<p>Let's start by writing down the profit functions:
<span class="math-container">$$
\pi_1 = p_1 q_1 - c_1 q_1 = (p_1 - c_1) q_1 = (p_1 - c_1)(a_1 + b_{11} p_1+b_{12} p_2)
$$</span>
and similarlily:
<span class="math-container">$$
\pi_2 = (p_2 - c_2)(a_2 + b_{21} p_1 +b_{22} p_2)
$$</span></p>
<p>In a <strong>simultaneous equilibrium</strong>, both firms simultaneously maximize their profits taking the strategies of the players as given.</p>
<p>Doing this will give you a best response of player 1 that will be a function of <span class="math-container">$p_2$</span>: <span class="math-container">$p_1 = p_1^\ast(p_2)$</span>. Similarly, the best response of player 2 will be a function of <span class="math-container">$p_1$</span>: <span class="math-container">$p_2 = p_2^\ast(p_1)$</span>.
In a Nash equilibrium, you are looking for a solution for both equations simultaneously:
<span class="math-container">$$
p_1 = p_1^\ast(p_2),\\
p_2 = p_2^\ast(p_1).
$$</span></p>
<p>In a <strong>sequential equilibrium</strong> on the other hand, you assume that one firm, say 1, chooses <span class="math-container">$p_1$</span> first and next firm 2 decides in <span class="math-container">$q_2$</span>, after being informed about the value of <span class="math-container">$q_1$</span> chosen by firm 1.  Firm 1, however, is smart enough, such that it can predict the optimal action that firm 2 is going to take in stage 2.</p>
<p>As such, we solve the game by backward induction, first solving for firm 2, giving <span class="math-container">$p_2 = p_2^\ast(p_1)$</span> and then we substitute this into the profit function of firm 1.
<span class="math-container">$$
\pi_1 = (p_1 - c_1)(a_1 + b_{11} p_1+b_{12} p_2^\ast(p_1))
$$</span>
This substitution conforms to the fact that player 1 knows that by choosing <span class="math-container">$p_1$</span> it will influence the price of firm 2. So firm 1 takes the future behaviour of firm 2 into account when choosing its optimal level of <span class="math-container">$q_1$</span>.</p>
<p>Maximizing <span class="math-container">$\pi_1$</span> with respect to <span class="math-container">$p_1$</span> will give an optimal value <span class="math-container">$p_1^\ast$</span> (which will no longer be a function of <span class="math-container">$p_2$</span>).</p>
<p>Then, to compute the equilibrium, you substitute this value into the best response for firm 2:
<span class="math-container">$$
p_2 = p_2^\ast(p_1^\ast).
$$</span></p>

CC BY-SA 4.0
2021-04-28T18:03:33.397Z
2021-04-28T18:03:33.397Z
tdm
<p>I am attempting to solve the following problem</p>
<p>The demand functions for two substitute goods, the production cost of which equals <span class="math-container">$c_1$</span> and <span class="math-container">$c_2$</span>,
are <span class="math-container">$q_1 = a_1 + b_{11}p_1 + b_{12}p_2$</span> and <span class="math-container">$q_2 = a_2 + b_{21}p_1 + b_{22}p_2$</span>.</p>
<p>How can I find equilibrium under simultaneous and sequential interaction in a price oligopoly</p>
<p>(the values of the coefficients <span class="math-container">$a_1,a_2,b_{11} ...$</span> can be selected by yourself.
Although, if I'm not mistaken, the sign matter)</p>
<p>I need help in understanding the difference between simultanious and sequential interaction solution</p>

CC BY-SA 4.0
2021-04-23T23:01:31.443Z
Please do show your attempt.
Herr K.
CC BY-SA 4.0
2021-04-24T11:11:37.133Z
@HerrK, first of all I have to figure out the condition for substitute goods
Michael
CC BY-SA 4.0
2021-04-24T11:51:30.127Z
What is a price oligopoly?
Michael Greinecker
CC BY-SA 4.0
2021-04-24T12:04:06.113Z
@MichaelGreinecker, well, it's just oligopoly, price under oligopoly
Michael
CC BY-SA 4.0
2021-04-24T12:15:18.140Z
Firms choose quantities?
Michael Greinecker
CC BY-SA 4.0
2021-04-24T12:21:57.063Z
@MichaelGreinecker, yes
Michael
CC BY-SA 4.0
2021-04-24T15:52:31.087Z
Start by first calculating prices as a function of quantities, and then profits as a function of quantities.
Michael Greinecker
CC BY-SA 4.0
2021-04-24T19:00:44.763Z
@MichaelGreinecker, ok, I made calculations, what's next?
Michael
CC BY-SA 4.0
2021-04-24T19:44:31.177Z
You calculate each firm's profit  maximizing quantity given the other firm's choice. Solve the resulting equations and you are done .
Michael Greinecker
CC BY-SA 4.0
2021-04-24T19:46:47.673Z
@MichaelGreinecker, what about simultanious and sequential interaction?
Michael
CC BY-SA 4.0
2021-04-24T20:32:50.410Z
That gives you the simultaneous version. For the Otter, the second one to player faces the same problem. You can substitute the solutution into the maximization problem of the first. But you should have seen this  before.
Michael Greinecker
<competitive-equilibrium><demand><oligopoly><simultaneous-equation-model>
Equilibrium with substitute goods
Michael
# Equilibrium with substitute goods

