Christian Bayer | Macro Paper Warehouse

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.

Monopsony Makes Firms Not Only Small but Also Unproductive: Why East Germany Has Not Converged

When employers face a trade-off between growing large and paying low wages — that is, when they have monopsony power — some productive employers will decide to acquire fewer customers, forgo sales, and remain small; these decisions have adverse consequences for aggregate labor productivity beyond the standard monopsony result that firms are too small. The paper documents that East German plants (compared to West German ones) face a steeper size-wage curve, invest less into marketing, and remain smaller, with the share of employment at plants with more than 249 employees standing at roughly 25% in East Germany versus 39% in West Germany in 2014 (and 31% versus 55% in manufacturing specifically). The steeper size-wage curve in East Germany is traceable to the historically determined underrepresentation of collective bargaining and union membership in small East German plants — a legacy of communist-era labor organization that caused union membership to collapse after reunification. The authors combine this evidence with a heterogeneous-plant model in which plants have product market power and choose how many customers to acquire subject to an upward-sloping size-wage schedule; two channels reduce aggregate productivity: a love-of-variety loss (fewer active plants means consumers bundle from a smaller variety of suppliers) and a compositional reallocation loss (labor is shifted from more productive to less productive plants, an effect exacerbated by product market power). When the model is calibrated to West Germany and the steeper East German size-wage trade-off is imposed, it predicts 10 percentage points lower aggregate labor productivity in East Germany — and for manufacturing, where East-West differences in plant size and the size-wage trade-off are particularly pronounced, the model predicts 18 percentage points lower productivity; in both cases the compression of the plant size distribution accounts for the largest share of the predicted productivity loss. The paper thus offers an explanation for why, more than thirty years after reunification, labor productivity and wages remain roughly 25% lower in the East German private sector despite uniform legal institutions across the two regions.

Solving discrete time heterogeneous agent models with aggregate risk and many idiosyncratic states by perturbation

This is a solution method for discrete-time heterogeneous-agent models with aggregate risk. It extends the perturbation approach of Reiter (2002, 2009) and complements the continuous-time work of Ahn, Kaplan, Moll, Winberry and Wolf (2017) by placing the dimensionality reduction at a new point in the pipeline: after the stationary equilibrium without aggregate risk has been solved, but before the nonlinear difference equation is linearized. Two reductions do the work. Value (or policy) functions are written as sparse expansions around their stationary-equilibrium counterparts using the discrete cosine transform, with only the largest coefficients allowed to move and the rest held at stationary values — the authors’ analogy is lossy video compression against a lightly compressed reference frame. The joint distribution over idiosyncratic states is factored into its marginal histograms, which vary freely, and a copula, which is held fixed at its stationary value. Because the deviation being zero exactly reproduces the stationary value function, the compression introduces no approximation error in the stationary equilibrium “irrespective of the degree of sparseness that is used in the calculation of the model dynamics.” The problem being solved is concrete: for a household problem with two assets and idiosyncratic income at 50 × 50 × 9 grid points, the distribution and value function are each vectors of 22,500 entries, the Jacobian blocks exceed 45,000 × 45,000, and each would occupy more than 7 GB stored densely. On the Krusell-Smith (1998) benchmark with the JEDC comparison-project calibration, the reduced method’s simulated log capital stock differs from the original Krusell-Smith algorithm’s by 0.0324% on average, exactly as the unreduced Reiter method does, and from the unreduced Reiter solution by 0.0003% on average; its Den Haan error is 0.0100% mean and 0.0191% max, against 0.0051% and 0.0131% for the Krusell-Smith algorithm, which remains the more accurate of the two. Run time is 0.38 seconds versus 91.61 for Krusell-Smith and 1.19 unreduced — “more than 240 times faster,” or 13 times faster once the 7.05 seconds for the stationary equilibrium are included. The scalability claim rests on a two-asset HANK model with 120,000 states and 240,000 controls that “is infeasible to solve for the aggregate dynamics… on the full histogram”: the fixed copula cuts states to 236 and DCT compression at 99.9999% energy cuts controls to 1427, giving a 5-minute solve plus 22 minutes for the stationary equilibrium on a laptop, with Den Haan errors of 0.033% mean and 0.092% max for capital. The costs are stated openly. The selection of retained coefficients is a heuristic, and the coefficients dropped “are only unimportant in the stationary equilibrium,” so robustness must be checked with Den Haan’s test rather than guaranteed. And the fixed copula is not innocuous for every shock: under TFP shocks the Jensen-Shannon distance between the reduced and unreduced joint distributions is a negligible 0.0005, but under idiosyncratic income-uncertainty shocks — which hit the joint distribution directly — the difference “attains a significant order of magnitude,” and recovering accuracy requires perturbing the copula’s own largest DCT coefficients (41 out of a possible 2100 in their example).