Monitoring, Endogenous Comparative Advantage, and Immigration
In Appendix A, we present a detailed solution of the last three stages of 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 (𝑎𝑚 ) and a worker who exerts productive effort (𝑎𝑤 ) and unproductive effort
(𝑑𝑤 ), produces 2√𝑎𝑤 𝑎𝑚 units of final good y.
𝑦(𝑎𝑤 , 𝑎𝑚 ) = 2√𝑎𝑤 𝑎𝑚
(𝐴 − 1)
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
𝑍(𝜃, 𝑞𝑚 , 𝑎𝑤 , 𝑑𝑤 ) ≡ 𝑎𝑤 +
𝑑𝑤
𝑡𝑚
(𝐴 − 2)
where 𝑡𝑚 denotes the training level obtained by a manager; 𝑎𝑤 and 𝑑𝑤 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) 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 𝜃 = 1 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.
1
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
1
1
2
2
owner), which are Π = 𝑝𝑦 − 𝑤𝑆, where her homothetic preferences are 𝑉𝑚 = 𝑅Π − 2𝑡 𝑎𝑚
− 2𝑏𝑞 𝑡𝑚
.A
𝑚
1
𝑚
1
2
2)
2
worker’s homothetic preferences are 𝑉𝑤 = 𝑅𝑍𝑤 − 2𝑡 (𝑎𝑤
+ 𝑑𝑤
− 2𝑏𝑞 𝑡𝑤
. Let’s denote with Λ the
𝑤
𝑤
sum of the worker’s and the manager’s post training utilities, where
Λ = 2𝑅𝑝√𝑎𝑚 𝑎𝑤 −
1
1
2
2)
2)
(𝑎𝑚
(𝑎2 + 𝑑𝑤
+ 𝑑𝑚
−
2𝑡𝑚
2𝑡𝑤 𝑤
(𝐴 − 3)
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, 𝑑𝑚 = 𝑑𝑤 = 0. 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
3 )1⁄4
3 )1⁄4
efforts are 𝑎𝑚 = 𝑅𝑝(𝑡𝑚 𝑡𝑤
and 𝑎𝑤 = 𝑅𝑝(𝑡𝑤 𝑡𝑚
. Therefore, the sum of the worker’s and the
manager’s post training utilities with optimal effort levels is
Λ = (𝑅𝑝)2 √𝑡𝑚 𝑡𝑤
(𝐴 − 4)
Stage 3. Production team choice
Both the worker and the manager maximize the above post training utility subject to their respective post∂Λ
training levels. Therefore, ∂𝑡 = (𝑅𝑝)2
𝑚
√𝑡 𝑤
√𝑡 𝑚
and
∂Λ
∂𝑡𝑤
= (𝑅𝑝)2
√𝑡 𝑚
.
√𝑡 𝑤
From here it should be obvious that
both the worker and the manager maximize their post-training utility only when the manager has exactly
2
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,
∂2 Λ
∂𝑡𝑚 ∂𝑡𝑚
> 0.
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 𝑡𝑤 teamed with a manager with training level 𝑡𝑚 .
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 Λm and Λw the respective post-training utilities of the manager and the worker. Their
equations are shown as follows
Λ𝑚 = 𝑅Π −
2
𝑎𝑚
2𝑡𝑚
Λ𝑤 = 𝑅𝑍𝑤 −
2
𝑎𝑤
2𝑡𝑤
(𝐴 − 5)
The worker (manager) maximizes her post-training utility subject to her productive effort levels, where
Π = 2p√am aw − waw and 𝑍 = waw. Therefore, the worker’s efficient productive efforts are aw =
Rwt w. In other words, the worker is willing to accept a wage that satisfies the following w =
aw
.
Rtw
We
3 )1⁄4
know from stage 4 that the optimal level of worker’s productive efforts is 𝑎𝑤 = 𝑅𝑝(𝑡𝑤 𝑡𝑚
.
Substituting the latter in the wage equation that the worker is willing to accept, we present the optimal
wage for a worker with training 𝑡𝑤 teamed with a manager with training 𝑡𝑚 in the following
𝑡𝑚 1⁄4
w = p( )
𝑡𝑤
(A − 6)
In an analogous way, we can show that the optimal wage that the manager will offer is exactly equal to
2
the above equation. In other words, the manager’s efficient productive efforts are am = [(Rp)2 𝑡𝑚
aw ]1⁄3.
3
From the above paragraph, we know that the worker’s efficient productive efforts are aw = Rpwt w.
