Optimal Fiscal Policy with Heterogeneous Agents: Le Grand & Ragot (2025)

What this is. The paper’s core results, the structural model it builds on, and the solution method: enough to know what it found and how, without reading all 50 pages. To replicate or extend, read the full source at doi.org/10.1086/734877 or the open preprint at hal.science/hal-05547657.

TL;DR

Le Grand and Ragot (2025) analyze optimal fiscal policy in a Bewley-Huggett-Aiyagari heterogeneous-agent model with capital accumulation, progressive labor taxation, a linear capital tax, and public debt. The government finances exogenous public spending via taxes and new debt. Three contributions:

First, in a simple analytical model, the steady-state optimal capital tax can be positive when credit constraints bind and non-CRRA utility generates a savings externality on post-tax factor prices, as with GHH or Decreasing RRA (DRRA) preferences. With CRRA utility, the Chamley (1986) and Judd (1985) zero-capital-tax result generalizes exactly (Corollary 1).

Second, a Stationary Ramsey Equilibrium (SRE) with positive capital and labor taxes requires three independent conditions: a non-first-best condition, the Straub and Werning (2020) stationarity condition, and a Laffer condition. Positive public debt additionally requires the savings-motive and public-spending threshold conditions in Result 1.

Third, in the quantitative model, for a given net present value (NPV) of a public spending shock, optimal public debt rises when shock persistence is low (the government borrows to smooth taxes) and falls when persistence is high (borrowing would require costly future taxes). The quantitative model, calibrated to the US via an inverse optimal approach, confirms these results.

Core results

Magnitudes are as reported; all results are from the source PDF.

#	Result	Locator	Magnitude
R1	Positive optimal capital tax when credit constraints bind for unemployed agents (GHH/DRRA utility)	Proposition 1 (eq. 29, p. 15); GHH case eqs. (33)-(34), p. 19; tractable example p. 24	Simple GHH example: $\tau^K = 6\%$, $\tau^L = 3\%$, $B > 0$ (parameters: $\alpha=0.3, \beta=0.7, \varphi=0.3, \delta=1, G=0.01$)
R2	Zero capital tax with CRRA separable utility, even with binding credit constraints	Corollary 1, p. 18	$\tau^K = 0$ for $U(c,l) = u(c) - v(l)$ with CRRA $u$; generalizes Chamley-Judd to incomplete markets with occasionally binding constraints
R3	Positive public debt is optimal when savings motive dominates public spending needs	Result 1 (eq. 39), p. 23	$B \geq 0$ iff $\bar{g}_1 \leq 0$ and $G \leq \bar{g}_{\text{pos}} Y_{FB}$; tractable example: $B > 0$ with $\alpha=0.3, \beta=0.7$
R4	Optimal public debt response to spending shock decreases in shock persistence at fixed NPV when impact debt is positive	Proposition 5 (p. 26); Figure 1 (panel 5, p. 53; discussion pp. 40-41)	$\partial \hat{B}_0/\partial \rho_G < 0$ at fixed shock size; at fixed NPV this result assumes $\hat{B}_0>0$; quantitatively debt rises for $\rho_G=0.1$ (1% of GDP shock) and falls for $\rho_G=0.99$ (0.02% of GDP shock)
R5	Capital tax rises at impact after a public spending shock (both persistence levels)	Figure 1 (panel 4, p. 53; discussion pp. 40-41)	Change in $\tau^K$ at impact is an order of magnitude larger than the change in the labor tax level; capital tax increases for both high- and low-persistence shocks
R6	Labor tax progressivity rises and labor tax level falls at impact after a spending shock	Figure 1 (panels 2-3, p. 53; discussion pp. 40-41)	Progressivity $\tau$ increases and labor tax level parameter $\kappa$ increases, which means the labor tax falls; both changes are much smaller than the capital tax response; public debt path differs markedly by persistence level (panel 5)
R7	A stationary Ramsey equilibrium with positive capital and labor taxes requires three independent conditions	Propositions 2-3, equations (35)-(38), text pp. 20-22	Non-first-best condition, Straub-Werning stationarity condition, and Laffer condition must all hold; positive public debt additionally requires the conditions in Result 1
R8	The linearized optimal capital path is stable only below a parameter threshold	Proposition 4, equation (42), text p. 25	$$
R9	Ex-ante social weights change the optimal capital-tax/labor-tax mix	Proposition 6, equations (46)-(47), text pp. 27-28	$(1-\beta)\frac{\tau^K}{\omega^A}=\frac{\omega^B}{\omega^A}-(1+\beta)\Lambda+\varphi(1+\beta)(1+\Lambda)\frac{\tau^L}{1-\tau^L}$; a higher weight on type B raises the capital tax relative to the labor tax
R10	Incomplete-market allocations have more persistent aggregate responses and larger welfare losses than first best	Figure 2, p. 54; text pp. 42-43	Following the same spending shock, output, capital, labor, and consumption are more volatile and persistent under incomplete markets; the equivalent-consumption welfare decline is larger, especially for the high-persistence shock
R11	Debt-response shape changes across four public-spending persistence levels	Figure 3, p. 55; text pp. 43-44	At equal spending-shock NPV, impact debt response decreases with persistence; $\rho_G=0.1$ gives an initial rise then monotone decline, $\rho_G=0.8$ an inverted U, and $\rho_G=0.95$ a J-shaped path
R12	Debt and tax responses persist under an alternative affine tax system	Section 5.4.1, text p. 44; Appendix A.9	Debt rises for a low-persistence shock and falls for a high-persistence shock; capital tax and progressivity rise at impact
R13	The fiscal-dynamics result persists under productivity-dependent welfare weights	Section 5.4.2, text pp. 44-45; Appendix A.10	Results are reported as qualitatively similar to the benchmark; the alternative weights apply to current productivity and instantaneous utility
R14	TFP shocks produce the same direction of debt response as spending shocks	Section 5.4.3, text p. 45; Appendix A.11	Debt rises on impact for a low-persistence TFP shock and falls when TFP-shock persistence is high, holding the cumulative TFP decline fixed
R15	Discount-factor shocks produce a distinct debt response	Section 5.4.3, text pp. 45-46; Appendix A.11	Greater patience raises capital and reduces debt; with very high shock persistence, debt can rise slightly on impact
R16	The general model links positive capital taxation to binding credit constraints	Equation (64), text p. 35	$\tau^K=\frac{\sum_f m^f\int_i\nu_i^f\ell^f(di)}{(1-\beta)\sum_f m^f\int_i u'(x_i^f)\ell^f(di)}$; $\tau^K>0$ when a positive mass of agents face binding constraints at the steady state

