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What are the objective functions of the players? I'm assuming each non-money producer has some love of variety utility, maybe separable in that and disutility of labor. Does their own good enter that utility function? If so, do they buy it from themselves at the price they set, or at their MC (with large n it probably doesn't matter much)?

What about the money producer? I'm assuming that the marginal cost of producing money is zero. What do they do with the units of good they buy in exchange for their money (if I'm understanding the set up directly)?

notsneaky: If we were actually solving the model for the equilibrium, we would need all that information. The equilibrium will depend on preferences and technology. But that would also be true if this were a Walrasian General equilbrium economy. Or a barter economy. All I am really trying to do here is specify those aspects of the model that make it a model of a monetary economy as opposed to a model with a Walrasian auctioneer, or whatever.

The simplest assumption about preferences would be to follow the New Keynesians and just assume symmetric Dixit-Stiglitz, so they want to consume all goods and have a taste for variety. You could put the nth good in the utility function along with the others, if you wanted. People like wearing shells or gold jewelry. Or if you added uncertainty in the repeated game, you wouldn't need to do that.

"Does their own good enter that utility function? If so, do they buy it from themselves at the price they set, or at their MC (with large n it probably doesn't matter much)?"

Yep, with large n it doesn't matter much. The most sensible assumption (that I implicitly made), was that it does ener their utility function and they "sell" it to themselves at MC. Just produce some extra for home consumption. (That's how we treat "leisure", anyway.)

You could assume the MC of producing money is zero. Or not. It won't matter much. You could assume the money-producer consumes the goods he buys, or else gives them as lump-sum in-kind transfers to the other players. Normally macroeconomists take the choices of the money-producer as exogenous, but public choice people would model them as Umaximising, or something.

So wouldn't it simplify the set up much if you just assumed that the money producer in second stage of the game just gives out a money endowment directly (that's where the "directly" above came from, I meant "correctly", but was thinking of this) rather than selling it?

Also if the players (can) choose P>MC I think it would make more sense to have the player charge themselves P, not MC for their own good. It's counterintuitive but it keeps the separation between players as producers and players as consumers straight.

And yes, I understand this is descriptive, not asking for any existence theorems or anything. Just want to get a feel for it and am also just thinking outloud.

notsneaky: your paragraphs:

1. Yes. Less general, and introduces an additional asymmetry that is not essential to the money producer, but it does simplify it.

2. Maybe. It's not fully rational. But it would simplify.

3. I think you are getting a feel for it. I just wanted to make all players as symmetric as possible though, except for the asymmetries that are essential to money.

OK, think I've the rules down. You guys ready for a game?

Ok. Let's assume that there's no disutility of labor so in the fourth stage, when production takes place, output is just produced to satisfy total demand. That's inline with the standard Bertrand assumption where after announcing a price firms stand ready to satisfy any demand. It's also inline with the Keynesian notion that output is demand determined.

By backward induction let's go to the third stage, when players announce their quantities. At this point the prices have been set and money has been doled out by the money producer. What stops a player from announcing an infinite demand? For example, suppose we have two players and they get utility from consuming just the other's good (like in the Diamond coconut model, more or less, just without the search). Money by itself has no utility value. For sake of argument say prices of both goods were set at 1 (any finite number will work) in previous stage. Player 1 has m1 money and will receive x21 extra units of it, where x21 is player 2's demand (for good 1). If she announces x12 (player 1's demand for good 2) greater than m1 + x21, what happens? The demand cannot be satisfied given the amount of money she has. Unless there's something else here, the worst thing that can happen she'll be rationed to just m1+x21 units of good 2. If she announces a demand less than m1+x21 she will wind up with fewer units of good 2 and money left over, which can't be optimal. So any x21>x12-m1 (that's a weak inequality) is 1's best response. So why not announce infinite demand?

Something else needs to be here. And of course you can't just require that the announced demand respects the budget constraint because that depends on the other player's choice (effectively, players are choosing each other's budget constraint).

notsneaky: hang on.

If MC is always zero, and if we have Dixit-Stiglitz preferences with a constant elasticity greater than one, and if n is large (not sure if that's really needed) we get infinite equilibrium output. Because MR is always positive, and MC is always zero. I think that's what you have rediscovered. We need to assume some sort of marginal disutility of producing goods, if people maximise utility.

Take the competitive limit, as elasticity approaches infinite. Full employment output should be infinite, if marginal disutility of production is always zero.

And once P and M have been set, if M/P is positive, we have to assume there is some demand to hold M at the end of the period, otherwise yes, equilibrium quantity demanded of the other n-1 goods will be infinite, because everyone will spend all his money, but it keeps on coming back to him.

(It's like MV=PY or M=kPY, implies infinite Y, if we assume M/P is strictly positive and V is infinite or k is zero. )

Thinking through it...

First comment: that may be (is) a separate problem. Specifically you need either disutility of labor (or some disutility of production) to make MC>0 or you need to get rid of Dixit-Stiglitz. I think the latter can be done; in the back of my head I had the usual linear demand functions assumed in imperfect-substitutes Bertrand models. I don't think the way this is done has all that much effect on what the model's trying to say.

Second comment: the above stuff about MC does not solve the problem of why agents wouldn't post infinite demands. There's actually two issues here I think. The first is just simply, why not? As long as the utility of money is zero, posting a higher demand has no negative consequence since the worse that can happen is that you just get rationed. So yes, either you need to assume Sidrauski utility (which is sort of cheating) or go immediately to a multi period model.

The second issue though is how the whole hot potato/multiplier occurs. Player 1 has some money and uses it to buy goods from player 2. Now player 2 has money from the CB plus the revenue so he spends that on Player 1's good. Etc. How many rounds does this go on for? It's like that Bevis & Butthead episode where they "sell" all their charity candy bars by passing a dollar back and forth between themselves, and eating the candy bars in turn (I've used that in class before). The way that standard models make that multiplier finite is by assuming some kind of leakage at each sub-round (savings, imports, reserves req etc) but here we don't have that. You could put in some kind of transaction cost which burns up some of the money holdings but then you'd need to specify optimal money management; a sub-sub-stage three. And that's getting complicated.

If it's a multi period game then there might be some benefit to holding money at the end of the period. But say that this desired "end of period" money stock is mm1 and mm2 for the players. So they only use m1-mm1 and m2-mm2 of the money they received from the money-producer and keep the rest for next player. But if V is (implicitly) infinite, then that doesn't matter. It only takes an epsilon of money to support infinite demands.

That should be "keep the rest for the next period", not player.


1. "Specifically you need either disutility of labor (or some disutility of production) to make MC>0 or you need to get rid of Dixit-Stiglitz."


"I think the latter can be done;..."

Agreed, it can. Basically you need elasticity to fall when aggregate Y increases, so MR goes to zero and then negative, when aggregate Y gets too big. The Nash equilibrium is at the level of aggregate Y where e=1. I went there once, and found it's best not to go there. Let's not go there.

2. You are correct. An *individual* player could not post infinite demand, because he would run out of money (hit the non-negativity constraint). But *all individuals* would do it, because the Nash equilibrium, with positive money, finite prices, and no desire to hold money at the end of the period (either from the utility of wearing money, or because you might need it next period in a repeated game), is infinite demand.

Simplest way to get money in, in a repeated game: assume half the individuals are producers on even-numbered periods and consumers in odd periods, and vice versa for the other half. Something like that. It's similar to having a 50% chance that demand for your product will be zero in any period, so if you don't already have money from the previous period, you can't buy anything.

And only the consumers get money from the money producer while producers hold it until the following period?

I got to think a bit more about it but it looks like that last comment is inching the idea more and more towards something like the Diamond Coconut model (where instead of even-odd, it's an endogenous mix of producers and consumers determined by the search process). But that model doesn't have money in it. I actually tried to think about how to put money into that model at one point but for reasons I can't remember right now gave up; it was too hard or I was too busy with other stuff or something.

And I think an individual player could post infinite demand, even if all other players post finite (or even zero) demands. Because there's no cost to that. I can say "I will buy one million widgets" but when the dust settles it turns out I end up holding enough money to only buy two. Well, I only get two, no skin off my back. I know that I will hit the non-negativity constraint but why should that bother me? It's sub optimal to post less than my expected money holdings can afford (because, with no desire to hold money at the end of the period, I just end up with useless pieces of paper) but that's the inequality sign going the other way.

So, as long as there's no reason to hold positive balances at the end of the period, even {me=infinite demand, everyone else = zero demand for my good} is an equilibrium strategy. Because, again, why not?

I assume the IOUs are also accepted as payment, equivalently to good n? If so, how do we get a recession?

This is the same question posed by your previous posts. In a simple Walrasian world with credit, why does the market interest rate deviate from the natural or full-employment rate? (These may or may not be the same.) And assuming something in the credit market does set the interest rate at the "wrong" level, what specific mechanism allows the central bank to correct that? In the money game with IOUs, player n no longer seems to have any special status.

The way I would frame the problem is: You cannot tell a coherent story about monetary policy or interest rates (or exchange rates or business cycles or...) without an explicit story about liquidity constraints. Historically, it was possible to represent liquidity constraints as a fixed stock of money as a reasonable first approximation. But more recently -- I mean for the past 30 years at least -- liquidity constraints no longer have even a minimal resemblance to a fixed M.

I think we need a new story for the IOU Game.

JW: "I assume the IOUs are also accepted as payment, equivalently to good n? If so, how do we get a recession?"

I was assuming they are not generally accepted as a medium of exchange. They return to the issuer, without circulating. Back later. Gotta do admin and then teach.

If everyone can issue them, it doesn't matter if they circulate to third parties or not, does it?

If I can always make my purchase of good z by issuing an IOU to the producer of z, then there is no reason for me to change my expenditure based on anything player n does. (Or no more so than on any other player, if money is in the utility function.)

But, will the producer of z always accept my IOUs? This is the big question! Does this kind of thought experiment help us answer it?

(I should be preparing for class myself...)

Nick O/T: Didn't you call Mark A. Sadowski the "world's greatest econ blog commentator" or something like that once? I forget the exact phrase you used (if indeed it was you!) and I can't find it.

JW: "Does this kind of thought experiment help us answer it?"

Short answer: no, it doesn't. But that wasn't the job this model was intended for. There are models that look at the consequences of what is used as money; there are models which look at the causes of what is used as money; (and very good models which manage to do both at the same time). This model is one of the first group. It doesn't explain why the nth good gets used as money. Or even why people use money. It just looks at the asymettries in the strategy space that are created between the n goods if we assume the nth good is used as money. I wanted to explore the relation between money and Bertrand/Cournot strategy space. AFAIK, nobody has explored that connection before.

If we take the NK model literally, as having n goods and no money, with all n firms playing Bertrand, it doesn't make logical sense.

Tom: here, where I added an edit to his comment.


Can I play Bertrand where as a producer I am setting a quantity per unit of account where that unit of account is not also a medium of exchange?

For instance we have an apple producer and orange producer. Unit of account is pounds of weight. Apple producer sets apples per pound to 10. Orange producer sets oranges per pound to 7. Assuming one pound of apples always trades for one pound of oranges, the relative price of apples per orange (or oranges per apple) is not entirely set by either the apple or orange producer.

Frank: yes. We can all set prices in grams of unobtainium. But since unobtainium doesn't exist, there is no demand or supply of unobtainium, and we only care about relative prices, that leaves prices indeterminate. If one vector of prices is an equilibrium to the game, if we double every element in that vector, that new vector is also an equilibrium.

We can also have a different good for MOA and MOE, where both goods actually exist, but I'm ignoring that possibility here. The model just assumes that MOA and MOE are the same good.


We can get an equilibrium condition where both producers are playing Bertrand and setting the quantity of goods relative to a fixed unit of account even if there is no medium of exchange, yes?

Then the problem with the New Keynesian model is that it is missing a unit of account (pounds, grams of unobtanium, etc.) not that it is missing a medium of exchange, yes?

"If we take the NK model literally, as having n goods and no money, with all n firms playing Bertrand, it doesn't make logical sense."

Is the unit of account in a New Keynesian model some measure of time (minutes, hours, days, years)?

An interest rate ( r or n ) is not a unitless measure.

Frank: you have forgotten the producer of the medium of account; there's three producers in total.

The NK model implicitly has an MOE and a MOA, but they need to be explicit.

Years of what? Person-years of (say) labour could be a unit of account. But years simpliciter, no.

I know r has the dimensions 1/time.


The medium of account doesn't have to be produced. It could something as simple as a large rock sitting between the apple grove and the orange grove. I as apple producer playing Bertrand will produce enough apples to equal the pound weight of that rock. You as orange producer playing Bertrand will do the same. Medium of account is the rock (which was not produced), unit of account is the pound weight of that rock, and there is no medium of exchange. One rock's worth of apples trades for one rock's worth of oranges even though you and I have the ability to independently determine how many apples and oranges constitute a rock's worth.

"Years of what? Person-years of (say) labour could be a unit of account. But years simpliciter, no."

Years of life for a fixed lifetime producer.

But I want it to be a produced good, so I can talk about the producer of money, and discuss how he is different from the producers of other goods.

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