This document presents a restatement of the S-curve hypothesis as an alternative to Kuznets' inverted U-curve model of economic development and income inequality. The author argues that Kuznets' model fails to account for structural transitions from agriculture to manufacturing and from manufacturing to services. When time-series data is used that captures both transitions, the relationship between per capita GNP and inequality forms an S-curve with two turning points. Empirical analysis of U.S. data from 1947-1990 supports the S-curve model and identifies a second turning point associated with the transition to a services economy.

1. A Restatement of the S-Curve Hypothesis
Romie Tribble, Jr*
Abstract
It is the contention of this paper that Kuznets’ inverted U-curve is really an S-curve
when one incorporates either times-series or cross-sectional data which allows one to
capture both the agriculture-to-manufacturing (ATM) and manufacturing-to-services
(MTS) structural transitions during the economic development process. With the use
of United States times-series data, this study demonstrates that Kuznets’ inverted U-
curve is a recurring quadratic reproduced by the inevitable shift in sectoral emphasis
occasioned by the ATM and MTS structural transitions. Such changes for high-level
per capita GNP countries is best explained graphically by an S-curve.
1. Introduction
There is growing empirical evidence that Kuznets’ inverted U-curve provides at best
an incomplete picture of the economic development process. While more vintage
studies, no doubt, provide empirical evidence in support of Kuznets’ inverted U-curve,
one is probably correct in suggesting that these studies were, no doubt, confined to a
cross-sectional database where the limited time span was largely confined to the ATM
structural transition or to a time period transpiring after the first critical turning point
in a manufacturing-based economy (Kuznets, 1955, 1963; Paukert, 1973; Ahluwalia,
1974, 1976, 1979; Randolph and Lott, 1993; Jha, 1996). A time-series database was
unavailable for such early studies and this is, perhaps, unfortunate since a careful
reading of Kuznets’ path-breaking 1955 paper was quite clear in defining the economic
development problem as a time-series problem within the context of the per capita
GNP–inequality relationship (Kuznets, 1955, pp. 1–3). More recent research, however,
has been afforded the availability of both extensive time-series as well as cross-
sectional databases with which to test Kuznets’ inverted U-curve hypothesis. The
majority of these studies have found instead a U-shaped per capita GNP–inequality
relationship both within the context of strictly time-series studies as well as studies
combining a mix of time-series and cross-sectional observations (Anand and Kanbar,
1986; Amos, 1988; Fields and Jakubson, 1990; Braun, 1991; Ram, 1991; Goetz and
Debertin, 1992). The later studies have generally relied on a much later time period,
thereby incorporating with a much greater likelihood the manufacturing-to-services
transition among a variety of countries with high-level per capita GNP (Ram, 1997).
In sum, these studies have generally concluded that Kuznets’ inverted U-curve is
largely unsubstantiated.
However, none of the recent studies seeks explicitly to explain the U-shaped curve
as an extension of a times-series database where a cubic formulation of the per capita
GNP–inequality relationship would indicate a second turning point associated with a
Review of Development Economics, 3(2), 207–214, 1999
© Blackwell Publishers Ltd 1999, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA
*Tribble: Spelman College, Atlanta, GA 30314, USA. Tel: 404-681-3643 (ext. 7581); Fax: 404-223-1449; E-
mail: rtribble@spelman.edu. The author wishes to acknowledge the suggestions, insights and critical
comments provided by William A. Darity, Jr., Samuel L. Myers, Jr, Jack H. Stone, John Virgo, as well as the
support of the University of Maryland’s African American Studies Program where this endeavor was
initially conceived and the Spelman College Bush Faculty Development Program.

2. MTS structural transition. Tribble (1996) embraces a cubic formulation of the per
capita GNP–inequality relationship and in doing so utilizes times-series data for
the US economy to demonstrate empirically that the U-shaped per capita GNP–
inequality relationship is merely an extension of the prior inverted-U curve depicting
the ATM structural shift of an earlier time period. In sum, Tribble’s S-curve identifies
two critical turning points in the per capita GNP–inequality relationship and suggests
that existing US times-series data depict the behavior of a high-level per capita GNP
country in the throes of the MTS structural transition. Secondly, Tribble’s 1996 paper
provides a theoretical basis for explaining the S-curve hypothesis as a viable extension
and restatement of Kuznets’ inverted U-curve hypothesis (Tribble, 1996, pp. 153–60).
