Development policies in LDC's with several ethnic groups -- a theoretical analysis

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FACULTY WORKING
PAPER NO. 1090

Dsvelopment Policies in LDC's with Several
Ethnic Groups — A Theoretical Analvsis

M. AH Khan

T. Datta Chaudhuri

THE UBRAHI fiH JUS

t|i 884

--SITY 0-

Coilege of Commerce and Suslness Adrninl3tra«:ion
Sureau of Economic and Business Research
University of Illinois. Urban 2-Ch ?--""'*!'-'"

:.m

BEBR

FACULTY WORKING PAPER NO. 1090
College of Commerce and Business Administration
University of Illinois at Urbana- Champaign
November , 1 984

Development Policies in LDC's with Several
Ethnic Groups - A Theoretical Analysis

M. Ali Khan, Professor
Department of Economics

T. Datta Chaudhuri
University of Calcutta

This is an extensively revised version of Johyi6 Hopkln6 Wo-liZying PapcA
Mo. 45 and represents research originally begun by the first author
while he was visiting the Department of Economics and the Math Center at
Northwestern University in 1978. A preliminary version was presented at
a seminar at the Development Research Center, IBRD in March 1979, and
some of the results of Section 2 were announced, without proof, in
Economic L<2Xt2AJ>, 2 (1979), 369-75.

Our first acknowledgement is to the insights of Amartya Sen and T. N.
Srinivasan but the research presented here has also benefitted from the
comments and suggestions of our Hopkin's colleagues: Bela Balassa,
Bruce Hamilton, Tatsuo Hatta, Lou Maccini, Eshi Motahar, Peter Newman,
Masa Nomura and Bob Schwab. Errors are, of course, solely ours.

The presence of economically and socially disadvantaged groups is a
common feature of most less-developed countries (LDC's). These "backward
classes" share a common religion, or belong to the same ethnic or tribal,
group or originate from a particular geographic region. In each instance
they form an easily-identifiable minority. The "untouchables" or "sched-
uled castes" in India readily come to mind but even a casual observer of
other South Asian, or indeed African and Latin American economies can
readily furnish his own examples.

Economic planners in LDC's have long been concerned with the econo-
mic and social advancement of such groups and various national plan
documents are replete with a variety of policy measures introduced to
help such communities. Such measures have typically taken the form of

specific employment quotas, minimum wage legislation, regional subsidies

2

and specially targeted development expenditures. A natural question

arises as to whether such policies accomplish what they are intended to
do and the impact they have on the welfare of other groups and on the
country as a whole. An analysis of these issues has been lacking in the
development literature. This literature, by and large, confines itself
to a two-sector setting with a homogeneous labor force whereas the very
nature of the problem calls for a general equilibrium analysis in a multi-
sectoral setting. Such an analysis is presented in this paper.

The plan of the paper is as follows. Section 1 presents the model
and Section 2 is devoted to checking out the viability of our equilibrium
concept; namely whether it exists, is locally unique and robust, and is
stable in terms of a reasonable adjustment process. Section 3 moves on

1

2

to a consideration of policy issues and Section 4 indicates several other
questions which our model can easily answer. Section 5 concludes the
paper and three appendices establish the results presented in the paper..

. 1. THE MODEL AND THE EQUILIBRIUM CONCEPT

We study a small, open economy with one urban city-center and n
rural sectors, some or all of which typically may be regarded as back-
ward. There are n ethnic groups, one associated with each of the n-rural
sectors whose members come to the urban center to find employment. The
volume of migration is controlled by expected wage equalization, a
seminal idea introduced into the development literature by Harris-Todaro
[10]. However, our multi sectoral setting allows us to extend the Harris-
todaro equilibrium condition by postulating that each migrant calculates
the probability of finding a city- job on the basis of the unemployment
rate specific to his own tribe rather than the aggregate unemployment
rate. This allows us to capture the fact that information about employ-
ment possibilities in the city flows to a village largely, or even sole-
ly, through those of its members who are already in the city. Further,
our extension also emphasizes the fact that during the period when he
is unemployed and looking for a city-job, a migrant has to fall back
for support on the employed members of his tribe or his region.

Our second departure from the original treatment of Harris-Todaro
lies in assuming, along with several recent studies, that urban wages
are not necessarily rigidly set but may be a general function of the
rural wage and the unemployment rate. A basis for such a function has

been succinctly set out in Sen [17, p. 55], and more thoroughly discussed
in a two sector, mobile capital context by Khan [13]. The only point
worth underscoring here is that the urban wage function may differ from
tribe to tribe reflecting different supply prices to the urban employer
or more generally, different institutional arrangements for each tribe.

In summary, then, the basic ingredients of our model can be simply
stated. There are n+l commodities and 2n + 1 factors of production,
n + 1 of which are commodity specific and non-shiftable and the remaining
n, i.e., the different types of laborers, flow freely between the urban
center and the rural sectors they come from. For any tribe, laborers are
allocated between the urban center and the rural sector on the basis of
the Harris-Todaro equality of expected returns. All factor returns are
endogenously determined and it is assumed that the supplies of each of the
2n + 1 factors and the prices of n + 1 commodities are exogenously given.
We shall devote the remainder of this section to a more formal presenta-
tion of the model and our equilibrium concept.

Let each of the n rural regions be indexed by i, and let ^ denote-
the population coming from region i. Let L. denote the rural employment,

L. the urban employment and \. the ratio of urban unemployed to urban

3
employed, all these variables pertaining to region i. We can thus write

^•r ■" ^^""S'^^'u = ^ 1=1, 2,. ..,n (1.1)

The technology available to the economy is given by

^•r = ^'r^^'r^^'r^ i=l,2,...,n (1.2a)

^u =^(hu'4n---^u'^) (l-2b)

where K. and K are the community-specific, non-shiftable factors of
production.

J. U

The endogenous urban wage of the i tribe is given by w. and its
endogeneity is brought out by

^iu " ^i^^r'^f'^i) (i =l,2,...,n) (1.3)

where ji- is a shift parameter. For details about the microeconomics
underlying the.Q(») functions, see Khan [13].

Labor migrates until expected returns are equalized and this can be
expressed as

^^.(w.^,X.,j-.) = w.^ = w.^d +X.) (i =l,...,n) (1.4)

with the probability of finding a job in the urban sector given by (1/1 +X.)

Finally, we shall assume that there is marginal productivity pricing
of labor in each sector. Thus

3F^>/9Li^ = ^•r/Pir (i=l,...,n) (1.5a)

V^'-iu " ^u/Pu (i=l,...,n) (1.5b)

where p. , p are the exogenously given prices for the n+1 commodities.

