Distributive Bias and
Labor Transfer in a
Two Sector Economy
John W. Mellor
Reprinted from the
Oxford Economic Papers
Vol. 33, No 3, November 1981
TECHNOLOGICAL CHANGE, DISTRIBUTIVE
BIAS AND LABOR TRANSFER IN A
TWO SECTOR ECONOMY
By UMA LELE and JOHN W. MELLOR*
SLOW growth in overall employment and unequal distribution of benefits
from the new foodgrain technologies continue to be two of the most pressing
current problems of many low income countries. There have been efforts to
increase employment rapidly, without substantial increase in the rate of
growth of food production, e.g. in India following the 1971 election.
However, such attempts have generally been accompanied by high rates of
inflation, particularly of food prices. This is because as much as 60 percent
of the increase in income of low income wage earners in developing
countries is spent on consumption of cereals alone (John W. Mellor and
Uma Lele 1973). And yet, the growth in food production in developing
countries has barely kept pace with the growth of population. The foodgrain
sector has thus not only been a slow generator of additional employment
and income; through inadequate supply of wage goods it has also constituted
a major constraint to the growth of nonagricultural employment.
The question of labor transfers has, of course, received extensive treat-
ment in development literature and especially in two-sector models' (most
notably by W. Arthur Lewis, Fei-Ranis, Jorgensen, Todaro and Harris). A
few formulations, such as those by Dixit and Hornsby, also deal with
increasing production of wage goods, but do not allow for technological
change.2 Various others treat the question of marketed surpluses of food,
but do not incorporate it formally in models of growth or relate it to labor
supply as a separate but interacting variable.3 The variations in the distribu-
tive bias of the different types of new technologies in foodgrain production
have, however, been extensively documented in the empirical literature.4
The critical role of the wage goods constraint in creating nonagricultural
employment has also been recognized by policymakers, but only implicitly.
Consequently, unlike Mainland China, few developing countries have had
the political will or the institutional mechanisms to mobilize the limited
*Uma Lele is Senior Economist, the World Bank and John W. Mellor is Director,
International Food Policy Research Institute, Washington, D.C., U.S.A. We are grateful to
Chandrashekhar Ranade for considerable assistance on the paper particularly in developing the
necessary proof. We also acknowledge the contribution of an anonymous reviewer in correcting
inaccuracies and improving clarity of presentation.
1 For a detailed review of two sector models see Mellor (1974).
2 See Mellor (1974).
3 See Mellor (1974).
4Mellor and Lele (1973). For a detailed analysis of several innovations in two major
locations in the Philippines, see Chandrashekhar G. Ranade (1977). See also, for India, C. H.
H. Rao (1975).
UMA LELE AND JOHN W. MELLOR
domestic food surpluses for consumption of wage earners without causing
the prices of food to rise in relation to those in the nonagricultural sector.
These price increases have discouraged decisionmakers from following a
policy of expanding employment.5 Similarly, few developing countries have
relied on rapidly increasing imports of cereals as a way of expanding
employment, partly arising out of a perception of inelastic demand for their
own exportable surpluses.
In agriculture, as the classic sector of diminishing returns, the production
increase necessary to release the wages good constraint is of course achieved
largely through technological change. Agricultural technologies, however,
vary substantially in their distributive bias. They therefore have important
implications for the generation of employment directly in the agricultural
sector. In addition, the different demand elasticities among various income
classes of food producers also affect the size of the marketable surplus of the
wage goods that is generated by the foodgrains sector. The initial employ-
ment effect, and the consequent size of the marketed surplus, thus in turn
affect the prices of food relative to nonfood output as well as the level of
real wages in the nonfoodgrain sector. These factors are thus crucial in
determining the rate at which the wages goods constraint is released and
off-farm employment is generated.
In this context we analyze the effect of alternative assumptions with
respect to distributive bias of technological change in the foodgrain sector
on (a) marketable surplus from that sector, (b) the rate of growth of
nonfoodgrain sector employment, (c) the price of foodgrain in relation to the
nonfoodgrain output and (d) the degree of factor intensity in the nonfood-
grain sector. We examine these relationships with the use of a two-sector
model similar to the large family of dualistic models so as to focus on the
critical role of food production in influencing labor transfers, and to analyze
the complex interactions of the food and the labor markets.
The distinguishing features of the two-sector model developed in this
paper are: (1) incorporation of biased technological change in the foodgrain
sector and (2) separation of the food and labor markets into two indepen-
dent but interacting markets. Rather than assuming that food moves com-
mensurately and automatically with labor, we assume the marketable sur-
plus of food to be influenced by the distribution of income and the different
price and income elasticities of demand of landowners and laborers in the
foodgrain producing sector for domestic consumption of foodgrains. Tech-
nologically induced changes in income distribution in the foodgrain sector
therefore affect the demand for food in the foodgrain sector, the marketable
surplus, the price of foodgrains in terms of nonfoodgrains output and the
rate of labor transfers to the nonfoodgrain sector.
5 For a critical analysis of such policies in India, see Lele (1971).
428 TECHNOLOGICAL CHANGE, DISTRIBUTIVE BIAS & LABOR TRANSFER
The model also provides results relating to the factor intensity in the
nonfoodgrain sector. It illustrates how the directions of change in these two
factors are influenced by the direction of distributive bias and the nature of
interaction between the food and the labor markets. These results are
substantially different from those in previous models.
The sharp differences between low and high income consumers in their
elasticities of demand for food are well documented. In India, for example,
cross-sectional estimates of income elasticities of demand indicate levels of
about 0.8 and 0.2 for bottom two and top two deciles respectively.6 On the
whole, income elasticities of demand for foodgrains are, however, observed
to be less than one and are assumed to be so in this model.7
In order to focus on the most important relationships from the point of
view of development policy, some additional assumptions have been made.
For instance, the sum of the absolute magnitudes of income elasticity of
demand (n) and the elasticity of budget share with respect to the change in
relative price of foodgrains (e) is assumed to be less than 1, as empirically
the absolute magnitude of e is usually expected to be small, i.e. closer to
zero than to 1.
In the labor market, the formulation assumes perfect mobility between
sectors so that, at equilibrium, the ratio between the wage rate in the
nonfoodgrain sector and the average labor income in the foodgrain sector is
constant. The average labor income in the foodgrain sector is determined by
the total labor income generated by the flow of labor in the foodgrain sector
divided equally among the total stock of labor. Per capital income of the
workers in the foodgrain sector then maintains a constant relationship to the
level of real wages in the nonfoodgrain sector. We assume an underemploy-
ment equilibrium in the foodgrain sector at a given wage W as depicted in
Fig. 1. The conditions of low productivity and the labor-leisure choices in
traditional agriculture which lead to such an underemployment equilibrium
have been well analyzed in the literature (Nakajima, 1961, Mellor, 1963 and
Sen, 1966). The assumption of underemployment equilibrium should not be
confused with an assumption of zero marginal productivity of labor.8 Rather
our assumption reflects the widely noted reality of highly elastic supply of
labor from agriculture, if the wage goods constraint is relaxed.
6 Mellor and Lele (1973). For the Philippines, Goldman and Ranade (1976) find that income
elasticity of demand for cereals, mainly rice, in the lowest income decile is 1.05 while it is 0.41
for the top decile.
7The results of the model remain unchanged irrespective of whether wage rate in the
nonfoodgrain sector is a multiple of or equal to the average labor income in the foodgrain
sector. It should be noted, much conventional wisdom to the contrary, that when the physical
environment dictates a short, peak work period, the wage rate in agriculture at that season may
be higher than that in nonagriculture at that or any other season, while concurrently the
average product or total yearly income is lower in agriculture than nonagriculture. For
empirical evidence, see Ranade (1977), p. 108.
SFor a full analysis of this important distinction, see Mellor (1963) and Sen (1966).
UMA LELE AND JOHN W. MELLOR
Labor employed (1 )
01A* 'A LA
Labor employed (1A)
0 1A* 1AL
FIG. 1. Equilibrium in foodgrain sector labor market.
1. Analytical framework
The production function for food grains, assumed to have constant returns
to scale and diminishing marginal rates of substitution, is as follows:
A = F(N, E)
aF a2 A a2A
FE=>O and <0
aE aN2 'aE
where A is the foodgrains output, and N and E are the levels of land and
labor inputs, respectively. Both land and labor are measured in efficiency
430 TECHNOLOGICAL CHANGE, DISTRIBUTIVE BIAS & LABOR TRANSFER
units such that N= xZ and E = ylA, where x and Z are respectively the
efficiency and the fixed amount of land, and y and 1A are respectively the
efficiency and the amount of labor employed. Both x and y are exogenously
given and depend upon technology t.9
It is assumed that technological change increases the efficiency of land
faster than that of labor, that is,
S= Az > A = (2)
dt x dt y
where Az and A. are rates of growth of the efficiency of land and labor
In the foodgrain labor market an equilibrium is reached at a constant real
wage (fW) equalizing the marginal physical productivity of labor and hence,
W= =yFE (3)
such that A < LA where LA is the total foodgrain labor force. Equilibrium in
the foodgrain sector labor market is shown in Fig. 1.
Then the relative share of foodgrain labor is
SL A E (4)
Further, the average income of laborers in the foodgrain sector is,
where r = proportion of foodgrain labor force in total labor force L, that is
r = LAL.
Marketed supply of foodgrains, M,, to the nonfoodgrain sector is the
difference between output and consumption in the foodgrain sector so that
where, C = constant consumption of foodgrains by landlords, and b = budget
share of foodgrains for laborers such that,
b = b(P, y) (7)
where P is the relative price of foodgrain output with the price of nonfood-
grain output as the "numeraire". Further,
ab p ab y
e<0O and -1<0
ap b ay b
9 For convenience, time and technology are denoted by the same variable t.
UMA LELE AND JOHN W. MELLOR
where e is the elasticity of budget share with respect to change in price and
Ti is income elasticity of demand for foodgrains. Note that the model thus
allows for different income elasticities of demand for landlords (assumed to
be equal to zero) and laborers (assumed to be less than one).
The production function for the nonfoodgrain sector is a Cobb-Douglas
linear homogeneous of the first degree as follows:
Q = K"L'- (8)
where, Q = nonfoodgrain output, K = exogenously given capital stock, L, =
labor input in the nonfoodgrain sector, and a = relative share of capital
In the nonfoodgrain sector laborers are employed at a wage rate W
equalling marginal productivity of labor, i.e.,
W= (1-a) = (1 -a) 1 (9)
L, (D"(1- r)-
Labor migrates from the foodgrain sector to the nonfoodgrain sector until
the wage rate in the nonfoodgrain sector is equal to a constant proportion 3
of per capital income of foodgrain laborers.
1 1 AWA
W = ) =pPlA where f3-1 (10)
S(1 r) LA
depending upon marginal productivities of labor in the two sectors.
Market demand for food in the nonfoodgrain sector, M,, is equal to the
budget share allocated to food consumption out of wage income by the
nonfoodgrain laborers, i.e. b(W/p) L. Thus in the foodgrain market,
equilibrium is attained when
M,=A-C-bSLA=b-L =MD, (11)
A-C- A--b=0 (12)
This describes the general equilibrium system. The formulation consists of
six predetermined variables, namely, capital (K), total labor (L), quantity of
land (Z), foodgrain wage (W), and efficiencies of land (x) and labor (y). It
can be shown that given these variables all the endogenous variables (lA, A,
SL, r, P, M, and W/P) can be uniquely determined. Note, given W, Z, x and
y, one can uniquely determine the labor input (lA), output (A) and the share
of labor (SL) from equation (3), (1) and (4) respectively (Fig. 1).
Further, differentiating (10) and (12) partially with respect to r, we get,
432 TECHNOLOGICAL CHANGE, DISTRIBUTIVE BIAS & LABOR TRANSFER
respectively, the following
P= P 1 +- a ... for labor market, and (13)
9r r 1-r
=-... for foodgrain market. (14)
All the terms on the right hand side of (13) are positive i.e. OP/ar>0, and
hence the price of foodgrain relative to nonfoodgrain output declines when
the proportion of population in the foodgrain sector declines; both with
respect to the labor market. This is explained by the fact that, ceteris paribus,
as the proportion of population in the foodgrain sector declines, per capital
income in that sector increases, and for the equilibrium in the labor market
to be maintained the adjustment has to come from a decline in the price of
foodgrain relative to nonfoodgrain output. Additionally, since r > 0 > e the
right hand side of equation (14) is negative. Therefore the price of foodgrain
relative to nonfoodgrain output increases as r declines with respect to the
foodgrain market. Again, this is explained by the fact that ceteris paribus, as
0 r* r-l
Proportion of labor in foodgrain sector (r)
FIG. 2. General equilibrium.
