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 is positive when credit constraints occasionally bind AND the utility function deviates from Constant Relative Risk Aversion (CRRA): for GHH or Decreasing RRA (DRRA) preferences, an externality of savings on post-tax factor prices creates a rationale for a positive capital tax. With CRRA utility, the Chamley (1986) and Judd (1985) zero-capital-tax result generalizes exactly (Corollary 1).

Second, the existence of a Stationary Ramsey Equilibrium (SRE) with positive capital tax and positive public debt requires three independent conditions: a non-first-best condition, the Straub and Werning (2020) stationarity condition, and a Laffer condition.

Third, 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 (the cost of future tax increases to retire debt is too large). A 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 decreasing in shock persistence at fixed NPV	Proposition 5 (p. 26); Figure 3 (p. 43)	$\partial \hat{B}_0/\partial \rho_G < 0$; quantitatively: debt rises for $\rho_G=0.1$ (1% of GDP shock), falls for $\rho_G=0.99$ (0.02% of GDP shock)
R5	Capital tax rises significantly at impact after a public spending shock (both persistence levels)	Figure 1 (panel 4, p. 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 level falls at impact after a spending shock	Figure 1 (panels 2-3, p. 41)	Progressivity $\tau$ increases and labor tax level $\kappa$ decreases; both changes are much smaller than the capital tax response; public debt path differs markedly by persistence level (panel 5)

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 result fails for CRRA utility because the externality cancels exactly. For public debt dynamics, shock persistence is the key driver of the optimal financing structure: transitory shocks call for borrowing (lower future taxes via smoothing) while persistent shocks call for front-loading adjustment (raising taxes now to avoid a highly distortionary persistent increase later).

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):

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

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.

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).

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):

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

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

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:

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

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.

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. 40).

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 is not human-verified or 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.