Job market signaling with imperfect competition among employers

Abstract

This paper studies a job market signaling model with imperfect competition among employers. In our basic model, workers are differentiated in productivity and preference over employers, both of which are workers’ private information. We conclude that if competition is sufficiently strong, a separating equilibrium exists. We also show that stronger competition among employers intensifies competition between workers; workers invest more in costly education to get attractive jobs, and social welfare decreases. When employers can observe worker’s preferences, wage discrimination strengthens competition among employers and makes workers better off.

Appendices

Appendix

1.1 Equilibrium analysis

1.1.1 Firms’ wage offers

For technical convenience, we solve case (i) and (iii) first, and then solve case (ii).

Case (i): Full competition First, we show that, when $k\le \frac{2}{3}\tilde{\theta }$, all type-$\tilde{\theta }$ workers will be hired by either firm. In equilibrium, all type-$\tilde{\theta }$ workers will consider their first best option as accepting the offer from the nearest firm and their second best the offer from the more distant firm.

Check the following analysis for details.

A worker would prefer to accept firm 1’s wage offer $w^1_{\tilde{\theta }}$ over firm 2’s offer if $w^1_{\tilde{\theta }}-kx \ge w^2_{\tilde{\theta }} -k(1-x)$. From this, we derive the market share of firm 1 $x^1_{\tilde{\theta }} =\frac{w^1_{\tilde{\theta }}-w^2_{\tilde{\theta }}}{2k}+ \frac{1}{2}.$ Then, the firm 1’s problem turns out to be

$$\begin{aligned} \max _{w^1_{\tilde{\theta }}<\tilde{\theta }} \;\;x^1_{\tilde{\theta }} (\tilde{\theta }-w^1_{\tilde{\theta }})=\left( \frac{w^1_{\tilde{\theta }} -w^2_{\tilde{\theta }}}{2k}+ \frac{1}{2}\right) (\tilde{\theta } -w^1_{\tilde{\theta }}). \end{aligned}$$

The first-order condition with respect to $w^1_{\tilde{\theta }}$ is

$$\begin{aligned} \frac{1}{2k}(\tilde{\theta }-w^1_{\tilde{\theta }}) -\left( \frac{w^1_{\tilde{\theta }}-w^2_{\tilde{\theta }}}{2k} +\frac{1}{2}\right) \le 0 \;\;\text {with equality if } w^1_{\tilde{\theta }}< \tilde{\theta }. \end{aligned}$$

(F.O.C.)

Therefore, in a symmetric solution where $w^1_{\tilde{\theta }}=w^2_{\tilde{\theta }}$, each firm’s wage offer would be $w(\tilde{\theta };k)=\tilde{\theta }-k$. With this wage offer, we can show that all workers’ individual rationality (IR) conditions are satisfied for any value of $k\le \frac{2}{3} \tilde{\theta }$.

$$\begin{aligned} w^1_{\tilde{\theta }}-kx^1_{\tilde{\theta }}=w^1_{\tilde{\theta }} -k\left( \frac{w^1_{\tilde{\theta }}-w^2_{\tilde{\theta }}}{2k} +\frac{1}{2}\right) =\tilde{\theta }-\frac{3}{2}k \ge 0. \end{aligned}$$

(A.1)

Therefore, for $k\le \frac{2}{3}\tilde{\theta }$, all workers would be covered by either firm and the wage schedule for a type-$\tilde{\theta }$ worker would be

$$\begin{aligned} w(\tilde{\theta };k)=\tilde{\theta }-k, \;\;\;\;k \le \frac{2}{3}\tilde{\theta }. \end{aligned}$$

(A.2)

Case (iii): Local monopsony Here, we show that when $k>\tilde{\theta }$, some workers in the middle will not be hired by either of the two firms. In equilibrium, all type-$\tilde{\theta }$ workers will consider only the options of accepting the offer from the nearest firm or being unemployed. Accepting the offer from the other firm would be dominated.

Suppose $x^{1}_{\tilde{\theta }}<\frac{1}{2}$, say the partial coverage condition. A worker would accept firm 1’s wage offer if $w^1_{\tilde{\theta }}-kx\ge 0$. So, firm 1’s market share turns out to be $x^1_{\tilde{\theta }}=\frac{w_{\tilde{\theta }}}{k}$, and its profit maximization problem is as follows.

$$\begin{aligned} \max _{w^1_{\tilde{\theta }}<\theta } x^1_{\tilde{\theta }}(\tilde{\theta }-w^1_{\tilde{\theta }}) =\frac{w^{1}_{\tilde{\theta }}}{k} \left( \tilde{\theta }-w^1_{\tilde{\theta }}\right) . \end{aligned}$$

From the first-order condition,

$$\begin{aligned} \tilde{\theta }-2w^1_{\tilde{\theta }}\le 0 \;\; \text {with equality if } w^1_{\tilde{\theta }}< \tilde{\theta }, \end{aligned}$$

(A.3)

we get the symmetric interior solution, $w^1_{\tilde{\theta }} =w^2_{\tilde{\theta }}=\frac{\tilde{\theta }}{2}$. Since the corresponding market share is less than $\frac{1}{2}$, $x^1_{\tilde{\theta }}=\frac{w_{\tilde{\theta }}}{k} =\frac{\tilde{\theta }}{2k}<\frac{1}{2}$, the solution is consistent with the partial coverage condition. Therefore, when $\tilde{\theta }<k$, the firm’s wage schedule for a type-$\tilde{\theta }$ worker is

$$\begin{aligned} w(\theta ;k)=\frac{\tilde{\theta }}{2}, \; k>\tilde{\theta }. \end{aligned}$$

(A.4)

The wage schedule does not depend on k, but only on $\tilde{\theta }$.

Case (ii): Limited competition Last, we identify the wage offers for an intermediate value of k such that $\frac{2}{3}\tilde{\theta }\le k<\tilde{\theta }$. In this case, all workers will be hired and paid wages which do not increase with the degree of competition. We argue that the competition between firms only works in a way that prevents a firm from covering those who prefer the opposing firm.

