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What about distributional effects and a declining marginal utility of consumption?

is this a question about the model or about the real world?

Jim:

1. There's declining MU of both present and future consumption. Declining MU of consumption explains why we want to save when income is temporarily high, and dissave when it's temporarily low, to smooth out our MU of C. It explains why the mpc out of *temporary* income is less than one. It doesn't say that the mpc out of permanent income is less than one.

2. suppose different people have different mpc's out of permanent income (though I can't see why they should, unless they have weird non-homothetic preferences [things don't scale up]). Why should that make the average of their mpc's less than one?

3. It's about both model and the world. I'm not sure about either.

Things don't scale up.

A low income person is very likely to spend all their income on things close to the base of Maslow's hierarchy of needs. A high income person will have all that covered with a fraction of their income. Saving is one of the things that happens with higher income, bingo: mpc < 1.

Some high income earners do have a high mpc. When they bitch about not being able to consume enough, they get mocked and ridiculed (I am thinking about the Chicago prof, do you need a link?).

Also, one of the things that happens with higher incomes is game playing. Some of the popular games include the stock market and related activities. This shows up as saving, more mpc < 1.

Just to add. I think mpc < 1 is an observed feature of the world. It needs to be bolted in to any model that needs it as a parameter, and any model that doesn't look like that should be rejected.

Jim: quoting myself (after fixing my typo): "To a first-order approximation, economies that are twice as big will have twice the consumption and twice the investment. Everything just scales up. The marginal propensity to spend -- consume plus invest -- is roughly one."

Canadian incomes have more than doubled over the last century. If you were right, so that consumption did not rise in proportion to income, we would have seen steadily increasing savings and investment as a fraction of income, and steadily falling real interest rates. If my assumption is exactly right, those fractions and the real rate of interest will stay exactly constant.

(And I'm not saying my assumption is exactly right, just that it is a reasonable approximation. That the marginal propensity to consume plus the marginal propensity to invest could be either greater or less than one, and one is a reasonable approximation.)

You are thinking micro cross-section; I am thinking macro time-series.

Animal spirits. In a deflationary environment, a commitment to spend $1/yr would not change any individual's expectations, so (pace the fallacy of composition) neither would aggregate expectations change. The sum of a finite number of zeros is zero.

In a deflationary environment mpc is strictly lower than in an inflationary environment, unless everyone's discount rate is 0 under all circumstances. The condition does not hold. Therefore mpc must be strictly less than one.

In a liquidity trap, the capital/output ratio is by definition higher than desired. (People wish to hold money in preference to the capital goods that they do have.) No investment.

Greg: why isn't output zero already?

I propose that the panglossian framework has crowded out any rational response to a demand failure:

Since there can't be anything wrong with, say, relative prices, income inequality, balance sheets, the production base, rent-seeking, etc, then we are left with talk therapy and goat sacrifices for remedies.

My essential point is that we're not in equilibrium. Other things are not equal.

Excuse me, I meant "a liquidity trap is not an equilibrium."

I didn't understand your addendum, but:

“If the desired capital/output ratio is independent of output, everything should just scale up. If output doubles, the desired capital stock should double, and so should desired investment, in the long run. And it would more than double in the short run, as firms tried to adjust the actual capital stock to the desired.”

That results in a contradiction.

If MPC = 1, that means S = 0.

And that means if I > 0, it must be financed by a current account deficit.

But if there is a current account deficit, the multiplier effect from MPC = 1 is diminished by MPM > 0.

So the multiplier can’t be infinite.

Perhaps you explain how your addendum repairs this contradiction, Nick.

Well, there are only so many people and so many tools. At any given point the economy can only produce what the combined effort of all the people using all the tools can make, and that must be a finite number, right?

The real ecomomy can't ever be infinite, so the multiplier can't be either. If everyone is employed and can fill all their needs with current production then how can the multiplier even work at all?

Why isn't the multiplier infinite? because logic tells us it isn't. Multiplier calculations never take into consideration the cost of each dollar spent. Government is highly inefficient, there is a huge handling fee before any money can be spent.

This seems like an easy question: because money is a good worth holding (for various reasons) and like all other goods it is a substitute for many other goods, when people get more of it they spend some amount of it but also keep some for all the reasons people normally hold money.

Don't forget the demographics. e.g. Japan's population/age profile is one of the reasons they've stayed in the trap.

@Nick, yes we are talking about different things. Another issue is the time discounted estimate of risk. There is no way I am going to believe the goat sacrificers will continue sacrificing goats forever. Something will happen to make them stop. Push the time horizon out far enough and the probability of Something becomes 1. So now you have a finite series.

For another take.

Krugman has a post on the subject. Care to comment?

I actually don't understand him. Why are the two curves different? The only thing I can think of is that the first curve is how people think the economy will respond and the second one is how it actually responds.


