# Calculating Nash Equilibrium prices for Bertrand duopolists

I am attempting to solve the following problem.

Suppose that firms' marginal and average costs are constant and equal to c and that inverse market demand is given by $P = a - bQ$ where $a,b > 0$.

Calculate the Nash Equilibrium prices for Bertrand duopolists, which choose prices for their identical products simultaneously.

Now I attempted to solve this problem and got $P_1 = P_2 = \frac{a+c}{2}$ where $P1, P2$ are prices.

My professor lists the answer as $P_1 = P_2 = c$.

Can someone please tell me where I messed up? Thanks!

## 1 Answer

The important thing to remember here is that Bertrand duopolists compete over price, not quantity. This means that in the game each player sets a price and the quantity sold is then determined by the demand curve. In this game the firm with the lowest price sells to the entire market and if both firms have the same price they each sell to half of the market. Now, we have to think about the logic that goes into determining equilibrium. It is easy to see that neither firm will ever set a price lower than $c$ because setting a price lower than marginal cost will give the firm negative profit. So we have to start that it must be the case that $p_1, p_2 > c$. Now, suppose that $p_1 > p_2 > c$. This cannot happen because firm 1 has an incentive to set a price below that of firm 2 and above $c$ so they will take the whole market. The same logic can be applied to the case where $p_2 > p_1 > c$. Now, consider the case where $p_1 > p_2 = c$. In this case firm 1 has the incentive to set there price equal to $c$ and split the market with firm 2, so this cannot be a Nash equilibrium. Again, we can apply the same logic to the case where $p_2 > p_1 = c$. So, the only candidate left for Nash equilibrium is scenario where $p_1 = p_2 = c$. This result is very interesting considering that with one firm we get a profit maximizing monopoly result but by adding just one additional firm we get the same pricing as perfect competition as long as the firms are playing the Bertrand price competition game.