Question

# Why would the second half of problem 3 (What is IC's optimal level of investment?) found on page 2 be equal to 1.

I am getting the result of 2/3 (two-thirds).

Econ 444 (Corporate Economics), Fall 2017 Problem Set #3

Due in class Wednesday, October 11 Problem 1. Bill’s is the only booth selling used records at the Coachella music festival.

As the monopoly seller of used records at the festival, Bill earns $700 in profit. This year,

Ted is considering setting up a competing booth at the festival. It would cost him $150

do so. However, Bill tells Ted that if he set up a booth, he would face a record price war.

After some research, Ted learns that if Bill follows through on his threat Bill would earn

$100 in profit while he himself would earn $25 (before accounting for the $100 set up cost).

Nevertheless, if Bill does not start a record price war, he would earn $300 while Ted would

earn $120.

1. Draw this game in extensive form. Be sure to fully label the game tree.

Solution.

T

Stay out

Enter

(0,700) B

Accommodate

Fight

(-125,100) (-30,300) 2. What is the subgame perfect equilibrium outcome?

Solution. In the subgame starting from Bill, Bill will accommodate. Suppose that

Bill accommodates, in the original game, Ted will stay out. Therefore the SPNE is

(Stay out, Accommodate if Enter). The outcome is that Ted stay out of the market,

so that Ted gets $0 and Bill gets $700. Econ 444: Problem Set #3 2 3. Would you advise Ted to set up his booth?

Solution. Observe that If Ted enters the market, Bill will accommodate and Ted earns

$-30.It is not profitable for him to do so due to the very high entry cost. He would be

advised to stay home.

4. Doing some more research, Ted learns that his setup cost will actually be $75, not

$150. How would your advice to Ted change given these lower costs?

Solution. Once Ted enters the market, Bill will accommodate, which gives Ted a

payoff of 45, even though he has threatened to start to a record price war. The new

SPNE is (Enter, Accommodate if Enter). Therefore, Ted would be advised to enter

due to the lower entry cost. Problem 2. Suppose Incumbent Coal (IC) invests in capital equipment KIC to mine coal.

Once IC makes its capital investment decision, it is committed to a certain level of production.

A potential entrant, Entrant Coal (EC), observes KIC and chooses how much capital KEC

it would like to invest in, which subsequently commits EC to a set level of production as

well. No other firms can enter the market. Each firm’s total revenue, in millions of dollars,

accruing from its respective capital investment is given by

T RIC = (4 − KIC − KEC )KIC T REC = (4 − KEC − KIC )KEC and the total cost of capital, in millions of dollars, for each firm is

T CIC = 2KIC T CEC = 2KEC 1. Write down the profit function for EC. What is EC maximizing over?

Solution. Given KIC , EC’s problem:

max(4 − KEC − KIC )KEC − 2KEC

KEC 2. Suppose IC’s level of capital investment is KIC . What EC’s is optimal level of capital

expenditure given IC’s choice? Econ 444: Problem Set #3 3 Solution. Take the first order condition of EC’s problem with respect to KEC to get

∗

KEC

(KIC ) = 2 − KIC

2 3. Write down IC’s profit function. What is IC’s optimal level of investment?

∗

Solution. Given KEC

(KIC ) = 2−KIC

,

2 IC’s problem: ∗

max(4 − KEC

(KIC ) − KIC )KIC − 2KIC

KIC Take the first order condition of IC’s problem with respect to KIC to get

∗

=3−2=1

KIC 4. Characterize the Subgame Perfect Nash Equilibrium (SPNE) strategy profile in market? What is the equilibrium level of investment in the market?

∗

∗

∗

Solution. Given KIC

= 1, KEC

(KIC

)= ∗

2−KIC

2 = 12 . Hence, the SPNE strategy profile ∗

∗

IC

(KIC )}, that is, {1, 2−K

, KEC

is {KIC

}. The equilibrium total level of investment in

2 the market is K ∗ = 1 + 1

2 = 32 . 5. Calculate each firm’s equilibrium profit. Does this market exhibit a first or second

mover advantage?

