Bertrand, Cournot, and the Simple Money Game

I'm writing this post mostly to try to get my own head straight. Read at your own risk.

Strategy space matters. The order of moves matters. What I call "the Simple Money Game" is a three-stage mixed Bertrand-Cournot general equilibrium game. Bertrand moves are made before Cournot moves. Each player is both a producer of one good and a consumer of many goods. The producer of one good plays Cournot, and the producers of the other goods play Bertrand. Consumers always play Cournot. That one good serves both as medium of account and as medium of exchange. "Money" is defined by the strategy space. Or maybe vice versa.

I think this Simple Money Game is similar to what many macroeconomists have at the back of our minds. It's similar to a New Keynesian model in particular. But I want to use it to look at the difference between money and other goods, and between the producer of money and the producers of other goods.

In Bertrand-Nash equilibrium, each producer sets the price of his good.

In Cournot-Nash equilibrium, each producer sets the quantity of his good.

We can also imagine a mixed Bertrand-Cournot-Nash equilibrium, where some producers set price (play Bertrand) and other producers set quantity (play Cournot).

Normally we talk about these games in a partial equilibrium context. But I want to talk about a general equilibrium context.

We cannot have a general equilibrium version of the Bertrand-Nash game. If there are n goods, there are only n-1 prices, so we cannot have n producers each setting his price. If there are only two goods, the price of apples in terms of bananas is by definition the reciprocal of the price of bananas in terms of apples. We cannot have the apple producer and the banana producer both setting the relative price of apples and bananas. If they try to set different prices, they simply create an opportunity for infinite arbitrage profits. One will give up, and play Cournot instead.

The "Simple Money Game" is a general equilibrium game with n players and n goods, with each player producing one good and consuming many goods. The producers of n-1 of those goods play Bertrand. Each sets a price of his good in terms of the nth good, and will accept payment only in the nth good. This means the nth good is both Medium of Account and Medium of Exchange. The nth good is money. The producer of the nth good plays Cournot. He decides what quantity of the nth good to sell.

We could say that each of the n players plays Cournot when they decide how much to buy with money, because they are choosing a quantity. But the n-1 producers of the non-money goods play Bertrand when they choose a price to sell their goods for money. In the normal ways we use the words (we never talk of "buying or selling money", forex markets aside) the n-1 players are both buyers and sellers. The nth player is only a buyer. Sellers play Bertrand; buyers play Cournot.

The order of moves matters. No player will announce how much he will buy before the seller announces the price, because this would give the seller an incentive to announce an infinite price.

One possible assumption is a two-stage game: all players announce their prices simultaneously in the first stage; and then announce their quantities simultaneously in the second stage. In this game, if the producer of money announces that he will sell less money than the other players expected, some of the other players may be unable to buy the quantities of goods they had announced they would buy. They simply run out of money (hit the non-negativity constraint), and cannot buy what they said they would buy. And this effect will snowball.

But instead I want to assume a three-stage game. All players announce their prices; next the producer of money announces his quantities; finally the other players announce their quantities. (You could say this is a variant on the Stackelberg game, with n-1 players being both leaders and followers, and the nth player coming in between.) [Update: I am implicitly assuming production takes place after buyers have placed their orders, and you might want to call this a fourth stage of the game.]

In this three-stage version of the Simple Money Game, we get a recession if the producer of money announces a smaller quantity of money than the other players had expected. Each agent knows he will sell fewer goods to the producer of money than he expected, so will revise down his own planned purchases of goods because of this, and he knows that other players will do the same, which means he will expect to sell even fewer goods, so he revises his planned purchases down because of this, and so on. The Nash equilibrium will have a smaller quantity of goods sold.

If this process happened in real time, it would sound similar to an Old Keynesian multiplier process, or an Old Monetarist hot potato process. But it all happens in virtual time in a one-shot Nash equilibrium game.

I think this three-stage version of the Simple Money Game is roughly what most of us macroeconomists have at the back of our minds.

We could have a repeated version of the Simple Money Game, where some or all of the goods are durable, with depreciation rates less than 100% per period. We would want to assume the nth good is durable.

Each producer of the n-1 non-money goods has n-1 market decisions each period. He decides the price of his own good. And he decides how much to buy of each of the other n-2 non-money goods. The producer of the nth good also has n-1 market decisions. He decides how much to buy of each of the n-1 non-money goods. (And every producer also has one non-market decision: he decides how much of his own good to produce for his own consumption).

In the repeated game, each producer of the non-money goods also has a long run money constraint. If he spends more money buying the n-2 other goods than he earns selling his own good, his stock of money will fall over time, and eventually hit the non-negativity constraint. In the long run he only has n-2 independent market decisions.

The producer of the money good has no such long run money constraint linking his market decisions. Each period he simply decides how much of the n-1 non-money goods to buy.

If we added uncertainty in the repeated game, the n-1 players would have an incentive to hold money even if holding it gave them no direct utility. It would reduce the risk they would hit the non-negativity constraint if they had unexpectedly few buyers for their good in one period.

All players also have a "production constraint" linking their market decisions, and their decision on how much of their own good to consume, to their total production. Each must produce enough both to sell and for his own consumption.

In Walrasian General Equilibrium theory, each agent has a budget constraint that is exactly like the production constraint of the producer of money in the Simple Money Game. He decides how much to buy of all the other goods, taking prices as given. He must produce enough of his own good to sell in exchange for what he buys, plus his own consumption. As others have said before: Walrasian General Equilibrium theory treats all goods as if they were money. But as I have shown, the n-1 players in the Simple Money Game who are not producers of money have a long run money constraint linking their market decisions in addition to their production constraint.

We could change the assumptions to allow each player to produce two goods. The second good would be an IOU, signed by the player, to deliver the nth good at some future date.

We could also change the assumptions so that prices could not always be changed in every period. That would allow recessions to last for many periods.

If we made both those changes, the Simple Money Game would be something like a New Keynesian model.

Comments

[notsneaky]

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)?

[Nick Rowe]

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.

[notsneaky]

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.

[Nick Rowe]

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.

[Tom Brown]

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

[notsneaky]

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).

[Nick Rowe]

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.

[Nick Rowe]

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. )

[notsneaky]

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.

[notsneaky]

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

[Nick Rowe]

notsneaky:

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."

Yes.

"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.

[Nick Rowe]

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.

[notsneaky]

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.

[notsneaky]

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?

[JW Mason]

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.

[JW Mason]

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.

[Nick Rowe]

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.

[JW Mason]

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...)

[Tom Brown]

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.

[Nick Rowe]

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.

[Frank Restly]

Nick,

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.

[Nick Rowe]

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.

[Frank Restly]

Nick,

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.

[Nick Rowe]

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.

[Frank Restly]

Nick,

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.

[Nick Rowe]

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.