Labor market efficiency: output as the measure of welfare

Abstract

We study the matching of workers to firms in which workers choose an observable and contractable effort after the match. If there are complementarities between a worker’s ability and a firm’s technology, positive assortative matching (PAM) is the only matching in any equilibrium and is the unique efficient matching. We investigate the effect of a policy that changes the matching of firms to workers from any matching to PAM, such as implementing a centralized clearing house. We characterize two sets of sufficient conditions on the production and cost functions under which the total output and welfare both increase. Under the first set of conditions, the increase in total output is an upper bound for the efficiency gain. In contrast, under the second set of conditions, the increase in total output is a lower bound for the efficiency gain. We identify a third set of conditions under which the total output decreases while welfare increases.

Appendix A Proofs

Proof of Lemma 3

We need to show that there is no blocking pair. Suppose $i^\prime , j^\prime$ and $\{w^\prime (i^\prime , j^\prime ,e)\}_{e \in [0,1]}$ is a blocking pair. First observe that if this wage schedule doesn’t induce the efficient effort then the same worker and firm can block with a different wage schedule that induces the efficient effort. Under this wage schedule the worker will choose an effort $e^\prime$ and will receive $w^\prime$.

Note that under the efficient effort firm j is indifferent between matching with worker $i=j$ and worker $i=j+1$:

$$\begin{aligned} \pi (i,i, e^*(i,i)) = \pi (i+1,i,e^*(i+1,i)). \end{aligned}$$

Moreover, worker $i+1$ is indifferent between matching with firm $j=i+1$ and firm $j=i$:

$$\begin{aligned} u(i+1,i+1,e^*(i+1,i+1)) = u(i+1,i,e^*(i+1,i)). \end{aligned}$$

Furthermore, because total surplus is strictly supermodular, we have:

$$\begin{aligned}{} & {} s^*(i,i) + s^*(i+1,i+1)> s^*(i,i+1) + s^*(i+1,i) \\{} & {} \quad \Leftrightarrow \pi (i,i,e^*(i,i)) + u(i,i,e^*(i,i)) + \pi (i+1,i+1,e^*(i+1,i+1)) \\{} & {} \qquad + u(i+1,i+1,e^*(i+1,i+1))> s^*(i,i+1) + s^*(i+1,i) \\{} & {} \quad \Leftrightarrow \pi (i+1,i,e^*(i+1,i) + u(i,i,e^*(i,i)) + \pi (i+1,i+1,e^*(i+1,i+1)) \\{} & {} \qquad + u(i+1,i,e^*(i+1,i))> s^*(i,i+1) + s^*(i+1,i) \\{} & {} \quad \Leftrightarrow u(i,i,e^*(i,i)) + \pi (i+1,i+1,e^*(i+1,i+1)) > s^*(i,i+1) \end{aligned}$$

Similarly we can show:

$$\begin{aligned} u(i+1,i+1,e^*(i+1,i+1)) + \pi (i+2,i+2,e^*(i+2,i+2)) > s^*(i+1,i+2) \end{aligned}$$

Adding these two inequalities and using the fact that $u(i+1,i+1,e^*(i+1,i+1)) + \pi (i+1,i+1,e^*(i+1,i+1))= s^*(i+1,i+1)$ we have:

$$\begin{aligned}{} & {} s^*(i+1,i+1) + u(i,i,e^*(i,i)) + \pi (i+2,i+2,e^*(i+2,i+2)) \\{} & {} \quad> s^*(i+1,i+2) + s^*(i,i+1) \\{} & {} \quad \Leftrightarrow u(i,i,e^*(i,i)) + \pi (i+2,i+2,e^*(i+2,i+2)) \\{} & {} \quad > s^*(i+1,i+2) + s^*(i,i+1) -s^*(i+1,i+1). \end{aligned}$$

Moreover, by supermodularity of $s^*$ (lemma 2) we have:

$$\begin{aligned} s^*(i, i+2) < s^*(i+1,i+2) + s^*(i,i+1) -s^*(i+1,i+1). \end{aligned}$$

Hence:

$$\begin{aligned} u(i,i,e^*(i,i)) + \pi (i+2,i+2,e^*(i+2,i+2)) > s^*(i,i+2). \end{aligned}$$

We can do the same process for any $i^\prime > i$. Hence no worker i and firm j were $i>j$ can block this CE.