Substituting the latter into the former, we find that the manager’s efficient productive efforts are am =
2
Rp(𝑡𝑚
𝑤t w )1⁄3 , but we know from stage 4 that the optimal level of manager’s productive efforts is 𝑎𝑚 =
3 )1⁄4
𝑅𝑝(𝑡𝑚 𝑡𝑤
. Hence, it follows that the manger should offer an optimal wage represented by w =
1⁄4
𝑡
p ( 𝑡𝑚)
𝑤
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
(𝑡𝑚 = 𝑡𝑤 = 𝑡). 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, 𝑡𝑚 = 𝑡𝑤 = 𝑡, we obtain the following manager’s indirect utility with optimal effort
levels who teams up with a worker of the same training levels: V𝑚 (𝑡, 𝑞𝑚 ) =
(𝑅𝑝)2 𝑡
2
−
𝑡2
.
2𝑏𝑞𝑚
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: V𝑤 (𝑡, 𝑞𝑤 ) =
(𝑅𝑝)2 𝑡
2
𝑡2
− 2𝑏𝑞 . Hence, in a symmetric
𝑤
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 V𝑦 (𝑡, 𝑞𝑦 ) =
V𝑤 V𝑚 (𝑡, 𝑞𝑚 ) = V𝑤 (𝑡, 𝑞𝑤 ), but this inequality is true only when 𝑞𝑤 = 𝑞𝑚 = 𝑞. 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
V𝑦 =
(𝑅𝑝)2 𝑡
𝑡2
−
2
2𝑏𝑞𝑦
(𝐴 − 7)
4
Maximizing V𝑚 (𝑡, 𝑞𝑚 ) subject to the manager’s training levels, we find the manager’s optimal training
levels 𝑡𝑚 = 12𝑏𝑞𝑚(𝑅𝑝)2 . Analogously, worker’s optimal training levels are 𝑡𝑤 = 12𝑏𝑞𝑤 (𝑅𝑝)2. Since 𝑡𝑚 =
𝑡𝑤 = 𝑡, the optimal level of training of an individual working in the Y sector is
1
𝑡 = (𝑅𝑝)2 𝑏𝑞𝑦
2
(𝐴 − 8)
Therefore, the indirect utility of an individual working in the simple sector with optimal training level is
1
𝑉𝑦 = 8 (𝑅𝑝)4 𝑏𝑞𝑦
(𝐴 − 9)
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.
5
In Appendix B, we present a detailed solution of the last three stages of the game 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 Π = 𝑥 − 𝑤𝐾, where her homothetic preferences are 𝑉𝑚 = 𝑅Π −
1
1
𝑎2
2𝑡𝑚 𝑚
−
1
𝑡2 .
2𝑏𝑞𝑚 𝑚
A
1
2
2)
2
worker’s homothetic preferences are 𝑉𝑤 = 𝑅𝐾𝑤 − 2𝑡 (𝑎𝑤
+ 𝑑𝑤
− 2𝑏𝑞 𝑡𝑤
. Let’s denote with Λ the
𝑤
𝑤
sum of the worker’s and the manager’s post training utilities, where
Λ = 2𝑅√𝑎𝑚 𝑎𝑤 −
1
1
2
2)
2)
(𝑎𝑚
(𝑎2 + 𝑑𝑤
+ 𝑑𝑚
−
2𝑡𝑚
2𝑡𝑤 𝑤
(𝐵 − 1)
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, 𝑑𝑚 = 0. 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 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 of a worker’s
unproductive efforts are 𝑑𝑤 = 𝑅𝑤𝑡𝑤
𝑒 1−𝜃 −1
.
𝑡𝑚
Thus, workers in the complex sector provide less
unproductive efforts when their country has more developed institutions. The optimal level of a
2 )1⁄3
manager’s and a worker’s productive efforts are 𝑎𝑚 = 𝑅(𝑡𝑤 𝑡𝑚
and 𝑎𝑤 = 𝑅𝑤𝑡𝑤 .
6
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 𝑡𝑚 with a worker with training 𝑡𝑤 under optimal efforts from the worker and the manager
Λ=
1
R2
w2tw
(3 {[(wt w )2 t m ]3 } − {
})
2
Ψ
where Ψ ≡ [
2
𝑡𝑚
2
2 +(𝑒 1−𝜃 −1)
𝑡𝑚
(𝐵 − 2)
] shows the quality of the monitoring ability of a manager with training tm and
national institutional development level 𝜃, where lim Ψ(θ) = 1.
