Search Costs and Wage Inequality

Appendix A: The optimization problem of the production sector

We consider the optimization problem of the representative firm in the production sector. From Eqs. (1)–(3) and (7), we know:

Since the firm cannot control the market tightness of the skilled labor market and the unskilled labor market, given θ _s and θ _u , we obtain first-order conditions as follows:

From Eqs. (A-4) and (A-5), it is easy to obtain d λ 1 d t = − r γ s f s ( θ s ) e − r t and d λ 2 d t = − r γ u f u ( θ u ) e − r t . Then, it is easy to obtain the job creation functions of the production sector, i.e. ( 1 − τ ) ∂ Y ∂ L s Y = w s + γ s f s ( δ s + r ) , and ( 1 − τ ) ∂ Y ∂ L u Y = w u + γ u f u ( δ u + r ) .

According to the production function, we can obtain the following job creation functions:

Furthermore, similar to Zenou (2008, 2009), we can figure out the following Bellman functions:

where i = s, u.

For Eq. (A-8), a vacancy is an asset owned by the firm. The expected return of the vacancy r I i V equals that the expected net gain f i ( θ i ) ( I i J − I i V ) from this job minus the search cost γ i to fill this vacancy. Actually, we can consider it in the following sense. To fill this vacancy, the firm needs to pay the search cost γ i , and then the filled job can make a gain I i J with the opportunity cost I i V . Furthermore, the vacancy cannot directly create any gain. Thus, when the matching process is unsuccessful, the vacancy will be virtually valueless. Thus, the expected net gain is f i ( θ i ) ( I i J − I i V ) − ( 1 − f i ( θ i ) ) × 0 − γ i . For Eq. (A-9), the expected return of the job r I i J equals the net gain created by labor ( 1 − τ ) Y i minus the wage to labor w _i and the loss induced by the destruction of the job δ i I i J .

Due to free entry, I i V = θ i I i V = 0 . Thus, we obtain: I s J − I s V = ( 1 − τ ) Y s − w s + θ s γ s r + δ s + θ s f s , and I u J − I u V = ( 1 − τ ) Y u − w u + θ u γ u r + δ u + θ u f u .

Appendix B: The proof of Proposition 1

First of all, following Beladi, Chaudhuri, and Yabuuchi (2008) and Konishi, Okuno-Fujiwara, and Suzumura (1990), we consider the dynamic adjustment process as follows:

where d _i  > 0 (i = 1, … , 8) denotes the adjustment speed and x ˙  = dx/dt.

Thus, Eqs. (B-1)–(B-8), we can get the following Jacobi matrix:

Then, we know | Γ | = L u Y ( 1 − u s ) L s ∏ i = 1 8 d i | J | > 0 . When the system is stable, the signs of coefficient matrices in Section 2, 3, and 5 are positive. Thus, | J | = u s ( f s + θ s f s ′ ) ( 1 − η ) λ Y ( 1 − η ) ( 1 − λ ) u u ( f u + θ u f u ′ ) [ ( ( 1 − τ ) Y 1 − u s + δ s + θ s f s u s ( f s + θ s f s ′ ) ( η − ( δ s + r ) f s ′ f s 2 ) γ s ( 1 − u s ) L s ( 1 − η ) λ − ( w s u L s + λ Y 1 − u s ) ) ( ( 1 − τ ) Y 1 − u u + δ u + θ u f u u u ( f u + θ u f δ u + θ u f u u u ( f u + θ u f u ′ ) ) ( η − ( δ u + r ) f u ′ f u 2 ) γ u ( 1 − u u ) L u ( 1 − η ) ( 1 − λ ) − ( w u u L u + ( 1 − λ ) Y 1 − u u ) ) − ( w s u L s + λ Y 1 − u s ) ( w u u L u + ( 1 − λ ) Y 1 − u u ) ] > 0, which implies that ρ s + ρ u < 1 , where ρ s = ( w s u L s + λ Y 1 − u s ) / ( ( 1 − τ ) Y 1 − u s + δ s + θ s f s u s ( f s + θ s f s ′ ) ( σ η − ( 1 + σ η ) ( δ s + r ) f s ′ f s 2 ) γ s ( 1 − u s ) L s λ ) , ρ u = ( w u u L u + ( 1 − λ ) Y 1 − u u ) / ( ( 1 − τ ) Y 1 − u u + δ u + θ u f u u u ( f u + θ u f u ′ ) ( σ η − ( 1 + σ η ) δ u + r f u 2 f u ′ ) γ u ( 1 − u u ) L u 1 − λ ) , and σ η = η 1 − η .

According to Eq. (20), we can obtain the following equations:

We can easily obtain that the signs of ∂ τ ∂ γ s , ∂ u s ∂ γ s , and ∂ u u ∂ γ s are positive and the signs of ∂ Y ∂ γ s , ∂ w s ∂ γ s , ∂ w u ∂ γ s , ∂ θ s ∂ γ s , and ∂ θ u ∂ γ s are negative, which implies that when skilled search costs increase, the output of the economy and the unskilled wage rate will decrease with a higher tax rate and a higher unemployment rate in the labor market.

From Eqs. (B-11) and (B-12), we know:

When | M R T S L s L u | = Y s Y u > ( δ s + r f s + η θ s ) ( 1 − ( δ u + r ) f u ′ f u 2 ) ( θ s + δ s + r f s ) ( η − ( δ u + r ) f u ′ f u 2 ) , an increase of skilled search costs will decrease skilled-unskilled wage inequality.

Appendix C: The proof of Proposition 2

Actually, it is easy to find that when the sign of ∂ u s ∂ γ s is positive, the dynamic adjustment system in Appendix B is stable. It is easy to figure out the sign of the change of the variables. All the signs are the same as those in Proposition 1, except the change of the wage rate.

According to Eqs. (C-3) and (C-4), we obtain: