Let there be two players. The apple producer buys bananas from the banana producer; and the banana producer buys apples from the apple producer. Each must place his order for how much fruit he wants to buy *before* observing the other's order. Each player's action depends on *his expectation of* the other player's action. The more fruit he expects to sell, the more fruit he chooses to buy.

The natural way to set this up would be to draw a diagram with "quantity of apples" on one axis, "quantity of bananas" on the other axis, draw the two reaction functions, and say that the Nash equilibrium is where those two curves cross.

**But notice that each axis represents two quite distinct quantities.** The apple axis is being used to represent both: the quantity of apples the banana producer demands; and the quantity of apples *the apple producer expects* the banana producer to demand. Only in Nash equilibrium are those two quantities the same, and the apple producer's expectations are correct. It's the same with the banana axis.

If the game were perfectly symmetric, so the two players are identical, we could flip the diagram over, swapping the axes, and it would look exactly the same. **More simply, we could delete one of the two reaction functions, and replace it with the 45 degree line.**

Take one of the two players. It doesn't matter which one. Put "quantity of fruit he demands" on the vertical axis, and "quantity of fruit he expects the other to demand" on the horizontal axis. Draw the reaction function. **The 45 degree line has a simple interpretation: it represents points at which expectations of the other player's choice are correct.** The point where the reaction function crosses the 45 degree line is the Nash equilibrium.

If we like, we could double the scale on both axes so the reaction function represents the *sum of *their quantities of fruit demanded as a function of *the sum* of their expectations of the quantity demanded by the other. Nash equilibrium is where the aggregate reaction function crosses the 45 degree line.

Re-label the vertical axis "planned aggregate expenditure" and the horizontal axis "expected aggregate income", and you've got the Keynesian Cross diagram. (We really ought to call the vertical axis "aggregate output demanded".)

Do I really need the assumption that the two players are identical so the game is symmetric? Well, yes and no. Start in Nash equilibrium, then suppose the apple producer gets more optimistic about his income and the banana producer gets more pessimistic about his income, but aggregate expected income stays the same. If the apple producer had a higher marginal propensity to consume (a steeper reaction function) than the banana producer, planned aggregate expenditure would increase, which would change the Nash equilibrium. So it only works if all the reaction functions have the same slope everywhere. (They can have different intercepts without messing anything up, but that's it.)

Or we could just wave our hands, say it doesn't matter much, and we've got bigger problems to worry about.

**But strictly speaking, the Keynesian Cross model is a representative agent model.** So you Old Keynesians shouldn't be throwing stones at representative agent models.

Adding a third player to the game is easy, if we stick with our assumption they are all identical, or keep waving our hands. Otherwise we have to draw a third axis. The carrot producer chooses how many apples to buy, and how many bananas to buy, as a function of how many carrots he expects the apple producer plus the banana producer to buy.

Is there any other way to interpret the Keynesian Cross diagram? **Could you really put "actual income" rather than "expected income" on the horizontal axis?** Well, **no**. You could have the apple producer move last, so he observes his sales of apples before choosing how many bananas to buy. Or you could have the banana producer move last, so he observes his sales of bananas before choosing how many apples to buy. **But you can't have everyone moving last. It doesn't make sense.**

Unless you introduce a "Keynesian auctioneer", who calls out an income for each player, asks them to submit written demands for goods, checks to see if the demands add up to the incomes he called out, declares the demands null and void if they don't add up right, and calls out a different set of incomes, and keeps repeating until they all add up right. And all this happens outside of real time, so we only ever observe the Nash equilibrium. Nobody *actually* moves until they know exactly what moves everyone else will choose.

Or you could assume that everyone knows everything about everyone else, and each player can solve the Keynesian auctioneer's problem so knows exactly what will happen. If you really insist that the economy is always and everywhere at the point where the Keynesian Cross model says it is, that's OK. **Provided you are ready to assume full-bore rational expectations plus common knowledge of everything.**

But it makes more sense to wave your hands, talk about how the game is repeated every period, and **how people learn from their mistakes**, and draw little zig-zag steps between the reaction function and the 45 degree line that converge to the Nash equilibrium if the slope of the reaction function is less than the slope of the 45 degree line, if the Nash equilibrium doesn't move too much before they learn where it is.

