If we had one central market, with a Walrasian auctioneer, where all goods are traded simultaneously for each other, none of this would matter. But we don't; so it does.
Partial equilibrium theorists don't need to know this stuff. People who believe prices are always at market-clearing levels don't need to know this stuff. Everyone else needs to know this stuff.
The world needs one person to prove existence of Walrasian general equilibrium, and a second person to check the first person did it right. Everyone else should learn this stuff instead.
If you are an old macroeconomist, or maybe French, you probably know this stuff. Otherwise, you probably don't. Because it all got swept away in the New Classical revolution. And then got swept under the carpet in New Keynesian "cashless" models (which aren't).
1. Let's count the number of markets.
Economics is supposed to be about things like goods getting traded in markets. So the least we can do is count the number of markets right.
If there is 1 good, there are 0 markets. Duh.
If there are 2 goods, there is 1 market, where people exchange the two goods.
If there are 3 goods, it gets complicated. In a barter economy, where each good can be traded for each of the other two goods, there are 3 markets. In a monetary economy, where one of the three goods is used as the medium of exchange, there are 2 markets.
If there are 4 goods: there are 6 markets in a barter economy; and 3 markets in a monetary economy where one of the goods is used as money. Count 'em.
If there are n goods (including money, if it exists): there are (n-1)n/2 markets in a barter economy; and (n-1) markets in a monetary economy.
I am assuming that in each market, two goods are traded. In a monetary economy, money is traded in all (n-1) markets, and each other good is only traded in one market. (Yes, there are more complicated cases, but let's keep this simple.)
Never ever use the words "the money market". All (n-1) markets are money markets.
2. How many excess demands are there?
In a monetary economy each good except money has one excess demand (or excess supply, if negative). But money has (n-1) excess demands: one excess demand for each market in which it is traded. Money can be in excess demand in the apple market (if there is an excess supply of apples), and in excess supply in the banana market (if there is an excess demand for bananas). So there are 2(n-1) excess demands, in total.
In a barter economy there are (n-1)n/2 markets, each with 2 excess demands, and so (n-1)n excess demands, in total. Each good has (n-1) excess demands.
In a Walrasian economy there is one market in which all n goods are traded at once. So there are n excess demands, one for each good. And the values of those excess demands must sum to zero. We call that "Walras' law". People plan to pay for what they buy.
In each market in a monetary economy the value of the excess demand for the non-money good, plus the excess demand for money in that particular market, must sum to zero. But that tells us absolutely nothing about cross-market restrictions. With n goods, and n-1 markets, n-2 of those markets can be clearing, but the n-1st market can have a $10 excess demand for xylophones matched by a $10 excess supply of money in the xylophone market.
3. Let's re-think consumer choice theory.
Suppose we live in a monetary economy, with n goods (including money) and n-1 markets. Suppose that prices are sticky, so markets don't always clear. If there is excess demand for apples, some buyers won't be able to buy as many apples as they want. If there is excess supply for bananas, some sellers won't be able to sell as many bananas as they want. People face quantity constraints, as well as their budget constraint. If they are rational, they will take those (actual and expected) quantity constraints into account when maximising utility.
Here is one way to think about it (Benassy, following Clower, interpreting Keynes):
In each of the n-1 markets, the consumer maximises utility, subject to the budget constraint, and subject to any quantity constraints in the other n-2 markets. If I can't buy as many apples as I want, I might buy more pears than I otherwise would. If I can't sell as much labour as I want, I might buy fewer bananas than I otherwise would (Keynesian consumption function).
What this means is that each individual has (up to) n-1 different consumer choice problems. Because in each of the n-1 markets, the quantity constraints are those that apply in the other n-2 markets. And that other n-2 is a different set of markets in each of the n-1 cases.
Yes, yes, yes, we could unify it all into just one consumer choice problem, where the individual maximises utility subject to the budget constraint and subject to any quantity constraints in all n-1 markets. But that gives us silly answers. There can never be an excess demand for apples, because if consumers know they can't buy more than 2 apples, they won't ever ask for more than 2 apples. Which is silly. The involuntarily unemployed worker still wants a job, even though he can't get one.
4. All general equilibrium analysis with sticky prices, and all stability analysis, is a complete waste of time, if it ignores this stuff.
