A model of search and matching with PES intermediation

Abstract

The paper is an extension of Morgan (A model of search, coordination and market segmentation. Department of Economics, SUNY-Buffalo Working paper, 1998). It shows that introducing a specialist matching intermediary, a public employment exchange, can improve the efficiency of the market under certain conditions. Coupled with search costs and incentive compatible lumpsum membership fees, agents participate in only one of the disjoint segments generated in such markets. A complete description is provided of the equilibrium matching of agents. In equilibrium, agents separate according to type into different segments viz., an informal economy, an economy operated by the intermediary, and an economy populated by relatively highly skilled agents only. Welfare properties of the equilibrium are provided.

Appendix

Proof of Proposition 1. Consider (9) and write

$$\begin{aligned} G({\tilde{v}},{\hat{v}};v_l)\equiv \beta \Bigg [\frac{\int _{\hat{v}}^{\tilde{v}}\mu ({\hat{v}},z)f(z)dz}{F({\tilde{v}})-F({\hat{v}})}-\frac{\int _{v_l}^{\hat{v}}\mu ({\hat{v}},z)f(z)dz}{F({\hat{v}})}\Bigg ]-R+c=0 \end{aligned}$$

for $v_{l}\le {\hat{v}} \le {\tilde{v}}$. $\mu ({\hat{v}},z)$ is differentiable, so $G(\tilde{v},\hat{v};v_l)$ is continuous and differentiable with respect to $\hat{v}$. The value of G(.) at $\hat{v}=v_{l}$ is

$$\begin{aligned} G(\tilde{v},v_l;v_l)=\beta \Bigg [\dfrac{\int _{v_l}^{\tilde{v}}\mu (v_l,z)f(z)dz}{F(\tilde{v})}- \dfrac{\int _{v_l}^{v_l}\mu (v_l,z)f(z)dz}{F(v_l)}\Bigg ]=\beta \Bigg [\frac{\int _{v_l}^{\tilde{v}}\mu (v_l,z)f(z)dz}{F(\tilde{v})}-\mu (v_{l},v_{l})\Bigg ]\, ; \end{aligned}$$

i.e., the benefit of the lowest type in $[v_l,\tilde{v}]$ in matching within the PES exceeds the match value generated by pairing with an identical agent of the same type. The value of G(.) at $\hat{v}=\tilde{v}$ is

$$\begin{aligned} G(\tilde{v},\tilde{v};v_l)=\beta \Bigg [\dfrac{\int _{\tilde{v}}^{\tilde{v}}\mu (\tilde{v},z)f(z)dz}{F(\tilde{v})-F(\tilde{v})}- \dfrac{\int _{v_l}^{\tilde{v}}\mu (\tilde{v},z)f(z)dz}{F(\tilde{v})}\Bigg ]=\beta \Bigg [\mu (\tilde{v},\tilde{v})-\frac{\int _{v_l}^{\tilde{v}}\mu (\tilde{v},z)f(z)dz}{F(\tilde{v})}\Bigg ]\, ; \end{aligned}$$

i.e., the benefit of the highest type in $[v_l,\tilde{v}]$ in matching with an identical agent of its own type exceeds the match value generated from matching within $[v_l,\tilde{v}]$.

Therefore,the net benefit of PES membership for a type $\hat{v}$ attains a maximum in $[v_l,\tilde{v}]$. If

$$\begin{aligned} \beta \Bigg [\frac{\int _{v_l}^{\tilde{v}}\mu (v_l,z)f(z)dz}{F(\tilde{v})}-\mu (v_{l},v_{l})\Bigg ]<R-c<\beta \Bigg [\mu (\tilde{v},\tilde{v})-\frac{\int _{v_l}^{\tilde{v}}\mu (\tilde{v},z)f(z)dz}{F(\tilde{v})}\Bigg ] \end{aligned}$$

then the Intermediate Value Theorem establishes the existence of at least one zero root; i.e., at least one solution ${\hat{v}}_1$ for (9), for given R and c. Then,

