A simple Keynesian-monetarist brain-teaser

Mostly for fun. Mostly for economics undergraduates. But I think there may be some serious lessons too.

Sometimes I like to really push a model, into an extreme limiting case. To see if I can still get my head around it. To see if my intuition is lining up with what the math is telling me should happen. To see if I really understand the model. To see if the model really makes sense.

Here's an example. The only pre-requisite is first-year macro.

Keynesians say that you get a recession when desired (national) saving exceeds desired investment (assume a closed economy) at full-employment output. Output, and hence income, drops until we get to some lower level of output where desired saving has fallen to equal desired investment.

What happens if desired saving is identically equal to desired investment at all levels of income (and at all rates of interest)?

Some may say the economy stays at full employment.

Some may say the equilibrium level of income is indeterminate.

I want to sketch a model where there is a third answer.

For simplicity, assume no exports, imports, government spending, taxes, or investment. The only newly-produced good is a non-storeable consumption good (or service). Haircuts, for example.

The standard Keynesian Cross model with the normal consumption function would then look like this:

1. Cd = a + bY    where a>0 and 0<b<1

2. Y = Cd

Cd is desired consumption. It's the quantity of haircuts demanded (per year, or whatever). Y is income from the production and sale of haircuts. The first equation is the consumption function, and the second is the equilibrium condition. The equilibrium condition assumes that output is demand-determined. There is no mention of supply -- the quantity of haircuts people want to sell. That's OK, provided we stay at or below full-employment income, and so there is never an excess demand for haircuts. (Strictly, 2 should be replaced by Y = min{Cd,Yf} where Yf is full-employment income).

And the solution for equilibrium income is:

3. Y = a/(1-b)

You've seen that before.

What happens to this model if we push it to the limit and assume a=0 and b=1? Imagine, just imagine, that people never want to save. They never want to add to their stock of assets, or reduce their stock of assets. Each year, they want to buy exactly the same number of haircuts as they sell, regardless of income, interest rates, or anything else. So the first equation gets modified to:

1' Cd = Y

The math is telling us that any level of income (at or below full employment) is an equilibrium. Any number for Y will satisfy both equations 1' and 2.

Is the math right? That sounds like a really daft question. "Of course the math is right, unless Nick made an algebra mistake!" OK, let me rephrase it: does what the math is saying make economic sense?

Obviously, negative values for Y satisfy the two equations, but don't make any economic sense. You can't produce and sell negative quantities of haircuts. Let's set that aside.

Can we conceive of an economy where the equilibrium number of haircuts produced and sold could be anywhere between zero and full-employment? Where the excess supply of haircuts could be anywhere between zero and the total number of haircuts people want to sell?

If this is a barter economy, it doesn't make any economic sense at all. If there are two unemployed hairdressers, both of whom want to sell a haircut, why don't they just do a swap? "You cut my hair and I'll cut yours".

Maybe they don't like the sort of styles the other can produce? But that isn't in the original model. The original model says the equilibrium number of haircuts equals a/(1-b). It says nothing about different styles of haircuts, and how a mismatch of supply and demand can cause structural unemployment. It says the quantity of haircuts produced depends only on the aggregate demand for haircuts. So forget about structural unemployment.

The only equilibrium that makes economic sense, in a barter economy, is full-employment.

Maybe this is not a barter economy. Maybe it's a monetary exchange economy. You can't do barter deals. The only trades allowed are simple exchanges of one haircut for money.

OK. So let's bring money into the model.

Assume there is a fixed stock of money M, and a simple money demand function where the stock of money people desire to hold is proportional to nominal income, and the equilibrium condition is that the desired stock of money Md equals the actual stock.

3. Md  = kPY

4. M = Md

And let's assume the price level is fixed at one. (Keynesians can't object to that, surely).

5. P = 1

If money were the only asset, my model would not make economic sense. Because all saving would have to be in the form of money, and equation 3 would tell us the desired stock of savings while equation 1 (or 1') tells us the desired flow of saving, and the two equations would be inconsistent. So let's suppose there's a second asset, antique furniture. Equation 1 (or 1') tells us people's desired flow of saving in the form of money + antique furniture. And equation 3 tells us how they desire to allocate that stock between the two assets. Introducing antique furniture lets 1 and 3 be consistent with each other.

What does the math tell us now? If we look at the system of equations 1', 2, 3, 4, and 5, the math is giving us a very clear answer:

6. Y = C = M/k

Assuming a fixed price level, the math is telling us there is a unique equilibrium level of income. And it's also telling us that recessions are caused by the demand for money at full-employment income being greater than the supply of money.

If you are a Keynesian, and believe that recessions are caused by desired saving at full employment income being greater than desired investment (which is zero in this model), can you get your head around this result? Does it make economic sense to you?

Maybe you think you can get your head around it. Because after all, it doesn't totally contradict your model. Your mental model is saying that, if desired saving is identically equal to desired investment, at all levels of income, equilibrium income could be anything between zero and full employment. So if some monetarist comes along and says "it's M/k", you might reply "whatever".

But now I've softened you up with that devious manoeuvre of the consumption function 1', let's give you a tougher one.

Scrap the weird consumption function 1'. Go back to the normal consumption function 1, where a>0 and 0<b<1. What determines the level of income?

The math won't help you now. The math is giving you two different answers: Y=a/(1-b) and Y=M/k. Except by sheer fluke, those two answers will be different. Which one do you believe? Which one makes economic sense?

Sure, you might try to weasel out of my question, the way John Hicks did with the ISLM model, by saying that desired consumption and/or the desired stock of money depend on the rate of interest (or the price of bonds, or the price of antique furniture), and so the rate of interest adjusts to make the two answers the same. But suppose, just suppose, they don't. Or that the price of antique furniture is also fixed, just like the price of haircuts.

Which answer do you think is right? The Keynesian answer a/(1-b), or the monetarist answer M/k? Or neither?

