The Optimum Quantity of Capital and Debt
📄 Summarized from the full manuscript · Human-reviewed for faithfulness before publication
In brief
What levels of capital, public debt, and capital versus labor taxes are best in the long run when households face uninsurable income risk? Unlike work comparing steady states, this paper solves for the whole path chosen at date zero, and proves the long run is independent of where the economy starts, with capital at its efficient level. The optimal long run holds debt near 1.1 times GDP, capital taxed near 21 percent, and labor near 50 percent. That reframes Aiyagari's positive capital tax: it does not shrink over-accumulated capital, since debt absorbs precautionary saving instead, but keeps households holding the efficient amount of capital alongside that debt.
What this paper finds — and why it matters
What are the optimal long-run levels of capital and government debt, and should capital be taxed at all, in a heterogeneous-agent, incomplete-markets economy of the kind studied by Aiyagari (1995)? Most of the prior literature answers a narrower question – which steady state maximizes welfare – but this paper instead solves the full dynamic Ramsey taxation problem, in which a planner commits at date zero to an entire path of linear labor and capital taxes and government debt to maximize the discounted present value of households’ lifetime utility, and derives three main theoretical results. First, exactly as under complete markets, the long-run pre-tax return to capital equals the rate of time preference – the capital stock satisfies the modified golden rule – even though, unlike the representative-agent case, genuine distributional concerns are present throughout. Second, and in sharp contrast to representative-agent Ramsey economies (where the steady state depends on the initial government debt level, since the planner smooths tax distortions relative to whatever fiscal burden it inherits), the long-run steady-state levels of capital, debt, and both tax rates in this incomplete-markets economy are independent of initial conditions – the same long-run policy is reached no matter where the economy starts. Third, building on this independence result, the authors develop a new Lagrangian computational method – solving for the known terminal steady state analytically first, then finding the transition path of taxes and debt that connects it to the calibrated initial economy via a system of first-order conditions – avoiding the essentially unverifiable global numerical search that would otherwise be required over hundreds or thousands of variables. Quantitatively, calibrating the model to U.S. income inequality with a unit Frisch elasticity of labor supply, the optimal long-run policy features a government debt level of about 1.1 times GDP, a capital income tax around 21 percent (positive, but low relative to most developed economies), and a labor income tax around 50 percent – a pattern of high debt, low capital taxation, and high labor taxation that the paper finds is robust across a wide range of alternative calibrations of labor supply and income-risk parameters. The paper also offers a reinterpretation of Aiyagari’s (1995) original finding of a positive long-run capital tax: rather than existing to correct households’ precautionary over-accumulation of capital, the tax is positive because the planner instead uses government debt to satisfy households’ demand for extra liquidity, which brings the capital stock itself back down to its efficient (modified-golden-rule) level; the capital tax’s remaining role is only to make households willing to hold exactly the optimal quantities of both capital and government debt simultaneously. Moving from a U.S.-calibrated initial steady state to the optimal transition path yields an average lifetime welfare gain of about 2.6 percent of consumption.
Summary of a working paper (SSRN/CEPR discussion paper), AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions – this paper has circulated as a working paper without a listed final journal publication.
Questions & answers
Q1. What three classic policy questions does the paper set out to answer, and in what kind of model?
The paper studies “the optimal levels of capital and government debt? Should capital be taxed and if yes, how much? What is the optimal extent of redistribution?” within “a heterogeneous agents, incomplete markets, Aiyagari (1995) economy,” in which households face idiosyncratic income shocks but no aggregate risk, are subject to exogenous credit (borrowing) constraints, and the only assets are physical capital and government debt (Introduction, p.1). The Ramsey planner “commits itself ex-ante to a path of linear labor and capital taxes and government debt to maximize agents’ discounted present value of lifetime utility” (p.1) – i.e., the paper solves the full dynamic optimal-taxation problem, not merely the steady state that maximizes welfare taken in isolation.
Q2. What is the paper’s first main theoretical result, and how does it relate to Aiyagari (1995)?
The paper proves “that it is optimal to equalize the pre-tax return on capital and the rate of time preference in the long-run, i.e. the capital stock satisfies the modified golden rule,” a result the paper notes directly “restates Aiyagari’s (1995) result under our assumption of exogenous instead of endogenous government spending as in Aiyagari (1995), clearing up a debate in the literature on whether endogenous government spending is a necessary condition in Aiyagari (1995)” (Introduction, p.1, incl. fn.1). The authors interpret this as an incomplete-markets analogue of the production-efficiency theorem: “if markets are incomplete distributional concerns are present, but we show that these do not interfere with efficiency in investment, reminiscent of the production efficiency result in Diamond and Mirrlees (1971)” (p.2).
