econs 305 - oligopoly - part 3 · econs 305 - oligopoly - part 3 eric dunaway washington state...

EconS 305 - Oligopoly - Part 3

Eric Dunaway

Washington State University

[email protected]

December 1, 2015

Eric Dunaway (WSU) EconS 305 - Lecture 33 December 1, 2015 1 / 49

Introduction

Yesterday, we looked at how �rms compete when they simultaneouslychoose quantities.

Today, we�ll �nish up our unit on Oligopoly by looking at a fewextensions of our models.


Asymmetric Firms

So far, in all of our analysis of oligopoly, we have assumed that the�rms are identical.

This makes the math quite easy, and allows us to have the samesolution for each �rm.Unfortunately, in the real world, this rarely happens.

Firms are typically di¤erent, either in the structure of their demandcurves or in the costs they face.

Let�s look at what happens when �rms face di¤erent costs.


Asymmetric Firms

What would happen if the �rms faced di¤erent marginal costs?

Now, they are no longer symmetric, so we should expect that theyproduce di¤erent quantities in Cournot competition.Intuitively, the �rm with lower costs should have a higher output leveland higher pro�ts.

The good news is that nothing changes mathematically.

We follow exactly the same steps we used before to solve the problem.


Asymmetric Firms

Let�s use the same inverse demand function that we have been usingthus far

p = 70� q1 � q2where now �rm 1 faces a constant marginal cost of MC1 = 10 and�rm 2 faces a higher constant marginal cost of MC2 = 20.

Which �rm should have the higher output level (quantity)?


Asymmetric Firms

p = 70� q1 � q2MC1 = 10 MC2 = 20

Let�s start with �rm 1�s best response function. First, we need theirmarginal revenue, which we get from the total revenue equation.

TR1 = pq1 = (70� q1 � q2)q1 = 70q1 � q21 � q1q2MR1 = 70� 2q1 � q2

(Note that it�s exactly the same as in the Cournot problem)

Next, we set �rm 1�s marginal revenue equal to its marginal cost.

MR1 = MC170� 2q1 � q2 = 10


Asymmetric Firms

70� 2q1 � q2 = 10

Now, we solve this expression for q1 to obtain �rm 1�s best responsefunction,

2q1 = 60� q2q1 = 30� 1

2q2

(Again, it�s exactly the same as in the Cournot problem)

Intuitively, nothing about �rm 1 has changed. They face the samemarginal revenue and same marginal cost as before, so they will notwant to change anything about their behavior.

What about �rm 2, though?


Asymmetric Firms

q1

q2

q1 = q2

BR1(q2)

30

60


Asymmetric Firms

p = 70� q1 � q2MC1 = 10 MC2 = 20

Now, we do the same thing for �rm 2. Starting with their marginalrevenue,

TR2 = pq2 = (70� q1 � q2)q2 = 70q2 � q1q2 � q22MR2 = 70� q1 � 2q2

(Again, it�s the same as in the Cournot problem. The marginalrevenue hasn�t changed)And then we set it equal to �rm 2�s marginal cost,

MR2 = MC270� q1 � 2q2 = 20


Asymmetric Firms

70� q1 � 2q2 = 20

Next, we solve this expression for q2 to obtain �rm 2�s best responsefunction,

2q2 = 50� q1q2 = 25� 1

2q1

which is di¤erent from what we saw in the Cournot problem.

Intuitively, �rm 2 is facing higher costs now. No matter what �rm 1produces, �rm 2 will want to produce less because of those costs.


Asymmetric Firms

q1

q2

q1 = q2

BR2(q1)50

25


Asymmetric Firms

Now, we do the same thing as before, and �nd our solution where thetwo best response functions intersect.

This is because if the best response function for �rm 1 and the bestresponse function for �rm 2 align, we are in equilibrium.

Graphically, we just �nd where our two best response functionsintersect in our �gure.


Asymmetric Firms

q1

q2

q1 = q2

BR2(q1)50

25

BR1(q2)

30

60


Asymmetric Firms

Notice this time that our intersection does not lay on the q1 = q2line, as it has before.