Putting all these facets together yields us our basic equilibrium
concept.

D.l: A Harris-Todaro Equilibrium is a n-tuple of the quadruple (w^ , xt,
L*^, L*^) » 0 such that for all i

(i) L*^ maximizes P^>F.^(L.^,K.^) - wU.^

(ii) l;^ maximizes P/^^h ^- • • 'L^r,Kj - p^lOL^^j
(iv) w*^(l+X*) =Q,.(w*^,x;,J-i) =w*^

2. EXISTENCE, LOCAL UNIQUENESS AND DYNAMIC STABILITY OF HARRIS-

TODARO EQUILIBRIA

In this section we present sufficient conditions under which Harris-
Todaro equilibria exist, are locally unique and stable in terms of an
"intuitively reasonable" adjustment process. These results are presented
as preliminary consistency tests of the viability of the model and as a
prelude to the more substantive, comparative static investigation pre-
sented in Section 3. The fact that such results are a prerequisite
for comparative static investigations is evident in the works of Bhag-
wati, Srinivasan, Calvo, Neary, Khan and others.

2.1 Existence of Equilibrium

The result of this section can be viewed as a generalization to a

multisectoral setting of the existence results in Srinivasan-Bhagwati

[18] and Calvo [6].

Let w denote a vector of rural wages (wi^,...,w ) and w^(i)
r ^ ir nr r

4.1.

denote w with the i component deleted. Let the excess demand for

J. L.

labor of the i tribe be denoted by (|).(w ) where

(}).(w ) = L. (w. ,p. ) + (1 +X.)L. (wt ,...,w ,p ) - =i. (2.1 .1)

If we focus on situations where the equality of expected wages is main-
tained, (1.4) allows us to write X. as a function of w. , say i2-.(w. ).
Using (1.3) we can rewrite (2.1.1) as

*iK) = L.^(w.^.p^^)+ (l+i,.(Wi^))L.^(wi^....,w^^,pJ - ^ (2.1.2)

We shall now assume the following

Assumption 2.1 .1 : For all i, and for all w. > 0, i^. (•) > w. .

Assumption 2.1.2: For all i, <^.{*) is a continuous, single valued func-
tion decreasing in w- .

Assumption 2.1.3: Lt c{). (w. ,w^(i) ) = «. and Lt (t).(w. ,w^(i) ) = -=£. (2.1

Assumptions 2.1.2 and 2.1.3 formalize the fact that the excess demand in
each market is a single-valued function of rural wages and that it is a
decreasing function of its own rural wage, unbounded from above and
bounded below by a negative number. It is clear that there exist inocuous
sufficient conditions on the technologies and the Q.{') functions which
imply Assumptions 2.1.2 and 2.1.3. Assumption 2.1.1 is adduced to insure
that in equilibrium the unemployment rate for each tribe is positive, or
to put it another way, there does not exist any tribe for which the urban
wage is lower than the rural wage. However, the reader should be clear in
the original Harris-Todaro setting of a rigid wage. Assumption 2.1.1 leads
us into difficulties. In such a context, a direct translation of this
assumption yields

Assumption 2.1.1': For all i, for all w.^ > 0, w.^ < fi(-) = T..

This assumption makes little sense since it keeps open the possibility
that each of the rigid wages T. be unbounded numbers, there being no
way of telling a priori what the equilibrium rural wage would be. Corol
lary 2.1 below presents a version of our existence result which takes
this difficulty into account.
We can now present

Theorem 2.1.1: under Assumptions 2.1.1 to 2.1.3, there exists at least
one Harris-Todaro equilibrium.

In the special case of exogenously-given urban wages, i.e., rz(») =
T. for all i, Theorem 2.1 can be specialized and stated in a way that
brings out its dependence solely on

Assumption 2.1 .4: F and for all i, F. give rise to continuously dif-
ferentiable, strictly decreasing demand functions for labor. The
production functions F. also satisfy Inada conditions, i.e., for

all i.

aF.^{0,K.^)/9L.^ = oo; SF.^(oo,K.^)/3L.^ = 0 (2.1.4)

We can now state

Corollary 2.1.2: under Assumption 2.1.4, for all i, there exist T*. such

^

that for all T, > T*, there exists at least one Harris-Todaro equi-
librium with exogenous wages.

Corolarry 2.1.2 avoids the difficulty inherent in Assumption 2.1.1'.
If for all i, T^. = Tt, one obtains competitive equilibria with no unem-
ployment; on the other hand, T. < T^ for even one i would generate a
negative value of a. and hence negate the existence of Harris-Todaro

8

., .. . 8
equi libna.

2.2 Local Uniqueness of Equilibrium

In this section we show that for "almost all" values of a^ = (a^. ,...,^)
Harris-Todaro equilibria are locally unique and continuous in ^. Such a
result is a minimal requirement for the validity of comparative-statics
exercises once we are guaranteed that equilibria exist. We shall need

Assumption 2.2.1: For all i, cj).(*) is a continuously differentiable func-
tion of w^.
r

Theorem 2.2.1: under Assumption 2.2.1, the set of ■£, for which Harrls-
Todaro equilibria are not locally unique and continuous In ^ Is
closed and of Lehesgue -measure zero.

Remark 2.2.2: Under the hypothesis of Theorem 2.2.1, the set of Harris-
Todaro equilibria is finite for all =£ in a closed set of Lebesgue
measure zero.

2.3 Stability of Equilibrium

In this section we ask whether a Harris-Todaro equilibrium, if dis-
turbed, will have tendencies to establish itself. We put forward an ad-
justment process and investigate conditions under which Harris-Todaro

9
equilibria are stable in terms of this process. These conditions revolve

around gross-substitutability and unfortunately, turn out to be fairly

strong. Nevertheless, they are worth having if only to provide a benchmark

against which the assumptions we make for comparative statics results are

to be judged. As Neary [15] has recently noted, a number of paradoxes in

trade theory have arisen as a result of the equilibria being unstable in

terms of natural adjustment processes.

The dynamic process ^ that we study is given by
Dw.^ = c^i{((L^.^+(l+X.)L.J/=£.) - 1}, i>\{0) > 0, (^.(0) = 0 (2.3.1a)

DA. = M(^.(-)/(l +^-)w.^) - 1}, ^P\{0) > 0, i'.{0) = 0 {2.3.1b)

where i runs from 1 to n and D denotes the differential operator.

