Optimal entry decision of unemployment insurance under partial information

Abstract

The aim of this paper is to study the optimal time for the individual to join an unemployment insurance scheme which is intended to protect workers against the consequences of job loss and to encourage the unemployed workers to find a new job as early as possible. The wage dynamic is described by a geometric Brownian motion model under drift uncertainty and the problem is a kind of two-dimensional degenerate optimal stopping problems which is hard to analyze. The optimal time of decision for the workers is given by the first time at which the wage process hits the free boundary which therefore plays a key role in solving the problem. This paper analyzes the monotonicity and continuity of the free boundary and derives a nonlinear integral equation for it. For a particular case the closed-form formula of free boundary is obtained and for the general case the free boundary is solved by the numerical solution of the nonlinear integral equation. The key in the analysis is to convert the degenerate problem into the non-degenerate one using the probability approach.