Abstract
In this paper we develop a new approach for finding optimal government policies in economies with heterogeneous agents. Using the calculus of variations, we present three classes of equilibrium conditions from government's and individual agent's optimization problems: 1) the first order conditions: the government's Lagrange–Euler equation and the individual agent's Euler equation; 2) the stationarity condition on the distribution function; and, 3) the aggregate market clearing conditions. These conditions form a system of functional equations which we solve numerically. The solution takes into account simultaneously the effect of the government policy on individual allocations, the resulting optimal distribution of agents in the steady state and, therefore, equilibrium prices. We illustrate the methodology on a Ramsey problem with heterogeneous agents, finding the optimal limiting tax on total income.
Highlights
Using the calculus of variations, we present three classes of equilibrium conditions from government's and individual agent's optimization problems: 1) the first order conditions: the government's Lagrange-Euler equation and the individual agent's Euler equation; 2) the stationarity condition on the distribution function; and, 3) the aggregate market clearing conditions
This paper provides a new approach for computing equilibria in which the stationary distribution of agents is a part of an optimal nonlinear, second-best government problem in a general equilibrium, Bewley type economy with heterogeneous agents
We study a steady state of an economy for which the optimal limiting government policy implies a convergence to that steady state
Summary
This paper provides a new approach for computing equilibria in which the stationary distribution of agents is a part of an optimal nonlinear, second-best government problem in a general equilibrium, Bewley type economy with heterogeneous agents. Definition 3 (Calculus of Variations Ramsey Problem) The Ramsey Problem in the calculus of variations is a generalized isoperimetric maximization problem (12), subject to the government budget constraint (14), with the individual policy function h given implicitly by the operator Euler equation F(h) = 0, the law of motion for the distribution function, λ, given implicitly by the operator equation L(λ, λ0 , h) = 0, the aggregate capital stock (13), the endogenous bounds of taxable activity, x(z) and x(z), for all values of z ∈ Z, and the free values of the government policy at the extreme lower and upper bounds. The government constructs its optimal limiting tax schedule by balancing the distributional effect on the individual savings function, h∗ , and the general equilibrium effect on the aggregate capital stock. The optimal limiting tax policy is a solution to the system of functional equations defined in Theorem 1, the functional equation for the stationary distribution λ, the two side conditions, and the Lagrange multiplier condition We solve this functional equations problem by the least squares projection method described in Appendix C. Alternative calibration for a low wealth dispersion economy based on Aiyagari (1994)
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