On the Social Desirability of Centralized Wage Setting when Firms are Run by Biased Managers

Nicola Meccheri

doi:10.1515/bejeap-2022-0321

Published by De Gruyter June 1, 2023

On the Social Desirability of Centralized Wage Setting when Firms are Run by Biased Managers

Nicola Meccheri

From the journal The B.E. Journal of Economic Analysis & Policy

https://doi.org/10.1515/bejeap-2022-0321

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Abstract

This paper compares the welfare outcomes obtained under alternative unionization regimes (decentralized vs. centralized wage setting) in a duopoly market in which shareholders delegate strategic decisions to biased (overconfident or underconfident) managers. In such a framework, the common tenet that consumer surplus and overall welfare are always higher under decentralized wage setting is completely overturned. Indeed, in the presence of centralized unionization (industry-wide union), firm shareholders always prefer to hire more aggressive or less conservative managers and, as a result, output (consumer surplus) and overall welfare are larger than in a decentralized wage setting structure.

Keywords: unionization structure; strategic delegation; biased managers; social welfare

JEL Classification: J51; L13; L22

Corresponding author: Nicola Meccheri, Department of Economics and Management, University of Pisa, Pisa, Italy, RCEA (Fellow) and GLO (Fellow), E-mail: nicola.meccheri@unipi.it

Acknowledgments

I am extremely grateful to the Editor Tobias Wenzel and to an anonymous referee for their comments and suggestions. I would like to thank also Cecilia Vergari, Luca Gori, participants at SIE 2022 (Torino) and seminar attendants at the University of Pisa for useful remarks. Usual disclaimers apply.

Appendix A

A.1 Common Tenet with Firms as Unit Profit-Maximising Decision Makers

This section shows that when firms are unit profit-maximising decision makers, that is, strategic decisions about output are not delegated to biased managers,^[23] the common tenet holds true: on the one hand, workers are better off under centralized wage setting whilst, on the other hand, firms, consumers and the society as a whole are better off under decentralized wage setting.

In order to obtain SPNE outcomes for this (benchmark) case in the presence of decentralized wage setting, we can use Eqs. (4) and (7) with k _i = k _j = 1, which leads to:

w i = w j = w ̃ D = θ ( 2 − γ ) 2 − θ γ

q i = q j = q ̃ D = 2 ( 1 − θ ) ( 2 + γ ) ( 2 − θ γ )

π i = π j = π ̃ D = 4 ( 1 − θ ) 2 ( 2 + γ ) 2 ( 2 − θ γ ) 2

W ̃ D = 4 ( 1 − θ ) ( 3 + θ − θ γ − θ γ 2 + γ ) ( 2 + γ ) 2 ( 2 − θ γ ) 2 .

Similarly, in order to get SPNE outcomes in the presence of centralized wage setting, we can use Eqs. (4) and (14) with k _i = k _j = 1, which leads to:

w i = w j = w ̃ C = θ

q i = q j = q ̃ C = 1 − θ 2 + γ

π i = π j = π ̃ C = ( 1 − θ ) 2 ( 2 + γ ) 2

W ̃ C = ( 1 − θ ) ( 3 + θ + θ γ + γ ) ( 2 + γ ) 2 .

By comparing equilibrium outcomes under decentralized and centralized wage settings, and considering that consumer surplus is proportional to output, the common tenet holds true since, for any θ, γ ∈ (0, 1), we get:

w ̃ D < w ̃ C ; q ̃ D > q ̃ C ; π ̃ D > π ̃ C ; C S ̃ D > C S ̃ C ; W ̃ D > W ̃ C .

Also notice that as far as the total wage bill is concerned, it is greater in a centralized structure unless unions are distinctly oriented towards wages, that is, for any θ < 2 − 4 − 2 γ γ .^[24] For instance, when θ = 1/2, i.e. unions attach the same weight to wages and employment, the total wage bill is always (i.e. for any γ ∈ (0, 1)) larger in a centralized structure.

