Discussion Paper No.

590

ENDOGENOUS HUMAN CAPITALACCUMULATION, COMPARATIVE ADVANTAGE
AND DIRECT VS. INDIRECT REDISTRIBUTION

Hisahiro Naito

July 2003

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Endogenous Human Capital Accumulation, Comparative
Advantage and Direct vs. Indirect Redistribution
Hisahiro Naito ∗†Institute of Social and Economic Research
Osaka University
andDepartment of Economics
University of California Irvine
Previous version August 30, 2002
Current version June 7, 2003

Forthcoming from Journal of Public Economics

Abstract
Recently, several papers have re-examined the so-called production efficiency theorem andthe Atkinson and Stiglitz theorem on commodity taxes in the optimal taxation literature.
Naito (1999) showed that indirect redistribution through production distortion or consumption distortion can Pareto-improve welfare and that the two theorems do not necessarily holdwhen different factors are imperfect substitutes and factor prices are endogenous. On theother hand, Saez (2001) argued that in the long run where human capital accumulation is
endogenous, the two theorems are still valid. This paper develops reasonable alternative models where individuals accumulate human capital based on their comparative advantage. The
present paper shows that the production efficiency theorem is not necessarily valid and thatindirect redistribution from the able to the less able such as tariffs and production subsidies
can increase efficiency even when skill accumulation is endogenous.

Keywords: Human capital accumulation, non-linear income taxation, and comparative advantage
JEL Number: H21, H23∗

I appreciate the participants of the Public Economics Research Group around Osaka and Daiji Kawaguchi forhis helpful information on the relationship between earnings and ability. I am very grateful for the comments from
two anonymous referees and professor Robin Boadway, the editor of the Journal. Their comments have been very
helpful for improving this paper. Of course, the author is responsible for all remaining errors.
†
Address: Institute of Social and Economic Research, Osaka University, Mihogaoka 6-1, Ibaraki City, Osaka,
Japan, postal code 567-0047
phone:81-6-6879-8581; fax: 81-6-6878-2766.
e-mail address: naito@iser.osaka-u.ac.jp

1

Introduction

Whether efficient income redistribution should be done through income taxation alone or should
be complemented with other measures such as production distortion or consumption distortionis one of the key issues whenever optimal public policies are discussed. With this regard, theproduction efficiency theorem (Diamond and Mirrlees, 1971), which states that production distortion is not optimal and the Atkinson and Stiglitz theorem on optimal commodity taxation(Atkinson and Stiglitz, 1976, 1980), which shows that commodity taxation is not necessary in
the presence of an optimal income tax system, are the most important results in public financeliterature.
However, in public finance literature researchers started examining those results. For example,
Cremer, Pestieau and Rochet (2001) showed that the Atkinson and Stiglitz theorem does not holdwhen individuals are different in ability and endowment. Saez (2003) showed that the Atkinson
and Stiglitz theorem does not hold when tastes are heterogenous. Naito (1999) showed that ina model similar to the model of Stiglitz (1982), if multiple goods are produced and factor prices
are endogenous, the Atkinson and Stiglitz theorem does not necessarily hold and the production
efficiency result does not either.
On the other hand, many of the previous studies on optimal income taxation have received
criticism that they did not focus on long term decisions such as human capital accumulation but
focused on the short term choices such as labor supply. As a result, it is sometimes argued thatthe result obtained in the short run model might not hold in the long run.
In particular, Saez (2003) made skill accumulation endogenous in the model of optimal taxation and analyzed several issues of public policy. He showed that Naito’s results are not valid
and that the production efficiency theorem and the Atkinson and Stiglitz theorem on commodity taxation are valid when human capital accumulation is endogenous. Since accumulation ofhuman capital has a strong effect on the economy in the long run and since the implications of
the production efficiency theorem and the Atkinson and the Stiglitz theorem are important, the1

contribution of Saez’s paper is substantial.
Despite such contributions, however, we believe that a further investigation would be needed.
In many previous analyses involving asymmetric information not only in public finance literature
but also in other literature, conclusions were not often robust in the sense that they criticallydepended on the structure of information and the timing of information revelation. Thus, itis worthwhile investigating the robustness of the result of Saez (2003) with another reasonable
set of assumptions. In particular, in this paper we will show that if higher ability persons havecomparative advantage in the sense that the relative return from accumulating skilled humancapital to unskilled human capital is higher than that of lower ability persons, the production
efficiency theorem does not hold. 1
To explain the intuition of the present paper, it would be useful to look at the differences
between the assumptions in Saez (2003) and those in Naito (1999). In Naito (1999), there are
two factors of production that are imperfect substitutes. In addition, from the beginning, eachindividual is attached to a particular labor market (the skilled labor market or the unskilledlabor market) but the government cannot observe whether each individual is attached to theskilled or unskilled labor markets. The main idea in Naito (1999) is that when the government
cannot observe an individual’s type, the government can affect different individuals differently byusing the response of the factor markets (Stolper and Samuelson theorem, Stolper and Samuelson(1941)). Since the income tax policy cannot discriminate the different types of agents attachedto different labor markets but a commodity tax and a tariff can, using a commodity tax (in acase of a closed economy), or a tariff (in a case of an open economy) with the response of factor
markets can increase the efficiency.
In Saez (2003), each job requires pre-determined skill levels. As a result, the income level1

In this paper, we only analyze the case of a small open economy due to the limitation of the space. As a result,
we only prove that the production efficiency theorem does not hold. In a small open economy, a commodity tax
cannot affect the producer prices and hence, factor prices. Therefore, the Atkinson and Stiglitz theorem holds in asmall open economy. On the other hand, the result is changed in a closed economy. In the previous version of thepaper, we proved that the Atkinson and Stiglitz theorem does not hold in a closed economy. See the section 5 inthe present paper for more discussion. Also, for proof, please see our previous version of the present paper (Naito,
2002).