2. Theory of the S-Curve
Much of the contemporary debate regarding the viability of Kuznets’ inverted U-curve
as a viable explanation of the economic development process have often overlooked
the empirical constraints under which Kuznets’ did his path-breaking theorizing. In
particular, Kuznets’ inverted U-curve was not premised on data-pertaining to the ATM
structural transition; but, instead refers to the time period just prior to the first critical
turning point where the economic integration of the traditional sector of agriculture
and the modern sector of manufacturing had begun in earnest. Hence, Kuznets’ data
did not aptly describe the structural shift in sectoral emphasis but merely assumed that
this shift had been made. More recent time-series data embrace not only the period
beyond the first critical turning point which Kuznets rightly anticipated, but also the
second critical turning point associated with the MTS transition which may not have
been anticipated as a structural shift in economic development emphasis. The S-curve
hypothesis, thereby, embraces multiple structural turning points in the economic devel-
opment process; hence, neither the inverted U-shape nor the U-shaped curve are accu-
rate representations of the secular nature of economic development, whether analyzed
within a strictly times-series and/or cross-sectional empirical context. Kuznets’ inverted
U-curve is a recurring quadratic reproduced by the inevitable structural change asso-
ciated with the shift in sectoral emphasis or in the emergence of a new, leading or alter-
native modern sector. In the future, the per capita GNP–inequality relationship will be
fitted not only to a cubic curve but also to a quortic or quintic curve as the economy’s
sectoral emphasis shifts for a third or fourth time.
Structural turning points emanate from distinctive changes in the share of income
accruing to upper- and middle-income groups or classes relative to lower-income
groups or classes. In particular, within the S-curve framework the first structural
turning point coincides with a subsequent decline in the relative share of income accru-
ing to the upper and middle classes while the share accruing to the lower classes
increases. This is the inevitable “trickle-down” effect which follows a prior period
(before the first critical turning point) when the share of income accruing to the upper
and middle classes increased while the share accruing to the lower class declined. This
train of events will, no doubt, be repeated when the economy begins the MTS struc-
tural shift which coincides with the second critical turning point. The shift in interclass
income distribution occasioned by the second critical turning point is also comple-
mented by a reallocation of upper- and middle-class resources and assets from the
traditional sector of economic emphasis (e.g. manufacturing) to the newly consolidated
modern sector (e.g. services). This resource and asset reallocation, as well as the
interclass redistribution of income, transpires irrespective of racial considerations, as
evidenced by the US economy. In keeping with Kuznets and Lewis, it is the savings
208 Romie Tribble, Jr
© Blackwell Publishers Ltd 1999

3. and investment propensities of the upper class, and secondarily, the middle class
which are in the forefront of the structural shifts in economic development
emphasis.1
The structural shifts in sectoral emphasis coinciding with designated critical turning
points in the S-curve hypothesis are associated with prior critical changes in intrasec-
toral and intersectoral inequality. Prior to the first critical turning point associated with
the ATM structural shift, the level of intrasectoral inequality in manufacturing exceeds
the level of intrasectoral inequality in agriculture. Moreover, the per capita income
growth in the manufacturing sector exceeds the level of such growth in the agricul-
tural sector (Kuznets, 1955, pp. 12–18). According to Lewis (1954), these findings on
intrasectoral and intersectoral inequality indicate that not only is agricultural produc-
tivity lagging behind manufacturing productivity but agricultural prices are falling
relative to manufacturing prices. In sum, prior to the first critical turning point, the
terms of trade favor manufacturing over agriculture and will continue to do so until
the surplus of agricultural labor is fully absorbed into the manufacturing sector.
According to Lewis, when this labor absorption is completed, the first critical turning
point is also achieved as wages in both the traditional and modern sectors will increase
as inequality begins a period of decline (Lewis, 1954, pp. 431–4). The terms-of-trade
advantage enjoyed by manufacturing will deteriorate as well.
Logic suggests that the second critical turning point is achieved when the level of
intrasectoral inequality in the newly emergent service sector begins to exceed the level
of intrasectoral inequality in the now traditional sector of manufacturing. Beyond this
second critical turning the per capita income growth in the service sector exceeds the
per capita income growth in the manufacturing sector. Hence, the service sector’s terms
of trade improves with respect to both manufacturing and agriculture (Tribble, 1996,
pp. 153–6).2
This disparity in sectoral income growth as well as in the terms of trade
will continue until the supply of surplus manufacturing labor is fully absorbed by the
service sector. This labor absorption prospect is far-reaching since manufacturing pro-
duction has been globalized to much greater extent than was the case with agriculture
during the ATM structural shift (Lewis, 1954, pp. 435–7).