Both Phillips curves look odd. They imply that 5% unemployment can be achieved when inflation approaches infinity. Perhaps, that's what Krugman's dream is, kind of Weimar/Zimbabwe solution ?

Hard to say.

RSJ: why should any of that stuff make mpspend /=1?

Greg: so if it's not an equilibrium, why are we still here?

JKH: "If MPC = 1, that means S = 0."

Nope. you are confusing MPC with APC.

Plus, I didn't say that MPC=1 I said MPspend = MPC+MPInvest = (approx) 1

Ed. The multiplier is a ratio. There are two ways to make the ratio infinite: make the numerator infinite; make the denominator infinitesimal. I'm working on the second.

joy: it doesn't matter, for my purposes, if the government is very wasteful. If they can increase AD by $1, it doesn't matter if all of that $1 is spent on totally useless stuff. And I'm not talking about (only) fiscal policy either.

jsalvatier: are you talking about the marginal propensity to hoard money, or the marginal propensity to save? If the former, we can print more money.

Roland: demographics would affect the APC. I'm talking about MPC. And the older people are the shorter is "permanent" (because they will die soon). So the mpc out of a temporary increase in income is higher for old people than for young people.

Jim: I already said that we can't assume goat sacrifices will continue forever. That's why I said I preferred monetary policy.

On Paul Krugman: one curve is an IS curve; the second curve is a Phillips Curve. We get an equilibrium where they cross. PK's curves cross twice (I think). So there are 2 equilibrium. If I spent the time putting them the "normal" way up, I could figure out which of the equilibria are stable.

It wasn't one of his clearest posts (must be the elections in the US getting to him), but it's not (obviously) logically wrong. It wasn't very clear to me either. But I can figure out what he must be trying to say. If I spent some time playing with his curves, I could make it clear.

MPspend = MPC+MPInvest

Nick, does anybody else use this expression, or is it your own formulation?

I don't think Krugman is looking for stability. I think he is looking for short term positive instability.

Maybe I'm missing something, but goat sacrifice sounds an awful lot like helicopter money. If the problem is that the interest rate is too high even at 0% and can't go lower except by increasing inflation, then doesn't goat money only work to the extent that it increases inflation (expected and/or actual)?

It occurs to me that large debt overhangs might throw a wrench in the works. The marginal propensity to pay debt (save) might be very high, so it eats up the permanent increase for some time.

For each person, all income is saved or spent on consumption.

y = c + s.

MPC and MPS are functions of y, idiosyncratic to that person.

MPC = dc/dy. MPS = ds/dy

Therefore MPC + MPS = dy/dy = 1 in all cases.

The multiplier theory has a control variable -- investment, together with a state variable (national income), and a feedback mechanism, in which national income is increased as a result of additional rounds of consumption.

the actual result is that Y(I) = MPC(I)*I + MPC(MPC(I)*I)*I + MPC(MPC(MPC(I)*I)*I)*I + ....

When everyone is trying to save, MPC(I) = 0 and MPS(I) = 1. So I needs to be large enough to make a difference -- to satiate the savings desires. You can imagine, say a savings desire shock that moves MPC(I) so that it is equal to 0 for I < I_0 and is equal to log(I/I_0) for I > I_0.

You can differentiate implicitly if you want, and get MPI(Y) = 1/(1-MPS(Y)), but the causation goes from I to Y in the multiplier. Y is the resulting outcome of the initial injection of investment.

You may not agree with the theory.

If you want to formulate some other multiplier theory in which some other external injection -- e.g. CB bond purchases -- can cause national incomes to go up, then go ahead and formulate the theory. Show how national incomes increase as a result of CB bond purchases.

Specify your forcing variable, the state function, and your feedback mechanism.

"In the stagnation regime, inflation is trapped at a low steady deflation level, consistent with zero net interest rates, and there is a continuum of consumption and output levels that may emerge."

I think he makes a common economist mistake. Underlying this result is the steady state sampling assumption, that inventory cycle changes don't matter. Can the stagnant state exist? No, Say's Law still applies. The trader will take his seemlessly useless inventory and simply lengthen the inventory cycle making it within range again. As a result he buys input less often, but will buy in larger quantities. The trader then occupies a greater share of the production spectrum, performing more processing inside the firm and gaining economies of scale but losing market variation at the consumer.

When economists think the yield curve shrank suddenly they are wrong; it really just lowered its dimensionality and shrank somewhat. So this quoted author looks at the manifold with the precision of yesteryear and sees it to be flat; but the economy has increased the economies of scale already, lowering its precision to match the flat manifold.

The bad equilibrium is an illusion.

