Last week I wrote a series of posts (starting with this and ending with this) that were mainly motivated by a single objective: to show how taking the accounting identity between savings and investment seriously can get someone, even a very fine economist, into serious trouble. That, I suggested, is what happened to Scott Sumner when, in a post about whether a temporary increase in government spending and taxes would increase GDP, he relied on the accounting identity between savings and investment to conclude that a reduction in savings necessarily leads to a reduction in investment. Trying to trace Scott’s mistake to misuse of an accounting identity led me a little further than I anticipated into the substance of the argument about how a temporary increase in government spending and taxes affects GDP, an argument that I am still not quite satisfied with, but which – you can relax — I am not going to discuss in this post. My aim in this post is merely to respond to one of Scott’s rejoinders to me, which is that he was just relying on a proposition – the identity of savings and investment – that is taught in just about every macro textbook, including textbooks by Paul Krugman and Greg Mankiw, two of the current heavyweights of the profession. If so, Scott observed, my argument is not really with him, but with the entire profession.

No doubt about it, Scott has a point, though I think that most textbooks and most economists have an intuitive understanding that the accounting identity is basically a fudge, and therefore, unlike Scott, generally do not rely on it for any substantive conclusions. The way that most textbooks try to handle the identity is to say that the identity really just refers to realized (ex post) saving and investment which must be equal, while planned (ex ante) investment and planned (ex ante) saving may not be equal, with the difference between planned investment and planned saving corresponding to unplanned investment (accumulation) of inventories. Equilibrium is determined by the equality of planned investment and planned saving, and any disequilibrium (corresponding to a divergence between planned saving and planned investment) is reflected in unplanned inventory accumulation (either positive or negative) which ensures that the identity between realized investment and realized saving is always satisfied. The usual fudge distinguishing between planned and realized investment and saving and postulating that unplanned inventory investment is what accounts for any difference between planned investment and saving is itself problematic, but it at least puts one on notice that there is a difference between an equilibrium condition and an accounting identity, while nevertheless erroneously suggesting that the accounting identity has some economic significance.

Not entirely coincidentally, Scott having got started on this topic by responding to a post by Paul Krugman, Krugman himself weighed in on the subject of accounting identities last week, enthusiastically citing a post by Noah Smith warning about the misuse of accounting identities in arguments about economics. Now the truth is that there is not too much in Krugman’s post that I disagree with, but there are certain verbal slips or misstatements that betray the confusion between accounting identities and equilibrium conditions that I am trying to get people to recognize and to stay away from. While avoiding any substantive error, Krugman perpetuates the confusion, thus contributing unwittingly to the very problem that motivated his post. Thus, his confusion is not just annoying to compulsive grammarians like me; it is also unnecessary and easily avoidable, and creates the potential for more serious mistakes by the unwary. So there is really no excuse for continuing to pay lip service to the supposed identity between savings and investment, regardless of how deeply entrenched it has become as the result of many decades of unthinking, rote repetition on the part of textbook writers.

Here’s Krugman:

Via Mark Thoma, Noah Smith has a terrific piece on how to argue with economists. All the points are good, but I’d like to focus on Principle 4, “Argument by accounting identity almost never works.”

What he’s referring to, I assume, is arguments like “since savings equals investment, fiscal stimulus can’t affect overall spending”, or “since the current account balance is equal to the difference between domestic saving and domestic investment, exchange rates can’t affect trade”. The first argument is, more or less, Say’s Law and/or the Treasury view. The second argument is what John Williamson called the doctrine of immaculate transfer.

This is pretty straightforward, though I don’t care for the examples that Krugman gives, displaying a conventional misunderstanding of Say’s Law. But Say’s Law is a whole topic unto itself. Nor can the Treasury view be dismissed as nothing more than the misapplication of an accounting identity. So I’m just going to ignore those two specific examples for purposes of this discussion. Back to Krugman.

Why are such arguments so misleading? Noah doesn’t fully explain, so let me put in a further word. As I see it, economic explanations pretty much always have to involve micromotives and macrobehavior (the title of a book by Tom Schelling). That is, when we tell economic stories, they normally involve describing how the actions of individuals, driven by individual motives (and maybe, though not necessarily, by rational self-interest), add up to interesting behavior at the aggregate level.

Again, nothing to argue with there, though the verb “add up” has just faintest whiff of an identity insinuating itself into the discussion.

And the key point is that individuals in general [as opposed to those strange creatures called economists who do care about “aggregate accounting identities?] neither know nor care about aggregate accounting identities.

Ok, now we are starting to have a problem. Individuals in general neither know nor care about aggregate accounting identities. Does that mean that those strange creatures called economist should know or care about aggregate accounting identities? I have yet to hear any cogent reason why they should.

Take the doctrine of immaculate transfer: if you want to claim that a rise in savings translates directly into a fall in the trade deficit, without any depreciation of the currency, you have to tell me how that rise in savings induces domestic consumers to buy fewer foreign goods, or foreign consumers to buy more domestic goods. Don’t tell me about how the identity must hold, tell me about the mechanism that induces the individual decisions that make it hold.

Here is where Krugman, after skating on the edge, finally slips up and begins to talk nonsense — very subtle nonsense, but nonsense nonetheless. What does it mean to say that an identity must hold? It means that, by the very meaning of the terms that one is using, the identity of which one is speaking must be true. It is inconceivable that an identity would not hold. If the difference between investment and savings (in an open economy) is defined to be identitically equal to the trade deficit, then talking about a mechanism that induces individual decisions to make it hold makes as much sense as saying that there must be a mechanism that induces individual decisions to make 2 + 2 equal 4. If, by the very meaning of the terms that I am using, the difference between investment and savings must equal the trade deficit (which, to repeat, is what it means to say that there is an identity between those magnitudes) there is no conceivable set of circumstances in which the two magnitudes would not be equal. If, in the very nature of things, two magnitudes could never possibly be different, it is nonsense to say that there is a mechanism of any kind (much less one describable in terms of the decisions of individual human beings) that operates to bring it about that the equality actually holds.

And once you do that, you realize that something else has to be happening — a slump in the economy, a depreciation of the real exchange rate, it depends on the circumstances, but it can’t be immaculate, with nothing moving to enforce the identity.

No, no! A thousand times no! If we are really talking about an identity, nothing has to be happening to enforce the identity. Identities don’t have to be enforced. Something that could not conceivably be otherwise requires nothing to prevent the inconceivable from happening.

When it comes to confusions about the macro implications of S=I, again the question is how the identity gets reflected in individual motives — is it via the interest rate, via changes in GDP, or what?

There are no macro implications of an identity; an identity has no empirical implications of any kind — period, full stop. If S necessarily equals I, because they have been defined in such a way that they could not possibly be unequal, then there is no conceivable state of the world in which they are unequal. Obviously, if S and I are equal in every conceivable state of the world, the necessary identity between them cannot rule out any conceivable state of the world. That means that the identity between S and I has no empirical implications. It says nothing about what can or cannot be observed in the real world at either the micro or the macro level.

Accounting identities are important; in fact, they’re the law. But they should inform your stories about how people behave, not act as a substitute for behavioral analysis.

I don’t know what law Krugman is referring to, but usually laws of nature tell us that some conceivable observations are not possible. Accounting identities don’t tell us anything of the sort. They are merely express certain conventional meanings that we are assigning to specific terms that we are using. How an accounting identity that could not be inconsistent with any conceivable state of the world can inform anything is a mystery, but I heartily agree that an accounting identity cannot be “a substitute for behavioral analysis.”

I have been rather (perhaps overly) harsh in my criticism on Krugman, but not to show that I am smarter than he is, which I certainly am not, but to show how easily habitual ways of speaking about macro lead to (easily rectifiable) nonsense statements. The problem is not any real misunderstanding on his part. Indeed, I would be surprised if, should he ever read this, he did not immediately realize that he had been expressing himself sloppily. The point is that macroeconomists have gotten into a lot of bad habits in describing their models and in failing to distinguish properly between accounting identities, which are theoretically unimportant, and equilibrium conditions, which are essential. Everything that Krugman said would have made sense if he had properly distinguished between accounting identities and equilibrium conditions rather than mix them up as he did, and as textbooks have been doing for three generations.

Savings and investment are equal in equilibrium, because that equality is a necessary and sufficient condition for the existence of an equilibrium. If so, being out of equilibrium means that savings and investment are not equal. So if we think that a real economy is ever out of equilibrium, one way to test for the existence of disequilibrium would be to see if actual savings and actual investment are unequal, notwithstanding the presumed accounting identity between savings and investment. That accounting identity is a product of the special definitions assigned to savings and investment by national income accounting practices, not by the meaning that our theory of national income assigns to those terms.

PS I will once again mention (having done so in previous posts on accounting identities) that all the essential points I am making in this post are derived from the really outstanding and unfortunately not very widely known paper by Richard G. Lipsey, “The Foundations of the Theory of National Income” originally published in *Essays in Honour of Lord Robbins* and reprinted in* Macroeconomic Theory and Policy: Selected Essays of Richard G. Lipsey*.

A “golden key” to close the debate!

Keep in mind, Krugman is writing to an informed but otherwise general audience. His format is a fairly small blog space owned by the NY Times. Compare the size of Krugman’s blog space with the space you needed to discuss one very obscure and technical point. Krugman’s value is in his ability to get peope’s intuition 99% right, as opposed to the nonsense on the WSJ op-ed page which makes people stupider.. The lay reader (who is afterall Krugman’s target audience) does not need to ponder extraordinarily fine distinctions. He’s not giving a seminar to fellow economists. His blog space is not NBER for the masses. His goal is to correct patent nonsense even if he sometimes has to smudge over very technical points. Economics for the guy reading the paper on the morning commuter train.

In any event, I think most economists who read Krugman’s piece already understood what he meant when he said that accounting identities were the law. It’s a good punchy line that will stick with the reader.

Once again, I never claimed any accounting identities implied anything about the multiplier, so I still think you are misunderstanding my position. i never claimed it told us anything about behavioral relationships. Nor do I believe any of those things. I must have worded my post poorly, as you seem to still have a misunderstanding of what I was claiming. Definitions can never tell us anything about causation.

At one point you say that most economists “unlike Scott” do not rely on it for any substantive conclusions. Then you describe the views of most economists, which are exactly the same views as I hold. Exactly. Not only do I hold these views, so does Noah Smith, and Paul Krugman, and Karl Smith, and Simon Wren-Lewis. We all agree that identities tell us nothing about behavior. We all agree that desired saving doesn’t equal desired investment. We all agree that actual saving equals actual investment. We all agree that actual equilibrium outcomes must be consistent with the equality of S and I. We all agree that that equality tells us nothing about the multiplier.

I think you are making a big mistake in assuming that when economists say S=I is an identity they are too sensible to really believe it. I really believe, it. I can’t imagine why anyone would claim something is an identity if they didn’t believe it. That would be a really serious error to make, which would leave students totally confused. Do you have evidence that they don’t believe it is an identity? Obviously there could never be any empirical evidence refuting the identity, as if measured saving doesn’t equal measured investment, we just assume the measurements were in error, Just like GDP=GDI, another identity.

I now think I should have used different language to criticize SWL, and then you would have understood me. I should have said that when disposable income falls, then both C and I must fall if there is consumption smoothing AND a balanced budget multiplier. That would have made exactly the same point, but without any reference to S=I. The only assumption would have been that GDP = C+I+G

Marcus,Thanks,but as Scott’s response below makes clear, the debate is far from closed.

Mr_RDES, You do have a point, but I am not sure that I find it compelling. If Krugman does not believe that there is an identity between investment and savings, but understands that the equality of savings and investment is a condition of equilibrium, he could have easily dispensed with the incoherent idea that there is or could be any mechanism operating to ensure that the identity is satisfied. He could have just as easily spoken in terms of an equilibrium condition. Speaklng about an identity is just confusion. The confusion need not be fatal, but it is greatly preferable to avoid it. And the reason that I am making a big deal about it is precisely because the confusion is rampant among economists.

Scott, First of all, I should have been more careful in describing your position, since you have already stated in earlier replies that I was not accurately representing your position. However, since, following your advice, I made Krugman the main target of criticism in this post, I sort of forgot that I have still not fully grasped your position.

Can you explain to me how you can assert that identities tell us nothing about behavior and that actual savings equals actual investment. Are you saying that the equality of actual savings actual investment is independent of the behavior of savers and investors? And are you saying that savers and investors are the same people or are different people?

Because I believe that it is incoherent to state that the income-expenditure model of national income requires savings to equal investment whether or not equilibrium obtains, I am prepared to give anyone who talks about an identity between saving and investment the benefit of the doubt and construe him to be speaking sloppily in referring to an identity when what he really means is that the equality of savings and investment is an equilibrium condition. And by the way, GDP is not identically equal to GDI either. The value of output is not the same thing as the value of payments to factors of production. Perhaps I will get back to you later about SWL, but it is getting too late for me to think straight.

I don’t think you are on such firm ground as you think you are, for you are sometimes saying “identity” to mean “definition” and other times to mean “something true in all states of the world”. These are not equivalent; the latter could be called an adding-up constraint. You betrayed yourself with your example, “2 + 2 = 4”, which is not in fact an identity in the sense of a definition. If only you had chosen “2 = 1 + 1”! That is an identity in some systems. But in those systems, you would have to write:

4 = 1 + (1 + (1 + 1)))

= (1 + 1) + (1 + 1)

= 2 + 2

to demonstrate the truth of the proposition that 2 + 2 = 4; it is something that is merely true in all states of the world, not by definition.

Personally, I think that Krugman is consistently using the “adding up constraint” meaning and that he has got it right.

“I now think I should have used different language to criticize SWL, and then you would have understood me. I should have said that when disposable income falls, then both C and I must fall if there is consumption smoothing AND a balanced budget multiplier. That would have made exactly the same point, but without any reference to S=I. The only assumption would have been that GDP = C+I+G”

1. Investment is a red herring. There was nothing to indicate that Wren-Lewis assumed investment as part of his model.

2. Even with investment, there was nothing to indicate that Wren-Lewis excluded the possibility of no fall in C.

3. Even with investment and Wren-Lewis arguing that there must be an actual negative change in C, then Wren-Lewis’ error is just that– arguing erroneously that C would fall, when it certainly need not. (Even *with* consumption smoothing!)

I’d use the analogy of supply and demand. The amount bought and the amount sold are identical. That fact can’t be used to tell us anything about what caused quantities or prices to change. It simply reflects the fact that purchases and sales are two sides of the same transaction. On the other hand the motives of those who engage in purchases and sales are different. Higher prices induce people to want to buy less, and induce other people to want to sell more. It’s quite possible that the market isn’t in equilibrium, that the quantity demand does not equal the quantity supplied. But even if not in equilibrium, it’s still true that amount bought and amount sold are identical.

By analogy, there are interest rates where the amount people desire to save doesn’t equal the amount others desire to invest. But there is no place for saving to go (in aggregate) other than into investment. So at the end of the day the amount saved must equal the amount invested. Anything that “looks like” saving that doesn’t result in investment is actually a loan to another person, business, or government, and hence isn’t aggregate saving.

Well, you’ve managed to confuse me. From where I sit, the problem is with the use of the term “equilibrium”. You have defined S and I in a certain way, and then give a proof that S is always equal to I.

As Phil Koop emphasized above, S=I is *not* a definition, it is the result of a proof. To use another example, show that the square of the length of the hypotenuse of a right triangle is the sum of the squares of the lengths of the other two legs. This is true for all concievable right triangles, but it is not the definition of a length, square, or right triangle.

You then very nicely showed how the fluctuations in inventories can help us see how S=I comes about. But there are in fact others. Suppose that there is no extra inventory. If people try to decrease S they will find nothing on the shelves, so they will be forced to save.

What you didn’t define is “equilibrium”. It sounded a lot to me that your definition was something like “intended S” = “intended I”. But why is this “equilibrium”, and how can you show that in fact S will equal “intended S”?

“But there is no place for saving to go (in aggregate) other than into investment.”

No. The excess savings can simply vanish as income falls.

Mitch,

The requirement that S=I has nothing to do with intent. It is unavoidable in a closed economy. Intent to save and intent to invest may shape the manner in which S and I maintain their balance. But there is no transaction within the closed economy which can create an S/I imbalance.

Once you pick any levels for C, I, and G, and T– at random, whatever– then you will find S=I, because private savings is by definition Y-T-C, and public savings is by definition T-G, so their sum S = Y-C-G = I.

If agents try to save, then Y may fall resulting in no change in savings. If agents try to save, inventories may rise, raising I. And so on.

Phil, You are right that “identity” has different meanings which I did not distinguish between. However, I don’t think that not distinguishing between those two meanings undermines my argument. The point is that any identity or tautologically true statement must be true in all states of the world, and is therefore devoid of empirical content. I have no problem with an adding up constraint which simply means that your model contains no errors of arithmetic. Doing arithmetic correctly is not enough to derive any meaningful statements about the real world. To do that you need either a dynamic model or at least a model of static equilibrium with which to perform a comparative statics exercise. An identity provides neither.

D R, If there is an increase in G in this period which requires taxes to be levied either now or in the future, the effect on consumption in this period (and next period) depends on what assumption is made concerning expected income in the following period as a result of government spending in this period. Unless you make such an assumption, in a Ricardian equivalence/consumption smoothing framework, you can’t say what happens to consumption in this period. The model is not closed without some way of deriving expected income in the next period. I am having trouble finding an intuitively satisfactory way of positing such a relationship.

Scott, Your analogy breaks down because supply and demand in a single market are indeed two sides of a single transaction. But factor payments (incomes) from which savings are drawn are undertaken in a different set of transactions from those in which expenditures on final output in which investments are made. So the strict identity that exists in equilibrium for a single market does not exist in a circular flow model of income and expenditure. The necessity for income to equal expenditure and savings to equal investment is a property of equilibrium not a necessary identity because you have established that two different terms apply to the same thing. I hate to keep saying this, but you need to read Lipsey’s article.

Mitch, You are obviously not alone; I have been confusing a lot of others, too. However, it seems that you are interpreting me in a way exactly the opposite of what I thought I was saying.

“You have defined S and I in a certain way, and then give a proof that S is always equal to I.”

I thought that I was saying that S does not always equal I, because S = I only in equilibrium, and equilibrium does not necessarily obtain at all times.

As I responded to Phil above, I don’t think that the distinction between a definition and a tautology matters for my purposes.

“You then very nicely showed how the fluctuations in inventories can help us see how S=I comes about. But there are in fact others. Suppose that there is no extra inventory. If people try to decrease S they will find nothing on the shelves, so they will be forced to save.”

I thought that I was saying that inventory fluctuations are not necessary or implied as the adjustment process by which a disequilibrium is transformed into a new equilibrium.

“What you didn’t define is “equilibrium”. It sounded a lot to me that your definition was something like “intended S” = “intended I”. But why is this “equilibrium”, and how can you show that in fact S will equal “intended S”?”

Equilibrium is a situation in which income and expenditure are equal and therefore in which there is no tendency for either to change. That definition is equivalent to an equality between savings and investment with no tendency for either to change. I thought that I was saying that the distinction between planned and realized saving was not implied or necessary for equilibrium to exist or to derinve an adjustment from a disequilibrium to a new equilibrium.

D R, Yes, in the model, that is one alternative means of adjustment. In the real world, you need some rather special conditions for that to be plausible.

I think that your answer to Mitch suffers from a failure to distinguish between what is possible within the model – it is possible for savings not to equal investment – and what is consistent with equilibrium. If savings is not equal to investment, then you are not in equilibrium and there is an adjustment process that restores equality between savings and investment, thereby bringing about a new equilibrium or restoring the old one.

David,

I am sorry. I fail to see how it is possible for savings and investment to be out of balance. Can you please provide a single example of how such an imbalance could come about?

I found this post rather a hard slog. You wrote: “the accounting identity [S = I] is basically a fudge”; I take it that means it is *not actually true*. But then you went on: “The way that most textbooks try to handle the identity is to say that the identity really just refers to realized (ex post) saving and investment which must be equal, while planned (ex ante) investment and planned (ex ante) saving may not be equal, with the difference between planned investment and planned saving corresponding to unplanned investment (accumulation) of inventories. Equilibrium is determined by the equality of planned investment and planned saving, and any disequilibrium (corresponding to a divergence between planned saving and planned investment) is reflected in unplanned inventory accumulation (either positive or negative) which ensures that the identity between realized investment and realized saving is always satisfied.” This made it appear that you think the identity is strictly true, since (I presume) S = *realized* savings and I = *realized* investment. As for planned savings (Sp) and planned investment (Ip), by the definition of ‘equilibrium’ they are equal *in equilibrium*, but may not be equal in disequilibrium. But then you called the distinction between *realized* and *planned* quantities a “fudge”; why? (It seems quite straightforward.)

Then you wrote: “Savings and investment are equal in equilibrium, because that equality is a necessary and sufficient condition for the existence of an equilibrium.” Here you seem to have jumped from using ‘savings’ to mean *realized* savings to using it to mean *planned* savings, and likewise for ‘investment’. That sort of equivocation should be avoided.

I wonder if your point is that, while Scott Sumner talked about S and I (which are necessarily identical), he should have been talking about Sp and Ip. If so, the point wasn’t very clearly expressed.

This link gives a very nice explanation of things, which is how I understand the meaning of the statement S=I:

http://wiki.answers.com/Q/Why_must_the_total_value_of_saving_in_the_economy_equal_the_total_value_of_investment

There is no need to invoke “equilibrium”. S = I is true not because they are defined to be the same, but for the same sort of reason that the Pythagorean Theorem holds. That is, if you define things a certain way, you can *show* with a mathematical proof that certain other relations must hold.

I should point out (though it is not stated in the link above) that all the terms here are flows, and the identity holds at every moment in time.

Now, according to you, S=I only holds in equilibrium. Inherently, this must mean that you don’t accept the definitions of the terms used in this proof. From what I am understanding (I may have misunderstood this), you are saying that your definition of I only coincides with this one in the case of “equilibrium”.

You invoke the concept of “tendency to change” as a definition of being out of “equilibrium”. It seems to me that you’ve thereby gone quite far afield, since you now need a model to determine the dynamics of these quantities. (E.g. people may be saving and investing what they want right now, but they might change their minds or die or something.)

So, it seems to me that to clarify what you are saying you have to (a) give the precise definition of I that you are using and (b) give the dynamical model that determines when these quantities change.

P.S. What happened to the “reply” button? Did you turn it off?

D R, Philo, and Mitch, You are all raising related issues that have to do with my attempt to say that there is no identity (of any kind) between savings and investment (either planned or realized). The equality of savings and investment, I assert, is a property of an equilibrium in the Keynesian income-expenditure model. In that model, properly understood, there is no identity between savings and investment. In the national income accounts the terms are defined so that measured savings must equal measured investment, because an explicit fudge factor is included to ensure that the two magnitudes will always turn out to be equal. But that is not the implication of the income-expenditure model, it is the result of the particular accounting definition chosen by the creators of the national income accounts.

D R, I am not sure what kind of example you are asking for. If you are asking for some kind of historical example, I would say that if you leave out the fudge factor in the national income accounts, measured savings probably never turns out to be equal to investment, so you simply have to look up the national income accounts and check to see that the fudge factor has never turned out to be zero. If you are referring to an example in terms of the income expenditure model, then you have to understand that in the simple form in which the model is normally taught, the model is solved for an equilibrium, and the condition for equilibrium is that savings equal investment. So to get the model to give you a case in which savings is not equal to investment, you have to set up the model in some dynamic form with a lag of at least one period between income and expenditure. If there is no lag in the model, you are positing that the model is always in equilibrium and that adjustments to changes occur instantaneously. In that case, the equality between savings and investment is implied by the assumption that the economy is always in a state of (macroeconomic) equilibrium. A simple way to introduce a lag is to say that expenditure in period n depends on income in period n-1. An increase in expenditure thus increases income in the following period, thereby further increasing consumption, and causing a further increase in income in the following period, and so on. During the entire adjustment process expenditure is greater than income in each period because of the lag which implies that investment is greater than saving. We are talking about planned investment and savings but there is no reason to assume that realized investment and saving are not equal to planned investment and saving.

Philo, I think that the above reply to D R addresses your concerns as well.

Mitch, Thanks for the link. It repeats exactly what is wrong with the way the income-expenditure model is presented in most textbooks, which is incoherent. The key point is that it illegitimately asserts or tacitly assumes that expenditure (E) and income (Y) are identical. They are not. Equality of income and expenditure is an equilibrium condition, not an identity. Treating them as identical makes the model incoherent, just as assuming that forces are identical would make a mechanical model incoherent by making it impossible to specify an equilibrium in terms of the equality of opposing (not identical) forces.

