Uniform Truncation

The main idea

The main idea is to start in the sequential representation (instead of the recursive one), to then construct a finite state-space representation (which can then be studied recursively !).

In heterogeneous-agent models, agents indeed differ according to their idiosyncratic history. An agent i has a period-t history \(y_{i}^{t}=\{y_{i,0},\ldots,y_{i,t}\}\), where \(y_{i,t}\) is the realization of an idiosyncratic risk at period \(t\). This risk can take \(k\) values (for instance it can be \(k\) productivity level).

Let \(h=(\tilde{y}_{-N+1},\ldots,\tilde{y}_{-1},\tilde{y}_{0})\) be a given history of length N.

The idea of the truncation method is to aggregate agents having the same truncated history \(h\) and to express the model using these groups of agents rather than individuals.

If there are \(k\) idiosyncratic states, then the nuber of histories is \(k^N\). This number can be large, as it grows exponentionally with \(N\). The refined truncation reduces singificantly this number as it becomes a linear function of \(N\).

In the truncated model, the agents’ aggregation assumes full risk-sharing within each truncated history, while the “true” Bewley model features wealth heterogeneity among the agents having the same truncated history \(h\). This simply comes from the heterogeneity in histories prior to the aggregation period (i.e., more than \(N\) periods ago).

This within-truncated-history is captured through additional parameters – denoted by “\(\xi_{h}\)” – which are truncated-history specific. This construction yields a consistent finite state-space representation.

The summary of the algorithm

To find the steady-state values of the Lagrange multipliers, we use the same algorithm as in LeGrand and Ragot (2022):

  1. Set a truncation length N
  2. Solve the steady-state allocation of the full-fledged Bewley model (using standard techniques like VFI or EGM).
  3. Consider the truncated representation of the economy for a truncation length N.
    1. find average state variables for each history
    2. Compute the relevant “\(\xi s\)” for the Euler equations to be consistent.

To use our truncation method in the presence of aggregate shocks, we further assume that:

  1. The parameters \(\xi_{h}\) remain constant and equal to their steady-state values.

We thus assume that the time-variation of within-history heterogeneity is small enough for it to have a second-order effect. This can be checked numerically:

  1. Change \(N\) to check that the dynamics is independant of \(N\).

References for detailed description

The algorithm with aggregate shock is presented here where we solve for optimal time-varying unemployment benefits.