𝜃→1
𝜕Λ
Maximizing B-2 over the wage (in other words, setting 𝜕𝑤 = 0), we find the optimal wage
1
𝑡𝑚 4 3
𝑤 = ( ) Ψ4
𝑡𝑤
(𝐵 − 3)
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 Λ as
Λ=
1
1
R2
(𝑡𝑚 𝑡𝑤 Ψ)2 (3 − Ψ 4 )
2
(𝐵 − 4)
The total post-training firm utility as represented by B-4 is strictly supermodular in worker’s and
manager’s training levels. Mathematically,
∂2 Λ
∂𝑡𝑚 ∂𝑡𝑚
> 0. 1 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.
1
For more on the properties of strictly supermodular functions and the complementarities on production see:
Costinot A. (2009) “An Elementary Theory of Comparative Advantage”, Econometrica, 77 (4), pp. 1165-1192.
7
Hence, in equilibrium, the fact that
∂2 Λ
∂𝑡𝑚 ∂𝑡𝑚
> 0 indicates that the manager and the worker must have the
same level of training, 𝑡𝑚 = 𝑡𝑤 = 𝑡. Therefore, we can now describe the optimal wage after matching as
3
𝑤 = Ψ4
(𝐵 − 5)
Let denote with Vm and Vw the respective indirect utilities of the manager and the worker before their
matching and their choice of efforts. Their equations are shown as follows
V𝑚 = 𝑅Π −
2
2
𝑎𝑚
𝑡𝑚
−
2𝑡𝑚 2𝑏𝑞𝑚
𝑉𝑤 = 𝑅𝐾𝑤 −
2
2)
2
(𝑎𝑤
+ 𝑑𝑤
𝑡𝑤
−
2𝑡𝑤
2𝑏𝑞𝑤
(𝐵 − 6)
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
2
3
𝑅𝑡𝑤 2
𝑡2
𝑉𝑚 = 𝑅 2 𝑡𝑤 3 −
−
2
Ψ
2𝑏𝑞𝑚
(𝐵 − 7)
𝑤 Ψ
𝑡2
𝑉𝑤 = 𝑅 2 𝑡 ( 1 − ) −
2
2𝑏𝑞𝑤
Ψ3
(𝐵 − 8)
Using the optimal wage of B-5, we can write the above equations as following
1
1
𝑡2
𝑉𝑚 = R2 Ψ 2 𝑡 −
2
2𝑏𝑞𝑚
(𝐵 − 9)
1
1
𝑡2
𝑉𝑤 = R2 Ψ 2 𝑡 −
2
2𝑏𝑞𝑤
(𝐵 − 10)
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 𝑉𝑥 = 𝑉𝑚 = 𝑉𝑤 ⇒ 𝑞𝑚 = 𝑞𝑤 = 𝑞. Hence, the indirect utility of an individual working in
the complex sector after the matching process is
1
1
𝑡2
𝑉𝑥 = R2 Ψ 2 𝑡 −
2
2𝑏𝑞
(𝐵 − 11)
8
Stage 2. Training
In this stage, in order to find the optimal training levels, we maximize 𝑉𝑥 from B-11 over the training
level. Hence, setting
𝜕𝑉𝑥
𝜕𝑡
= 0, we obtain the optimal training level of an individual who works in the
complex sector
2
𝑡𝑥∗ =
2
𝑅2 Ψ1⁄2 𝑏 2(𝑒 1−𝜃 −1) +𝑡𝑥∗
( ∗ 2 1−𝜃 2
2
𝑡𝑥 +(𝑒
−1)
) 𝑞𝑥
(𝐵 − 12)
This is equation (14) in the main text. Substituting 𝑡𝑥∗ from B-12 into B-11, we find the indirect utility of
an individual with optimal trainings who works in the complex sector
1
𝑉𝑥 = 𝑅 2 𝛹 3⁄2 𝑡𝑥∗
4
(B − 13)
This is equation (15) in the main text.
9
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 𝑞 ∗ ∈ [𝑞𝑚𝑖𝑛 , 𝑞] exists, when 𝑉𝑥 (𝑞∗ ) = 𝑉𝑦 (𝑞 ∗ ), then
this 𝑞 ∗ is unique. In the 2nd step, we prove the existence of 𝑞 ∗. Let's start with Step 1.
