**UPDATE for the sake of traffic directed here by Krugman and DeLong**: I've written a couple things at this point so the summary is:

1. This post (you're here. you don't need a link)

2. Henry Hazlitt is often promoted as a place for people to start with economics. I don't think that's wise.

3. The estimable Gene Callahan has an older post on this same argument from Rothbard, but I think his criticism doesn't go far enough - explanation here.

4. There is a legitimate problem with the Keynesian cross, but New Keynesianism addresses it.

5. One of my commenters makes my point much more succinctly - I reproduce that with an explanation of how it relates to my point here.

*****

He writes:

"Now, though I cannot seem to find a reference, I have a vague memory that it was Murray Rothbard who observed [<<< DPK: Well we're clearly off to a bad start] that the really neat thing about this argument is that you can do exactly the same thing with any accounting identity. Let’s start with this one:No, no, no, no.

Y = L + E

Here Y is economy-wide income, L is Landsburg’s income, and E is everyone else’s income. No disputing that one.

Next we observe that everyone else’s share of the income tends to be about 99.999999% of the total. In symbols, we have:

E = .99999999 Y

Combine these two equations, do your algebra, and voila:

Y = 100,000,000 L

That 100,000,000 there is the soon-to-be-famous “Landsburg multiplier”. Our equation proves that if you send Landsburg a dollar, you’ll generate $100,000,000 worth of income for everyone else."

The Keynesian multiplier has two sides: income and expenditure. The economy's income today is the economy's expenditure yesterday. If you had a lot of income today but there was not a lot of expenditure today, then tomorrow you will have less income.

Landsburg drops the expenditure side of that equation. In effect, he swaps:

Y = C + I

for

Y = wL + rK

Except his income earners are himself and everybody else.

Wonder of wonders, when you set income equal to income you get a forty-five degree line.

So by botching the income/expenditure distinction Landsburg gives us a triviality.

*Even Say did better than that*. Say at least kept income and expenditure separate. His mistake (initially) was assuming they were always equal. But on top of that triviality, Landsburg adds a genuine error.

The slope of the consumption component of the

__expenditure side__of the Keynesian cross (the side that Landsburg is missing) is not strictly speaking determined by the consumption share of output, it's determined by the marginal propensity to consume. When we consider the economy as a whole, of course, we can think in terms of the consumption share of output (investment equals savings and all that). Normally we

*are*talking about the economy as a whole, so it doesn't matter all that much that we make that distinction (when I TAed an intro macro class I always taught it as MPC, though).

Now as I highlighted above, we can't even really call Landsburg's 0.99999999 figure an MPC because it's not expenditure - it's income. But Landsburg is

*treating*"L + E" like expenditure, so let's at least do it correctly for him. Either it's all consumed or part of it is not consumed. Landsburg seems to want E to consume all of it's income (hence the 0.99999999 coefficient) and since he is furnishing that dollar out of L to E presumably he is consuming all of his income, or 0.00000001 of total income. So actually the MPC for the economy is 1, and not 0.99999999, implying an infinite multiplier. I buy something from you for one dollar, you spend that whole dollar buying something from someone else for one dollar, etc. etc. and the dollars keep racking up. You can think of it as shifting the consumption curve (which is parallel to the 45-degree line) up such that there is no longer any stable equilibrium and consumption bounces off to eternity.

This brings me back to my initial point: Keynes made the multiplier go because he had income

*and*expenditure. When you have income and income you get infinite solutions because you just have a tautology. When Landsburg mixed up MPC and income share (and implicitly assumed an MPC of 1 even though he mistakenly called it an MPC of 0.99999999, which gave him a very large equilibrium solution), he was essentially repeating the tautology under the guise of having performed the same exercise.

**A very minor additional problem posed by Landsburg**

So Landsburg confuses a couple things - he doesn't have an expenditure side so he has to smuggle it in later, and when he smuggles it in later he confuses MPC and income share which concealed the fact that he was only dealing in a single identity (if he had realized his implicit MPC was 1 and not 0.99999999 he might have picked up on what was going wrong).

Econ 101 students ought to know better because of course the Keynesian cross is pretty uniformly taught with both income and expenditure from the get-go!