Overall (paper’s conclusion). The key friction for positive optimal capital taxation is an occasionally binding credit constraint: it introduces a price externality of savings that the planner corrects with a positive capital tax. The positive-capital-tax result does not hold with separable CRRA utility, where the externality cancels exactly. For public debt dynamics, shock persistence is the key driver of the optimal financing structure: transitory shocks call for borrowing to smooth taxes, while persistent shocks call for front-loading the adjustment through lower debt because borrowing would require costly future taxes.

Theory / model

The economy runs in discrete time. A continuum of $F$ ex-ante types of heterogeneous agents face idiosyncratic productivity risk. A representative firm produces using Cobb-Douglas technology. The government has access to a linear capital tax, a nonlinear (HSV) labor tax, and public debt. Aggregate uncertainty enters only through an exogenous public spending path (an MIT shock), so the economy is otherwise deterministic at the aggregate level.

Production. The net-of-depreciation production function is (p. 7, eq. 1):

$$Y_t = F(K_{t-1}, L_t) = K_{t-1}^\alpha L_t^{1-\alpha} - \delta K_{t-1}. \tag{1}$$

Factor prices satisfy $\tilde{w}_t = F_{L,t}$ and $\tilde{r}_t = F_{K,t}$.

Tax instruments. The labor tax follows Heathcote, Storesletten, and Violante (2017) (HSV). An agent earning pre-tax labor income $\tilde{w} y l$ pays (p. 8, eq. 2):

$$T_t(\tilde{w}yl) := \tilde{w}yl - \kappa_t(\tilde{w}yl)^{1-\tau_t}, \tag{2}$$

where $\kappa_t$ governs the level of labor taxation and $\tau_t \in [0,1]$ governs progressivity ($\tau_t = 0$ is a linear tax; $\tau_t = 1$ is full income redistribution). The capital tax $\tau_t^K$ is linear and applied to all interest-bearing assets. Defining post-tax factor prices (p. 9, eqs. 4-5):

$$w_t := \kappa_t(\tilde{w}_t)^{1-\tau_t}, \qquad R_t := 1 + r_t = 1 + (1-\tau_t^K)\tilde{r}_t, \tag{4,5}$$

the government budget constraint in post-tax prices simplifies to (p. 9, eq. 6):

$$G_t + R_t B_{t-1} + w_t \sum_{f=1}^F m^f \int_i (y_{i,t}^f l_{i,t}^f)^{1-\tau_t} \ell^f(di) \leq F(K_{t-1}, L_t) - (R_t - 1)K_{t-1} + B_t. \tag{6}$$

