The Marginal Propensity to Consume in Heterogeneous Agent Models

With a representative agent or complete markets, there is a single common MPC that is roughly the size of the interest rate, so such models “are unable to speak to” the empirical evidence on MPCs (Introduction, p. 1). The authors note the MPC is “a central concept in modern macroeconomics” because, in models with nominal rigidities, it governs the size of fiscal multipliers, the transmission mechanism of monetary policy, the amplification of aggregate shocks, and portfolio choice between risky and safe assets – so the usefulness of heterogeneous-agent models for these questions is “closely tied to their ability to reproduce the evidence on MPCs” (p. 1).

"(i) hand-to-mouth households, (ii) precautionary saving, and (iii) ex-ante heterogeneity" (Introduction, pp. 2-3). Poor hand-to-mouth (zero-wealth or constrained) households consume the majority of any extra liquidity; wealthy hand-to-mouth households in two-asset models, who hold their wealth in illiquid form, respond similarly. Precautionary saving – whether from prudence (u’’’ > 0) or from an occasionally binding constraint even without prudence – makes the consumption function concave in wealth, so MPCs are elevated even for non-hand-to-mouth low-wealth households. Ex-ante heterogeneity in impatience, elasticity of substitution, returns, or behavioral biases affects both the shape of the wealth distribution and the consumption function conditional on wealth.

The model is a standard quarterly, discrete-time Aiyagari-Bewley-type economy with CRRA (log) utility, a no-borrowing constraint, and an income process combining AR(1) and IID components estimated from the PSID; the discount factor is chosen so that mean model wealth matches mean US net worth (4.1 times mean annual earnings, excluding the top 5% of the wealth distribution) (Section 2.1, pp. 4-5). Calibrated this way, “the average quarterly MPC in the baseline precautionary saving model is 4.6%, nearly one order of magnitude larger than the certainty MPC” of 0.5% (Section 2.1, p. 7), with only about 2% of model households classified as hand-to-mouth versus 14% in the data (p. 5).

Decomposing the average MPC into a certainty term, a “Borrowing Constraints” term computed from a version of the model with deterministic average income (so the consumption function is concave only because of the constraint together with impatience, βR < 1), and a residual “Precautionary Savings + Income Risk” term, the paper finds that 68% of the baseline gap is explained by borrowing constraints even absent any income risk, with the remaining 38% attributable to precautionary saving from income uncertainty (Section 2.1, “Decomposition Relative to Certainty Benchmark,” p. 9; the two shares need not sum to 100% because of an interaction term). The authors call this “perhaps surprising[]” since it shows a strong force toward a concave consumption function even without uncertainty.

Yes, but at a steep cost: targeting mean liquid wealth (defined as bank accounts and directly held stocks/bonds net of credit-card debt, only 0.56 times mean earnings) raises the average quarterly MPC to 14%, and targeting median liquid wealth raises it to 33% (Section 2.2, “Target for Wealth-Income Ratio,” pp. 10-11). However, the paper stresses these calibrations “necessitate abstracting from essentially the entire stock of aggregate assets owned by the household sector” – the mean-liquid-wealth calibration ignores 85% of wealth in the (top-5%-excluded) sample, or 98% of total wealth – which “limits [their] usefulness in general equilibrium models with capital” (p. 11).

Both have only modest effects once the discount factor is recalibrated to hold aggregate wealth fixed: lowering the interest rate from 1% to 0% barely changes the MPC, and raising it to 5% raises the average quarterly MPC by about half a percentage point (Section 2.2, “Interest rate,” Table 2, p. 10). Raising relative risk aversion from 1 to 6 actually lowers the average MPC, from 4.6% to 3.0%, because the stronger precautionary motive that steepens the consumption function is more than offset by the resulting reduction in the share of low-wealth households once wealth is re-targeted (Section 2.2, “Curvature in Utility,” p. 12).

These extensions can generate large average MPCs – for example, 19% with a wide dispersion in discount factors, or over 20% with a wide dispersion in the elasticity of intertemporal substitution (IES) – because a subset of very patient (or high-IES) households ends up holding the bulk of aggregate wealth (up to 88% held by the top 10%), which lets the model match mean wealth while still allowing a large fraction of households to be impatient and hand-to-mouth (Section 3.1, pp. 14-19). But the paper shows these same calibrations reproduce the “missing middle” problem seen with liquid-wealth targeting: for example, with the largest discount-factor dispersion considered, median wealth is roughly ten times smaller than in the data, and 75% of households hold less than $50,000 in wealth versus 38% in the data (Section 3.1.1, p. 16). The paper also isolates that, with Epstein-Zin preferences, it is specifically heterogeneity in the IES, not in risk aversion holding the IES fixed, that drives these large-MPC results (Section 3.1.2, pp. 17-18).

