This post was inspired by Steve Williamson's post on Roger Farmer. Steve has a model(pdf) that he sees as capturing the essence of what Roger is saying. Leave aside the question of whether Steve's model is a good interpretation of Roger. I like Steve's model because it's clean and simple. But Steve's model is a model of a barter economy. I want a version with monetary exchange.
Caveat: I'm a neophyte at search models. The last one I built that was even vaguely like this was in 1987. But I'm going to post this anyway.
Search and monetary exchange are intimately connected. Matching a worker to a job is non-trivial. Matching a product to a consumer is non-trivial. Barter exchange would require we solve both matches at the same time. The worker would need to fit the job, and the good produced by the worker would need to fit the worker's preferences. A double-coincidence of matches that's almost impossible to solve. That's why we use monetary exchange, so we can separate those two matching problems and solve them one at a time.
But then a monetary disequilibrium can make that matching problem harder again. I can find people who want to sell what I would like to buy. I can find people who would like to buy what I want to sell. But they aren't the same people, so we can't barter. And unless those who would like to buy are actually able and willing to part with money, there's no effective match of buyer and seller.
So I'm going to sketch my own model, with search, money, and keynesian unemployment. It's based on Roger's idea, which I will explain later.
Representative agent model, large number of small agents, continuous time.
Each agent owns a farm. He works to produce food, but cannot consume his own food. All food must be eaten at the point of production, so they can't do simple barter. Agents are anonymous, so there's no credit. So they must use monetary exchange.
Each agent spends a fraction L of his time working, then a fraction U of his time standing at the farm gate waiting for a shopper, then the remaining fraction V of his time shopping. L+U+V=1. (U is the unemployment rate, and V is the vacancy rate.) Work to produce food, then sell the food for money, then take the money and go shopping for food to eat, then go back to work, and repeat.
Search is random. A shopper has a probability k per period of finding a farm, and a probability kU of finding a farm with the owner waiting at the gate to sell food. Each agent has a probability V of shopping, so the number of matches per agent per period will be F=kUV. If a shopper carries M/P in real money balances, where P is the price of food, the amount of food purchased and consumed per agent per period will be C=FM/P=kUVM/P.
Technology: simply assume C=L.
So we can rewrite the agent's resource constraint as C+U+F/kU=1
Preferences: assume each agent maximises a utility function W(C,F), where F is the frequency of shopping trips. Ignore discounting. Just maximise the average flow of utility per unit of time.
I need to explain why F appears in the utility function. If F did not appear, there would not be an interior solution; people would do just one shopping trip carrying a massive amount of money and buy everything at once for all time. But nobody wants to do that in the real world. You want to smooth out your shopping over time (because the milk goes off if you buy a whole month's supply at once). You trade off frequency of shopping trips with total time spent shopping. If our agent goes shopping 1% less frequently he spends 1% less time shopping, so he can spend more time producing, and so can sell more food and consume more food. In equilibrium he is just indifferent at the margin between the utility of the extra consumption and the disutility of the drop in shopping frequency.
[Update: you can also think of F as the velocity of circulation of money. But F is a choice variable in my model, it's not assumed fixed.]
The agent's problem is to maximise W(C,F) subject to C+U+F/kU=1.
A utility function that gives tractable results is W=CF. This yields F/C=kU as the solution to the agent's problem. Intuitively, a higher unemployment rate makes shopping easier (because it's easier to find a seller) so the agent chooses to shop more frequently. Equivalently, an agent will choose to go shopping with real money balances M/P=1/kU. Think of that as a money demand function, for agents who are shopping. But since only a fraction V of agents are shopping at any time, the average money demand per agent will be m/P=VM/P.
Assuming that money demand equals the exogenous money supply, after a little algebra we can solve for the equilibrium unemployment rate as U=1/(1+2km/P). This result seems reasonable. At one extreme, as m/P goes to zero, the unemployment rate approaches 100%. At the other extreme, as m/P gets very large, the unemployment rate approaches 0%.
But how is P determined?
That's where Roger Farmer comes in. Roger says that P will be indeterminate within some range. That's because when a buyer meets a seller in a search model, the buyer's reservation price will be strictly above the seller's reservation price. If the buyer turns down the deal he will have to keep searching for another seller, which is costly. If the seller turns down the deal he will have to keep waiting for another buyer, which is costly. If they don't do a deal they lose the time they have already invested in search and waiting. It's like a market with one seller and one buyer — bilateral monopoly/monopsony — where any P between the two reservation prices will let both gain from the trade.
This means that m/P will be indeterminate over some range. And this means that U, the unemployment rate, will be indeterminate over some range. Which is what makes it a Keynesian model, as well as a monetarist model. There is no unique natural rate of unemployment. There is a continuum of natural rates.
It's like a model where small menu costs can make prices sticky. Only here, you only need infinitesimally small menu costs to make prices sticky, within some range. So a change in the nominal money supply can have real effects, within some range.
And in this particular model, unless my intuition is misleading me, I think the range of equilibrium unemployment rates can be anywhere from 0% to 100%.
