<p>Just looking at the setup, everything is symmetric. This means that if they were to move simultaneously then we should have <span class="math-container">$p_1 = p_2$</span>, and the payoffs for the two firms the same.</p>

<p>According to your calculation, <span class="math-container">$(p^*_1,p^*_2)=(\frac{a}{2},\frac{a}{4})$</span>. Since <span class="math-container">$Q_1 = Q_2 \Rightarrow \pi_1 &gt; \pi_2$</span>. So it's quite clear that in this case, the first mover has the advantage.</p>

<p>Intuitively, you can figure out which mover has the advantage by simply asking, would the first mover get higher payoff if he/she moves second?</p>

<p>Mathematically, if strategies are strategic substitute, then you can show that the first mover will have the advantage. Reference <a href="https://www.jstor.org/stable/2555388" rel="nofollow noreferrer">here</a>.</p>

<p>Added later: one interesting way (I think) to look at this is this is just like Cournot competition, just with price and quantity switched.</p>

CC BY-SA 4.0
2019-09-18T18:10:09.973Z
Did you verify that p1 < p2? It seems puzzling that Firm 2 would need to offer a different price, given perfect complementarity with identical firms.  Perhaps my math is out, but with the OP's corrected demand function I find that p1 = p2 = a/3.
heh
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2019-09-18T23:51:17.730Z
Can verify by plugging things in. $\pi_1(a/3, a/3)=a^2/9$. But $\pi_1(a/2, a/4)=a^2/8$.
Art
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2019-09-18T23:52:03.197Z
P1>P2, you mean?
Art
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2019-09-19T14:14:52.660Z
On the first round of this game, Firm 1 would solve its profit maxing problem to find that $p^*_1 = a - p_2 /2$.  If we assume that both demand functions are common knowledge, then Firm 1 will anticipate that $p^*_2 = a - p_1 /2$ because the firms are identical.  Plugging this in to Firm 1's optimal decision yields $p^*_1 = a/3$, which in turn leads Firm 2 to choose $p^*_2 = a/3$.
heh
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2019-09-19T14:15:42.610Z
In other words, I believe the OP has the wrong set of prices, and was wondering if you had checked his answer or taken it as given.
heh
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2019-09-19T14:37:44.857Z
Might wanna check your work. $p_2^* = (a-p_1)/2$. This means firm 1 maximizes $p_1\cdot(a - p_1 - (a-p_1)/2) = p1 \cdot (a - p_1 - a/2 + p_1/2) = p_1 \cdot (a/2 - p_1/2) = a \cdot p_1/2 - p_1^2/2$. FOC is $p_1 = a/2$.
Art
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2019-09-19T14:51:22.837Z
$Q_1 = a - (p_1+p_2)$, so $\pi_1 = p_1 a - p_1^2 - p_1 p_2$. Then, $\frac{d\pi_1}{dp_1} = a - 2p_1 - p_2$. Setting to zero yields $p^*_1 = \frac{a - p_2}{2}$. Identical firms, so $p^*_2 = \frac{a - p_1}{2}$. 

Firm 1 anticipates this by assumption, therefore$p^*_1 = \frac{a - \frac{a - p_1}{2}}{2}$ which simplifies to $p^*_1 = \frac{2a - a + p_1}{4}$ and leads to $p^*_1 = \frac{a}{3}$.

Then Firm 2 takes this as given, yielding $p^*_2 = \frac{a}{3}$
heh
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2019-09-19T14:55:21.557Z
See, that's the whole point. You need to solve for $p_2$ as a function of $p_1$ first. Then you solve $p_1$. It's not true that $p_1^*=(a-p_2)/2$ since $p_1$ is the first mover. Let's end our discussion here :)
Art
CC BY-SA 4.0
2019-09-18T16:14:12.177Z
2019-09-19T00:08:35.667Z
2019-09-19T00:08:35.667Z
Art
<ul>
<li>There are two goods, <span class="math-container">$1$</span> and <span class="math-container">$2$</span> produced by two firms at zero marginal costs. The goods are perfect complements. The demand for each goods is: <span class="math-container">$Q_1=Q_2=a-(p_1+p_2)$</span>. The prices are set sequentially, with the leader firm (<span class="math-container">$1$</span>) credibly committing to a price which the follower takes as given. Determine the equilibrium prices, quantities and profits. </li>
</ul>

<p>I understand that in standard Stackelberg games of prices, the follower has a second mover advantage. For the above question, I found that the reaction curves turned out to be downward sloping, i.e. prices are strategic substitutes. <span class="math-container">$(p^*_1,p^*_2)=(\frac{a}{2},\frac{a}{4})$</span>. 
Would the second mover advantage still hold? Evaluating the leader's profit function at Bertrand prices would show that the leader has an incentive to set a price above <span class="math-container">$p^B_1$</span> since prices are strategic substitutes and goods are perfect complements (evaluating <span class="math-container">$\frac{\partial\pi_1^L}{\partial p_1}$</span> at bertrand prices). 
Is this the correct method to evaluate second mover/first mover advantage? </p>

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2019-09-18T15:50:37.790Z
What do you mean by $Q_1,Q_2=a-(p_1,p_2)$?
Herr K.
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2019-09-18T15:52:08.600Z
@HerrK. I'm sorry for the error. I've edited the question.
S.Rana
CC BY-SA 4.0
2019-09-18T15:57:36.010Z
The left hand side is still unclear. Did you mean $Q_1=Q_2=\cdots$ or $Q_1+Q_2=\cdots$?
Herr K.
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2019-09-18T15:58:35.193Z
@HerrK. Edited! Thanks for pointing it out.
S.Rana
CC BY-SA 4.0
2019-09-18T16:33:51.487Z
I don't think Stackelberg guarantees second-mover advantage. One way to think about turn-advantage in a sequential game is to ask whether the outcome would change if the turn order changed. I notice you changed the demand function - you may want to rework the optimal prices you found. It would be odd for them to be different in this context (identical firms whose revenues are coupled via perfect complementarity).
heh
<game-theory><self-study><industrial-organisation><dynamic-programming>
Sequential Price Competition for Perfect Complements
S.Rana
# Sequential Price Competition for Perfect Complements

