An endogenous gridpoint method for distributional dynamics

This paper introduces the Distributional Endogenous Gridpoint Method (DEGM), a novel numerical technique for solving the distributional dynamics that arise in heterogeneous agent macroeconomic models. The core problem is how to efficiently update the distribution of agents over the state space as the economy evolves. The dominant existing approach — the “lottery method” of Young (2010) — discretizes the state space and represents policy functions as lotteries over nearby gridpoints, producing a transition matrix that is linear in optimal policies. This linearity renders the lottery method incapable of capturing nonlinear effects in distributional dynamics, a limitation that becomes quantitatively significant for higher-order perturbation solutions.

DEGM extends Carroll’s (2006) endogenous gridpoint method from individual optimization to the distributional level. Rather than discretizing the density and integrating forward, DEGM works directly on the cumulative distribution function (CDF). The key insight is that when the policy function is monotone — as savings functions typically are — the endogenous gridpoints generated by the policy function trace out exact points on the post-policy CDF without requiring integration. Specifically, if A*_{i,j} = a*(A_i, Y_j) are optimal asset choices from grid point A_i at income Y_j, then the CDF values at those endogenous points are known analytically as F_t(A_i | Y_j). An interpolant using shape-preserving splines constructed through these points allows evaluation of the updated CDF at any point without integration. The income transition step is handled separately via standard quadrature over the discretized income process.

The paper demonstrates DEGM’s performance with two applications. First, in the Aiyagari (1994) economy, DEGM converges to the stationary equilibrium an order of magnitude faster than the lottery method in terms of gridpoints. At nk=40 gridpoints, the lottery method deviates from the benchmark capital stock by 1.72% and the wealth Gini by 2.24% (for nh=5), while DEGM deviates by only 0.09% and 0.12% respectively. Both methods converge to the same solution as the number of gridpoints increases, but DEGM reaches this limit far faster.

Second, the authors introduce a Krusell-Smith style model with aggregate investment risk (capital depreciation shocks calibrated following Barro, 2006, as a 0.4% quarterly probability of 7.5% capital destruction causing a 10% annual GDP drop) as a new baseline for studying aggregate nonlinearities with household heterogeneity. This model overcomes the near-linearity of aggregate capital dynamics in the original Krusell-Smith specification. Using a third-order perturbation solution with DEGM, aggregate investment risk lowers the capital stock by 5 to 11 basis points and increases wealth inequality by up to 11 basis points relative to the non-stochastic steady state, depending on idiosyncratic income risk calibration. The lottery method systematically mispredicts these effects: it always predicts a decrease in wealth inequality in the presence of investment risk, while DEGM predicts an increase. At third order, the lottery method predicts wealth Gini changes of +2.0 bp (persistent calibration) and -149.7 bp (transitory calibration), while DEGM predicts +10.7 bp and +2.1 bp respectively.

The mechanism for increased inequality under investment risk is heterogeneous: for less wealthy households the substitution effect dominates (they reduce saving more in response to risky returns), while for wealthy households the income effect is stronger and precautionary saving motives dominate. The lottery method, by making the distributional transition matrix linear in policies, zeros out the second derivative of the transition matrix with respect to the policy function, missing the term capturing how the density at the pre-image of each asset level is affected nonlinearly. DEGM’s cubic spline interpolant captures all nonlinearities up to third order, enabling economically meaningful results that qualitatively differ from lottery-method predictions on wealth inequality.

Q: What is the fundamental numerical problem that DEGM solves? A: Evolving the distribution of agents forward over time in heterogeneous agent models requires evaluating a Kolmogorov forward equation, which naively demands numerical integration. The lottery method avoids integration by discretizing the state space and expressing transitions as a linear matrix operation, but this forces the distributional dynamics to be linear in optimal policies. DEGM avoids integration by exploiting policy function monotonicity: the endogenous policy gridpoints are the interpolation nodes, so the CDF update requires only interpolation, not integration. This preserves nonlinear effects up to the order of the splines used.

