I won’t bother with another encomium to Scott Sumner. But how many other bloggers are there who could touch off the sort of cyberspace fireworks triggered by his series of posts (this, this, this, this, this and this) about Paul Krugman and Simon Wren-Lewis and their criticism of Bob Lucas and John Cochrane? In my previous post, after heaping well-deserved, not at all overstated, praise upon Scott, I registered my own perplexity at what Scott was saying. Thanks to an email from Scott replying to my post (owing to some technical difficulties about which I am clueless, his comment, and possibly others, to that post weren’t being accepted last Friday) and, after reading more of the back and forth between Scott and Wren-Lewis, I now think that I finally understand what Scott was trying to say. Unfortunately, I’m still not happy with him.
Excuse me for reviewing this complicated multi-sided debate, but I don’t know how else to get started. It all began with assertions by Lucas and Cochrane that that old mainstay of the Keynesian model, the balanced-budget multiplier theorem, is an absurd result because increased government spending financed by taxes simply transfers spending from the private sector to the public sector, without increasing spending in total. Lucas and Cochrane supported their assertions by invoking the principle of Ricardian equivalence, the notion that the effect of taxation on present consumption is independent of when the taxes are actually collected, because the expectation of future tax liability reduces consumption immediately (consumption smoothing). Paul Krugman and Simon Wren-Lewis pounced on this assertion, arguing that Ricardian equivalence actually reinforces the stimulative effect of government spending financed by taxes, because consumption smoothing implies that a temporary increase in taxation would cause current consumption to fall by less than would a permanent increase in taxation. Thus, the full stimulative effect of a temporary increase in government spending is felt right away, but the contractionary effect of a temporary increase in taxes is partially deferred to the future, implying that a temporary increase in both government spending and taxes has a net positive immediate effect.
[See update below] Now this response by Krugman and Wren-Lewis was just a bit opportunistic and disingenuous, the standard explanation for a balanced-budget multiplier equal to one having nothing to do with the deferred effect of temporary taxation. Rather, it seems to me that Krugman and Wren-Lewis were trying to show that they could turn Ricardian equivalence to their own advantage. It’s always nice to turn a favorite argument of your opponent against him and show that it really supports your position not his. But in this case the gambit seems too clever by a half.
Enter Scott Sumner. Responding to Krugman and Wren-Lewis, Scott tried to show that the consumption-smoothing argument is wrong, and the attempt to turn Ricardian equivalence into a Keynesian argument a failure. I don’t know about others, but it did not occur to me on first reading that Scott’s criticism of Krugman and Wren-Lewis was so narrowly focused. The other problem that I had with Scott’s criticism was that he was also deploying some very strange arguments about the alleged significance of accounting identities, which led me in my previous post to make some controversial assertions of my own denying Scott’s assertion that savings and investment are identically equal as well as the equivalent one that income and expenditure are identically equal.
So what Scott was trying to do was to show that consumption smoothing cannot be an independent explanation of why an equal temporary increase in government spending and in taxes increases equilibrium income. Krugman and Wren-Lewis were suggesting that it is precisely the consumption-smoothing effect that produces the balanced-budget multiplier. Here’s Wren-Lewis:
Both make the same simple error. If you spend X at time t to build a bridge, aggregate demand increases by X at time t. If you raise taxes by X at time t, consumers will smooth this effect over time, so their spending at time t will fall by much less than X. Put the two together and aggregate demand rises.
This is not your parent’s proof of the balanced-budget multiplier, in which consumption decisions are based only on current income without consideration of future income or expected tax liability. It’s a new proof. And it drove Scott bonkers. So what he did was to say, let’s see if Wren-Lewis’s proof can work on its own. In other words, let’s assume that the standard argument for the balanced-budget theorem — that all government spending on goods and services is spent, but part of a tax cut is spent and part is saved, so that an equal increase in government spending and taxes generates a net increase in expenditure, leading in turn to a corresponding increase in income — is somehow false. Could consumption smoothing rescue an otherwise disabled balanced-budget multiplier
This was a clever idea on Scott’s part. But implementing it is not so simple, because if you are working with the simple Keynesian model, you can’t help but get the balanced-budget multiplier automatically. (A balanced-budget multiplier of 1 is implied by the Keynesian cross. In the world of IS-LM, you must be in a liquidity trap to get a multiplier of 1. Otherwise the multiplier is between 0 and 1.) At this point, the way to proceed would have been for Scott to say, well, let’s assume that something in the Keynesian model changes simultaneously along with the temporary increase in both government spending and taxes that exactly offsets the expansionary effect of the increase in spending and taxes, so that in the new equilibrium, income is exactly where is started. So, let’s say that initially Y = 400, and G and T then increase by 100. The balanced-budget multiplier says that Y would rise to 500. But let’s say that something else also changed, so that the two changes together just offset one another, resulting in a new equilibrium with Y = 400, just as it was previously. At this point, Scott could have introduced consumption smoothing and determined how consumption smoothing would alter the equilibrium.
