The effects of match uncertainty and bargaining on labor market outcomes: evidence from firm and worker specific estimates

Abstract

In this paper we examine wage dispersion in labor markets across currently employed workers. We argue that differences in the potential productivity of a match (typically assumed to be known in the previous literature) generates a surplus between the minimum wage the worker is willing to accept and the maximum wage the firm is willing to offer for the job. Existence of this surplus leads to wage dispersion due to negotiating over the amounts extracted by each agent. Our objective is to estimate the surplus extracted by each firm-worker pair and the effect of the net extracted surplus on the wage, for each firm-worker pair using the two-tier stochastic frontier model. An empirical application finds that, on average, firms paid workers less than their expected productivity. More specifically, at the mean, the net effect of productivity uncertainty leads to equilibrium wages which are 3.33% below the expected productivity of matches.

Appendix

1.1 Derivations of selected equations

Derivation²³ of Eq. 5:

Beginning with the definition of the composed error term ɛ₁ = v − u, the marginal distribution of this is, following Kumbhakar and Lovell (2000),

$$ f\left(\varepsilon_{1}\right)=\left({1\mathord{\left/ {\vphantom {1\sigma_{u}}}\right. } \sigma_{u}} \right)\left(\Upphi \left(-{\varepsilon_{1} \mathord{\left/ {\vphantom {\varepsilon_{1}\sigma_{v}}}\right. }\sigma_{v}} -{\sigma_{v} \mathord{\left/ {\vphantom {\sigma_{v} \sigma_{u}}} \right. } \sigma_{u}} \right)\hbox{exp} \left\{{\varepsilon_{1}\mathord{\left/{\vphantom {\varepsilon_{1} \sigma_{u}}} \right.} \sigma_{u}} +{\sigma_{v}^{2}\mathord{\left/{\vphantom {\sigma_{v}^{2} 2\sigma _{u}^{2}}} \right.} 2\sigma_{u}^{2}} \right\}\right). $$

(A.1)

The three component error may then be written as ɛ = ɛ₁ + w, which implies that ɛ₁ = ɛ − w, yielding the following joint distribution, $g(\varepsilon,w)=g(\varepsilon_{1},w)\cdot \vert d\varepsilon_{1} /d\varepsilon\vert= g(\varepsilon_{1},w)=f(\varepsilon_{1})\cdot f(w)$. Upon integrating out w one obtains the marginal distribution of ɛ. This is done below.