Stop looking at the math! The math can't help you with this one. You have to think! You're an economist anyway, not a mathematician. At least, that's what you are supposed to be.

There is no level of income at which people can consume what they desire and hold the amount of money they desire, given that level of income. They can't do both the things they want. Which one do they do?

Suppose you insist that the Keynesian answer is right. OK, try this.

If actual output is at the Keynesian answer, but the monetarist answer is less than the Keynesian answer, this is what is happening in the economy. At the current level of income, there is an excess supply of haircuts. People want to sell more haircuts, but can't, because nobody wants to buy any more haircuts than they are currently buying. So there's unemployment. And at the current level of income, people's actual saving, of money plus antique furniture, equals their desired saving. Both are zero, They are spending all their income, which is what they want to do, and not trying to accumulate assets. But they hold less money than they want to hold. They want to sell antique furniture to get more money. But they can't, because everyone else is trying to do the same thing, so there are no willing buyers for antique furniture. They want to sell more haircuts but can't; they want to sell more antique furniture but can't. There aren't enough buyers of haircuts and antique furniture.

Do you think there's any chance some people might say "Sod it! I don't want to cut my consumption below what it is now, but I really want to hold more money. And I can't get more money by selling antiques, because there are no buyers. And I can't get more money by selling more haircuts, because there are no buyers. So I'm just going to have to cut back on my consumption for a bit, even though I don't want to, because that's the only way I can get hold of more money."?

Because if some people do say that, the Keynesian answer is wrong.

That doesn't mean the monetarist answer is right. The truth might be somewhere in between. If you can't get to your desired choice on two variables at the same time, you generally want to sacrifice a bit of both.

Put it another way. Ultimately, you can't separate the consumption function from the money demand function. We buy consumption goods with money. At that very instant when we decide how much to buy, we are trading off extra consumption against extra holdings of money. The "bond" market doesn't always clear, for everyone. And we can't instantly costlessly continuously access the "bond" market anyway. If we could, why would anyone ever want to hold a positive stock of money?

If all markets are continuously clearing and perfectly frictionless, people can sell and buy as much as they want of everything. They can satisfy all of their choices simultaneously. But that's not the Keynesian vision. The Keynesian vision is one of market frictions and non-market-clearing, where people can't always instantly buy and sell what they want. There are involuntarily unemployed workers. And because they can't sell as much labour as they want, that means their income is lower than they want, and that impacts their choice of how much to consume. And the trouble with Keynesians is that they aren't Keynesian enough. They assume there is a frictionless "money market" where people can always instantly get as much money as they want to hold. If there were, they wouldn't want to hold any. Those same frictions that explain Keynesian unemployment also explain why people use money to buy and sell goods, and why people hold money. And those frictions mean people might not be able to satisfy their consumption choices and money demand choices simultaneously.

(BTW. The equations 1 and 2 define a vertical IS curve, because desired consumption is perfectly interest-inelastic. Equations 1' and 2 define an IS curve that is very thick and covers the whole space. Equations 3,4, and 5 define a vertical LM curve. And the example I talked about at the end was one where the vertical LM curve was to the left of the vertical IS curve.)

Comments

[jsalvatier]

Does anyone know of a good paper or book with a mathematical model exemplifying monetary disequilibrium? It would be nice (but not necessary) if it used something like rational expectations. I've tried to work one out on my own but my relevant math skills are not quite up to par.

[JW Mason]

The only equilibrium that makes economic sense, in a barter economy, is full-employment.

I think there are some other assumptions here you're not making explicit. Suppose people are myopic. They form their expectations of this period, by looking at last period. So I producing fewer haircuts than I could, and I am consuming fewer haircuts than I'd like. But, when I make my plans for this period of how many haircuts to buy, I look at the number of haircuts I sold last period. I assume that's all I can afford, so I decide to buy the same number as last period, even tho I'd really like more. And everyone else is forming their plans the same way, so I don't unexpectedly sell any more than I did last period, either. So every point is an equilibrium, just like the math says.

Of course if there were some benevolent central planner who knew all our desires for haircut consumption and capacities for haircut production (call him the auctioneer), then he could match us all up and this problem wouldn't exist. But we are talking about a decentralized market economy, aren't we? In which case, there can be coordination problems, even with just one good.

[rogue]

How about we introduce a supply equation?

X = Cd + Sd

Haircut supply (X) equals quantity that haircutters desire to consume (Cd) plus quantity they want to deliver in order to save (Sd). Then:

Y – Sd = Y’
Y – Cd = Y’

So we deduct the excess quantity of haircuts from income Y (which is also quantity desired for consumption Cd) to get to new income level.

Y’ = Cd’ + Sd’

New lower income level is now equal to new lower desired quantity of haircuts for consumption plus new even higher quantity of haircuts economy desires to deliver in order to save.

Sd’ = Y – Y’

The even higher quantity of haircuts economy now desires to deliver (Sd')is equal to the loss of income that was due to the original excess capacity for haircuts.

If Y' is low enough and Sd' high enough, we could reach equilibrium with Cd' = 0

If Y = Cd to begin with, where does this leave us?

[rsj]

"Except by sheer fluke, those two answers will be different. Which one do you believe? Which one makes economic sense?"

Perhaps I am missing something, but wouldn't k adjust to make them equal? There is no "economic" definition of K -- it is just a plug item, right? It doesn't appear in any of the other equations, nor does it enter into any notion of utility or investment. Nor is there is any relationship between k and interest rates, consumption, or anything else -- is there?

[edeast]

Hi Nick,
"There is no level of income at which people can consume what they desire and hold the amount of money they desire, given that level of income. They can't do both the things they want. Which one do they do?"

Wouldn't this economy recurse into nothing. What gives to stop the recession; k,P,M? and what is wrong with Hicks answer, the zlb? Interest rates have to mediate the competing desires.