Finally, the cubic formulation of the per capita GNP–inequality relationship as
embodied in the S-curve identifies distinct phases of income inequality behavior before
and after the first critical turning point.These distinct phases of income inequality have
important implications for explaining changes in the rate of growth of both aggregate
and interclass inequality as economic growth transpires. The structural transition is a
period where inequality increases at an increasing rate at both the aggregate and inter-
class level. Here, of course, one finds an abundance of surplus labor in the traditional
sector as well as an increase in the share of national income allocated to profits rela-
tive to wages. During the phase of structural integration, inequality increases at a
decreasing rate as surplus labor from the traditional sector is being rapidly depleted
and income divergence between the modern and traditional sectors is rapidly declin-
ing. Beyond the first critical truing point, one observes inequality declining at an
increasing rate as the phase of structural diffusion witnesses an expanding domestic
market fueled by the increase in the share of national income allocated to wages
relative to profits. During this phase both aggregate and interclass inequality are
declining. Finally, prior to the second critical turning point, inequality declines at a
decreasing rate as the phase of structural dissolution accelerates manufacturing’s
decline as upper- and middle-class resources and assets begin the shift from manufac-
turing to services. The rate of improvement in both aggregate and interclass inequal-
ity slows during this phase as well.
A RESTATEMENT OF THE S-CURVE HYPOTHESIS 209
© Blackwell Publishers Ltd 1999

4. 3. S-Curve Model
The S-curve hypothesis as a cubic formulation of the per capita GNP–inequality rela-
tionship can be expressed as
GINI = b0 + b1PGNP + b2PGNP2
+ b3PGNP3
+ m,
b1 > 0, b2 < 0, b3 > 0,
|b1| > |b2| > |b3|. (1)
The phases of inequality associated with the S-curve can be distinguished as follows:
Phase 1 f¢(PGNP) > 0 income inequality
(ATM) f≤(PGNP) > 0 increases at an increasing rate
Phase 2 f¢(PGNP) > 0 income inequality
f≤(PGNP) < 0 increases at a decreasing rate
f¢(PGNP) = 0 *first critical turning point
f≤(PGNP) < 0 achieved
Phase 3 f¢(PGNP) < 0 income inequality
f≤(PGNP) > 0 decreases at an increasing rate
Phase 4 f¢(PGNP) < 0 income inequality
f≤(PGNP) < 0 decreases at a decreasing rate
f¢(PGNP) = 0 *second critical turning point
f≤(PGNP) < 0 achieved
Phase 1 f¢(PGNP) > 0 income inequality
(MTS) f≤(PGNP) > 0 increases at an increasing rate.
With respect to class income share and interclass income inequality, the S-curve can
be expressed as:
UYS = a0 + a1PGNP + a2PGNP2
+ a3PGNP3
+ m,
a1 > 0, a2 < 0, a3 > 0,
|a1| > |a2| > |a3|.
MYS = q0 + q1PGNP + q2PGNP2
+ q3PGNP3
+ m,
q1 > 0, q2 < 0, q3 > 0,
|q1| > |q2| > |q3|.
LYS = l0 + l1PGNP + l2PGNP2
+ l3PGNP3
+ m,
l1 > 0, l2 < 0, l3 > 0,
|l1| > |l2| > |l3|.
Moreover:
a1 > q1 > l1: increase in interclass income inequality
a2 < q2 < l2: decline in interclass income inequality
a3 > q3 > l3: increase in interclass income inequality.
The above theoretical predictions for aggregate class income share and aggregate
interclass inequality will hold as well when income is conditioned by race. Changes in
racial income inequality within the S-curve framework can be depicted as
a1
w
> a1
b
, q1
w
> q1
b
, l1
w
> l1
b
,
where interracial inequality increases, and as
210 Romie Tribble, Jr
© Blackwell Publishers Ltd 1999

5. a1
w
< a1
b
, q1
w
< q1
b
, l1
w
< l1
b
,
where interracial inequality decreases.
4. Empirical Findings
The data for per capita GNP were derived from the Economic Report of the President
and are expressed in nominal US dollars. The data for the United States GINI
coefficient as well as aggregate class and racial class income shares were derived
from the US Census report entitled “Income at Selected Positions and Percentage
Share of Aggregate Income in 1947–90 Received by Each Fifth and the Top Five
Percent of Families and Unrelated Individuals, by Race of Householder”. The data
for 1988–90 derive from Census Bureau updates published in 1989, 1990, and 1991.
