The Ramsey steady-state conundrum in heterogeneous-agent economies

The conundrum is that researchers solving Ramsey optimal-taxation problems in Aiyagari-type models routinely assume an interior Ramsey steady state exists, without proving it, even though “optimal tax policies derived from the analyses depend critically on the validity of such an ’existence assumption.’” The authors open by quoting Aiyagari’s (1995) own admission that “it seems quite difficult to guarantee that a solution to the optimal tax problem converges to a steady state,” and note the problem was first flagged by Chen, Chien, and Yang (2019). Because Aiyagari’s famous conclusion that capital should be taxed in the long run rests on exactly this unverified existence assumption, the paper sets out to check it directly rather than assume it.

They prove that, under the standard power (CRRA) utility function with risk-aversion parameter sigma >= 1 – the normal empirical range – no interior Ramsey steady state exists in the standard Aiyagari model; if a Ramsey steady state exists at all, it must be non-interior, with aggregate consumption converging to zero. This is Proposition 2: “Under the parameter condition sigma >= 1, there is no interior Ramsey steady state; and the only possible Ramsey steady state (if it exists) must be non-interior with C_t to 0.” Aiyagari’s own analysis relies on a single Ramsey first-order condition (with respect to capital) and the assumption that the associated multiplier converges; the authors instead derive the additional first-order conditions with respect to individual wealth, labor supply, and consumption, and show the “interior, convergent-multiplier” assumption is inconsistent with those additional conditions (Section 4.2). For the complementary case sigma < 1, they show an interior steady state may exist, “but if it exists, it must have a divergent Ramsey multiplier and the failure of the MGR” – the modified golden rule condition Aiyagari relied on to derive a positive capital tax.

Because the market interest rate in any competitive equilibrium of the standard Aiyagari model is always below the household time discount rate (Q > beta, i.e., r < 1/beta), the planner has “a dominant incentive to front-load consumption by borrowing more cheaply in the short run as a trade-off for low consumption in the long run, because the future utility cost of debt financing is heavily discounted by a time discount factor beta that is lower than the market discount rate 1/r.” This arbitrage incentive “never disappears unless r = 1/beta,” a condition that is infeasible in the standard model because household asset demand would need to go to infinity to bid the rate up that far. So the planner keeps issuing debt and front-loading consumption, requiring an ever-rising labor tax to finance the growing debt burden, until (for sigma >= 1) the only sustainable limit is a labor tax of 100% and consumption of zero (Introduction; Section 4.2).

When sigma >= 1, households’ saving response to the anticipated future tax increase reinforces the government’s ability to issue more debt (the income effect dominates), which is exactly what lets the planner keep front-loading consumption all the way to the non-interior limit; when sigma < 1, the opposite happens. “Individuals are willing to hold more government debt in the short run in anticipation of a high labor tax rate in the long run when sigma >= 1. This facilitates the government’s strategy of front-loading consumption.” But when sigma < 1, “the substitution effect dominates the income effect,” so the expected future tax increase “will lead to a reduction rather than an increase in individuals’ current saving,” which “significantly limits the government’s ability to increase debt” – the planner’s front-loading intent persists but cannot be financed, leaving an interior steady state in which the Ramsey multiplier diverges instead (Section 5.1).

Because the non-existence result alone doesn’t reveal why or under what conditions an interior steady state might exist, the authors add an “ad hoc wealth-pooling technology” that lets individuals who have shared the same truncated (kappa-period) shock history pool their wealth, shrinking the wealth distribution’s state space and making the model analytically tractable, while the model converges back to the standard Aiyagari model as kappa goes to infinity. This modified model “allows for an interior Ramsey steady state when sigma >= 1, but it converges to the standard Aiyagari model in the limit as the risk-sharing technology in our model is gradually eliminated” – letting the authors trace out exactly how and when the interior steady state disappears as risk-sharing effectiveness declines (Section 5, drawing on the truncation method of Le Grand and Ragot (2022)).

