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

Authors: Yong Hyun Shin; Ho-Seok Lee
Journal: Advances in Mathematical Physics
 (2015)

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## 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.


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## Body

## 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 w(t) was modeled as the following arithmetic Brownian motion: (1)dwt=gdt+σdBt,where g is a positive constant and Bt is a standard Brownian motion. If Q is the schooling preference net of education costs, an individual’s objective is to maximize the expected value V with the schooling duration τ: (2)V≔E∫0τe-rtQdt+∫τ∞e-rtwτdt,where r is the constant discount rate and Q 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 offerw(t), which follows a geometric Brownian motion with coefficients modulated by a two-state Markov process as follows:(3)dwt=μΘtwtdt+σΘtwtdBt,w0=w>0,where (Bt)t≥0 is a standard Brownian motion under a given probability space (Ω,F,P) and (Θt)t≥0 is a two-state Markov process defined on (Ω,F,P) and can take value 1 or 2. We also assume that (Θt)t≥0 is independent of (Bt)t≥0 and has the following generator:(4)Λ=-λ1λ1λ2-λ2,λ1>0,λ2>0.For each regime i,(i=1,2), μi is a constant drift and σi is a constant volatility. We allow that schooling preference (net of education costs), denoted by Qt=QΘt, is also regime switching.We assume the following as in Fan [1] (see also verifications therein). An individual will stop schooling at time τ 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 τi that maximizes the expected value Vi(w), which consists of the present value of lifetime earnings and schooling preference (net of education costs) for each regime i,(i=1,2),(5)Viw≔maxτiE∫0τie-rtQΘtdt+∫τi∞e-rtwτidt∣Θ0=i,w0=w=maxτiE∫0τie-rtQΘtdt+e-rτiwτir∣Θ0=i,w0=w,where r is the discount rate and τi can be considered as an optimal stopping time of education. We denote w(τ1):=w¯1, w(τ2):=w¯2 and we only consider the case where(6)w¯1<w¯2.w¯i is called the reservation wage in labor economics.Remark 1.
For each regimei,(i=1,2), we define the functions fi(·) and G(·): (7)fim≔-12σi2mm-1-μim+r+λi,Gm≔f1mf2m-λ1λ2.Let n1<0 and n2>0 be two real roots of the quadratic equation f2(n)=0. We also consider the quartic equation G(m)=0, which has four real roots m1<m2<0<m3<m4.Theorem 2.
Under assumption (6), for each regime i,(i=1,2), the value function Vi(·) is derived as follows: (8)
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. The values of A3,A4,C1,C2,w¯1, and w¯2 are expanded in the proof.Proof.
From (5), for 0<w<w¯1, we obtain the following coupled Bellman equations:(9)L1V1w-Q1=λ1V2w,L2V2w-Q2=λ2V1w,where the differential operator Li is given by (10)Li•=-12σi2w2∂2•∂w2-μiw∂•∂w+r+λi•, for each regime i,(i=1,2). Thus the solutions to (9) are of the forms (11)V1w=A3wm3+A4wm4+a,V2w=f1m3λ1A3wm3+f1m4λ1A4wm4+b, where (12)a=r+λ2Q1+λ1Q2r+λ1r+λ2-λ1λ2,b=r+λ1Q2+λ2Q1r+λ1r+λ2-λ1λ2.
Forw¯1≤w<w¯2, we have the equation(13)L2V2w-Q2=λ2wr,since V1(w)=w/r. Thus the solution to (13) is of the form (14)V2w=C1wn1+C2wn2+λ2rr-μ2+λ2w+Q2r+λ2.
Now we determineA3,A4,C1,C2,w¯1, and w¯2 using the C1-conditions of V1 and V2 at w=w¯1 and w=w¯2. The C1-condition of V1(w) at w=w¯1 implies(15)A3=1/rm4-1w¯1-am4m4-m3w¯1m3,A4=1/r1-m3w¯1+am3m4-m3w¯1m4.The C1-condition of V2(w) at w=w¯2 yields(16)C1=n2-1r-μ2/rr-μ2+λ2w¯2-n2Q2/r+λ2n2-n1w¯2n1,C2=1-n1r-μ2/rr-μ2+λ2w¯2+n1Q2/r+λ2n2-n1w¯2n2.From the C1-condition of V2(w) at w=w¯1 we obtain the equations(17)f1m3λ1A3w¯1m3+f1m4λ1A4w¯1m4+b=C1w¯1n1+C2w¯1n2+λ2rr-μ2+λ2w¯1+Q2r+λ2,(18)m3f1m3λ1A3w¯1m3+m4f1m4λ1A4w¯1m4=n1C1w¯1n1+n2C2w¯1n2+λ2rr-μ2+λ2w¯1.Plugging (15) into the LHS of (17) implies(19)LHSof17=f1m3λ11/rm4-1w¯1-am4m4-m3+f1m4λ11/r1-m3w¯1+am3m4-m3+b=m4-1f1m3+1-m3f1m4rλ1m4-m3ξw¯2+m3f1m4-m4f1m3λ1m4-m3a+b,and plugging (16) into the RHS of (17) implies(20)RHSof17=n2-1r-μ2/rr-μ2+λ2w¯2-n2Q2/r+λ2n2-n1ξn1+1-n1r-μ2/rr-μ2+λ2w¯2+n1Q2/r+λ2n2-n1ξn2+λ2rr-μ2+λ2ξw¯2+Q2r+λ2=1rr-μ2+λ2r-μ2n2-1ξn1+1-n1ξn2n2-n1+λ2ξw¯2+Q2r+λ2n1ξn2-n2ξn1n2-n1+1,with the relationship(21)w¯1=ξw¯2,ξ∈0,1.From (19) and (20), we obtain(22)w¯2=m3f1m4-m4f1m3/λ1m4-m3a+b-Q2/r+λ2n1ξn2-n2ξn1/n2-n1+11/rr-μ2+λ2r-μ2n2-1ξn1+1-n1ξn2/n2-n1+λ2ξ-m4-1f1m3+1-m3f1m4/rλ1m4-m3ξ.
Also plugging (15) into the LHS of (18) implies(23)LHSof18=m3f1m3λ11/rm4-1w¯1-am4m4-m3+m4f1m4λ11/r1-m3w¯1+am3m4-m3=m3m4-1f1m3+m41-m3f1m4rλ1m4-m3ξw¯2+m3m4f1m4-f1m3λ1m4-m3a,and plugging (16) into the RHS of (18) implies(24)RHSof18=n1n2-1r-μ2/rr-μ2+λ2w¯2-n2Q2/r+λ2n2-n1ξn1+n21-n1r-μ2/rr-μ2+λ2w¯2+n1Q2/r+λ2n2-n1ξn2+λ2rr-μ2+λ2ξw¯2=1rr-μ2+λ2r-μ2n1n2-1ξn1+n21-n1ξn2n2-n1+λ2ξw¯2+n1n2Q2ξn2-ξn1r+λ2n2-n1,with condition (21). From (23) and (24), we obtain(25)w¯2=m3m4f1m4-f1m3/λ1m4-m3a-n1n2Q2ξn2-ξn1/r+λ2n2-n11/rr-μ2+λ2r-μ2n1n2-1ξn1+n21-n1ξn2/n2-n1+λ2ξ-m3m4-1f1m3+m41-m3f1m4/rλ1m4-m3ξ.From (22) and (25), we derive the following algebraic equation with respect to ξ(26)m3f1m4-m4f1m3/λ1m4-m3a+b-Q2/r+λ2n1ξn2-n2ξn1/n2-n1+11/rr-μ2+λ2r-μ2n2-1ξn1+1-n1ξn2/n2-n1+λ2ξ-m4-1f1m3+1-m3f1m4/rλ1m4-m3ξ=m3m4f1m4-f1m3/λ1m4-m3a-n1n2Q2ξn2-ξn1/r+λ2n2-n11/rr-μ2+λ2r-μ2n1n2-1ξn1+n21-n1ξn2/n2-n1+λ2ξ-m3m4-1f1m3+m41-m3f1m4/rλ1m4-m3ξ,and if there exists a unique solution ξ∈(0,1) to (26), we can sequentially determine w¯2, w¯1, A3, A4, C1, and C2.


