Thomas Winberry | Macro Paper Warehouse

A method for solving and estimating heterogeneous agent macro models

This is a computational method for solving and estimating heterogeneous-agent macro models with aggregate shocks, built around one substitution: the infinite-dimensional cross-sectional distribution is replaced by a flexible parametric family, so its moments become finite-dimensional endogenous state variables, and the resulting system is then perturbed around its stationary equilibrium. The stated motivating gap is adoption rather than accuracy: of the many existing algorithms, “none are as general, efficient, or easy-to-use as the standard perturbation methods routinely employed to solve representative agent models using prepackaged toolboxes like Dynare. Due to this challenge, heterogeneous agent models have yet to reach widespread adoption, particularly among central banks and policy institutions.” The method has three steps — globally accurate projection approximations in the individual states (Chebyshev collocation for the firm’s value function, an exponential-polynomial density adapted from Algan, Allais and Den Haan for the distribution), computation of the stationary equilibrium without aggregate shocks, and a locally accurate Taylor expansion in the aggregate state, implemented directly in Dynare with a published code template. Demonstrated on the Khan and Thomas (2008) model of heterogeneous firms with fixed capital adjustment costs, a first-order solution takes 35 seconds for a degree-2 approximation of the distribution and 47 seconds for degree 4, most of that in the stationary equilibrium. Accuracy is reported by degree: degree 2 already gives steady-state aggregates “virtually indistinguishable” from a fine histogram (output 0.498 against 0.498, capital 1.006 against 1.007), while degree 1 is visibly wrong (capital 1.200), and higher degrees are needed only for the shape of the distribution, not the aggregates — cross-sectional covariance and variance of log capital do shift with the degree of approximation, though “these differences are small and do not generate differences in the dynamics of aggregate variables.” Business cycle moments are conventional: output volatility 2.14 percent, consumption 0.48 times as volatile as output, investment 3.86 times, all highly correlated because TFP is the only shock. Three generalizations establish the claimed breadth. A second-order approximation is a one-line Dynare change and shows quantitatively small sign- and state-dependence, consistent with Khan and Thomas; mass points from occasionally binding constraints are handled by adding the mass at the constraint as an extra distributional parameter, which is how the Krusell and Smith (1998) household model is solved; and approximate aggregation is shown not to be required, with a volatile-enough investment-specific shock breaking it. The efficiency claim is externally sourced: in Terry’s comparison project, an implementation of this method “solves and simulates the model in 0.098% of the time of the Krusell and Smith (1998) method.” That speed is what makes the paper’s estimation exercise possible: 10,000 Metropolis-Hastings draws in 44 minutes and 32 seconds, estimating the parameters of neutral and investment-specific productivity shocks conditional on three values of the fixed-cost upper bound. As frictions rise from 0.01 to 1, the estimated volatility of investment-specific shocks rises from 0.0056 to 0.0077 while the other parameters barely move, so “matching the aggregate investment data with larger adjustment frictions requires more volatile shocks.” The limits are stated rather than implied: the local approximation is “not well suited for capturing nonlinearities that are global in nature,” the parametric family fails for fat-tailed distributions, the estimation conditions on frictions rather than estimating them jointly, and the resulting variance decompositions differ only slightly across the friction values considered.