Equation (2.3.1a) states that rural wages for the i tribe go up if

J. L

there is positive excess demand in the i labor market and go down if
the excess demand is negative. 4).(*) specifies the speed of adjustment of
the rural wage. Equation (2.3.1b) relates the rate of change of unemploy-

J. u

ment of the i tribe to the discrepancy between expected wages. If the

J. U

expected urban wage of the i tribe is greater than the rural wage, i.e.,
^,- {•)/(! +A-j) > w. , there is increased migration to the city and the un-
employment rate rises. Depending on the context, it may be useful to re-
gard X^. as a flow and, in such a case, DA. can be interpreted as the rate
of change of this flow, i.e., migrants come to the city or leave it at an
increased rate.

Before presenting our results, we shall need some further notation.
Let

i ^ llo^ilLll and e^' = (1 +x)^i^2_Mli

w 3 log w. A '■ ■ "' d\

Thus the e's refer to elasticities which pertain to the urban wage.
We shall need the following assumptions

Assumption 2.3.1 : For all i, and for all w. , A., 1 >e^ > 0 and e^ < 0,

ir I — w — A —

Assumption 2.3.2: The Hessian of F given by H = [s^F^j/3L^.^3L .^] is

10

2 2

syrmetric and negative definite. In addition 9 F. /9L. < 0 for

all i.
Assumption 2.3.3: For all i and j, labor of the i tribe is a gross

.th

where nlj^j = 9 log L^.jj/9 log w.^

substitute for labor of the j tribe, i.e., n"^ > 0 for all i ^ j

J -

A justification of Assumption 2.3.1 hinges on the microfoundation of
Tz. (•) for which the reader is again referred to Khan [13]. Assumption
2.3.2 is innocuous. In the sequel, reference to stability of equilibrium
is to be understood to mean stability in terms of the adjustment process
T given by (2.3.1). We can now state

Theorem 2.3.1: under Assumption 2.3.1 to 2.3.3, locally unique Harris-
Todaro equilibria are partially locally asymptotically stable in

the sense that w (t) locally converges to w if \(t) is held fixed

r

at A and vice versa.

Theorem 2.3.1 is, unfortunately, only a partial answer to the stability
question. A motivation for a more complete answer can be had by examining
the matrix A given in Figure 3. A represents the matrix of partial deriva-
tives of equations 2.3.1 if we disregard the functions <^A*) and ^.{').
Assumptions 2.3.1 to 2.3.3 guarantee, as well as become clear in the proofs
that the diagonal blocks have eigenvalues with negative real parts. Thus,
for complete local asymptotic stability, we have to ensure that when the
off-diagonal blocks are brought into the picture, they do not upset any-
thing. In other words, the diagonal blocks dominate the off-diagonal
blocks. This is precisely the force of the next assumption. Let I. de-
note L.^/=e. and £.^ denote L.^(l +X.)/^..

11

As

sumption 2.3.4: There exist positive numbers d-, and dp such that

(i) d^ Max{

f irx , ^i i
^ir^^-r ^ Viu

' .l.<<^ > '2^f^''<^^^

(ii) d^ Max{|e,^ - 1 | } > dT Maxfle"" - 1|}

w

We can now state

Theorem 2.3.2: under Assumption 2.3.1 to 2.3.4, Harris-Todaro equilibria
are locally unique and locally asymptotically stable.

Under Assumption 2.3.1, Assumption 2.3.4(ii) is automatically fulfilled
with d-j = d2 = 1 . It is easy to check that given Assumptions 2.3.1 to 2.3.3,
a sufficient condition for the validity of Assumption 2.3.4 is simply

In,., I > (^iu/^>)^^^^^,eJ|) for all i.

It is also easy to check that in a setting with one rural sector. Assumptions
2.3.1 to 2.3.3 insure that Assumption 2.3.4 is redundant.

We conclude this subsection with the remark that Theorem 2.3.2 strengthens
the local uniqueness result given as Theorem 2.2.

3. DEVELOPMENT POLICIES

In this section we analyze four policy issues; namely, immigration
policy, sector-specific capital inflows, minimum wage legislation and em-
ployment quotas. In keeping with the basic thrust of our paper we shall be
primarily concerned with tracing out the differing welfare-effects of a
particular policy on the variety of groups in our stylized economy.

12

The model can be usefully thought of in terms of Figure 2 which brings
out the fact that the only linkage between any two tribes occurs through
the city. This rather straightforward observation is of some consequence
for the analysis to follow because it emphasizes the essential block-dia-
gonal structure of the model. If we ignore this and follow the route we

have taken so far by working with the independent variables (w-.X. ),•=•]»

13
we end up with the A matrix we used in our stability analysis, the struc-
ture of whose inverse is far from evident. With such variables the model
is reduced to an extent that is useful for the purposes of Section 2 but
which obscures its essential simplicty in the context of comparative static
analysis.

In this section we shall work with the independent variables
^'■ir''"iu''^i ^i=l ^"^ reduce the model to its primal form:

L. + (1 +\.)l. = ^.

i = 1 , . . . , n (3.1a)

Pir(9Fi/3L^.^)(l+X.) = Pij(9F^/8L.J i=l,...,n (3.1b)
Pu^^V^S-u^ = ^^(P^>(9F.^/^L.^),A.,J^) i=l,...,n (3.1c)
Total differentiation of the above system of equations yields

2n

A

A

I--.

I D

'.1

v

(3.2)

13

where all submatrices except D are block-diagonal such that

A. =
1

ir

i.

lU

(l-e')/n. 1-e^

b. =
1

14

Z.
lu

0

"^1 = (?'-l)'^-=+ (3.3)

^ju

and where for any variable z, z denotes the proportional change dz/z, T
denotes transpose, and

(hr'^i)'

"1

v.. =

(i,, e;^-(l-e;)p.^-(l-e^)(Vp.^))^
(^wPir"^^"<^ir-Pu-(VPiu))

with p.^ = a log K.^/3 log w.^ and p.^ = a log K^/d log w.^

(3.4a)
(3.4b)
(3.4c)
(3.4d)

The equation system (3.2) should be seen as a formalization of
Figure 2 with the central linking role of the urban center being
parametrized by the elements of D. Rewriting (3.2) in terms of the in-
verse, we obtain

a(2nx 2n) , 3(2nxn)

y(n X 2n) I o(nx n)

(3.5)

with the order of each submatrix as indicated in the brackets. The quali-
tative information on the entries in 6 is crucial for signing the entries
in a, 3 and y and such information is furnished in the following Lemma.

14

Lemma 3.0: under Assumptions 2.3.1 to 2.3.3, 6 . . <_ 0 for all i,j.

