The point is to keep trying until you get it right. I am sorry to say that I got it wrong last time, so I’m taking another shot at it.
Let’s consider, as does Simon Wren-Lewis a two-period model. The first period is in underemployment equilibrium. Let’s say that consumption in period 1 is given by the equation
C0 = 100 + bY0,
where b represents the marginal propensity to consume out of income.
Let’s say that investment is a fixed amount:
I0 = 100.
The expenditure (aggregate demand) equation is thus
E0 = 200 + bY0.
The equilibrium is determined by applying the equilibrium condition E0 = Y0, which gives us
Y0 = 200/(1-b).
Now the case that I posited in my previous post involved b = 0, reflecting income smoothing. This is tricky, because we have to make an assumption about what households expect their income to be in the next period, which can be assumed to be long relative to the initial period, though for simplicity I’m going to let the two periods be equal in length. If households expect income in the next period to reflect full employment, presumably they would try to increase their consumption now, spending more and increasing equilibrium income now, so there is an inherent inconsistency in the model which needs to be resolved, but I am not going to worry about that either. Let’s just take the model at face value.
In this equilibrium, note that consumption, C0, is 100, investment, I0, is 100, and saving, S0, is also 100.
What happens if the government immediately tries to intervene to raise income by increasing government spending, G0, from 0 to 100, and imposes taxes, T0, of 100 to finance its spending? The increased spending is only for this period and the taxation is only for this period, not the next one; in period 1, government spending and taxation go back to zero. What this does is to cause the consumption function to be revised as households choose a uniform level of consumption to be maintained for both periods, reflecting the liability to pay taxes this period, but no obligation to pay taxes next period.
Expecting income next period of 200, households would have chosen to consume 100 this period and 100 next period. But with a tax liability of 100 this period, households will choose, instead of consuming zero this period and 100 next period, to consume 50 this period and 50 next period. They have to borrow 50 this period to be able to pay their tax liability in order to have 50 left over for consumption. Next period, they will have to repay the loan of 50, and will have only 50 left over for consumption (income remaining at 200 with consumption equal to 50 and investment equal to 50, the loan repayment of 50 corresponding, it seems to me, to exports to shipped to foreigners). So the new consumption equation is
C’0 = 50 + bY0, where b is again equal to 0.
Now adding government spending and taxes, we have G = 100 and T = 100, so our new expenditure equation becomes
E’0 = 250 + b(Y’0 – 100).
But since b = 0, this reduces to
E’0 = 250 = Y’0.
We still have I0 = S0 = 100. Since b = MPC = 0, (1-b) = MPS = 1. The increase in income from 200 to 250 is just enough to generate another 50 in savings to offset the 50 in borrowing required to keep consumption level at 50 in period 0 and period 1.
The increase in government spending and taxes of 100 in period 0 raises the period-0 equilibrium (as compared with the case with no government spending and taxes) is 50, so the multiplier is .5.
Of course, this is not a full-equilibrium solution. A full equilibrium should have Y1 also equal to 250 instead of 200, which means that consumption could have been increased by 25 in both periods, but I haven’t worked that solution out yet.
The reason why in this post I arrive at a result different from the result in my previous post is that I made a simple flunk-the-quiz mistake in the previous post, reducing the expenditure curve by 100 to reflect the reduction in disposable income from taxes as if it were a permanent reduction in disposable income rather than a one-period reduction in disposable income. So instead of assuming the MPC was 0 as I wanted to do, I was assuming, for purposes of the effect of taxes on consumption, an MPC of 1. Yikes! My assertion that everything depended on a positive MPC was entirely wrong. In a simple Keynesian model, you get a balanced-budget multiplier of 1 provided that the MPC is less than 1. That was a pretty bad blunder on my part, and I apologize. Scott, himself, seemed to perceive that something was amiss in a comment on the previous post, so I hope that we are now converging toward a solution.
Again my apologies for hastily posting my previous post without checking my work more carefully. I had better get some rest now.
Uneasy Money 
David,
So in essence because of the initial underemployment equilibrium individuals can borrow 50 to “consumption smooth” without affecting I because the economy has unused savings that can be drawn upon by these individuals to increase their consumption in period 1. Is that a correct understanding of your post ?
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This horse has been whipped until it dropped, then whipped until it died, and now David Glasner is whipping the ground in which the horse has been buried.
Meanwhile, Bernanke is all but conceding he is not stimulating enough, and yet he is not stimulating enough. Out there in the real world. Bernanke’s presentation today is remarkable.
