Monetary Policy and Heterogeneity: An Analytical Framework I

THANK is a tractable heterogeneous-agent New-Keynesian model that captures analytically core micro-heterogeneity channels of quantitative-HANK: cyclical inequality and risk; self-insurance, pre-cautionary saving, and realistic intertemporal marginal propensities to consume. I use it to elucidate key transmission mechanisms and dynamic properties of HANK models. Countercyclical inequality yields aggregate-demand ampliﬁcation and makes determinacy with Taylor rules more stringent; but solving the forward guidance puzzle requires procyclical inequality: a Catch-22. Solutions include combining inequality with a distinct risk channel, with compensating cyclicalities; I provide evidence that disposable income inequality was procyclical in the last two, Great and COVID recessions, while risk is countercyclical. Alternative policy rules also solve the Catch-22, e.g. price-level-targeting or, in the model version with liquidity, setting nominal public debt. Optimal policy with heterogeneity features a novel inequality-stabilization motive generating higher inﬂation volatility—but is unaffected by risk, insofar as the target efﬁcient equilibrium entails no inequality.


Introduction
A spectre is haunting Macroeconomics-the spectre of Heterogeneity.Some of the world's leading policymakers have been asking for research on it, and its other name, "Inequality", in connection with stabilization, monetary and fiscal policies.For until recently, research on these two topics has been, with few exceptions, largely disconnected.Yet a burgeoning field emerged as a true synthesis of these two lodes: heterogeneous-agent (HA) and New Keynesian (NK), leading to HANK. 1  The vast majority of contributions consists of quantitative models, involving heavy machinery for their resolution, the price to pay to achieve the realism conferred by matching the micro data. 2  Yet given that much of the post-2008 bad press of existing DSGE models refers to their being too complex and somewhat black-box, it seems important to build simple tractable representations of these models to gain analytical insights into their underlying mechanisms and make their policy conclusions sharper and easier to communicate.The two, quantitative and analytical approaches are thus strongly complementary and reinforce each other.
With this paper, I wish to suggest a tractable HANK model, THANK, focusing on the distinction between cyclical inequality and cyclical risk and their importance for understanding the workings of HANK models.To achieve this, I first outline THANK as a projection of rich-heterogeneity quantitative HANK along several key dimensions.I then exploit the tractability to elicit in closed form the distinct roles of cyclical inequality and risk for the model's dynamic properties: determinacy of interest rate rules, curing the forward guidance (FG) puzzle, amplification and fiscal multipliers; and optimal monetary policy.
THANK is a three-equation model isomorphic to the textbook representative-agent (RANK) model, which it nests; yet it captures several dimensions that the recent quantitative literature finds important for the study of macro fluctuations with heterogeneity.First, it features a key aggregatedemand (AD) amplification, the "New Keynesian cross" present in any HANK model where some households are constrained hand-to-mouth.Heterogeneity shapes aggregate outcomes through cyclical inequality: how the distribution of income between constrained and unconstrained changes over the cycle, e.g. who suffers more in recessions.This originates in the TANK model in Bilbiie (2008Bilbiie ( , 2020)), generalizes to the subsequent rich-heterogeneity literature (Auclert (2019)), and has recently been investigated empirically (Patterson (2023)).
Second, my analytical model incorporates precautionary, self-insurance saving, and the distinction between liquid and illiquid assets, staples of quantitative HANK models, e.g.Kaplan et al (2018).Third, it also encompasses risk; I propose a decomposition of cyclical income risk into one part due to cyclical inequality and one related to (conditional) skewness: cyclical variations in the likelihood of ending up in the bad state. 3I show that this distinction is key for some of the theoretical model properties.Finally, the version with liquidity allows an analytical solution for the key "intertemporal MPCs" (iMPCs) that Auclert, Rognlie, and Straub (2023) introduced in a quantitative HANK, extended here to study the novel interaction with cyclical inequality.To the best of my 1 The abbreviation is due to Kaplan, Moll, and Violante (2018); the opening sentence is a paraphrase of Marx and Engels. 2 Overwhelming evidence was long available for the failure of an aggregate Euler equation, for a high fraction of households having zero net worth and a high marginal propensity to consume MPC, "hand-to-mouth".Important work clarified the link with liquidity constraints: some "wealthy" households behave as hand-to-mouth if wealth is illiquid (Kaplan and Violante (2014)), perhaps because it consists of a mortgaged house (Cloyne, Ferreira, and Surico (2015)).
3 Relatedly, the model delivers key stylized statistical properties of idiosyncratic income emphasized by a large empirical literature: autocorrelation, cyclical variance, negative skewness, and leptokurtosis.knowledge, to this date, THANK is the only tractable framework that simultaneously captures all of these important features of the rich micro-heterogeneity models and allows gaining insights into their mechanisms.This burgeoning literature is reviewed in detail in Appendix A, including my own past work.
Armed with this representation, the paper studies analytically the role of cyclical inequality for the model's dynamic properties-in particular in relation to cyclical risk.Aggregate dynamics depend chiefly on the cyclicality of inequality, that is on the constrained agents' income elasticity to aggregate income χ.AD-amplification occurs with countercyclical inequality, when χ > 1: a demand increase leads to a disproportionate increase in constrained agents' income and a further demand expansion; intertemporally, this also delivers compounding in the aggregate Euler equation.Conversely, procyclical inequality χ < 1 leads to AD-dampening and Euler-equation discounting.
Understanding determinacy properties when inequality is cyclical is crucial for being able to even solve quantitative models, and for policy: what rules or institutions can anchor expectations when distributional mechanisms are at play.The determinacy properties of Taylor rules reflect the above intuition, materializing into a modified Taylor principle.When inequality is countercyclical, the central bank needs to be possibly much more aggressive than the "Taylor principle" (increasing nominal interest more than inflation) to rule out indeterminacy.Whereas in the discounting, procyclical-inequality case, the Taylor principle is sufficient but not necessary: for a large region there is determinacy even under a peg, undoing the Sargent-Wallace result.
A Catch-22 for HANK models thus emerges: AD-amplification relative to RANK and multipliersthat much of the literature uses HANK models for-require countercyclical inequality χ > 1.Yet ruling out the forward guidance puzzle-that the later an interest rate cut takes place the larger its effect today (Del Negro, Giannoni, and Patterson, 2023)-requires the opposite: procyclical inequality χ < 1; evidently, χ > 1 aggravates the puzzle.
The paper then proposes solutions to the Catch-22, consisting of both model ingredients and alternative policy rules.A key contribution is emphasizing the different role for transmission and the decomposition of two cyclical channels: income inequality and, as previously emphasized by others and reviewed below, risk.My model includes a novel formalization of the latter that can also give rise to dampening/amplification, but through a distinct mechanism: if uninsurable risk increases in expansions, precautionary saving leads agents to cut back demand; if it falls in expansions (is countercyclical), agents dis-save and augment their demand.The Catch-22 is resolved in a model where both inequality and risk are cyclical, as long as their cyclicalities have the opposite sign: so that if e.g.risk is countercyclical and yields AD amplification, inequality is procyclical enough to cure the FG puzzle.Figure 1 zooms in on the last two U.S. recessions to show 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. 44 The reason is that transfers are highly countercyclical.Below, I review existing evidence on the cyclicality of inequality and risk, e.g.Heathcote et al (2023) and Guevenen et al (2014).Other modelling choices circumvent the Catch-22 altogether (e.g.Auclert et al assume proportional incomes i.e. acyclical inequality and risk), or augment the model with information frictions that impart a separate source of Euler discounting.The analysis of optimal monetary policy further underscores the importance of distinguishing the different roles of inequality and risk.In the benchmark I study, whereby fiscal policy is in charge of fixing long-run inequality distortions and there is no liquidity, risk is irrelevant for optimal policy, while cyclical inequality is instead a key input.Complementary to quantitative studies that address phenomenal technical challenges, I calculate optimal policy analytically in THANK, approximating aggregate welfare to second-order to derive a quadratic objective function for the central bank.This encompasses a novel inequality motive, implying optimally tolerating more inflation volatility when more households are constrained.While inequality is of the essence for optimal policy, risk is not-insofar as the policymaker shares society's first-best (perfect-insurance) objective.Risk does matter for implementation: with countercyclical inequality and idiosyncratic risk, the interest-rate rule that implements optimal discretionary policy may entail cutting real rates, when in RANK it would imply increasing them.Furthermore, optimal policy under commitment ensures determinacy regardless of heterogeneity and inequality-cyclicality and, while affected by similar inequality considerations, amounts to a form of price-level targeting.