Here is the “proof” given by your link:

“Savings is just what people earn minus what they spend and what they pay in taxes. Lets call Savings (S), “what they earn” Income (Y), “what they spend” Consumption (C) and “what they pay in taxes” Taxes (T).”

Let’s ignore taxes and government spending. Note that savings is the part of income not spent. But we are not told what constitutes income, we are just told that we will designate income by the symbol “Y.”

“So now:

S = Y – C – T (Equation 1)

Looking at the economy as a whole, the income of a nation (Y) is either spent by people (C), spent by government (G) or spent by businesses as investment (I).”

Here is where income (Y) is illegitimately identified with expenditure (which I designate as “E”). Reference is made to what is spent out of income as if what is spent, i.e., C + I + G, were the same as what is earned. But spending out of income is clearly not the same what is earned. Income consists in the payments made by business firms to factors of production (workers, owners of productive resources and capital), in other words earnings. Those payments are made in a different set of transactions from the set of transactions in which households spend on consumption and businesses spend on investment. Unlike the purchases and sales of goods in a single market, which are two sides of the same transaction, payments by businesses to households in return for productive services do not directly correspond to (are not carried out simultaneously with) the purchases by households of products sold by business firms. There is no identity.

“Now:

Y = C + G + I (Equation 2)”

No! No! No!

The correct definition is

E = C + G + I (Equation 2)

“If we assume that the government doesn’t spend more or less than it taxes, then G = T, or:

G – T = 0 (Equation 3)

Substituting the right side of Equation 2 into Equation 1, we have:

S = Y – C – T = (C + G + I) – C – T = I + (G – T) (Equation 4)”

No! No! No!

Since E is not identical to Y, you can’t get to equation 4 without first writing explicitly

E = Y (Equation 4) not an identity, but an equilibrium condition

Only after introducing equation 4 do we get what used to be Equation 4, but is now Equation 5

S = Y – C – T = (C + G + I) – C – T = I + (G – T) (Equation 5)

Without the extra equation, the model literally makes no sense because you can’t solve for Y if E is identical to Y.

David,

It seems to me that you have your definitions wrong. The only sense in which expenditures on current production does not equal current production is the change in inventories.

Any “spending” outside of E must constitute a swap of assets, and not add to current income.

D R, I am not sure that I follow what you are saying, I said nothing about current production, which indeed is a third category and is not identical to either income or expenditure. I agree that the difference between expenditure on final output and current production corresponds to changes in inventories. I was distinguishing between income (factor payments by businesses to households) and expenditure (payments by households to businesses and by businesses to businesses for finished products). Income and expenditure as I have defined them are clearly not identical. Expenditure on current output can easily differ from current factor payments because there is a lag between expenditure on current output and the distribution of receipts by business to the factors of production that made the output. If there are no lags, then there is no disequilibirum and the equilibrium condition E = Y is always satisfied. But if you allow for the possibility of lags and the possibility of disequilibrium, then E cannot be continuously equal to Y.

David, I don’t agree. Maybe we could make progress if you gave me an example of actual saving that DOES result in an increase in aggregate saving, but doesn’t result in more investment. Where does this actual saving go?

BTW, I agree with DR that attempts to save may just reduce income. But of course in that case actual saving hasn’t changed, and thus S=I still holds.

Again, economists define saving as the resources that go into investment. I’ll take your word for it that Lipsey uses a different definition, but he certainly doesn’t “prove” I am wrong, because it’s impossible to disprove an identity. At best you can disagree with how a person has defined some terms. But my definition is the conventional definition, so that sort of attempt to refute my argument won’t get you very far.

You seem to be saying “Sumner is right if one uses his definition of S and I. And Sumner does use the conventional definition. But I prefer to define the terms differently.”

“I was distinguishing between income (factor payments by businesses to households) and expenditure (payments by households to businesses and by businesses to businesses for finished products).”

Aren’t businesses owned by households? If someone gives me money in exchange for a good, does it matter for aggregate income whether I intend to later transfer that money to someone else? If expenditure is giving pieces of green paper to others, and income is getting pieces of green paper from others, and the only way to get green pieces of paper is to have someone give them to you, how could an expenditure not produce an exactly equivalent amount of income?

Also, what role do cash balances play in all this? Maybe I’m just confused about definitions, but if savings is the portion of income not consumed, what happens if everyone decides to hold all their savings as mattress cash? Or maybe I’m flow-stock confused…

You begin to make yourself clearer. By my reckoning, you have answered one of the two questions I have posed to you in my comment.

Now we won’t set E = Y, and therefore

S = Y – C – T [definition of Y]

= Y – E + E – C – T [add and subtract E]

= (-R) + (C + G + I) – C – T [definition of R and E]

= I + (G – T) – R

or

S + R + (T – G) = I

where we have defined “retained earnings” R = E – Y as the difference between the amount corporations receive and what they pay out to consumers. We still have “savings” equals “investment”, but here “savings” includes the savings done by corporations (and, actually, by the government), rather than just by people. You are correct, and this is unsurprising, it seems to me.

I would still reiterate that this has nothing to do with “equilibrium” Companies can go on retaining earnings forever, just as people go on saving forever, if we’re willing to let the stockpile of retained earnings grow (and we are). Likewise the government can always be borrowing or saving if the economy is growing.

As I see it, you have never properly defined what you mean when you use the word “equilibrium”. You used it above to assert that R = 0 holds in equilibrium, but I see no sense in which that is meaningful.

The discussion in various posts has failed to distinguish between (a) a solution to some model (I think this is the sense in which you say R=0 is “equilibrium”) (b) what physicists call a “steady state”, which designates a situation in which flows are unchanging over time, and (c) true “equilibrium”, which is a situation in which the *levels* are unchanging. (To use an example from physics, a car going a constant 60 mph on a highway is a steady state, not an equilibrium.)

It’s true that businesses do not get to count the profit from sales of their excess production, but any increase in inventories still must count as income to the business. (Purchases out of inventories are actually just asset swaps.)

If Ip is investment net of inventory changes and P is current production, then

P = C+I+G

E = C+Ip+G, and

Y = E + (I-Ip) = (C+Ip+G) + (I-Ip) = C+I+G = P

National income is still equal to national production.

Scott, I reproduce (with slight notational changes) the following example from p. 11 of Lipsey’s paper “The Foundations of the Theory of National Income.”

Let C(t) = aY(t-1)

I(t) = I*

Y(t) = C(t) + I(t)

The equilibrium condition is that Y(t) = Y(t -1) = Y*. In this formulation E(t) = Y(t)

Assume that in period 0 a changes from .9 to .8 and that I*= 100. Then we have the following period by period sequence of consumption, saving, and income.

Period Y(t-1) C(t)=aY(t-1) I* Y(t)

-2 1000 900 100 1000

-1 1000 900 100 1000

0 1000 800 100 900

+1 900 720 100 820

+2 820 656 100 756

+3 756 606 100 706

… … … … …

∞ 500 400 100 500

There are two possible definitions of savings, one is:

S(t) = Y(t-1) – C(t)

The other is:

σ(t) = Y(t) – C(t).

According to the first definition, which could be called planned savings, savings in period 0 is 200, and gradually declines to 100 when a new equilibrium is reached. According to the second definition, which could be called realized savings, savings in each period (month) is 100. What accounts for the difference between planned and realized savings? At the start of each month (or the end of the previous month after all other transactions were executed), households receive a check for their earnings in the previous month, 1000, which is sufficient to finance their consumption expenditures and their savings (purchases of bonds) in the upcoming month. Up to period 0, they purchased 100 in bonds at the beginning of each month and spent the remaining 900 in the course of the month, ending up with a zero cash balance on the last day of the month. In period zero, they try and succeed to purchase 200 in bonds, reducing consumption spending to 800. As a result, earnings in period 0 fall to 800. Households planned to add 200 to their accumulated bond holdings, and succeeded in doing so. The reduction in realized savings reflects the reduction in the cash payment received at the end of the month when they receive 900 instead of the 1000 they had received in previous months. But they are also planning to reduce consumption again in the upcoming month.

You said:

“BTW, I agree with DR that attempts to save may just reduce income. But of course in that case actual saving hasn’t changed, and thus S=I still holds.”

Whether actual saving has changed depends on how you define saving. More than one definition is possible. My point is that you seemed to be trying to derive an empirical implication about the real world from a definition, a definition simply being a convention about the meaning of a word.

“Again, economists define saving as the resources that go into investment.” Whoa, where did that come from? In a simple Fisherian two period barter model, there is no distinction between saving and investment. Once you leave that model behind, there is a huge distinction between saving which is undertaken by a different set of people from the ones who are investing. Showing how savings are transformed into investment in a money economy is a theoretical problem, which you seem to be dismissing with the wave of your hand. If you follow that approach, you will wind up doing macro using a representative agent model. I suggest that you reconsider. We are talking about the meaning of savings in the income-expenditure model. The model has no necessary identity between savings and investment. Savings may be necessary for investment, just as eggs are necessary for chickens. That doesn’t establish an identity between savings and investment any more than there is an identity between eggs and chickens.

“At best you can disagree with how a person has defined some terms. But my definition is the conventional definition, so that sort of attempt to refute my argument won’t get you very far.” If you want to define savings as identical to investment, you are entitled to do so; what we are arguing about is what implications, if any, follow from that definition. I am claiming that an identity has no empirical implications. Do you disagree?

alexanderamon, Obviously there is a circular flow in which spending is transformed into income and income is transformed into expenditure. That would be true if there were no leakages from the circular flow at any point and no injections to the flow at any point. But what if, all of a sudden, people decide that they want to spend more and take some of their accumulated cash and start to spend it? At the instant of the first purchase expenditure has risen. Has income also risen at that instant even though factor payments are fixed an nobody knows what profits will be for the entire period? It seems to me obvious that there has to be an adjustment process that brings income and expenditure back into balance. Positing that income and expenditure are identical makes it logically impossible to say anything about the adjustment process. If you do that, all you can say is that it just happens.

Cash balances are lurking in the background, but are not made explicit in the simplest Keynesian income-expenditure model, sometimes referred to as the Keynesian cross, which, for the most part, is what we have been talking about in this thread.

Mitch, Glad that you have detected signs of progress. I don’t think that I ever suggested that properly interpreting the income-expenditure model would lead to any surprising substantive results. It mainly leads to avoiding certain incoherent statements about the identity of savings and investment like the necessity for unplanned inventory adjustment to take place when the model is in disequilibrium. You seem to have a problem with the notion of equilibrium in this model. I don’t see why, and I don’t see why your problem, whatever it is, has anything to do with my point that neither expenditure and income nor savings and investment are identical. That said, equilibrium in the income-expenditure model means that barring some change in the parameters of the model, consumption, investment, income and savings will not change. That is a property of the model, it does not mean that the model is an accurate or even an acceptable representation of the real world.

D R You said:

“It’s true that businesses do not get to count the profit from sales of their excess production, but any increase in inventories still must count as income to the business. (Purchases out of inventories are actually just asset swaps.)”

The income-expenditure model on its face is talking about income and expenditure, i.e., payments of money, and you are telling me not focus on money payments because of an accounting convention. Production to inventory does not necessarily give rise to any income payment. The payment of income will take place after the inventory is sold off. You are saying that the only possible interpretation of the model is one that conforms to a particular accounting convention. Who made the accountants the ultimate arbiters of the interpretation of a model? Now I am not saying that you should necessarily avoid the accounting convention. The problem arises when you try to interpret the model and draw an empirical implication from the model based on an accounting convention. The typical implication is that in disequiibrium, there has to be an economic mechanism of some kind (e.g., unplanned inventory investment takes place in disequilibrium) operating to ensure that the accounting convention is satisfied at all times. That is just nonsense.

David,

Income rises at the time the asset is produced, because that is when wealth is increased. That is, income is a matter of production, not money.

Nominal income is production valued at market prices– not the dollars exchanged in the market. That means if I produce something, income rises even if I don’t trade it for money.

It’s a different question how nominal income is distributed (profits, wages, etc.) We can try to figure out, say, based on current market prices, how much of current production a worker can afford as a result of compensation. But to the extent that the worker does not get compensated, the business retains the income.

In short, GDP breaks down production by who takes possession of the current production, and GDI breaks down production by who profits from the current production.

@Scott: I know you claim that your definition of savings as identical to investment is standard convention, and is the same definition that economists like Krugman and Wren-Lewis use, but posts like this contradict that assertion:

http://krugman.blogs.nytimes.com/2012/02/06/zero-bounds-and-butter-mountains-wonkish/

Why is Krugman drawing a Keynsian cross between savings and investment if he believes they are identical always (and not just in equilibrium)?

JSR,

Because there is a difference between savings and investment schedules (savings and investment desired at each rate of interest) and quantities saved and invested.

Say I decide to save more at the current rate of interest, so I consume fewer goods, *simultaneously* increasing actual investment in the form of higher inventories. So now there is actual investment in excess of the amount desired. The disequilibrium is between planned and unplanned investment, not between actual savings and actual investment.

I’m pretty sure that’s why he keeps using the word “incipient” to describe the excess supply of savings– because it can’t actually happen as such. Savings will always equal investment, but that doesn’t mean that desired savings must always equal desired investment.

David, You said;

“At best you can disagree with how a person has defined some terms. But my definition is the conventional definition, so that sort of attempt to refute my argument won’t get you very far.” If you want to define savings as identical to investment, you are entitled to do so; what we are arguing about is what implications, if any, follow from that definition. I am claiming that an identity has no empirical implications. Do you disagree?”

I think this gets to the gist of our dispute. I only claim one implication for the S=I identity. I will explain that implication below:

If someone assumes S=I as an identity, as Wren-Lewis explicitly does, then any example they provide must be consistent with that identity. They cannot argue that saving falls and investment doesn’t fall. Wren-Lewis didn’t explicitly argue that investment doesn’t fall, but he assumed the change in Y was identical to the change in C+G, which implies investment didn’t change in any model where S=I is an identity.

So I’m not claiming that S=I tells us anything about how these variables change in the real world, except that any change in S should equal the change in I, at least if you are one of those economists who believes S=I is an identity. I understand that you aren’t one of those economists. But Wren-Lewis is. So using his definitions, his argument against Cochrane was flawed. I agree that if you define S and I differently, then they don’t have to be equal. But in that case his argument against Cochrane would have been totally incoherent. Cochrane could have said “So you showed G rose more than C fell, why should I care?”

All I am saying is that if one defines S=I, by definition, then if one argues S changes, once must also argue that I changes. Do you disagree? I am saying nothing more. That’s my only “empirical claim.”

JSR, Those schedules do not represent actual S&I, they represent desired S and desired I at various interest rates or income levels. And Wren-Lewis explicitly said S=I is an identity, so you are also wrong about him.

DR, Finally we agree about something!

Scott,

I think we agree on quite a lot. For example:

“Wren-Lewis didn’t explicitly argue that investment doesn’t fall, but he assumed the change in Y was identical to the change in C+G, which implies investment didn’t change in any model where S=I is an identity.”

… is something we agree on. I am trying to understand why you fail to follow through on the implications– specifically, that if I does not change and there is consumption smoothing, then C must not fall, resulting in a BBM of 1.0.

Why do you not argue, “Wren-Lewis said consumption fell, but it did not!”

…. I mean, we agree that it applies to any *closed economy* model, but I took it for granted that’s what you meant.

DR, I think we identify WLs mistake in different places. You assume what he “really meant” is different from what I assume he really meant. I believe he really did mean to assume C fell in the example refuting Cochrane, and he messed up the rest of the analysis.

Scott,

Why do you call it an assumption that C fell? It seems clear that “spending at time t will fall by much less than X” follows from the assumption of consumption smoothing.

Maybe you could help educate me on an important piece of this argument — when Wren-Lewis referred to “consumption smoothing”, I thought he meant by that: “If taxes increase by X in period 1, consumers will smooth the consumption decrease over time, i.e. they wilil decrease their consumption by X/n for n periods”.

Am I mistaken in my understanding of consumption smoothing?

A quick question here — I’ve been taking consumption smoothing to mean “If taxes go up by X in period 1, consumers will smooth their spending reduction over time, i.e. C will not decrease by X in period 1, it will decrease by X/n for n periods”.

Am I mistaken in my understanding of what consumption smoothing is?

D R You said:

“Income rises at the time the asset is produced, because that is when wealth is increased. That is, income is a matter of production, not money.”

You are free to define income in that way if you want to, but it is not how income is defined in the income-expenditure model. In the income-expenditure model, there are flows of income paid by businesses to households in exchange for productive services (Y) and flows of expenditure from households to businesses in exchange for real goods and services consumed by households and expenditure by businesses to businesses for final investment goods (E). There is also a third flow (usually left implicit) of output valued at market prices (Q). The flows need only be equal in equilibrium. Households base their consumption decisions on the payments they receive from businesses not on unrealized capital gains.

“Nominal income is production valued at market prices– not the dollars exchanged in the market. That means if I produce something, income rises even if I don’t trade it for money.”

Again, a perfectly respectable definition, but you are imposing it on the income-expenditure model, and the model does not hang together logically if you insist that income is identical to expenditure. If you are willing to restrict the model to equilibrium states, you can say that income always equals expenditure in equilibrium, but you cannot infer anything about the adjustment process that brings about equilibrium or that moves the model from one equilibrium to another in response to a parameter change.

“It’s a different question how nominal income is distributed (profits, wages, etc.) We can try to figure out, say, based on current market prices, how much of current production a worker can afford as a result of compensation. But to the extent that the worker does not get compensated, the business retains the income.”

P.H. Wicksteed famously (well maybe not anymore but it was still famous when I was a student) proved a little over a hundred years ago that paying factors their marginal product under a constant returns production technology would exhaust the product. In a perfectly competitive equilibrium, the production technology is locally equivalent to constant returns to scale, so Wicksteed’s proof also works for a perfectly competitive equilibrium. According to you, Wicksteed’s proof was a pointless exercise, because income is product and the two are identically equal. Obviously P.H. Wicksteed believed that it was meaningful to distinguish between income and output. On this point (and many others), I am happy to stand with Wicksteed, a truly admirable person and scholar.

JSR and D R, This is where it gets really confusing. The distinction between planned investment and unplanned investment (including unplanned inventory accumulation or decumulation) is an attempt to get around the impossibility of solving the income expenditure model if expenditure is treated as identical to income. So a new distinction has to be introduced between planned and unplanned investment and between planned and unplanned expenditure. But that distinction is itself based on an empirical assumption about the existence of inventories and an assumption that inventories are allowed to fluctuate. There is nothing in the logic of the model that requires disequilibrium to be associated with unplanned changes in inventories. That is only one of several possible lags, and not necessarily the most plausible, that can account for a disequilibrium. You just have to accept that income and expenditure are being defined in a way that allows them to be different rather than posit as a dogma that income and expenditure are metaphysically identical.

Scott, I think that in my exercises in calculating what happens when you work through a two-period income-expenditure model with an underemployment equilibrium in the first period and then you recalculate the equilibrium on the assumption that G and T go up in the first period, you can show that consumption-smoothing/Ricardian equivalence leads to an increased Y in the first period, compared to the equilibrium without the increase in G and T. The increase in Y is less than the increase in G, because C falls but does not fall by as much taxes increased. C goes down, but it does not go down by as much as it would have without consumption smoothing. I does not change and S does does not change compared to the no increase in G and T equilibrium. I think that this result is consistent with Wren-Lewis’s argument? What is your problem with it?

About Wren-Lewis’s agreement that I is identical to S, I believe that it reflects the inconsistency in the conventional treatment of I and S in the textbooks, and it illustrates the logical problems caused by that unnecessary and incoherent complication in an otherwise straightforward model (which doesn’t mean that I endorse it as an analytical tool).

Scott and DR, Isn’t there an ambiguity in saying that Wren-Lewis says that consumption fell. It depends on what you take as a baseline, the initial equilibrium with no increase in G and T, or the equilibrium with increased G and T, but no consumption smoothing.

David,

Are you arguing that businesses do not record inventory increases as assets?

As for Wren-Lewis, the answer is yes. Furthermore Wren-Lewis left it ambiguous as to whether consumption fell with respect to *any* particular baseline– zero is less than any positive number.

In fact, with respect to the former baseline, consumption cannot fall. Assuming there is slack in the economy to handle the increase in G, then the change in C must equal the change in Yd=Y-T=Y-G=C. Consumption smoothing implies that the change in both C and Yd must be zero.

But if you decide to *add* investment into Wren-Lewis’ discussion, then there isn’t necessarily anything pinning down consumption. One requires details about *how* the smoothing works– that is, how agents smooth consumption and not merely that they do so. Otherwise, I can say consumption *rises* and investment along with it. (Say, consumption rises by 100 times the increase in G, and investment rises by 400 times the increase in G, so that Yd rises by 500 times the rise in G– a BBM of 501.0) So one might say that Wren-Lewis should have spelled out the how… but it’s *irrelevant* because Wren-Lewis’ example doesn’t have investment any more than it has foreign trade.

D R, What i am asserting is that how businesses enter increases in inventory in their books is irrelevant to the logic of the income-expenditure model. What matters is when they make payments to owners of factors of production. There is no reason to believe that adding production to inventory necessitates a contemporaneous payment to any factor of production.

What do you mean? It’s not necessary to pay workers for their production? That doesn’t matter to aggregate income– it just means higher profits and lower wages.

I mean that the revenue with which they finance the payments to factors accrues at the time of sale not at the time of production.

What difference does it make? That’s a question of finance, not income. Factory workers don’t sit around waiting for cars to sell so they can buy groceries. Their wages accrued as they worked the line, even if they don’t receive payment until week’s end.

If there is insufficient revenue at the end of the week, then the employer borrows. So what?

In any given quarter, workers are paid for their labor whether their product sells or just sits in a warehouse. If they produce more widgets than get sold, then the employer has less cash/more debt and more widgets than if they had all sold.

In fact, you can pay me three widgets for every widget I produce, and if I produce ten widgets, then my income is 30 widgets and your income is -20 widgets.

Similarly, if you pay me one widget for every two I produce, and if I produce ten widgets, then my income is five widgets and your income is five widgets.

The number of widgets you sell and when you sell them is irrelevant, except insofar as you have to make good on what you owe me.

That is just my point. Payments to factors of production in a given period don’t have to match either the receipts of business in the period or the value of output in the period. There are three distinct, but closely related flows, payments to factors of production, the value of output and expenditures by households and businesses on final output. Over time the value of the three flows tends toward an equilibrium in which they match each other and are stable in the absence of a parameter change, but in response to a parameter change, a new equilibrium must be reached and during the adjustment process from the old to the new equilibrium the flows cannot all match each other, otherwise there could be no adjustment process.

There is nothing to adjust. The employer and employee agree to terms how the employee will be compensated for production. The employee produces. The employee gets some share of production, and the employer gets some share of production. There can be no disequilibrium, because all shares are necessarily accounted for.

Sales just don’t enter into national income except to see who is demanding the widgets and to determine their market value.

Do you see? If you pay me $5/hour to make widgets for you, it doesn’t matter how many I produce in any period of time. My income is $5 per hour worked, and yours is however many widgets I made, less $5 per hour worked.

D R, You said:

“There is nothing to adjust. The employer and employee agree to terms how the employee will be compensated for production. The employee produces. The employee gets some share of production, and the employer gets some share of production. There can be no disequilibrium, because all shares are necessarily accounted for.”

I don’t think that you are keeping track of the definitions. Income is defined as payments to factors of production paid by business firms. I assume that, by employer, you mean a business firm. A business firm employs factors of production to produce output, and uses the receipts that it receives from consumers to make payments to the factors of production that it employs. But a business firm is not itself a factor of production. So when you say that the employee get some share of production and the employer get some share of production, that is not necessarily correct. The business firm has to distribute its receipts to other factors of production or to its owners. Those distributions take place over time and do not necessarily coincide with the production of output by the firm. Your identification of the receipts of the business firm with factor income is therefore not logically implied by the definitions of the income expenditure model or any other economic axiom or principle or theorem. You might say that everything that belongs to the business firm belongs to its owners, i.e., households, so that it is appropriate to impute any undistributed revenue accruing to firm to the owners. That might or might not be true, but that is a theoretical behavioral assertion about the nature of the legal relationship that obtains between businesses — sole propietorships, partnerships, corporations, etc. It is not an identity. Business firms are not identical to their owners. An owner of Apple stock does not automatically take into account each sale of an ipad in making decisions about how much to consume in any given period. The simple income-expenditure model is based on a naïve assumption that consumption decisions by households depend on the current flow of factor payments paid to them by business firms. The current flow of factor payments accruing to households is a flow that can be measured. The current rate of production is another flow that can be measured. And the current rate of expenditure on final output is a third flow that can be measured. There is no identity, based on the definitions of the income-expenditure model guaranteeing that those measured flows are always equal to each other.