No – temptation (Gul and Pesendorfer [2001]) behaves like an endogenous discount factor that falls with wealth, generating a large average MPC (19% with a high temptation parameter) but “suffers from the same missing middle problem that plagues versions with ex-ante heterogeneity” (Section 3.2.1, pp. 19-20). Present bias (instantaneous gratification, following Laibson, Maxted, and Moll [2021]) has, if anything, the opposite problem: once recalibrated to match the same aggregate wealth, it has “a negligible effect on the average MPC,” because a higher required discount factor to hit the wealth target offsets the larger mass of very-low-wealth households the model produces (Section 3.2.2, p. 20).

Yes – a “spender-saver” model (in the spirit of Campbell and Mankiw [1989]) with a small share (15% in the paper’s example) of very impatient households and the rest calibrated to hit aggregate wealth can simultaneously deliver a 17% average quarterly MPC, 14.8% hand-to-mouth households, and a median wealth (0.96) reasonably close to the data (1.54) (Section 3.4, p. 22). The authors flag a major drawback, however: this extreme, bimodal form of heterogeneity makes the model’s response to targeted fiscal transfers implausibly concentrated, producing “an intertemporal MPC function with a steep peak coinciding with the time of the disbursement and little subsequent propagation” – a poor fit to the dynamic pattern of MPCs found empirically (Section 3.4, p. 22; elaborated in Section 5.2).

Households can save in a low-return liquid asset or a higher-return illiquid asset, with portfolio rebalancing available only at random arrival times and subject to a fixed transaction cost; because the welfare cost of holding most wealth illiquidly is second-order relative to the first-order gain from the higher illiquid return, many households optimally hold substantial illiquid wealth alongside minimal liquid wealth, making them “wealthy hand-to-mouth” – unable to smooth small income shocks despite positive net worth (Section 4, pp. 23-24). Combining the 14% poor hand-to-mouth with a further 27% wealthy hand-to-mouth households (2019 SCF), the paper notes “in total 41% of US households are hand-to-mouth” (Section 4, p. 23).

The baseline two-asset model, calibrated to match mean total net worth (4.1), the total hand-to-mouth share (41%), and the poor-hand-to-mouth share (14%), generates an average quarterly MPC of 16.1% – about five times the comparable one-asset figure – and does so without the missing-middle problem: median net worth is 1.1, much closer to the empirical 1.56, because middle-of-distribution households hold most of their wealth illiquidly, where it barely affects the MPC (Section 4.1, pp. 25-26). Achieving this fit requires a sizable annualized gap between the illiquid and liquid returns – about 8 percentage points in the baseline calibration – and the paper cross-checks that most of the cross-sectional variation in MPCs tracks liquid wealth specifically, consistent with recent empirical evidence, while the MPC is comparatively flat as a function of illiquid wealth or total net worth (Section 4.1, p. 27, Figure 4).

Citing Jordà, Knoll, Kuvshinov, Schularick, and Taylor [2019], the paper notes the estimated real return gap between stocks/housing and T-bills since 1950 is only 5-6%, though this excludes tax advantages of housing and retirement accounts and employer 401(k) matching, and excludes the direct utility flow from owner-occupied housing (estimated elsewhere by the authors at 3-4% of financial-return equivalent) (Section 4.3, pp. 28-29). Adding a commitment/temptation motive for holding illiquid assets to the two-asset model can match the same hand-to-mouth and MPC targets with a much smaller return gap – as low as 4 percentage points in one calibration – because it independently raises households’ desire to hold wealth illiquidly (Section 4.3, pp. 29-30).

Because of the consumption function’s concavity, the MPC out of small windfalls exceeds the MPC out of large windfalls, and the MPC out of losses exceeds the MPC out of equally-sized gains; these asymmetries are modest in the baseline one-asset model but much stronger in models with more households in the concave region of the consumption function (liquid-wealth calibrations, ex-ante heterogeneity models, and two-asset models) (Section 5.1, Table 7, pp. 31-32). On timing, the paper also confirms, across models, that “the annual MPC is much smaller than four times the quarterly MPC” – cumulating with (1+m0)^4 - 1 would overstate the true annual MPC by 35% in the baseline calibration – because the time profile of MPCs declines, sometimes sharply, after the initial quarter (Section 5.2, p. 33, building on the annual-vs-quarterly comparison first noted in Section 2.2).

The average quarterly MPC out of a $500 illiquid windfall is 1.4%, and out of a $5,000 illiquid windfall is 2.4% – roughly an order of magnitude smaller than the corresponding liquid-wealth MPCs, matching the broad pattern in empirical estimates of housing-wealth and stock-wealth MPCs (which the paper cites as clustering between roughly 0 and 2.5%) (Section 5.3, pp. 33-34, citing Carroll, Otsuka, and Slacalek [2011]; Mian, Rao, and Sufi [2013]; Di Maggio, Kermani, and Majlesi [2020]).