (no mathematical anslysis needed).

(Answer) Entry of China decreases the price of silver due to promoting competition in the market.

You can formally show this by solving Cournot game with three firms, including Peru, Mexico, and

China.

3.3 Cournot in the desktop computer industry

In 2010, global sales of Windows-based desktop computers were 351 million units, whereas average

price was $605. The leading market shares were as follows: HP, 17.9%; Acer, 13.9%; Dell, 12%;

Lenovo, 10.9%; Asus, 5.4%; others, 39.9%. Suppose that the Cournot model provides a good

approximation of the industry’s behavior. Suppose moreover that inverse demand is given by

P = a − bQ; each firm has constant marginal cost; and that, from previous studies, the price

elasticity of market demand is estimated to be -0.5 (at the market equilibrium level).

(a) Based onc= the observable data, determine the coefficient b in the demand function.

(Answer) The price elasticity of market demand is ε ≡ dQdP ·

PQ = −1

b605351 = −0.5 using Q = a

b −Pb .

The estimation of b is b = 1210351 . a = P + bQ = 1815.

(b) Estimate each firms marginal cost and margin.

(Answer) From the lecture slide, we have the formula for markup, which is P−MCiP = si

|ε| = 2si. For

marginal cost, we have MCi = (1− 2si)P .

Firm HP Acer Dell Lenovo Asus others

Marginal Cost $388.41 $436.81 $459.8 $473.11 $539.66 $122.21

Markup 35.8% 27.8% 24% 21.8% 10.8% 79.8%

(c) [Hard] Suppose that Dell’s marginal cost decreases by 5%. Based on your answer to the previous

question, estimate the impact of this cost decrease on equilibrium price and on Dell’s market

share.

(Answer) Dell’s marginal cost becomes 436.81. From the lecture slide, Q∗D ≡na−

∑ci

(n+1)b = 6∗1815−2397.017∗ 1210

351

=

6∗1815∗351−2397.01∗3517∗1210 ≈ 351.953. P ∗D ≡ a − na−

∑ci

n+1 = a+∑ci

n+1 = 1815+2397.017 ≈ 601.716. The 5%

6

decrease in Dell’s cost leads to about 0.716 reduce in equilibrium price. Dell’s market share becomesP ∗D−MCD

P ∗D|εD| =

P ∗D−436.81P ∗D

· 3511210 ·

PDQD≈ 0.1359. Note that the elasticity is not 0.5 now. |εD| ≈ 0.496.

The 5% decrease in Dell’s cost leads to about 1.6 percentage point increase in Dell’s market share.

4 Collusion

4.1 Repeated Prisoner’s Dilemma

Consider the repeated game with the following payoff matrix.

Firm 2

C D

Firm 1C (πC , πC) (πL, πD)

D (πD, πL) (πN , πN )

Both firms have discount factor δ.

(a) Show that the trigger strategy is a subgame perfect Nash equilibrium if

δ >πD − πCπD − πN

(Answer) To avoid deviations, we need πc + δπc + δ2πc + · · · = πC1−δ ≥ πD + δπN

1−δ = πD + δπN +

δ2πN + · · · . From this inequality, you may prove the statement.

(b) The above inequality implies that collusion would be easier (i.e., the above condition is more

likely to hold) if the following three cases hold. Give an intuitive reason why these are the

case.

1. Deviation payoff πD is lower.

2. Cooperation payoff πC is higher.

3. πN is lower.

(Answer) When the deviation payoff πD is smaller, each firm has lower incentive to deviate from the

trigger strategy. If cooperation payoff is higher, firms prefer to maintain the cartel (i.e., follow the

trigger strategy) rather than deviating from the strategy. Finally, if πN is lower, the punishment

from the deviation is more harsh because firms continue to obtain this payoff forever once the

deviation occurs.

4.2 Collusion among Asymmetric Firms

Consider a market with 1 million consumers. Each consumer is willing to purchase one unit so

long as the price is less than or equal to u. Two firms simultaneously set prices in each period

and consumers purchase from the firm with the lowest price; if both firms set the same price, then

7

consumers are equally distributed between the two firms. Firm H has cost cH , firm L has cost cL,

where cL < cH < u. Firms interact over an indefinite number of periods and the discount factor is

given by δ.

