Optimal Monetary Policy with Heterogeneous Agents: Discretion, Commitment, and Timeless Policy

The paper systematically revisits the canonical New Keynesian consensus on optimal monetary policy (Clarida, Galí, and Gertler, 1999; Woodford, 2003; Galí, 2015) inside a heterogeneous-agent economy, studying policy under discretion, then under commitment, then for stabilization, exactly paralleling that earlier literature’s structure. The model is “a canonical one-asset heterogeneous-agent New Keynesian (HANK) economy with wage rigidity, which represents a minimal departure from the representative-agent New Keynesian (RANK) model” (Introduction). Households face idiosyncratic labor-productivity risk and trade a single bond subject to a borrowing constraint; labor unions set nominal wages subject to Rotemberg adjustment costs, giving a forward-looking wage Phillips curve; and the only policy instrument is the path of the nominal interest rate, which the paper studies both non-linearly and via sequence-space perturbation methods extended to Ramsey (optimal-policy) problems (Introduction; Section 2).

Proposition 2’s targeting rule shows the planner trades off an “aggregate labor wedge” (the usual stabilization objective) against a “distributive pecuniary effect” term equal to the cross-sectional covariance between household wealth and marginal utility of consumption, which is always negative since marginal utility falls with wealth. Formally, the discretionary targeting rule sets the aggregate labor wedge equal to Omega^D_t times the covariance term, and “since the marginal utility of consumption falls with household wealth,” this covariance is strictly negative, so “optimal monetary policy under discretion targets a negative aggregate labor wedge, which is associated with an overheated economy” (Section 3, equation 25 and surrounding discussion). In the isoelastic-preferences case, this is restated as an output-gap rule with two multiplicative wedges: the familiar “desired markup” wedge from monopolistic competition, and a new “desired redistribution” wedge strictly greater than one that further motivates the planner to overheat the economy (equation 26).

Because in the Markov-perfect equilibrium the public rationally anticipates the planner’s incentive to raise output above its natural level, so the attempt to stimulate becomes futile and manifests instead as elevated steady-state inflation – a novel, quantitatively dominant source of Barro-Gordon inflationary bias. Proposition 3 derives the stationary inflationary bias as the sum of a standard “Markup” term (from inefficiently low employment) and a “Redistribution” term (proportional to the wealth-marginal-utility covariance); the paper reports that “quantitatively, the contribution of the novel redistribution motive is over 4 times larger than that of markups in our calibration exercise” (Section 3, Proposition 3). Crucially, the paper notes the markup-correcting employment subsidy that eliminates inflationary bias in RANK “is no longer sufficient to address inflationary bias in HANK” (Section 3, concluding discussion), since it does nothing about the redistribution channel.

Yes: Proposition 5 shows that the stationary Ramsey plan under commitment features exactly zero long-run inflation in both HANK and RANK, fully eliminating the long-run component of discretionary inflationary bias. The logic is that in any stationary competitive equilibrium the real interest rate and real allocation are pinned down by real forces, so the planner’s only remaining choice is how to split a given real rate between the nominal rate and inflation; “since maintaining non-zero inflation is costly due to nominal rigidities while adjusting the nominal rate is not, the planner finds it optimal to exclusively use the nominal interest rate in the stationary Ramsey plan while promising to keep inflation at zero” (Section 4.2). The authors caveat that this zero-inflation benchmark could change if inflation and the nominal rate had asymmetric effects across households or under other frictions not modeled here.

Even a planner who sets policy with commitment “from time 0 onwards” still behaves time-inconsistently exactly at time 0, because the standard Ramsey problem’s optimality conditions force the initial multipliers on every forward-looking constraint to equal zero – a condition inconsistent with the non-zero multipliers a stationary Ramsey plan generically features. Concretely: “the optimality conditions for the standard Ramsey problem require the initial conditions theta_0 = 0 and phi_0(a,z) = 0 for all (a,z) because initial inflation pi^w_0 and lifetime value V_0(a,z) respectively are free. But any stationary Ramsey plan will generically feature theta_ss =/= 0 and phi_ss(a,z) =/= 0” (Section 4.3). In HANK this time-0 problem has two distinct sources – the familiar forward-looking Phillips curve (as in RANK, following Barro and Gordon 1983) and, newly, the forward-looking individual Bellman equations that appear as planning constraints once households are heterogeneous – so even initializing the economy exactly at its own stationary Ramsey allocation, the planner still finds it optimal to deviate and generate inflation at time 0.