“Sales just don’t enter into national income except to see who is demanding the widgets and to determine their market value.”

I agree. Sales measures expenditure on final output C + I, or equivalently, receipts by business firms for final output sold. Income measures payments to factors of production. They are not the same!

“If you pay me $5/hour to make widgets for you, it doesn’t matter how many I produce in any period of time. My income is $5 per hour worked, and yours is however many widgets I made, less $5 per hour worked.”

Who am I? In the world of the income-expenditure model, if I am employing you, then I am a business firm. The residual left over to me after I pay you your monthly wage also has to be transferred from me as a business firm to my other self as a household. You can posit that such transfers are instantaneous and automatic, but that is an empirical proposition, not a logically entailed implication of the definitions that we are working with. For certain kinds of employment contracts and business practices, the assumption seems reasonable; for others, especially the currently dominant corporate business organization, it doesn’t seem at all reasonable, to me at any rate. If that is the assumption you want to make, fine. But then you are making an assumption about how the world works not working with an identity.

“The simple income-expenditure model is based on a naïve assumption that consumption decisions by households depend on the current flow of factor payments paid to them by business firms. The current flow of factor payments accruing to households is a flow that can be measured.”

To the extent that such payments are out of balance with production, there is a corresponding imbalance going the other way. If you fail to pay me my wage this quarter, then yes, *my* income is lower this quarter, but *your* income is higher this quarter by exactly the same amount.

“You can posit that such transfers are instantaneous and automatic, but that is an empirical proposition, not a logically entailed implication of the definitions that we are working with.”

I don’t care if the transfers are instantaneous, laggy, or never take place at all. Your income this week is what I produce this week, less what you pay me this week. It doesn’t matter if what you pay me this week is for what I produced this week, last week, in the Stone Age, or next year. It doesn’t even matter if I never have or will produce anything. Between us, our income rises by what I produce. It doesn’t matter one bit if your business self pays your household self. That would only transform business income into household income, but do nothing for the aggregate.

D R, I said:

“The simple income-expenditure model is based on a naïve assumption that consumption decisions by households depend on the current flow of factor payments paid to them by business firms. The current flow of factor payments accruing to households is a flow that can be measured.”

You replied:

“To the extent that such payments are out of balance with production, there is a corresponding imbalance going the other way. If you fail to pay me my wage this quarter, then yes, *my* income is lower this quarter, but *your* income is higher this quarter by exactly the same amount.”

Again, you are not identifying who I am. But if I am a producer and an employer, then I must be a business firm. If I am business firm, then the revenue I receive is not income, but receipts, which can be either retained within the firm or distributed to factors of production including the owners of the firm. You are insisting that income to households in a given period must equal the value of output, and that is only true if you insist that all the output produced by a business firm must somehow be transferred to the factors of production in the same period. There is no physical mechanism of transfer, but you are asserting that this transfer must take place by definition. You are free to insist on such a definition, but it does not accord with the definitions of the model or standard economics in which, as I pointed out earlier, the exhaustion of the value of the product in competitive equilibrium is regarded as a theorem to be proved rather than a definition. Remember that In the simple consumption function of the income-expenditure model the amount consumed in a given period is a treated as a function of (Y) which corresponds to the income of households. It is not a function of the receipts or the output of business firms.

I said:

“You can posit that such transfers [i.e., from firms to households] are instantaneous and automatic, but that is an empirical proposition, not a logically entailed implication of the definitions that we are working with.”

You replied:

“I don’t care if the transfers are instantaneous, laggy, or never take place at all. Your income this week is what I produce this week, less what you pay me this week. It doesn’t matter if what you pay me this week is for what I produced this week, last week, in the Stone Age, or next year. It doesn’t even matter if I never have or will produce anything. Between us, our income rises by what I produce. It doesn’t matter one bit if your business self pays your household self. That would only transform business income into household income, but do nothing for the aggregate.”

Again, I am not sure who I am. Income is paid by business firms to households, you can define income in a different way if you want to, but if you do, you are deviating from standard usage. My income is what I receive as payment for my productive services or for the services of factors of production that I own. You are just positing an alternative definition in which the income of one factor of production is the residual left over from the value of output after all the other factors have been paid. So you start with an a priori identity between output and income and you ensure that the definition holds by defining the income of one factor not as payments for productive services but as the residual of output after all other factors have been paid. In the income expenditure model, income is defined in terms of payments to factors and output is defined as the value of output. Those two independent definitions don’t have to yield identical results. You are asserting that two things are identical, but the identity is the result of how you choose to define your terms.

Perhaps to take this discussion, now seemingly in an infinite loop, in a different direction, let me ask you, what, in you view, characterizes a disequilibrium state in the income-expenditure model?

“If I am business firm, then the revenue I receive is not income, but receipts, which can be either retained within the firm or distributed to factors of production including the owners of the firm.”

Who is talking about revenue receipts? You are. A business may sell assets for money which may also be retained or distributed, but this has no bearing on aggregate income.

“You are insisting that income to households in a given period must equal the value of output”

No, I am not. Please do not misrepresent my position. I am insisting that aggregate income in a given period must equal aggregate output.

“Remember that In the simple consumption function of the income-expenditure model the amount consumed in a given period is a treated as a function of (Y) which corresponds to the income of households. It is not a function of the receipts or the output of business firms.”

That is true only if you assume from the outset that there is no business income, no government income, and no foreign-sector income. A more typical model is that C = C(Y-T) and make a simplifying assumption that all income is distributed either to households or to government. But that is a question of how income is distributed, and doesn’t change the fact that aggregate income must equal aggregate output.

“My income is what I receive as payment for my productive services or for the services of factors of production that I own.”

Note that you are talking about *your* income. It doesn’t matter who *you* are. Whatever production is not distributed to *you* must be retained or distributed to someone else. This is a question of how income is distributed, and does not change the fact that aggregate income must equal aggregate output.

“…the residual of output after all other factors have been paid.”

Yes. That is correct. In the end, there can be no residual, because someone is producing something of value and the production must be owned/enjoyed by someone. Payments are just transfers of wealth and do not increase aggregate income; purchases are just asset swaps and do not increase aggregate income.

The GDP/GDI “statistical discrepancy” is not that kind of residual. It is a measurement problem. It’s like taking a ruler which has 1/16th inch marks on one side and 1mm marks on the other. I can maybe measure something to 1/32nd of an inch and then to 0.5mm. No matter how hard I try, I’m probably not going to get exactly equivalent results. That doesn’t mean that the something I’m measuring is not of consistent length.

“Perhaps to take this discussion, now seemingly in an infinite loop, in a different direction, let me ask you, what, in you view, characterizes a disequilibrium state in the income-expenditure model?”

What do you mean? Can income and expenditures be out of balance? Sure. There is nothing keeping producers from continuing to produce a product which is not selling; so long as there are inventories to be sold, there is nothing keeping from consumers from buying a product which is discontinued.

D R, You said:

“Who is talking about revenue receipts? You are. A business may sell assets for money which may also be retained or distributed, but this has no bearing on aggregate income.”

Actually, I am only talking about receipts received for final output, which though not the same as final output, is closely related to final output.

I said:

“You are insisting that income to households in a given period must equal the value of output.”

You replied:

“No, I am not. Please do not misrepresent my position. I am insisting that aggregate income in a given period must equal aggregate output.”

I am sorry if I misunderstood and misstated what you were saying. I hope that you were not suggesting that I did so deliberately.

I said:

“In the simple consumption function of the income-expenditure model the amount consumed in a given period is a treated as a function of (Y) which corresponds to the income of households. It is not a function of the receipts or the output of business firms.”

You replied:

“That is true only if you assume from the outset that there is no business income, no government income, and no foreign-sector income. A more typical model is that C = C(Y-T) and make a simplifying assumption that all income is distributed either to households or to government. But that is a question of how income is distributed, and doesn’t change the fact that aggregate income must equal aggregate output.”

I don’t agree. Y accrues to households. If you subtract T from Y you get the disposable income. There is no “income” to the government apart from T which is paid to the government out of household income Y. Similarly, there is nothing in the definitions corresponding to business income. There is just the receipts of businesses which corresponds to C + I and perhaps some portion of G.

I said:

“My income is what I receive as payment for my productive services or for the services of factors of production that I own.”

You replied:

“Note that you are talking about *your* income. It doesn’t matter who *you* are. Whatever production is not distributed to *you* must be retained or distributed to someone else. This is a question of how income is distributed, and does not change the fact that aggregate income must equal aggregate output.”

Again you are defining income as another word for the value of output. You are giving two names to the same magnitude. I am saying that income represents payments to factors of production for their services. I believe that my usage of the term “income” corresponds better both to its usage in every-day language and in economic theory, and for sure in the income-expenditure model. It is most certainly not a fact “that aggregate income must equal aggregate output.” That is how you choose to define “income.” Now I agree, and I have said before, that in the national income accounts total output is treated as identical with total income, which means that aggregate income in the end must equal aggregate output. But this does not follow from the income-expenditure model, it is a convention imposed by the national income accounts,

You said:

“The GDP/GDI “statistical discrepancy” is . . . a measurement problem. It’s like taking a ruler which has 1/16th inch marks on one side and 1mm marks on the other. I can maybe measure something to 1/32nd of an inch and then to 0.5mm. No matter how hard I try, I’m probably not going to get exactly equivalent results. That doesn’t mean that the something I’m measuring is not of consistent length.”

Again, to try to explain what I am saying. It is possible to conceive of income as being something different from output. If so, two different lengths are being measured, not one. The national income accounts start with the premise that the two are the same and therefore it must turn out that they are defined in such a way that they will be the same, except for the “statistical discrepancy.” But the identity between the two lengths is the result of an accounting convention, not a fact of nature or an implication of economic theory.

I said:

“Perhaps to take this discussion, now seemingly in an infinite loop, in a different direction, let me ask you, what, in you view, characterizes a disequilibrium state in the income-expenditure model?”

You replied:

“What do you mean? Can income and expenditures be out of balance? Sure. There is nothing keeping producers from continuing to produce a product which is not selling; so long as there are inventories to be sold, there is nothing keeping from consumers from buying a product which is discontinued.”

What I meant was something else: can you give me a simple (numerical) example of a disequilibrium and provide an economic interpretation of what is going on and how equilibrium would be achieved?

David,

I’m going to cut though a lot of this, because there are serious issues, and side issues, and I don’t want the important ones lost. If you want to come back later to anything I don’t address, I’ll be happy to respond.

First, Y does not accrue to households alone. That is flat-out wrong. That can be an assumption you make, but it is plainly untrue in practice. Businesses do retain earnings.

You are entirely wrong about investment.

If a business produces nothing in a period, but sells one consumption widget out of inventory, it has traded one asset for another and has no net income. This transaction increases C but lowers I by exactly the same amount, because inventory changes are part of investment. So the business produced nothing, nothing was added to GDP, nothing was added to GDI, and yet expenditures rose.

It’s possible to “conceive of income as being something different from output”– that is true. If I only count income as pennies I find in the street, them GDP will be much much larger than income. But GDP must still equal GDI.

*Of course* it’s an accounting convention. The whole point of breaking down GDP into C G and I is to see who winds up with the stuff produced. The whole point of breaking down GDI into compensation, net taxes, net operating surplus, and consumption of fixed capital is to see who profits from producing exactly the same stuff.

If GDP failed to equal GDI, then we’d be screwing up the accounting.

Now, if you don’t think that consumers consume c+mpc(GDP-Taxes) then that’s totally ok. I mean, I don’t. Disposable income is not actually equal to Y-T, and consumers do not consume c+mpc(Disposable Income) anyway– though it’s a pretty decent approximation. But however you write up your closed-economy model, aggregate income had better equal aggregate production because any *other* income you see within the economy must be offset by some corresponding loss.

A numerical example, you ask? OK…

DR, Inc.– the sole producer of anything in the USDR– having no pre-existing inventory, but all 100 DR-bucks in the economy, produces 100 widgets in a given period. DR, Inc. pays its employees DR$80. DR, Inc. supplies all 100 widgets to market at a price of DR$1 apiece. The employees desire to purchase 90 of these widgets at such a price, but having only DR$80, borrow DR$10 from DR, Inc. (at what I am sure is a reasonable 10 percent rate of interest all due with principal next period)

However, DR, Inc. now has DR$100 (the original DR$100, less DR$80 in wages, less DR$10 in loans, plus DR$90 in sales receipts) plus DR, Inc. now has 10 widgets, and an IOU for DR$10– a net income of DR$20.

The employees of DR, Inc now have DR$0 (they started with nothing, plus DR$80 in wages, plus DR$10 in loans, minus DR$90 in expenditures) plus the employees have 90 widgets and a debt of DR$10– a net income of DR$80.

Between DR, Inc. and the employees of DR, Inc., net income is DR$100– exactly the market value of all widgets produced.

However, supply and demand did not balance. 100 widgets were supplied, and only 90 were demanded.

In the next period, DR, Inc. pays its employees DR$72 to produce 80 widgets and pays a DR$20 dividend. DR Inc. supplies all 90 widgets (80 produced, plus 10 inventory) to market at a unit price of DR$1. Again, 90 widgets are demanded and sold. Having spent DR$90 out of DR$92 income, there is still the issue of the IOU– needing another DR$9 (DR$10 in principal, and DR$1 in interest, less DR$2 after consumption). DR, Inc. kindly rolls over the DR$9 in debt.

DR, Inc. now has DR$100 (the original DR$100, less DR$72 in wages, DR$20 in dividends, plus DR$90 in sales, DR$1 in interest income, and DR$1 net borrowing) plus zero widgets (the 10 in inventory having been sold) plus an IOU for DR$9. The net income of DR, Inc this period is a loss of DR$11.

The employees now have once again DR$0 (starting with nothing, plus DR$72 in wages, plus DR$20 dividends, plus DR$9 in loans, less DR$90 in spending, less DR$11 in debt service) plus 180 widgets (starting with 90, plus this 90), less DR$1 net borrowing (a DR$10 debt retired, and DR$9 acquired.) Their net income is DR$91

Between DR, Inc. and employees, net income in this period is DR$80– exactly the market value of all widgets produced.

However, in this period, supply and demand did balance. 90 widgets were supplied and 90 were demanded.

J R, I am sorry to have taken so long to respond to your last comment, but other posts have generated a lot of comments. Combined with other commitments and not feeling very well for a week, responding to your comments got lost in the shuffle.

You said:

“Y does not accrue to households alone. That is flat-out wrong. That can be an assumption you make, but it is plainly untrue in practice. Businesses do retain earnings.”

I agree that businesses to retain earnings, but if we define Y as payments to factors, which, to me seems a very natural definition and a widely accepted meaning of the term, then retained earnings by business correspond to one of the factors can cause income to differ from output.

You said:

“If a business produces nothing in a period, but sells one consumption widget out of inventory, it has traded one asset for another and has no net income.”

Whether the business has any income from selling from its inventory of widgets produced last year depends on the price at which it sells the widget. If it sells it for more than the cost incurred to produce it, a profit accrues in the current period which represents earnings that will either be retained or distributed to factors of production in the current period. You might include the profit on the inventory as output in the current period, depending on accounting conventions. On the other hand if the inventory is sold for less than the cost of production, the loss must be deducted from earnings. In an extreme case like wine aging in vats, most of the value of output is generated after production has taken place.

You said:

“It’s possible to “conceive of income as being something different from output”– that is true. If I only count income as pennies I find in the street, them GDP will be much much larger than income. But GDP must still equal GDI.”

Well, there is a big difference between payments to factors of production and picking up pennies in the street, so I don’t think that your counterexample is very compelling. Whether GDP must equal GDI depends on how we define output and income.

You said:

“The whole point of breaking down GDP into C G and I is to see who winds up with the stuff produced. The whole point of breaking down GDI into compensation, net taxes, net operating surplus, and consumption of fixed capital is to see who profits from producing exactly the same stuff.”

Actually, in the income-expenditure model, the point of breaking expenditure into C, G, and I is that we have behavioral theories about C and I which we deploy in constructing a model explaining the determination of E, Y, and Q. We can also see how to use G as a policy instrument to alter equilibrium Y by the desired amount. Similarly breaking Y down into C, S, and T allows us to use T as an alternative policy instrument for changing equilibrium Y by the desired amount.

Thanks for providing me with your numerical example. But it wasn’t quite what I was looking for. I was looking for an interpretation of disequilibrium in terms of a simple Keynesian cross version of the income-expenditure model. Perhaps your example could be translated into those terms, but it would require a non-trivial effort on my part to do the translation.

I may come back to this, but let me state again in case this isn’t clear:

You can model whatever you durn well wish. If you wish to model C as a constant, you are free to do so. If you wish to model C as equal to Y, or a million times Y, you are free to do so. If you wish to model C as zero you are free to do so.

That doesn’t mean that the resulting equilibrium conditions have any relevance to anything. Nevertheless, accounting identities must still hold.

You are free to “define Y as” whatever you want. But GDP must still equal GDI. Then again, you may define “GDP” as something other than the usual sense, but you can’t expect anyone to understand what you are saying.

From http://www.bea.gov/national/pdf/nipaguid.pdf

“Gross domestic product … is the market value of the goods and services produced by labor and property located in the [region].”

“Gross domestic income … measures output as the costs incurred and the incomes earned in the production of GDP.”

Note that neither of these definitions have anything to do with whether the production is actually sold. The costs and incomes in GDI are associated with *production* not sale.

“In theory, GDP should equal GDI, but in practice, they differ because their components are estimated using largely independent and less than perfect source data.”

As far as I can tell, you are using the same words but talking an entirely different language than what is usual. So I have no hope of understanding you.

You had better come up with your own example of a model, complete with definitions of every variable. In the mean time, I don’t see any point in trying to argue. Once we settle on a common language, we can move on.

D R, I am afraid that we are not making much progress.

You said:

“You can model whatever you durn well wish. If you wish to model C as a constant, you are free to do so. If you wish to model C as equal to Y, or a million times Y, you are free to do so. If you wish to model C as zero you are free to do so.

That doesn’t mean that the resulting equilibrium conditions have any relevance to anything. Nevertheless, accounting identities must still hold.”

That seems to me entirely backwards. Accounting identities are simply definitions and nowadays we don’t view definitions as embodying fundamental truths, merely as simply conventions that are convenient or useful. A theory, on the other hand, may be true or false, depending on how good a theory it is. But it does attempt to give us some knowledge about the world. The very fact that it could be disproved by being inconsistent with observations provides it with scientific status not pertaining to any definition.

Actually the definition of gross domestic income you cite is not far off from how I would define it.

“the incomes earned in the production of GDP”

The difference is that I treat earnings as being paid to factor owners, which seems to me a perfectly natural way to define the term, so I don’t think it is very hard to understand. To satisfy the accounting identity, earnings paid to factors of production sometimes need to be supplemented by imputations to factors that correspond, at best, to bookkeeping entries that never actually reach the pockets of factor of production.

What I meant by an example of a model in disequilibrium was simply a numerical explanation of what is happening in the simplest version of the income expenditure model you can think of, something like C = 100 + .9Y, and I = 100, which has an equilibrium corresponding to Y = 2000. How can such a model be understood if equilibrium does not obtain?

“That seems to me entirely backwards. Accounting identities are simply definitions and nowadays we don’t view definitions as embodying fundamental truths, merely as simply conventions that are convenient or useful. A theory, on the other hand, may be true or false, depending on how good a theory it is.”

I don’t see why what I said is backwards. I am in full agreement with everything you say after that. It simply doesn’t matter what *theory* you put forward to explain how GDP and GDI break down into “expenditures” and “income” respectively. Your theory may or may not be an accurate reflection of the real world. Your theory may or may not have any predictive power whatsoever.

But you cannot make GDP fail to equal GDI without taking at least one of “GDP” or “GDI” to be something other than its generally accepted meaning.

“Actually the definition of gross domestic income you cite is not far off from how I would define it… The difference is that I treat earnings as being paid to factor owners, which seems to me a perfectly natural way to define the term, so I don’t think it is very hard to understand.”

So you define GDI as something other than its generally accepted meaning? Good for you. Then *you* will find that “GDI” need not equal GDP. And if I define “GDI” as payments to musicians performing “Wär Gott nicht mit uns diese Zeit” then in turn *I* will find that “GDI” need not equal GDP.

If you like the decimal systems more and declare it “perfectly natural” that the foot be defined as ten inches, then you probably shouldn’t go around telling people that their yardsticks are six inches short of three feet.

“What I meant by an example of a model in disequilibrium was simply a numerical explanation of what is happening in the simplest version of the income expenditure model you can think of, something like C = 100 + .9Y, and I = 100, which has an equilibrium corresponding to Y = 2000. How can such a model be understood if equilibrium does not obtain?”

I have no idea because you have not defined C or Y or I, so I have no idea how to interpret what you write. Sorry to be pedantic, but you have already declared that you don’t agree with the usual definition of GDI, so I am loathe to presume to understand what you are saying.

Of course, if C = 100+0.9Y and I = 100, then sure *if* Y=C+I, then Y = 2000. It doesn’t matter if C is the number of worms in my back yard and I is the dollar value of the Sun as determined by my aunt, and Y is the number of notes in “Freue dich, erlöste Schar.”

Still, if you *define* equilibrium in your model as Y=C+I, then only Y=2000 will do. If I *define* equilibrium as Y=C, then only Y=1000 will do. What is the connection between Y and C+I which makes your equilibrium condition of any interest?

Dear,

I am a quant researcher in an asset management company in Japan. I think wise people would easily understand that an identity brings no important information, if he/she spent 5 more minutes pondering on the concept of identity. But they tend to have no motivation, because an identity is “always-right-thing”, it looks there’s no reason to spend more time on this matter to understand, instead they say “it’s time to use it!”

For this discussion, I think, it helps having some examples of several types of applied equations, not only an identity. While everyone is using the word “identity”, what really in his/her mind might be slightly different concepts. And I also think that non-macroeconomics examples would be good, because everyone can be cool. I am a layman on economics, however, I think that I can contribute a bit to the discussion. I’ve just thought up some examples on this matter that might be useful to clear confusion or ambiguity. I hope this helps a bit.

By the way, forgive me for my not-so-good English!

1. “Two-car train” examples

There is a two-car train. The two cars have the same weight without passengers. Suppose we are interested in the weight with passengers. For simplicity, suppose every passenger have the same weight.

Two-car train-Example 1: ordinary (mere) equation, not identity

We define W1 as the weights of Car1 (with passengers), W2 as the weights of Car2 (with passengers). Then the equation

(a) W1 = W2

may hold or not, depending on the numbers of the passengers on each cars. So, this is not an identity. With this equation, we can have an argument like this:

1. At the beginning, there are same number of passengers in Car1 and Car2. So W1 =W2 hold (an equilibrium)

2. Put 50 more passengers on the Car1. Then equation (a) doesn’t hold any more, now W1 > W2 hold.

3. Being packed so closely, the passengers tolerance for individual distance was violated, the passengers on Car 1 feel uneasy. Some of them began to move to Car2.

4. Then, W1 tend to decrease and W2 tend to increase over time, so things will be move toward equilibrium.

This is a kind of equilibrium argument. This particular example is nothing important, but it makes sense.

Two-car train-Example 2: Identity

We define Wx as the total weight of the wheels and motors. Wy is defined as the total weight of everything else. Note that the passenger’s weights are included in Wy.

Wx + Wy is the weight of the whole train including passengers. But we know the weight of the whole train is also equal to W1 + W2. Now we have an identity

(b) Wx + Wy = W1 + W2

Now it is clear that it does not make sense to argue like this; “Let the left side fixed and increase the right side by adding (weight of) 50 passengers, then what will happen next?”. Because if we put 50 passengers in Car 1 or 2, then Wy increase in the same amount by definition, so we can’t let the left side “fixed”. It would be a logical contradiction. It’s similar to say: “What the history of the music might have been, if Leopold Mozart died in 1780 while father of Amadeus Mozart lived until 1787?” Because Leopold was Amadeus’s father, this doesn’t make sense.

A Comment on “Two-car train” examples

The point is that in (b) the left side and the right side correspond to the same thing in reality (this always happens when an identity is applied to the real world); i.e. total weight of the whole train. We just have two different expressions of the same thing and put them on the each side of the equation. On the other hand, in equation (a) in Example 1, the left side and the right side correspond to two different things in reality, that makes it possible to think about a change that occurs in one side only and wonder “what dynamics would makes this equation hold?”.

