Feder's (1982) model of dualistic growth is derived in levels, suitable for time-series analysis and (i) extended to contexts where aggregate input data are unavailable (ii) sectoral externalities and productivity differentials are generalised in a two- and three-sector (agriculture-manufacturing-services) context.

Norman Gemmell

Tim Lloyd

English

Research Papers in Economics

An extended 'Feder' model of dualistic growth 
Norman Gemmell Tim Lloyd
University of Nottingham University of Nottingham
Abstract
Feder’s (1982) model of dualistic growth is derived in levels, suitable for time−series
analysis; and (i) extended to contexts where aggregate input data are unavailable; (ii) sectoral
externalities and productivity differentials are generalised in a two− and three−sector
(agriculture−manufacturing−services) context.
Citation: Gemmell, Norman and Tim Lloyd, (2002) "An extended 'Feder' model of dualistic growth." Economics Bulletin, Vol.
15, No. 9 pp. 1−6
Submitted: August 15, 2002.  Accepted: October 23, 2002.
URL: http://www.economicsbulletin.com/2002/volume15/EB−02O00004A.pdf1
1.   Introduction
In a much-cited contribution to the growth literature, Feder (1982) proposed a model of
growth for developing countries (LDCs) that recognised the importance of dualism – in
his case, technology differences between sectors.  Feder incorporated sectoral
disequilibrium in the form of a productivity differential, and externality spillovers
between two sectors, into a neoclassical growth model, using an export/non-export
distinction.  This approach underlies most subsequent investigations of dualistic growth,
though an agriculture-manufacturing distinction has more commonly been adopted (see
Feder 1986, Hwa 1989, Dowrick 1990 and Dowrick and Gemmell 1991).
1  Below we use
this agriculture-manufacturing distinction but the principles can be applied to any
dualistic structure.
Feder specifies two sectoral production functions:
A = F(Ka, La; M) (1)
M = G(Km, Lm) (2)
where A (M) is agricultural (manufacturing) output, Ki (Li) is capital (labor) in sector i (=
a, m denoting agriculture and manufacturing respectively), and the term in M in (1)
captures externalities.
2  Lacking data on sectoral inputs (for a cross-section of countries),
Feder (1982) shows that the two production functions can be solved in first differences to
express aggregate (A+M) output growth in terms of aggregate input growth and the
growth rate of the externality-originating sector (M).  For Feder’s two-sector case and
cross-country data this provides a clever solution to his data deficiencies.
Given the weaknesses of cross-section growth regression methods and the growing
availability of time-series data for LDCs, an interesting issue is whether the Feder model
can be applied in a time-series context.  As Engle and Granger (1987) showed, long-run
output relationships are revealed through data in levels, while short-run behavior is
captured by relationships in first differences.  In a time-series context therefore, Feder’s
method of first differencing equations (1) and (2) and applying the resulting growth
equation to period-averaged data would be inappropriate since it cannot capture the
equilibrium relationship embodied in non-stationary data.
This paper shows how the dualistic model can be adapted to a time-series context by
solving the model in levels.  We also extend the model in three ways.  Firstly, we
generalise externalities to allow spillovers between both sectors. Indeed, for many
countries a large fraction of output is generated in the service sector, so that significant
interactions may be missed if this sector is ignored.  We therefore extend the Feder
model to three sectors in an appendix (since this is analytically straightforward and yields
empirical rather than analytical insights). Thirdly, as we show, the model in levels
requires data on the (aggregate) stock, rather than the flows, of inputs.  Since suitable
                                                
