Uninsured Idiosyncratic Risk and Aggregate Saving

“The first is to provide an exposition of models whose aggregate behavior is the result of market interaction among a large number of agents subject to idiosyncratic shocks” – in which, unlike representative-agent models, individual dynamics and uncertainty do not coincide with aggregate dynamics and uncertainty, even though aggregate variables themselves are unchanging in steady state (Introduction, p. 659). The second goal is “to use such a model to study the quantitative importance of individual risk for aggregate saving,” motivated by conjectures in the literature (e.g., Modigliani [1988], Zeldes [1989a]) that precautionary saving might be a quantitatively important source of aggregate capital accumulation (p. 660).

Following Schechtman and Escudero [1977], this is the dynamic problem of an individual facing uncertain earnings and a constant return on assets who chooses consumption and asset accumulation to maximize expected discounted utility subject to a borrowing limit (Section III, p. 665). Under the paper’s conditions (bounded relative risk aversion, among others), this problem yields a single-valued, continuous asset-demand function and a Markov transition law for total resources whose support is bounded, guaranteeing a unique, stable stationary distribution of assets that varies continuously with the borrowing limit, wage, and interest rate (pp. 665-669, Proposition 5 of Aiyagari [1993a]).

A “fixed” borrowing limit is an arbitrary cap (such as zero, i.e., no borrowing at all) not derived from the model’s own budget constraint; the “present value” borrowing constraint, by contrast, equals the largest debt the agent could repay for certain out of the worst possible future earnings stream, wl_min/r, and is implied by present-value budget balance together with nonnegative consumption when r > 0 (Section III, pp. 665-666). The distinction matters for at least two results: it determines whether the model’s steady state is guaranteed to have a positive interest rate (guaranteed under the present-value constraint, not necessarily under a fixed constraint), and, in the paper’s government-debt reinterpretation, it determines whether Ricardian debt neutrality holds – debt neutrality holds only if the borrowing limit is adjusted for changing tax obligations, i.e., only under the present-value form (Section III, “Some Alternative Interpretations,” pp. 673-674).

Average household assets, Ea, are shown to tend to infinity as the interest rate r approaches the time preference rate λ from below, and to be effectively unbounded (or undefined via an explosive Euler-equation argument) at or above λ, because with an infinite horizon and repeated shocks a consumer who is not strictly impatient relative to the return on assets will accumulate unboundedly large precautionary assets to buffer against a run of bad shocks (Section III, pp. 668-669, with the formal supermartingale argument in footnote 20). Consequently, for values of r below λ, average assets under idiosyncratic uncertainty are always at least as large as under certainty, and the model’s key general-equilibrium feature is that Ea rises steeply as r approaches λ from below, which acts to check how far idiosyncratic risk alone can push the interest rate down (pp. 669-671).

The steady-state condition is K(r) = Ea(r): the per capita capital that firms demand at interest rate r, from the marginal-product-of-capital condition r = f₁(k,1) − δ, must equal the per capita assets that households supply at that same rate, aggregated from individual optimal saving behavior (Section III, “General Equilibrium,” pp. 670-672, Figure IIb). Because the elasticity of the K(r) curve matters as much as the shape of Ea(r) in determining whether uninsured risk mainly moves the interest rate or aggregate saving, the paper stresses that “in a partial equilibrium analysis, by choosing an interest rate close enough to the rate of time preference, one can generate arbitrarily large precautionary saving in excess of the certainty case” – but in general equilibrium the actual increase in saving is a genuinely quantitative question, not something assumable by picking r (p. 672).

By reinterpreting individual asset holdings a_t as real money balances (m_t − 1)/p, the borrowing limit as 1/p, and r as the interest paid on money, the same individual optimization problem becomes a Bewley [1983]-style monetary model whose steady-state equilibrium condition is that per capita money holdings equal unity (i.e., per capita “assets” are zero) (Section III, “Some Alternative Interpretations,” pp. 674-675). This reformulation, the paper notes, clarifies why monetary equilibria cannot support an interest rate arbitrarily close to the rate of time preference: there is a specific interest rate r# at which average assets are exactly zero, and no lowering of the price level (raising the borrowing limit) beyond the level consistent with r# can push the equilibrium interest rate any higher (p. 675).

The model period is one year; the utility discount factor is 0.96 (CRRA utility, with relative risk aversion μ ∈ {1, 3, 5}); the Cobb-Douglas capital share is 0.36 and the depreciation rate is 0.08 (Section IV, p. 675). Idiosyncratic log-earnings follow a first-order autoregressive process approximated by a seven-state Markov chain, with coefficient of variation σ ∈ {0.2, 0.4} and serial correlation ρ ∈ {0, 0.3, 0.6, 0.9}, chosen with reference to PSID- and NLS-based estimates in Abowd and Card [1987, 1989] and Heaton and Lucas [1992] suggesting a plausible range of roughly 20-40 percent coefficient of variation in annual earnings (Section IV, pp. 675-676); the borrowing limit is set to zero (no borrowing), which the paper notes is the assumption most favorable to generating a large precautionary-saving effect on aggregate capital (p. 676).