Agents. Each agent $i$ of type $f$ maximizes expected discounted utility (p. 10, eq. 7):

$$\max_{\{c,l,a\}} \; \mathbb{E}_0 \sum_{t=0}^\infty \beta^t U(c_{i,t}^f, l_{i,t}^f), \tag{7}$$

subject to the budget constraint (eq. 8) and a borrowing limit $a_{i,t}^f \geq -\underline{a}$:

$$c_{i,t}^f + a_{i,t}^f = R_t a_{i,t-1}^f + w_t(y_{i,t}^f l_{i,t}^f)^{1-\tau_t}. \tag{8}$$

Denoting by $\beta^t \nu_{i,t}^f \geq 0$ the multiplier on the credit constraint, the consumption Euler equation is (eq. 10):

$$U_c(c_{i,t}^f, l_{i,t}^f) = \beta \mathbb{E}_t\!\left[ R_{t+1} U_c(c_{i,t+1}^f, l_{i,t+1}^f) \right] + \nu_{i,t}^f. \tag{10}$$

The labor supply first-order condition is (eq. 11):

$$-U_l(c_{i,t}^f, l_{i,t}^f) = (1-\tau_t) w_t y_{i,t}^f (y_{i,t}^f l_{i,t}^f)^{-\tau_t} U_c(c_{i,t}^f, l_{i,t}^f). \tag{11}$$

The remaining equilibrium conditions in the general environment are the original government budget (text p. 9, equation 3), household feasibility and borrowing/nonnegativity constraints (text p. 10, equation 9), market clearing (text pp. 11-12, equations 12-13), and the first-best benchmark (text p. 12, equations 15-16):

$$G_t+(1+\tilde r_t)B_{t-1}\leq \sum_{f=1}^F m^f\int_i T_t(\tilde w_t y_{i,t}^f l_{i,t}^f)\ell^f(di)+\tau_t^K\tilde r_t(B_{t-1}+K_{t-1})+B_t. \tag{3}$$ $$a_{i,t}^f\geq-\underline a,\qquad c_{i,t}^f\geq0,\qquad l_{i,t}^f\geq0. \tag{9}$$ $$A_t=K_t+B_t=\sum_{f=1}^F m^f\int_i a_{i,t}^f\ell^f(di),\qquad \sum_{f=1}^F m^f\int_i y_{i,t}^f l_{i,t}^f\ell^f(di)=L_t. \tag{12}$$ $$\sum_{f=1}^F m^f\int_i c_{i,t}^f\ell^f(di)+G_t+K_t=K_{t-1}+F(K_{t-1},L_t). \tag{13}$$ $$\max_{(c_{i,t}^f,l_{i,t}^f,L_t,K_t)_{t\geq0}} W_0\quad\text{subject to}\quad \sum_{f=1}^F m^f\int_i c_{i,t}^f\ell^f(di)+G_t+K_t=K_{t-1}+F(K_{t-1},L_t),\quad \sum_{f=1}^F m^f\int_i y_{i,t}^f l_{i,t}^f\ell^f(di)=L_t,\quad K_{-1}\text{ given}. \tag{15,16}$$

Social welfare and Ramsey problem. The government is a utilitarian planner with type-specific Pareto weights $\omega^f$. Aggregate social welfare is (eq. 14):

$$W_0 = \sum_{f=1}^F m^f \omega^f \left( \mathbb{E}_0 \sum_{t=0}^\infty \beta^t \int_{i \in I^f} U(c_{i,t}^f, l_{i,t}^f) \ell^f(di) \right). \tag{14}$$

A Ramsey Equilibrium (RE) is the competitive equilibrium with the highest $W_0$ over all fiscal policies satisfying the government budget constraint. A Stationary Ramsey Equilibrium (SRE) is an RE in which aggregate quantities, prices, fiscal policy, and public spending are all constant. At the SRE the planner’s FOC for public debt implies the modified golden rule $1 + F_K = 1/\beta$, a condition first derived in the context of optimal capital taxation under incomplete markets by Aiyagari (1995); this pins down the long-run capital stock independently of the SWF weights $\omega^f$.

Simple model and the capital tax condition. Section 3 studies a simplified environment with deterministic productivity fluctuations (Woodford 1990): two agent types (employed and unemployed) alternating each period, a linear labor tax $\tau^L$, and a zero borrowing limit. Binding credit constraints affect only the unemployed at the SRE.