$$ \begin{aligned} f\left(\varepsilon \right)=&\int\limits_{0}^{\infty }\frac{1}{\sigma_{u}} \left(\Upphi \left(-\frac{\varepsilon_{1} }{\sigma_{v}} -\frac{\sigma_{v}}{\sigma_{u}} \right)\hbox{exp} \left\{\frac{\varepsilon_{1}}{\sigma_{u}}+\frac{\sigma_{v}^{2} }{2\sigma_{u}^{2}} \right\}\right)\frac{1}{\sigma_{w}} \hbox{exp} \left\{-\frac{w}{\sigma_{w}} \right\} dw\\ =&\frac{1}{\sigma_{u} \sigma_{w}} \left[\hbox{exp} \left\{\frac{\varepsilon}{\sigma_{u}}+\frac{\sigma_{v}^{2} }{2\sigma_{u}^{2}} \right\}\int \limits_{0}^{\infty}\Upphi \left(\frac{w}{\sigma_{v}} -\left(\frac{\varepsilon}{\sigma_{v}} +\frac{\sigma_{v}}{\sigma_{u}} \right)\right)\hbox{exp} \left\{-w\left(\frac{1}{\sigma_{w}}+\frac{1}{\sigma_{u}} \right)\right\} dw\right]\\ =&\frac{-1}{\sigma_{u}+\sigma_{w}} \left[\hbox{exp} \left\{\frac{\varepsilon}{\sigma_{u}}+\frac{\sigma_{v}^{2} }{2\sigma_{u}^{2}} \right\}\int \limits_{0}^{\infty}\Upphi \left(\frac{w}{\sigma_{v}} -\left(\frac{\varepsilon}{\sigma_{v}} +\frac{\sigma_{v}}{\sigma_{u}} \right)\right)d\left(\hbox{exp} \left\{-w\left(\frac{1}{\sigma_{w}}+\frac{1}{\sigma _{u}} \right)\right\}\right) \right]\\ =&\frac{-\hbox{exp} \left\{\alpha \right\}}{\sigma_{u}+\sigma_{w}} \left[\left. \Upphi \left({w\mathord{\left/ {\vphantom {w \sigma_{v} }} \right. } \sigma_{v}}+\beta \right)\hbox{exp} \left\{-w\lambda \right\}\right|_{0}^{\infty} -\int \limits_{0}^{\infty}\phi \left({w\mathord{\left/ {\vphantom {w \sigma_{v}}} \right. } \sigma_{v}} +\beta \right)\hbox{exp} \left\{-w\lambda \right\}dw \right]\\ =&\frac{\hbox{exp} \left\{\alpha \right\}}{\sigma_{u}+\sigma_{w}} \left[\Upphi \left(\beta \right)+\hbox{exp} \left\{-\alpha \right\}\hbox{exp} \left\{\frac{\sigma_{v}^{2}}{2\sigma_{w}^{2}} -\frac{\varepsilon}{\sigma_{w}} \right\}\int \limits_{0}^{\infty }\frac{1}{\sigma _{v}} \phi \left(\frac{w}{\sigma_{v}} -\left(\frac{\varepsilon}{\sigma_{v}} -\frac{\sigma_{v} }{\sigma_{w}} \right)\right) dw \right]\\ =&\frac{\hbox{exp} \left\{\alpha \right\}}{\sigma_{u}+\sigma_{w}} \Upphi \left(\beta \right)+\frac{\hbox{exp} \left\{a\right\}}{\sigma_{u}+\sigma _{w}} \int \limits_{-b}^{\infty}\phi \left(z\right)dz =\frac{\hbox{exp} \left\{\alpha \right\}}{\sigma_{u}+\sigma_{w}} \Upphi \left(\beta \right)+\frac{\hbox{exp} \left\{a\right\}}{\sigma_{u}+\sigma _{w}} \Upphi \left(b\right)\\ &\hbox{where}\,\,\alpha =\frac{\varepsilon}{\sigma_{u}}+\frac{\sigma _{v}^{2}}{2\sigma_{u}^{2}};\;\quad \beta =-\left(\frac{\varepsilon}{\sigma_{v}}+\frac{\sigma_{v} }{\sigma_{u}} \right);\;\quad\lambda =\frac{1}{\sigma_{u}} +\frac{1}{\sigma_{w}} \\ & \quad a=\frac{\sigma_{v}^{2}}{2\sigma _{w}^{2}} -\frac{\varepsilon }{\sigma_{w}};\quad b=\frac{\varepsilon}{\sigma_{v}} -\frac{\sigma_{v}}{\sigma_{w}}. \end{aligned}$$

(A.2)

Derivation of Eqs. 7 and 8:

$$ \begin{aligned} f\left(u\left|\varepsilon \right. \right)=&\frac{f\left(u,\; \varepsilon \right)}{f\left(\varepsilon \right)}=\frac{\left({\hbox{exp} \left\{a\right\}\mathord{\left/ {\vphantom {\hbox{exp} \left\{a\right\} \sigma_{u}\sigma_{w}}} \right. } \sigma _{u} \sigma_{w}} \right)\hbox{exp} \left\{-\lambda u\right\}\Upphi \left({u\mathord{\left/ {\vphantom {u \sigma_{v}}} \right. } \sigma_{v}} +b\right)}{\left({1\mathord{\left/ {\vphantom {1 \left(\sigma_{u} +\sigma_{w} \right)}} \right. } \left(\sigma_{u}+\sigma_{w} \right)} \right)\left[\hbox{exp} \left\{a\right\}\Upphi \left(b\right)+\hbox{exp} \left\{\alpha \right\}\Upphi \left(\beta \right)\right]}\\ =&\frac{\lambda \hbox{exp} \left\{a\right\}\hbox{exp} \left\{-\lambda u\right\}\Upphi \left({u\mathord{\left/ {\vphantom {u \sigma_{v}}} \right. } \sigma_{v}} +b\right)}{\left[\hbox{exp} \left\{a\right\}\Upphi \left(b\right)+\hbox{exp} \left\{\alpha \right\}\Upphi \left(\beta \right)\right]}\\ =&\frac{\lambda \hbox{exp} \left\{-\lambda u\right\}\Upphi \left({u\mathord{\left/ {\vphantom {u \sigma_{v}}} \right. } \sigma_{v}}+b\right)}{\chi_{1}} \end{aligned} $$