THANK: An Analytical HANK Model
This section outlines THANK, an analytical model that captures several key channels of complex HANK.While related to several studies reviewed above, the exact model is to the best of my knowledge novel to this paper and its companion Bilbiie (2020), which uses a special case (with this paper as its reference for the full model), focusing on AD amplification of monetary and fiscal policies through a "New Keynesian Cross" and on using it as a one-channel approximation to richer HANK.
A unit mass of households j 2 [0, 1] discount the future at rate β, derive utility from consumption C nominal return i t > 0. Households participate infrequently in financial markets and freely adjust their portfolio when they do.When they do not, they receive only the payoff from previously accumulated bonds.Denote these two states, for ease and anticipating what equilibrium we will focus on, S from "savers" for participants and H from "hand-to-mouth" for non-participants.
The exogenous change of state follows a Markov chain: the probability to stay type S and H is respectively s and h, with transition probabilities 1 s and 1 h; later on, I assume that s is a function of aggregate activity.I focus on stationary equilibria whereby the mass of H is the standard unconditional probability λ, and 1 λ is the mass of S: that is the ergodic distribution in the idiosyncratic dimension around which we approximate the model with respect to aggregate shocks.TANK is nested with permanent idiosyncratic shocks s = h = 1 and λ fixed at its initial free-parameter value; the other non-oscillating extreme is the iid case s = 1 h = 1 λ: being S or H tomorrow is independent of today's state.
A particularly useful benchmark that I refer to as oscillating THANK abstracts from risk and focuses on inequality only.There is deterministic switching every period, so there is no risk; but agents oscillate between the two states, across which income inequality is still cyclical.This is nested in my model when s = h = 0 with agents alternating states and λ = 1 2 . 5Key assumptions on the asset market structure simplify the equilibrium and afford an analytical solution; the precise combination of assumptions is novel, although subsets have been used by existing literature, notably the seminal monetary-theory contributions of Lucas (1990) and Shi (1997). 6 First, households belong to a family whose utilitarian equally-weighted intertemporal welfare its head maximizes facing limits to risk sharing.Households can be in two states or "islands", with all participants on island S and all non-participants on H.The family head can transfer all resources across households within the island, but can only transfer some resources between islands.
In face of idiosyncratic risk there is thus full insurance within type, after idiosyncratic uncertainty is revealed, but limited insurance across types.At the beginning of the period, the family head pools resources within the island.The aggregate shocks are revealed and the family head determines the consumption/saving choice for each household in each island.Then households learn their nextperiod participation status and must move to the corresponding island, taking bonds with them.There are no transfers after the idiosyncratic shock is revealed, which is taken as a constraint for the consumption/saving choice.The model thus incorporates a specific notion of liquidity: bonds are liquid in that they can be used to self-insure before idiosyncratic uncertainty is revealed.
The flows across islands are as follows.The total measure of households leaving island H each period who participate next period is (1 h) λ; the rest λh stay.Likewise, a measure (1 s) (1 λ) leaves island S for H at the end of each period.Total welfare maximization implies that the family head pools resources at the beginning of the period in a given island and implements symmet- 5 The detailed outline is in Appendix C.1 for completion.This limit case is akin to Woodford (1990), abstracting from the endogenous income distribution, essential here.It is also related to the model class in Chapter 22 of Ljungqvist and Sargent (2018), ignoring the limited commitment part; I am grateful to Kurt Mitman for suggesting the latter connection.
6 Closer to this literature, this way of reducing heterogeneity and eliminating the wealth distribution as a state variable also extends Challe and Ragot (2011), Challe et al (2017), Heathcote and Perri (2018) and Bilbiie and Ragot (2020).NK models with two switching types were studied by Curdia and Woodford (2016)  ric consumption/saving choices for all households in that island.Denote by B j t+1 per-capita, real beginning-of-period-t + 1 bonds on island j = S, H: after the consumption-saving choice, and also after changing state and pooling.The end-of-period-t (after the consumption/saving choice but before agents move across islands) per capita real values are Z j t+1 .We have the following relations: Rescaling by the population masses and using (1): (2) The program of the family head is: subject to the laws of motion for bond flows (2) and budget constraints.Y j t are post-tax incomes, which for S also include dividends D t from holding firm shares; these can thus be regarded as completely-illiquid across islands, "immobile" assets.All households receive the real return on their respective bond holdings, with i t 1 nominal interest and π t net inflation, and face positive constraints on new bonds (4).
The Kuhn-Tucker conditions with complementary slackness are: The key is the Euler equation ( 5), governing the bond-holding decision of S self-insuring against the risk of becoming H and taking into account that bonds can be used when moving to the H island. Equation ( 6) determines the bond choice of agents in the H island; both bond Euler conditions are written as complementary slackness conditions.With this market structure, the Euler equations ( 5) and ( 6) are of the same form as in fully-fledged incomplete-markets model of the Bewely-Huggett-Aiyagari type.In particular, the probability 1 s measures the uninsurable risk to switch to a bad state next period, risk for which only bonds can be used to self-insure-thus generating a demand for bonds for "precautionary" purposes. 7o obtain the simple equilibrium representation, I focus on equilibria where the constraint of H always binds and ( 6) is in fact a strict inequality, whatever the reason: for instance, the shock is a "liquidity"/impatience shock making them want to consume more today (decreasing β in (6) to a low enough value β H ); or their average income in H is lower enough than in S, e.g. if profits are high enough; or simply because a technological constraint prevents them from accessing asset markets.
I consider two equilibria, according to whether liquidity is supplied or not: as a benchmark, the zero-liquidity limit reminiscent of Krusell, Mukoyama and Smith (2011)8 ; and then in section 5, an equilibrium with government-provided liquidity.In the former case, we assume that even though S's demand for bonds is well-defined (their constraint is not binding), the supply is zero so there are no bonds held in equilibrium.Under these assumptions the only equilibrium condition from this part of the model is the Euler equation for bonds of S, (5) holding with equality.The H's constraint binding and zero-liquidity implies that they are hand-to-mouth C H t = Y H t .Because transition probabilities are independent of history and there is full insurance within type, all agents who are H in a given period have the same income and consumption.
The rest of the model is purposefully kept exactly like the TANK version in Bilbiie (2008Bilbiie ( , 2020)), nested here with s = 1.The λ households who are "hand-to-mouth" H make a labor supply decision determining their income Y H t = W t N H t + T H t , where W is the real wage, N H hours and T H t fiscal transfers to be spelled out.The remaining 1 λ agents also work, and receive profits from the firm shares they are assumed to hold, net of taxes Y S t = W t N S t + 1 1 λ D t + T S t ; this provides a simple mapping to the total factor income inequality data in Figure 1.The hours choice delivers the standard intratemporal optimality condition for each j: as risk aversion and ϕ U j NN N j /U j N as the inverse labor supply elasticity, and small letters logdeviations from a symmetric steady-state (to be discussed below), we have the labor supply for each j: ϕn j t = w t σ 1 c j t .Assuming identical elasticities across agents, the same holds on aggregate ϕn t = w t σ 1 c t .
The supply, firms' side is standard and outlined for completion in Online Appendix O.A.3.A notable assumption as a benchmark is the standard NK optimal sales subsidy inducing marginal cost pricing.This policy is redistributive: since steady-state profits are zero D = 0, it taxes the firms' shareholders and results in the full-insurance, symmetric steady-state used here as a benchmark C H = C S = C. Loglinearizing around it, profits vary inversely with the real wage d t ( ln (D t /Y)) = w t , an extreme form of a general property of NK models.This series of assumptions is not necessary for the results and can be easily relaxed, but makes the algebra more transparent.Firms' optimal pricing under Rotemberg costs implies the loglinearized Phillips curve: where u t are cost-push shocks that I abstract from until studying optimal policy in Section 6.To obtain maximum tractability and closed forms, I first focus on the simplest special case: nested in (7) above with myopic firms (β = 0), used previously in a different context in Bilbiie (2019).Online Appendix O.A.3 microfounds this assuming that firms pay a Rotemberg cost relative to yesterday's market average price index, rather than to their own individual price (the latter leads to ( 7)).That is, firms ignore the impact of today's price choice on tomorrow's profits.While over-simplified, this nevertheless captures the key supply-side NK trade-off between inflation and real activity and allows us to isolate and focus on the essence of this paper: AD.All the results reassuringly generalize to the standard Phillips curve (7), as I show in Online Appendix O.C.The government conducts fiscal and monetary policy.The former consists of a simple endogenous redistribution scheme: taxing profits at rate τ D and rebating the proceedings lump-sum to H who thus receive τ D λ D t per capita; this is key for the cyclical-inequality channel.In the version with liquidity, the government also supplies liquid nominal bonds and levies lump-sum uniform taxes/transfers on households.The central bank sets the nominal interest rate i t .
Market clearing implies for equilibrium in the goods and labor market respectively With uniform steady-state hours N j = N by normalization and the fiscal policy assumed above inducing C j = C, loglinearization around a zero-inflation steady state delivers y t = c t = λc H t + (1 λ) c S t and n t = λn H t + (1 λ) n S t .

