Exploiting MIT shocks in heterogeneous-agent economies: the impulse response as a numerical derivative

This paper proposes a new way to compute the equilibrium of heterogeneous-agent models with aggregate uncertainty: rather than building a recursive, linearized law of motion for a high-dimensional state such as the cross-sectional wealth distribution – the strategy behind existing linearization methods including Reiter (2009, 2010), Childers (2017), and Ahn, Kaplan, Moll, Winberry, and Wolf (2017) – the authors solve, using entirely standard nonlinear methods, a single deterministic perfect-foresight transition path that follows a one-time, unexpected (“MIT”) shock away from the model’s non-stochastic steady state. Under the working assumption that the true stochastic equilibrium is well approximated by a linear system, that one nonlinear transition path is literally the model’s numerical derivative at each future time horizon, so the response to any sequence of aggregate shocks – and to any number of independent shocks, with computation time rising only linearly in the number of shocks – can be recovered by scaling and summing copies of this single impulse response, with no analytical differentiation and no explicit treatment of the distribution as a linearized state. The only nontrivial numerical tool the method requires is value-function iteration: once to solve the model’s non-stochastic steady state, and again, backward over a finite horizon, to solve for the transition path, with the passage of calendar time serving as the sole state variable added along the way. Applied to a standard Aiyagari-style economy with valued leisure and two aggregate technology shocks, the method reproduces conditional-moment results close to Dynare’s linearization- and second-order-perturbation-based solutions in a representative-agent benchmark, delivers accurate impulse responses (including for distributional statistics like the Gini coefficient and the hand-to-mouth share) in the heterogeneous-agent case, and, in an extension with a consumption externality and deficit-financed transfers, shows that Ricardian equivalence breaks down and fiscal transfers have real stabilizing effects – all without added numerical difficulty. The paper reports that its own scalability and additivity checks are satisfied “with flying colors” in this application, but is explicit that the whole approach rests on the assumption that linearization is in fact a good approximation for the model at hand, offers no formal stability or determinacy characterization of the transition path it computes, and, unlike recursive methods, provides no separate goodness-of-fit metric for that assumption.

Timo Boppart | Macro Paper Warehouse

Exploiting MIT shocks in heterogeneous-agent economies: the impulse response as a numerical derivative