Solution. In terms of IC:

1

∗

∗

∗

∗

πIC = (4 − KEC

− KIC

)KIC

− 2KIC

= $ M illion

2

In terms of EC:

1

∗

∗

∗

∗

πEC = (4 − KEC

− KIC

)KEC

− 2KEC

= $ M illion

4

Note πIC > πEC , so there exists a first mover advantage.

6. Suppose now that IC can easily adjust its level of capital at any moment (e.g. it is not

committed to its investment ex-post). How would this effect equilibrium firm profits? Econ 444: Problem Set #3 4 Solution. If IC cannot commit to its investment ex-post, then this become a simultaneous game where EC’s best response function is

KEC (KIC ) = 2 − KIC

2 and IC’s best response function is

KIC (KEC ) = 2 − KEC

2 Solve these two equations simultaneously to get equilibrium output such as

∗

∗

KEC

= KIC

= 2

3 and individual firm’s equilibrium profits as

∗

∗

πEC

= πIC

= 4− 2

4

2 2 2

− ) − 2 · = $ M illion

3 3 3

3

9 Hence, EC’s profit increases; while IC’s profit decreases.

7. How much would the incumbent IC be willing to pay for the ability to commit to a

level of capital. In other words, what is the value of commitment?

Solution. The value of commitment is the difference of profits with and without the

first mover’s advantage, that is,

1 4

1

− = $ M illion

2 9

18

Problem 3. Suppose there are two stores in state college that sell Windbreakers and compete on prices. University Windbreakers (UW) is located on the corner of College and

University. Atherton Windbreakers (AW) is located on the corner of College and Atherton.

The street is 1 mile long. Both stores buy their merchandise from the same distributor,

so their marginal cost for windbreakers is $10 a piece. However, there is one strange thing

about State College. The wind always blows to the East. That means it is easier for the

town’s 100 customers, who are uniformly located along the street, to walk in that direction.

As a result, a customer located at point x along the street receives the following utility when

purchasing from UW and UA, respectively

ux,U W = V − PU W − 2x

ux,AW = V − PAW − 8(1 − x)

where V = $100. Econ 444: Problem Set #3 5 1. At what point x along the street are customers indifferent between purchasing from

UW or AW given prices PU W and PAW ?

Solution. Solve

V − PU W − 2x∗ = V − PAW − 8(1 − x∗ )

to get

x∗ =

1

8 + PAW − PU W

10 2. What is each firm’s demand function?

Solution.

DU W = x∗ · 100 = 80 + 10PAW − 10PU W

DAW = (1 − x∗ ) · 100 = 20 + 10PU W − 10PAW

3. Write down each firm’s profit function. What is each firm maximizing over?

Solution. In terms of UW,

max DU W · (PU W − 10)

PU W equivalently,

max 80 + 10PAW − 10PU W · (PU W − 10)

PU W In terms of AW,

max DAW · (PAW − 10)

PAW equivalently,

max 20 + 10PU W − 10PAW · (PAW − 10)

PAW 4. Write down each firm’s best response function. How does each firm’s optimal price

respond to a rise in its rival’s price? Are prices strategic complements or substitutes

in this case?

Solution. Take the first order condition of UW’s profit-maximization problem with

respect to PU W to get UW’s best response function:

PU W = 18 + PAW

2 Econ 444: Problem Set #3 6 Similarly, take the first order condition of AW’s profit-maximization problem with

respect to PAW to get AW’s best response function:

PAW = 12 + PU W

2 Hence, prices are strategic complements since a firm’s price is increasing in its rival’s

price.

5. Characterize the equilibrium strategy profile in this market. What are the equilibrium

prices? Which firm sets a higher price?

Solution. Solving two best response functions together to find the equilibrium strategy

profits, that is,

∗

{PU∗ W , PAW

} = {16, 14} So UW sets a higher price in the equilibrium.

6. Write down each firm’s equilibrium profits and market shares. How would these change

if customers found it equally costly to travel in each direction (e.g. if the wind no longer

blew only to the east)?