**It is not obvious how players actually solve for the location of the Nash equilibrium. They probably use some sort of trial-and-error process.** Of course, if all agents really were identical, and knew they were identical, it would be easy.

*But for God's sake don't start talking about saving vs investment and planned vs actual and income vs expenditure and inventories and profits and stuff like that. It only confuses things, and diverts attention from how people revise their expectations about the actions of others. And don't even think about accounting identities. The only accounting identity you need is that "quantity of apples bought" is the very same thing as "quantity of apples sold", so it's best just to talk about "quantity of apples bought-and-sold".*

**There's what we choose to do, and what we expect others to choose to do. There's aggregate quantity demanded, and expected aggregate quantity demanded. A thing, and people's belief about that very same thing. Those are two different things.**

But if you are paying attention, you will notice I have talked only about demand, and have said nothing about supply. I have talked about how many bananas the apple-producer wants to buy, and have said nothing about how many bananas the banana producer **wants** to sell. The quantity **actually bought-and-sold** is not the same thing as the quantity demanded (what the buyer **wants** to buy), nor the same thing as the quantity supplied (what the seller **wants** to sell). A sensible assumption to make is that quantity actually bought-and-sold is equal to whichever is less: quantity demanded; quantity supplied.

Q = min{Qd;Qs}, or "the short side rule", as it's known in the trade.

If the apple producer wants to buy 10 bananas, but the banana producer only wants to sell 6 bananas, then only 6 bananas get bought-and-sold.

The Keynesian Cross model ignores the supply side. It assumes that quantity bought-and-sold is always equal to quantity demanded. Implicitly it must be assuming that quantity demanded is always less than quantity supplied. It's a model of the determination of quantities bought-and-sold when there is excess supply for all goods.

For each good there are now 6 things: Q, Qd, Qs, expected Q, expected Qd, and expected Qs. The Keynesian Cross simplifies this down to two things, by implicitly assuming that Qd < Qs, so that Q=Qd, and that everyone knows this, so we are left with only two things for each good: Qd; and expected Qd.

If you want to bring the supply side in then each player now has two reaction functions: his demand function, and his supply function. In Walrasian equilibrium prices adjust until all curves cross at the same point. In the Keynesian Cross equilibrium they don't, and there is excess supply of everything, so the short side rule tells us to ignore the supply curves.

"Could you really put "actual income" rather than "expected income" on the horizontal axis? Well, no."

Sure you can. I have a theory that expenditure is a function of actual income, even if that's only because expected income is itself a function of actual income.

Your example is a little artificial, as it looks at a one-off game where neither producer has anything to go on. Instead, let's say we are looking at income for the year and fruit trades take place several times a day. Both producers will adjust their expenditure over time to correct for expectation errors. Towards the end of the year, there may still be an expectation error, but the actual outcome to date is going to be an important factor.

Posted by: Nick Edmonds | February 21, 2015 at 10:44 AM

I'm an apple produce and if I thought the banana producer would buy 10 apples I would order 10 bananas and produce 20 apples. But I only think he will order 5 so I only order 5 and produce 15.

I'm a banana producer and if I thought the apple producer would buy 10 bananas I would order 10 apples and produce 20 bananas. But I only think he will order 5 so I only order 5 and produce 15.

So 15 apples and 15 bananas get produced and the market clears even though both apple producers and banana producers would be better off producing 20 and trading 10. In this model how would expectations adjust to get to the better equilibrium ?

I like the idea that economic models need to help us understand "how people revise their expectations about the actions of others" but don't we sometimes need to bring in other factors (or at least more complexity) to explain how we get not just an equilibrium but to the optimal equilibrium ?

