Firm financed training and pareto improving firing taxes

Abstract

This paper analyses how the under-investment in firm financed training caused by hold up can justify the introduction of firing taxes in a laissez-faire economy with search frictions and risk neutral agents. In particular, we show two main results. First, the introduction of a firing tax for newly hired workers combined with hiring subsidies, always acts as a Pareto improving policy. Second, with no hiring subsidies, the introduction of a firing tax for the newly hired always increase the welfare of employed while its impact on the welfare of unemployed depends on the returns to training. Hence, policy implications are derived.

Appendices

Appendix 1: The local maximum

Firms maximize their expected profit with respect to training and threshold productivity, i.e. $MaxJ_{{\varepsilon^{d} ,h}}^{0} \left( {\varepsilon^{e} ,h} \right)$. The equilibrium wage at entry does not vary when firms make their optimal choices, so that the expected profit is

• (i)
  $$J_{0} \left( {\varepsilon^{e} ,h} \right) = \frac{{\varepsilon^{e} - h - w_{0} + \sigma }}{1 + r} - \frac{{\left[ {F\left( {\varepsilon^{d} } \right) - \beta \left( {1 - F\left( {\varepsilon^{d} } \right)} \right)} \right] \cdot T}}{1 + r} - \frac{{\left( {1 - \beta } \right)\left[ {1 - F\left( {\varepsilon^{d} } \right)} \right] \cdot U}}{1 + r} + \frac{(1 - \beta )}{(r + s)(1 + r)}\int\limits_{{\varepsilon^{d} }}^{{\varepsilon_{u} }} {\left( {\varepsilon^{\prime } + ah^{\alpha } } \right)dF\left( {\varepsilon^{\prime } } \right)}$$

  First order conditions are given by Eqs. (11) and (12). To find a local maximum the Hessian matrix must be semi definite negative so that:

  $$\frac{{(1 - \beta )^{2} \alpha (1 - \alpha )h^{\alpha - 2} [1 - F(\varepsilon^{d} )]f(\varepsilon^{d} ) - (1 - \beta )[f(\varepsilon^{d} )]^{2} \alpha^{2} a^{2} h^{2\alpha - 2} }}{{(r + s)^{2} (1 + r)^{2} }} > 0$$

  that is:

• (ii) $\frac{{(1 - \alpha ) \cdot \left[ {1 - F\left( {\varepsilon^{d} } \right)} \right]}}{{a \cdot \alpha \cdot f\left( {\varepsilon^{d} } \right)}} > h^{\alpha }$.

Appendix 2: Existence and uniqueness of equilibrium

Existence Define the four component vector $z \equiv \left( {h,\varepsilon^{d} ,\theta ,U} \right)$. An equilibrium is a fixed point of the correspondence $\varPsi$ from $z$ to $z$, with $\varPsi (z) = \left( {h(z),\varepsilon^{d} (z),\theta (z),U(z)} \right)$, where $\theta (z)$ is defined implicitly by Eq. (13), $U(z)$ is defined by Eq. (14) and $h_{{}} (z)$ and $\varepsilon^{d} (z)$ are chosen to maximize $J_{0} \left( {\varepsilon^{e} ,h} \right)$. The correspondence $\varPsi$ is a best response relation. Thus it is convex valued and has a closed graph. $\varPsi$ maps a convex compact set into itself, so Kukutani’s fixed point theorem implies that $\varPsi$ has a fixed point, which is an equilibrium.

Proof To prove the continuity of functions in vector $z$ is straightforward. The optimal training choice is a continuous function and takes values in the range $0 \le h \le \left[ {\frac{{\alpha a\left( {1 - \beta } \right)}}{(r + s)}} \right]^{{\frac{1}{1 - \alpha }}} \equiv \bar{h}$; then the set of possible values of training is compact. Market tightness is also continuous and assumes values in the range: $0 \le m^{ - 1} (\theta ) < \left[ {\frac{{\left( {1 - \beta } \right)\varepsilon^{e} }}{{k\left( {1 + r} \right)}} + \frac{{\left( {1 - \beta } \right)\left( {\varepsilon + a\bar{h}^{\alpha } } \right)}}{{k\left( {1 + r} \right)\left( {r + s} \right)}}} \right]$. The value of being unemployed can be written as $U = U\left( {h,\varepsilon ,\frac{{r\left( {1 - \beta } \right)}}{\beta k}U} \right)$, with $\theta = \frac{{r\left( {1 - \beta } \right)}}{\beta k}U$, so that $0 < U \le U\left( {\overline{h} ,\varepsilon_{U} } \right) \equiv \overline{U}$. Finally $\underline{\varepsilon } \equiv \varepsilon (0,\overline{h} ) < \varepsilon_{{}}^{d} < \varepsilon (\overline{U} ,0) \equiv \overline{\varepsilon }$, where $\left[ {\varepsilon_{L} ,\varepsilon_{U} } \right]$ is the finite support of the distribution of the idiosyncratic productivity and $h_{\hbox{max} }$ is the maximum amount of training.

