The 45 degree line means Y=min{Yd,Ys}

(I'm disagreeing with Roger Farmer on the interpretation of the 45 degree line. I say the 45 degree line is not a supply curve. This post explains what I think it is.)

Here's the Keynesian Cross diagram:

How should we interpret the green 45 degree line?

To keep it simple, imagine an economy where all goods are services, so we can forget all that nonsense about desired vs undesired inventory accumulation. It's a haircut economy. You can't cut your own hair. People cut each others' hair, for money. Haircuts get produced-and-bought-and-sold all at the same time.

On the horizontal axis we have the quantity of haircuts actually produced-and-bought-and-sold, Y.

On the vertical axis we have the quantity of haircuts demanded Yd, which means the quantity of haircuts people want to buy, which may or may not be the same as the quantity they actually buy.

On the vertical axis we also have the quantity of haircuts supplied Ys, which means the quantity of haircuts people want to sell, which may or may not be the same as the quantity they actually sell.

The green 45 degree line tells us that the actual quantity of haircuts produced-and-bought-and-sold will equal whichever is less: quantity demanded; or quantity supplied. You don't get a haircut produced unless there is both a willing buyer and a willing seller. So the economy will be where the demand curve cuts the 45 degree line, or where the supply curve cuts the 45 degree line, whichever is less.

If some sellers of haircuts, or some buyers of haircuts, could make you an offer you can't refuse, the economy would not be on the 45 degree line. If some sellers felt an obligation to sell more than they wanted to sell, or if some buyers felt an obligation to buy more than they wanted to buy, the economy would not be on the 45 degree line. If people had false beliefs about how much they were able to sell, or about how much they were able to buy, the economy would not be on the 45 degree line. The 45 degree line tells us something substantive about the economy.

The red demand curve tells us that the quantity of haircuts demanded will be an increasing function of the quantity of haircuts people are able to sell, if they cannot sell all they want to sell.

The blue supply curve tells us that the quantity of haircuts supplied will be an increasing function of the quantity of haircuts people are able to buy, if they cannot buy all they want to buy.

This economy is in semi-equilibrium at Y'. At Y', the quantity of haircuts bought-and-sold is equal to the quantity demanded, but is less than the quantity supplied. Yd=Y<Ys. People can buy as many haircuts as they want to buy, but want to sell more haircuts than they are able to sell. If they could sell 10 more haircuts per year, they would want to buy (say) 6 more haircuts per year. They would increase their expenditure if they were able to increase their income, but they can't, so they won't. There's a demand-side multiplier from shifts in the demand curve.

Now suppose something causes the red demand curve to shift up, so it intersects the 45 degree line at Y*. The economy would now be in full equilibrium at Y*, where Yd=Y=Ys. People are buying the quantity they want to buy, and selling the quantity they want to sell. This is what the economy looks like in full equilibrium:

Now suppose something causes the red demand curve to shift up even further. Or something causes the blue supply curve to shift down. The economy will now be in semi-equilibrium where the supply curve cuts the 45 degree line, so Yd>Y=Ys. There is an excess demand for haircuts. People are selling as many haircuts as they want to sell, but unable to buy as many haircuts as they want to buy. If they could buy 10 more haircuts per year, they would want to sell (say) 6 more haircuts per year. They would increase their income if they were able to increase their expenditure, but they can't, so they won't. There's a supply-side multiplier from shifts in the supply curve.

When Yd=Y<Ys, people can't sell as much as they want, so the demand curve slopes up, but can buy as much as they want, so the supply curve is horizontal.

When Yd>Y=Ys, people can sell as much as they want, so the demand curve is horizontal, but can't buy as much as they want, so the supply curve slopes up.

This is all just old Barro-Grossman/Malinvaud etc. stuff. Roger and I both learned it at the same time in Peter Howitt's advanced macro class. But should we be teaching it in first year?

Comments

[Min]

If the vertical axis represents min(Yd, Ys) that's how it should be labeled.

Now, assuming that the units of each axis are the same, then the green 45 degree line represents the equation, Y = ??? I put in the question marks because the vertical axis is not well defined. Apparently.

(I find it incredible that economists would not agree what the green line means, BTW.)