[edeast]

err, you answered my hicks question in the last paragraphs, with frictions in the ad hoc "bond" and "money" markets. but wouldn't frictions factor into these instantaneous interest rate determinations.

[Nick Rowe]

jsalvatier: I ought to, but I don't. Can anyone else think of a paper or book along those lines?

JW: I totally agree. I have slid all that under the rug. My only defence: I wanted to keep it simple; everybody else does it too; and in other older posts I have tried to bring some of that stuff out from under the rug and tried to deal with it.

"Of course if there were some benevolent central planner who knew all our desires for haircut consumption and capacities for haircut production (call him the auctioneer), then he could match us all up and this problem wouldn't exist. But we are talking about a decentralized market economy, aren't we? In which case, there can be coordination problems, even with just one good."

Yes indeed. Very nicely put.

But in a barter economy, with only one good, except I can't reach to cut my own hair, my offer to sell a haircut *is* an offer to buy a haircut. It doesn't take much coordination since any two unemployed hairdressers can coordinate themselves. And it's precisely because we can't get all the people together to do a deal in a complex economy that we use money instead.

rogue: I'm not immediately following your equations. I may try again later.

rsj: a monetarist would reply "Perhaps I am missing something, but wouldn't a and b adjust to make them equal? There is no "economic" definition of a and b -- they are just plug items, right? They don't appear in any of the other equations, nor does they enter into any notion of utility or investment. Nor is there is any relationship between a and b and interest rates, money, or anything else -- is there?"

edeast: "What gives to stop the recession; k,P,M?"

Why not a or b?

"and what is wrong with Hicks answer, the zlb? Interest rates have to mediate the competing desires."

Can we always safely assume interest rates *will* adjust to mediate the conflicting desires? Why doesn't P always instantly adjust to mediate the competing desires to sell haircuts and buy haircuts?

[rsj]

"There is no "economic" definition of a and b -- they are just plug items, right? "

No, B is the propensity to consume out of income and A represents some minimum (required) consumption. A and B have economic meaning, whereas K does not. Interest rates also have an economic meaning. Everything in your model has an economic meaning except for k.

Now if you have an economic meaning for k -- something relating it to utility or time preference or technology, etc, then fine -- perhaps by specifying this meaning we can figure out what would change in the model.

[edeast]

k could be an indicator of the amount of friction in the monetary economy, or the taxes thing.

[Kevin Donoghue]

"Does anyone know of a good paper or book with a mathematical model exemplifying monetary disequilibrium?"

I'm probably misunderstanding the question, because the Barro-Grossman (1977) paper which Nick often mentions seems like the obvious answer. But Nick didn't give that reply.

[Nick Rowe]

Damn! It just ate my comment! Trying again.

rsj: a and b are the *desired* (autonomous and marginal) propensities to consume out of income. k is the *desired* propensity to hold (a stock of) money out of income. People *want* to do things. a, b, and k tell us what they *want* to do. If k=1/12, that means people want to hold one month's income as money. Keynes said that b is a psychological constant. OK, let's say that k is a psychological constant too. (Neither is, of course, but you can't "privilege" (sorry for the pomo BS) one parameter over another.)

[Nick Rowe]

Kevin: (You meant B&G 197*1* IIRC). Maybe. But it doesn't really work through the hot potato disequilibrium process. IIRC, it just jumps to the new equilibrium.

[rogue]

Sorry, I have a correction on equation 3 of my previous comment. this is what i meant:

Since Y = CD, subtracting excess quantity of haircuts (due to desire to save, or Sd) from either income (Y) or desired consumption (Cd)nets to a new lower income level:

Y – Sd = Y’
or CD – Sd =Y’

This new lower income level Y' now leads to new lower desired quantity of haircuts for consumption (Cd') plus new even higher quantity of haircuts economy desires to deliver in order to save (Sd'):

Y’ = Cd’ + Sd’

The even higher quantity of haircuts economy now desires to deliver (Sd')could be taken to equal to the loss of income that was due to the original excess capacity for haircuts:

Sd’ = Y – Y’

But then you can also say that the lower consumption desired (Cd') may also be equal to the loss of income:

Cd’ = Y – Y’

Or the loss in income manifests itself in both the change in consumption and saving patterns:

Y - Y' = delta Cd + delta Sd

[rogue]

My non-math point is that some people may want to supply more than they demand so they can earn higher income and save. The overcapacity leads other people to have less income, and hence, compensate by either consuming less, or working harder, thereby compounding the supply glut.

sorry for the original math mixup.

[Nick Rowe]

rogue: I *think* this is the model you may have in mind:

Cd(t) = a + bY(t-1)

Y(t) = C(t)

Consumption today depends on last period's income.

Or, equivalently, (echoing JW Mason a bit)

Cd = a + bE(Y)

Y = Cd

E(Y) = Y(t-1)

People base their current consumption decisions on their expected income for this period (they won't know what it actually is until everyone has actually bough haircuts), and they base their expectations on last period's actual income.

[Kevin Donoghue]

Nick, yes I meant A General Disequilibrium Model of Income and Employment (1971). Of course by "disequilibrium" Barro and Grossman actually mean an equilibrium with involuntary unemployment (or repressed inflation). I not sure how one would "work through the hot potato disequilibrium process" without telling some quite complicated story in which the auctioneer generates a sequence of prices converging to a full-employment Walrasian equilibrium. Maybe that's the sort of thing jsalvatier is after.

[edeast]

How does M differ from P? Sticky prices/wages etc. Can you assume that M=1. So that eqn 6: Y = C = 1/kP. Does 1/kP make sense or different sense from M/k?

[edeast]

Say your M was gold, and it was easier to change prices than to go mine some more.