Uniqueness of the interior solution If condition (ii) is met, the maximized expected profit can be represented as function of the equilibrium value of being unemployed: $J_{0} (U) = MaxJ_{{_{{\varepsilon^{d} ,h}}^{0} }} \left( {\varepsilon^{e} \left( U \right),h(U),U} \right)$. $J_{0} (U)$ is continuous and decreasing with respect to the value of being unemployed, $\frac{{dJ_{0} }}{dU} < 0$, so that $\frac{{d\theta \left( {J_{0} } \right)}}{dU} < 0$. Considering the positive monotonic relation between the equilibrium market tightness and workers’ outside option, Eq. (14), and the continuous positive relation between $\theta$ and $J_{0} (U)$, Eq. (13), it is verified that: $\frac{{dU\left( {J_{0} ,\theta \left( {J_{0} } \right)} \right)}}{dU} < 0$. Thus there can be only one equilibrium value of $U$and, as a consequence, there is only an equilibrium value of $\theta$. Two different solutions to $\left( {h,\varepsilon^{d} } \right)$ would have to give the same maximal $J_{0} (\varepsilon^{e} ,h)$ and the same $U_{{}}$. However this result is not possible since the sum $J_{0} + E_{0}$ is increasing in $h$ at the equilibrium.

A corner solution A corner equilibrium with no layoffs is also possible for some parameters. When $\varepsilon^{d} = \varepsilon_{L}$, the equilibrium values of training and market tightness are given by Eqs. (12)–(13):

$$h_{\hbox{max} } = \left[ {\frac{\alpha a(1 - \beta )}{r + s}} \right]$$

$$m^{ - 1} (\theta ) = \frac{(1 - \beta )}{k(1 + r)}\left[ {\left( {\left( {\varepsilon^{e} - h_{\hbox{max} } } \right) - rU + \left( {\varepsilon^{e} + ah^{\alpha } } \right)} \right)} \right]$$

while the value of being unemployed is determined by Eq. (14). We note that in Eq. (14), for $\theta = 0$, $U = 0$, and for $\theta \to + \infty$, $U = U_{\hbox{max} }$. In Eq. (13) for $U = 0$, $\theta > 0$, and for $U > 0$, $\frac{\partial \theta }{\partial U} < 0$. This implies a decreasing relationship between job creation condition and workers’ outside option when $\varepsilon^{d} = \varepsilon_{L}$ and $h_{{}} = h_{\hbox{max} }$. Thus, there exists a candidate equilibrium $\left\{ {\varepsilon_{L} ,h_{\hbox{max} } ,\theta ,U} \right\}$.

Appendix 3: Comparative statics with hiring subsidies

Totally differentiating Eqs. (11)–(13), and using the equilibrium budget constraint, $\sigma = F\left( {\overline{\varepsilon }^{d} } \right)T$, one obtains:

$$\frac{d\theta }{dT} = \frac{1}{\varTheta }\left[ {\frac{{\left( {1 - \beta } \right)}}{{\left( {1 + r} \right)}}\beta \frac{dh}{dT} + \frac{{\left( {1 - \beta } \right)}}{{\left( {1 + r} \right)}}\left[ {F\left( {\overline{\varepsilon }^{d} } \right) - F\left( {\overline{\varepsilon }^{d} } \right)} \right] + \frac{{\left( {1 - \beta } \right)f\left( {\overline{\varepsilon }^{d} } \right)T}}{{\left( {1 + r} \right)}}} \right]$$

