I have had occasion to make many references in the past to Richard Lipsey’s wonderful article “The Foundations of the Theory of National Income” which was included in the volume *Essays in Honour of Lord Robbins*. When some 40 years ago, while a grad student at UCLA, I luckily came upon Lipsey’s essay, it was a revelation to me, because it contradicted what I had been taught as an undergrad about the distinctions between planned (ex ante) investment and savings, and realized (ex post) investment and savings. Supposedly, planned investment and planned savings are equal only in equilibrium, but realized investment and savings are always equal. Lipsey explained why the ex ante/ex post distinction is both incorrect and misleading. In this post I want to begin to summarize some of the important points that Lipsey made in his essay.

Lipsey starts with a list of seven erroneous propositions commonly found in introductory and intermediate textbooks. Here they are (copied almost verbatim), grouped under three headings:

I The Static Model in Equilibrium

1 The equilibrium of the basic Keynesian model is given by the intersection of the aggregate demand (i.e., expenditure) function and the 45-degree line representing the accounting identity *E* ≡ *Y*.

II The Static Model in Disequilibrium

2 Although people may try to save different amounts from what people try to invest, savings can’t be different from investment; realized (ex post) savings necessarily always equals realized (ex post) investment.

3 Out of equilibrium, planned savings do not equal planned investment, so it follows from (2) that someone’s plans are being disappointed, and there must be either unplanned savings or dissavings, or unplanned investment or disinvestment

4 The simultaneous fulfilment of the plans of savers and investors occurs only when income is at its equilibrium level just as the plans of buyers and sellers can be simultaneously fulfilled only at the equilibrium price.

III The Dynamic Behavior of the Model

5 Whenever savers (households) plan to save an amount different from what investors (business firms) plan to invest, a mechanism operates to ensure that realized savings remain equal to realized investment, despite the attempts of savers and investors to make it otherwise. Indeed, this mechanism is what causes dynamic change in the circular flow of income and expenditure.

6 Since the real world, unlike the simple textbook model, contains a very complex set of interactions, it is not easy to see how savings stay equal to investment even in the worst disequilibrium and the most rapid change.

7 The dynamic behavior of the Keynesian circular flow model in which disequilibrium implies unintended investment or disinvestment can be shown by moving upwards or downwards along the gap between the expenditure function and the 45-degree line in the basic Keynesian model.

Although some or all of these propositions are found in most standard textbook treatments of national income theory, every one of them is wrong.

Let’s look at proposition 1. It says that the equilibrium level of income and expenditure is determined algebraically by the following two relations: the expenditure (or aggregate demand) function:

*E = E(Y) *+ *A*

and the expenditure-income accounting identity

*E ≡ Y*.

An accounting identity provides no independent information about the real world, because there is no possible state of the world in which the accounting identity does not hold. It therefore adds no new information not contained in the expenditure function. So the equilibrium level of income and expenditure must be determined on the basis of only the expenditure function. But if the expenditure function remains as is, it cannot be solved, because there are two unknowns and only one equation. To solve the equation we have to make a substitution based on the accounting identity *E ≡ Y.* Using that substitution, we can rewrite the expenditure function this way.

*E = E(E) *+ *A*

If the expenditure function is linear, we can write it as follows:

*E = bE *+ *A*,

which leads to the following solution:

*E = A/*(1 – *b*).

That solution tells us that expenditure is a particular number, but it is not a functional relationship between two variables representing a theory, however naïve, of household behavior; it simply asserts that *E *takes on a particular value.

Thus treating the equality of investment and savings as an identity turns the simply Keynesian theory into a nonsense theory.

The point could be restated slightly differently. If we treat the equality of investment and savings as an identity, then if we follow the usual convention and label the vertical axis as *E*, it is a matter of indifference whether we label the horizontal axis *Y* or *E*, because *Y* and *E* are not distinct, they are identical. However we choose to label the horizontal axis, the solution of the model must occur along the 45-degree line representing either *E = Y* or *E = E*, which are equivalent. Because, the equality between *E* and itself or between *E* and *Y* is necessarily satisfied at any value of *E*, we can arbitrarily choose whatever value of *E* we want, and we will have a solution.