Cyclical Income Risk and Inequality in THANK
A keystone to this paper's analysis is to distinguish income inequality and risk, and their cyclicality.I define income inequality as the income ratio in the two states Γ t Y S t /Y H t ; this is proportional to the unconditional variance of log income (and also to the Gini and entropy, see Online Appendix O.B.1): var ln Importantly, in the model's equilibrium inequality is cyclical: it depends on aggregate output Γ (Y t ).
In the data and in quantitative HANK models alike, income risk is cyclical.Other analytical HANKs model it as either unrelated (Acharya and Dogra (2020)) or differently related (Challe et al (2017); Holm (2021); Ravn and Sterk (2020); Werning (2015)) to liquidity constraints and handto-mouth status.To capture a cyclical risk component that is distinct from cyclical inequality and further differentiate from the cited papers, I assume that the probability of becoming constrained depends on current aggregate demand 1 s (Y t ). 9 If the first derivative of 1 s (.) is positive 9 In a model with endogenous unemployment risk like Ravn and Sterk or Challe et al, this happens in equilibrium through search and matching.This is also related to Werning's Section 3.4, where nevertheless it is unconditional probabilities (and population shares) that are cyclical, and depend on next period's aggregate (I study that version of my model too in Appendix B).Here, to capture purely idiosyncratic (as opposed to "aggregate") variation, λ is invariant.s 0 (Y t ) > 0, the probability is higher in expansions; insofar as being constrained leads on average to lower income, this makes income risk procyclical.Conversely, s 0 (Y t ) < 0 makes risk countercyclical.
A precise definition of "income risk" is notoriously controversial.The literature often employs the conditional variance of idiosyncratic (log) income, found to be countercyclical in the data by Storesletten, Telmer, and Yaron (2004).This is easily calculated in my two-state model as: The simplest oscillating THANK benchmark s = (h =) 0 is especially useful because it partials out risk: the variance ( 9) is nil with agents alternating states every period.But inequality is still arbitrarily cyclical Γ t (Y t ) .As we will see momentarily, this is key for the model's dynamics, which we now solve for: first focusing on inequality and then enlarging the scope to broader income risk.

Cyclical Inequality and Aggregate Demand in THANK
We derive an aggregate Euler-IS equation by taking an approximation (in the aggregate-shocks dimension) around the ergodic idiosyncratic distribution with relative shares given by (1).To isolate the role of cyclical inequality, we first approximate around a symmetric steady-state with Γ = 1 and C H = C S .Start from the individual self-insurance Euler equation ( 5): To find the aggregate(d) counterpart, we need a theory connecting the distribution c j t to aggregates c t or y t .Once idiosyncratic uncertainty is revealed and asset markets clear, this part-one of many possible examples of the distribution-aggregate feedback-is exactly the TANK model in Bilbiie (2008Bilbiie ( , 2020)), for simplicity.I summarize its main implications here and refer to Online Appendix O.B for a complete derivation and to those papers for a thorough discussion.In equilibrium, individual consumptions/incomes are related to aggregate income by: where Inequality is procyclical (∂γ/∂y > 0) iff χ < 1 and countercyclical iff χ > 1.The composite parameter χ is the model's keystone: a sufficient statistic that in this specific model depends on fiscal redistribution 1 τ D /λ and labor market characteristics ϕ.It is important to stress that this is but one possible simple theory of the income distribution-to-aggregate two-way feedback. 10istributional considerations make χ different from 1.In RANK, such considerations are absent since one agent works and receives all the profits.When aggregate income goes up, labor demand, real wages and marginal cost increase.This decreases profits, but because the same agent incurs both the labor gain and the profit loss, the redistribution of income across factors is neutral.Take now the case with heterogeneity and countercyclical inequality χ > 1. 11 If demand goes up, the wage goes up, H's income increases and so does their demand.Thus aggregate demand increases by more than initially, shifting labor demand and increasing the wage even further, and so on-a (New) Keynesian cross.With χ < 1, inequality is procyclical and the AD expansion is instead smaller than the initial impulse, as H recognize that this will lead to a fall in their income.
Replacing the individual (11) in the self-insurance equation ( 10), we obtain the aggregate Euler-IS: The AD interest elasticity is the TANK one (Bilbiie, 2008), σ 1 λ 1 λχ , reflecting the described New Keynesian Cross logic. 12I confine attention throughout to λχ < 1, so this stays positive; the cited paper also analyzes the "inverted Keynesian cross" region with λχ > 1.
The key novelty of THANK's aggregate Euler-IS equation relative to TANK is that it is characterized by compounding δ > 1 iff inequality is countercyclical χ > 1 and discounting δ < 1 if procyclical χ < 1.The stand-alone role of cyclical inequality for Euler compounding/discounting is illustrated most sharply in the oscillating THANK no-risk benchmark.Even in that extreme case, my model implies Euler discounting-compounding with, replacing s = 0 and λ = 1 2 in (13): This illustrates most clearly that risk is not necessary for Euler discounting/compounding: cyclical inequality is sufficient, combined with a self-insurance, liquidity motive.
In RANK and TANK, good future income news imply a one-to-one demand increase today as households (who can) substitute consumption towards the present and, with no assets, income adjusts.Discounting occurs when procyclical inequality meets self-insurance: When good news about future aggregate income arrive, households recognize that in some states they will be constrained and not benefit fully from it.They self-insure, increasing their consumption less than if there were no transition to another state; the saving demand increase cannot be accommodated (there is no asset), so income falls accordingly.Countercyclical inequality leads to compounding instead: good aggregate income news boost today's demand because they imply less need for self-insurance.Since future income in states where the constraint binds over-reacts to good aggregate news, households demand less saving.With zero savings in equilibrium, households consume more than one-to-one and income increases more than without transition.
These intertemporal effects are strongest in the deterministic, oscillating THANK case, but the 11 This occurs e.g. with fiscal redistribution of profits that is not skewed towards H, τ D < λ and upward-sloping labor supply ϕ > 0. The benchmark imlicitly used by Campbell and Mankiw's (1989) is χ = 1, which occurs when profits are uniformly redistributed τ D = λ or labor is infinitely elastic ϕ = 0; income inequality is then acyclical.See also Bilbiie (2008, footnote 14). 12The direct effect scales down by 1 λ, but the indirect increases with λ at rate χ; with χ > 1 the latter dominates, delivering amplification.λχ is akin to an aggregate-MPC slope of a planned-expenditure line.As shown in Bilbiie (2020) the aggregate MPC out of aggregate income combines the two out-of-own-income MPCs weighed by the elasticities to the aggregate mpc = ( 1 same intuition applies in the general case s > 0 (see also Proposition 3 in the companion paper Bilbiie (2020)).The extent of self-insurance is now proportional to the level of risk 1 s: it vanishes in the TANK (1 s = 0) limit, and increases as 1 s approaches λ.Yet even the more general THANK version still focuses on cyclical inequality: while it embeds idiosyncratic risk (intimately related to binding liquidity constraints), this is by construction acyclical.To substantiate this and clarify the role of risk and its interaction with inequality, we now turn to the cyclicality of risk.