But this does raise a ([very] minor) additional problem: if people consume a lot of their income then we can get similar outrageously high multipliers! Wouldn't the implication be that policymakers should encourage citizens to consume as much of their income as they can?

Of course not. Because every Econ 101 textbook I know of teaches aggregate supply before they teach aggregate demand. You can gussy that up with as much optimization as you want, but at the end of the day an Econ 101 student can still understand how that pins the multiplier down with the imperative of investment.

**One more point**

Take the consumption share of income, use that as an MPC for convenience sake, and get a multiplier. What is it - four? five? That's still outrageously high.

I've seen people actually cite this as proof that the Keynesian model is wrong.

Guess what folks: you have to factor in crowding out. As Keynes wrote in chapter 10 of the

*General Theory*: "

*if the propensity to consume in various hypothetical circumstances is (together with certain other conditions) taken as given and we conceive the monetary or other public authority to take steps to stimulate or retard investment, the change in*".

**the amount of employment will be a function of the net change**in the amount of investmentAnother way of putting this is that when the government spends money you can't just multiply that by the multiplier from the Keynesian cross! Or, put another way, you have to multiply government expenditure by the multiplier from the Keynesian cross, and subtract out the reduction in investment multiplied by the multiplier from the Keynesian cross (and of course also subtract out any reduction in consumption through taxation, etc., multiplied by a here-unspecified multiplier that will be different from "the multiplier"). In a sense it's misleading to call the empirical government spending multiplier "the multiplier" and to also call the Keynesian cross multiplier "the multiplier", but if you think about what the Keynesian cross is doing it's not all that hard to keep straight.

Thanks for that very detailed post. Rothbard's point had always struck me as ludicrous / unfair, but I never took the time to really try to think through in what way(s) exactly this is so. Now all I have to do is study your post for a while to get the answer.

ReplyDeleteAssuming it's right... although I think it is. It took me a little while to figure out what was going wrong.

DeleteI'll wait till Bob Murphy has weighed in so that I will know what the correct position is.

DeleteYou have no idea what kind of self restraint I am exercising right now ;-)

DeleteI've asked DeLong if I'm thinking about this right... see what he says too.

DeleteI think you're reading a bit much into what the example says. The very general approach is the same in both cases.

ReplyDeleteHere's all I take from it:

1. Write an identity, any identity: A=B+G-X^2

2. Posit some kind of plausible relationship (presumably based on empirical evidence): B=A/2

3. Combine 1 with 2: A=2*(B+G-X^2)

I have no idea what the letters stand for, it doesn't matter. The lesson? The equation produced in 3 doesn't tell us anything about the

causalrelationship between the RHS variables and A...Whoops. #3 should be A=2*(G-X^2).

DeleteUmmm... right. Equations don't give you the causal story. Who says they do?

DeleteI think you're missing something edarniw. It's important that Landsburg misses the income/expenditure distinction and that he misses the MPC/income share distinction and that he completely fails to discuss investment (it's an implicit MPC=1 economy) and therefore fails to bring in any supply considerations.

All these points I belabored matter precisely because when Landsburg tosses all that out he has only a few equations, and then he rebukes the Keynesians cross for not telling us anything about the causal relationships when it is Landsburg that is throwing out (it seems without even realizing it) every element of the story that gives a causal structure to the equations.

If his only point is that a one-to-one mapping exhibits explosive growth when you introduce an exogenous shift, then he doesn't have much of a point.

re:

DeleteUmmm... right. Equations don't give you the causal story. Who says they do?Are you saying no one commits this mistake? I don't think it's prevalent, but the error is made, especially if you're not acquainted with the stuff.

re:

I think you're missing something edarniw. It's important that Landsburg misses the income/expenditure distinction and that he misses the MPC/income share distinction and that he completely fails to discuss investment (it's an implicit MPC=1 economy) and therefore fails to bring in any supply considerations.It really seems like your trying to reconcile the examples with each other in a very specific way.

