Roger Farmer always has done interesting and different stuff. We need economists like that. But it's risky of course. What I'm trying to do here is articulate something that makes me uneasy about his recent line of macro theorising. Like his simple model here with Konstantin Platanov (pdf).
Consider a simple model with three goods: Apples, Bananas, and Mangoes. Mangoes are used as money. So there are two markets: the apple market (where apples are traded for money); and the banana market (where bananas are traded for money. And there are two prices: the price of apples in terms of money; and the price of bananas in terms of money. Something like my old minimalist model.
If both prices are perfectly flexible, and instantly adjust to clear the two markets, this model has a unique equilibrium.
Now let's change the model. Assume the apple market has costly search. It is difficult for individual buyers and sellers of apples to find each other. So when they do meet there will be strictly positive gains from trade to that particular pair of individuals. It's not like a textbook competitive market where each can credibly threaten to switch to another trading partner if the other raises/lowers his price by a penny from the market-clearing price. Because it's costly to find another trading partner. So there is a range of prices within which each would prefer to accept that price rather than search for another partner.
But the banana market is textbook.
Economists usually assume that in these cases of bilateral monopoly/monopsony the apple price will be determined by relative bargaining power, so if the two have equal relative bargaining power (for example) they will split the difference and agree on a price in the middle of that range. The Nash Bargaining Solution is one way to model cases like this.
But Roger Farmer does not want to follow that path. OK, let's follow Roger.
What we now have is a model with multiple equilibria. A continuum of equilibria. There is a range of equilibria. Mathematically, we have four interdependent endogenous variables: Qa, Qb, Pa, Pb; but we are one equation short of a solution. We can, if we wish, pick any one of those 4 endogenous variables and assume it is exogenous, or add any extra equation we feel like adding to the system, (within reason, because we must ensure there is some solution within the allowed range of equilibria).
There is nothing wrong mathematically, for example, with assuming the price of bananas is exogenous. That lets us solve for the other three endogenous variables. The price of apples adjusts to clear the banana market at that exogenous price of bananas. This may lead to the apple market having more sellers than buyers, or more buyers than sellers, but so be it.
But does it make economic sense to make that assumption?
It doesn't feel right to me, and what I'm trying to do here is articulate why it doesn't feel right to me.
The root of the indeterminacy problem is in the apple market, not in the banana market. Think about an individual apple buyer meeting an individual apple seller in the forest where both have been searching. How likely is it they will reason: "Well, we had better agree on a price of $1 for this apple, because if everyone else thinks just like us, and agrees on a price of $1 per apple, then we can solve the model and know that the supply of bananas will equal the demand for bananas at a price of $2 per banana, and we also know that everybody expects the price of bananas will be $2, so that validates everyone's exogenous expectation."
It seems to me more plausible they will instead reason: "Well, the normal price of apples is $1, and I don't want to get ripped off, and I don't want to rip him off, and I might get mad if he tries to rip me off, and he might get mad if I try to rip him off, so either one of us might walk away from the trade if the price is above or below $1, even though we would be better off accepting a take-it-or-leave-it offer slightly above or below $1."
I think we tend to see norms developing precisely in those markets where individuals face bilateral monopoly/monopsony. But those norms are local norms about those particular markets.
General equilibrium theory is not wrong when it says that all prices are determined simultaneously by everything. But at the same time it is true that individuals set prices following their own individual interests, which depend on their outside opportunities, which depend on all the other individuals.
Now Roger does not talk about apples and bananas. His labour market is my apple market; and his stock market is my banana market. Roger closes his model by adding a belief function about stock prices. I would instead close his model by adding a norm about wages.
But, as always, I might have misunderstood Roger, or missed seeing something. And I really wish I could articulate my own unease better.
I don't know about your unease, but you've articulate mine quite nicely.
The standard NK model feels similarly wrong. There, we are not two meeting in the woods. It's just me alone, having been given my randomly-timed permission to change prices. And I am supposed to reason, "I had better pick the price that's on the knife-edge path between inflationary and deflationary equilibria. Because if I don't, and everyone else thinks like me, prices will either rise or fall without bound. And that would be... well, not actually harmful to me, in this world... but inelegant!"
So if I have understood that right, we still have no dynamic model (except RBC) in which individual behavior is really coherent. Even if you believe in both Calvo and Confidence faeries.
Posted by: Michael Margolis | June 21, 2016 at 06:01 AM
Michael: (Sorry, your comment got stuck in our spam filter, and I had to fish it out.)
Thanks.
I think the standard NK model is OK in that regard. The individual firm, when touched by Calvo's fairy's wand, does set the price that maximises its expected profit, given what it expects other firms to do. It's a Bertrand Equilibrium.
Posted by: Nick Rowe | June 21, 2016 at 08:58 AM
Hi Nick,
Thanks for fishing me out, and for the answer.
The bothersome question about the NK model is whether the price chosen corresponds to the only Bertrand equilibrium?
If I've understood Cochrane's 2011 JPE piece correctly (which I may not have, I am exploring way beyond my training here) in general it does not. And this gets half-hidden in most presentations by sticking the word "bounded" discretely into the prose. There is a unique bounded equilibrium, plus an infinity of explosive equilibria, and authors mention in passing that the one they analyze is the former.
The matter of how firms might coordinate on the bounded one seems to me as bothersome as how we might agree on the generally expected wage in Farmer's model.
Posted by: Michael Margolis | June 21, 2016 at 10:37 AM
Michael: you are right, I think. I've done a couple of posts on a related issue, about the NK modellers just assuming an eventual convergence to the natural rate of output.