Cyclical Inequality and Risk
To gauge the connection between the cyclicality of inequality and risk, consider the cyclicality of the risk measure (9).A first-order approximation around steady state Γ delivers: The first component of ( 15) is due to cyclical inequality: when χ > 1, inequality is countercyclical and so is risk because income at the bottom overreacts, increasing variance in both expansions and recessions.This "intensive" margin operates even when the second channel is absent, i.e. with constant s or symmetric distribution s = 1 2 .However, locally around no-level-inequality Γ = 1, the case we just studied, variance is acyclical: it is proportional to ln Γ. Idiosyncratic risk itself is still cyclical to higher orders and away from steady state, but this has locally no first-order effect on the variance, on precautionary saving, and thus on Euler discounting-compounding.A more extreme case is TANK, s = 1: there is no transition and no risk in levels.Away from these special cases, risk is cyclical to first order through the inequality channel χ 6 = 1 only if there is level inequality Γ > 1; this is true even when the second channel is off, e.g. with acyclical (pure) risk s Y = 0.
The second component of ( 15) is related to the cyclicality of conditional skewness, in a manner reminiscent of Mankiw (1986). 13I derive this formally in Online Appendix O.B, but the intuition is simple: skewness is negative when s > 1 2 (there is left-tail risk).When 1 s (Y) is decreasing with aggregate activity s Y < 0, it becomes more likely to draw from the left tail in recessions; hence, (skewness) risk is countercyclical: upward income movements become less likely and downward more likely in a recession.This "extensive" margin operates even with acyclical inequality χ = 1: risk is cyclical only through the second channel s Y 6 = 0, and only if there is level inequality Γ > 1 and skewness s 6 = 1 2 .While the precise decomposition ( 15) is model-specific, the general idea, mechanisms and their equilibrium implications transcend this model.Thus, the model nests several scenarios that are useful to distil the roles of risk and inequality through their levels and cyclicalities, and their interactions, governed by Γ, s, χ and s Y .To reflect this, I now turn cyclical risk back on in the local 13 Online Appendix O.B derives higher moments: formally, conditional skewness (1 2s) / p s (1 s) < 0 when s > 1 2 .Direct evidence on this conditional skewness in levels is not yet available; Guvenen et al ( 2014) and a large literature following them document countercyclical skewness in growth rates.Bilbiie, Primiceri and Tambalotti (2022) compute the skewness of income growth rates for a quantitative estimated version of THANK and show that the cyclicality is close to the one documented in the data by Guvenen et al. dynamics, both by allowing s to depend on the cycle and approximating around an unequal steady state with Γ > 1. Risk is cyclical through both channels, see ( 15), and matters to first order-the loglinearized aggregate Euler-IS becomes, see Appendix B: where 1 s s+(1 s)Γ 1/σ > 1 s is the inequality-weighted transition probability measure of risk.This illustrates the two effects of income risk cyclicality corresponding to the decomposition (15) above.The first comes from income inequality, which around a steady-state with Γ > 1 and s > 0 also makes risk cyclical, captured by Γ 1/σ 1 (δ 1) s.When δ > 1 there is an additional compounding force that increases with the inequality level.The second effect encapsulates "pure" (independent on cyclical inequality) cyclical risk through its key determinant: the elasticity s Y Y/ (1 s), the second term in (15) above, which determines the novel parameter η.Dampen- ing/amplification of both current and future shocks occurs through η even with acyclical inequality δ = 1.Procyclical risk η < 0 implies dampening and Euler discounting: interest rate cuts or good news generate an expansion today, which increases the probability of moving to the bad state and triggers precautionary saving, curtailing the expansion.Conversely, countercyclical risk generates amplification and compounding: the aggregate expansion reduces the probability of moving to the bad state and mitigates the need for insurance, magnifying the initial expansion.
This formalization of cyclical risk has similar reduced-form implications for aggregate demand to the cyclical-inequality channel, but the underlying economic mechanism is different.Furthermore, while η is related to Acharya and Dogra's (2020) different formalization of risk with CARA preferences leading to a P(seudo-)RANK, the implications are different in important respects.In my model, η captures the cyclicality of conditional skewness, a key element of the reviewed evidence; whereas Acharya and Dogra's PRANK relies on symmetry (normal shocks), abstracting from skewness to focus on variance.Werning's (2015) general non-linear model contains both channels but without the distinction, decomposition and discussion of their differential effects on transmission.Ravn and Sterk (2020)'s model delivers something akin to η from search and matching, but abstracts from the role of cyclical inequality.Finally, novel to my framework is the contemporaneous amplification ("multiplier") of current interest rate changes; this is a consequence of risk depending on current aggregate demand, whereas in the mentioned contributions it depends on future activity. 14he pure-risk channel captured by η operates only if there is long-run level inequality Γ > 1, the literal risk of moving to a lower income level.Whereas the cyclical-inequality channel relies only on the cyclicality of income when constrained χ.Both channels capture precautionary saving: the former, through the effect of uncertainty and the third derivative of the utility function (η is proportional to prudence σ); the latter, through the effect of constraints, a separate source of concavity in the consumption function, a manifestation of the general results in Carroll and Kimball (1996).Inspecting ( 16) reveals three endogenous wedges relative to RANK, corresponding to each channel; in a recent contribution, Berger, Bocola and Dovis (2023) provide a valuable data "accounting" of related wedges.Bilbiie, Primiceri and Tambalotti (2022) quantify each channel's contribution to estimated business-cycle amplification through heterogeneity and conclude that cyclical-risk (with long-run inequality level) is the most quantitatively significant.Such exercises may help disentangle the signs of δ 1 and η which, as we will see next, are crucial for the model's properties.

THANK Analytics: Determinacy, Puzzles, and Amplification
This section exploits tractability to conduct a pencil-and-paper analysis elucidating the dynamic properties of the model, with a particular focus on the roles of cyclical inequality and risk in shaping the determinacy properties, the FG puzzle, and conditions for amplification-multipliers.