We shall supply a proof of this lemma in Appendix III. However, the
reader should note that 6 = (D-CA" B)" and that CA~ B is a diagonal
matrix with a typical entry

I. (e? - e"')/A.n. ; A. = lA.i = i. {^ - el) - i- (1 -eM/n-
lu^ A w^' 1 ir' 1 ' i' ir^ \' lu^ w" ir

Once we have 6, the remaining matrices are easy. By straightforward
calculations, the reader can derive the following.

-Y =

c A-^

^ri ..

n n

^11^1^^

^nl^l^l^

^In^A'n^

^nnS^'

(3.6a)

whe»

-1

re c.AT ' = ((e,^ - eM, e^il. - e,^£. n- )/A.n-
11 ^^ X w'» w lu X ir ir^' i 'r

(3.6b)

-6 =

b^A-1

bA-1

n n

-1

^llh^

<5nlMn^

n I n n

^inh^l^

5 b A"""
nn n n

(3.7a)

T

where b.A'^ = (ili,(l-e;[), -i^i,(T " <)/^>) M.

a

>-l

= A"' (I -By) = a.

(3.7b)
(3.8a)

15

A^ (I + b-,6^^c^A^ )'

^^Vm^i^^

'''^^'^^'^^'/n

n n nn n n

(3.8b)

where

a

ij

TJ

^i^j^jr

^iu(^-<)(^i-w)

-'J'-<?^i-'>r

^iu(^-4)(^^ju-^xW

^-u^^-^w^^^^-^^jr^jr^/^T

(3.8c)

a

ii A.

r,. ^' A.n. iu^^X w^^
'ir 1 ir

(3.8d)

In the sequel we shall denote the ij element of a particular matrix
a., by a. .(ij). We shall also use the hypothesis of Lemma 3.0 as a Standing
Hypothesis for the discussion to follow.

3.1 Population Growth

In this subsection we discuss the positive and normative effects of
the growth in the numbers of a particular tribe. This is a problem of
some importance for development theory. Firstly, attempts at population

16

control in the LDC's meet the greatest resistance when they are perceived
to be principally directed at a particular ethnic group. It is interesting
to see, albeit in our simple stylized setting, whether there is in fact
an economic rationale for this resistance. Secondly, there is substantial
emigration especially from South Asian economies to the Middle Eastern
Gulf States. This emigration has tended to be region-specific partly as
a result of information flows and partly as a consequence of the costs of
migration becoming lower for the group whose members are already there.
It is of interest to see what such emigration implies for the other groups
in the economy.

The algebraic groundwork for the relevant comparative static exercise
has already been laid out above. We just have to read off the various
entries. Thus L.^/=£\ = ct.^.(ll), X.//j = a.j(21 ) and ^^j£^ = ^ij^^^*
After again reminding the reader that we are assuming the hypothesis of
Lemma 3.0, we can collect these results as

Proposition 3.1.1: An increase in the population of a particular tribe
leads to an increase in the urban employment of all tribes. It
leads to a decrease in rural employment as well as the unemploy-
ment rates of all other tribes. Finally, it leads to an increase
in rural employment and unemployment rate of its own tribe if and
only if

Z. 5. .Cef -e"";

1 + : > 0 (3.1.1)

A.n .
1 ir

Thus, an increase in the population of the i tribe leads to in-
creased urban employment of all groups and to a decrease in the rural
work force of all other groups. Put differently, and somewhat boldly,

Punjabi emigration to the Middle East leads to an increase in Pathan or

15
Baluchi urban unemployment rates.

17

These results are interesting and at first seem counter- intuitive.
As population of a particular group increases, one would expect labor of
that group to become cheaper and to be more intensively employed in the .
city. Given the overriding assumption of labor substitutabil ity one would
expect labor of all other groups to be substituted against, leading to
their increased urban unemployment and to increased rural employment rates,
This is contrary to what the results suggest. The reason is that these
are only the first round effects and equilibrium is not attained at this
stage. As the labor of all other groups is let off from the city, it too
becomes cheaper. Since the urban wages are determined from rural wages
and urban unemployment rates, they fall to an extent that nullifies the
first round effects leading to Proposition 3.1.1.

Some special cases of Proposition 3.1.1 may be worth pointing out
briefly. If the group whose population increases has a fixed rigid wage
in the urban center, i.e., e, = e =0, (3.1.1) is automatically fulfilled
and thus the elasticities of substitution for all other sectors are
strictly irrelevant. In this case, the urban and rural employment rates
of all other tribes are unaffected. It is worth reminding the reader
that it is precisely this case that was studied by Harris-Todaro in a
two-sector context.

Another special case is of interest. This is a situation when there

is an exogenously given proportional wage differential between the urban

and rural wages of the tribe whose numbers increase. In this case e = 1
^ w

and (3.1.1) applies only to rural employment since the own unemployment
rate is independent of population growth.

18

We can now use the results of Proposition 3.1.1 to discuss the norma-
tive effects of population growth of a particular tribe. As any reader
of Bhagwati [3] knows, population growth may well be immiserizing in
terms of aggregate welfare, given the variety of distortions in our model.
It is of some interest to see how the above result could be sharpened and
also generalized to the welfare of the variety of groups in our
model .

Consider first aggregate welfare given by

n

W„ = (T p. X 0+ p X . (3.1.2)

A -^i ir ir ^u u

It is easy to check that

m^/U. =-Jwj>Lj^(9Xj/3a.) (3.1.3)

This is intuitively appealing as it says that aggregate welfare will go
up if the size of urban unemployment of each group, weighted by its
shadow wage, goes down. We have seen from Proposition 3.1.1 that this
is indeed so for all j ^ i. Thus the reader can write for himself neces-
sary and sufficient conditions for population growth to be immiserizing
in terms of aggregate welfare. It should be noted that for the special
case e = 1, population growth is never immizerizing.

The situation as regards regional welfare is much more clearcut.

when

J _

we consider the value of regional output Ui = p^-^X. as an index
• K jr jr

/J

of welfare. Then ^ — = w. ^^ shows clearly that with population growth
in the i tribe, j regional welfare falls. Further if the necessary and

19

sufficient condition in Proposition 3.1.1 is not satisfied, then the wel-
fare of all the rural regions taken together falls. Thus inflow of re-
fugees in a particular region can reduce aggregate regional welfare.

It is interesting to note that if the necessary and sufficient con-
dition of Proposition 3.1.1 does not hold, then the welfare of the working

class as a whole given by }v^^l. + Tw. L . , improves and that of the rural

^ 3^ 3^ ^ ju ju

landlords given by Ip^Y^i ' l^ir'"ir worsens.

