When do Endogenous Portfolios Matter for HANK?

Almost all of the HANK literature takes household asset portfolios as exogenously fixed, a modeling choice the authors trace directly to the solution methods in common use. “The vast majority of this Heterogeneous-Agent New Keynesian literature studies settings where agents hold asset portfolios that are exogenously fixed… This assumption is natural given that these models are either solved with first-order perturbation methods, or by assuming a perfect-foresight economy hit by unanticipated ‘MIT’ shocks: in either case, the portfolio choice is indeterminate” (Introduction). But “agents that perceive aggregate risk and that can invest their wealth in different types of assets have a well-defined optimal portfolio in the neighborhood of the steady state,” and allowing for these “zeroth-order” portfolios (a term from Devereux and Sutherland 2011) “may mute some of the redistribution channels highlighted in the existing literature, and perhaps overturn some of its main conclusions” (Introduction).

When markets are complete with respect to aggregate risk (at least as many assets as shocks, with a spanning condition satisfied), optimal portfolios must satisfy a simple risk-sharing condition: each household’s expected marginal utility, conditional on any realization of aggregate shocks, must move in the same proportion relative to its steady-state value (Section 2.2, Proposition 1; equation 1 in the Introduction). Formally, E[u’(c_0,i)]/E[u’(c_ss,i)] = lambda_0 for all households i, where lambda_0 – shown in Proposition 2 to be a relative stochastic discount factor delivering second-order risk premia from a first-order solution – is common across households but can vary by shock realization. “This equation can be used to test for the optimality of exogenous portfolios, and, when this test fails, to solve for the portfolios and the lambda_0 that imply (1) while clearing all asset markets” (Introduction).

The paper shows that solving for endogenous portfolios and impulse responses jointly “boils down to a simple modification of sequence-space Jacobians,” using the same steady-state objects (marginal utilities, MPCs, the “fake news” distributional perturbations) as the exogenous-portfolio calculation (Section 3.2). Concretely, the corrected Jacobian is H_X + H_corr_X, where H_corr_X captures both the direct effect of a shock on household transfers via equation (14) and the indirect effect through the shared multiplier lambda; “the corrected Jacobians… can then be used in place of the uncorrected Jacobians” to solve the model exactly as in the exogenous-portfolio case (Section 3.2). The authors note that “code posted on GitHub illustrates the simplicity of this implementation by reproducing the results presented in this paper” (Introduction, fn. 3).

The application is a deliberately simple HANK economy: households face idiosyncratic labor-income risk and choose between two assets – real risk-free bonds and stocks that are claims on monopolistically-competitive firms’ profits – subject to a zero borrowing constraint, with log utility, sticky wages and equal-rationed labor, and a central bank that sets the real interest rate exogenously (Section 3.1). The calibration has no government bonds in the initial steady state, so the natural exogenous-portfolio benchmark is that “all agents hold 100% stock portfolios” (Section 3.1, “Exogenous vs endogenous portfolios”). The authors argue the model’s simplicity is not disqualifying because it shares two features “widely shared in the literature”: monetary policy has aggregate effects close to a representative-agent model (following Werning 2015), while deficit-financed fiscal policy has powerful effects via the “intertemporal Keynesian cross” of Auclert, Rognlie and Straub (2024) (Section 3.1).

Endogenous and exogenous portfolios deliver identical impulse responses, because the shock leaves every household’s consumption unchanged, so the risk-sharing test is trivially (if vacuously) satisfied. “A shock to epsilon^G, with sigma_B = sigma_r = 0, implies dc_it = 0 for all agents i. Exogenous and endogenous portfolios deliver the same solution, even if additional assets are available to complete markets. Portfolios are undetermined” (Proposition 4, Section 4.1). The underlying mechanism is a unit government-spending multiplier with no consumption response for any agent (citing Haavelmo 1945 and Auclert, Rognlie and Straub 2024), which also means stock and bond returns coincide even on impact, so “even though they perceive the aggregate risk, agents here are still indifferent between all portfolios” (Section 4.1).

Endogenous and exogenous portfolios again deliver the same aggregate impulse responses, because the exogenous 100%-stock allocation is itself already the optimal, complete-markets-consistent portfolio for every household. “A shock to epsilon^r, with sigma_G = sigma_B = 0, implies dc_it = -c_it * sum_(s>=0) dr_(t+s)/(1+r) for all agents i. Optimal portfolios are 100% stocks. Hence, exogenous and endogenous portfolios deliver the same solution” (Proposition 5, Section 4.2). The result rests on log utility, equal rationing in the labor market, and the 100%-stock allocation jointly implying that every agent’s Euler equation holds with equality – “a result initially derived by Werning (2015)” – so that “given the risk premium on stocks, any agent finds that increasing or reducing exposure to the stock market would lower their expected utility” (Section 4.2).