Nick,

I thought multiplier theory was based on “exogenous” injections of disposable income, originating from investment, or government spending, (or exports?) That income then gets multiplied into a corresponding increment of consumption spending. The math works in such a way as to equate the initial injection to cumulative saving in equilibrium. The original income injected ends up being saved by the non-injecting sector and consumption plays catch up in order to balance against that saving according to MPC. E.g. it catches up to the initial income created and saved to offset an investment injection, or it catches up to the initial income created and saved by the non government sector to offset a government expenditure injection. There is no money left over to account for additional investment in either case, because any money left over from consumption must be left as part of cumulative saving in order to reach balance against the original injection. Any additional investment would create new income separate from the multiplier stream, so there is no MPInvest as part of that process. I asked if your (MPspend = MPC+MPInvest) was original. It certainly looks odd to me as part of a multiplier process, and investment embedded in a single multiplier process looks inherently destabilizing to me. Your MPInvest should be treated as part of a separate autonomous injection process, or the math for the multiplier become destabilizing.

In other words, MPC and MPS fully account for the multiplication of the original income change. Your MPinvest is something beyond that process, creating a new income stream that is separate from the originating income stream and its follow up multiplier process. You need to specify how the MPinvest function comes about and what it depends on and how it relates to the original income stream. E.g. if the original income stream and multiplier revolves around an investment injection, your MPinvest becomes very suspect because the multiplier simply resolves the initial income corresponding to the initial investment into an ultimately required saving component. New MPinvest can’t flow from that same saving component since it is already required to resolve the balance between prior investment and saving. You may be confusing stocks and flows there.

OK guys. This is disappointing me. You didn't get what I was trying to say. My fault? Your fault? Who knows. Let me try again.

Start with the first year textbook Keynesian Cross diagram. Desired Aggregate Expenditure on the vertical axis, and income on the horizontal. Standard 45 degree line showing the semi-equilibrium condition that AE=Y.

The AE curve is an Engel curve. It shows AE as a function of Y. AE=F(Y,X), where X is everything else that might affect AE, other than Y. So X in my model is monetary policy, fiscal policy, goats, whatever.

The derivative of F with respect to Y (Fy) , the slope of the AE curve, is called the marginal propensity to spend. The textbook model assumes the marginal propensity to spend is positive, but less than one. And assumes the intercept of the AE curve is positive. So we get one equilibrium, and it's stable.

The effect of a change in X is given by dY/dX= Fx[1/(1-Fy)]. That term [1/(1-Fy)] is the generic multiplier. It's what multiplies the immediate "first-round" effect of any change in AE caused by a change in X.

Why should Fy be less than 1? Why should the multiplier be finite?

In the simplest textbook model, investment is assumed exogenous wrt Y, and consumption is the only part of AE that depends on Y, and the mpc is less than one. So Fy is less than 1. Both assumptions are questionable.

The modern theory of the consumption function (from Irving Fisher through Milton Friedman up to New keynesian Euler equation approaches) tell us that the mpc out of a temporary change in Y will be much less than 1, approaching zero for very transitory changes in Y (except for borrowing-constrained people, where the mpc will be 1). But the mpc out of permanent income is not required to be less than 1. It may be less than 1, but doesn't have to be. It depends on the utility function, rate of time preference, rate of interest, whatever. And it will be a lot bigger, and closer to 1, than the mpc out of a temporary change in income.

And, by the way, an mpc=1 does not mean that savings=0. It means that the *change* in savings if income changes will =0.

Plus, it is unreasonable to assume that Investment is exogenous wrt Y. The paleo-Keynesian multiplier-accelerator model incorporated this effect of Y on I. Since investment is a flow, and capital a stock, and the desired stock of capital is assumed to depend positively on Y, the precise relationship between I and Y depends on what you assume about depreciation and the adjustment process of the actual to the desired capital stock. But generally you get an investment function of the form I=H(Y,K) where Hy is positive. (But since K depends on lagged I it's a dynamic relationship, and I'm skating over those dynamics).

So the marginal propensity to spend out of permanent income, Fy, is the marginal propensity to consume out of permanent income plus the marginal propensity to invest out of permanent income. And nothing in theory says the sum of those two has to be less than one, let alone much less than one.

So the AE curve doesn't have to be flatter than the 45 degree line. It might be coincident with the 45 degree line over some range. It might even be steeper where the two curves cross, so the equilibrium is unstable.

If you ignore the dynamics of the investment accelerator, and just look at the long run investment relationship, where gross investment just replaces depreciation, and so is proportional to the desired capital stock, which in turn is proportional to income, a good benchmark assumption is that everything scales. Fy=1. And since AE=Y today (otherwise we wouldn't be here today), that means the AE curve coincides with the 45 degree line. And a very small change in X, as long as Fx is positive, will create an indefinitely large increase in Y, till be hit capacity, and the model is no longer valid.

One point of clarification:

“And, by the way, an mpc=1 does not mean that savings=0. It means that the *change* in savings if income changes will =0.”

I didn’t say savings.

