Optimal Policies

Main idea

Finding the solution of a Ramsey program in a heterogeneous-agent model is difficult, as the planner must internalize the effect of its instrument (for monetary policy, fiscal policy or any other policy) on the whole whole distribution of wealth. In other words, the first-order condition of the planner are difficult to derive.

The main idea is to use the finite state-space representation (uniform or refined) truncation) to solve for optimal policy in this environment.

There is a large (but finite) set of Lagrange multipliers on each constraint in the truncated representation, including Euler equations.

The important methodological point is to construct the truncated representation of the model for each value of the instrument.

Algorithm

Steady State

The difficult part is to find the steady state of the model.

The following algorithm is consistent and rapide enough:

  1. Set a truncation length N and guess values for the planner’s instruments.

  2. Solve the steady-state allocation of the full-fledged Bewley model with the previous instrument values, using standard techniques.

  3. Construct the truncated representation of the economy for a truncation length N, notably the \(\xi\).

    1. Derive the first-order conditions (FOCs) of the planner in the truncated representation.

    2. Solve for the joint distribution of wealth and Lagrange multipliers.

    3. Analyze whether the values of instruments are above or below their optimal value (using theFOCs of the planner)

  4. Change the instruments’ values accordingly (or stop if their value is close enough to the optimal value), and redo the process from Step 2.

  5. Increase the truncation length N, and restart from Step 2 until increasing N has no impact on the instruments’ values.

Note that we solve the full-fledged Bewley model and we construct the truncated model at each step. Hopefully, finding the values of Lagrange multipliers is very fast, as it is only linear algebra. It is thus inverting matrices.

Aggregate shocks

The set of dynamics equations of the truncated representation, together with the set of first order conditions of the planner form a dynamics system.

Once the steady states is found, one can use a first or second order approximation of the optimal dynamics around the steady state, using DYNARE for instance. Other techniques (like sequence jacobian) can be used to compute transitions.

References

In this paper, a simple case is solved (the optimal provision of a public good) to compare the solution with other solution methods (using transitions).

See other papers using these methods here.