Posted by: Nick Rowe | June 21, 2016 at 07:06 PM
In that model, I think that selecting an equilibrium with the belief function determines unique values for all the other variables including the nominal wage. Therefore, a belief about the real price of capital is also simultaneously a belief about everything else. Like you, I 'd probably prefer to view it as a belief about wages that implies a belief about capital value, rather than the other way round, but I think they really amount to the same thing. (There might be some difference when it comes to way expectation errors are represented.)
Posted by: Nick Edmonds | June 22, 2016 at 04:36 AM
Nick E: "In that model, I think that selecting an equilibrium with the belief function determines unique values for all the other variables including the nominal wage. Therefore, a belief about the real price of capital is also simultaneously a belief about everything else."
That's right. But a *plausible* belief function about (e.g.) Pb might imply a very implausible (implicit) belief function about Pa. It seems to me to be best to think about beliefs in the same market where prices are proximately indeterminate. Why should the individual buyer and seller of labour care about whether the price they agree on validates (makes rational) someone else's beliefs about stock prices?
Posted by: Nick Rowe | June 22, 2016 at 07:57 AM
Can a belief function about Pb be plausible if it implies something implausible about Pa?
Posted by: Nick Edmonds | June 22, 2016 at 08:52 AM
Nick E: fair point. I'm still trying to articulate this.
Posted by: Nick Rowe | June 22, 2016 at 09:33 AM
I confess I find it quite difficult to understand why a high unemployment would be a stable equilibria with Farmer's matching function. If firms were to try to cut wages slightly, I can't see why they would be unable to recruit. From the little search theory I know, employment is limited to successful meetings, but with Farmer there seems no limit to the number of successful meetings an individual firm can engineer, only at the aggregate level. So, with unemployed labour, each firm seems incentivised to try to pay less. Maybe there's more there that I'm missing because I don't know the literature. But it's odd, because in his model it looks like even 99.99% unemployment would be a stable equilibrium.
Posted by: Nick Edmonds | June 22, 2016 at 11:02 AM
Nick E: there's a gap between the supply price of labour and the demand price of labour. It's worth $5 to me and $10 to you, so any $5 < W < $10 is feasible. It's pure bargaining. Who sets W? You are assuming the firm sets W, on a take-it-or-leave-it basis, in which case you are right.
Posted by: Nick Rowe | June 22, 2016 at 11:09 AM
Yes, but I think here we have workers who will work for $5 and firms that will hire at $10, but where no bargain is being struck. I thought that was explained in a matching model by saying that those meetings weren't happening because of the search friction, but I don't see that limitation here, especially as we might have virtually the entire workforce unemployed. Labour is only worth $10 to the firm when it costs them to go find the guy who'll work for $9.
Posted by: Nick Edmonds | June 22, 2016 at 12:05 PM
If Roger Farmer is correct in that there exists a "belief function" for the stock market then prices are not perfectly flexible as you assume for the banana market. Then we have one market with search frictions and one with "sticky" prices. So the sensible solution should include adjustments in the prices of both markets where the final prices depend on the speed of adjustment for these prices?
In which case Roger is more or less right after all since beliefs about the appropriate stock market price will have (some) influence on equilibrium wages.
Posted by: Hugo André | June 23, 2016 at 11:54 AM
Hugo Andre: "If Roger Farmer is correct in that there exists a "belief function" for the stock market then prices are not perfectly flexible as you assume for the banana market."
No. If we draw partial equilibrium supply and demand curves for the stock market (banana market), the stock price (banana price) is always where S and D curves cross. But those two curves will shift if the wage (price of apples) changes, so Roger says the wage (price of apples) changes to shift those curves until they cross at that exogenous price of stocks (bananas).
Posted by: Nick Rowe | June 25, 2016 at 10:10 AM
I think you're going to get better understanding if you explicitly put T (time) into the system.... and forget about equilbirium.
Equilibrium is a snare and a delusion in economics. I mean, for god's sake, look at any commodity market. Do you see equilibrium? No, you see wild swings.
A good model should *predict* those wild swings, not handwave them away. I can generate the oil price swings with a simple supply-demand model with a time lag in (a) supply creation, and (b) demand destruction, along with a depletion model for supply destruction and a classical demand curve for demand creation. (High price now --> supply in three years and demand destruction in 1 year, supply destruction happens only with depletion).
The majority of oil companies and oil users really do look at today's oil price and make decisions which will stick for years based on it, which is why this model works surprisingly well...
But a good oil market model also needs to model storage, hedging, and momentum traders, all of which are more sophisticated actors using more complex models of future oil prices.
Nobody is using an equilibrium model because they don't work.
These damn equilibrium models are misleading...
Hugo wrote:
"If Roger Farmer is correct in that there exists a "belief function" for the stock market "
Several, really; traders are heterogeneous. Though they can be classed into groups for modeling purposes.
"So the sensible solution should include adjustments in the prices of both markets where the final prices depend on the speed of adjustment for these prices?"
Yes, yes, except there are no "final" prices. The most common result is a price which swings back and forth in an industry-specific cycle.
Roger Farmer is definitely on the right track but hasn't gone far enough away from equilibria...
Posted by: Nathanael | June 27, 2016 at 01:38 AM
"Now Roger does not talk about apples and bananas. His labour market is my apple market; and his stock market is my banana market. Roger closes his model by adding a belief function about stock prices. I would instead close his model by adding a norm about wages.
But, as always, I might have misunderstood Roger, or missed seeing something. And I really wish I could articulate my own unease better."
You understood well. A standard S&M model is closed with an equation that describes how the surplus of a match is divided between a firm and a worker, usually generalized Nash bargaining.
To have one degree of freedom and indeterminacy, Roger Farmer arbitrary dropped this last equation.
Posted by: M. | June 27, 2016 at 04:00 PM