Financial Heterogeneity and the Investment Channel of Monetary Policy

Motivated by the observation that aggregate investment is one of the most cyclically responsive components of GDP, this paper asks which firms drive that response: those most affected by financial frictions, or those least affected? Combining a high-frequency-identified measure of monetary policy shocks with quarterly Compustat data, the paper finds that firms with lower default risk – measured by lower leverage and a higher structural “distance to default” – invest significantly more in response to an expansionary monetary shock than high-default-risk firms, with a one-standard-deviation lower leverage ratio associated with roughly a one-quarter larger investment semielasticity and a one-standard-deviation higher distance to default with roughly a one-half larger one; these differential responses persist for up to three years and are accompanied by low-risk firms gaining relatively more access to external finance and facing smaller increases in borrowing costs than high-risk firms after an expansionary shock. To interpret this finding, the paper embeds heterogeneous firms subject to collateralized borrowing and endogenous default risk into an otherwise standard New Keynesian model with sticky-price retailers, calibrated to match firm-level investment, borrowing, and lifecycle facts, and reproduces the empirical pattern: low-risk firms are more responsive because they face a comparatively flat marginal cost-of-investment-finance curve, so the direct effect of a lower real interest rate on the marginal benefit of capital passes through to investment largely undamped, whereas high-risk (net-worth-constrained) firms sit on the upward-sloping segment of that curve, where a monetary expansion’s cash-flow, collateral-value, and default-probability channels only partially offset the higher financing cost of additional capital. The paper’s decomposition further shows that, unlike in household-consumption models (Kaplan, Moll and Violante 2018; Auclert 2019), where indirect general-equilibrium income effects dominate the direct interest-rate effect, both direct and indirect price effects contribute quantitatively to firm investment responses, because firms are more interest-sensitive than consumption-smoothing households. Because low-risk firms drive most of the aggregate response, the model implies the aggregate potency of monetary policy depends on the prevailing cross-sectional distribution of default risk: a counterfactual low-net-worth initial distribution generates an aggregate capital response about 33% smaller than the calibrated steady-state distribution, even though every constrained firm remains more responsive to monetary policy than it would be absent financial frictions altogether – consistent with the amplification logic of Bernanke, Gertler and Gilchrist (1999).

New Keynesian Economics with Household and Firm Heterogeneity

This paper extends the Heterogeneous-Agent New Keynesian (HANK) research program, which over the past decade has studied how idiosyncratic income risk and borrowing constraints reshape monetary and fiscal transmission through household consumption, to the investment side of the economy by building a parallel canonical model of heterogeneous firms. The authors develop a firm model in the tradition of Khan and Thomas (2013) and Ottonello and Winberry (2020), in which firms face idiosyncratic productivity risk and choose investment subject to capital adjustment costs, a collateral constraint on borrowing, and a non-negativity constraint on dividends. They define a firm-level “marginal propensity to invest” (MPI) – the response of investment to a one-time, unexpected increase in cash flow – as the direct analogue of the household marginal propensity to consume (MPC), and show that financially constrained firms have a strictly positive, sometimes greater-than-one MPI (because new capital can be pledged as collateral for further borrowing), while unconstrained firms follow the standard Q-theory prediction of a zero MPI. Because there is no broad quasi-experimental evidence on MPIs comparable to what exists for MPCs, the authors calibrate the distribution of MPIs indirectly, obtaining an unweighted average MPI of 0.50 but a much smaller size-weighted MPI of 0.05, reflecting a strong negative correlation between firm size and financial constraint in their calibration. This distinction matters for policy transmission: a uniform, deficit-financed transfer to firms stimulates aggregate investment strongly, governed by the large unweighted MPI, while the indirect general-equilibrium effect of higher aggregate demand on cash flows – and hence on investment – operates through the much smaller size-weighted MPI, so that in their calibrated response to a monetary policy shock, this indirect “cash flow” channel accounts for only about 13% of the total investment response, with the bulk still coming from the traditional direct effect of interest rates on firms’ discount rates. When heterogeneous households and heterogeneous firms are combined in the same model, the authors document “aggregate demand complementarities”: because a transfer to firms raises aggregate demand and therefore household income, which stimulates further consumption by high-MPC households, the total output response to a firm transfer is roughly 20% larger than in a model with heterogeneous firms alone. The paper closes by systematically mapping the concepts of the heterogeneous-household literature onto their firm-side analogues – target buffer-stock assets versus target capital, precautionary saving versus constraint-avoidance behavior, MPCs/iMPCs versus MPIs/iMPIs – to organize an agenda of open questions, chief among them the need for direct empirical estimates of firm-level MPIs to discipline these new mechanisms the way quasi-experimental MPC estimates have disciplined the household side.