Nick Rowe: "The green 45 degree line tells us that the actual quantity of haircuts produced-and-bought-and-sold will equal whichever is less: quantity demanded; or quantity supplied. You don't get a haircut produced unless there is both a willing buyer and a willing seller. So the economy will be where the demand curve cuts the 45 degree line, or where the supply curve cuts the 45 degree line, whichever is less."

Well, that is an interpretation of the green line. But if supply and demand are curves, and not simple quantities, then that interpretation does not make sense. And, in fact, you show a blue supply curve and a red demand curve. In that case, min(Yd, Ys) decidedly does not correspond to the green line, but to a different curve composed of portions of each of those curves.

FWIW, I have come up with a different interpretation of the green line. Let the vertical axis represent (Yd OR Ys), depending on which one we are depicting. Then the green line stands for the equation, Y = Yd or for the equation, Y = Ys, whichever one we are depicting. If I understand you correctly, the amount actually sold, given the supply curve, the demand curve, and the green line, is where the green line intersects the curve, min(Yd, Ys). In the diagram where the red curve is less than the blue curve, then the red curve is equal to the min(Yd, Ys) curve.

Surely the meaning of the green line is not a secret. Whoever first came up with it knew what it meant.

[Nick Rowe]

Min: take a standard supply and demand diagram. We have P on the vertical axis, and three things on the horizontal axis: Q, Qd, and Qs. And we also assume (usually) that Q=min{Qd,Qs}. Qd is a function of P, and Qs is a function of P. And if Qd or Qs depend on X as well as P, then a change in X will shift the demand or supply curves.

But in the KC diagram, Qd is a function of Q, (and I have added a supply curve to the diagram, where Qs is also a function of Q).

But Marshall, who first drew that supply and demand diagram, had a different interpretation. We don't have to stick with the interpretation of the guy who first drew it. We might disagree about the best way to interpret it.

[Min]

Hmmm. I guess we should not be talking about the intersection of the green line with the red curve or blue curve, since the actual amount sold cannot be more than the amount supplied or demanded. The proper term is osculation, where they "kiss". :)

[Kevin Donoghue]

"But should we be teaching it in first year?"

No. Here's what you should be teaching:

Definition: c is a fixed point of the function f if and only if f(c) = c.

Let Y be expected income and a monotonically increasing function C (with C(0) > 0 and 0 < C'(Y) < 1) be planned expenditure, assumed for simplicity to depend on Y alone. The sole purpose of the 45-degree line is to pick out the unique fixed point.

Then present a simple model with goods, labour and some sort of asset whose supply is given exogenously. Work out the Walrasian equilibrium where W and P are optimal and then the 'Keynesian' equilibrium where W and/or P are above the optimum, i.e. the stock of assets is undervalued (a simple, explicit version of the Barro-Grossman model).

That's basically what Robert Hall was doing when he showed Paul Krugman the World's Smallest Macromodel. I'd be surprised if anyone can do better.

[Nick Rowe]

Kevin: I would delete W, make goods and labour the same thing, insist that the asset is the medium of exchange, and add a (quantity constrained) supply function too.

Max U(Yd,Ys,M/P) st P(Ys-Yd)=deltaM st Yd=Ybar or Ys=Ybar where Ybar=min{Yd,Ys}.

FOC: Ud = Um or Us = -Um

That's basically what I taught in Cuba, where the supply constraint was biting hard, in the 1990's. But the Cubans could handle the math/abstraction.

[Kevin Donoghue]

Nick, you sometimes remind me of Krugman's story about the mock festschrift for Rudi Dornbusch. But I guess you can't really be the student who proposed to simplify the goods-and-money model by eliminating the goods, leaving just the money.

[Min]

Nick Rowe: "Min: take a standard supply and demand diagram. We have P on the vertical axis, and three things on the horizontal axis: Q, Qd, and Qs."

No, you have Q on the horizontal axis.

As you point out, "Qd is a function of P, and Qs is a function of P." Which means that on the PxQ plane we can draw the curves, Q = Qd(P) and Q = Qs(P). Which we do. We may label the curves, "Qd" and "Qs", but that does not mean that the horizontal axis is anything different from Q.

Nick Rowe: "But Marshall, who first drew that supply and demand diagram, had a different interpretation. We don't have to stick with the interpretation of the guy who first drew it. We might disagree about the best way to interpret it."

But if the interpretation is so different that the curves are different for any given circumstance, then we really have different graphs.

