Mostly for fun, and teaching. But I think it might matter.

A village had two cobblers. The right-handed cobbler was best at making right shoes. The left-handed cobbler was best at making left shoes. So the villagers would buy one shoe from each cobbler.

Other villages were some distance away, so the cobblers had monopoly power and faced downward-sloping demand curves for their shoes. Each cobbler set a price to maximise individual profits, taking the other cobbler's price as given. The cobblers were in a Bertrand equilibrium, except the two goods are strict complements, rather than substitutes as is normally assumed.

Then one day the two cobblers fell in love, got married, and set the price for a pair of shoes to maximise their joint profits. They set a price that was lower than the price of a pair of shoes when they were single. The cobblers were happy because their profits were higher. And the other villagers were happy too, because the price of shoes was lower.

Is this (one reason) why firms exist?

Math appendix:

Marginal Revenue is (1-(1/E))P where E is Elasticity of demand and P is Price. A profit-maximising firm wants Quantity sold where Marginal Revenue equals Marginal Cost, and so sets price at P = [1/(1-(1/E))]MC.

The demand function for shoes is Q_{R} = Q_{L} = D(P_{R}+P_{L}), where (P_{R}+P_{L}) is the price of a pair of shoes. If P_{R}=P_{L} initially, a 1% cut in P_{R}, holding P_{L} constant, will cause only a 0.5% cut in (P_{R}+P_{L}), so the elasticity of demand for right shoes (or left shoes) is half the elasticity of demand for pairs of shoes. So when the two cobblers merge, elasticity doubles, and they set Price as a smaller markup over Marginal Cost.

In the initial Bertrand equilibrium, a small cut in P_{R}, holding P_{L} constant, will have a negligible effect on profits from right shoes (since we are starting at a point where P_{R} maximises profits from right shoes). But it will increase the demand for left shoes, and so increase profits from left shoes. So if the two cobblers collude, they would want to collude to cut prices. And each individual cobbler has an incentive to cheat on the collusive agreement by raising price above the joint-profit-maximising price. It's the opposite of the standard case where the two goods are substitutes.

We could relax the assumption of strict complementarity. Results would be similar in any example where the two goods are complements rather than substitutes. Because if one firm cuts price it increases (rather than reduces) the other firm's demand and profits. Mergers and collusion will result in lower rather than higher markups of price over marginal cost.

[I vaguely remember reading about a river in Germany(?) where all the jurisdictions along the river collected a toll on passing boats. And how the sum of the individual revenue-maximising tolls was bigger than the joint revenue-maximising toll. But that's all I can remember. **Update**: Alex Tabarrok on Twitter tells me it's the "Rhine River Problem", originally from Cournot. In Cournot equilibrium firms set quantities taking other firms' quantities as given (in Bertrand equilibrium firms set prices taking other firms' prices as given). The Cournot equilibrium for the two cobblers is a corner solution where both sell zero shoes (because selling slightly fewer right than left shoes means left shoes have a zero price so the right cobbler gets all the revenue).]

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