Short or long-term contract? Firm’s optimal choice

Abstract

This article studies the behaviour of a firm searching to fill a vacancy. The main assumption is that the firm can offer two different kinds of contracts to the workers, either a short-term contract or a long-term one. The short-term contract acts as a probationary stage in which the firm can learn about the worker. After this stage, the firm can propose a long-term contract to the worker or it can decide to look for another worker. We show that, if the short-term wage is fixed endogenously, it can be optimal for firms to start a working relationship with a short-term contract, but that this policy decreases unemployment and welfare. On the contrary, if the wage is fixed exogenously, this policy could be optimal also from a welfare point of view.

Appendices

Appendix 1: Search equilibrium on the SL policy with w _O exogenous

Proof of Proposition 2: First, observe that the set X of workers that the firm accepts in SL contract is always an interval (x ⁺, 1], some x ⁺. Indeed, by the definition of undominated equilibrium, ${\frac{x}{1-\delta (1-\beta )}}>\delta \Uppi^{SL}(x)$. Hence for any $x^{\prime }>x,{\frac{x^{\prime}}{1-\delta (1-\beta )}}\geq \delta \Uppi^{SL}$.

When w _O is fixed exogenously, by Proposition 1, all the workers with $x\in ({\frac{w_{O}}{\gamma }},1]$ will always reject the STC.

Thus, the search problem faced by the firm may be rewritten, from (6) above, as:

$$ \max \Uppi^{SL}={\frac{\kappa \left[ q\delta (1-\beta )\left( \int_{0}^{x^{-}}xf(x)dx-w_{0}\right) \right] +q\delta ^{2}(1-\beta )^{2}\left[ (\int_{x^{-}}^{x^{+}}xf(x)dx)-Ew_{sl}\right]}{\kappa \left[ 1-\delta (1-qE)-q\delta ^{2}(1-\beta )(1-\int_{x^{+}}^{1}f(x)dx)\right]}} $$

or

$$ \max \Uppi^{SL}={\frac{\kappa \left[ q\delta (1-\beta )\left( \int_{0}^{x^{-}}xf(x)dx-w_{0}\right) \right] +q\delta ^{2}(1-\beta )^{2}\left[ C(\int_{x^{-}}^{x^{+}}xf(x)dx)-D\right]}{\kappa \left[ 1-\delta +qE\delta -q\delta ^{2}(1-\beta )(1-\int_{x^{+}}^{1}f(x)dx) \right]}} $$

(21)

where $\kappa =(1-\delta (1-\beta )), Ew_{sl}={\frac{\gamma (\int\nolimits_{x^{-}}^{x^{+}}xf(x)dx)+\alpha \delta (1-\beta )w_{O}}{1+\alpha \delta (1-\beta )}}, x^{+}={\frac{w_{O}}{\gamma}}, E$ is the probability to meet a worker in the set [0, x ⁺).

$$ C={\frac{1-\gamma +\alpha \delta (1-\beta )}{1+\alpha \delta (1-\beta )}}\hbox{ and }D={\frac{\alpha \delta (1-\beta )w_{O}}{1+\alpha \delta (1-\beta )}}. $$

The first-order conditions with respect to x ⁻ are given by

$$ \begin{aligned} &-x^{-}f(x^{-})Cq\delta ^{2}(1-\beta )^{2}(1-\delta (1-\beta ))\left[ \left( 1-\delta +qE\delta -q\delta ^{2}(1-\beta )(1-\int\limits_{x^{-}}^{1}f(x)dx)\right) \right]\\ &\quad+q\delta ^{2}(1-\beta )f(x)(1-\delta (1-\beta ))\\ &\qquad\left[ q\delta ^{2}(1-\beta )^{2}\left[ C(\int\limits_{x^{-}}^{x^{+}}xf(x)dx)-D\right] +(1-\delta (1-\beta ))\left[ q\delta (1-\beta )\left( \int\limits_{0}^{x^{+}}xf(x)dx-w_{0}\right) \right] \right]=0 \end{aligned} $$

or

$$ x^{-}={\frac{q\delta ^{2}(1-\beta )\left( \int\limits_{x^{-}}^{x^{+}}xf(x)dx)-\int\limits_{x^{-}}^{1}f(x)dx-{\frac{D}{C}} \right) +{\frac{\kappa }{C}}\left[ q\delta \left( \int\limits_{0}^{x^{+}}xf(x)dx-w_{0}\right) \right] } {\left[ 1-\delta +qE\delta -q\delta ^{2}(1-\beta )\right] }} $$