2

represents the amount of skill the individuals acquired. People have a heterogenous ability toacquire skill. However, since such heterogeneity of ability is incorporated in the utility function
as a difference of disutility to acquire skill, there is not any room for such heterogenous ability tointeract with the market reaction. Thus, the heterogeneity of abilities is intrinsically independentof the external environment of the economy. When the heterogeneity of abilities is independentof the response of the factor market, it is essentially equivalent to assuming that the dimensionof factors used for production is one. In such a case, changes of factor prices due to government
policy cannot increase the economic efficiency.
The key idea of the present paper is that in the presence of comparative advantage in accumulating different types of human capital, individuals with different abilities will be affecteddifferently by the responses of factor markets even when skill accumulation is endogenous. In
such a case, a policy that introduces inefficient production but affects the factor prices differently
for different factors can indirectly redistribute from the less able to the able. Although such a
policy cause a distortion, it has only the second order effect, but such an indirect redistribution
has the first order effect on welfare. Thus, it can increase the social welfare.
For illustration, consider a situation where there are two types of human capital: skilled humancapital and unskilled human capital and where those who have higher ability have comparativeadvantage in accumulating skilled human capital. Comparative advantage in accumulating skilledhuman capital for the able means that the relative benefit from accumulating skilled humancapital to unskilled human capital for individuals with high ability is higher than for the lessable. We could think that training, knowledge and experience in white collar jobs are skilledhuman capital and those in blue collar jobs are unskilled human capital. In such a situation, adecrease of the return from skilled human capital and an increase of the return from unskilledhuman capital will hurt the able relatively more and give relatively more benefit to the less able.
The intuition of this paper is that when individual ability is not observable to the social planner
but the social planner is interested in redistribution from high ability individuals to low ability3individuals, then a policy that will change the returns from skilled and unskilled human capitaldifferently might be useful for an efficiency reason.
The crucial assumption in the present paper is the presence of comparative advantage inhuman capital accumulation. Whether such an assumption is reasonable or not is an interesting
empirical question. Earlier literature of the human capital theory assumed that earning could be
explained completely once it is conditioned by human capital level. Earlier empirical evidences
showed that there is a strong correlation between earnings and the level of human capital andindicated that ability does not matter for explaining earnings once they are conditioned by thehuman capital levels. On the other hand, recent literature of labor economics and self-selection
emphasizes that ability can also increase earning and play a systematic role for explaining earnings
even after it is conditioned by the human capital level. This literature points out that even in
an extreme case when human capital does not increase the productivity at all, if ability canincrease the productivity and if higher ability agents tend to acquire more skills, there will be acorrelation between human capital level and earnings. In the standard signaling literature, it iscommonly assumed that a higher ability person would get more benefit from acquiring skill. In
addition, recently, Dinardo and Tobias (2001) and Tobias (2003) examined whether the returns
from schooling are higher for high ability individuals than for low ability individuals by using a
non-parametric method. They found that the returns are higher for high ability individuals thanfor low ability individuals. This suggests that assuming the presence of comparative advantage
is not unrealistic as an approximation of the reality.
At this point, one might wonder about the difference between Naito (1999) and the present
paper. In the case of Naito (1999), each type of worker is attached to a different labor market.
As a result, skilled workers can supply only skilled labor and unskilled workers can supply only
unskilled labor. However, in the present paper, both high ability persons and low ability persons
have options to accumulate both types of human capital or either type of human capital. Thus,
the reasoning that an increase of the return from unskilled human capital and a decrease of the4return from skilled human capital always increases efficiency is not obvious.
The organization of this paper is as follows. In section 2, we present the model in a small
open economy and analyze the production efficiency theorem by Diamond and Mirrlees (1971)
when two factors are imperfect substitutes. In section 3, we analyze the same issue when two
factors are perfect substitutes. In section 4, we shall give the implications and in section 5 wewill give a brief conclusion.

2

The model

The economy is small and open and there are two output goods: good 1 and good 2. Good 1 isskilled human capital intensive good and good 2 is unskilled human capital intensive good. We
assume that there are two types of human capital in this economy: skilled human capital andunskilled human capital. In this economy, there is a continuum of agents and all agents haveidentical, additive separable utility functions with respect to consumption, skilled human capital
investment and unskilled human capital investment. We index all individuals’ ability by i where
i takes any value from one to two. We assume that the utility function of the type i agent has
the following form:u(c1i , c2i ) − fs (hsi ) − fu (hui )where u(c1i , c2i ) is strictly increasing with each argument and strictly concave and fs (hsi ) andf (hui ) are strictly increasing and strictly convex. c1i and c2i are the consumption of good 1 andgood 2 by agent i. We assume that the labor supply is fixed and it is normalized to one. hsi
and hui are the levels of skilled and unskilled human capital of individual i. hsi and hui can beinterpreted as the knowledge levels, years of education, experience and training for each type ofskill. In addition, to illustrate our point, we assume that fs (hsi ) and f (hui ) have the following
functional forms:2
fs (hsi ) = (hsi )γs and fs (hui ) = (hui )γu
2

Our main results can hold in more general functional forms.

5where γs and γu measure the curvature of the disutility functions of skilled and unskilled humancapital accumulation respectively and they are strictly greater than one. Given the amount ofskilled human capital and unskilled human capital of individual i, we assume that the earning ofindividual i is determined as follows:
earningi = gs (i) × ws × hsi + gu (i) × wu × hui(1)

where ws and wu are the returns from one efficient unit of skilled and unskilled human capital,
respectively. (1) means that when individual i accumulates hsi units of skilled human capitaland hui units of unskilled human capital, the efficient unit of skilled human capital and unskilledhuman capital are gs (i) × hsi and gu (i) × hui and the total return from skilled human capital andunskilled human capital are gs (i) × ws × hsi and gu (i) × wu × hsi , respectively. Let gs (i) × ws
and gu (i) × wu be wis and wiu . gs0 (i)/gs (i) and gu0 (i)/gu (i) measure the absolute advantage of anagent with ability i +  over agent i in accumulating skilled human capital and unskilled humancapital, respectively. We assume that agents who have higher ability have absolute advantage
in accumulating both skilled human capital and unskilled human capital: gs0 (i)/gs (i) > 0 and
gu0 (i)/gu (i). 3 Also, as we discussed in the introduction, we assume that agents who have higher
ability have comparative advantage in accumulating skilled human capital than unskilled humancapital. Thus, we assume that
gs0 (i)g 0 (i) γu> u
gs (i)
gu (i) γs(2)

The assumption (2) has a clear economic meaning. Consider a situation where the disutility
functions of accumulating skilled human capital and unskilled human capital have the same
degree of curvature (γs = γu ). In this case, (2) means that an agent whose ability is higher
will have a larger rate of increase of wis , the return from accumulating skilled human capital,
than that of wiu , the return from accumulating unskilled human capital. When the curvature of
the disutility functions are different, (2) says that the condition of the comparative advantage
3

The assumption of the absolute advantage is not necessary. The assumption of the absolute advantage is asufficient condition that guarantees that agents who have higher i will receive higher utility. As long as agents withhigher ability can receive higher utility the assumption of the absolute advantage is not necessary.

6

must be adjusted by the ratio of the curvatures of marginal disutility of skilled human capital
accumulation and unskilled human capital accumulation. 4
At this point, note that (1) is different from the assumptions in Saez in several ways. In
Saez, he assumed that heterogeneity of individuals is incorporated in the utility function, notthe earning equation. Thus, once earning is conditioned by the human capital level, individuallevel heterogeneity of ability does not play any systematic role for explaining earnings. On theother hand, in (1) even after conditioned by the level of human capital, heterogeneity of ability
plays a systematic role for explaining earnings and it induces higher earning for agents withhigher ability. In addition, the relative return from one efficient unit of skilled human capital tounskilled human capital is higher for the agent who has higher ability than for the agent who has
lower ability. As we discussed in the introduction, this interaction term between heterogeneity ofability and the return of human capital plays a crucial role in the present paper.
As for the objective of the government, we assume that the social planner will maximize thefollowing utilitarian social welfare function:
Z 2{u(c1i , c2i ) − fs (hsi ) − fu (hui )}ni di .

(3)

1

As for prices, we normalize the producer price and the consumer price of good 1 to one.
Let p2 , q2 and p∗2 be the consumer price and the producer price and the international price ofgood 2, respectively. As the purpose of this section is to examine whether introducing production
distortion can increase the social welfare or not, we consider imposing a tariff on good 2. Althougha tariff introduces not only a production distortion but also a consumption distortion, the firstorder effect of consumption distortion on welfare can be ignored as we will demonstrate. Let σ
The reason that the terms of gs0 /gs and gu0 /gu need to be adjusted by the curvature of the marginal disutility
is as follows. For illustration, consider a condition that gs0 /gs and gu0 /gu must satisfy when agents with ability
i +  and agents with ability i have the same degree of comparative advantage under the assumption of γs > γu .
The assumption of γs > γu implies that the marginal disutility of skilled human capital changes faster than themarginal disutility of unskilled human capital when the amount of skilled and unskilled human capital respectively
changes at the same rate. Note that the marginal disutility per return of skilled and unskilled human capital must
be equal at the margin. This implies that in order that agents with ability i +  and agents with ability i have thesame degree of comparative advantage, gs0 /gs must be smaller than gu0 /gu .
47

be a size of a tariff on good 2. Then, we will have
p2 = q2 = p∗2 + σ.