(a) What is the Nash equilibrium of the one-shot game (that is, if firms were to set prices only

once)

(Answer) pH = pL > cH cannot be equilibrium as in the symmetric case. Firm H cannot charge

less than cH . Firm L will charge cH −ε where ε is small positive number to guarantee itself a profit

as close as possible to (cH − cL) × 1 million. ε should not be equal to 0, so there is no pure Nash

equilibrium in a strict sense. Let us ignore this technical issue throughout this practice questions3.

We define the equilibrium as the limit4.

In the Nash equilibrium, firm L charges the price slightly lower than cH and firm H charges the

price which is equal to cH .

(a) Suppose that firms engage in a collusive agreement, whereby they set p = u in each period

and revert to the one-shot equilibrium derived in part (a) if any firm ever deviates from

the collusive equilibrium. Determine the lowest value of the discount factor δ such that this

collusive equilibrium is stable.

(Answer) The discount factor δ is assumed to be common for two firms. For firm H, the following

equation should hold. The optimal deviation is letting its price be sightly lower than u in some

period.

1

2(u− cH) +

1

2δ(u− cH) +

1

2δ2(u− cH) + · · · = 1

2

u− cH1− δ

≥ (u− cH) + δ0 + · · · ,

which is equivalent to δ ≥ 12 . For firm L,

1

2(u− cL) +

1

2δ(u− cL) +

1

2δ2(u− cL) + · · · = 1

2

u− cL1− δ

≥ (u− cL) + δ(cH − cL) + δ2(cH − cL) · · · ,

which is equivalent to δ ≥ 12 ·

u−cLu−cH . From u−cL

u−cH > 1, the lowest value of δ is 12 ·

u−cLu−cH .

5 Horizontal Mergers

5.1 Mergers and the HHI index

Consider an industry with four competitors who simultaneously set output levels (Cournot model).

Market demand is given by Q = 50− 14 p. Firms 1 and 2 have a cost function given by C = 120+30 q.

Firms 3 and 4 have a cost function given by C = 80 + 50 q.

(a) Determine the equilibrium market shares of each firm. Determine the value of the concentration

index H.3See Blume, A. (2003) “Bertrand without Fudge” Economic Letters for the detail.4See Exercise 5.1 of Tirole, J. (1988) The Theory of Industrial Organization, MIT Press for details.

8

(Answer) In the lecture slide, we derived Cournot equilibrium with Ci(qi) = ciqi + Fi and p =

a − bQ. Here, a = 200 and b = 4. The market supply is Q∗ =na−

∑cj

(n+1)b = 32. For each firm,

qi =a−nci+

∑j 6=i cj

(n+1)b . We have q1 = q2 = 212 , q3 = q4 = 11

2 . Then, s1 = s2 = 2164 , s3 = s4 = 11

64 . The

index H is 2(2164

)2+ 2

(1164

)2= 1124

4096 = 2811024 ≈ 0.27441.

Suppose that firms 1 and 4 merge, forming Firm 1&4; and that the marginal cost of the newly

formed firm is equal to c1&4 = c1 = 30.

(b) Determine the new equilibrium market shares of each firm. Determine the new value of the

concentration index H.

(Answer) The market supply is Q∗ =na−

∑cj

(n+1)b = 49016 . We have q1&4 = q2 = 190

16 , q3 = 11016 . Then,

s1&4 = s2 = 1949 , s3 = 11

49 . The index H is 2(1949

)2+(1149

)2= 843

2401 ≈ 0.3511.

(c) Compare this value of H with what you would get by simply considering the initial market

shares. Justify the difference in value.

(Answer) H rises because the number of firms decreases and market share of each firm rises.

6 Market Structure (Entry/Exit)

6.1 Customs Union

Consider the industry for product X in country A. Demand is given by p = a−Q/SA, where p is

price, Q total output, and SA a measure of market size. Suppose that firms compete a la Cournot

and have costs C = F + c q.

(a) Determine the equilibrium number of firms. [Hint: The profit for each firm is given by πn =(a−c)2SA

(n+1)2− F when there are n firms in the market. Use this formula.]

(Answer) You can ignore the integer constraint in this problem. By solving πn = 0, you can get the

number of firms in free-entry

nA = (a− c)√SAF− 1

Consider now the industry for product X in country B. Everything is identical to country A,

except that market size is twice that of country A: SB = 2SA.