1 Matsuyama (1992) provides an alternative endogenous dualistic growth model based on learning-
by-doing in manufacturing.  Dowrick (1990) also includes a service sector and Ram (1986)
replicates the Feder model for government-private sectors.
2 M represents externalities here rather than an input since firms in agriculture are assumed to
ignore manufacturing outputs in their profit maximising decisions; i.e. ‘they are not reflected in
market prices’ (Feder, 1982, p.61).2
capital stock data are often unavailable in LDCs
3, we demonstrate that one additional
assumption allows aggregate inputs to be eliminated, which reduces data requirements
and overall complexity.  Though this combines ‘fixed resource’ and ‘resource growth’
effects, we argue that empirically short-run dynamics are likely to approximate the ‘fixed
resource’ case.
2.  Adapting and Extending the ‘Feder’ Model
To solve the Feder model in levels consider a linear approximation of (1) and (2) such
that
4:
M K L A a a a a a a γ β α ϕ+ + + = (1’)
A K L M m m m m m m γ β α ϕ+ + + = (2’)
where 0 < αi, βi < 1; γi > 0 (i = a, m). Here, two-way spillovers (externalities), γi, are
possible, marginal products are measured by αi and βi, and  i ϕ  translates inputs into
outputs.  Adopting Feder’s (1982, p.61) assumption of marginal productivity differences
between sectors:
δ β β α α+ = = 1 a m a m / / 0 > δ (3)
and adding (1’) and (2’) yields:
A M } K L ){ / ( K L M A Y m a m m m m a a γ γ β α δ δ β α ϕ+ + + + + + + = + = 1( 4 )
where  m a L L L + = , and  m a K K K + = , ( m a ϕ ϕ ϕ+ = ). Substituting for  m m m m K L β α+
from (2’), it can be shown that:
{} M
) (
K L ' A
m
a
a a
m 





− +
− +
+ +  


 


− +
+
+ =
γ δ
γ δ
β α
γ δ
δ
ϕ
1
1 1
1
1
(5)
where   


 


− +
+
=
m
) (
'
γ δ
δ ϕ
ϕ
1
1
.  Equation (5) is the equivalent in levels of the first difference
expression derived by Feder, though in his case aggregate output growth appears on the
left-hand-side (LHS).  Writing A, rather than Y on the LHS of (5) avoids collinearity
problems in empirical tests of Feder’s model for which Ram’s (1986) application was
                                                
3  Such constraints can sometimes be overcome by summing (and depreciating) data on aggregate
investment over a number of years prior to the period of investigation.  Unfortunately such data
are unavailable and/or unsuitable in many LDCs, though Nehru and Dhareshwar (1993) have
produced some capital stock estimates to 1990.
4 The model can also be solved for the Cobb-Douglas case, but the resulting expressions for sector
outputs (equivalent to (5) and (6) below) are not as directly comparable with Feder’s (1982)
results.  For time-series analysis of a growing economy, the linear approximation is likely to be
less restrictive than it would be in cross-section since, empirically, production function shifts (due
to technical progress) can be expected to dominate movements along production functions.3
criticised.
5  The parameters in (5) reduce to those in Feder’s (1982), equation (9), for
0 = m γ , and with Y on the LHS.
The term 






− +
− +
m
a ) (
γ δ
γ δ
1
1 1  captures the effects on agriculture of manufacturing expansion,
for given factor endowments, and may be positive or negative.  Externalities have
positive effects on agriculture; productivity differentials reduce agricultural output for
1 < a γ ,
6 as expansion of manufacturing, conditional on fixed (total) endowments,
competes inputs away from agriculture.
The terms   


 


− +
+
m
a
γ δ
δ α
1
) 1 (  and   


 