Let’s assume that 𝑞 exists. Using (14) and (6), when 𝑉𝑥 (𝑞∗ ) = 𝑉𝑦 (𝑞∗ ), there exists a 𝑞 ∗, such that for any
2
2(𝑒 1−𝜃 −1) +𝑡𝑥∗
𝑞 > 𝑞 ∗, 𝑉𝑥 (𝑞∗ ) ≥ 𝑉𝑦 (𝑞 ∗ ), this 𝑞 ∗ is unique. 𝑉𝑥 (𝑞∗ ) ≥ 𝑉𝑦 (𝑞 ∗ ) only if Ψ (√
2
2(𝑒 1−𝜃 −1) +𝑡𝑥∗
Let 𝐿 ≡ Ψ (√
2
2
2
𝑡𝑥∗ +(𝑒 1−𝜃 −1)
) 𝑎𝑛𝑑 𝐷 ≡ 𝑝. Then,
𝜕𝐿
𝜕𝑞
> 0 𝑎𝑛𝑑
𝜕𝐷
𝜕𝑞
2
2
2
𝑡𝑥∗ +(𝑒 1−𝜃 −1)
) ≥ 𝑝 (𝐶 − 1)
= 0. 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, 𝑞 ∗ is unique.
We start the second step with the proof of 𝑞 ≤ 𝛾𝑚𝑎𝑥 . Suppose that the opposite is true. Therefore, 𝑞 ∗ >
𝑞𝑚𝑎𝑥 . In terms of C-1, this implies that 𝑉𝑥 (𝑞𝑚𝑎𝑥 ) > 𝑉𝑦 (𝑞𝑚𝑎𝑥 ) ∀ 𝑞 ∗ ∈ [𝑞𝑚𝑖𝑛 , 𝑞𝑚𝑎𝑥 ]. Hence, no individual
will be employed in the complex sector, which indicates that the relative price of the simple good
approaches zero (𝑝 → 0). This implies that 𝑞 ∗ < 𝑞𝑚𝑎𝑥 . But, this contradicts our assumption that 𝑞 ∗ >
𝑞𝑚𝑎𝑥 . Hence, 𝑞 ∗ ≤ 𝑞𝑚𝑎𝑥 . In an analogous way, one can show that 𝑞 ∗ > 𝑞𝑚𝑖𝑛 .
Proof of Corollary 1.
To prove all parts of corollary 1, we must find an expression for 𝑞 ∗. 𝑉𝑥 (𝑞) = 𝑉𝑦 (𝑞 ∗ ) implies that
3
2Ψ 2 (𝑞∗ )𝑡𝑥∗ (𝑞∗ )
𝑞 =
𝑏𝑅 2 𝑝4
∗
From (C-2) and (14) it is easy to verify that
(𝐶 − 2)
𝜕𝑞∗
𝜕𝜃
< 0,
𝜕𝑞∗
𝜕𝑏
<0
∀𝜃 ∈ [0,1).
Proof of Proposition 2.
We prove part 1 & 2 of proposition 2 using 2 lemmas. Then, part 3 of proposition 2 follows.
Lemma 1. ∀𝜃 ∈ [0,1)∃ 𝑞(𝜃) where 𝑡𝑥 is convex in 𝑞 if and only if 𝑞 > 𝑞(𝜃).
Lemma 2. 1) ∃ 𝑞0 ∈ [𝑞𝑚𝑖𝑛 , 𝑞𝑚𝑎𝑥 ], such that 𝑡𝑥 (𝑞0 ) = 𝑡𝑦 (𝑞0 ) and 2) 𝑞0 < 𝑞 ∗ ∀𝜃 ∈ [0,1)
10
Proof of Lemma 1.
1
Dividing both sides of (14) with 𝑡𝑥 , we get
theorem,
𝜕𝑡𝑥∗
𝜕𝑞
𝜕𝑓
⁄𝜕𝑞
= − 𝜕𝑓
⁄𝜕𝑡 ∗
𝑥
; hence,
𝜕𝑡𝑥∗
𝜕𝑞
𝑓(𝑡𝑥∗ , 𝑞)
≡1=
2
2
𝑅2 Ψ2 𝑏 2(𝑒 1−𝜃 −1) +𝑡𝑥∗
( ∗ 2 1−𝜃 2
2𝑡𝑥∗
𝑡𝑥 +(𝑒
−1)
) 𝑞𝑥 . From the implicit
> 0. Thus, the optimal training levels of an individual working in the
complex sector is increasing in her natural ability level ∀𝜃 ∈ [0,1).