(b) Determine the equilibrium number of firms in market B. How does it relate to the number of

firms in country A? What is the relation between firm size in country A and country B.

9

(Answer) In market B,

nB = (a− c)√SBF− 1

= (a− c)√

2SAF− 1

=√

2(a− c)√SAF− 1

≈√

2nA

The equation implies that the number of firms in market B is less than the twice of that in market

A. In other words, the number of firms increases less than proportionally. Regarding the firm size,

we use the individual firm’s production quantity as the firm size, which is given by

q =S(a− c)n+ 1

.

The firm size in market A and B is

qA =SA(a− c)nA + 1

qB =SB(a− c)nB + 1

=2SA(a− c)√

2(nA + 1)=√

2qA.

Suppose that countries A and B, initially in autarky, decide to sign a trade agreement, so that

they become effectively a single market of size S = SA + SB.

(c) Does economic integration lead to firm entry or to firm exit? Also, what does it imply for the

average size of each firm? Explain the intuition for the result.

(Answer) Let SA+B = SA + SB. Under economic integration,

nA+B = (a− c)√SA + SB

F− 1

=√

3(a− c)√SAF− 1

and the total number of firms before integration is

nA + nB = (1 +√

2)(a− c)√SAF− 2

>√

3(a− c)√SAF− 1

(if (a− c)√

SAF is large enough, which would be the case usually.) Therefore, economic integration

10

Figure 1: Game Tree for 7.1 (a)

leads to firm exit. The average firm size is

qA+B =SA+B(a− c)nA+B + 1

=3(a− c)√3(nA + 1)

=√

3qA.

Intuitively, bigger market size means both higher demand, which is good for firms. However, eco-

nomic integration leads to more firms in the market, so that competition is more fierce. As a result,

some firms find it not profitable to continue their business and thus exit the market.

7 Entry Deterrence

7.1 Movie Release Date

Two movie studios, A and B, must decide when to release their blockbuster movies. There are two

options: November or December. In the November release date there are enough viewers to sell

tickets valued at a total of $500 million. In the December release date, the total is $800 million. If

only one of the studios opens during a given release date, then it takes the whole pie. If two studios

open during the same weekend, then they share the pie (equally).

Since studio A has a close relationship with various movie theater chains, it makes its choice

ahead of its rival. Moreover, studio A can commit to its choice and the choice is observable by

studio B.

(a) Describe the game formally and determine its Nash equilibrium (or equilibria, if there is more

than one).

(Answer) The game is expressed in the following matrix. For B’s action, the left letter shows the

B’s action when A takes N and the right one shows the B’s action when A takes D.

A\B NN ND DN DD

N 250,250 250,250 500,800 500,800

D 800,500 400,400 800,500 400,400

The Nash equilibria are (D,NN), (D,DN) and (N,DD). See Figure 1 for the game tree.

(b) Suppose that, before studio A makes its choice, studio B has the opportunity to write a contract

with advertisers for the week before the release date. Suppose that the cost of advertising is

11

Figure 2: Game Tree for 7.1 (b)

the same in November and December; and that, if studio B wants to change the date after

signing the initial contract, it must pay a penalty for breach of contract equal to $120 million.

Determine the subgame-perfect equilibrium of the new game.

(Answer) (D,(N,N),(D,N,D,D)) is the subgame-perfect equilibrium of the new game. See Figure 2

for the game tree. Use backward induction.

7.2 Entry Deterrence

Consider a homogeneous product industry with P = 200 − Q. There is currently one incumbent

firm and one potential competitor (entrant). The incumbent’s marginal cost is 40 and the entrant’s

marginal cost is 20. Entry into the industry requires a sunk cost of F.

(a) Determine the incumbent’s optimal output in the absence of potential competitor.

(Answer) The output is 80 and the price is 120.

(b) Suppose the entrant takes the incumbent’s output choice as given. Show that the entrant’s

equilibrium profit is decreasing in the incumbent’s output.

(Answer) Let qI be the incumbent’s output. The profit for the entrant can be written as

π =

(220− qI

2

)(180− qI

2

)− F

by solving the 2nd stage decision problem for the entrant in Stackelberg game.