(A.3)

where χ₁ = Φ (b) + exp{α − a}Φ (β). Similarly,

$$ \begin{aligned} f\left(w\left|\varepsilon \right. \right)=&\frac{f\left(w,\; \varepsilon \right)}{f\left(\varepsilon \right)} =\frac{\left({\hbox{exp} \left\{\alpha \right\}\mathord{\left/ {\vphantom {\hbox{exp} \left\{\alpha \right\} \sigma_{u} \sigma_{w}}} \right. } \sigma_{u} \sigma_{w}} \right)\hbox{exp} \left\{-\lambda w\right\}\Upphi \left({w\mathord{\left/ {\vphantom {w \sigma_{v}}} \right. } \sigma_{v}}+\beta \right)}{\left({1\mathord{\left/ {\vphantom {1 \left(\sigma_{u} +\sigma_{w} \right)}} \right. } \left(\sigma_{u}+\sigma_{w} \right)} \right)\left[\hbox{exp} \left\{a\right\}\Upphi \left(b\right)+\hbox{exp} \left\{\alpha \right\}\Upphi \left(\beta \right)\right]}\\ =&\frac{\lambda \hbox{exp} \left\{\alpha \right\}\hbox{exp} \left\{-\lambda w\right\}\Upphi \left({w\mathord{\left/ {\vphantom {w \sigma_{v}}} \right. } \sigma_{v}}+\beta \right)}{\left[\hbox{exp} \left\{a\right\}\Upphi \left(b\right)+\hbox{exp} \left\{\alpha \right\}\Upphi \left(\beta \right)\right]}\\ =&\frac{\lambda \hbox{exp} \left\{-\lambda w\right\}\Upphi \left({w\mathord{\left/ {\vphantom {w \sigma_{v}}} \right. } \sigma_{v}}+\beta \right)}{\chi_2} \end{aligned} $$

(A.4)

where χ₂ = Φ(β) + exp {a − α}Φ (b) = exp {a − α}χ₁.

Derivation of Eqs. 9 and 10:

