# How to solve the Bertrand model when marginal costs are different and not constant?

Find the equilibrium in the Bertrand model with two firms, with total costs given by:

$$TC_1(q_1) = \alpha q_{1}^2$$

$$TC_2(q_2) = \beta q_{2}^2$$

Inverse demand is given by

$$P = A - Q$$,

where $$Q = q_1 + q_2$$.

Here $$\alpha, \beta, A > 0$$ are parameters.

Here I will show a numerical example with a method I was taught in Intermediate Microeconomics, but it only seems to work when both firms have the same constant marginal cost:

$$TC_i(q_i) = 4q_i$$, $$i \in \{1,2\}$$.

$$P = 100 - Q$$.

Set $$P = MC$$

$$\implies 100-Q = 4 \implies Q^\star = 96 \implies q_i^\star = \frac{Q^\star}{2} = 48$$

So in this example the equilibrium is given by

$$P^\star = 4, q_1^\star = 48, q_2^\star = 48$$.

The profits of the firms are

$$\Pi_i^\star = 4 \cdot 48 - 4 \cdot 48 = 0$$.

However, in the case of the two quadratic functions, how do I even set $$P = MC$$ since both marginal costs are different?

And even if $$\alpha = \beta$$, any marginal cost function would be in terms of a single $$q_i$$, not the big $$Q$$.

• If I'm not mistaken, when $\alpha\ne\beta$ there is no pure strategy Nash equilibrium. There are mixed ones, but they are difficult to find/prove. If $\alpha=\beta$, you can find a range of pure strategy Nash equilibria. See Chapter 5 of Vives (1999) for a thorough discussion on both cases. There is also an answer here for a mixed strategy Nash equilibrium in the case of constant but asymmetric MCs. Mar 27 at 4:46