When Duopolies are the Same as Monopolies

Hugo Sonnenschein's 3 page pager

The great economic theorist Hugo Sonnenschein died this summer. While he wasn’t a “price theorist” by any stretch of the term, he had an immense influence on economic theory as a whole, and economics is worse off without him.

Sonnenschein is most famous within economics for the Sonnenschein conjecture (often lumped into the Sonnenschein–Mantel–Debreu theorem). The conjecture basically says (in price theory language), even if we assume that individuals have nice downward-sloping demand curves, the aggregated market demand curves can look like anything. Market demand can go up, down, all over the place, and anything can happen. Sonnenschein also did lots of other ground-breaking high theory (from the Coase conjecture to monopolistic competition), in addition to being in charge of the University Chicago during the 90s. An awe-inspiring career.

My favorite Sonnenschein paper is only three pages long(!) on Cournot’s theories of duopoly and complementary monopolies. I have a soft spot for theory papers that deal with the history of thought. It also helps that it’s the only one of his papers I really understand. But it’s good, period. I hope people should read it and McCloskey’s exposition in her textbook.

I’ll lay the idea out here with no math. The paper tells us something profound about the relationship between price and quantity in economic theory.

Cournot Duopoly

How are duopolies and complementary monopolies “the same”? Let’s start with the more famous model, Cournot duopoly. (It’s a highlyoverrated model, but that’s for another newsletter).

Cournot’s duopoly model has two identical firms producing identical goods. The two firms don’t directly choose their price. Instead, they compete on quantity. It’s sometimes just called quantity competition. Each firm chooses its quantity, and then consumers bid up prices to clear the market. The market is in equilibrium if each seller maximizes profit, given what the other firm is doing. Today, this equilibrium is just called a Nash equilibrium, but you'll still sometimes hear of a Cournot-Nash equilibrium.

In this duopoly market, the price ends up being between the monopoly price and the competitive price. People also seem to like that as you increase the number of firms beyond two, the Cournot oligopoly price goes down to the competitive price.

The price with two sellers is below the monopoly price because there is an externality between the two sellers. With two sellers, Armen and Bengt, anytime Armen increases his quantity, that drives down the price that Bengt receives, hurting Bengt. This is sometimes called a pecuniary externality since Bengt gets hurt “through prices.”

We know that surplus may not be maximized anytime there is an externality, although it still could be.

Complementary Monopolies

The less well-known model by Cournot is about complementary monopolies. Let’s take McCloskey’s example of end-to-end railroads. Suppose that a railroad trip between me in Kennesaw, GA and Josh in Oxford, MS is broken up with a stop in Birmingham, AL. I am the only operator on the GA side, making me a monopoly, and Josh operates the MS side, making him a monopoly. Assume we don’t collude.

There is some demand for trips from Kennesaw to Oxford, but two different railroads operate the two legs of the trip. Our services are complementary, so consumers care about the total price (my price plus Josh’s price).

Unlike the duopoly model, we choose our prices. We set out prices at the same time, and then consumers decide how many trips to make. Again, the market is in equilibrium if our prices are optimal for each of us, given what the other is doing.

As Cournot pointed out, this arrangement will result in higher prices than if Josh and I merged into one company and became a single monopolist.

Cournot knew each of these results in 1838. Sonnenschein’s insight was that they are identical when viewed in the right light. The “it’s the same picture” meme above has the formal argument.

To understand how two models are the same, we can again consider the externalities between the two sellers.

Like for the duopoly case, Josh and I don’t achieve the jointly optimal price because there is an externality between us. When I raise my price, that pushes down people’s willingness to pay for Josh’s stretch. Therefore, since I’m not thinking about my externality on Josh, I raise my price more than would be jointly optimal. Josh does the same thing, and the price goes higher than the monopoly price. If the price is higher, there are fewer trips between Kennesaw and Oxford.

It wouldn’t just be better for Josh and me to merge, thus monopolizing the railroad market 😱; it would be better for consumers. More sellers are not necessarily better!

Notice that in both cases, we are talking about one person’s strategy affecting another person’s profit. We are outside of the world of perfect competition where competition is so fierce that no one can influence prices and no one generates an externality. In a future newsletter, I’ll write on when more competition, such as having more sellers in a market, can hurt consumers.

Speaking of monopolies, I have a piece out this week in RealClearMarkets on the Epic v. Apple lawsuit. The ruling made me worried about the future of antitrust. Check it out!