A.2 Oligopoly with n Firms and Quantity Competition

In the case of oligopoly with n firms competing in quantities with homogeneous product, taking inverse demand (25) into account, the first-order condition of the manager i’s choice leads to:

(A1) q i = 1 2 k i − w i − ∑ j ≠ i q j .

At the equilibrium, Eq. (A1) is satisfied for each of the n firms. Summing the Eq. (A1) derived for the n firms, we obtain:

(A2) ∑ i = 1 n q i = 1 2 k i − w i + ∑ j ≠ i ( k j − w j ) − ∑ i = 1 n ∑ j ≠ i q j .

Defining with Q = ∑ i = 1 n q i total output, we have that ∑ i = 1 n ∑ j ≠ i q j = ( n − 1 ) Q , and solving (A2) with respect to Q, we obtain:

(A3) Q = 1 n + 1 k i − w i + ∑ j ≠ i ( k j − w j ) .

By considering that ∑ j ≠ i q j = Q − q i , substituting (A3) into (A1) and solving with respect to q _i, we find the equilibrium quantity as a function of wages and manager types:

(A4) q i = 1 n + 1 n ( k i − w i ) − ∑ j ≠ i ( k j − w j ) .

At the second stage, wages are set by unions. Consider first the case with decentralized wage setting, where each firm has its own union. Union i maximises V i = w i θ q i 1 − θ with respect to w _i. Taking (A4) into account and solving for the first-order condition, we obtain:

(A5) w i = θ n n k i − ∑ j ≠ i ( k j − w j ) .

At the equilibrium, Eq. (A5) is satisfied for each of the n unions. Summing the Eq. (A5) derived for the n unions, we obtain:

(A6) ∑ i = 1 n w i = θ n k i + ∑ j ≠ i k j + ( n − 1 ) W

where W = ∑ i = 1 n w i . Solving (A6) with respect to W and taking into account that ∑ j ≠ i w j = W − w i , we can substitute in (A5) which leads to the equilibrium wage set by union i as a function of manager types:

(A7) w i = θ n − θ n + θ k i − ∑ j ≠ i k j .

At the first stage, firm owners choose the type of managers to be hired. By substituting in the profit equation (with inverse demand given by Eq. (24)) for (A4) and (A7), as well as for the equilibrium values of ∑ j ≠ i q j and ∑ j ≠ i w j , and taking into account that in the symmetric equilibrium k _i = k _j = k, profit maximisation leads the SPNE value of the manager type. Then, by substituting back, we obtain also the SPNE values of wages, output and profits, as all reported in Section 4.2.

Consider, instead, the case with centralized wage setting (industry-wide union). At the second stage, the union sets wages to maximise V = w θ ( q i + ∑ j ≠ i q j ) 1 − θ , where (by considering w _i = w _j = w) q _i is given by (A4) and ∑ j ≠ i q j is obtained as Q − q _i by using (A3) and (A4). Thus, subgame equilibrium wages as a function of manager types are given by:

(A8) w = θ n k i + ∑ j ≠ i k j .

As for the decentralized wage setting case, at the first stage, firm owners choose the manager type to maximise profits. By substituting in the profit equation for equilibrium output and wages, profit maximisation (taking into account that in the symmetric equilibrium k _i = k _j = k) leads to the SPNE value of the manager type. By substituting back, we obtain also the SPNE values of wages, output and profits with centralized wage setting, as reported in Section 4.2.

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Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/bejeap-2022-0321).

Received: 2022-08-23

Accepted: 2023-05-21

Published Online: 2023-06-01

On the Social Desirability of Centralized Wage Setting when Firms are Run by Biased Managers

Abstract

Acknowledgments

A.1 Common Tenet with Firms as Unit Profit-Maximising Decision Makers

A.2 Oligopoly with n Firms and Quantity Competition

References

Supplementary Material

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