Now, Farmer has employment on the X axis. That is not just a different interpretation, that is a different graph. On his graph the green line stands for Y = X.

BTW, you have not addressed my idea that both the supply and demand curves should not intersect the green line, since Ys >= Y and Yd >= Y.

[Min]

I wrote: "Let the vertical axis represent (Yd OR Ys), depending on which one we are depicting. Then the green line stands for the equation, Y = Yd or for the equation, Y = Ys, whichever one we are depicting."

Let me put this a different way. We have two graphs, one in the YxYd plane and one in the YxYs plane, depending on the vertical axis. In one graph the green line stands for the equation, Y = Yd, and in the other plane it stands for Y = Ys. Since we distinguish Yd and Ys by color, we can conveniently juxtapose the two graphs, as long as we keep straight about when Y = Yd and Y = Ys.

If we had a third graph where the vertical axis stood for min(Yd, Ys), then the green line would stand for the equation, Y = min(Yd, Ys), and we could juxtapose that graph on the other two, as well.

[Min]

Oops!

I wrote that Farmer put employment on the vertical axis. It's expenditure.

Was that a Freudian slip or a Keynesian slip? ;)

[Nick Rowe]

Kevin: my best laugh of the day!

Min: Yep. you can draw lots of graphs on the same axes, as long as the units are the same.

Typepad has been playing up, and preventing me from posting comments. Same on Mark Thoma's blog.

[Livio Di Matteo]

Nick: That was an interesting piece of information-you took macro with Peter Howitt. So did I in 1985-86. Great class.

[Adam P]

Nick, I agree with your conclusion but not your explanation.

It seems to me when you draw your figure above (the first one) with a supply curve, a demand curve and a 45-degree line then the interpretation of the 45-degree line is just the accounting identity that actual expenditure equals actual income. The curves Yd and Ys show combinations of desired demand/supply for a given level of income.

Thus, while it's true that the economy finds itself at the intersection of the 45-degree and the lower of Yd and Ys it is not true that the economy can ever be off the 45-degree line.

For example, you say "If some sellers of haircuts, or some buyers of haircuts, could make you an offer you can't refuse, the economy would not be on the 45 degree line." I don't think that's right, if the sellers can force people to buy then that shifts the Yd curve. Demand is what it is, coerced or not.

[Nick Rowe]

Livio: You too! Macro was great at Western.

Adam: Suppose I want to buy 10 apples, but the store only wants to sell me 6, so I actually buy 6. I want to say that my quantity demanded was 10. I want to be able to distinguish both quantity demanded and quantity supplied from quantity actually bought. The accounting identity says only that quantity bought = quantity sold. But I think that's so trivial a semantic truth, that I would rather just say there is one quantity-bought-and-sold.

I expect we could define demand your way, if we wanted, but I don't think it would be useful to do so. And an imaginary economy in which the rules were different, so Q=max{Qd,Qs}, would be very different from an economy in which Q=min{Qd,Qs}.

[JKH]

"And an imaginary economy in which the rules were different, so Q=max{Qd,Qs}, would be very different from an economy in which Q=min{Qd,Qs}."

That feels like an indicator of hangover from the negative money post, Nick.

Both of which could be head-hurting.

But is there any connection?

[Nick Rowe]

JKH: Not really! A world in which Q=max{Qd,Qs} would be a very weird world! Too weird even for me. The law would say that if someone wanted to sell you something, you must buy it. And if someone wanted to buy something from you, you must sell it. And God only knows how prices would get determined!

Accounting tells us that expenditure and income (i.e. quantity bought and quantity sold) are the same thing, so let us give that thing one name/symbol: Q. We also have Qd and Qs, so that is three variables in all. We know that Qd is a function of Q and other things: Qd=D(Q,X), and Qs is a function of Q and other things Qs=S(Q,Z). And so we need some third equation to tell us how Q depends on Qd and Qs. And it's Q=min{Qd,Qs}, if we assume that all exchanges are voluntary. So when we draw those three equations as graphs, that's the picture we get.

[JKH]

Nick,

Maybe you've covered this, but can you expand a bit on the first principals logic of how you view Qd and Qs as partial functions of Q?

And can you reconcile "time period accounting" in doing that?

I.e. I gather you would view Q as ex post accounting but Qd and Qs as ex ante relative to such time period consistency?