Existence hinges on a feasibility condition, phi(1 - beta) < 1, where phi is the ratio of the minimum aggregate wealth needed for an unconstrained (full-self-insurance) allocation to aggregate consumption in that allocation; when it holds and sigma >= 1, a unique interior Ramsey steady state exists with the modified golden rule satisfied, a zero long-run capital tax, and a labor tax and debt-to-output ratio that rise with phi. Proposition 4 gives explicit formulas for the optimal labor tax tau_n and debt-to-output ratio B/Y as functions of phi and the model’s structural parameters, and states that tau_n < 1 exactly when the feasibility condition holds; when it does not hold, “the only possible Ramsey steady state is non-interior with C to 0 and tau_n to 1.” The authors show phi rises with the persistence of idiosyncratic shocks and with kappa (weaker risk-sharing), so the feasibility condition becomes harder to satisfy – and eventually fails – as the modified model approaches the standard Aiyagari model, which is exactly why the standard model has no interior steady state at all under sigma >= 1 (Section 5.1).

Because in the interior, unconstrained steady state, the sole source of allocative inefficiency in the Aiyagari model – binding individual borrowing constraints – is “fully addressed by a sufficient supply of government debt,” so there is no remaining role for capital taxation to correct: “the Ramsey planner never uses capital taxation to reduce the burden of labor taxation,” even when the required labor tax rate is close to 100%. The authors describe this as “reminiscent of the classical result in representative-agent models” where capital should not be taxed once other instruments can address the relevant distortion, and contrast it directly with Aiyagari’s conclusion: “this result is in sharp contrast to the Ramsey steady state considered by Aiyagari (1995),” where a positive capital tax was used precisely to force the modified golden rule to hold given an assumed convergent multiplier (Section 5.1, Conclusion).

Yes: as the truncation length kappa is extended from 0 to 10 in a calibrated two-state-shock version of the modified model, the optimal debt-to-output ratio and labor tax rate both rise sharply (labor tax from about 5.5% toward nearly 100%) while aggregate consumption, capital, and labor all fall toward zero – illustrating the interior steady state converging to the non-interior one as the model approaches the standard Aiyagari case. The calibration uses sigma = gamma = 2, delta = 0.1, alpha = 0.35, a low time-discount factor beta = 0.65 “to make the mechanism sharper,” and a symmetric two-state Markov employment process (Section 5.2). The authors note total tax revenue itself first rises (for kappa below about 4) and then falls back toward zero as the tax base collapses – consistent with, but distinct from, a conventional Laffer-curve mechanism, since “the government’s dominant concern is to achieve consumption equality by relaxing everyone’s borrowing constraint” rather than to maximize revenue per se.

Yes – if the Ramsey planner is assumed to maximize only the steady-state welfare of the competitive equilibrium (rather than the time-zero present value of the full dynamic path), the incentive to front-load consumption vanishes entirely, and the unconstrained allocation is no longer optimal even when it is feasible. Proposition 6 shows that under steady-state welfare maximization, “unconstrained allocation is not optimal regardless of IES [sigma], even if unconstrained allocation is feasible”; instead the planner sets consumption unequal across employed and unemployed households and leaves low-income individuals’ borrowing constraints strictly binding, because without time discounting there is no arbitrage benefit to exploiting the gap between the market rate and the discount rate (Section 6). This confirms that the paper’s central mechanism – front-loading driven by the wedge between beta and the market discount rate – is specifically a feature of dynamic, time-zero Ramsey analysis.

They conclude that results in the sizable literature – both theoretical and numerical – that assume without proof the existence of an interior Ramsey steady state “may be dubious and must be interpreted with caution,” and they name specific numerical papers whose existence assumptions their analysis calls into question. They single out Acikgoz, Hagedorn, Holter, and Wang (2018), who “directly assum[e] the existence of a unique interior Ramsey steady state… without proof” under the same power-utility specification used here, and Dyrda and Pedroni (2023), who “claim to find an interior Ramsey steady state under a risk aversion parameter sigma = 2, which we prove to be incorrect.” The authors close by proposing two possible ways positive capital taxation could still be rationalized in richer settings – an explicitly binding natural borrowing limit, or relaxing the assumption that factors are paid their marginal product – while stressing these are avenues for future work, not results established in this paper (Conclusion).