Decentralized union-oligopoly bargaining when wages signal strength

Abstract

This paper analyzes decentralized wage bargaining in a unionized oligopoly industry. The novel features of the proposed model are that firms are subject to incomplete information concerning their cost and that wages may signal firms’ private information. The potential for signaling exerts an upward shift on the equilibrium wage profile which mitigates the externality that has been shown to weaken unions’ bargaining power in decentralized wage bargaining.

Appendix: Supplement to the proof of Proposition 3

Here we show that, evaluated at $z=\theta , w=w_B$, one has $\partial _z \log N<0$, and thus $\partial _z N<0$, for all $\theta$, assuming that $\bar{\theta }\ge \frac{2}{11} \alpha$. The proof is prepared by two auxiliary results:

(1) Observe that, evaluated at $z=\theta , w=w_B$,

$$\begin{aligned} \partial _z E(L)&= \frac{1}{6} \int _0^\alpha \left( -1+\frac{4}{3}\right) dF(\theta _2) = \frac{1}{18} \end{aligned}$$

(32)

$$\begin{aligned} \partial _z E(L^2)&= \int _0^\alpha 2L \partial _z L dF(\theta _2) = \frac{1}{9} E(L) \end{aligned}$$

(33)

$$\begin{aligned} E(L)&= \frac{1}{36}\left( 9(1-\delta )+7 \bar{\theta }-16 \theta _1\right) \end{aligned}$$

(34)

(2) Therefore, evaluated at $z=\theta , w=w_B$,

$$\begin{aligned} 2E(L)-(w_B-\delta )&= \frac{1}{36}\left( 9(1-\delta )+11 \bar{\theta }- 20 \theta _1\right) \quad \text {(by (34))} \nonumber \\&> \frac{1}{36}\left( 18 \alpha + 11 \bar{\theta }-20 \theta _1 \right) \quad \text {(because }\alpha < (1-\delta )/2) \nonumber \\&\ge \frac{1}{36}\left( 18 \alpha + 11 \bar{\theta }-20 \alpha \right) \quad \text {(because }\theta _1 \le \alpha ) \nonumber \\&= \frac{1}{36}\left( 11 \bar{\theta }- 2 \alpha \right) \ge 0 \quad \left( \text {because } \bar{\theta }\ge \frac{2}{11}\alpha \right) . \end{aligned}$$

(35)

Hence, evaluated at $z=\theta , w=w_B$,

$$\begin{aligned} \partial _z \log N&= \frac{w'(\theta )}{w(\theta )-\delta } + \frac{\partial _z E(L^2)}{E(L^2)} + \frac{\partial _z E(L)}{E(L)}\\&= -\frac{1}{3(w_B(\theta )-\delta )} + \frac{E(L)}{9 E(L^2)} + \frac{1}{18 E(L)} \quad \text {(by (32, 33))} \\&< -\frac{1}{3(w_B(\theta )-\delta )} + \frac{E(L)}{9 E(L)^2} + \frac{1}{18 E(L)} \\&= -\frac{1}{3(w_B(\theta )-\delta )} + \frac{1}{6 E(L)}\\&= -\frac{1}{3}\left( \frac{1}{(w_B(\theta )-\delta )}- \frac{1}{2 E(L)}\right) <0. \end{aligned}$$

There, the first inequality is based on Jensen’s Inequality concerning a continuous variation of the random variable $\theta _2$ for the convex function $(\cdot )^2$, and the last inequality follows from the fact that $w_B(\theta )-\delta < 2 E(L)$, as shown in (35) above.