Job matching and coalition formation with utility or disutility of co-workers.

*(English)*Zbl 1016.91070Under the assumption that co-workers may generate utility or disutility in the workplace the model of the job matching market introduced in [A. S. Kelso jun. and V. P. Crawford, Econometrica 50, 1483-1504 (1982; Zbl 0503.90019)] is investigated. It is shown that this assumption may affect the emptyness of the core.

Suggested model of a labor market is \({\mathcal M}= (F, W, f, u)\), where \(F= \{1, \dots, n \}\) is the set of firms, \(W= \{1, \dots, m \}\) is the set of workers, \(f =(f_{1}, \dots ,f_{n})\) (\(f_{i}\) is the production function for the firm \(i \in F\)) and \(u= \{ u_{1}, \dots, u_{m} \}\) (\(u_{j}(S_{i},i)\) is the utility function for the worker \(j\) if he is hired by the firm \(i\) and his co-worker in the firm \(i\) is \(S_{i}\)). The profit function \({\pi}_{i} : 2^{W} \times \mathbb{R}^{nm}_{+} \to \mathbb{R}\) for the firm \(i\) is determined by \({\pi}_{i}(S_{i},p) =f_{i}(S_{i})- \sum_{j \in S_{i}} p_{ij}\) (\((p_{i1}, \dots, p_{im})\) is the wage offers to workers by the firm \(i\)) and the surplus function \(v_{j}: W_{j} \times \mathbb{R}^{nm}_{+} \to \mathbb{R}\) (\(W_{j}= \{ S\subset W \mid j \in S \}\)) for the worker \(j\) if he is hired by the firm \(i\) is defined by \(v_{j}(S_{i}, p)= p_{ij}+ u_{j}(S_{i}, i)\). The Marshallian aggregate surplus in the labor market \(\mathcal M\) is determined by \[ V= \max_{S_{0},S_{1}, \dots, S_{n} \in {\mathcal P}(W)} \sum_{i \in F} (f_{i}(S_{i})+ \sum_{j \in S_{j}} u_{j}(S_{i}, i)), \] where \(S_{0}\) is the set of all unemployed workers. The coalition form game for the labor market \(\mathcal M\) is determined by \[ w(T)= \max_{ {\mathcal S} \in {\mathcal P}(T)} \sum_{i \in T \cap F} (f_{i}(S_{i})+ \sum_{j \in S_{j}} u_{j}(S_{i}, i)), \] where \(T \subset F \cup W\).

Some sufficient conditions for a nonempty core are established, competitive equilibrium and the tax/subsidy system are studied. The example of the labor market with an empty core for the case when money are not available in the market is designed.

Suggested model of a labor market is \({\mathcal M}= (F, W, f, u)\), where \(F= \{1, \dots, n \}\) is the set of firms, \(W= \{1, \dots, m \}\) is the set of workers, \(f =(f_{1}, \dots ,f_{n})\) (\(f_{i}\) is the production function for the firm \(i \in F\)) and \(u= \{ u_{1}, \dots, u_{m} \}\) (\(u_{j}(S_{i},i)\) is the utility function for the worker \(j\) if he is hired by the firm \(i\) and his co-worker in the firm \(i\) is \(S_{i}\)). The profit function \({\pi}_{i} : 2^{W} \times \mathbb{R}^{nm}_{+} \to \mathbb{R}\) for the firm \(i\) is determined by \({\pi}_{i}(S_{i},p) =f_{i}(S_{i})- \sum_{j \in S_{i}} p_{ij}\) (\((p_{i1}, \dots, p_{im})\) is the wage offers to workers by the firm \(i\)) and the surplus function \(v_{j}: W_{j} \times \mathbb{R}^{nm}_{+} \to \mathbb{R}\) (\(W_{j}= \{ S\subset W \mid j \in S \}\)) for the worker \(j\) if he is hired by the firm \(i\) is defined by \(v_{j}(S_{i}, p)= p_{ij}+ u_{j}(S_{i}, i)\). The Marshallian aggregate surplus in the labor market \(\mathcal M\) is determined by \[ V= \max_{S_{0},S_{1}, \dots, S_{n} \in {\mathcal P}(W)} \sum_{i \in F} (f_{i}(S_{i})+ \sum_{j \in S_{j}} u_{j}(S_{i}, i)), \] where \(S_{0}\) is the set of all unemployed workers. The coalition form game for the labor market \(\mathcal M\) is determined by \[ w(T)= \max_{ {\mathcal S} \in {\mathcal P}(T)} \sum_{i \in T \cap F} (f_{i}(S_{i})+ \sum_{j \in S_{j}} u_{j}(S_{i}, i)), \] where \(T \subset F \cup W\).

Some sufficient conditions for a nonempty core are established, competitive equilibrium and the tax/subsidy system are studied. The example of the labor market with an empty core for the case when money are not available in the market is designed.

Reviewer: Vladimir G.Skobelev (Donetsk)

##### MSC:

91B40 | Labor market, contracts (MSC2010) |

91A12 | Cooperative games |

Full Text:
DOI

##### References:

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