Capital Income Taxation and Self-Fulfilling Aggregate Instability

Q1. What is the fiscal increasing returns mechanism that overturns the Schmitt-Grohé-Uribe result?

When the government adjusts the capital income tax rate to balance the budget, higher labor input raises output and the capital tax base, allowing a lower tax rate; under endogenous capital utilization, this triggers an additional channel in which a lower after-tax depreciation cost induces firms to utilize capital more intensively, further raising the effective capital stock and output — generating fiscal increasing returns to scale and a factor share redistribution from capital to labor that together make indeterminacy possible. In log-linearized terms, the effective output-labor elasticity in the equilibrium aggregate production function exceeds unity for tax rates in the interval (τ̄, τ̂) where τ̄ = ρ/(ρ+δ) and τ̂ is the Laffer-curve peak, and this greater-than-unity elasticity is the formal condition for indeterminacy (Corollary 1). With a constant utilization rate as assumed in prior work, both the factor share redistribution and fiscal increasing returns effects vanish, the effective output-labor elasticity falls below unity, and indeterminacy becomes impossible — confirming that endogenous capital utilization is the essential ingredient.

Q2. What are the formal conditions for indeterminacy under the baseline capital tax rule?

Proposition 1 establishes that the fiscal policy with capital income taxation induces indeterminacy of equilibrium if and only if the long-run capital income tax rate τk lies strictly in the open interval (τ̄, τ̂), where τ̄ = ρ/(ρ+δ) and τ̂ is the unique Laffer-curve peak. Under the standard calibration (ρ = 0.04, δ = 0.1, α = 0.3), this interval is (0.286, 0.717) — a wide range covering the effective capital income tax rates of the U.S., UK, and Japan. The determinant of the Jacobian of the linearized dynamic system is positive and the trace is negative over this interval, implying that both eigenvalues are negative, which is the condition for indeterminacy with one predetermined variable (capital) and one jump variable (marginal utility of income).

Q3. How do capital depreciation allowances serve as a stabilization device?

When the taxable capital income base is reduced by a fraction γ ∈ [0, 1] of depreciation expenses, the effective degree of fiscal increasing returns to scale decreases strictly with γ, and the lower bound of the indeterminacy interval τ̄D strictly rises with γ; with full depreciation allowances (γ = 1), the quadratic equation characterizing the lower bound has a repeated unit root, the indeterminacy interval becomes empty, and multiplicity of equilibria is entirely impossible regardless of the capital income tax rate. Corollary 2 formalizes this result analytically. The intuition is that depreciation allowances reduce the procyclicality of the effective tax burden on capital, so the after-tax return to capital responds less strongly to activity, weakening the self-fulfilling loop. Partial allowances — even well below γ = 1 — can sufficiently shrink the indeterminacy region to require implausibly high tax rates for instability.

Q4. What is the role of public debt and what new result does the model deliver?

Contrary to the established view that public debt can serve as an automatic stabilizer that exempts balanced-budget fiscal policy from beliefs-driven instability (Schmitt-Grohé and Uribe 1997, Huang et al. 2018), this paper shows that public debt can be destabilizing when capital income taxes adjust to balance the budget: a higher public debt-to-GDP ratio expands the indeterminacy region, and this destabilizing effect is amplified when capital depreciation allowances are low. Figure 5 in the paper illustrates numerically that raising the public debt-to-GDP ratio from an average of 0.975 (US/UK average) to 1.429 (Japan) dramatically widens the indeterminacy region, particularly at low depreciation allowance rates. This novel result — that public debt destabilizes rather than stabilizes under capital income tax adjustment — constitutes a third main contribution of the paper alongside the indeterminacy result and the stabilization role of depreciation allowances.

Q5. What are the quantitative results for the US, UK, and Japan?

Under the calibrated depreciation allowance rate of approximately 0.2 (the GDP-weighted European average from D’Erasmo et al. 2017, also consistent with the US), all three large economies lie within the indeterminacy region at their current effective income tax rates; stabilization requires either raising the depreciation allowance rate to 0.76 for all three, or reducing income tax rates by 47% for the US, 52% for the UK, and 39% for Japan from their calibrated levels while holding depreciation allowances at 0.2. Less dramatic combination policies also work: for the US, a 10% income tax cut combined with raising the depreciation allowance to 0.67 would suffice, as would a 5% tax cut combined with raising the allowance to 0.70. These calculations are calibrated to effective factor income tax rates from Mendoza et al. (1994) updated to 1996 and public debt-to-GDP ratios from OECD Economic Outlook (2014).

Q6. How does the paper relate to and contribute to the broader indeterminacy literature?

The paper’s mechanism — fiscal increasing returns arising from the interaction of optimal capital utilization and capital income taxation — is novel relative to both strands of the indeterminacy literature: unlike Benhabib-Farmer-style models that require the aggregate production function to have increasing returns as a primitive assumption, and unlike the Schmitt-Grohé-Uribe labor-tax indeterminacy that also does not require increasing returns but found capital taxation immune, this paper shows that increasing returns can emerge endogenously from a constant-returns-to-scale production technology via fiscal policy, requiring no externalities or other non-standard features. The mechanism provides a policy-based micro-foundation for aggregate increasing returns that resolves the empirical criticism of the prior indeterminacy literature; it also distinguishes the result from Huang et al. (2018), who showed that endogenous capital utilization under labor income tax adjustment raises indeterminacy likelihood but leaves the production function at constant returns to scale.

Key Concepts

fiscal increasing returns : the mechanism in this paper whereby higher economic activity lowers the capital income tax rate (via a higher tax base), raises the after-tax return to capital, and induces greater capital utilization and further output expansion; operationally defined by the effective output-labor elasticity exceeding unity in the equilibrium aggregate production function.

equilibrium indeterminacy : the existence of multiple rational-expectations equilibria converging to the same steady state, arising from the fiscal increasing returns mechanism and permitting self-fulfilling sunspots fluctuations unrelated to economic fundamentals; characterized by both eigenvalues of the Jacobian being negative (both predetermined structure of the dynamic system).

capital depreciation allowance : the fraction γ ∈ [0, 1] of capital depreciation costs deductible from the taxable capital income base; the stabilization device the paper identifies, which works by attenuating the procyclical component of the effective capital tax burden and thereby reducing the fiscal increasing returns to scale.

factor share redistribution : in this paper, the shift of the effective factor income share from capital to labor that results from endogenous capital utilization interacting with the capital tax rule; contributes to indeterminacy by raising the effective output-labor elasticity above the share of capital in the production function.

Summary of a forthcoming paper, AI-assisted. Draft pending human review. See the linked original for the authoritative claims and full conditions.