First, we argue that when $\frac{2}{3}\tilde{\theta }\le k<\tilde{\theta }$, firms’ wage offers which do not cover all type-$\tilde{\theta }$ workers are not sequentially rational. In a symmetric equilibrium, to be in the partial-coverage case, a firm’s market share should be less than $\frac{1}{2}$. Let’s suppose $x^{1}_{\tilde{\theta }}=\frac{w^1_{\tilde{\theta }}}{k}<\frac{1}{2}$, which implies $w^1_{\tilde{\theta }}<\frac{k}{2}$. By substituting this market share into the first-order condition of the profit maximization in the partial-coverage regime expressed in (A.3), we get

$$\begin{aligned} \tilde{\theta }-2w^1_{\tilde{\theta }}>\tilde{\theta }-k\ge 0. \end{aligned}$$

This implies that, at any value of the wage offer $w^1_{\tilde{\theta }}<\frac{k}{2}$, the firm has incentive to raise the wage level. Hence, the partial-coverage wage schedule such that $w^1_{\tilde{\theta }}<\frac{k}{2}$ is not sequentially rational.

Second, to cover all workers in a symmetric solution, the market share for each firm should be $\frac{1}{2}$. We can easily see that, if the wage offer is equal to $\frac{k}{2}$, all workers would accept the offer and the marginal worker at $x=\frac{1}{2}$ receive zero surplus. Now, let’s suppose the other firm’s wage offer $w^2_{\tilde{\theta }}=\frac{k}{2}$ and check if firm 1 has any incentive to increase it wage offer to some level greater than $\frac{1}{2}$.

Firm 1 solves

$$\begin{aligned} \max _{w^1_{\tilde{\theta }}<\tilde{\theta }} \left( \frac{w^1_{\tilde{\theta }}-w^2_{\tilde{\theta }}}{2k} +\frac{1}{2}\right) (\tilde{\theta }-w^1_{\tilde{\theta }}) =\left( \frac{w^1_{\tilde{\theta }}}{2k}+ \frac{1}{4}\right) \left( \tilde{\theta }-w^1_{\tilde{\theta }}\right) \end{aligned}$$

subject to the full-coverage condition $w^1_{\tilde{\theta }}\ge \tfrac{k}{2}$. The first-order condition for this problem is as follows.

$$\begin{aligned} \frac{\tilde{\theta }-w^1_{\tilde{\theta }}}{2k} -\left( \frac{w^1_{\tilde{\theta }}}{2k}+\frac{1}{4}\right) \le 0 \Leftrightarrow \tilde{\theta }-2w^1_{\tilde{\theta }} -\frac{k}{2} \le 0 \;\;\text {with equality if } w_{\tilde{\theta }}> \tfrac{k}{2}. \end{aligned}$$

From this condition, we can conclude that at any value of $w_{\tilde{\theta }}> \tfrac{k}{2}$, the firm has incentive to decrease the level of the wage offer.

$$\begin{aligned} \tilde{\theta }-2w^1_{\tilde{\theta }}-\frac{k}{2}<\frac{3k}{2}-2w^1_{\tilde{\theta }}-\frac{k}{2} =k-2w^1_{\tilde{\theta }}< 0. \end{aligned}$$

This implies that there is no incentive to increase the wage offer from $w^1_{\tilde{\theta }}=\frac{k}{2}$. So, for $\frac{2}{3}\tilde{\theta }\le k<\tilde{\theta }$, the sequentially rational wage schedule is

$$\begin{aligned} w(\tilde{\theta };k)=\tfrac{k}{2}, \;\;\;\;\tfrac{2}{3} \tilde{\theta }\le k<\tilde{\theta }. \end{aligned}$$

(A.5)

1.1.2 Pooling equilibrium

In the pooling equilibrium, both types take the same education level, $e=0$. So, $E(\theta |\mu )=\tilde{\theta }=q\theta _l+(1-q) \theta _h=2-q$. By substituting this in Eq. (3.1), we get,

$$\begin{aligned} w^{*}(\tilde{\theta }=2-q;k)=\left\{ \begin{matrix} (2-q)-k&{}\text {if} &{}k \le \frac{2}{3}(2-q) \\ \frac{k}{2}&{}\text {if} &{} \frac{2}{3}(2-q)< k \le (2-q)\\ \frac{(2-q)}{2}&{}\text {if} &{} (2-q) < k. \end{matrix}\right. \end{aligned}$$

The following proposition identifies a pooling equilibrium in our basic model.

Proposition 6

(Pooling equilibrium) In the basic model, for any $q\in (0,1)$ and $k > 0$, there exists a pooling equilibrium consisting of a strategy profile $\left\langle w(e;k),\right.$$\left. e(\theta ,x;k) \right\rangle$ and supporting beliefs $\mu (\theta |e)$. Specifically, for given $q\in (0,1)$ and $k > 0$, a pooling equilibrium is a triple, $\left\langle w(e;k),e(\theta ,x;k),\right.$$\left. \mu (\theta |e) \right\rangle$, which satisfies $w(e=0;k)=w^{*}(2-q;k)$, $w(e>0;k)=w^{*}(2-r;k)$, $e(\theta ,x;k)=0$, $\mu (\theta =1|e=0)=q$, and $\mu (\theta =1|e>0)=r>q$.

Cho and Kreps (1987) define the concept of intuitive criterion in a signaling game with a single sender and a single receiver. Therefore, it is inevitable to modify the concept for our model with multiple receivers. In Cho and Kreps (1987), they first define the best response for the receiver to the sender’s deviation given the receiver’s beliefs which assign probability zero to types in the set of senders who would be worse off with same deviation. Then they say an equilibrium fails the Intuitive Criterion if there is a type who would be better off by deviating to off-equilibrium strategy based on the best response for the receiver to the sender’s deviation.

Similarly, we first define the ‘set of best response strategy profiles’ for multiple receivers in which all receivers’ strategies are best responses to each other’s as well as to the sender’s signaling strategy. One may ask why a receiver’s strategy should be a best response to the other receivers’ strategies. Suppose a receiver’s ‘best response’ strategy does not have to be a best response to the other receivers’ strategies. In our model, the worst case scenario for the sender (worker) who deviates from a pooling equilibrium strategy to any off-equilibrium strategy is that all receivers (firms) believe one of the other receivers would offer high enough wage to hire the sender (worker) and just offer zero wages. Then, no sender can expect higher utility by deviating from any pooling equilibrium strategy. Thus, all pooling equilibria survive the intuitive criterion. Therefore, to define a reasonable concept of intuitive criterion, in any ‘best response strategy profiles,’ all receivers’ strategies need to be best responses to each other’s strategies.