So the only reasonable way to interpret the equality between investment and saving, so that you can derive a solution to the simple Keynesian model is to treat *E* and *Y* as distinct variables that may differ, but will always be equal when the economy is in equilibrium.

So the only coherent theory of income is

*E = E(Y) *+ *A*

and, an equilibrium condition

*E = Y.
*

*E* and *Y* do not represent the same thing, so it makes sense to state a theory of how *E *varies in relation to *Y*, and to find a solution to the model corresponding to an equilibrium in which *E* and *Y* are equal, though they are distinct and not necessarily equal.

But the limitation of this model is that it provides us with no information about how the model behaves when it is not in equilibrium, not being in equilibrium meaning that *E* and *Y* are not the equal. Note, however, that if we restrict ourselves to the model in equilibrium, it is legitimate to write *E* ≡ *Y, *because the equality of *E* and Y is what defines equilibrium. But all the erroneous statements 2 through 7 listed above all refer to how the model.

The nonsensical implications of constructing a model of income in which expenditure is treated as a function of income while income and expenditure are defined to be identical has led to the widespread adoption of a distinction between planned (ex ante) investment and savings and realized (ex post) investment and savings. Using the ex ante/ex post distinction, textbooks usually say that in equilibrium planned investment equals planned savings, while in disequilibrium not all investment and savings plans are realized. The reasoning being that is that if planned saving exceeds planned investment, the necessity for realized savings to equal realized investment requires that there be unintended investment or unintended dissaving. In other words, the definitional identity between expenditure and income is being used to tell us whether investment plans are being executed as planned or being frustrated in the real world.

Question: How is it possible that an identity true by definition in all states of the world can have any empirical implications?

Answer: It’s not.

In my next installment in this series, I will go through Lipsey’s example showing how planned and realized saving can indeed exceed planned and realized investment over the disequilibrium adjustment induced by a reduction in planned investment relative to a pre-existing equilibrium.

UPDATE (2/21/2015]: In the second sentence of the paragraph beginning with the words “An accounting Identity provides,” I wrote: “It therefore adds information not contained in the expenditure function,” which, of course, is the exact opposite of what I meant to say. I should have written: “It therefore adds NO NEW information not contained in the expenditure function.” I have now inserted those two words into the text. Thanks to Richard Lipsey for catching that unfortunate mistake.

If choices are– a) the Keynesian Cross is incoherent or b) the basic identity of macroeconomics and arithmetic are incorrect, then I choose option a.

I don’t see much value of translating the notion that firms only produce what they expect they can sell into some notion that income adjusts until planned saving equals planned investment. (I will grant, however little value i put on the notion, I had never considered it incoherent.)

In my view, the Keynesian cross “equilibrium” is no equilibrium worth the name anyway.

My understanding of the identity between saving and investment is not due to consideration of the Keynesian Cross.

“The accounting identities equating aggregate expenditures to production and of both to incomes at market prices are inescapable, no matter which variety of Keynesian or classical economics you espouse. I tell students that respect for identities is the first piece of wisdom that distinguishes economists from others who expiate on economics. The second?… Identities say nothing about causation.”

– James Tobin

David, Lipseys comments on your previous post rightly promoted this article.

Keynes in correspondence with Myrdal grew to be dismissive of the Ex ante Ex Poste distinction – introduced by Myrdal. – Clausde Gnos has writted extensively on this.

http://www.tandfonline.com/doi/abs/10.1080/0953825042000225625?journalCode=crpe20

Indeed his first drafts of the General Theory applied the ‘swedish’ method, then he realised it was wrong and took a new approach which is the one we see published. Here income and expenditure are in constant flux and all that matters is the entrepreneurs expectations as to what is profititable at all future points in time (what Keynes termed the ‘flux’).

In the published draft the Kahn identities are true – like all accounting identities, at all times, but they are not causative and they dont imply equilibrium. Instead it is the unexpected profits/losses – saving and dissaving and running up and down of inventory which are. Indeed this was Keynes break with Hawtry as he saw inventory changes as the effect not the cause of income changes.

Athough I find Gnos’s various writing deconstructing the foundations of national income accounting etc. in the 1930s and 40s compelling I find his style less developed in terms of how accounting identities and causative forces should be reconstructed meaningfully and mathematically.