12

8 Product Differentiation

8.1 Logit Demand Model

There are N consumers in the market. Consider the following demand system:

q1(p1, p2) =exp(αp1 + βx1)

exp(αp1 + βx1) + exp(αp2 + βx2)N

q2(p1, p2) =exp(αp2 + βx2)

exp(αp1 + βx1) + exp(αp2 + βx2)N

where α < 0. Derive the price elasticity matrix (i.e., the own- and cross-price elasticity).

(Answer) For the notational convenience, define δj ≡ αpj +βxj for j = 1, 2 and D ≡ exp(δ1) +

exp(δ2). I also define sj = exp(δj)/D. Then,

∂q1∂p1

p1q1

=α exp(δ1) ·D − exp(δ1) · α exp(δ1)

D2N × D

exp(δ1)· p1N

=αD − exp(δ1)

Dp1 = α(1− s1)p1,

∂q2∂p1

p1q2

=− exp(δ2) · α exp(δ1)

D2N × D

exp(δ2)· p1N

=−α exp(δ1)

Dp1 = −αs1p1,

∂q1∂p2

p2q1

=− exp(δ1) · α exp(δ2)

D2N × D

exp(δ1)· p2N

=−α exp(δ2)

Dp2 = −αs2p2,

∂q2∂p2

p2q2

=α exp(δ2) ·D − exp(δ2) · α exp(δ2)

D2N × D

exp(δ2)· p2N

=αD − exp(δ2)

Dp2 = α(1− s2)p2.

Those results can be summarized in a matrix.(ε11 ε12

ε21 ε22

)=

(α(1− s1)p1 −αs1p1−αs2p2 α(1− s2)p2

)

8.2 Bertrand Competition with Differentiated Products

Consider the Bertrand competition in a differentiated products market. There are two firms (firm

1 and 2) and the demand function is given by

D1(p1, p2) =1

2+p2 − p1

2t

D2(p1, p2) =1

2+p1 − p2

2t,

13

where t > 0. Remember that this demand system is coming from the location model.

The marginal cost of production is constant and same for both firms. Denote the marginal cost

by c.

(a) Derive the own- and cross-price elasticity under this demand system.

(Answer)

ε11 =∂D1

∂p1

p1D1

=

(− 1

2t

)(p1

12 + p2−p1

2t

)= − p1

t+ p2 − p1.

ε12 =∂D2

∂p1

p1D2

=1

2t

(p1

12 + p1−p2

2t

)=

p1t+ p1 − p2

.

You can calculate ε21, ε22 in a similar manner. Then(ε11 ε12

ε21 ε22

)=

(− p1t+p2−p1

p1t+p1−p2

p2t+p2−p1 − p2

t+p1−p2

).

(b) Find the Nash equilibrium of this game.

(Answer) The profit of firm 1 is

π1(p1) = D1(p1, p2)(p1 − c).

The FOC is

∂π1(p1)

∂p1=∂D1(p1, p2)

∂p1(p1 − c) +D1(p1, p2)

= − 1

2t(p1 − c) +

1

2+p2 − p1

2t

= 0,

which leads to t+c = 2p1−p2. By solving the profit maximizing problem of firm 2, t+c = 2p2−p1.

The Nash equilibrium is p∗1 = p∗2 = t+ c.

(c) How t affects the equilibrium price. Explain intuition.

(Answer) t represents degree of the perceived differentiation between firm 1’s products and firm 2’s

products by consumers. When t is higher, prices are higher. Product differentiation can reduce the

price competition. Firms have market power and obtain positive profit.

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9 Vertical Relationship

9.1 Intermediate Input Supplier and Final Good Producer

Consider a market for an automated agricultural machine. Imperial Heavy Industries (downstream

firm) is the monopolist in this market. Because an automated agricultural machine is very sophis-

ticated equipment, the Imperial Heavy Industries needs to procure the special parts from Tsukuda

Manufacturing (upstream firm) to produce an automated agricultural machine. Denote the market

demand for an agricultural machine by P = a−Q. The production technology of automated agri-

cultural machines is given by Q = m2 where m is special parts produced by Tsukuda Manufacturing.

The marginal cost of producing special parts is constant and given by c (< a2 ). Assume that the

Imperial Heavy Industries does not pay any additional cost for production other than purchasing

special parts.