𝜕𝑓
⁄𝜕𝑞
Let 𝑔(𝑡𝑥∗ , 𝑞) ≡ 𝐴 = − 𝜕𝑓
⁄𝜕𝑡 ∗
𝑥
and
𝜕2 𝑡𝑥∗
𝜕𝑞2
where 𝐴 is a constant. Thus,
< 0 if 𝑡𝑥∗ < 𝑞(𝜃), where; lim 𝑞(𝜃) = 0 ;
𝜃→0
𝜕𝑞(𝜃)
𝜕𝜃
𝜕2 𝑡𝑥∗
𝜕𝑞2
𝜕𝑔
⁄𝜕𝑞
= − 𝜕𝑔
⁄𝜕𝑡 ∗
𝑥
; and
𝜕2 𝑡𝑥∗
𝜕𝑞2
> 0 if 𝑡𝑥∗ > 𝑞(𝜃),
< 0. In order to complete the proof of lemma 1,
we have to show the existence of 𝑞(𝜃). This is done by substituting 𝑞(𝜃) into (14) and combining it with
(15). Thus, 𝑉𝑥 [𝑞(𝜃)] 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 lim [𝑞(𝜃)] = 0 𝑖𝑚𝑝𝑙𝑦
𝜃→0
0;
𝜕𝑡𝑥∗
𝜕𝑞
> 0;
𝜕2 𝑡𝑥∗
𝜕𝑞2
> 0 only if 𝑞 > 𝑞(𝜃), while
𝜕2 𝑡𝑥∗
𝜕𝑞2
𝜕𝑞(𝜃)
𝜕𝜃
<
< 0 only if 𝑞 < 𝑞(𝜃), then 𝑡𝑥∗ is strictly convex in
𝑞 ∀ 𝑞 > 𝑞(𝜃) and 𝑡𝑥∗ is strictly concave in 𝑞 ∀ 𝑞 < 𝑞(𝜃).
Proof of Lemma 2
Assume that the first part of Lemma 2 is true. Then, ∃ 𝑞0 ∈ [𝑞𝑚𝑖𝑛 , 𝑞𝑚𝑎𝑥 ], such that 𝑡𝑥 (𝑞0 ) = 𝑡𝑦 (𝑞0 ).
From (14) we know the optimal training level of an individual working in the complex sector. We also
1
2
know that the optimal training level of an individual working in the simple sector is 𝑡𝑦 = 𝑅 2 𝑏𝑞𝑦 . Hence,
from setting 𝑡𝑥 = 𝑡𝑦 , we can find 𝑞0 . With the help of (C-2), we find that 𝑞0 < 𝑞 ∗ . 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 𝛾0 , and in the second step we proof the existence of 𝑞0 . Let us start
with step 1.
If there exists a 𝑞0 ∈ [𝑞𝑚𝑖𝑛 , 𝑞𝑚𝑎𝑥 ], such that 𝑡𝑥 (𝑞0 ) = 𝑡𝑦 (𝑞0 ), then 𝑡0 is unique. The inequality 𝑡𝑥 (𝑞0 ) ≥
𝑡𝑦 (𝑞0 ) is true since
𝜕𝑡𝑥 (𝑞0 )
𝜕𝑞
> 0, which is the training level first-order condition of utility optimization of
11
individuals working in the complex sector, and
𝜕𝑡𝑦 (𝑞0 )
𝜕𝑞
= 0, which is the optimal ability level of an
individual working in the simple sector.
To prove the existence of 𝑞0 , we assume that 𝑞 ∗ < 𝑞𝑚𝑖𝑛 . From the proof of Lemma 2, it is straightforward
that 𝑡𝑥 (𝑞𝑚𝑖𝑛 ) > 𝑡𝑦 (𝑞𝑚𝑖𝑛 ) ∀ 𝑡0 ∈ [𝑞𝑚𝑖𝑛 , 𝑞𝑚𝑎𝑥 ]. 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 (𝑝 → ∞). But, this contradicts our assumption
that 𝑞0 < 𝑞𝑚𝑖𝑛 . Analogously, one can show that 𝑞0 < 𝑞𝑚𝑎𝑥 .
