Income and Wealth Heterogeneity in the Macroeconomy

Whether the representative-agent abstraction can be defended not by arguing that its assumptions hold, but by showing that a model with realistic heterogeneity produces nearly the same aggregate time series. The paper sets up the choice explicitly. “Most of dynamic general equilibrium macroeconomic theory relies heavily on the representative-agent abstraction… At first glance, the representative-agent assumption appears to be inconsistent with a serious treatment of microfoundations. There are two circumstances, however, under which the representative-agent construct would be a reasonable modeling strategy.” The first — that the theoretical conditions for a representative consumer roughly hold in the data — is dismissed: “This view, however, is hard to defend; one problem is that it is difficult to justify the assumption that there are complete insurance markets for consumers’ idiosyncratic risks.” The second is the paper’s subject: “that the aggregate variables in theoretical models with a more realistic description of the microeconomic environment actually behave like those in the representative-agent models. In this paper we begin the exploration of this second possibility.”

A stochastic growth model with a measure-one population of infinitely lived consumers, a two-state aggregate productivity shock, a two-state individual employment shock, and exactly one asset — capital — with a lower bound at zero. Output is Cobb-Douglas in capital and labour. Each agent is endowed with one unit of time yielding labour input when employed and none when unemployed. Aggregate productivity is good or bad on a first-order Markov chain; individual shocks are correlated with the aggregate but satisfy a law of large numbers, so “the only exogenous source of aggregate uncertainty in the economy is the aggregate productivity shock” and “the number of agents who are unemployed always equals u_g in good times and u_b in bad times.” Markets are incomplete by assumption — “there is only one asset — capital. This asset plays the twin roles of being a store of value for the individual agent and a means of self-insurance against the income shocks” — with holdings restricted to be non-negative, which the authors call the borrowing constraint and describe as “generous,” citing Aiyagari’s result that with a zero lowest income realization a negative bound is equivalent to a zero one when loans must be repaid. The aggregate state is the pair (Γ, z) with Γ the measure over capital and employment status, and its law of motion H is part of the equilibrium object.

The distribution Γ is infinite-dimensional and enters the individual’s problem because future prices depend on future aggregate capital, which — since savings decisions do not aggregate — depends nontrivially on all moments of the current distribution; the solution is to bound agents’ perceptions and then verify the bound barely binds. The authors state the restriction precisely: “Since current prices depend only on the total amount of capital and not on its distribution, limiting agents to a finite set of moments is restrictive only as far as future prices are concerned.” Their approach: “assume that agents are boundedly rational in their perceptions of how Γ evolves over time and to increase the sophistication of these perceptions until the errors that agents make because they are not fully rational become negligible.” A candidate equilibrium is a Markov process H_I in a class G that is a fixed point in two senses — decision rules are optimal given H_I, and H_I is the best approximation in G to the behaviour of m that the aggregated decision rules generate — subject to the additional requirement that the fit be “close to perfect.” The honest statement of what is sacrificed is the paper’s own: “the calculated object satisfies all the standard equilibrium conditions except the agents’ ability to make perfect forecasts.” The iterative procedure is six steps: choose I, guess a functional form and parameters, solve the consumer’s problem, simulate N agents over T periods, re-estimate on the stationary region, and stop if parameters have converged and fit is satisfactory — otherwise raise I or change the functional form.

Quarterly, with β = 0.99, δ = 0.025, relative risk aversion of 1, and capital share 0.36; shocks set so aggregate fluctuations roughly match postwar U.S. magnitudes. Productivity takes z_g = 1.01 and z_b = 0.99, unemployment rates u_g = 0.04 and u_b = 0.1, with “the fluctuations in the macroeconomic aggregates hav[ing] roughly the same magnitude as the fluctuations in observed postwar U.S. time series.” The joint process for productivity and employment is chosen so that “the average duration of both good and bad times is eight quarters and so that the average duration of an unemployment spell is 1.5 quarters in good times and 2.5 quarters in bad times.” Simulations use 5,000 agents over 11,000 periods with the first 1,000 discarded, typically starting from equal asset holdings; the authors report results “are not sensitive to changes in the initial wealth distribution.” The consumer’s problem is solved by approximating the value function on a coarse grid with cubic spline and polynomial interpolation.