Here endogenous portfolios matter a great deal: the impact transfer multiplier falls from 0.2 under exogenous portfolios to 0.08 under endogenous portfolios (a 60% reduction), while the cumulative multiplier falls more modestly from 0.77 to 0.53 (30%), because deficit-financed transfers disproportionately raise poor agents’ consumption and violate the risk-sharing condition. “With 100% stock portfolios, deficit-financed fiscal transfers disproportionately raise the consumption of poor agents, and therefore disproportionately lower their expected marginal utility, violating condition (1)” (Section 4.3). Because the stock market booms on impact (proportional to the cumulative transfer multiplier), “optimal portfolios reduce the stock market exposure of poor agents and raise the exposure of rich agents… [and] since rich agents have lower marginal propensity to consume out of capital gains than poor agents, optimal portfolios reduce the aggregate transfer multiplier” (Section 4.3). The authors flag that the implied optimal portfolios are extreme – poor agents near the borrowing constraint take short stock positions “including some with thousands of times their net worth” – so this exercise is explicitly “an upper bound of how much endogenous portfolios can shrink the transfer multiplier” (Section 4.3).

Adding plausible portfolio constraints – stocks restricted to between -100% and 200% of net worth – brings the deficit-financed-shock output path “virtually identical to the result with our baseline, exogenous portfolios,” because most households end up pinned at their short-sale or leverage constraint rather than at their unconstrained optimum (Section 5). The compression is “most dramatic for the low-asset, high-MPC agents, whose consumption is most responsive to portfolio returns,” which is exactly why the general-equilibrium multiplier moves back toward the exogenous-portfolio benchmark (Section 5). This result is central to how the paper qualifies its headline finding: the large attenuation in Q7 depends on allowing implausibly large gross positions.

With both a monetary shock and a deficit-financed fiscal shock present (two shocks, but still only a stock and a bond, so markets are incomplete with respect to aggregate risk), impulse responses to the two shocks become coupled, but in this particular case they remain “virtually identical to those with exogenous portfolios,” because deficit-financed shocks are much harder to hedge than monetary shocks and so optimal portfolios stay close to the 100%-stock benchmark (Section 6.1). The paper also shows that adding more distinct deficit-financed shocks of varying persistence pulls the incomplete-markets impulse response back toward the exogenous-portfolio benchmark when the added shocks are more persistent (harder to hedge with the available assets), but pushes it further away when they are less persistent – “incomplete markets do not have a monotonic relationship with the exogenous portfolio or complete markets impulse responses” (Section 6.3).

In the baseline calibration, risk premia recovered from the method are small: about 24 annualized basis points for the monetary policy shock (matching the standard consumption-CAPM formula given log utility) and essentially zero for the deficit-financed fiscal shock, because the fiscal shock moves both consumption and returns only modestly (Section 4.4, Table 2). The authors are explicit that this reflects the model’s simplicity – with CRRA utility and risk aversion of 1 the baseline “is subject to a standard equity premium puzzle” – and note the method should extend readily to richer models with cyclical idiosyncratic risk that can generate empirically realistic premia (Section 4.4, Conclusion).

When the model is reconfigured so that households hold only nominal bonds (Huggett-style private IOUs) and high-MPC agents are, by construction of the exogenous benchmark, heavily levered in nominal debt, optimal portfolios move in the opposite direction from Q7: they shrink these agents’ nominal exposure and substantially dampen the monetary transmission mechanism. In the exogenous, 100%-nominal-portfolio economy the covariance between MPCs and net nominal positions is -0.61, which the authors describe as “vastly out of” the empirically estimated range of roughly -0.11 to 0.07 from Auclert (2019); optimal portfolios bring this covariance to a more reasonable 0.05, and correspondingly the exogenous-portfolio economy’s very large response to a monetary shock (a 6.5% output contraction with 5% deflation on impact, driven by a debt-deflation Fisher spiral) is muted toward the response of an economy with fully sticky prices (Section 7). The authors note the implied portfolios – low-asset agents holding positive nominal bonds, high-asset agents as the nominal liability holders – accord qualitatively with evidence that “it is indeed rich and middle-class households that tend to hold the biggest mortgages” (Doepke and Schneider 2006), citing this as a point in favor of the endogenous-portfolio model’s realism on this dimension (Section 7).

Endogenous portfolios “do not always make a difference,” and when they do, the effect is to lower the impact response of consumption to shocks – but this finding hinges specifically on whether high-MPC agents are allowed to take large gross positions. “When they do [make a difference], they tend to lower the impact effect of shocks on consumption… However, we found that the optimal portfolios underlying these results tend to be rather extreme… When we add reasonable short sales and leverage constraints, the large reduction in the aggregate deficit-financed multiplier goes away” (Conclusion). The authors caution against over-generalizing in either direction: “our results should not be taken to mean that endogenous portfolios can never generate plausible portfolio distributions across agents, or that they can never make a significant difference when they do,” and they flag the method’s generality – directly applicable to any model solved with the sequence-space Jacobian method – as useful for future work on cross-country, cross-asset-class, and cross-maturity portfolio questions (Conclusion).