The Tale of the Two Cobblers (collusive price cuts)

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).]

Comments

[Frances Woolley]

A jargon-filled way of making this point would be to talk about "internalizing pecuniary externalities" and then mumbling something about Coase.

A variation on this theme is the story of Disneyland/Disneyworld. Disney corp owns Disneyland, but not the land adjacent to it. Hence the area around Disneyland is filled with hotels, the market for hotels in the Anaheim area is competitive, and it's possible to get a hotel close to Disneyland at a reasonable rate. When Disney Corp built Disneyworld, they didn't want the same thing to happen again, so they built Disneyworld in what was, I think, a swamp in the middle of nowhere, and purchased all of the land for a long way around.

Disneyland: theme park with market power surrounded by hotels with little market power. Disneyworld: theme park market power leveraged into hotel market power..

Your post imagines a situation where Disneyworld has market power in theme park market, and there's another, entirely separate, corp that owns all of the land around Disneyworld and has market power in the hotel market. I think that kind of a situation seems counterintuitive precisely because, as you say, there's such a strong incentive for the two entities to merge to form one big firm, hence these kinds of situations are rarely observed in the real world.

[Nick Rowe]

Frances: "I think that kind of a situation seems counterintuitive precisely because, as you say, there's such a strong incentive for the two entities to merge to form one big firm, hence these kinds of situations are rarely observed in the real world."

Exactly! It's like all "origin" fables: the point of the fable is to explain why the "before" scenario is unlikely.

(Biggest flaw with my fable here is that transactions costs would also be an explanation for why shoes are sold in pairs. It's hard to get a perfectly matching pair of shoes if you order right and left separately. Your Disney example works better in that respect. But it's easier to do the math with my shoe example.)

[Matthew Young]

QR = QL

This was not true before the merger. The buyer bought the shoes at different rate and quantities, the elasticities were different.

So a better look than price, or quantity is household inventory of shoes. Once we have a figure on household shelf cost, we can count the aggregate savings after the merger. I use transaction rate and quantity. The least elastic shoe, the right, comes four per purchase. The most elastic, the left, is one shoe per purchase.

Example: Normally I have two conditions in inventory, I have one pair of shoe in my closet, and my next transaction will buy a second pair, or I have two pairs, and I delay my next purchase. My closet holds 2 purchases, maximum, four shoes.

In the other case, I am buying one left shoe at a time and four right shoes, that would be a typical difference in elasticity. In my closet I will have either one or two left shoes, and either 1,2,3,4.5 right shoes. Because, if I have one left and one right, soon I will buy the next batch of four. So, using all the assumptions of the model (counting shelf costs is proportional to shoe costs), I need about twice the inventory space.

[Majromax]

> This was not true before the merger. The buyer bought the shoes at different rate and quantities, the elasticities were different.

Not systematically. The representative agent has equal elasticities and quantities for left and right shoes.

A priori, left and right shoes see equal wear and suffer equal risk of loss or damage. So for an individual agent, the utility of a new single shoe is very low if it would make one's collection more lopsided, since having multiple left shoes (for example) but only one right shoe protects just against an unlikely coincidence of loss or damage.

Because the utility function has such a sharply discontinuous derivative at perfectly matched pairs, we can further expect this representative-agent truth to hold for most individual agents.

[Nick Rowe]

Majro: yep. The indifference curves are L-shaped.

[C Trombley]

It seems like this is similar to "double marginalization". Usually you see double marginalization in supply chains, like a monopolistic cobbler and a monopolistic shoelace maker. The moral is the same - the cooperative solution is more efficient for everyone.

[Nick Rowe]

CT: Yes. Alex Tabarrok on Twitter mentioned the "double marginalisation" problem. One slight difference though is that the two cobblers sell direct to the public. If the right cobbler sold right shoes to the left cobbler, who then sold pairs of shoes to the public, we would have bilateral monopoly/monopsony between the right and left cobblers.

[Matthew Young]

I was thinking the same thing, Nick.

What if our society had a left leaning problem, they really did wear one shoe out faster than the other? Then we get intermediaries, distributors willig to run around and swap excess inventories for deficit inventories. We get, at an extra cost, a more granular flow of shoe with the additional step in distribution.

[rsj]

Nick,

This is pretty common once you notice the amount of "bundling" that happens. Eg. E.g. I can get internet + cable for $100 or just internet for $90 and cable TV for $50. If I get both internet access and my mobile access from Verizon, it's cheaper than getting each separately. I can get deals on hotel + airfare that are cheaper than individual hotel and airfare.

[Nick Rowe]

rsj: I think you're right. But in those particular examples there *may* be an alternative explanation. Because they are non-rival goods (except for the hotel and airfare example), so have zero marginal cost, so it may be a sorta price discrimination strategy? Like if there's an imperfect (or ideally negative) correlation between people's willingness to pay for the two goods, you can get more revenue by bundling them? (I remember studying the economics of "tied sales" ages ago, but my memory of this is very fuzzy.)