(Asked by: username_0 on 2019-09-18)

* There are two goods, $1$ and $2$ produced by two firms at zero marginal costs. The goods are perfect complements. The demand for each goods is: $Q\_1=Q\_2=a-(p\_1+p\_2)$. The prices are set sequentially, with the leader firm ($1$) credibly committing to a price which the follower takes as given. Determine the equilibrium prices, quantities and profits.

I understand that in standard Stackelberg games of prices, the follower has a second mover advantage. For the above question, I found that the reaction curves turned out to be downward sloping, i.e. prices are strategic substitutes. $(p^\*\_1,p^\*\_2)=(\frac{a}{2},\frac{a}{4})$.
Would the second mover advantage still hold? Evaluating the leader's profit function at Bertrand prices would show that the leader has an incentive to set a price above $p^B\_1$ since prices are strategic substitutes and goods are perfect complements (evaluating $\frac{\partial\pi\_1^L}{\partial p\_1}$ at bertrand prices).
Is this the correct method to evaluate second mover/first mover advantage?

**Tags:** game-theory, self-study, industrial-organisation, dynamic-programming
> **Comments:**
> **username_1**: What do you mean by $Q_1,Q_2=a-(p_1,p_2)$?  
> **username_0**: @HerrK. I'm sorry for the error. I've edited the question.  
> **username_1**: The left hand side is still unclear. Did you mean $Q_1=Q_2=\cdots$ or $Q_1+Q_2=\cdots$?  
> **username_0**: @HerrK. Edited! Thanks for pointing it out.  
> **username_2**: I don't think Stackelberg guarantees second-mover advantage. One way to think about turn-advantage in a sequential game is to ask whether the outcome would change if the turn order changed. I notice you changed the demand function - you may want to rework the optimal prices you found. It would be odd for them to be different in this context (identical firms whose revenues are coupled via perfect complementarity).  

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## 1 answers

### Answer by username_3 on 2019-09-18. Score: 1 | Accepted answer

Just looking at the setup, everything is symmetric. This means that if they were to move simultaneously then we should have $p\_1 = p\_2$, and the payoffs for the two firms the same.

According to your calculation, $(p^\*\_1,p^\*\_2)=(\frac{a}{2},\frac{a}{4})$. Since $Q\_1 = Q\_2 \Rightarrow \pi\_1 > \pi\_2$. So it's quite clear that in this case, the first mover has the advantage.

Intuitively, you can figure out which mover has the advantage by simply asking, would the first mover get higher payoff if he/she moves second?

Mathematically, if strategies are strategic substitute, then you can show that the first mover will have the advantage. Reference [here](https://www.jstor.org/stable/2555388).

Added later: one interesting way (I think) to look at this is this is just like Cournot competition, just with price and quantity switched.

> **Comments:**
> **username_2**: Did you verify that p1 < p2? It seems puzzling that Firm 2 would need to offer a different price, given perfect complementarity with identical firms.  Perhaps my math is out, but with the OP's corrected demand function I find that p1 = p2 = a/3.  
> **username_3**: Can verify by plugging things in. $\pi_1(a/3, a/3)=a^2/9$. But $\pi_1(a/2, a/4)=a^2/8$.  
> **username_3**: P1>P2, you mean?  
> **username_2**: On the first round of this game, Firm 1 would solve its profit maxing problem to find that $p^*_1 = a - p_2 /2$.  If we assume that both demand functions are common knowledge, then Firm 1 will anticipate that $p^*_2 = a - p_1 /2$ because the firms are identical.  Plugging this in to Firm 1's optimal decision yields $p^*_1 = a/3$, which in turn leads Firm 2 to choose $p^*_2 = a/3$.  
> **username_2**: In other words, I believe the OP has the wrong set of prices, and was wondering if you had checked his answer or taken it as given.  
> **username_3**: Might wanna check your work. $p_2^* = (a-p_1)/2$. This means firm 1 maximizes $p_1\cdot(a - p_1 - (a-p_1)/2) = p1 \cdot (a - p_1 - a/2 + p_1/2) = p_1 \cdot (a/2 - p_1/2) = a \cdot p_1/2 - p_1^2/2$. FOC is $p_1 = a/2$.  
> **username_2**: $Q_1 = a - (p_1+p_2)$, so $\pi_1 = p_1 a - p_1^2 - p_1 p_2$. Then, $\frac{d\pi_1}{dp_1} = a - 2p_1 - p_2$. Setting to zero yields $p^*_1 = \frac{a - p_2}{2}$. Identical firms, so $p^*_2 = \frac{a - p_1}{2}$.   Firm 1 anticipates this by assumption, therefore$p^*_1 = \frac{a - \frac{a - p_1}{2}}{2}$ which simplifies to $p^*_1 = \frac{2a - a + p_1}{4}$ and leads to $p^*_1 = \frac{a}{3}$.  Then Firm 2 takes this as given, yielding $p^*_2 = \frac{a}{3}$  
> **username_3**: See, that's the whole point. You need to solve for $p_2$ as a function of $p_1$ first. Then you solve $p_1$. It's not true that $p_1^*=(a-p_2)/2$ since $p_1$ is the first mover. Let's end our discussion here :)  

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