I don't think you can conclude it's an MPC=1 economy. All that's assumed is that income=income of steve + income everyone else. 1. nothing is said about expenditure and 2. some of those other individuals could be businesses or government employees (in which case they're subsequent expenditure would not go under "C").

edarniw -

DeletePeople make this mistake but I would think the solution is to introduce supply constraints and diminishing returns and depreciation, which make investment necessary, which pins down the multiplier. I don't think the solution is to say - as Landsburg has said - that "the reasoning is invalid". It's not invalid at all. He is simply wrong when he says that.

re: "

I don't think you can conclude it's an MPC=1 economy."If the MPC is not 1 then he has derived his result inaccurately and his identification of consumption share with MPC is extremely confusing.

You resolve that for me and I'm happy to remove this part of the post.

re:

"1. nothing is said about expenditure"Right - that's the whole problem.

If you assume that the MPC in Steve's model is 0.99999999 then isn't the multiplier 100,000,000? If so then his result would hold. Maybe that is what he meant, but if so he should really have stated that assumption.

ReplyDeleteRight - although it's not clear why his MPC is 0.99999999 by his logic. But that's what I get into in the "very minor problem" section. It's still not a real problem if you have an MPC that big unless you ignore everything else we know in economics (and why would you do that?).

DeleteHe states in the comments section:

Delete"So the point is a serious one: Of *course* when you give me a dollar, there’s no reason to think equation E = .99999999Y still holds, which invalidates the reasoning. And equally of course, when the govt spends an extra dollar, there’s no reason to think the equation C=.8Y still holds, which invalidates *that* reasoning."

That was in response to NickJ's argument which was completely different from mine (and IMO not convincing).

DeleteAnd the fact that it's a coefficient on Y does not make it an MPC.

Yes, this is a terrible argument.

ReplyDeleteSomething that I have noted in economics blog posts is a lack of temporal subscripts. Yes, if we understand that we are talking about a particular interval of time, then we do not need subscripts. But "if you send Landsburg $1" means going from the equation, Y0 = 100,000,000 * L0 to Y1 = L1 + E1, where L1 = L0 + 1. Obviously we cannot solve for Y1, based only on this information.

ReplyDeleteIf this argument is meant as a refutation of Keynes, it is silly. Keynes was a better mathematician than to make such an error.

The subscripts are less important when you keep expenditures and income distinct because you're just solving for an equilibrium and you can effectively ignore the dynamics. When you do what Landsburg did - and just use income and income - I agree it starts to get a little confusing without the subscripts.

DeleteEven then, we're just skipping "find where Y1 = Y2" and going right to "find Y-bar". It shouldn't be that confusing I hope.

It appears that it confused some people. ;)

DeleteTo expand on that a bit, if you are assuming equilibrium, then sending $1 to Landsburg is a small perturbation, such that Y1 = Y0 + 1 and the relation that Y = 100,000,000 * L no longer holds, or sending $1 to Landsburg stands for sending $1 to everybody, and Y1 = 100,000,000 * Y0. (According to the pragmatics of language, the second case should not apply, because then there is a reason of singling out Landsburg, and he is not a stand-in for everybody. But we are talking about math. :) )

DeleteHave you seen Landsburg's followup yet?

ReplyDeleteYes. It's just stuff he's said in the comment section before. As far as I know, my criticism still holds.

DeleteDaniel, if you're going to bring in supply side and crowding out constraints, OK, but then I think that destroys the textbook case for using an MPC to figure out the multiplier on government spending. That's the whole point. Neither Rothbard nor Landsburg was trying to endorse the typical Keynesian multiplier.

ReplyDeleteI can't think of how you would teach the textbook Keynesian cross without supply and and crowding out. Have you ever taught it without that???

DeleteThe textbook use of the MPC critically relies on exogenous investment spending. That relies on supply. When we talk about government spending multipliers we are assuming an exogenous G shock that does not change I. That means an after-crowding-out-has-been-taken-into-effect shock (hence the Keynes quote). These are all part of the Keynesian cross instruction as far as I'm aware. This is why empirically we don't get spending multipliers on the order of four or five.

DeleteIf I am missing how we teach the Keynesian cross, please tell me where I'm wrong.

No, of course not Daniel, but that's because I'm an Austrian economist who thinks the typical Keynesian case for a government multiplier is stupid.

