From New Monetarist Economics:
I guess I shouldn't be surprised. David Levine's piece on Keynesian
economics appears to have generated plenty of heat. See for example the comments
section in my post linking to Levine. I'm imagining an angry mob dressed
like the Pythons, as in the photo above, running through the streets of Florence
looking for Levine. Each has a copy of the General Theory, and they're
aiming to inflict torture by taking turns reciting it to David, until he
renounces his heretical writings.
What drew my attention to Levine's piece initially were blog posts by Brad DeLong and Nick Rowe. If Levine's piece were a prelim question, I'm afraid we would have to fail both Brad and Nick. Brad can't quite get off the ground, as he doesn't understand that Levine's model is indeed a monetary economy and not a barter economy. Nick achieves liftoff, and we can give him points for recognizing the double coincidence problem and that the phone is commodity money. But then he stalls and crashes, walking off in a huff complaining that Levine doesn't know what he's talking about. Levine has posted an addendum to his original post, which I think demonstrates that he does in fact have a clue.
In any case, I thought Levine's example was interesting, and I'd like to follow John Cochrane's suggestion of filling in some of the spaces, which will require some notation, and a little algebra. First, adding to David's addendum, let's generalize what he wrote down. This is just a version of an economy with an absence of double coincidence of wants. If any two people in this world meet, it will never be the case that each can produce what the other wants. It's roughly like Kiyotaki and Wright (1989), except with 4 goods instead of 3. And of course there are some very old versions of the double coincidence problem in the work of Jevons and Wicksell, for example. Brad DeLong, who reads the old stuff assiduously, perhaps missed those things.
Let's first imagine a world with T types of people, indexed by i = 1,2, ..., T. There are many people of each type. Indeed, for convenience assume that there is a continuum of each type with mass 1. A person of type i can produce one indivisible unit of good i at a utility cost c, and receives utility u from consuming one indivisible unit of good i + 1 (mod T) (i.e. T + 1 (mod T) = 1 ). We need n >= 3 for a double coincidence problem, and n will matter for some elements of the problem, as we'll see. A key feature of the problem will be that each person can meet only one other person at a time to trade - that's a crude way to capture the costs of search and exchange. We could allow for directed search, and I think that would make no difference, but we'll just cut to the chase and assume that each person of type 1 meets with a type 2, each type 2 meets with a type 3, etc., until the type T - 1 people meet with the type Ts. Further, we'll suppose that, as in David's example, good 1 is perfectly durable and costless to store, while all the other goods are perishable - they have infinite storage costs. Assume that u - c > 0 (with some modifications later).
A key element of the problem is that the indivisibility of goods fixes the prices - indeed, in a Keynesian fashion - so long as we only permit these people to trade using pure strategies. That is, David assumes that when two people meet they both agree to exchange one unit of a good for one unit of some other good, or exchange does not take place. But let's do something more general. Suppose that 2 people who meet can engage in lotteries. That is, what they agree to is an exchange where a good is transferred with some probability, in exchange for the other good with some probability. Then, the probabilities play the role of prices. That is, with indivisible goods, we can think about an equilibrium with lotteries as a flexible price equilibrium, and the Levine equilibrium, where one thing always trades for one other thing, as a sticky price equilibrium.
This sounds like it's going to be hard, but it's actually very easy. Work backwards, starting with a meeting between a type T - 1 and a type T. Trade can only happen if type T-1 has good 1, which is what type T consumes, so suppose that's the case. We have to assume something about how these two would-be trading partners bargain. The simplest bargaining setup is a take-it-or-leave-it offer by the "buyer," i.e. the person who is going to exchange something he or she doesn't want for something the "seller" produces. The buyer has one unit of good 1, which is of no value to him/her, so the buyer is willing to give this up with probability one. Since u > c, the seller is willing to produce one unit of good T in exchange, so the optimal offer for the seller is in fact the Levine contract - one unit of good 1 in exchange for one unit of good T. And the same applies to the meetings where types 2, 3, ..., T-1 are the buyers.
But, the type 1 people - these are the producers of the commodity money in this economy - are different. Unlike the buyers in the other meetings, they have to produce on the spot. And, since they make a take-it-or-leave it offer, they are in a position to extract surplus from sellers - and they do it. So, the trade they agree to is an exchange where each type 2 person produces one unit of good 2 and gives it to a type 1 person, and the type 1 person agrees to produce good 1 with probability p(1), where
So, in equilibrium, only a fraction c/u of each of types 2, 3,..., T gets to consume, and all the type 1s - the money producers - consume. There is a welfare loss from this commodity money system, in that the money producers are extracting seignorage from everyone else. In the fixed price equilibrium, where everyone has to trade one unit of a good for one unit of another good, the type 1s are worse off, and everyone else is better off.