∗

Solution. Given {PU∗ W , PAW

} = {16, 14}, we could solve equilibrium output and prof∗

its such that DU∗ W = 60 and DAW

= 40, that is, the market share of UW is 60% and

∗

= 160.

the market share of AW is 40%. Besides, πU∗ W = 360 and πAW If consumers find it equally costly to travel in each direction, then two firms should

split the market equally. As a result, the profit and market shares of UW decrease;

while the profit and market share of AW increase instead.

7. Calculate consumer’s surplus in this market. How does it compare to the competitive

case? Can you say something about dead weight loss in this model?

Solution.

1

1

CS H = 100 · (82.8 + (84 − 82.8) · 0.6 + (86 − 82.8) · 0.4) = $8380

2

2

In a competitive equilibrium, PU W = PAW = 10. At these prices, market shares are then

x∗ = 0.8 1 − x∗ = 0.2 Econ 444: Problem Set #3 7 consumer surplus is now higher at

1

1

CS C = 100 · (88.4 + (90 − 88.4) · 0.8 + (90 − 88.4) · 0.2) = $8920

2

2

and firm profits are zero. Dead weight loss is then the difference in total surplus between

the competitive case and the duopoly case. We can write that as

DW L = CS C − CS H − ΠH = 8920 − 8380 − 520 = $20

So in this case, the DWL is a small relative to consumer surplus, or even firm profits. In

other words, the market is quite close to efficient.

Problem 4. Two ice cream shop owners, IScream Left (IL) and IScream Right (IR) are

considering setting up ice cream stands along the Ocean City board walk, which is exactly

one mile long. To do so, they need to make two decisions: (1) Where do they want to

locate their stands along the boardwalk (2) How much do they want to charge for an ice

cream. They know that there are 1,000 potential customers uniformly distributed along the

boardwalk and that each will buy at most one ice cream. They also know that the utility

a customer located at point 0 ≤ x ≤ 1 along the boardwalk gets when she buys ice cream

from IL or IR, respectively, is

ux,IL = V − pIL − t(a − x)2

ux,IR = V − pIR − t(1 − a − x)2

where a and 1 − a are the respective locations of IL and IR along the boardwalk (assume

0 ≤ a ≤ 21 ). (Note this means that we are forcing IL and IR to locate symmetrically along

the boardwalk. This is a simplification. The assumption that a ≤ 1

2 is without loss of generality.) Let V = 110 and t = 2, and suppose both IL and IR face a unit cost of c = $2

for each ice cream.

1. Suppose IL locates at a and sells at price pIL while IR locates at 1 − a and sells at

price pIR . What is the location x of a consumer that is indifferent between buying

from either stand?

Solution. The indifferent consumer is one that has an equal utility of buying from

either stand (ux,IL = ux,IR ). Write

V − pIL − t(a − x)2 = V − pIR − t(1 − a − x)2 Econ 444: Problem Set #3 8

x∗ = pIR − pIL

1

+

2(1 − 2a)t 2 2. At these prices and locations, what is the demand for each stand?

Solution. At these prices and locations, all consumers to the left of x∗ buy from IL,

and all consumers to the right of x∗ buy from IR. Then

p −p

p − p

1

1

IR

IL

IR

IL

DIL = x∗ · N =

+

· N = 1000

+

2(1 − 2a)t 2

4(1 − 2a) 2

p − p

p −p

1

1

IL

IR

IL

IR

+

· N = 1000

+

DIR = (1 − x∗ ) · N =

2(1 − 2a)t 2

4(1 − 2a) 2

3. If IL and IR locate at a and 1 − a, respectively, and IR sets its price at pIR , what

should IL charge for Ice cream?

Solution. IL’s optimal price conditional on IR’s price, and each stand’s location, is

determined by IL’s best response function. To find IL’s best response function, we first

maximize IL’s profit with respect to price, holding constant its rival’s price and both

stands’ locations.

max πIL = maxpIL (pIL − c)DIL

pIL IL’s first order condition is then

pIR − 2pIL + c 1

+ =0

(1 − 2a)2t

2

and its best response function (solving for pIL ) is then

pIL = 1

1

· (pIR + c + (1 − 2a)t) = · (pIR + 2 + 2(1 − 2a))

2

2 4. Similarly, what is IL’s optimal price if IR sets its price at pIR ?