UMA LELE AND JOHN W. MELLOR
the proportion of population in the foodgrain sector declines and per capital
income in that sector increases, the wage rate in the industrial sector also
increases, raising effective demand for foodgrain and their price relative to
nonfoodgrain output. These opposite phenomena lead to the unique values
of P and r given the predetermined variables and the values of IA, A and SL
as shown in Fig. 2. Then, finally, W/P and Ms can be determined from
equations (10) and (6). Stability of this equilibrium is shown in Appendix A.
II. Sensitivity analysis
The technological change affects first the efficiencies of land and labor.
Change in the efficiencies would in turn affect the relative share of labor
depending upon the nature of substitution between land and labor as
dSL 1 ( dy 1
=(o- 1) (15)
dt SL dt y
where o- is the elasticity of substitution between land and labor. This
equation implies that the relative share of labor would decrease, remain
constant, or increase depending whether o- is less than, equal to, or greater
The sensitivity of each of the endogenous variables such as foodgrain
labor, price of foodgrains in relation to nonfoodgrain output, marketed
surplus and real wages with respect to effect of technological change on
labor's share is shown in the following sensitivity matrix. It also shows the
sensitivity of these variables to population growth and growth of nonfood-
grain capital separately.
The most interesting results are obtained in the case of an increase in
foodgrain output that is accompanied by a change in relative factor shares.
The results obtained for a constant labor share are reinforced when labor's
share declines as a result of an increase in foodgrain output. In the case of
W/P, the real wage rate in the nonfoodgrain sector, the effect of increased
foodgrain output accompanied by decline in labor's share directly depresses
per capital income of the labor force in the foodgrain sector while decline in
labor's share causes a decrease in the proportion of population in the
foodgrain sector. This latter phenomenon acts to increase per capital income
of the existing population in the foodgrain sector. Thus, the direction of
change of the equilibrium level depends upon the relative magnitudes of
these opposite influences.
When an increase in foodgrain output is accompanied by an increase in
labor's relative share, the effect on the proportion of the labor force in
o This relation can be derived by using equations (3) and (4).
434 TECHNOLOGICAL CHANGE, DISTRIBUTIVE BIAS & LABOR TRANSFER
Increase in foodgrain output (A) Growth of
when relative share of labor (SL)
Endogenous when relative share of labor (S Capital stock Population
variable Increases Constant Decreases (K) (L)
Proportion of +
foodgrain labor in
total labor (r)
Price of + +
Real wage in + + +
Marketable + + +
a See Appendix B for the mathematical steps in deriving the sensitivity matrix on the basis
that 0< q <1, e <0 and 0 < <1. Negative (positive) sign means decline (increase) in that
variable. "" means the direction of change in that endogenous variable is indeterminate.
agriculture (r), on the price of foodgrains relative to nonfoodgrains (P) and
on marketable surplus (M,) may take either sign. If labor's relative share
increases only slightly, relative to the increase in foodgrain output, the effect
of increased foodgrain output on r, P and M, will be greater relative to that
of increased labor's share. However, if the labor's share increases substan-
tially as a result of the increase in foodgrain output, the effect on r, P and M,
may be opposite to that when increased foodgrain output is not accom-
panied by changing labor share.
These interactions are discussed in the dynamic analysis in the next
section. The preceding discussion does suggest that in the context of growth
the most interesting results in the sensitivity matrix are those relating to
labor's share in foodgrain output. They show that with an increased labor
share, as exemplified by production increases in a traditional foodgrain
sector, the marketed surplus of foodgrain may decline and the real wage in
the nonfoodgrain sector may increase. Converse changes may be expected
when technological change decreases labor's share in foodgrain output. The
factor shares in the foodgrain sector are thus of crucial importance in the
growth of the nonfoodgrain sector in a dualistic economy.