The planner’s first-order condition (FOC) linking post-tax interest and wage rates at the SRE is Proposition 1 (p. 15, eq. 29):

$$\underbrace{1 - \beta R}_{\text{smoothing wedge}} = \underbrace{\frac{F_L - w}{w}}_{\text{labor wedge}} \cdot \underbrace{\frac{\sigma_u - \sigma_e + \varsigma^l_{c,e}}{\sigma_e + \frac{1}{\varphi_e} - \varsigma^l_{c,e} + \varsigma^c_{l,e}}}_{\text{net distributional gain}}, \tag{29}$$

where $\sigma_e, \sigma_u$ are the inverses of the intertemporal elasticity of substitution (IES) for employed and unemployed agents, $\varphi_e$ is the Frisch elasticity of labor supply, and $\varsigma^l_{c,e}$, $\varsigma^c_{l,e}$ are cross-derivative terms that vanish for separable utility. The smoothing wedge equals $\beta(1+F_K-R) = (1-\beta)\tau^K$, so a positive smoothing wedge is equivalent to a positive capital tax.

The savings and labor-supply first-order conditions that decompose these wedges are equations 30-31 (text pp. 16-17):

$$1-\beta R=\Xi\left(\sigma_u-\sigma_e+\varsigma^l_{c,e}\right). \tag{30}$$ $$\frac{F_L-w}{w}=\Xi\left(\sigma_e+\frac{1}{\varphi_e}-\varsigma^l_{c,e}+\varsigma^c_{l,e}\right). \tag{31}$$

For separable CRRA utility, $\sigma_u = \sigma_e$, the numerator vanishes, and hence $\tau^K = 0$ (Corollary 1). The capital tax is positive when the IES differs between employed and unemployed agents (DRRA utility, so $\sigma_u > \sigma_e$) or when the utility is non-separable in a suitable way (GHH, KPR).

The elasticity definitions used in Proposition 1 are equations 27-28 (text p. 14):

$$\sigma_e=-c_e\frac{U_{cc}(c_e,l_e)}{U_c(c_e,l_e)},\qquad \sigma_u=-c_u\frac{U_{cc}(c_u,0)}{U_c(c_u,0)}. \tag{27}$$ $$\varphi_e=l_e\left(\frac{U_{ll}(c_e,l_e)}{U_l(c_e,l_e)}\right)^{-1},\qquad \varsigma^l_{c,e}=l_e\frac{U_{cl}(c_e,l_e)}{U_c(c_e,l_e)},\qquad \varsigma^c_{l,e}=c_e\frac{U_{cl}(c_e,l_e)}{U_l(c_e,l_e)}. \tag{28}$$

For the GHH existence analysis, Propositions 2-3 define the non-first-best and stationarity thresholds (text pp. 20-21, equations 35-38):

$$\bar g_1:=\frac{1-\beta}{\beta}\frac{\alpha}{1/\beta+\delta-1}-\frac{1-\beta}{1+\beta}\frac{1-\alpha}{\varphi+1}. \tag{35}$$ $$\bar g_{La}:=\frac{1-\alpha}{\varphi}\left(1+\frac{1-\beta}{1+\beta}\frac{1}{1+\varphi}+\frac{\varphi}{1+\varphi}\right)(1-\bar\tau^L_{La})^{1+\varphi},\quad \bar\tau^L_{La}:=\frac{1}{1+\varphi}-\frac{1}{1-\alpha}\frac{\varphi}{1+\varphi} \frac{\bar g_1}{1+\frac{1-\beta}{1+\beta}\frac{1}{1+\varphi}+\frac{\varphi}{1+\varphi}}. \tag{36,37}$$ $$\bar g_{SW}:=\bar g_1+(1-\alpha)\left(1+\frac{1-\beta}{1+\beta}\frac{1}{1+\varphi}+\frac{\varphi}{1+\varphi}\right) \left(1-\frac{1}{1+\varphi(1+\beta)}\right)^\varphi. \tag{38}$$

The positive-debt threshold (text p. 23, equation 39) is

$$\bar g_{pos}=\frac{1+\beta}{1-\beta}(1+2\varphi)(-\bar g_1),\qquad B\geq0\ \text{iff}\ \bar g_1\leq0\ \text{and}\ G\leq\bar g_{pos}Y_{FB}. \tag{39}$$