Extremely good by every measure the paper applies, and it reports several rather than one. The estimated log-linear laws are log k’ = 0.095 + 0.962 log k with R² = 0.999998 and error standard deviation 0.0028% in good times, and log k’ = 0.085 + 0.965 log k with R² = 0.999998 and 0.0036% in bad times, on 10,000 simulated observations; a nonlinear flexible functional form gave “virtually identical results.” Beyond statistical fit, the paper converts the error into the terms that matter for the agent: “forecasts of prices 25 years ahead (given the sequence of future aggregate productivity shocks) have maximum errors of less than 0.1 percent,” and “from a welfare perspective, superior prediction techniques would only lead to vanishingly small increases in the agent’s utility.” The authors draw the strongest conclusion they think is warranted, and its form is notable: “The accuracy is so high that we find it very hard to argue on the basis of the ‘irrationality’ of the agents in our model that our approximate equilibrium is a less satisfactory economic model than an exact equilibrium.” They also acknowledge that including more moments must improve forecasts statistically, since “strict aggregation does not obtain.”

The paper raises the objection itself, offers a theoretical reason to doubt it, and then checks it empirically. “Like most numerical procedures, the present one does not provide bounds on how far the approximate equilibrium deviates from an exact equilibrium. In particular, one might imagine that there are self-fulfilling approximate equilibria: because agents perceive a simple law of motion, they behave accordingly. However, there is nothing in the theoretical link between these perceptions and the aggregate savings behavior that is suggestive of self-fulfilling equilibria.” The check: “When we include a higher moment — we tried various dispersion measures for capital — in the perceived law of motion, our approximate equilibrium fixed point remains virtually identical to the simpler case in terms of both the aggregate processes and individual behavior.”

Because almost all capital is held by agents with essentially the same marginal propensity to save, so moving capital among them has no aggregate effect — not because the distribution is stable. The logical condition is stated first: “aggregation would obtain if all agents had the same propensity to save out of wealth, because then changes in the distribution of the total stock of capital would have no aggregate effects.” The paper then shows the decision rule is nearly linear with a common slope: “the marginal propensities to consume are almost identical for agents with different employment states and levels of capital. As the agent’s wealth increases, the slope increases toward one; a slope of one would amount to exact permanent income behavior.” And it stresses that the distribution itself is not static: “In the stationary state, the distribution of capital does move around significantly: for example, the standard deviation, skewness, and kurtosis of the capital stock distribution all display substantial variation in the simulated aggregate time series. Most of the capital, however, is held by agents with essentially the same savings propensity. Very few agents — the very poorest ones — have a much lower propensity, and the capital that they hold is negligible.”

Because one asset provides very effective insurance in utility terms even though it provides poor insurance of consumption, so most agents are saving for intertemporal rather than precautionary reasons. The paper separates the two senses of “effective” explicitly: “Self-insurance in our model is not very effective in terms of smoothing individual relative to aggregate consumption; for example, the unconditional standard deviation of individual consumption is about four times that of aggregate consumption, and the unconditional correlation of the consumption of any two agents is very close to zero. However, in utility terms, agents in our stationary equilibria are insured well enough that the marginal propensity to save out of current wealth is almost completely independent of the levels of wealth and labor income, except at the very lowest levels of wealth.” The reason there is enough capital to make this work is structural: “The availability of enough capital is automatic here since our model has a neoclassical production function with high marginal returns to capital at low levels of capital and a realistic capital/output ratio.”

Robust to a large battery of changes, with the discount factor the only lever that visibly degrades the fit, and even then only under extreme settings. “The basic finding from these experiments is that it is extremely difficult to find exceptions to the approximate aggregation result. For example, if individual shocks are more volatile or more persistent (alternatively, if borrowing is more restricted or if agents are more risk averse), the aggregate economy responds by accumulating just enough extra capital to provide most agents with a large enough buffer that the shocks do not hurt them much in utility terms.” The discount factor works in the direction Bewley’s theory predicts — “higher discount rates strengthen (and lower discount rates weaken) the aggregation result” — but “large decreases in β are necessary in order for the goodness of fit to significantly worsen; for example, the percentage standard deviation of the regression error is only about 10 times higher for a β as low as 0.67.” The valued-leisure extension is the sharpest test, because prices then depend on total work effort as well as capital, so “with significant dependence of individual work effort on wealth, thus, aggregation might fail. However, it turns out that even in a formulation with large wealth effects, the relation between wealth and effort is almost linear for most agents.” A companion paper extends the result to an economy with a riskless bond as a second asset.