The Macroeconomic Dynamics of Labor Market Polices

This paper builds a dynamic macroeconomic model with rich worker heterogeneity, firm monopsony power and putty-clay adjustment frictions in order to trace the time path of the effects of the federal minimum wage and the Earned Income Tax Credit, rather than comparing steady states. The putty-clay structure — capital is CES ex ante but Leontief once installed — reconciles the small short-run employment elasticities documented in the minimum-wage literature with the large long-run elasticities of substitution estimated by Katz and Murphy (1992) and Card and Lemieux (2001), and it means firms adjust their input mix only as old capital depreciates. The paper’s key result is that the welfare impact of these policies “cannot be inferred from evidence on their short-run effects alone”: both an increase in the minimum wage and an expansion of the EITC make low-wage workers better off in the first few years, but at longer horizons small minimum-wage increases and EITC expansions of any size continue to help them while sufficiently large minimum-wage increases hurt them, because firms substitute away from workers whose wage is pushed well above the efficient level. Quantitatively, in the baseline parameterisation (a targeted wage markdown of 0.75 and within-education-group substitution elasticity ϕ = 4), an $8.50 minimum wage raises non-college employment by about 0.8% in the long run while a $15 minimum wage lowers it by about 12.1%, with employment falling for all non-college workers initially earning under $11 — 26% of non-college workers; a budget-equivalent EITC instead raises non-college employment by 5.9%, and only about one-fifth of the long-run employment decline from a $15 minimum wage materialises in the first two years. The authors also find that combining the EITC, or the broader U.S. tax and transfer system, with a moderate minimum-wage increase supports low-wage workers better than either policy alone, “because doing so more effectively offsets firms’ monopsony power,” and that a minimum wage fixed in nominal terms — as U.S. federal policy is — has small long-run real effects because inflation and productivity growth erode it before firms find it worth re-tooling.

When inequality matters for macro and macro matters for inequality

This paper argues that two standard excuses for relying on representative-agent macro models – that heterogeneous-agent models with aggregate shocks are computationally intractable, and that realistic household heterogeneity does not matter much for aggregate dynamics anyway – are both weaker than commonly believed. To address the first, the authors extend Michael Reiter’s discrete-time linearization approach to continuous time: they solve a model’s stationary equilibrium fully nonlinearly using the finite-difference methods of Achdou et al. (2015), take a first-order Taylor expansion of the full discretized equilibrium system around that steady state using automatic differentiation, and solve the resulting large linear system of stochastic differential equations by standard techniques, exploiting continuous time’s tendency to generate sparse transition matrices; a companion model-free dimensionality-reduction method, adapted from the engineering “model reduction” literature, lets the computer – rather than the researcher – identify the low-dimensional information in the cross-sectional distribution needed to forecast prices accurately, generalizing the “approximate aggregation” logic of Krusell and Smith (1998) beyond cases where a hand-picked set of moments happens to work. On the standard Krusell-Smith (1998) business-cycle model, the method is roughly 1,500 times faster and about three times more accurate (by Den Haan’s 2010 error metric) than the best-performing algorithm in the JEDC comparison project, though its accuracy – being a local, linear approximation – degrades as the size of aggregate shocks grows. To address the second excuse, the authors apply their (open-sourced) toolbox to a two-asset incomplete-markets model, calibrated to match the U.S. joint distribution of income, wealth, and marginal propensities to consume, in which “wealthy hand-to-mouth” households arise endogenously from a costly-to-access illiquid asset. This richer model jointly reproduces two features of aggregate consumption dynamics – sensitivity to predictable income changes and relative smoothness – that have long challenged representative-agent and simple spender-saver benchmarks, and, in an extension with capital-skill complementarity, shows that aggregate productivity shocks can generate substantial, shock-specific swings in income and consumption inequality, providing what the authors call “a striking counterexample to the main result of Krusell and Smith (1998).”