Now consider $i < j$. First observe that:

$$\begin{aligned}{} & {} u(i+1,i+1,e^*(i+1,i+1)) +\pi (i,i, e^*(i,i)) \\{} & {} \quad =u(i+1,i,e^*(i+1,i)) + \pi (i+1,i,e^*(i+1,i) = s^*(i+1,i) \end{aligned}$$

Hence $i+1,j=i$ can’t block. Moreover, by supermodularity of total surplus:

$$\begin{aligned}{} & {} s^*(i+2,i+1) + s^*(i+1,i)> s^*(i+2,i) +s^*(i+1,i+1) \\{} & {} \quad \Leftrightarrow u(i+2,i+2,e^*(i+2,i+2)) +\pi (i+1,i+1, e^*(i+1,i+1)) \\{} & {} \quad \quad +u(i+1,i+1,e^*(i+1,i+1)) +\pi (i,i, e^*(i,i)) \\{} & {} \quad> s^*(i+2,i) +s^*(i+1,i+1) \\{} & {} \quad \Leftrightarrow u(i+2,i+2,e^*(i+2,i+2)) +s^*(i+1,i+1) +\pi (i,i, e^*(i,i)) \\{} & {} \quad> s^*(i+2,i) +s^*(i+1,i+1) \\{} & {} \quad \Leftrightarrow u(i+2,i+2,e^*(i+2,i+2)) +\pi (i,i, e^*(i,i)) > s^*(i+2,i)\\ \end{aligned}$$

Hence $i+2,j=i$ can’t block.

Note that each worker gets at least zero utility and each firm makes positive profit, firms are best replying, and workers are choosing the utility maximizing effort level. $\square$

Proof of Lemma 4

Suppose not, then there exists at least one pair of worker-firm such that the worker is not choosing the efficient effort. Therefore, there is another contract that induces the efficient effort and increases the surplus of the match. Hence, there is a wage schedule that induces the efficient effort, gives the worker a strictly higher utility, and strictly increases the firm’s profit. Therefore, a blocking pair exists, a contradiction. $\square$

Proof of Theorem 1

Suppose not, then there exists at least two pairs of worker-firm in a contracting equilibrium such that $i < i^\prime$, $j > j^\prime$, ${\hat{\mu }}(i) = j$, and ${\hat{\mu }}(i^\prime ) = j^\prime$. By Lemma 2, the total surplus is strictly supermodular in the types of firms and workers, given the efficient effort for every firm-worker pair. Since the total surplus is strictly supermodular, if we match i with $j^\prime$ and $i^\prime$ with j and let them choose the efficient effort, the total surplus increases:

$$\begin{aligned}{} & {} s(i,j,e^*(i,j)) +s(i^\prime ,j^\prime ,e^*(i^\prime ,j^\prime )) < s(i,j^\prime ,e^*(i,j^\prime )) +s(i^\prime ,j,e^*(i^\prime ,j)) \end{aligned}$$

Hence, either $(i,j^\prime )$ or $(i^\prime ,j)$ is a blocking pair. $\square$

Proof of Lemma 5

The following algorithm with n steps proves the result. Do the following step for $i = n, n-1,\ldots , 1$: In step i: Select the following two pairs of matched worker-firm pairs: The pair that has firm i as the firm and the pair that has worker i as the worker. If these are different pairs, use meet and join on these two pair, otherwise go to the next step.