HANK, Taylor, and Sargent-Wallace
I now solve the model under further assumptions delivering a one-equation representation that may be useful in different contexts where reducing dimensionality is necessary for closed-form solutions.In particular, first, the nominal rate i t follows a Taylor rule (we study other policies momentarily): The model is completed by adding the simple aggregate-supply, Phillips-curve specification (8); all the results carry through with the forward-looking (7) as I show in Online Appendix O.C.With this simple RANK-isomorphic HANK we first revisit a classic determinacy result and derive a HANK Taylor principle.Replacing ( 8) and ( 17) in (13), THANK collapses to one equation: captures the effect of good news on AD, and the elasticity to interest rate shocks.
There are three channels shaping this key summary statistic.First, the "pure AD" effect through δ coming from cyclical inequality, operating even with fixed prices or fixed real rate i t = E t π t+1 .Second, a supply feedback cum intertemporal substitution: the inflationary effect (κ) of good income news triggers ceteris paribus a fall in the real rate and intertemporal substitution towards today, the magnitude of which depends on static amplification/dampening 1 λ 1 λχ .Finally, all this demand amplification generates inflation and real rate movements.When policy is active φ > 1, a higher real rate and a contractionary effect today ensue, the strength of which depend on cyclical-inequality.These considerations drive Proposition 1 (the case with NKPC ( 7) is in Online Appendix O.C).  18) has a determinate, locally unique rational expectations equilibrium if and only if (as long as λχ < 1): .
The Taylor principle φ > 1 is sufficient for determinacy if and only if inequality is procyclical so δ 1.
The proposition follows by recalling that the determinacy requirement is that the root ν be inside the unit circle; in the discounting case δ < 1, the threshold φ is evidently weaker than the Taylor principle, while in the compounding case it is stronger.With countercyclical inequality and δ > 1, a sunspot increase in future aggregate income generates a disproportionate increase in income in the H state and thus incentives to dis-save and a further demand boost today, making the sunspot selffulfilling even with φ = 1.The central bank needs to do more to counteract this.The opposite holds in the discounting case: the Taylor principle is sufficient for determinacy.In the oscillating THANK extreme, the threshold is φ = 1 + (χ 1) 2 κσ .The Taylor threshold φ > 1 reappears for either of χ = 1 (acyclical inequality), s ! 1 (TANK), or κ !∞ (flexible prices).The determinacy region squeezes rapidly with countercyclical inequality because of a complementarity with idiosyncratic risk apparent from φ = 1 + (χ 1)(1 s) κσ(1 λ) .The threshold depends on price stickiness because policy responds to inflation, but the relevant amplification goes through real demand's equilibrium response, and price stickiness modulates the relationship between the two.If instead policy responds to real activity i t = π t + φ c c t , the determinacy threshold is φ c > (χ 1)(1 s) (1 λ)σ and no longer depends on price stickiness because policy then acts directly on demand.Figure 2 plots the threshold φ as a function of λ (for λ < χ 1 ) for different 1 s, with procyclical inequality in the left panel and countercyclical in the right.The parametrization assumes κ = 0.02, σ = 1, and ϕ = 1.In the countercyclical-inequality case, the threshold increases with λ and does so at a faster rate with higher risk 1 s.The required response can be large: for the calibration used in Bilbiie (2020) to match Kaplan et al's HANK aggregates (χ = 1.48, λ = 0.37, 1 s = 0.04) it is φ = 2.5 and can approach 5 for other calibrations therein.With procyclical inequality, the Taylor principle is sufficient but not necessary for determinacy.For a large region, there is determinacy even under a peg φ = 0, undoing the Sargent-Wallace result, namely if and only if φ < 0 or: With enough discounting, the sunspot is ruled out by the economy's endogenous forces, unlike in RANK where ν 0 = 1 + κσ 1; as we shall see, ( 19) also rules out the forward guidance puzzle.Finally, a simple analogous derivation reveals the threshold when adding cyclical risk: with the different intuition discussed above for AD amplification/dampening through η. 15 3.2 A Catch-22 for HANK: No Puzzle, No Amplification?
We are now in a position to state the Catch-22: the closed-form conditions for amplification in THANK are the opposite of those needed to solve the FG puzzle.To state this formally, we introduce two policy shocks: discretionary exogenous changes in interest rates i t in the Taylor rule i t = φπ t + i t ; and public spending: the government buys an amount of goods G t with zero steadystate value (G = 0) and taxes all agents uniformly to finance it. 16Straightforward derivation delivers the aggregate Euler-IS, starting with cyclical inequality only, extending (13), with ζ ϕσ/ (1 + ϕσ): Together with the static PC π t = κc t + κζg t and the AR(1) E t g t+1 = µg t , this delivers Proposition 2 (see Online Appendix O.C.2 for the general case with NKPC (7)).
Proposition 2 A Catch-22 for HANK: 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: whereas the forward-guidance puzzle is ruled out ( The oscillating THANK delivers the sharpest version of the Catch-22; the Euler equation is: 15 Ravn and Sterk (2020) show that SaM delivers η > 0 endogenously, making the Taylor principle insufficient.In works subsequent to this paper's determinacy Proposition 1: Acharya and Dogra (2020) derived a Taylor principle and Auclert et al (2018) provided numerical simulations in a quantitative HANK on the role of cyclical risk for determinacy. 16The redistribution of the taxation financing spending is essential for the multiplier, see Bilbiie (2020) in TANK: I sidestep it assuming uniform taxation.See Bilbiie, Monacelli, and Perotti (2013) with amplification/multipliers for χ > 1 but discounting and curing the FG puzzle for χ < 1.
In the general case, the first part of Proposition 2 pertains to amplification, the focus of the majority of quantitative HANK.Kaplan et al (2018) show that HANK yields monetary policy amplification through indirect effects; see also Auclert (2019), Gornemann et al (2015), and Debortoli and Galí (2018).This occurs only with countercyclical inequality.The THANK fiscal multiplier is: With fixed prices κ = 0 and proportional incomes χ = 1, one recovers the benchmark zeromultiplier derived in RANK by Bilbiie (2011) and Woodford (2011).Positive multipliers occur with countercyclical inequality χ > 1 through the "new Keynesian cross".17If the stimulus is persistent (µ > 0), there is an extra kick as agents, expecting higher demand and income, also self-insure less.
The second part of Proposition 2 pertains to solving the FG puzzle.The condition is (19), i.e. determinacy under a peg, found by iterating (21) forward with φ = 0 to obtain: The response to a T periods ahead interest cut is, for any T 2 (t, T): which is decreasing in T iff ν 0 < 1.Then, since ν T 0 E t c t+ T vanishes as T !∞, solving the equation forward yields a unique solution: the earlier determinacy with a peg result.The condition ν 0 < 1 rewritten as 1 δ > κσ 1 λ 1 λχ captures a simple intuition.To cure the puzzle, the left-side HANK discounting needs to dominate the right-side AS-compounding of news that is the source of trouble in RANK.This entails jointly idiosyncratic uncertainty 1 s > 0 and procyclical inequality χ < 1 σκ 1 λ 1 s < 1 (in oscillating THANK, this is χ < 1 σκ 2 ).One implication is an interpretation of McKay et al (2016), where inequality is procyclical since profits are distributed disproportionately to low-productivity households: "as if" τ D > λ here; see Hagedorn et al (2019) for a quantitative illustration.

Catch-22 Solutions
The foregoing analysis identifies a challenge for ensuring determinacy and avoiding puzzles in HANK with countercyclical inequality.What modelling choices can circumvent this?This section lists possible solutions: some novel to this paper, others proposed in subsequent work; they pertain to both model ingredients and to alternative policy rules.

Procyclical inequality, countercyclical risk
Proposition 2 purposefully abstracts from cyclical risk.Turning it back on can in theory resolve the Catch-22, providing amplification without the puzzle.
Proposition 3 THANK with cyclical inequality and risk resolves the Catch-22 if and only if one channel is procyclical "enough" when the other is countercyclical; more precisely, by direct inspection of (16): The first condition delivers discounting, while the second simultaneously yields static amplification in ( 16); in Appendix B I show that the latter extends to the fiscal multiplier too.The Catch-22 is thus resolved when the two channels coexist and move in opposite directions with the right relative strengths.In the procyclical-inequality case χ < 1, risk needs to be countercyclical "enough" η > 0 in a way made specific by (24); in other words, the "risk channel" needs to be strong enough, as modulated by both its level and cyclicality, which depends on long-run inequality-the average consumption loss incurred when changing state. 18 When ( 24) is violated it may, depending on parameter configurations, result in either (i) no puzzle, no amplification (e.g. when risk is not countercyclical enough); or (ii) amplification, but also the FG puzzle (e.g. when risk is too countercyclical).When risk and inequality are both countercyclical THANK delivers further amplification and aggravates the puzzle, while determinacy conditions become even more stringent, see (20).Finally, whether the Catch-22 applies also depends on whether other amplification/dampening channels operate, see below.Such other channels notwithstanding, it is important to gauge what the data suggests about the cyclicalities of inequality and risk.
A large literature measures income risk and finds that it is unambiguously countercyclical (suggesting η > 0): prominent examples include Storesletten et al's (2004) countercyclical-variance estimates and Guvenen et al's (2014) measure of (growth-rates) skewness cyclicality.The same is true for inequality in labor earnings: see Heathcote et al (2010) for early evidence, and Patterson (2023) for evidence controlling for MPCs, a counterpart to the model's χ, but pertaining to earnings only.
The picture is more nuanced for ex-post inequality in disposable income, including taxes and transfers.Since the latter are countercyclical, overall disposable income inequality, the true data counterpart of the model's key object, is less countercyclical and may even become procyclical, as illustrated in Figure 1 for the two last recessions.A recent analysis of inequality dynamics taking into account transfers is Heathcote et al (2023); their Figure 17 shows that, although earnings inequality was countercyclical as usual, disposable income inequality was procyclical in the COVID recession, due to unprecedented transfers to the bottom.A similar if mitigated picture emerges also for the Great Recession, especially for bottom inequality, see also Perri and Steinberg (2012).
One caveat to interpreting this evidence through the model's lens is necessary.None of the available evidence isolates and disentangles the two channels in a model-equivalent way, i.e. provide identified estimates for Γ, χ, s, and η.In that regard, the paper provides theoretical restrictions 18 Note that for the interval to be non-empty with χ < 1, we need λ < 1 s.The exact condition is and is always satisfied in the oscillating THANK limit; in the iid limit 1 s !λ, it only requires some steady-state inequality Γ > 1.With χ > 1 the condition requires procyclical risk and 1 s < λ. to inform that measurement and reasons for its importance for macro transmission.19

Other solutions: model ingredients or policy rules
Both the Catch-22 and the FG puzzle are model properties contingent upon modelling choices but also on the Taylor rule and related to determinacy.I next review some modelling assumptions and policy rules that ensure determinacy and sidestep the Catch-22.

Model ingredients circumventing the Catch-22
One route is to abstract altogether from cyclical inequality and risk, e.g.Auclert et al (2023).This allows emphasizing other important channels that operate even in a "proportional incomes" benchmark, achieved therein by assuming sticky wages with uniform hours worked, and no profits.In section 5, I show how to blend that novel channel with cyclical inequality (this paper's focus) to gain further insights into determinacy and resolving the Catch-22.
A class of extensions of my model that resolve the Catch-22 adds deviations from full-information rational expectations, obtaining a separate source of Euler discounting that, if strong enough, can undo the compounding due to countercyclical inequality or risk.Such contributions include Pfauti and Seyrich (2023) using cognitive discounting; Elias-Gallegos (2023) using dispersed information; and Meichtry (2022) using sticky information.