[Nick Rowe]

Kevin: what I learned from Peter Howitt: you have to imagine that there are 2 auctioneers. The first auctioneer does a tatonnement on prices. The second auctioneer does a tatonnement on quantities. The second auctioneer works much more quickly than the first. So, the P-auctioneer calls out a randome vector of prices. The Q-auctioneer calls out a random vector of maximum quantities that each agent will be allowed to buy or sell (rations). Agents then submit their demands and supplies, given those prices, and given those rations. The second auctioneer then revises what he allows each agent to buy or sell based on what other agents are selling or buying. If there's a bigger supply than demand, he increases the ration of what agents can buy, and reduces the ration of what they can sell. And so on.

In the standard "disequilibrium" model, like B&G, the Q-auctioneer works infinitely quickly compared to the P-auctioneer. So the Q-auctioneer converges to a solution before the P-auctioneer has even adjusted prices. That solution is a Keynesian "equilibrium". It's the B&G "equilibrium". It's where Cd=Y.

And all this may be happening in real time, so stocks of everything are changing during that Q-process, if "false trading" is allowed.

And the hot potato process is what is happening during the convergence to that Q-auctioneer's solution.

Yes, it's hard to model!

[Nick Rowe]

edeast: if P were perfectly flexible, then M/P adjusts to equilibrate the demand and supply of money. Money is then neutral, even in the short run.

[rsj]

Nick,

OK, that makes sense, but it assumes that money is fundamentally different from "wealth", which does not conform to our experience. I.e. no one holds 12 months of income as money, but they do hold 12 months of income as bonds. You would have a similar effect if you put wealth into the utility function as if you put money into the utility function. But if you put in wealth, then CB asset purchases don't do anything (to increase wealth), whereas interest rate policy can increase wealth. You see where I am going here? I.e. diagnosing the problem as one of wealth in the utility function versus money in the utility function makes a big difference for the prescribed solution.

[Nick Rowe]

rsj: money is wealth (at least to the individual who holds it), but it *is* very different from other forms of wealth. See my comment in reply to scepticus on the ZMP post.

[rsj]

Money is *only* different from wealth if you put a term of the form, say, u(M/p) into your utility function. But if you put a term of the form u(W/p) into your utility function, you get similar weird behavior but with wealth instead of money.

I.e. only by assuming that money is unique can you conclude that it is unique.

If you assume the consumer is maximizing sum{ B^k(log C_k + log(1-L_k) + (1/a)(qK)^a)} -- where q is the price of capital goods in terms of consumption goods, and L_k is labor, then your consumption Euler equation becomes:

C_{n+1} = BR_nC_n/(1 - BR_nC_n(qK)^{a-1})

which gives completely different dynamics than the standard (deterministic) equation: C_{n+1} = BR_nC_n

But if you can put money into the utility function, then why not put in wealth? I.e. a demand for wealth for its own sake, independent of future consumption, just as you assume there is a demand for money independent of consumption.

[Nick Rowe]

rsj:

1. If a second fridge was dropped by helicopter, and at the same time the price of fridges halved, it would make a real difference to you. Not so with money.

We use money to do the shopping. The stock we want to hold should presumably bear some relation to the sizes of the flow in and the flow out. Proxied by current income. Just like the inventory of cars in a dealer's lot. We save, in general, for intertemporal consumption smoothing. That presumably depends on current income relative to expected future income.

2. To repeat myself: "Stop looking at the math! The math can't help you with this one. You have to think!"

Sorry, but that's what I felt! Bugger Euler!

If you want to hold more fridges (or antique furniture) there is only one way to do so: go to the fridge market and try to buy more fridges. If you want to get more money, you can go to *any* market and either buy less other stuff or sell more other stuff. An excess demand for money will have economy-wide effects. An excess demand for fridges...?

[Jacques René Giguère]

Nick:
"An excess demand for money will have economy-wide effects. An excess demand for fridges...?"

Part of the problem is people (and most economists including the publishers of price index) still not understanding the difference between "economy-wide micro" effects (excess demand for fridges meaning excess demand for fridge-producing factors) and macro effects.
Maybe if we stopped teaching micro and macro in the same curriculum to the same students and declared that micro is henceforth "gloubnology" and macro is "kapostics", then gloubnologists like Mulligan would no more interfere in "kapostics" than in cardiology...

[Nick Rowe]

Jacques: Yep. In general equilibrium theory, anything which affected fridges would affect everything else in the whole economy. If some accident destroyed half the stock of fridges, the ripples would be felt across the whole economy. If some accident destroyed half the stock of money, the ripples would be felt across the whole economy too. But I don't see any way that the former would cause a massive recession, while I think that would be the likely result of the latter.

It's hard.

[rsj]

Nick, with respect, you are confusing money with income.

And -- I would add -- attempting to "count" money rather than looking at income, which is what you should consider.

The decision to purchase capital is a capital transaction, and need not be an income transaction at all. Money is not some incompressible fluid. You don't know anything about the number of *newly produced* capital goods sold just by summing over each individual's savings demands. The missing piece is expected profitability of firms in the future -- and it could be the distant future, where there are no futures markets that allow us to agree on how many consumption goods will be demanded and at what price.

If households believe that the profitability of firms will be higher in the future, then the price of capital goods relative to consumption goods will go up in the current period and more new capital goods will be sold in the current period -- each individual household can borrow to bid up the price of capital without giving up any present consumption opportunities, as capital transactions are not income transactions. As they do this, they create income for someone else in the economy so that, in aggregate, an income adjustment clears savings demanded with savings supplied.

When the expected future profitability of firms declines, the price of capital goods must also decline, relative to consumption goods, and there will be less savings. It will turn out, ex-post, that households that do save are merely buying capital goods from each other, depriving each other of income, again, an income adjustment equates expectation of future profitability with savings demands. Perhaps if we had futures markets for consumption goods, then these markets would determine expectations of future profitability, so that everyone would remain on their optimal consumption plan. But we don't have these markets.