For the simple two-state employed/unemployed economy, the planner’s problem and its implementability conditions are stated in text pp. 13-14, equations 17-26. These equations are the simple-model specification used for the propositions, not an estimated regression:

$$\max_{\{c_{e,t},l_{e,t},c_{u,t},a_{e,t},a_{u,t}\}_{t\geq0}} \sum_{t=0}^{\infty}\beta^t\left[U(c_{e,t},l_{e,t})+U(c_{u,t},0)\right]. \tag{17}$$ $$c_{e,t}+a_{e,t}=R_ta_{u,t-1}+w_tl_{e,t},\quad c_{u,t}+a_{u,t}=R_ta_{e,t-1}. \tag{18,19}$$ $$U_c(c_{e,t},l_{e,t})=\beta R_{t+1}U_c(c_{u,t+1},0),\quad U_c(c_{u,t},0)\geq\beta R_{t+1}U_c(c_{e,t+1},l_{e,t+1}),\quad \text{equality in the second condition if }a_{u,t}>0. \tag{20,21}$$ $$-U_l(c_{e,t},l_{e,t})=w_tU_c(c_{e,t},l_{e,t}),\quad F(A_{t-1}-B_{t-1},l_{e,t})+B_t\geq G_t+B_{t-1}+(R_t-1)A_{t-1}+w_tl_{e,t}. \tag{22,23}$$ $$A_t=a_{e,t}+a_{u,t},\quad a_{e,t},a_{u,t}\geq0,\quad c_{e,t},c_{u,t}>0,\quad l_{e,t},l_{u,t}\geq0. \tag{24,25,26}$$

GHH utility. For the Greenwood-Hercowitz-Huffman utility function (p. 19, eq. 32):

$$U(c,l) := u\!\left(c - \chi^{-1} \frac{l^{1+1/\varphi}}{1+1/\varphi}\right), \tag{32}$$

where $u$ has constant IES $1/\sigma$ and $\varphi > 0$ is the Frisch elasticity. In the log-GHH case (IES = 1), Proposition 1 reduces to a simple relation between the capital and labor taxes (eq. 34):

$$(1-\beta)\tau^K = \frac{\tau^L}{1-\tau^L} \varphi(1+\beta). \tag{34}$$

For a GHH utility with constant inverse IES $\sigma$, the unsimplified wedge condition is equation 33 (text p. 19):

$$1-\beta R=\frac{F_L-w}{w}\varphi\sigma\left(1+\beta(\beta R)^{1/\sigma-1}\right). \tag{33}$$

The capital tax is thus positive whenever the labor tax is positive, and increases with the discount factor $\beta$ and the Frisch elasticity $\varphi$.

In the extension with two ex-ante types, the social objective and its capital-tax condition are equations 45-47 (text pp. 27-28). Define $\Lambda=\frac{\Omega^B(y^B)^{\varphi+1}}{\Omega^A(y^A)^{\varphi+1}}$, the relative labor income of the always-employed type. Proposition 6 gives:

$$\omega^A\sum_{t=0}^{\infty}\beta^t\left[\log\left(c_{e,t}^A-\chi^{-1}\frac{(l_{e,t}^A)^{1+1/\varphi}}{1+1/\varphi}\right)+\log(c_{u,t}^A)\right] +\omega^B\sum_{t=0}^{\infty}\beta^t\log\left(c_{e,t}^B-\chi^{-1}\frac{(l_{e,t}^B)^{1+1/\varphi}}{1+1/\varphi}\right). \tag{45}$$ $$\frac{1-\beta R}{\omega^A}=\frac{\omega^B}{\omega^A}-(1+\beta)\Lambda+\frac{F_L-w}{w}\varphi(1+\beta)(1+\Lambda), \tag{46}$$ $$(1-\beta)\frac{\tau^K}{\omega^A}=\frac{\omega^B}{\omega^A}-(1+\beta)\Lambda+\varphi(1+\beta)(1+\Lambda)\frac{\tau^L}{1-\tau^L}. \tag{47}$$

Method

The paper applies two computational methods and an identification strategy.

Factorization approach. In the general model of Section 4, the Ramsey program is solved using the factorization method of Marcet and Marimon (2019). This writes the Lagrangian of the sequential Ramsey problem so that the discounted sum collapses to a single-period term embedding forward-looking constraints (the agents’ Euler equations). The resulting first-order conditions (FOCs) are derived in Appendix A.6. The capital tax FOC (eq. 60, p. 32) equates the net distributive gain of a capital tax to the cost imposed on savings incentives:

$$\sum_{f=1}^F m^f \int_i \hat{\psi}_{i,t}^f a_{i,t-1}^f \ell^f(di) = \sum_{f=1}^F m^f \int_i \lambda_{i,t-1}^f u'(x_{i,t}^f) \ell^f(di), \tag{60}$$

where $\hat{\psi}_{i,t}^f := \mu_t - \psi_{i,t}^f$ is the net value to the planner of reallocating one unit from agent $(i,f)$ to public funds, $\mu_t$ is the shadow value of government resources, and $\lambda_{i,t}^f$ is the Lagrange multiplier on agent $(i,f)$‘s Euler equation. The public-debt FOC implies the modified golden rule at the steady state:

$$\mu_t = \beta(1+\tilde{r}_{t+1})\mu_{t+1}, \qquad \Rightarrow \quad 1 + F_K = \frac{1}{\beta}. \tag{59}$$