ReplyDeleteThe question is, if I go and look up the original Samuelson textbook reference, will those caveats be in the textbook? Then, harder, if I go look up Krugman and Wells' textbook discussion, will your points about time-subscripts and crowding out, be in there? I am confident they won't be in Samuelson, but not as sure about Krugman/Wells.

Time subscripts I don't think are relevant here. Undergrad textbooks ignore the dynamics. Who cares? You're solving for an equilibrium. How could that possible matter?

DeleteYes, I am positive Krugman will have supply and crowding out in his textbook and I've never even read the thing.

I feel like I'm taking crazy pills - of course kids get taught supply.

I just posted this at Landsburg. This is all "a lot about nothing":

ReplyDeleteSigh. This is worthless

If Y = E + L and E= .999999Y then L = .000001Y

then

Y-E =L means

Y(1-.999999) =L which means that

Y = L/(.000001) which means that

Y = .000001Y/.000001

and all this means is that Y=Y. Hmmm.

Excellent - had to share in a new post.

DeleteThank you!

DeleteIt seems you proved that an accounting identity is indeed an identity! Can I play, too?

ReplyDeleteY = C + I + G

C = 0.8Y

I + G = 0.2Y

Y - C = I + G

Y(1-0.8) = I + G

Y = (I + G)/(1-0.8) = 0.2Y / (0.2) = Y

Right, you can pull the identity back out if you want to. You put it in there, so nothing's stopping you from pulling it back out.

DeleteThe question is, what is the theoretical content of your third equation? It has none.

That there is a difference between your second and your third equation is PRECISELY why it's wrong to talk in terms of shares of income and why you have to talk in terms of MPC. If we stick to talking in terms of MPC, it would be easy to see that your second and third equations just restate your first equation, so you had nothing other than your first equation all along.

When it's an MPC relation, then your second equation adds something new - a behavioral relation.

I'm not sure if you're directing this at Landsburg or me. If you're directing it at Landsburg I assume you already know all this. If you're directing it at me and don't understand my counter-argument let me know and I'll try to say it differently.

It was directed at malcolm, who seems to be saying that an accounting identity is worthless because it is an identity.

DeleteLandsburg's logic is perfectly clear. He is saying that if you combine an accounting identity with a bad assumed model (E=0.999999Y or C=0.8Y regardless of government policy), then you cannot trust the derived results.

I don't follow your logic. I guess you are trying to say that C=0.8Y is not as bad an assumption as E=0.999999Y. If so, you miss Landburg's point. Unless you are saying that C=0.8Y is a very good model even when government policy changes. If so, then you have a fundamental disagreement with Landsburg.

Your second and third equation essentially repeated your first equation. Of course you're just going to get your first equation out of it... that's elementary.

DeleteA model requires discriminating behavioral assumptions to do any more than that.

If we assume MPC = 0.99999999 it will work, it will get a huge multiplier, and that's fine... except that there's no good reason to expect an MPC like that based on what we know about supply, diminishing returns, depreciation, population growth, etc. etc. See my section labeled "A very minor additional problem posed by Landsburg" for this point.

Now, on top of all this Landsburg (and Rothbard) are extremely confusing in discussing income vs. expenditure and MPC vs. income share. My contention is that this confusion on the very basic building blocks of the model is what leads them to miss the point that I'm making.

An MPC of 0.99999999 with an enormous multiplier isn't illogical in the sense that we normally think of logic. The problem there is garbage-in-garbage-out which is why I called it a "very minor" problem that is basically solved by noting what's garbage according to economic theory and what's not. That's why I prefer to focus on the other problems.

To see why MPC=0.99999999 leading to an enormous multiplier is not logically a problem, consider what it implies. It implies that exogenous investment is essentially zero. The only way that could happen is if one way or another we were post-scarcity - if optimization was not constrained by the productive capacity of the economy. In that circumstance it's reasonable to think that we would have huge multipliers because we could keep producing and producing and producing with no supply constraints or scarcity inducing exogenous investment.

DeleteAll the components of the model - low exogenous investment and high multipliers - hang together just fine logically and in the context of economic theory in a given set of circumstances. The reason why it seems so odd is that those circumstances don't apply to the world we live in.

We are going in circles.

