Economist Writing Every Day

This post solves for the equilibrium quantity of production with quadratic total cost under Cournot and Stackelberg competition.

Say that there are two firms. They produce the exact same quality and type of goods and sell them at the same price. Let’s also assume that the market clears at one price. Finally, let’s assume increasing marginal costs.

Let’s say that they face the following demand curve:

The firms have a total cost of:

The marginal cost is the derivative with respect to the choice variable for each firm, or their respective quantities produced:

The total revenue is just the price times the quantity sold.

This is all standard fare for economic modeling. You’re free to make different assumptions. You can even adopt different slopes in the demand curve to reflect goods with different characteristics.

Cournot Competition

If you imagine a lengthy production process, or otherwise that they physically attend the same market, then it’s reasonable to assume that they don’t know one another’s choice of quantity produced.

We know how firms maximize profit: They produce the quantity at which the marginal revenue equals the marginal cost. But, what is marginal revenue? The derivative of total revenue with respect to the choice variable:

Now we can set the marginal revenue equal to marginal cost and solve for the optimal level of output:

Notice that the optimal level of output depends on the production decision of the other firm. These are called response functions. If we solve for the quantities at which they intersect, then we are solving for where both firms are producing the best response to one another. This is known as a Pure Strategy Nash Equilibrium (PSNE).

Luckily, in many applications, one or more of the above terms are zeros, which makes things much simpler.

The general process for solving for the Cournot equilibrium is:

1. Set MR=MC to find the response functions.
2. Find where the response functions intersect.

Stackelberg Competition

But what if one of the firms knows what the other firm will produce. This would be like doing your market analysis before you enter the market and is not unreasonable. Firm 1 already exists in the market and chooses their level of output first. Firm 2 chooses their level of output second. They both still sell in the same market.

This case is slightly more conceptually intensive. Firm 1 sees that firm 2 will enter and so produces in anticipation of how firm 2 will respond. How will firm 2 respond? As it turns out, firm 2 will respond in exactly the same manner as in the Cournot market.

Firm 1 will consider this anticipated response before choosing their own optimal level of production. Mathematically, we substitute firm 2’s level of output into firm 1’s total revenue function (they are forward looking).

Now, with the total revenue function that includes the anticipated response by firm 2, we can take the derivative with respect to the firm 1’s choice variable to get the marginal revenue.

Now we can set firm 1’s marginal revenue equal to marginal cost and solve for the optimal level of output:

Now that we know how much firm 1 produces, we simply plug it into firm 2’s optimal response to get their level of production. If you graph q1 against any of the parameters, then you’ll see that q1 is linear (affine) in I, x1, & x2 and nonlinear in the other terms.

Again, things get easier if any one of these terms is zero.

This is how we solve all simultaneous (Cournot) and finitely sequential (Stackelberg) games. In this particular case, we have a demand function along with revenue and costs. But setting the marginal revenue equal to the marginal cost is the same as taking the derivative of the payoff function, then setting it equal to zero to solve for the response function. The take away is that, given complete enough information by the two firms about the relevant variables, we and they can identify the outcome of competition.

However, a super important caveat is that this analysis requires zero chance of other firms entering the market. As the expected number of additional entrants increases, we get closer and closer to the perfectly competitive equilibrium.