I don't give a damn if it's dynamic and stochastic.
Because we live in a monetary exchange economy, not an economy where all goods can be traded at once in one big Walrasian market.
Start in full market-clearing equilibrium. Now double all prices in terms of money. So there is an excess demand for money. In a monetary economy that will disrupt all n-1 markets. Money flows both into our pockets and out of our pockets. No individual can increase the flow into his pocket, unless other people are willing to increase the flows out of their pockets, which they won't be. But nothing can stop every individual reducing the flow out of his pocket, by buying less of all other goods. So trade in all goods gets disrupted. (We call that a "recession".)
But in a barter economy, or Walrasian economy, where "money" means only "the good in terms of which prices are measured", there is nothing to prevent mutually beneficial trade in all other goods, because their relative prices are unaffected. Everyone wants more money, but can't get any more, but trade in all other goods continues as before.
Worthwhile Canadian Initiative 
notsneaky, do you allow for zero or negative marginal utility?
“Suppose wage is $10 per hour, and price of apples is $1 each. Workers start the period with $0 money. They want to sell 10 hours of labour and buy 100 apples. But they can only sell 6 hours of labour, so only buy 60 apples. There is an excess supply of 4 hours labour. What is the excess demand for apples? The “notional” excess demand for apples is 40. The “constrained” (or “effective”) excess demand for apples is 0. (The Keynesian consumption function is a constrained demand function.) The words “notional” and “constrained/effective” are from Clower, IIRC. Walrasian demands are notional demands. They ignore quantity constraints in other markets.”
I think a better way to say that is Q demanded without MOE = 100 apples and Q demanded with MOE = 60 apples.
Suppose H=K=a=r=1. The price that the clearing house pays for each good is two units of money, m=2. The price it sells each good back to the agents at is one unit of money, nx=ny=1. Agents anticipate that there’ll be excess demand and the good will be rationed. The notional (Walrasian) demand for X by agent A is (1/2)mX/nx=X. For X by B is (1/2)ma*Y/nx=Y. By symmetry (and by maximizing utility) X=Y. So total Walrasian demand for X is 2X. Supply of X is X, so 2X > X.
Each agent does have enough money to pay for their respective demands (agent A has X units of money and so does agent Y) but just not enough X has been produced to satisfy both demands.
Why doesn’t agent A produce more X? They know the good will be rationed and that they’ll get half of whatever’s produced. They could increase production but that would cost them more in terms of disutility of labor (in fact, with excess demand they’re already producing more than under the Walrasian equilibrium, essentially for strategic reasons)
notsneaky:
“U_A= ln(cxA)+ln(cyA)-(r/K)X
U_B= ln(cxB)+ln(cyB)-(r/H)Y”
I find myself fumblingly following your Cobb-Douglas mathematical logic. You are introducing some concepts that are new to me, slowing my comprehension and necessitating some mental realignment. That takes time.
I think one strength of your math model is that both capital and labor are traded in the central exchange. Your model also adds time (or sequences) to Nick’s mechanical model of money at the center of markets.
Have you considered placing this entire chain of mathematical logic into a blog post of your own? [If you have a blog (of your own), it is not linked via the comment attribution (as are Nick’s and my comments).]
Thanks for sharing this logic and the sequential development of it.
TooMuchFed
Marginal utility of what? In terms of X the marginal utility is (k/X)-(r/A) where k, constant, depends on whether there is excess demand or not.
Roger, Cobb-Douglas is easy because it just means that agents always spend a constant share of their income on each good. So here, each agent will always want to spend one half their money on one good and one half on the other good. It’s about as simple as you can get.
I don’t think capital is being trade, everything is produced with labor, just it has to be produced before any kind of exchange takes place.
I used to have a blog once upon a time
To explain the above more simply. Suppose agent A produces 3 units of X and agent B produces 3 units of Y (these are in fact Nash-equilibrium quantities for the game with excess demand). A brings 3 units of X to the clearing house and gets 6 dollars for it. B gets 6 dollars for their Y. The Cobb-Douglas utility just means that A wants to spend 3$ on buying back X and 3$ on buying Y, same for B. The buy-back price of X and Y is 1$ per unit. So total demand for X (and Y) is 3$ from A and 3$ from B, so demand for 6 units. But only 3 have been produced. So the clearing house just splits the 3 units among them, 1.5 and 1.5.