The reader can now provide for himself necessary and sufficient con-
ditions under which tribal welfare Wt(=P- X. +w. L. ) and the urban

T' '^jr jr ju ju

n
capitalists' welfare U (= p X - I w- L. ) improves or worsens.

U UU ■i=lJ'^J'^

3.2 Region- Specific Capital Inflows

The importance of the study of region-specific capital inflows needs
little justification. In the context of South-East Asian economies, the
problem has attained even more significance once one views the introduc-
tion of the Green Revolution technology with its attendant changes in ir-
rigation methods and use of fertilizers as a capital inflow. Huge capital
investments were made in order to increase the effective area under culti-
vation and its fertility. The socio-economic implications of this techno-
logy and what it has achieved have been extensively discussed elsewhere
but the general equilibrium repercussions of inflow of capital has been
seldom modelled. The following proposition brings out the nature of
such repercussions in the context of our model and that is without even
introducing essential complications like inequality of land holding and

20

indebtedness of the rural peasantry.

Proposition 3.2.1: An increase in the inflow of capital in the i

region will increase rural employment and decrease urban employ-
ment for members of the i region if

eh.
= ^_^_ i^_ > 0 (3.2.1)

1 T\ A ir =

ir

Satisfaction of this condition also ensures an increase in rural
employment of all other regions, a decrease in the urban em-
ployment of their members and a rise in their unemployment rates.

So under the above mentioned condition total rural employment in the
economy increases and total urban employment falls. The decrease in urban
employment of other regions is because of second round effects dominating
the first round ones as explained in the context of population growth in
Section 3.1. It is worth pointing out that if the members of the i
region face rigid urban wages, i.e., e,^, = e^ = 0, then the above proposi-

W A

18

tion is automatically true.

We observe readily that under (3.2.1) the welfare of the working
class as a whole deteriorates, that of the landlords improves and that
the regional welfare of all the regions improves. Moreover, following

Calvo [6], if we assume that the members of the i tribe only are

20

unionized in the urban sector then their unemployment rates remain un-
changed and combining this with (3.2.1) and (3.1.3) we get the result
that the overall welfare of the economy deteriorates as a result of
such capital inflow. The reader can provide conditions for himself for

21

the more general case.

Since (3.2.1) is a sufficient condition only, it is worthwhile to
look at conditions under which capital flows to any rural region will
lead to a decrease in employment there. It is important, because, in
the context of labor surplus LDC's,such capital inflows are thought of
as undesirable since they tend to be labor replacing.

.th

Proposition 3.2.2: An increase in the inflow of capital m the i
region will reduce rural employment there if

K. < 0 and (3.2.2)

.6.. . .

(l-e^)(l-e^)-^Z. + ri-e^;6..e'' - (1-e^) > 0 (3.2.3)

W A L. 1 A 11 w w

1

We conclude this subsection with the observation that a change in

■f" h

p. , the price of the i rural commodity, has identical qualitative
effects as a change in K. , the capital stock of the i region. This
can be seen on inspection of (3.4b and c).

3.3 Minimum Wage Legislation

The policy issue we consider now is a change or a repeal of the
minimum wage laws. It is worth reminding the reader that we consider a

J. u

situation where the urban, minimum wage of the i tribe is greater than

the market-determined wage.

Let w. = j-^ be the institutionally fixed minimum wage for the i

tribe and note that in this case e^ = e,^ = 0 and e^ = 1 .

w A J-

Proposition 3.3.1 :

a) An increase in the urban minimum wages of the i tribe will

22

lead to a decrease in their urban employment and an increase in
their unemployment rate. It will increase rural employment if
and only if

\6 \ > 1 (3.3.1)

ii

b) It necessarily leads to a decrease in urban employment of
all other tribes and an increase in their rural employment
and urban unemployment rates.

The above result is interesting in that it gives conditions under

which an increase in the urban wage rate of a particular ethnic group

n
will increase the Overall unemployment rate I X.. Further, if (3.3.1)

i=l ^

is not satisfied, then the level of unemployment of the i tribe will
increase. This condition in a two sector setting reduces to the elasti-
ticity of labor demand in the urban sector being greater than unity which
is precisely the Corden-Findlay [7] result. But since theirs is a two-
sector setting they do not have any proposition like 3.3.1b.

Given Proposition 3.3.1 and the various welfare functions discussed
in Section 3.1 the reader can supply for himself as detailed a welfare
analysis as he desires; the results do not merit a case by case treat-
ment here.

3.4 Employment Quotas

The last policy issue we consider is in connection with the govern-
ment fixing the quota of urban employment of a particular tribe. In a
country like India where the labor force is heterogeneous, one of the ways

23

the government has been trying to bring the backward and socially handi-
capped people classified under "scheduled castes and tribes" to the fore-
front is by directly increasing their opportunities for employment in

21
the urban centers. Of course, employment quotas for socially backward

groups are not only found in India but in many other LDC's.

Recall that the size of the urban unemployed of the i tribe is

■f" h

given by A.L. and let the i region be the one whose members enjoy an

urban employment quota. Let k. = ' ^^ be the unemployment rate speci-

^ ^i

fied initially by the government. Such a quota leads to two modifications
in the equation system (3.2) namely (i) the submatrix A. has zero in place
of i.^ leading to A. = |A. | = ii.^(]-e^) and (ii) ((1-k.)^^: - k^.k^.) re-
places ^. in u . .

th
Proposition 3.4: An increase in the employntent quota of the i tribe

will increase urban employment and decrease both rural employment
and urban unemployment rate of all tribes.

Since, with a rise in employment quota the unemployment rate falls
for the i tribe, there is an increased tendency for them to migrate
which reduces their rural employment. Intuition suggests that since the
urban employers have to employ a higher number of i tribal people,
they will substitute against the members of the other tribes and their
employment levels will fall. But this fall in employment causes an excess
supply of rural labor leading to a fall in their rural wage rates and
hence urban wage rates and eventual increased employment. Since their
unemployment rates drop too, increased migration takes place causing a

24

fall in their rural employment. The reader can readily see that given
the welfare functions in Section 3.1, the overall workers' welfare will
rise and that of landlord's fall. Regional value of production and
hence regional welfare will fall too. Again, conditions can be derived
to sign overall welfare, urban capitalists' welfare and tribal welfare.

4. OTHER POLICY ISSUES

In the introduction we pointed out that the analytical structure of
this paper could handle a wide range of policy issues other than the ones
we have already discussed. Our purpose in this section is to briefly
point out the different situations in which this framework can be use-
ful.