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I deeply appreciated this as a lay person (or, at least, I deeply appreciated what I could follow). The question I have is whether the implication of this is that an elevated propensity to consume using your income (rather than pay taxes, service debt, or save) will screw this whole pooch.
In other words, this model seems to imply that a recovery is BETTER served in the near term by the paying off of debt rather than the consumption of more stuff, which seems to run counter to the idea that consumer spending is good for a recession. Am I reading this wrong?
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Rob, It’s not because of the underemployment equilibrium that they “can” borrow, it’s why they want to borrow once they are hit with a temporary tax that will expire when the next period arrives. Investment is assumed to be already planned and in the pipeline. The planned expenditures (private consumption, private investment and government spending) generate enough income to produce the savings that allow investment to be financed at full employment. It’s a very simplified model, but it accommodates the Lucas/Cochrane objection to the standard Keynesian model in the sense that if the increase in government spending and taxes were permanent there would be no effect on income because households would reduce consumption by the same amount as their expected future tax liability.
Benjamin, Sorry to annoy you, but the point of this blog is to try to improve my understanding of economics and economic policy. It’s not a propaganda platform for any movement or cause, including market monetarism. Take it or leave it; it’s your choice. But I hope that you choose wisely. Sometimes, the road to understanding is bumpy. And I’m sure there will be still be something left to say about Bernanke and the FOMC, so stay tuned.
Lewis, Thanks for the kind words, which are especially welcome at this moment. Actually, I don’t think that you are drawing the correct inference from the model, which is not that debt is bad per se or that debt in inconsistent with a recovery. The trick is to know how much debt you can afford to pay back. That is tricky, because in an uncertain world we cannot know how much income we will have in the future on which to draw for purposes of debt service. There is no simple formula to use for making that calculation.
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David,
I see that they are borrowing it because they want to implement consumption smoothing. What I was driving at is why it is that they are able to borrow this 50 without messing up investment spending. In a healthy economy with no underemployment and no increase in the money supply then if individuals increase borrowing they can only do so only by cutting off investment elsewhere (and of course causing interest rates to increase). My point was that there must be unused money balances in the system at the underemployment equilibrium that allows additional borrowing without impacting interest rates and therefore investment
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Something is making me a bit uncomfortable. If MPC=1, Y is infinite. That bothers me because while I think MPC=1 is empirically unrealistic, it’s also not entirely absurd and so should not lead to an absurd result.
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Rob, Ok, your intuition is right. For the Keynesian cross result, you need to assume that planned investment is not deterred by an increase in the interest rate. That assumption can be satisfied if there is absolute liquidity preference (a liquidity trap) so that people are willing to lend their idle cash balances at the slightest increase in interest rates.
PrometheeFeu, The way to think of an MPC equal to one is that the expenditure curve is parallel to the 45-degree line in the Keynesian cross. There is no intersection and hence no equilibrium. The system explodes when there is any spending injection because nothing leaks out in the form of savings, so expenditure and income keep growing. Of course resource constraints prevent output from growing without limit, so the increase in expenditure would be matched by increasing prices. The other way to salvage the model is to assume that there is no autonomous component to spending in which case the expenditure curve coincides with the 45-degree line and every point on the 45-degree line is an equilibrium. Sorry that’s just the way the model works.
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@David Glasner:
OK, that makes a lot more sense. In a way, MPC=1 is almost like infinite velocity. As soon as income flows in your wallet as income, it immediately flows back out as consumption into another person’s wallet as income again and immediately flows out of their wallet as consumption again etc…
I know we’re doing this one step at a time, but it sounds to me that it’s important to look at the next period. (All in period 1)
C = 50 + bY
I = 100
E = Y = 150 + bY
b = 0
Y = 150
It sounds as though the boost you get in year 0 is something you just pay for in year 1. Though obviously, that’s 2 partial equilibria where agents are not correctly anticipating the income in year 1.
Let me try the general equilibria:
C = k + bY
b = 0
C = k
Y0 = k + I + G
Y1 = k + I
Y0 – Y1 = G
G = T
Y0 – Y1 = T
So it does look as though whatever stimulus you get in year 0, you just pay back in year 1.
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OK, I must have messed something up. My above result is pretty absurd.
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If C = k+b(Y-T) and b<1 then:
Y0-T0 = k+b(Y0-T0)+I+G0-T0 = k+I+b(Y0-T0) = (k+I)/(1-b)
Y1-T1 = k+b(Y1-T1)+I+G1-T1 = k+I+b(Y1-T1) = (k+I)/(1-b)
so disposable income doesn't change from one period to the next, and therefore neither does C.
