ChristianoBufferStock.tex

¹Contact: ccarroll@jhu.edu, Department of Economics, 440 Mergenthaler Hall, Johns Hopkins University, Baltimore, MD 21218, http://www.econ2.jhu.edu/people/ccarroll, and National Bureau of Economic Research.    

1 Introduction

Larry,

You wrote:

I’m with Bob Hall, who has been heard to remark that the only context in which that kind of general equilibrium really needs to be insisted upon is in the treatment of a medium-run model of the entire world. (The short run is not well captured by such models because all sorts of frictions, discontinuities, market failures, expectational errors, capital market frictions, and other inconveniences intervene. The long run is best analyzed with growth models.) Given the freedom with which capital moves around the world these days, even the United States is arguably best modeled as a small open economy (as in Bernanke’s “Global Savings Glut” argument, Greenspan’s “conundrum” and similar arguments).

The reason that this matters is that when you constrain the model to match a capital/output ratio of 3 or 4, and you assert that all agents have the same time preference rate, you end up with a model in which most of the individual microeconomic agents also have wealth-to-income ratios in the range of 3-4.

But people that rich are not the ones whose consumption is likely to react violently to a change in unemployment risk. They have enough wealth to ride out even quite a long unemployment spell.

Instead, you need a model that is calibrated to match the behavior of the typical consumer, not the mean holder of wealth.

The reason that this is not a distinction without a difference is that, when microeconomic risk is properly accounted for, the consumption function is not linear. (Carroll and Kimball (1996) prove the concavity of the consumption function in quite generic circumstances: HARA utility (which encompasses all the usual specific functional forms for utility usually considered).)

This point is illustrated in figure 1, taken from Carroll (2001), which depicts the optimal level of consumption (c) as a function of ‘cash-on-hand’ x for a consumer with CRRA utility and a microeconomically realistic calibration of uncertainty.

The lower “No Borrowing” consumption function reflects optimal behavior for a consumer who is prohibited from borrowing at all. The higher, “Can Borrow Up to 0.3” function is optimal for a consumer who can borrow an amount up to 30 percent of a year’s permanent income.

In either case, though, the point is that the response of spending to a transitory income shock (which, in this framework, translates directly into an incremental amount of x) depends critically on the consumer’s cash-on-hand. At low levels of cash-on-hand, the marginal propensity to consume is (optimally) 1: Each extra dollar in resources translates one-for-one into an extra dollar of spending. Carroll (2004) proves that as x approaches infinity, the MPC approaches the value that it takes in a perfect foresight model.

So, average consumption behavior will depend on how consumers are distributed across different levels of x.

Returning to Aiyagari (1994) versus Carroll (1992), the key difference is that I calibrated my model so that it matched the wealth of the median consumer, while Aiyagari (1994) calibrated his model so that it matched mean wealth rather than the median. Thus, his model effectively puts most consumers way way way out to the right in the figure, while mine puts them much closer to the steep part of the consumption function.

Fast-forwarding to today, people working on the microfoundations of aggregate consumption behavior like Kaplan (2012) or Kaplan and Violante (2011) are increasingly taking the view that I got it right (from the standpoint of macroeconomic analysis), because if you want to study consumption then you want to study the typical consumer not the typical wealth holder. (The two will be identical only if consumption is strictly proportional to wealth, c = γx for some γ – which is what Kimball and I proved is NOT TRUE except in a set-of-measure-zero set of knife-edge cases).

Returning, now, to the essence of your question, which I will reframe as follows: “Suppose we calibrate a [Bewley-Carroll] model to exhibit fluctuations in unemployment risk of the magnitude that we typically see in recessions. Does that model imply that those fluctuations in risk should result in fluctuations in spending that are large enough to make an important contribution to business cycle dynamics?”

That’s a question that I think has not been fully addressed by the existing literature. But there are good reasons to think that the answer is “easily.”

The first important point to make here is that your quantitative intuitions are likely not to be very good on such a question, because they are likely mostly formed from linear approximations around steady states. The key insight is that the effect of uncertainty on behavior is nonlinear, and even low-probability events can have a profound effect.

This point is illustrated in figure 2, taken from Carroll and Toche (2009), which shows a comparison of the consumption function that is optimal in a perfect foresight model c(me) with the consumption function that is optimal when that model is modified to introduce the possibility that the consumer might become permanently unemployed.¹

As you can see, the effect on the level of consumption is both large and nonlinear, being larger for people at lower levels of wealth (because their consumption is greater risk since they have lower resources to buffer their consumption against unempoyment).

Another piece of the answer is contained in Carroll, Slacalek, and Tokuoka (2013). That paper asks and answers two questions:

1. What modifications to a standard [Bewley/Carroll] model are needed to permit that model to do a reasonably good job of matching the distribution of wealth in the U.S. Answer: A modest amount of heterogeneity in the time preference rate is sufficient. Specifically, if annual time preference rates are uniformly distributed in the interval [β - 0.02,β + 0.02], the model-implied distribution of wealth is remarkably close to the empirical distribution
2. Does the model that matches the distribution of wealth have different implications than one that gets the mean right a la Aiyagari (1994) but fails to match the distribution. Answer: Yes. For a version of the model that matches the distribution of total net worth, the aggregate Marginal Propensity to Consume is about 0.23 (that is, a manna-from-heaven stimulus check of $1 to every American would result in $0.23 of extra spending in the next year). For a version of the model that matches the distribution of liquid assets, the model-implied MPC is about 0.45. (The micro empirical literature on the MPC typically measures it as being somewhere in the range of 0.2 to 0.7).

A final, closely related, question is how to think about the distinction between “expenditures” (in the NIPA sense) and “consumption” (in the model). The main difference between these is in the treatment of durable goods. NIPA personal consumption expenditures include, e.g., purchases of new motor vehicles, but those are not the kinds of nondurable spending that the theoretical model is designed to model.

The beginnings of an answer are in Carroll and Dunn (1997). That paper explicitly incorporates unemployment risk into an sS model of the purchase of durable goods. The main quantitative problem with the model is that when we change the unemployment probability (in exactly the kind of Markov-switching way you describe above), the impact of a change from the low-unemployment to the high-unemployment state is much too LARGE. (That is, expenditures in the model move even more than NIPA durable goods expenditures move over the business cycle). Roughly speaking, this is because, if the economy moves from expansion to recession, almost all of the people who were just about to buy a car (they were on the margin) decide that they want a bigger precautionary buffer and so they put off their car purchase for 6-9 months. But if EVERYBODY who was about to buy a car puts it off, spending goes to zero. Of course, it would not be too hard to damp this effect by, e.g., saying that some kinds of people are in jobs where unemployment risk does not change much when a recession comes along, and some people have to buy a new car because they totalled their old one, and so on. But the takeaway is that, far from the effects of fluctuating uncertainty being too small to make much difference, the problem is the opposite.

References