(4)

As for the equations determining the returns from skilled and unskilled human capital, we
assume the standard two sector Heckscher-Ohlin model. In this economy, there are two sectors.
Sector 1 is the skilled human capital intensive sector and it produces good 1. Sector 2 is theunskilled human capital intensive sector and it produces good 2. Each sector uses both skilled andunskilled human capital. Consumers (workers) are perfectly mobile between two sectors. Whenan agent who has hsi units of skilled human capital and hui units of unskilled human capital works
in sector k, it means that sector k uses gs (i) × hsi units of skilled human capital and gu (i) × huiunits of unskilled human capital. Each sector behaves as a price taker and maximizes its profit.
Let F k (Hks , Hku ) be the production function in sector k = 1, 2 where Hks and Hku are the total
amount of skilled human capital and unskilled human capital used in sector k. We assume that
F k (Hks , Hku ) exhibits constant returns to scale and it is concave with respect to both arguments.
Let ck (ws , wu ) be the cost function in sector k to produce one unit of output in sector k when
the returns of one efficient unit of skilled human capital and unskilled human capital are ws andwu , respectively. When both good 1 and good 2 are produced at the equilibrium, ws and wu are
determined
1 = c1 (ws , wu ) and q2 = c2 (ws , wu ),(5)

From the Stolper -Samuelson theorem, ∂ws /∂q < 0 and ∂wu /∂q > 0.
The output of both goods are determined from the following factor market equilibrium conditions:
∂c1 1
∂c2 2
y +
y =
s
∂w
∂ws

Z 2

gs (i) × hsi × ni di, and

1

∂c2 2
∂c1 1
y +
y =
u
∂w
∂wu

Z 2

gu (i) × hui × ni di(6)1

Although the output of both goods can be calculated from equation (6), it is more useful towork on the production possibility frontier for analytical reasons. Let H s and H u be the total
amount of skilled human capital and unskilled human capital in this economy and define a8

production possibility frontier as Γ(H s , H u ). Since the production functions are concave and thefactor intensity of the two sectors are different, the production possibility set is convex. Let the
producer price of good 1 and good 2 be 1 and q2 . Then, the output of good 1 and good 2 aredetermined as the solution of the following constrained maximization problem:
max y1 + q2 y2

s.t. (y1 , y2 ) ∈ Γ(H s , H u ) = 0

Thus, the output of good 1 and good 2 can be thought as a function of q2 , H s and H u . Let
Y (q, Hu , Hu ) be the output function of good 2. At the optimum, the slope of production possibility set is equal to the relative producer price of good 2. Thus, we obtain Yq ≡ ∂Y /∂q2 > 0.
The Rybcyzynski theorem shows that YH u ≡ ∂Y /∂H u > 0 and YH s ≡ ∂Y /∂H s < 0.
The purpose of the social planner is to maximize the utilitarian social welfare function. Given
the additive separable utility function and the utilitarian social welfare function, the social planner
wants to redistribute income from those who have higher ability to those who have lower ability.
On the other hand, since the social planner cannot observe individual ability but rather individual
earning, the social planner needs to design a non-linear income tax system T (R) to redistributeincome where T (R) is a tax liability function and R is pre-tax income.
Before designing an income tax system, it is useful to consider the problem of designing thenon-linear income tax system in two steps. The first step is to know how an individual i will
choose skilled human capital and unskilled human capital to generate pre-tax income, R. The
second step is to know, given an after-tax-income schedule of X = R − T (R), how each individual
chooses pre-tax income.
The first stage of the problem can be solved considering the following programming problem:
min fs (hsi ) + fu (hui )(7)s.t. R = wis × hsi + wiu × hui
where wis = gs (i) × ws and wiu = gu (i) × wu
Let the minimized value of the above problem be Z(wis , wiu , R). Z(wis , wiu , R) is the minimized
9

disutility to generate the pre-tax income R for an agent whose net returns from skilled humancapital and unskilled capital are wis and wiu , respectively. We denote the solution of the above
problem as hsi (wis , wiu , R) and hui (wis , wiu , R). For the analysis later, it is useful to calculate compensated human capital supply. Consider the following dual problem of (7):E(wis , wiu , V ) ≡ max wis hsi + wiu hui
st. fs (hsi ) + fu (hui ) ≤ V
Let the solution of the above problem be ehji (wis , wiu , V ) where j = s, u. Then, from the dual
relationship, we will havehji (wis , wiu , E(wis , wiu , V )) ≡ ehji (wis , wiu , V )j=s,uBy taking derivative on both sides, we will have the Slutsky equation for hsi and hui :∂hji∂hji s∂ehji∂hji∂hji u
∂e
hji
+
h
=
and
+
h
=
∂wis
∂R i
∂wis
∂wiu
∂R i
∂wiu

j=s,u

Note that the indifference curve of fs (hsi ) + fu (hui ) is strictly concave. Therefore, ∂ ehsi /∂wis > 0,∂ehui /∂wu < 0, ∂ ehui /∂wiu > 0 and ∂ ehui /∂wis < 0. This relationship means that if an individual
maximizes his earnings holding the total disutility constant, an increase of the net return fromskilled human capital will increase the supply of skilled human capital and an increase of the
return of unskilled human capital will decrease the supply for skilled human capital. As forthe properties of Z, let the Lagrangian multiplier of the disutility minimization problem be αi .
Then, fs0 (hsi ) = αi wis , fu0 (hui ) = αi wu , Zwis ≡ ∂Z/∂wis = −αi hsi , Zwu ≡ ∂Z/∂wu = −αi hui andZR ≡ ∂Z/∂R = αi .
Let X(R) be the after-tax income schedule that the government designed. Then, at the second
stage of the problem, given Z(wis , wiu , R) and X(R), each individual i will maximize his utility:
max U (p2 , X(R)) − Z(wis , wiu , R)
{R}

where U (p2 , x) is the indirect utility function from the consumption of two goods when the
consumer price of good 2 is p2 and the after tax income is x. The objective of the social planner10

is to design a schedule of X(R) to maximize the social welfare. On the other hand, the Revelation
Principle shows that without loss of generality we can focus on the incentive compatible revelation
mechanism. Thus, let (Rj , Xj ) be the pre-tax income and after tax income when an agent
announces that his type is j. Then define v(i) and vb(j; i) as follows:
v(i) = max U (p2 , Xj ) − Z(wis , wiu , Rj ){j}vb(j; i) = U (p2 , Xj ) − Z(wis , wiu , Rj )v(i) is the maximized utility given the schedule of (Rj , Xj ) and vb(j; i) is the indirect utility when
agent i announces that he is type j. The incentive compatibility condition implies that the typei agent has an incentive to announce that he is type i:i = arg max vb(j; i){j}Assuming the differentiability of (Xj , Rj ), the first order condition of the incentive compatibility
condition is∂bv (j, i)∂U ∂x ∂Z ∂R
=
−
=0
∂j
∂x ∂j
∂R ∂j
j=i
On the other hand, by using the above first order condition, we have dv/di = −Zwis × (dwis /di).
Since αi is the Lagrangian multiplier of the required income constraint in the disutility minimization problem (7), from the FOC of the minimization problem we obtain00w j hj
dv
g (i)
g (i)
= αi Ri { s θsi + u θui } where θji = i i
digs (i)
gu (i)
Ri(8)

Because of the assumption from the absolute advantage, dv/di > 0. (8) has a clear economic
meaning. It means that the slope of the value function v(i) is proportional to the weighted
average of the absolute advantage of skilled human capital accumulation and unskilled humancapital accumulation. For analytical reasons, it is useful to eliminate αi in the above equation.
Using the first order condition for hsi and hui , we can rewrite (8) as follows:
0

0

g (i)
g (i)
dv
= s fs0 (hsi )hsi + u f 0 (hui )hui .
digs (i)
gu (i)
11(9)

Given (9), as Mirrlees (1971) pointed out, it is more useful to assume that the social planner
controls vi and Ri . 5 Then, xi is defined by the following relationship:v(i) = U (p2 , Xi ) − Z(wis , wiu , Ri ).

(10)

Let x(R, v, p2 , wis , wiu ) be the solution that solves (10) about X. Obviously, ∂x/∂v = (Ux )−1 ,∂x/∂Ri = ZR /Ux and ∂x/∂p = −(Up )/(Ux ), ∂x/∂p = −(Up )/(Ux ) ,∂x/∂wis = Zwis /(Ux ) and∂x/∂wu = Zwu /(Ux ).
Finally, the government budget constraint implies thatZ 2

Z 2
ni {Ri − xi }di + σ{1c2i ni di − Y (p∗2 + σ, H s , H u )} = 0

1

The problem of the social planner is to solve the following constrained optimization program:
Z 2
W (σ) = max{Ri ,vi } 10v(i)ni di .
0

dv
g (i)
g (i)
st.
= s fs0 (hsi )hsi + u f 0 (hui )huidigs (i)
gu (i)
Z 2
Z 2
ni {Ri − xi (Vi }di + σ{
c2i ni di − Y (p∗2 + σ, H s , H u )} = 0
11

σ is given.
In the above programming problem, W (σ) is the maximized social welfare for given σ. Also notethat hsi and hui are functions of (Ri , wis , wiu ) and that wis and wiu are the functions of σ.
Our interest is to know whether a change of σ from 0 will increase the social welfare or not.
Analytically, by calculating dW/dσ, and evaluating at σ = 0, we can check whether introducing
a distortion in production side (and consumption side too) can increase the social welfare. Let µi
and λ be the Lagrangian multiplier of the incentive compatibility constraint and the government5

One might think that the local incentive compatibility constraints are not sufficient for the global incentive
compatibility constraints. On the other hand, the literature of the mechanism design shows that a single crossing
property (SCP) and the monotonicity constraints are sufficient conditions for local incentive compatibility constraints to satisfy the global incentive compatibility constraints. (Fudenberg and Tirole, 1992). In this paper, we
assume that the monotonicity constraints are always satisfied. This assumption is equivalent to assuming that
there is no bunching. Many of the previous papers assumed that there is no bunching at the optimum. (Konishi∂hs∂2Z
1995, Naito 1998). As for SCP, we can check it by examining ∂R∂i
> 0. This is true as long as ∂Ri > 0.

12

budget constraint. By using the envelope theorem, we obtaindW=−
dσ σ=0Z 2µi
1d[f 0 (hsi )hsi (gs0 /gs )] dhsi dwis
dhsi dwiu
{
+
}di
dhsi
dwis dσ
dwu dσd[f 0 (hui )hui (gu0 /gu )] dhui dwiu
dhui dwiu
{
+
}di
dhui
dwiu dσdwiu dσ
1
Z 2
Z 2
∂xi
∂xi ∂wis∂xi ∂wu∗su−−)ni di+ λ{c2i ni di − Y (p2 + σ, H , H )} + λ(−∂p∂wis ∂σ∂wu ∂σ
1
1
Z 2

−

µi

After several calculations, we can obtain the following equation (See Appendix):dW=−
dσ σ=0Z 2no∂ehs ∂wis∂ehsi ∂wiu
00µi γs (gs /gs ) − γu (gu /gu ) fs0 (hsi )[ is+]di > 0∂wi ∂σ∂wiu ∂σ1(11)

Because of the property of the compensated supply function of hsi , ∂ ehsi /∂wis > 0 and ∂ ehsi /∂wiu <0. From the Stolper-Samuelson theorem, ∂wis /∂σ < 0 and ∂wiu /∂σ > 0. From the assumption on00comparative advantage, γs (gs /gs ) − γu (gu /gu ) > 0. As for the sign of the Lagrangian multiplierof the incentive compatibility constraint, the standard argument shows that µi ≥ 0 for all i.(SeeAppendix). Thus, we obtain dW/dσ > 0.

Proposition 1 Suppose that at the zero distortion on production and the consumption in a small
open economy the social planner sets the income tax structure to maximize the social welfare
function in an endogenous skill accumulation model . Then an introduction of a tariff (export
subsidy) on an unskilled-labor-intensive good will increase the social welfare.
The above equation (11) has several implications. For an illustration, consider a situation
where the disutility functions of skilled and unskilled human capital accumulation have the same
00degree of curvature, i.e. γs = γu ≡ γ. Then, (11) shows that if (gs /gs ) = (gu /gu ), dW/dσ = 0.
In other words, if there is no comparative advantage and if higher ability individuals are as goodat accumulating skilled and unskilled human capital as lower ability individuals, then there is
no welfare gain from changing the returns of skilled and unskilled human capital. Second, notethat (∂ ehsi /∂wis )(∂wis /∂σ) and (∂ ehsi /∂wiu )(∂wiu /∂σ) measure how changes of returns from eachtype of human capital change the compensated supply of skilled human capital. Also note that13

γ × fs0 (hsi ) = fs00 (hsi )hsi + fs0 (hsi ) and that fs00 (hsi )hsi + fs0 (hsi ) is related with a change of v̇. In
addition, note that µi measures how the social welfare increases when the incentive compatibility
is relaxed. This implies that the term after the integration measures how a compensated changeof the returns from skilled and unskilled human capital changes the slope of v̇ and increases thesocial welfare. Also the calculation needed for obtaining the equation shows that the effect ofconsumption distortion on welfare is zero, because as long as σ is small, such a distortion is of
the second order.6
The intuition of the above proposition is as follows. In a situation where higher ability
individuals have comparative advantage in accumulating skilled human capital and lower ability
individuals have comparative advantage in accumulating unskilled human capital, a decrease of
the return from skilled human capital and an increase of the return from unskilled human capitalwill hurt higher ability individuals and benefit lower ability individuals. If the social planner isinterested in redistributing income from high ability individuals to low ability individuals, suchchanges of the returns from skilled and unskilled capital can indirectly redistribute income. On
the other hand, starting from zero distortion, the deadweight loss of the production distortionis of the second-order but the welfare gain of relaxing the incentive problem has the first-ordereffect. As a result, introducing the production distortion increases the social welfare.

3

Extension: A case of Perfect substituteIn the previous section, we have assumed that two types of human capital are imperfect substitutes
in order to assume differentiability of the human capital accumulation functions. As a result,people always accumulate both types of human capital. In reality, however, people sometimes
accumulate only one type of human capital and, as a result, the choice of human capital becomes
discrete. The purpose of this section is to analyze the welfare effect of direct versus indirect
redistribution when human capital accumulation is endogenous and different types of human
6

R2
R2
i
This can be easily checked from λ 1 c2i ni di = λ 1 (− ∂x)ni di in the Appendix.
∂p14

capital are perfect substitutes. 7
In this section, because the assumption that two types of human capital are perfect substitutes,
we assume the following utility function for agent i:
u(c1i , c2i ) − as hsi − au hui
where u(c1i , c2i ) is strictly increasing with each argument and strictly concave.
As in the previous section, we assume that following comparative advantage condition holds:

gs0 (i)g 0 (i)> u
gs (i)
gu (i)(12)

The economic meaning of the above equation is the same as before. When two types of skill accumulation are perfect substitutes in the disutility function, the agent always solves the following
constrained disutility minimization problem:
Z(wis , wiu , R) ≡ min as hsi + au hui
st R = wis hsi + wiu hui
where wis = gs (i) × ws and wiu = gu (i) × wu
In the above problem, for an agent with ability i, if as /au < wis /wiu he will accumulate only skilledhuman capital and if as /au > wis /wiu , he will accumulate only unskilled human capital. Notethat because of the assumption of comparative advantage (12), wis /wiu is an increasing functionof i. Let i∗ be i that satisfies (ws × gs (i))/(wu × gu (i)) = as /au . Then, agents whose ability isgreater than i∗ accumulate only skilled human capital and agents whose ability i is less than i∗
accumulate only unskilled human capital. We assume that such i∗ is located within 1 and 2.8
7

