Efficiency Wages in Cournot-Oligopoly

Marco de Pinto; Laszlo Goerke

doi:10.1515/bejeap-2018-0236

Published by De Gruyter July 20, 2019

Efficiency Wages in Cournot-Oligopoly

Marco de Pinto and Laszlo Goerke

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

https://doi.org/10.1515/bejeap-2018-0236

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Abstract

In a Cournot-oligopoly with free but costly entry and business stealing, output per firm is too low and the number of competitors excessive, assuming labor productivity to depend on the number of employees only or to be constant. However, a firm can raise the productivity of its workforce by paying higher wages. We show that such efficiency wages accentuate the distortions occurring in oligopoly. Specifically, excessive entry is aggravated and the welfare loss due to market power rises.

Keywords: oligopoly; efficiency wages; excessive entry; welfare

JEL Classification: D43; J31; L13

Acknowledgements

We are grateful for helpful comments by an anonymous referee, Florian Baumann, Michael Neugart, Alexander Rasch, Philipp Weinschenk and Ulrich Zierahn. We also thank participants of the European Society of Population Economics annual meeting 2018 (held in Antwerp) and seminar participants in Trier for inspiring discussions and Andreas Jung for his excellent research assistance.

Appendix

Throughout the Appendix, we consider the case of a linear production function, i. e. F′=1. For simplicity, we suppress the respective underline notation (as employed in the main text).

A.1 Derivation of Eqs. (19) and (20)

Using (18), we can rewrite (15) as:

(22)xopt=n∗+1nopt+1x∗=(nopt+1)1.5nopt+1x∗=(nopt+1)0.5x∗,

which is identical to Eq. (19). To compute (20), note that in case of a linear production function, the social planner’s objective can be rewritten as:

(23)W=0.5(q−p)nx+nπ=qnx−nwl−0.5(nx)2−nk=(n∗(γ)+1)x∗nx−0.5(nx)2−nk,

where we have used (1), (3) and (10). Combining (23) with (22) and (9) yields,

(24)Wopt=(n∗+1)x∗noptxopt−0.5(noptxopt)2−noptk=(nopt+1)xoptnoptxopt−0.5(noptxopt)2−noptk=(noptxopt)2+xoptnoptxopt−0.5(noptxopt)2−noptk=xoptnoptxopt−noptk+0.5(noptxopt)2=((xopt)2−k)nopt+0.5(noptxopt)2=((xopt)2−(x∗)2)nopt+0.5(noptxopt)2.

A.2 Derivation of Proposition 2 Part (I)

Differentiating (10) with respect to γ yields:

(25)dn∗dγ=−1x∗[dw∗dγe(w∗)−γ+w∗ddγe(w∗(γ))−γ].

The second derivative in square brackets can be expressed as:

(26)ddγe(w∗(γ))−γ=e(w∗)−γ[−ln(e(w∗))−γe(w∗)de(w∗(γ))dγ].

Inserting de(w∗(γ))/dγ=de/dw×dw∗/dγ into (26) and substituting the result into (25) yields:

(27)dn∗dγ=−1x∗[dw∗dγe(w∗)−γ−w∗e(w∗)−γ(ln(e(w∗))+γe(w∗)dedwdw∗dγ)]=−1x∗e(w∗)−γ[dw∗dγ−γdedwdw∗dγw∗e(w∗)−w∗ln(e(w∗))]=−1x∗e(w∗)−γ[dw∗dγ(1−γdedww∗e(w∗))−w∗ln(e(w∗))].

Using the Solow-condition implies:

(28)dn∗dγ=1x∗e(w∗)−γw∗ln(e(w∗))>0.

Differentiating (18) with respect to γ yields:

(29)dnoptdγ=23(n∗+1)−1/3⏟<1dn∗dγ⏟>0>0.

This shows that:

(30)d(n∗−nopt)dγ=dn∗dγ−dnoptdγ>0,

which proves the first part of Proposition 2.

A.3 Derivation of Proposition 2 Part (II)

From (9) and (19), we obtain dx∗/dγ=0 and dxopt/dγ>0. This immediately proves the second part of Proposition 2.

A.4 Derivation of Proposition 2 Part (III)

Using (11) and (20), we can write the welfare loss as:

(31)W∗−Wopt=−((xoptxopt−x∗x∗)nopt+0.5((noptxopt)2−(n∗x∗)2))=−((xopt+x∗)(xopt−x∗)nopt+0.5(noptxopt+n∗x∗)(noptxopt−n∗x∗)).

From (18) and (19), we find:

(32)xoptx∗=(nopt+1)0.5,

(33)n∗=(nopt+1)1.5−1.

This leads to:

(34)noptxopt−n∗x∗=x∗(noptxoptx∗−n∗)=x∗(nopt(nopt+1)0.5−(nopt+1)1.5+1)=x∗+x∗(nopt(nopt+1)0.5−(nopt+1)1.5)=x∗−x∗(nopt+1)0.5=−(xopt−x∗).

Given (32), (33) and (34), we can rewrite (31) as:

(35)W∗−Wopt=−(xopt−x∗)((xopt+x∗)nopt−0.5(noptxopt+n∗x∗))=−(xopt−x∗)(0.5xoptnopt+x∗nopt−0.5n∗x∗)=−(xopt−x∗)x∗(0.5(nopt+1)0.5nopt+nopt−0.5(nopt+1)1.5+0.5)=−(xopt−x∗)x∗(0.5(nopt+1)0.5nopt+nopt+1−0.5(nopt+1)1.5−0.5)=−(xopt−x∗)x∗((nopt+1)(1+0.5(nopt+1)−0.5nopt−0.5(nopt+1)0.5)−0.5)=−(xopt−x∗)x∗((nopt+1)(1−0.5(nopt+1)−0.5)−0.5)

Defining Θ≡(nopt(γ)+1)[1−0.5(nopt(γ)+1)−0.5]−0.5>0, we can calculate:

(36)W∗−Wopt=−(xopt(γ)−x∗)x∗Θ(γ),

(37)d(W∗−Wopt)dγ=−x∗(dxoptdγ⏟>0Θ+(xopt−x∗)dΘdnopt⏟>0dnoptdγ⏟>0)<0,

which proves the last part of Proposition 2.

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Published Online: 2019-07-20

Efficiency Wages in Cournot-Oligopoly

Abstract

Acknowledgements

Appendix

A.1 Derivation of Eqs. (19) and (20)

A.2 Derivation of Proposition 2 Part (I)

A.3 Derivation of Proposition 2 Part (II)

A.4 Derivation of Proposition 2 Part (III)

References

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