Macroeconomic Fluctuations with HANK & SAM: an Analytical Approach

Because the existing literature’s reliance on numerical solutions leaves its results untied to the deep structure of the economy. The complaint is specific: “so far the HANK literature has relied mostly on insights from calibrated models which are solved numerically. Therefore, little is known about the generality of the results and how they relate to the deeper structure of the economy. This paper addresses this issue.” The intended payoff is stated in three areas — “insights into the local and global determinacy of equilibria, the conduct of monetary policy, and the impact of interactions between heterogeneity and market frictions on short- and long-run macroeconomic outcomes” — and the authors position the exercise explicitly as an analogue of Clarida, Galí and Gertler’s three-equation treatment of the standard New Keynesian model.

Nominal rigidity in the goods market, search and matching in the labour market, incomplete insurance in the financial market. Firms are monopolistically competitive with quadratic Rotemberg price adjustment costs, “as a result, macroeconomic outcomes are affected by changes in the nominal interest rate set by the central bank.” The labour market runs on a Cobb-Douglas matching function with costly vacancy posting, an exogenous constant separation rate, Nash-bargained wages, and a non-negativity constraint on vacancies; the point of the friction is that “job prospects are uncertain which exposes households to idiosyncratic income risk.” Financial markets are incomplete “in the sense that households cannot fully insure against this income risk. Instead, they attempt to self insure through savings in risk free bonds or in equity.” The three together are what the authors name HANK&SAM. The labour-market friction also delivers involuntary unemployment, aligning the model with the Gertler-Trigari, Blanchard-Galí strand of the New Keynesian literature.

By three assumptions that collapse the wealth distribution to a point while preserving incomplete markets and MPC heterogeneity. The assumptions: households cannot short equity and face a borrowing constraint permitting only employed households to borrow; households differ in both labour productivity and the returns available from equity investment; and there are two types, one with zero on-the-job productivity but equity market access, the other with positive productivity but no equity returns. In equilibrium “firms are owned by capitalists who drop out of bond and labor markets, while workers hold no equity and are either employed or unemployed. Moreover, in equilibrium all households end up consuming their income period-by-period and the real interest rate has to satisfy the Euler equation of employed workers. Consequently, the equilibrium features inequality in outcomes but the wealth distribution is degenerate, which enables an analytical characterization.” A footnote is careful that consuming one’s income is an equilibrium outcome, not an assumed rule: “since we allow for debt, the Marginal Propensity to Consume (MPC) out of an idiosyncratic temporary income shock does not equal one for the employed agents and the capitalists in the model. Only for the borrowing-constrained agents (the unemployed) it holds that MPC=1.” The authors also note their route differs from earlier tractable models in which “agents are unable to borrow,” whereas here “the employed households — who end up pricing the bonds — are in principle able to borrow but choose voluntarily not to do so.”

A precautionary-saving term in the employed workers’ Euler equation whose size and direction of movement are determined inside the model by labour market tightness, which is what creates a feedback loop from goods demand back to labour demand. The Euler equation is c⁻ᵘ = βE[(R/Π)c’⁻ᵘ(1 + ω(1 − η’)((ϑ/w’)⁻ᵘ − 1))]; the bracketed term “derives from lack of insurance against unemployment risk.” The paper walks through three cases to isolate what endogeneity adds. With exogenous constant risk the term is a constant and “does not have an important effect on macroeconomic outcomes, apart from the steady-state real interest rate falling below the pure rate of time-preference.” With exogenously countercyclical risk, “demand contracts in recessions relative to booms because of the increase in the precautionary saving motive,” but “if the earnings risk is exogenous, there is no further feedback to the demand side.” Only when transition rates and wages are endogenous does the wedge “introduce a feedback mechanism”: “A contraction in demand will now be reflected in lower vacancy postings which impact on the job finding rate and on wages and thereby induces a feedback mechanism which sets the HANK&SAM model apart from both models with complete markets, models with exogenous earnings uncertainty…, and models with flexible prices.” The transmission is spelled out in a footnote: higher precautionary saving pushes the real rate down, an active nominal rule then makes inflation fall, optimal price setting requires real marginal costs to fall, which requires lower real wages or lower hiring costs, and lower hiring costs require firms to hire less, cutting the job finding rate further.

