In a Bertrand competition with differentiated goods where firms set the prices sequentially, we have the following demand functions:

q1 is quantity of goods demanded for firm 1 q2 is quantity of goods demanded for firm 2 p1 and p2 are prices of goods for firm 1 and firm 2.

q1 = 16 - 2*p1 + p2 q2 = 16 - 2*p2 + p1

The marginal cost is 4. No fixed costs.

The profit function for firm 1 is: TR1 = p1*(16 - 2*p1 + p2) - (16 - 2*p1 + p2)*4

The profit function for firm 2 is: TR2 = p2*(16 - 2*p2 + p1) - (16 - 2*p2 + p1)*4

Firm 1 sets the price first, firm 2 sets the price after. It's a squential game.

How can I write the strategies of firm 2?

Isn't it just S(p1) = 6 + 1/4*p1 ?

How is p1 = p2 = 8 a Nash equilibrium?

  • $\begingroup$ Please consider formatting mathematical notations using MathJax. $\endgroup$
    – Herr K.
    Mar 20 '19 at 22:59
  • $\begingroup$ It's about Bertrand competition, how can it be off-topic in economics? It's just about economics. $\endgroup$
    – Coder88
    Mar 29 '19 at 20:51
  • $\begingroup$ "This question does not meet the standards for homework questions as spelled out in the relevant meta posts. For more information, see our policy on homework question and the general FAQ." $\endgroup$
    – Herr K.
    Mar 29 '19 at 21:44

You have the profit function of firm 2 in terms of prices $p_1$ and $p_2$. Then, you can find $p_2^*(p_1)$, the optimal reaction of firm 2 for any observed price $p_1$. Firm 1 anticipates this reaction. So you can plug $p_2^*(p_1)$ into the profit function of firm 1 and maximize it with respect to $p_1$.


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