Update: I forgot to solve for output (consumption, employment). Here's the solution: C=k(m/P)/[1+2k(m/P)]. I think that shows that output is increasing in m/P, but at a decreasing rate as you approach "full employment" (where C=1) [Correction: C=0.5 at "full employment", thanks himaginary]. You might want to interpret that as an AD function. As m/P increases, agents spend more time shopping and working, and less time unemployed waiting for a buyer.
Worthwhile Canadian Initiative 
I like this. Even then, some comments.
Barter economies do not exist. Never have, never will. Barter does exist, but has always been a rather insignificant side show. Coordination problems in non-monetary economies are solved, but not by trade. They are solved by division of labor and production based upon age, gender, caste, whatever (just like we still do, to a large extent, in a monetary economy). And upon ‘parties’. In many economies, weddings, funerals, ‘the Potlatch’ and the like are a non insignificant way to distribute goods (and prestige, and connections, well, the whole gift exchange thing). In present day South Africa, your funeral may very well be the largest expenditure of your entire life.
If you think about this, the ‘mutual coincidence of wants’ is not a problem of non-montary economies. These economies have solved this problem (see above). It’s a problem of monetary economies. This problem sometimes shows as a liquidity problem. You want to buy a house, you have the income, but you don’t have the money. A mortgage solves this liquidity problem. It sometimes shows as a search problem. Habits and instititions and the like have, of course, solved much of that problem (the monthly pay check, pension funds, the grocery store, the entire theory of ‘whole sale’ is in effect based upon they way in which this solves parts of the problem). Farmers do not stand along the road – they have established routines to sell their products. Even then, these solutions are weridly expensive. Banks, insurance companies, grocery stores, malls, cash registers, whatever. Together, the macro costs of markets and the macro costs of money will easily ben 15% of GDP
And if you think even deeper, it turns out that the essence of prices is stickyness. If they weren’t sticky (sometimes only for a minute, sometimes for fifteen years like my mortgage interest rate), they would not exist, in a market economy. In a market economy, you agree upon a price – which is impossible without stickyness (and indeed, in my definition of a market, post transaction ‘shadow prices’ which can only be known ‘the morning after’ are anathema to real market behavior: the very essence of a market is that people know about prices (or at least about the algorithm used to define a price) before making a transaction. And that’s why money exist. It’s the sticky price par excellence.
“In a market economy, you agree upon a price – which is impossible without stickyness (and indeed, in my definition of a market, post transaction ‘shadow prices’ which can only be known ‘the morning after’ are anathema to real market behavior: the very essence of a market is that people know about prices (or at least about the algorithm used to define a price) before making a transaction.”
I’ve never thought about it like that, but this is a very good point. You are in escence correct. Maybe that is why people (at least in West) are rather reluctant to haggle. If the price is fuzzy, they can’t compare it to other prices and calculate their budget constraint. The Walrasian auctioneer in fact abstracts from a core feature of markets.
No long run neutrality of money?
Merijn: IIRC, you have made that argument about barter before. You are starting to convince me. OK, what it means is that the transactions costs of barter must be so high (with a few exceptions), that trade without money is dominated by non-market methods of economic organisation. But I still want to think about barter as a theoretical possibility, because it helps me understand monetary exchange better, by way of contrast. People use mopney, and almost never use money, so there must be some really big problem with barter that money overcomes.
Bill: welcome back! We are missing you, and Scott too.
Yes, money will be (or may be) non-neutral in this model. That’s because the price level is indeterminate. The LRAS curve is vertical, but very thick. So a shift in the AD curve can lead to a change in P or a change in Y. Anything can happen.
I didn’t have inflation in the model, but if I had included it the LR Phillips Curve would be thick too, and it would slope the wrong way. A higher inflation tax might reduce m/P, and cause increased unemployment. Money is essential in this model. People need it to trade. So when you tax it there’s less trade.
Merijn, what about Wall Street? Isn’t that built on extremely fluid markets & spot prices?
Nick, did you mean k to be a frequency (number of farms found) rather than a probability? Otherwise, C=L is a contradiction.
Wonks Anonymous, prices change every tick, but they are still known ex-ante (in the barbarous jargon of the trade: “previsible” or “adapted.”)
Phil: I’m not sure. I was trying hard to keep my head straight on this. I’m still a bit muddled about the units. Let’s see.
L has to be a pure number (unitless). And so C has to be a pure number as well. OK. Instead of C being apples per year, it has to be unitless. So we need to think of C as the fraction of potential output if people spent all their time working. So if people spend 60% of their time working, C = 60% of the output that could be produced if people spent all their time working. Yep.
Now, can’t k be both a frequency and a probability? I’m muddled (again). Dunno. What should k be?
“There must be some really big problem with barter that money overcomes.”
What about equity?
Say that the economy must use debt, or credit, because production requires time, or because a farmer can’t barter wheat for bread as the baker wont have the bread until he gets the wheat; and the farmer gets a lot of wheat once a year, but needs to eat bread each day, etc. There are all sorts of time coincidence problems in addition to problems of coincidence of wants. Not to mention time involved when traveling from one market to another.
