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1
Economics and Politics, Wiley 31(3), 465-484.
Anti-immigration Policy in the Destination Country and Skilled-Unskilled Wage Inequality
in a Source Economy with or without Unemployment♠
♠♠
♠
Sarbajit Chaudhuri
Dept. of Economics, University of Calcutta
56A, B.T. Road, Kolkata 700 050, India.
Address for communication: Sarbajit Chaudhuri, 23 Dr. P.N. Guha Road, Belgharia, Kolkata
700 083, India. Tel: (91)98305-30963 (M), 91-33-2557-5082 (C.U.); E-mail:
sceco@caluniv.ac.in ; sarbajitch@yahoo.com
(April 20, 2019)
Abstract: This paper is devoted to theoretically examine the consequence of the anti-
immigration policy adopted in the destination country on the skilled-unskilled wage inequality in
a source nation using a couple of two-sector, specific factor general equilibrium models in both
the presence and absence of unemployment. Emigration requires incurring some capital cost for
professional skill formation on the part of the every prospective emigrant that adds to the
opportunity cost of emigration. The authority of the destination country determines the number
of visas to be granted and hence directly controls the magnitude of skilled emigration from the
source country. In the migration equilibrium, the expected skilled wage income abroad is equal
to the opportunity cost of emigration. In both the presence and absence of unemployment of
unskilled labour, the outcome of the policy on the wage inequality crucially hinges on both the
magnitude of the fixed cost of emigration and the technological factors. In the specific-factor
Harris-Todaro model, the degree of imperfection in the unskilled labour market is an additional
factor. Finally, some policy recommendations have been made for protecting the interest of the
poor unskilled workforce.
Keywords: Anti-immigration policy; Source economy; Destination country; Skilled labour;
Unskilled labour; Wage inequality; General equilibrium model.
JEL classification: D58, I24, I28, J31, J61.
♠
♠♠
♠
The author is thankful to two anonymous referees of this journal for their interesting and constructive
comments on the original paper that have helped him in improving its quality. Thanks are also due to Puja
Putatunda, Rohan Kanti Khan, Sushobhan Mahata, Soumyajit Mondal and Rupak Bardhan, all graduate
students at the Department of Economics, University of Calcutta, for their comments that have helped the
author in rewriting the first part of this revised paper. The author is, however, solely responsible for
remaining errors, if any.
2
Anti-immigration Policy in the Destination Country and Skilled-Unskilled Wage Inequality
in a Source Economy with or without Unemployment
1. Introduction
Two recent economic policies that US has recently undertaken are worth mentioning. First, the
country has imposed high tariffs on the imports of certain commodities that are exported by
China and some other developing countries. Second, the US president, Mr. Donald Trump has
announced to resorting to stricter immigration policies by lowering the country-specific quotas of
H-1B and H-4 Visas issued by the country with a view to reduce the number of high-skilled
immigrants citing the reason to preserve more white-collar jobs for the native skilled workers.
1
These two could be viewed as anti-globalization policies that restrict freer international trade and
mobility of labour across countries. The justification for resorting to such policies could be
provided in terms of the political economy arguments. The understanding is that the anti-
globalization policies would
make US to be the most powerful nation in the world that can dictate
terms to other countries in framing their own policies, both domestic and international that would
ultimately benefit US. This might be a strategic move on the part of the US president, Mr. Trump for
arousing a cheap nationalist sentiment among the electorates in a bid to win the reelection and
remain in power for the second consecutive term.
What the consequence of the anti-immigration policy could be on the skill-based wage inequality
in a source developing economy is a very pertinent question. Conventional wisdom suggests that
1
In this context it is worth mentioning a couple of striking facts that have emerged from a report by
Lopez and Radford (2015) on the US immigrants. According to their report (see Table 16), 13.4 per cent
of US population in 2015 were foreign born. Besides, while the percentage of all US born people having
advanced educational degrees was only 11.4, the figure for all foreign-born people was 12.6 percent. The
figures for people having graduate degrees were 19.5 percent and 17.1 per cent, respectively. The
corresponding educational figures for immigrants from South and East Asian countries were 22.5 and
28.7 per cent, respectively. From these statistics, it is clear that the skilled workers without US origin,
especially with South and East Asian including Indian origin constitute a major proportion of the skilled
workforce in US and have contributed significantly to economic growth in the country.
3
a reduction in the magnitude of ‘brain drain’ of skilled labour from a developing country should
improve the skilled-unskilled wage inequality in the source country. Increased availability of
skilled labour should drag the skilled wage down thereby improving the wage inequality.
Nevertheless, the outcome may not necessarily be as much straightforward as it appears. Because
the anti-immigration policy is a very recent issue, its impact on the developing countries cannot
be readily empirically studied. However, attempts could be made to theoretically predict the
possible outcomes under alternative scenarios on the basis of which the government can
undertake measures so that the possible detrimental effect on the problem of inequality could be
minimized. The objective of this paper is to find out the apparently missing link between the
anti-immigration policy undertaken by the US and the phenomenon of increasing economic
inequality despite global prosperity.
2
2. A Survey of the Existing Literature, Motivation and Results in Brief
A large part of the literature deals with the welfare impact of ‘brain drain’ both on the destination
and source nations. Some notable works are Bhagwati and Rodriguez (1975), Rivera-Batiz
(1982), Djajic (1986, 98), Quibria (1988, 1997), Michael (2003, 2011) and Djajić and Michael
(2013) etc.
The second strand of literature discusses the issues like optimal immigration, human capital and
immigration decision etc. Djaji¢ et al. (2012) where they have built up a game theoretic model of
interaction between the host and source countries to find out the optimum level of possible
migration of skilled worker that the source country government could attain through a choice of
the number of workers that would be allowed to take skill upgrading training. Besides, Stark et
al. (1998), Chau and Stark (1999), and Fan and Stark (2007) have studied skill formation and
international migration by paying attention to skill acquisition incentives created by the prospect
2
As per findings of Pimentel et al. (2018), Quereshi (2018), World Inequality Report (2018) etc. the
ongoing process of globalization although increased the rates of economic growth across countries, the
income and wealth inequality has sharply increased. The benefits of higher rates of economic growth have
been pocketed by only a very few super rich people while the poorer section of the population has been
largely left out.
4
of international migration. Djajić and Michael (2009) discuss the case of temporary migration in
terms of a game-theoretic model between host and source countries for deriving an optimal
duration of a guest-work permit. They have found that from the destination country perspective,
the duration of temporary migration should be as long as possible in the Nash equilibrium.