2. Some additional comments and examples:

A moral of these examples is that when we are defining variables (or terms), if we go like

Let A denote “Something”

Let B denote a part of the “Something” and let C denote “everything else” (i.e. C: = A-B)

Then we can easily come up with an identity:

(c) A = B + C

With this equation, an equilibrium argument leads to meaningless words. If we really want to do equilibrium arguments, an example of remedy is to re-define term C. If we define newC as part of original C or an idealized version of original C ( then original C = newC + D ) then

(d) A = B + newC

is not an identity any more. In fact A = B + newC + D is an identity. If we are lucky, D is proved to have some convenient property, like “converge to zero over time” or “have some periodicity”, which might make equilibrium argument fruitful.

Note, it is possible to say in (b) that: let the left side (then also right side) fixed, and add 20 passengers on Car 1, what should happen?. An answer for this is “remove 20 passengers form Car2”. This makes sense, but it is not equilibrium of left side and right side. This can be argued only with right side alone (“what should happen if we want the value of one component increase while total value fixed?”)

A typical example of identity in Mathematics is

(e) （Ⅹ+ Y）^2 = X^2 + 2XY + Y^2

This looks like identity, and every one can see that it doesn’t convey any deep sense. But in applications, we often use some “names”, like

let’s define DTF as (X+Y)^2 , RCF as X^2 + 2XY, PPC as Y^2, then the equation looks like

(f) DTF = RCF + PPC

and it can have misleading appearance that it has some deeper meaning beyond definitions. Note, “An Identity doesn’t convey important information” doesn’t mean that each term that make up the identity, e.g. DTF, RCF, PPC, doesn’t convey important information. In fact they may bring very important information themselves. The statement only means that the trueness of the equality doesn’t convey important information. In “Leopold Mozart’s son’s father is Leopold”, the two (father and son) are important figures in the history of music (at least Amadeus =the son is), but trueness of this sentence is nothing more than “one’s son’s father is himself” which convey no important information on the history of music.

It is very important to notice that in physics, some non-identities hold very precisely (if not perfect) in real world. An example is a Newton’s equation

(g) F = M*A.

Here, F stands for force, M stands for mass, A stands for acceleration. This is not an identity. We don’t know why this hold. We can’t prove this equation mathematically. Some characteristic of this universe makes it hold. Even if this equation does not hold, there would be no logical contradiction. In fact, by an Einstein theory, we know that it won’t hold when speed is close to the speed of light. This is a law, expressed in the form of equation. This kind of equations should not be confused with identities.

This is end of my note. I hope this helps.

Tokio Sato

DR, My position is that definition given in the national income accounts is not necessarily “the generally accepted meaning.” And for purposes of evaluating or understanding a theory we should adopt the definition that fits the theory not impose a definition derived from outside the theory, even if it is the “generally accepted” definition, which in our case does not seem to me to be the case. Your example of an alternative definition of GDI is not really relevant because it is not derived on the theory of national that we are discussing. I maintain that my definition is derived from that theory. Everyone agrees that when we are talking about length we are talking about a single magnitude and are just choosing between different units of measurement when we go from feet to yards or to meters. We are discussing whether output is inherently the same magnitude as income. You assert that they are inherently the same thing. I say they are not inherently identical though they are very closely related to each other. Therefore under some conditions a measurement of one need not necessarily give you the same number that you get if you measured the other at the same time. It’s true that if they differ we can identify the reason for the difference (e.g., undistributed business profits), but that doesn’t mean that for purposes of analysis it is not useful to keep track of two separate magnitudes rather than treat them as identical.

Concerning my numerical example, I was asking you to apply your understanding of the terms C, Y, and I, and explain to me what is going on in the model if it is not in equilibrium.

Tokio, Many thanks for your comment. I believe it captures very nicely what I have been trying to say. And perhaps I shall come back to it and comment on it further when I am not quite so pressed for time.

“We are discussing whether output is inherently the same magnitude as income. You assert that they are inherently the same thing. I say they are not inherently identical though they are very closely related to each other.”

Right. I don’t know what kind of *aggregate* income exists that is not tied to production of real goods and services. In the aggregate, the only way in which we can be said to have greater wealth is to have produced something. If the market price of Apple stock rises, that is not aggregate income. It just means that someone traded relatively more cash for fewer shares. It means that at the higher price that the owners of Apple stock can demand a larger share of output and non-owners can demand less.

It may be that you want to *model* the capital gains as increasing demand for consumption goods (or investment goods) but that doesn’t mean that the capital gains count as aggregate income.

“Concerning my numerical example, I was asking you to apply your understanding of the terms C, Y, and I, and explain to me what is going on in the model if it is not in equilibrium.”

I am not sure I understand the question.

We have no prices, so we have defined aggregate demand (Y_d) in terms of production (Y), specifically Y_d(Y) = C_d(Y)+I_d(Y) = 200+0.9*Y.

That is, there is excess supply of Y-Y_d(Y) = 200-0.1*Y. Perhaps Y=2000 and there is no excess supply. Or, perhaps Y is not equal to 2000 and there is excess supply, and thus a disequilibrium between supply and demand.

As a result, producers find themselves with more of the good they produced and less of what they hoped to trade for the goods.

Obviously, I meant “excess supply of Y-Y_d(Y) = 0.1*Y-200”

D R, You said:

“I don’t know what kind of *aggregate* income exists that is not tied to production of real goods and services. In the aggregate, the only way in which we can be said to have greater wealth is to have produced something.”

The verb “tied” is important. I totally agree that income is “tied” to output. One is not possible without the other, but they are not identical. Eggs are tied to chickens, and chickens are tied to eggs; eggs cause chickens, and chickens cause eggs. But chickens and eggs are not identical to each other.

You are still not getting my question, which if there is a disequilibrium, the equations Y = C + I, C = c(Y), I = i(Y) are not being solved. I am not asking for an interpretation in terms of supply equals demand, just tell me how to interpret Y, C, and I so that the equations of the model are not being solved. Is there some interpretation that corresponds that you can attach to the equations that allows you to interpret what is happening in terms of those equations.

The way I interpret them is to say that expenditure is unequal to income and savings is unequal to investment (in disequilibrium) output might or might not be equal to income, but expenditure has to be different from income if the model is not in equilibrium.

“You are still not getting my question, which if there is a disequilibrium, the equations Y = C + I, C = c(Y), I = i(Y) are not being solved.”

My problem with your question is the part where you say “there is a disequilibrium” What, in your model, is supposed to equilibrate? Those equal signs are filled with ambiguity. I don’t know which of your equations are definitions, which are identities, and which are equilibrium conditions.

For example, let us take as given the function c(.). Let Y be the total amount of production in the economy. Suppose consumers purchase consumer goods in the amount of c(Y). Then there is actual (ex-post) investment equal to Y-c(Y). There is no way around this unless you introduce into the model another class of expenditure.

What else could Y-c(Y) represent? If something is produced but not counted in consumption, it is either…

1) … a planned investment, and should be counted in I; or

2) … an unplanned increase in inventories, and should be counted in I; or

3) … something which was destroyed after being produced, but before it could be counted as inventories. Like what? A service? Perhaps a massage? In that case it shouldn’t count in Y unless it counted in C.

I have no understanding of what Y-c(Y) could possibly represent except investment (or some other class of expenditure.) Thus…

a) i(Y)=Y-c(Y) for all possible Y; or

b) your model is misspecified; or

c) you have to correct my understanding by make clear what each equals sign means (definition, identity, or equilibrium condition)

“The verb “tied” is important. I totally agree that income is “tied” to output. One is not possible without the other, but they are not identical. Eggs are tied to chickens, and chickens are tied to eggs; eggs cause chickens, and chickens cause eggs. But chickens and eggs are not identical to each other.”

Oh good goddess. If you think they are not IDENTICAL, then for crying out loud please give me one example of a type of income which counts as such in the aggregate (i.e., does not count as a loss elsewhere) but is not production. Otherwise, give me one example of a type of production which does not count as income in the aggregate.

I don’t think I’m asking for much. You say they are not the same. I just want a single example of the difference between production and aggregate income.

Until you do that, I fear I will be unable to imagine why you think there is any difference.

D R, You asked:

“What, in your model, is supposed to equilibrate? Those equal signs are filled with ambiguity. I don’t know which of your equations are definitions, which are identities, and which are equilibrium conditions.”

Fair enough. It’s hard to insert identity signs instead of equal signs, but in this case, it’s obviously necessary. E stands for expenditure, Y stands for income (factor payments) received by households, and Q stands for output.

E ≡ C + I

Y ≡ C + S

C = c + mpc*Y (c is a constant, mpc is the marginal propensity to consume)

I = I* (I* is a constant)

E = Y (equilibrium condition), which is equivalent to I = S

E = Q (assumption that inventories are constant, so that expenditure equals, but not identically, output)

You have investment i(Y) = Y – c(Y), actually I think it is i(Y) ≡ Y – c(Y). If that is true for every possible value of c, I and Y, how do you solve for equilibrium Y? By treating investment as identically equal to the difference between income and consumption, you no longer have an equilibrium condition. It seems that you are trying to define the equilibrium condition in terms of a distinction between planned and unplanned investment, so that the equilibrium conditions is that unplanned investment is zero in equilibrium, but there is nothing in the model that distinguishes between planned and unplanned investment. There is nothing in the model that requires such a distinction to be introduced, and there is no reason why the model can’t be understood perfectly well with no inventories or with constant inventories as I have assumed.

I have given you my explanation of income, which is factor payments from business firms to households. It seems quite straightforward to me and it allows for the value of output to differ from income by the amount of retained earnings retained by businesses and profits (losses) on inventories. You don’t like that definition even though it seems to me perfectly clear and consistent with traditional usage in economics (though, admittedly, not national income accounting).

“Fair enough. It’s hard to insert identity signs instead of equal signs, but in this case, it’s obviously necessary. E stands for expenditure, Y stands for income (factor payments) received by households, and Q stands for output.”

Thanks. I appreciate the understanding. I still need clarification, however. You have specified a couple of identities, but I don’t understand how you arrive at them. Do they hold by definition? If so, then it seems like C is a type of expenditure. Or is it a factor payment? How can it be both?

What is the significance of I? How does I differ from C?

What happens to Q-Y? Does it simply vanish? How is that not effectively business savings?

I don’t understand your “assumption” that E=Q. If you follow through on that assumption, then Q and Y are not independent in any way.

Q = E = I+C

Q-Y = I+C-Y = I-(Y-C) = I-S

I is not equal to S only because you have defined S as household savings, rather than aggregate savings. Which is fine, but doesn’t address “S=I” in the common sense.

“It seems that you are trying to define the equilibrium condition in terms of a distinction between planned and unplanned investment, so that the equilibrium conditions is that unplanned investment is zero in equilibrium”

YES. That is precisely my argument.

“I have given you my explanation of income, which is factor payments from business firms to households. It seems quite straightforward to me and it allows for the value of output to differ from income by the amount of retained earnings retained by businesses and profits (losses) on inventories.”

RIGHT. But if there are retained earnings, you have to count them somewhere. You don’t get to call S household savings and then say that S is not equal to I. You’re just redefining S.

In case it’s not abundantly clear, one does not accidentally build a factory. Unplanned investment is an unplanned increase in inventories. (One may plan to increase inventories– essentially buying one’s own product.) In this sense, an unplanned increase in inventories is the result of an excess of supply.

If you don’t want to count planned inventory increases in demand, then any increase in inventory is the result of excess supply.

This is a follow up of my previous comment.

It seems that sometimes identities are regarded as constraints. I hope this comment makes it clear that it is not the case. In my previous comment I said that the left side and the right side correspond to the same thing in an identity. Maybe that’s enough to believe that an identity is not a constraint (of dynamics). It is similar to that “Tom is a football player” affects how he lives but “Tom is Tom” does not.

Still, I think it is beneficial to look at the issue in a different light. To do that, I will borrow terms “degree of freedom”, “dimension” form math and introduce “redundancy” concept.

Suppose, there are two dynamic models (or worlds) X, Y and two set of researchers for each model.

Model-X

Model-X is driven by two variables (or factors): X1, X2. The state of this system is determined when the value of X1 and X2 are given. The X-researches seek to understand the dynamics of (X1, X2). They record the value of (X1, X2) every month, like (-1, 5), (2, 7), (3, 4).

Model-Y

Model-Y is similar to model X but it is a 3-variable model.