Badly, and in both tails: the bottom is not poor enough and the top is not rich enough. “Too few agents hold low levels of wealth, and the concentration of wealth among the richest agents is far too small. On the basis of data in Wolff (1994) and Díaz-Giménez, Quadrini, and Ríos-Rull (1997), for example, the poorest 20 percent of the population have about zero wealth on average, whereas the richest 5 percent of the population hold roughly half of all the wealth. In contrast, the benchmark model predicts that the poorest 20 percent hold (on average) 9 percent of total wealth whereas the richest 5 percent hold (on average) 11 percent: there is significant skewness, but not nearly as much as in the data.” Table 1 gives the full comparison: the benchmark’s top 1 / 5 / 10 / 20 / 30 percent hold 3, 11, 19, 35 and 46 percent of wealth, with no households at negative wealth and a Gini of 0.25, against data values of 30, 51, 64, 79 and 88 percent, 11 percent negative, and a Gini of 0.79. A footnote extends the indictment beyond this model: the same failure holds “for similar frameworks that have richer processes for labor income than we do… for a given, realistic, labor income distribution, there is far too little skewness in wealth.”

A social-insurance-like floor plus borrowing at the bottom, and a small, symmetric, slowly moving amount of heterogeneity in patience at the top — interpreted as imperfectly inherited “genes” rather than as an economic choice. For the bottom, “we assume that unemployed agents receive income too. We set their income to a number that is about 9 percent of the average employed wage,” and the capital floor is moved below zero, “with maximum allowable borrowings being set at about half of average annual earnings” (specifically, an unemployment wage of 0.07 and a borrowing limit of −2.4, “which is stricter than an always pay back constraint”). For the top, the authors note the options — “it seems necessary either to make rich agents have higher propensities to save or to give them higher returns on saving (or both)” — and choose the former via a Markov discount factor on {0.9858, 0.9894, 0.9930}, with 80 percent of the invariant distribution at the middle value and 10 percent at each extreme, no immediate transitions between extremes, and an average duration of 50 years at the extremes to “roughly match the length of a generation.” The discipline imposed on the experiment is explicit: differences in discount factors must not be large, and their distribution must be symmetric around its mean. The reading is deterministic in an uncomfortable way, and the paper says so: “In the stochastic-β model, poor agents are poor because they (their dynasties) have chosen to be poor. This choice, in turn, is based purely on (exogenous) genetics.” The economic point is the amplification: “the equilibrium interaction between agents with different degrees of patience forces very large differences in wealth from seemingly small differences in patience.”

It matches the Gini and the share with negative wealth, understates the extreme top, and approximate aggregation still holds with errors about twice as large. The stochastic-β model’s shares for the top 1 / 5 / 10 / 20 / 30 percent are 24, 55, 73, 88 and 92 percent, with 11 percent of agents at negative wealth and a Gini of 0.82, against data values of 30, 51, 64, 79, 88, 11 and 0.79. The paper reports the remaining mismatch and attributes it to its own design constraints: the model “predicts somewhat too low a concentration in the extreme upper tail of the wealth distribution and too high a concentration in the middle; these shortcomings reflect the restrictions of symmetry and so forth that we impose on the discount factor distribution.” Aggregation: log k’ = 0.100 + 0.961 log k with R² = 0.999991 and error 0.0056% in good times, 0.095 + 0.961 log k with R² = 0.999985 and 0.0077% in bad, so “the percentage standard deviations of the regression errors do go up by about a factor of two, but nonetheless the fit is still remarkably good.” The explanation is the same as before, now sharper: “although marginal propensities differ more across consumers in the stochastic-β model than in the benchmark model, almost all the wealth is held by well-insured consumers. Therefore, only the rich agents matter for determining the aggregates.”

