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.

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Notes

  1. 1.

    See OECD (2009) for a more detailed information of the evolution of temporary contracts in OECD countries.

  2. 2.

    Additional to this, a recent paper by Ederveen and Thissen (2007) for the new EU member states finds that the impact of the rigidity of labor market instituions on unemployment is mixing.

  3. 3.

    The implications of exogenous wages on economic performance have already been analysed by Zagler (2005). In his model, endogenous salaries obtained from the bargaining process between individual firms and unions could generate a distorted remuneration system that pays too much to the innovation sector and too little to the existing stock of knowledge. Zagler (2005) suggests that optimal salaries may be obtained from centralized wage pacts and not by government policy.

  4. 4.

    See also Paolini (2007).

  5. 5.

    It can be interpreted as including the value of leisure and home production, net of search costs. This wide notion of unemployment income also justifies the assumption that benefits are related to the type of worker.

  6. 6.

    We give the definition of contract, a, in the next section.

  7. 7.

    Given that M > N, evidently v > u.

  8. 8.

    Notice that this is only a hypothetical wage, as the firm will extend the contract only to workers in σ.

  9. 9.

    Remember, however, that all the workers with \(x \in ({\frac{w_{o}}{\gamma }},1]\) will always reject the STC.

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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.

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Correspondence to Dimitri Paolini.

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A previous version of the paper has been circulated as “Search and the Firm’s Choice of the Optimal Labor Contract”, CRENoS 07-08.

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.

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Paolini, D., Tena, J.D. Short or long-term contract? Firm’s optimal choice. Empirica 39, 1–18 (2012). https://doi.org/10.1007/s10663-010-9152-1

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Keywords

  • Search
  • Temporary employment
  • Short-term wage

JEL Classification

  • J31
  • J41
  • J64