Income and Wealth Heterogeneity in the Macroeconomy

In a calibrated stochastic growth model where a continuum of infinitely lived households face partially uninsurable employment risk and can self-insure only by holding aggregate capital, the macroeconomic aggregates turn out to be almost perfectly described by just two numbers — the mean of the wealth distribution and the aggregate productivity shock. The paper calls this approximate aggregation and states it carefully: “all aggregate variables — consumption, the capital stock, and relative prices — can be almost perfectly described as a function of two simple statistics,” so that “the distribution of aggregate wealth is almost completely irrelevant for how the aggregates behave in the equilibrium.” With a log-linear law of motion in the mean alone, the fitted rules are log k’ = 0.095 + 0.962 log k in good times and 0.085 + 0.965 log k in bad, both with R² = 0.999998 and regression-error standard deviations of 0.0028% and 0.0036%; price forecasts 25 years ahead have maximum errors under 0.1 percent. The mechanism is not that the distribution is stable — its standard deviation, skewness and kurtosis all “display substantial variation” — but that the marginal propensity to save is nearly independent of wealth except at the very bottom, and the agents whose propensities differ hold a negligible share of capital. The paper is equally clear that this benchmark fails as a model of inequality: the poorest 20 percent hold 9 percent of wealth against roughly zero in the data, the richest 5 percent hold 11 percent against roughly half, and the Gini is 0.25 against 0.79. The fix is a stochastic discount factor taking three values 0.9858, 0.9894 and 0.9930 with 50-year average duration at the extremes, read as imperfectly inherited patience; that model reproduces the data’s Gini (0.82 against 0.79) and its 11 percent of households with negative wealth, though it still understates the extreme upper tail. Approximate aggregation survives — R² = 0.999991 and 0.999985, with errors roughly twice as large. What does not survive is permanent-income behaviour: because impatient households face a large wedge between their discount rate and the market return, they consume hand-to-mouth, and while they “matter little for what happens to the evolution of economywide wealth,” their consumption is a large share of the total, pushing the aggregate consumption-output correlation to 0.825 against 0.691 in the complete-markets benchmark. Precautionary saving is small in the benchmark (0.6 percent of the capital stock) but rises to 6.7 percent with risk aversion of five. The methodological claim is stated as an enabling one — this “opens the possibility of characterizing a large class of new macroeconomic models in which heterogeneity in income and wealth plays a key role” — and the robustness claim is stated as an empirical regularity of their experiments, not a theorem: “it is extremely difficult to find exceptions to the approximate aggregation result,” while “like most numerical procedures, the present one does not provide bounds on how far the approximate equilibrium deviates from an exact equilibrium.”

Anthony A. Smith, Jr. | Macro Paper Warehouse

Income and Wealth Heterogeneity in the Macroeconomy