Integrating household Euler equations in the general model gives the steady-state capital-tax identity (text p. 35, equation 64):

$$\tau^K=\frac{\sum_{f=1}^F m^f\int_i\nu_i^f\ell^f(di)}{(1-\beta)\sum_{f=1}^F m^f\int_i u'(x_i^f)\ell^f(di)}. \tag{64}$$

The quantitative Ramsey program also transforms labor supply and progressivity and imposes the implementability and market-clearing conditions (text pp. 29-31, equations 48-55):

$$l_t=\left(\chi(1-\tau_t)w_t\right)^{1/(1/\varphi+\tau_t)},\qquad \tilde\tau_t=\frac{(1/\varphi+1)(1-\tau_t)}{1/\varphi+\tau_t}. \tag{48,49}$$

Here $x_{i,t}^f=c_{i,t}^f-\chi^{-1}(l_{i,t}^f)^{1+1/\varphi}/(1+1/\varphi)$. The planner maximizes weighted discounted utility (50), subject to the transformed government budget (51), individual budget and Euler equations (52-53), borrowing, complementary-slackness and nonnegativity conditions (54), and asset/labor clearing (55). The planner’s marginal social value of liquidity and public funds (text pp. 31-32, equations 56-58) is:

$$\max \sum_{f=1}^F m^f\omega^f\sum_{t=0}^{\infty}\beta^t\int_i u(x_{i,t}^f)\ell^f(di). \tag{50}$$ $$G_t+T_t+r_tA_{t-1}+\left(\frac{1}{\tilde\tau_t}+\frac{1}{1/\varphi+1}\right) \frac{l_t^{1/\varphi+1}}{\chi}\sum_{f=1}^F m^f\int_i(y_{i,t}^f)^{\tilde\tau_t}\ell^f(di) =F(A_{t-1}-B_{t-1},L_t)+B_t-B_{t-1}. \tag{51}$$ $$x_{i,t}^f=(1+r_t)a_{i,t-1}^f-a_{i,t}^f+ \frac{1}{\chi\tilde\tau_t}l_t^{1/\varphi+1}(y_{i,t}^f)^{\tilde\tau_t}. \tag{52}$$ $$u'(x_{i,t}^f)=\beta\mathbb{E}_t[(1+r_{t+1})u'(x_{i,t+1}^f)]+\nu_{i,t}^f. \tag{53}$$ $$a_{i,t}^f\geq-\bar a,\quad \nu_{i,t}^f(a_{i,t}^f+\bar a)=0,\quad \nu_{i,t}^f\geq0,\quad x_{i,t}^f\geq0,\quad l_{i,t}^f\geq0. \tag{54}$$ $$A_t=\sum_{f=1}^F m^f\int_i a_{i,t}^f\ell^f(di),\quad L_t=l_t\sum_{f=1}^F m^f\int_i(y_{i,t}^f)^{(1/\varphi+1+\tilde\tau_t)/(1/\varphi+1)}\ell^f(di). \tag{55}$$ $$\psi_{i,t}^f=\omega^f u'(x_{i,t}^f)-\left(\lambda_{i,t}^f-(1+r_t)\lambda_{i,t-1}^f\right)u''(x_{i,t}^f),\quad \hat\psi_{i,t}^f=\mu_t-\psi_{i,t}^f,\quad \hat\psi_{i,t}^f=\beta\mathbb{E}_t\left[(1+r_{t+1})\hat\psi_{i,t+1}^f\right]. \tag{56,57,58}$$

The remaining main-text Ramsey first-order conditions set the unit labor-supply level and progressivity (text pp. 32-33, equations 61-63). In the displays, integrals are over type-f histories as in the source:

$$\frac{1+1/\varphi}{\chi\tilde\tau_t}l_t^{1/\varphi+1} \sum_{f=1}^F m^f\int_i\hat\psi_{i,t}^f(y_{i,t}^f)^{\tilde\tau_t}\ell(di)\ell^f(di) =\mu_t\sum_{f=1}^F m^f\int_i\left[\frac{l_t^{1/\varphi+1}}{\chi}(y_{i,t}^f)^{\tilde\tau_t} -(y_{i,t}^f)^{(1/\varphi+1+\tilde\tau_t)/(1/\varphi+1)}F_{L,t}l_t\right]\ell^f(di). \tag{61,62}$$ $$0=\frac{l_t^{1+1/\varphi}}{\chi\tilde\tau_t}\sum_{f=1}^F m^f\int_i\hat\psi_{i,t}^f(y_{i,t}^f)^{\tilde\tau_t} \left(-\frac{1}{\tilde\tau_t}+\log y_{j,t}^f\right)\ell(di) -\mu_t\frac{l_t}{1/\varphi+1}\sum_{f=1}^F m^f\int_i\log y_{j,t}^f \left[\frac{l_t^{1/\varphi}}{\chi}(y_{i,t}^f)^{\tilde\tau_t} -(y_{i,t}^f)^{(1/\varphi+1+\tilde\tau_t)/(1/\varphi+1)}F_{L,t}\right]\ell(di). \tag{63}$$

Truncation method. For the quantitative model, the paper uses the truncation approach of LeGrand and Ragot (2022a) with the refinement of LeGrand and Ragot (2022b), both building on heterogeneous-agent-bewley-model traditions. The method aggregates agents by their recent idiosyncratic productivity histories of length $N$, replacing the full infinite-dimensional distribution with a finite number of “representative histories.” Histories that are more frequently visited are given longer truncation lengths (refined truncation), reducing the state space from exponential to linear in the maximum truncation length.

Inverse optimal approach. The Social Welfare Function (SWF) weights $(\omega^f)_{f=1,\ldots,F}$ are identified from the observed US fiscal system via an inverse optimal approach following Bourguignon and Amadeo (2015) and Heathcote and Tsujiyama (2021). Given the calibrated steady-state fiscal parameters $(\tau^K, \kappa, \tau, B)$, the model’s FOCs at the SRE are solved for the unique $(\omega^f)_{f=1,\ldots,F}$ consistent with optimality. With $F=3$ agent types, the identification reduces to inverting a 3x3 matrix of FOC constraints.

Public debt dynamics. In the simple log-GHH model, capital is the unique state variable in the linearized dynamics. The optimal capital path after a public spending shock of initial size $\hat{G}_0$ and persistence $\rho_G$ is (Result 2, eq. 41):

$$\hat{K}_t = \rho_K \hat{K}_{t-1} + \sigma_K \hat{G}_t, \qquad \rho_K > 0, \; \sigma_K < 0, \tag{41}$$

from which the closed-form public debt impulse response function follows (eq. 44):

$$\hat{B}_t = \hat{G}_0 \!\left(\Theta^K \rho_K^t - \Theta^G \rho_G^t\right). \tag{44}$$

The impact response $\hat{B}_0 = \hat{G}_0(\Theta^K - \Theta^G)$ can be positive or negative depending on $\rho_G$. Proposition 5 (p. 26) proves that $\partial \hat{B}_0/\partial \rho_G < 0$ at fixed $\hat{G}_0$ and, more importantly, also at fixed NPV of public spending. The intuition: when persistence is low, the planner borrows to smooth the large transitory shock and retires debt with a small future tax increase; when persistence is high, the capital stock falls persistently, making future tax increases very costly, so the planner front-loads fiscal adjustment without issuing new debt.

Empirical specifications

The quantitative model (Section 5) is calibrated to the US and solved numerically. The paper does not estimate a regression: fixed effects and regression standard errors do not apply. Its quantitative evidence comes from the calibrated equilibrium, inverse-optimal recovery of three social-welfare weights, and deterministic MIT-shock simulations.

Parameters and calibration targets. The period is a quarter. Technology is Cobb-Douglas: $F(K,L) = K^\alpha L^{1-\alpha} - \delta K$ with $\alpha = 0.36$ (capital share) and $\delta = 0.025$ (depreciation, corresponding to 10% annually), following Krueger, Mittman, and Perri (2018). The discount factor $\beta$ is set to match an annual capital-to-output ratio of 2.7. The GHH utility has Frisch elasticity $\varphi = 0.5$ (recommended by Chetty et al. (2011) for the intensive margin in heterogeneous-agent models) and scaling $\chi = 0.05$ to generate a steady-state labor supply of roughly $1/3$.

Ex-ante heterogeneity. Three agent types ($F=3$) are distinguished by their ex-ante productivity processes, corresponding to educational attainment: high-school or less, some college, and at least a bachelor’s degree, with average productivity levels of 0.8, 1, and 2 and population shares of $1/3$ each (2022 CPS data, footnote 28, p. 38). Each type follows an AR(1) log-productivity process:

$$\log y_t^f = \rho_y^f \log y_{t-1}^f + \varepsilon_t^f, \qquad \varepsilon_t^f \overset{\text{iid}}{\sim} \mathcal{N}(0,(\sigma_y^f)^2),$$

discretized with five idiosyncratic states per type using Rouwenhorst (1995), yielding 15 productivity levels and 455 truncated histories total after refinement.