These two lines of work, however, do not discuss the consequence of international migration of
high-skill labour on the skill-based wage inequality in the developing countries. Nevertheless,
there exists a separate vast theoretical literature on relative wage inequality that discusses various
aspects of the problem.
3
However, there are only a few papers in this literature that exclusively
connect international mobility of labour to relative wage inequality in the developing nations.
For example, Marjit and Kar (2005) using a two-sector, specific factor full-employment model
and Chaudhuri (2004) in terms of
2 3
×
Harris-Todaro (HT, hereafter) type model with urban
unemployment of unskilled labour emphasize the importance of the distributive shares of the
intersectorally mobile factor in influencing the direction of change in the wage inequality.
Yabuuchi and Chaudhuri (2007) have also reached the same broad conclusion. On the other
hand, Beladi et al. (2008) have shown how the difference in the intersectoral factor intensities
between skilled labor and capital instead of the distributive shares of the mobile factor(s) could
determine the change in wage inequality in terms of a two-sector HT type model with three
inputs, two types of labour and capital, and perfect mobility of all the inputs across sectors.
Finally, Chaudhuri (2008) using a three-sector specific factor HT type model with
unemployment of unskilled labour has found that distributive factor shares count only when the
urban wage and the rural unskilled wages are positively correlated. However, if they are not
3
A few early notable works are those of Acemoglu (1998), Feenstra (2004), Zhou and Trefler (2005) and
Marjit et al. (2004) etc. that study the roles of trade liberalization and technological change on the wage
inequality. The most recent literature includes works like Pan and Zhou (2013), Pi and Zhou (2012,
2014), Anwar and Sun (2015), Pi and Chen (2016), Chaudhuri et al. (2018), Pi and Zhang (2017) etc. that
have paid attention to the role of governmental behaviors e.g., public inputs, pollution control,
institutional arrangements including capital market distortion, and economic policies like education
subsidy policy in affecting the relative wage inequality.
5
interconnected, the outcomes of international migration on wage inequality may not depend on
distributive shares of the mobile factor.
The common feature of these works is that the magnitude of skilled emigration has been
assumed to be exogenous to both the host and the source countries. Although the magnitude of
immigration allowed from a particular source country is a policy variable of the host country, the
underlying assumption is that all high-skill workers are equally desirous to emigrate. This is true
only if emigration is costless.
4 , 5
In the present paper, by introducing a fixed cost of emigration
that is financed through taking a loan from the capital market at the competitive interest rate, we
endogenously determine the number of high-skill people who want to emigrate. The opportunity
cost of emigration is the sum of the local high-skill wage and the interest cost on capital. Of
these desirous people only a fraction (may be all) actually succeeds to emigrate after obtaining
visas which are distributed among them by the host country embassy through a lottery. Note that
the number of visas to be granted is exogenously determined by the authority of the host country.
Hence, in our model, how many people will be unable to emigrate despite their willingness is
endogenously determined through equating the expected foreign skill income with the
opportunity cost of emigration.
In above setting, the consequence of an exogenous fall in the emigration quota on the skill-based
wage inequality has been examined both in the presence and absence of unemployment of
unskilled labour. The result that we obtain is that the consequence on the wage inequality
depends on both the fixed cost of emigration and the technological factors. This result does not
change despite the presence of unemployment of unskilled labour in the HT case. However, in
this case the degree of imperfection prevailing in the unskilled labour market is an additional
factor that also influences the wage inequality in this model. Finally, a couple of development
policies have been suggested that can lessen the gravity of the possible unfavourable
consequence of the anti-immigration policy on the skill-based wage inequality.
4
In positive economics, workers desire to emigrate only because of earning differentials between the two
countries.
5
See relevant portion of discussion on cost of emigration in Section 3.
6
3. The Model
Let us consider a two-sector, specific factor full-employment model for a small open developing
economy with three inputs, unskilled labour, skilled labour and capital. Here ‘capital’ means
‘working capital’. It is extremely crucial to point out that like many other trade models, no
distinction has here been between ‘working capital’ and ‘physical capital’. Both of them are
called ‘capital’. In all the papers where capital has been assumed to be perfectly mobile between
manufacturing and agricultural sectors have made this assumption either explicitly or implicitly.
It is easily understandable that the nature of physical capital used in those two sectors cannot be
the same. Hence, by the input ‘capital’ one means ‘working capital’
.
6
The underlying idea is that
there exists a technology, which can instantaneously convert ‘working capital’ into ‘physical
capital’ fully costlessly.
The endowments of the two types of labour are exogenously given at
L
and
S
. The fixed capital
endowment of the economy is
K
. The returns to the three factors of production are
,
S
W W
and
r
,
respectively.
In this model, the intersectorally mobile factor of production is capital while unskilled labour and
skilled labour are specific to sector 1 (low-skill sector) and sector 2 (high-skill sector),
respectively. The other standard assumptions of a specific factor competitive general equilibrium
model like constant-returns-to-scale technologies of production with positive but diminishing
marginal productivity to each factor etc. hold. Sector 1 and sector 2 are the exports sector and the
import-competing sector of the economy, respectively. Owing to the small open economy
assumption, the commodity prices,
i
P
s are given internationally. Although we call ours a
developing economy, we in the first model do not consider any characteristics of the developing
countries e.g. factor market distortion, rural-urban migration and urban unemployment etc.
6
Many of the works, for example, Gupta (1997a, b), Gupta (1993), Chaudhuri and Gupta (2014), Khan
(1980, 1982), Chandra and Khan (1993), Grinols (1991), Chaudhuri (2016, 2014, 2005) and Chaudhuri et
al. (2018) etc. although deal with dissimilar problems have made this crucial assumption.
7
However, we will subsequently include these features to examine the robustness of our results
obtained in the full-employment case with no market distortion.
The usual price-unit cost equality conditions are as follows.
1 1 1
L K
Wa ra P
+ =
` (1)
2 2 2
S S K
W a ra P
+ =
(2)
Here
ji
a
is the amount of the
j
input required to produce one unit of output of the
i
th commodity
with
, ,
j L K S
=
and
1, 2
i
=
.
Let
i
X
denote the level of output of the
i
th sector. Hence, the full-employment condition for
unskilled labour is given by the following.
1 1L
a X L
=
(3)
Before proceeding further it should be clearly spelt out that we in this model fully concentrate on
the emigration issue of skilled labour only and hence completely ignore the aspect of emigration
of unskilled labour.
