Monetary Policy and Heterogeneity: An Analytical Framework

THANK is “a three-equation model isomorphic to the textbook representative-agent (RANK) model, which it nests,” but adds two household types – participants (“savers,” S) and non-participants (“hand-to-mouth,” H) – who switch states via a Markov chain with transition probabilities s and h, plus a family-insurance structure with full insurance within type and limited insurance across types (Section 2, pp. 3-4). TANK is nested as the special case of permanent idiosyncratic states (s=h=1); the “oscillating THANK” benchmark used throughout for its sharpest analytical results sets s=h=0 so households alternate deterministically between S and H every period, eliminating risk entirely while inequality remains “arbitrarily cyclical.” The model’s assumptions on the asset market – a family head who pools resources within an island before the aggregate shock is known but cannot fully insure across islands – are described as “novel” in combination, though subsets trace to Lucas (1990) and Shi (1997) (Section 2, p. 4, fn. 6).

χ is the elasticity of hand-to-mouth households’ income to aggregate income (y^H_t = χy_t), which the paper calls “the model’s keystone: a sufficient statistic” linking the income distribution to aggregates (Section 2.2, p. 8). Equilibrium income inequality γ_t is procyclical (rises when aggregate income rises) if and only if χ<1, and countercyclical if and only if χ>1. In this specific model χ depends on the degree of fiscal redistribution of profits (τ_D/λ) and on labor-supply elasticity (φ); the paper stresses this is “but one possible simple theory” of the distribution-aggregate feedback (p. 8, fn. 10). In RANK, by contrast, distributional considerations are absent altogether “since one agent works and receives all the profits” so income gains to labor and losses to profits net out for that single agent (p. 8).

Substituting the individual income rule into each household’s self-insurance Euler equation yields an aggregate Euler-IS equation ct = δE_t c_{t+1} - σ(…)（i_t - E_tπ_{t+1}), where the compounding/discounting coefficient δ exceeds 1 if and only if inequality is countercyclical (χ>1) and is below 1 if inequality is procyclical (χ<1) (Section 2.2, eq. 13, pp. 8-9). The economics: with countercyclical inequality, good news about future aggregate income disproportionately raises H’s future income, so households anticipate needing less self-insurance and dis-save today, amplifying today’s demand beyond one-for-one; with procyclical inequality, households recognize they may end up constrained in the future and so self-insure more today, dampening the demand response below one-for-one. The paper shows this compounding/discounting survives even in the risk-free oscillating benchmark (eq. 14), demonstrating that “risk is not necessary for Euler discounting/compounding: cyclical inequality is sufficient” (p. 9).

Under a simple Taylor rule i_t=φπ_t, THANK collapses to a single forward-looking equation, and Proposition 1 shows the model is determinate if and only if φ exceeds a threshold φ* that depends on the compounding/discounting parameter δ: “The Taylor principle φ>1 is sufficient for determinacy if and only if inequality is procyclical,” i.e. δ≤1 (Section 3.1, p. 13). With countercyclical inequality (δ>1), the threshold is strictly above 1 and can be large – the paper reports φ*≈2.5, rising toward 5 under some calibrations matching Kaplan, Moll and Violante (2018) (p. 13). With procyclical inequality, the Taylor principle is sufficient but not even necessary: determinacy can hold “even under a peg φ=0… undoing the Sargent-Wallace result,” provided discounting is strong enough (eq. 19, p. 13-14).

Proposition 2 states that “in THANK with cyclical inequality, there is amplification of monetary policy relative to RANK and the fiscal multiplier on consumption is positive if and only if χ>1, whereas the forward-guidance puzzle is ruled out… only if χ<1” (Section 3.2, p. 14). That is, exactly the countercyclical inequality that produces the demand amplification and positive fiscal multipliers most quantitative HANK models are built to generate is the same condition that makes the forward guidance puzzle worse rather than better; conversely, the procyclical inequality that cures the puzzle also eliminates amplification and multipliers. The oscillating-THANK special case gives this its sharpest form (eq. 22, p. 14): the same χ that switches the sign of the multiplier switches the sign of the puzzle’s presence.

