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

Dimitri Paolini; Juan de Dios Tena

doi:10.1007/s10663-010-9152-1

Empirica

February 2012, Volume 39, Issue 1, pp 1–18 | Cite as

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

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Dimitri PaoliniEmail author
Juan de Dios Tena

Original Paper

First Online: 13 January 2011

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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.

Keywords

Search Temporary employment Short-term wage

JEL Classification

J31 J41 J64

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Notes

Acknowledgments

We would like to thank F. Bloch, L. Deidda, S. Perelman, V. Vannetelbosch, and especially G. Bloise and T. Pietra for their helpful comments and discussions. Remaining errors are ours. Financial support of the Italian MIUR is gratefully acknowledged.

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.

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Authors and Affiliations

Dimitri Paolini
- 1
- 2
Email author
Juan de Dios Tena
- 3
- 4

1.CRENoS and DEIR, Università degli Studi di SassariSassariItaly
2.CORE, Université catholique de LouvainLouvain-la-NeuveBelgium
3.CRENoS and DEIR, Università di SassariSassariItaly
4.Universidad Carlos IIIMadridSpain

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

Abstract

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