Deletemalcolm's second and third equations repeated his first equation. That is what an identity is!

Take an identity. Plug in a simplistic model (i.e., a model that does not hold when government policy chnages). Derive a multiplier.

Landsburg's point is that the simplistic model invalidates the derived multiplier.

As far as I can tell, all you are saying is that the C=0.8Y model makes more sense than the E=0.999999Y model.

But unless you are claiming that C=0.8Y is a very good model that holds even when government policy changes, then it is a poor choice to derive a multiplier. Which is Landsburg's point. Accounting identity + poor model = untrustworthy multiplier derivation.

No, no, no.

DeleteMalcolm's second and third equation are behavioral laws. Landsburg established the consumption function for E. Malcolm adds that if Landsburg is spending his dollar clearly his personal MPC is 1 so his contribution to the total MPC is 0.00000001, so the economy's MPC is 1.

That is NOT an accounting identity. That is a behavioral law added to an identity that together produce a trivial result.

Your coefficient on G+I is not an MPC by definition (I and G are not C). If you want to assume an MPI relation with Y, fine, but that's something different. You got them by repeating the accounting identity (if C is 0.8 of the total then the rest has to be I and G).

This is exactly why it's important to keep MPC and income share separate. This is why I spent so much time on that point while Malcolm just condensed it.

No, no, no to you, too.

DeleteI don't see what any of that has to do with Landsburg's point.

1) Take an accounting identity. Landsburg's examples were Y = C + I + G and Y = L + E. Those are both identities.

2) Now make an assumption. Landsburg's examples were C = 0.8Y and E=0.999999Y. Those are both assumptions or models. They are both poor models at times when policy changes.

3) Combine (2) with (1) to derive a multiplier. But the multiplier is not credible since (2) is a poor model.

Rather than trying to confuse the issue by going off on a tangent, you should try to follow the logic there. If you think the logic has a problem, then why not specifically state where you think that the logic is wrong?

1. Strictly speaking, Y = L + E is not an accounting identity, it's a definition. Y = C+I+G is an accounting identity because it takes both sides of the ledger (income and expenditure) into account. That's just semantics when it speaks to what we are calling it, but the exclusion of the expenditure side leads to other problems later (i.e. - not recognizing the point I'm about to make in #2).

Delete2. No, they are both good models. The latter just doesn't bear any particular resemblance to the current economy and also doesn't jive with everything else we know from economic theory. So it's garbage-in-garbage-out but there's nothing at all wrong with the logic.

3. See my response to 2.

I am not confusing the issue at all.

So, aside from semantics about mathematical identities, you agree with Landsburg's point -- garbage-in-garbage-out. Good thing you did not confuse the issue at all.

DeleteAs I've said before, if his only point is that a one-to-one mapping exhibits explosive growth when you introduce an exogenous shift, then he doesn't have much of a point.

DeleteI think you are missing the deeper problems.

If you think he is following the same logic as the Keynesian model, I think you are missing the deeper problem. He is dropping several of the key elements that ensure a sensible result. You don't get to pick out parts of a theory show that in isolation it's possible to put garbage in and get garbage out, and then say there are problems with the theory.

JohnW,

DeleteI don't know why you're not seeing this because it's been explained to you several times, but SL's equation is just an accounting identity that doesn't even have a multiplier! You derive the Keynesian multiplier by creating a behavioral equation about consumption and then plug it into the identity to explain how it holds.

Behavioral vs. accounting identity. That's the mistake here and it's rather brow-raising that a professor with a Mathematics Ph.D misses this.

RJ: Just where did JohnW say that he has a doctorate in mathematics?

DeleteSteve Landsburg was who I was referencing.

DeleteI can't say for certain if JohnW has one or not.

My mistake then.

Delete"Because every Econ 101 textbook I know of teaches aggregate supply before they teach aggregate demand."

ReplyDeleteMaybe things have changed since I've read ECON 101 textbooks. I recall that McConnell and Brue teach the Keynesian Cross before getting to aggregate supply. Case & Fair too. Have they changed?

Landsburg is a contrarian troll. I have never read anything by him that wasn't easily debunked. If this is now appearing on his blog, Slate must have gotten tired of him.