The authors extend Marcet and Marimon’s (2019) recursive-multiplier approach – and Woodford’s (1999) timeless perspective – to continuous-time heterogeneous-agent economies by adding time-0 “timeless penalties” for each forward-looking constraint, defining a timeless Ramsey problem that Proposition 6 proves is time-consistent at the stationary Ramsey plan: “the planner has no incentive to deviate from the stationary Ramsey plan in the absence of shocks.” Under this timeless plan, inflation is set to zero both in the long run (as under standard commitment) and, now, also in the short run at time 0, whereas the standard Ramsey plan and discretionary policy both still generate inflation on impact (Section 4.3, discussion of Figure 1). The authors are candid that “from a time-0 perspective, the standard Ramsey plan attains higher welfare than the timeless Ramsey plan,” but argue the timeless benchmark is nonetheless valuable because it avoids Woodford’s concern that repeatedly re-solving the standard Ramsey problem each time new information arrives is an impractical guide to policy, because it isolates the pure stabilization motive from time-0 considerations, and because the paper’s sequence-space perturbation methods are only valid approximations around the timeless plan.

The timeless inflation penalty (Proposition 7) generalizes the familiar RANK inflation penalty but is now additionally shaped by distributional considerations even under the correct employment subsidy, while the new timeless distributional penalty (Proposition 8) specifically penalizes the welfare gains that indebted, high-marginal-utility households would otherwise realize from discretionary redistribution. On the inflation penalty: with the appropriate employment subsidy in RANK, “no inflation penalty is required because no time consistency problem emerges,” but in HANK “the time consistency problem on inflation does not disappear” even with that subsidy, because changes in aggregate activity have distributional consequences that the inflation penalty must also account for (Section 4.4). On the distributional penalty: because in RANK “the nominal interest rate is sufficient to fully correct the representative household’s consumption-savings decision,” no separate penalty is needed there, but in HANK “the Ramsey planner finds that households privately consume too much or too little,” so the planner values making forward-looking promises about lifetime values – promises the paper shows must satisfy a novel “promise-keeping Kolmogorov forward equation” that tracks how these implicit commitments move with households as they transition across wealth and income states (Proposition 8).

No – the paper proves that optimal stabilization policy under the timeless Ramsey plan always features inflation “overshooting” in both RANK and HANK: whatever the sign of inflation on impact of a shock, it must later swing to the opposite sign so that the time-integral of inflation following any shock is exactly zero. This follows directly from the boundary conditions on the inflation penalty (theta_0 = lim_{T to infinity} theta_T = theta_ss) combined with its law of motion, which together imply integral_0^infinity pi^w_t dt = 0 “in response to any shock. That is, if inflation is positive on impact in response to a shock, it must turn negative at some point in the future, and vice versa” (Section 4.6, equation 50).

Yes: even with the correct employment subsidy and no cost-push shocks, a HANK planner generally finds it suboptimal to close both the inflation gap and the output gap simultaneously, because doing so is feasible but the planner instead chooses to trade off aggregate stabilization against distributional considerations – so “Divine Coincidence consequently fails even with the appropriate employment subsidy and in the absence of cost-push shocks.” In RANK, with the right subsidy and demand/TFP shocks, the targeting rule collapses to a zero aggregate labor wedge at all times – the standard Divine Coincidence benchmark of Blanchard and Galí (2007) – and only cost-push shocks break it. In HANK the same targeting rule (Proposition 9) contains an extra redistribution term (proportional to the wealth-marginal-utility covariance) plus inflation and distributional penalty terms, so “the aggregate labor wedge that makes equation (49) hold in response to a shock need not be zero” even absent cost-push shocks (Section 4.6).

In the calibrated model, the timeless Ramsey planner still leans against a demand (discount-rate) shock in HANK, but less forcefully than in RANK: the on-impact output-gap response is only dampened by about 50% relative to a Taylor-rule benchmark, inflation is nearly fully stabilized, and the optimal interest-rate path is hump-shaped rather than immediate, reflecting the value of promised overshooting under commitment. In RANK, the planner “raises the interest rate by about 50 basis points to lean against the 50 basis point discount rate shock” and achieves the full Divine Coincidence outcome; in HANK, “the on-impact output gap response under optimal policy is only dampened by 50% relative to the Taylor rule case,” while “the inflation gap…is stabilized almost entirely” (Section 5.2). A decomposition traces the departure from Divine Coincidence specifically to the redistribution wedge rather than to the (unchanged) markup wedge or to the timeless penalty terms, which “contribute little to the optimal on-impact response of the output gap” (Section 5.2).