Repeated matching, career concerns, and firm size

Abstract

I propose a two-period matching model of firms and managers to show that managerial career concerns may not guarantee assortative matching in the labor market for managers. In the model, firms compete for managerial talent, and managers are concerned about their reputations. The market updates managers’ reputations whenever their performance is publicly disclosed, which leads to rematching in a subsequent period. I show that some talented managers sit out the market in an earlier period to secure their reputations in a later period. The size distribution of firms—by influencing the wage distribution of managers—is a key determinant of early sitting out: managers’ sitting out may happen under a Power-law distribution of firm size, whereas it never happens under a uniform distribution. The model highlights the roles of firm size distributions and the effects of labor markets on incentive provision within firms.

Appendices

Appendix

Whenever there is no confusion, I omit the subscript for time index throughout the Appendix.

A.1.: Proof of Lemma 1

Proof

As in Terviö (2008), the number of sorting constraints is substantially reduced because the sorting constraint $SC\left(i,k\right)$ is implied by $SC\left(i,j\right)+SC(j,k)$. To derive the market value of managers, consider the sorting constraint $SC\left(i,i+\epsilon \right), \epsilon >0$ and normalize it by $-\epsilon$ to find the slope of the market value.

$$\frac{Y\left(S\left[i\right],\gamma [i]\right)-Y\left(S\left[i\right],\gamma \left[i-\left(-\epsilon \right)\right]\right)}{-\epsilon }\le \frac{\omega \left(i\right)-\omega \left(i-\left(-\epsilon \right)\right)}{-\epsilon }\Rightarrow \underset{\epsilon \to 0}{{\text{lim}}}{\omega }^{\mathrm{^{\prime}}}\left(i\right)=-{Y}_{\gamma }\left(S\left[i\right],\gamma [i]\right){\gamma }^{\mathrm{^{\prime}}}[i],$$

where ${Y}_{\gamma }$ is the partial derivative with respect to reputation $\gamma (=\gamma \left[i\right])$ and ${\gamma }{\prime}\left[i\right]=\frac{d}{di}{H}^{-1}\left(M-i\right)$. Use the inverse function theorem,

$${\gamma }^{\mathrm{^{\prime}}}\left[i\right]=-\frac{1}{h\left({H}^{-1}\left(M-i\right)\right)}=-\frac{1}{h\left(\gamma \left[i\right]\right)}<0.$$

Because of the (No rents to the least talented manager) condition, the last matched manager $\gamma [1]$ receives his per-period reservation utility, ${w}_{0}$, in this benchmark. Thus, the market value for manager $\gamma [i]$ for $i\le 1$ is:

$$\omega \left( i \right) = \omega \left( 1 \right)\underbrace {{ - \int\limits_{i}^{1} {Y_{\gamma } \left( {S\left[ j \right],\gamma [j]} \right)\gamma ^{{\prime }} [j]dj} }}_{{{\text{reputation}}\,{\text{premium}}}},$$

where $\omega \left(1\right)={w}_{0}$. All managers $\gamma \left[i\right], i\in (1,M]$ receive ${w}_{0}$. Because ${\Pi }_{i}^{P}={\Pi }_{i}^{NP}=0$, $\omega \left(i\right)\ge {w}_{0}$ for $i\le 1$ and $\omega \left(i\right)={w}_{0}$ for $i>1$, which satisfies managers’ participation constraints.

A.2.: Manager ${\varvec{i}}$ with ${\varvec{\pi}}\left({\varvec{i}}\right)>0$ is always matched in period 2 for any ${\varvec{M}}>1$

Proof

Let $\gamma =\gamma [i]$ with $\pi \left(i\right)>0$ be given, and let ${H}_{c}$ denote a conjectured period 2 reputation distribution. By sitting out, manager $\gamma [i]$ maintains his reputation $\gamma$, but the rank will be ${i}_{\phi }$ under the reputation distribution ${H}_{c}$. For manager $\gamma [i]$ to consider sitting out (instead of participation), the manager must be matched in period 2 with certainty; that is, ${i}_{\phi }<1, {\text{where}} M-{i}_{\phi }={H}_{c}(\gamma [{i}_{\phi }])$ is required.

Let $\lambda \equiv \underset{i}{{\text{sup}}}\{i|\pi \left(i\right)>0\}<1$. Recall $\gamma \left[\lambda \right]={H}^{-1}\left(M-\lambda \right)$. Manager $\gamma \left[{\lambda }_{\phi }\right]$ is matched in period 2 if

$$M-{H}_{c}\left(\gamma \left[{\lambda }_{\phi }\right]\right)<1\iff M<1+{H}_{c}\left({H}^{-1}\left(M-\lambda \right)\right),$$

(9)

where I use $\gamma \left[{\lambda }_{\phi }\right]=\gamma \left[\lambda \right]={H}^{-1}\left(M-\lambda \right)$. If $H={H}_{c}$, then inequality (10) is equivalent to $0<1-\lambda$. Thus, it is always satisfied for any $M$ because $\lambda <1$.

Recall that ${\int }_{0}^{1}\gamma dH={\int }_{0}^{1}\gamma d{H}_{c}$ because the measure of good type managers is fixed. However, after period 1 production, matched managers’ reputations change. Hence, ${H}_{c}$ is a mean-preserving spread of $H$; thus, $H$ second-order stochastically dominates ${H}_{c}$. This suggests that ${H}_{c}^{-1}\left(M-1\right)<{H}^{-1}\left(M-1\right)$: the manager with rank $i=1$ under the distribution ${H}_{c}$ has a lower reputation than the manager with rank $i=1$ under the distribution $H$.

Depending on where $\lambda$ is located in period 1, there are two possibilities: $\lambda >{\lambda }_{\phi }$ or $\lambda \le {\lambda }_{\phi }$. If $\lambda >{\lambda }_{\phi }$, then manager $\gamma \left[{\lambda }_{\phi }\right]$ is ranked higher under ${H}_{c}$ than under $H$. In this case, ${H}_{c}\left(\gamma \left[{\lambda }_{\phi }\right]\right)=M-{\lambda }_{\phi }$ is greater than $M-\lambda$, thereby satisfying inequality (10) for any $M$. Therefore, manager $\gamma \left[{\lambda }_{\phi }\right]$ is matched in period 2 with certainty. On the other hand, if $\lambda \le {\lambda }_{\phi }$, then manager $\gamma \left[{\lambda }_{\phi }\right]$ is ranked the same as or lower than the rank under $H$. In this case, ${H}_{c}\left(\gamma \left[{\lambda }_{\phi }\right]\right)=M-{\lambda }_{\phi }$ is less than $M-\lambda$. However, because ${H}_{c}^{-1}\left(M-1\right)<{H}^{-1}\left(M-1\right)$ and $\lambda <1$, and I know $\gamma \left[{\lambda }_{\phi }\right]=\gamma \left[\lambda \right]$, regardless of production technology, manager $\gamma \left[{\lambda }_{\phi }\right]$ can never be ranked at 1 under ${H}_{c}$, i.e., ${\lambda }_{\phi }<1$. Thus, regardless of $M$, manager $\gamma \left[{\lambda }_{\phi }\right]$ is matched in period 2 with certainty.

