Monetary Policy with Heterogeneous Agents: Insights from TANK models

Whether a simple two-agent model can stand in for a full heterogeneous-agent model when the object of interest is aggregate behaviour. The authors state the purpose as assessing “the merits of HANK models for our understanding of an economy’s aggregate behavior, relative to a simpler alternative that assumes the existence of two types of consumers.” The motivation is practical as much as theoretical: solving HANK models “requires the use of nontrivial computational techniques, given the need to keep track of the distribution of wealth, and the hurdles arising from the presence of occasionally binding borrowing constraints,” and that reliance on numerical methods “often presents a challenge when it comes to understanding the mechanisms underlying some of the findings, and may thus limit their use in the classroom or as an input in policy institutions.”

The gap in average consumption between constrained and unconstrained households, and the dispersion of consumption within the unconstrained group. Working from a framework related to Werning (2015), the authors show these two dimensions “explain the differential behavior of a HANK economy relative to its RANK counterpart,” and that “these two dimensions of heterogeneity are captured in a simple way by two ‘wedges’ that appear in an Euler equation for aggregate consumption, and whose behavior can be traced in response to any aggregate shock, allowing us to assess their quantitative significance.” HANK models generally account for both; TANK models capture only the first.

In three ways. First, the fraction of constrained agents is fixed in TANK, whereas in HANK it “is endogenous and may vary over time, as a result of the interaction of aggregate shocks and the distribution and composition of wealth at any point in time.” Second, TANK “assume[s] away the impact on agents’ current decisions of the likelihood of being financially constrained in the future,” which in HANK is a source of time-varying precautionary saving. Third, TANK needs no tracking of the wealth distribution at all — its equilibrium conditions “can be reduced to a system of difference equations isomorphic to of the standard NK model.”

That TANK approximates the aggregate dynamics of a canonical HANK model well, both qualitatively and quantitatively, for monetary and non-monetary shocks. The reason is specific and empirical rather than assumed. On one hand, “for standard calibrations of a HANK model, consumption heterogeneity between constrained and unconstrained households fluctuates significantly in response to aggregate shocks, and its fluctuations are well captured by a TANK model.” On the other, “consumption heterogeneity within the subset of unconstrained households remains roughly constant, since those agents are able to limit consumption fluctuations by borrowing and saving.” So “ignoring this second form of heterogeneity –as in a TANK model– is largely inconsequential for determening the behaviour of aggregate variables.”

As a standard Bewley-Aiyagari-Huggett economy with the same New Keynesian supply side as the TANK model. Idiosyncratic income follows an AR(1) with persistence 0.966 and standard deviation 0.017, following McKay et al. (2016) and Auclert (2016). The only asset is a riskless one-period nominal bond in zero net supply, and each agent faces an exogenous borrowing limit equal to a fraction of steady-state output. In the baseline that limit is 100 percent of output, which “implies that 21% of the agents are borrowing constrained in steady state”; tighter (50 percent) and looser (200 percent) limits imply constrained fractions of 36 percent and 11 percent respectively.

Table 1 reports the impact real-interest-rate elasticity of output with the RANK response normalised to one, and TANK tracks HANK closely across most specifications. With constant transfers, “the response of output to a change in the real interest rate in HANK is about 70% larger than in a RANK economy,” while “the output response is 64% larger than in RANK” in TANK (1.70 against 1.64). Under the other transfer regimes the two stay close — 1.23 against 1.24, and 0.97 against 1.00. Tightening the borrowing limit raises both dramatically and together (HANK 5.30, TANK 5.74, implied constrained share 0.36). The widest gap between the two is under the loosest borrowing limit, 1.23 against 1.46, with an implied constrained share of 0.11 — which the authors nonetheless include in their claim that TANK is a “good approximation to the richer HANK model” under alternative borrowing-limit parameters.

Not as a general matter — the paper is explicit that the sign is conditional. It shows analytically “that households’ heterogeneity may amplify or dampen the effects of aggregate shocks depending on the size of constrained agents and the cyclicality of fiscal transfers, among other factors, but independently of the magnitude of nominal rigidities, or the way monetary policy is conducted.” The dependence on fiscal transfers is not incidental: transfer policy is among “the main drivers of the differences between HANK and RANK models,” which is why the same economy can show an elasticity of 1.70 or 0.97 relative to RANK depending only on how transfers respond.

It adds a third term to the central bank’s loss function and breaks the divine coincidence. For a utilitarian central bank — one “assigning equal weight to the utility of all the agents” — a second-order approximation to welfare around an efficient steady state with no inequality yields a quadratic loss in inflation, the output gap, and a consumption-heterogeneity index. “The only difference with respect to the loss function of a standard RANK model is the presence of the term” in that index, which “indicates that fluctuations in consumption heterogeneity generate welfare losses for the central bank.” Intuitively, “a benevolent central bank would like to spread to costs of fluctuations equally across all the agents, which implies no fluctuations in consumption heterogeneity.”

Because the heterogeneity index responds to the natural level of output, not only to the output gap. The constraint linking them means that “as long as” the coefficient on natural output is non-zero, “fluctuations in the natural level of output leads to fluctuations in the heterogeneity index.” Consequently “a central bank that wishes to stabilize output at its natural level would have to tolerate some fluctuations in heterogeneity,” and conversely. In the authors’ words, a trade-off emerges “whenever fluctuations in (the natural level of) output are not proportionally distributed across households” — at which point “it is not possible to simultaneously stabilize inflation, the output-gap and the heterogeneity index —i.e. the ‘divine coincidence’ does not hold.”

Small, and the paper says so plainly rather than resting on the qualitative result. “However, we find that for standard calibrations of the TANK model, the optimal policy still implies minimimal fluctuations in inflation and the output-gap, and is thus nearly identical to the one that would prevail in a RANK model.” The novelty is therefore the existence and structure of the trade-off, not a quantitatively different policy prescription.

The comparison is confined to aggregates, and the manuscript is explicitly unfinished. Section 4 restricts attention “to the responses of aggregate variables to aggregate shocks,” so nothing here claims TANK reproduces HANK’s distributional or welfare detail — only that the aggregate wedges are well approximated. The robustness claim is likewise bounded: the finding is “robust to alternative calibrations of the borrowing limits, and specifications of the fiscal transfers,” which are the dimensions varied in Table 1. Table 2 adds a further check the record should credit: under an alternative transfer policy with differing tax progressivity, “the same results also hold”. The title page carries the note Preliminary and Incomplete.