Q3. What is the paper’s second main theoretical result, and why does it matter for tractability?
The paper shows that “the long-run steady-state allocations and policies are independent of initial conditions. In particular, the long-run level of government debt is uniquely determined and does not depend on the initial value of debt or capital,” and likewise for steady-state capital and labor tax rates (Introduction, p.1). This is explicitly contrasted with representative-agent, complete-markets Ramsey economies (as in Lucas 1990 and Chari and Kehoe 1999), where “the steady-state Ramsey planner solution depends on initial conditions, such as the initial government debt level,” because the planner smooths tax distortions over time relative to whatever debt burden it inherits, following the logic of Barro (1979) (p.2). The independence result matters computationally: because the long-run (terminal) steady state can be characterized analytically without reference to initial conditions, the authors can treat it as a known terminal condition and solve only for the transition path connecting it to the calibrated initial economy, rather than searching jointly over the transition and the terminal point – “this is the first paper to use a Lagrangian approach to compute optimal public policies within the class of Bewley/Huggett/Aiyagari models” (Introduction, pp.3-4).
Q4. Why, economically, is the long-run steady state independent of initial conditions when markets are incomplete?
Under complete markets, lowering taxes today and issuing more debt to finance the cut carries only a cost (higher future distortions to service the extra debt), so it is optimal to keep taxes and debt roughly constant from early on; under incomplete markets, the same policy carries an additional benefit – “more debt has a welfare-enhancing element, as it improves households’ ability to smooth consumption in response to income shocks” (Introduction, p.2, citing Heathcote 2005). The planner therefore lowers taxes and raises debt for two reasons rather than one (lower current distortions plus more liquidity) while still facing only the one cost (higher future distortions), and the optimal debt level balances these forces at the margin; because the marginal liquidity benefit of debt does not depend on the initial fiscal position, the resulting long-run debt level does not either.
Q5. Does the paper confirm Aiyagari’s (1995) interpretation of why the optimal capital tax is positive?
No – the paper argues the standard interpretation is “inaccurate”: rather than the capital tax correcting households’ “over-accumulation of capital due to a precautionary savings motive,” the planner instead “issues as much debt as is necessary to enhance consumption smoothing such that capital demand satisfies the modified golden rule,” so government debt, not a distorted capital stock, is what absorbs the extra precautionary saving (Introduction, p.2). “The capital income tax rate is positive, as Aiyagari’s (1995) arguments are valid, and is such that the private sector is willing to absorb the optimal levels of both capital and government debt. Thus there is no need to implement a higher than efficient capital stock in order to achieve better consumption smoothing, simply because debt can be used instead to prevent the over-accumulation of capital” (p.2). The paper further notes that if either instrument – issuing debt, or taxing labor to adjust after-tax wages – were unavailable to the planner, “the capital stock will not satisfy the modified golden rule” (p.2, incl. fn.3), underscoring that the clean efficiency result depends on the planner having both instruments available.
Q6. What does the paper find quantitatively for the optimal long-run capital tax, labor tax, and debt level?
In the benchmark calibration – aimed at U.S. income inequality with a Frisch elasticity of labor supply equal to one – “the long-run taxes on capital and labor are around 21 and 50 percent respectively. The optimal long-run level of government debt equals 1.1 times GDP” (Introduction, p.3). The paper describes this pattern – “government debt is high, capital is taxed at a low rate, and labor income is taxed at a high rate when compared to current U.S. values” – as “robust across various different alternative calibrations,” including low and high Frisch elasticities, low and high income elasticities of labor supply, low and high income inequality, and a version with permanent income differences, “although the precise numbers do depend on the details of the calibration” (p.3).
Q7. What keeps optimal government debt from growing without bound, given that debt provides valuable liquidity?
Because the government is assumed to always honor its debt (no default premium restricts borrowing), “distortionary taxes is the only element keeping debt from becoming infinitely large and thus maximizing the liquidity services” (Introduction, p.3). The authors note the resulting debt level, while high, “is not unrealistically high as some countries (Japan) have a debt to GDP ratio as high as 2,” and attribute part of the finding to the standard Aiyagari model’s omission of channels – such as endogenous human capital accumulation (citing Wu 2021) – that would make high labor taxes more distortionary than they are in this model; “the lesson here is that within the Aiyagari model distortionary taxes do not severely limit the level of government debt” (p.3).
Q8. Does the paper’s Ramsey steady state always exist, in contrast to the “immiseration” possibility found in related work?
The paper explicitly engages this question: it adopts an assumption (Assumption 1) that the optimal solution converges to a steady state for any initial conditions, and shows this converges to positive, finite steady-state Lagrange multipliers on households’ savings decisions, in contrast to Straub and Werning (2020), who show that in a different (two-agent, capitalist-worker) model without government debt, “the optimal solution does not converge to a steady state if the intertemporal elasticity of substitution is below one,” because the relevant multiplier would have to be negative, a contradiction (Section 3, pp.14-15). The authors state that “in contrast, in our incomplete markets model, we prove the existence of positive steady-state Lagrange multipliers on households savings decisions, such that the arguments in Straub and Werning (2020) do not extend to our model,” while noting they do not attempt a full theoretical characterization of exactly when a steady state exists (as Straub and Werning do for their model), leaving that “to future research” (p.15). The paper similarly reports that an upper bound it imposes on capital taxation binds only in the very first transition period, not indefinitely – unlike a related possibility Straub and Werning identify in the Chamley (1986) representative-agent model when initial debt is close to the peak of a Laffer curve (p.15, incl. fn.15).
Q9. What does the paper find about the welfare gains from moving to the optimal policy?
Computing the transition from an initial steady state calibrated to the U.S. economy to the optimal policy path, the authors find “the welfare gain including the full transition period to the new steady state is 2.6% in consumption” (Introduction, p.3; Section on welfare, pp.24 ff.), which they describe as “a substantial welfare gain, especially because we here have 10-year time periods, and the welfare gain must be viewed as an average welfare gain over 10 [years].” The welfare gain is heterogeneous across households: the paper reports it “is decreasing in the asset level and increasing in labor [income],” so lower-wealth, more labor-income-dependent households benefit disproportionately from the transition to the optimal policy relative to wealthier households.
Key terms in this paper
Definitions below follow the paper's own usage.
- Modified golden rule (long-run capital efficiency)
- The paper's Theorem/main result (Section 3.2) that in the incomplete-markets Ramsey problem, exactly as in the complete-markets case, the long-run pre-tax return on capital equals the rate of time preference: "1 = beta(1 + F_K(K,N) - delta)" (their eq. 20), so capital is efficiently accumulated in the long run "reminiscent of the production efficiency result in Diamond and Mirrlees (1971)" even though distributional concerns (absent in the representative-agent case) are present throughout.
- Independence of the long-run steady state from initial conditions
- The paper's second main theoretical result (Section 3): the long-run steady-state levels of government debt, capital, and capital/labor tax rates chosen by the Ramsey planner do not depend on the initial capital stock, initial government debt, or initial cross-sectional wealth distribution -- in sharp contrast to representative-agent, complete-markets Ramsey problems (as in Lucas 1990; Chari and Kehoe 1999), where the steady state depends on initial debt because the planner smooths distortions relative to whatever initial fiscal burden it inherits.
- Debt, not capital taxation, absorbs the precautionary-saving motive
- The paper's reinterpretation (Section 1) of Aiyagari's (1995) positive long-run capital income tax: rather than existing "to correct households' over-accumulation of capital due to a precautionary savings motive," the tax is positive because the Ramsey planner issues exactly as much government debt as needed to satisfy households' precautionary demand for liquidity, which brings the capital stock itself back down to the modified golden rule level; the capital tax's role is then only to make households "willing to absorb the optimal levels of both capital and government debt" simultaneously, not to shrink an otherwise-excessive capital stock.
- Lagrangian transition algorithm exploiting a known terminal steady state
- The paper's core computational innovation (Section 1): rather than numerically searching for the transition path and terminal steady state of the Ramsey problem jointly -- an "unwieldy," possibly intractable high-dimensional search, since it is hard to verify a found optimum is global -- the authors exploit the independence-of-initial-conditions result to pin down the terminal (long-run) steady state analytically first, and then solve only for the transition path connecting the known initial calibration to that known terminal point via a system of first-order (Lagrangian) conditions, describing this as "the first paper to use a Lagrangian approach to compute optimal public policies within the class of Bewley/Huggett/Aiyagari models."
- Benchmark optimal long-run policy (high debt, low capital tax, high labor tax)
- The paper's benchmark numerical result (calibrated to U.S. income inequality with a Frisch labor-supply elasticity of one): the optimal long-run policy has a capital income tax around 21 percent, a labor income tax around 50 percent, and government debt equal to about 1.1 times GDP -- "high" government debt, a "relatively low" capital tax compared to most developed economies, and a "high" labor tax compared to current U.S. values -- a pattern the paper reports is robust across low- and high-Frisch-elasticity, low- and high-income-elasticity, low- and high-inequality, and permanent-income-difference variants of the calibration, moving to which from an initial U.S.-calibrated steady state yields a welfare gain of about 2.6 percent of consumption including the full transition.