A.3.: The derivation of the posterior distribution of reputation

Recall that $H\left(\gamma \left[i\right]\right)=M-i$. For notational convenience, I omit the rank $i$ whenever there is no confusion, but keep $i$ in $H\left(\gamma \left[i\right]\right)$ to use the matching function ${\mu }_{1}\left(i\right)$. The derivation is similar with Anderson and Smith (2010) and Anderson (2015). Let $\tau (s|\gamma )$ denote the probability that reputation $\gamma$ is updated to at most $s$, conditional on ${\mu }_{1}\left(i\right)\ne u$ for $\gamma \left[i\right]$ ($=\gamma$). Then, the posterior distribution of matched managers is:

$$\widehat{H}\left(z\right)={\int }_{0}^{z}{\int }_{{\mu }_{1}\left(i\right)\ne u}\tau \left(s|\gamma \right)dH(\gamma [i])dsss$$

Recall that the updated reputation is $\gamma \frac{{f}_{g}\left(x\right)}{{f}_{\gamma }(x)}$ for performance $x$, and that $\frac{{f}_{g}\left(x\right)}{{f}_{\gamma }(x)}$ is monotone increasing in $x$. Thus, if there is some performance $x$ that updates $\gamma$ to $s$, then it is unique. Let $\frac{{f}_{g}\left(x\right)}{{f}_{\gamma }(x)}\equiv l\left(x\right)$. Because $l\left(x\right)$ is monotone increasing, it is invertible; thus,

$$s=\gamma \frac{{f}_{g}\left(x\right)}{{f}_{\gamma }(x)}=\gamma \times l\left(x\right)\Rightarrow x={l}^{-1}\left(\frac{s}{\gamma }\right).$$

Therefore, the above posterior is rewritten as:

$$\widehat{H}\left(z\right)={\int }_{0}^{z}{\int }_{{\mu }_{1}\left(i\right)\ne u}{F}_{\gamma }\left({l}^{-1}\left(\frac{s}{\gamma }\right)\right)dH(\gamma [i])ds.$$

When matching takes place in period 1, every manager $\gamma$ computes the expected posterior based on his conjectured matching ${\mu }_{c}$ (which is the same as ${\mu }_{1}$ in equilibrium).

A.4.: Quantile, quantile density, and density quantile functions

For a strictly increasing cumulative distribution function $G(S)$, let $S\left[i\right]$ denote a quantile function such that.

$$S\left[i\right]={G}^{-1}\left(1-i\right), i\in \left[\mathrm{0,1}\right].$$

The quantile density function is defined as (Parzen 1979):

$$\frac{dS\left[i\right]}{di}={S}^{\mathrm{^{\prime}}}\left[i\right]=-\frac{1}{g\left(S\left[i\right]\right)}.$$

The reciprocal of the quantile density function, $g(S\left[i\right])$, is called a density quantile function (Gilchrist 2000).

The following lemma is useful for Proposition 1.

Lemma 2

$\pi \left(i\right)>0$ is equivalent to

$$\begin{aligned} & F_{\gamma \left[ i \right]} \left( m \right)E\left[ {S\left[ {i_{x} } \right]\gamma \left[ {i_{x} } \right] - S\left[ {i_{\phi } } \right]\gamma \left[ {i_{\phi } } \right] - \left( {V\left( {i_{x} } \right) - V\left( {i_{\phi } } \right)} \right)\left| x \right\rangle m} \right] \\ & < \overline{F}_{\gamma \left[ i \right]} \left( m \right)E\left[ {S\left[ {i_{\phi } } \right]\gamma \left[ {i_{\phi } } \right] - S\left[ {i_{x} } \right]\gamma \left[ {i_{x} } \right] - \left( {V\left( {i_{\phi } } \right) - V\left( {i_{x} } \right)} \right)|x < m} \right], \\ \end{aligned}$$

(10)

where ${\overline{F} }_{\gamma \left[i\right]}\left(m\right)=1-{F}_{\gamma \left[i\right]}(m)$ and $V\left(i\right)={\int }_{i}^{1}\frac{\gamma \left[l\right]}{g\left(S\left[l\right]\right)}dl$ denotes firm $S[i]$’s share of the match surplus.

A.5.: Proof of Lemma 2

Proof

To compare ${\Pi }_{i}^{P}$ with ${\Pi }_{i}^{NP}$ in period 1, manager $\gamma [i]$ conjectures period 1 matching. Let ${\mu }_{c}$ denote a conjectured period 1 matching function, and let $1$ denote an indicator function over the set $\{\gamma [i]|{\mu }_{c}\left(i\right)=u {\text{for}} \gamma [i]\in [\mathrm{0,1}]\}$ where $1$ equals 1 if a manager is unmatched in period 1, or zero if matched. The posterior distribution of reputation among matched managers is characterized as in A.3:

$${\widehat{H}}_{c}\left(z\right)={\int }_{0}^{z}{\int }_{{\mu }_{c}\left(i\right)\ne u}{F}_{\gamma }\left({l}^{-1}\left(\frac{s}{\gamma }\right)\right)dH(\gamma [i])ds.$$

Therefore, given ${\mu }_{c}$, the reputation distribution at the beginning of period 2 is:

$$H_{c} \left( \gamma \right) = \underbrace {{\int _{0}^{\gamma } \hat{h}_{c} \left( z \right)dz}}_{{{\text{matched}}}} + \underbrace {{\int _{0}^{\gamma } h\left( z \right)1dz}}_{{{\text{unmatched}}}},$$

where ${\widehat{h}}_{c}$ and $h$, respectively denote the probability density function of ${\widehat{H}}_{c}$ and $H$. Let ${h}_{c}$ denote the probability density function of ${H}_{c}$. Period 2 matching is assortative, so, ${\mu }_{2}\left(i\right)\ne u$ if $\gamma [i]\ge {H}_{c}^{-1}(M-1)$ and ${\mu }_{2}\left(i\right)=u$ if $\gamma [i]<{H}_{c}^{-1}(M-1)$. Thus, given ${\mu }_{c}$, the market value in period 2, ${\omega }_{c}$, is characterized as follows (Lemma 1).

$$\omega _{c} \left( i \right) = w_{0} \underbrace {{ - \int _{i}^{1} Y_{\gamma } \left( {S\left[ j \right],\gamma [j]} \right)\gamma ^{\prime } \left[ j \right]dj}}_{{{\text{reputation}}\,{\text{premium}}}},$$

where ${\gamma }{\prime}\left[j\right]=\frac{\partial }{\partial j}{H}_{c}^{-1}\left(M-j\right)=-\frac{1}{{h}_{c}\left({H}_{c}^{-1}\left(M-j\right)\right)}=-\frac{1}{{h}_{c}(\gamma \left[j\right])}<0$. Recall that ${Y}_{\gamma }\left(S\left[j\right],\gamma [j]\right)=\left({m}_{g}-{m}_{b}\right)S[j]$. Thus, the reputation premium is:

$$\begin{aligned} - \mathop \int \limits_{i}^{1} Y_{\gamma } \left( {S\left[ j \right],\gamma \left[ j \right]} \right)\gamma^{\prime}\left[ j \right]dj & = \left( {m_{g} - m_{b} } \right)\mathop \int \limits_{i}^{1} S\left[ j \right]\left( { - \gamma^{\prime}\left[ j \right]} \right)dj \\ & = \left( {m_{g} - m_{b} } \right)\left( {\mathop \int \limits_{i}^{1} S^{\prime}\left[ j \right]\gamma \left[ j \right] - \left( {S\left[ j \right]\gamma \left[ j \right]} \right){\prime} dj} \right) \\ \end{aligned}$$

(11)

The second equality is because of the product rule: ${\left(S\left[j\right]\gamma \left[j\right]\right)}{\prime}=S\left[j\right]{\gamma }{\prime}\left[j\right]+{S}{\prime}\left[j\right]\gamma \left[j\right]$. Use the fundamental theorem of calculus:

$${\int }_{i}^{1}{S}^{\mathrm{^{\prime}}}\left[j\right]\gamma \left[j\right]-{\left(S\left[j\right]\gamma \left[j\right]\right)}^{\mathrm{^{\prime}}}dj={\int }_{i}^{1}{S}^{\mathrm{^{\prime}}}\left[j\right]\gamma \left[j\right]dj+\left(S\left[i\right]\gamma \left[i\right]-S\left[1\right]\gamma \left[1\right]\right).$$