The problem is the treatment of time and causation. Myrdals response to Keynes criticism was to refer to everything as momentary points of zero time But Keynes (flow input output) and Myrdal (instantaneous time) are reconcilable in a modern flow input output – continuous time mathematical model. Then you can treat accounting identities as a measure of state – as a state machine, and any change of state as a change to a variable with an external cause.

so for example I=E becomes I=E + delta I. an equation familiar and logical in form to anyone taught BASIC at school in the 1970s or 80s. In such a state approach there is no distinction between ex poste and ex ante, this really only applies in discreet time. Matious Graseeli of the Fields institute (maths research) http://www.debtdeflation.com/blogs/2013/07/05/matheus-grasselli-on-mathematics-for-good-economics/

has done some good work on this so far only in monetary theory, using lebesque integration, but there is no reason why it cannot be used to look at AD/AS and reconstruct macro based on on strict verifiable and computable accounting definitions

I just don’t get this. You can’t have a theory of national income based purely on accounting identities alone, sure. But using accounting identities in the theory is simply to help set it out more clearly. The identities define the space of possible solutions. In the Keynesian cross, the solution space is not the whole two dimensional area defined by the axes – it’s the one dimensional 45 degree line. We then need a theory to tell us where we are on the line. Drawing it in two dimensions is just to help explain the theory part, which appeals to the decisions taken by individuals in relation to their own income and expenditure.

Whatever the merits of that theory, whether we define income and expenditure so that we are necessarily on that line, or whether we define it so that we can access the full two dimensional space doesn’t really seem to change anything.

Bill, You cannot prove a theory is incoherent by imposing a definition on the theory when there is another possible definition under which the theory is not incoherent, so choices a and b are not exhaustive. You are free to reject the Keynesian cross, and I don’t subscribe to it as a useful theory, but it is a key ingredient of standard macroeconomic theory. But the argument that the equality between savings and investment is analogous to the equality between purchases and sales is not valid, as Hawtrey, Robertson, Haberler, and Lutz all explained in the 1930s in their critical reviews of the General Theory. So it is you who are taking Keynes’s side in this argument.

Ramanan, I have the utmost respect and admiration for James Tobin as an economic theorist. But it is not clear what to me exactly what Tobin meant by “respect for identities.” I totally agree with the statement “identities say nothing about causation.”

Andrew, Thanks for this comment, which is difficult for me to comprehend on a quick read, but I will try to digest more thoroughly and look at your links over the next few days.

Nick, The 45-degree line is one-dimensional in two a two-dimensional space, unless you define Y and E to be the same thing. If you do that, you a one-dimensional line in a one-dimensional space and your theory has collapsed.

Let’s call E “demand” and Y “supply”. Equilibrium is E=Y. What is the confusion?

How about this?

Let’s say my theory is that, in equilibrium, each household i will be spending an amount given by the function: Ei = ai + bi Yi, where E, a, b and Y may be different for each i. I can aggregate this to get: sum(E) = sum(a) + avg(b) sum (Y). (b is a weighted average, weighted by the size of Y). Clearly Ei and Yi are not in general equal.

This is just a theory about the set of households. On its own it is not enough. Sum(E) and sum(Y) may not be equal. There may be other non-household agents involved. To close this off, I need to impose the condition that they equal, which is equivalent to saying that these households comprise the entire economy and represent a closed system. Even if that condition is that they are always equal in all possible states, not simply that they are equal in equilibrium, it is still necessary for me to include the condition in order to determine sum(E) and sum(Y). So it’s not meaningless.

Nick, You are describing an equilibrium system. What happens when the a’s or the b’s change. There will be an instantaneous change in expenditure which will not necessarily be matched by an instantaneous change in Es without an exactly equal change in Ys if there are lags built into the system, so that the model has to go through a series of disequilibrium steps before arriving at the new equilibrium. If there are no lags built into the system so that all adjustments are instantaneous, then the system is always in equilibrium. But then you can’t speak about what the system is like outside of equilibrium by referring to a distinction between ex ante and ex post.

David, Lipsey’s “erroneous propositions” are simply textbook definitions that he doesn’t like. Again, economists define S as being equal to I, that’s what an identity is–a definition. Ditto for income/expenditure. If you don’t like those definitions that’s fine, but it would make more sense for you to say you’d prefer a different definition of savings and investment, than to say you think the definition is wrong. Yes, it’s wrong in terms of your preferred definition, but your preferred definition is wrong in terms of my preferred definition. Or Paul Krugman’s preferred definition.