It raises the steady-state capital stock — always — and changes second moments only slightly in the benchmark, more with high risk aversion or valued leisure. “The lack of full insurance always raises the steady-state aggregate capital stock since capital has the additional value of insuring against risk. The amount of precautionary savings for the benchmark model is about 0.6 percent (calculated as the percentage increase in the capital stock when insurance markets are closed), which is quite small. However, with more risk-averse agents, the amount of precautionary savings rises significantly: with a degree of relative risk aversion of five, the amount of precautionary savings is 6.7 percent.” The real business cycle version gives less precautionary saving, “since varying leisure allows a complementary way of adjusting consumption in response to shocks.” On second moments the verdict is deliberately modest: “The incomplete-markets economies have second-moment properties that are different from their representative-agent counterparts, but not by large amounts… simulated realizations show that the two market structures lead to virtually indistinguishable time series, except for the difference in means.” Mean capital moves from 11.54 to 11.61 in the benchmark, the consumption-output correlation from 0.691 to 0.701, investment volatility from 0.031 to 0.030, and fourth-order output autocorrelation from 0.486 to 0.481. The summary: “in the models we examine that do not have preference heterogeneity, the market structure matters only marginally for aggregate time-series behavior.”

The aggregate consumer departs visibly from permanent-income behaviour, and the consumption-output correlation is the statistic that shows it. In the stochastic-β model mean capital is 11.78 and the consumption-output correlation 0.825, against 0.691 in the complete-markets benchmark. The mechanism turns on an asymmetry between wealth shares and consumption shares: “Although these impatient consumers have very little capital and, thus, matter little for what happens to the evolution of economywide wealth, their consumption behavior is a more significant part of the total; the most patient (high-β) consumers are much richer but consume more only by the amount of the interest payments on their assets.” The wedge is what does the work: the interest rate sits slightly below the most patient agents’ discount rate, while “the difference between the interest rate and the discount rate of the least patient agents, however, is larger. This larger wedge leads to a stronger departure from permanent income behavior.” Concretely, the correlation between the aggregate consumption of the least patient agents and aggregate output averages 0.90; for the most patient it is 0.61. The contrast with complete markets is exact: “In a complete-markets setting in which all consumers have the same discount rate, consumers exhibit permanent income behavior regardless of the degree of their impatience… if the discount rate is lowered, the stationary equilibrium interest rate adjusts so that the two remain very close.”

That their behaviour mimics a binding borrowing constraint but is not caused by one, and the two may be empirically hard to separate. “An important related point is that the more impatient consumers in the incomplete-markets model do not smooth consumption and thus also may give the impression of being up against a borrowing constraint. Although the borrowing constraint does indeed limit consumption possibilities, the chief reason for impatient agents’ rush to consume is precisely that they are more impatient than others. Empirically, it is not obvious how to distinguish a binding borrowing constraint from a lower than average degree of patience.” A footnote supplies a bound on how large the market-structure effect could be in a closely related economy: with two permanently distinct discount factors, where the complete-markets version does aggregate and only the patient group is active in the stationary state, “the consumption-output correlation is .691 in the complete-markets version and .865 in the incomplete-markets version,” and the authors infer that “the effects of changing the market structure are likely similar for the stochastic-β model.”

That a two-statistic summary suffices for the aggregates, but that a representative consumer reproducing those aggregates may not exist within the usual class — and it lists four specific reasons the result might not generalize. On the first finding: “a low-dimensional object — the total capital stock and the value of the aggregate productivity shock — seems to be sufficient for characterizing the stochastic behavior of all the macroeconomic aggregates… Hence, one need track only the evolution of the ‘aggregate budget’.” On the second, the answer is conditional: constructing an aggregate consumer whose maximization reproduces these series “is sometimes, but not always, an easy task,” and for the preference-heterogeneity extensions “it seems necessary to move outside the class of models with a single infinitely lived agent with time-additive preferences. In particular, in representative-agent models with time-additive preferences, it seems difficult to obtain the departures from permanent income behavior that we observe.” Four open directions are named: life-cycle savings motives combined with incomplete markets (noting that complete-markets life-cycle models already behave much like the benchmark, so “life cycle considerations alone do not lead to significant departures from permanent income behavior”); endogenous borrowing constraints as in Kiyotaki and Moore, which “might ascribe a more important role to the distribution of wealth”; forcing consumers to use their own production technologies so that returns to saving differ across agents; and fixed costs in capital accumulation, which following Banerjee and Newman “may also lead to a more fundamental dependence of the aggregates on the distribution of resources across agents.” The closing note is programmatic: the methodological findings suggest “it now seems within our ability to begin using equilibrium models to analyze the interrelation between business cycles, inequality, and economic policy,” with household data the way to discriminate among competing hypotheses about the wealth distribution.