(Asked by: username_0 on 2021-04-23)

I am attempting to solve the following problem

The demand functions for two substitute goods, the production cost of which equals $c\_1$ and $c\_2$,
are $q\_1 = a\_1 + b\_{11}p\_1 + b\_{12}p\_2$ and $q\_2 = a\_2 + b\_{21}p\_1 + b\_{22}p\_2$.

How can I find equilibrium under simultaneous and sequential interaction in a price oligopoly

(the values of the coefficients $a\_1,a\_2,b\_{11} ...$ can be selected by yourself.
Although, if I'm not mistaken, the sign matter)

I need help in understanding the difference between simultanious and sequential interaction solution

**Tags:** competitive-equilibrium, demand, oligopoly, simultaneous-equation-model

> **Comments:**
> **username_1**: Please do show your attempt.  
> **username_0**: @HerrK, first of all I have to figure out the condition for substitute goods  
> **username_2**: What is a price oligopoly?  
> **username_0**: @username_0Greinecker, well, it's just oligopoly, price under oligopoly  
> **username_2**: Firms choose quantities?  
> **username_0**: @username_0Greinecker, yes  
> **username_2**: Start by first calculating prices as a function of quantities, and then profits as a function of quantities.  
> **username_0**: @username_0Greinecker, ok, I made calculations, what's next?  
> **username_2**: You calculate each firm's profit  maximizing quantity given the other firm's choice. Solve the resulting equations and you are done .  
> **username_0**: @username_0Greinecker, what about simultanious and sequential interaction?  
> **username_2**: That gives you the simultaneous version. For the Otter, the second one to player faces the same problem. You can substitute the solutution into the maximization problem of the first. But you should have seen this  before.  

---

## 1 answers

### Answer by username_3 on 2021-04-28. Score: 1

Let's start by writing down the profit functions:
$$
\pi\_1 = p\_1 q\_1 - c\_1 q\_1 = (p\_1 - c\_1) q\_1 = (p\_1 - c\_1)(a\_1 + b\_{11} p\_1+b\_{12} p\_2)
$$
and similarlily:
$$
\pi\_2 = (p\_2 - c\_2)(a\_2 + b\_{21} p\_1 +b\_{22} p\_2)
$$

In a **simultaneous equilibrium**, both firms simultaneously maximize their profits taking the strategies of the players as given.

Doing this will give you a best response of player 1 that will be a function of $p\_2$: $p\_1 = p\_1^\ast(p\_2)$. Similarly, the best response of player 2 will be a function of $p\_1$: $p\_2 = p\_2^\ast(p\_1)$.
In a Nash equilibrium, you are looking for a solution for both equations simultaneously:
$$
p\_1 = p\_1^\ast(p\_2),\\
p\_2 = p\_2^\ast(p\_1).
$$

In a **sequential equilibrium** on the other hand, you assume that one firm, say 1, chooses $p\_1$ first and next firm 2 decides in $q\_2$, after being informed about the value of $q\_1$ chosen by firm 1. Firm 1, however, is smart enough, such that it can predict the optimal action that firm 2 is going to take in stage 2.

As such, we solve the game by backward induction, first solving for firm 2, giving $p\_2 = p\_2^\ast(p\_1)$ and then we substitute this into the profit function of firm 1.
$$
\pi\_1 = (p\_1 - c\_1)(a\_1 + b\_{11} p\_1+b\_{12} p\_2^\ast(p\_1))
$$
This substitution conforms to the fact that player 1 knows that by choosing $p\_1$ it will influence the price of firm 2. So firm 1 takes the future behaviour of firm 2 into account when choosing its optimal level of $q\_1$.

Maximizing $\pi\_1$ with respect to $p\_1$ will give an optimal value $p\_1^\ast$ (which will no longer be a function of $p\_2$).

Then, to compute the equilibrium, you substitute this value into the best response for firm 2:
$$
p\_2 = p\_2^\ast(p\_1^\ast).
$$


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