Or does that all just drop away by assuming continuous time and continuous accounting?

As an amateur, I remember the Keynesian cross type of graph, but was always puzzled a bit by it.

[Nick Rowe]

JKH: "I gather you would view Q as ex post accounting but Qd and Qs as ex ante relative to such time period consistency?"

If we use the ex post/ex ante distinction, that would be correct. But I prefer to make the distinction between: Q the amount we actually buy-and-sell this period: Qd (demand) the amount we want to buy this period; Qs (supply) the amount we want to sell this period.

For example:

1. if people want to sell 10 haircuts per day (Qs=10), but people want to buy 6 haircuts per day (Qd=6), then only 6 haircuts get bought-and-sold per day (Q=6). (And the hairdressers sit there unemployed hoping for another 4 customers who never come).

2. if people want to buy 10 haircuts per day (Qd=10), but people want to sell 6 haircuts per day (Qs=6), then only 6 haircuts get bought-and-sold per day (Q=6). And 4 customers get turned away with no haircut.

"As an amateur, I remember the Keynesian cross type of graph, but was always puzzled a bit by it."

Take Roger Farmer and me. We have almost the same economics education, but we interpret that same picture very differently. So it's not surprising it puzzles you! I learned it in high school, but it wasn't till my PhD I at least thought I understood it! Now I learn that Roger understands it very differently!

[JW Mason]

This is interesting. But….

The Keynesian cross has income on the x-axis. now, it is true that income is always equal to output, ex post. but it makes a difference. because the AD curve only makes sense if we think of it as expenditure as a function of income. So I don't see how you can put it on the same graph. The question is how much people are spending on goods and services, vs. how much they are receiving in payment for goods and services (which must be the same, obviously). I don't understand what it means to put a supply curve on there. Output is not a function of income.

So I guess I disagree with both you and Roger Farmer.

Now one thing you do here that I like, is get rid of the inventories story, which always bugs me. The logic of demand-determiend output does not depend on unwanted inventory accumulation. Better to say as you do:

If people had false beliefs about how much they were able to sell, or about how much they were able to buy, the economy would not be on the 45 degree line.

That's the way I explain it. The income on horizontal line is expected income. So if you are at a point off the 45 degree line, actual spending is different from expected income, so realized income is different from what was expected, so people will change their spending decisions next period.

[JW Mason]

Sorry, the sentence "So I don't see how you can put it on the same graph" is not supposed to be there.

[Nick Rowe]

JW: "I don't understand what it means to put a supply curve on there. Output is not a function of income."

Suppose there is excess demand for output, so goods are rationed. If you are only able to buy $1,000 worth of goods a month, would you wish to work the extra hours to earn more than $1,000 income, when you can't buy anything with that extra income, but only save it, hoping you might be able to spend it in future, when rationing is lifted?

This is the exact counterpart of the standard keynesian case, where there is excess supply, so you are constrained in how many hours you can work, and can only spend more than your income by dissaving, hoping that the rationing on hours worked will end in future.

In Cuba in the 1990's this seemed to me to be very real. Low output meant low rations which meant no point in working the extra hour which meant low output. The supply-side multiplier.

Yep, in a more realistic model Yd would depend on expected Y, which is not necessarily the same as actual Y. In the olden days, we would draw those zig-zag paths to the new equilibrium, implicitly assuming adaptive expectations Ye=Y(t-1). But the simple KC model, where we are always on the intersection of the demand curve and 45 degree line, implicitly assumes rational expectations. The representative agent jumps straight to the Nash Equilibrium, even though he doesn't know what other agents' demand functions are.

[JW Mason]

In the olden days, we would draw those zig-zag paths to the new equilibrium, implicitly assuming adaptive expectations Ye=Y(t-1).

It's still the olden days, in my classroom.

[Nick Rowe]

Start in equilibrium, where Yd=Y=Ys.

1. Now impose a lump sum tax. Every individual cuts Yd by $1 (income effect on consumption), but notices he can sell $1 fewer goods than before (he is rationed when selling goods), so cuts Yd some more. Standard demand-side multiplier.

2. Starting back in equilibrium, now impose a lump sum transfer. Every individual cuts Ys by $1 (income-effect on consumption of leisure), but notices he can now buy $1 less goods than before (he is rationed when buying goods), so cuts Ys some more. Supply-side multiplier.

[JW Mason]

Let's think about this. In the Old Keynesian story, we have (expected) income determines expenditure, expenditure determines output, and output determines (realized) income. In your story, it's (expected) spending determines income, income determines output, and output determines (realized) expenditure? No, I don't think that works. In an economy where supply constraints bind, it is not the case that if everyone was more optimistic about what their rations coupon would buy, there would end up being more goods int he store t accommodate the larger number of coupons earned. Supply constraints come from, well, the supply constraint. The Keynesian cross is supposed to be illustrating the process of circular causation where income determines spending and spending determines income. But there isn't any such circular causation on the supply side, is there?

I'm not sure I understand what you're saying.

[Nick Rowe]

JW: "It's still the olden days, in my classroom."

Good. You still teach the very short run. (You have already read it, but others reading this might not.)

[JW Mason]

I guess I understand the logic but I'm not convinced this describes any real economy, even Cuba in the 1990s. Production in Cuba was limited because they were no longer getting cheap oil from the USSR, stuff like that. Optimism about the value of rationing tickets can't make up for that.

The point of the Keynesian cross model is to show how a desire to spend creates the income corresponding to that spending. But nobody would say, in Cuba, that desire to earn more money would have created stuff to spend it on. (Well ok, in the long run in some sense, maybe, but not at the time scale this kind of thing is supposed to represent.)

But it's an interesting thought exercise.

[JW Mason]

Nick, do you really think this model captures the reasons for shortages in Cuba?

[Nick Rowe]

JW: It is hard for us to imagine, because we live in capitalist economies where monopoly power (monopolistic competition) is so prevalent. We are often rationed in what we can sell at prevailing prices, but hardly ever rationed in what we can buy at prevailing prices. Suppose you faced a binding ration. Why would you ever work more than you needed to work to buy the goods you were allowed to buy? All you can do with the extra money is leave it accumulating in your savings account.

Desired expenditure is determined by permanent income, in demand-constrained economies.

Desired income is determined by permanent expenditure, in supply-constrained economies.

[JW Mason]

Again, I see the logic (I think) but I don't think this is what we mean by supply constraints in the real world. The situation you are describing is one where more goods could be produced, if people wanted to work more. But people don't want to work -- not because work is unpleasant -- but because they don't think there will be anything to spend their earnings on. So the supply constraints here are a coordination failure. If some outside agency decided to produce a bunch of stuff, there would be more to buy, people would choose to buy more, and we would see a supply-multiplier -- the increase in output would be even less than before, and we would move back toward a world of demand constraints.

I don't think that describes real supply-constrained economies. I don't believe that if the Cuban government in 1992 had announced a major new program of hospital building and road construction, that would have increased the amount of consumer goods available for sale. Do you?

[JW Mason]

Oh oops -- this shows how hard I'm finding it to get into this -- public goods won't help. It has to be a major program of producing consumer goods FOR SALE. Which then, hypothetically, might lead to a multiplier of even more consumer goods, as people fin fit worthwhile to work more. Which, hm, doesn't sound so implausible. Huh.

[Nick Rowe]

JW: "Nick, do you really think this model captures the reasons for shortages in Cuba?"

Yes, in part. Certainly much better than the demand-side version of the Keynesian Cross. My Cuban students seemed to think so (they just looked blankly at demand-side models, until I told them they were models of capitalist economies). It's the "monetary overhang" model. Instead of an excess demand for money, there's an excess supply of money.

[Nick Rowe]

JW: Or, raise the price level, to reduce M/P. Or, cut real transfer payments. Or, cut G.

[JW Mason]

Like Krugman's babysitting co-op. Except now, no one is participating because they don't think it's worthwhile to earn tokens, instead of not wanting to spend them.

[Nick Rowe]

JW: Yep. And it's not worthwhile earning the tokens because they won't be able to spend them all. If the stock of tokens is too big, it causes a recession. "I already have plenty of tokens that are hard to spend, so why would I want to earn more by babysitting for you?"

My "anti-NK model" is another example. Cutting interest rates causes a recession in a supply-constrained world.

[Min]

Nick Rowe: "We also have Qd and Qs, so that is three variables in all. We know that Qd is a function of Q and other things: Qd=D(Q,X), and Qs is a function of Q and other things Qs=S(Q,Z). And so we need some third equation to tell us how Q depends on Qd and Qs."

Then we should have a function, Q = F(X,Z). Right?

[Roger Farmer]

Nick
This is all very interesting, but I don't think it has much to do with the General Theory. In Keynes, unemployment occurs in equilibrium where the 'aggregate supply price' equals the 'aggregate demand price' (his words not mine). The income expenditure model, as several people pointed out on my blog, is indeed Samuelson. But that model does get at the of Chapter 3 of the GT.

In Chapter 3 of the GT, its employment on the horizontal axis and GDP in wage units on the vertical axis. The model in my AD-AS paper makes sense of this without invoking sticky prices.

Roger

[Nick Rowe]

Min: yep. And That function Q=F(X,Z) would be the solution to Q=min{D(Q,X),S(Q,Z)}.

Roger: If we assume a production function Y=L, where L is employment of hairdressers, then I do have employment on the horizontal axis, and GDP in wage units on the vertical axis. Or, if we assume a production function Y=log(L), then if I draw my graph on a log scale (on both axes) that is equivalent to having employment on the axes.

What Keynes in Chapter 3 calls the "aggregate supply function" simply isn't very interesting. It's what "classical" economists call the "labour demand function" W/P = MPL(L), with W=1 (it's the numeraire), plus the production function Y=F(L), and we can solve the two together to get P.Y = some function of L, which Keynes calls the "aggregate supply function. It's substantively identical to the SRAS function drawn in second year textbook models which assume sticky wages, except that W is the numeraire, so we have P/W on the vertical axis. It does not represent the output that would be produced if suppliers of labour could sell the amount of labour they would like to sell.

What is interesting about chapter 3 is the "aggregate demand function", which is decidedly non-"classical". Because it says that the amount of output demanded is a function of the amount of labour actually sold, and hence output demanded is a function of the amount of output actually sold.

In short, Keynes' "aggregate supply function" is just the "classical" labour demand curve plus the "classical" production function, plus a lot of algebra. Except for the algebra, there's nothing new there. It is Keynes' "aggregate demand function" that is new and exciting.

[JW Mason]

This is all interesting and I was wrong to think that it does not describe any real economy. It might be relevant to Cuba in the 1990s, as you say. But I don't think it describes the sort if supply constraints we observe in modern capitalist economies. When think demand is above potential, it is definitely not because people quitting work because they don't think there will be anything to buy with the money they would earn.

This model suggests a symmetrical relationship, where output is low when demand exceeds supply, and when supply exceeds demand. That's not what we think happens in the economies we live in. Nominal output rises continuously with demand, real output rises continuously also, altho perhaps approaching an asymptote. (I have doubts about the existence of long run potential output, actually, but that's not relevant here.) There's no tipping point where a further increase in the number of babysitting tokens causes the amount of babysitting performed to decline.

I think this asymmetry exists for two reasons. First, because people expect sales to be demand constrained. When you find you can sell more or less of your labor, as a worker, or products, as a producer, you adjust your expectations about your longer term income and what level of spending you can afford. Since you don't generally expect your spending to be limited by supply, any supply constraints you do happen to encounter don't change your belief about the value of earning money. (I think the fact that economic units are more specialized in sales than in purchases contributes to this.) Second, because in real economies there is also a financing constraint. Because credit is limited, it is often the case that you must make a sale before you are able to make a purchase. There is no equivalent case in the supply-constrained world where you must commit to a purchase before you are able to make a sale. So demand constraints have both the funding and expectations channels to work through, while supply constraints have only expectations.

The fact that prices are more flexible upwards than downwards might contribute to the difference too. But I'm wary of that argument.

It seems to me that what we call supply constraints in the real world are a different phenomenon, they have to so with limited substitution possibilities between different inputs.

[Nick Rowe]

JW: I tend to agree. Except maybe WW2 rationing in the UK? Must ask Mum.

[Min]

JKH: "I gather you would view Q as ex post accounting but Qd and Qs as ex ante relative to such time period consistency?"

Nick Rowe: "If we use the ex post/ex ante distinction, that would be correct. But I prefer to make the distinction between: Q the amount we actually buy-and-sell this period: Qd (demand) the amount we want to buy this period; Qs (supply) the amount we want to sell this period."