− +
+
m
a
γ δ
δ β
1
) 1 (  capture the effects of expansion in L and K
respectively.  Since  1 0 < < a a ,β α , and  0 > δ , larger values of δ imply lower
agricultural outputs - a larger productivity differential implies more of any resource
increase is drawn into manufacturing.  The externality,  m γ , also affects agriculture
positively via factor accumulation, which boosts manufacturing output.
Given data on aggregate capital and labor stocks and sector outputs, equation (5) may be
interpreted as capturing the long-run or steady-state relationship and is readily estimated
using standard time-series methods.  With cointegration between A and M, for example,
an error correction representation identifies short and long-run effects of expansion of
aggregate inputs and sectoral productivity.  Without such data, the model can be solved
to eliminate aggregate inputs.  This requires two further assumptions. Firstly, the Feder
(1982, 1986) assumption: that sectoral marginal products of labor are proportional to
average economy-wide labor productivity: ) / ( L Y a α α= . Secondly, we make a
symmetric assumption for capital productivity:  ) / ( K Y a β β= . This allows substitution
for  a α  and  a β  in (5), giving:
M A
m
a






− + − +
− + + +
+ =
γ β α δ
β α γ δ
ϕ
)) ( 1 )( 1 (
1 ) )( 1 (
"( 6 )
where   

 
 − + − =
+δ
γ
β α φ ϕ
1 ) ( 1 ( / "
m .  In addition to the effects identified earlier, larger
values of α or β imply larger agricultural output since a larger fraction of any endowment
expansion goes to agriculture if the sector’s capital or labor productivity is relatively
high.  An additional spillover from manufacturing arises here because manufacturing
output expansion raises Y/L  or Y/K, thereby raising agricultural marginal products,
a a β α ,.
Like equation (5), (6) cannot be regarded as a conventional reduced form relationship
since both A and M are potentially endogenous.
7  This suggests that vector autoregressive
                                                
5 See the ‘comments’ on Ram (1986) in American Economic Review, 79 (1989).
6 If there are no externalities  ) 0 ( = = m a γ γ , (5) shows that manufacturing competes resources
away from agriculture due to the productivity differentials, captured by the term –1/(1+δ) in (5).
7 This problem also afflicted Feder’s applications, even though endowments were included, since
one sector’s output is a RHS variable.4
(VAR) methods  - which treat all variables as potentially endogenous - may provide a
valuable model testing tool, especially when aggregate input level data are unavailable.
Of course, if (6) is estimated empirically the parameter on M captures the combined
effects of externalities and productivity differentials for given resources, and the effects
of resource expansion.  This necessarily limits the conclusions that can be drawn from
any observed inter-sectoral linkages.  The VAR approach however allows separate
identification of short- and long-run effects.  Since in the short-run factor endowments
are likely to be relatively fixed, short-run time-series evidence from regressions on (6)
should approximate the effects of expansion in M identified in (5).  The difference
between short- and long-run parameters will then mainly capture the effects of resource
growth.  If inter-sectoral externalities take some time to come through, short-run effects
of manufacturing expansion will primarily reflect resource competition as captured by
the parameter element –1/(1+δ) in (5) and (6).
Finally, an appendix available from the authors presents three-sector equivalents of
equations (5) and (6) above, which reveal analogous relationships between agriculture,
manufacturing and services.  Though interactions become more complex, as with two
sectors, expansion of either manufacturing or services can have positive or negative net
effects on agriculture depending on the size of sector (marginal) productivity
differentials, inter-sectoral externalities and sectoral competition for resources.
3.  Conclusions
This paper has derived Feder’s (1982) model of dualistic growth in levels rather than
first differences suitable for time-series applications.  We incorporated multiple inter-
sectoral externalities, and demonstrated that, with one additional assumption,
aggregate inputs can be eliminated.  In this latter case, though effects associated with
given resource endowments cannot be separated from endowment expansion effects,
short-run estimates from time-series regressions are likely to approximate the fixed
endowment case.5
References
Dowrick, S.J. (1990) “Sectoral change, catching up and slowing down: OECD post-war
economic growth revisited” Economics Letters 31, 331-335.
Dowrick, S.J. and N. Gemmell (1991) “Industrialisation, catching-up and economic
growth: A comparative study across the world's capitalist economies” Economic
Journal 101, 263-275.
Engle, R.F. and C.W.J. Granger (1987) “Cointegration and error-correction:
Representation, estimation and testing” Econometrica 55, 251-276.
Feder, G. (1982) “On exports and economic growth” Journal of Development Economics
12, 59-73.
Feder, G. (1986) “Growth in semi-industrial countries: A statistical analysis” in
Industrialisation and Growth by H.B. Chenery, S. Robinson, and M. Syrquin,
Eds., Oxford University Press:New York.
Hwa, E-C (1989) “The contribution of agriculture to economic growth: Some empirical
evidence” in The Balance Between Industry and Agriculture in Economic
Development.  Vol. 2, Sector Proportions by J. Williamson and V.R.
Panchamurtchi, Eds., The World Bank:New York.
Matsuyama, K, (1992) “Agricultural productivity, comparative advantage and economic
Growth” Journal of Economic Theory 58, 317-334.
Nehru, V. and A. Dhareshwar (1993) “A new database on physical capital stock:
Sources, methodology and results” Revista Analisis de Economico 8, 37-59.
Ram, R. (1986) “Government size and economic growth: A new framework and some
evidence from cross-section and time-series data” American Economic Review
76, 191-203.6
Appendix
In a more detailed appendix we derive a three-sector version of the model in Section 2.
Here we present the main results.  Let A, M and S represent output in agriculture,
manufacturing and services respectively;  s m a L L L L + + = and  s m a K K K K + + =  represent
inputs of labor and capital respectively.  Sector production functions are given by:
S M K L A s
a
m
a a a a a γ γ β α+ + + = (A1)
S A K L M s
m
a
m m m m m γ γ β α+ + + = (A2)
A M K L S a
s
m
s s s s s γ γ β α+ + + = (A3)
Now, let  i
a
i
a
i δ
β
β
α
α
+ = = 1 s m i , = (A4)
Following the procedure of Section 2, a three-sector version of equation (5) is obtained
as:
() S M K L A
s
a
s
m
a
m
m
s
m
s
s
a
s
m
a
m
s
m
s
m
s
a
s
m
a
m
s
a
m
a
a a










