Optimal Capital Income Taxation with Incomplete Markets, Borrowing Constraints, and Constant Discounting

The paper directly challenges the policy conclusion Lucas (1990) drew from Chamley’s (1986) zero-capital-tax theorem, arguing instead that with incomplete insurance markets the optimal capital income tax is strictly positive even in the long run. “Lucas [1990] took this route and, using a representative agent model, has argued that for the U.S. economy there is a significant welfare gain from switching to this policy… eliminating the capital income tax can result in a welfare gain across steady states of over 5 percent of consumption” (p. 1). Against this, “we show that for the Bewley [1986] class of models with incomplete markets, heterogeneous agents and borrowing constraints the optimal tax rate on capital income is positive even in the long run. If we regard such models as providing a good description of reality, then we need to reassess the presumed welfare gains of reducing the capital income tax to zero. The presumed welfare gains may well turn into losses” (p. 2). The authors are explicit that their claim is stronger than “different from zero”: “the result in this paper is not just that the capital income tax rate is different from zero in the long run, but that it is always positive for the type of environment/market structure considered” (p. 2).

Each of a continuum of agents splits one unit of time between market work n_t, taxed at rate tau_n, and home work (1-n_t), which produces theta_t * H(1-n_t) of untaxable home goods, where theta_t is an idiosyncratic, Markov-persistent productivity shock; agents also hold a single asset a_t subject to a_t >= 0 (no borrowing) and save/consume to maximize expected discounted utility (Section II, “The Environment,” p. 6, and “Government,” pp. 7-8). “A crucial assumption here is that while market work can be taxed, home work cannot be taxed” (p. 7) – this asymmetry is what allows labor-income taxation to distort the market/home margin and is essential to the government’s tax-instrument problem. The household’s decision problem reduces, after substituting out the labor choice, to a standard “income fluctuation problem” in assets a_t and the shock theta_t (p. 9), with a unique stationary cross-sectional distribution of assets given any constant after-tax prices (citing Schechtman and Escudero 1977 and Clarida 1987, 1990).

The government chooses time paths for government consumption G_t, the after-tax wage w-bar_t, and the after-tax return r-bar_t to maximize discounted utilitarian social welfare (private plus public consumption utility) subject only to the economy’s resource constraint (Section II, “The Optimal Tax Problem,” p. 12, equation 2.11). “Note that in the above problem, the only constraint (aside from non-negativity constraints) is the resource constraint (2.10). The government budget constraint need not be included as an additional constraint since the individual decision rules automatically satisfy individual budget constraints, which together with the resource constraint implies the government budget constraint” (p. 12) – a standard primal-approach simplification. The capital income tax rate at any date is then recovered residually as tau_k = 1 - r-bar/r, the wedge between the after-tax and pre-tax returns (p. 13, Section III).

Yes: Theorem 1 establishes existence of a solution to the optimal tax problem, though it leaves open whether market production stays positive or whether the solution converges to a steady state. “Theorem 1. A solution to the optimal tax problem exists” (p. 17), proved by showing that a feasible policy involving zero market production and full autarky always exists, and then using continuity and compactness arguments to establish that an optimum exists relative to that lower bound (p. 17). The authors are careful to flag what the theorem does not deliver: it “does not guarantee that in the solution, market production is necessarily positive… or that, in the long run, it is bounded away from zero” (p. 17), and the paper’s subsequent results are stated under an explicit convergence assumption (Assumption 1, p. 18) that factor prices, capital, consumption, and labor all converge to strictly positive, finite limits.

Proposition 1 shows that the limiting pre-tax return on capital must equal the rate of time preference, r_t -> rho = (1-beta)/beta (p. 18), proved from the planner’s first-order conditions on capital and government consumption (equations 4.1a-b) together with the fact that the shadow value of the resource constraint, lambda_t, converges to a strictly positive, finite limit. The intuition given is a variational argument: “imagine that the government increases investment at date t by one unit… and decreases government consumption by one unit… As a consequence, the resource constraint and the government budget constraint continue to be satisfied at date t… Further, individuals are unaffected by these changes” (pp. 18-19); reversing the extra unit of capital into higher government consumption one period later, appropriately discounted, and requiring the marginal costs and benefits to balance at an optimum delivers exactly the modified-golden-rule condition on the pre-tax return.

With incomplete markets, aggregate asset demand A(r-bar) rises without bound as r-bar approaches rho from below, while the economy’s total asset supply (capital plus government debt) is bounded above – so no equilibrium can sustain r-bar arbitrarily close to rho, and the equilibrium after-tax return must always be strictly less than rho. “More crucially, asset demand A(.) tends to infinity as r-bar tends to rho from below. The intuition is that when r-bar equals rho, the individual would like to maintain a smooth marginal utility of consumption profile. However, since there is some probability of receiving a sufficiently long string of bad theta’s, the only way to maintain a smooth marginal utility of consumption profile is to have infinite assets” (p. 15). Because capital is bounded by a maximum sustainable stock and government debt is bounded by bounded tax revenue, “it is not possible to support as an equilibrium an interest rate that is arbitrarily close to the time preference rate” (p. 4), so “with incomplete markets the steady state equilibrium value of r-bar is always less than rho… regardless of the values of r, tau_n or G” (p. 16).