So…
Y0 = (k+I)/(1-b)+G0
Y1 = (k+I)/(1-b)+G1 = (k+I)/(1-b)
Y0+Y1 = 2(k+I)/(1-b)+G0
So you still get a multiplier of 1.0 regardless of b. And there is no "pay back" the next year. The additional output is just not sustained. It's just G0-for-free in year zero. Though not exactly free… someone actually had to work more.
Now, if you impose the tax in the second period… T1=G0
Y0-T0 = k+b(Y0-T0)+I+G0 = k+I+G0+b(Y0-T0) = (k+I+G0)/(1-b)
Y1-T1 = k+b(Y1-T1)+I+G1-T1 = k+I-T1+b(Y1-T1) = (k+I-G0)/(1-b)
so now disposable income does change from one period to the next, and so does C.
C0 = k+b(k+I+G0)/(1-b) = k/(1-b)+(I+G0)b/(1-b)
C1 = k+b(k+I-G0)/(1-b) = k/(1-b)+(I-G0)b/(1-b)
Y0 = (k+I+G0)/(1-b)
Y1 = (k+I-G0)/(1-b)+T1 = (k+I-bG0)/(1-b)
Y0+Y1 = 2(k+I)/(1-b)+(1-b)G0/(1-b) = 2(k+1)/(1-b)+G0
That is, the short-run multiplier is 1/(1-b) but the long-run multiplier is 1.
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Oh, and in the latter case, b=0 implies that C0=C1… that is, there is consumption smoothing.
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David-
I look forward to your positive contributions to the discussion of Bernanke’s monetary policy. While Americans are unemployed, and GDP is 13 percent below trend, unit labor costs are falling and inflation is coming in a madam, zip, zero.
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@D R:
OK, thanks. I see it now. Y0 – Y1 = T0 is still true (assuming T1 = 0) but it doesn’t mean what I said it meant above. (I had one food in the land of partial equilibria and one foot in the land of general equilibria).
And G0=T1 with b=0 basically shows that under a consumption smoothing assumption Ricardian equivalent is right: taxes now or taxes later is the same thing and the multiplier is still 1.
It’s now all clear in my head.
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David,
I just want to thank you for your efforts and your blog, even if doofuses like myself will probably require a few years to figure it all out. Keep up the good work!
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David, If both Y-T and C fall by 50, I don’t see where there is any consumption smoothing.
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Scott, I understand consumption smoothing to mean that once the tax is imposed consumption in both periods is equal, not that consumption post-tax is equal to consumption pre-tax. Is that what you mean?
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David, Consumption smoothing means that C changes by less than disposible income. If it changes by exactly the same amount as disposible income, then you aren’t smoothing consumption at all.
What’s the income in the next period?
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Scott, We are talking about a 2-period model in which there is a Keynesian unemployment equilibrium in the first period. We then perform a comparative statics experiment in which we increase government spending in the first period but not the second. Consumption in the first period is based on expected income in the second period and consumption smoothing implies that households want consumption in both periods to be equal. You have to make an assumption about what expected income is in the second period, presumably (but necessarily) expected income in the second period is equal to the level of income consistent with full employment. Do you have a problem with this set up?
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I did the math in partial equilibria and DR did the math in general equilibria.
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“And G0=T1 with b=0 basically shows that under a consumption smoothing assumption Ricardian equivalent is right: taxes now or taxes later is the same thing and the multiplier is still 1.”
It makes sense that there is no difference, because b=0 implies that C=k regardless of… well… anything… and so is always smooth whether or not you assume consumption smoothing. (With C=k, consumers cannot fail to smooth consumption, so the additional assumption is meaningless.)
That means consumption does not change even in response to a rise in permanent income. So the model is pretty strange. Holding C and I constant in a closed economy must imply a multiplier of exactly 1.0 on G– irrespective of consumption smoothing– because Y=(k+I)+G.
Obviously, b=0 is a degenerate case– not terribly interesting, and surely invalid when the economy is at full capacity for production.
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D R and PrometheeFeu, I have not gone through your math, but I got into trouble by not recalling that any change in expected (permanent) disposable income causes a recalculation of k. Are you repeating my mistake? The one period mpc (b) is equal to zero, but k changes. So C and k do change in response to a change in expected (permanent) income, i.e., the consumption function shifts up or down even though the slope of the function with respect to current income is 0. You can’t calculate the multiplier for a change in government spending in the current period only in the conventional way.
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