Thursday, 19 January 2012

Consumption smoothing and the balanced budget multiplier

Only for economists

                In some of the debate following this post, and then this, there often seems to be a big distinction drawn between models based on consumption smoothing, and the old fashioned balanced budget Keynesian multiplier. Even Paul Krugman felt it necessary to say that he ‘never said a word about the balanced budget multiplier’. Now of course the models are different. However I want to suggest that in the context of fiscal expansion in a recession caused by demand deficiency, the balanced budget multiplier story can be retold in a manner consistent with consumption smoothing.
                The most basic model of consumption smoothing involves two periods. Let period 1 be a demand deficient recession, and so output is determined in a Keynesian manner by aggregate demand. Period 2, which is much longer, is Classical, and nothing changes in period 2. (If having a two period model of unequal lengths is a worry, think of period 2 as being divided into a large number of sub-periods of equal length to period 1, but where every sub-period is Classical.) We keep monetary policy neutral by assuming the real interest rate is constant. Optimising consumers will then spend a fixed proportion of their permanent income in period 1: that is consumption smoothing. Let’s call this proportion c, which could be quite small. There is no investment, and the economy is closed.
                The government now increases government spending by G in period 1 only, and taxes rise by the same amount in period 1. The ‘direct’ or ‘first round’ effect is that consumption in period 1 falls by less than G, because the impact of higher taxes on consumption is smoothed via permanent income. That is as far as I needed to go in my ‘Mistakes’ post to make the point I wanted to make. However it is obviously not the end of the story, because higher output implies higher income. What happens to output eventually (call the answer Y)? Well consumption rises/falls by Y-G times the fixed proportion c, so we solve Y=G+c(Y-G), which of course implies Y=G, a multiplier of one. This is not only the same result as given by the Keynesian balanced budget multiplier, but the mechanics are identical. Consumption does not change at all: higher period 1 income offsets the higher taxes. We do not need to worry about any knock on effects in period 2, because permanent income ends up unchanged. So the simple Keynesian balanced budget multiplier need not be considered some ancient fossil that we are forced to teach undergraduate students, but a simple expression of what consumption smoothing implies in a particular context.
                We could get to the same result using consumption smoothing alone, by noting that consumption in period two is tied down by (classical) Y and permanent G. Second period consumption and the Euler equation then fixes period 1 consumption, as real interest rates are unchanged by assumption. So any change in government spending in period 1 leads to an equal increase in output.
Woodford, in section 2 of the paper noted by Krugman and myself, does something with similarities to this, but with more elegance. My period 2 becomes the steady state, a steady state in which (given the usual assumptions) the real interest rate equals the rate of time preference. With real interest rates fixed at this value in all periods, consumption is equal in all periods, so any temporary change in government spending leads to an equal temporary change in output. We get a multiplier of one. With this benchmark, it is then intuitive to see how the multiplier will fall if real interest rates are not constant but rise. Equally, if we are at a zero lower bound, the multiplier will be greater than one because higher output generates inflation, which reduces real rates.
Now I am not trying to say here that the simple, most basic Keynesian multiplier apparatus is in any sense ‘as good as’ consumption smoothing. In fact, if I was writing an introductory macro textbook, I would start with the two period consumption model and consumption smoothing, and mention the current income Keynesian consumption function only in passing. (My reasons for doing this are explained here.) All I want to suggest is one way of reinterpreting the balanced budget multiplier that is consistent with consumption smoothing. I think it is also nice that all this stuff ends up with the same result, a multiplier of one. If someone wants to argue that the multiplier is zero they need some additional argument, and as I suggested here, I have yet to see one that seems appropriate to the current situation.


  1. So you "defeated" Sumner's criticism by assuming that the economy doesn't have investment but does have savings.

    That's fine as far as your model goes, but no one should mistake your model for looking anything like the real world.