Q: How does DEGM handle the borrowing constraint and the resulting mass point? A: Savings policy functions are typically weakly monotone: constant at the borrowing constraint for sufficiently poor households, then strictly monotone above a threshold. DEGM accommodates this by starting the endogenous grid at the EGM solution corresponding to the borrowing constraint (the threshold a_j above which the policy is strictly monotone), restoring strict monotonicity on the relevant domain. The mass point at the borrowing constraint is captured by evaluating F_t(a_j, Y_j). Echoes of the borrowing constraint diminish as the number of income states increases, and in practice 10 income gridpoints are sufficient to smooth them.

Q: How much faster does DEGM converge relative to the lottery method for the stationary equilibrium? A: In the Aiyagari economy with nk=40 asset gridpoints, the lottery method’s capital stock deviates from the benchmark by 1.72% and the wealth Gini by 2.24% (nh=5), while DEGM deviates by only 0.09% and 0.12% respectively — roughly a 20-fold improvement in accuracy for the same gridpoints. At nk=80, the lottery method still shows 0.56%/0.78% deviations while DEGM shows 0.03%/0.00%. Although for a fixed number of gridpoints the lottery method is faster in wall-clock time (0.35s vs 0.82s at nk=40, nh=20), DEGM is faster for a given level of accuracy because it requires far fewer gridpoints.

Q: Why does the lottery method fail at higher-order perturbations? A: The lottery method constructs its transition matrix as a piecewise linear function of the optimal policy a*, so its second derivative with respect to a* is zero. As a result, it misses the second term in the second-order derivative of the end-of-period CDF: the term involving the derivative of the density at the pre-image of each asset level times the squared linear policy effect. This missing nonlinearity becomes quantitatively important at second and third order. DEGM’s cubic hermitian spline interpolant captures all nonlinearities up to third order, allowing it to correctly represent how the distribution responds nonlinearly to aggregate shocks.

Q: What does the paper find about the effect of aggregate investment risk on the capital stock and wealth inequality? A: Using a third-order perturbation solution with DEGM, aggregate investment risk lowers the capital stock by 5 to 11 basis points from the non-stochastic steady state, depending on whether income risk is persistent or transitory (DEGM third-order: -4.7 bp persistent, -11.4 bp transitory). Wealth inequality increases by up to 11 basis points (DEGM third-order: +10.7 bp persistent, +2.1 bp transitory). The lottery method diverges dramatically at third order, predicting Gini changes of +2.0 bp and -149.7 bp for the persistent and transitory calibrations respectively, compared to DEGM’s +10.7 bp and +2.1 bp.

Q: What is the mechanism through which aggregate investment risk increases wealth inequality? A: The mechanism operates through heterogeneous saving responses across the wealth distribution. For less wealthy households, capital income is a small share of total income, so the substitution effect of risky returns dominates: higher investment risk reduces their incentive to save. For wealthy households, capital income is central, so the income effect is stronger and precautionary saving motives intensify. A capital depreciation shock upon realization compresses the wealth distribution, but the risk of such a shock increases inequality on average because it disproportionately reduces saving among poorer households.

Q: How do the authors extend DEGM to handle aggregate risk and higher-order perturbations? A: The authors follow Reiter (2009) in including the distribution and value functions in the state space, defining a nonlinear difference equation over these objects. Higher-order perturbation of this system proceeds using the algorithms of Andreasen et al. (2018) and Levintal (2017), with second-order terms solved via a generalized Sylvester equation using Kim et al.’s (2008) doubling algorithm. The implementation handles up to 3,200 variables at second order and 220 variables at third order. For the second-order solution, the Bayer-Luetticke (2020) state-space reduction and its refinement in Bayer et al. (2024) yield results identical to the full unreduced system.