It can go either way in theory, and the paper argues countercyclicality is the empirically relevant case on three separate grounds. The tension is stated plainly: in a recession “worsening labor market conditions stimulate precautionary saving amongst employed workers because they perceive a higher risk of unemployment. However, as the labor market deteriorates, wages are also likely to fall which implies a smaller income loss in case of unemployment which accordingly implies less precautionary saving.” In log-linearized form the sign is Θ^F ≡ ωη[(ϑ/w)⁻ᵘ − 1] − χµω(1 − η), the first term countercyclical, the second procyclical in the wage elasticity χ. The three arguments: Storesletten, Telmer and Yaron’s PSID evidence that “idiosyncratic labor market income risk is strongly countercyclical”; Guvenen, Ozkan and Song’s finding, on a 10 percent sample of US working-age males, that “the countercyclicality of earnings risk derives from increased left-skewness in recessions… rather than from a countercylical variance,” which matches the paper’s emphasis on discrete unemployment risk; and an indirect check on comovement plus a direct evaluation of Θ^F, both in Section 5.

The sign of the correlation between the real interest rate and labour market tightness, which is negative under complete markets, acyclical risk, or procyclical risk, and can turn positive only under strongly countercyclical risk. The logic: under complete markets “the propensity to save will be high in booms and low [in] recessions. Therefore, when jobs are easy to find because the economy is doing well, real interest rates will tend to be low because of high desired savings and vice versa,” and procyclical earnings risk “induces an even stronger negative covariance.” By contrast, with strongly countercyclical risk “(employed) agents have an incentive to save for precautionary reasons in recessions, which can induce a positive comovement.” The log-linearized Euler equation gives the threshold: tightness and the real rate comove positively when Θ^F exceeds (1 − βRρ_A)/(βRρ_A µχ). In US data — the Federal Funds rate minus a six-month moving average of CPI inflation against the vacancy-unemployment ratio using Barnichon’s composite Help Wanted index, logged and detrended on a linear trend estimated through end-2007 — “these two variables comove positively, indicating that real interest rates are low when jobs are hard to find and vice versa. This points towards dominance of the countercyclical endogenous earnings risk channels.” The authors describe the whole empirical section as “not meant as a formal test of the model, but rather as a cursory, indicative comparison.”

CPS transition rates, a consumption drop on job loss from the literature, two risk aversion values and two wage elasticities — and the countercyclical term dominates in every combination but one. Calibration is monthly: the average job finding rate is 25.2 percent and the job loss rate 2 percent per month, measured from CPS data between January 1990 and August 2019. The consumption drop on job loss is 20 percent following Karabarbounis and Chodorow-Reich, with 5 percent as an alternative; risk aversion is 0.5 or 2. The wage semi-elasticity comes from Gertler, Huckfeldt and Trigari’s −0.16 for job stayers, chosen because “Θ^F captures the expected wage of those currently employed, in the event they remain employed,” with −1.5 as a deliberately extreme alternative sitting between their estimates for new hires from unemployment (−0.164) and job switchers (−2.085). The authors’ own regression of real average hourly earnings for production and nonsupervisory workers on the job finding rate and a trend gave at most χ̂ = 0.03, “which corresponds to about ∂ln w/∂u = −1.5.” Result: “The countercyclical effect of unemployment risk clearly dominates the procyclical effect of wage risk (i.e. Θ^F > 0). Only when we assume both a small consumption drop (5 percent) and a very elastic wage… do we find that the effect of wage risk slightly dominates. Given that these values are relatively unlikely in the light of most studies in the literature, we conclude that countercyclical earnings risk is probably the more relevant case.” The hedge — “probably” — is the paper’s own.