This is a direct result of the asymmetry between the �rms.

Mathematically, we can solve our system of two equations and twounknowns to �nd that intersection point

q1 = 30� 12q2

q2 = 25� 12q1

Rearranging terms a bit,

q1 +12q2 = 30

12q1 + q2 = 25


Asymmetric Firms

q1 +12q2 = 30

12q1 + q2 = 25

Let�s multiply the botton equation by �2

q1 +12q2 = 30

�q1 � 2q2 = �50and now add the two equations together,

q1 +12q2 � q1 � 2q2 = 30� 50

�32q2 = �20

q�2 =403� 13.33


Asymmetric Firms

q�2 =403� 13.33

Lastly, we can plug q�2 into �rm 1�s best response function to get ouranswer

q�1 = 30�12q�2 = 30�

12

�403

�=703� 23.33

Wow, �rm 1 produces quite a bit more than �rm 2. Their costadvantage lets them take a much larger share of the market!


Asymmetric Firms

q1

q2

q1 = q2

BR2(q1)50

25

BR1(q2)

30

60

23.33

13.33


Asymmetric Firms

We can also calculate pro�ts for each �rm. This �rst requires that wecalculate the price, though

p� = 70� q�1 � q�2 = 70�703� 403=1003� 33.33

Starting with �rm 1,

TR1 = p�q�1 = 33.33(23.33) = 777.59

TC1 = MC1 � q�1 = 10(23.33) = 233.33π�1 = TR1 � TC1 = 777.59� 233.33 = 544.26

and for �rm 2,

TR2 = p�q�2 = 33.33(13.33) = 444.29

TC2 = MC2 � q�2 = 20(13.33) = 266.66π�2 = TR2 � TC2 = 444.29� 266.66 = 177.63


Asymmetric Firms

As we can see, �rm 1 had both a higher output and pro�t level than�rm 2 had.

In fact, �rm 1 did even better than it did under the standard Cournotmodel, since �rm 2 reduced their output by a lot.

Firm 2 ended up worse o¤, produce much less and earning much lessin pro�ts.


Stackelberg Competition

The Stackelberg model was developed in the mid twentieth centryby German economist Heinrich Freiherr von Stackelberg.

He started with the Cournot model, and let one �rm move �rst (Theleader) and the rest of the �rms act as followers.

In our model, we will assume that �rm 1 is the Stackelberg leader�rm and gets to move �rst, setting its level of output.

Then, �rm 2 is able to observe �rm 1�s output level and respond withits own output level.



Why does one �rm get to move �rst?

Typically, the leader �rm (�rm 1 in our case) is much larger than theother �rms, or it has some kind of control over the market that theother �rms wait to see what the leader does.We are going to see that the leader �rm gets a huge advantage frommoving �rst.



Remember from our Game Theory unit that we want to use backwardinduction to �nd an equilibrium.

We start at the bottom of the game tree and �gure out the bestresponse for �rm 2.Then, we substitute that best response in for �rm 1 and solve it again.

Since we are dealing with a whole spectrum of quantities, we reallycan�t draw the game tree in this case, but mathematically, we aregoing to follow the same process.



Let�s return to our same example,

p = 70� q1 � q2MC = 10

In order to use backward induction, we want to solve for �rm 2�s bestresponse function. To get this, we �rst need his marginal revenue,which we get by applying the power rule to �rm 2�s total revenue.

TR2 = pq2 = (70� q1 � q2)q2 = 70q2 � q1q2 � q22MR2 = 70� q1 � 2q2



Next, we set �rm 2�s marginal revenue equal to its marginal cost, andsolve for q2 to get �rm 2�s best response function.

MR2 = MC

70� q1 � 2q2 = 10

2q2 = 30� q1q2 = 30� 1

2q1

Notice that this best response function is identical to what we foundunder Cournot competition.

This should be the case for the follower. From their perspective,nothing has changed. They are going to observe a quantity from �rm 1and then optimally respond to it.



If we were working with a game tree, now we would substitute theseresults up the tree and get a reduced form game.

