Heterogeneity and Aggregate Fluctuations: Insights from TANK Models

The question is narrow by design: what do the propagation channels in HANK models do to aggregate fluctuations, relative to a representative-agent New Keynesian model, and can a tractable model reproduce them? The stated goal is twofold — “we want to understand the mechanisms through which heterogeneity, in the form of idiosyncratic income risk, affects aggregate fluctuations and the transmission of shocks in HANK models,” and “we want to investigate whether there are simple tractable models that can capture reasonably well the mechanisms that we identify as most relevant in richer models.” Two exclusions are flagged in footnotes rather than buried: firm heterogeneity is set aside (“our focus in the present paper is on models with household heterogeneity”), and so is the distributional question — “HANK models may also be used for understanding the distributional impact of shocks or alternative policy interventions. Our focus here is exclusively on its implications for aggregate fluctuations.” A third exclusion appears in §7: “our analysis has refrained from normative considerations, such as the implications of heterogeneity for the optimal design of monetary policy,” with citations to work showing that “stabilizing inflation is no longer optimal in the presence of inequality.”

Idiosyncratic income risk under incomplete markets is treated as the defining feature of a HANK model, and a tractable model is defined precisely as one that abstracts from it. The authors are explicit: “We view the presence of idiosyncratic income shocks (in the absence of complete markets) as a defining feature of HANK models, relative to tractable models like RANK or TANK. It is also a main factor behind its nontrivial equilibrium dynamics, since it gives rise to a time-varying wealth distribution which becomes an infinite-dimensional state variable.” Everything else — binding borrowing constraints, the liquid/illiquid distinction — is called “non-essential, in the sense that one can make alternative assumptions in their regard, without altering what we view as the defining assumption.” A tractable model is then “one that abstracts from the presence of idiosyncratic income risk, which allows one to assume a small number of household types, each of which is made of a time-invariant set of households that are identical, both ex-ante and ex-post,” whose linearized equilibrium conditions “can be solved for analytically, arguably rendering them more suitable for use in the classroom or for communication with policy makers.” This definitional choice is what makes the paper’s headline result meaningful: it is idiosyncratic risk specifically, not “heterogeneity” loosely, that is being tested.

Through a single additional term in the aggregate Euler equation — the risk shifter — and the gap from RANK “depends exclusively on current and expected future values” of it. Log-linearizing and aggregating the household Euler equation gives ĉ_t = E_t{ĉ_{t+1}} − (1/σ)r̂_t − ((σ+1)/2)v̂_t, where v̂_t is a consumption-weighted average of individual consumption risk in deviation from its mean. The authors flag the modelling assumption that makes this first-order: “We assume that due to the presence of idiosyncratic income risk, variations in v_t are of the same order of magnitude as variations in aggregate variables resulting from aggregate shocks. This is in contrast with a model with a representative household, in which by construction var_t{c_{t+1}} is of second order.” Iterating forward, aggregate consumption “depends inversely on current and anticipated values of the risk shifter which capture the extent of precautionary savings.” The gap between the two models is therefore exactly the discounted sum of expected future risk shifters — and, being endogenous, “cannot be solved in closed form, so we need to (numerically) solve for the equilibrium of the HANK-I model to evaluate quantitatively the size of that gap.”

Because the responsiveness of individual consumption to idiosyncratic income is concentrated among poor households, and poor households carry little weight in aggregate consumption. The paper uses an approximation from its companion work: v_t(j) ≈ ψ_t(j)²σ²_ξ, where ψ_t(j) is the elasticity of individual consumption with respect to the innovation in the idiosyncratic income process and σ²_ξ its variance. Then ψ_t(j)² “is decreasing and (strongly) convex in the level of consumption, capturing the fact that the consumption of households with less liquid wealth and closer to the natural debt limit is more responsive to shocks that change that wealth (i.e. their MPCs are higher).” The conclusion follows arithmetically: the derivative of ψ² with respect to a shock “is quantitatively significant only for poor households. Since the weight of those households in aggregate consumption is small, the dynamic response of the aggregate risk shifter becomes muted, thus accounting for the small gap.” Quantitatively, a 25 basis point reduction in the real rate on impact (one percentage point annualized) produces effects that are “stronger on impact, and more persistent” in HANK-I than RANK, but “the difference is, however, quantitatively very small”: simulated output volatility is larger by a factor of 1.22 with correlation 0.97.

An expansionary monetary shock compresses wealth dispersion through the interest rate exposure channel while widening income dispersion through the income distribution channel, and with dividends shared at all (θ > 0) the two “work in opposite directions and thus tend to neutralize” each other. The interest rate exposure channel is that a rate cut “implies a redistribution from (rich) lenders to (poor) borrowers, which reduces wealth dispersion.” The income distribution channel works through the markup: differentiating household income gives dY_t(j)/Y = [Ξ_t(j) − θ(1 − Ξ_t(j))(σ + φ)]dy_t + θ(1 − Ξ_t(j))(1 + φ)da_t, so an expansion raises a household’s income “if and only if Ξ_t(j) > θ(σ+φ)/(1 + θ(σ+φ))” — that is, “income is redistributed from poor/low productivity to rich/high productivity households.” Turning off the income distribution channel by setting θ = 0 leaves a “tiny gap” in the output response, so its independent contribution is small; in that case the interest rate exposure channel alone amplifies output relative to RANK “albeit in a small amount given the low weight of poor households’ consumption in the aggregate.”

Nothing at all in RANK, and a very small positive amount in HANK-I, entirely through redistribution from rich to poor. With the central bank holding the real rate constant, “a one percent positive technology shock has no effect on output in the RANK model since the central bank keeps the real rate constant, thus preventing aggregate demand from increasing.” In HANK-I the shock “raises dividends while reducing labor income by the same amount” at unchanged output, which “raises the income of households with productivity below the mean (Ξ_t(j) < 1), for which dividends account for a larger share of their income, while lowering it for the remaining households.” The reduction in consumption risk for the poor outweighs the increase for the rich, precautionary saving falls, and demand expands — “again, the effect is quantitatively small because the reduction in precautionary savings affects mostly households with a low consumption share to begin with.” The clean test is the θ = 0 counterfactual, whose response “overlaps perfectly with the zero response associated with RANK, for in that case there is no income redistribution and hence no change in consumption.”

It generates persistence, but the paper judges the amount beyond the first period to be quantitatively small. Responding to purely transitory shocks, “while the presence of idiosyncratic risk generates significant persistence in the output response, the effect beyond the initial period is quantitatively small. In other words, the endogenous response of the wealth distribution to an aggregate shock has quantitatively small implications for aggregate output.” A footnote registers the counterargument against their own reading rather than omitting it: “tiny quantitative differences that are highly persistent may end up having significant cumulative effects, often described by means of a cumulative multiplier statistic. That statistic may capture differences that grow very fast in percent terms with the horizon if the reference response in the denominator gets close to zero.”

Not necessarily, and the paper calls its own finding a seeming paradox: for monetary shocks it does not amplify significantly, for technology shocks it does. With ψ̄ = 2 and 30% of households constrained in steady state, “the presence of a binding borrowing constraint does not amplify significantly the response of output” to an expansionary monetary shock — despite the large Keynesian-cross slope that the hand-to-mouth share creates. Two offsetting factors are named: intertemporal substitution by unconstrained households is weaker because “the latter account only for a fraction 1 − λ^H_t of all households,” and “because of the tighter borrowing constraint the distribution of wealth across households in HANK-II is less dispersed than in HANK-I. As a result, the interest rate exposure channel is more muted.” For technology shocks the arithmetic flips, because “the absence of a monetary policy response neutralizes the interest rate exposure channel, as well as the intertemporal substitution by unconstrained households. As a result, the higher multiplier associated with the presence of hand-to-mouth households ends up becoming the key factor.” The summary is stated as a conditional, not a general claim: amplification “depends on the nature of the shock as well as on the strength of potential offsetting effects (including an eventual endogenous response of monetary policy, not modeled here).”

It misses the interest rate exposure channel because its hand-to-mouth households hold no assets, and it gets the sign of the income distribution channel backwards because their consumption is pure labour income. In TANK-I the constrained simply consume their wage bill, so aggregate consumption is λ^H(M_t/M)Y_t + (1 − λ^H)C^U_t. The first defect: “(20) fails to capture the interest rate exposure channel revealed by (19), since hand-to-mouth households are not indebted in TANK-I.” The second: the equation “points to a negative relation between aggregate consumption and the markup, for a given initial level of output, due to the negative effect of a higher markup on labor income… Thus, the sign of the income distribution channel at work in HANK-II… is reversed.” Empirically, setting λ^H = 0.30 to match HANK-II’s steady state, the monetary output response is “highly amplified in TANK-I… almost trebling the effect on impact,” and for technology shocks “the difference is even starker since the sign of the output response in TANK-I is reversed relative to HANK-II, due to the fall in labor income.” Verdict: “one can hardly view TANK as providing a reasonable approximation to HANK.” There are also two defects the authors classify as intrinsic to any TANK model rather than fixable: a constant rather than time-varying hand-to-mouth share, and no first-order precautionary saving.

Three modifications — permanent indebtedness, below-average productivity, and dividend receipt on the HANK rule — and the result tracks HANK-II closely on monetary shocks. The hand-to-mouth are assumed “permanently against the borrowing constraint introduced in HANK-II, i.e. B^H_t = −ψ̄Y for all t”; to have productivity Ξ^H < 1; and to receive dividends under the same rule as in HANK-II. Calibration sets λ^H = 0.30, Ξ^H = 0.56 and Θ^H = 0.78, “thus matching the steady state values of λ^H_t and Ξ^H_t in HANK-II,” with ψ̄ = 2 as in HANK-II. The conditions under which the approximation should work are stated as an explicit list rather than asserted: it holds if variations over time in λ^H_t and Ξ^H_t are small, if the gap between actual and assumed debt levels is small, and “if the impact of the shock on aggregate precautionary savings is small (as we showed to be the case in the context of HANK-I).” Results: the monetary output response “matches closely”; for the technology shock “the match is also reasonably good, especially in comparison with TANK-I, which even fails to get the sign right.” Simulated volatility ratio is 1.10 for monetary shocks with correlations “very close to unity”; for technology shocks the ratio is 1.97, but the authors decline to treat that as damning — “it is not clear that the latter is much meaningful since the absolute impact of the shock is tiny in the two cases.”

It also reproduces the effect of a policy-relevant change in the environment, and the decomposition of the consumption response — except for a residual it structurally cannot contain. Tightening the borrowing limit to ψ̄ = 0.8, which raises the constrained share from 0.30 to 0.50, “shifts down the impulse response, i.e. it dampens the impact of monetary policy” in HANK-II, and “a similar downward shift is observed in the case of TANK-II”; the amplification of the technology response is also captured. The paper then decomposes the aggregate consumption response into a hand-to-mouth term, a RANK term, and a residual arising from variation in λ^H_t plus the risk and composition shifters. That residual “is absent in TANK-II, since the latter assumes subsets of unconstrained and hand-to-mouth households that are time invariant in size and composition, and displays no precautionary savings.” For monetary shocks the decomposition is similar across models, “suggesting that not only the TANK-II model is successful in approximating the aggregate properties of HANK-II but also in capturing the underlying mechanisms.” For technology shocks, “given the small output response to the shock, the residual component (shown as ‘other’ in the Figure) is relatively more important, even though still small in absolute terms. That component cannot be captured by the TANK-II model.”

It reverses the direction of the correction: monetary shocks are amplified and the technology-shock output response turns negative, because wealthy hand-to-mouth households can smooth cash-on-hand by liquidating equity, which severs the tight link between dividends and their consumption. In HANK-III households hold liquid bonds and equity through intermediaries, equity adjustment carries a cost and cannot go short, so three groups coexist: unconstrained, wealthy hand-to-mouth (borrowing constraint binds, equity positive), and poor hand-to-mouth (both bind). Calibration sets ω = 0.002 for an annualized equity premium of 0.8 percent and a steady-state equity return of 1.0071, total assets of 3.20 times annual GDP, χ₂ = 2, χ₀ = 2.55, χ₁ = 9.60, giving 30 percent constrained of which 25 percent are wealthy and 5 percent poor hand-to-mouth, and liquid and illiquid assets of 0.25 and 2.9 times annual GDP, “close to the values reported in Kaplan et al. (2018).” The mechanism is stated precisely: the equity-withdrawal term in the constrained household’s budget means that “in HANK-III the decline in dividends does not directly impact their cash-on-hand unless it is reflected in lower withdrawals from the equity account. In other words, households’ ability to manage their cash-on-hand through the adjustment of their portfolio weakens the income distribution channel. As a result the relative importance of the interest rate exposure channel is enhanced.” For technology shocks, output falls, because “poor hand-to-mouth households do not benefit from the higher dividends, while the wealthy hand-to-mouth cannot freely convert dividends into available cash-on-hand.”

It is flagged as counterfactual and attributed to the constant-real-rate assumption, with the fix shown later in the paper rather than promised. The authors write that this finding “contrasts with existing evidence” and “should not be held against the empirical merits of HANK-III since it hinges critically on our (counterfactual) assumption of a constant real rate,” citing Galí (1999) and Basu et al. (2006), and adding: “In section 6 below we show how the sign of that response switches from negative to positive when we assume a more realistic monetary policy response.” The constant-real-rate assumption itself is defended on methodological grounds at the outset — it is chosen “so that the response of aggregate variables to different shocks is not affected by any particular assumption regarding the monetary policy rule, which would generally lead to different paths of the real rate across environments.”