The paper decomposes cyclical income risk into a component coming from cyclical inequality and a separate component from conditional skewness (cyclical variation in the probability of landing in the constrained state), and Proposition 3 shows “THANK with cyclical inequality and risk resolves the Catch-22 if and only if one channel is [countercyclical while the other is procyclical, and strongly enough]” (Section 4.1, pp. 15-16). Intuitively, if risk is countercyclical (generating amplification through a precautionary-saving channel independent of inequality) while inequality is procyclical enough to generate sufficient Euler discounting, the model can have both amplification/multipliers and a cured forward guidance puzzle simultaneously; if both channels move the same way, the tension is not resolved and can even be worsened.

Using realtimeinequality.org data (Blanchet et al., 2023) on the last two U.S. recessions, the paper reports that “inequality in disposable income (post taxes and transfers) has been procyclical – although inequality in factor income (labor earnings plus capital income) was strongly countercyclical, as was ex-ante income risk” (Introduction, p. 3, Figure 1), attributing the wedge between factor-income and disposable-income cyclicality to strongly countercyclical government transfers (p. 3, fn. 4). This is offered as suggestive, not definitive, evidence that the empirically relevant case may combine procyclical (disposable-income) inequality with countercyclical risk – precisely the combination that Proposition 3 shows can resolve the Catch-22.

Proposition 4 shows a Wicksellian price-level-targeting rule i_t=φ_p p_t “leads to local determinacy even when [inequality and risk compounding together would otherwise violate it],” so the model “delivers amplification without also aggravating the FG puzzle even when both inequality and risk are countercyclical” (Section 4.2, p. 17), because even a small response to the price level (rather than its growth rate) anchors long-run price-level expectations in a way a pure interest-rate peg cannot. In the model’s extension with government-bond liquidity, Proposition 6 shows an analogous result for a nominal-debt rule following Hagedorn (2020): “The THANK model with a well-defined demand for liquid bonds” achieves determinacy under this fiscal/monetary combination as well (Section 5.3, p. 20), giving the paper two distinct policy-based escapes from the Catch-22 that do not require assuming away cyclical inequality or risk.

In the version of THANK with government-bond liquidity, Proposition 5 derives closed-form expressions for the intertemporal MPCs – the response of consumption today to an income shock realized T periods in the future – extending the iMPC concept introduced by Auclert, Rognlie, and Straub (2023) to embed the cyclical-inequality channel (Section 5.1-5.2, pp. 18-19). The paper states this is, to its knowledge, the first analytical derivation of iMPCs that simultaneously captures cyclical inequality alongside the self-insurance/liquidity motive that the original iMPC concept was built around, allowing the paper to connect its closed-form determinacy results to the numerical iMPC-based determinacy diagnostics used in the quantitative-HANK literature.

Proposition 7 shows that the Ramsey welfare-maximization problem in THANK (in the version without liquidity) reduces to minimizing a discounted quadratic loss in inflation, the output gap, and a new term: inequality itself, π_t²+α_yy_t²+α_γγ_t², where α_γ is proportional to (χ-1)² (Section 6, p. 21-22). This inequality-stabilization motive is present whenever χ≠1 regardless of the sign of cyclicality, and it raises the effective weight on output stabilization relative to RANK. Under discretion, the paper derives the standard tradeoff-implying targeting rule (eq. 38-39) and shows that “optimal policy in THANK requires greater inflation and lower output volatility than in RANK” (p. 22); it further shows that the interest-rate rule implementing optimal discretionary policy can require cutting rates in response to shocks that would call for a rate increase in RANK, when inequality is countercyclical enough (p. 23). Under commitment, optimal policy amounts to a form of price-level targeting (eq. 40) that “delivers determinacy regardless of heterogeneity” (p. 23).

Because the Ramsey problem’s welfare approximation is taken around the perfect-insurance, no-inequality efficient (flexible-price) equilibrium as its target, “idiosyncratic risk and its cyclicality are irrelevant for optimal policy, insofar as the target flexible-price equilibrium is the first-best… without inequality” (Section 6, p. 22). However, the IS curve that determines which interest rate actually implements the optimal allocation is not itself independent of cyclical inequality (it enters through δ), and while the paper notes the interest rate implementing optimal policy in this liquidity-free benchmark is also independent of risk cyclicality, it flags that “this is no longer the case – and risk then matters – in the model with liquidity, where the interest rate has direct distributional consequences,” left to future work (p. 22, fn. 23).