AFAIK in modern macro when economists say “disequilibrium” they usually don’t refer the case where Y doesn’t equal E, but rather the case where actual Y doesn’t equal the natural rate of Y.

One final point. I think this discussion was triggered by my claim that consumption smoothing implies C will be more stable that GDP, and hence that S and I will be less stable than GDP. That’s true of investment even if S does not equal I, isn’t it?

Nat, Sorry I missed your oomment when I responded earlier. I have no problem with E = Y as an equilibrium condition, whether we call E “demand” or “expenditure” and Y “supply” or “income.” The problem arises when you define them to be identical.

Scott, I really don’t think that the point is that he doesn’t like the definitions, but that the definitions lead to incoherent or mistaken interpretations of what the model is saying. If you define E and Y to be identical, then it is not easy to formulate a theory of how to solve the model for a unique equilibrium value of income and expenditure. How do you solve the equation x = y for a unique value of x if you define x to be identical to y? If x and y are identical they are equal for all values of x. Similarly, the insistence that savings and investment are identical leads to an interpretation of how the simple income-expenditure model behaves out of equilibrium that is clearly inconsistent with the behavior of the model when you introduce a lag and get a period by period adjustment taking the model from the old equilibrium to a new equilibrium. This is a simple textbook version of the adjustment to a parameter change, and it requires no distinction between ex ante and ex post.

I think that, depending on the context, economists might consider a situation with E not equal to Y as a disequilibrium, even though they might also regard a situation with actual Y not equaling the natural rate of Y as a disequilibrium of another sort. The latter might be considered as a condition of full equilibrium and the former a condition for short-period, or even a temporary, equilibrium.

I was not disagreeing with your conclusion that consumption tends to be more stable than GDP; I was just raising an issue with the reasoning that used to arrive at the conclusion.

Have you read anything by Wynne Godley, David Glasner?

“Thus treating the equality of investment and savings as an identity turns the simply Keynesian theory into a nonsense theory.”

I actually don’t see this.

The Keynesian theory is about path adjustment toward something defined as “equilibrium” according to some sort of behavioral function.

Actual saving and actual investment must be equal at every moment along that path.

I’d be extremely surprised to find that somebody can show Keynes didn’t understand this.

The problem lies in the use of the word “equilibrium” as the anchor point for some notion of equivalence between saving and investment. It is an anchor point for the end of the path that is determined by the behavioral function. That is all.

Again, after second reading, on this:

“So the only reasonable way to interpret the equality between investment and saving, so that you can derive a solution to the simple Keynesian model is to treat E and Y as distinct variables that may differ, but will always be equal when the economy is in equilibrium.”

I disagree.

I think it’s just an algebraic solution as to where the economy will end up based on where investment will END UP plus the multiplier effect. The economy is always on the 45 degree line, but the cross changes where it is. All points on the cross other than the intersection are essentially meaningless, because the graph only includes the assumed end game for investment. You would need a continuous movie of investment iteration to see all of the end points in sequence.

That doesn’t require treating E and Y as distinct.

Blue Aurora, Sorry, no I have not. Can you tell me briefly why you think I might be interested in him?

JKH, The nonsense results from treating expenditure and income as identical, which means that the theory collapses to the 45-degree line. The behavioral function says that expenditure is a function of itself, so you nothing by which to explain how expenditure (consumption) varies. A solution is just specified arbitrarily. Keynes obviously didn’t understand it, and Hawtrey, Robertson, and Haberler and others all pointed it out to him, but he seems to have had a blind spot on that point.

From what I’ve read, Aggregate Expenditure is not set equal to Income/Output through an identity relation. Rather, equality between these is the equilibrium criterion in the model.

And how do we reach that equilibrium? Through changes in Output/Income brought about by firms’ responses to unplanned changes in their inventory. Aggregate Expenditure can certainly deviate from total Income/Output in a given period. For example, a sudden rise in autonomous expenditure, which results in inventory (production from prior periods) depletion.

It seems like the issue here stems from the treatment of AE = Y as an identity rather than an equilibrium criterion.