4.1 Manpower Planning

One of the assumptions we have made throughout the paper is that the
cost per man to the urban employer is identical to the wage that is paid

out. Suppose this is not so and the urban employer also bears some addi-

22
tional training costs \l) that are not passed on to the employee. In this

case, (1.5b) is replaced by p -r-. — = w. + ii^.(')» where i|;(«) typically

U oL . 1 U I

depends on r. and the quit rate q. which would itself depend on X.. It
is easy to check that our methods can easily handle this case.

4.2 Urbanization

An extensive analysis of the effects of capital inflow in the rural
sector has been made in Section 3.2. The positive and normative effects
of capital inflow in the urban sector can also be arrived at along the

25

same lines. This is important since LDC's have gone through extensive
urbanization in the recent past. Equally important is an analysis of
excessive pressure on the urban center as a result of migration. This
has been of some concern to urban planners because of the resource waste
associated with it, e.g., time waste due to congestion, high maintenance
costs of rapidly deteriorating sanitage and sewerage systems, etc. Our
framework can accommodate analysis of such issues.

At any point in time the total number of people in the urban center

n
is 5^ L. (1 +X.). Let the cost associated with the population pressure

be given by a function

C = (j)

J,^u(i^\-)

where ({>' > 0, c{)" > 0

(4.2.1)

Total welfare is now given by

Wa = Pn^n ' .^/ir^•r " ^

I L. (1 +X.)

(4.2.2)

The policy measures we have talked about can now be evaluated in the

light of this modified welfare function and the costs associated with

urbanization will play a crucial role in dictating the choice of poli-
cies.

4.3 Wage Subsidies

Wage subsidies have been discussed at length in Khan [13] in the
context of a two sector model with intersectorally mobile capital. We
leave it to the reader to discuss the consequence of such subsidies in

26

the context of our model here. It is worth pointing out, however, that in
the case of w. = r. , the effects of wage subsidies are identical to
changes in j". , a problem already discussed above in section 3.3.

4.4 Sub-Optimal Tariff Policy

Finally, it is worth pointing out the relevance of our model to ques-
tions dealing with the positive and normative effects of tariffs in a
multi sectoral economy riddled with distortions. In particular, one can
study this problem under a variety of assumptions pertaining to the dis-
bursement of tariff revenue.

5. CONCLUDING REMARKS

In this paper we have presented a model of a small, open economy which
can be used to study the effects of various policy changes on the distri-
bution of income between landlords, laborers, capitalists and on dif-
ferent regional groups. We obtain worthwhile comparative-static results
in a setting for which the existence, uniqueness and local stability of
equilibria cannot be deduced routinely from corresponding results pertain-
ing to the general competitive models as set out in Debreu [9]. It is
thus satisfying that the hypothesis 0 < e^ < 1, e\ < 0, for example, has

— W — A —

a role to play in the existence, uniqueness, stability and comparative
static results.

Our basic model can be seen as a multi sectoral generalization of the

Ricardo-Viner model, one that is somewhat different from the generalization

n
presented by Jones [12]. If we let X^^ = F^( J L.^,Kj, and fi.(w.^,A.,Xj) =

27

w. i w in equations (2.2b) and (2.3a) respectively, we obtain Jones'
model in [12]. As such we have presented a multi commodity trade model
that may also be well -suited for answering trade- theoretic questions
particularly for LDC's.

A'l

APPENDIX I: PROOF OF THEOREM 2.1.1
We introduce the formal proof by sketching the underlying ideas;

■f" h

focus on the i market and assume that w (i) is fixed. Given Assumptions
2.1.2 and 2.1.3, we can find the value of w. tht will clear that i market,
i.e., equate ({).(w. ,w (i)) to zero. Figure 1 illustrates the procedure
for doing this. Given the boundary conditions, we can find m, in at which
(j).{',w (i)) is respectively positive and negative and an application of
Bolzano's Theorem (see [1, p. 73]) yields the result. We thus get a map-
ping ip from w (i) to w. . Doing this for all the markets, we obtain a
mapping ijj(0 = (ij^-i (•)>.• •»i|^p(')) which takes the set of rural wages into
itself. A fixed point of such a mapping, if it existed, would give us a
Harris-Todaro equilibrium. However, the problem is that we cannot prove
the existence of such a fixed point through any of the usual theorems
since we cannot limit i|i(') to a compact set. This is because the bounds

m, m vary from market to market and depend on w (i). To overcome this
~ r

difficulty, one has to choose bounds 1/m and m for all markets right from
the start and restrict each (f).(') to the closed interval [l/m,m]. If,
in this interval, we do not find a wage that clears that market, we choose
one that minimizes absolute value of excess demand, i.e., in Figure 1,
A' when the interval is AA' or B when it is BB' . We then proceed as
above, and find a fixed point of the mapping rp. As m is allowed to get
larger and larger, we can generate a sequence of fixed points. We now
bring in the boundary conditions embodied in Assumption 2.1.3 to show
that there exists a Harris-Todaro equilibrium for large enough m. As-

A-2

sumption 2.1.1 is used to generate non-negative equilibrium values of X..
In conclusion, it is worth mentioning that it is the absence of a direct
analogue of Walras' Law for Harris-Todaro equilibria that necessitates
the use of an argument different from the one conventionally used to prove
the existence of competitive equilibria.
We can now present

Proof of Theorem 2.1.1: We shall need the following notation, additional
to that provided in the text.

M = {x e R I (1/m) £ x <_ ml, m a positive integer;

M"^' = TT M; C.:M" ^ M; ijj.iM" -^ M and i|;:M" -> m"
k=l ^ ^

C.(w^) = {z £ M I 4).(z,w^(i)) > 0}U{l/m}

'^^{vJ^) = (w.^ £ ^i^\^ I ^ir '^■'"■'"T'zes (|)^(w^) over C.(w^)}

C.(w ) is nonempty and bounded for all w e M and for all i. Given
continuity of tf).(») in w , it is also closed.

C.(') is a continuous correspondence over M . To show this, we need

n 1

only consider the restriction of C.(') to M ' . We use C.(«) or C.(w (i))
to denote this restriction. The upper-semi-continuity of C.(«) is straight-
forward. For lower semi-continuity, let w^ (i) -^ w9 e C(w°(i)). We have
to construct a sequence (wV } such that wV^ -> w^ and wV (i) for all v.
The araument revolves around three cases.

A- 3

Let 4).(w°) > 0 and let

1 ' r

wV^ = w°^ - (1/v) if w°^ =

m

.0

= w. + (1/v) otherwise

V V,

There exists v such that v >v implies (f).(w. ,w (i)) > 0. Suppose not,

i.e., for all "'^, there exists v"^ > v"^ such that 4).(w.'^,w "^(i)) ^ o.