Posted by: Market Fiscalist | February 21, 2015 at 10:59 AM

Please let me explain why I think you begin with a fundamentally flawed presentation:

I think you expect that the zero, zero point for both that apple and banana axis is in the lower left corner of the chart.If so, then both expectations for apples are on one line and both expectations for banana sales are on the second line.

This results both expectations (buyer and seller) laying on either the apple axis or the banana axis. Any line between the two axis would have no meaning. The 'no meaning' condition would persist even if both players chose the same point on each axis.

I must be missing an important concept in your presentation. Sorry

Posted by: Roger Sparks | February 21, 2015 at 11:02 AM

"But for God's sake don't start talking about saving vs investment and planned vs actual and income vs expenditure and inventories and profits and stuff like that. It only confuses things, and diverts attention from how people revise their expectations about the actions of others."

It only confuses things if you talk about these things in a confused way. Sadly many do. But really there's no reason why a model with a consumption good and a capital good need be any more confusing than one with apples and bananas. In fact it's a good deal easier to see why a firm making a capital good might have a problem choosing its level of output. As Solow remarked, the world has its reasons for not being Walrasian.

Somewhere in Keynes's correspondence you can find the marvellous complaint: "What I'm trying to say is really very simple and cannot possibly require all this exegesis."

Posted by: Kevin Donoghue | February 21, 2015 at 11:22 AM

Nick E: "I have a theory that expenditure is a function of actual income, even if that's only because expected income is itself a function of actual income."

No. You have a theory that expenditure is a function of expected income. And expected income is a function of past actual income, via some sort of learning process. Just like me.

MF: 1. your two reaction functions lie on top of one another. MPC=1 in your example. You have a continuum of Nash equilbria.

2. Why would you produce more bananas than you think you will sell?

3. "So 15 apples and 15 bananas get produced and the market clears..." No it doesn't. Because by your assumption A buys 5 bananas and B buys 5 apples.

4. "Market clearing" does not mean "quantity demanded = quantity produced". It means "quantity demanded = quantity supplied". And "quantity supplied" does NOT mean "quantity produced". And it does NOT mean "quantity sold". It means "quantity you WANT to sell".

Roger: sorry, but I don't have a clue what you are saying.

Yes, It's normal to have both axes start at zero. Otherwise I would have to draw the 45 degree line somewhere else, so it starts at {0,0} and has a slope of one.

Posted by: Nick Rowe | February 21, 2015 at 11:23 AM

Kevin: I tend to agree. But a reasonable hypothesis about the demand for a capital good is that it depends on expectations about future (and distant future) demand for the good that capital good helps produce. (Of course, once you switch to the Permanent Income Hypothesis, the same can be said about demand for a consumption good too.)

Posted by: Nick Rowe | February 21, 2015 at 11:29 AM

" Why would you produce more bananas than you think you will sell?"

I was assuming that both fruit growers consume 10 of their own fruit and sell the rest. So in my equilibrium they both expect to sell and do sell 5 bananas , but would rather sell (and order) 10 units of fruit, but don't because of expectations.

Posted by: Market Fiscalist | February 21, 2015 at 11:34 AM

"You have a theory that expenditure is a function of expected income. And expected income is a function of past actual income"

Therefore, expenditure is a function of actual past income, right? And actual past income can mean income in the same period, just from earlier in that period.

I grant you though that you cannot ever quite get rid of a bit of expected element. On the last day of the year, maybe you know your income from the previous 364 days, but still have to form an expectation for the last day. For me that's good enough to label your axis "Actual income". There's another variable there, which is the expectation error, but it's small.

btw, I'm not saying you can't label it "Expected income", nor am I expressing a view on what actually determines expenditure. I'm just disputing the idea that it makes no sense to label it "Actual income".

Posted by: Nick Edmonds | February 21, 2015 at 11:47 AM

Nick E: "There's another variable there, which is the expectation error, but it's small."

It's not obvious that the last day's expectation error will be small. (depends on the learning process, whether the equilibrium shifts during the year, etc.). More importantly, the sum of the expectations errors is not the same as the expectations error on the last day, and we only get the Nash equilibrium for annual expenditure if the sum of the expectations errors over all days in the year is zero (and the consumption function is linear).

Posted by: Nick Rowe | February 21, 2015 at 12:05 PM

NicK:

You answered my question. Thanks.

In this case, the 45 degree line is a mechanical reference line that allows correlation between two not-parallel scales. It is used to translate one scale to a second scale. In your case, apples to bananas.

As you say, we can place the predictions of two buyers on the 45 degree scale. One buyer will predict high and one low, resulting in two points on the 45 degree line. Now, we find the Nash Equilibrium someplace between these two points but on the 45 degree line.

Thanks for the answer.

Posted by: Roger Sparks | February 21, 2015 at 12:11 PM

While it is true that you could have four axes, one for actual apples, one for expected apples, one for actual bananas, and one for expected bananas, is that necessary? Why can't you make do with two axes one for each fruit, and let the function curves distinguish between expected and actual?

Text: "Only in Nash equilibrium are those two quantities the same, and the apple producer's expectations are correct."

If we let the curves distinguish between expected and actual, the intersection of the curves represents the point where the expect and actual are the same, which is what we want to find. :)

Posted by: Min | February 21, 2015 at 03:27 PM

OK. Suppose that we are interested in equlibrium conditions, so that expected apples = actual apples and expected bananas = actual bananas. We start with four dimensions and because of the two equations, end up with two dimensions, a plane. Introduce another equation, apples = bananas, and we do get a line.

So far, so good. But now let us add the carrot guy. That gives us six dimensions. It apparently gives us six equations, but apples = carrots and bananas = carrots, then apples = bananas, so there are only five pertinent equations. We still get a line.

Likewise, for every person that we add, we only add two pertinent equations, and we still get a line. That, I gather, is your point. :)

Posted by: Min | February 21, 2015 at 03:49 PM

Oops! That last should be, we add two dimensions but only two pertinent equations, and we still get a line.

Posted by: Min | February 21, 2015 at 03:50 PM

"The Keynesian Cross model ignores the supply side. It assumes that quantity bought-and-sold is always equal to quantity demanded. Implicitly it must be assuming that quantity demanded is always less than quantity supplied. It's a model of the determination of quantities bought-and-sold when there is excess supply for all goods."

IOW, when there is a "general glut". Or, as John Stuart Mill said, an insufficient supply of money.

Posted by: Min | February 21, 2015 at 03:56 PM

"But strictly speaking, the Keynesian Cross model is a representative agent model."

I'll take your word for it. However, to generate a line for any number of agents, all we need is for the equations relating the quantities to be linear.

Posted by: Min | February 21, 2015 at 04:09 PM

There is no decision of any type in any realm (economics or other) that is not (at least partially) a function of expectation.

Name one.

(Somehow, accounting has survived that)

Posted by: JKH | February 21, 2015 at 04:30 PM

I was imagining a function where, eg, daily expenditure was x = a + b [ye + y(-1) - ye(-1) ], where ye is expected income for the day and y is actual income for the day. So each day's expenditure adjusts for previous day's expectation error. Expenditure for the year is then X = 365a + b ( Y + DE ), where Y is the sum of the ys and DE is the difference between the expectation errors for the last day of this year and the last day of last year. Even with large daily errors, this will be small compared with Y.

Posted by: Nick Edmonds | February 22, 2015 at 04:32 AM

Nick E: "daily expenditure was x = a + b [ye + y(-1) - ye(-1) ]"

Doesn't that create an explosive system, where the errors get bigger over time? (I haven't had coffee yet.)

Min: "I'll take your word for it. However, to generate a line for any number of agents, all we need is for the equations relating the quantities to be linear."

Don't trust me, especially on math, but:

y1 = a1 + b1x1

y2 = a2 + b2x2

does not give us (y1+y2) = F(x1+x2) I think. Unless b1=b2.