Proof of the first and the second part of Proposition 2
Now, we can prove part 1 & 2 of Proposition 2. 𝑡𝑥 is concave in 𝑞 only when 𝑞 < 𝑞(𝜃). From (17)
1
2
2
𝑅2 Ψ2 𝑏 2(𝑒 1−𝜃 −1) +𝑡 ∗
lim { 2 ( ∗ 2 1−𝜃 𝑥2
𝑡𝑥 →0
𝑡𝑥 +(𝑒
−1)
) 𝑞𝑥 } = 0. In the region where 𝑡𝑥 is strictly concave in 𝑞, 𝑡𝑥 never intersects
𝑡𝑦 . In Lemma 2, we showed the existence of 𝑞0 such that 𝑡𝑥 (𝑞0 ) = 𝑡𝑦 (𝑞). Thus, 𝑡𝑥 must be convex in 𝑞
at 𝑞0 . Moreover, we demostrated that 𝑡𝑥 is convex in 𝑞 ∀𝑞 > 𝑞0 . Since, 𝑡𝑦 is concave in 𝑞 ∀𝑞 > 0, then
𝑡𝑥 > 𝑡𝑦 ∀𝑞 > 𝑞0 . We showed in the proof of the second part of Lemma 2 that 𝑞0 < 𝑞 ∗. This implies that
𝑞𝑥 is convex in 𝑞 ∀𝑞 > 𝑞 ∗ and 𝑡𝑥 > 𝑡𝑦 ∀𝑞 > 𝑞 ∗.
Proof of Proposition 3.
We have to prove that 𝐼𝑥 (𝑞) ≥ 𝐼𝑦 (𝑞) ∀𝑞 > 𝑞 ∗. Let’s first find 𝐼𝑦 (𝑞) 𝑎𝑛𝑑 𝐼𝑥 (𝑞). 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 𝐼𝑦 (𝑎) = 𝑝𝑎. 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 𝐼𝑦 (𝑎), we can
obtain 𝐼𝑦 (𝑡) = 𝑝2 𝑅𝑡. In the second stage we found the optimal training levels of an individual working in
1
the simple sector. Substituting it into 𝐼𝑦 (𝑡), we obtain 𝐼𝑦 (𝑞) = 2 𝑝4 𝑅 3 𝑏𝑞. In the first stage we found the
optimal skill level for an individual working in the simple sector. Substituting it into 𝐼𝑦 (𝑞), we can obtain
1
the income of an individual working in the simple sector: 𝐼𝑦 (𝑞) = 16 𝑝8 𝑅 7 𝑏2
12
(𝐶 − 5)
𝑒 1−𝜃 −1
) 𝑑𝑤 ].
𝑡𝑚
In the complex sector, the worker income is 𝐼𝑤 (𝑎) = 𝑤𝐾 = 𝑤 [𝑎𝑤 + (
In the fourth stage,
we found the optimal effort and distortion levels exerted from a worker employed in the complex sector.
Substituting the optimal effort and distortion levels as indicated in the fourth stage into 𝐼𝑤 (𝑎), we obtain
𝐼𝑤 (𝑡) =
𝑤 2 𝑅𝑡𝑤
.
Ψ
In the third stage, we showed that the most skilled managers match with the most skilled
workers. Hence, 𝑡 = 𝑡𝑤 = 𝑡𝑚 , and moreover an individual working in the X sector is indifferent on being
3
a worker or a manager. In addition, we found that the optimal wage after the matching process is 𝑤 = Ψ 4
Thus, 𝐼𝑥 (𝑡) = 𝐼𝑤 (𝑡) = 𝐼𝑚 (𝑡) = 𝑅Ψ 1⁄2 𝑡. In the second stage, we determined the optimal training level of
an individual working in the complex sector. Substituting it into 𝐼𝑥 (𝑡) we get
2
5
1
2(𝑒 1−𝜃 − 1) + 𝑡𝑥∗ 2
𝐼𝑥 (𝑞) = 𝑅 2 Ψ2 𝑏 ( ∗ 2
) 𝑞𝑥
2
𝑡𝑥 + (𝑒 1−𝜃 − 1)2
(𝐶 − 6)
Thus, using C-5 and C-6, where in C-5, we substitute 𝑞𝑦 with 𝑞 ∗ and in C-6 we substitute 𝑞𝑥 with 𝑞 ∗, the
inequality 𝐼𝑥 (𝑞) ≥ 𝐼𝑦 (𝑞) stands for all 𝑞 > 𝑞 ∗ because we show that 𝐼𝑥 (𝑞) ≥ 𝐼𝑦 (𝑞) ∀𝑞 is equivalent to
2
1 > Ψ(𝑞 ∗ ), which is true since (𝑒 1−𝜃 − 1) > 0.
Proof of Proposition 4.
The proof of uniqueness and existence of (𝑞 ∗ )𝑗 is exactly the same as the proof of uniqueness and
existence of 𝑞 ∗ in a closed economy of Proposition 1 (see the 1st and 2nd step of the proof of prop. 1).