Three: the intended steady state, a zero-lower-bound liquidity trap, and an unemployment trap — and even the first two behave differently than under complete markets. Proposition 1 states the three, conditional on the Taylor principle holding and δ_θ ≥ 0, with a necessary condition for the unemployment trap that the Euler-equation schedule be steeper than the Phillips-curve schedule at the intended steady state, which in turn requires the steady-state wedge to be decreasing in the job finding rate. For the intended steady state, the wedge exceeds unity regardless of its slope, so the real rate is 1/(βΘ^SS(η)) < 1/β: “the steady-state real interest rate under incomplete markets derives from the precautionary saving motive induced by idiosyncratic unemployment risk. Since the job finding rate in the intended steady state depends on the interest rate rule, economic policy is a co-determinant of the long-run real interest rate.” The central bank can replicate the complete-markets levels of unemployment and inflation by choosing tightness and the intercept appropriately, but only conditionally — “this is not possible, however, without the use of fiscal policy if wages depend on market incompleteness, as they will in general,” because wages also enter the Phillips curve.

Because the steady-state real rate is no longer pinned at 1/β; it depends on the job finding rate through the risk wedge. In the standard model “the liquidity trap steady state is characterized by deflation because (R/Π)^CM = 1/β regardless of whether the ZLB is binding or not,” and the paper notes the empirical awkwardness of this: “although inflation has been moderate in the aftermath of the financial crisis, no country has experienced persistent deflation.” Here Π^LT = βΘ^SS(η^LT) > β. With no policy response to tightness (δ_θ = 0) and a price-stability target the trap remains deflationary, but “when δ_θ > 0, however, inflation may be positive or negative in the liquidity trap. In particular, steady-state inflation is likely to be positive if (ϑ/w(η^LT))⁻ᵘ ≫ 1 and wages are not too responsive to the job finding rate, i.e. when the endogenous risk wedge is sufficiently countercyclical.” The authors draw out what this means: “purely expectational liquidity traps can arise even in inflationary environments.”

An expectations-driven steady state in which firms stop posting vacancies entirely, requiring simultaneously incomplete markets, sticky prices and sufficiently countercyclical risk. The mechanism: “expectations of poor labor market conditions may trigger such an increase in desired savings (and therefore lower goods demand) that firms’ reductions in hiring sends the economy on a downward spiral towards an outcome where firms no longer want to hire because of lack of demand for their goods.” The necessary condition is that the Euler-equation schedule be steeper than the Phillips-curve schedule; the authors are honest that “sufficient conditions are harder to state but can easily be checked in applications to a given model.” Its likelihood “is higher when monetary policy reacts little to inflation and/or labor market tightness, and when hiring costs are limited,” and sufficiently aggressive policy rules out the trap “because the central bank neutralizes the impact of deteriorating labor market conditions through interest rate cuts.” The extremity of a literally zero job finding rate is acknowledged and relaxed: a variant with some frictionless hiring yields the same picture “apart from the limit point displaying a low but positive job finding rate.” The secular-stagnation comparison is where the policy sting lies. Against Hansen’s combination of slow technical progress and population aging, and Eggertsson and Mehrotra’s deleveraging, “the unemployment trap can occur in our model purely because of expectations and thus does not rely on sudden changes in population growth, technological progress or financial tightening. Furthermore, while the nominal interest rate may be low in the unemployment trap, its root cause does not derive from the ZLB on nominal interest rates. Therefore, the ongoing discussions about re-design of monetary policy to prevent secular stagnation by avoiding the ZLB may be in vain.”

Because the central bank must now offset a second self-fulfilling channel — demand stimulated by falling unemployment risk — on top of the standard nominal one. Proposition 2 gives the determinacy condition for the intended steady state under two auxiliary assumptions (risk-neutral capitalists, and a normalization of the inflation coefficient that the paper shows is inconsequential since the tightness coefficient is unrestricted); Appendix A3 relaxes the risk-neutrality assumption “and show[s] that it has no material consequences.” With inelastic real wages the condition separates cleanly into five terms and the paper reads each: stickier prices make indeterminacy more likely and flexible prices guarantee determinacy; less substitutability across goods acts symmetrically to stickier prices, and perfect competition guarantees determinacy; “a larger endogenous risk wedge unambiguously demands more aggressive monetary policy to ensure local determinacy,” and when the wedge is zero “the equilibrium is always locally determinate”; more aggressive response to tightness makes indeterminacy less likely; and larger labour adjustment costs make determinacy more likely, because “when it is costly for firms to adjust on the labor margin, they are more likely to adjust prices which neutralizes the feedback mechanism.” The intuition for the Taylor-principle failure is stated directly: “when endogenous earnings risk is countercyclical, expectations of higher inflation may be self-fulfilling even if the central bank were to stabilize the direct impact of inflation on the real interest rate since demand (and thus inflation) is also stimulated by a decline in unemployment risk. Thus, monetary policy needs to be even more aggressive to rule out expectational equilibria.” The paper distinguishes itself here from the rule-of-thumb-household literature on precisely the mechanism: “in these models there is no idiosyncratic risk and hence no precautionary saving motive. In our model, the precautionary motive, coupled with endogenous risk, is the key source behind the breakdown of the Taylor principle.”

The sticky-price term multiplies the risk wedge in the determinacy condition, so below a threshold the other frictions are literally irrelevant and above it all of them matter at once. “As long as monetary policy dominates the endogenous risk wedge, Θ^F < βδ_θ, the sticky-price wedge and the labor market wedge are irrelevant and the intended equilibrium is locally determinate. However, once the endogenous risk wedge dominates the monetary policy effect, Θ^F > βδ_θ, the three wedges all matter and market incompleteness, nominal rigidities and risk aversion become complements, making local indeterminacy increasingly likely in combination.” The paper also identifies three stabilizing channels of wage flexibility, all pushing towards determinacy: it lowers Θ^F directly, it creates a marginal cost channel because wages fall when tightness is low, and it creates a discounting channel arising from the incomplete-markets Euler-equation discount βR < 1 — the last of which “arises only under incomplete markets, but does not require job risk to be endogenous,” and which the authors link to the forward guidance puzzle literature. If wage flexibility is high enough that Θ^F < 0, “local determinacy is guaranteed as long as δ_θ ≥ 0.”

Yes, and under the ordinary Taylor principle — but the rule has no purchase on unemployment there. With the vacancy non-negativity constraint binding, the job finding rate is trivially pinned at zero and the log-linearized Euler equation reduces to 0 = δ_π Π̂_s − E_s Π̂_{s+1}, so “the steady state is locally determinate if and only if δ_π > 1… Thus, the unemployment trap is determinate under a standard Taylor rule which responds more than one-for-one to inflation.” As the introduction puts it: “Around this steady state, the monetary policy rule determines the rate of inflation, but has no grip on unemployment.”

Countercyclical risk amplifies the effect on job finding, and under an explicit condition greater price stickiness amplifies it too — the reverse of the standard model. Proposition 3 gives closed-form solutions. Higher productivity lowers marginal costs, raises vacancies and improves job finding; the response “depends positively on the endogenous risk wedge,” so countercyclical risk amplifies it and procyclical risk stabilizes it. The amplification-from-stickiness condition is ρ_A βR Θ^F > δ_θ/(1 − α) + µβ(1 − ρ_A βR)χ, i.e. “when endogenous earnings risk is sufficiently countercyclical relative to the stabilizing effects of monetary policy responses to labor market slack and wage flexibility.” The mechanism is the feedback loop again: “the costlier it is to adjust prices, the more hiring responds to variations in demand, and variations in hiring is the core of the amplification mechanism.” Both limbs of the conditional are reported — “when condition (54) is instead violated, higher price stickiness is stabilizing” — and two further stabilizers are identified: elastic real wages, which both moderate the precautionary swings and flatten the marginal cost response, and high vacancy posting costs, “because they lead firms to load more of the adjustment on prices than on hiring.”

Under the same condition, yes — and the paper offers a local projection on US data as indicative support rather than proof. In the standard model with a stationary productivity process “higher productivity implies lower marginal costs which means that inflation falls.” Here inflation can move either way, and rises exactly when the amplification condition holds, because “the demand stimulus from higher productivity is sufficiently strong.” The evidence: regressing 400 times the quarterly log change in the CPI on TFP log growth (times 100) using Fernald and Wang’s TFP series with four lags from 1980, “depending on whether one controls for movements in factor utilization or not, higher TFP either leaves inflation unchanged or gives rise to higher inflation.” The claim made from this is deliberately weak: “While the empirical results come with a fair amount of uncertainty, they do suggest that a positive inflation response is not simply an odd feature of our model.” A footnote adds that “the result holds also for the core PCE and here the positive response holds regardless of the TFP measure,” and for a sample starting in 1984.