## 3. Numerical Examples

The drift termμΘt 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 μ1=μ2 in our numerical examples. On the other hand, we use σ1≤σ2 and Q1≤Q2. 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 Figures1(a) and 1(b), we plot the reservation wage w¯i against the transition intensity λi. Figures 1(c) and 1(d) state that a higher wage offer volatility σi yields a higher reservation wage w¯i. 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 σ1(resp.,σ2) has an impact on w¯2(resp.,w¯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).Figure 1
Numerical implications.
(a)
Reservation wage with respect toλ1 if r=0.03, μ1=0.02, σ1=0.2, Q1=10, μ2=0.02,σ2=0.4,Q2=15,λ2=0.2
(b)
Reservation wage with respect toλ2 if r=0.03,μ1=0.02,σ1=0.2,Q1=10,λ1=0.2,μ2=0.02,σ2=0.4,Q2=15
(c)
Reservation wage with respect toσ1 if r=0.03,μ1=0.02,Q1=10,λ1=0.2,μ2=0.02,σ2=0.4,Q2=15,λ2=0.2
(d)
Reservation wage with respect toσ2 if r=0.03,μ1=0.02,σ1=0.2,Q1=10,λ1=0.2,μ2=0.02,Q2=15,λ2=0.2
(e)
Reservation wage with respect toQ1 if r=0.03,μ1=0.02,σ1=0.2,λ1=0.2,μ2=0.02,σ2=0.4,Q2=15,λ2=0.2
(f)
Reservation wage with respect toQ2 if r=0.03,μ1=0.02,σ1=0.2,Q1=10,λ1=0.2,μ2=0.02,σ2=0.4,λ2=0.2
(g)
Reservation wage with respect tor if μ1=0.02,σ1=0.2,Q1=10,λ1=0.2,μ2=0.02,σ2=0.4,Q2=15,λ2=0.2



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*Source: 475279-2015-11-24.xml*