Problems on Duopoly

Problems on Duopoly

Notation used in these problems:

Q=total output

P=market price (the good is homogeneous, so both firm’s sell at this price)

QA= firm A’s output

QB=firm B’s output

= profits

These problems are all based on a market demand function:

P=300-Q.

For simplicity there are no costs of production.

The industry is initially a monopoly which maximizes  (at $22,500) by setting P=150,

Q=150 (=QA)

Then a second firm (B) enters. We want to figure out what some duopoly theories will predict about the new equilibrium.

I. Cournot

The Cournot model will predict: Q=200 (QA=QB=100) and P=100.

Industry =$20,000 shared equally by A and B.

a. Can you figure out analytically why the model predicts this equilibrium?

b. Explain why total industry profits decline after B enters.

II. Conjectural Variation

We saw in class that independent decision making would lead B to believe A will cut back if B increases output (and vice versa). If firms understand this and act upon it, the equilibrium will be different from Cournot (where B acts as if A maintains a fixed QA). In the example in class, whenever B produced +1 unit A reacted by cutting back ½ unit; when A cut back by 1 unit, B reacted with a +1/2 unit increase in QB.

If A and B come to understand these reactions and act on them, the industry

equilibrium will be Q=240, P=60, total =14,400.

a. Can you figure this one out analytically?

b. explain why total industry profit is lower than under naïve Cournot rivalry (where neither A nor B figures out that the rival reacts to its

output decision).

c. Does the result suggest that A or B should not use knowledge of the rival’s response? Explain why or why not.

III. Stackelberg

In the preceding problems, firms are equally knowledgeable. But firms usually try to get an edge on rivals; sometimes they succeed. The Stackelberg model predicts what happens when one rival (say A) knows more about the interactions between them than the other does. You will find this model discussed in ch. 7 of the text. In this application, suppose A understands that QB will go up ½ for each -1 of QA. But B goes along (incorrectly) believing QA is fixed.

The equilibrium now is: QA=150,QB=75. So Q=225, P=75, and =$16,875.

a. Again, can you figure this one out analytically? (If you got IIa you’ll get this. If not…)

b. Should A to use its information advantage by competing with B (as in this example), or should A try to cooperate with B?

Answers to ‘Problems on Duopoly

Preface. In general, analytic solutions to problems like these involve defining a profit

function and then finding the maximum of that function. In this case the problem is

made simpler by the absence of costs: this makes maximizing  equivalent to maximizing

revenue. So, generally, you start with

R(evenue)A=PQA, and RB=PQB.

Then use the demand function [P=300-Q= 300-(QA+QB)] instead of P, so that

RA=[300-QA-QB]QA=300QA-QA2-QBQA,

RB= [300-QA-QB]QB=300QB-QA2-QBQA.

In these problems A and B find the maximum of these revenue functions independently, and

the market equilibrium requires that these independent decisions are simultaneously fulfilled.

I. Cournot

a. You want to find the QA and QB which simultaneously solves the first-order conditions:

dRA/dQA=300-2QA-QB=0QA=1/2(300-QB)

dRB/dQB=300-2QB-QA=0QB=1/2(300-QA)

The answers given last week are the solution to these two simultaneous equations in the 2 unknowns (QA and QB). Then find P=300-(100+100)=100. So industry =100200=20,000.

b. When B comes in Q goes up (long run as well as short run: A never cuts back to exactly offset B’s output). ANY increase in output beyond Q=150 will dissipate industry profits.

II. Conjectural Variation

a. Focus on the derivatives of the last term in each of the revenue functions. In general they would be:

d(QBQA)/dQA=[QB+QAdQB/dQA], and

d(QAQB)/dQB=[QA+QBdQA/dQB].

In the Cournot model the terms like dQA/dQB=0, because B believes QA will not change when QB changes. However, the example in class shows that in fact dQA/dQB and dQB/dQA are both really = -1/2. If A and B both come to understand this, then the first order conditions would be

dRA/dQA=300-2QA-[QB-1/2QA]=300-1.5QA-QB=0QA=(1/1.5)(300-QB)

dRB/dQB=300-2QB-[QA-1/2QB]=300-1.5QB-QA=0QB=(1/1.5)(300-QA).

The simultaneous solution to these is QA=QB=120, or Q=240P=60.

b. The key here, just as in Ib, is ‘what happens to output?’ when B and A understand the rival reactions. And the answer is ‘output is greater than when they don’t understand.’ Why? Consider B. Under Cournot, B thinks its demand is limited to customers not currently served by A (and vice versa). Here B understands that, in addition to those new customers, B will get some customers from A because A will cut back output. In other words, B faces a greater demand in this case than under Cournot rivalry. Accordingly, it’s rational for B to produce more than it would under Cournot. But anything that makes it sensible for a firm to produce more carries the industry further away from the  maximizing P.

c. Each firm should use its knowledge, even though it seems to make them poorer. The reason is that, in these problems, they are not colluding. Collectively they would be better off not to use their knowledge. But unless A has reason to believe that B will refrain from using its knowledge (and vice versa), each had better look out for its own interest as best it can - and that means using all the info you have on the assumption your rival will do the same.

III. Stackelberg

a. Here A’s first-order condition is the same as in II (where both A and B are knowledgeable), but B’s first-order condition is the same as in I (where both mistakenly believe there’s no output response). So, from II

QA=(1/1.5)(300-QB),

while from I

QB=(1/2)(300-QA).

Again, you have 2 equations in the 2 unknowns (QA, QB). QA=150, QB=75 are the solutions to these.

b. Note that A is doing even better here than under Cournot rivalry (A=$11,250 here v. only $10,000 under Cournot). In fact A is doing as well as it would if A and B split the market at the monopoly price. But this is still rivalry, and cooperation yields more TOTAL , not only in these examples, but generally. The fact that rivalry always dissipates some potential monopoly rents is a key to the next part of the course, which focuses on cooperation. Here this fact allows A to do better by offering to cooperate with B so that both A and B are better off than they are now (at the expense, of course, of consumers). For example, suppose they can agree to ‘Let’s both of us cut back Q by 10%’. Then QA=135, QB=67.5Q=202.5, P=97.5A=$13,162.5, B=$6581.25. This beats independent rivalry for both.

The way a firm like A could use its knowledge (legalities and other problems to be discussed aside) is to extract a better than even split of the rents available by cooperating.

Here A could tell B upfront ‘I’m going to get more than ½ the output, because I can just stand pat and get as much  that way as by cutting a deal with you.’

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