The skilled wage in the destination country,
S
W
is significantly higher than that in the source
country,
S
W
.The decision to emigrate to developed countries arises because of this wage
differential. If an emigrant reaches the destination country, he will find a high-skilled job with a
decent salary. Let
V
S
stand for the number of high-skill people aspiring to go abroad. However,
after applying for a passport and a visa, there is no guarantee that he will get the visa and
ultimately be able to emigrate. How many high-skill people would be allowed to emigrate,
*
S
is
a policy variable of the destination country. We assume that the skilled labour force is
homogenous. Because of the inter-country wage differential, all of them will be interested to
emigrate provided emigration is costless. It is also assumed that they apply for their respective
visas at the specified time. The selection then takes place through an unbiased lottery.
8
Before proceeding further it is extremely crucial to discuss about a vocational training cost that
each desirous emigrant has to incur from his own pocket to enable himself a prospective
candidate for emigration. Because this cost plays a pivotal role in our model, it would be of
much help if it could be validated by facts and figures. In this connection, some recent empirical
findings are worth mentioning. Jain et al. (2019), Azam et al. (2013) and Bleakley and Chin
(2004, 2010) have pointed out that acquiring spoken English fluency increases the wage
premium by approximately 33 per cent and that English fluency is increasingly viewed as a key
skill for success and labour mobility across countries. Jain et al. (2019) have reported that this
vocational training involves substantial cost and that a full subsidy towards the cost of training
among a randomly selected college students in Hyderabad, India has raised the participation rate
to the training program by 23 per cent. In other words, the own price elasticity of demand for the
training course in spoken English is – 2.3 (Jain et al, 2019).
It may be mentioned that in order to work in an IT company located in India, good
communicative power in the national language, Hindi might be acceptable although for working
in an MNC in US, a good command over English is an absolute necessity. Hence, keeping all the
above discussion in mind, it is not unreasonable to assume that learning the international
language through private language coaching centers involves a substantial cost. In the absence of
any public assistance for a vocational training course like this, a desirous emigrant has to bear
the cost out of his own pocket. Furthermore, the process for obtaining a passport and a visa and
travel to the destination country involve some more cost that a prospective emigrant has to incur.
Let the cost that a desirous emigrant has to incur be
b
. In the absence of any own fund, he has to
take a loan from the capital market at the prevailing interest rate,
r
to meet this cost that he has to
repay with interest before leaving the country. Alternatively, this could be viewed as a part of
opportunity cost of emigration.
7
It is to be noted that in our static model skill acquisition takes
7
Note that the total opportunity cost of a desirous emigrant is the sum of domestic skilled wage and this
capital cost i.e.
( )
S
W rb
+
.
9
place instantaneously.
8
Note that because of this training program the skilled workforce does no
longer remain homogeneous. The high-skill workers who have undergone such a training
program,
V
S
have better communicative skill in the international language relative to the
remaining people,
( )
V
S S
−
although this additional skill is not required for working in the
domestic high-skill sector (sector 2). Because the endowment of high-skill labour and the
number of visas to be issued,
*
S
are known and because the procedure of granting visas is an
unbiased lottery, the probability of getting a visa and going abroad is given by
*
( ).
V
S
p
S
=
Hence,
the probability of acquiring a visa and going abroad,
p
is trivially positively correlated to the
number of visas to be issued,
*
S
by the authority of the destination country at a given
V
S
.
The interest cost has to be added to the domestic skilled wage,
S
W
to obtain the opportunity cost
of emigration. On the other hand, the expected skilled wage of a prospective emigrant in the
destination country is
*
( )
S
V
S
W
S
. In the international migration
equilibrium
, we must have the
following.
*
( )
S S
V
S
W rb W
S
+ =
(4)
In other words, in the international migration equilibrium, the expected income of the marginal
emigrant in the foreign country must be equal to his opportunity cost of emigration. Because
S
W
and
r
are endogenously determined, the number of skilled workers desiring to emigrate,
V
S
also
comes out from the general equilibrium of the system.
Although
V
S
number of high-skilled workers are willing to go abroad, ultimately, only
*
S
of
them succeed. The remaining
( *)
S S
−
workers are fully employed in the source country at the
8
Ideally, the aspect of skill formation should be dealt in terms of a dynamic model. For example,
Chaudhuri et al. (2018) have used a two-period general equilibrium model to examine the impact of an
education subsidy on the skilled-unskilled wage inequality both in the presence and absence of
unemployment and labour market imperfection.
10
domestic skilled wage rate,
S
W
. The full-employment condition for skilled labour in the source
economy is then given by
2 2
( *)
S
a X S S
= −
(5)
The full-employment condition of capital is the following.
1 1 2 2K K V
a X a X bS K
+ + =
(6)
Note that
V
bS
is the amount of capital (working capital) that the aspiring emigrants borrow from
the capital market for enabling themselves ready for leaving the source country.
9
Hence,
( )
V
K bS
−
is the effective amount of capital, which is used for domestic production of the two
commodities.
Using equations (3) and (5), equation (6) can be rewritten as follows.
1 2
1 2
( ) ( )( *)
K K
V
L S
a a
L S S bS K
a a
+ − + =
(6.1)
As we can see, this system consists of six independent equations, (1) – (5) and (6.1) to solve for
the six endogenous variables, namely
1 2
, , , ,
S
W W r X X
and
V
S
. This system does not satisfy the
decomposition property. The equilibrium values of
, ,
S
W W r
and
V
S
are simultaneously solved
from equations (1), (2), (4) and (6.1). Once factor prices are known, the factor-coefficients,
ji
a
s
are also known because these are functions of factor prices. Then,
1
X
and
2
X
are obtained from
equations, (3) and (5), respectively. Note that the equilibrium values of all the endogenous
variables come out as functions of commodity prices, factor endowments and the policy
parameter
*
S
.
9
Each desirous candidate has to go through the vocational training course for making him ready for
emigration.
11
4. Comparative statics
A stricter immigration policy in this model means a lower number of visas,
*
S
granted by the
destination country with a view to curb immigration of high-skill labour.
10
This lowers the
probability of obtaining a visa at the initial number of people applying for visas,
V
S
and hence
the expected income of the marginal emigrant in the destination country.
