Employer Incentives for Providing Informal On-the-job Training in the Presence of On-the-job Search

Abstract

We analyze the provision of informal general training in a frictional labor market in which employers cannot commit to training levels and workers cannot commit to stay. We demonstrate that employers’ training decisions are driven by both an investment motive, to improve productivity, and a compensation motive, to increase employee retention. The investment motive decreases with higher wages, while the compensation motive increases. In our calibration exercises, the former dominates, which creates a negative relationship between wages and training. Furthermore, in contrast to recent studies missing the compensation motive, lessening the search frictions raises overall training levels due to enhanced compensation motives, approaching Becker’s result for a frictionless labor market.

Appendix: Details of the Numerical Algorithm

In this subsection, I describe an alternative way of solving the model by converting the fixed point problem in the functional space into a fixed point problem in a finite dimensional vector space.

Taking the derivative of Eqs. 14 and 17 with respect to ϕ yields

$$\begin{array}{@{}rcl@{}} \lefteqn{[r+\delta+\rho+\lambda[1-F(\phi)]+\mu_{i}x^{*}_{i}]\frac{dE_{i}(\phi, x^{*}(\phi))}{d\phi}}\\ &&=y_{i}+\mu_{i}x^{*}_{i}\frac{dE_{i + 1}(\phi, x^{*}_{i + 1}(\phi))}{d\phi}+\mu_{i}(E_{i + 1}(\phi, x^{*}_{i + 1})-E_{i}(\phi, x^{*}_{i}))\frac{dx^{*}_{i}(\phi)}{d\phi},~~\text{and} \end{array} $$

(31)

$$\begin{array}{@{}rcl@{}} \lefteqn{[r+\rho+\delta+\lambda[1-F(\phi)]+\mu_{i}x^{*}_{i}]\frac{dJ_{i}(\phi)}{d\phi}}\\ && =-y_{i} -c^{\prime}(x^{*}_{i})\frac{dx^{*}_{i}}{d\phi}+ \mu_{i}\frac{dx^{*}_{i}(\phi)}{d\phi} (J_{i + 1}(\phi)-J_{i}(\phi))+\mu_{i}x^{*}_{i}\frac{dJ_{i + 1}(\phi)}{d\phi}+\lambda\frac{dF(\phi)}{d\phi}J_{i}(\phi). \end{array} $$

(32)

Plugging (31) into (11) yields

$$\begin{array}{@{}rcl@{}} \lefteqn{c^{\prime}(x^{*}_{i}(\phi)) = \mu_{i}(J_{i + 1}(\phi)-J_{i}(\phi))}\\ && + \lambda \frac{dF(\phi)}{d\phi}\left[\frac{y_{i}-c(x^{*}_{i})+\mu_{i}x^{*}_{i}(\phi)\frac{dE_{i + 1}}{d\phi}}{\mu_{i}(E_{i + 1}(\phi,x^{*}_{i + 1}(\phi))-E_{i}(\phi,x^{*}_{i}(\phi)))}+\frac{dx^{*}_{i}(\phi)}{d\phi}\right]^{-1}J_{i}(\phi).~~~~~ \end{array} $$

(33)

Equation 33 shows how $x^{*}_{i}(\cdot )$ proceeds to achieve the first order condition. By taking the derivative of Eq. 18 with respect to ϕ, I obtain

$$ \sum\limits_{i = 1}^{n}\left[(\lambda_{u}u_{i}+\lambda G_{i}(\phi))\frac{dJ_{i}(\phi)}{d\phi}+\lambda\frac{dG_{i}(\phi)}{d\phi}J_{i}(\phi)\right]= 0.~~~ $$

(34)

