# Sequential Price Competition for Perfect Complements

• 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?

• What do you mean by $Q_1,Q_2=a-(p_1,p_2)$? – Herr K. Sep 18 '19 at 15:50
• @HerrK. I'm sorry for the error. I've edited the question. – S.Rana Sep 18 '19 at 15:52
• The left hand side is still unclear. Did you mean $Q_1=Q_2=\cdots$ or $Q_1+Q_2=\cdots$? – Herr K. Sep 18 '19 at 15:57
• @HerrK. Edited! Thanks for pointing it out. – S.Rana Sep 18 '19 at 15:58
• 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 Sep 18 '19 at 16:33

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.

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

• 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 Sep 18 '19 at 18:10
• 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 Sep 18 '19 at 23:51
• P1>P2, you mean? – Art Sep 18 '19 at 23:52
• 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 Sep 19 '19 at 14:14
• 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 Sep 19 '19 at 14:15