Policy rules circumventing the Catch-22
Other solutions to the Catch-22 rely on determinacy with a peg and draw on results in RANK that extend to HANK.One solution is the "Wicksellian" policy rule of price level targeting which yields determinacy in RANK (Woodford (2003); Giannoni ( 2014)).This is especially powerful in HANK, as emphasized in Proposition 4; see the proof and discussion in Online Appendix O.C.3.

Proposition 4 Wicksellian rule in HANK:
In the THANK model, the Wicksellian rule i t = φ p p t with φ p > 0 leads to local determinacy even when δ + η > 1.Thus, the model delivers amplification without also aggravating the FG puzzle even when both inequality and risk are countercyclical.
Intuitively, the puzzle's source is indeterminacy under a peg and a Wicksellian rule provides determinacy under a "quasi-peg": some, albeit small response to the price level anchors long-run expectations.A similar logic would apply to a money-supply rule, which yields price-level determinacy.This paper assumes throughout a passive-Ricardian fiscal policy.A different route to determinacy is to resort to an active, non-Ricardian fiscal rule (Leeper (1991); Woodford (1996); Cochrane (2017)).In incomplete-market economies, yet a different fiscal policy can deliver determinacy (Hagedorn (2020)); to study it in my model, we need to turn to the equilibrium with liquidity.In the version with liquidity in the form of government bonds used for precautionary saving, I derive the savings demand and study the intertemporal propagation and determinacy implicationswith a focus on the interaction with cyclical inequality.

Savings-Liquidity Demand
Denote by B N t+1 the total nominal quantity of government bonds outstanding at the end of each period.In nominal terms, B N t+1 = (1 + i t 1 ) B N t P t T t , and in real terms: where R t = 1+i t 1 1+π t is the gross interest rate.The bond market clears B t+1 = λZ H t+1 + (1 λ) Z S t+1 .Recall now that Z H t+1 = 0, so that B t+1 = (1 λ) Z S t+1 and using the flow definitions: and for S similarly B S t+1 = sZ S t+1 = s 1 λ B t+1 .The respective budget constraints imply: where Ŷj t is j's disposable (net of taxes) income.Savers hold all bonds for next period; because bonds are liquid a fraction 1 s λ of the payoff, including interest, accrues to next period's hand-to-mouth.In Appendix C, I derive the steady-state demand for liquidity-bonds, or savings function: under log utility σ = 1, normalizing Y = 1; ( 27) is an increasing convex function of R under standard restrictions.The Appendix explores this analytically in detail, but is is worth noticing that as βR ! 1 debt becomes proportional to (1 λ) (Γ 1) and thus tends to zero with no steady-state income inequality Γ = 1.The condition for a positive liquidity demand B > 0 is: this is strictly true when βR ! 1 and implies more general restrictions on 1 s, λ, β, see appendix C. Essentially, (28) requires some idiosyncratic risk and liquidity; it is violated e.g. in TANK.
18 This version embeds a distinct amplification channel orthogonal to the NK Cross, the intertemporal Keynesian cross of Auclert et al (2023) (see also Hagedorn et al (2018)), and allows a novel analytical solution for their key summary statistics, the iMPCs.Loglinearizing ( 26) around a zero-liquidity steady state with R = β 1 delivers (see Appendix C for details): where b t is in shares of steady-state Y. Aggregating (29), we have: The iMPCs are the partial derivatives of aggregate consumption c t with respect to aggregate disposable income ŷt+k at different horizons k, keeping fixed everything else, i.e. taxes and public debt.To find them, we solve the equilibrium dynamics of private liquid assets b t by replacing (29) into the loglinearized self-insurance Euler equation ( 10); this yields the key equation relating the demand for liquid assets to individual incomes, and thus to (cyclical) income inequality: where . An important object for the model's dynamic propertiesin particular, the key determinant of the iMPC's persistence-is the stable root of this (liquid-)asset accumulation equation, x b = 1 2 Θ q Θ 2 4β 1 ; the general case is analyzed in Appendix C.
Here, we focus instead on distilling the role of cyclical inequality; the intuition is clearest in the oscillating THANK case s = 0, with agents saving when they expect lower income and vice versa: The consumption function follows by substituting this into (30), delivering Proposition 5. (33)

Proposition 5
The iMPCs for oscillating THANK in response to a time-T disposable income shock are: This illustrates the key points transparently.With acyclical inequality χ = 1 ( ŷj t = ŷt , Auclert et al's case) a current income shock induces agents to self-insure, saving in liquidity to maintain higher future consumption.While a future shock makes them consume in anticipation, depleting liquid savings.The second point concerns the additional role of the novel channel of cyclical inequality: higher income cyclicality when constrained χ makes agents consume more out of news and less out of past and current aggregate income.When self-insuring, agents take into account how the aggregate shock affects income in each state and change their asset demand and equilibrium liquidity consequently.The cyclicality of inequality thus skews the temporal path of the iMPCs and provides an additional degree of freedom for matching it.Even this simplest s = 0 case can then match the two key iMPCs matched by Auclert et al, the contemporaneous dc 0 /d ŷ0 = 0.55 and one-year-after dc 1 /d ŷ0 = 0.15 with β = 0.95 annually and χ = 1.47.
The expressions for THANK with s > 0 are still analytical and convey the same intuition, but are more tedious (see Proposition 8, Appendix C). Figure 3 plots the iMPCs for THANK, expression (52) in Appendix C, along with TANK and the data from Fagereng et al.In THANK, I match the two target MPCs with λ = 0.33, s = 0.82 (0.96 quarterly) and χ = 1.4. 20The intertemporal path is remarkably in line with both the data and Auclert et al's quantitative HANK: the effect dies off after a few years, whereas TANK misses this intertemporal amplification altogether, the iMPCs being: One important application of the analytical iMPCs is to clarify the connection between quantitative determinacy results and my THANK Taylor principle, underscoring again the role of cyclical inequality.In quantitative HANK, Auclert et al (2023) showed subsequently that the Taylor principle is sufficient when the sum of iMPCs (out of an income shock far into the future) is larger than 1, which makes the model "explosive" (stable forward) and thus determinate.I show this analytically 20 This is coincidentally close to the calibration in Bilbiie (2020) matching general-equilibrium statistics with the zero-liquidity model.Figure A1 in the Appendix provides a comparison of different calibrations, and iMPCs in response to future shocks.See Cantore and Freund (2021) for a subsequent, simpler analytical iMPC calculation with portfolio adjustment costs.
This holds in the general model, see Appendix C. Thus, the requirement for Taylor principle sufficiency and for the sum of iMPCs to be larger than one is the same: procyclical inequality.

A nominal-debt rule as a Catch-22 solution
We are now ready to formulate the route to determinacy inherent in this model version (and the larger incomplete-market class of which it is an example, see Hagedorn, 2020), even in cases with countercyclical inequality and risk whereby the Taylor principle fails and the FG puzzle is normally aggravated.When the government chooses the quantity of nominal debt, the price level is determined without an interest-rate rule.If nominal taxes are set to balance the budget intertemporally for any price level (making policy passive-Ricardian, no fiscal theory), the central bank sets freely the nominal interest rate that clears the liquid-bond market with no need to respond to any endogenous variable.Local determinacy prevails, as shown in Proposition 6. 21 Proposition 6 A nominal debt rule: The THANK model with a well-defined demand for liquid bonds ((28) holds) leads to local determinacy even when δ > 1 under the nominal-debt quantity rule: Intuitively, condition (28) requires that H agents receive a fraction of savings larger than the interest income, β 1 1 s λ > β 1 1 = r; more generally, it requires that H agents receive positive net income from liquidity in steady state, see Appendix C. The proposition generalizes to b N t+1 = φ b p t with φ b < 1 so that real debt b t+1 = (φ b 1) p t falls when the price level increases.Furthermore, it can be easily shown that it holds with forward-looking Phillips curve and rules out the FG puzzle even in the "amplification" region, sidestepping the Catch-22.See Hagedorn (2020) for a general version of those arguments and Hagedorn et al (2018) for an analysis of fiscal multipliers and the earliest illustration of how a quantitative HANK with this policy rule sidesteps the Catch-22.