  2. John Schultz, quite often textbook models abstract from capital accumulation, so 'I' drops out of the model. You can add it back in without altering anything important. Since the post is explicitly addressed to economists the danger that any reader will confuse the model with reality is slight.

    I like two-period models. Infinite-horizon models like Woodford's make for very tedious reading.

  3. Well, I think this is useful but you've let Sumner steer the discussion away from the original topic, which is their (Sumner, Cochrane, etc.) error in applying Ricardian Equivalence to a temporary fiscal stimulus. He is the one who brought up BBM, and has managed to twist himself up over it, distracting all his readers from his failure to respond to your original post. Yes, I know it really does come down to the BBM and this is a good post, but this is a debate, not a seminar.

  4. John Schultz,

    The result would still apply in a model with investment. The point is that first period income increases so consumption can be maintained without a fall in savings, so there's no need for investment to fall.

    1. "The point is that first period income increases ..."

      That seems to be assuming what is to be proven, that income increases, which strikes me as circular reasoning.

      Yes, if you think that the Keynesian model is correct, then Wren-Lewis is simply demonstrating that this is how the Keynesian model functions when a temporary stimulus occurs.

      However, I thought the whole point of this exercise was to convince people who don't believe that the Keynesian model is correct that even using their models that fiscal expansion can stimulate output (at least in the short run).

      Kinda like Krugman tried to do here:

    2. No, the point of the exercise was that Lucas and Cochrane were trying to say that the Keynesian model must be wrong, because any increase in government spending would be offset by taxation. This claim can be refuted via consumption-smoothing.

      Then Sumner claimed that consumption-smoothing was not relevant and somehow invalidated the Keynesian model, because he associated consumption smoothing with a decrease in saving, which he argued implied a decrease in investment. Simon's post above shows why that claim is wrong.

      The point is, if you want to argue that the Keynesian model is wrong, you have to come up with a valid objection. For example, If we assume prices are flexible, then we can infer that we are already at the natural rate of output, and hence changes in demand are irrelevant. Nobody seems willing to make that argument, so instead we get silly claims that demand can't change based on accounting identities.

      As for the Krugman link, I think he may be slightly wrong on this. He is correct that temporary increases in government spending should increase demand (even with Ricardian Equivalence). However, in a flexible price model that would not expand the economy. It would just lead to inflation, unless the government spending was crowded out by an increase in interest rates.

    3. "This claim can be refuted via consumption-smoothing."

      Only if total output increases due to the increase in government expenditure (and equal taxation)!

      "For example, If we assume prices are flexible, then we can infer that we are already at the natural rate of output, and hence changes in demand are irrelevant."

      That essentially boils down to: the aggregate supply curve is vertical.

      "Nobody seems willing to make that argument, so instead we get silly claims that demand can't change based on accounting identities."

      Uh, no. It seems to me that the (neo)classical economists are basically asserting that either (a) the aggregate supply is completely vertical or (b) additional government expenditure can't move aggregate demand out, as Wren-Lewis assumes, because the private sector will perfectly compensate for it. That is if G, T goes up, then C goes down by an equal amount.

    4. Here is the key assumption that WL makes:

      "Let period 1 be a demand deficient recession, and so output is determined in a Keynesian manner by aggregate demand."

      That argument won't convince people who don't believe aggregate demand determines output in a Keynesian manner, even in a recession. Such people might also take argument with the phrase "demand deficient recession," which again seems to be assuming your outcome. They might argue that the economy "wants" to be in recession due to a previous bubble, etc.

    5. Well, let them make the argument. Ultimately, it's an empirical question. There seems to be plenty of evidence that the economy behaves as we would expect based on a Keynesian model (e.g., evidence that money is not neutral in the short-run) as well as evidence that prices are sticky (e.g., Krugman recently posted about Mike Mussa's paper on the effects of changes in nominal exchange rates on the real exchange rate:

      which basically disproves the idea of flexible prices).

      As far as I can tell, Lucas and Sumner accept the possibility of a demand-deficient recession (not sure about Cochrane), which is why they rely on silly arguments about fiscal policy being unable to affect demand.