Besides the reason mentioned in the previous section, conducting a welfare analysis when individual behavior
includes a discrete choice is useful from a theoretical standpoint as well. In many important economic situations
such as the choice of location to live, the choice of technology by firms and labor market participation, decisions
made by consumers or firms include discrete choices. Until very recently, a welfare analysis that includes discrete
choices was rare. As far as the author knows, only Boadway and Cuff (2001) started to investigate this issue very
recently. They analyzed an optimal taxation problem when some individuals are bunched at the bottom. Anotherpurpose of this section is to contribute to such a literature as well.
8This assumption is not so restrictive as the following reason. For example, if i∗ is greater than 2, all agents

15

Given such i∗ , Z(wis , wiu , R) is
R) for i∗ ≤ i ≤ 2
wis
RZ(wis , wiu , R) = au ( u ) for 1 ≤ i < i∗ .
wi
Z(wis , wiu , R) = as (Let X(R) be an after-tax income schedule that the government designs. Then, each agent chooses
his best R to maximize U (p2 , X(R)) − Z(wis , wiu , R). Once R is chosen, an agent chooses his
optimal skill type and accumulates human capital to generate pre-tax income R. Let ve(i) be themaximized value given the schedule X(R):ve(i) ≡ max U (p2 , X(R)) − Z(wis , wiu , R).
R

For the analysis of the optimal schedule of X(R), we assume that the schedule of X(R) is acontinuous function. Although it is possible that the optimal schedule of X(R) is not continuous,
the tax schedules of almost of all developed countries are continuous. When X(R) is a continuous function, it is straightforward to show that ve(i) is continuous with respect to i from thetheory of the maximum (Berg 1963). In addition, there is an interesting property on ve(i) in theneighborhood of i∗ that turns out to be crucial for our result. The following lemma shows thatproperty of ve(i).
Lemma 1 When i increases, the graph of ve(i) has a counter-clockwise kink at i∗ .
Proof. Let ves (i) be the maximized utility of an agent with ability i given the tax schedule when
he can accumulate only skilled human capital. Also, let veu (i) be the maximized utility of anagent with ability i when he can accumulate only unskilled human capital. By the definition, thegraph of ve(i) is the upper envelope of ves (i) and veu (i) and i∗ is at the intersection between ves (i)and veu (i). This implies that there is a counter-clockwise kink at i∗ (See also Figure 1).

will accumulate only unskilled human capital. However, the production needs both skilled and unskilled humancapital. As a result, the return from skilled human capital will start to increase and the return from unskilledhuman capital will start to decrease. This implies that i∗ will start to decrease. This process will continue until
some agents start to accumulate skilled human capital.

16

Now consider the problem of designing a nonlinear income tax system. As in the previous
section, we define v(i) as follows:
v(i) = max U (p2 , Xj ) − Z(wis , wiu , Rj )j

By using the same technique as in the previous section, we can calculate dv(i)/di for i in (1, i∗ )and (i∗ , 2).
dv
g 0 Ri= as sfor i ∈ (i∗ , 2)digs gs (i)wsdv
g 0 Ri= au u () for i ∈ (1, i∗ )digu gu wu

(13)
(14)

Next we will check a single crossing property of the utility function U (p2 , X) − Z(R, ws , wu , R).
The marginal rate of substitution between X and R is
1 as
for i ∈ (i∗ , 2)Ux gs ws
1 au=
for i ∈ (1, i∗ )Ux gu wu

MRS(R,x) =Thus the MRS(R,X) is a decreasing function of i and a single crossing property is satisfied.
This means that the local incentive compatibility and the monotone condition of R are sufficientconditions for the global incentive compatibility (Fudenberg and Tirole, 1991). We assume thatthe monotonicity constraint is not binding.
As in the previous section, it is useful to think that the government controls v(i) and Ri andthat Xi is defined from the following relationship:v(i) = U (p2 , X) − Z(wis , wiu , Rj )Finally for analytical convenience, rewrite the first order condition of (13) and (14) :v̇ s =

gs0 s s
a hi
gs

and

v̇ u =

gu0 u ua hi .
gu

Based on the setup, the purpose of the government is to solve the following programming problem:

17

Z i∗
W (σ) = max

Z 2uv (i)ni di +1

gs0 s sa hi
gsg0v̇ u = u au hui
gu
s

st.

v̇ =

Z 2
i∗

i∗

v s (i)ni difor i∗ < i ≤ 2(IC1)

for 1 < i < i∗(IC2)v s (i∗ ) = v u (i∗ )(BD1)Ris∗ = Riu∗

(BD2)

{Ris − x(Ris , vis , wis , wiu )}ni di

Z i∗+{Riu − x(Riu , viu , wis , wiu )}ni di
1
Z 2
+σ{
ni c2i di − Y (p∗2 + σ, H s , H u )} ≥ 0

(RC)

1

where
s

Z 2

H =
i∗

hsi gs (i)ni di and H u =Z i∗

hui gu (i)ni di

1

The above programming problem deserves several comments. First, (IC1) and (IC2) are thelocal incentive compatibility constraints. Second, (BD1) comes from the assumption that the tax
schedule that the government designs is continuous and, as a result, the utility level of the agents
must be continuous. (BD2) comes from the assumption that individual i∗ chooses only one R.
Now let µsi ,µui and λ be the Lagrangian multipliers of (IC1),(IC2) and (RC). Let β1 and β2 be
the Lagrangian multipliers of (BD1) and (BD2). The first order conditions can be calculated
and we will write them in the Appendix to save the space. Then, what we need to know is theeffect of increasing σ from zero on the social welfare, which is equivalent to dW/dσ. By using
the envelope theorem, we have (See Appendix)∂i∗dW=dσ σ=0∂σ

g0
g0µsi∗ as hsi∗ s − µui∗ au hui∗ u
gs
gu


(15)

From the FOC of vis∗ and viu∗ , we have µsi∗ = µui∗ . In addition, as we show in the Appendix µsi
and µui are always non-negative. Furthermore as hsi∗ (gs0 /gs ) and au hui∗ (gu0 /gu ) are the right hand18

slope of vis and the left hand slope viu at i∗ . From Lemma 1, the slope of vis is steeper than the∗slope of viu at i∗ . Since ∂i∂σ > 0, we have dW/dσ > 0.

Proposition 2 Consider a small open economy where individuals accumulate human capitalendogenously and different types of human capital are perfect substitutes. Suppose that the social
planner designs a nonlinear income tax system to maximize the utilitarian social welfare function
without any production distortion and that there is no-bunching at the switching point i∗ . Then,
introducing a tariff on an unskilled human capital intensive good will increase the social welfare.
At this point, it would be useful to consider the economic meaning of (15). Figure 1 shows
the graph of ve(i) , ves (i) and veu (i). When the government increases the tariff σ from zero, thegraph of ves (i) will shift downward and the graph of veu (i) shifts upward. As a result, i∗ will
increase. Also, notice that from (IC1) and (IC2), the slope of ves (i) increases and the slope ofveu (i) decreases.
In the mechanism designs problem, v̇, the slope of the value function, is related with how thecompensation schedule must be sensitive with unobserved ability. When v̇ is higher, it means thatthe social planner needs to give higher utility to those with higher ability. With redistributive
social welfare function, the social planner wants to give higher utility to agents with lower ability.
Thus, when v̇ is high, the level of utility that the social planner can give to the agents with lower
ability is limited since the amount of the resource is limited. In such a situation, if the government
can make v̇ smaller exogenously, it is possible to increase the social welfare and changing σ canbe a good policy tool for changing v̇.
When σ increases, the change of v̇ is not the same for all individuals however. As Figure 1
shows, all individuals whose ability is lower than i∗ will experience a decrease of v̇ and all individuals whose ability is greater than i∗ will experience an increase of v̇ except the neighborhoodof i∗ . But, as the analysis in the Appendix shows, the effect of a change of v̇ for those agents isof the second order and can be replicated by the adjustment of the nonlinear income system. On

