Monitoring, Endogenous Comparative Advantage, and Immigration

Monitoring, Endogenous Comparative Advantage, and Immigration

In Appendix A,we present a detailed solution of the last three stagesof our game in the Y sector using the method of backward induction under the assumption of complementarities in production between a worker and a manager, but in the absence of moral hazard.

Appendix A

The production function exhibits constant returns to scale, as described in the following equation. In the Y sector, the final good is produced as a result of matched efforts of a manager and a worker. We follow the assumption of complementarities in production. A firm with a manager who provides one unit of labor exerting effort () and a worker who exerts productive effort () and unproductive effort (), produces units of final good y.

In the Y sector, managers exert the amount of effort that maximizes firms’ profit, while workers exert the amount of effort that maximizes their income defined in the performance measure, where the latter is defined as follows

where denotes the training level obtained by a manager; and respectively denote the amount of productive and unproductive efforts of a worker w paired with a manager m. The above equation of the performance measure is similar to equation (2)[D1]in the main text (the equation of the performance measure in the presence of moral hazard in the case of the X sector), but now under the assumption that in this sector the manager has perfect information regarding the training abilities of a worker, and she can perfectly observe or/and verify worker’s efforts.

The solution of the sub-game perfect Nash equilibrium of only the last three stages of the game is presented as follows.

Stage 4. Production

In the simple sector, in the last stage, a manager endowed with tm units of training, who provides am productive efforts, pairs up with a worker endowed with tw units of training, who supplies aw productive and dw unproductive efforts. Manager’s income comes from her firm’s profit (since she is the firm’s owner), which are , where her homothetic preferences are . A worker’s homothetic preferences are . Let’s denote with the sum of the worker’s and the manager’s post training utilities, where

Both, the manager and the worker maximize the above aggregate post-training utilities over their respective productive and unproductive effort levels. The optimal unproductive levels for both the worker and the manager are equal to zero,This is because, the manager is also the owner of the firm and her income generates only from the firm’s profit, and therefore, she has no incentives to provide unproductive efforts. At the same time, the worker has also no incentives to provide unproductive efforts because there is no moral hazard in this section, and therefore, her manager can perfectly observe or/and verify her unproductive and productive efforts. The optimal level of a manager’s and worker’s productive efforts are and . Therefore, the sum of the worker’s and the manager’s post training utilities with optimal effort levels is

Stage 3. Production team choice

Both the worker and the manager maximize the above post training utility subject to their respective post-training levels. Therefore, and . From here it should be obvious that both, the worker and the manager maximize their post-training utility only when the manager has exactly the same level of training as the worker assuring a positive assortative matching between both parties of the firm.

Another way to arrive at the same conclusion is the fact that the post-training firm utility as represented by A-4 is strictly supermodular in worker’s and manager’s training levels. Mathematically, The property of supermodularity of the post-training firm utility is based on the assumption of complementarity of production between a worker and a manager of the same firm.

In the following analysis of this stage of the game we find the optimal wage of a worker with training level teamed with a manager with training level .

Prior to the establishments of the firms, potential owners announce an optimal wage to maximize the total utility of the production team and potential workers decide to either accept it or reject it. Both partners of the team have perfect information regarding their own and their partner’s training levels. Hence, let us denote with and the respective post-training utilities of the manager and the worker. Their equations are shown as follows

The worker (manager) maximizes her post-training utility subject to her productive effort levels, where and . Therefore, the worker’s efficient productive efforts are . In other words, the worker is willing to accept a wage that satisfies the following . We know from stage 4, that the optimal level of worker’s productive efforts is . Substituting the latter in the wage equation that the worker is willing to accept, we present the optimal wage for a worker with training teamed witha manager with training in the following

In an analogous way, we can show that the optimal wage that the manager will offer is exactly equal to the above equation. In other words, the manager’s efficient productive efforts are . From the above paragraph, we know that the worker’s efficient productive efforts are . Substituting the latter into the former, we find that the manager’s efficient productive efforts are , but we know from stage 4, that the optimal level of manager’s productive efforts is . Hence, it follows that the manger should offer an optimal wage represented by in order to maximize her post-utility training level.