Q: What is the state-space reduction procedure and how much does it compress the system? A: The full system uses 402 states and 412 controls (persistent calibration). A copula representation of the distribution reduces this to 213 states and 412 controls; adding DCT compression of the value function gives 213 states and 98 controls; further adding a factor representation from the first-order solution yields 111 states and 98 controls — a 75% reduction. The R-squared-like IRF statistic remains 1.00 across all reductions, and ergodic moments are identical (capital: 25.54, Gini: 0.61 for the persistent calibration).

Q: Does DEGM produce different first-order impulse responses than the lottery method? A: For first-order perturbations, DEGM and the lottery method converge to the same solution as the number of gridpoints increases, but DEGM converges faster. For the first-order dynamics of the wealth distribution (wealth Gini IRFs), DEGM reaches convergence with nk=40 gridpoints while the lottery method requires nk=160. For aggregate capital stock IRFs, both methods converge quickly at first order. Quantitative differences become significant only at second and higher orders.

Q: What calibration is used for the investment risk model? A: Capital depreciation deviates from its steady-state value by a shock with second moment sigma_delta = 0.005 and third moment tau_delta = 0.012. This corresponds to a 0.4% quarterly probability that a disaster destroys 7.5% of the capital stock and causes a 10% drop in annual GDP, consistent with the evidence in Barro (2006). The model is solved under both a persistent income calibration (beta=0.98, rho=0.98, sigma_epsilon=0.14, implied Gini=0.66) and a transitory income calibration (beta=0.99, rho=0.88, sigma_epsilon=0.18, implied Gini=0.42).

Distributional Endogenous Gridpoint Method (DEGM): A numerical method for evolving the joint CDF of agents over the state space by constructing an interpolant at endogenous gridpoints A*_{i,j} = a*(A_i, Y_j) — the optimal policy values — at which CDF values are known analytically as F_t(A_i | Y_j), thus updating the distribution through interpolation rather than integration and preserving nonlinearities up to the order of the spline.

Lottery Method (LM): Young’s (2010) standard technique that replaces the continuous distribution with a discrete counterpart and represents optimal policy functions as probability weights over nearby gridpoints, yielding a single transition matrix A* such that f_{t+1} = f_t * A*. The transition matrix is linear in optimal policies, which zeroes out the second derivative of the distributional dynamics with respect to policies and causes systematic misprediction of distributional dynamics under higher-order perturbation.

Kolmogorov Forward Equation (Distributional Dynamics): The law of motion for the joint CDF F_t(a, y) describing how the distribution of households over assets and income evolves given optimal policies and the income transition process. In DEGM, this equation is split into a sub-period for asset choices (where endogenous gridpoints allow integration-free updating) and a sub-period for income transitions (handled by quadrature over the discretized income process).

Higher-Order Perturbation Solution: A Taylor expansion of the model’s nonlinear equilibrium conditions around the non-stochastic steady state beyond first order. Second-order solutions capture precautionary motives and mean deviations from the steady state; third-order solutions additionally capture asymmetric effects of shocks, requiring DEGM’s nonlinear distributional representation to produce accurate results.

Aggregate Investment Risk (Capital Depreciation Shocks): Shocks to the aggregate capital depreciation rate calibrated following Barro (2006) as a 0.4% quarterly probability of a disaster that destroys 7.5% of the capital stock and causes a 10% annual GDP drop. Proposed as a replacement for near-linear Krusell-Smith aggregate productivity shocks to generate genuine nonlinearities in aggregate capital dynamics while remaining equally parsimonious.

State-Space Reduction: A sequence of compression techniques — copula representation of the wealth distribution, discrete cosine transform (DCT) compression of the value function, and factor representation from the first-order solution — that reduce the Reiter (2009) system from 402 states and 412 controls to 111 states and 98 controls (a 75% reduction) with no measurable loss of accuracy in impulse responses or ergodic moments.

Shape-Preserving Interpolation: Interpolation methods (linear spline or piecewise cubic hermitian splines) that maintain the monotonicity of the CDF when constructing the interpolant from endogenous gridpoints. Cubic hermitian splines additionally preserve differentiability, making the distributional dynamics smooth enough for third-order perturbation and capturing all nonlinear effects that the lottery method misses.