(22)

To check that the solution is unique, observe that the left-hand side of Eq. 22 is increasing in x ⁻, with range (0,1). On the other hand, the right-hand side of Eq. 22 is decreasing in x ⁻, falling from

$$ {\frac{q\delta ^{2}(1-\beta )\left( \int\nolimits_{x^{-}}^{x^{+}}xf(x)dx)-\int\nolimits_{x^{-}}^{1}f(x)dx-{\frac{D}{C}} \right) +{\frac{\kappa}{C}}\left[ q\delta \left( \int\nolimits_{0}^{x^{+}}xf(x)dx-w_{0}\right) \right]} {\left[ 1-\delta +qE\delta -q\delta ^{2}(1-\beta )\right] }} $$

to ${\frac{q\delta ^{2}(1-\beta )\left( -{\frac{D}{C}}\right) +{\frac{\kappa }{C}}\left[ q\delta \left( \int\nolimits_{0}^{x^{+}}xf(x)dx-w_{0}\right) \right] }{\left[ 1-\delta +qE\delta -q\delta ^{2}(1-\beta )\right] }}$. Hence a unique solution to (22) exists.

Appendix 2: Search equilibrium on the SL policy with w _O endogenous

Proof of Proposition 3: As in the Proof of Proposition 1, the search problem faced by the firm may be rewritten as

$$ \max \Uppi^{SL}={\frac{\kappa \left[ q\delta (1-\beta )(1-\gamma )\left( \int\nolimits_{0}^{1}xf(x)dx\right) \right] +q\delta ^{2}(1-\beta )^{2}\left[ (1-\gamma )(\int\nolimits_{z}^{1}xf(x)dx)\right] } {\kappa \left( 1-\delta (1-q)-q\delta ^{2}(1-\beta )(1-\int\nolimits_{z}^{1}f(x)dx)\right)}} $$

where κ = (1 − δ(1 − β)) with $w_{SL}=w_{O}=\gamma x$.

The first-order conditions with respect to z are given by

$$ \begin{aligned} &-zf(z)q\delta ^{2}(1-\beta )^{2}(1-\gamma )\left[ (1-\delta (1-\beta ))\left( 1-\delta +q\delta -q\delta ^{2}(1-\beta )(1-\int\limits_{z}^{1}f(x)dx)\right) \right]\\ &\quad+q\delta ^{2}(1-\beta )f(z)(1-\delta (1-\beta ))\\ &\qquad\left[ q\delta ^{2}(1-\beta )^{2}(1-\gamma )\left[ (\int\limits_{z}^{1}xf(x)dx)\right] +\left( 1-\delta (1-\beta )\right) \left( q\delta (1-\beta )(1-\gamma )\left( \int\limits_{0}^{1}xf(x)dx\right) \right) \right] =0 \end{aligned} $$

or

$$ z={\frac{q\delta}{1-\delta +q\delta -q\delta ^{2}(1-\beta )}}\left[ \delta (1-\beta )\left( \int\limits_{z}^{1}(x-z)f(x)dx\right) +\kappa \left( \int\limits_{0}^{1}xf(x)dx\right) \right] $$

(23)

To check that the solution is unique, observe that the left-hand side of Eq. 23 is increasing in z, with range (0,1). On the other hand, the right-hand side of Eq. 23 is decreasing in z, falling from

$$ {\frac{q\delta}{1-\delta +q\delta -q\delta ^{2}(1-\beta )}}\left[ \delta (1-\beta )\left( \int\limits_{0}^{1}(x-z)f(x)dx\right) +\kappa \left( \int\limits_{0}^{1}xf(x)dx\right) \right] $$

to ${\frac{q\delta}{1-\delta +q\delta -q\delta ^{2}(1-\beta)}}\left[ \kappa \left( \int\nolimits_{0}^{1}xf(x)dx\right) \right].$ Hence a unique solution to (23) exists.