Fiscal calibration. The capital tax rate $\tau^K = 36\%$ is taken from Trabandt and Uhlig (2011), using the Mendoza, Razin, and Tesar (1994) methodology on US data prior to 2008. The labor tax progressivity $\tau = 0.18$ is from Heathcote, Storesletten, and Violante (2017). The level $\kappa$ is chosen to match a public-spending-to-GDP ratio of $G/Y = 17\%$.

Inverse optimal identification. At the calibrated steady state, the planner’s FOCs identify the SWF weights as $\omega^1 = 13.1\%$, $\omega^2 = 81.6\%$, $\omega^3 = 5.3\%$ for the three types (p. 40). These weights are positive, consistent with a sensible SWF, and sum to 100% by normalization.

MIT shock specification. The public spending shock enters as (eq. 40):

$$\hat{G}_t = \begin{cases} \hat{G}_0 & t = 0 \\ \rho_G \hat{G}_{t-1} & t > 0, \end{cases} \tag{40}$$

with $\rho_G \in (-1,1)$. Two persistence values are studied: $\rho_G = 0.1$ (low persistence, initial shock = 1% of GDP) and $\rho_G = 0.99$ (high persistence, initial shock = 0.02% of GDP), calibrated to the same NPV of public spending (Panel 1, Figure 1, p. 53; discussion pp. 40-41).

For the analytical model, the stability bound and closed-form capital impulse response (text pp. 25-26, equations 42-43) are:

$$|\rho_K|<1\quad\Longleftrightarrow\quad\alpha\leq\frac{1}{1+(1-\beta)(1+\varphi)}. \tag{42}$$ $$\hat K_t=\sigma_K\hat G_0\frac{\rho_K^{t+1}-\rho_G^{t+1}}{\rho_K-\rho_G}. \tag{43}$$

Robustness. Results hold under an affine tax system (Appendix A.9, linear labor tax plus lump-sum transfer as in Dyrda and Pedroni (2022)) and under a productivity-dependent SWF that assigns weights to instantaneous rather than intertemporal utility (Appendix A.10). Results for TFP shocks and discount factor shocks are reported in Appendix A.11 and are qualitatively similar to the public spending shock results.

Datasets used

Dataset	Role in paper	Wiki page
US Current Population Survey (CPS) 2022	Calibration of average earnings for three education groups (high-school, some college, bachelor+); sets relative productivity levels 0.8, 1, 2	[no page yet]
Trabandt and Uhlig (2011) capital tax estimates	Sets steady-state capital tax target $\tau^K = 36\%$ (Mendoza-Razin-Tesar 1994 methodology, US pre-2008)	no page yet
Heathcote, Storesletten, and Violante (2017) estimates	Sets labor tax progressivity target $\tau = 0.18$	no page yet

Sample: calibrated to US steady-state fiscal data circa 2007; productivity AR(1) processes estimated to target US income risk moments. Dynamics are first-order perturbations around the calibrated SRE; not estimated from time-series data.

When to read the full paper

Read the source (or HAL preprint) if you are: studying the analytical conditions for existence of a stationary Ramsey equilibrium in heterogeneous-agent models (Propositions 2-3 and Appendices A.3-A.5); building or comparing quantitative Ramsey optimal policy models for the US (the calibration and truncation method details are in Sections 5 and Appendix A.7-A.8); or working on the question of whether capital taxes should rise or fall in response to public spending shocks (the key quantitative IRFs are in Figures 1-3). The replication code at doi.org/10.7910/DVN/ZMIFAZ reproduces all tables and figures.

Attribution and rights

Source: peer-reviewed, Journal of Political Economy 133(7), July 2025, pp. 2320-2369. Published by the University of Chicago Press; paywalled. An open preprint is available at hal.science/hal-05547657 under CC BY-NC-ND 4.0.

This page was extracted by an LLM (claude-sonnet-4-6) on 2026-06-26 and checked against the source PDF by paper-verifier (gpt-6-luna) on 2026-10-04; it has not been independently reproduced. Redistribution of the VOR is not permitted (paywalled); this page contains extracted summaries only.

Le Grand, François, and Xavier Ragot. “Optimal Fiscal Policy with Heterogeneous Agents and Capital: Should We Increase or Decrease Public Debt and Capital Taxes?” Journal of Political Economy 133, no. 7 (2025): 2320-2369. DOI: 10.1086/734877.