(12)

Let ${i}_{x}$ and ${i}_{\phi }$ denote, respectively, a rank in period 2 upon period 1 performance of $x$ and rank in the case the manager sits out. Notice that, given ${\mu }_{c}$ (which includes potential sitting out by some managers), individual manager $\gamma [i]$’s participation or sitting out does not change the distribution of managers’ reputations in period 2. Hence, manager $\gamma [i]$ compares the expected payoff from participation with that from sitting out based on the distribution ${H}_{c}\left(\gamma \right)$.

$${\Pi }_{i}^{P}={\int }_{\underline{x}}^{\overline{x}}{\omega }_{c}\left({i}_{x}\right){f}_{\gamma [i]}\left(x\right)dx, {\Pi }_{i}^{NP}={\omega }_{c}\left({i}_{\phi }\right),$$

where ${\omega }_{c}(i)$ is involving ${H}_{c}\left(\gamma \right)$. In equilibrium, all managers’ conjectured matching is consistent; thus, ${\mu }_{c}={\mu }_{1}$. Therefore, ${\omega }_{c}\left(i\right)={\omega }_{2}(i)$.

Recall that ${f}_{g}$ and ${f}_{b}$ are crossing at $x=m$; thus, $\gamma \left[{i}_{m}\right]=\gamma \left[i\right]=\gamma \left[{i}_{\phi }\right]$. For $x>m, \gamma [{i}_{x}]>\gamma [i]$, and for $x<m,\gamma \left[{i}_{x}\right]<\gamma [i]$. The value of sitting out can be written as follows:

$$\pi \left(i\right)={\Pi }_{i}^{NP}-{\Pi }_{i}^{P}={\omega }_{2}\left({i}_{\phi }\right)-{\int }_{\underline{x}}^{\overline{x}}{\omega }_{2}\left({i}_{x}\right){f}_{\gamma [i]}\left(x\right)dx.$$

$$= \underbrace {{\int _{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} }}^{m} \left( {\omega _{2} \left( {i_{\phi } } \right) - \omega _{2} \left( {i_{x} } \right)} \right)f_{{\gamma [i]}} \left( x \right)dx}}_{{{\text{down}}\,{\text{side}}}} - \underbrace {{\int _{m}^{{\bar{x}}} \left( {\omega _{2} \left( {i_{x} } \right) - \omega _{2} \left( {i_{\phi } } \right)} \right)f_{{\gamma [i]}} \left( x \right)dx}}_{{{\text{upside}}}}.$$

Here, $\pi \left(i\right)>0$ if the downside potential is greater than the upside potential. Note that

$${\int }_{m}^{\overline{x}}\left({\omega }_{2}\left({i}_{x}\right)-{\omega }_{2}\left({i}_{\phi }\right)\right){f}_{\gamma [i]}\left(x\right)dx=\left({m}_{g}-{m}_{b}\right){\int }_{m}^{\overline{x}}{f}_{\gamma [i]}\left(x\right){\int }_{{i}_{x}}^{{i}_{\phi }}S\left[l\right]\left(-{\gamma }^{\mathrm{^{\prime}}}\left[l\right]\right)dldx.$$

Use (12) and (13),

$${\int }_{m}^{\overline{x}}{f}_{\gamma \left[i\right]}\left(x\right){\int }_{{i}_{x}}^{{i}_{\phi }}S\left[l\right]\left(-{\gamma }{\prime}\left[l\right]\right)dldx={\int }_{m}^{\overline{x}}{f}_{\gamma \left[i\right]}\left(x\right){\int }_{{i}_{x}}^{{i}_{\phi }}{S}{\prime}\left[l\right]\gamma \left[l\right]-{\left(S\left[l\right]\gamma \left[l\right]\right)}{\prime}dldx={\int }_{m}^{\overline{x}}{f}_{\gamma \left[i\right]}\left(x\right)\left({\int }_{{i}_{x}}^{{i}_{\phi }}{S}{\prime}\left[l\right]\gamma \left[l\right]dl+S\left[{i}_{x}\right]\gamma \left[{i}_{x}\right]-S\left[{i}_{\phi }\right]\gamma \left[{i}_{\phi }\right]\right)dx$$

$$={\int }_{m}^{\overline{x}}{f}_{\gamma \left[i\right]}\left(x\right)\left(S\left[{i}_{x}\right]\gamma \left[{i}_{x}\right]-S\left[{i}_{\phi }\right]\gamma \left[{i}_{\phi }\right]\right)dx+{\int }_{m}^{\overline{x}}{f}_{\gamma \left[i\right]}\left(x\right){\int }_{{i}_{x}}^{{i}_{\phi }}{S}{\prime}\left[l\right]\gamma \left[l\right]dldx$$

$$= \underbrace {{\int _{m}^{{\bar{x}}} f_{{\gamma [i]}} \left( x \right)\left( {S\left[ {i_{x} } \right]\gamma \left[ {i_{x} } \right] - S\left[ {i_{\phi } } \right]\gamma \left[ {i_{\phi } } \right]} \right)dx}}_{{{\text{total}}\,{\text{size}}\,{\text{of}}\,{\text{the}}\,{\text{surplus}}}} - \underbrace {{\int _{m}^{{\bar{x}}} f_{{\gamma \left[ i \right]}} \left( x \right)\int _{{i_{x} }}^{{i_{\phi } }} \frac{{\gamma \left[ l \right]}}{{g\left( {S\left[ l \right]} \right)}}dldx}}_{{{\text{slope}}\,{\text{(firm's}}\,{\text{share)}}}}.$$

The last equality uses that ${S}{\prime}\left[l\right]=-\frac{1}{g\left({G}^{-1}\left(1-l\right)\right)}=-\frac{1}{g\left(S\left[l\right]\right)}.$ Because $\gamma \left[l\right]$ is continuous, and $g\left(S\left[l\right]\right)$ is strictly positive and continuous, this implies that $\frac{\gamma \left[l\right]}{g\left(S\left[l\right]\right)}$ is continuous. Thus, $\frac{\gamma \left[l\right]}{g\left(S\left[l\right]\right)}$ is integrable. Here, the slope effect is determined by its density quantile function $g(S\left[i\right])$. Technically, ${S}{\prime}\left[j\right]\gamma \left[j\right]$ represents the slope of trace of $S\left[j\right]\gamma \left[j\right]$ for plane $\gamma \left[j\right]$ and captures how fast firm size changes with rank when $\gamma \left[j\right]$ is fixed. The downside potential can be similarly written as follows. For simplicity, normalize it by $\left({m}_{g}-{m}_{b}\right)$.