In each step i, worker i will match with firm $j=i$ as the result of join operation. Hence, after n steps, the matching is positive assortative matching. $\square$

Proof of Theorem 2

 

• 1. By taking cross partial derivatives of production function at the efficient effort, we have:

  $$\begin{aligned} v^*_{i j}= & {} v_{i j} + v_{i e} e_j^* + v_{j e} e_i^* + v_{e e} e_i^* e_j^* + v_e e^*_{i j} \\ \Rightarrow v^*_{i j}= & {} s^*_{i j} + c_{e e} e_i^* e_j^* + v_e e^*_{i j}. \end{aligned}$$

  Recall that, $s^*_{i j} < 0>$. By Assumption 1, $c_{e e}>0, v_e >0$. Given Assumption 2 or 3, $e_i^* e_j^* >0$. Shahdadi (2021) proves that, under Condition 1, the argmax (the efficient effort) is strictly supermodular, i.e., $e^*_{i j} >0$. Therefore, the total output at the efficient effort is supermodular. Because the total profit, total surplus, and efficient effort are supermodular, reducing search friction in the labor market results in an increase of the total profit, total surplus, and total effort. By Lemma 5, after a finite sequence of these random meetings, the matching is PAM. Note that if we start from any matching different from PAM, some of these random meetings result in a new worker-firm pairs; hence, the total profit, total surplus, and total effort increases by moving to PAM. We can conclude that PAM has the highest total surplus, highest total output, and highest total labor.

• 2. Because the argmax (the efficient effort) is strictly supermodular, by eliminating all frictions in the labor market, the total effort increases. Given Assumption 2, the efficient effort is increasing in firm’s technology. Recall that, starting from any matching, there is a finite sequence of meet and join operations such that the final matching is PAM. Therefore, it is enough to prove the result for implementing meet and join operations once, i.e., for a reduction of search friction. Consider two pairs of worker-firm pairs, $(i,j), (i^\prime , j^\prime )$, where $i < i^\prime , j > j^\prime$. Implementing meet and join operations are equivalent to assigning better worker $i^\prime$ to the better firm j. The better workers, matched with the better firm, works more than the other worker matched with the better firm. Moreover, this increase in labor is more than the decrease in labor in the lower firm after implementing meet and join operations. Stated differently, the labor at the higher firm was higher at the initial matching, compared with the labor at the lower firm. Moreover, the increase in labor at the higher firm is greater than the decrease in the lower firm. Because the cost function is convex, the total cost of effort increases as the result of implementing meet and join operations.

$\square$

Proof of Theorem 3

• 1. By taking cross partial derivatives of production function at the efficient effort, we have:

  $$\begin{aligned} v^*_{i j}&= v_{i j} + v_{i e} e_j^* + v_{j e} e_i^* + v_{e e} e_i^* e_j^* + v_e e^*_{i j} \\ \Rightarrow v^*_{i j}&= s_{i j} + s_{i e} \left( \frac{- s_{e j}}{s_{e e}}\right) + s_{j e} \left( \frac{- s_{e i}}{s_{e e}}\right) + v_{e e} \left( \frac{- s_{e i}}{s_{e e}}\right) \left( \frac{- s_{e j}}{s_{e e}}\right) \\&\quad - v_e \left( \frac{s_{e e}^2 s_{e i j} + s_{e e e } s_{e i} s_{e j} - s_{e e i} s_{e j} s_{e e} - s_{e e j} s_{e i} s_{e e} }{s_{e e}^3}\right) \\ \Rightarrow v^*_{i j}&= \left( \frac{-1}{s_{e e}^3}\right) \big (s_{i j} (-s_{e e}^3) + 2 s_{i e} (s_{e j})(s_{e e}^2) + v_{e e} (s_{e i})( s_{e j})( - s_{e e}) \\&\quad + v_e s_{e e}^2 s_{e i j} + v_e s_{e e e } s_{e i} s_{e j} - v_e s_{e e i} s_{e j} s_{e e} - v_e s_{e e j} s_{e i} s_{e e} \big ) \\ \Rightarrow v^*_{i j}&= \left( \frac{-1}{s_{e e}^3}\right) \big ( ( s_{e i} s_{e j})(v_e s_{e e e } + 2 s_{e e}^2 - v_{e e} s_{e e} ) \\&\quad +(s_{e e}^2 )(v_e s_{e i j} - s_{i j} s_{e e}) - ( v_e s_{e e})( s_{e e i} s_{e j} + s_{e e j} s_{e i}) \big ) \\ \end{aligned}$$

  Shahdadi (2021) proves that, under Condition 2, $e^*_{i j} <0$. Under Condition 2, the total output at the efficient effort is supermodular. Hence, PAM has the highest total surplus, highest total output, and lowest total labor.