The standard sign, amplified by countercyclical risk. Proposition 4 gives the solutions; the job finding response to a contractionary shock is negative, “as in the standard NK model,” and “this decline is amplified by the presence of countercyclical risk… provided that there is some persistence in the monetary policy shocks.” The intuition is the feedback loop run forward: “the boom in demand created by a monetary expansion reduces idiosyncratic risk, creating a further boom in demand. Hence, market incompleteness will tend to provide a more powerful role for monetary policy shocks in the case where the endogenous earnings risk is countercyclical.”

No, on the numerical evidence the paper provides — and the paper first explains why the question is genuinely open. The conjecture against the paper is stated fairly: “when the model implies amplification, the downward pressure on real interest rates in a recession might be thought to stimulate investment in real capital which, in turn, neutralizes the amplification. Going against this conjecture, however, depressed goods demand (in bad times) also means lower return on capital investment which magnifies the amplification. Hence, it is ex ante unclear.” Since closed-form solution is impossible with capital, the answer is numerical, in an extension with Cobb-Douglas production, capitalist-owned capital, risk-averse capitalists and a constant real wage: “We find that the introduction of capital accumulation preserves the amplification mechanism, or even makes it slightly stronger. In other words, it is the demand effect that dominates in our simplified setting.” Two further appendix results are flagged: with endogenous earnings risk “negative supply shocks may bring the economy to the ZLB, unlike in the standard NK model,” and the channel “may overturn the property of the standard NK model that positive supply shocks tend to be contractionary at the ZLB.” Extensions the paper declines to attempt — positive bond supply, a positive borrowing limit for the unemployed — are named as requiring “a full-blown numerical approach, which we deliberately avoided in this paper in order to gain insights from analytical formulas.”

Against Acharya and Dogra on both MPC heterogeneity and whether the risk-output link is structural; against TANK models on the absence of a precautionary motive; and against Werning on whether the wedge is derived or postulated. Acharya and Dogra achieve tractability with CARA utility and no occasionally binding constraints, and the difference matters: “in their setting, Marginal Propensities to Consume (MPCs) are the same across households, whereas in ours they differ strongly,” and MPC heterogeneity “is often emphasized as a central property which differentiates HANK models from representative-agent NK models.” The second difference is methodological: “they postulate a reduced-form relation between income risk and aggregate output, whereas we model this relation structurally, via SAM frictions.” The paper credits the overlap where it exists — “focusing on the importance of the cyclicality of income risk and equilibrium determinacy, Acharya and Dogra (2019) do confirm our results” — and states its own increment: “We go further in also considering the impact on long-run equilibria and on the amplification of shocks.” On Werning, the wedge “similarly highlight[s] such a wedge in an analytical ‘aggregated’ Euler equation, but does not model explicitly how it is determined in equilibrium.” And on two-agent New Keynesian models, the paper gives a precise equivalence rather than a blanket contrast: “When earnings risk is acyclical, either because the two forces exactly cancel out or because earnings risk is exogenous, the model’s implications for aggregate fluctuations are similar to those of two-agent New Keynesian models.”

Five results, offered as having “a more general nature” than the assumptions that produced them — a belief the paper states as such. The conclusion lists: the endogenous earnings risk wedge itself, arising from an interaction between goods and labour demand that “is missing in NK models and in HANK models without unemployment risk or other sources of endogenous idiosyncratic and uninsurable income risk”; the condition for countercyclicality; amplification of productivity and monetary shocks with complementarity between nominal rigidities and incomplete markets; possible failure of the Taylor principle; and the unemployment trap, which arises “when endogenous earnings risk is sufficiently countercyclical unless monetary policy is very aggressive.” On generality the authors are explicit about the status of the claim: “The assumptions that we made might be thought to be strong, but they enable us to generate a series of results that we believe have a more general nature,” and earlier, “It goes beyond the purpose of the paper to demonstrate robustness in a fully fledged setting but we do show how the introduction of capital accumulation and other extensions do not materially change our results.” The degenerate wealth distribution is the assumption most obviously at stake, and it is named as such rather than buried.