<p>You have the profit function of firm 2 in terms of prices <span class="math-container">$p_1$</span> and <span class="math-container">$p_2$</span>.
Then, you can find <span class="math-container">$p_2^*(p_1)$</span>, the optimal reaction of firm 2 for any observed price <span class="math-container">$p_1$</span>.
Firm 1 anticipates this reaction. So you can plug <span class="math-container">$p_2^*(p_1)$</span> into the profit function of firm 1 and maximize it with respect to <span class="math-container">$p_1$</span>.</p>

CC BY-SA 4.0
2019-03-20T21:39:01.373Z
2019-03-20T21:39:01.373Z
Bayesian
<p>In a Bertrand competition with differentiated goods where firms set the prices sequentially, we have the following demand functions:</p>

<p>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.</p>

<p>q1 = 16 - 2*p1 + p2
q2 = 16 - 2*p2 + p1</p>

<p>The marginal cost is 4. No fixed costs.</p>

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

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

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

<p>How can I write the strategies of firm 2?</p>

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

<p>How is p1 = p2 = 8 a Nash equilibrium?</p>

CC BY-SA 4.0
2019-03-20T22:59:36.740Z
Please consider formatting mathematical notations using [MathJax](https://math.meta.stackexchange.com/q/5020/70119).
Herr K.
CC BY-SA 4.0
2019-03-29T20:51:40.183Z
It's about Bertrand competition, how can it be off-topic in economics? It's just about economics.
Coder88
CC BY-SA 4.0
2019-03-29T21:44:18.707Z
"**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."
Herr K.
<game-theory><nash-equilibrium><dynamic-games>
Sequential Bertrand game with differentiated goods, how to write the strategies of firm 2
Coder88
# Sequential Bertrand game with differentiated goods, how to write the strategies of firm 2

(Asked by: username_0 on 2019-03-20)

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?

**Tags:** game-theory, nash-equilibrium, dynamic-games

> **Comments:**
> **username_1**: Please consider formatting mathematical notations using [MathJax](https://math.meta.stackexchange.com/q/5020/70119).  
> **username_0**: It's about Bertrand competition, how can it be off-topic in economics? It's just about economics.  
> **username_1**: "**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."  

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## 1 answers

### Answer by username_2 on 2019-03-20. Score: 1 | Accepted answer

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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