But that is not what Scott did. Instead, he relied on arguments from irrelevant accounting identities, as if an accounting identity can be used to predict (even conditionally) the response of an economic variable to an exogenous parameter change. Let’s now go back to a more recent restatement of his argument against Wren-Lewis (a restatement with the really bad title “It’s tough to argue against an identity”). Here’s Scott responding to Paul Krugman’s jab that Lucas and Cochrane had committed “simple fail-an-undergraduate-level-quiz errors.”
First recall that C + I + G = AD = GDP = gross income in a closed economy. Because the problem involves a tax-financed increase in G, we can assume that any changes in after-tax income and C + I are identical.
By after-tax income, Scott means C + S, because in equilibrium, E (expenditure) ≡ C + I + G = Y (income) ≡ C + S + T. So if G = T, then C + S = C + I. Scott continues:
Suppose that because of consumption smoothing, any reduction in after-tax income causes C to fall by 20% of the fall in after-tax income. Then by definition saving must fall by 80% of the decline in after-tax income. So far nothing controversial; just basic national income accounting.
It is not clear what accounting identity Scott is referring to; the accounting identities of national income accounting do not match up with the equilibrium conditions of the Keynesian model. But the argument is getting confused, because there are two equilibria that Scott is talking about (the equilibrium without consumption smoothing and the one with smoothing), and he doesn’t keep track of the difference between them. In the equilibrium without consumption smoothing, Y is unchanged from the initial equilibrium. Because after-tax income must be less in the new equilibrium than in the old one, taxes having risen with no change in Y, private consumption must be less in the new equilibrium than the old one. By how much consumption fell Scott doesn’t say; it would depend on the assumptions of the model. But he assumes that in the equilibrium with consumption smoothing, consumption falls by 20%. Presumably, without consumption smoothing, consumption would have fallen by more than 20%. But here’s the problem. Instead of analyzing the implications of consumption smoothing for an increase in government spending and taxes that would otherwise fail to increase equilibrium income, while reducing disposable income by the amount of taxes, Scott simply assumes that consumption smoothing leaves Y unchanged. Let’s follow Scott to the next step.
Now let’s suppose the tax-financed bridge cost $100 million. If taxes reduced disposable income by $100 million, then Wren-Lewis is arguing that consumption would only fall by $20 million; the rest of the fall in after-tax income would show up as less saving. I agree.
Again, Scott is assuming a solution to a model without paying attention to what the model implies. The solution of a model must be derived, not assumed. The only assumption that Scott can legitimately make is that Wren-Lewis would agree that without consumption smoothing the $100 million bridge financed by $100 million in taxes would not change Y. The effect on Y (and implicitly on C and S) of consumption smoothing must be derived, not assumed. Next step.
But Wren-Lewis seems to forget that saving is the same thing as spending on capital goods.
I interrupt here to protest emphatically. There is simply no basis for saying that saving is the same thing as spending on capital goods, just as there is no basis for saying that eggs are chickens, or that chickens are eggs. Eggs give rise to chickens, and chickens give rise to eggs, but eggs are not the same as chickens. Even I can tell the difference between an egg and a chicken, and I venture to say that Scott Sumner can, too. Now back to Scott:
Thus the public might spend $20 million less on consumer goods and $80 million less on new houses. In that case private aggregate demand falls by exactly the same amount as G increases, even though we saw exactly the sort of consumption smoothing that Wren-Lewis assumed. But Wren-Lewis seems to forget that saving is the same thing as spending on capital goods. Thus the public might spend $20 million less on consumer goods and $80 million less on new houses. In that case private aggregate demand falls by exactly the same amount as G increases, even though we saw exactly the sort of consumption smoothing that Wren-Lewis assumed.
Scott has illegitimately assumed a solution to a model after introducing a change in the consumption function to accommodate consumption smoothing, rather than derive the solution from the model. His numerical assumptions are therefore irrelevant even for illustrative purposes. Even worse, by illegitimately asserting an identity where none exists, he infers a reduction in investment that contradicts the assumptions of the very model he purports to analyze. To say “in that case private aggregate demand falls by exactly the same amount as G increases, even though we saw exactly the sort of consumption smoothing that Wren-Lewis assumed” is simply wrong. It is wrong precisely because saving is not “the same thing as spending on capital goods.” I know this is painful, but let’s keep going.
Those readers who agree with Brad DeLong’s assertion that Krugman is never wrong must be scratching their heads. He would never endorse such a simple error. Perhaps investment was implicitly assumed fixed; after all, it is sometimes treated as being autonomous in the Keynesian model. So maybe C fell by $20 million and investment was unchanged. Yeah, that could happen, but in that case private after-tax income fell by only $20 million and there was no consumption smoothing at all.