Let $\varDelta =\varTheta \times [0,1]$ denote the entire space of type and location. D is a non-empty subset of $\varDelta$ and $\pi ^i$ is Firm i’s profit function. We use a new notation $\tilde{\mu }$ for the belief on the location as well as the type. In the following definition, for any $D \subseteq \varDelta$, $\tilde{\mu }(D | e)$ identifies the probability that the true state is in the set of D for a given signal e. And for any $\delta \in \varDelta$, $\tilde{\mu }(\delta | e)$ is the probability density at the $\delta$ given signal e. We shall borrow other notations from Fudenberg and Tirole (1991).

Definition 1

(The set of best responses) Let $BR(\tilde{\mu },e)$ be the set of all pure-strategy profile for n receivers $(w^i_{BR})_{i=1}^n$ where each receiver’s strategy $w^i_{BR}$ is the best response to the other receivers’ best response strategy profile $w^{-i}_{BR}$ and signal e for a given belief $\tilde{\mu }$.

$$\begin{aligned} BR(\tilde{\mu },e)=\left\{ (w^i)_{i=1}^n |\forall i,\; w^i\in \arg \max _{\omega ^i}\int _{\delta \in \varDelta }\tilde{\mu }(\delta | e)\pi ^i(\omega ^i,w^{-i};\delta ) d\delta \right\} . \end{aligned}$$

Then, for a non-empty subset D of $\varDelta$, let

$$\begin{aligned} BR(D,e)=\bigcup _{\tilde{\mu }:\tilde{\mu }(D|e)=1}BR(\tilde{\mu },e). \end{aligned}$$

With this concept of the set of best response, we formally define the intuitive criterion with multiple receivers. $u^{*}(\delta )$ is the pooling equilibrium payoff of the sender with $\delta \in \varDelta$.

Definition 2

(Intuitive criterion with multiple receivers (n-receivers)) Fix a vector of equilibrium payoffs $u^{*}(\cdot )$ for the sender. For each strategy e, let J(e) be the set of all $\delta$ such that

$$\begin{aligned} u^{*}(\delta )>\max _{\mathbf {w}\in BR(\varDelta ,e)}u(e,w,\delta ). \end{aligned}$$

(A.6)

If for some e there exists a $\delta ' \in \varDelta$ such that

$$\begin{aligned} u^{*}(\delta ')<\min _{\mathbf {w}\in BR (\varDelta \backslash J(e),e)}u(e,w,\delta '), \end{aligned}$$

(A.7)

then the equilibrium fails the Intuitive Criterion.

With this definition of the intuitive criterion with multiple receivers, we check if any pooling equilibrium survives it or not, for a given value of k and q.⁶

Proposition 7

(The intuitive criterion) If $k < \bar{k}$, all pooling equilibria characterized in Proposition 6 fail the Strong Intuitive Criterion.

Later, we show that $k < \bar{k}$ if and only if there exists a separating equilibrium. So, Proposition 7 shows that, if there exists a separating equilibrium, all pooling equilibria characterized in Proposition 6 fail the intuitive criterion. This justifies our focus on a separating equilibrium.

Proof of Proposition 7

Let us denote the pooling equilibrium payoff with $e=0$ by $u^{*}_h(\delta )=u^{*}_l(\delta )$ for any $\delta$.

We are not interested in the case with $J(e)=\{(\theta ,x) \in \varTheta \times [0,1]| \theta =2 \}$, since if it is,

$$\begin{aligned} u^{*}_h(\delta )=&u^{*}_l(\delta )<\min _{\mathbf {w} \in BR(\varDelta \backslash J(e),e)}u(e,\mathbf {w},\delta '=(1, x))\\ \le&\min _{\mathbf {w}\in BR(\varDelta \backslash J(e),e)} u(e,\mathbf {w},\delta =(2, x))\le \max _{\mathbf {w} \in BR(\varDelta ,e)}u(e,\mathbf {w},\delta =(2, x)) \end{aligned}$$

so we cannot have any equilibrium fails the Intuitive Criterion for any give k or e. So, from now on, $J(e)=\{(\theta ,x) \in \varTheta \times [0,1]| \theta =1 \}$

We fix the belief on $x\sim U[0,1]$. So, there is only one possible best response strategy profile $(w_{BR}^1,w_{BR}^2)$ for any value of k and e in $BR(\varDelta \setminus J(e),e)$, which is as follows.

$$\begin{aligned} w_{BR}\equiv w_{BR}^i(e;\mu (\theta =\theta _h|e)=1,k)=\left\{ \begin{matrix} 2-k&{}\text {if} &{}k \le \frac{4}{3} \\ \frac{k}{2}&{}\text {if} &{} \frac{4}{3}< k \le 2\\ 1&{}\text {if} &{} 2 < k, \end{matrix}\right. \end{aligned}$$

(A.8)

Moreover, this profile $(w_{BR}^1,w_{BR}^2)=(w^{BR},w^{BR})$ is not only in the set of best response $BR(\varDelta ,e)$, but also Firms’ strategy profile that grants the low type worker the highest possible utility for any value of e and k.

Therefore, in order to check if a pooling equilibrium fails the Intuitive Criterion, it is enough to check if there exists any $e>0$ such that $u_l^{*}(\delta )>u_l(e;w_{BR})$ and $u_h^{*}(\delta )<u_h(e;w_{BR}).$

Let us point out $q>0 \Leftrightarrow \tfrac{2}{3}(2-q)<\frac{4}{3}$. Then, we now have several distinct intervals.

For $k\le \tfrac{2}{3}(2-q)$, with any value of $e>q$, $u_l^{*}(\delta )=(2-q)-k-kx>u_l(e;w_{BR})=2-k-kx-e.$ Then, with any value of $q<e<2q$, $u_h^{*}(\delta )=(2-q)-k-kx<u_h(e;w_{BR}) =2-k-kx-\tfrac{e}{2}.$ Hence, with any value of $q<e<2q$, we can prove that all pooling equilibria fail the intuitive criterion. Similarly, for $\tfrac{2}{3}(2-q)<k < \min \{2-q,\frac{4}{3}\}$, with any value of $2-\frac{3k}{2}<e<4-3k$, we can prove that the pooling equilibria fail the intuitive criterion.