where, as in “Appendix 3”, $\varTheta = \frac{\eta \cdot \theta }{m(\theta )} + \frac{{r + \left[ {1 - F\left( {\varepsilon_{f}^{d} } \right)} \right]}}{1 + r} \cdot \frac{\beta k}{r(1 - \beta )}$. Given that $\overline{\varepsilon }^{d} = \varepsilon^{d}$ and evaluating job creation at T = 0, it reduces to: $\left. {\frac{d\theta }{dT}} \right|_{T = 0} = \frac{1}{\varTheta }\frac{(1 - \beta )}{(1 + r)}\beta \frac{dh}{dT}$. From Eq. (14) we have $\frac{dU}{dT} = \frac{{\beta^{2} k}}{\varTheta \cdot r(1 + r)}\frac{dh}{dT}$ so that, on the job destruction condition, will be:

$$\left. {\frac{{d\varepsilon_{{}}^{d} }}{dT}} \right|_{T = 0} = \frac{ - (r + s)}{{\left[ {\underbrace {{1 - \frac{{f\left( {\varepsilon_{f}^{d} } \right)a^{2} \alpha^{2} \left( {1 - \beta } \right)h^{2\alpha - 1} }}{{\left( {1 - \alpha } \right)\left( {r + s} \right)}}}}_{ > 0} + (r + s) \cdot \frac{{\beta^{2} k}}{\varTheta \cdot r(1 + r)} \cdot \frac{{f\left( {\varepsilon^{d} } \right)a\alpha (1 - \beta )h^{\alpha } }}{(1 - \alpha )(r + s)}} \right]}} < 0$$

Training is inversely related to the threshold productivity level, when firing cost varies, i.e. $\left. \frac{dh}{dT} \right|_{T = 0} = - \frac{{f\left( {\varepsilon^{d} } \right)}}{{(1 - \alpha )\left[ {1 - F\left( {\varepsilon^{d} } \right)} \right]h^{\alpha - 2} }} \cdot \frac{{d\varepsilon^{d} }}{dT} > 0$ and, as a consequence, $\left. {\frac{d\theta }{dT}} \right|_{T = 0} > 0$ and $\left. \frac{dU}{dT} \right|_{T = 0} > 0$.

Appendix 4: Comparative statics without hiring subsidies

With no hiring subsidies, the introduction of firing costs implies:

$$\frac{d\theta }{dT} = \frac{(1 - \beta )}{\varTheta (1 + r)}\left[ {\beta \frac{dh}{dT} - F\left( {\varepsilon^{d} } \right)} \right]$$

Totally differentiating Eqs. (11)–(13), and setting σ = 0, one obtains:

$$\left. {\frac{{d\varepsilon^{d} }}{dT}} \right|_{T = 0} = \frac{{ - (r + s)\left[ {\frac{{\varTheta \cdot r(1 + r) + (r + s)\beta kF\left( {\varepsilon^{d} } \right)}}{\varTheta \cdot r(1 + r)}} \right]}}{{\left[ {\underbrace {{1 - \frac{{f\left( {\varepsilon^{d} } \right)a^{2} \alpha^{2} (1 - \beta )h^{2\alpha - 1} }}{(1 - \alpha )(r + s)}}}_{ > 0} + (r + s) \cdot \frac{{\beta^{2} k}}{\varTheta \cdot r(1 + r)} \cdot \frac{{f\left( {\varepsilon^{d} } \right)a\alpha (1 - \beta )h^{\alpha } }}{(1 - \alpha )(r + s)}} \right]}} < 0$$

where $\varTheta = \frac{\eta \cdot \theta }{m(\theta )} + \frac{{r + \left[ {1 - F\left( {\varepsilon_{f}^{d} } \right)} \right]}}{1 + r} \cdot \frac{\beta k}{r(1 - \beta )}$ and $\left. \frac{dh}{dT} \right|_{T = 0} = - \frac{{f\left( {\varepsilon^{d} } \right)}}{{(1 - \alpha )[1 - F\left( {\varepsilon^{d} } \right)]h^{\alpha - 2} }} \cdot \frac{{d\varepsilon^{d} }}{dT} > 0.$

The sign of the effect on market tightness is not clear a priori.