$$ \begin{aligned} E\left(u\left|\varepsilon \right. \right)=&\int \limits_{0}^{\infty }u\frac{\lambda \hbox{exp} \left\{-\lambda u\right\}\Upphi \left({u\mathord{\left/ {\vphantom {u \sigma_{v}}} \right. } \sigma_{v}}+b\right)}{\chi_{1}} \; du \\ =&\frac{-1}{\chi_{1} \lambda} \left[\int \limits_{0}^{\infty}\Upphi \left({u\mathord{\left/ {\vphantom {u \sigma_{v}}} \right. } \sigma_{v}}+b\right)\; d\left(\hbox{exp} \left\{-\lambda u\right\}\right)+\lambda \int \limits_{0}^{\infty }\Upphi \left({u\mathord{\left/ {\vphantom {u \sigma_{v}}} \right. } \sigma_{v}}+b\right) \, d\left(u\,\hbox{exp} \left\{-\lambda u\right\}\right)\right]\\ =&\frac{1}{\chi_{1} \lambda} \left[\Upphi \left(b\right)+\int \limits_{0}^{\infty}\hbox{exp} \left\{-\lambda u\right\}\phi \left({u\mathord{\left/ {\vphantom {u \sigma_{v}}} \right. } \sigma_{v}}+b\right)\; {du\mathord{\left/ {\vphantom {du \sigma_{v}}} \right. } \sigma_{v}}+\lambda \int \limits_{0}^{\infty}u\,\hbox{exp} \left\{-\lambda u\right\}\phi \left({u\mathord{\left/ {\vphantom {u \sigma_{v}}} \right. } \sigma_{v}}+b\right) \; {du\mathord{\left/ {\vphantom {du \sigma_{v}}} \right. } \sigma_{v}} \right]\\ =&\frac{1}{\chi_{1} \lambda} \left[\Upphi \left(b\right)+\frac{\hbox{exp} \left\{\alpha \right\}}{\hbox{exp} \left\{a\right\}} \left[\int \limits_{0}^{\infty}\phi \left({u\mathord{\left/ {\vphantom {u \sigma_{v}}} \right. } \sigma_{v}} +b+\sigma_{v} \lambda \right) \; {du\mathord{\left/ {\vphantom {du \sigma_{v}}} \right. } \sigma_{v}} +\lambda \int \limits_{0}^{\infty}u\phi \left({u\mathord{\left/ {\vphantom {u \sigma_{v}}} \right. } \sigma_{v}}+b+\sigma_{v} \lambda \right) \; {du\mathord{\left/ {\vphantom {du \sigma_{v}}} \right. } \sigma_{v}} \right]\right]\\ =&\frac{1}{\chi_{1} \lambda} \left[\Upphi \left(b\right)+\frac{\hbox{exp} \left\{\alpha \right\}}{\hbox{exp} \left\{a\right\}} \left[\int \limits_{-\beta}^{\infty}\phi \left(z\right)\; dz+\sigma_{v} \lambda \int \limits_{-\beta}^{\infty}z\phi \left(z\right) dz+\lambda \sigma_{v} \beta \int \limits_{-\beta}^{\infty}\phi \left(z\right)\, dz \right]\right] \\ =&\frac{1}{\chi_{1} \lambda} \left[\Upphi \left(b\right)+\hbox{exp} \left\{\alpha -a\right\}\Upphi \left(b\right)+\sigma_{v} \lambda\,\hbox{exp} \left\{\alpha -a\right\}\left[\phi \left(-\beta \right)+\beta \Upphi \left(\beta \right)\right]\right]\\ =& \frac{1}{\lambda}+\frac{\sigma_{v} \left[\phi \left(-\beta \right)+\beta \Upphi \left(\beta \right)\right]}{\chi_{2}}. \end{aligned} $$

(A.5)

The derivation for E(w|ɛ) follows similarly as:

$$ \begin{aligned} E\left(w\left|\varepsilon \right. \right)&=\int \limits_{0}^{\infty }w\frac{\lambda\,\hbox{exp} \left\{-\lambda w\right\}\Upphi \left({w\mathord{\left/ {\vphantom {w \sigma_{v}}} \right. } \sigma_{v}}+\beta \right)}{\chi _{2}} \; dw \\ &=\frac{-1}{\chi_{2} \lambda} \left[\int \limits _{0}^{\infty}\Upphi \left({w\mathord{\left/ {\vphantom {w \sigma_{v}}} \right. } \sigma_{v}}+\beta \right)\; d\left(\hbox{exp} \left\{-\lambda w\right\}\right)+\lambda \int \limits _{0}^{\infty }\Upphi \left({w\mathord{\left/ {\vphantom {w \sigma_{v}}} \right. } \sigma_{v}}+\beta \right) \, d \left(w\,\hbox{exp} \left\{-\lambda w\right\}\right)\right] \\ &=\frac{1}{\chi_{2} \lambda} \left[\Upphi \left(\beta \right)+\int \limits_{0}^{\infty}\hbox{exp} \left\{-\lambda w\right\}\phi \left({w\mathord{\left/ {\vphantom {w \sigma_{v}}} \right. } \sigma_{v}}+\beta \right)\; {dw\mathord{\left/ {\vphantom {dw \sigma_{v}}} \right. } \sigma_{v}}+\lambda \int \limits _{0}^{\infty}w\,\hbox{exp} \left\{-\lambda w\right\}\phi \left({w\mathord{\left/ {\vphantom {w \sigma_{v}}} \right. } \sigma_{v}}+\beta \right) \; {dw\mathord{\left/ {\vphantom {dw \sigma_{v}}} \right. } \sigma_{v}} \right]\\ &=\frac{1}{\chi_{2} \lambda} \left[\Upphi \left(b\right)+\hbox{exp} \left\{a-\alpha \right\}\left[\int \limits _{0}^{\infty}\phi \left({w\mathord{\left/ {\vphantom {w \sigma_{v}}} \right. } \sigma_{v}}+\beta+\sigma _{v} \lambda \right) \; {dw\mathord{\left/ {\vphantom {dw \sigma _{v}}} \right. } \sigma_{v}}+\lambda \int \limits_{0}^{\infty}w\phi \left({w\mathord{\left/ {\vphantom {w \sigma_{v}}} \right. } \sigma_{v}}+\beta +\sigma_{v} \lambda \right) \; {dw\mathord{\left/ {\vphantom {dw \sigma_{v}}} \right. } \sigma_{v}} \right]\right]\\ &=\frac{1}{\chi_{2} \lambda} \left[\Upphi \left(b\right)+\hbox{exp} \left\{a-\alpha \right\}\left[\int \limits_{-b}^{\infty}\phi \left(z\right)\; dz+\lambda \sigma_{v} \int \limits_{-b}^{\infty }z\phi \left(z\right) dz+\lambda \sigma_{v} b\int \limits _{-b}^{\infty}\phi \left(z\right)\, dz \right]\right] \\ &=\frac{1}{\chi_{2} \lambda} \left[\Upphi \left(\beta \right)+\hbox{exp} \left\{a-\alpha \right\}\Upphi \left(b\right)+\sigma_{v} \lambda\, \hbox{exp} \left\{a-\alpha \right\}\left[\phi \left(-b\right)+b\Upphi \left(b\right)\right]\right]. \end{aligned} $$