For $q>\frac{2}{3}$ and $(2-q)<k < \frac{4}{3}$ and $x\in [0,\frac{2-q}{2k}]$, with any value of $e>1+\frac{q}{2}-k$, $u_l^{*}(\delta )=\frac{2-q}{2}-kx>u_l(e;w_{BR})=2-k-kx-e.$ Then, with any value of $e<2+q-2k$ and $x\in [0,\frac{2-q}{2k}]$, $u_h^{*}(\delta )=\frac{2-q}{2}-kx<u_h(e;w_{BR})=2-k-kx-\tfrac{e}{2}$.

For $q>\frac{2}{3}$ and $(2-q)<k < \frac{4}{3}$ and $x\in [\frac{2-q}{2k},\frac{1}{2}]$, with any value of $e>2-k-kx$, $u_l^{*}(\delta )=0>l_h(e;w_{BR})=2-k-kx-e.$ Then, with any value of $e<4-2k-2kx$ and $x\in [\frac{2-q}{2k},\frac{1}{2}]$, $u_h^{*}(\delta )=0<u_h(e;w_{BR})=2-k-kx-\tfrac{e}{2}$. Hence, with any value of $1+\frac{q}{2}-k<e<4-3k$, we can prove that the pooling equilibria fail the intuitive criterion when $2-q<k<\frac{3}{2}-\frac{q}{4}$.

Therefore, if $k < \min \{\frac{3}{2}-\frac{q}{4},\frac{4}{3}\}$, all pooling equilibria fail the Intuitive Criterion. $\square$

1.1.3 Workers’ signaling strategies in a separating equlibrium

Derivation of Eqs. 3.3and  3.4:

When $k\le \frac{2}{3}$, from the incentive compatibility conditions for low type workers,

$$\begin{aligned} u(e_l;\theta _l,x)\ge u(e_h;\theta _l,x) \Leftrightarrow \left( \theta _{l}-k-kx\right) \ge \left( \theta _{h}-k-kx-\frac{e_h}{\theta _{l}}\right) , \end{aligned}$$

we get $e_h\ge \theta _{l}(\theta _{h}-\theta _{l})$. Similarly, for high type workers,

$$\begin{aligned} u(e_h;\theta _h,x)\ge u(e_l;\theta _h,x) \Leftrightarrow \left( \theta _{h}-k-kx-\frac{e_{h}}{\theta _{h}}\right) \ge \left( \theta _{l}-k-kx \right) , \end{aligned}$$

we get $e_h\le \theta _{h}(\theta _{h}-\theta _{l})$.

Hence, if $k\le \frac{2}{3}$, the sequentially rational signaling strategy is $e_l=0$ and

$e_h\in \left[ \theta _{l}(\theta _{h}-\theta _{l}),\theta _{h} (\theta _{h}-\theta _{l})\right] =\left[ 1,2 \right]$.

When $\frac{2}{3} < k \le 1$, for low type workers, we get $e_h\ge \theta _{l}(\theta _{h}-\tfrac{3}{2}k)$, and for the high type workers, we get $e_h \le \theta _{h}(\theta _{h}-\tfrac{3}{2}k)$. Hence, if $\frac{2}{3} < k \le 1$, the sequentially rational signaling strategies are $e_l=0$ and $e_h\in \left[ \theta _{l} (\theta _{h}-\tfrac{3}{2}k),\theta _{h}(\theta _{h}-\tfrac{3}{2}k)\right] =\left[ 2-\tfrac{3}{2}k,4-3k \right]$.

When $1 < k \le \frac{5}{4}$, for low type workers with $x \in \left[ 0,\frac{\theta _{l}}{2k}\right]$, we get $e_h\ge \theta _{l}\left( \theta _{h}-\tfrac{\theta _{l}}{2}-k\right)$. For low type workers with $x \in \left( \frac{\theta _{l}}{2k}, \frac{1}{2} \right]$, we get $e_h\ge \theta _{l}\left( \theta _{h}-k-kx\right)$. Similarly, for high type workers with $x \in \left[ 0,\frac{\theta _{l}}{2k}\right]$, we get $e_h \le \theta _{h}\left( \theta _{h}-\tfrac{\theta _{l}}{2}-k\right)$. For high type workers with $x \in \left( \frac{\theta _{l}}{2k}, \frac{1}{2} \right]$, we get $e_h \le \theta _{h}\left( \theta _{h}-k-kx\right)$. Hence, if $1 < k \le \frac{5}{4}$, the sequentially rational signaling strategies are $e_l=0$ and $e_h\in \left[ \theta _{l} \left( \theta _{h}-\tfrac{\theta _{l}}{2}-k\right) ,\theta _{h} \left( \theta _{h}-\frac{3}{2}k\right) \right] =\left[ \tfrac{3}{2}-k,4-3k\right]$.

When $\frac{5}{4} < k \le \frac{4}{3}$, for low type workers with $x \in \left[ 0,\frac{1}{2k}\right]$, we get $e_h\ge \tfrac{3}{2}-k$. For low type workers with $x \in \left( \frac{1}{2k}, \frac{1}{2} \right]$, we get $e_h\ge 2-k-kx$. Similarly, for high type workers with $x \in \left[ 0,\frac{1}{2k}\right]$, we get $e_h \le 3-2k$. For high type workers with $x \in \left( \frac{1}{2k}, \frac{1}{2} \right]$, we get $e_h \le 4-2k-2kx$. Hence, if $\frac{5}{4} < k \le \frac{4}{3}$, $e_h$ should be in $\left[ \tfrac{3}{2}-k,4-3k \right]$. However, $\tfrac{3}{2}-k>4-3k$, so there is no symmetric separating equilibrium when $\frac{5}{4} < k \le \frac{4}{3}$.

When $\frac{4}{3} < k \le 2$, from the I.C. condition of a high type worker with $x \in \left( \frac{1}{2k}, \frac{1}{2} \right]$, we get $e_h \le k-2kx.$ Hence $e_h$ should be zero to satisfy this condition. Therefore, there is no symmetric separating equilibrium if $\frac{4}{3} < k \le 2$.