All of the above can work just as well in a pure credit economy as in a commodity money economy. That is true in any economy in which you make a distinction between cash-flow and income. Yes, the cash always "goes" somewhere whenever households buy something (e.g a bond), but that does not mean that _income_ is constant.

[Joe]

Nick,

I still can't get my head around the idea of savings ever being above investment. Wouldn't the interest rate automatically adjust within 10 minutes to make the two equal? Are interest rates sticky? How could the loanable funds market fail to clear? Or, is it the case that when Keynesians use the word 'savings' they include the concept of 'money hoarding' within that term?

Best.

[Joe]

Nick,

Let me give you an example of what I men. The following quote is from mark blaug's history of economic thought, from the chapter on say's law. He distinguishes the simple quantity theory mechanism of Locke, cantillon, and Hume, from the interest rate mechanism of Thornton and wicksell......

"in a monetary economy, therefore, savings and investment cannot be always and necessarily equal to the supply and demand for loans. But in equilibrium this will be true because equilibrium is given by the condition that people are satisfied with heir cash holdings. It is evident then that a consistent interpretation of classical economics Implies the denial of the proposition that planned saving is identically equal to planned investment. This kind of statement is simply a keynesian translation of the language of loanable funds."

But how can the loanable funds market not be in equilibrium? How on earth can wicksell and Keynes say that?

[Jacques René Giguère]

Nick:
It is hard indeed when general equilibrium theorist don't understand they are gloupnologists deluding themselves thinking they are kaposticians.Especially if we give them a Nobel.
At least Marie Curie was a master in both chemistry and physics when she got her two prizes...
The more I think of it,the more I really think nobody should study both micro and macro for at least 40 years ,the way Moses led the escaped Hebrews in the desert forget their old ways before entering Canaan.

[Kevin Donoghue]

"...is it the case that when Keynesians use the word 'savings' they include the concept of 'money hoarding' within that term?"

The short answer is, yes, that is the case. Saving is defined by S=Y-C. If your model determines Y and C then it also determines S. The confusions arise from the fact that in verbal discussions, C might refer to a function, or an equilibrium value (one point on that function) or even C as per the national accounts (again a single value but not necessarily an equilibrium value). Consequently 'S' can also have several meanings. Sometimes the confusion is cleared up by distinguishing between ex-ante and ex-post saving; and sometimes that just adds to the confusion.

One of the virtues of mathematicians is that they are very clear about the difference between a function f and a value f(x). Economists not so much, though if they are French they are usually trustworthy in that respect. So I differ from Nick. I say: look at the math! But look like you were taught in a French school!

[Nick Rowe]

rsj: "Nick, with respect, you are confusing money with income."

No I'm not! There is income even in a barter economy, where there is no money. Plus, money is a stock; income is a flow.

Simplify, simplify, simplify.

There are no newly-produced capital goods in my model. The only newly-produced good is haircuts, which are a consumption good. There is also a fixed stock of antique furniture, which was all produced 100 years ago. People inherited it from the grandparents, and nobody can remember how to produce it any more. Income is the number of haircuts produced and sold each period. (The nice thing about haircuts is that production and sale are pretty well identical).

(Someone *could* argue that the flow of enjoyment from owning antique furniture should also be included in income, and they would have a point, but that's not how income is normally defined, and it would also break my nice production function which says income=output of haircuts=employment.)

And there is a stock of money. That stock is fixed.

There are no firms. Just self-employed hairdressers.

There's a market for haircuts. Since you can't cut your own hair, everybody both buys and sells haircuts. And it's a monetary exchange economy, so you buy and sell haircuts for money.

There's a market for antiques. You can buy or sell antiques for money.

There is no market in which antiques can be exchanged directly for haircuts. You have to sell your antiques for money, then take that money and buy a haircut.

There are no other goods, and no other markets.

Each individual has only two choices to make:
1. How many haircuts to buy in the haircut market (I'm assuming less than full-employment, so you can always buy as many as you like, but can't choose to sell more, because quantity of haircuts traded is demand-determined, by the short-side-rule).
2. How many antiques to *try* to buy or sell in the antique market, to add or subtract from his stock of antiques.

Now, can you restate your point within that simple model.

[Nick Rowe]

(I've just had an interesting thought. Well, interesting to me, anyway. I've implicitly assumed no theft of haircuts. Nobody skips without paying. So production=sales. Suppose that 10% of haircuts were stolen. There's a 10% chance the hairdresser will go into a dream just as she has finished cutting your hair, so you can slip out the door quickly, and don't have to pay money. How would that affect the equilibrium level of employment and output? Assume it has no effect on the supply side of haircuts.)

[Nick Rowe]

Joe: as Kevin says, anything an individual does with his (disposable) income, except spend it on newly-produced consumption goods, is defined as "saving". S=Y-C. In my model, an individual's flow of saving equals the increase in his stock of antiques plus the increase in his stock of money. (With a more complex model, add in increase in his stock of bonds plus his flow of expenditure on newly-produced investment goods, etc.)

Mark Blaug's statement there is a little bit misleading though. Even without money, saving, as currently defined, doesn't mean just a supply of loans (demand for bonds). It can also be a demand for antique furniture, or land, or whatever. And this, by the way, is where many Keynesians go very wrong in their thinking. They think that an excess demand for antiques, or land, can cause a recession. Nope. It's an excess demand for the medium of exchange (money).

[anon]

Nick Rowe, theft doesn't really change things, assuming that folks are risk neutral and there is no heterogenouity. The price charged for a haircut rises by about 11% and expected spending and income are unchanged.

Assuming risk aversion, the possibility of 'theft' amounts to a transaction cost, so agents exchange fewer haircuts and are made worse off. Under heterogeneouity, the market tends to collapse due to adverse selection.