$$\begin{aligned} & \frac{1}{{\left( {m_{g} - m_{b} } \right)}}\mathop \int \limits_{{\underline {x} }}^{m} f_{\gamma \left[ i \right]} \left( x \right)\left( {\omega_{2} \left( {i_{\phi } } \right) - \omega_{2} \left( {i_{x} } \right)} \right)dx = \mathop \int \limits_{{\underline {x} }}^{m} f_{\gamma \left[ i \right]} \left( x \right)\mathop \int \limits_{{i_{\phi } }}^{{i_{x} }} S\left[ l \right]\left( { - \gamma^{\prime}\left[ l \right]} \right)dldx \\ & = \mathop \int \limits_{{\underline {x} }}^{m} f_{\gamma \left[ i \right]} \left( x \right)\left( {S\left[ {i_{\phi } } \right]\gamma \left[ {i_{\phi } } \right] - S\left[ {i_{x} } \right]\gamma \left[ {i_{x} } \right] - \mathop \int \limits_{{i_{\phi } }}^{{i_{x} }} \left( { - S^{\prime}\left[ l \right]} \right)\gamma \left[ l \right]dl} \right)dx \\ & = \mathop \int \limits_{{\underline {x} }}^{m} f_{\gamma \left[ i \right]} \left( x \right)\left( {S\left[ {i_{\phi } } \right]\gamma \left[ {i_{\phi } } \right] - S\left[ {i_{x} } \right]\gamma \left[ {i_{x} } \right]} \right)dx - \mathop \int \limits_{{\underline {x} }}^{m} f_{\gamma \left[ i \right]} \left( x \right)\mathop \int \limits_{{i_{\phi } }}^{{i_{x} }} \frac{\gamma \left[ l \right]}{{g\left( {S\left[ l \right]} \right)}}dldx \\ \end{aligned}$$

Therefore, $\pi \left(i\right)>0$ can be written as:

$$\begin{aligned} & \mathop \int \limits_{{\underline {x} }}^{m} f_{\gamma \left[ i \right]} \left( x \right)\left( {S\left[ {i_{\phi } } \right]\gamma \left[ {i_{\phi } } \right] - S\left[ {i_{x} } \right]\gamma \left[ {i_{x} } \right]} \right)dx - \mathop \int \limits_{{\underline {x} }}^{m} f_{\gamma \left[ i \right]} \left( x \right)\mathop \int \limits_{{i_{\phi } }}^{{i_{x} }} \frac{\gamma \left[ l \right]}{{g\left( {S\left[ l \right]} \right)}}dldx \\ & > \mathop \int \limits_{m}^{{\overline{x}}} f_{\gamma \left[ i \right]} \left( x \right)\left( {S\left[ {i_{x} } \right]\gamma \left[ {i_{x} } \right] - S\left[ {i_{\phi } } \right]\gamma \left[ {i_{\phi } } \right]} \right)dx - \mathop \int \limits_{m}^{{\overline{x}}} f_{\gamma \left[ i \right]} \left( x \right)\mathop \int \limits_{{i_{x} }}^{{i_{\phi } }} \frac{\gamma \left[ l \right]}{{g\left( {S\left[ l \right]} \right)}}dldx \\ & \Leftrightarrow F_{\gamma \left[ i \right]} \left( m \right)E\left[ {S\left[ {i_{\phi } } \right]\gamma \left[ {i_{\phi } } \right] - S\left[ {i_{x} } \right]\gamma \left[ {i_{x} } \right]{|}x < m} \right] - \mathop \int \limits_{{\underline {x} }}^{m} f_{\gamma \left[ i \right]} \left( x \right)\mathop \int \limits_{{i_{\phi } }}^{{i_{x} }} \frac{\gamma \left[ l \right]}{{g\left( {S\left[ l \right]} \right)}}dldx \\ & > \overline{F}_{\gamma \left[ i \right]} \left( m \right)E\left[ {S\left[ {i_{x} } \right]\gamma \left[ {i_{x} } \right] - S\left[ {i_{\phi } } \right]\gamma \left[ {i_{\phi } } \right]{|}x > m} \right] - \mathop \int \limits_{m}^{{\overline{x}}} f_{\gamma \left[ i \right]} \left( x \right)\mathop \int \limits_{{i_{x} }}^{{i_{\phi } }} \frac{\gamma \left[ l \right]}{{g\left( {S\left[ l \right]} \right)}}dldx. \\ \end{aligned}$$

Let $V\left(i\right)\equiv {\int }_{i}^{1}\frac{\gamma \left[l\right]}{g\left(S\left[l\right]\right)}dl, i\in \left[\mathrm{0,1}\right]$ denote firm $S[i]$’s share. Then, from rank ${i}_{x}$ to ${i}_{\phi }$,

${\int }_{{i}_{x}}^{{i}_{\phi }}\frac{\gamma \left[l\right]}{g\left(S\left[l\right]\right)}dl=V\left({i}_{x}\right)-V\left({i}_{\phi }\right)$Then, $\pi \left(i\right)>0$ is equivalent to

${F}_{\gamma \left[i\right]}\left(m\right)E\left[S\left[{i}_{x}\right]\gamma \left[{i}_{x}\right]-S\left[{i}_{\phi }\right]\gamma \left[{i}_{\phi }\right]-\left(V\left({i}_{x}\right)-V\left({i}_{\phi }\right)\right)|x>m\right]<{\overline{F} }_{\gamma \left[i\right]}\left(m\right)E\left[S\left[{i}_{\phi }\right]\gamma \left[{i}_{\phi }\right]-S\left[{i}_{x}\right]\gamma \left[{i}_{x}\right]-\left(V\left({i}_{\phi }\right)-V\left({i}_{x}\right)\right)|x<m\right].$

A.6.: Proof of Proposition 1

Proof

The proof consists of three steps. Step 1) I first derive a firm size threshold above which no distortion takes place in equilibrium. Step 2) For firms that are below the threshold, I then constructively find period 1 matching that involves a discontinuity due to the benefit of sitting out. Step 3) Lastly, I check if the outcome of Step 2 satisfies the conditions in the definition of equilibrium.

Step 1) Let $\pi \left(i\right)>0$ and $[\eta , \lambda ]$ be given, where $\eta =\underset{i}{{\text{inf}}} \{i|\pi \left(i\right)>0\}$ and $\lambda =\underset{i}{{\text{sup}}} \{i|\pi \left(i\right)>0\}$. Because $\gamma \left[0\right]=1$, any realized performance does not change the reputation level and its rank in period 2. Moreover, individual manager $\gamma \left[0\right]$’s participation or sitting out does not change the distribution of reputation; thus, $\pi \left(0\right)=0$. This implies that $0<\eta \le \lambda$. Because $\gamma \left[1\right]$ has an upside but no downside potential for participation, $\pi \left(1\right)\le 0$, implying that $\eta \le \lambda <1$.

Claim 1:

$\pi (i)$ is continuous.

Proof of Claim 1

Because $\pi \left(i\right)=-{\int }_{{i}_{\phi }}^{1}{Y}_{\gamma }\times {\gamma }{\prime}\left[j\right]dj+{\int }_{\underline{x}}^{\overline{x}}{\int }_{{i}_{x}}^{1}{Y}_{\gamma }\times {\gamma }{\prime}\left[j\right]dj{f}_{\gamma \left[{i}_{x}\right]}\left(x\right)dx$, it is sufficient to show that $\int {Y}_{\gamma }\times {\gamma }{\prime}[j]dj$ is continuous. Recall that the distribution functions $G$ and $H$ are smooth and strictly increasing, and $\widehat{H}$ is piecewise smooth. Thus, the corresponding quantile function $S[j]$ is smooth and $\gamma [j]$ (at the beginning of period 2) is piecewise smooth. Therefore, ${Y}_{\gamma }\times {\gamma }{\prime}[j]$ is Riemann integrable over its domain. Then, the integral of ${Y}_{\gamma }\times {\gamma }{\prime}[j]$, (i.e., $\int {Y}_{\gamma }\times {\gamma }{\prime}[j]dj$), is differentiable; thus, $\int {Y}_{\gamma }\times {\gamma }{\prime}[j]dj$ is continuous (Rudin 1976, theorem 6.10 and 6.20). Q.E.D.