19the other hand, there are some individuals who experience the first order change of v̇. Individuals whose ability is in (i∗ , i∗ + ∂i∗ /∂σ) will switch from accumulating skilled human capital tounskilled human capital. Since the graph of v(i) has a counter-clockwise kink at i∗ , individualsin (i∗ , i∗ + ∂i∗ /∂σ) will experience the first order decrease of v̇. This implies that the government
needs less ability-sensitive compensation schedules for those agents. Because this change of v̇ has
the first order effect, it will increase the social welfare.
(15) can be interpreted in terms of the marginal tax schedule as well. Note that (∂Z/∂Rm )/Ux
is equal to 1−Tim where Tim is the marginal tax rate of income of those who accumulated m = s, u
type of skill and his ability is equal to i. From the FOC of Ris and Riu ,λnTis = µsi × as

u 0∂hsi gs0uuu ∂hi gu
and
λnT
=
µ
×
a
i
i
∂Ris gs∂Riu gu

Thus, Since (∂h/∂R) × R = h, we havedW∂i∗=λn(Ris∗ Tis∗ − Riu∗ Tiu∗ ).
dσ σ=0∂σ
Tis∗ and Tiu∗ are the marginal tax rates of individuals just above i∗ and just below i∗ , respectively.
When σ increases, the individual just above i∗ who initially accumulated skilled human capital willswitch from accumulating skilled human capital to unskilled human capital. Since the marginal
tax rate of those who accumulated skilled human capital is higher than the marginal tax rate forthose who accumulated unskilled human capital around i∗ , the marginal tax rate will decrease.9
Thus, Ris∗ Tis∗ − Riu∗ Tiu∗ is the earning that is affected by a change of the marginal tax rates. Since
this change of the marginal tax rate is of the first order, it can increase the social welfare.

4

Discussion

In the above two sections, we have shown that indirect redistribution through an increase of the
return from unskilled human capital and a decrease of the return from skilled human capital
9

Readers still might wonder why the marginal tax rate for those who accumulated skilled human capital ishigher than those who accumulated unskilled human capital around i∗ . The reason is around the right hand sideof i∗ , the marginal return from ability is higher at the right hand side of i∗ than at the left hand side of i∗ becausei∗ is a switching point.

20

would increase the social welfare. One natural question at this point would be why such changesof returns do not cause the adverse effect on human capital accumulation and, if they cause it,
why we can ignore it. The answer to such a question is that it will cause the adverse effect onhuman capital accumulation but a redistributive income tax system also causes such an incentive
problem. In a circumstance where each individual’s comparative advantage is not observableand human capital accumulation is endogenous, the redistributive income taxation necessarily
introduces adverse incentive effects on human capital accumulation. In such a situation, thequestion is not whether redistributive change of returns from different types of skill causes theadverse incentive effect but whether it can mitigate the existing incentive problem caused byincome taxation. Given that income taxation is subject to asymmetric information due to theunobservability of comparative advantage at individual levels, redistributive changes of returnfrom different types of skill mitigates the asymmetric information problem since the government
can affect agents with different types of comparative advantage differently. As a result, theredistributive changes of returns from different types of skill will increase the economic efficiency.
At this point we should emphasize that the assumption of comparative advantage plays acrucial role in our analysis. This implies that empirical studies that examined the returns fromhuman capital accumulation for individual with different abilities such as Dinardo and Tobias(2001) and Tobias (2003) are important. In addition, the results in the empirical studies and theresult in this paper can have important implications for public policy. For example, it might bepossible that encouraging skilled human capital accumulation through government funding does
not necessarily increase the social welfare if comparative advantage in human capital accumulation
exists and individuals who get the benefit most from the government funding are individuals withhigher ability.
In this paper, we examined the production efficiency theorem in a small open economy setting.
It is worth mentioning that in a small open economy setting, the Atkinson and Stiglitz theorem
holds since a commodity tax cannot affect the factor prices due to the factor equalization theorem

21

(Samuelson, 1949). On the other, it is possible to extend the intuition of the present paper to aclosed economy setting by using the two sector-two factor general equilibrium model (Harberger,
1962). In this case, we can prove that the Atkinson and Stiglitz theorem does not hold since
a commodity tax can affect the producer prices and factor prices in a closed economy. Morespecifically, we can prove that imposing a commodity tax on skilled human capital intensive good
will increase the social welfare.10

5Conclusion

In this paper, we have examined whether indirect redistribution such as tariffs and production
subsidies can complement income taxation in the long run where human capital accumulation is
endogenous. For that purpose, I developed two models where individuals can choose the amount ofboth skilled and unskilled human capital based on their comparative advantage. In the first model,
we assumed that skilled human capital and unskilled human capital are imperfect substitutes andthat individuals accumulate both skilled and unskilled human capital. In the second model, we
assumed that skilled human capital and unskilled human capital are perfect substitutes and that
individuals accumulate only one type of human capital. Assuming that individuals with higher
ability have comparative advantage in accumulating skilled human capital, we have shown thatindirect redistribution such imposing a tariff on an unskilled human capital intensive good canincrease the efficiency and complement an income tax system. This suggests that the validityof the production efficiency theorem depends on how the process of human capital accumulation
is modelled. The result of this paper also suggests that empirical studies such as Dinardo andTobias (2001) and Tobias (2003) that showed the returns from human capital were different
among individuals with different abilities have important implications for public policy.
10

For a formal proof, see our previous version of the paper (Naito, 2002).

22

Appendix
Derivation of equation (11)Let µi and λ be the Lagrangian multiplier of the incentive constraint and the resource constraint.
Then, the Lagrangian function is
Z 2

Z 2

dv
− fs0 (hsi )hsi (gs0 /gs ) − fu0 (hui )hui (gu0 /gu )di]+
di
1
1
Z 2
Z 2
+λ
ni {Ri − xi (vi )}di + σ{ni c2i di − y2 (p∗2 + σ, H s , H u )}W (σ) =

v(i)ni di +

µi [

1

1

By using the integration by parts, we can obtain
Z 2
Z 2
dv
0 s s 0W (σ) =
vi ni di +µi di −
µi fs (hi )hi (gs /gs )di −µi fu0 (hui )hui (gu0 /gu )di
di
1
1
1
1
Z 2
Z 2
+λ
ni {Ri − xi }di + σ
ni c2i di − σY (p∗2 + σ, H s , H u )}
1
1
Z 2
Z 2
Z 2
Z 2
=
vi ni di + µ2 v2 − µ1 v1 −u̇i vi di −µi fs0 (hsi )hsi (gs0 /gs )di −µi fu0 (hui )hui (gu0 /gu )di
1
1
1
Z 12

Z 2
+λ
ni {Ri − xi }di + λσni c2i di − Y (p∗2 + σ, H s , H u )
Z 2

1

Z 2

1

Therefore, the first-order-conditions are∂xi∂c2i ∂xi
+ λσ
=0
∂vi
∂xi ∂vi00d[fs0 (hsi )hsi (gs /gs )] ∂hui
d[fu0 (hui )hui (gu /gu )] ∂hui∂xi
∂c2i ∂xi−µi− µi+ λni − λni+ λσni
=0
dhsi
∂Ri
dhui
∂Ri
∂Ri
∂xi ∂Rini − u̇i − λniµ1 = 0 and µ2 = 0
By using the envelope theorem, we obtainZ 20dWd[fs0 (hsi )hsi (gs /gs )] dhsi dwis
dhsi dwiu
=−
µi
{
+
}di
dσ σ=0dhsi
dwis dσdwiu dσ1Z 20dhui dwiu
d[f 0 (hu )hu (g /gu )] dhui dwis−
µi u i is u
{ s
+
}di
dhi
dwi dσdwiu dσ
1
Z 2
Z 2
∂xi
∂xi ∂wis∂xi ∂wu∗su+ λ{c2i ni di − Y (p2 + σ, H , H )} + λ(−−−)ni di∂p∂wis ∂σ∂wu ∂σ11Note that −∂xi /∂p2 = (Up2 )/(Ux ). From the Roy’s identity, (Up2 )/(Ux ) = −c2i . Therefore,
R2
R2
∂xi
∂xi
∂xi
i
s
u
λ 1 c2i ni di = λ 1 (− ∂x∂p )ni di. In addition, ∂ws = zwi /Ux and ∂wu = zw /Ux and ∂Ri = ZRi /Ux .
i

23

∂xi
∂xi
s
u
Using the definition of Zwis and Zwu , ∂w
s = −αi hi /Ux , ∂w u = −αi hi /Ux and ZR /Ux = αi /Ux .
i

On the other hand, the FOC of Ri at σ = 0 is that
0

−µi

0

d[fs0 (hsi )hsi (gs /gs )] ∂hsi
d[fu0 (hui )hui (gu /gu )] ∂hui−µ+ λni = λni αi /Ux
i
dhsi
∂Ri
dhui
∂Ri

Thus, dW/dσ becomesZ 2
0
dWd[f 0 (hs )hs (g /gs )] dhsi dwis
dhsi dwiu=−µi s i is s
{ s
+
}di
dσ σ=0dhi
dwi dσdwiu dσ1
Z 20
d[fu0 (hui )hui (gu /gu )] dhui dwis
dhui dwiu
−
µi
{
+
}di
dhsi
dwis dσdwiu dσ1
Z 200sd[fs0 (hsi )hsi (gs /gs )] ∂hsi
d[fu0 (hui )hui (gu /gu )] ∂huis ∂wi
− λy2 +
[−µi
−
µ
+
λn
]h
di
i
i
i
dhsi
∂Ri
dhui
∂Ri
∂σ
1
Z 2
0
0
u
d[fs0 (hsi )hsi (gs /gs )] ∂hsi
d[fu0 (hui )hui (gu /gu )] ∂hui
u ∂w
+
[−µi
−
µ
+
λn
]
h
di
i
i
i
i
dhsi
∂Ri
dhui
∂Ri
∂σ
1
Z 2
Z 2
s
∂wu
s ∂wi
− λy2 +
λni hidi +λni hui
di
∂σ
∂σ
1
1
R2
R2
R2
R2
R2
s
u
∂ws
s ∂w
u ∂wu
s ∂ws
Note that 1 λni hsi ∂σi di+ 1 λni hui ∂w∂σ di = 1 λni hi i ∂σ di+ 1 λni hi ∂σ di. 1 λni hi i ∂σ di+u ∂wu
1 λni hi ∂σ d is a change of total earning due to a tariff when levels of human capital of all indi-

R2

viduals are fixed. On the other hand, from perfect competition, for given level of human capitalof all individuals, the total revenue of the firm should be equal to the total payment to facR2
R2
tor owners. Thus, y1 + (p∗2 + σ)y2 = ws 1 λni hsi idi + wu 1 λni hui di always holds. Let Q(σ)
be the total revenue of firms when all human capital level of all individuals are fixed. Then,
R2
R2
s
u ∂wu
dQ/dσ = 1 λni hsi i ∂w
∂σ di + 1 λni hi ∂σ di. By definition of Q(σ)Q(σ) = max y1 + (p∗2 + σ)y2

s.t. (y1 , y2 ) ∈ Γ(H s , H u ) = 0

H s and H u are fixed.
R2
R2
s
s ∂wi
u ∂wu
From the envelope theorem, dQ
dσ = y2 . Therefore , −λy2 + 1 λni hi ∂σ di + 1 λni hi ∂σ di = 0.
Note that from the definition of e
hsi and e
hui , we havefs0 (e
hsi )

eu
es
es
∂e
hsi
0 e u ∂ hi
0 e s ∂ hi
0 e u ∂ hi
+
f
(
h
)
=
0
and
f
(
h
)
+
f
(
h
)
=0
u i
u i
u i
∂wis
∂wis
∂wiu
∂wiu
24

By using the Slutsky equation for hsi and hui and the above equation, we haveZ 2  00 s s
dW
fs (hi )hi
fu00 (hui )hui∂ehs ∂wis00=−µi [ 0 s + 1](gs /gs ) − [ 0 u + 1](gu /gu ) fs0 (hsi ) isdidσ σ=0fs (hi )fu (hi )∂wi ∂σ1Z 2  00 s s
fs (hi )hi
fu00 (hui )hui∂ehs ∂wiu
00µi [ 0 s + 1](gs /gs ) − [ 0 u + 1](gu /gu ) fs0 (hsi ) iu
di−fs (hi )fu (hi )∂wi ∂σ1Z 2  00 s s
fs (hi )hi
fu00 (hui )hui∂ehs ∂wis∂ehsi ∂wiu00=−µi [ 0 s + 1](gs /gs ) − [ 0 u + 1](gu /gu ) fs0 (hsi )[ is+]difs (hi )fu (hi )∂wi ∂σ∂wiu ∂σ1
From the condition of the comparative advantage, the inside of the large bracket is positive.
∂ehs ∂ws∂e
hs ∂wuAlso, both ∂wis ∂σi and ∂wiu ∂σi are positive. Thus, we have dW
dσ σ=0 > 0 .
iiProof of µi ≥ 0∂c2i ∂xi
i
From the FOC of vi , we will have ni − u̇i − λni ∂x∂vi + λσ ∂xi ∂vi = 0. Thus, we haveni − λni

∂xi
= µ̇i
∂vi

i
at σ = 0. By integrating both sides and using the definition of ∂x∂vi and µ1 = 0, we will haveZ i
ni {1 −1λ} = µiUx

From the first order condition of the revelation problem, Ux (p2 , X)X 0 (i) = ZR R0 (i). This meansthat the sign of X 0 (i) and R0 (i) are the same. Since v(i) is strictly increasing, X 0 (i) and R0 (i)must be increasing. When X 0 (i) is increasing, Uλx is increasing. This implies that if at some i∗∗ ,1 − λ/Ux = 0, then for any i > i∗∗ , 1 − λ/Ux < 0. However, µ2 = 0 from the FOC of v2 . Thisimplies that µ1 is initially strictly positive until i∗∗ and then it begins to decrease and reaches
to zero at i = 2. Therefore, µi ≥ 0 for all 1 ≤ i ≤ 2.

Proof of Proposition 2

25

The Lagrangian is:
Z i∗
L=

Z 2

u

v (i)ni di +
1

i∗

Z i∗

s

v (i)ni di +µui {v˙u − au hui (gu0 /gu )}di +

1

Z 2
i∗

µsi {v˙s − as hsi (gs0 /gs )}di+β1 {vis∗ − viu∗ } + β2 {Ris∗ − Riu∗ }Z i∗
Z 2{Ris − x(Ris , vis , wis , wiu )}ni di+λ{Riu − x(Riu , viu , wis , wiu )}ni di + λ
∗
1
i
Z 2
+λσ{
ni c2i di − Y (p∗2 + σ, H s , H u )}1

By using the integration by parts, we obtain
Z i∗

Z 2

Z i∗

Z i∗

v (i)ni di +µui au hui (gu0 /gu )di
i∗
1
1
Z 2
Z 2
µs2 v2s − µsi∗ vis∗ −µ˙si vis di −µsi as hsi (gs0 /gs )di + β1 {vis∗ − viu∗ } + β2 {Ris∗ − Riu∗ }
∗
∗
i
i
Z i∗
Z 2
u
u u
s
u
+λ
{Ri − x(Ri , vi , q, wi , wi )}ni di + λ{Ris − x(Ris , vis , wis , wiu )}ni di
∗
1
i
Z 2
+λσ{
ni c2i di − Y (p∗2 + σ, H s , H u )}

L=

u

s

v (i)ni di + µui∗ viu∗ − µu1 v1u −1µ˙ui viu di −1

The first order condition for vis , v2s , vis∗ , Ris , Ris∗ , viu , viu∗ , v1u , Riu and Riu∗ are∂x∂c2i ∂xi
vis : ni − µ˙si − λni s + λσ
=0∂vi
∂xi ∂visv2s : µs2 = 0vis∗ : −µsi∗ + β1 = 0
Ris : −µsi × as

∂hsi gs0∂x∂c2i ∂xi+ λni − λni+ λσni=0ss∂Ri gs∂Ri∂xi ∂Ri
Ris∗ : β2 = 0∂x∂c2i ∂xi
viu : ni − µ˙si − λni s + λσ
=0∂vi
∂xi ∂vis
viu∗ : µui∗ − β1 = 0
v1u : µu1 = 0
Riu : −µui × au

∂hui gu0∂x∂c2i ∂xi+ λni − λni+ λσni=0uu∂Ri gu∂Ri∂xi ∂Riu
Riu∗ : β2 = 0
26

Now we characterize those first order conditions. First, note that at σ = 0,µsi = µsi∗ +Z i∂xnj (1 − λ)dj for i ∈ (i∗ , 2) and µui∗ = µu1 +
∂v
∗
j
i

Thus, since µu1 = 0, µsi =Ri∂x
1 nj (1 − λ ∂vj )djZ i∗
nj (1 − λ
1

∂x
)dj
∂vj

(16)

∂x
for i ∈ (i∗ , 2). Note that ∂v= 1/(Ux ) . A single
jcrossing property and the monotonicity of Ris and Riu guarantee that xi is increasing. This implies∂xthat ∂v
is increasing and the inside of the integral is a decreasing function of i. Since µs2 = 0,jµui and µsi are non-negative.
Now we examine dW/dσ and evaluate at σ = 0. From the envelope theorem,
dW
∂i∗
=
dσ
∂σ

0



g
v u (i∗ )ni∗ − v s (i∗ )ni∗ + µ˙u i∗ v u (i∗ ) + µui∗ v˙iu∗ − µ˙u i∗ v u (i∗ ) − µui∗ au hui ugug0−µs i∗ v(i∗ ) − µsi∗ v̇is∗ + µ̇si∗ v(i∗ ) + µsi∗ as hsi s + β1 {v˙s i∗ − v̇iu∗ } + β5 R˙u i∗ − β5 Ṙs i∗
gs−λ{Ris∗ − x(Ris∗ , vis∗ , q, wis∗ , wiu∗ )}ni∗ + λ{Riu∗ − x(Riu∗ , viu∗ , q, wis∗ , wiu∗ )}ni∗
( Z
)
( Z
)
Z 2
Z i∗
s
u
2
2
∂ws
∂x
∂wu
∂x
s s ∂hi 0
u u ∂hi 0+−µi ag di − λ−µi ag di − λg (i)ni dis ni gs (i)di + ∂σu u∂σ∂wis s∂wiu ui∗i∗ ∂wi
i∗
1 ∂w
Z 2
Z 2
Z ∗
∂x ∂q
∂x ∂q i∗su
+λni c2i di − λY (p2 + σ, H , H ) +−ni di −ni di∂q ∂σ∂q ∂σ 11i∗∂qFrom the Roy’s identity, c2i = − ∂x∂q ∂σ . Thus,

( Z
)

Z 2
s
2
0
0
s
g
g
∂w
∂x
∂h
µsi∗ as hsi s − µui∗ au hui u +−µsi as is gs0 di − λn g (i)di
s i s
gs
gu
∂σ
∂wi
i∗
i∗ ∂w
( Z
)
Z i∗
u
2
∂wu
∂h
∂x
+
−
µui au iu gu0 di − λn g di − λY (p∗2 + σ, H s , H u )
u i u
∂σ
∂w
∂w
∗
i
1
i

∂i∗
dW
=
dσ
∂σ



Now we need to calculate the inside of the integral. Note that from the definition of hsi and hui ,we have
s
u
∂hui
∂hsi
s ∂hi
u ∂hi
=
−h
and
=
−h
i
i
∂wis
∂Ris
∂wiu
∂Riu

This implies that−µsi assuu∂hsi 0s s s ∂hi 0
u u ∂hi 0u u u ∂hi 0
g
=
µ
a
h
g
and
−
µ
a
g
=
µ
a
h
g
i
i
i
i
i
∂wis s∂Ris s∂wiu u∂Riu u27

By using the FOC of Ris and Riu ,∂hsi 0∂xg = hsi gs {λni − λni}∂Ris s∂Ris∂x∂hui 0gu = hui gu {λni − λni}µui au hui
u
∂Ri
∂Riu
µsi as hsi

Thus, dW
dσ is


0
0
∂i∗
dW
s s s gsu u u gu=µi∗ a hi − µi∗ a hi
dσ
∂σ
gs
gu
Z 2

Z 2
s
∂w
∂x
∂x
s
+
hi gs {λni − λni}di − λs gs ni di
∂σ
∂Ris
i∗
i∗ ∂wi
(Z
)
Z i∗
2
∂wu
∂x
∂x
u
+
hi gu {λni − λni}di − λu gu ni di
∂σ
∂Riu
i∗
1 ∂wi
− λY (p∗2 + σ, H s , H u )sNext, we need to calculate λ ∂w
∂σ

∂wu
s
i∗ hi gs (i)ni + λ ∂σ

R2

R i∗
1

hui gu (i)ni di. From the argument of theprevious section, we have∂ws
λy2 = λ
∂σ

Z 2
i∗

∂w
hsi gs (i)ni + λu Z i∗∂σhui gu (i)ni di.

1

∂x
∂x
∂x
u ∂x
Third, we will show that hsi ∂R
s = − ∂w s and hi ∂Ru = − ∂w u . From the definition of Z, we
i

i

i

i

have
∂Z
∂Z
= as /wis and= −as hsi (1/wis ) for i ∈ (i∗ , 2)
s
∂Ri
∂wis
∂Z
∂Z
= au /wiu and= −au hui (1/wis ) for i ∈ (1, i∗ )
∂Riu
∂wiu
∂x
∂x
∂x
∂x
∂x
u ∂x
s ∂x
Thus, by using the definition of ∂Rs , ∂w s , ∂Rs , ∂w s , we can check that hi ∂Rs = − ∂w s and hi ∂Ru =
i

i

∂i∗
dW
=
dσ σ=0
∂σ



i

i

i

∂x
− ∂w
u.
i

Therefore, dW/dσ is
g0g0µsi∗ as hsi∗ s − µui∗ au hui∗ u
gs
gu



0

0

From the FOC of vis∗ and viu∗ , we have µsi∗ = µui∗ . In addition, as hsi ggss and au hui gguu are the rightside slope of vis and the left side slope viu at i∗ From Lemma 1, the slope of vis is steeper than the∗dWslope of viu at i∗ . Since ∂i∂σ > 0, dσ > 0.

28

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29

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30

Figure 1: The Graph of v(i)

V
Vs

Vu

I*

i