      The idea that the economy wants to be in recession because of a previous bubble sounds like the Austrian theory that was disproved by Milton Friedman.

      The fact is, I would be quite interested to hear a convincing neoclassical explanation of the current recession. But the more I listen to people such as Lucas, Prescott, Cochrane, Fama, Kocherlakota etc. the more I think the emperor has got no clothes. And the more I read Keynesians like Krugman, Wren-Lewis, and to some extent Mankiw, the more I think they are on the right side of the argument.

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  7. I have just one big difficulty; namely how to reconcile the idea of consumption smoothing with that of the paradox of drift? I am afraid that we cannot have both.

  8. Yep. Nice clear exposition of the New keynesian result.

    Period one really needs to be short. Because if Y increases in period one, inflation will rise too. So inflation at the beginning of period 2 will be higher than it would otherwise have been. If inflation is "too high" at the beginning of period two (higher than target), then the central bank will need to raise r to bring inflation back down to target, and so period two won't be classical. C and Y would therefore have to be below the classical levels at the beginning of period two, which would reduce C and the multiplier in period one.

    It is best to interpret period one as a period when inflation is falling below target, and the central bank either can't or won't loosen monetary policy enough to bring it back to target. So the higher rate of inflation in period one, caused by the higher Y, caused by the higher G, is something that is desired.

    The best fiscal policy, for a multiplier bigger than one, would be to increase G in period one and at the same time announce a cut in G in period two. Holding Y=C+G constant in period two (the central bank adjusts r to keep Y and inflation constant) C would be higher than otherwise in period two, which by consumption-smoothing would increase C in period one.

  9. "So the simple Keynesian balanced budget multiplier need not be considered some ancient fossil that we are forced to teach undergraduate students, but a simple expression of what consumption smoothing implies in a particular context."

    I think I've already made the same point earlier than this post (although somewhat more crudely, maybe) here;)

  10. The MPC is quite flexible changing with time and if the old calculations with the balanced budget multiplier lead to the same result with the MPC replaced by consumption smoothing then they must be equivalent, in which case one should be able to express consumption smoothing in terms of MPC.

    Therefore it makes sense to assert that what consumption smoothing does is to lead consumers to change their aggregate MPC through a transformation of the form bY=a(Y+ΔY) or a = (bY/(Y+ΔY) where a is the new MPC in response to an one off event, like a lump some tax.

    As an example, let's set b = .5, I = 50 and G = 50 and Y = 200

    That is, Y = .5(200) + 50 + 50 = 200

    Now let's give an autonomous increase to G of 100, fully financed with a corresponding tax increase.

    Then Y = .5(300-100) + 50 + 150 = 300

    Now let's change b into a

    in which case we have

    Y = .5(300-100)/(300-100+100)(300) + 50 + 150 = (100/300)300 + 50 + 150 = 100 + 50 + 150 = 300

    leading to the same result as expected from the way that we have calculated a.

    Does this make economic sense? In my opinion it does. Faced with an one off increase in taxes, consumers change their MPsC in order to maintain consumption at the same level.

  11. Sorry about this, I have to sober up.

  12. Going back to my penultimate post if I may. I only realized that I stuffed up the examples very badly when on reflection it occurred to me that the value for the truncate multiplier could never be one at the end of the first iteration.

    After a good night's sleep the corrected examples follows:

    Now let's give an autonomous increase to G of 100, fully financed with a corresponding tax increase.

    Then Y = .5(200-100) + 50 + 150 = 250

    Now let's change b into a with a = (.5(200))/(200-100)= 1

    in which case we have

    Y = (1*(200-100)) + 50 + 150 = + 50 + 150 = 100 + 50 + 150 = 300

    If we keep iterating Y = .5(200-100) + 50 + 150 = 250

    eventually we will get Y = 300 or a multiplier of one.

    What a few drinks at the local RSL club can do.


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