This analysis suggests not only that change in factor shares may be a
particularly important feature of current "green revolution" agricultural
technology, but also helps remove a growing anomaly in the perception of
UMA LELE AND JOHN W. MELLOR
Japanese economic history. Recent downward revision of estimates of the
growth rate for agricultural output in the early Meiji period are consistent
with retention of the earlier estimates of growth in nonagricultural employ-
ment if one takes into account the acceleration in agricultural marketing
associated with change in agricultural technology (See Thomas Smith 1959
and James Nakamura 1966). The yield increasing agricultural technology
associated with the Meiji period shifted factor shares away from labor as
compared to the highly labor-intensive methods of production increase in
the preceding Tokugawa period (Sen 1966). Thus we see agriculture's
contribution to overall Japanese growth as arising from the effect of tech-
nological change on both the level of output and the change in factor shares
arising from that increased output.
III. Dynamic analysis
The dynamic analysis involves the simultaneous effect of change in factor
shares through change in factor efficiencies, population and capital stock on
nonfoodgrain employment, real wages, terms of trade and marketable
surplus. These results are presented in the following equations.
drl dSL 1 dA 1 dLl\ /d l1 dL1\
--=cc c2 3 -Cc4 (16)
dt r dt SL \dt A dt L dtQ dt L c 6
dW, 1 dK 1 dL 1(17)
dt W a\dtK dt L,
dP 1 dSL 1 /dAl dL l\ dA 1 dQ 1\
_= d,-- d2 -- 3-d -t-d4 (
dt P dt S, dt A dtL L d3t A dt (18)
= el--e (19)
dt dt r
where c's, d's and e's are all positive given that 0 < l < 1 and 0 < j e < 1.1
From equation (16), the influence of various factors on the rate of growth
of nonfoodgrain employment can be derived. For example, the greater the
rate of growth of foodgrain output, the faster the rate of growth of
nonfoodgrain employment. The rate of growth of employment in the non-
foodgrain sector is inversely related to the rate of change of labor's share in
Technological change in the foodgrain sector which increases labor's share
in output dampens the rate of growth of nonfoodgrain employment. This
occurs through: (1) decreasing the marketed supply of foodgrain, and (2)
increasing the level of wages in the nonfoodgrain sector required to with-
draw labor from foodgrain production. Technological change that reduces
1 See Appendix C for derivation of the table.
436 TECHNOLOGICAL CHANGE, DISTRIBUTIVE BIAS & LABOR TRANSFER
labor's share of foodgrain output may increase the growth of nonfoodgrain
employment. Equation (19) shows the identity between the rate of growth of
nonfoodgrain employment and marketable surplus. Thus it can be seen that
the same factors shown on the right hand side of equation (16) determine in
the same manner the rate of growth of marketable surplus.
Equation (17) shows that there is a monotonically increasing relation
between the capital-labor ratio in the nonfoodgrain sector and per capital
income in the foodgrain sector. Also, since a <1 the capital-labor ratio
increases more rapidly than the rate of growth of per capital income. It is
S *-A W
interesting to note here that since Y= -per capital income in the
foodgrain sector may increase, not only because of an increase in foodgrain
output, but also because of an increase in labor's share or a decline in the
labor force in the foodgrain sector. It, therefore, seems highly probable that
the capital-labor ratio in the nonfoodgrain sector would rise over time, for
even if foodgrain output increases only as rapidly as the population growth,
and even if labor's share does not increase, just the withdrawal of popula-
tion from the foodgrain sector would cause an increase in per capital income
of foodgrain sector laborers. However, the faster foodgrain production
grows and the more labor augmenting technological change in the foodgrain
sector, by keeping the capital-labor ratio in the nonfoodgrain sector from
rising as rapidly as it would otherwise, the more likely is the comparative
advantage to continue in the production and export of labor-intensive
commodities in a dualistic economy such as that depicted here.
Equation (18) shows that the movement of relative prices of food and
nonfoodgrain output is dependent upon the relative share of labor and
growth of foodgrain production relative to that of population and nonfood-
grain output, and may move in either direction depending upon the mag-
nitudes of these several parameters and variables. It should be noted that
the relative prices between sectors are determined by the price and income
elasticities on the one hand and by the factor shares in the foodgrain sector
and average propensities to consume of the two income classes on the other
hand. However, it can be seen that a foodgrain output increase accompanied
by a reduced factor share to labor will certainly turn the relative price
against the foodgrain sector.