What Scott is saying is that if you were to assume that savings is not the same as investment, so that investment remains at its original level, then C + I goes down by only $20. Then in equilibrium, given that G = T, C + S, private after-tax income also went down by $20 million, in which case consumption accounted for the entire reduction in Y, which, if I understand Scott’s point correctly, contradicts the very idea of consumption smoothing. But the problem with Scott’s discussion is that he is just picking numbers out of thin air without showing the numbers to be consistent with the solution of a well-specified model.
Let’s now go through the exercise the way it should have been done. Start with our initial equilibrium with no government spending or taxes. Let C (consumption) = .5Y and let I (investment) = 200.
Equilibrium is a situation in which expenditure (E) equals income (Y). Thus, E ≡ C + I = .5Y + 200 = Y. The condition is satisfied when E = Y = 400. Solving for C, we find that consumption equals 200. Income is disposed of by households either by spending on consumption or by saving (additional holdings of cash or bonds). Thus, Y ≡ C + S. Solving for S, we find that savings equals 200. Call this Equilibrium 1.
Now let’s add government spending (G) = 100 and taxes (T) = 100. Consumption is now given by C = .5(Y – T) = .5(Y – 100). Our equilibrium condition can be rewritten E ≡ C + I + G = .5(Y – 100) + 200 + 100 = .5Y + 250 = Y. The equilibrium condition is satisfied when E = Y = 500. So an increase in government spending and taxes of 100 generates an increase in Y of 100. The balanced budget multiplier is 1. Consumption and saving are unchanged at 200. Call this Equilibrium 2.
Now to carry out Scott’s thought experiment in which the balanced-budget multiplier is 0, we have to assume that something else is going on to keep income and expenditure from rising to 500, but to be held at 400 instead. What could be happening? Perhaps the increase in government spending causes businesses to reduce their planned investment spending either because the government spending somehow reduces the expected profits of business, by reducing business expectations of future sales. At any rate to reduce equilibrium income by 100 from the level it would otherwise have reached after the increase in G and T, private investment would have to fall by 50. Thus in our revised model we have E ≡ C + I + G = .5(Y – 100) + 150 + 100 = .5Y + 200 = Y. The equilibrium condition is satisfied when E = Y = 400. The increase in government spending and in taxes of 100 causes a reduction in investment of 50, and therefore generates no increase in Y. The balanced budget multiplier is 0. Consumption and savings both fall by 50 to 150. Call this Equilibrium 2′.
Now we can evaluate the effect of consumption smoothing. Let’s assume that households, expecting the tax to expire in the future, borrow money (or draw down their accumulated holdings of cash or bonds) by 10 to finance consumption expenditures, planning to replenish their assets or repay the loans in the future after the tax expires. The new consumption function can be written as C = 10 + .5(Y – T). The revised model can now be solved in terms of the following equilibrium condition: E ≡ C + I + G = 10 + .5(Y – 100) + 150 + 100 = .5Y + 210 = Y. The equilibrium condition is satisfied when E = Y = 420. Call this equilibrium 3. Relative to equilibrium 1, consumption and savings in equilibrium 3 fall by 30 to 170, and the balanced budget multiplier is .2. The difference between equilibrium 2′ with a zero multiplier and equilibrium 3 witha multiplier of .2 is entirely attributable to the effect of consumption smoothing. However, the multiplier is well under the traditional Keynesian balanced-budget multiplier of 1.
Scott could have avoided all this confusion if he had followed his own good advice: never reason from a price change. In this situation, we’re not dealing with a price change, but we are dealing with a change in some variable in a model. You can’t just assume that a variable in a model changes. If it changes, it’s because some parameter in the model has changed, which means that other variables of the model have probably changed. Reasoning in terms of accounting identities just won’t do.
Update (1/17/12): Brad DeLong emailed me last night, pointing out that I was misreading what Krugman and Wren-Lewis were trying to do, which was pretty much what I was trying to do, namely to assume that for whatever reason the balanced-budget multiplier without consumption smoothing is zero, so that an equal increase in G and T leads to a new equilibrium in which Y is unchanged, and then introduce consumption smoothing. Consumption smoothing leads to an increase in Y relative to both the original equilibrium and the equilibrium after G and T increase by an equal amount. So I withdraw my (I thought) mild rebuke of Krugman and Wren-Lewis for being slightly opportunistic and disingenuous in their debating tactics. I see that Krugman also chastises me in his blog today for not checking my facts first. My apologies for casting unwarranted aspersions, though my rebuke was meant to be more facetious than condemnatory.
Uneasy Money 