When $2 < k$, from the I.C. condition of a high type worker with $x \in \left( \frac{1}{2k}, \frac{1}{k} \right]$, we get $e_h \le 2-2kx.$ Hence, $e_h$ should be negative to satisfy this condition. Therefore, there is no symmetric separating equilibrium if $2 < k$.

1.1.4 Robustness of the existence results

Here, we discuss how robust our results to the functional form of the cost function $c(e;\theta )$. Note that the analysis on the firms’ wage offers does not depend on the workers’ cost function. So, the analysis on Sect. 1 still hold. Now, consider the workers’ signaling behavior. We can repeat the analysis in Sect. 1 with a cost function $c(e;\theta )$ which is different from the linear cost function $\frac{e}{\theta }$. For example, with a quadratic cost function $\frac{e^{2}}{\theta }$, we can derive the similar equations to Eqs. 3.3 and 3.4 as follows.

$$\begin{aligned} e(\theta =\theta _{l},x;k)&=0, \quad \text {if} \quad k \le \bar{k}. \end{aligned}$$

(A.9)

$$\begin{aligned} e(\theta =\theta _{h},x;k)&\in \left\{ \begin{matrix} \left[ \sqrt{\theta _{l}(\theta _{h}-\theta _{l})} ,\sqrt{\theta _{h}(\theta _{h}-\theta _{l})} \right] &{}\text {if} \quad k \le \tfrac{2}{3}\theta _l\\ \left[ \sqrt{\theta _{l}(\theta _{h}-\tfrac{3}{2}k)}, \sqrt{\theta _{h}(\theta _{h}-\tfrac{3}{2}k)}\right] &{}\text {if} \quad \tfrac{2}{3}\theta _l<k \le \theta _l\\ \left[ \sqrt{\theta _{l}\left( \theta _{h}-\tfrac{\theta _{l}}{2}-k\right) }, \sqrt{\theta _{h}\left( \theta _{h}-\frac{3}{2}k\right) }\right] &{}\text {if} \quad \theta _l<k \le \bar{k}. \end{matrix}\right. \end{aligned}$$

(A.10)

With either the quadratic or the linear cost function, a separating equilibrium exists when $k \le \bar{k}$. Actually, this result stays the same for any functional form of cost function if it can be rewritten as $c(e;\theta )=\frac{\tilde{c}(e)}{\theta }$. Therefore, the existence of the separating equilibrium does not depend on the functional form of the cost function with respect to the signaling level e.

One may expect that, when the cost function is quadratic in e, there exists a separating equilibrium even higher value of k, $k\ge \bar{k}$, since the signaling cost is negligible around $e=0$, $\lim _{e\rightarrow 0}\frac{\partial c}{\partial e}=0$. Even though, with the quadratic cost function, the high type worker’s marginal cost of signaling is negligible when the level of education goes to zero, $\lim _{e\rightarrow 0}\frac{\partial c(e,\theta _h)}{\partial e}=0$, she cannot credibly signal her productivity by taking any positive level of education since the low type worker’s marginal cost of ‘mimicking’ is also negligible, $\lim _{e\rightarrow 0}\frac{\partial c(e,\theta _l)}{\partial e}=0$.⁷

Of course, the absolute levels of signaling with the quadratic cost function are different from the levels with the linear cost function. However, in any separating equilibrium from the Pareto optimal to Pareto pessimal one, the total cost of signaling stays the same as the case with the linear cost function. The intuition behind this result is as follows. In a separating equilibrium, by taking a certain level of costly education, the high type worker deters the low type worker from mimicking her: The high type worker’s signaling behavior incurs the high-enough ‘total cost’ of mimicking for the low type worker, so eliminates the incentive to deviate from the equilibrium. This total cost is not determined by the functional form of the cost function, but by the equilibrium wage difference of the different types of workers, $w^{*}(\theta _h;k)-w^{*}(\theta _l;k)$.

1.1.5 Welfare analysis

The following equations are the complete characterizations of the equilibrium outcomes in the basic model with privately known location.

• The complete characterization of workers equilibrium payoffs

  $$\begin{aligned} u(e^D;\theta _h,x)=&\left\{ \begin{matrix} \theta _h-k-kx-\tfrac{\theta _l(\theta _h-\theta _l)}{\theta _h} =\tfrac{3}{2}-(1+x)k&{}\text {if} &{}k \le \frac{2}{3}, \\ \theta _h-k-kx-\tfrac{\theta _l\left( \theta _h-\frac{3}{2}k\right) }{\theta _h} =1-\left( \tfrac{1}{4}+x\right) k&{}\text {if} &{} \frac{2}{3}< k \le 1,\\ \theta _h-k-kx-\tfrac{\theta _l\left( \theta _h-\frac{\theta _l}{2}-k\right) }{\theta _h} =\tfrac{5}{4}-\left( \tfrac{1}{2}+x\right) k&{}\text {if} &{} 1<k\le \tfrac{5}{4}. \end{matrix}\right. \end{aligned}$$

  (A.11)
  $$\begin{aligned} u(e^D;\theta _l,x)=&\left\{ \begin{matrix} \theta _l-k-kx&{}\text {if} &{}k \le \frac{2}{3},\\ \tfrac{k}{2}-kx&{}\text {if} &{} \frac{2}{3}< k \le 1,\\ \tfrac{\theta _l}{2}-kx&{}\text {if} &{} 1<k\le \tfrac{5}{4}\; and \; x \le \frac{1}{2k}. \end{matrix}\right. \end{aligned}$$

  (A.12)

• The complete characterization of the workers’ surplus:

  $$\begin{aligned} WS(k,q)&\equiv E_\theta \left[ \int _0^1u(e;\theta ,x,k)dx\right] \nonumber \\&= \left\{ \begin{matrix} \left( (1-q)\theta _h+q\theta _l-k\right) -\frac{1}{4}k-(1-q) \tfrac{\theta _l(\theta _h-\theta _l)}{\theta _h} &{}\text {if} &{}k \le \frac{2}{3},\\ \left( (1-q)(\theta _h-k)+q\frac{k}{2}\right) -\frac{1}{4}k-(1-q) \tfrac{\theta _l\left( \theta _h-\frac{3}{2}k\right) }{\theta _h} &{}\text {if} &{} \frac{2}{3}< k \le 1,\\ \left( (1-q)(\theta _h-k)+q\frac{\theta _l}{k}\frac{\theta _l}{2}\right) -\left( (1-q)\frac{1}{4}k+q\frac{\theta _l^2}{4k}\right) -(1-q) \tfrac{\theta _l\left( \theta _h-\frac{\theta _l}{2}-k\right) }{\theta _h} &{}\text {if} &{} 1<k\le \tfrac{5}{4}. \end{matrix}\right. \end{aligned}$$