Optimal Policy in THANK
THANK is also useful for studying optimal monetary policy analytically, in the version without liquidity.This provides a benchmark that helps elucidate some key mechanisms operating in the rich-heterogeneity quantitative-HANK studies featuring several additional relevant channels, such as Bhandari et al (2021).I build on Woodford's (2003, Ch. 6) analysis in RANK.In Appendix D, I spell out the full Ramsey problem and derive a linear-quadratic problem equivalent to it under certain conditions, taking a second-order approximation to aggregate welfare around a flexibleprice equilibrium that is efficient.The target equilibrium of the central bank is the socially-desirable, 21 Given the steady-state demand for bonds (27), determinacy of the steady-state price level is immediate: the proof is exactly as in Hagedorn (2020).In a nutshell, monetary policy chooses steady-state i, which given π determines R and steady-state real B. The fiscal authority's choice of nominal B N (and its growth rate) then determines P (and steady-state π).perfect-insurance equilibrium induced by a fiscal policy generating zero profits to first order under flex prices, following the TANK analysis in Bilbiie (2008, Proposition 5).This delivers Proposition 7.
Proposition 7 Solving the welfare maximization problem is equivalent to solving: s.t. ( 7),( 12), and ( 13), where the optimal weights on output and inequality stabilization are, respectively: Several results are worth emphasizing.While the weight on output (gap) stabilization α y is the same as in RANK, there is an additional term pertaining to (consumption or) income inequality. 22his affects the central bank's stabilization tradeoff, adding a redistribution motive.However, idiosyncratic risk and its cyclicality are irrelevant for optimal policy, insofar as the target flexible-price equilibrium is the first-best with perfect insurance, without inequality; thus, the aggregate implications of the distributional channel for optimal policy in THANK happen to be the same as in TANK.
This is different from Challe (2020), which abstracts from inequality altogether but where an isomorphism occurs between RANK and a different analytical HANK with cyclical risk through search and matching.The common point is that my framework too features irrelevance of income risk, but in THANK relative to TANK.Both the optimal allocation and the interest rate policy instrument are radically different, and depend crucially on the cyclicality of inequality here.Furthermore, as we will see, in my framework the cyclicality risk does not matter for implementation either.
An important observation concerns the interest rate, which is both residual to the policy problem and unaffected by risk cyclicality.Recalling that we approximate around the efficient equilibrium Γ = 1, the IS curve ( 13) is not a constraint: as in RANK, it merely determines i t once we found the optimal allocation (y t , π t ).And since the IS curve approximated around Γ = 1 is also independent of cyclical risk, so will the interest rate that implements optimal policy. 23onsider for simplicity only shocks that drive no wedge between inequality and aggregate output gap, which stay proportional: (12) holds; the analysis of shocks that do drive a wedge is relevant for capturing further mechanisms in richer HANK, but beyond the scope of this paper and pursued in follow-up work.We can simplify the problem by replacing (12), obtaining the per-period loss: The inequality motive thus amounts, in my benchmark THANK relative to RANK, to a higher weight on output stabilization that increases with λ.Importantly, this holds regardless of whether inequality is counter-or pro-cyclical, as long as it is cyclical: the extra stabilization motive is proportional to (χ 1) 2 .The simple intuition is based, as in TANK, on the key role of profits which are eroded by inflation volatility.With higher λ, less agents receive profits; the weight on inflation falls, and vanishes in the λ ! 1 limit, where there is no rationale for stabilizing profit income.
We can now study optimal policy in THANK starting with discretion, or Markov-perfect equilibrium.The ability to study this more realistic, time-consistent policy is an appealing feature of the tractable framework, since computing Markov-perfect optimal policies in quantitative models is cumbersome.This is obtained by solving (36) assuming that the central bank lacks commitment and treats expectations parametrically, without internalizing its actions' effect on them; this amounts to re-optimizing every period subject to (7) with fixed expectations at the decision time t.The problem being mathematically identical to RANK, we go directly to the solution: This targeting rule under discretion requires engineering an aggregate demand decrease for a given inflation increase.Assuming AR(1) cost-push shocks E t u t+1 = µu t , the equilibrium is: Optimal policy under discretion implies that both output and inflation deviate from target: a tradeoff between inflation and output stabilization.Since α is increasing in λ, it follows directly that optimal policy in THANK requires greater inflation and lower output volatility than in RANK.
One instrument rule implementing this equilibrium is found by using the aggregate IS (13): Unlike in RANK, the instrument rule implementing optimal policy may be passive φ d < 1 with enough compounding δ > µ 1 , i.e. with countercyclical enough inequality: optimal policy requires a real rate cut in THANK when in RANK it would require an increase.Whereas with procyclical inequality δ < 1, the required instrument rule is even more active than in RANK.However, it is independent of the cyclicality of risk in this benchmark.Optimal (timeless-)commitment policy, the time-inconsistent Ramsey equilibrium, requires committing to the different targeting rule, by similar arguments as in RANK (Woodford, 2003, Ch. 7): It is straightforward to show that commitment to (40) delivers determinacy regardless of heterogeneity.The difference from RANK is still captured by the inequality motive shaping α, but optimal commitment policy still amounts to price-level targeting, like in RANK.THANK, a tractable HANK model with two types and two assets, captures analytically several key channels of quantitative HANK models and allows elucidating their distinct and interacting roles, in particular cyclical income inequality and risk.I illustrate this by investigating these channels' roles for the dynamic properties of HANK models: determinacy, amplification, multipliers, resolving the forward guidance puzzle, and optimal monetary policy.
The key channel is cyclical inequality: whether the income of constrained hand-to-mouth agents comoves more or less with aggregate income.This channel is already the main focus of TANK (Bilbiie ( 2008)) but interacts with self-insurance and risk in THANK, as in quantitative HANK.Procyclical inequality delivers discounting in the aggregate Euler equation, which makes the Taylor principle unnecessary for determinacy and can cure the forward guidance puzzle.Conversely, countercyclical inequality generates Euler-equation compounding, making the Taylor principle insufficient for determinacy and aggravating the puzzle.This is a Catch-22, for countercyclical inequality is precisely the condition for amplification and multipliers in HANK, which is what many studies focus on, exploiting a New Keynesian cross inherent therein.
The paper proposes a decomposition of cyclical variations in income risk into one component related to cyclical inequality, and one due to cyclical skewness-variations in the probability of the bad, constrained state.The Catch-22 can be resolved if one channel is procyclical enough when the other is countercyclical-so the former delivers Euler-discounting without mitigating the amplification provided by the latter.If however both channels are countercyclical, determinacy conditions become very stringent and the puzzle is aggravated.The available evidence points to risk being countercyclical but inequality in disposable income being mildly procyclical in the last two recessions; the theory developed here can guide further measurement.
Policy rules can also circumvent the Catch-22, even when both inequality and risk are countercyclical.For example, a Wicksellian rule of price-level targeting resolves this tension by making THANK determinate and puzzle-free.This virtue is shared by a rule setting nominal debt proposed by Hagedorn (2020), as I show analytically in my model's version with liquidity.
Optimal monetary policy, solved for analytically in THANK, requires an inequality objective, atop stabilizing inflation and output around an efficient perfect-insurance equilibrium.Regardless of risk, optimal policy implies tolerating more inflation as a result of distributional concerns when inequality is cyclical.While timeless-optimal commitment policy still amounts to price-level targeting, even though along the adjustment path there it still entails accepting more inflation.
To date and to the best of my knowledge, THANK is the only tractable framework to capture all these channels found to be key in the complementary, rich-heterogeneity HANK: cyclical inequality, precautionary self-insurance saving, intertemporal MPCs, and features of idiosyncratic income uncertainty and risk (cyclical variance and skewness, and kurtosis).
As models of the economy as a whole become larger and more complex, with many sectors, frictions, and sources of heterogeneity, the quest for tractable representations seems important for entropic reasons.It is my hope that this framework is thus useful for policymakers and central banks, for communicating to the larger public, for students and colleague economists from other fields seeking to enter the fascinating realm of macro fluctuations and stabilization policy in a world where heterogeneity and inequality are of the essence.

B Cyclical Risk in THANK
The self-insurance equation when the probability depends on aggregate demand is: where j = 0 for current and j = 1 for future.We loglinearize this around a steady-state with inequality; this requires imperfect steady-state redistribution, i.e. the subsidy does not completely undo market power, generating zero profits.I focus on a steady state with no subsidy: the profit share is D/C = 1/ε, and W N/C = (ε 1) /ε.Under the same redistribution scheme as before, the consumption shares are: 27 Other modifications of the NK model that solve its puzzles include changing the information structure (Garcia-Schmidt and Woodford (2019), Gabaix (2019), Angeletos and Lian (2017), Farhi and Werning (2019), Woodford (2018)), pegging interest on reserves (Diba and Loisel (2017)), wealth in the utility function (Michaillat and Saez (2017), Hagedorn (2018) where I restrict attention to cases with positive real interest-rate r.Let r t be the ex-ante real rate for brevity, the steady-state value of the probability s (Y) = s and its elasticity relative to the cycle s 0 (Y)Y 1 s(Y) .

