Abstract
In labour theory, equilibrium is described in terms of mean variables, which is limited and can be misleading. In this article, we model the labour market as a closed Markovian network and find the steady state distribution of unemployment and advertised vacancies. We determine the stochastic equilibrium distribution for two different types of matching functions and allow for both unemployed and on the job search. In general cases, where probabilities cannot be analytically computed, we find restrictions that must hold for all matching processes. Our modelling is applicable to most economic markets with frictions.
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Acknowledgements
We thank Professor C Pissarides for his comments and suggestions on an earlier version of this paper. We also thank an anonymous referee for his or her interesting and useful comments. We acknowledge the support of DGYCIT Grants: BFM2002-02189 and BFM2003-08204.
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Appendix A
Appendix A
Proof of Theorem 1
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The ergodic theorem establishes that Equation (1), together with the normalizing condition (2), have a unique solution. Since the queueing model is reversible, rather than solving (1) and (2) directly, we state the detailed balance equations at each node, namely
We solve (A.1) recursively to get,
and, given (2),
and
Proof of Corollary 2
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The marginal equilibrium probabilities are
The conditional distributions are easily obtained from p kj , p k· and p ·j . The moments follow directly from the properties of the binomial distribution.
Proof of Theorem 3
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When V⩽L, equations (6) and (7) are always satisfied. When V>L, we differentiate (5) with respect to z and y at z=y=1, and get
and
Note that E[v a]=P z ′(1, 1) and E[v f]=P y ′(1, 1). Adding (A.2) to (A.3) we get (6). We then solve (A.2) and (A.3) to obtain E[v a] and E[v f]. Taking second-order derivatives in (5) with respect to z and y at z=y=1, we get
Note that
and
Adding (A.4), to (A.5) and (A.6), and taking into account (6), we get (7). Solving (A.4), (A.5) and (A.6) we get Cov(v a, v f).
Proof of Theorem 4
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We proceed as we did in the proof of Theorem 1. The equilibrium Chapman–Kolmogorov equations at each node are
Equations (A.7) can be solved recursively to get
and, given the normalizing condition (2),
Proof of Corollary 5
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It can be done easily by following the same steps we took in the proof of Corollary 2.
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Rodrigo, A., Vazquez, M. & Carrera, C. Markovian networks in labour markets. J Oper Res Soc 57, 526–531 (2006). https://doi.org/10.1057/palgrave.jors.2602015
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DOI: https://doi.org/10.1057/palgrave.jors.2602015