Abstract
We propose a regime switching model of schooling choice as a job search process. We adopt a two-state Markov process and the derived coupled Bellman equations are solved by seeking the root of an auxiliary algebraic equation. Some numerical examples are also considered.
1. 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].
2. 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 is a standard Brownian motion under a given probability space and is a two-state Markov process defined on and can take value or . We also assume that is independent of and has the following generator:For each regime , is a constant drift and is a constant volatility. We allow that schooling preference (net of education costs), denoted by , 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 ,where is the discount rate and can be considered as an optimal stopping time of education. We denote , and we only consider the case where is called the reservation wage in labor economics.
Remark 1. For each regime , we define the functions and : Let and be two real roots of the quadratic equation . We also consider the quartic equation , which has four real roots
Theorem 2. Under assumption (6), for each regime , the value function is derived as follows: The values of , and are expanded in the proof.
Proof. From (5), for , we obtain the following coupled Bellman equations:where the differential operator is given by for each regime . Thus the solutions to (9) are of the forms where For , we have the equationsince . Thus the solution to (13) is of the form Now we determine , and using the -conditions of and at and . The -condition of at impliesThe -condition of at yieldsFrom the -condition of at we obtain the equationsPlugging (15) into the LHS of (17) impliesand plugging (16) into the RHS of (17) implieswith the relationshipFrom (19) and (20), we obtainAlso plugging (15) into the LHS of (18) impliesand plugging (16) into the RHS of (18) implieswith condition (21). From (23) and (24), we obtainFrom (22) and (25), we derive the following algebraic equation with respect to and if there exists a unique solution to (26), we can sequentially determine , , , , , and .
3. Numerical Examples
The drift term 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 in our numerical examples. On the other hand, we use and . That is, we assume that, in regime , schooling preference (net of education costs) and wage offer volatility are higher than those in regime .
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 has an impact on 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).
(a) Reservation wage with respect to if , , , ,
(b) Reservation wage with respect to if
(c) Reservation wage with respect to if
(d) Reservation wage with respect to if
(e) Reservation wage with respect to if
(f) Reservation wage with respect to if
(g) Reservation wage with respect to if
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
Yong Hyun Shin gratefully acknowledges the support of the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2058027).