− −
+ −
+










− −
+ −
+ + 





− − =
+ +
+ +
+ +
+ +
−
+ +
δ
γ
δ
γ
δ
γ
δ
δ
γ
δ
γ
δ
γ
δ
δ
γ
δ
γ γ γ
β α
1 1
1 1
1
1 1
1 1
1 1
1 1
1 1
1 (A5)
Eliminating inputs is again achieved by assuming that the marginal productivity of labor
and capital in agriculture is proportional to average productivity in the economy as a
whole, such that, ) L / Y ( a α α=  and  ) K / Y ( a β β= .  The three-sector equivalent to
equation (6) is then derived as:
()
()
()
()
S M A
a
m m
s
m
s
s
a m
a
m m
m
a m
s
a
s
s
a
s
s
m
s












−  

 
 − + − +
+  

 
 − + + +
+










−  

 
 − + − +
−  

 
 + + + +
=
+
+
+
+
γ β α δ
γ γ β α δ
γ β α δ
γ β α δ
δ
γ
δ
δ
γ
δ
γ
1
1
1
1
1
) ( 1 1
1
) ( 1 1
1 1
(A6)

A new database on physical capital stock: Sources, methodology and results”

Agricultural productivity, comparative advantage and economic

Cointegration and error-correction: Representation, estimation and testing”

Government size and economic growth: A new framework and some evidence from cross-section and time-series data”

Growth in semi-industrial countries: A statistical analysis”

Industrialisation, catching-up and economic growth: A comparative study across the world's capitalist economies”

On exports and economic growth”

Sectoral change, catching up and slowing down: OECD post-war economic growth revisited”

The contribution of agriculture to economic growth: Some empirical evidence”

An extended 'Feder'' model of dualistic growth

http://www.accessecon.com/pubs/EB/2002/Volume15/EB-02O00004A.pdf

An extended 'Feder'' model of dualistic growth

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