In Walrasian equilibrium where prices (trade ratios) adjust so that supply equals demand, A would only want to produce 2 units of X and same for Y. This is because producing the good is costly in terms of disutility of labor. But suppose B produces only 2 units of Y. They’ll get 4$. Then X can do better than producing 2 units by producing just a bit more and shifting the share of money in their favor, getting more than 1/2 of each good. Y of course knows this and thinks the same, so they respond by increasing their production a bit more as well. X increases back in response. Y increases. They both increase until each is producing 3, which is “over production”
“Marginal utility of what?”
All the services.
See my comment above at 2:10.
“All workers want 70 units of each different service but no more.”
TMF, in the model, the marginal utility of good x is 1/x, the marginal utility of y is 1/y and the marginal utility of leisure is r (more or less, the marginal disutility of labor is -r)
“The “supply” and “demand” curves that economists draw are necessarily hypothetical imaginary constructs. In the real world we have transactions, and only transactions… credits and debits must balance at all times (both in money, and in goods, or “stock” as accountants prefer to call it). Conjecture about what else might have happened must necessarily remain unmeasurable, although there are various ways to estimate what the local region of supply and demand curves might look like (i.e. estimate the slope).”
At market level they aren’t necessarily curves. They can be any polynomial shape you can draw in a straight line.
Those are the SMD conditions that Steve Keen keeps going on about. There are no market demand curves.
“TMF, in the model, the marginal utility of good x is 1/x, the marginal utility of y is 1/y”
Quick glance.
It appears 0 and negative marginal utilities of good x and good y are not possible in that model. Is that correct?
If you mean marginal utility of consumption of either x or y then yes that is correct.
Bob, you realize that none of that actually makes any sense?
notsneaky: I’m still trying to get my head around your model.
Let me work with your 3.13 example, which clarifies things a lot. I have two problems with it:
1. Why does the apple-producer take apples to the market, and then buy back his own apples? Why not just keep the apples he wants to eat? (If all markets are clearing, and there are no taxes or subsidies or transactions costs, it won’t make any difference, of course).
2. In your example, they start out with no money, and the clearing house starts out with no fruit. They are selling fruit to the clearing house at $2 each, and buying fruit from the clearing house at $1 each. So, OK, I can see why there would be an excess demand for fruit, that the clearing house cannot satisfy. And this explains my question 1 above, because it’s like a subsidy. But it’s like a fruitbroker/marketmaker who has no inventory of fruit setting his bid price above his offer price. He is wide open to arbitrage losses. And the Walrasian auctioneer always has the same bid and offer prices.
Bob’s presumably been reading that Steve Keen paper on perfect competition. Steve Keen actually does sometimes do some interesting stuff. But his paper on perfect competition is not an example of that. It’s just plain wrong. Here is Chris Auld on the subject, but it’s off-topic for this post.
And too Much Fed is still stuck on an example where people are satiated in consumption, which is why he thinks there’s excess supply. He doesn’t get that they would stop wanting to work if they are satiated in consumption (or are living in the Garden of Eden), even though I have explained it to him. It’s also off-topic.
Well, why do agents take their goods to the Walrasian auctioneer and let him compute the trade ratios? Why do the coconut-pickers in Diamond’s Island model just not eat their own coconuts rather than waste their time searching for someone to trade with? It’s an abstraction. Just a way to have a model with disequilibrium prices and excess money demand/supply. I guess what might make it more confusing is that the clearing house is actually playing two roles here, one as a money-issuing central bank and one as a non-Walrasian auctioneer. But we could separate out these two functions.
Yes, but see 1. above. The way to think of the “sell to” prices, m and am, is that they’re just a way of carrying out monetary policy in a one period model. The way to think of the “buy back” prices is that they’re the sticky, disequlibrium prices that exist because… because we assumed prices are sticky.
We could add an extra move by the clearing house, either before agents decide how much to produce or after, right before they trade in their production (depending on what we want to focus on), where, taking the “buy back” prices nx and ny as given (prices have already been set in advance) it decides on the “sell to prices” m and am.
Then we can add in shocks, timing, strategic considerations and get most of the issues of monetary macro.