Because $\pi \left(i\right)$ is continuous, provided that $\pi \left(i\right)>0$ for some $i,$ it implies that $\eta < \lambda$. Thus, $\pi \left(i\right)\ge 0$ for all . Notice that for any $i\in [\eta , \lambda ]$, if firm $S[i]$’s sorting price is greater than or equal to the participation price, then there is no distortion, and the matching is positive assortative. Formally,

$$\forall i\in \left[\eta , \lambda \right], {\omega }^{SC}\left(S\left[i\right],i\right)\ge {\omega }^{PC}\left(i\right)={w}_{0}+\pi \left(i\right),$$

where.

$${\omega }^{SC}\left(S\left[i\right],i\right)=\underset{j>i}{{\text{max}}}Y\left(S\left[i\right],\gamma [i]\right)-Y\left(S\left[i\right],\gamma [j]\right)+{\omega }_{1}(j).$$

Here, the reason why the max operator is taken over j is because the sorting constraint must check for all possible matches $j>i$. Recall that, in the static assignment, ${\omega }^{SC}\left(S\left[i\right],i\right)=\underset{j\to {i}^{+}}{{\text{lim}}}Y\left(S\left[i\right],\gamma [i]\right)-Y\left(S\left[i\right],\gamma [j]\right)+{\omega }_{1}\left(j\right)$ is determined by the marginal sorting constraint, and the market value is the sorting price for all managers but the last matched manager.

However, when $\pi \left(i\right)>0$ is added to the participation price for some managers, then, ${\omega }^{SC}\left(S\left[i\right],i\right)$ may not be determined by $j$ close to $i$. Because $\pi \left(i\right)$ is continuous, for $j\to {i}^{+}$, $\pi \left(j\right)$ is close to $\pi \left(i\right)$. Thus, it can still satisfy the marginal sorting constraint. The extra added value $\pi \left(i\right)$, however, may exceed the productive improvement, $Y\left(S\left[i\right],\gamma [i]\right)-Y\left(S\left[i\right],\gamma [k]\right)$, for some distant $k$. Rearrange ${\omega }^{SC}\left(S\left[i\right],i\right)$. Then, the sorting price dominates the participation price if

$$S\left[i\right]\ge \underset{j>i}{{\text{max}}}\frac{\pi \left(i,j\right)}{\left({m}_{g}-{m}_{b}\right)\left(\gamma \left[i\right]-\gamma \left[j\right]\right)}\equiv {S}^{*}\left[i\right], \forall i\in \left[\eta , \lambda \right], j\in \left(i,M\right],$$

(14)

where $\pi \left(i,j\right)={\omega }^{PC}\left(i\right)-{\omega }_{1}(j)= \pi \left(i\right)+{\int }_{j}^{1}{Y}_{\gamma }\left(S\left[l\right],\gamma [l]\right)\times {\gamma }{\prime}\left[l\right]dl$. Here, the market price ${\omega }_{1}(j)$ is based on the equilibrium matching. The interpretation for ${S}^{*}[i]$ is the minimum firm size threshold that equalizes the sorting price to the participation price for manager $\gamma [i]$ whose $\pi \left(i\right)>0$. Thus, for any $i\in \left[\eta , \lambda \right]$, if $S\left[i\right]\ge {S}^{*}[i]$, the sorting price always dominates the participation price, and there is no distortion in matching, which is characterized as follows:

${\mu }_{1}\left(i\right)=i, \forall i\in \left[\mathrm{0,1}\right], {\text{and}} {\mu }_{1}\left(i\right)=u,\forall i\in (1,M]$ if $S\left[i\right]\ge {S}^{*}[i]$ for any $i\in \left[\eta , \lambda \right]$,

where ${S}^{*}[i]$ is defined in (14).

Step 2) A distortion occurs only if, for some $i\in [\eta , \lambda ]$, $S\left[i\right]<{S}^{*}[i]$. Because $\pi (i)$ is continuous, and $Y\left(S\left[i\right],\gamma [j]\right)$ is continuous and is increasing with firm size, when $S\left[i\right]<{S}^{*}[i]$, one can find $k>i$ such that the sorting constraint is reversed.

$$Y\left(S\left[i\right],\gamma [i]\right)-Y\left(S\left[i\right],\gamma \left[k\right]\right)+{\omega }_{1}\left(k\right)<{w}_{0}+\pi \left(i\right).$$

(15)

The following claim shows that a firm’s willingness to pay is decreasing with rank $j$.

Claim 2

For $j< k$, $\frac{\partial }{\partial j}\left(Y\left(S\left[j\right],\gamma [j]\right)-Y\left(S\left[j\right],\gamma [k]\right)\right)<0$.

Proof of Claim 2

Recall that as $j$ increases, both firm size $S[j]$ and manager reputation $\gamma [j]$ decrease. Using the multivariable chain rule,

$$\begin{aligned} & & \frac{d}{dj}\left( {Y\left( {S\left[ j \right], \gamma \left[ j \right]} \right) - Y\left( {S\left[ j \right],\gamma \left[ k \right]} \right)} \right) \\ & = \frac{{\partial Y\left( {S\left[ j \right], \gamma \left[ j \right]} \right)}}{\partial S\left[ j \right]}\frac{dS\left[ j \right]}{{dj}} + \frac{{\partial Y\left( {S\left[ j \right], \gamma \left[ j \right]} \right)}}{\partial \gamma \left[ j \right]}\frac{d\gamma \left[ j \right]}{{dj}} - \frac{{\partial Y\left( {S\left[ j \right],\gamma \left[ k \right]} \right)}}{\partial S\left[ j \right]}\frac{dS\left[ j \right]}{{dj}} \\ & = \left( {\frac{{\partial Y\left( {S\left[ j \right], \gamma \left[ j \right]} \right)}}{\partial S\left[ j \right]} - \frac{{\partial Y\left( {S\left[ j \right], \gamma \left[ k \right]} \right)}}{\partial S\left[ j \right]}} \right)\frac{dS\left[ j \right]}{{dj}} + \frac{{\partial Y\left( {S\left[ j \right], \gamma \left[ j \right]} \right)}}{\partial \gamma \left[ j \right]}\frac{d\gamma \left[ j \right]}{{dj}} < 0. \\ \end{aligned}$$

The last inequality is due to $\frac{\partial Y\left(S\left[j\right], \gamma [j]\right)}{\partial S\left[j\right]}>\frac{\partial Y\left(S\left[j\right], \gamma [k]\right)}{\partial S\left[j\right]}$ for $j<k$, and $\frac{\partial Y\left(S\left[j\right], \gamma [j]\right)}{\partial \gamma \left[j\right]}>0$, but $\frac{dS\left[j\right]}{dj}<0$ and $\frac{d\gamma \left[j\right]}{dj}<0$. Q.E.D.

Claim 2 implies that firm $S[j]$’s productive impact from a manager’s reputation is decreasing with $j$. Because the size is in reverse order, the firm’s willingness to cover the manager’s value of sitting out is decreasing with $j$ (i.e., as its size decreases).

Claim 3

If firm $S\left[j\right]$ deviates to manager $\gamma \left[k\right] {\text{for}} k>j$ in equilibrium, then for all firms $S\left[l\right],l\in \left(j,k\right)$ deviate to manager $\gamma [l+\left(k-j\right)]$ from manager $\gamma [l]$.

Proof of Claim 3

Suppose not. Then, by switching manager $\gamma [k]$ with manager $\gamma [l]$, firm $S\left[j\right]$ is strictly better off, the aggregate match output increases without making firm $S\left[l\right]$ worse off by transferring $\left(Y\left(S\left[l\right],\gamma \left[l\right]\right)-{\omega }_{1}\left(l\right)\right)-\left(Y\left(S\left[l\right],\gamma \left[k\right]\right)-{\omega }_{1}\left(k\right)\right)>0$ from the improved match output $\left(Y\left(S\left[j\right],\gamma \left[l\right]\right)-\omega \left(l\right)\right)-\left(Y\left(S\left[j\right],\gamma \left[k\right]\right)-{\omega }_{1}\left(k\right)\right)>0$ (and similarly for managers $\gamma [l]$ and $\gamma [k]$), thus contradiction. Therefore, if $S\left[j\right]$ deviates to manager $\gamma \left[k\right], k>j$ in equilibrium, then all firms $S\left[l\right],l\in \left(j,k\right)$ deviate to manager $\gamma [l+\left(k-j\right)]$ from manager $\gamma [l]$. Q.E.D.

