Optimal Monetary Policy According to HANK

The authors note that while a large HANK literature has shown uninsurable idiosyncratic risk can “dramatically change the positive effects of monetary policy,” the normative question – how policy should optimally be conducted, and how it affects risk and inequality – was “less well studied,” partly because “solving for the Ramsey optimal policy involves choosing the evolution of an infinite dimensional state variable (the wealth distribution) … subject to an infinite number of constraints” (Introduction, p. 1). Rather than the numerical approach of contemporaneous work (Bhandari et al. 2018), the authors “instead take an analytical approach,” exploiting CARA utility and normally distributed idiosyncratic shocks so that “the infinite dimensional distributions of consumption, hours worked and wealth can be summarized by their cross-sectional averages,” while the effect of inequality on welfare is “also summarized by a finite dimensional sufficient statistic” (p. 1-2).

Households in a perpetual-youth (Blanchard-Yaari) structure face CARA utility over consumption and (dis-utility of) labor, receive idiosyncratic normally distributed shocks to labor disutility, and can only self-insure via a riskless actuarial bond or by adjusting hours (Section 2, pp. 4-6). Proposition 1 shows equilibrium consumption is $c^s_t(i) = C_t + \mu_t x^s_t(i)$, where $x^s_t(i)$ is demeaned cash-on-hand and $\mu_t$ is a common marginal propensity to consume (MPC) that is “increasing in current and future interest rates and decreasing in current and future wages” (p. 8), because both channels affect households’ ability to smooth shocks via borrowing or future labor supply. Under complete markets, by contrast, consumption would be fully insured against these shocks ($\partial c/\partial \xi = 0$); market incompleteness is what gives interest-rate and wage paths a role in determining how much individual income shocks pass through to individual consumption.

Proposition 2 shows the planner’s period felicity function factors as $U_t = u(y_t, n_t; \bar\xi),\Sigma_t$, where $u(\cdot)$ is the notional flow utility of a representative agent and $\Sigma_t$ – increasing in the within-cohort dispersion of consumption – is “the welfare cost of inequality”; $\Sigma_t = 1$ in a riskless economy and $\Sigma_t > 1$ whenever there is dispersion, so “higher $\Sigma_t$ reduces welfare” (Section 4, p. 11). The law of motion for $\Sigma_t$ (eq. 22) shows that “higher cash-on-hand risk … and a higher pass-through $\mu_t$ both tend to increase consumption inequality,” and that inequality “inherits the slow moving dynamics of wealth inequality” through a lagged $\Sigma_{t-1}$ term (p. 12) – meaning today’s policy choices have persistent effects on tomorrow’s inequality.

First, the income risk channel: “if this risk is countercyclical, then the central bank has an incentive to raise output in order to lower risk, while the opposite is true if risk is procyclical” (Introduction, p. 2). Second, the self-insurance channel: lower current and future interest rates make it “easier for households to borrow in response to an unfavorable shock,” and higher future wages make it cheaper (in disutility terms) to work off debt, both lowering the pass-through from income shocks to consumption ($\mu_t$) “at any level of income risk” (p. 2-3). The authors stress this second channel is “of course absent in RANK” and is also shut down in an important class of existing HANK models that impose a zero-liquidity limit, since with zero liquidity the pass-through from income to consumption risk is mechanically fixed at 1 regardless of policy (Remark 1, p. 15).

Because households enter date 0 with an existing, non-degenerate wealth distribution, “an unexpected fall in interest rates benefits poor debtors, reducing their interest payments and increasing their consumption … [while] lower interest rates reduce the interest income of rich savers, reducing their consumption,” so that a surprise rate cut mechanically compresses consumption dispersion via the unhedged interest rate exposure (URE) channel (Auclert 2019) (Introduction, p. 3). Crucially, “this channel … only operates for unexpected changes in interest rates”: the paper shows formally that an anticipated cut would be pre-empted by savers saving more and debtors borrowing more in advance, so what reduces inequality is specifically the surprise component of $\mu_0$ relative to its previously expected value $E_{-1}\mu_0$ (Section 4, p. 12-13, eq. 23).

With Omega = 0 – a calibration in which the income-risk and self-insurance channels exactly cancel for $t>0$ – Proposition 3 shows that for all dates after the initial one, the HANK and RANK target criteria are “almost identical,” reflecting that “monetary policy cannot affect consumption risk at dates $t>0$” (p. 18-19). But the date-0 target criterion differs from RANK in three ways: it is optimal to create an output boom and positive inflation at date 0 even absent any shock (since a surprise rate cut lowers $\Sigma_0$ via the URE channel); output tracks its flexible-price level less than one-for-one at all dates ($\delta < 1$); and the criterion puts less weight on inflation relative to output (Section 5.4.1, p. 19-20). The authors are explicit about the resulting inconsistency: “the desirability of exploiting households’ unhedged interest-rate exposure for redistribution … makes the Ramsey plan time-inconsistent … the continuation of a Ramsey plan is not a Ramsey plan” (p. 19).

The authors show it is specifically about initial wealth dispersion: “if there is an equalization of asset positions across all households at the beginning of date 0 … the planner is unable to affect $\Sigma_t$ at any date (up to first-order),” and “divine coincidence holds” thereafter – optimal policy simply implements zero inflation and tracks flexible-price output at all dates (Section 5.4.1, p. 22). This confirms the time-inconsistency result is not an artifact of the countercyclical-risk assumption but follows directly from the existence of savers and debtors with different net asset exposures at the moment a policymaker gains the ability to surprise them.

Proposition 4 extends the result: with countercyclical risk, “an increase in output $y_t$ reduces the consumption risk faced by households at any date, not just at date 0,” so monetary policy deviates from the RANK target criterion at every date, not only date 0 (Section 5.4.2, p. 23-24). In steady state, this means the flexible-price level of output sits strictly above the efficient level (a “negative labor wedge,” i.e., an effective subsidy to output), and the planner optimally lets output track this inefficiently high flexible-price level rather than eliminate the subsidy, “because the cost in terms of increased consumption inequality is too high” (p. 25). Following a negative productivity shock, optimal HANK policy therefore “lets output decline” by less than RANK and “implements a lower path of nominal interest rates, curtailing the fall in output … [even though] this entails higher inflation and output above its efficient level,” because “inequality is already higher in recessions and so is the benefit from a reduction in inequality” (Introduction, p. 3; Section 5.4.2, p. 24-26).

Introducing nominal bonds reintroduces the Fisher channel: “the presence of nominal debt means that unanticipated higher inflation reduces consumption inequality” by compressing the real value of nominal wealth dispersion at date 0, in addition to the reduction already achieved via the surprise rate cut (Section 6, p. 26-27). Quantitatively, in the paper’s baseline (flat Phillips curve, $\kappa = 0.01$) calibration this effect is small, since “even a large cut in real interest rates … generates only a small increase in inflation”; but with a steeper Phillips curve ($\kappa = 0.5$), the planner “creates a larger increase in inflation in the economy with nominal debt,” yielding a correspondingly larger reduction in inequality, and the effect is reinforced (not merely mechanical) because the planner actively exploits the added channel (p. 27-28).

The model is calibrated at an annual frequency with a standard deviation of idiosyncratic income shocks of 0.5 (matched to Guvenen et al. 2014), a relative-prudence coefficient of 3, a Frisch elasticity of 1/3, a Phillips-curve slope of 0.01, a 10 percent steady-state markup, and a survival probability of 0.85 following Nisticò (2016) (Section 5.2, p. 15-16). The authors explicitly distinguish their mechanism from prior HANK optimal-policy work that imposes a zero-liquidity limit (e.g., Bilbiie 2019a; Ravn and Sterk 2017; Challe 2020): “this assumption rules out the self-insurance channel because in equilibrium households do not borrow or lend and hence they spend all their income on consumption,” so “our analysis shows that this assumption rules out an important channel through which monetary policy affects inequality” (Introduction, p. 4).

“A utilitarian planner trades off the benefits of lower inequality against the costs of pushing up output and inflation above their efficient levels. In recessions, inequality is already high – in the relevant case with countercyclical risk – so the marginal benefit of reducing inequality is particularly high. Consequently, optimal monetary policy is more accommodative in recessions relative to a RANK benchmark” (Conclusion, p. 29). The authors summarize the four channels together – income risk, self-insurance, URE, and Fisher – as jointly implying that “expansionary monetary policy can reduce consumption inequality,” but the paper’s framing throughout is that this benefit must always be weighed against the efficiency costs of moving output and inflation away from their RANK-optimal paths, and that the URE channel specifically introduces a form of time inconsistency with “no […] counterpart” in RANK (Abstract; Conclusion, p. 29).