The data for this study covers the time period 1947–1990. The share of income accru-
ing to this top fifth of the income distribution constitutes the upper-class share while
the share going to the second and third quintiles constitute the middle-class share.
Finally the share allocated to the fourth and fifth quintiles comprise the lower-class
share.
The estimates for the S-curve are provided in Table 1 (Tribble, 1996, p. 161). As
hypothesized, the first term is positive, the second term negative and the third term
positive.The second critical turning point with respect to the MTS structural shift tran-
spires at a PGNP level between $10,000 and $11,000 and an associated GINI coefficient
between 0.3754 and 0.3554, respectively (Tribble, 1996, p. 160).3
The aggregate interclass income distribution is also consistent with S-curve predic-
tions. Prior to the first critical turning point, economic growth is associated with
A RESTATEMENT OF THE S-CURVE HYPOTHESIS 211
© Blackwell Publishers Ltd 1999
Table 1. Empirical Estimates of the United States S-Curve: Aggregate,Aggregate Class and Aggre-
gate Racial–Class Parameters
A lnPGNP (lnPGNP)2
(lnPGNP)2
R2
F
GINI 124.20 42.16 -4.74 0.177 0.8040 54.17
(2.071) (2.172) (2.274)
US -12,148 4,118.9 -462.9 17.3 0.7859 44.04
(2.055) (2.142) (2.228)
MS -7,641 2,421.5 -254.2 8.89 0.6015 18.12
(2.002) (1.949) (1.896)
LS 18,651 -6,142 674.5 -24.68 0.7570 37.37
(3.376) (3.804) (3.736)
WUS -8,358 2,884.12 -328.9 12.4 0.7696 40.73
(1.449) (1.532) (1.616)
WMS -5,291.7 1,661.8 -172.3 5.95 0.6096 18.73
(1.376) (1.324) (1.272)
WLS 15,129 -4,496.3 550.3 -20.19 0.7407 34.27
(3.014) (3.078) (3.143)
BUS -21,651 7,217.9 -779.8 29.52 0.6653 23.85
(1.809) (1.859) (1.909)
BMS -12,553.6 4,025.6 -428.7 15.2 0.3269 5.82
(1.303) (1.286) (1.268)
BLS 35,077.8 -11,499.1 1,256.5 -45.76 0.7357 33.40
(4.373) (4.431) (4.489)

6. an increase in income inequality as the share accruing to the upper class exceeds the
combined share going to the middle and lower classes. Here, one can speak of income
inequality increasing at an increasing rate. Indeed, the share of income going to the
lower class is actually negative. Beyond the first critical turning point, the share accru-
ing to the lower class exceeds the share accruing to middle and upper classes who both
experience negative income growth. This finding supports the S-curve’s contention of
income declining at an increasing rate. However, beyond the second critical turning
point where the MTS structural shift begins in earnest, the income share going to the
upper class again exceeds the combined share allocated to the middle and lower
classes. Here, one observes income inequality increasing at an increasing rate as well
as a shift in the terms of trade away from manufacturing and agriculture to the emer-
gent service sector.
This aggregate interclass income distribution is fundamentally replicated when one
examines the racial interclass distribution. This replication is also consistent with the
theoretical expectations of the S-curve. However, one must, certainly, note that changes
in per capita GNP have a much greater quantitative impact on income inequality
within the black population relative to the white population. Second, the size of the
parameter estimates also substantiate the contention of some that income inequality
is much greater within the black population. Prior to the first critical turning point,
income inequality increases at an increasing rate for both races but the pace of this
increase is higher within the black population. A similar outcome is observed beyond
the second critical turning point associated with the MTS structural shift. Moreover,
this latter trend in interclass inequality for both races substantiate the S-curve’s con-
tention that the service sector’s terms of trade improve vis-à-vis manufacturing and
agriculture.
Fourth, the empirical findings for racial interclass inequality indicate that
a1
w
< a1
b
, q1
w
< q1
b
, l1
w
> l1
b
.
Hence, the S-curve findings for the United States economy point to a narrowing of
the racial income gap between whites and blacks similarly situated in the upper and
middle classes in narrowing, while the racial income gap within the lower class is actu-
ally widening. Thus, the S-curve findings for the United States economy is one with a
long-run tendency for an increase in both aggregate and racial class inequality (Tribble,
1996, p. 163).