  (A.13)

• The complete characterization of the firms’ surplus:

  $$\begin{aligned} FS(k,q)&\equiv \sum _{i\in \left\{ 1,2\right\} }E_{\theta }\left[ \int _0^{x^i} x(\theta -w^i)dx\right] \nonumber \\&= \left\{ \begin{matrix} k, &{}k \le \frac{2}{3} \\ (1-q)k+q(\theta _l-\frac{k}{2}), &{} \frac{2}{3}< k \le 1\\ (1-q)k+q\frac{\theta _l}{k}\left( \theta _l-\frac{\theta _l}{2}\right) , &{} 1<k\le \tfrac{5}{4} \end{matrix}\right. \end{aligned}$$

  (A.14)

1.2 Comparison with publicly known location

Proof of Proposition 4

To identify the sequentially rational wage schedules, we compare the firms’ willingness to pay for a worker with $(\theta ,x)$, which is exactly $\theta$, to a worker’s willingness to sell, which is the minimum level of a wage offer a worker with $(\theta ,x)$ would accept, by investigating the worker’s individual rationality (I.R.) conditions for both firms. Without loss of generality, we assume $x\le \tfrac{1}{2}$.

Equation ($I.R.^1$) and ($I.R.^2$) are a worker $(\theta , x)$’s I.R. conditions for Firm 1 and Firm 2, which are stronger with a higher transportation cost k. A worker $(\theta , x)$ are willing to accept the wage offers only when Firm 1 and Firm 2’s offers are greater or equal to kx and $k(1-x)$ respectively. With these, we identify the firms’ equilibrium wage schedules at the last stage in three different rages of x. Remember, in equilibrium, no firm would offer any wage higher than the willingness to pay $\theta$. That is, in equilibrium, $w^1\le \theta$ and $w^2\le \theta$.

First, if $kx > \theta$, or equivalently $x>\frac{\theta }{k}$, the worker’s willingness to sell for either Firm 1 or Firm 2, kx or $k(1-x)$, is greater than the firms’ willingness to pay, $\theta$. Hence, the worker will not be hired by any firm. Therefore, when $k>2 \theta$, $x \in \left( \frac{\theta }{k} ,1-\frac{\theta }{k} \right)$ is excluded from the market.

Second, if $kx \le \theta$ and $k(1-x) > \theta$, or equivalently $x\le \frac{\theta }{k}$ and $x<1-\frac{\theta }{k}$, the worker $(\theta , x)$ is not willing to accept any wage offer from Firm 2 such that $w^2\le \theta$. So, there is no competition between firms. Hence, Firm 1 acts like a local monopsonist, and offers the wage which covers the worker’s transportation cost $w_\theta ^1=kx$.

Finally, if $kx \le \theta$ and $k(1-x) \le \theta$, or equivalently $1-\frac{\theta }{k}< x\le \frac{\theta }{k}$, either firms’ willingness to pay is greater than or equal to the worker’s willingness to sell. Hence, a worker $(\theta ,x)$ would accept Firm 1’s offer if $w_\theta ^1-kx-\frac{e_\theta }{\theta }\ge w_\theta ^2-k(1-x)-\frac{e_\theta }{\theta }$. We combine this with ($I.R.^1$) to get $w_\theta ^1=kx$ if $\theta -k(1-2x)\le kx$, and $w_\theta ^1=\theta -k(1-2x)$ if $\theta -k(1-2x)> kx$.

From the analysis above, we identify the sequentially rational wage schedules for $x\in [0,\frac{1}{2}]$ and $k\le 2$,⁸

$$\begin{aligned} w(x, e<e_h)&=\left\{ \begin{matrix} kx &{} \text {if} &{}x\le 1-\frac{1}{k} \text { and } k>1\\ \theta _{l}-k(1-2x) &{} \text {if} &{}1-\frac{1}{k}<x\le \frac{1}{2},\\ \end{matrix}\right. \\ w(x, e\ge e_h)&=\theta _{h}-k(1-2x). \end{aligned}$$

So, in a separating equilibrium, high type workers always enjoy the competitive wages $w(x, e\ge e_h)>kx$, while low type workers do only when k is sufficiently small, in other words, only when the market is sufficiently competitive for low type workers too. The competitive wage, $\theta -k(1-2x)$, increases in $\theta$ and x and decreases in k. From this fact, we can say that, in a separating equilibrium, the competitive wage would be higher for the worker with higher productivity and also higher in the market with higher competition (lower k). We can also conclude that the competition is severer for a worker in the middle relative to a worker at the either ends. To hire a worker who are more likely take a given wage offer of the other firm, a firm should make a more favorable offer for the worker.

Next, we find the corresponding Pareto dominant signaling strategies for both types of workers by simply checking the workers’ incentive compatibility conditions. When $k\le 2$,⁹

$$\begin{aligned} e(x, \theta =1)&=0 \end{aligned}$$

(A.15)

$$\begin{aligned} e(x,\theta =2)&=\left\{ \begin{matrix} \theta _{h}-k(1-x) &{} \text {if} &{}x\le 1-\frac{1}{k}\\ \theta _{h}-\theta _{l} &{} \text {if} &{}1-\frac{1}{k} <x\le \frac{1}{2}.\\ \end{matrix}\right. \end{aligned}$$

(A.16)

In a separating equilibrium, low type workers do not take any education, and high type workers at different locations choose different levels of, which are non-negative education, increasing in x and decreasing in k. If competition is severer with higher x or lower k, high type workers have more incentive to take costly education to extract the competition rent. $\square$

We summarize the equilibrium outcome as follows.

• Worker’s payoff:

  $$\begin{aligned} u(e;\theta _h,x)=&2-k(1-2x)-kx-\tfrac{e}{2} =\left\{ \begin{matrix} 1-(1-x)\tfrac{k}{2} &{} \text {if} &{}x\le 1-\frac{1}{k}\\ \tfrac{3}{2}-(1-x)k &{} \text {if} &{}1-\frac{1}{k}<x\le \frac{1}{2}.\\ \end{matrix}\right. \end{aligned}$$

  (A.17)
  $$\begin{aligned} u(e;\theta _l,x)=&\left\{ \begin{matrix} 0 &{} \text {if} &{} x\le 1-\frac{1}{k}\\ 1-(1-x)k &{} \text {if} &{}1-\frac{1}{k}<x\le \frac{1}{2}. \end{matrix}\right. \end{aligned}$$

  (A.18)

• Workers’ surplus: producer surplus:

  $$\begin{aligned} WS(k)=&q\times 2\int ^{\tfrac{1}{2}}_{0} u(e;\theta _l,x,k)dx +(1-q)\times 2\int ^{\tfrac{1}{2}}_{0} u(e;\theta _h,x,k)dx \nonumber \\ =&\left\{ \begin{matrix} q\left( 1-\tfrac{3}{4}k\right) +(1-q)\left( \tfrac{3}{2}-\tfrac{3}{4}k\right) =\tfrac{3}{2}-\tfrac{1}{2}q-\tfrac{3}{4}k &{} \text {if} &{}k \le 1 \\ q\left( \tfrac{1}{4}k-1+\tfrac{1}{k}\right) +(1-q)\left( \tfrac{1}{2} -\tfrac{1}{4}k+\tfrac{1}{2k}\right) \\ =\tfrac{1}{4}k-\tfrac{1}{2}-\tfrac{1}{2}q +\tfrac{1}{k} &{} \text {if} &{} 1 < k \le 2. \end{matrix}\right. \end{aligned}$$

  (A.19)

• Firms’ surplus: consumer surplus:

  $$\begin{aligned} FS(k)=&q\times \varPi (w;\theta _l,k)+ (1-q)\times \varPi (w;\theta _h,k) \nonumber \\ =&\left\{ \begin{matrix} q \tfrac{k}{2}+(1-q) \tfrac{k}{2}=\tfrac{k}{2} &{} \text {if} &{}k \le 1 \\ q\left( 2-\tfrac{k}{2}-\tfrac{1}{k}\right) +(1-q) \tfrac{k}{2}=2q-\tfrac{q}{k} +\tfrac{k}{2}-kq &{} \text {if} &{} 1 < k \le 2. \end{matrix}\right. \end{aligned}$$

  (A.20)

• Total surplus: social welfare

  $$\begin{aligned} SW(k)=&WS(k)+FS(k) \nonumber \\ =&\left\{ \begin{matrix} \left( \tfrac{3}{2}-\tfrac{1}{2}q-\tfrac{3}{4}k\right) +\left( \tfrac{k}{2}\right) =\tfrac{3}{2}-\tfrac{1}{2}q -\tfrac{1}{4}k &{} \text {if} &{}k \le 1 \\ \left( \tfrac{1}{4}k-\tfrac{1}{2}-\tfrac{1}{2}q +\tfrac{1}{k} \right) + \left( 2q-\tfrac{q}{k} +\tfrac{k}{2}-kq \right) \\ =\tfrac{3}{2}q-\tfrac{q}{k}-kq +\tfrac{3}{4}k+\tfrac{1}{k}-\tfrac{1}{2} &{} \text {if} &{} 1 < k \le 2. \end{matrix}\right. \end{aligned}$$

  (A.21)

Continuous type

In the basic model, we assume that there are only two types of workers; the high and the low type. In this section, we investigate the duopsony model with a continuous type space of workers, i.e., the type space $\varTheta \equiv (\underline{\theta }, \overline{\theta })$ is a connected subset of the set of positive real numbers, $\mathbb {R}_{++}$.

Since we are focusing on a separating equilibrium, we assume a worker with a different value of $\theta$ chooses a different level of education $e(\theta )$. That is, in a separating equilibrium, $\theta \ne \theta '$ if and only if $e(\theta ) \ne e(\theta ')$.

Firms’ problems are exactly the same as that in the basic model. After observing a worker’s education level $e(\theta ')$, the firms update their beliefs as $\mu (\theta =\theta ' | e=e(\theta '))=1$ . Hence, the sequentially rational wage schedule for a worker who takes the education level $e(\theta ')$ turns out to be

$$\begin{aligned} w(e(\theta ');k)=\left\{ \begin{matrix} \theta '-k &{} \text {if} &{} \theta '\ge \frac{3}{2}k \\ \frac{k}{2} &{} \text {if} &{} k<\theta '\le \frac{3}{2}k\\ \frac{\theta '}{2} &{} \text {if} &{} \theta '<k. \end{matrix}\right. \end{aligned}$$

(B.1)

Now, in order to investigate the existence of a separating equilibrium, we need to check if there is a signaling strategy $e(\theta )$ which is incentive compatible and sequentially rational for every $\theta$.¹⁰

In analysis, we first show that if $\underline{\theta }<\frac{3}{2}k$, there is no separating equilibrium. Then, we show that, if k is sufficiently small, so that $\underline{\theta }\ge \frac{3}{2}k$, there exists a separating equilibrium.

First, for some k, let’s assume $\underline{\theta }<\frac{3}{2}k$, so that there is $\theta '\in \varTheta$ which satisfies $k< \theta '< k\frac{3}{2}$. Then for a sufficiently small $\epsilon >0$ we have $\varTheta _\epsilon \equiv (\theta '-\epsilon , \theta '+\epsilon ) \subset \varTheta$. Then, from Eq. (B.1), we can conclude the sequentially rational wage offer for $\theta \in \varTheta _\epsilon$ is $\frac{k}{2}$.

To satisfy incentive compatibility, the following should be satisfied.

$$\begin{aligned} \forall \theta \in \varTheta _\epsilon ,\;\; U \left( \theta ;\theta , k \right) = \max _{\hat{\theta }} U \left( \hat{\theta } ;\theta , k \right) = \frac{k}{2} - kx - \frac{e(\hat{\theta })}{\theta }. \end{aligned}$$

From the envelope theorem, for any $\theta \in \varTheta _\epsilon$, $-\frac{e'(\theta )}{\theta }=0.$

Therefore, the level of education should be constant in $\theta$, which cannot be a separating signaling strategy. In other words, if there is some $\theta \in \varTheta$ which satisfies $k< \theta < \frac{3}{2}k$, there cannot be any separating equilibrium.