Posted by: Nick Rowe | February 22, 2015 at 07:43 AM

Nick, are you sure you're not knocking down a strawman here?

Yes, every model is in some sense a model involving representative agents. I don't think that's the criticism coming from the old school Keynesians, because a representative agent model, to be accepted by the mainstream, needs more than that. It needs to assume that prices and quantities traded are the solution of some utility maximization/profit maximization process, without going into the messy details of how those prices and quantities are arrived at. And that part is really hard. If, for example, you knew how to maximize the profitability of a firm, you would make a lot of money as the manager of that firm. The fact that, in real life, people are *paid* enormous sums to work full time at managing the firm (rather than just managing the employees) at least tells me that its not something that is easy to do for the general public. It certainly isn't costless or feasible in zero time. And firm managers often fail to maximize profits. These are generally smart people who have some background in economics, and still they fail spectacularly to set the right prices and quantities.

It's not clear that by following some simple decision rules in real time, that we will end up at the mathematical solution describing a utility maximization/profit maximization equilibria (the solution of which is an NP-complete problem). But if we look at household or firm behavior, they certainly appear to be following some simple decision rules.

Here, I am describing real life behavior of firms and entrepreneurs. You can't throw a rock out your window without hitting a small business owner who is doing something incredibly stupid and who doesn't want your advice on his price-setting. You can't open the newspaper and not read about some bone-headed corporate move. And how many of your friends are actually optimizing their consumption in any meaningful way? Many wouldn't even save for retirement unless the government/employers "nudged" them to automatically contribute to their 401Ks, in which case they passively save whatever amounts are pre-set for them and never bother even changing those ratios until they are in their late 40s.

Now it's possible to come up with representative agent models that do not assume that the economy is always at a set of prices and quantities that maximize utility, but that the agents behave according to simple rules and the above drawbacks become non-issues. You don't care about the optimizing equilibria because you ignore it. The economy ends up wherever it ends up if everyone follows their behavioral rule. In this case, the prices and quantities are _also_ the result of two lines crossing, or some algebra, but it is describing the solution of behavioral rather than optimization equilibria.

Those models can't get published. Only the optimization equilibria models get published.

That is the great divide. Pointing out that behavioral models can also be solved with algebra doesn't really do anything to respond to this criticism. It's a valid criticism.

Posted by: rsj | February 22, 2015 at 09:12 AM

rsj: "Nick, are you sure you're not knocking down a strawman here?"

I wasn't really trying to knock down the Keynesian Cross here. Hope it didn't read like that. I was trying to explicate it, and also use it to explicate a more general point about equilibria in all economics models.

Taken at face value, I dislike the KC model. But I also think it tells us something very important about positive feedback processes in macro.

Posted by: Nick Rowe | February 22, 2015 at 09:52 AM

Nick E: to answer my own question: I think it depends on how those expectations are formed. And if those expectations are formed in a way that is vaguely rational, then yes, we should expect to see them average out close to zero in a long enough time period.

Posted by: Nick Rowe | February 22, 2015 at 10:28 AM

Possibly the problem is entirely with me, but I have a doctorate from a "select" school and am a professor at such a school and though I have tried to understand what is being argued here I am lost. Why not set down a summary statement and even try to be a little clearer and avoid excessive jargon? (I really do not think the problem is with me alone.)

Posted by: ltr | February 22, 2015 at 11:44 AM

Moi: "I'll take your word for it. {That the Keynesian cross is a representative agent model}. However, to generate a line for any number of agents, all we need is for the equations relating the quantities to be linear."

Don't trust me, especially on math, but:

y1 = a1 + b1x1

y2 = a2 + b2x2

does not give us (y1+y2) = F(x1+x2) I think. Unless b1=b2.