Claim 3 is based on complementarity. Thus, if firm $S\left[j\right]$ is matched with manager $\gamma [k]$, then it is efficient to assign firm $S\left[l\right]$ to the managers $\gamma [l+\left(k-j\right)]$ assortatively until every firm hires one manager.

Now I find a period 1 matching provided that there is $i\in [\eta ,\lambda ]$ such that $S\left[i\right]<{S}^{*}[i]$. Due to complementarity, the assignment starts from the top. Because $\pi \left(0\right)=0$ as $\gamma \left[0\right]=1$, and $\pi (i)$ is continuous, the sorting price dominates the participation price for low $i$. Thus, ${\mu }_{1}^{*}\left(j\right)=j$ for $0\le j\le \eta$. Let ${i}^{*}\equiv \underset{i}{\mathrm{inf }}\{i\in \left[\eta , \lambda \right]\left|S\left[i\right]<{S}^{*}\left[i\right]\right\}$ and

$${k}^{*}\equiv \underset{k}{{\text{inf}}} \left\{k|Y\left(S\left[{i}^{*}\right],\gamma [{i}^{*}]\right)-Y\left(S\left[{i}^{*}\right],\gamma [k]\right)+{\omega }_{1}^{*}\left(k\right)<{w}_{0}+\pi \left({i}^{*}\right)\right\}.$$

As in Step 1, ${\omega }_{1}^{*}\left(k\right)={w}_{0}-{\int }_{k}^{1+A}{Y}_{\gamma }\left(S\left[l\right],\gamma [l]\right)\times {\gamma }{\prime}\left[l\right]dl$, where $A$ denotes the measure of managers who sit out in equilibrium. Because the measure ${k}^{*}-{i}^{*}$ managers sit out (Claim 3), $A={k}^{*}-{i}^{*}$. Thus, ${k}^{*}$ is implicitly determined by:

$${k}^{*}=\underset{k}{{\text{inf}}} \left\{k|Y\left(S\left[{i}^{*}\right],\gamma [{i}^{*}]\right)-Y\left(S\left[{i}^{*}\right],\gamma [k]\right)<\pi \left({i}^{*}\right)+{\int }_{k}^{1+(k-{i}^{*})}{Y}_{\gamma }\left(S\left[l\right],\gamma [l]\right)\times {\gamma }^{\mathrm{^{\prime}}}\left[l\right]dl\right\}.$$

Given (15), ${k}^{*}$ always exists. By the definition of ${i}^{*}$, for all $j\in (\eta ,{i}^{*})$, the sorting price dominates the participation price; thus, ${\mu }_{1}^{*}\left(j\right)=j$. Because firm $S[{i}^{*}]$ is strictly better off by matching with manager $\gamma [{k}^{*}]$, ${\mu }_{1}^{*}\left({k}^{*}\right)={i}^{*}$. Due to Claim 3, for $j\in ({i}^{*},1]$, firm $S[j]$ matches with manager $\gamma \left[j+\left({k}^{*}-{i}^{*}\right)\right]$. Due to the assumption that $\pi \left(i\right)$ is single peaked in the neighborhood of $\left[\eta , \lambda \right]$, there is no other deviation for firm $S\left[j\right], j>{i}^{*}$. Therefore, ${\mu }_{1}^{*}\left(j+\left({k}^{*}-{i}^{*}\right)\right)=j$ for $j\in \left[{i}^{*},1\right]$, which is equivalent to ${\mu }_{1}^{*}\left(i\right)=i-({k}^{*}-{i}^{*})$ for $i\in \left[{k}^{*}, 1+\left({k}^{*}-{i}^{*}\right)\right]$.

The market price is determined similarly as in Lemma 1. The difference comes from the discontinuity in matching due to the value of sitting out. First, consider $i<\eta$. As in Lemma 1, the market value is determined by the binding marginal sorting constraint and the sorting price dominates the participation price for $i<\eta$. Thus, for manager $\gamma [i]$, the market value starts from manager $\gamma [\eta ]$’s pay, and the reputation premium is accumulated up to $i<\eta$.

$${\omega }_{1}^{*}\left(i\right)={\omega }_{1}^{*}\left(\eta \right)-{\int }_{i}^{\eta }{Y}_{\gamma }\left(S\left[l\right],\gamma [l]\right)\times {\gamma }^{\mathrm{^{\prime}}}\left[l\right]dl\mathrm{ for }i\in \left[0,\eta \right].$$

For $i\in [\eta ,{i}^{*}]$, because the matching is ${\mu }_{1}^{*}\left(i\right)=i$, the market price is determined by sorting price, which must be greater than or equal to the participation price. Recall that firm $S\left[{i}^{*}\right]$ compares manager $\gamma [{i}^{*}]$ with manager $\gamma [{k}^{*}]$ and that a subset of managers $\gamma [j]$ for $j\in [{i}^{*}, {k}^{*}]$ sits out. Thus, for $i\in [\eta ,{i}^{*}]$, starting from the baseline manager pay ${\omega }_{1}\left({i}^{*}\right)$, the reputation premium is accumulated from ${i}^{*}$ up to $i$.

$${\omega }_{1}^{*}\left(i\right)={\omega }_{1}^{*}\left({i}^{*}\right)-{\int }_{i}^{{i}^{*}}{Y}_{\gamma }\left(S\left[l\right],\gamma [l]\right)\times {\gamma }^{\mathrm{^{\prime}}}\left[l\right]dl\ge {w}_{0}+\pi \left(i\right)\mathrm{ for }i\in \left[\eta ,{i}^{*}\right].$$

Because a subset of managers $\gamma [i]$ for $i\in [{i}^{*}, {k}^{*}]$ sits out, ${\omega }_{1}^{*}\left(i\right)={w}_{0}+\pi \left(i\right)$, and all of them are indifferent between participation and sitting out. For $i\in \left[{k}^{*},1+\left({k}^{*}-{i}^{*}\right)\right]$, starting from the last matched manager $\gamma \left[1+\left({k}^{*}-{i}^{*}\right)\right]$ whose market value is ${\omega }_{1}^{*}\left(1+\left({k}^{*}-{i}^{*}\right)\right)={w}_{0}+\pi (1+\left({k}^{*}-{i}^{*}\right))$, the reputation premium is accumulated from $1+\left({k}^{*}-{i}^{*}\right)$ up to $i$

$${\omega }_{1}^{*}\left(i\right)={\omega }_{1}^{*}\left(1+\left({k}^{*}-{i}^{*}\right)\right)-{\int }_{i}^{1+\left({k}^{*}-{i}^{*}\right)}{Y}_{\gamma }\left(S\left[l\right],\gamma [l]\right)\times {\gamma }{\prime}\left[l\right]dl {\text{for}} i\in \left[{k}^{*},1+\left({k}^{*}-{i}^{*}\right)\right].$$

Because ${w}_{0}$ is sufficiently high, ${\omega }_{1}\left(1+\left({k}^{*}-{i}^{*}\right)\right)\ge 0$. Lastly, for the market values to be well-defined at rank $\eta , {i}^{*}$ and ${k}^{*}$,

$${\omega }_{1}^{*}\left(\eta \right)={\omega }_{1}^{*}\left({i}^{*}\right)-{\int }_{\eta }^{{i}^{*}}{Y}_{\gamma }\left(S\left[l\right],\gamma [l]\right)\times {\gamma }^{\mathrm{^{\prime}}}\left[l\right]dl,$$

${\omega }_{1}^{*}\left({i}^{*}\right)={w}_{0}+\pi ({i}^{*})$, and ${w}_{0}+\pi \left({k}^{*}\right)={\omega }_{1}^{*}\left(1+({k}^{*}-{i}^{*})\right)-{\int }_{{k}^{*}}^{1+({k}^{*}-{i}^{*})}{Y}_{\gamma }\times {\gamma }{\prime}\left[l\right]dl$.