By splitting the hand-to-mouth into poor and wealthy and calibrating the wealthy group’s dividend receipt to match HANK-III’s equity withdrawals, after which the income distribution channel turns out to be very weak. Poor hand-to-mouth consume labour income net of debt service; wealthy additionally “cash-in some dividends from their holdings of stocks.” Since the wealthy in TANK-III cannot adjust equity, the strategy is to “calibrate Θ^W so that it captures the change in individual withdrawals F^W_t in response to a unit increase in aggregate dividends D_t, as implied by HANK-III” — the ratio of impact responses, averaged across the two shocks because it differs slightly between them. Calibration: λ^P = 0.05, λ^W = 0.25, Ξ^P = 0.43, Ξ^W = 0.66, implying average hand-to-mouth productivity Ξ^H = 0.62, with Θ^W = 0.58. The resulting λ^H Ξ^H − λ^W Θ^W = 0.04 > 0 gives “a negative relation between the average markup and aggregate consumption, given output, in contrast with our calibrated TANK-II model” — but since that difference is near zero, “the income distribution channel is, however, very weak quantitatively,” and because λ^H Ξ^H = 0.18 is small, “similar results are obtained in a version of the TANK-III model that makes the extreme assumption of Θ^W = 0.” TANK-III “is able to capture, at least qualitatively,” both the amplification of monetary shocks and the sign reversal for technology shocks. The decomposition again matches well for monetary shocks and less well for technology shocks, “in particular given the substantial role of the residual component” — on which the authors say plainly, “We plan to investigate further the reasons for the larger gap in this particular case.”

Under a conventional Taylor rule the models converge dramatically, and under strict inflation targeting all of them — RANK, every TANK, every HANK — deliver an identical output path equal to natural output. With φ_π = 1.5 and φ_y = 0.5/4 = 0.125 as in Taylor (1993), and with nominal rather than real bonds so unexpected inflation matters, “the assumption of an endogenous response reduces even further the gap between the predictions of RANK, TANK and HANK models regarding the aggregate output response to both monetary policy and technology shocks, thus strengthening the view of a limited role for the presence of idiosyncratic income risk as a factor shaping aggregate fluctuations.” The explanation rests on two properties: “the natural level of output, y^n_t, is invariant to the presence of idiosyncratic income risk,” and the shared New Keynesian Phillips curve “displays the divine coincidence property, namely, stabilization of inflation implies stabilization of the output gap, and viceversa.” The limiting case is stated as a proposition: under strict inflation targeting (π_t = 0 for all t) “all the HANK, TANK and RANK models considered above generate an identical equilibrium output path, which corresponds to that of natural output,” with the proof following directly from the Phillips curve. The authors immediately bound it: “the fact that equilibrium output is identical across models under strict inflation targeting does not mean that this is also the case for other variables, including the real interest rate.” And they note in §7 that the result fails if heterogeneity affects natural output — “to the extent that the presence of heterogeneity affects the natural level of output, the irrelevance proposition found above will no longer obtain.”

Both directions are reported. Supporting: Berger et al. (2023), using CEX consumption data, “find that wedges capturing deviations from perfect risk sharing only account for 7% of output fluctuations”; Bayer et al. (2024) and Bilbiie et al. (2023), estimating medium-scale heterogeneous-agent models, “conclude that household heterogeneity does not fundamentally alter our understanding of the causes and consequences of aggregate fluctuations”; and McKay and Wolf (2023), who “argue that many of the redistributive channels at work in HANK economies operate in opposite directions, and tend to offset each other.” Cutting against: Fagereng et al. (2021), on Norwegian tax-registry data, “estimate large MPCs out of lottery wins (about 0.5), that persist for several years,” a finding that “could be rationalized by certain HANK models, but is inconsistent with representative agent models… and with TANK models — where the MPC falls abruptly after one period.” The authors offer the opposing estimates rather than only the convenient ones: studies of the 2008 rebate “find a smaller MPC on impact (below 0.3) that remain positive for at most few months. This latter finding can be matched by the simple TANK models discussed above.” On fiscal shocks specifically they concede an open question: “Determining whether simple TANK models can account for the empirical evidence on iMPCs remains an open question.”

Explicitly conditional on precautionary saving being unresponsive to aggregate shocks, and silent on several margins the authors name. The first caveat: “our main result applies to environments where idiosyncratic income risk and the associated precautionary savings motive play a limited role for the transmission of aggregate shocks. This is the case in the HANK models considered above, where the ‘risk-shifter’ is largely insensitive to aggregate shocks.” The paper names what would break this — “larger fluctuations in the ‘risk-shifter’ would naturally arise in the presence of countercyclical income risk[,] an aspect that has been ignored in our analysis, but that has been shown to be empirically relevant” — and cites Bilbiie (2023) as an example where “the precautionary savings motive plays a more prominent role.” Second, only monetary and technology shocks are considered. Third, heterogeneity is not allowed to operate through the supply side, and the authors flag that HANK economies with segmented labour markets or heterogeneous firms are left “for future research.” The concluding formulation is correspondingly hedged: “For each HANK model considered, we have found a suitably specified and calibrated RANK or TANK model that captures reasonably well its implications for aggregate output and the main channels through which aggregate shocks are transmitted.”