And I think Bob’s referring to Keen’s “exposition” of the Sonneschein-Mantel-Debreau “anything goes” theorem. It’s just as wrong as Keen’s imperfect competition stuff.
Bob, the SMD theorem is really just about the uniqueness of the general equilibrium – it’s hard to guarantee that there’s only one (though every example of multiple equilibria I’ve seen tend to be fairly contrived). It doesn’t say anything about market demand not existing or anything like that. Keen also confuses the SMD theorem with the Gorman Form Representation theorem (which is about when you can write aggregate demand as a function of aggregate wealth, rather than a vector of individual wealths, IIRC). I have no idea what “any polynomial shape you can draw in a straight line” means.
Hi Nick
Fascinating post. I often wonder about what I would have learned had I done a PhD in an earlier era, what got swept off the curriculum to make room for what we study now. So thanks for giving us a glimpse. Sometimes, when I am feeling particularly rowdy, I go to the library and take out an old, unfashionable economics book and attempt to read it. A few weeks ago I read the first hundred or so pages of Franklin Fisher’s “Disequilibrium Foundation of Equilibrium Economics.” I thought it was very interesting but had a hard time seeing how it fit into modern economics. I know people call the searching and matching models disequilibrium, but they don’t resemble the model Fisher writes down (at least to me). You mention stability in passing in this post (thought I don’t get the reference), what do you think of that stuff?
notsneaky:
1. OK. Let’s run with Diamond’s tabu on eating your own coconuts (and it works fine for haircuts, since you can’t cut your own hair).
2. OK. Let’s think of it this way. Your central clearing house is also a central bank, that prints or burns money, and uses the money it prints or burns to finance percentage subsidies or taxes on sales (or purchases) of goods.
“And I think Bob’s referring to Keen’s “exposition” of the Sonneschein-Mantel-Debreau “anything goes” theorem. It’s just as wrong as Keen’s imperfect competition stuff.”
Ah. You are probably right.
Sam: Thanks!
I haven’t read Franklin Fisher (or if I did, I don’t remember). Brad DeLong knows the stuff I am talking about in this post, and has also, IIRC (but don’t trust my memory), done a post on Franklin Fisher. He could give a better answer than me. The two key questions are these: when Fisher does his stability analysis, does he assume agents take quantity constraints into account, and is he talking about a monetary exchange economy with n-1 markets? If no to both, ignore it. If yes to both, read it.
“Self-employed service providers, who sell one service and buy many. Start in competitive equilibrium. Then M doubles, holding all P’s fixed (or all P’s halve, holding M fixed). There’s an excess supply of money. Everybody wants to buy more services, but nobody wants to sell more, so they fail to buy more. They can’t increase the flow OUT of their pockets. But they can reduce the flow INTO their pockets. They decide to sell less services (“buy” more leisure). Output falls. But that makes the excess demand for services even worse, and causes a supply-side multiplier effect, where output falls still further.”
“The reason it rarely happens in capitalist countries is because equilibrium is monopolistically competitive, not perfectly competitive, so *****firms***** always want to sell more at current prices, since P > MC. That’s why a loosening of monetary policy causes output to temporarily increase in New Keynesian models, which assume monopolistic competition.”
“And too Much Fed is still stuck on an example where people are satiated in consumption, which is why he thinks there’s excess supply. He doesn’t get that they would stop wanting to work if they are satiated in consumption (or are living in the Garden of Eden), even though I have explained it to him. It’s also off-topic.”
Nope. When firms are added, the scenario changes.
There are service providers who are all firms. Workers buy services from firms and work for firms. All workers want 70 units of each different service but no more.
Start in competitive equilibrium where 70 units of each different service are supplied. Every entity runs a balanced budget. There are no savings. Profit is zero for each firm. M = 10,000.
I believe Q demanded = Q bought = Q sold = Q supplied for each service here.
Then one firm becomes more productive. It does not increase output although it would rather do that. Instead, the firm reduces hours worked becoming profitable. It saves 50 for a really long period of time (there is the excess demand for money making this on-topic). All P’s are fixed. It pays its workers 50 less. The workers maintain a balanced budget. They spend less on services.
“Then one firm becomes more productive. It does not increase output although it would rather do that. Instead, the firm reduces hours worked becoming profitable. ”
Why you need JG and strong demand.