The proof of the first part of Proposition 4 is simple. (𝑞 ∗ )𝐻 < (𝑞 ∗ )𝑂 since
𝜕𝑞∗
𝜕𝑡
< 0 and
𝜕𝑞∗
𝜕𝑏
< 0 regardless
of the country index (see equation C-2 in the proof of corollary 1). But 𝑏 𝐻 > 𝑏 𝑂 by assumption. This
implies that 𝑡 𝐻 > 𝑡 𝑂 ∀𝑞 > (𝑞 ∗ )𝐻 . The argument for the existence of the latter inequality comes directly
from the first and second part of Proposition 2.
Proof of Proposition 5.
The proof of uniqueness and existence of (𝑞 ∗ )𝑗 , is analogous to 𝑞 ∗.
13
Proof of the first part of Proposition 5.
(𝑞 ∗ )𝐻 < (𝑞 ∗ )𝑂 since
𝜕𝑞∗
𝜕𝜃
< 0 regardless of the country index. But we assumed that 𝜃 𝐻 > 𝜃 𝑂 . Hence,
(𝑞 ∗ )𝐻 < (𝑞 ∗ )𝑂 .
Proof of the second part of Proposition 5.
We start with the proof of the inequality (𝑞 ∗ )𝑂 ≥ 𝑞𝑚𝑖𝑛 . Assume that (𝑞 ∗ )𝑂 < 𝑞𝑚𝑖𝑛 . We know that
(𝑞 ∗ )𝐻 < (𝑞 ∗ )𝑂 . Thus, (𝑞 ∗ )𝑂 < 𝑞𝑚𝑖𝑛 & (𝑞 ∗ )𝐻 < (𝑞 ∗ )𝑂 . But, if both (𝑞 ∗ )𝐻 and (𝑞 ∗ )𝑂 are strictly lower
than 𝑞𝑚𝑖𝑛 , then no one enters into the simple sector, implying that the relative price of the simple good
(𝑝) goes to infinity. Therefore, (𝑞 ∗ )𝐻 and (𝑞 ∗ )𝑂 must be strictly higher than 𝑞𝑚𝑖𝑛 . But we assumed that
(𝑞 ∗ )𝑂 < 𝑞𝑚𝑖𝑛 . Consequently, (𝑞 ∗ )𝑂 ≥ 𝑞𝑚𝑖𝑛 . Analogously (𝑞 ∗ )𝐻 ≤ 𝑞𝑚𝑎𝑥 .
Proof of Corollary 2.
The proofs of all inequalities of Corollary 2 are analogous with the proofs of Propositions 2 and 3.
Proof of Proposition 6.
The proof of part 1 and 2 of Proposition 6 is analogous to that of Proposition 5, and the proof of part 3 of
Proposition 6 are analogous with those of Propositions 2 and 3. The proof of part 4 of Proposition 6:
We drop the superscript (j) when necessary for notation simplicity. We first prove that 𝑞 ∗ exists. Then, we
show that also ∃ 𝑞̃ such that 𝐼𝑥 (𝑞̃)𝐻 > [𝐼𝑥 (𝑞̃)𝑂 + 𝑐] ∀𝑞 > 𝑞̃.
The proof of the existence of 𝒒∗:
3
We know from Proposition 1 that 𝑉𝑥 (𝑞) ≥ 𝑉𝑦 (𝑞) only if:
2Ψ2 (q∗ )t∗x (q∗ )
R2 bq∗
≥𝑝4
(𝐶 − 7)
The left hand side of (C-7) is increasing in 𝑞 and approaches zero when 𝑞 approaches zero, while the right
hand side of (C-7) is constant. Let’s suppose that 𝑞 ∗ does not exist. Thus, the right hand side of (C-7)
must be strictly higher than one. Hence, no one enters into the complex sector in each country, meaning
that the relative price of the simple good approaches zero. Thus, the right hand side of (C-7) is strictly less
than one. This contradicts our assumption that 𝑝 > 1. Thus, 𝑞 ∗ must exist. From Proposition 3, we know
that 𝐼𝑥 (𝑞)𝐻 ≥ 𝐼𝑥 (𝑞)𝑂 ∀𝑞 > (𝑞 ∗ )𝑂 ; since 𝑞 ∗ exists, then 𝑞̃ must also exist for 𝑐 < (𝑞 ∗ )𝑂 − (𝑞 ∗ )𝐻 . Since
14
𝑉𝑥 (𝑞)𝑗 is positive, then there must exist a 𝑞̃ such that for any positive value of 𝑐, 𝑉𝑥 (𝑞̃)𝐻 = [𝑉𝑥 (𝑞̃)𝑂 +
𝑐] , where 𝑞̃ > 𝑞 ∗. Thus, 𝑉𝑥 (𝑞)𝐻 > [𝑉𝑥 (𝑞)𝑂 + 𝑐] ∀𝑞 > 𝑞̃. Hence, ∃ 𝑞̃ such that 𝐼𝑥 (𝑞)𝐻 > [𝐼𝑥 (𝑞)𝑂 +
𝑐] ∀𝑞 > 𝑞̃ and 𝐼𝑥 (𝑞) ≥ 𝐼𝑦 (𝑞) ∀𝑞 > 𝑞̃. Thus, the individuals who obtain the highest level of income are
those who work in the complex industry in H. Thus, the flow of international labor migration will be from
O to H.