Secondly, let’s now assume $\varTheta \subseteq [0,k]$ or $\theta <k$ for all $\theta \in \varTheta$. With the corresponding wage schedule from Eq. (B.1), $w(e(\theta );k) =\frac{\theta }{2}$, the incentive compatibility condition requires

$$\begin{aligned} \forall \theta \in \varTheta ,\;\; U\left( \theta ;\theta , k \right) =\max _{\hat{\theta }} U\left( \hat{\theta } ;\theta , k \right) =\frac{\hat{\theta }}{2} - kx - \frac{e(\hat{\theta })}{\theta }. \end{aligned}$$

After applying the envelope theorem, we get the following equation.

$$\begin{aligned} e(\theta )=\frac{\theta ^2}{4}+M \end{aligned}$$

(B.2)

with some constant $M \in \mathbb {R}$.

Let’s consider a worker at a certain location, $x=\frac{\theta }{2k}$, which is the marginal type worker who is not excluded from the market. This worker’s utility by taking education of $e(\theta )$ and accepting $w(\theta ;k)$ is

$$\begin{aligned} U\left( \theta ;\theta , k \right) =\frac{\theta }{2}-kx -\frac{e(\theta )}{\theta }=\frac{\theta }{4}-kx-\frac{M}{\theta } =-\frac{\theta }{4}-\frac{M}{\theta }. \end{aligned}$$

(B.3)

To be individually rational, Eq. (B.3) should be non-negative. It implies, $M\le -\frac{\theta ^2}{4}.$

Combine this with Eq. (B.2), we get $e(\theta )=\frac{\theta ^2}{4}+M\le \frac{\theta ^2}{4} -\frac{\theta ^2}{4}=0.$ However, the education level cannot be negative. Hence, the signaling strategy (B.2) is not valid. Therefore, we conclude that, if $\varTheta \subseteq [0,k]$, there cannot be any separating equilibrium.

The following proposition summarizes the above analysis.

Proposition 8

(Non-existence) If $\underline{\theta }<\frac{3}{2}k$, then there is no separating equilibrium.

This proposition says if there is some type of workers for whom the market is not sufficiently competitive, $\theta < \frac{3}{2}k$, then there is no separating equilibrium. This directly implies that there is no separating equilibrium if $\varTheta =\mathbb {R}_{++}$.

Finally, let’s assume $\underline{\theta }\ge \frac{3}{2}k$. The incentive compatibility requires

$$\begin{aligned} \forall \theta \in \varTheta ,\;\; U\left( \theta ;\theta , k \right) =\max _{\hat{\theta }} U\left( \hat{\theta } ;\theta , k \right) =\hat{\theta }- k - kx - \frac{e(\hat{\theta })}{\theta }. \end{aligned}$$

The envelope theorem gives us that

$$\begin{aligned} e(\theta )=\frac{1}{2}\theta ^2+M \end{aligned}$$

(B.4)

where $M\in \mathbb {R}$.

Now, we check the workers individual rationality. The utility of a worker $(\theta ,x)$ is

$$\begin{aligned} U\left( \theta ;\theta , k \right) =\theta -k-kx-\frac{e(\theta )}{\theta } =\frac{\theta }{2}-k-kx-\frac{M}{\theta }. \end{aligned}$$

(B.5)

For any worker $(\theta ,x) \in \varTheta \times \left[ 0,\tfrac{1}{2} \right]$, Eq. (B.5) should be nonnegative.

$$\begin{aligned} \frac{\theta }{2}-k-kx-\frac{M}{\theta }\ge \frac{\theta }{2} -\frac{3k}{2}-\frac{M}{\theta }\ge 0. \end{aligned}$$

Combining with nonnegative constraint for Eq. (B.4), we get the condition for the constant M.

$$\begin{aligned} \forall \theta , \ \tfrac{1}{2}\theta ^2 -\tfrac{3k}{2}\theta \ge M \ge -\tfrac{1}{2}\theta ^2. \end{aligned}$$

Because $\tfrac{1}{2}\theta ^2 -\tfrac{3k}{2}$ is increasing in $\theta$ and $-\tfrac{1}{2}\theta ^2$ is decreasing in $\theta$, we get

$$\begin{aligned} \tfrac{1}{2}\underline{\theta }^2 -\tfrac{3k}{2} \underline{\theta } \ge M \ge -\tfrac{1}{2}\underline{\theta }^2. \end{aligned}$$

(B.6)

Therefore, if $\underline{\theta }\ge \frac{3}{2}k$, we can identify a separating equilibrium with some constant M which satisfies (B.6). Moreover, we can identify the Pareto efficient separating equilibrium with $M=-\tfrac{1}{2}\underline{\theta }^2$.

We summarize the above analysis in following propositions.

Proposition 9

(Pareto efficient separating equilibrium) If $\underline{\theta }\ge \frac{3}{2}k$, or equivalently if $\varTheta \subseteq [\frac{2}{3}k,\infty )$, then there is a unique Pareto efficient separating equilibrium consisting of a strategy profile $\left\langle w(e;k),e(\theta ,x;k) \right\rangle$ and supporting beliefs $\mu (\theta |e)$. Specifically, for a given k and $\varTheta \subseteq [\frac{2}{3}k,\infty )$, a Pareto efficient separating equilibrium is a triple, $\left\langle w(e;k),e(\theta ,x;k),\mu (\theta |e) \right\rangle$,

$$\begin{aligned} w(e;k)&=\sqrt{2\left( e+\frac{1}{2}\underline{\theta }^2\right) } -k,\;\;\; e(\theta ,x;k)=\frac{1}{2}\theta ^2-\frac{1}{2} \underline{\theta }^2, \\ \forall \tilde{\theta }&\in \varTheta , \;\; \mu \left( \theta =\tilde{\theta }; e=\frac{1}{2}\tilde{\theta }^2-\frac{1}{2}\underline{\theta }^2\right) =1. \end{aligned}$$

Proof of Proposition 5

Proposition 8 and 9 prove the Proposition 5. $\square$