Plugging (24), (26), and (32) into (34) yields

$$\begin{array}{@{}rcl@{}} {\sum\limits_{i}^{n}}\left[(\lambda_{u}u_{i}+\lambda G_{i}(\phi))\frac{-y_{i}-c^{\prime}(x_{i}^{*})\frac{dx^{*}_{i}}{d\phi}+\mu_{i}\frac{dx^{*}_{i}}{d\phi}(J_{i + 1}(\phi)-J_{i}(\phi))+\mu_{i}x^{*}_{i}(\phi)\frac{dJ_{i + 1}(\phi)}{d\phi}} {r+\delta+\rho+\lambda(1-F(\phi))+\mu_{i}x_{i}(\phi)}\right.\\ +\left.\lambda\frac{\mu_{i-1}x^{*}_{i-1}(\phi)g_{i-1}(\phi)}{\rho+\delta+\lambda(1-F(\phi))+\mu_{i}x^{*}_{i}}J_{i}(\phi)\right]~~\\ +{\sum\limits_{i}^{n}}\left[\frac{(\lambda_{u}u_{i}+\lambda G_{i}(\phi))\lambda J_{i}(\phi)}{r+\rho+\delta+\lambda(1-F(\phi))+\mu_{i}x^{*}_{i}}+\frac{(\lambda_{u}u_{i}+\lambda G_{i}(\phi))\lambda J_{i}(\phi)}{\rho+\delta+\lambda(1-F(\phi))+\mu_{i}x^{*}_{i}}\right]\frac{dF(\phi)}{d\phi}= 0. \end{array} $$

(35)

Equation 35 shows how F evolves with ϕ to maintain the equal profit. Consequently, the steady state equilibrium is described as follows.

• (i) Given U _n , our equilibrium selection rule, $U_{n}=E_{n}(\underline {\phi }, 0)$, yields $\underline {\phi }$.

• (ii) Given $\{U_{i}\}_{i = 1}^{n-1}$ and $\underline {\phi }$, the initial condition $U_{i}=E_{i}(\underline {\phi }, x^{*}_{i}(\underline {\phi }))$ together with (14) yields $x^{*}_{i}(\underline {\phi })$, which in turn determines $J_{i}(\underline {\phi })$ for each i using (17).

• (iii) Given $\{U_{i}=E_{i}(\underline {\phi }, x^{*}_{i}(\underline {\phi })), u_{i}\}_{i = 1}^{n}$, the system of the differential Eqs. 22, 31, 32, 33 and 35 with $F(\underline {\phi })= 0$, $\{G_{i}(\underline {\phi })= 0\}_{i = 1}^{n}$, and the initial conditions in (i) and (ii) yields $\{E_{i}, J_{i}, x^{*}_{i}, G_{i}\}_{i = 1}^{n}$ and F.

• (iv) The terminal condition $F(\overline {\phi })= 1$ determines $\overline {\phi }$.

• (v) Given $\{x^{*}_{i}, G_{i}\}_{i = 1}^{n}$ and $[\underline {\phi }, \overline {\phi }]$, (25) and (27) jointly restore $\{u_{i}\}_{i = 1}^{n}$.

• (vi) Given $\{E_{i}, J_{i}, x^{*}_{i}, G_{i}\}_{i = 1}^{n}$ and $[\underline {\phi }, \overline {\phi }]$, (13) restores $\{U_{i}\}_{i = 1}^{n}$.

One advantage of this alternative prescription is that it pins down the fixed point problem in a functional space into the fixed point problem in a finite dimensional vector space. In particular, if I assume n number of different types, I need to solve for the system of 4 × n differential equations in (iii). Note that $x^{*}_{n}= 0$ reduces one equation, but F requires one more equation. From (i), (ii) and (iii), I also have 4 × n initial conditions. Thus, I just need to solve for the fixed point of $\{U_{i}\}_{i = 1}^{n}$ in n dimensional vector space. The monotonicity condition and (12) should be checked to ensure the existence of the optimal decision. The existence of the equilibrium fixed point of $\{U_{i}\}_{i = 1}^{n}$ is not proved analytically. Thus, the proof is replaced with the numerical examples.