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
Now, consider a "demand shock." That is, suppose that all the type T people receive utility u* from consuming, where u* < c, and everyone else is the same as before. A point I want to make here is that the Keynesian failure can come from anywhere in the chain - it need not come from the money producers. If we consider the flexible price, or lottery, equilibrium, now the type T - 1 buyers have to do something different in order to get the type T sellers to produce. It is still best for the buyers in these meetings to offer their commodity money (good 1) with certainty, but the take-it-or-leave-it offer the buyer makes involves the seller producing with probability
Note that I'm putting "demand shock" in scare quotes. Why? Because, in spite of the fact that the comparative static experiment involved a decline in the utility type Ts receive from consuming, it affects everything a type T does, in particular his or her labor supply. Why work if you don't like to eat? This illustrates why terms like "demand shock" and "demand deficiency" have no meaning in a properly specified general equilibrium model. This is a standard criticism of IS/LM models that goes back to at least the mid-1970s. For example, the IS curve is shifting because the behavior of some consumers changed, but those consumers are the same people who are supplying labor in the labor market, and holding money in the money market. Why don't we take account of that? Why indeed. Spelling these things out in the model means you don't miss that, which could be very important.
The next step is easy, as Levine already did it. If prices are fixed (all trade is one thing for one other thing), then the "demand shock" will shut everything down. The flexible price lottery equilibrium is, as far as I can tell, Pareto efficient, so that's a useful benchmark. So note first that having this economy shut down - in this extreme example - will sometimes be efficient, if (4) does not hold. Thus, in that case, the fixed price equilibrium is actually OK. But that's not what interests us. Suppose u* < c, but (4) holds. Then clearly the fixed price equilibrium is not Pareto efficient. But how would we fix it? David goes through some possibilities, but the key message is that, if the government is going to intervene in this world in a good way, it has to redistribute. Somehow the government has to move surplus to type Ts from everyone else, so that the type T's are willing to trade. If the shocks are causing some inefficiency, we can't correct the problem through some blunt policy which says the government should just buy some stuff, and it really doesn't matter what. Indeed, it does matter, and this crude model is an illustration of that fact.
As well, note that a typical justification for thinking about the sticky price equilibrium rather than the flexible price equilibrium, is that pricing is hard for the people in the model to figure out. Indeed, that's the case here. People sometimes argue that mixed strategies (as in the flexible price equilibrium) are very difficult to implement in practice. But that doesn't let the government off the hook. If it wants to correct the incorrect pricing - the prices are the wrong ones in the sticky price equilibrium - they have to do so by replicating the flexible price equilibrium, and that involves lotteries. That's just an example of a general problem in Keynesian economics.
So, you might wonder why we would worry about a commodity money economy, if that's not the type of world we currently live in. Well, it's not so hard to extend the idea to a fiat money economy. Some things change in an interesting way, but the basic idea stays intact. We're going to work in an overlapping generations framework. Samuelson's OG model is not used so much anymore, but it was a standard workhorse for monetary economics at the University of Minnesota until about the mid-1980s. For this example, it works nicely.
The people that live in this world look much like the people in the commodity money economy, except they each live for two periods. They can produce an indivisible unit of the current perishable consumption good when young at a cost c, and receive utility u from consuming an indivisible unit of consumption good when old. Each period, a continuum of two-period-lived people with unit mass are born. In period 1, there is a continuum of old people who live only one period. The initial old each receive utility u from consumption of one unit of the consumption good, and each has one unit of indivisible fiat money. We'll make the bold assumptions that fiat money cannot be counterfeited, and that it is perfectly durable. Each period, each young person is matched with one old person.
We'll suppose, as in the commodity money economy, that in a meeting between a young person and an old person with money, the old person makes a take-it-or-leave-it lottery offer to the young person. This is actually easier to analyze than the commodity money equilibrium, as the initial old people want to give up their money no matter what - they're not like the commodity money producers who have a cost of producing money. So, here the flexible price equilibrium and the fixed-price equilibrium are the same thing. Each period, every old person exchanges one unit of money for one good produced by a young person, every young person receives utility -c + u > 0, every initial old person receives utility u, and money circulates forever.
Now, suppose that sometime in the future, in period T, utility from consuming is lower for all the people who are born that period, i.e. they receive u* from consuming when old, and u* < c, just as before. Again, this is easier than in the commodity money case, as this economy will not shut down under flexible pricing. Letting p(i) denote the probability a young person produces in a trade with an old person, we get
As in the commodity money economy, everything shuts down if everyone has to trade at fixed prices. In period T, the young will not accept money, and so by induction no one will. Here, just as with commodity money, the problem in the fixed-price economy is not a monetary problem - it's that the prices are wrong. It's always puzzled me, for example, why Mike Woodford thinks of his models as prescriptions for how central banks should behave, as the relative price distortions that exist in those models look like problems for the fiscal authority to work on. I haven't worked out the details, but I think that a policy that would work in the fixed price equilibrium is to simply replicate the flexible price equilibrium with a sequence of taxes on old agents (random confiscations of money) and subsidies for the young (random transfers of money). You can do something similar in a Woodford model with consumption taxes (see this paper by Correia et al.).
We could also think about unanticipated preference shocks in this model. For example, suppose the utility of consumption for old persons is a random draw, which they learn when they are young. With probability q they receive u*, and with probability 1 - q, they receive u. Then, we can construct an equilibrium in which the young produce with probability s* when their utility when old will be u*, and produce with probability s when their utility from consuming when old will be u. For an equilibrium to exist requires
As in all the previous cases, if prices are fixed, then this economy shuts down because of these demand shocks. There is always a positive probability that the young next period will not accept money, so it is not valued in equilibrium and there is no trade. Again, the problem is that the prices are wrong. A fix for this is for the government to step in, if it can, and replicate the allocation that was achieved under flexible prices. What should work is that, when a bad shock is realized (the young learn that their utility from consumption when old is u*), the government taxes money away from old people, at random, and gives it to young people, again at random. Note that this doesn't involve running a deficit - it's a tax/transfer scheme with taxes = transfers. Again, the cure is redistribution. Further, note that an optimal allocation has cycles - it's not optimal to smooth the cycle completely, even if that were possible (and I'm not sure it is).
So, I think this is an interesting example. It's obviously special, and we wouldn't take it to the U.S. Treasury and tell them about it, hoping to influence their decisions. The message is that whatever anyone thinks they know about "Keynesian" ideas, and the "Keynesian" policies derived from those ideas, they should reconsider. There's nothing obvious about that stuff. We can write down coherent models in which Keynesian phenomena occur, and the optimal policies don't seem to look like anything like that Paul Krugman recommends. And it's not because his IS/LM model is "right." Far from it. We understood that long ago.