ReplyDeleteIt is a little understandable that math people get monomaniacal about the equations, but economists shouldn't make that mistake. I think the person who made the point about time series, and the author making the point about equilibrium got closest to discarding this truly silly discussion.

ReplyDeleteThe mathematical model tries to approximate what is happening at a point in time. At this time a high multiplier implies, non-mathematically:

- A person with impending bills and extra time to do more work. He gets some money, most of it quickly goes out, and he still has some free time, and has more bills.

- If that money goes to another person in the same sad shape, much of it moves along again.

- If that money goes to someone in fine shape, the money stalls for a time, maybe a long time.

The more people in rough shape (bills and not enough work), the higher the chance the money keeps moving, and quickly at that. Of course this isn't an infinite cycle, hopefully in a pleasant, livable society, more and more people get to be in the third group.

Some idiot playing gotcha with the equations trying to describe the economy assuming that everything is static (or already in equilibrium) isn't worth a minute of time, unless that minute helps someone else understand how idiotic it is.

This isn't complicated, it isn't even math.

One quibble:

ReplyDelete"The economy's income today is the economy's expenditure yesterday."

This is incorrect. Correct would be to say: "The economy's income today is the economy's expenditure today." It is just double entry bookkeeping. Each transaction is recorded twice as income (seller) and expenditure (buyer). In each transaction income = expenditure, you just sum them up, there is no today-tomorrow. Economy's income today most certainly is not the economy's expenditure yesterday, both income and expenditure vary from day to day.

Quibble accepted.

DeletePresumably it's OK to say "expenditure today is a function of income yesterday", and that's the intertemporal link in the chain.

Yep!

DeleteAnother way to make everyone rich.

ReplyDeleteY=C+I+G

Assume: I is exogenous

Government revenue =tY where t is the tax rate then:

G=tY +Gd where Gd is deficit spending that is assumed exogenous.

Also assume C=Co +CmYd – a pretty common Keynesian consumption function where: Cm=marginal propensity to consume, Co is subsistence consumption and Yd = disposable income = Y-tY = Y(1-t)

Then:

Y=Co +CmY(1-t) +Gd +tY +I

do the math and:

Y= (Gd+Co+I)/((1-Cm)(1-t))

Where the Keynesian multiplier is 1/((1-Cm)(1-t))

So all you need do is make the tax rate t =99.9999999% and voila – infinite Y!

Note: this works even if the government runs a balanced budget (Gd=0)and Cm changes in response to a change in tax policy.

While this might be a "whole lot of nothing", to paraphrase Malcolm's words, I still have a question on modeling mathematically (even if it might be very simple maths) Keynesian theory.

ReplyDeleteWhy does J.M. Keynes, particularly in Chapters 8 to 10 of

The General Theory, use the formulation of MPC < 1?If you read his book properly, one of the big things he's really talking about is actually the stabilization of investment.

Correct me if I'm wrong, but he also seems to leave it to the reader to attach his "Marginal Propensity" concept to other parts of the macro-economy (i.e., Marginal Propensity to Invest, Marginal Propensity to Import, et cetera).

If Keynes wanted to make sure even the most careless of readers wouldn't take away the wrong message of mindless consumption spending at all costs, why didn't he explicitly spell out the other marginal propensities?

Why didn't he put the following thing in his book, which might have been more helpful?

MPC + MPI = 1

Thanks!

ReplyDelete"all of it's income" should be all of its income. Please delete this comment after typo corrected.

Thanks!

ReplyDelete"all of it's income" should be all of its income. Please delete this comment after typo corrected.

I think you are giving his argument way too much credit. When discussing spending/income in the context of the Multiplier Effect you look at how much was spent/earned over specific period of time. So, in effect the discussion is a conversation about rates. It is always preferable to discuss rates using calculus. However, this discussion can be expressed algebraically provided that you recognize that you are dealing with degrees of change and provide a coefficient that demonstrates how changing one variable will change the rate of other variables. The only thing Landsburg shows is that Landsburg makes .0000001Y. There is nothing to indicate that the initial relationship will hold if his income increases by $X.

ReplyDeleteWho are you talking about Old Odd jobs? I never said any of that.

ReplyDelete