Monday, October 18, 2010

Household size heterogeneity and the representative agent

A large, but shrinking, proportion of models in macroeconomics assume the existence of a representative agent. While there is clear evidence that households are heterogeneous, under some conditions aggregation may still hold. But even if it does not, what matters is whether it makes a quantitative difference.

Christos Koulovatianos, Carsten Schröder and Ulrich Schmidt address the question from a different angle. They ask whether differences in household size matter. They show it does not, theoretically, if the utility functions exhibit household-size economies (beyond subsistence consumption) that are invariant with income. I think that what the authors want to say here is that household size does not matter as long as preferences are such that consumption demand is linear in household size, which in fact does not imply that preferences are heterogeneous. Household size is just an argument in the household utility function. Or: household utility can be aggregated for individual utilities if the decisions rules can be aggregated, which is the case when they are linear.

Anyway, beyond these semantic issues, Koulovatianos, Schröder and Schmidt then provide survey evidence documenting that preferences have indeed the required property, namely that households perceived that the income necessary to maintain a given standard of living is linear in the number of its members. What is interesting here is that the survey covers many countries, possibly preempting the criticism I mentioned the other day.

On a separate note, this paper is particularly painful to read. It runs for 89 pages, and it is very confusing (even more than my post) because the authors do not use the right terminology. Also, they need to learn the virtues of conciseness and precision. I hope they did not send it to a journal in such bad shape. And of course they assume that all households have the same time preference, against which there is ample evidence and which matters much more as it has strong implications for savings.


Christos Koulovatianos said...

This paper analyzes multi-dimensional heterogeneity. The dimensions of heterogeneity are two: (a) heterogeneity with respect to available resources (rich vs. poor), and (b) heterogeneity with respect to household size.

The theoretical analysis regarding household-size economies answers a particular question: "How can we control for one dimension in order to examine the role of the other dimension if aggregation is present?" This is an important question given the conceptual complexity two dimensions of heterogeneity bring to the table.

Household size economies are easy to define conceptually: give a one-member household a certain level of permanent income forever; then identify the permanent income of a multimember household that makes the material comfort (utility) of individuals in that household equal to this of the one-member household (assume that individuals in multimember households have a common utility function within the household); these two permanent incomes are called equivalent incomes.

Ratios of equivalent incomes capture household-size economies. In practice, multi-member households need lower per-capita income compared to one-member households because more than one people can share goods within the household (e.g. common space in a house, furniture, etc.).

Now, the requirement of exact aggregation implies that equivalent incomes are linearly related to each other. This is a property describing a necessary pattern that captures both dimensions of heterogeneity, i.e. rich vs. poor and household size: it says that if family-type subsistence needs are subtracted from household income, the resulting discretionary incomes entail household-size economies that are the same across the rich and the poor.

That demand functions (expressed as decision rules of household wealth) are linear parallel in wealth across different households (the intercepts can differ) is an additional property of aggregation.

Corollary 1 says that falsifying the whole theoretical construct of a representative consumer for an economy with underlying household-size heterogeneity collapses into empirically rejecting a single equation: this linear relationship across equivalent incomes given by equation (16) in the paper.

But equivalent incomes have a lot of controlling inside their definition. They are permanent incomes. So, how can one identify real-world objective data where different households always consume their permanent income? This is a very difficult task, so the paper proposes that the controlling can be done by survey respondents who think about hypothetical households in terms of permanent incomes.

Figure 1 is the key result of the paper. In each panel of Figure 1, five distinct groups of respondents (totally independent groups of about 400 respondents each) evaluate a multi-member household's equivalent income in relation to a fixed predetermined one-member-household income that has been given to them. The survey provides five income levels for rich vs. poor spanning a wide income range from the real-world one-member-household income distribution.

Corollary 1 says: if I take the answers of these five independent respondent groups and I join them together in one picture, then the five averages of the equivalent incomes stated by the five groups should all be on a straight line. Each panel of Figure 1 shows exactly this, a statement validated by specification tests appearing in Table 1. These are 7 tests (7 panels each corresponding to one multi-member family type), all unable to falsify the representative consumer (at least strongly).

Christos Koulovatianos said...

