IO Chapter 7 - Oligopoly

Now game theory becomes more relevant, since you have a few firms, and they are concerned not only about their actions, but the actions of their rivals.

Bertrand Model (another one we won't spend a lot of time on):

Assume 2 firms, homogeneous product, simultaneous price setting, same MC (constant) and linear demand. (How realistic do you think these assumptions are?)

Because the product is homogeneous, the lower priced competitor will make all the profit.

If p₁ > p₂ then D₁ = 0, p₁ = p₂ then D₁ = D₂ or each gets half the demand or ½ D(p).

Can we model this using algebra? How about making a game grid out of this?

But in order to know how to price, each firm has to know the other firm's price plan.

The bottom line is that both firms will end up setting p=MC, because to set their price any higher is to risk losing all their business.

In the Bertrand model merely having a second competitor leads the market from a monopolistic (inefficient) one to an efficient one, where p=MC.

This is a rather oversimplistic model, since 1. firms rarely play only one time period. 2. more generally firms do not have to make simultaneous decisions and can often follow or lead. 3. most firms try their hardest to convince consumer that their products are NOT homogeneous. 4. Firms may face capacity constraints, so they may be unable to provide the entire market, even if they capture it all.

All four of these factors change the problem enough so that the Bertrand model, although simple, is not very useful. It may though be useful in cases where we believe that price is difficult to change, but quantity is relatively easy to adjust.

What happens if 4 is true (limited capacity.)

Then we are back to a model which is quite similar to the dominant firm model.

Another oligopoly model is the Cournot model.

Ex: The demand function is given by: p = 105 - 5 (q₁ + q₂) where Q= q₁ + q₂ and C(q) = 5q (both firms face the same cost function.)

max p = (105 - 5 (q₁ + q₂)) q₁ - 5q₁

d p/dq₁ = 105 - 10q₁ - 5q₂ - 5q₁ = 0

Simplifying we get: q₁ = 10 - ½ q₂

This is a game because the first firm's solution is a function of the second firm's decision. This formula is known as a reaction function. Once firm one has an idea of what firm 2 will produce, it can decide on its own production. So the next question is, are the firms likely to be able to second guess each other's decisions. If so, will the solution be a Nash equilibrium?

We can plot this equation, with q₂ and q₁ on the X and Y axes respectively.

We can also solve for the second firm's reaction function, which turns out to be identical to the first (except q₁ and q₂ are reversed), since the cost functions for both firms are identical.

So firm 2's reaction function will be q₂ = 10 - ½ q₁

This can also be graphed.

Because there are unlimited 'moves' in this case, it is better to draw the game grid as a graph, rather than as a list of choices and outcomes. But we could do that as well, for a certain number of outcomes.

We see that there is an equilibrium point. This it turns out is the solution to the Cournot model.

If we substitute the second equation into the first, we find that

10 - ½ (10 - ½ q₁) = q₁

10-5 + 1/4 q₁ = q₁

5 = 3/4 q₁

q₁ = 20/3

It will also be the case that q₂ = 20/3

p = 105 - 5(40/3) = 105 - 200/3 = 115/3

So how do the monopoly, perfect competition and Cournot duopoly fare in terms of p and q outcomes?

Monopoly: p = 105 - 5q

(105 -5q)q - 5q

105 -10q -5 =0

100 = 10q

q=10

p = 105 -50

p=55

PC

p = 5

5 = 105 - 5q

100 = 5q

q=20

We see that the Cournot duopoly is between PC and the monopoly in terms of pricing and quantity supplied to the market.

Comparative Statics:

This is an exercise where economists change the specification of the model and see how the solution changes. (In a way we have already done this, by showing, what would happen if first we had one firm, then a second firm entered the market, and then multiple firms were competing. But we can also change other variables. For instance, what if there is an increase in demand, or costs.)

We'll do the example of an increase in demand, since the book does the example of increased costs.

Suppose demand suddenly becomes more inelastic for some reason. How might that change the problem?

The reaction function will shift inward (as it also does when the cost increases) and the quantity supplied by each firm will be reduced (although in this example the price will not change. What do you think will happen when the costs increase. (Check at home what happens if C(q) = 7.5q instead of 5q.

Another interesting example is where the two firms are not identical. Suppose they have differing cost functions (or as the book discusses, suppose they face exchange rate variations, which lead to differing costs at different times.)