11
The skilled wage,
S
W
is greater than the unskilled wage,
W
. The absolute wage gap is then
( )
S
W W
−
. The skilled−unskilled wage gap improves (worsens) in absolute terms if
( )
S
W W
−
falls (rises). On the other hand, the wage inequality improves (deteriorates) in relative terms if
.0)()
ˆˆ
(><−
AS
WW
Here
'^ '
means proportional change e.g.
ˆ
( )
dZ
Z
Z
=
.
For examining the impact of a stricter anti-immigration policy on the relative wage inequality
after totally differentiating equations (1), (2), (4) and (6.1) and after simplification the following
expression can be obtained.
12
2
2 1
2
ˆ ˆ
*
( ) ( )[ ( )( )]( )
ˆ*
( )
[ ]
S V K
K K
S V
W W S a S
b
K a S
S
θ θ
−= − −
∆
−
(7)
10
It has already been mentioned that in this paper, we solely concentrate on emigration of skilled labour.
However, the aspect of international migration of unskilled labour can be carried out separately. For
example, Beladi et al. (2008) deals with international migration of both skilled and unskilled labour
although the aspect of migration is completely exogenous.
11
The number of high-skill workers applying for visas,
V
S
will also undergo a change. However, the
probability of receiving a visa,
*
( )
V
S
S
definitely falls.
12
This expression has been derived in Appendix 2.
12
In (7),
ji
θ
denotes the distributive or allocative share of the
j
th input in the
i
th sector for
, ,
j L S K
=
and
1, 2
i
=
e.g.
1 2
1 2
1 2
( ); ( )
K K
K K
ra ra
P P
θ θ
= =
etc. Note that
1 1 2 2
[( ) 1 ( )]
L K S K
θ θ θ θ
+ = = +
.
Recall that
ji
a
is the amount of the
j
input required to produce one unit of output of the
i
th
commodity. ‘^’ denotes proportional change. Finally,
∆
is a negative term that has been defined
in Appendix 1. In Appendix 1 it is also shown that
0
∆ <
under the sufficient condition that
2
2
( )
K
S
a
b
a
≥
. The other notations have already been defined.
From (7) the following results readily follow.
ˆ ˆ
( ) 0
ˆ
*
S
W W
S
−
<
if, (i)
2 1
K K
θ θ
>
; and, (ii)
2
2
( )
K
S
a
b
a
≥
or if, (i)
2 1
K K
θ θ
<
; and, (ii)
b
is small enough. (8)
ˆ ˆ
( ) 0
ˆ
*
S
W W
S
−
>
if, (i)
2 1
K K
θ θ
<
; and, (ii)
2
2
( )
K
S
a
b
a
≥
or if, (i)
2 1
K K
θ θ
>
; and, (ii)
b
is small enough.
The results, as presented in (8), can be summarized in the form of the following proposition.
Proposition 1: Anti-immigration policy in the host country accentuates the relative wage
inequality in the source country either if (i)
2
2
( )
K
S
a
b
a
≥
and, (ii)
2 1
K K
θ θ
>
.; or if (i)
b
is small
enough; and, (ii)
2 1
K K
θ θ
<
. On the contrary, the policy improves the wage inequality either if (i)
2
2
( )
K
S
a
b
a
≥
and, (ii)
2 1
K K
θ θ
<
.; or if (i)
b
is small enough; and, (ii)
2 1
K K
θ θ
>
.
13
We explain proposition 1 as follows. If
*
S
decreases by 1 unit, the skilled endowment of the
economy,
( *)
S S
−
rises by 1 unit. Consequently,
2
X
increases by
2
1
( )
S
a
units. This implies that
the demand for capital rises by
2
2
( )
K
S
a
a
units. On the other hand, a decrease in
*
S
lowers the
probability of obtaining a visa by an aspiring emigrant which in turn lowers his expected income
in the foreign country. This eventually lowers the number of prospective emigrants,
V
S
.
Consequently, the aspiring emigrants require borrowing a lower amount of capital that in turn
raises the supply of productive capital,
( )
V
K bS
−
. Therefore, there are two opposite forces
working on the determination of direction of change in the return to capital,
r
. Our analysis
finds that the return to capital,
r
falls under the sufficient condition that
2
2
( )
K
S
a
b
a
≥
.
13
Consequently, the two wages,
S
W
and
W
increase under this sufficient condition because the
capital costs decrease in both the sectors. Now, if
2 1
K K
θ θ
>
, the saving on capital cost will be
higher in sector 2 than that in sector 1. This means that the rate of increase of the skilled wage
must be greater than that of the unskilled wage. Consequently, the skill-unskilled wage
inequality worsens. Conversely, when
1 2
K K
θ θ
>
,
W
rises at a higher rate than
S
W
. Thus, the
relative wage movements go in favour of unskilled labour.
Therefore, both the magnitude of the fixed cost of emigration and the technologies of production
are the determining factor behind the direction of change in the skill-based wage inequality that
results from the anti-immigration policy adopted in the destination country.
13
This has been proved in Appendix 2.
14
5. Consequence of anti-immigration policy on wage inequality in a Specific-factor Harris-
Todaro structure
So far, we have confined our analysis to a small open economy case where there are no
distortions and only one type of labour i.e. skilled labour is employed in the high-skill sector. Let
us now consider two types of labour, both skilled and unskilled in the production of the high-
skill commodity (good 2). Besides, we also add some developing country flavor so that we can
theoretically predict the outcome of the stricter anti-immigration policy in the developed
countries on a small open developing economy having some typical characteristics like rural-
urban migration, existence of factor market distortions, problem of involuntary unemployment
etc.
Let us think about a two-sector, specific factor small open Harris-Todaro (HT, hereafter) type
dual economy. Sector 1 is the rural sector that produces a low-skill exports good using unskilled
labour and capital while sector 2 is the urban sector that produces a high-skill good by means of
skilled labour, capital and unskilled labour. This is the import-competing sector of the economy.
Unskilled workers employed in the rural sector earn a competitive low wage,
W
while their
compatriots in the urban sector earn an institutionally fixed wage,
*
W
with
*
W W
>
.
14
Because
the urban sector (sector 2) uses three inputs, two types of labour and capital, we make a
simplifying assumption that the unskilled labour-output ratio in this sector,
2
L
a
is technologically
fixed.
15
All markets except the urban market for unskilled labour are perfect. All other standard
assumptions including CRS technologies hold.
16
The two commodity prices are given by the
small open economy assumption.
14
The institutionally fixed wage could be due to strict implementation of the government’s minimum
wage legislation, collective bargaining of unskilled labour unions etc. See Khan (1980) and Chaudhuri
(2016) in this context.