$$\begin{aligned} G(\tilde{v},{{\hat{v}}_1};v_l)\equiv \beta \Bigg [\frac{\int _{{\hat{v}}_1}^{\tilde{v}}\mu (\hat{v},z)f(z)dz}{F(\tilde{v})-F({{\hat{v}}_1})}-\frac{\int _{v_l}^{{\hat{v}}_1}\mu (\hat{v},z)f(z)dz}{F({{\hat{v}}_1})}\Bigg ]-R+c \equiv 0 . \end{aligned}$$

${\hat{v}}_1$ is unique if G(.) is monotonic with respect to $\hat{v}$ on $[v_l,\tilde{v}]$.

$$\begin{aligned} \frac{\partial G(\tilde{v},\hat{v};v_l)}{\partial \hat{v}}&= \beta \frac{\partial }{\partial \hat{v}}\Bigg [\dfrac{\int _{\hat{v}}^{\tilde{v}}\mu (\hat{v},z)f(z)dz}{F(\tilde{v})-F(\hat{v})}- \dfrac{\int _{v_l}^{\hat{v}}\mu (\hat{v},z)f(z)dz}{F(\hat{v})}\Bigg ]\\&= \beta \Bigg [\frac{[F(\tilde{v})-F(\hat{v})]\frac{\partial }{\partial \hat{v}}\int _{\hat{v}}^{\tilde{v}}\mu (\hat{v},z)f(z)dz + f(\hat{v})\int _{\hat{v}}^{\tilde{v}}\mu (\hat{v},z)f(z)dz }{[F(\tilde{v})-F(\hat{v})]^{2}}\Bigg ]\\&\times \dfrac{[F(\hat{v})]\frac{\partial }{\partial \hat{v}}\int _{v_l}^{\hat{v}}\mu (\hat{v},z)f(z)dz + f(\hat{v})\int _{v_l}^{\hat{v}}\mu (\hat{v},z)f(z)dz }{F(\hat{v})^{2}}\Bigg ]. \end{aligned}$$

Keeping in mind that the function $\mu$ is twice continuously differentiable and increasing in its domain for both its arguments, and then using Leibniz’s rule to differentiate under the integral.⁶ Furthermore, the supermodularity of the $\mu$ function ensures that $\mu _{12}>0$, $\forall$ $v\in$ $[v_l,v_u]$. We can further write $\frac{\partial G(\tilde{v},\hat{v};v_l)}{\partial \hat{v}}$ as

$$\begin{aligned} \frac{\partial G(\tilde{v},\hat{v};v_l)}{\partial \hat{v}}= & {} \frac{\beta }{[F(\tilde{v})-F(\hat{v})]^{2}}\Bigg [[F(\tilde{v})-F(\hat{v})]\int _{\hat{v}}^{\tilde{v}} \frac{\partial \mu (\hat{v},z)}{\partial \hat{v}}f(z)dz-f(\hat{v})[F(\tilde{v})\\&-F(\hat{v})]\mu (\hat{v},\hat{v}) +f(\hat{v})\int _{\hat{v}}^{\tilde{v}}\mu (\hat{v},z)f(z)dz\Bigg ]\\&-\frac{\beta }{F(\hat{v})^{2}}\Bigg [F(\hat{v})\int _{v_l}^{\hat{v}} \frac{\partial \mu (\hat{v},z)}{\partial \hat{v}}f(z)dz+f(\hat{v})(F(\hat{v})\mu (\hat{v},\hat{v})\\&-f(\hat{v})\int _{v_l}^{\hat{v}}\mu (\hat{v},z)f(z)dz\Bigg ].\\ \frac{\partial G(\tilde{v},\hat{v};v_l)}{\partial \hat{v}}= & {} \beta \Bigg [\frac{\int _{\hat{v}}^{\tilde{v}}\mu _{1}(\hat{v},z)f(z)dz}{F(\tilde{v})-F(\hat{v})}-\frac{\int _{v_l}^{\hat{v}}\mu _{1}(\hat{v},z)f(z)dz}{F(\hat{v})}\Bigg ]\\&+ \frac{\beta f(\hat{v})}{F(\tilde{v})-F(\hat{v})}\Bigg [ \frac{\int _{\hat{v}}^{\tilde{v}}\mu (\hat{v},z)f(z)dz}{F(\tilde{v})-F(\hat{v})} - {\mu (\hat{v},\hat{v})}\Bigg ]\\&+\frac{\beta f(\hat{v})}{F(\hat{v})}\Bigg [\frac{\int _{v_l}^{\hat{v}}\mu (\hat{v},z)f(z)dz}{F(\hat{v})} - {\mu (\hat{v},\hat{v})}\Bigg ]\\&= \beta [A_1-A_2]+\beta f(\hat{v})[B_1-B_2]+ \beta f(\hat{v})[C_1-C_2]>0. \end{aligned}$$

