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Duopoly and Cournot Equilibrium
*December 12, 2012*

*Posted by tomflesher in Micro, Teaching.*

Tags: Cournot, duopoly, equilibrium, intermediate microeconomics, market week

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Tags: Cournot, duopoly, equilibrium, intermediate microeconomics, market week

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A few days ago, we discussed perfectly competitive markets; yesterday, we talked about monopolistic markets. Now, let’s expand into a case in between – a **duopolistic**, or two-seller, market. This is usually called a **Cournot problem**, after the economist who invented it.

We’ll maintain the assumption of **identical goods**, so that consumers won’t be loyal to one company or the other. We’ll also assume that each company has the same costs, so we’re looking at **identical firms** as well. Finally, assume that there are a lot of buyers, so the firms face a **market demand** of, let’s say, Q^{D}(P) = 500 – 2P, so P = 250 – Q^{D}(P)/2. Since the firms are producing the same goods, then Q^{S}(P) = q_{1}(P) + q_{2}(P).

Neither firm knows what the other is doing, but each firm knows the other is identical to it, and each firm knows the other knows this. Even though neither firm knows what’s going on behind the scenes, they’ll assume that a firm facing the same costs and revenues is rational and will optimize its own profit, sothey can make good, educated guesses about what the other firm will do. Each firm will determine the other firm’s likely course of action and compute its own **best response**. (That’s the one that maximizes its profit.)

Now, let’s take a look at what the firms’ profit functions will look like.

Recall that Total Profit = Total Revenue – Total Cost, and that Marginal Profit = Marginal Revenue – Marginal Cost. Companies will choose quantity to optimize their profit, so they’ll continue producing until their expected Marginal Profit is 0, and then produce no more. Firm 1’s total revenue is Pxq_{1} – revenue is always price times quantity. Keeping in mind that price is a function of quantity, we can rewrite this as (250 – Q^{D}(P)/2)xq_{1}. Since Q^{D}(P) = q_{1} + q_{2}, this is the same as writing (250 – (1/2)(q_{1} + q_{2}))q_{1}. Then, we need to come up with a **total cost** function. Let’s say it’s 25 + q_{1}^{2}, where 25 is a **fixed cost** (representing, say, rent for the factory) and q_{1}^{2} is the **variable cost** of producing each good. Then, Firm 1’s profit function is:

Profit_{1} = (250 – (1/2)(q_{1} + q_{2}))q_{1} – 25 – q_{1}^{2}

or

Profit_{1} = (250 – q_{1} /2 – q_{2}/2)q_{1} – 25 – q_{1}^{2}

or

Profit_{1} = 250q_{1} – q_{1}^{2}/2 – q_{1}q_{2}/2 – 25 – q_{1}^{2}

The marginal profit is the change in the total profit function if Firm 1 produces one more unit; in this case it’s easier to just use the calculus concept of taking a derivative, which yields

Marginal Profit_{1} = 250 – q_{1} – q_{2}/2 – 2q_{1} = 250 – 3q_{1} – q_{2}/2

Since the firms are identical, though, firm 1 knows that firm 2 is doing the same optimization! So, q_{1} = q_{2}, and we can substitute it in.

Marginal Profit_{1} = 250 – 3q_{1} – q_{1}/2 = 250 – 5q_{1}/2

This is 0 where 250 = 5q_{1}/2, or where q_{1} = 100. Firm 2 will also produce 100 units. Total supplied quantity is then 200, and total price will be 200. We can figure out each firm’s profit simply by plugging in these numbers:

Total Revenue = Pxq_{1}^{2} = 200×100 = 20,000

Total cost = 25 + q_{1}^{2} = 25 + 100×100 = 25 + 10,000 = 10,025

Total Profit = 9,075

This was a bit heavier on the mathematics than some of the other problems we’ve talked about, but all that math is just getting to one big idea: it’s rational to produce when you expect your marginal benefit to be at least as much as your marginal cost.