It said it means saving as a flow is zero – the infinitesimal calculus being the limit.

Saving is a flow. Savings is the stock.

Correct?

"Plus, it is unreasonable to assume that Investment is exogenous wrt Y."

Nick, I'm just trying to get this clear. What you are saying is not how the multiplier is normally taught or presented in textbooks is it? You are doing something quite different here?

"Fy, is the marginal propensity to consume out of permanent income plus the marginal propensity to invest out of permanent income. And nothing in theory says the sum of those two has to be less than one"

If the sum is greater than one, does that mean a current account deficit, or something else?

JKH: "Saving is a flow. Savings is the stock."

I'm talking about flows, in both cases. mpc=1 does not mean the flow of saving is zero. mpc=1 means that a change in Y does not change the flow of saving.

[Edited by NR to correct typo correctly spotted by JKH.]

"mpc=0 (I think you mean 1?) means that a change in Y does not change the flow of saving"

Right. I'm starting at the level of the "injection" which is itself a change in income, and looking at the flow of saving and income propagated within that level. I.e. I'm isolating the injection (change in income) that is the basis for what I thought was the usual multiplier mechanism.

Correct?

Thanks, Nick. Looks like this is beyond me.

JKH: Damn! Yes. I meant "mpc=1". (I will go back and edit my comment).

Patrick: "It occurs to me that large debt overhangs might throw a wrench in the works. The marginal propensity to pay debt (save) might be very high, so it eats up the permanent increase for some time."

I ought to do a post on this, sometime. I disagree.

There are two possibilities:

1. The debtor will not default. In this case, consumption is low, but it doesn't mean the mpc is low. The mpc could be 1. You are paying the debt anyway, and if you get $1 more permanent income, you consume $1 more.

2. The debtor will default. Here, an extra $1 of income all goes to paying down the debt, and reduces the amount of default. But that puts an extra $1 of income in the creditor's pocket. It's just a transfer. It's the creditor's mpc that matters.

Nick @ 10:01 pm:

On Paul Krugman: one curve is an IS curve; the second curve is a Phillips Curve.

Both curves are functions in the unemployment-inflation coordinates, thus Phillips curves. The first one has axes swapped for some reason.

An IS curve is plotted in the Y-i coordinates.

Still unclear why Krugman thinks we need infinite inflation to get 5% unemployment.

Re. MPC/MPS. I am accustomed to RSJ's exposition where MPC+MPS=1, trivially.

But I am not a macro guru by a long shot.

Off topic: I disagree with those who thought Krugman's post was muddled or confusing. To me it seemed lucid and, in fact, compelling. A far more plausible reason why "the Fed's message is self-contradictory" than Nick's.

Regarding the actual subject of your post, the whole question seems to be a trivial or tautological aspect of the "first year textbook Keynesian Cross diagram." If the premise is that we are in a "liquidity trap", then of course the slope of the demand curve must be less then unity - that is the only way to achieve our unfortunate equilibrium. Equally, the whole point of the Keynesian policy recommendation is to induce a disequilibrium by raising this slope above unity. That is what we are trying to achieve, not an assumption excluded by the model!

So why won't a proposal to sacrifice one goat per year in perpetuity be enough? That is not a theoretical question but an empirical one. Try it and see! Take a poll in one of your classes - promise to start sacrificing goats and ask your students whether they are feeling a burning itch to go out there and start spending.

Let's switch discussions about Paul Krugman to my new post.

vjk: "Re. MPC/MPS. I am accustomed to RSJ's exposition where MPC+MPS=1, trivially."

Of course it's true, trivially. But that doesn't say anything about whether MPC is less than 1. And it doesn't say anything about whether the MPSpend (as opposed to the MPS (i.e. MPSave) is less than 1. There's more to spending than consumption. There's investment too. RJS assumes investment is exogenous wrt income. I don't.

Phil: "If the premise is that we are in a "liquidity trap", then of course the slope of the demand curve must be less then unity - that is the only way to achieve our unfortunate equilibrium."

First off, the standard textbook exposition assumes the slope of the AE curve is less than unity whether or not we're in a liquidity trap.

Second, we might be in an unstable equilibrium (in this space). Or the AE curve might run along the 45 degree line, so it's a metastable continuum of equilibria.

All these reasoning only will make sense if the duration of the liquidity trap was infinite; but a liquidity trap will not last forever - then sacrificing a goat every year will not rise the income in $1 all years (only until liquidity trap ends); then, because the rise of your permanent income will be less then $1/year, your consumption will also rise than $1/year, then your propension to consume will be less then 1, and we have again the less-than-infinitum multiplier

Miguel: I think you are right. I was only looking at the worst case scenario. But if we are going to escape in (say) 10 years, a commitment to raise the price level permanently beginning in year 2020 would have to replace the commitment to goat sacrifices after 2020.

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