If you are sticking to a single time period, then if the vertical axis represents Yd, Ys, or min(Yd, Ys) and the horizontal axis represents Y, and the green line represents the equation, Y = min(Yd, Ys), then neither the demand curve nor the supply curve can intersect the green line, since Yd >= Y and Ys >= Y. Neither curve can have a section where Y > Yd or Y > Ys, i. e., to the right of the green line. The graphs you show, where they do intersect the green line, are impossible.

[JW Mason]

I also prefer thinking ex ante and ex post. But I'm not sure it makes any difference for the issues here.

Min: I would say that a segment with Y > Yd shows us that IF Y were at that level, the amount demanded would be less than the amount produced. Which, yes, means that output will not be at that level. But there is nothing impossible about posing the hypothetical.

[Nick Rowe]

Min: assume there exists a Keynesian auctioneer, who can't do math, and is trying to solve for the solution Y by trial and error.

1. He makes a guess for Y. Call it Y".
2. He makes a guess about whether Ys or Yd will be bigger than Y".

He then announces his guesses to the agents, and asks each of them how much they would want to buy or sell, and adds them up across all agents to get Yd and Ys.

If the preliminary answer shows that his second guess is wrong, he changes his second guess and asks his question again, and adds up their answers again to get Yd and Ys.

The answers he gets, for all values of Y", tell us the blue and red curves.

[Tom Brown]

Nick, did either you or Roger ask David Laidler about the 45 deg line?
O/T: Sadowski is somewhat mystified by your explanation to me:
http://www.themoneyillusion.com/?p=26213#comment-320452 but Glasner agrees w/you:
http://uneasymoney.com/2014/02/20/whos-afraid-of-says-law/#comment-52175

[Tom Brown]

... BTW, my question back to you on the OT: Why would demand go up?

[Tom Brown]

re: 45 degree line & Laidler's take: does anything ever get resolved in econ? Does anybody every say "Ah! You're right, I had that thing in their backwards! Thanks!" ... like they do in the engineering world? :D ... so I still don't know whether to believe you or Roger. Nothing is resolved, right?

[Errorr]

Whoa that is a lot of math and jargon in the comments.

Why can't we just say that every seller needs a buyer and thus since the transactions are discrete events the Actual economy will always be on the 45° line.

I just always assumed it was a shorthand reminder that no economy exists that is not on the 45° line.

[Min]

JW Mason: "I would say that a segment with Y > Yd shows us that IF Y were at that level, the amount demanded would be less than the amount produced. Which, yes, means that output will not be at that level. But there is nothing impossible about posing the hypothetical."

But you are not having the green line represent the equation, Y = min(Yd, Ys). And if that is the case, then Yd and Ys must be above the green line, or touching it. Nick is not just reinterpreting the green line, he is making a different graph.

[Min]

Nick Rowe: "assume there exists a Keynesian auctioneer, who can't do math, and is trying to solve for the solution Y by trial and error.

1. He makes a guess for Y. Call it Y".
2. He makes a guess about whether Ys or Yd will be bigger than Y".

He then announces his guesses to the agents, and asks each of them how much they would want to buy or sell, and adds them up across all agents to get Yd and Ys.

If the preliminary answer shows that his second guess is wrong, he changes his second guess and asks his question again, and adds up their answers again to get Yd and Ys.

The answers he gets, for all values of Y", tell us the blue and red curves."

First, it is not at all clear what the "auction" has to do with real life. ;)

Second, it is not at all clear that the auction yields Yd and Ys instead of Yd" and Ys".

Third, it does not change the relations, Yd >= Y and Ys >= Y, even if Y" > Yd or Y" > Ys. The curves should still not cross the green line.

[Nick Rowe]

Min: take a standard supply and demand diagram, with price on the axis. If we assume you can't force people to buy more than they want, or sell more than they want, does that mean we can never draw supply and demand curves crossing?

[Min]

Nick, I think that I have a simple way to explain the problem.

Both you and Farmer label the horizontal axis Y. He labels the vertical axis X. But you label it Y. To be sure, you are concerned with different kinds of Y on the different axes, but they are still Y, not X.

On his graphs the green line stands for the equation, Y = X. On yours it stands for the equation, Y = min(Yd, Ys). Now it may be that when Y = X, X = min(Yd, Ys). Then you may interpret the green line in that light.