[Nick Rowe]

anon: dunno. If we assume the price of paid-for haircuts stays the same, and if we assume that the demand for money depends only on income/expenditure of paid-for haircuts (you don't need to hold money if you steal the stuff you want), then M=kPY says that actual Y will rise 10% above paid-for Y, which stays the same. (And if P rises by 10%, as you assume, total output would stay the same).

What does Y=a/(1-b) say? Not so obvious.

Now, in a model where demand=supply, you would be right about transactions costs causing fewer trades. But this is not a model where demand=supply. There is excess supply. The disincentives for suppliers are irrelevant, at the margin.

[Joe]

I suppose then I should ask the following question... Is the price of money determined by the price level, as monetarists say, or is it determined by interest rates, as Keynesians say?

I could put that another way... Is the demand for money determined by interest rates or income?

No textbook, blog, paper, or economist has ever answered these questions to my satisfaction. They pick and choose which one of those it is according to the situation.

Sometimes i'm told that monetary disequilibrium is a problem because prices, and thus the purchasing power of money, fail to adjust in the money market. In this money market the vertical axis is 1/p. Other times, the vertical axis is the interest rate in the money market and monetary disequilibrium is a problem because the return to capital (natural rate) does not equal the actual interest rate.

The first is problematic because it has no use for interest rates (Hume, fisher, Friedman, and now I suppose sumner). The second is problematic because 1) it doesn't explain prices and 2) doesn't include money (Thornton, wicksell, Keynes, hicks, wood ford). I think I got that right.....

Could you imagine if modern biological science had two totally different and incompatible and competing systems that explain how the human body works, or in modern physics to have two totally separate systems of how the universe, matter, and reality work!

[Nick Rowe]

Joe: let me give you the fairly orthodox, textbook answer.

Money is traded in every market (it's the medium of exchange). Money usually does not have a price of its own (it's the medium aka unit of account).

In the market for apples, the price of money is 1/the price of apples.

In the market for bananas, the price of money is 1/the price of bananas.

In the market for labour, the price of money is 1/the price of labour (aka the wage)

In the market for land, the price of money is 1/the price of land.

In the market for bonds, the price of money is 1/the price of bonds. And the price of a bond that pays $1 next year is 1/(1+i) where i is the nominal interest rate.

The demand for money is: proportional to the prices of apples, bananas, labour, etc,; an increasing function of real income; and a decreasing function of the rate of return on assets like bonds and land.

Start in equilibrium, where Md=Ms. Then *permanently* double the stock of money.

In the long run, when all prices are flexible, the price of apples, bananas, labour, and land, will double too. The price of a bond that pays $1 next year will stay exactly the same. So the rate of interest stays the same. The rate of return on land stays the same too, because the price of land has doubled but rents on land have doubled too. Real income and employment stay the same.

In the short run, some prices are fixed. The usual assumption is that apples, bananas and labour, and maybe rents on land, are sticky. So they can't adjust. So something else has to give. The usual story is that bond prices rise (interest rates fall), land prices rise (the rate of return on land falls), and real income rises. And they rise enough that the demand for money increases enough to equal the new supply.

Does that help?

[Kevin Donoghue]

Joe: "Is the demand for money determined by interest rates or income?"

I see nothing wrong with the textbook answer I learned back in the days when I had a full head of hair. It's a function of both.

I suspect the following will come across as a bit patronising. Apologies for that. I've just enjoyed a good dinner and several glasses of wine, which always makes me want to tutor the world. What the hell, it's a blog.

Keynes remarked somewhere or other, that the only person who learns anything from a mathematical model is the one who works it out from scratch. In that spirit, I suggest that the way to understand macro is: (1) write down a utility function for a representative household and a production function for a representative firm; (2) solve for the full-employment equilibrium; (3) assuming the money-wage (and the price of goods also if you like) to be above the full-employment equilibrium value and treated as given constants, solve for the Keynesian equilibrium.

Contrary to what Joan Robinson and Nick Rowe suggest, mathematics and logical reasoning are not incompatible.

[Kevin Donoghue]

Nick,

I wouldn't be too happy explaining a recession by excess demand for money or other assets. The whole point of constructing a model AFAIAC is to show how exogenous variables determine endogenous variables. There has to be a story about why the (notional) excess demand doesn't show up as (effective) excess demand. I can certainly accept "M/P (or M/W) is too low."

Now, if A is antiques and V(A,M/P) is a function nested in a utility function U(C,N,V(.)), I'm open to argument as to whether I should accept "V/P (or V/W) is too low."

It’s Saturday night on my side of the pond so with that I’ll bid you a good night.

[Nick Rowe]

Joe:

Simplest textbook story. The one I tell first year students.

1. The demand for money depends on the price level, real income, and (negatively) on the rate of interest.

2. The rate of interest adjusts very quickly, income adjusts more slowly, and the price level very slowly.

3. In the very short run the interest rate is the only thing that adjusts, so it has to adjust a lot. In the medium run, both income and interest rates can adjust, so they both adjust a bit. In the long run, the price level is the only thing that adjusts.

Maths and monopolistic Competition are the two things where I really agree with Joan Robinson!

[Jacques René Giguère]

Nick:
"They think that an excess demand for antiques, or land, can cause a recession. Nope. It's an excess demand for the medium of exchange (money)." But anttuque merchants take a cut. They give themselves that cut as payment for the value added ( gathering things in a convenient place and so forth.) Keynesians don't go wrong on that issue. They assume that buying antiques or land still deliver part of the purchase price as income.
Joan Robinson, the great unacknowledged...

Kevin Donghue:
"One of the virtues of mathematicians is that they are very clear about the difference between a function f and a value f(x). Economists not so much, though if they are French they are usually trustworthy in that respect. So I differ from Nick. I say: look at the math! But look like you were taught in a French school!"
Being Québécois, I did all my studies in French and I have no idea what you are talkiing about...

[Kevin Donoghue]

I was just being facetious really; Bénassy and Malinvaud use mathematics as I think it should be used, but I’ve no good reason for claiming that credit for this belongs to their schoolteachers.