— ^v ir r —

V . V -
Lt (l).(w.'^,w "^(i)) = (t).(w ) £ 0, a contradiction. We can now easily

construct the required sequence.
Let (|). (w ) = 0 and w? > (1/m). For each v consider the sign of (J).(w? ,w^(i))
Since <^^{') is a single-valued function, it cannot be zero. If it is positive, let
^ir "^ ^ir* I ^ it is negative, let wV^ = x^ where ({)^. (x^,w^(i ) ) = 0. Given that
(j).(*) is a continuous function and that 0.(0, w (i)) = <», we are guaranteed by
Bolzano's Theorem (see [1, p. 73]), that x^ exists. The only question is whether
X e M. We can assert that there exists v such that v >^ v implies x^ e M. Suppose

not, i.e., for all v"^, there exists v^ > v"^ such that x "^ i M. Certainly x "^ < w?

— ^ -^ — ir

V . V . V.

for all j. If not, <^.{x '^,w^"^(i)) = 0 implies (i).(w? ,w^"^(i)) > 0, given that
4).(-) is a decreasing function of w?^. This case has been disposed of earlier.
Thus X is a bounded sequence and by the Bolzano-Weierstrass Theorem, (see [1,

p. 43]), it must have a convergent subsequence x -^ . Let Lt x ^^ be x. Then

k->oo

v., v.,

Lt <j).(x ,w (i)) = <}).(x,w°(i)). Since (j). is a single valued function

, X = w. ,
ir

A- 4

v.,
0 1 k

and since w. > 1/m, for all large enough k, x "• e M. We have our required con-
tradiction. We also have a sequence to complete the argument for lower semi-con-
tinuity in this case.

The only case remaining is when (f)-(w ) < 0 or ^-{vi ) = 0 and w. = 1/m. In
this case, let w. = 1/m for all v.

Given that C(') is a continuous correspondence and that <t>'{') is a continuous
function, we can appeal to a Theorem given in Debreu, (see [8], p. 19]), to assert
that '];.(•) is an upper semicontinuous correspondence. Given that 4).(0 is a single-
valued function, upper semi-continuity reduces to continuity of i|;.(').

ip is thus a continuous mapping of M into itself. Since M is easily seen to
be a nonempty, convex, compact subset of R , we can apply Brouwer's Fixed Point
Theorem (see [8, p. 26]), to assert the existence of w e M . Let A. = £.(w. )
and w^^^ = (l+x'J')w"^ . Under Assumption 2.1, X^ > 0.

If 4) (w ) = 0 for all i, the proof is finished, but this will, of course, not

*
be the case for arbitrary m. We will now show that there exists m such that for

all m >_ m ,

1/m* < w"^ < m*. (A. 1.1)

ir

*
We first provide an argument for the upper bound. Suppose not, i.e., for all m.,

* m.

there exists m. > m. and an i such that w.^ > m.. Since i is between 1 and n, there
J - J ir - J

m .
at least one i for which w."^ is becoming arbitrarily large. By construction

0

m . m
0 'o

Ml . Ill . _

(j). (w.'^ ,w "^(i )) > 0. But we know from Assumption 2.1.3 that there exists j such
iirro'—

m . m .
for all j > 3,4). (w."^ ,w "^(i ) < 0, a contradiction. The argument for the lower

1 ' 1 r' r ' 0

0 0

A- 5

bound is similar. Suppose for all m., there exists m. _> m . such that for some Tq

m . m . m .

w."^^ = (1/m.)- By construction, this implies that (f). (w."^ ,w "^(i )) <_ 0. But,

^0*^ 0 0

again we know from Assumption 2.1.3 that there exists j such that for all

m mj
J > 3A^ (w^.^^,w^(i)) > 0, a contradiction,

0 0

* * •

Now letw =w ,X. =A. ,w. =w. ,L. =L. (w) and L. = L. (w. ), giving
r r 1 1 lu lu iu lu u' ir ir ir" ^ ^

us our Harris-Todaro equilibrium. Certainly for all i, 4)-(w ) = 0. If not, we
contradict (A.l .1) Q.E.D.

Remark A1 : If we do not assume that ({){•) is a decreasing function of w. , the proof
fails on several counts, most important of which is probably the failure of lower
semi-continuity of the correspondence C.(w (i)). Figure 4 illustrates this. Of
course, if there is only one market, i.e., only one equation 2.1.1, we need rely only
on Balzano's Theorem and the simple argument given in the Idea of Proof will suf-
fice. In this case, we do not need to assume that ({)(•) is a decreasing function of
w. if we do not insist on a unique equilibrium.

Remark .A2: It is well to remind the reader that Assumption 2.1.2 does not
guarantee a unique Harris-Todaro equilibria.

A-6

APPENDIX II: PROOF OF THEOREM 2.2.1

The proof of Theorem 2.2.1 is a consequence of Sard's Theorem
[9], which in a informal version states that the set of values taken
by a differentiable function at points at which its derivatives are zero, is
"small," i.e., of Lebesgue measure zero. We can now present

Proof of Theorem 2.2.2: Define a mapping $ from the strictly positive orthant of
R , R, ,j into R , where

$.(w ) = L. (w. /p. ) + (1+2. (w. ))L. (w ) (i=l,...,n).

■V r ir ir ^ir^ i ir lu r

Then a Harris -Todaro equilibrium can be characterized by J" ( =£) . Given Assumption
2-2.1 a routine application of Sard's Theorem, [9, p. 388], allows us to deduce
that the set C of 3^ in r" for which the Jacobian of $ has rank smaller than n is
of Lebesgue measure zero. Since the Jacobian of J is a continuous function of
w , 4)~ (C) is a closed set. Hence C is a closed set. For any ^ ^ C, the inverse
function theorem, [ 1, p. 144], applies and the corresponding Harris-Todaro equi-
libria are locally unique, and continuous in ^. Q.E.D.

Proof of Remark 2.2.2: As in Debreu [9], it can be shown that for a compact
subset K of r", ^~ (K) is compact. This allows us to denote that the set of
Harris-Todaro equilibria is finite for all =£ in a closed set of Lebesgue measure
zero.

A- 7

APPENDIX III

In this section, we provide the proofs of the results in Section 2.3. We begin
with a

Lemma A.I : Under Assumptions 2,3.1 to 2.3.3, the matrices A,(-B) in A, (see Figure 3
are stable Metzler matrices, such a matrix being one with diagonal elements nega-
tive, off-diagonal elements non-negative and the real parts of whose eigenvalues
are negative.