Firm 1 would just pick which of its strategies gave the highest payo¤,knowing how �rm 2 is going to respond.

We can still do this mathematically.

Firm 1 needs to calculate its marginal revenue. Starting with totalrevenue,

TR1 = pq1 = (70� q1 � q2)q1 = 70q1 � q21 � q1q2



Before �rm 1 calculates its marginal revenue, it will want toincorporate what it knows about �rm 2. It knows that �rm 2�squantity is going to be based o¤ its best response function.

q2 = 30�12q1

This gives �rm 1 an advantage. It can �nd its optimal quantity basedon this information. We can substitute this best response function infor q2 in the total revenue equation.

TR1 = 70q1 � q21 � q1q2 = 70q1 � q21 � q1�30� 1

2q1

�= 70q1 � q21 � 30q1 +

12q21 = 40q1 �

12q21

and, applying the power rule,

MR1 = 40� q1Eric Dunaway (WSU) EconS 305 - Lecture 33 December 1, 2015 26 / 49


A little bit of intuition on this step:

There are some subtle di¤erences between how the variable q1 behavesbefore and after we apply the power rule.Before we apply the power rule (The Total Revenue Level), q1 is just aregular variable, and �rm 1 wants as much information about thevariable as possible. This is why the �rm wants to substitute the bestresponse function in at this level.After we apply the power rule, q1 is now an equilibrium variable.

Think about it as before and after the �rm makes its decision. Firm 1needs the information about the best response function before itchooses its optimal level of output. Thus, we need to insert the bestresponse function before we apply the power rule.



From here, we just need to set �rm 1�s marginal cost equal to itsmarginal revenue.

MR1 = MC

40� q1 = 10

q�1 = 30

Then, we plug this value in to �rm 2�s best response function

q�2 = 30�12q�1 = 30�

12(30) = 15



Firm 1 produces double the amount of �rm 2!

In actuality, �rm 1 is producing the monopoly market quantity. This isno coincidence, as �rm 1 will use its market power to take fulladvantage of its pro�ts.After taking the lion�s share, �rm 2 is left with the scraps, andproduces half as much as �rm 1.



To �nd the market price, we just plug our equilibrium values back into the inverse demand curve,

p� = 70� 30� 15 = 25

And now, we can calculate the total revenue, total costs, and totalpro�ts for each �rm:

TR1 = p�q�1 = 25(30) = 750

TC1 = 10q�1 = 10(30) = 300

π�1 = TR1 � TC1 = 750� 300 = 450TR2 = p�q�2 = 25(15) = 375

TC2 = 10q�2 = 10(15) = 150

π�2 = TR2 � TC2 = 375� 150 = 225



Let�s compare this with our results from yesterday.

For �rm 1, the pro�t level is the same as in the cartel. This alsomeans that by being the leader, �rm 1 receives more pro�ts than itwould by moving simultaneously in Cournot Competition.

This is known as �rst mover advantage.

For �rm 2, it�s bad. Its pro�t level is lower than both the cartel andCournot levels.

The price is lower than both the cartel and Cournot levels, which isnice for the consumers.

This also implies that total pro�ts are less under Stackelberg than theyare for both the cartel and Cournot.More consumer surplus!


Welfare Comparisons

Let�s look at some welfare comparisons before we move on to our lasttopic for the day.

For simplicity, I will just be showing the results graphically, but all ofthese values for consumer surplus, producer surplus, and dead weightloss could easily be calculated using triangle and rectangle formulas.

I have included the actual values on the �gures. Practice calculatingthem on your own.


Welfare Comparisons

Q

p

70

70

40

40

30

30

25

45 60

10

Cartel / MonopolyCournot

Stackelberg

Bertrand / Competitive

D

S


Welfare Comparisons

Q

p

70

70

40

40

30

30

25

45 60

10

Cartel / Monopoly

D

S

CS450

PS900 DWL

450


Welfare Comparisons

Q

p

70

70

40

40

30

30

25

45 60

10

Cournot

D

S

CS800

PS800

DWL200


Welfare Comparisons

Q

p

70

70

40

40

30

30

25

45 60

10

Stackelberg

D

S

CS1012.5

PS675

DWL112.5


Welfare Comparisons

Q

p

70

70

40

40

30

30

25

45 60

10

Bertrand / Competitive

D

S

CS1800


Welfare Comparisons

As we can see, with increased competition, the consumer surplus risesand both the producer surplus and dead weight loss fall.

