Peter Diamond’s Paradoxes
How unrealistic models can illuminate the world
Peter Diamond has some very elegant papers on paradoxes. They are not necessarily explaining how the world works — indeed, you should not rely on them to make predictions about anything. Rather, they are pointing out that if you make certain assumptions about the world, you are led to highly unrealistic descriptions of the world. Something is wrong; you have to change something.
His 1971 article in the JET, “A Model of Price Adjustment”, is an excellent example of this. Let’s construct a simple, but plausible, model of the economy. We will assume that there is one good in this world, and all firms face identical costs. There are many consumers, who may have all manner of individual demand functions. Search occurs in discrete periods. Each consumer visits one firm in each period, and decides whether or not they want to buy. If it’s below a certain price, known only to them, they buy — otherwise they do not. This price is known as a reservation price. Each time they search, they incur a cost. People are aware of everything, and able to form expectations, although firms are not able to price discriminate. Under these conditions, the price will converge to the same price which would be charged by a monopolist.
Wait what. Surely we have a lot of firms with identical costs. Is there not any clearer set-up for perfect competition, and price falling to the competitive price? The trouble is that searching is costly. People expect firms to charge them the jointly profit maximizing price — and expecting this, it isn’t worthwhile incurring the cost of searching. If the cost of searching is zero, it jumps discontinuously to the competitive price — but any amount of search cost is sufficient for it to be sold at monopoly price. Because the monopoly price can be achieved only restricting the quantity sold, then searching having costs will have impacts on welfare above and beyond what you would naively expect. In the labor market, (so, flip buyer and seller) we would have unemployment and a wage rate below market clearing.
You might be saying, “Okay Nicholas, but we know that in the real world firms do not all price at the monopoly price. Surely your model must be bunkum!”. And certainly, firms do not converge to the monopoly price. The point is that trying to explain how and why sheds light on the real world. Let us state the big three conditions.
Identical cost of production
Sequential search
Search is costly
We can then see how relaxing any one of the constraints leads to different outcomes. Perhaps firms facing different costs of production means that consumers can threaten to defect to different firms, or maybe you can observe the prices of multiple firms at once. If consumers can credibly commit to boycotting a firm which charges too high a price, as in this paper (Bagwell and Ramey 1992), you can also have prices go to the competitive price, without needing to relax the three assumptions. Identifying why it differs, though, is clarifying.
The Diamond coconut model is another beautiful little model. Drawn from the last pages of his 1982 paper, “Aggregate Demand Management in Search Equilibrium”, we imagine a tropical island in which the only good produced is coconuts. One cannot consume the coconuts which one themselves picks – it is necessary to trade for them with another person. There’s no credit or money, or really negotiating ahead of time. Thus, whether it is optimal to climb the tree depends upon what other people do. The important thing is that there is no one equilibrium level of unemployment. There are multiple fully rational equilibriums of unemployment.
The load bearing assumption is the no-credit assumption, but it’s still interesting to think about. We would predict that credit market expansions might have an effect on unemployment above and beyond the impact on production. Reductions in the cost of searching should also greatly impact unemployment. Above all, we should not be so confident that we are indeed at the natural rate of unemployment. Perhaps we are. Or, perhaps we are what people believe to be the natural rate. There is no market on the rate of unemployment – it is the result of many interlocking markets, which need not converge toward perfect efficiency. (The existence of general equilibrium does not mean that we will converge to it, even given infinite time. Scarf 1960 gives some examples where a simple set-up of endowments and indifference curves leads to eternal loops. I urge caution in this – many cranks read any criticism of general equilibrium and think “aha! All of economics is wrong!”, which is entirely wrongheaded. This attitude is overwhelmingly found in Great Britain, btw –
I haven’t been able to find a paper on this (though, I haven’t really looked), but it seems to me that the Diamond (1971) model supports two equilibriums, if not more. In the world before, we had buyers sequentially searching for a seller to sell them a good. In the labor market, however, sellers of labor go sequentially to buyers of labor, and we reach the monopsony price. Who’s to say which should be the more powerful? The sellers of a good could go from buyer to buyer too, in which case it would be sold at the monopsony price, same as in the labor market. If both parties are sequentially matching with parties, then the price which it settles at should be ambiguous. Either way, we needn’t assume that markets with many buyers and sellers converge to the competitive price over time.
Perhaps this is an explanation for the belief that inflationary episodes allow businesses to raise prices and extract rents. Imagine that we are in an equilibrium where businesses believe other firms won’t raise their prices, so it isn’t profitable for them to raise their prices. An inflationary shock changes people’s beliefs about what other firms will do, and allows firms to extract rents. If equilibriums are dependent upon people’s expectations, then let's take expectation shocks seriously.