By assuming the existence of labor and food markets as two separate but
interacting markets in a dualistic economy, the model highlights the adverse
effect of the wages good constraint on growth of employment in the
non-agricultural sector in a situation of traditional low productivity agricul-
ture faced in many developing countries. Further, it demonstrates the
UMA LELE AND JOHN W. MELLOR
relationship of increased agricultural production and especially of factor
shares with growth of employment in the nonagricultural sector. This it does
by showing that technological change which increases labor's share in
agriculture may well lead to a decline in the marketed surplus of foodgrains
and an increase in the real wages in the nonfood sector. On the other hand,
in situations of biased technological change even if the direct employment
effect of new technology in agriculture is limited, by generating a marketed
surplus of foodgrains, such technological change may relax the wages goods
constraint, thus facilitating an increase in employment in the nonagricultural
World Bank, Washington, D.C.
International Food Policy Research Institute, Washington D.C.
APPENDIX A: STABILITY CONDITIONS
Let us hypothesize that the terms of trade increase over time if demand for the marketable
surplus exceeds its supply,
P= H[MD-M] (A.1)
such that H' > 0 and that labor migrates to the nonfoodgrain sector when the demand price for
nonfoodgrain sector labor exceeds its supply price.
S= G[--AWI (A.2)
such that G'<0.
A necessary and sufficient condition for local stability of the system (A.1) and (A.2) are that1
aP aO aP a OP 9t
-+-<0 and ----->0 (A.3)
OP ar aPOr r OP
Differentiating equations (12) and (10) with respect to P and r we get
aP ar SLbA e SLA ra r (1
-+-= H' + G'PSLA [1+ 1 -<0 (A.4)
aP Or r P rL 1-r r
aP ai P a H WbA I ra
-----= -H'G'- S'"b-S)be a1+-r\ >0 (A.5)
O8Pr raP r2 L \1
When -q >0, e <0, H'>0 and G'<0. Note that these are sufficient conditions for the system to
satisfy (A.3) and hence they are the sufficient conditions for local stability of the system.
APPENDIX B: TO DERIVE SENSITIVITY MATRIX
The effect of changes in exogenous variables x, y, K or L on endogenous variables IA, r, P,
M, and (W/P) can be determined as follows: let 0 = t, K or L. Note that change in t,
technological change, implies change in x and y.
SThese conditions are derived by using the theoretical discussion in P. A. Samuelson,
Foundations of Economic Analysis, New York 1947 pp. 266-67.
438 TECHNOLOGICAL CHANGE, DISTRIBUTIVE BIAS & LABOR TRANSFER
Differentiate (3) logarithmically with respect to 0 and note Az > AL. Then
1A 1 A--AL+ L > 0 when 0 = t.
80 IA 0 when 0 =K or L.
Further, substitute the value of P from equation (10) in (12) and then differentiate (12) partially
with respect to 0. After rearranging terms,
r 1 SLbe,(O) + h,2(0)-SL[r-b(n -e)l-- A (B.2)
80 r || t o -II
where e and -t are, respectively, the elasticity of budget share with respect to price and income
elasticities of demand for foodgrains by laborers; t,(0) and #2(0) are functions of 0; and
AI =Sb[ -(1 + -r)] >0, (B.3)
since O<71<1 and e<0.
Differentiating (10) logarithmically with respect to 0 and then rearranging terms gives the
following two equations:
aP e 9 I ra 9r
a =(0 ) --+ +- ar (B.4)
ao P 0 'A 1,-r) t'
8(WIP) 0 alA 0 Or 0
aO (W/P) aO AA 8 r
where 13(0) is the function of L.