[Nick Rowe]

Jacques: "But anttuque merchants take a cut. They give themselves that cut as payment for the value added ( gathering things in a convenient place and so forth.) Keynesians don't go wrong on that issue. They assume that buying antiques or land still deliver part of the purchase price as income."

If we have a model where there is a stock demand for money, and also flow demands for newly-produced goods, and we want to be able to specify those demand functions separately, (like in ISLM, for example) we also need some other non-money non-produced asset in the model. The common assumption is to call that other asset "bonds". But bonds are only one part of all the non-money assets we own. To force people's heads away from the bonds metaphor, and to make them realise that it is just a metaphor, I call my alternative non-money asset "antique furniture". (It's also a clearer example, since antiques are concrete goods). But "land" would do as well.

Now, when people trade antiques, antique dealers take commissions. And those commissions get counted as part of GDP. Similarly, when people trade bonds, bond dealers take commissions, which are also counted as part of GDP. But those commissions are ignored in standard macro models. That's not what they have in mind when macroeconomists talk about increases in output and employment. They aren't thinking of bond traders' commissions.

Actually, the fact that there are transactions costs and hence commissions on trading the non-money non-produced asset strengthens my story. It means the "bond" market is not the perfectly frictionless ideal that is assumed.

Interesting to think about this stuff.

Kevin: you lost me on you 02.57 comment. You might be on to something, but I can't figure it out.

[Adam P]

"non-money non-produced asset in the model. The common assumption is to call that other asset "bonds". "

I think this is a fundamental misunderstanding, in general equilibrium bonds (even government ones) are a produced asset.

For bonds issued by firms this is easy to see, since the firm is assumed to have already maximized the present value of its profit stream it *can't* pay back the bond with interest if it doesn't invest the proceeds in something that yields at least the interest rate it's paying.

For most firms that means investing the borrowed money in a real asset, for banks it means lending the money out at a higher rate but the borrower is in the same position.

The same is true of government issuance, the government spends the money either on something that directly facilitates higher future real income or it facilitates someone else doing this.

In the end we always have:

In aggregate, the assumption that the economy started on the production possibilities frontier, and not inside it, implies that any increase in the aggregate bond stock *must* be backed by an increase in productive capacity or it will be defaulted on in real terms (which includes inflated away).

Nick's antique furniture as savings medium example does not show what he thinks it does.

[Nick Rowe]

Adam: It's very good to see your comments.

Let me come at your comment sideways, rather than directly.

Assume all "bonds" are issued to finance new investment by firms. (That's roughly what you have in mind, I think, since we are ignoring government, and if all agents are identical there won't be any borrowing or lending between households).

Let's replace "bonds" with "bonds plus stocks", so we don't have firms defaulting if something unexpected happens. So bonds plus stocks finance new capital goods.

For simplicity, let's assume the households own the capital goods directly, rather than owning paper claims of bonds plus stocks on firms' capital goods. Firms rent capital goods from households.

Capital goods depreciate.

There are some sort of adjustment costs so that one unit of a consumption good cannot be costlessly transformed into one unit of the investment good. So they can have different prices. Standard Jorgensen(?) sort of assumption.

Now, take the limit of that model as the depreciation rate goes to zero, and the adjustment costs go to infinity. Capital goods become land. It doesn't wear out, and you can't make any more.

Now we've got a model where labour plus land produce consumption goods, in some sort of production function. Let there be two consumption goods: haircuts; plus household furnishing services.

Take the limiting case where the production functions imply that haircuts require labour-only, and household furnishing services require land only.

Re-name "land" "antique furniture".

I think that means that my antique furniture model is economically equivalent to a limiting case of a model with bonds.

[Adam P]

Nick,

Fine with your limiting case but, first of all, that means your example isn't as general as advertised. Perhaps in such a limiting case only an excess demand for money can cause a recession but you usually want to claim that's true in all economies, not just your very specific limiting case.

Second of all, you said "like in ISLM, for example" and my point is that ISLM is *exactly the opposite* limiting case! ISLM assumes linear produciton technonlgies, no adjustment costs at all. Jorgensen was trying to reconcile that with reality after all.

So I still think that your example doesn't show what you think it does because it doesn't apply to econimies that are not in your limiting set.

[Nick Rowe]

Adam: "Fine with your limiting case but, first of all, that means your example isn't as general as advertised."

Fair point. My model has consumption only, and no investment. I'm toying with the idea of doing a similar extremely simple model, but the exact opposite limiting case. The only produced good is an infinitely-lived consumer durable. They don't produce haircuts; they only produce furniture. But I don't want firms holding stocks of unsold furniture in the model. Because that complicates it.

"ISLM assumes linear produciton technonlgies, no adjustment costs at all. Jorgensen was trying to reconcile that with reality after all."

If the technology really were linear C+I=Y=F(K,L)=C+Kdot and there were no adjustment costs, the IS curve should be horizontal. And it's not usually drawn that way. I see Jorgensen as trying to reconcile the ISLM logically with itself, rather than empirically. (OK, I'm not really disagreeing with you, since there's still the empirical question, and that's what J saw himself as doing, I'm just offering a second complementary perspective.)

[Adam P]

"If the technology really were linear C+I=Y=F(K,L)=C+Kdot and there were no adjustment costs, the IS curve should be horizontal"

I don't think that's true, can you explain?

[Nick Rowe]

Adam: The price of capital goods should always equal the price of consumption goods, since one unit of output can be transformed into either one unit of consumption or one unit of capital goods, regardless of the mix produced. There's a demand for the stock of capital as a function of the rate of interest, and an infinitesimal change in the desired stock causes an infinitely large change in the flow demand for investment, so the investment demand curve is perfectly interest-elastic, given the existing stock of capital. The slope of the IS goes to infinity when the interest elasticity of demand for investment goes to infinity. A tiny fall in r below MPk would cause an infinite demand for investment, and hence an infinite demand for newly-produced goods.