5. Concluding Remarks
The empirical findings for the United States economy confirm that the per capita
GNP–inequality relationship is best explained by the S-curve hypothesis rather than
Kuznets’ inverted U-curve or the proposed U-curve espoused by others. Indeed, the
findings of the study indicate that Kuznets’ inverted U-curve is a recurring quadratic
reproduced by the structural shifts inherent in the economic development process. On
a more somber note, however, the findings arouse concerns that economists may have
to accept, as others do, the relativity of political democracy as well as the status quo
with respect to the “trickle down” implications of contemporary economic develop-
ment strategies, whether they be of the western or non-western type. Moreover, these
findings suggest limitations on the quantitative extent of the actual “trickle down”
redistributive mechanisms.As mentioned above, the “trickle up” thesis inherent in the
U-curve is theoretically plausible; yet, the empirical evidence is not available to sub-
stantiate such a contention.
212 Romie Tribble, Jr
© Blackwell Publishers Ltd 1999

7. The structural shift inherent in MTS transition has caused income inequality to
increase as found with both the S-curve and the associated aggregate class and racial
class income share models. Moreover, the empirical findings suggests that income
inequality is increasing at an increasing rate as the share of income accruing to the
upper and middle classes increases, while the share of the lower classes decreases in
absolute terms. As the upper class shifts its resources and assets to the service sector
and away from manufacturing, the findings on income inequality noted above support
this study’s theoretical contention that terms of trade have shifted decidedly in favor
of services and against both manufacturing and agriculture. This resource and asset
reallocation by upper-class residents transpires irrespective of race. Without a funda-
mental redistribution of resources and assets, it is unlikely that the terms of trade will
shift favorably toward the middle and lower classes until there is a decisive change in
prevailing market forces. Today’s globalization of markets makes predicting such a
decisive change considerably more problematic.
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A RESTATEMENT OF THE S-CURVE HYPOTHESIS 213
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8. Notes
1. The advocates of the U-shaped relationship between per capita GNP and inequality have,
perhaps, in their decision to embrace a refutation of Kuznets’ inverted U-curve, overlooked the
theoretical implications of adopting such a stance. A U-shaped relationship indicates a “trickle
up” process of economic development where the initial phase of structural transition entails an
increase in the share of income accruing to lower and, perhaps, the middle class while the share
accruing to the upper class would actually decline. Moreover, these changes in interclass income
shares would initially coincide with a shift of lower- and, perhaps, middle-class resources and
assets from the traditional sector of economic emphasis to the newly consolidated modern
sector. Beyond the first critical turning point, however, income “trickles” upward as the share of
income accruing to the upper-income group would increase relative to share allocated to the
lower and, perhaps, middle classes. However, beyond the second critical turning point during the
MTS structural shift, the interclass income distribution observed prior to the first critical turning
point would be replicated and economic development would depend critically on the savings
and investment propensities of the lower and, perhaps, the middle classes.Although the U-curve
has much theoretical appeal, contemporary empirical evidence from the United States economy
does not substantiate it.
2. Recent time-series empirical evidence for the United States economy indicates that as the
rate of growth of the service sector’s share of GNP has increased relative to both manufactur-
ing and agriculture, its terms of trade vis-à-vis both sectors has improved. Moreover, as the
growth rate of the service-sector share of GNP has increased, the rate of growth of United States
inequality as measured by the GINI coefficient has also grown at an increasing rate. Surpris-
ingly,an increase in the growth rate of manufacturing share of GNP has no statistically significant
impact on the rate of growth of United States inequality. This is, perhaps ample evidence that
the MTS structural transition is not completed. Finally, as the service sector’s terms of trade with
both manufacturing and agriculture have improved, the level of inequality as measured by the
GINI coefficient has increased beyond the second critical turning point.
3. Ram’s 1991 paper generates estimates of the per capita GNP associated with the critical
turning point of the quadratic form of the per capita GNP-inequality relationship consistent with
the S-Curve estimation of the critical turning point per Capita GNP. Ram’s findings point to the
MTS structural transition; hence, those findings constitute a logical anticipation of the S-Curve
hypothesis. A similiar conclusion can be drawn regarding her 1997 paper from a cross-sectional
perspective as regards its role as a precursor of the S-Curve hypothesis.
214 Romie Tribble, Jr
© Blackwell Publishers Ltd 1999