(A.6)

Thus,

$$ E\left(w\left|\varepsilon \right.\right)=\frac{1}{\lambda} +\frac{\sigma_{v} \left[\phi \left(-b\right)+b\Upphi \left(b\right)\right]}{\chi_{1}}. $$

(A.7)

Derivation of Eqs. 11 and 12:

$$ \begin{aligned} E\left(e^{-u} \left|\varepsilon \right. \right)&=\int \limits_{0}^{\infty}e^{-u} \frac{\lambda e^{-\lambda u} \Upphi \left({u\mathord{\left/ {\vphantom {u \sigma_{v}}} \right. } \sigma_{v}}+b\right)}{\chi_{1}} du =\frac{\lambda}{ \chi_1} \int \limits_{0}^{\infty }e^{-\left(1+\lambda \right)u} \Upphi \left({u\mathord{\left/ {\vphantom {u \sigma_{v}}}\right. } \sigma_{v}}+b\right) du\\ &=\left(\frac{-\lambda}{\chi_{1} \left(1+\lambda \right)} \right)\int \limits_{0}^{\infty}\Upphi \left({u\mathord{\left/ {\vphantom {u \sigma_{v}}} \right. } \sigma_{v}}+b\right) d(e^{-\left(1+\lambda \right)u}). \end{aligned} $$

(A.8)

Using integration by parts, we get

$$ \begin{aligned} E\left(e^{-u} \left|\varepsilon \right. \right)&=\left(\frac{-\lambda}{\chi_{1} \left(1+\lambda \right)} \right)\left[\left. \Upphi \left({u\mathord{\left/{\vphantom {u \sigma_{v}}} \right. }\sigma_{v}} +b\right)e^{-\left(1+\lambda \right)u} \right|_{0}^{\infty} -\int \limits_{0}^{\infty}e^{-\left(1+\lambda \right)u} \phi \left({u\mathord{\left/ {\vphantom {u \sigma_{v}}} \right. } \sigma_{v}}+b\right) {du\mathord{\left/ {\vphantom {du \sigma_{v}}} \right. } \sigma_{v}} \right]\\ &=\left(\frac{\lambda}{\chi_{1} \left(1+\lambda \right)} \right)\left[\Upphi \left(b\right)+e^{\alpha -a+.5\sigma_{v}^{2} -\sigma_{v} \beta} \int \limits_{0}^{\infty}\phi \left({u\mathord{\left/ {\vphantom {u \sigma_{v}}} \right. } \sigma_{v}}+\left(b+\sigma_{v} \left(1+\lambda \right)\right)\right) {du\mathord{\left/ {\vphantom {du \sigma_{v}}} \right. } \sigma_{v}} \right], \end{aligned} $$

