A Regime Switching Model of Schooling Choice as a Job Search Process

where r is the constant discount rate andQwas also assumed to be constant in Fan [1]. However, this abstraction may not be able to fully capture reality. Specifically, in an economic recession, the desire to obtain education may rise; for example, graduate program enrollment increases in an economic recession (Fan [1]), but the cost of education, for example, tuition fees, may decline (Binder [2]). In addition, wage offer volatility may have a higher value in an economic recession (Gronau [3]) than in an economic expansion. Therefore, we extend the model by allowing wage offer dynamics and schooling preference (net of education costs) to bemodulated by a two-state Markov process. Since an arithmetic Brownian motion allows for a negative wage offer, we utilize a geometric Brownian motion as the wage offer process in this paper. Our methodology to derive the closed-form solutions is in line with that used in the real option framework with regime switching, such as Dixit and Pindyck [4], Guo et al. [5], and Bensoussan et al. [6].


Introduction
In the course of schooling as a job search process, the role of uncertainty that gives rise to an option was studied by Fan [1].Fan [1] used an analytic formulation, in which the wage offer process () was modeled as the following arithmetic Brownian motion: where  is a positive constant and   is a standard Brownian motion.If  is the schooling preference net of education costs, an individual's objective is to maximize the expected value  with the schooling duration : where  is the constant discount rate and  was also assumed to be constant in Fan [1].However, this abstraction may not be able to fully capture reality.Specifically, in an economic recession, the desire to obtain education may rise; for example, graduate program enrollment increases in an economic recession (Fan [1]), but the cost of education, for example, tuition fees, may decline (Binder [2]).In addition, wage offer volatility may have a higher value in an economic recession (Gronau [3]) than in an economic expansion.Therefore, we extend the model by allowing wage offer dynamics and schooling preference (net of education costs) to be modulated by a two-state Markov process.Since an arithmetic Brownian motion allows for a negative wage offer, we utilize a geometric Brownian motion as the wage offer process in this paper.Our methodology to derive the closed-form solutions is in line with that used in the real option framework with regime switching, such as Dixit and Pindyck [4], Guo et al. [5], and Bensoussan et al. [6].

The Model
We assume that a risk-neutral individual receives a wage offer (), which follows a geometric Brownian motion with coefficients modulated by a two-state Markov process as follows: where (  ) ≥0 is a standard Brownian motion under a given probability space (Ω, F, P) and (Θ  ) ≥0 is a two-state Markov process defined on (Ω, F, P) and can take value 1 or 2. We also assume that (Θ  ) ≥0 is independent of (  ) ≥0 and has the following generator: For each regime , ( = 1, 2),   is a constant drift and   is a constant volatility.We allow that schooling preference (net of education costs), denoted by   =  Θ  , is also regime switching.
We assume the following as in Fan [1] (see also verifications therein).An individual will stop schooling at time  to accept a wage offer, which will be his constant wage throughout the remainder of the individual's infinite lifetime.Further, this stopping process is irreversible.Then an individual's objective is to choose optimally the schooling duration   that maximizes the expected value   (), which consists of the present value of lifetime earnings and schooling preference (net of education costs) for each regime , ( = 1, 2),   () fl max where  is the discount rate and   can be considered as an optimal stopping time of education.We denote ( 1 ) :=  1 , ( 2 ) :=  2 and we only consider the case where is called the reservation wage in labor economics.

Numerical Examples
The drift term  Θ  in the wage offer dynamics in (3) is the exponential growth rate of the wage offer and can be regarded as depending on an individual's skill and ability but not on labor market conditions or macroeconomic conditions.Therefore, we set  1 =  2 in our numerical examples.On the other hand, we use  1 ≤  2 and  1 ≤  2 .That is, we assume that, in regime 2, schooling preference (net of education costs) and wage offer volatility are higher than those in regime 1.
In Figures 1(a) and 1(b), we plot the reservation wage   against the transition intensity   .Figures 1(c) and 1(d) state that a higher wage offer volatility   yields a higher reservation wage   .These results are consistent with those of Fan [1] in the sense that the wage offer volatility yields an option value to schooling.An interesting result, however, is that  1 (resp.,  2 ) has an impact on  2 (resp.,  1 ) and the schooling decision in a regime is dependent on the wage offer volatility in the other regime.The role of schooling preference (net of education costs) is illustrated in Figures 1(e) and 1(f).We see that more preference in education provides an individual more incentive to postpone starting work.Again, the schooling decision in a regime is dependent on schooling preference (net of education costs) in the other regime.Lastly, an individual's opportunity cost of schooling increases with the discount rate and this is explored in Figure 1(g).