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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 $Q$.

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  • $\begingroup$ 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. $\endgroup$
    – Herr K.
    Mar 27 at 4:46

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