$A_1-A_2>0$ because $\mu _1(.)$ is increasing in $[v_l,\tilde{v}]$. $B_1$ represents a type $\hat{v}$’s average payoff of matching with any type in $[\hat{v},\tilde{v}]$. $B_2$ is the match value generated by matching with itself in $[\hat{v},\tilde{v}]$. Since, any average type in $[\hat{v},\tilde{v}]$ is higher than $\hat{v}$, $B_1-B_2>0$. Similarly, $C_1$ represents a type $\hat{v}$’s average payoff of matching with any type in $[v_l, \hat{v}]$. $C_2$ is the match value generated by matching with itself in $[v_l, \hat{v}]$. Since, any average type in $[v_l, \hat{v}]$ is lower than $\hat{v}$, $C_1-C_2<0$. Further, $B_1$ and $C_1$ are the average normalized payoff of $\hat{v}$ with z, for $\hat{v}\le z \le \tilde{v}$ and $v_l\le z \le \hat{v}$ respectively. The latter is smaller than the former because the strictly supermodular function $\mu (\hat{v},z)$ possesses strictly increasing differences in $(\hat{v},z)$. This implies that $\mid {B_1-B_2}\mid >\mid {C_1-C_2}\mid$. Therefore, $G(\tilde{v},\hat{v};v_l)$ is a strictly increasing function of $\hat{v} \in$ $[v_l,\tilde{v}]$. $\square$

Proof of Remark 1

Since $v_u$ chooses $\tilde{v}$ to optimize its net expected payoff from matching in the higher segment, the reservation type of $v_u$ cannot be lower than $\tilde{v}$. Suppose $\rho (v_u;\tilde{v},v_u)>\tilde{v}$. Then the expected value of a match generated for $v_u$ in $[\tilde{v},\rho (v_u;.)]\bigcup [\rho (v_u;.),v_u]$ (call it $E_2$) is

$$\begin{aligned} E_2=\Big (1-F(\rho (v_u;.))\Big ) \Bigg [\frac{-c+\beta \int _{\rho (v_u;.)}^{v_u}\mu (v_u,z)f(z)dz}{1-F(\rho (v_u;.))}\Bigg ]-c \Big (F(\rho (v_u;.))-F(\tilde{v}))\Big ). \end{aligned}$$

The first term in $E_2$ is the expected payoff for $v_u$ of matching with any type at least as high as its reservation type, multiplied by the probability that a type $v_u$ actually encounters a v $\in$ $[\rho (v_u;.),v_u]$. The second term in $E_2$, is the expected search cost due to a type $v_u$ rejecting any type v $\in$ $[\tilde{v},\rho (v_u;.)]$.