(A.9)

and using the change of variable, $z=\frac{u}{\sigma_{v}} +\left(b+\sigma_{v} \left(1+\lambda \right)\right)\Rightarrow dz={du\mathord{\left/ {\vphantom {du \sigma_{v}}} \right. } \sigma_{v}}$, we have

$$ {E\left(e^{-u} \left|\varepsilon \right. \right)=\left(\frac{\lambda}{\chi_{1} \left(1+\lambda \right)} \right)\left[\Upphi \left(b\right)+e^{\alpha -a+.5\sigma_{v}^{2} -\sigma_{v}\beta} \Upphi \left(\beta -\sigma_{v}\right)\right]}. $$

(A.10)

For the derivation of Eq. 12 we follow the same procedure as follows:

$$ \begin{aligned} E\left(e^{-w}\left|\varepsilon \right. \right)&=\int \limits_{0}^{\infty}e^{-w}\frac{\lambda e^{-\lambda w} \Upphi \left({w\mathord{\left/{\vphantom {w \sigma_{v}}} \right. } \sigma_{v}}+\beta \right)}{\chi_{2}} dw =\left({\lambda \mathord{\left/ {\vphantom {\lambda \chi_{2}}} \right. }\chi_{2}} \right)\int \limits_{0}^{\infty}e^{-\left(1+\lambda \right)w} \Upphi \left({w\mathord{\left/ {\vphantom {w \sigma_{v}}} \right. }\sigma_{v}}+\beta \right) dw \\ &=\left(\frac{-\lambda}{\chi_{2} \left(1+\lambda \right)} \right)\int \limits_{0}^{\infty}\Upphi \left({w\mathord{\left/ {\vphantom {w \sigma_{v}}} \right. } \sigma_{v}}+\beta \right) \; de^{-\left(1+\lambda \right)w}. \end{aligned} $$

(A.11)

Using integration by parts

$$ \begin{aligned} E\left(e^{-w} \left|\varepsilon \right. \right)&=\left(\frac{-\lambda}{\chi_{2} \left(1+\lambda \right)} \right)\left[\left.\Upphi \left({w\mathord{\left/ {\vphantom {w \sigma_{v}}} \right.} \sigma_{v}}+\beta \right)e^{-\left(1+\lambda \right)w} \right|_{0}^{\infty} -\int \limits_{0}^{\infty}e^{-\left(1+\lambda \right)w} \phi \left({w\mathord{\left/ {\vphantom {w \sigma_{v}}} \right. } \sigma_{v}}+\beta \right) \; {dw\mathord{\left/ {\vphantom {dw \sigma_{v}}} \right. } \sigma_{v}} \right] \\ &=\left(\frac{\lambda}{\chi_{2} \left(1+\lambda \right)} \right)\left[\Upphi \left(\beta \right)+e^{a-\alpha -b\sigma_{v} +.5\sigma_{v}^{2}} \int \limits_{0}^{\infty}\phi \left(\frac{w}{\sigma_{v}}+\left(\beta+\sigma_{v} \left(1+\lambda \right)\right)\right) {dw\mathord{\left/ {\vphantom {dw \sigma_{v}}} \right. } \sigma_{v}} \right]. \end{aligned} $$

(A.12)

Finally, using the change of variable, $z=\frac{w}{\sigma_{v}} +\left(\beta+\sigma_{v}\left(1+\lambda \right)\right)\Rightarrow dz={dw\mathord{\left/ {\vphantom {dw \sigma_{v}}} \right. } \sigma_{v}}$, we have

$$ E\left(e^{-w}\left|\varepsilon \right. \right)=\left(\frac{\lambda }{\chi_{2} \left(1+\lambda \right)} \right)\left[\Upphi \left(\beta \right)+e^{a-\alpha -b\sigma _{v}+.5\sigma_{v}^{2}} \Upphi \left(b-\sigma_{v}\right)\right]. $$

(A.13)