15
See footnote 16.
16
Note that sector 2 is the high-skill sector where only a limited number of unskilled labour is needed in
certain low-skill posts e.g. sweepers, peons, security guards etc. However, because the firms in sector 2
are registered units, the unskilled workers must be paid at least the minimum wage as stipulated by the
government,
*
W
which is, however, higher than the competitive unskilled wage,
W
prevailing in sector 1.
15
The usual unit cost-price equality conditions are given by the following two equations.
1 1 1
L K
Wa ra P
+ =
(9)
2 2 2 2
*
L S S K
W a W a ra P
+ + =
(10)
Skilled labour and capital inputs are fully utilized. The two full-employment conditions are once
again are the following.
2 2
( *)
S
a X S S
= −
(11)
1 1 2 2K K V
a X a X bS K
+ + =
(12)
Here
V
bS
is the amount of capital that is used by
V
S
number of skilled workers for the purpose of
prospective emigration.
On the other hand, due to the existence of urban-rural wage differential, many of the rural
unskilled workers migrate to the urban for getting higher paid jobs in this sector. However, not
all the migrated workers get employment because the availability of urban jobs is limited. Hence,
some of the unskilled workers remain unemployed in the urban sector and the probability of
getting a job for a prospective rural unskilled migrant becomes less than unity. This gives rise to
the concept of expected urban wage. In the migration equilibrium, the expected urban wage is
equal to the rural unskilled wage. The HT migration equilibrium condition and the unskilled
labour endowment equation are given by the following two equations, respectively.
2 2
2 2
*( )
L
L U
a X
W W
a X L
=
+
(13)
1 1 2 2L L U
a X a X L L
+ + =
` (14)
Here
U
L
is the level of urban unemployment of unskilled labour. Using (14), the HT migration
equilibrium condition can be rewritten as follows.
Hence, we allow factor substitution only between skilled labour and capital. The number of such posts
increases only if the industry expands and vice versa. This would probably justify why
2
L
a
has been
assumed to be technologically given. However, the other two inputs, capital and skilled labour display
constant returns to scale between themselves.
16
1 1 2 2
*
( )
L L
W
a X a X L
W
+ =
(13.1)
Finally, the equality condition between the opportunity cost of each skilled emigrant and the
expected skilled income in the host country is written once more as follows.
*
( )
S S
V
S
W rb W
S
+ =
(15)
In this general equilibrium system, there are seven independent equations, (9) – (12), (13.1), (14)
and (15) to solve for the seven unknowns, namely
1 2
, , , , ,
S V
W r W S X X
and
U
L
.
Using (11), equations (13.1) and (12) can be, respectively rewritten as follows.
2
1 1
2
*
( )( )( *)
L
L
S
aW
a X S S L
W a
+ − =
(13.2)
2
1 1
2
( )( *)
K
K V
S
a
a X S S bS K
a
+ − + =
(12.1)
Multiplying (13.2) and (12.1) by
1
K
a
and
1
L
a
, respectively and subtracting we get
2 2
1 1 1 1 1
2 2
*
( )( )( *) ( )( *) ( )
L K
K L L V K L
S S
a aW
a S S a S S a bS a L a K
W a a
− − − − = −
(16)
Sector 1 and sector 2 use two common inputs, unskilled labour and capital. Although there is an
intersectoral differential between the two unskilled wage rates, the two sectors can be classified
in terms of factor intensities in value sense. On a par with the dual economy literature, we
assume that sector 2 is more capital-intensive vis-à-vis sector 1 with respect to unskilled labour
in value sense.
17
This means that
2 1
2 1
*
K K
L L
a a
W a Wa
>
or
2 1
2 1
K K
L L
θ θ
θ θ
>
.
17
See Chaudhuri and Mukhopadhyay (2009) for further on the factor intensity classification of sectors in
a dual HT economy.
17
The equilibrium values of
, ,
S
W r W
and
V
S
are simultaneously solved from equations (9), (10),
(15) and (16). Once factor prices are known, the factor coefficients are also obtained.
2
X
is
obtained from (11).
1
X
is then determined from either equation (13.2) or (12.1). Finally,
U
L
is
solved from (14).
Although unskilled workers in an HT model earns different wages in different sectors, the
average wage of all unskilled workers in the economy is equal to the rural sector wage,
W
because in migration equilibrium the expected urban wage income for the marginal worker
leaving the rural sector is equal to
W
. This is easily seen from equation (13.1) that
1 1 2 2
*
( )
L L
Wa X W a X
W
L
+
=
(13.3)
This is known as the ‘envelope property’ of the HT structure. See Chaudhuri and Mukhopadhyay
(2009) and Chaudhuri (2017) in this context.
The relative wage inequality is again given by
ˆ ˆ
( )
S
W W
−
.
Differentiating (9), (10), (15) and (16) and simplifying the following expression can be
derived.
18
12
2 1 2
2
ˆ ˆ *
( ) ( )[ ( )( )][ (1 )
ˆ*
[ ]
S S L V K
K K L
A S V
W W S a S
b
K a S
S
ε λ θ θ θ
−= − − −
Ω
(17)
Ω
is a term that has been defined in Appendix 2. In Appendix 2 it is also shown that
0
Ω <
under
the sufficient condition that
2
2
( )
K
S
a
b
a
≥
.
From expression (17), the following proposition readily follows.
Proposition 2: The anti-immigration policy in the destination country worsens the skilled-
18
The detailed derivations are available in Appendix 2.
18
unskilled wage inequality in the developing country either if (i)
2
2
( )
K
S
a
b
a
≥
; and, (ii)
2 1 2
(1 )
K K L
θ θ θ
> −
.; or if (i)
b
is small enough; and, (ii)
2 1 2
(1 )
K K L
θ θ θ
< −
. The wage inequality,
however, improves either if (i)
2
2
( )
K
S
a
b
a
≥
and, (ii)
2 1 2
(1 )
K K L
θ θ θ
< −
.; or if (i)
b
is small enough;
and, (ii)
2 1 2
(1 )
K K L
θ θ θ
> −
.
As explained previously, if the number of visas granted by the destination country decreases two
opposite forces on the return to capital,
r
arise. The net result of these two opposite effects would
be a decrease in the return to the return to capital,
r
under the sufficient condition that
2
2
( )
K
S
a
b
a
≥
.
Now as
r
falls both the skilled and the average unskilled wage i.e.