So assume there would be debt, allowing everyone to purchase inputs on credit. But there is also risk in your ability to repay the debt. If the farmer kept silos of wheat in storage in the event that he might have a bad harvest, then imagine the huge carrying costs involved, not to mention wastage as that would be output that no one could consume. It would be a tax on the productive economy.
But we all agree to denominate debts in terms of money, and allow money to extinguish debts, then we can keep some coins as a buffer. The farmer can sell all of his wheat and hoard coins while other people eat.
“I think that shows that output is increasing in m/P, but at a decreasing rate as you approach “full employment” (where C=1). ”
As C=FM/P=kUVM/P, and M/P=1/kU at the equilibrium, it seems that C always equals to V at the equilibrium.
Since C+U+V=1, maximum value of C seems to be 1/2, not 1.
“Intuitively, a higher unemployment rate makes shopping easier (because it’s easier to find a seller) so the agent chooses to shop more frequently.”
Conversely, lower unemployment seems to cause collapse of the economy in this model. When U=0, F=kUV becomes zero, so C=FM/P also becomes zero unless M/P goes to infinity (If M/P goes to infinity, C approaches 1/2 as stated above).
Wonk Anonymous;
“Merijn, what about Wall Street? Isn’t that built on extremely fluid markets & spot prices?”
Precisely. Maybe it’s why those ” markets ” are so unstable and useful only to the parasitical organisms of that particular ecosystem.
As per Paul Volcker; “The only useful innovation of the finacial industry in the last 30 years is the ATM”
Nick, when you say “Barter is costly”, I tend to think not of frictional costs, but of the coincidence of wants.
Picture a Hayekian Triangle which describes the production of the economy leading from primary resources on the left side (peak) to consumption on the right (height, it’s a right triangle). Time is on the base.
Workers work in one specific slice of the Hayekian Triangle, their job, employement, profession, whatever. The idea is that they have a Thing that they do. Workers in modern economies specialize. We farm wheat, we work in retail, we assemble widgets. Workers are scattered all along the triangle.
Everybody needs a diverse basket of consumption though, so everyone wants a piece of the right-hand height. So we need a method to convert earnings from prior production work to consumable goods, which is a time problem. We also need a method to translate the pay from specialized work into a diverse basket of consumable goods and services.
The answer to both problems is money.
himaginary: well-spotted!
(You need to distinguish M and m. m=VM. m is exogenous, M is endogenous. m is the average amount of money per agent; M is the amount of money per agent who is shopping.)
Yes, you are right. Agents spend equal time shopping V as working C. And as m/P goes to infinity, C approaches 0.5. I hadn’t noticed that. And F approaches zero as m/P goes to infinity.
There’s presumably some welfare-maximising level of m/P, and U.
(I wonder if I have got the units right, as Phil asks above?)
Merijn, what about Wall Street? Isn’t that built on extremely fluid markets & spot prices?
The memoirs of Alan Greenspan provide the answer to that one. A young Alan Greenspan goes to work at a Wall Street firm. They have a currency trading desk which is a significant moneymaker, which puzzles Greenspan. His economics background says that’s an incredibly speculative business.
The traders reply that they make money on the bid-and-ask spread on their transactions, not on currency levels themselves. Their desk supplies liquidity sure enough, but makes its money on volume through the spread. As such its a really reliable moneymaker, in essence its like hooking up a water meter on the currency market.
Nick,
Interesting post.
I would add one note. You wrote,
“Search and monetary exchange are intimately connected. Matching a worker to a job is non-trivial. Matching a product to a consumer is non-trivial. Barter exchange would require we solve both matches at the same time. The worker would need to fit the job, and the good produced by the worker would need to fit the worker’s preferences. A double-coincidence of matches that’s almost impossible to solve. That’s why we use monetary exchange, so we can separate those two matching problems and solve them one at a time.”
Indeed. Unfortunately, in much of the work in which search is introduced in the labor market, the importance of monetary exchange is neglected. In other words, the Walrasian auctioneer is ‘shot’ only in the labor market. (Shooting the Walrasian auctioneer is an explicit reference to a Farmer paper.) An under-appreciated point in understanding the dichotomy of search is that monetary exchange is used even in the case in which there is a double coincidence. For example, grocery stores do not pay their employees in whole or in part in groceries despite the fact that those workers clearly use some amount of their money income to purchase groceries — perhaps even at the same store.
Josh: thanks! I agree there’s something strange about a search model without monetary exchange. If matching is hard, we will use money to buy and sell. So if you think matching is important, there ought to be money in the model.
Not sure if you are 100% right on the groceries though. When my daughter went tree planting in Northern Ontario, the only source of supply was the employer. A double coincidence created by transportation costs. I think they subtracted the cost of any supplies from her pay cheque.
Nick,
Admittedly, I learned this insight from Ross Starr (2003):
“University of California faculty whose children are enrolled at the University pay fees in money, not in kind; Ford employees buying a Ford car pay in money, not in kind; Albertson’s supermarket checkout clerks acquiring groceries pay in money, not in kind. This observation suggests that the focus on the absence of double coincidence of wants – as distinct from transaction costs – as an explanation for the monetization of trade may miss a significant part of the underlying causal mechanism.”
Examples to the contrary, I think, are exceptions to the rule.