Proof: The necessary conditions for profit maximization in the urban sector give

us (8F /8L. ) = w. , (i=l,...,n). Differentiating these equations with respect to
w. , we obtain

[a^F /3L. 8L. ][aL. /3w. ] = H[3L. /8w. ] = I
■■ u iu ju-"- ju lu-" *- ju lu-"

where I is an identity matrix of order n. Under Assumption 2.3.2, H is symmetric
negative definite. Hence H" is symmetric and negative definite. Thus by Assump-
tion 2.3.2 [3L ./9w. ] is a stable Metzler matrix. Now by a Theorem in Arrow, [2,
p. 8], the class of stable Metzler matrices is closed under pre or post multipli-
cation by a non-null, non-negative diagonal matrix. It can be easily checked that
the class of symmetric, stable matrices is closed under addition of a negative,
diagonal matrix. These facts along with Assumption 2.3.3, allow us to complete the
proof. Q.E.D.

A-8

Proof of Theorem 2.3.1 : Linearizing the differential equations (2.3.1) around a
locally unique Harris-Todaro equilibrium, we obtain

Dw^ = hA(w^-w*) + hB(X-A*)

D^ = gC(w^-w*) + gD(A-X*)

where h and g are diagonal, matrices with typical entries H!(0) and GI(0) and A, B,
C, D are square matrices constituting A, (see Figure 3). Using again, the theorem
in Arrow quoted in the proof of lemma B.l above, we can show that all the eigen-
values of hA and gD have negative real parts. Application of the Theorem in [11 ,
p. 181], completes the argument. Q.E.D.

Proof of Theorem 2.3.2: The theorem is an easy consequence of two theorems in
Okuguchi, [16], on matrices with dominant diagonal blocks. Adopt the Minkowski
norm for a matrix with the absolute value norm for vectors. Then local uniqueness

of Harris-Todaro Equilibria follows from Theorem 1 in [16] and local asymptotic
stability from Theorem 3 in [16], and the Theorem in [n , p. 181]. Note that A and
D are stable, Metzler matrices, the former by virtue of Lemma B.l, as is required
for Theorem 3 in [16]. Q.E.D.

Proof of Lemma 3.0: Using the arguments in the proof of Lemma Al , it is easy to
show that D is a stable Metzler matrix. Sine CA~ B is a diagonal matrix, 5 is
stable Metzler matrix. Now by a theorem in Arrow [2, p. 7], the inverse of a
stable, Metzler matrix has all non-pcsitiv'2 entries. Q.E.D.

D-1

^^.(•,w^(i))

-=£,

FIGURE 1

D-2

FIGURE 2

D-3

a;

E I—

3

E E

3

E

I— r<

3

- r<

O)

- E

13
E

13
E I—

cr

E 3

3

3
E E

c:

E 2

O)

+

c

(U
•r-

+

3
E

(U

- ^

CO
C9

II

<

D-4

M<J>,-(-.w^(i))

FIGURE 4

F-1

FOOTNOTES

For example see the Five Year Plan documents of the Government of
India. Indeed, in terms of general objectives of the Indian Five Year
Plans, the following excerpt from Chapter XXXIV of the Third Five Year
Plan is illuminating. "... besides ensuring rapid and sustained growth
for the economy as a whole, at least during the next two or three Plans,
measures for advancing the economic and social interests of scheduled
tribes, scheduled castes and other weaker sections of the community
should be so intensified, that they do, in fact, reach a level of well-
being comparable with that of other sections of the population."

For example, see the reports cf the Committee on the Welfare of
Scheduled Castes and Scheduled Tribes published yearly by the Lok Shabha
Secretariat, Government of India. Together with general developmental
expenditures, there exists regulations for the reservation of posts
for the members of scheduled castes and tribes in the public sector.
The Committee notes "... that the present policy of the Government in
reserving a percentage of vacancies occurring every year for Scheduled
Castes and Scheduled Tribes is not only equitable but is also in the

overall interest of the Administration."

3
Note that this is a somewhat unconventional way of defining the

unemployment rate. In [ 6 ] and [19], for example, this rate is defined

as the ratio of the unemployed to the total urban labor force. This

translates to X./{l+A.) in our notation.

^Also Table 1 in [14].

F-2

5
One such set of sufficient conditions are the Inada conditions (see

2.1.4 below) and Assumptions 2.3.1 and 2.3.2 below.

This was first pointed out to the authors by T. N. Srinivasan.

The reader can usefully compare Corollary 2.1.2 with the existence
theorems given in Srinivasan-Bhagwati [18] and Calvo [6].

o

These facts are exploited in a two-sector, mobile capital setting
to present an example of non-existence of a Harris-Todaro equilibrium

in Khan [15].

9
It must be emphasized that the results presented in this section

are subject to all the blemishes of the tatonnement stability theory of
competitive equilibria. We have in mind particularly an adjustment pro-
cess not based on maximizing behavior of individual agents, presence of
an auctioneer and no trading out of equilibrium.
^°Also Table 1 in [16].
For a precise definition of local asymptotic stability of equi-
libria, see, for example, [11, pp. 185-86].

Theorem 2.3.1 does not need e^ < 1 for its validity.

w — "^

13

See Figure 3.

The only exception to this is X. which is given by dX.|(l +X.) to

allow for situations when X. may be zero.

15
Pathans, Baluchis and Punjabis are all ethnic groups of Pakistan.

Note that K. is a parameter.

Unlike subsection 3.1, no necessary and sufficient condition can be

provided.

18

If we follow Stiglitz [19] and assume e = 0, then also the above

F-3

proposition is automatically true.

1 9

If other tribes also are unionized, then there might exist linkages

between trade unions, an analysis of which is outside the scope of our

study here.

Following Calvo [6], this implies e^ = 1 and e, = 0.

21
See footnote 2.

22
This has been considered in a two-sector setting in Stiglitz [19].

R-1

REFERENCES

[1] Apostal, T. M., Mathematical Analysis (Reading, Mass.: Addison-

Wesley Publishing Co., 1967.
[2] Arrow, K. J., "Price Quantity Adjustments in Multiple Markets with

Rising Demands." In K. J. Arrow, Samuel Karl in, Patrick Suppes

(eds.). Mathematical Methods in the Social Sciences (Stanford,
Calif.: Stanford University Press, 1960).

[3] Bhagwati, J. N., "The Generalized Theory of Distortions and Wel-
fare." In J. N. Bhagwati, et al (eds.). Trade, Balance of Payments
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[4] Bhagwati, J. N. and T. N. Srinivasan, "On Reanalyzing the Harris-
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