Proof of Proposition 7.
Proof of part 1) of Proposition 7:
From Proposition 6, 𝐼𝑥 (𝑞)𝐻 > [𝐼𝑥 (𝑞)𝑂 + 𝑐] ∀𝑞 > 𝑞̃ since 𝜃 𝐻 > 𝜃 𝑂 . Hence, only those individuals of O
with 𝑞 > 𝑞̃ will immigrate in H. No one who works in the y sector has incentives to immigrate in H since
𝑗
𝐼𝑦 (𝑞)𝐻 = 𝐼𝑦 (𝑞)𝑂 ∀ 𝑞 > 0, because
𝜕(𝐼𝑦 )
𝜕𝜃𝑗
= 0. Thus, only individuals who work in the complex sector
𝑗
could emigrate in H because
𝜕(𝐼𝑦 )
𝜕𝜃𝑗
> 0. However, not all individuals of O that work in the complex sector
will emigrate to H. There would be some of them whose income difference because of immigration is
lower than the fixed costs of immigration. These individuals will work in the complex sector in O. The
natural ability levels of such individuals are 𝑞 ∗ < 𝑞 𝑂 < 𝑞̃. Thus, all individuals with 𝑞 > 𝑞̃ will emigrate
in H. The highest ability individuals of O will choose to emigrate in H because
𝜕𝑞
𝜕𝑡𝑥
> 0 ∀𝑞 > 𝑞 ∗ and
moreover ∀ 𝑞 > 𝑞̃.
Proof of part 2) of Proposition 7:
We know from part 3) of Proposition 6 that (𝑞 ∗ )𝐻 < (𝑞 ∗ )𝑂 . Therefore, 𝑞 ∗ < (𝑞 ∗ )𝑂 . This implies that
𝐼𝑥 (𝑞)𝐻 > 𝐼𝑥 (𝑞)𝑂 ∀𝑞 > 𝑞 ∗. Also using part 4) of Proposition 6, we know that 𝑞 ∗ < 𝑞̃ and we also know
that 𝐼𝑥 (𝑞)𝐻 > [𝐼𝑥 (𝑞)𝑂 + 𝑐] ∀𝑞 > 𝑞̃. But, when c approaches zero then 𝐼𝑥 (𝑞)𝐻 > 𝐼𝑥 (𝑞)𝑂 ∀𝑞 > 𝑞 ∗, which
is true for all 𝑞 > (𝑞 ∗ )𝑂 . Consequently, there must exist a threshold level of 𝑐̅ that satisfies the following
𝐼𝑥 (𝑞)𝐻 − [𝐼𝑥 (𝑞)𝑂 + 𝑐̅] = 0 such that all individuals of O with 𝑞 > 𝑞̅ find it beneficial to immigrate in H.
15
Proof of Corollary 3.
We know that if
𝑏 𝐻 < 𝑏 𝑂 and 𝜃 𝐻 > 𝜃 𝑂 , it is quite possible that 𝑝𝐻 = 𝑝𝑂 in autarky. Thus, both
countries will not engage in international trade with each other at all. However, if both countries enter a
common labor market area, then according to Proposition 7, for sufficiently low fixed immigration costs
(𝑐̅), the following inequality holds 𝐼𝑥 (𝑞)𝐻 > [𝐼𝑥 (𝑞)𝑂 + 𝑐̅] ∀𝑞 > 𝑞̅. Therefore, there would exist
emigration only from O to H, implying that the relative price of the simple good increases because of the
increase in the world supply of the complex good due to emigration of individuals of O, with (𝑞 ∗ )𝐻 <
𝑞 < (𝑞 ∗ )𝑂 , toward H. Thus, immigration changes the relative world price of the simple good (𝑝𝑂 < 𝑝𝐻 ).
Therefore, after immigration, H will have a comparative advantage in the complex good, while O will
have a comparative advantage in the simple good. In other words, immigration creates trade.
16