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.)
Worthwhile Canadian Initiative 
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.)
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.
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.
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.
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).
“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.