S
W
and
W
increase (see
equations (9) and (10)). The impact on wage inequality depends on the rates of proportional
increase in the two wages, which in turn now depends on production technologies. If
2 1
2 1
( ) ( )
K K
S L
θ θ
θ θ
>
i.e. if
2 1 2
(1 )
K K L
θ θ θ
> −
, the saving on unit capital cost would be higher in sector 2
relative to that in sector 1. Consequently, the relative wage inequality moves against unskilled
labour. In the opposite case i.e. where, the skilled-unskilled wage inequality improves. However,
if the fixed cost,
b
is sufficiently low,
r
rises and the two wages,
W
and
S
W
fall. Accordingly, the
condition under which the wage inequality deteriorates (improves) would exactly be the
opposite.
A comparison between proposition 1 and proposition 2 shows that the capital intensity
conditions are slightly different in the two cases. In the HT case, the high-skill sector apart from
capital and sector-specific input, skilled labour uses unskilled labour in production.
19
Besides,
the unskilled labour market facing this sector is imperfect and the unskilled wage is exogenously
given. Sector 2 is more capital-intensive vis-à-vis sector 1 in value sense. Additionally, the
unskilled labour-output ratio,
2
L
a
and consequently the distributive share of unskilled labour in
19
Note that in the HT case, both capital and unskilled labour are perfectly mobile intersectorally.
19
this sector i.e.
2
2
2
*
( )
L
L
W a
P
θ
=
is are given. Therefore, the distributive share of skilled labour in
sector 2 i.e.
2 2 2
( (1 ))
S K L
θ θ θ
= − −
while in the full-employment case we had
2 2
( (1 )).
S K
θ θ
= −
Thus, the degree of imperfection prevailing in the unskilled labour market plays an important
role in determining the direction of change in the wage inequality in the HT case.
6. Effect on urban unemployment of unskilled labour
Because we have now considered a specific factor HT economy, it is possible for us to examine
the consequence of the anti-immigration policy on the open unemployment of unskilled labour in
the urban sector of the source economy.
Subtracting equation (14) from equation (13.1) we get
2 2
*
( 1)
L U
W
a X L
W
− =
(18)
Differentiating equations (18) and (11) it is possible for us to derive the following expression.
20
*
2
1 2 2
1 2 1 2
2
ˆ
* *
( ) ( )[ ( )( )][ ( ) ] ( )
ˆ
*
*
( 1)
[ ]
U L V K L
SK L K K S
S V LU
L S a S S
b S
K a S S S
S
λλ
θ θ θ θ
λ
= − − −
Ω −
≤
(19)
From equation (19), the following proposition is imminent.
Proposition 3: Anti-immigration policy in the developed nation works favorably on the urban
unemployment problem of unskilled labour if (i)
2
2
( )
K
S
a
b
a
≥
; and, (ii)
*
2
1 2 2
1 2
( ) ( )
L K L
SK
K S LU
S
θ θ λ
θ θ λ
≥.
We verbally explain proposition 3 in the following fashion.
20
This has been derived in Appendix 3.
20
Let us consider the case where the condition,
2
2
( )
K
S
a
b
a
≥
is satisfied. In the migration equilibrium,
the expected urban wage for a prospective rural migrant equals the actual rural wage. How a
stringent anti-immigration policy, through a reduction in international migration of skilled
labour, produces an excess supply of capital at the preexisting level of its demand has already
been explained. Consequently, the return to capital,
r
falls and consequently the two
endogenously determined wages,
W
and
S
W
increase (see equations (9) and (10)).
The excess supply of capital through a Rybczynski effect expands the capital-intensive (with
respect to unskilled labour) urban sector in terms of both output and employment of unskilled
labour. Hence the expected urban wage for a prospective rural unskilled migrant,
)},/(1/*{
33
XaLW
LU
+
rises as the probability of getting a job in this sector rises for every
unskilled worker. This paves the way for fresh migration from the rural sector to the urban
sector. This is the centrifugal force that drives the rural unskilled workers to move away from the
rural sector. If the rural sector wage remains unchanged, the number of new jobs created in the
urban sector falls short of the number of new unskilled migrants into the urban sector. This is the
standard result as obtained in the two sector mobile capital HT model of Corden and Findlay
(1975). In such a situation, the level of urban unemployment of unskilled labour unambiguously
rises. Nevertheless, in the present case the competitive rural wage has also increased. This
creates a centripetal force that prevents rural workers from migrating into the urban sector. Thus,
there are clearly two opposite effects working on determination of the size of the unemployed
urban unskilled workforce. If the latter effect outweighs the former, the level of urban
unemployment falls. This happens under the second condition as stated in proposition 3.
7. Two Alternative Scenarios
As discussed in section 1, empirical evidence suggests that acquisition of a good communicative
ability in English involves a substantial cost on the part of every desirous emigrant that is
incurred through taking a loan from the capital market at the existing interest rate,
r
. Let us now
21
think about a public subsidy policy that brings down the effective cost to zero i.e. leads to a
situation where,
0
b
≃
. In this hypothetical case, all the skilled workers will be equally desirous
to emigrate. Hence,
V
S S
=
although the actual magnitude of emigration,
*
S
is still the policy
variable of the destination economy. First consider the full-employment case. A decrease in
*
S
raises the net endowment of skilled labour,
( *)
S S
−
of the source country. This depresses the
skilled wage,
S
W
that in turn raises the return to capital,
r
(see equation 2) because the demand for
capital rises. Sector 2 expands and draws capital from sector 1 causing the latter to contract. The
unskilled wage,
W
falls as well because the demand for unskilled labour has fallen (see equation
1). The skilled-unskilled wage inequality worsens (improves) if sector 2 is less (more) capital-
intensive relative to sector 1 in the Jones-Neary (1984) sense i.e. if
2 1
( )
K K
θ θ
< >
. This is
standard result in the wage inequality literature (see e.g. Chaudhuri (2004) and Marjit and Kar
(2005) among others).
We now consider the HT case. In sector 2, both
*
W
and
2
L
a
(defined earlier) are given.