But once you are off the green line, it matters that the vertical axis is different. Because then X =/= min(Yd, Ys). And that means that you cannot draw the same graphs as Farmer.

The problem is not your interpretation of the green line, it is your changing the dimensions of the graph.

[Nick Rowe]

OK. I have now checked my old lecture notes, and heard back from David Laidler (and I asked Peter Howitt too).

My memory was wrong, and Roger's was right. David calls the 45 degree line an AS curve. (Peter asks why it matters what we call it).

But is this a semantic dispute over the meaning of "supply"? I define "supply" (strictly "quantity supplied") as the quantity sellers *want* to sell, or *offer8 to sell, and not what they actually sell.

If you ask someone how much he wants to sell, there is a very big difference between these two answers:

1. "I want to produce and sell 100, but since buyers only want to buy 60, I will only produce and sell 60."

2. "Whatever, I don't care, I just want to sell whatever you want to buy, as long as it is less than or equal to 100."

Only if we define "supply" the first way can we talk about "excess supply".

The English language doesn't work for "supply". We should do like the French, and talk about how much sellers *offer* to sell.

[Min]

Nick Rowe: "The English language doesn't work for "supply". We should do like the French, and talk about how much sellers *offer* to sell."

In English does it make sense to apply the phrase, "willing and able" to both supply and demand? As in, how much one is willing and able to sell or willing and able to buy?

[JW Mason]

maybe WW2 rationing in the UK?

At first I thought, no, that doesn't work. The signature of this kind of supply constraint is the combination of rationing with unemployment, and that doesn't fit WWII Britain.

But then I thought, what if it's a two-stage process? First, we have a big fall in supply, or big increase in demand, for some basket of goods. Small shifts of course are dealt with by the price mechanism, but for really big shifts that may not work so well -- because of inflation, distributional effects, whatever -- so rationing has to be introduced. Initially, there's no adverse effect on labor supply -- in effect, people are being forced to save more than they would choose, but that should have only second-order effects on the perceived value of the wage. But now suppose rationing continues long enough that people doubt whether there will ever be sufficient goods available for sale. Now income in excess of available consumption goods no longer appears to be a form of saving. So now you do get a fall in labor supply, -- which reduces current output so that rationing must be even stricter. This is a kind of secondary shortage on top of the original shortage, and this secondary shortage might even persist after the original supply or demand shift is reversed. A simple solution, in that case, would be to expropriate the excess savings -- but better not do that if you want to be able to use rationing again in the near future. More interestingly, if the government can somehow procure a short-term increase in the supply of consumption goods, that may increase the perceived value of the wage enough to bring labor supply back up to its old level -- a kind of supply-side pump priming.

In this case, we could say that there is a "corridor of stability" on the supply side as well as the demand side, and we just happen to have remained within the corridor. We've never had a large enough and persistent enough excess of desired purchases over desired sales for the process to become self-sustaining. But you could.

I wonder if there's episodes in the history of the Soviet economy we could interpret this way?

[Nick Rowe]

Min: "In English does it make sense to apply the phrase, "willing and able" to both supply and demand? As in, how much one is willing and able to sell or willing and able to buy?"

No. Greg Mankiw defined supply/demand as the quantity people are willing and able to sell/buy. When I was Canadian co-author, I crossed out all the "and able" bits. Otherwise he contradicted himself when he talked about price controls; "with a binding price floor, sellers are unable to sell as much as they are willing and able to sell" doesn't make sense.

JW: what you describe there is exactly the idea. It's not so much unemployment, as people working less than they normally would otherwise want to. My guess is that it would apply in the old Soviet Union.

[Nick Rowe]

JW: Aha! Read this! The supply-side multiplier at work!

[Tom Brown]

Nick, thanks for letting us know what Laidler and Howitt say: semantics or not I feel better that at least one thing was "resolved" here ... to some small extent anyway. :D It'll be interesting to see if anything more substantial gets resolved.

[Min]

Nick Rowe: "Greg Mankiw defined supply/demand as the quantity people are willing and able to sell/buy. When I was Canadian co-author, I crossed out all the "and able" bits. Otherwise he contradicted himself when he talked about price controls; "with a binding price floor, sellers are unable to sell as much as they are willing and able to sell" doesn't make sense."

Many thanks, Nick. :) Although I expect that a general semanticist could rescue Mankiw with able1 and able2. ;)