Figures 2 and 3 show 42 more tests, but these refer to pilot studies where each respondent evaluates equivalent incomes referring to all five one-member household incomes given to them by the survey. These 42 tests are not the perfect thing to do for testing the validity of equation (16), but they lend support to the findings of Figure 1 (evidence from very different countries).

A comment about the importance of aggregation: aggregation implies that agents following simple rules of thumb for forecasting (such as tracking macroeconomic variables alone) may end up with perfectly informed forecasts about economic behavior in each and every corner of a complex economy of substantially heterogeneous agents. In brief, economic systems and economic forecasts are not as chaotic as we might think sometimes.

I wrote these two comments because I know that all this is a rather long and complicated stream of theoretical verifications and tests in the paper. My co-authors and I are still struggling with communicating this work...

Slides of the paper can also be found through this link:

Christos Koulovatianos said...

Two clarification issues regarding the economic logician’s posting:

1. The key null hypothesis of the paper tested is not that "income necessary to maintain a given standard of living is linear in the number of its members". Instead, the right thing to say is "If family-type subsistence needs are subtracted from household income, the resulting family-type permanent discretionary incomes entail household-size economies that are the same across the rich and the poor."

2. The model behind Theorem 2 of the paper does not imply "that all households have the same time preference". It assumes that all households have the same *choice-independent* rate of time preference.

To see this distinction think of a momentary utility function with respect to both consumption and time, denoted by v(c,t). Then,

rate of time preference = -v_12/v_1

where v_1 is the partial derivative with respect to the first argument (c), and v_12 is the cross derivative with respect to c and t.

Now, the multiplicatively separable form between c and t

v(c,t) = exp(rho(t))*u(c)

is the only way to make the rate of time preference choice-independent (it is easy to verify that rate of time preference = -v_12/v_1 = rho'(t)) in this case, and a bit more involved to verify that this multiplicatively separable form is is also necessary (there is an argument in a proof in the paper for this).

In Theorem 2 the functions we obtain are of the form

v(c,t) = exp(rho(t))*u(c,t)

where the rate of time preference is not choice-independent:

rate of time preference = rho'(t) - u_12/u_1

Now in the expression v(c,t) = exp(rho(t))*u(c,t) v's and u's are different across individuals, but rho(t) is common across all households. So, the rate of time preference is heterogeneous across households in the underlying model. In the thought experiment that respondents of the survey we drop this heterogeneity, but heterogeneous rates of time preference are consistent with aggregation.

VIlfredo said...

I have seen this paper presented twice, several years apart (I think it was the same paper...). After neither of the presentations I understood what the paper is about. And I still do not. EL is making an honest effort here to do what the authors apparently cannot: give a statement in a few paragraphs of what the research question is, why it matters, what the method is and what the result is.

EL tries that, obviously being confused about the paper as well. An author responds with very long answers that do not address the concerns, just as it happened in those two presentations, and we are none the wiser.

Folks: rewrite this paper. Force yourself to not exceed 25 pages. Even if you think everything is important, loose some weight and put the extra stuff in an appendix.

VIlfredo said...

Christos: EL raises broad questions, you respond at lengths about details. This is the problem with your paper.

Christos Koulovatianos said...

Dear VIlfredo:

Yes, point taken, and many thanks! We'll try to do our best...

In a nutshell though: what you must have seen years ago was the comprehensive theoretical work that reveals the structure of aggregation; now it is the evidence part. The point of the paper as it stands now is that we find evidence of a world which possesses smiple structure where individual consumer behavior shows up on the aggregate. It was not clear whether such evidence would be out there.

Thanks again!

Anonymous said...

Christos, this is very good advice from Vilfredo you got here. I saw you present this Summer, I presume this paper, but I must confess I did not understand what it was about. With EL's summary I have a better picture, but it is still not clear.

Imagine you were to write a short piece like on VoxEU. What you would write there is what you need to focus on. The rest goes in the appendix, or another paper.

Anonymous said...

I have seen this paper presented in front of a theory audience and I must say that it is a very difficult topic even for theorists. It is easier to present such a paper in front of theorists though, rather than in front of empirical people. I saw the paper as it is now and I am a bit surprised by the comments: it is no longer than 28 pages and technical matters appear in the appendix. And there is an enormous amount of work and evidence put in this paper. I wonder why you are being so negative.

Anonymous said...

@Anonymous: perhaps because you're posting this comment under the veil of anonymity ...