“The differences between the saving rates with and without insurance are quite small for moderate and empirically plausible values of σ, ρ, and μ. However, for high values of σ, ρ, and μ, the presence of idiosyncratic risk can raise the saving rate quite significantly by up to seven percentage points,” and in the most extreme case considered (σ = 0.4, ρ = 0.9, μ = 5) the saving rate rises by “almost fourteen percentage points” relative to the full-insurance benchmark saving rate of 23.67 percent (Section V, “Aggregate Saving,” pp. 678-679, Tables IIA-IIB). The paper also notes that approximating a difference-stationary (rather than trend-stationary) earnings process by letting ρ approach one while σ grows without bound (holding the implied variance of earnings changes fixed) would “depress the return to capital and increase the saving rate enormously” (p. 679).

For CRRA preferences, relative prudence (RP) equals (μ + 1) and the equivalent precautionary premium (EPP) equals RP times half the squared coefficient of variation of consumption; increases in earnings variability, persistence, or risk aversion each raise the EPP (directly, or by shifting the aggregate asset-demand curve to the right), and in every case the equilibrium interest rate falls and the saving rate rises, consistent with Table II (Section V, “Aggregate Saving,” pp. 679-680). The paper flags a scope caveat: Kimball’s concepts were originally developed for a single consumer with a two-period horizon and no borrowing constraint, whereas the paper’s Ea curve reflects aggregation across a large population of infinite-horizon, potentially-constrained consumers (footnote 40, p. 679).

Using an approximate welfare-loss formula, μσ_c²/2 (relative risk aversion times half the squared coefficient of variation of consumption), the paper contrasts actual consumption variability under optimal asset trading with the variability an agent would experience holding a fixed, non-traded quantity of assets equal to the per capita amount (Section V, “Importance of Asset Trading,” pp. 679-680). In one example (μ = 3, σ = 0.4, ρ = 0.6), the coefficient of variation of consumption with a fixed asset position would be 0.35, versus an actual value of 0.17 under optimal trading – “by optimally accumulating and depleting assets, consumption variability is cut in half, yielding a welfare benefit of about 14 percent of per capita consumption, or about 8 percent of per capita GNP” – a result the paper contrasts explicitly with representative-agent models (citing Cochrane [1989]), in which such consumption-smoothing gains from asset trading cannot arise because there is no idiosyncratic risk to smooth (p. 680).

The model qualitatively reproduces several features of U.S. data: consumption is much less dispersed than income, wealth is much more dispersed than income, and all of the model’s cross-section distributions are positively skewed (median below mean), with Lorenz curves showing consumption the most equally distributed, then net income, then gross income, then capital, with saving the most unequally distributed (Section V, “Cross-Section Distributions and Inequality Measures,” pp. 680-681). However, the model cannot generate the observed degree of inequality: in one parameterization (μ = 5, ρ = 0.6, σ = 0.2) the model’s Gini coefficients for net income and wealth are 0.12 and 0.32, respectively, versus roughly 0.4 and 0.8 in U.S. data (p. 681). The paper attributes part of this gap to measurement mismatches (the model’s infinitely-lived “household” implicitly spans generations linked by bequests, unlike survey households) and to the model’s restriction to a single source of inequality – different histories of labor endowment shocks – while abstracting from other sources such as differences in human capital (footnote 45, p. 681).

Aiyagari suggests the framework may help resolve asset-pricing puzzles that Mehra and Prescott [1985] argued “cannot be ‘accounted for by models that abstract from transactions costs, liquidity constraints and other frictions absent in the Arrow-Debreu set-up,’” though doing so would require generalizing the model to include aggregate (not just idiosyncratic) dynamics and uncertainty – “a very hard problem computationally,” since the cross-section asset distribution would then be a full state variable evolving stochastically, which the paper flags as unresolved (Section VI, pp. 681-682). He also notes a substantive policy divergence from complete-markets theory: whereas Chamley [1986] shows the optimal long-run capital-income tax is zero under complete markets, a companion paper (Aiyagari [1994], the “Optimal Capital Income Taxation” working paper) shows the optimal long-run capital tax is strictly positive once idiosyncratic shocks and incomplete markets are introduced, implying that welfare gains from eliminating capital taxation computed in complete-markets settings (e.g., Lucas [1990]) “may well turn out to be welfare losses in an incomplete markets model” (p. 682).