Consider the following two stage game:

• Stage 1: Tsukuda manufacturing decides the price for special parts w.

• Stage 2: Given the parts price w, Imperial Heavy Industries decides the production quantity

for automated agricultural machine.

(a) Write down the profit maximization problem in stage 2 and solve it.

(Answer) Since m = 2Q, the problem is

maxQ,m

(a−Q)Q− wm = (a−Q)Q− 2wQ

The FOC is

a− 2Q− 2w = 0

Thus,

Q =a− 2w

2,

or,

w =a

2−Q.

(b) Derive the demand function for special parts.

(Answer) The demand for m will be

m = 2Q = a− 2w.

(c) Write down the profit maximization problem in stage 1 and solve it.

(Answer) From the demand function given in (b), profit maximization problem for Tsukuda becomes

maxw

(w − c)m(w) = (w − c)(a− 2w).

15

The FOC is

−4w + 2c+ a = 0.

The optimal price is

w∗ =2c+ a

4.

Note that m∗ = a2 − c,Q

∗ = a4 −

c2 , and P ∗ = 3a

4 + c2 .

(d) Suppose that Imperial Heavy Industries and Tsukuda Manufacturing merged into a single firm.

Solve the profit maximization problem in this case.

(Answer) The firm needs two units of special parts for one unit of Q, so the marginal cost of the

single firm is 2c. The maximization problem of the single firm is

maxQ

(a−Q)Q− 2cQ.

The FOC is

a− 2Q− 2c = 0.

The optimal quantity is Qs = a2 − c. P

s = a−Qs = a2 + c.

Note that you can compare the price for automated agricultural machine with vertical integration

(as in (d)) and without vertical integration (as in (a)–(c)), and will find that the price under

vertical integration is lower than the price without vertical integration. You can see this fact from

P ∗−P s = a4 −

c2 > 0. This is the elimination of double marginalization due to vertical integration.

See the lecture slide for the details.

9.2 One-Upstream and Two-Downstream firms

Consider the market in which there exists one upstream manufacturer (firm A) and two downstream

retailers (firm 1 and firm 2). Firm 1 and 2 buy the same wholesale goods from firm A and sell to

consumers. The market demand is given by P = a − Q. The production cost of firm A is given

by the constant marginal cost cM . Assume for simplicity that firm 1 and 2 do not incur any retail

costs. Consider the following two stage game:

• Stage 1: Firm A sets the price for wholesale product. Denote the price by w.

• Stage 2: Given the wholesale price w, firm 1 and 2 compete in Cournot fashion.

(a) Solve the Cournot competition in the stage 2 (given w).

(answer) Firm 1’s problem is

maxq1

(a− q1 − q2)q1 − wq1.

The FOC is a− 2q1− q2−w = 0. You can have the similar FOC for firms as a− q1− 2q2−w = 0.

By solving these, we have

q1 = q2 =(a− w)

3.

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(b) Using the results above, derive the wholesale demand.

(answer) There are two firms that purchase the wholesale goods. Therefore, the wholesale demand

is

q1 + q2 =2(a− w)

3.

(c) Solve the stage 1.

(answer) The manufacture’s problem is

maxw

(w − cM )2(a− w)

3.

The FOC is2(a− w)

3+−2

3(w − cM ) = 0.

Solving this, we have

w =a+ cM

2.

A Cournot Competition with n firms

Each firm has Ci(qi) = ciqi + F . The inverse demand function is P = a − bQ. The profit

maximization problem of each firm is

maxqi

πi(qi) = qi(a− bQ− ci)− Fi.

The FOC isdπi(qi)

dqi= a− bQ− ci − bqi = 0⇒ qi =

a− cib−Q.

Taking summation across i yields

Q =na−

∑j cj

b− nQ⇒ Q∗ =

na−∑

j cj

(n+ 1)b.

Then,

q∗i =a− cib−Q∗

=a− nci +

∑j 6=i cj

(n+ 1)b.

17

In the equilibrium, we have

π∗i =1

b

(a− nci +

∑j 6=i cj

(n+ 1)b

)2

− Fi,

P ∗ = a− bQ∗

=a+

∑j cj

n+ 1,

which are shown in the lecture slide.

18