Differentiating both the sides of the marketable surplus equation (6) with respect to 0 and
then rearranging the terms gives:
SM, / lA 0 laP O\
-- =04(0) + SLA(1-b b) -I- SLAbs--P
where 1(4(0) is a function of 0. Substituting (B.2) and B.4) in the above equation and then
rearranging terms gives:
=4() + A2(0)+ SLA(1 r) SAb(B.6)
Different Values of qi's
Value of 0 41(0) 02(0) #3(0) 04(0)
S=t 0 rSzAz rSLAL 0 Szz -SLL
O=K a 0 0 0
O=L 1-a -SLb(q-l1) -1 ASLb(q-1)
UMA LELE AND JOHN W. MELLOR 439
Substitute tib's for different 0 and (B.1) in (B.2), (B.4), (B.5) and (B.6). Then,
When 0= t:
t -IAI= -rSzAz-rSLAL--SL[r-b(t-e)]AZ-AL + A)L
at r Sz
=-rSzAz -rSLAL-[r-rSz -SLb(1 -) ]Z [ -AL + AL
= L(-1)-[r-SLb(t-e)][Az AL]. (B.7)
Since C>0, from (12) we get, r-SLb>O. Further, since O
r--lqSLb>0 and r-(q -e)SLb >0. Using these inequalities and (B.1) in (B.7) we get
ar 1< < dSL 1 <
--<0 when 1, tha w hen wth hen --=0 (B.8)
t r > dt SL >
This gives the first three elements in the first row of the sensitivity matrix. Using (B.8) the first
three elements of the remaining row of the sensitivity matrix can be derived from (B.4), (B.5)
When 0 = K:
ar K aSLbe aP K aSLbnq a(W/P)
OK r Al < K P |Ad aK
K ar K M, or 1\
--->0 and = SLAb >0.
(W/P) OK r aK \(K r
These inequalities give the fourth column of the sensitivity matrix.
When = L:
-L AI= SLbe(1-a)-SLb( 1)= -aSLbe+SLb[1- (- e)]>0,
OP L =ra \/Or L\
aL P 1-r aL r
O(W/P) L ar L\ aM, Or 1
-(W/P -1-- <0 and = -SLAb--
aL (W/P) \(L r aL aL r
From these inequalities the last column of the sensitivity matrix is derived.
APPENDIX C: TO DERIVE GROWTH RATES OF r, P, W AND M~
Equation (10) and (12) can be written respectively as follows:
SLA Q 1
PS= (1-a) (1-=_a) =KW
rL L (1-r)
A C-SLA b = 0. (C.2)
Substitute the value of P from (C.1) in (C.2) and then differentiate (C.2) totally with respect to
t. After rearranging the terms,
drl 1 1 dA 1
dt- = SLbe [la + a2- a+D SLb qaI +Srb( -1)t2-(r-SLb)
dt r ID| I dt A
440 TECHNOLOGICAL CHANGE, DISTRIBUTIVE BIAS & LABOR TRANSFER
dSL 1 dAl dL1 dQ1 dL1
a"l dt SL, 2 dt t dtL a dt Q dt L'
(DI= SLb r > 0.
IDrl S E)a-1 SLb(l Tl+e)a2+SLbea3-(r- SLb) dA 1]
dt r [D Ii dt A
= Cla C22 C33 C4, (C.3)
where C, >0(i = 1,...4) because O< <1 00.
Differentiating (C.1) logarithmically with respect to t and then substituting (C.3),
dP 1 1
d --= -at-a2 +3+ 1-- [SLb(Tl -e)az -SLb(1 -TI +e)a2
dt P +DI(1-r)
1 SLb-q SLb(1 -l)
+Sebsea3l C4= 2
1-r (1-r)IlD (1-r)IDI a
SLb, l 1
(a2 a3) C4
= da d2a2 d(a2 a3) d4, (C.4)
where all di's>0.
Differentiating the marketable surplus equation (6) with respect to t,
dM, dA 1dPl1
=A(1-SLb) -ASLba, -ASLb( --l1)a2- ASLbe d
dt dt A \dtP)
+ ASLb( 1dr- 1 (C.5)
Substituting (C.4) in (C.5) and rearranging the terms it can be shown that
dM, A dr 1 dr
A(1-r)- ASLb' = e e2 (C.6)
dt A \dtr dt r
where el and e2>0.
Finally, differentiating (C.1) logarithmically with respect to t,
dW 1 dK 1 dL1 ()
dt W \dt K dt L
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