Solution. Just like IL, IR will also be maximizing its profit with respect to its own

price. Solving IR’s problem

max (pIR − c)DIR

pIR we get that

pIR = 1

1

· (pIL + c + (1 − 2a)t) = · (pIL + 2 + 2(1 − 2a))

2

2 Econ 444: Problem Set #3 9 5. Holding their locations on the boardwalk fixed at a and 1 − a, respectively, what is the

equilibrium price of ice cream?

Solution. In any equilibrium, both firms best response conditions must be satisfied.

To find the pair of prices for which this is true, we must solve this two equations

system. We do this by substituting one best response function into the other. When

we do this we get that

pIL = pIR = c + (1 − 2a)t = 2 + 2(1 − 2a)

Alternatively, note that in this case, because we assume symmetry in both costs and

location, both stands’ best response functions are identical. When this is true, there

will be a symmetric solution and both stands will set the same price. When we do

this, we can simplify the problem by solving just one equation:

1

p = (p + c + (1 − 2a)t) ⇒ pIL = pIR = c + (1 − 2a)t = 2 + 2(1 − 2a)

2

6. Given your solution from above, where on the board walk would IL and IR set up their

stands?

Solution. We now know what the price of ice cream in the market for each 0 ≤ a ≤ 1

2 . Now we have to solve for the locations a, 1 − a that maximize both firms’ profits. Note

that, in contrast to the pricing problem above, IL and IR now choose their location a

to maximize profit. Let’s start with IL’s problem:

max πIL = (pIL − c)DIL = (c + (1 − 2a)t − c)500 = (c + 2(1 − 2a) − c)500

a We know that at each a the price of ice cream will be the same at both stands.

Consequently, each stand will collect one half of the demand.

Observe that as a increase, IL moves toward the center of the boardwalk, IL’s profit

decreases. Consequently, it is optimal for IL to set a = 0. This is also true for IR

(try it!). Consequently, IL and IR will set up their stands at opposite ends of the

boardwalk, as far apart as possible.

Importantly, this is a dominant strategy. Regardless of IR’s location, IL will always

locate at the far end of the boardwalk a = 0, and vice versa. As a result, this is a Nash

Equilibrium. Econ 444: Problem Set #3 10 7. How much ice cream does each stand sell? How much does each stand make in profit?

Solution. We know from above that each stand will sell 500 units - half the total

demand. We also know that each shop will locate its stand at each respective edge of

the boardwalk. AT a = 0 the market price for ice cream will be:

p = c + t = 2 + 2 = $4

and each shop’s profit will subsequently be

π = (4 − 2) ∗ 500 = $1000 8. Do these equilibrium prices and locations constitute a Subgame Perfect Nash Equilibrium (SPNE)? Explain why or why not.

Solution. The prices and locations we have solved for thus far do constitute an SPNE

outcome. This is because in each stage (subgame) of this two stage game both firms

are playing a Nash equilibrium. We were able to get this solution by using backward

induction. First we solved for the Nash equilibrium prices conditional on locations (the

second stage of the game). Then, stepping back, we solved for the Nash equilibrium

in location choice. Problem 5. (True/False). For each statement, assert whether it is true or false and

explain your answer.

1. In a quantity game, e.g. Cournot, quantities are strategic substitutes.

Solution. True. Just as in the simultaneous Cournot game, a firm optimally responds

to an increase in its rival’s production by reducing its quantity produced. This follows

from firms’ best response functions. 2. Firms playing a Subgame Perfect Nash equilibrium (SPNE) in a finitely repeated Betrand game must play a Nash equilibrium in each sub-games.

Solution. True. By definition, a Sub-game perfect Nash equilibrium (SPNE) of a

dynamic game necessitates a Nash equilibrium in each sub-game. Econ 444: Problem Set #3 11 3. Two Betrand competitors on a linear city will choose to locate as far away from each

other as possible.

Solution. True. Notice how in problem 4 the firms will choose to locate as far from

each other as possible to reduce price competition pressure. This is called the principle

of maximal differentiation.

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