However, if $\rho (v_u;\tilde{v},v_u)=\tilde{v}$, then $v_u$’s expected value of matching in $[\tilde{v},v_u]$ is

$$\begin{aligned}&\Big (1-F(\rho (v_u;.))\Big ) \Bigg [\frac{-c+\beta \int _{\rho (v_u;.)}^{v_u}\mu (v_u,z)f(z)dz}{1-F(\rho (v_u;.))}\Bigg ]\nonumber \\&\quad -c \Big (F(\rho (v_u;.))-F(\tilde{v}))\Big )+c \Big (F(\rho (v_u;.))-F(\tilde{v}))\Big )>{E_2}. \end{aligned}$$

Therefore $\rho (v_u;\tilde{v},v_u)=\tilde{v}$. $\square$

Proof of Proposition 2

Consider (16) and write

$$\begin{aligned} G_1(\tilde{v},v_u;\hat{v})=\frac{\beta \int _{\tilde{v}}^{v_u}[\mu (v_u,z)-\mu (\tilde{v},z)]f(z) dz}{F(v_u)-F(\tilde{v})}-\frac{\beta \int _{{\hat{v}}_1=h(\tilde{v};.)}^{\tilde{v}}[\mu (v_u,z)-\mu (\tilde{v},z)]f(z) dz}{F(\tilde{v})-F({\hat{v}}_1=h(\tilde{v};.))}. \end{aligned}$$

Consider the compact interval $[\hat{v},v_u]$ on $\mathbb {R}$. $G_{1}(\tilde{v},v_u;\hat{v})$ ⁷ is defined on $[\hat{v},v_u]$. Evaluating $G_{1}(.)$ at its end points of the domain yields

$$\begin{aligned} G_{1}(\hat{v},v_u;\hat{v})= \frac{\beta \int _{\hat{v}}^{v_u}[\mu (v_u,z)-\mu (\hat{v},z)]f(z)dz}{F(v_u)-F(\hat{v})}-\frac{\beta \int _{{\hat{v}}_1=h(\tilde{v};.)}^{\hat{v}}[\mu (v_u,z)-\mu (\hat{v},z)]f(z)dz}{{F(\hat{v}})-F({{\hat{v}}_1=h(\tilde{v};.)}})>0. \end{aligned}$$

The second term in the above expression in $G_{1}(\hat{v},v_u;\hat{v})$ is smaller than the first term since $v_u$ is the highest type in the matching set specified in the first term above.

Also,

$$\begin{aligned} G_{1}(v_u,v_u;\hat{v})= \frac{\beta \int _{v_u}^{v_u}[\mu (v_u,z)-\mu (v_u,z)]f(z)dz}{F(v_u)-F(v_u)}-\frac{\beta \int _{{{\hat{v}}_1=h(\tilde{v};.)}}^{v_u}[\mu (v_u,z)-\mu (v_u,z)]f(z)dz}{F(v_u)-F({{\hat{v}}_1=h(\tilde{v};.)})}=0. \end{aligned}$$

$\mu (v_u,z)$ is continuous on $[v_l,v_u]$, so $G_{1}(\tilde{v},v_u;\hat{v})$ is both continuous and bounded on $[\hat{v},v_u]$, since functions continuous on a closed interval are also bounded in that interval. By Weierstrass’s Theorem, $G_{1}(.)$ must attain a maximum in $[\hat{v},v_u]$. That is, there exists an agent ${\tilde{v}^{*}}$ such that

$$\begin{aligned} G_{1}(\hat{v},v_u;\hat{v}) > G_{1}({\tilde{v}}^{*},v_u;\hat{v}) \ge G_{1}(v_{u},v_{u};\hat{v})\,\, \mathrm{for}\,\, {\tilde{v}}^{*} \in [\hat{v},v_u]. \end{aligned}$$

Additionally, ${\tilde{v}^{*}}$ is unique. Since the reservation value for any type in this matching model is unique for every type in $[v_l,v_u]$ by construction, $\rho (v_u;{\tilde{v}^{*}},v_u)={\tilde{v}^{*}}$, is the only value of v $\in$ $[\hat{v},v_u]$ that maximizes the function $G_{1}(.)$. $\square$