Since the tax rate is tau_k = 1 - r-bar/r, Proposition 1 (r -> rho) together with Proposition 2 (r-bar < rho) directly implies a strictly positive limiting capital income tax rate, while Proposition 3 shows this whole apparatus collapses to Chamley’s zero-tax result once idiosyncratic risk is switched off.* “Proposition 1 says that in the long run the pre-tax return to capital must equal the rate of time preference. Therefore, to show that the capital income tax is strictly positive even in the long run we need to show that r-bar* … < rho. This is shown in Appendix A (part 2)… The proof is by contradiction, i.e., we rule out r-bar* >= rho by showing that per capita assets go to infinity” (p. 19). Proposition 3 then shows that “under complete markets the capital income tax rate is zero,” because eliminating idiosyncratic risk (setting theta_t = E(theta) for all agents) makes “the model in section II… a special case of that in Chamley [1986],” so the representative-agent Euler equation gives r-bar* = rho exactly and the tax rate is zero (p. 20).

The model period is one year, with discount factor beta = 0.96, CRRA utility with risk aversion mu in {3, 5}, a Cobb-Douglas production function with capital share alpha = 0.36 and depreciation delta = 0.08, and a labor supply function with elasticity lambda in {2, 1.5, 1} calibrated so that per capita market work equals 1/3 when the labor tax is 0.35, matching Lucas’s (1990) estimate that both the U.S. labor and capital income tax rates are about 0.36 (Section V, “Model Specification and Parameterization,” pp. 22-23). The idiosyncratic labor-endowment shock follows an AR(1) in logs approximated by a seven-state Markov chain, with coefficient of variation sigma_e in {0.2, 0.4} and serial correlation rho_e in {0, 0.6}, values chosen to match PSID/NLS-based estimates of earnings and hours variability from Kydland (1984), Abowd and Card (1987, 1989), and Heaton and Lucas (1992) (Section V, pp. 23-25); government consumption is set to 20% of gross market output at the benchmark labor tax rate (p. 26).

Rather than solving the full dynamic Ramsey problem from specific initial conditions (which the authors describe as computationally very hard), the paper constructs a “locus” of steady-state tax-rate pairs (tau_n, tau_k) that are consistent with the modified golden rule – the necessary long-run condition from Propositions 1-2 – and checks whether the observed U.S. pair lies on or near that locus. “If the observed labor and capital income tax rates are quite far from the locus then one can conclude that the actual tax policy is quite far from being long run optimal… Therefore, a minimum condition for not dismissing the observed tax rates as being long run optimal is that they lie on the locus. Of course, the observed tax rates being on the locus does not imply that actual tax policy is long run optimal” (p. 21) – resolving that stronger question would require solving the full dynamic path, which “is very hard” because the cross-sectional asset distribution is an infinite-dimensional state variable (p. 22, fn. 19).

With low risk aversion, elastic labor supply, and low, i.i.d. earnings variability (Table 2), the model’s long-run optimal capital tax rates are close to zero (0.4-0.7%) even as the labor tax varies from 30% to 40% – but with higher risk aversion, less elastic labor supply, and more variable, persistent earnings (Tables 3 and 4), optimal capital tax rates rise to 25-32% and then to 44-45%, bracketing the roughly 36% figure Lucas (1990) used for the U.S. economy. “In Table 2, labor supply is fairly elastic, the idiosyncratic shock has relatively low variability and is i.i.d. over time. The capital income tax rates are quite close to zero suggesting that the results of Chamley [1986] and Lucas [1990] would continue to hold (approximately) for this economy… However, Tables 3 and 4 below show that modest changes in parameter values are sufficient to generate quite large values for the capital income tax rate; values that cluster around the actual value from the data… One could even conclude that the observed tax rate is too low rather than too high” (pp. 26-27).

Higher risk aversion, more persistent earnings shocks, and less elastic labor supply each independently raise desired precautionary asset holdings, and therefore raise the capital tax needed to hold the capital stock at the modified-golden-rule level – with the labor supply elasticity mattering the most. “A higher risk aversion coefficient leads individuals to desire to accumulate a larger quantity of assets requiring a higher tax on capital income… a high persistence in earnings implies a much larger variability in the consumer’s permanent income which is the relevant measure for precautionary saving… A lower elasticity of labor supply makes total (market plus non-market) earnings more variable also leading to larger desired asset holdings… The influence of the labor supply elasticity is quite strong. For example, if the labor supply elasticity is reduced from 1.5 to 1 with the other parameter values as in Table 3, the capital income tax rates rise from around 28 percent to around 58 percent” (p. 28). The authors also note these quantitative patterns are consistent with their own earlier finding (Aiyagari 1992) that high risk aversion, earnings variability, and persistence substantially raise aggregate precautionary saving and depress the equilibrium return on capital even with no taxes at all (p. 28).

The paper’s headline claim is a reassessment, not a precise recommendation: the observed U.S. capital income tax rate cannot be dismissed as far from long-run optimal, and under some reasonable parameterizations it may even be too low rather than too high, directly undercutting the Chamley-Lucas case for zero capital taxation. The quantitative exercise is explicitly framed as bounding plausibility rather than pinpointing an optimum: “the goal of this section is to suggest that the observed labor and capital income tax rates in the U.S. economy cannot be easily dismissed as being far from optimal relative to a reasonably parameterized version of the model” (p. 20). The theoretical results themselves rest on an explicit maintained assumption – that the solution to the government’s dynamic optimal-tax problem actually converges to a steady state with strictly positive, finite limiting prices and quantities (Assumption 1, p. 18) – which the authors acknowledge is “quite difficult to guarantee” in general and is, in this class of models, also assumed rather than proved by Chamley (1986) and Lucas (1990) themselves (p. 17, fn. 16).