The state of this system is determined when the value of Y1, Y2 and Y3 are given. The Y-researches seek to understand the dynamics of (Y1, Y2,Y3). They record the value of (Y1, Y2, Y3) every month, like (8, 5, 3), (12, 6, 1), (17, 4, -2).

One day, an X-researcher said:

“We often refer to the value of X1 minus X2 and calculate the value with calculator each time. I think it is convenient if we call this value “X3” and calculate every month and record it with X1 and X2.”

After that, X-researchers’ records looks like (-3, 4,-7), (2, 5, -3), (8, 6, 2). Now, Model-X looks like Model-Y. An apparent difference is that X-researchers have an identity X3 = X1 – X2 or X1 = X2 + X3.

Does the identity constrain something? Will Model-X change its dynamics to accommodate itself to the identity? Definitely not. Because “X3” is just a convenient term introduced to describe phenomenon on model-X easily. Model-X can have any state “on its own will” regardless of the value of “X3”. The model-dynamics in “after-X3-eara” is the same as the dynamics in “before-X3-eara”. From the X-researchers point of view, if they cast aside the convenience of the term and take trouble to say “X1 minus X2” or “difference between X1 and X2” each time, they can do and express anything without “X3”.

This situation can be summed up as follows:

“The variable-system (X1, X2, X3) has redundancy”.

Here, redundancy means that some of the variables can be expressed with other variables, so that in fact we can do well without them. Model-X is essentially a two dimensional system, or in other word, Model-X’s degree of freedom is 2. We can create (define) as many variables as we like, for example, X4=X1-3*X2, X5=5*X1+12*X2. Still, variable-system (X1,X2,…,X5) has degree of freedom 2. Variable system’s degree of freedom can’t exceed model’s intrinsic dimension. If number of variables exceeds model’s degree of freedom, the variable-system will have redundancy. If a new variable represents something new that cannot be expressed by other variables, then introduction of that variable increase the variable-system’s degree of freedom. Even if variable-system’s degree of freedom is less than model’s, a new variable defined (expressed) with old-variables will not increase the variable-system’s degree of freedom.

Redundancy allows us to express things simply, but with a cost. The cost is that it also increases the possibility that we end up saying some nonsense. Basically we should restrain ourselves not to control too many variables at once in our thought experiment. A safety rule is that if the variable-system’s degree of freedom is M, we better make assumptions on less-than-or-equal-to M variables. (To be precise, “We better make assumptions on a sub-group of variables that have no redundancy”. Because there can be redundant sub group that are very small)

In the case of Model-X, the model’s degree of freedom (or dimension) is 2, so we can make assumptions on less-than-or-equal-to 2 variables at once.

Because there are one to one relations between (X1, X2), (X1, X3) and (X2, X3), the three 2-dimentional systems can be treated equal basis. If we choose (X2, X3) and said something about X2 and X3, we also said about something about X1 implicitly. If we try to add a comment on X1, we have a big chance to mess up our argument.

For example,

1. We believe X2 is likely to go up strongly.

2. We also believe X3 have weak momentum. X3 is not likely to have strong movement.

3. Finally, suppose X1 is strongly stable. If X2 goes up while X1 and X3 have no movement, the identity: X1 = X2 + X3 will break. Because the identity should be obeyed by the system, the weakest variable X3 (because it have “weak” momentum) will give in at last, and X3 will go down eventually.

This is a faulty reasoning. When we stated 1 and 2 about X2 and X3, we stated about X1 too. That is, because X1 = X2 + X3, we have implicitly said that “X1 will likely to go up (with X2)”. So we can’t say “X1 is strongly stable” in the first place.

On the other hand, for Y-researchers it poses no problems to make assumption on Y1, Y2, and Y3 at once, for example: 1. Y2 is likely to go up, 2. Y3 is not likely to have strong movement, 3. Y1 is strongly stable.

The moral of this is that:

1. An identity is a sign of redundancy in the variable system.

2. An identity doesn’t constrain the system-dynamics, it constrain us (or researchers) in the way we handle our reasoning.

Note that it is possible to say: 1. We believe X2 will go up, 2. We believe X3 will go down 3. We believe X1 will go up. Because the third statement (combined with 1, 2) is as same as “We believe that X2 will go up more than X3 will go down”, there is no contradiction. But maybe the latter expression: “We believe that X2 will go up more than X3 will go down” is more straightforward and it is easier to see if it is consistent with other assumptions. This means that we can break the safety rule but we must be very careful for consistency to be maintained.

In general, if there is a function Z1 = G(Z2, Z3,…, Zn), at least 1 freedom is lost. And if G is not a one-to-one function, not all the sub-variable-systems can be treated equal basis.

I used “degree of freedom” and “dimension” in the same meaning, “dimension” is simple but I thought “freedom” may convey some appealing feeling.

The above examples are very simplified. In real application, the problem of finding the degree of freedom and to know which combination of variables is appropriate to make assumptions on them can be very complicated, I think.

D R, C + I is the definition of expenditure. On the other hand, income (factor payments) can be disposed of by the households that earn income either by spending on consumption, C, or by saving, S.

I is spending by business firms on final output , as opposed to spending on raw materials or on factor payments. The difference between C and I is that only households spend on C and only businesses spend on I.

Business savings is an accounting definition that is not encompassed within the income expenditure model. All savings and consumption is undertaken by households. Business saving does not vanish, but changes in business saving imply a difference between expenditure and income.

The equilibrium for the model can be stated as E = Y or equivalently I = S. So, yes, any difference between E and Y is exactly matched by an equal difference between I and S. And the reason that they are defined that way is to allow us to make the distinction between and equilibrium condition which does not always have to be satisfied and an accounting definition which can never not be satisfied.

I am defining S in a way that is in accord with the income expenditure model, which is not the same way that the accounting definitions of the system of the national income accounts have defined it. Applying national income accounting definitions make it impossible to distinguish between equilibrium and disequilibrium in the model.

Planned inventory accumulation counts as investment in the income expenditure model. The point is that it is not necessary to posit unplanned inventory accumulation to account for disequilibrium in the income expenditure model. There can be disequilibrium even if there is no unplanned inventory accumulation.

I will respond to Tokio Sato’s comment in a separate comment.

David,

I’m sorry, but you have brought us precisely nowhere. We started out with the identity S=I, and you say that if you use a different definition of S and a different definition of I, then the S=I identity need not hold.

You clearly fail to see the absurdity in employing such an argument and therefore there can be no use whatsoever in arguing beyond this point.

Good luck.

This is some additional and intuitive comments from a graphical point of view. Maybe less important and less interesting.

Model-X’s records, (X1, X2), can be displayed graphically in “X-Y (2-D)

plots”, i.e. dots in a sheet. Model-Y’s records; (Y1, Y2, Y3) can be

displayed in “X-Y-Z (3-D) plots”, i.e. dots in a cubic space. So,

Model-Y’s records have “thickness” compared to the Model-X’s. In a

sense, Model-Y is richer in information. Then what about Model-X

equipped with X3? Isn’t it 3-D?

Recall that

Xn = a1*X1 + a2*X2 + …. + a(n-1)*X(n-1) + b

is an equation for (n-1)-dimensional hyper-plane in the n-dimensional space. So, the identity of Model-X : X3 = X1 – X2 can be regarded as an equation for a plane.

This means that with the aid of X3 we can embed Model-X’s records in 3-D

space, but all the dots are on the same 2-D plane. Because Model-X has

essentially 2-dimensional information, even if we equipped it with a

redundant variable to add one dimension, its values (X1, X2, X3) only

fill 2-dimansional area, which has no thickness at all. By defining new

variables, we can fatten Model’s appearance but cannot increase its

intrinsic information.

Model-X with X3 is a, in a sense, quasi-3-D model. The identity reveals

the fact that all the dimensions are not fully used by the model. Note,

it is not like that the identity restricts the output of the model to

the plane, but in fact the output has 2-dimensional shape in the first

place.

D R, I am sorry that we don’t seem to be able to work out our differences here. I hope to try again in a future post. In the meantime, I think that Tokio Sato’s post below actually helps to frame the issue more usefully. I think the point is that a model requires at one degree of freedom to be able to solve it for a unique equilibrium. If you have more than one degree of freedom you can get multiple solutions, if you have no degrees of freedom, the model is over-determined, and hence, unsolvable. Thus, you have to choose your definition in a way that doesn’t over-determine the system.

Tokio, Does my response to D R above capture the point that you are trying to make?

I noticed this post this morning. I’ll comment on this within a week.

My comments were aimed to clear what attitude was appropriate for identities. It seems that David-DR argument is mostly about definitions. If definitions are not fixed, you can’t tell if an equity is an identity or not at all. So my previous comments are not so useful. But I try to clear some points.

My notation in this comment is as follows.

Y, C, I, S are defined as GDP versions (i.e. national income accounting definition)

David Glasner’s versions are indicated as Ydg, Cdg, … etc, when you declare your own definitions.

For “definition” and “identity”, I’ve invented some unusual notations, “=d=” and “=i=”.

A =d= B indicate A is defined as B

A =i= B indicate A = B is an identity.

Note, A =d= B implies A =i= B, but the converse is not always true.

First, let’s look at the definition structure of GDP version variable-system.

Y =d= measured in real world (according to the GDP rule book)

C =d= measured in real world (according to the GDP rule book)

I =d= Y – C

S =d= Y – C

They (GDP officers?) may do some measurements for I and S, but their aim is to re-produce the value of Y -C, so these are essentially the definitions.

It is clear by definition that

Y =i= C + I

I =i= S

No issue so far, I think.

This is some kind of an observation system. We have dedicated variable-system for this purpose such as Y, C and I. Values of some variables come from the real world. We process them according to the definitions of other variables to get their values. In a sense, this variable system is driven by real-world-born figures. With this device we can observe the dynamics of some real world phenomenon. We have concrete definitions for the figures that come from real-world. That concrete definitions are the variable-system’s link to the real-world. Without them the variable system is just an abstract mathematical structure.

Once this statistical or empirical variable system is established, you can think of a model version of it. You can develop models for Y and C. This means that basically what you try to modelize are “come from realty”-parts (with your interpretation of real-world dynamics in mind). Other part is just a mathematical structure. Of course, you can re-define the mathematical part, but you better do this while you stay with empirical version. Because re-define the mathematical part without empirical meaning can be dangerous, it is likely that you will be not so sure about the meaning of the re-defined variables.

When you substitute model-Y and model-C to the variable system, you have models of I or S. Now, the variable-system is driven by the model output.

Note it is important to realize that you have to proceed in the order of “real-world-based-definitions first, model next”. You can’t develop a model if you don’t know what kind of phenomenon to be approximated (or described, mimicked) by the model.

Even if you change the definitions of Y and C (different measurements in the real world or output of some models), as long as you keep the definition structure (i.e. values of Y and C come from somewhere, I and S are calculated according to the definitions),

Y =i= C + I

I =i= S

still hold. So, these identities are “inherent” while you stay in this variable-definition-structure.

Next, let’s look at your model. Based on the available comments so far, it seems

Idg =d= measured in real world (spending by business firms on final output)

Edg =d= C + Idg (As for C, I reckon you are satisfied with the GDP version.)

Sdg =d= “defined in a way that is in accord with the income expenditure model”

Ydg =d= C + Sdg

( You said that “Y” = “C” + “S” is an identity in your version, and

you want to use C and Sdg as “C” and “S”, then “Y” can’t be the GDP

version any more.)

I see a problem here. You broke the rule of “real-world-based-definitions first, model next”. You should define Sdg in “measured in real world”-style first. After that you can talk about models. Otherwise no one can understand the meaning of your model.

You need an explicit real-world type definition (to clarify the differences between your version and the GDP version explicitly is enough). Maybe “accord with the income expenditure model” may suggests something, but again there will be issues like

“What C means in the income expenditure model?”

“What is the definition of the income expenditure model in the first

place?”

Then again you will be at the start point.

By the way, if you want “Y” to be the GDP version, you should live with S instead of Sdg. Because if Y =i= C + S should hold, then the (Y, C, S)-variable system’s dimension is 2, this means that only 2 of them can be defined freely.

Maybe what I called “variable-system” and what I called “model” are both usually referred to as “model”. But I need discrimination between them to explain my view.

I pretty much agree with Tokio Sato here.

Of course, I’ve been saying the problem is definitional for almost three months now. https://uneasymoney.com/2012/01/31/krugman-on-mistaken-identities/#comment-4063

So I don’t expect much, but I do appreciate the confirmation that I’m not out of my mind here.