Stage 2. Training

We showed in the previous subsection that due to the assumption of complementarities in production, there exists a positive assortative matching process between a manger and a worker who work together in the production process of the same firm, such that each manager has equal training with her worker. In this game a symmetric equilibrium assures that two individuals that work in the same firm (team) must be indifferent between being the manager and the worker of the firm. From the manager’s optimal effort level from stage 5, the optimal wage of A-6, and the fact that in this setup as shown in stage 4, , we obtain the following manager’s indirect utility with optimal effort levels who teams up with a worker of the same training levels: . In an analogous way, we find the following worker’s indirect utility with optimal effort levels who teams up with a manager of the same training levels: . Hence, in a symmetric equilibrium, both individuals who are working in the same team and have the exact training levels must be indifferent between being a manager or a worker in the same firm. This implies that , but this inequality is true only when .In other words, we can write the indirect utility of an individual working in the Y sector with optimal effort levels and after the matching process, where a manager and a worker of the same firm have identical training levels as

Maximizing subject to the manager’s training levels, we find the manager’s optimal training levels . Analogously, worker’s optimal training levels are . Since , the optimal level of training of an individual working in the Y sector is

Therefore, the indirect utility of an individual working in the simple sector with optimal training level is

Note that the above equation is exactly the same as equation (6) in the main text.From this point, following the same analysis as in the main text, it should be obvious to the reader that all the results of the paper remained unchanged. In other words, the assumption of having just one individual working as a self-employed in the Y sector is introduced only for simplicity in order to avoid additional notation in the paper.

In Appendix B, we present a detailed solution of the last three stages of thegame in the complex sector using the method of backward induction.

Appendix B

Stage 4. Production

In the complex sector, in the last stage, a manager endowed with tm units of training, who provides am productive efforts, pairs up with a worker endowed with tw units of training, who supplies aw productive and dw unproductive efforts. Manager’s income comes from her firm’s profit (since she is the firm’s owner) which are , where her homothetic preferences are . A worker’s homothetic preferences are . Let’s denote with the sum of the worker’s and the manager’s post training utilities, where

Both, the manager and the worker maximize the above aggregate post-training utilities over their respective productive and unproductive effort levels. The optimal unproductive levels for the manager are equal to zero, This is because, the manager is also the owner of the firm and her income generates only from the firm’s profit, and therefore, she has no incentives to provide unproductive efforts. However, the worker has incentives to provide unproductive efforts because there is moral hazard in the complex sector and her objective is to maximize her wage, but. But the manager cannot perfectly observe or/and verify worker’s productivity in terms of worker’s effort despite the fact that the manager can perfectly observe worker’s training levels as represented by equation (4) in the main text. The optimal level ofa worker’s unproductive efforts are . Thus, workers in the complex sector provide less unproductive efforts when their country has more developed institutions. The optimal level of a manager’s and a worker’s productive efforts are and .

Stage 3. Production team choice

Prior to the establishments of the firms, potential owners announce an optimal wage to maximize the total utility of the production team and potential workers decide to either accept it or reject it. Both partners of the team have perfect information regarding their own and partner’s training levels as represented by equation (4) in the main text. Substituting the optimal productive and unproductive efforts that we found in the above stage 4 into B-1, we obtain the total post-training utility derived from matching a manager with training with a worker with training under optimalefforts from the worker and the manager

where shows the quality of the monitoring ability of a manager with trainingtm and national institutional development level , where .

Maximizing B-2 over the wage (in other words, setting ), we find the optimal wage

It should be obvious from B-3 that the optimal wage is higher in more developed national institutions, or/and the manager’s training levels. Substituting the optimal wage of B-3 into B-2, we can write as

The total post-training firm utility as represented by B-4 is strictly supermodular in worker’s and manager’s training levels. Mathematically, This property of the post-training firm utility is not dependent on the existence of moral hazard in this section, but it is based on the assumption of complementarity of production between a worker and a manager of the same firm. In Appendix A, we arrived at the same conclusion under complementarity in production, but in the absence of moral hazard.

Hence, in equilibrium, the fact that indicates that the manager and the worker must have the same level of training, . Therefore, we can now describe the optimal wage after matching as

Let denote with and the respective indirect utilities of the manager and the worker before their matching and their choice of efforts. Their equations are shown as follows

Substituting the optimal effort levels that we found in stage 4, for the manager and the worker, we obtain their indirect utility with their optimal productive and unproductive efforts after the matching process

Using the optimal wage of B-5, we can write the above equations as following

Thus, in a symmetric equilibrium if two individuals with the same training create a team, each of them should be indifferent between being a worker or the manager (owner) of the team. Combining B-9 with B-10, implies that . Hence, the indirect utility of an individual working in the complex sector after the matching process is

Stage 2. Training

In this stage, in order to find the optimal training levels, we maximize from B-11 over the training level. Hence,setting , we obtain the optimal training level of an individual who works in the complex sector

This is equation (14) [D2]in the main text. Substituting from B-12 into B-11, we find the indirect utility of an individual with optimal trainings who works in the complex sector

This is equation (15)[D3] in the main text.

In Appendix C, we present the proofs of all propositions and corollaries.

Appendix C

Proof of Proposition 1:

We follow 2 steps. In the 1st step, we show that if a exists, when , then this is unique. In the 2nd step, we prove the existence of . Let's start with Step 1.

Let’s assume that exists. Using (14) and (6), when , there exists a , such that for any , , this is unique. onlyif

Let . Then, . Therefore, the left hand side of C-1 is increasing in the natural ability levels, while the right hand side of A-1 is constant. Thus, is unique.

We start the second step with the proof of . Suppose that the opposite is true. Therefore, . In terms of C-1, this implies that . Hence, no individual will be employed in the complex sector, which indicates that the relative price of the simple good approaches zero . This implies that . But, this contradicts our assumption that . Hence, . In an analogous way, one can show that .

Proof of Corollary 1.

To prove all parts of corollary 1, we must find an expression for . implies that

From (C-2) and (14) [D4]it is easy to verify that ,

Proof of Proposition 2.

We prove part 1 & 2 of proposition 2 using 2 lemmas. Then, part 3 of proposition 2 follows.

Lemma 1.where is convex in if and only if .

Lemma 2. 1), such that and 2)

Proof of Lemma 1.

Dividing both sides of (14) [D5]with , we get From the implicit theorem, ; Hhence, . Thus, the optimal training levels of an individual working in the complex sector is increasing in her natural ability level .

Let where is a constant. Thus, and if , and if , where; . In order to complete the proof of lemma 1, we have to show the existence of . This is done by substituting into (14) [D6]and combining it with (15). [D7]Thus, exists and is strictly higher than zero. Since, we know that the indirect utility with optimal training levels is strictly convex in individuals’ ability level, and only if , while only if , then is strictly convex in and is strictly concave in .

Proof of Lemma 2

Assume that the first part of Lemma 2 is true. Then, , such that . From (14) [D8]we know the optimal training level of an individual working in the complex sector. We also know that the optimal training level of an individual working in the simple sector is. Hence, from setting , we can find.With the help of (C-2), we find that . This concludes the proof of the second part of Lemma 2. The proof of the first part of Lemma 2 consists of two steps. In the first step, we prove the uniqueness of , and in the second step we proof the existence of . Letus start with step 1.

If there exists a , such that, then is unique. The inequality is true since, which is the training level first-order condition of utility optimization of individuals working in the complex sector, and , which is the optimal ability level of an individual working in the simple sector.

To prove the existence of , we assume that . From the proof of Lemma 2, it is straightforward that . Hence, no individual will invest to optimize her abilities in the y sector, meaning that no individual will be employed in the x sector. Mathematically, this means that the relative price of the simple sector goes to infinity . But, this contradicts our assumption that . Analogously, one can show that .

Proof of the first and the second part of Proposition 2

Now, we can prove part 1 & 2 of Proposition 2. is concave in only when . From (17) [D9]. In the region where is strictly concave in , never intersects . In Lemma 2, we showed the existence of such that . Thus, must be convex in at . Moreover, we demostrated that is convex in Since, is concave in , then . We showed in the proof of the second part of Lemma 2 that . This implies that is convex in and .

Proof of Proposition 3.

We have to prove that . Let’s first find . In the x sector we assumed that each firm consists of one individual. Thus, the income of each individual is equal to her firm’s profit. In the fourth stage, we found the optimal profits for a firm operating in the simple sector. Substituting the optimal effort levels as indicated in the fifth stage into , we can obtain . In the second stage we found the optimal training levels of an individual working in the simple sector. Substituting it into , we obtain In the first stage we found the optimal skill level for an individual working in the simple sector. Substituting it into , we can obtain the income of an individual working in the simple sector: