Revisiting Nash Wages Negotiations in Matching Models

Transcription

1 Introduction Revisiting Nash Wages Negotiations in Matching Models S. Amine 1 S. Baumann 2 P. Lages Dos Santos 2 F. Valognes 3 1 CIRANO University of Quebec en Outaouais 2 CERENE University of Le Havre 3 CREM University of Caen New Directions in Welfare 2011 Congress OECD Paris, France, 6-8 July 2011

2 Introduction Literature on Matching Models Bargaining Solutions Objective Literature on Matching Models Diamond (1971): A Model of Price Adjustment Mortensen (1982): Property rights and efficiency in mating, racing and related games Pissarides (2000): Equilibrium Unemployment Theory

3 Introduction Literature on Matching Models Bargaining Solutions Objective Bargaining Solutions Symmetric Nash Bargaining solution [Nash, 1950] is usually applied on these models. Some authors suggest that the Nash solution is an unreasonable model of pairwise negotiations : Siegal and Fouraker (1960) Nydegger and Owen (1974) Different kinds of bargaining solutions : Kalai-Smorodinsky solution (1975) Egalitarian bargaining solution (1997) Equal sacrifice solution (1985)

4 Introduction Literature on Matching Models Bargaining Solutions Objective Kalai-Smorodinsky Solution Few authors applied the KS-solution to the labour market analysis. Gerber and Upmann (2006) analyze a classic bargaining problem between a labour union and an employers federation through the Nash and KS solutions. Laroque and Salanié (2004) stress the effect of the minimum wage on the employment in the case of wage bargaining between firms and workers.

5 Introduction Literature on Matching Models Bargaining Solutions Objective Objective Comparison between the standard Nash solution and the Kalai-Smorodinsky solution Reflection about the choice of the solution applied

6 (1) Introduction Standard hypothesis of matching model (Pissarides, 2000) : Large exogenous number of workers. Large endogenous number of firms. Firms are identical and offer a single job. r : Rate of time preference. s : Exogenous job destruction rate. Friction on labour market represented by m(v, U). U : Number of employable unemployed workers. V : Number of vacant jobs. θ : Market tightness. q(θ) : Probability for a firm to meet an employable worker. p(θ) : Probability for an employable worker to find a job.

7 (2) Introduction Standard hypothesis of matching model (Pissarides, 2000) : w : current wage b : unemployment benefits y : productivity of a worker U 1 : worker utility U 2 : firm utility d 1 : expected utility of an unemployed worker d 2 : expected value for a vacant job

8 Introduction (1) According to the usual Belmann s equation : ru 1 (w, θ) = w s(u 1 d 1 ) (1) rd 1 (w, θ) = b + p(u 1 d 1 ) (2) ru 2 (w, θ) = y w s(u 2 d 2 ) (3) rd 2 (w, θ) = c + q(θ)(u 2 d 2 ) (4)

9 Introduction (2) The Pareto-curve is defined as the set of all pair (w, θ) such that U(w, θ) is Pareto efficient. Hence, it is the set of all (w, θ) for which worker s and firm s indifference curves are tangent to each other, i.e which satisfies: dw dθ = dw U1 =u 1 dθ U2 =u 2 p θ s(b w) (r + p(θ))(r + p(θ) + s) = q θ s(y w + c) (r + q(θ))(r + q(θ) + s) (5)

10 The Nash solution (1) Introduction (A1) Individual rationality (IR): u 1 d 1 and u 2 d 2, i.e. φ(s, d) d. (A2) Pareto optimality (PO): For u S and û S, if û u, then û = u. (A3) Symmetry (SYM): If d 1 = d 2 and if {(u, v) : (v, u) S} = S, then u = v if (S,d) is symmetric. (A4) Invariance with respect to linear utility transformations (ILUT): If T is obtained from S by a linear transformation, then the solution (u1, u 2 ) will be transformed by the same function. If T = {(α 1 u 1 + β 1, α 2 u 2 + β 2 ) : (u 1, u 2 ) S} and h = (α 1 d 1 + β 1, α 2 d 2 + β 2 ), then φ(t, h) = (α 1 φ 1 (S, d) + β 1, α 2 φ 2 (S, d) + β 2 ). (A5) Independence of irrelevant alternatives (IIA): For all closed and convex set T S, if φ(s, d) T, then φ(t, d) = φ(s, d).

11 The Nash solution (2) Introduction β (0 < β < 1) represents the workers bargaining strength Nash : max w,θ (U 1 d 1 ) β (U 2 d 2 ) 1 β (6) The global surplus, noted S, is divided between the two agents according to the Nash rule: { U1 d Nash : 1 = β(u 1 d 1 + U 2 d 2 ) = βs U 2 d 2 = (1 β)(u 1 d 1 + U 2 d 2 ) = (1 β)s

12 The Nash solution (3) Introduction φ N w b (w, θ) = (1 β) r + s + p(θ) = β y w + c r + s + q(θ) w b r + s + p(θ) y w + c r + s + q(θ) = β 1 β (7)

13 The KS solution (1) Introduction (A5 ) Individual monotonicity (IM): considering two sets S and T with S T and the disagreement point of the two sets d, if (u1, u 2 ) is the solution of (S, d) and if (u 1, u 2 ) is the solution of (T, d), then u 1 u 1 and u 2 u 2. Replacing IIA by this monotonicity axiom allows players to make interpersonal comparison of utility gains and hence more in accord with the literature on the job matching.

14 The KS solution (2) Introduction The Kalai-Smorodinsky solution is the unique solution that satisfies IR, PO, SYM, ILUT and IM. The KS curve is given by the function φ KS : φ KS = (U 2 d 2 )(U max 1 d 1 ) (U max 2 d 2 )(U 1 d 1 ) = 0

15 The KS solution (3) Introduction on the ideal point: For the firm: a maximum profit results from a minimum wage w ( w = b) For the worker: the ideal wage is equal to his productivity (ŵ = y) p(ˆθ) = 1 and q( θ) = 1

16 The KS solution (4) Introduction KS:(U 2 U2 min )(U1 max U1 min ) (U2 max U2 min )(U 1 U1 min ) = 0 ) ( y w + c ) ( ŵ b r + s + q(θ) r + s + p(ˆθ) ( y w + c r + s + q( θ) ) ( ) w b = 0 r + s + p(θ) (8) ŵ b r + s + p(ˆθ) y w + c r + s + q( θ) = q θ r + q(θ) p θ r + p(θ) (9)

17 Introduction Pareto, Nash and KS curves Figure: Nash and Kalai Smorodinsky Solutions

18 Bargaining Solutions Introduction For a fixed θ, we can determine the wage according to the two bargaining solutions : We denote by Ψ(θ) = r + s + q(θ) r + s + p(θ). ( ) 1 β y + c + bψ(θ) β w N = ( ) (10) 1 β 1 + Ψ(θ) β ( ) y b + c y + c + bψ(θ) y b w KS = ( ) (11) y b + c 1 + Ψ(θ) y b

19 Introduction Bargaining Solutions (2) The wage resulting from the Nash solution is higher than the one from the KS solution under a condition: w N > w KS if y b + c y b > 1 β β Proposition 1 In the literature, the symmetric Nash solution, in which the negotiation power between the firm and the worker is equal, is usually applied. However, the KS solution points out that this hypothesis is not relevant if the cost of a vacant job is positive. Beside, the KS solution would enable to determinate the real negotiation power of each agent. This power is then stronger for the firm to the detriment of the worker.

20 Introduction : calibration The matching function is represented by a Cobb-Douglas function: M(V, U) = V 1/2 U 1/2, which gives q(θ) = θ 1/2. We retain the following standard parameters values : β = 0.5; c = 0.3; r = 0.05; s = 0.15; y = 1 b c r s y w N w KS Table: Impacts on the wage according to the bargaining solutions

21 Introduction : calibration (2) Proposition 2 Through the calibration of this model, we obtain that the parameters have the same impacts on the equilibrium wage, whatever the solution chosen. However, the variation of Kalai on the wage is stronger than in the Nash solution, except for unemployment benefits.

22 Introduction Discussion on public policies Application of the other bargaining solutions