The consumers are able to take advantage of the in�ghting amongthe �rms.

At the perfectly competitive level, both the producer surplus and deadweight loss are eliminated, and the consumers reap all of the surplusfrom the market.


Price Match Guarantee

To conclude this unit, I wanted to talk about an interestingapplication: the price match guarantee.

Several companies in the US o¤er to match their competitors prices,guaranteeing that their customers will always get the best deals!

But does this actually make consumers better o¤?

Let�s go back to Bertrand competition and see what happens whenboth �rms o¤er a price match guarantee.



A few things to remember about Bertrand competition:

The best response for each �rm is to undercut the other �rm by onepenny (as long as the price is between marginal cost and the monopolyprice).The equilibrium price is equal to marginal cost.



Now, we implement the price match guarantee.

The consumer now is able to receive the lowest price between the two�rms regardless of which �rm he buys from.Let�s see how the best response functions change. We�ll look at �rm1�s perspective.



Using the same three segments as before.

If �rm 2�s price is above the monopoly price, it is best for �rm 1 torespond with the monopoly price.This is due to pro�ts being able to be increased by lowering the price.Both �rm 1 and �rm 2 get half of the market since �rm 2�s customersjust price match down to the monopoly price.



If �rm 2�s price is between marginal cost and the monopoly price, �rm1 has 3 options.

It can charge a price higher than �rm 2�s price, and then �rm 1�scustomers will just price match to �rm 2�s price, and both �rms gethalf of the market.It can charge a price equal to �rm 2�s price, and both �rms get half ofthe market.It can charge a price lower than �rm 2�s price, and then �rm 2�scustomers will price match to �rm 1�s price, and both �rms get half ofthe market.

As opposed to the typical Bertrand model, undercutting is now theworst option. Firm 1 doesn�t gain any customers, but it loses pro�tsby charging a lower price!

Firm 1�s best response is to charge any price between �rm 2�s priceand the monopoly price.



If �rm 2�s price is below marginal cost, �rm 1 is best to just not sellthe product.

If it sold the product at any price above �rm 2�s price, customers couldmatch it to �rm 2�s price and �rm 1 would take a loss.If it undercut �rm 2, it would also take a loss.



It�s hard to plot how the best functions are going to look, but we candescribe them.

If �rm 2 starts with a price between marginal cost and the monopolyprice, �rm 1 will charge at least as high of a price as �rm 2, if nothigher.

If �rm 1 charges a price higher than �rm 2�s, �rm 2 will respond byraising their price to at least as high as �rm 1�s price, if not higher.This "price creep" will continue until the market �nds equilibrium withboth �rms charging the monopoly price!



Price Match Guarantees lead to higher prices?

Yes they do.By preserving their customer base regardless of their price, �rms lose allincentive to undercut one another.They are able to bring in much higher pro�ts at expense of theconsumers. They also create signi�cant dead weight loss.


Summary

When a �rm is able to set their quantity �rst, they can claim a largeportion of the market at the expense of the other �rm.

Competition increases consumer surplus and lowers dead weight loss.

Price Match Guarantees are meant to look like they help consumers,but really just discourage competition.


Preview for Friday

Risk and Uncertainty

How do insurance premiums work?

Perlo¤, Chapter 17.


Assignment 7-5 (1 of 1)

1. Consider a two �rm duopoly that faces an inverse demand curve of

p = 150� q1 � q2

and constant marginal costs of MC = 60.

a. If �rm 1 behaves as a Stackelberg leader (�rst mover) and �rm 2behaves as a Stackelberg follower (second mover), what are theequilibrium quantities, prices and pro�ts for each �rm? (Hint: Theywill not be equal!)


econs 305 - oligopoly - part 3 · econs 305 - oligopoly - part 3 eric dunaway washington state...

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