journal of economic theory68,149 173(1996)
article no.0008
A General Two-Sector Model of Endogenous Growth with Human and Physical Capital: Balanced Growth and Transitional Dynamics*
Eric W.Bond and Ping Wang
Pennsylvania State University,University Park,Pennsylvania16802
and
Chong K.Yip
Georgia State University,Atlanta,Georgia30303;and
Chine University of Hong Kong,Shatin,NT,Hong Kong
Received June28,1993;revid January24,1995
We examine a two-ctor endogenous growth model with general constant-
朗诵培训return-to-scale production technologies governing the evolution of human and
physical capital.We prove the existence,uniqueness,and saddle-path stability of
the balanced growth equilibrium.A dual approach drawing on techniques from
international trade theory is ud to provide complete characterization of the
transitional dynamics of consumption,goods and education outputs,human and
physical capital inputs,and the relative price of human capital investment.We
investigate the long-run effects of changes in time preference and factor taxation,
and show the emergence of instability or indeterminacy when factor taxes are too
distortionary.Journal of Economic Literature Classification Number:D90. 1996
Academic Press,Inc.
1.Introduction
Recently,an increasing number of economists,led by Romer[18], Lucas[14],Rebelo[17],and Stokey[22],have prented models of economic growth in which the rate of economic growth is an endogenously *We thank John Boyd III,Greg Huffman,Ron Jones,Bob King,Paul Romer,Roy Ruffin, Karl Shell,Alan Stockman,Henry Wan,two anonymous referees,and an associate editor,as well as minar participants at Cornell,Indiana,Michigan State,and Rochester and the Midwest Mathematical Economics Conference for helpful comments and suggestions on earlier drafts of the paper.Part of this paper was written while the cond author was visiting the Federal Rerve Bank of Dallas and the Purdue University.Needless to say,the usual disclaimer applies.
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0022-0531Â96 18.00
仪式英文Copyright 1996by Academic Press,Inc.
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150BOND,WANG,AND YIP
determined variable.This new growth literature has analyzed the deter-minants of the rate of growth
along a balanced growth path,in order to provide explanations of the diver economic development experiences that have been obrved.Due to the nonstationary nature of the problem, however,its transitional dynamics is generally unexplored.In this paper, we completely characterize the balanced growth path and the transitional dynamics of a two-ctor model of endogenous growth with human and physical capital.We analyze a model in which endogenous growth aris from the fact that the two-ctor technology exhibits constant returns to scale in the reproducible inputs.This model rves as a uful first step for understanding the transitional dynamics of two-ctor models of endogenous growth and rves as a uful benchmark against which other models which include elements of increasing returns to scale can be com-pared.Moreover,our analysis enables us to investigate parameter pertur-bations as well as to evaluate factor tax and subsidy policies both in the long-run balanced growth equilibrium and along the short-run transition path.
evereWe analyze a two-ctor model of endogenous growth,consisting of a goods ctor and an education ctor,in which each ctor produces the respective type of ,physical or human)under conditions of con-stant returns to scale.The static version of the two-ctor production model we consider has a long tradition in the theory of international trade and was introduced into the exogenous growth literature by Uzawa[24, 25].The analysis of the static two-ctor production m
odel is substantially simplified by the fact that it has a block recursive ,the factor returns can be determined as a function of output prices alone,independ-ently of factor supplies,as in the factor price equalization property of trade models),under the assumptions of perfect competition in goods and factor markets and the equalization of factor rewards across ctors.Jones[11] illustrates how this structure leads to``magnification effects'':an increa in the price of one ctor's output leads to a more than proportional increa in the price of the factor ud intensively in that ctor(the Stolper Samuelson theorem)and an increa in the quantity of a factor leads to a more than proportional expansion of the output of the good that us that factor intensively(the Rybczynski theorem).We show that the two-ctor endogenous growth model can be analyzed using the properties of static two-ctor models,combined with an intertemporal no-arbitrage condition which requires an equality between the net rates of return on physical and human capital.1This approach allows us to obtain conditions under which there exists a unique balanced growth equilibrium with a positive common 1We are thus accounting for the``capital gain''on each reproducible capital stock as described in Shell et al.[20]within the exogenous growth framework.
growth rate,and also highlights the role played by the relative factor intensities of the two production ctors in determining the transitional dynamics of the system.This can be en most clearly by takin
g a dual approach that focus on the relative price of the education Âlearning output (in units of goods),which is the ratio of the respective costate variables in the dynamic optimization problem and will equal the value of an increment of human capital.The intertemporal no-arbitrage condition,which links the capital gain on human capital to the difference between the rentals on capital and the wage rate,thus becomes a differential equation determining the relative price of human capital investment and its dynamic adjustment in the neighborhood of the balanced growth equilibrium.The role of factor intensities in the transitional dynamics aris becau of the Stolper Samuelson effect.If the education ctor is labor intensive,an increa in the price of human capital rais the wage rate and makes human capital investment more attractive,so the capital gain on human capital will fall to restore equality of rates of return to the two factors,implying a stable price adjustment process.If the education ctor is capital intensive,however,the increa in the price of educational output will reduce the wage rate,requiring an increa in the capital gain on human capital.
We show that the model will exhibit saddle-path stability for either fac-tor intensity ranking of the ctors.In the ca where the education ctor is capital intensive,we show that the price of education output must jump immediately to its steady-state value.The output adjustment process (at fixed prices)is stable in this ca and results in convergence to the balanced growth path capital Âla
bor ratio.In contrast,the output adjustment process at fixed prices is unstable for the ca in which the education ctor is labor intensive becau of the Rybczynski effect :an increa in the relative supply of human capital results in an increa in the rate of human capital accumulation and a decrea in the output of the goods ctor.2In this ca,saddle-path stability results from adjustments in the price of educa-tion output along the transition path.For each factor intensity ranking,an unstable adjustment process in prices or quantities is offt by adjustment in the other variable and the system is saddle-path stable.The relative growth rates of consumption and factor supplies along the transition path will differ between the two cas,however.We also show that the stability of this adjustment process may be upt by factor taxation if the factor taxes are too distortionary.
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TWO-SECTOR MODEL OF ENDOGENOUS GROWTH 2
There is an interesting comparison between the instability of the quantity adjustment process described in this paper and that obtained in the two-ctor growth model of Uzawa
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贝乐学科英语价格[24,25]where only capital accumulation is endogenous.In the Uzawa model,instability may ari whe
n the investment good is capital intensive relative to the consumption good.This is analogous to our ca in which the education ctor is labor intensive,since each factor is ud intensively in its own production.
152BOND,WANG,AND YIP
The intertemporal no-arbitrage condition is also uful for analyzing the effects of factor taxation,since it leads to a natural decomposition of the effects of factor taxation into static effects(the location of existing factor supplies across ctors)and dynamic effects(whether to invest in physical or human capital).We consider three types of factor taxes:taxes on physi-cal capital,taxes on human capital in the goods ctor,and subsidies to human capital in the education ctor.We show that each type of factor tax will reduce the rate of growth while the subsidy will increa it,regard-less of the ctoral factor intensities.The effects of factor taxation on the capitalÂlabor ratio in the balanced growth path is,however,dependent on the factor intensity ranking.
Our approach differs from that of other recent work in this area,which has generally approached the problem by using simulation with specific functional forms.For example,King and Rebelo[12]investigate the two-ctor neoclassical exogenous growth model with Cobb Douglas prod
uc-tion functions,Mulligan and Sala-i-Martin[15]calibrate a two-ctor endogenous growth model with externalities using Cobb Douglas produc-tion technologies,and Xie[26]and Benhabib and Perli[1]have ud explicit functional forms to analyze the Lucas[14]framework.3Our results are thus more general than tho using explicit functional forms to analyze endogenous growth without externalities,and rve as a benchmark against which to compare results from models with externalities.Note that the approach taken in this paper could also be extended to models with externalities.
Section2of the paper prents a general two-ctor endogenous growth model and proves the existence and uniqueness of the balanced growth equilibrium.Section3shows that the system will be saddle-path stable and characterizes the transitional dynamics in the neighborhood of the balanced growth path.Section4provides applications of the model illustrating the effects of parameter and policy changes on the balanced growth equilibrium and the transition path.Section5concludes the paper.
cust2.A Two-Sector Model of Endogenous Growth
In this ction we prent a two-ctor model of endogenous growth in which there are two reproduc
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ible factors of production,physical capital (K)and human capital(H).We then establish conditions under which 3In particular,Benhabib and Perli[1]show that in the Lucas[14]model with human capital externalities,one may obtain a saddle-path stable,balanced growth equilibrium if the subjective rate of time preference is greater than the maximal rate of human capital accumula-tion and the intertemporal elasticity of substitution is sufficiently high.
there exists a unique balanced growth path in which consumption and factor stocks grow at a common (constant)rate,and in which relative prices are constant.
2.1.The Model
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The two ctors will be referred to as the goods ctor,X ,and the educa-tion ctor,Y .The goods are produced according to a constant returns to scale technology X =F (sK ,uH )=uHf (k x ),where s and u are the shares of physical and human capital,respectively,allocated to the goods ctor and k x #(sK )Â(uH )is the capital Âlabor ratio in the goods ctor.Goods may be either consumed,C ,or added to the capital stock.The evolution of the stock of physical capital is thus given by
北京英语K =uHf (k x )&$K &C ,(1)
where $is the rate of depreciation of physical capital.In the education ctor,Y ,the production function is also assumed to exhibit constant returns to scale,with Y =G ((1&s )K ,(1&u )H )=(1&u )Hg (k y )and k y #[(1&s )K ]Â[(1&u )H ].Note that the educational process is assumed to require both physical capital (libraries,laboratories,etc.)and human capital (faculty and students).The output of the education ctor rais the supply of effective labor units by adding to the stock of human capital,
H =(1&u )Hg (k y )&'H ,(2)
where 'is the rate of depreciation of human capital.The output per unit labor functions,f and g ,are assumed to be strictly increasing and strictly concave.Both products and factors are allowed to be dispod freely.This production structure can be thought of as a generalization of King and Rebelo [13]and Rebelo [17].
A reprentative agent's optimization problem is given as
max C ,s ,u ,K ,H 0(C )=| 0C (t )1&_1&_e &\t dt (P)
subject to (1),(2),H (0)=H 0>0,and K (0)=K 0>0,where \is a constant subjective rate of time preference,and _#(0,1)_(1, )is the inver of the constant intertemporal elasticity of substitution.The
utility functional form is chon becau it can yield a balanced growth path under some standard regularity conditions.4
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TWO-SECTOR MODEL OF ENDOGENOUS GROWTH 4We exclude the ca of log-linear utility to ensure homogeneity property of the utility function,which will simplify the analysis of the dynamics of the system below.For similar treatment,e Huffman [8].