• 2. Because the argmax (the efficient effort) is submodular, the total effort decreases. Under Assumption 3, the efficient effort is decreasing in firm’s technology.

  Recall that, starting from any matching, there is a finite sequence of meet and join operations such that the final matching is PAM. Therefore, it is enough to prove the result for implementing meet and join operations once. Consider two pairs of worker-firm pairs, $(i,j), (i^\prime , j^\prime )$, where $i < i^\prime , j > j^\prime$. Implementing meet and join operations are equivalent to assigning better worker $i^\prime$ to the better firm j. The better worker, matched with the better firm, works less than the other worker matched with the better firm. Moreover, this decrease in labor is more than the increase in labor in the lower firm after implementing meet and join operations. Stated differently, the labor at the higher firm was lower at the initial matching, compared with the labor at the lower firm. Moreover, the decrease in labor at the higher firm is greater than the increase in the lower firm. Because the cost function is convex, the total cost of effort decreases as the result of implementing meet and join operations.

$\square$

Proof of Theorem 4

• 1. Recall that:

  $$\begin{aligned} v^*_{i j}= & {} \left( \frac{-1}{s_{e e}^3}\right) \big ( ( s_{e i} s_{e j})(v_e s_{e e e } + 2 s_{e e}^2 - v_{e e} s_{e e} ) \\{} & {} + (s_{e e}^2 )(v_e s_{e i j} - s_{i j} s_{e e}) - ( v_e s_{e e})( s_{e e i} s_{e j} + s_{e e j} s_{e i}) \big ) \end{aligned}$$

  Shahdadi (2021) proves that, under Condition 2, $e^*_{i j} <0$. Under Condition 3, the total output at the efficient effort is submodular. Hence, PAM has the highest total surplus, lowest total output, and lowest total labor.

• 2. Eliminating frictions in the labor market results in PAM. By the first part of the theorem, PAM has the highest total surplus and lowest total output. The proof for a reduction of search friction follows from the proof of the first part of the theorem.

$\square$

Proof of Theorem 5

In a contracting equilibrium, if $i< i^\prime$ and $j< j^\prime$ where (i, j) and $(i^\prime ,j^\prime )$ are feasible matches, then $(i,j^\prime )$ and $(i^\prime ,j)$ are not part of a stable matching; i.e., it is not an equilibrium outcome, because either (i, j) or $(i^\prime ,j^\prime )$ is a blocking pair.

We want to show that for any given labor market with friction, if the total surplus in a contracting equilibrium is strictly supermodular in worker’s type and firm’s type, then the profit of the firm is strictly supermodular in worker’s type and firm’s type. Define ${\hat{r}}(i,j)= {\hat{w}}(i) - c(e^*(i,j))$ as the rent of worker i when he/she is matched with firm j, in a given contracting equilibrium $<\{{\hat{w}}(i,j,e^*(i,j))\}_{e \in [0,1]},{\hat{\mu }}>$. Consider $i< i^\prime$ and $j< j^\prime$, where (i, j) and $(i^\prime ,j^\prime )$ are part of the induced matching in the contracting equilibrium. By revealed preferences, firm $j^\prime$ prefers its current match compared with the worker assigned to firm j:

$$\begin{aligned}{} & {} \pi (i^\prime ,j^\prime , {\hat{r}}(i^\prime ,j^\prime )) \ge \pi (i,j^\prime , {\hat{r}}(i,{\hat{\mu }}(i))) \\{} & {} \pi (i,j, {\hat{r}}(i,j)) \ge \pi (i^\prime ,j, {\hat{r}}(i^\prime ,{\hat{\mu }}(i^\prime ))) \\{} & {} \quad \Rightarrow \pi (i^\prime ,j^\prime , {\hat{r}}(i^\prime ,j^\prime ))+ \pi (i,j, {\hat{r}}(i,j)) \ge \pi (i,j^\prime , {\hat{r}}(i,j)) + \pi (i^\prime ,j, {\hat{r}}(i^\prime ,j^\prime )) \end{aligned}$$

Therefore, the profit at the efficient effort level is strictly supermodular. Similar argument proves the second part of this theorem. $\square$