B.1 Current aggregate demand
For the case in text (with s (Y t )), the loglinearized self-insurance equation is: Replacing c S t , c H t and using the notation for δ, η and s we obtain ( 16) in text.For the case with government spending, the aggregate Euler equation becomes: This can be used to compute fiscal multipliers in the case with cyclical risk, i.e. the fixed-real-rate multiplier of a one-time transitory (E t g t+1 = 0) increase in g t is: The first term is the NK cross, further amplified by the cyclical-risk multiplier (1 η) 1 ; the second term is novel, due exclusively to cyclical risk.With countercyclical risk η > 0 an increase in public demand decreases risk and precautionary saving, boosting private demand with elasticity η; this generates further expansionary rounds, decreasing risk further, etc., the (1 η) 1 multiplier.The amplification condition in Proposition 3 λ χ 1 1 λχ + η > 0 is sufficient condition for a positive multiplier when χ < 1, since ζ < 1.When χ > 1, the positive-multiplier condition is more stringent λ

B.2 Future aggregate demand
For the case with s (Y t+1 ), mimicking the implications of the other tractable contributions emphasizing cyclical income risk reviewed in text, the aggregate Euler-IS is: which replacing individual consumption levels as function of aggregate becomes or, using the notation in text: There is discounting if risk is procyclical enough η < 1 s 1 λχ (1 χ).But the contemporary AD elasticity to interest rates is unaffected by the cyclicality of risk (this is thus isomorphic to Acharya and Dogra's different formalization of cyclical risk based on CARA utility).

C Liquidity in THANK C.1 Oscillating THANK Model: Detailed Outline
This Appendix spells out the full details of the oscillating THANK model where agents oscillate between states.For the sake of consistency, I outline it as a special case of THANK with s = h = 0.An alternative representation solves the individual agent's problem (instead of a family head problem) and leads to the exact same Euler equation under the stated assumptions.The program of the family head is now: subject to the following constraints.H have a binding liquidity constraint and do not save, and S can take the asset knowing they transition next period for sure; using the same notation for beginning-and end-of-period bond holdings as in THANK, the flows from the latter into the former are: i.e. no saving for H and all of S 0 s end of period savings become H 0 s beginning-of period assets.The budget constraints for the respective households are thus: with the same notation as before,where post-tax incomes are Y j t (including dividends for S); H households receive the gross real return R t 1 = 1+i t 1 1+π t on bond holdings.The Euler equation for bonds is: S agents save in an attempt to smooth consumption relative to tomorrow's different state, where they transition with certainty.In the equilibrium with liquidity, the supply of bonds is B t+1 = 1 2 Z S t+1 and using the flow definitions, B H t+1 = Z S t+1 = 2B t+1 ; the respective budget constraints imply:

C.2 THANK: Steady-State Demand for Liquidity/Precautionary Saving
The two budget constraints (after asset market clearing) for the case with liquidity (26), evaluated at the steady state deliver, respectively, with Y j denoting income of agent j: where we already imposed the steady-state budget T = (R 1)B and assumed uniform taxation.The To derive the steady-state savings function (demand for liquidity), replace ( 43) in ( 44): which rewritten and under the particularly tractable case of log utility σ ! 1 becomes: Using SS inequality Γ = Y S /Y H to write: we finally obtain (normalizing Y = 1 without loss of generality) ( 27) in text: Notice that SS inequality is determined in the model: using symmetric SS hours and the expression for the real wage w = m 1 where m is the steady-state markup with arbitrary subsidy m 1 λ D = 1 m 1 + 1 τ D 1 λ (m 1) Y. Replacing both of these: so there is SS inequality Γ > 1 iff m > 1 and τ D < λ.
Start by noticing that to have a self-insurance motive C S > C H the standard condition from incompletemarket models applies: R < β 1 .When is SS debt positive?A positive numerator and denominator require, respectively: where the latter implies that H get positive income from liquidity net of taxes ((43)).For an equilibrium interest rate to exist we thus need With large enough Γ this always satisfied as the RHS is negative β 1 1 < (1 s) (Γ 1).
An intuitive interpretation of condition (28) in text is that positive long-run liquidity, self-insurance saving requires the present discounted value of the risk of becoming constrained (infinite discounted sum of 1 s at β) to be larger than the unconditional long-run probability of being constrained. 28  Oscillating THANK.To derive the steady-state demand for liquidity, we use post-tax income and the steady-state budget T = (R 1)B, assuming uniform taxation, to rewrite the budget constraints: 28 Recall that the non-oscillation condition 1 s < λ stems from the law of motion λ t+1 = (1 s) (1 λ t ) + hλ t with positive root if 1 s < h ! 1 s < λ. (1 + R) 1 + (βR) 1 (1 + Γ) .( 47)

C.3 Derivation of analytical iMPCs
For analytical convenience, for this section I derive results loglinearizing around a long-run steadystate with zero liquidity B = 0, implying R = β 1 and no consumption inequality C H = C S , so (55) and the corresponding equation for S agents become ( 29), where we imposed asset market clearing.
where the roots of the characteristic polynomial of (48) are x b = 1 2 Θ q Θ 2 4β 1 and (βx b ) 1 , with 0 < x b < 1 as required by stability whenever β > 1 1 s λ .Substituting (49) in (30) delivers the aggregate consumption function, the key equation for calculating the analytical iMPCs in Proposition 8: Downloaded from https://academic.oup.com/restud/advance-article/doi/10.1093/restud/rdae066/7699676 by guest on 07 July 2024 Fig. 1: Income inequality (top/bottom 50%) in the last two recessions; realtimeinequality.orgdata (Blanchet et al, 2023)A different class of solutions consists of alternative policy rules that solve the Catch-22 regardless of how countercyclical inequality and risk are: they preserve amplification while ruling out the FG puzzle by determining the price level.I show that a Wicksellian price-level targeting rule achieves this in THANK.A debt quantity rule(Hagedorn (2020) andHagedorn et al (2018)) has the same virtue.In my model's version with liquidity, I prove analytically both this and a version of Auclert et al's numerical determinacy criterion based on iMPCs but extended to embed the cyclical inequality channel; for the latter, I thus derive novel analytical expressions for the iMPCs.The analysis of optimal monetary policy further underscores the importance of distinguishing the different roles of inequality and risk.In the benchmark I study, whereby fiscal policy is in charge of fixing long-run inequality distortions and there is no liquidity, risk is irrelevant for optimal policy, while cyclical inequality is instead a key input.Complementary to quantitative studies that address phenomenal technical challenges, I calculate optimal policy analytically in THANK, approximating aggregate welfare to second-order to derive a quadratic objective function for the central bank.This encompasses a novel inequality motive, implying optimally tolerating more inflation volatility when more households are constrained.While inequality is of the essence for optimal policy, risk is not-insofar as the policymaker shares society's first-best (perfect-insurance) objective.Risk does matter for implementation: with countercyclical inequality and idiosyncratic risk, the interest-rate rule that implements optimal discretionary policy may entail cutting real rates, when in RANK it would imply increasing them.Furthermore, optimal policy under commitment ensures determinacy regardless of heterogeneity and inequality-cyclicality and, while affected by similar inequality considerations, amounts to a form of price-level targeting.

Liquidity, Inequality, and Intertemporal MPCs in THANK
R I P T ). Downloaded from https://academic.oup.com/restud/advance-article/doi/10.1093/restud/rdae066/7699676 by guest on 07 July 2024 Denoting steady-state inequality Γ C S /C H , we loglinearize around a steady state: R I P T Downloaded from https://academic.oup.com/restud/advance-article/doi/10.1093/restud/rdae066/7699676 by guest on 07 July 2024 Downloaded from https://academic.oup.com/restud/advance-article/doi/10.1093/restud/rdae066/7699676 by guest on 07 July 2024 The equilibrium dynamics of private liquid assets b t are found by replacing these individual budget constraints (29) into the loglinearized self-insurance Euler equation for bonds (10), with ŷjE t b t+2 Θb t+1 + β 1 b t =Finding the derivatives of b t+k with respect to ŷt requires a model of how individual disposable incomes are related to aggregate, such as this paper's.Furthermore, since the calculation of iMPCs keeps fixed by definition all the other variables (in particular taxes, their distribution, and thus public debt), the partial derivatives of individual disposable incomes with respect to aggregate disposable income are respectively d ŷH t = χd ŷt and d ŷS t = 1 λχ 1 λ d ŷt .29Solving the asset dynamics equation taking this into account delivers: