Optimal Monetary Policy According to HANK: Acharya, Challe & Dogra (2023)

What this is. The paper’s core results, the CARA-normal HANK model, the welfare-based LQ planning problem, and the HANK target criterion with its defining equations: enough to understand what it found and how, without reading all 42 pages. To replicate or extend the analysis, read the full source at doi.org/10.1257/aer.20200239.

TL;DR

The paper derives optimal monetary policy analytically in a heterogeneous-agent New Keynesian (HANK) model where households face uninsurable idiosyncratic income risk. CARA preferences and normally distributed idiosyncratic shocks (from Acharya and Dogra (2020)) make the model analytically tractable: the economy aggregates linearly, and the planner’s welfare can be expressed as a function of aggregate output and a scalar measure of consumption inequality. The key finding is that the HANK planner’s loss function differs from the representative-agent (RANK) loss function of Galí (2015) and Woodford (2003) in two ways. First, the planner puts more weight on stabilizing economic activity relative to inflation (calibrated Upsilon = 1.76 vs 1 in RANK). Second, the planner also cares about stabilizing the level of output (not just the output gap), because output stabilization reduces consumption risk when income risk is countercyclical (calibrated delta = 0.6 < 1). Following productivity or markup shocks, the HANK planner therefore cushions output declines by accepting positive inflation on impact, in contrast to RANK’s divine coincidence (zero inflation and zero output gap following productivity shocks). The analysis builds on the second-order LQ approximation approach of Benigno and Woodford (2005) to handle the inefficient HANK steady state. McKay and Wolf (2022) use a related LQ approach but their planner is not Pareto optimal.

Core results

Magnitudes are as reported; * = 5%, ** = 1%. Locators point into the source PDF. R3 and R4 magnitudes are approximate figure readings (Figures 4-5, pp. 1768, 1771). R5 is a qualitative irrelevance result.

#	Result	Locator	Magnitude
R1	HANK loss function puts weight Upsilon(Omega) > 1 on output activity, scaling up the weight on output relative to inflation compared to RANK	Proposition 3, eq. (33), Figure 3, p. 1763-1765	In calibration: Upsilon = 1.76; relative weight on price stability roughly halved relative to RANK (epsilon/(kappa * Upsilon) vs epsilon/kappa)
R2	HANK target criterion introduces output level, weighted (1-delta) against the output gap (weighted delta), reducing weight on the price level relative to RANK	Proposition 4, eq. (36), p. 1766	In calibration: delta = 0.6 (vs 1 in RANK); roughly equal weight on output level stabilization and output gap stabilization
R3	Following a fall in productivity, HANK planner prevents output from falling to the flexible-price level, accepting positive inflation on impact; RANK achieves divine coincidence (zero inflation, zero output gap)	Proposition 5, Figure 4, pp. 1767-1768	HANK: output gap approx. +0.2 pp and inflation approx. +0.025 pp at date 0; both reverse after period T. RANK: output gap = 0, pi = 0 for all t
R4	Following a positive markup shock, HANK allows a larger inflation increase and a smaller output decline than RANK, because a large output decline increases consumption inequality	Proposition 6, Figure 5, pp. 1769-1771	HANK: output approx. -0.2 pp vs RANK -0.3 pp at impact; inflation approx. +0.04 pp vs RANK +0.01 pp at impact
R5	Acyclical income risk (Omega = 0): HANK optimal policy = RANK optimal policy, even though inequality exists, because its evolution does not depend on output	Lemma 2, p. 1766	When Omega = 0: Upsilon = delta = 1; identical target criteria and optimal {yhat, pi} paths in HANK and RANK
R6	Unequally distributed profits add dividend stabilization to HANK loss function; when only 10% of households hold dividends (stockholder fraction eta^d = 0.1), the planner raises output approx. 0.1 pp in response to a positive markup shock	Proposition 7, Figure 6, pp. 1773-1775	K(eta^d) is increasing in profit concentration; eta^d = 0.1: output raised approx. +0.1 pp above equal-distribution baseline

Overall (paper’s conclusion). HANK differs from RANK because monetary policy can stabilize consumption inequality. When income risk is countercyclical (the empirically relevant case), the HANK planner puts some weight on stabilizing the level of output in addition to the output gap, tolerates higher inflation after adverse shocks, and implements interest rates that fall by less than in RANK. Extensions to unequal profit distribution and initial wealth inequality (URE channel via Auclert (2019)) reinforce the same qualitative conclusion: both motives lead optimal monetary policy to put more weight on output stabilization relative to RANK.

Theory / model

The model is a Bewley-Huggett economy with a New Keynesian production side. Households follow a perpetual-youth (Blanchard-Yaari) life-cycle with constant per-period survival probability $\vartheta$. Population is normalized to 1. The date $s$ problem of household $i$ born at date $s$ is (p. 1746, eq. 1):

$$\max_{\{c_t^s(i),\, l_t^s(i),\, a_{t+1}^s(i)\}} E_s \sum_{t=s}^\infty (\beta\vartheta)^{t-s}\, u\!\left(c_t^s(i),\, l_t^s(i);\, \xi_t^s(i)\right) \tag{1}$$

subject to a budget constraint (simplified, eq. 5, p. 1748):

$$c_t(i) + w_t l_t(i) + q_t a_{t+1}(i) = w_t \xi_t^s(i) + (1-\tau_t^a) a_t(i) + D_t - T_t, \tag{5}$$

where $w_t$ is the posttax real wage, $q_t = \vartheta/R_t$ is the bond price, $a_t(i)$ is real actuarial bond holdings, $D_t$ are dividends, and $T_t$ are lump-sum taxes. Each household faces i.i.d. idiosyncratic disutility-of-labor shocks $\xi_t^s(i) \sim N(\bar{\xi},\sigma_t^2)$. Agents have CARA preferences (p. 1747, eq. 4):

$$u(c, l; \xi) = -\frac{1}{\gamma} e^{-\gamma c} - \rho\, e^{\frac{1}{\rho}(l - \xi)}, \tag{4}$$

with coefficient of absolute risk aversion $\gamma$ and Frisch labor supply parameter $1/\rho$. CARA utility with normal shocks enables linear aggregation: equilibrium individual consumption and labor supply are linear in demeaned cash-on-hand (Proposition 1, eqs. 15-16, p. 1750):

$$c_t^s(i) = \mathcal{C}_t + \mu_t\, x_t^s(i), \qquad l_t^s(i) = \rho\ln w_t - \gamma\rho\, c_t^s(i) + \xi_t^s(i), \tag{15-16}$$

where $x_t^s(i) = (1-\tau_t^a)a_t^s(i) + w_t[\xi_t^s(i)-\bar{\xi}]$ is demeaned cash-on-hand and $\mu_t$ is the marginal propensity to consume (MPC) out of cash-on-hand. Aggregate consumption $\mathcal{C}_t$ evolves as (eq. 17):

$$\mathcal{C}_t = -\frac{1}{\gamma}\ln\beta R_t + \mathcal{C}_{t+1} - \frac{\gamma\mu_{t+1}^2 w_{t+1}^2 \sigma_{t+1}^2}{2}, \tag{17}$$

where the last term is a precautionary savings motive absent in RANK. The MPC satisfies (eq. 18):

$$\mu_t^{-1} = 1 + \gamma\rho w_t + \frac{\vartheta}{R_t}\,\mu_{t+1}^{-1}. \tag{18}$$

Lower real interest rates reduce $\mu_t$, facilitating self-insurance (the self-insurance channel). Intermediate goods producers face Rotemberg quadratic price adjustment costs; the goods market clears at $y_t = c_t$. The nonlinear IS equation is (eq. 21, p. 1751):

$$y_t = y_{t+1} - \frac{1}{\gamma}\ln\beta\!\left(\frac{1+i_t}{\Pi_{t+1}}\right) - \frac{\gamma}{2}\,\mu_{t+1}^2 w_{t+1}^2 \sigma_{t+1}^2. \tag{21}$$

The Phillips curve is standard (linearized, eq. 30, p. 1760):

$$\pi_t = \beta\pi_{t+1} + \kappa(\hat{y}_t - \hat{y}_t^e) + \frac{\varepsilon}{\Psi}\,\hat{\varepsilon}_t, \tag{30}$$

where $\kappa = \frac{\varepsilon}{\Psi}\frac{1+\gamma\rho}{\rho/y}$ and $\hat{y}_t^e = \frac{1+\rho/y}{1+\gamma\rho}\hat{z}_t$ is the flexible-price (productively efficient) level of output.

Welfare decomposition. The social welfare function is the sum of average household lifetime utilities. By Proposition 2 (eq. 26, p. 1754), the period $t$ felicity can be written as:

$$U_t = u(c_t, n_t;\bar{\xi}) \times \Sigma_t, \tag{26}$$

where $\Sigma_t \geq 1$ is the welfare cost of consumption inequality: it equals 1 under complete markets and exceeds 1 whenever consumption dispersion is positive (higher $\Sigma_t$ reduces welfare since $u(\cdot) < 0$). The dynamics of $\Sigma_t$ are (eq. 27, p. 1755):

$$\ln\Sigma_t = \frac{\gamma^2}{2}\mu_t^2 w_t^2 \sigma_t^2 + \ln(1-\vartheta + \vartheta\Sigma_{t-1}). \tag{27}$$

Consumption inequality is driven by within-period consumption risk $\mu_t^2 w_t^2 \sigma_t^2$ (idiosyncratic variance passed through the MPC) plus the accumulated effect of preexisting wealth inequality inherited from $\Sigma_{t-1}$.

How monetary policy affects inequality. Linearizing eq. 27 and using assumptions on the cyclicality of $\sigma_t$, the two channels by which monetary policy affects consumption risk become explicit (eq. 31, p. 1760):

$$\hat{\Sigma}_t = \Lambda\hat{\mu}_t - \gamma y(\Theta-1)\hat{y}_t + \beta^{-1}\tilde{\beta}\,\hat{\Sigma}_{t-1},$$

where $\Lambda = \gamma^2\mu^2 w^2\sigma^2 > 0$ and $\Theta = 1 - \Lambda\varphi/\gamma$ with $\varphi = \partial\ln\sigma_t^2/\partial y_t$ measuring income-risk cyclicality. The first term captures the self-insurance channel (lower $\mu_t$ reduces consumption risk); the second captures the income-risk channel (when risk is countercyclical, $\Theta > 1$, higher output lowers $\Sigma_t$). Lemma 1 (eq. 32, p. 1762) combines both channels into a single sufficient statistic $\Omega$, the cyclicality of consumption risk:

$$\hat{\Sigma}_t = -\gamma y\,\Omega\!\left[\hat{y}_t - \varkappa(\Omega)\hat{y}_t^e\right] + \beta^{-1}\tilde{\beta}\,\hat{\Sigma}_{t-1}, \tag{32}$$

where $\Omega = \frac{\Lambda}{1-\Lambda} + \frac{\Theta-1}{1-\Lambda} \geq \Omega^c = \frac{\Lambda}{1-\Lambda} > 0$ when risk is acyclical or countercyclical. When $\Omega = 0$ (acyclical risk and no self-insurance), monetary policy cannot affect consumption risk, and HANK optimal policy coincides with RANK (Lemma 2).

The HANK planner’s LQ problem. The planning problem is to minimize the second-order approximation to social welfare losses over sequences $\{\hat{y}_t, \pi_t\}_{t=0}^\infty$, subject to the linearized Phillips curve (30). Proposition 3 (eq. 33, p. 1763) states:

$$\min_{\{\hat{y}_t,\,\pi_t\}_{t=0}^\infty} \frac{1}{2}\sum_{t=0}^\infty \beta^t \left\{\Upsilon(\Omega)\left[\hat{y}_t - \delta(\Omega)\hat{y}_t^e\right]^2 + \frac{\varepsilon}{\kappa}\,\pi_t^2\right\}, \tag{33}$$

where $\Upsilon(\Omega) > 1$ and $\delta(\Omega) \in (0,1)$ when $\Omega \geq \Omega^c > 0$ (acyclical or countercyclical income risk). In RANK ($\sigma = 0 \Rightarrow \Omega = 0$), $\Upsilon = \delta = 1$ and eq. (33) reduces to the standard RANK problem (eq. 34):

$$\min_{\{\hat{y}_t,\,\pi_t\}_{t=0}^\infty} \frac{1}{2}\sum_{t=0}^\infty \beta^t \left\{(\hat{y}_t - \hat{y}_t^e)^2 + \frac{\varepsilon}{\kappa}\,\pi_t^2\right\}. \tag{34}$$

The HANK loss function has two differences from RANK. First, the weight on economic activity is scaled by $\Upsilon(\Omega) > 1$, implying a lower relative weight on price stability. Second, the planner targets $\hat{y}_t - \delta(\Omega)\hat{y}_t^e$ rather than the output gap $\hat{y}_t - \hat{y}_t^e$: since $\delta(\Omega) < 1$, the planner aims to close the gap between $\hat{y}_t$ and $\delta(\Omega)\hat{y}_t^e < \hat{y}_t^e$, placing positive weight on stabilizing the level of output as well as the output gap.

The HANK target criterion (Proposition 4, eq. 36, p. 1766) for all $t \geq 0$:

$$\left[1 - \delta(\Omega)\right]\hat{y}_t + \delta(\Omega)\!\left(\hat{y}_t - \hat{y}_t^e\right) + \frac{\varepsilon}{\Upsilon(\Omega)}\,\hat{p}_t = 0, \tag{36}$$

where $\hat{p}_t$ is the log price level. In RANK the target criterion is $(\hat{y}_t - \hat{y}_t^e) + \varepsilon\hat{p}_t = 0$ (eq. 37), i.e., flexible price level targeting. The HANK criterion places weight $(1-\delta)$ on the output level $\hat{y}_t$, weight $\delta$ on the output gap, and a lower weight $\varepsilon/\Upsilon < \varepsilon$ on the price level.

Method

The planner’s problem is solved via the linear-quadratic (LQ) approach, following Benigno and Woodford (2005). A naive LQ approach (maximizing a quadratic approximation to welfare subject to linear constraints) does not yield first-order accurate approximations at an inefficient steady state, as is natural in HANK models with uninsurable income risk. To address this, the paper uses a second-order approximation of the constraints to eliminate first-order terms from the welfare approximation, yielding the LQ problem (33) that is first-order accurate. This mirrors Benigno and Woodford (2005)‘s approach in RANK, generalized here to an economy where the steady state features consumption inequality ($\Sigma > 1$).

The linearized model consists of three main equations (p. 1753): the IS equation (23), MPC recursion (24), and the Phillips curve (25). Log-linearizing around the zero-inflation steady state and using eq. (20) to eliminate wages:

$$\hat{y}_t = \Theta\hat{y}_{t+1} - \frac{1}{\gamma y}\!\left(\hat{i}_t - \pi_{t+1}\right) - \frac{\Lambda}{\gamma y}\,\hat{\mu}_{t+1}, \tag{23}$$ $$\hat{\mu}_t = -\gamma\mu wy\!\left[(1+\gamma\rho)\hat{y}_t - \hat{z}_t\right] + \tilde{\beta}\!\left(\hat{\mu}_{t+1} + \hat{i}_t - \pi_{t+1}\right), \tag{24}$$

where $\hat{i}_t = \ln(1+i_t) - \ln R$, $\Theta = 1 - \Lambda\varphi/\gamma$, $\Lambda = \gamma^2\mu^2 w^2\sigma^2$, and $\tilde{\beta} = \vartheta/R$. These reduce to the standard RANK IS curve when $\Lambda = 0$ (no idiosyncratic risk). The Ramsey plan that solves problem (33) can be implemented by the following interest rate rule (eq. 38, p. 1770):

$$i_t = i_t^\star + \phi\pi_t + \phi_{\text{gap}}(\Delta y_t - \Delta\hat{y}_t^e) + \phi_y\,\Delta y_t, \tag{38}$$

where $\phi_{\text{gap}} = \phi\frac{\Upsilon(\Omega)}{\varepsilon}\delta(\Omega)$ and $\phi_y = \phi\frac{\Upsilon(\Omega)}{\varepsilon}[1-\delta(\Omega)]$. The HANK rule reacts more strongly to changes in output growth and the output gap (relative to $\pi_t$) than the RANK rule where $\Upsilon = 1, \delta = 0$.

Empirical specifications

The paper’s main results are analytical (propositions and closed-form expressions for $\Upsilon$ and $\delta$); calibration and impulse response functions (IRFs) are illustrative. Parameters are set to match US annual aggregate and micro targets:

• Annual frequency, real interest rate $r = 4\%$, steady-state output $y = 1$.
• Survival probability $\vartheta = 0.85$ (following Nistico 2016 and Farhi-Werning 2019).
• Standard deviation of income in steady state $w\sigma(1-\gamma\rho\mu w) = 0.5$, in line with Guvenen, Ozkan, and Song (2014).
• Cyclicality of income risk $\varphi = -5.76$, consistent with Storesletten, Telmer, and Yaron (2004) who find the standard deviation of (log) income rises from 0.12 in expansions to 0.21 in recessions.
• Phillips curve slope $\kappa = 0.1$, elasticity of substitution $\varepsilon = 10$ (10% steady-state markup).
• Coefficient of relative risk aversion $\gamma c = \gamma$ and Frisch elasticity $\rho/y$ calibrated so that median household CRA = 2 and Frisch = 1/3 in steady state.
• Persistence of shocks: $\varrho_z = 0.95^4$ (productivity) and $\varrho_\varepsilon = 0.9^4$ (markup), from Bayer, Born, and Luetticke (2020).
• Shock standard deviations: $\sigma_z = 0.012$ and $\sigma_\varepsilon = 0.034$ (Bayer, Born, and Luetticke 2020).

With these parameters, the baseline calibration yields $\Omega = \Omega^c + (\Theta-1) > 0$ (countercyclical consumption risk) and the welfare parameters $\Upsilon = 1.76$ and $\delta = 0.6$. IRFs are constructed as the one-standard-deviation impulse response to a date-0 productivity or markup shock, holding the initial wealth distribution at its Ramsey steady state. Figures 4 and 5 (pp. 1768-1770) compare HANK optimal policy (blue) to RANK optimal policy (red dashed) and to the non-optimal policy that sets $\hat{y}_t = \hat{y}_t^e$ and $\pi_t = 0$ (black dotted). Section V calibrates the unequal dividends extension with $\mathcal{D}_y < 0$ (negative output elasticity of dividends in the baseline) and stockholder fractions $\eta^d \in \{0.1, 0.5, 1\}$ (Figure 6, p. 1775).

Datasets used

The paper is theoretical with calibrated parameters; no primary dataset is used directly. The calibration matches moments from the following published studies:

Dataset / Source	Role in paper	Wiki page
Guvenen, Ozkan, and Song (2014), SIPP administrative records	Target for steady-state standard deviation of income (set to 0.5)	No page yet
Storesletten, Telmer, and Yaron (2004), PSID earnings data	Calibration target for cyclicality of income risk (phi = -5.76)	No page yet
Bayer, Born, and Luetticke (2020) HANK calibration	Shock persistence and standard deviation parameters	No page yet

Replication code and data are available at doi.org/10.3886/E184261V1.

When to read the full paper

Use the original if you are: deriving the formal welfare approximation and the exact expressions for $\Upsilon(\Omega)$ and $\delta(\Omega)$ (online Appendices E.2-E.3); studying extensions to hand-to-mouth households (Appendix H), persistent idiosyncratic risk (Appendix I), or demand shocks (Appendix J); checking the proof of divine coincidence breakdown (Proposition 5, Appendix F); or comparing the URE channel under the non-utilitarian planner to the baseline (Propositions 8-9, Appendix D.4). The locators above point to the key propositions and figures.

Attribution and rights

Source: peer-reviewed, American Economic Review 113(7), July 2023. This distillation was extracted by an LLM on 2026-06-24 and is not human-verified or independently reproduced. The article is paywalled; this page contains extracted summaries only (extract-only redistribution).

Acharya, Sushant, Edouard Challe, and Keshav Dogra. “Optimal Monetary Policy According to HANK.” American Economic Review 113, no. 7 (July 2023): 1741-1782. DOI: 10.1257/aer.20200239. Replication data: doi.org/10.3886/E184261V1. All rights reserved, American Economic Association. This page is a distillation by the Institute for Automated Research (LLM-extracted, not human-verified, not reproduced).