It's not quite as straightforward as that, because if output changes then either labour or capital or both must be unemployed, and that may change the Marginal Revenue Product of capital goods. But that extra complication would tend to make the IS curve *upward*-sloping, because the value of the MPk would tend to fall in a recession (you can't sell the extra output from an extra unit of capital, or there's a larger ratio of employed capital to employed labour, if workers are unemployed), and this recession would lower the desired capital stock.

[Nick Rowe]

Not sure if that was as clear as it should have been. Trying again.

If the desired capital stock is a sensible negative function of the rate of interest, then the flow demand for investment should be perfectly interest-elastic. So IS is horizontal.

If we add adjustment costs, a la J., then the supply-price of capital goods is an increasing function of the flow of investment, so the desired capital stock depends on the rate of interest and also negatively on the relative price of capital goods, and thus negatively on the flow of investment. So investment demand is not perfectly interest-elastic.

The complication: in a recession, either labour or capital is unemployed (or both).

If only labour is unemployed, then the rise in the K/L ratio reduces the MPK and lowers the desired capital stock, so the rate of interest would need to be *lower* in a recession to prevent investment demand falling further.

If only capital is unemployed, then an extra machine would likely sit idle, so its effective MPK would be zero. (Firms can't sell the extra output from an extra machine).

Both those effects tend to make the IS curve slope up.

(I had trouble posting the last comment. If it happens to you, back arrow, forward arrow, then click Post again.)

[Kevin Donoghue]

You might be on to something, but I can't figure it out.

What I have in mind is a setup where utility is a function U(N, C, V) of leisure, consumption and a portfolio composed of money and antiques. V(A, M/P) is just some function that converts the two components of the portfolio into a scalar. Households derive utility from contemplating their stock of treasure, as measured by that index.

What if some of the antiques are destroyed? Maybe the resulting rise in the price of antiques restores equilibrium with no change at all in money wages. But I think the equilibrium money wage could fall. Presumably what I need to do, in order to tell this story in respectable economese, is rig the utility function so that leisure and treasure are complements, which doesn’t seem such a stretch. Leisure isn’t quite so satisfying when you’re looking at that sad empty space where the Louis XV silver tureen used to be. If wages are sticky downwards then we get involuntary unemployment, resulting from the shrinkage in the stock of antiques.

Is that not an instance of “excess demand for antiques causing a recession” (or at any rate creating an output gap), or do you have something else in mind when you say such a thing is impossible?

BTW, I won’t be all that surprised if it turns out that the above reasoning is badly flawed. The fact that I’m not sure I’m right, because I haven’t actually summoned up the energy to start mucking about with an explicit utility function, reinforces my belief that we need mathematics to check our intuitions. It’s just too easy to tell a story that sounds reasonable.

[Nick Rowe]

Kevin: Take your utility function, and let "M" stand for "bling". M is some sort of jewelry we wear, but only get utility because it flaunts our wealth, so our utility depends on the real value of bling we are wearing, not the physical amount M. And then assume we have a barter economy. If the price of bling in terms of goods is fixed at some non-market clearing value, the rest of the economy could keep on trading as before, even though there's an excess demand for bling.

[Kevin Donoghue]

Nick,
My understanding was that we were talking about a monetary economy. Let the money be beautiful by all means. The antiques are beautiful too. But they're not money. Bling is the only money. Yet, unless my reasoning is incorrect (which is all too likely) the destruction of antiques creates a typical Keynesian output gap.

[Nick Rowe]

Kevin: OK. I misunderstood you. A destruction of part of the stock of antiques would increase desired saving (reduce desired consumption) and so cause a deficit, from the Keynesian perspective. The parameter a or b falls. But, at the same time, it might reduce the demand for money, and so cause an expansion, from the monetarist perspective. The parameter k falls.(Holding all prices constant in both cases).

[rsj]

"Each individual has only two choices to make:
1. How many haircuts to buy in the haircut market (I'm assuming less than full-employment, so you can always buy as many as you like, but can't choose to sell more, because quantity of haircuts traded is demand-determined, by the short-side-rule).
2. How many antiques to *try* to buy or sell in the antique market, to add or subtract from his stock of antiques.

Now, can you restate your point within that simple model."

No, I can't. This is because I am relying on fluctuations in investment to generate fluctuations in output.

At a minimum, I need some concept of capital goods which can be produced.

The quantity of new capital goods produced and sold in a given period is equal to aggregate savings in that period.

But the total quantity of capital goods is the capital stock. There is no market for newly produced capital goods. There is a just a market for capital, in which pre-existing and newly produced capital are indistinguishable.

Anyone with capital, including the capital goods producers, is a potential seller, and will sell at some price. Anyone is also a potential buyer. Someone buying capital from this market does not even know whether they are buying new or used capital goods.

When someone uses their income to buy used capital goods rather than (new) consumption goods, they are decreasing the income of someone else. So whether the decision of individuals to purchase capital results in more aggregate savings or a decline in income is still an open question.

Instead, look at how much capital the capital goods producers will choose to sell. That will be a function of the relative price of capital to consumption, right?

And the key wrinkle I am introducing is to point out that expectations of future resale value influence this price, and can cause it to deviate from the optimal price:

If people believe that the capital goods will worth a (too small) amount of consumption goods in the next period, then the price of capital goods in terms of consumption goods will fall today, and so there will be less savings today. If that quantity of savings is less than the savings demanded, then an income adjustment will clear savings demanded and savings supplied.

But the belief that the capital good is worth less next period has nothing to do with how much people want to save or not save in the current period. It's just a prediction and so is not influenced by preferences.

And it can be a self-fulfilling prediction.

If there was a dated commodities market so that you can buy capital tomorrow, and this dated commodities market was simultaneously clearing, then you wouldn't have this problem.