Step 3) To show that this is indeed an equilibrium, the equilibrium above must satisfy the (SC) and (PC) constraints for all firms and (both matched and unmatched) managers, and there is no individual firm or manager or pair of firm and manager that can block the above assignment. By construction, for all $i$ such that ${\mu }_{1}^{*}\left(i\right)\ne u$, ${\omega }_{1}^{*}(i)$ satisfies manager $\gamma [i]$’s participation constraint, and firm $S[{\mu }_{1}^{*}\left(i\right)]$’s sorting constraint. If manager $\gamma [i]$ asks ${\omega }_{1}^{*}\left(i\right)+\zeta ,\zeta >0$, this cannot be an equilibrium, as the firm $S[{\mu }_{1}^{*}\left(i\right)]$ will find another manager $\gamma [i]$ at price ${\omega }_{1}^{*}(i)$. If firm $S[{\mu }_{1}^{*}\left(i\right)]$ offers ${\omega }_{1}^{*}\left(i\right)-\zeta ,\zeta >0,$ this cannot be an equilibrium, as manager $\gamma [i]$ will not accept the contract, and will find another firm $S[{\mu }_{1}^{*}\left(i\right)]$ that is willing to pay ${\omega }_{1}^{*}(i)$. Therefore, there is no profitable deviation by any firm or any matched manager. Moreover, for manager $\gamma \left[i\right], i\in [{i}^{*},{k}^{*}]$, ${\omega }_{1}^{*}(i)$ is the same as the participation price, ${\omega }_{1}^{*}\left(i\right)={\omega }^{PC}(i)$; thus, they are indifferent between participation and sitting out. But the participation price is greater than the sorting price that a firm is willing to pay. Thus, manager $\gamma \left[i\right], i\in [{i}^{*},{k}^{*}]$ cannot be better off without making firms worse off.

To check that there is no block that Pareto improves the matching found above, let $u\left(i\right)=Y\left(S\left[{\mu }_{1}^{*}\left(i\right)\right],\gamma [i]\right)-{\omega }_{1}^{*}(i)$ denote firm $S\left[{\mu }_{1}^{*}\left(i\right)\right]$’s equilibrium payoff. Because

$${\omega }_{1}^{*}\left(i\right)+u\left(j\right)=Y\left(S\left[j\right],\gamma [i]\right) {\text{for}} j={\mu }_{1}^{*}(i) {\text{and}}$$

$${\omega }_{1}^{*}\left(i\right)+u\left(j\right)>Y\left(S[j],\gamma [i]\right) {\text{for}} j\ne {\mu }_{1}^{*}\left(i\right),$$

there is no block either. Therefore, the matching function and the market value function characterized above constitute an equilibrium.

A.7.: Proof of Proposition 2

Proof

Recall that $\pi \left(i\right)>0$ can be written as follows:

$$\begin{gathered} F_{\gamma \left[ i \right]} \left( m \right)E\left[ {S\left[ {i_{x} } \right]\gamma \left[ {i_{x} } \right] - S\left[ {i_{\phi } } \right]\gamma \left[ {i_{\phi } } \right] - \left( {V\left( {i_{x} } \right) - V\left( {i_{\phi } } \right)} \right)\left| x \right\rangle m} \right] \hfill \\ < \overline{F}_{\gamma \left[ i \right]} \left( m \right)E\left[ {S\left[ {i_{\phi } } \right]\gamma \left[ {i_{\phi } } \right] - S\left[ {i_{x} } \right]\gamma \left[ {i_{x} } \right] - \left( {V\left( {i_{\phi } } \right) - V\left( {i_{x} } \right)} \right)|x < m} \right], \hfill \\ \end{gathered}$$

where the expectation operator is with respect to ${f}_{\gamma \left[i\right]}\left(x\right)$.

To express the above condition with respect to the density quantile function, plug $V\left(i\right)$ and rearrange the terms,

$${F}_{\gamma \left[i\right]}\left(m\right)E\left[S\left[{i}_{x}\right]\gamma \left[{i}_{x}\right]-S\left[{i}_{\phi }\right]\gamma \left[{i}_{\phi }\right]-\left({\int }_{{i}_{x}}^{1}\frac{\gamma \left[l\right]}{g\left(S\left[l\right]\right)}dl-{\int }_{{i}_{\phi }}^{1}\frac{\gamma \left[l\right]}{g\left(S\left[l\right]\right)}dl\right)|x>m\right]$$

$$<{\overline{F} }_{\gamma \left[i\right]}\left(m\right)E\left[S\left[{i}_{\phi }\right]\gamma \left[{i}_{\phi }\right]-S\left[{i}_{x}\right]\gamma \left[{i}_{x}\right]-\left({\int }_{{i}_{\phi }}^{1}\frac{\gamma \left[l\right]}{g\left(S\left[l\right]\right)}dl-{\int }_{{i}_{x}}^{1}\frac{\gamma \left[l\right]}{g\left(S\left[l\right]\right)}dl\right)|x<m\right]$$

$$\iff S\left[{i}_{\phi }\right]\gamma \left[{i}_{\phi }\right]-{\int }_{{i}_{\phi }}^{1}\frac{\gamma \left[l\right]}{g\left(S\left[l\right]\right)}dl>E\left[S\left[{i}_{x}\right]\gamma \left[{i}_{x}\right]-{\int }_{{i}_{x}}^{1}\frac{\gamma \left[l\right]}{g\left(S\left[l\right]\right)}dl\right].$$

(16)

Let $D\left(i\right)\equiv {\int }_{i}^{1}\frac{\gamma \left[l\right]}{\gamma \left[i\right]}\frac{g\left(S\left[i\right]\right)}{g\left(S\left[l\right]\right)}dl$. Here, $D\left(i\right)$ represents the firm $S[i]$’s share of the total match surplus, normalized by $\frac{\gamma \left[i\right]}{g\left(S\left[i\right]\right)}$. The slope effect can be written as: ${\int }_{{i}_{x}}^{1}\frac{\gamma \left[l\right]}{g\left(S\left[l\right]\right)}dl=D\left({i}_{x}\right)\frac{\gamma \left[{i}_{x}\right]}{g\left(S\left[{i}_{x}\right]\right)}$. Use $\gamma \left[{i}_{x}\right]=\gamma \left[i\right]\frac{{f}_{g}\left(x\right)}{{f}_{\gamma \left[i\right]}\left(x\right)}$ and observe that

$$\begin{aligned} E\left[ {S\left[ {i_{x} } \right]\gamma \left[ {i_{x} } \right] - \mathop \int \limits_{{i_{x} }}^{1} \frac{\gamma \left[ l \right]}{{g\left( {S\left[ l \right]} \right)}}dl} \right] & = \mathop \int \limits_{{\underline {x} }}^{{\overline{x}}} \left( {S\left[ {i_{x} } \right]\gamma \left[ {i_{x} } \right] - D\left( {i_{x} } \right)\frac{{\gamma \left[ {i_{x} } \right]}}{{g\left( {S\left[ {i_{x} } \right]} \right)}}} \right)f_{\gamma \left[ i \right]} \left( x \right)dx \\ & = \gamma \left[ i \right]\mathop \int \limits_{{\underline {x} }}^{{\overline{x}}} \left( {S\left[ {i_{x} } \right]\frac{{f_{g} \left( x \right)}}{{f_{\gamma \left[ i \right]} \left( x \right)}} - \frac{{D\left( {i_{x} } \right)}}{{g\left( {S\left[ {i_{x} } \right]} \right)}}\frac{{f_{g} \left( x \right)}}{{f_{\gamma \left[ i \right]} \left( x \right)}}} \right)f_{\gamma \left[ i \right]} \left( x \right)dx = \gamma \left[ i \right]E_{g} \left[ {S\left[ {i_{x} } \right] - \frac{{D\left( {i_{x} } \right)}}{{g\left( {S\left[ {i_{x} } \right]} \right)}}} \right], \\ \end{aligned}$$

where ${E}_{g}$ denotes an expectation operator under a good type manager.