Nevertheless, both
S
W
and
W
fall and
r
rises. Besides, sector 2 expands while sector 1 contracts
due to increase in effective skilled endowment and scarcity of capital, respectively. The direction
of change in the relative wage inequality once more depends on the rates of decrease in the two
wages. In the full-employment case, sector 2 uses only two inputs, skilled labour and capital
while in the HT case, sector 2 uses an additional factor, unskilled labour. Hence, the distributive
share of capital in the present case would be,
2 2 2
( 1 )
K S L
θ θ θ
= − −
. The wage inequality improves
if and only if
1 2 2
(1 )
K S L
θ θ θ
> − −
. In the opposite case, the wage inequality worsens. Because
2
2
2
*
( )
L
L
W a
P
θ
=
, in the HT case, in addition to the technological factor, the institutional factor of
unskilled labour market also plays a decisive role.
Finally, we have seen that due to a fall in emigration of skilled labour, sector 2 expands both in
terms of output and employment of unskilled labour. (Note that
2
L
a
is fixed (Footnote 17)). This
raises the expected unskilled wage income. On the other hand, the unskilled wage in sector 1 has
fallen. Both these factors together raise the magnitude of migration of unskilled labour from the
22
rural sector (sector 1) to the urban sector (sector 2). As per the standard HT migration
mechanism, the number of new migrants exceeds the number of new jobs created in the urban
sector. Consequently, the unemployment problem of unskilled labour in the urban sector
unequivocally aggravates.
In view of the above discussion, we can come to the conclusion that if emigration to foreign
countries is made costless by resorting to a subsidy policy towards developing communicative
English skill, we obtain the standard results that are available in the theoretical on skilled-
unskilled wage inequality
We will now consider another scenario where vocational training apart from capital uses skilled
labour more intensively relative to sector 2. Vocational training services sector, call it sector
V
,
produces a non-traded services that is used by desirous emigrants. Under competitive conditions,
we will have another zero-profit equation. Sector 2 and sector
V
together now form a Heckscher-
Ohlin Subsystem (HOSS). The price of the services,
b
is now determined by the demand and
supply forces. Without resorting to any mathematics, let us intuitively think about the possible
consequence on wage inequality. The anti-immigration policy lowers the migration quota from
the source country and hence the probability to be able to migrate as well. Consequently, the
demand for services
V
falls due to a decrease in the expected skilled wage abroad. Given the
supply of this services, its price,
b
must fall which produces a Stolper-Samuelson effect in the
HOSS, leading to a fall in the skilled wage,
S
W
and an increase in the return to capital,
r
. The
unskilled wage,
W
also falls (see equation 1). The effect on wage inequality again depends on the
Jones and Neary (1984) type factor-intensity condition involving the distributive shares of capital
in sector 1 and sector 2. Although some secondary adjustments e.g. supply by sector
V
, net
change in its price,
b
etc. the final outcome in a stable equilibrium would be what we have stated
above because primary effects will outweigh secondary effects.
21
Results would be exactly
opposite, if instead of skilled labor-intensive, sector
V
is capital-intensive vis-à-vis sector 2.
Hence, even if we endogenize, vocational training, no counterintuitive results could be obtained.
21
This will follow from the Walrasian static stability condition for the non-traded services sector.
23
8. Concluding remarks and policy implications
In view of some anti-globalization policies undertaken by a couple of powerful developed
nations very recently, this piece has theoretically examined the possible outcome of the anti-
immigration policy in the destination country on the relative wage inequality in a source
economy with the help of two two-sector general equilibrium models, both with and without
unemployment. There are three inputs of productions: unskilled labour, skilled labour and
capital. Capital is perfectly mobile across sectors while skilled labour is specific to one of the
two sectors in both the models. In the second model with Harris-Todaro type unemployment,
unskilled labour is used in both the sectors and is perfectly mobile between the sectors although
the unskilled wage rates differ across sectors.
22
In both the full-employment and the Harris-
Todaro models with specific inputs, the number of high-skill emigrants from the source country
is exogenous. However, the number of skilled workers desiring to emigrate is endogenously
determined. The exogenously given numbers of visas are randomly allocated among the
prospective emigrants through a lottery. Hence, the probability of getting a visa and being able to
emigrate is less than unity. However, emigration requires a bit of professional skill formation,
the cost of which is financed by taking a loan of the amount,
b
from the capital (working capital)
market. In the international migration equilibrium, the expected skilled wage abroad and the
opportunity cost of emigration (sum of domestic skilled wage and interest cost on loan) are
equal.
The important results that we have obtained and their policy implications are as follows. The
anti-immigration policy, captured through a reduction in the number of visas granted, worsens
(improves) the skill-based wage inequality if
2
2
( )
K
S
a
b
a
≥
and if the capital content of the high-skill
sector,
2
K
θ
is greater (less) than that of the low-skill sector,
1
K
θ
. Hence, in the case where
2 1
( )
K K
θ θ
>
for improving wage disparity, a tax on capital use in the high-skill sector may be
imposed and/or a capital subsidy may be given to the low-skill sector. A capital tax on sector 2
22
Because we are only interested in the international migration of skilled labour, no attention has been
paid to migration of unskilled labour.
24
leads to a substitution of the dearer input, capital by the cheaper input, skilled labour.
Consequently,
2
K
a
and hence
2
K
θ
fall. A capital subsidy policy does produce exactly the opposite
effect. On the other hand, the government may tinker with the market mechanism by introducing
a capital tax and/or a capital subsidy policy on the appropriate sectors depending on whether
2
2
( )
K
S
a
a
is greater or less than the fixed cost of an emigrant,
b
, so that the wage inequality does not
at least deteriorate. Finally, in the HT case, apart from appropriate capital tax and/or capital
subsidy policy to the appropriate sectors, a suitable employment subsidy or a tax policy should
be adhered to deal with imperfection in the unskilled labour market in sector 2. The employment
subsidy policy will also help to improve the employment scenario of unskilled labour.
As a final point, it should be mentioned that we have developed a simple tractable analytical
structure for a small open economy that might be useful for theoretically predicting the possible
outcomes of the anti-immigration policy in the destination country on the skilled-unskilled wage
inequality in the source country under different scenarios. The policy is only a very recent issue.
Hence, what actually will happen to the relative wage distribution between the two types of
labour can only be observed through empirical studies in the future. However, our suggested
policies can be readily implemented for overcoming the possible undesirable consequence of the
international policy on the poorer section of the working population.
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Appendix 1: The full-employment case
Differentiating equations (1), (2), (4) and (6.1) we obtain three expressions that can be arranged
in the following matrix form.