They way you have explained things, with two agents you start with four dimensions, one for expected quantities and one for actual quantities for each agent. Then you add three linear equations to derive a line, going from four dimensions down to one. Two of the equations just say that for each agent the actual and expected quantities are equal, because of equilibrium. The third equation relates the quantities for each agent. You assume symmetry, which gives you a representative agent model. All I am saying is that you still get a line if the relationship is linear, without assuming symmetry. That remains so as you add more agents. (Adding a third agent gives you six dimensions but six equations. The reason that you do not get a point is that one of the equations is redundant, by assumption.)

I think that symmetry is what makes for the representative agent. Right?

Posted by: Min | February 22, 2015 at 01:36 PM

Oops! In my previous note, the section beginning, "Don't trust me," and ending "Unless b1 = b2." is by Nick Rowe.

The section beginning, "The way you have explained things," until the end is by me.

Sorry.

Posted by: Min | February 22, 2015 at 01:41 PM

rsj: "every model is in some sense a model involving representative agents."

Puleaze! How about a model with Mom, Pop, Sis, and Bro? Who is the representative agent?

Posted by: Min | February 22, 2015 at 01:47 PM

Min said: "IOW, when there is a "general glut". Or, as John Stuart Mill said, an insufficient supply of money."

I do not completely agree with that.

Assume the apple producer wants to eat 10 apples and eat 10 bananas and wants to supply 20 apples with 20 hours of work. Assume the banana producer wants to eat 10 apples and eat 10 bananas and supply 20 bananas with 20 hours of work.

For both the apple producer and banana producer, eating more than 10 apples or eating more than 10 bananas will give the producer a bellyache.

The apple producer supplies 20 apples. The apple producer eats 10 apples and sells 10 apples for 10 bananas to eat.

The banana producer supplies 20 bananas. The banana producer eats 10 bananas and sells 10 bananas for 10 apples to eat.

This is expected to go on for 50 years. At year 5, a mutation of both the apple tree and the banana tree means the apple tree can produce 25 apples with 20 hours of work and the banana tree can produce 25 bananas with 20 hours of work. The apple producer supplies 25 apples, eats 10 apples, and tries to sell 15 apples. The apple producer can only sell 10 apples. The apple producer puts 5 apples into inventory. They rot there.

The banana producer supplies 25 bananas, eats 10 bananas, and tries to sell 15 bananas. The banana producer can only sell 10 bananas. The banana producer puts 5 bananas into inventory. They rot there.

How long will it be before both producers realize each one should only want to produce 20 apples/bananas and sell 10 apples/bananas while working fewer hours? That means "productivity growth" went towards fewer hours worked and not increased output. It also means quantity want to sell is below potential quantity to sell.

Posted by: Too Much Fed | February 22, 2015 at 08:07 PM

ltr: this post was aimed at teachers of economics, and those who have already seen the Keynesian Cross model, and are familiar with Nash equilibrium. For those people, I was just putting two familiar things together.

Are you unfamiliar with the Keynesian Cross model? Or Nash equilibrium? or both?

The Wikipedia entry on the Keynesian Cross, for example, is a fairly good representation of how it is commonly taught. But it's crap. And this post is about why it's crap.

Posted by: Nick Rowe | February 22, 2015 at 10:32 PM

@TMF:

> That means "productivity growth" went towards fewer hours worked and not increased output. It also means quantity want to sell is below potential quantity to sell.

As far as I can tell, you've aptly described a situation of general glut while attempting to escape the concept. Your mature-treeconomy is not operating at the production possibilities frontier, so both more apples and more bananas could be produced without a tradeoff.

Posted by: Majromax | February 23, 2015 at 11:38 AM

Majromax said: "Your mature-treeconomy is not operating at the production possibilities frontier, so both more apples and more bananas could be produced without a tradeoff."

Something like that.

I am actually trying to say the most definition of economics, unlimited wants/needs and limited resources does not have to be true.

That means real AS can get above the level of real AD without there being an aggregate demand shock that needs to be corrected.

Posted by: Too Much Fed | February 24, 2015 at 11:22 PM