Thus, inequality (16) is written as:

$$\Leftrightarrow \gamma \left[ {i_{\phi } } \right]\left( {S\left[ {i_{\phi } } \right] - \frac{{D\left( {i_{\phi } } \right)}}{{g\left( {S\left[ {i_{\phi } } \right]} \right)}}} \right) > \gamma \left[ i \right]E_{g} \left[ {S\left[ {i_{x} } \right] - \frac{{D\left( {i_{x} } \right)}}{{g\left( {S\left[ {i_{x} } \right]} \right)}}} \right].$$

Because $\gamma \left[i\right]=\gamma \left[{i}_{\phi }\right]=\gamma$, the condition above reduces to expression (8):

$$S\left[ {i_{\phi } } \right] - \frac{{D\left( {i_{\phi } } \right)}}{{g\left( {S\left[ {i_{\phi } } \right]} \right)}} > E_{g} \left[ {S\left[ {i_{x} } \right] - \frac{{D\left( {i_{x} } \right)}}{{g\left( {S\left[ {i_{x} } \right]} \right)}}} \right].$$

(17)

The left-hand side of (17) represents the difference between the quantile function of firm size and the reciprocal of the density quantile function of firm size (multiplied by the firm $S[i]$’s share $D\left(i\right)$) when the manager sits out. The right-hand side is the expected difference between the quantile function and the reciprocal of the density quantile function when the manager participates and is a good type. The above condition implies that the manager’s share (in the case of participation) increases slowly even if he is a good type: as firm size increases, $D\left({i}_{x}\right)/g\left(S\left[{i}_{x}\right]\right)$ increases faster than its size. I apply condition (8) for different distributions of firm size—uniform and Pareto. Provided that $D\left(i\right)=1$ for all $i$, the following cases show which distributions can potentially satisfy condition (17).

Case 1) uniform distribution over (0,1).

$$g\left(S\right)=1,S\left[i\right]=1-i, {S}^{\mathrm{^{\prime}}}\left[i\right]=-1=-\frac{1}{g\left(S\left[i\right]\right)}.$$

Plug these into (17):

$${E}_{g}\left[1-{i}_{x}-1\right]={E}_{g}\left[-{i}_{x}\right]<-{i}_{\phi }.$$

Therefore, under the uniform distribution assumption, (17) is equivalent to ${E}_{g}\left[{i}_{x}\right]>{i}_{\phi }$.

Case 2) Pareto distribution.

$$g\left(S\right)=\frac{1}{\beta }{S}^{-\frac{1+\beta }{\beta }},S\left[i\right]={i}^{-\beta }, {S}^{\mathrm{^{\prime}}}\left[i\right]=-\beta {i}^{-\beta -1}=-\frac{1}{g\left(S\left[i\right]\right)}.$$

Plug these into (17):

$${E}_{g}\left[{i}_{x}^{-\beta }-\beta {i}_{x}^{-\beta -1}\right]={E}_{g}\left[\frac{1}{{i}_{x}^{\beta }}\left(1-\frac{\beta }{{i}_{x}}\right)\right]<\frac{1}{{i}_{\phi }^{\beta }}\left(1-\frac{\beta }{{i}_{\phi }}\right).$$

When $\beta =1$ (a Zipf’s law), condition (17) is written as.

$${E}_{g}\left[\frac{1}{{i}_{x}^{\beta }}\left(\frac{\beta }{{i}_{x}}-1\right)\right]>\frac{1}{{i}_{\phi }^{\beta }}\left(\frac{\beta }{{i}_{\phi }}-1\right),$$

For any $i<1$ because $\frac{\beta }{{i}_{x}}>1$ and $\frac{\beta }{{i}_{\phi }}>1$. Notice that $\frac{1}{{y}^{\beta }}\left(\frac{\beta }{y}-1\right)$ is strictly convex decreasing in $y\in (\mathrm{0,1})$ for any $\beta >0$ because.

$$\frac{\partial }{\partial y}\left(\frac{1}{{y}^{\beta }}\left(\frac{\beta }{y}-1\right)\right)=-\beta {y}^{-2-\beta }\left(1+\beta -y\right)<0,$$

$$\frac{{\partial }^{2}}{\partial {y}^{2}}\left(\frac{1}{{y}^{\beta }}\left(\frac{\beta }{y}-1\right)\right)=\beta \left(1+\beta \right)\left(2+\beta -y\right){y}^{-3-\beta }>0.$$

Thus, by Jensen’s inequality, the inequality is always satisfied for $\beta =1.$

A.8.: Proof of Proposition 3

Proof

Conditional on the manager $\gamma \left[i\right]$’s participation, firm $S\left[j\right]$ offers the contract satisfying the (IR) constraint:

$$\frac{1}{2}E\left[{w}_{x}|i\right]+\frac{1}{2}E\left[{v}_{x}|i\right]-c\ge {\omega }_{t}\left(i\right)-c,$$

where the contract ensures that the manager’s expected payoff of exerting effort for firm $S\left[j\right]$ is greater than or equal to his payoff of working for other firms who will pay ${\omega }_{t}\left(i\right)$ in expectation.

Moreover, the contract satisfies the (IC) constraint:

$$\frac{1}{2}E\left[{w}_{x}|i\right]+\frac{1}{2}E\left[{v}_{x}|i\right]-c\ge E\left[{v}_{x}|i\right].$$

Thus, the minimum incentive compatible payment must give ${v}_{x}=0$ for all $x$, and the binding (IR) wage satisfies $\frac{1}{2}E\left[{w}_{x}|i\right]={\omega }_{t}\left(i\right)$. Since ${\omega }_{t}\left(i\right)\ge {w}_{0}$ for any $i$ and $t$, and $c<{w}_{0}$, the payoffs of all participating managers who will exert effort are ensured to be positive.

Since all firms design the contract to elicit effort, knowing that he will exert effort upon participation, manager $\gamma \left[i\right]$ participates if:

$$\begin{array}{*{20}c} {\frac{1}{2}E\left[ {w_{x} {|}i} \right] + \frac{1}{2}E\left[ {v_{x} {|}i} \right] - c + {\Pi }_{i}^{P} \ge w_{0} + {\Pi }_{i}^{NP} .} \\ \end{array}$$

(18)

The market value, ${\omega }_{t}\left(i\right)$ in period $t=1$, is determined by (18) and the (SC) constraint:

$$Y\left(S[j],i\right)-{\omega }_{t}\left(i\right)\ge Y\left(S\left[j\right],k\right)-{\omega }_{t}\left(k\right) \forall i, j, k,$$

as in the proof of Proposition 1.