1 1
2 2
2 1
1 2 3 3
ˆ
0
0 0
0
ˆ
0 0
ˆ
ˆ
*
0 1
ˆ
- ˆ
*
L K
K S
S
KV V
W
r
S
D D W
C C C
D S
S
θ θ
θ θ
λ
=
(A.1)
1 1
1 1
[ ( )] 0;
K KL LK
C S S
λ
= + >
1 1 2 2
2 1 2
[ ( ) ( )] 0;
K KL LK K KS SK
C S S S S
λ λ
= + + + >
2 2
3 2
[ ( )] 0;
K KS SK
C S S
λ
= + >
(A.2)
1 2
( ) 0; ( ) 0;
S
S S
Wrb
D D
W rb W rb
= > = >
+ +
2
3
2
*
[( )( )] 0.
K
S
a S
Da K
= >
Solving (A.1) using (A.2) and after simplifying we obtain the following expressions.
1 2 2
2
ˆ
*
( ) ( )( )[ ( )( )]
ˆ*
( 1)
[ ] 0
S VL K K
S V
W S a S
b
K a S
S
θ θ
= −
∆
≤
<
if
2
2
( )
K
S
a
b
a
≥
(A.3)
1 2 2
2
ˆ*
( ) ( )( )[ ( )( )]
ˆ*
( 1)
[ ] 0
L S V K
S V
Sar S
b
K a S
S
θ θ
= − −
∆
≤
>
if
2
2
( )
K
S
a
b
a
≥
; (A.4)
29
1 2 2
2
ˆ
*
( ) ( )( )[ ( )( )]
ˆ*
( 1)
[ ] 0
K S V K
S V
SaW S
b
K a S
S
θ θ
= −
∆
≤
<
if
2
2
( )
K
S
a
b
a
≥
; (A.5)
2
2
ˆ122
( ) ( ) ( )( ) { ( ) }
ˆ
1 3 1 2 3 2 2 1 2 1
*2
) (
[ ]
0
K
S
S r a
V S K
D b C
a
if b
C C
L L K S K S
W rb
S
a
a
SS
θ
θ θ θ θ θ θ
=
≥
− − + +
∆ +
>
(A.6)
where
22
1 2 3 2 2 1 2 1
2
[ ( )( ) ( )]
[ ]
SK
L KV K S K S
S S
ra
b C C C
W rb a
θ
θ λ θ θ θ θ
∆ = − − + −
+
(A.7)
Using (A.2) and simplifying from (A.6) it can be found that that
0
∆ <
if
2
2
( )
K
S
a
b
a
≥
(A.8)
Subtracting (A.5) from (A.3) and (A.11) from (A.9) we respectively obtain the following two
expressions.
2 2
1 2 1 2 2 1
2 2
ˆ ˆ * *
( ) ( )[ ( )( )]( ) ( )[ ( )( )]( )
ˆ*
[ ] [ ]
S S V S V
K K
L K K S K K
A S V A S V
W W S S
a aS S
b b
K a S K a S
S
ε ε
θ θ θ θ θ θ
−= − − − −
∆ ∆
=
(7)
Appendix 2: The HT case
Differentiating equations (11), (12), (17) and (18) and arranging in a matrix notation we obtain
the following.
1 1
2 2
2 1
1 2 3 4
ˆ
0
0 0
0
ˆ
0 0
ˆ
ˆ
0 1 *
ˆ
- ˆ
*
L K
K S
S
V
W
r
D D S
W
F F F F
HS
S
θ θ
θ θ
=
(A.13)
where:
30
1 2
( ) 0; ( ) 0;
S
S S
Wrb
D D
W rb W rb
= > = >
+ +
1 1 *
1 1 1 2
[ ( ) ] 0
K L KL LK L
F S S
λ λ λ
= + + >
1 1 2 2
2 1 1 1 2
[ ( ) ] 0
L K KL LK SK L K KS
F S S S S
λ λ λ λ λ
= + + + >
*
1 2 1 2
( ) 0
L K K L
λ λ λ λ λ
=−>
2 2
3 1 2
[ ] 0
SK L K KS
F S S
λ λ λ
= + >
(A.14)
4 1
( ) 0
V
L
bS
F
K
λ
= >
*
( ) 0
*
S
H
S S
λ
= >
−
1 2 4 2
1 2 3 2 2 1 2 1
2
[ ][( ) ] [ ( ) ]
[ ] 0
(+) (+)
L S K
L K S K S
S S
r F ab F F F
W rb a
θ θ
θ θ θ θ θ
Ω = − − + +
+
<
(A.15)
if
2
2
[ ( )]
K
S
a
ba
≥
Solving (A.13), using (A.14) and (A.15) and simplifying the following results can be obtained.
11 2 2
2
ˆ
*
( ) ( )( )[ ( )( )]
ˆ*
( 1)
[ ] 0
S L VL K K
S V
W S a S
b
K a S
S
λ
θ θ
= −
Ω
≤
<
if
2
2
[ ( )]
K
S
a
ba
≥
(A.16)
1 2 1 2
2
ˆ
*
( ) ( )( )[ ( )( )]
ˆ*
( 1)
[ ] 0
K S L V K
S V
SaW S
b
K a S
S
θ θ λ
= −
Ω
≤
<
if
2
2
( )
K
S
a
b
a
≥
(A.17)
;
Subtracting (A.17) from (A.16) we get
31
12
1 2 1 2
2
ˆ ˆ *
( ) ( )[ ( )( )][ ]
ˆ*
[ ]
S S L V K
L K K S
A S V
W W S a S
b
K a S
S
ε λ θ θ θ θ
−= − −
Ω
(A.20)
or,
12
2 1 2
2
ˆ ˆ *
( ) ( )[ ( )( )][ (1 )
ˆ*
[ ]
S S L V K
K K L
A S V
W W S a S
b
K a S
S
ε λ θ θ θ
−= − − −
Ω
(17)
Appendix 3: Effect on urban unemployment of unskilled labour
Differentiating equations (18) and (11) and simplifying we arrive at the following expression.
*
2
2
*
ˆ
ˆ ˆ ˆ
( ) ( ) *
*
L
U SK S
LU
S
L W S W S
S S
λ
λ
= − + −
−
( (A.21)
Using (A.16) and (A.17) and simplifying further from (A.21) it entails
*
2
1 2 2
1 2 1 2
2
ˆ
* *
( ) ( )[ ( )( )][ ( ) ] ( )
ˆ
*
*
( 1)
[ ]
U L V K L
SK L K K S
S V LU
L S a S S
b S
K a S S S
S
λλ
θ θ θ θ
λ
= − − −
Ω −
≤
(19)
