Endowment Structure, Industrialization and Post-industrialization: A Three-Sector Model of Structural Change. Abstract

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1 Endowent Structure, Industrialization Post-industrialization: A Three-Sector Model of Structural Change Justin Lin Zhaoyang Xu China Center for Econoic Research Peing University Deceber 006 (Preliinary Draft Abstract The paper ais to provide a unified fraewor to characterize both the stage of industrialization of post-industrialization during the history of econoic developent The updating of endowent structure tends to increase the relative output of the sectors with higher capital intensity, then the gross copleentarity between the labor-intensive service goods the capital-intensive copound substitute goods between agricultural goods industrial goods (industrial goods is ore capital-intensive than agricultural goods, would initially lead to the flow of factors fro agriculture to industry, later fro agriculture industry to services In addition, our odel establishes the consistency of sectoral change in allocation of factors with the asyptotically constant interest rate aggregate growth rate in the setup with three sectors Our odel can be viewed as a natural extension of the wor of Aceoglu Guerriieri (006 in the sense that it introduces ore general setup thus has potential to explain ore interesting phenoenon in the sectoral change Keywords: endowent structure, industrialization post-industrialization, structural change

2 I Introduction Recently, econoists researching structural change have paid ore attention to the puzzle whether Kuznets facts, the systeatic change in relative iportance between different sectors, in particular, agriculture, anufacturing services, can be consistent with Kaldors facts, the relative constancy of the real interest, the aggregate growth rate, the capital-output ratio the share of labor or capital incoe in GDP The latest research on this thee are three independent papers by Kongsaut, Rebelo Xie (00, Ngai Pissarides (006 Aceoglu Guerriieri (006 Figure : Structural Change in USA Resource: Ngai Pissarides ( Structural Change in Korean Agriculture Industry Services Figure : Structural Change in Korean Resource: OECD industrial data Although it is valuable to research the consistency of Kuznets facts with Kaldors facts, the essential objective faced by the theory of structural change is to explain the crucial phenoena of non-balanced growth between different industrial sectors its ipact on the whole econoy This insistency on the essential objective is of the essence in the sense that there still exists soe very iportant phenoenon of structural change failed to be explained by the current theory To illustrate this point, let us loo at the epirical facts Figure reveals the history of structural

3 change for postwar US: a falling eployent share of agriculture, a rising share of services a rising share before the 960s but decreasing share afterwards in industry Figure shows the siilar trend of structural change in Korean In addition, the historical OECD evidence presented by Kuznets (966 Maddison (980 also provides alost copletely the sae picture of structural change as that of US Korean In these epirical facts, what ipresses us ost is not the rising share of services or the falling share of agriculture, but the hup-shaped, that is, initial rising later falling, share of industry We ephasize this hup-shaped share of industry in respect that it corresponds to the two ipressive stages during the econoic developent, that is, the stage of industrialization ainly characterized by the assive flow of production factor fro agriculture to industry before the pea of the hup-shaped share, of post-industrialization ainly characterized by the assive flow of production factor into services behind that pea For the USA, the turning point fro industrialization to post-industrialization eerged in the 960s; as for Korean, it is in the 990s that Korean began its transition fro industrialization to post-industrialization; as far as our China is concerned, it is well nown that we still be in the stage of industrialization Figure, Structural change predicted by Kongsaut, Rebelo Xie (00 Most literature on structural change are of two-sector odel thus siply can but focus on one stage of structural change of industry, either industrialization or post-industrialization There are ainly two class of literature, one of which ephasizes the role of non-hoothetic preferences consistent with Engel s law of or hierarchy of needs in inducing in structural change is of the ainstrea (Murphy, Shleifer Vishny (989, Matsuyaa (99, Echevarria (997,

4 Laitner (000,Kongsaut, Rebelo Xie (00, Caselli Colean (00, Gollin, Parente Rogerson (00,Matsuyaa (005; another of which is proposed by Bauol (967, ephasizing structural change resulting fro differential productivity differences between different sectors The forer literature explain industrialization by assuing that consuer has higher elasticity of incoe for industrial goods relative to agricultural goods, post-industrialization by assuing consuer has higher elasticity of incoe for service goods to industrial goods The later literature assue that the labor-intensive service goods the capital-intensive anufacturing goods are gross substitutes, thus the higher growth rate of productivity in capital-intensive sector will result in the flow of production factor fro the progressive sector, ie anufacturing, into the relatively laggard sector, ie, services There is soe latest literature on structural change odel with three or even ore sectors, but they erely depict the flow of production factor fro agriculture into services, all failed to characterize the hup-shaped eployent or capital share of industry in the whole econoy That is, in these odels, we can not see the stage of industrialization or of post-industrialization For exaple, in the paper by Kongsaut, Rebelo Xie (00, their Geary-Stone utility function results in a rising share of services, falling share of agriculture constant share of industry (see figure ; in Ngai Pissarides s paper (006 that extend Bauol s odel (967, the labor or capital eployent share is possible to be decreasing, constant or increasing, but ipossible to be initially rising then falling (see figure This paper constructs a three-sector odel of structural change with hup-shaped eployent share of industry The ain idea is inspired by Bauol s thoughts Our critical assuptions are the gross substitutity between the industrial goods the agricultural goods, the gross copleentarity between the service goods the copound goods of the industrial goods the agricultural goods The substitutity of industry agriculture results in the flow of factor fro lower-growing agriculture into higher-growing industry, the copleentarity of the copound industry services lead to the flow of factor fro the higher-growing copound industry into lower-growing services The net result of these two effects will bring about a hup-shaped eployent share of industry While the direction of structural change is of Bauol s style in our odel, the ain econoic force for structural change is the differences of the three sectors in their capital intensity proposed first by one of our authors Justin Lin (997 odeled first by Aceoglu Guerriieri (006 Technical progress leads to the updating of endowent structure, that is, the larger stoc of capital per capita, which conduces to the growth of capital intensive sectors thus cause sectoral change in allocation of factors cobined with the effect of the substitutiy copleentarity between different goods Our odel is highly related to the wor of Aceoglu Guerriieri (006 in the sense that we also use the differences in capital intensity the updating of endowent structure as the ain econoic force for structural change Also, our odel is related to that of Ngai Pissarides (006 in the sense that we all use the sae echanis proposed by Bauol (967 But, our odel iproves greatly on their wor in the following aspects:, we extend the two-sector odel of Aceoglu Guerriieri (006 into a three-sector odel, including the consistency of structural change in allocation of factors with the asyptotically constant interest rate aggregate growth rate;, we introduce heterogeneous elasticity of substitute between different sectors, which is a natural eaningful extension for the odel of Ngai Pissarides

5 (006;, the ost iportant, our odel can predict a hup-shaped eployent share of industry therefore be able to explain both industrialization post-industrialization in a unified fraewor The rest of the paper is organized as follows Section specifies the basic assuptions set up, giving the first order optial conditions for our dynaic syste Section deterines the direction of structural change for different sectors gives the dynaic functions representing structural change Section 4 proves the existence of balanced growth paths their local stability Section 5 undertaes a siple calibration to chec the hup-shaped eployent share of industry Section 6 concludes II The Basic Model In our econoy, there are three sectors with Cobb-Douglas technologies = AK L ( i =,,, ( i i i i i i where we assue that the capital intensity across different sectors is different, specifically, we assue > > (A This assuption eans that the first sector with subscript represents industry with the highest capital intensity, the second sector with subscript represents agriculture with the less capital intensity, the third sector with subscript represents services with the least capital intensity or highest labor intensity The industrial goods agriculture goods are produced copetitively using constant elasticity of substitution (CES production function with elasticity of substitution between the two goods equal toε > : ε ε ε ε ε ε M = γ + ( γ ( Here above-unit elasticity of substitution represents the gross substitutability between the industrial goods the agricultural goods, for concreteness, we denote M as the copound goods the sector producing copound goods as the copound sector The final goods is produced by cobing the above copound goods with the service goods with an elasticity of substitution < : = φ + M ( φ ( Substituting equation ( into (4 iediately iplies a two-level CES production function for the final goods which has been used by Krusell, Ohanian, Rios-Rull, Violante (000 Caselli Colean (006

6 To eliinate the effect of differences in productivity growth on structural change, we assue the productivity or technology of the three sectors all grow at the sae rate, ie, t A( t = e A(0 ( i =,, (4 i i In addition, we assue that there is an exponential population growth nt Lt ( = e L(0 (5 All factor arets are copetitive, the aret clearing for the two factors iplies K = K + K (6 M M L= L + L, (7 where K respectively denote the capital labor stoc used to produce both the M LM industrial agricultural goods, therefore KM = K+ K LM = L+ L We assue that all households have constant relative ris aversion (CRRA preferences over total household consuption (rather than per capita consuption, thus it iplies that the econoy adits a representative agent with CRRA preference: θ C ρt e dt (8 0 θ where Ct ( is aggregate consuption at tie t, ρ is the rate of tie preferences θ 0 is the inverse of the interteporal elasticity of substitution or the coefficient of relative ris aversion The budget constraint for the econoy or the social planner is: K = C (9 According to the above setup, the social optiization proble can be characterized by a syste of optial control with objective function (8 subject to the budget constraint (9, the aret clear conditions (6 (7 The Hailton function of this syste can be written as: θ ε ε ε C ε ρt ε ε H = e + λ φ( γ + ( γ + ( φ C θ, (0 + μ( K K K K + υ( L L L L then the first order optial conditions can be written as : H C θ e ρt = λ = 0 ( C

7 M ε = λφγ( ( μ 0 M H K K H M ε = λφ ( γ ( ( μ 0 K M K = ( = ( = λ( φ( μ = 0 H K K M ε = λφγ ( ( ( ν = 0 M H L L H M ε λφ( γ ( ( ( ν L M L = = 0 (4 (5 (6 = λ( φ( ( v = 0 H L L, (7 H λ = = μ K ; (8 the transversality condition is: li λ( tkt ( = 0 (9 t III Structural Change Let us introduce the following notations for the fractions of capital labor in different sector: K K K,, =, = = K K K M l L L L,, = l = l = L L LM Obviously, we have = ( (0 l = l ( l, ( therefore, it is sufficient to characterize the structural change of the whole econoy as long as we describe the dynaic behaviors of capital fractions in any two sectors In this setup, we will focus on, the capital fraction of industry relative to the copound sector,, the fraction

8 of capital in services Based on the, we then could depict the dynaics of industry s capital fraction relative to the whole econoy in ters of equation (0 Lea we can denote labor fractions of industry relative to the copound sector of services relative to the whole econoy as the express of their capital fractions, that is, l l as the expression of : l ( = ( + ( ( ( l ( + ( ( = ( Proof Cobing equations ( (5, we obtain u v L = K + (, (4 then cobing equation ( (6 yields u v L = K (5 At last, cobing the above two equations using the definitions of l gives l l =, (6 reforing it gives ( To obtain equation (, firstly cobing equations (4 (7 yields u v L = K, (7 then applying the siilar process in the obtainent of equation (6 gives = l which can be easily refored into ( = l ( l l (8 by using = ( l = l ( l At last, substituting the expression of l in ( into (8 after soe transforation, we

9 get ( Based on the lea, we can get the following iportant results Lea In equilibriu, we have the following inequality > l l > Proof Reforing the equation (6 gives l = ( l (, (9 which is obviously larger than zero because we have assued that is strictly larger than that is strictly saller than one in ters of its definition As long as equation (8 > l, the following inequality will be naturally established based on the ( l ( <, l( ( which directly iply < l because we have assued that > The econoic eaning of lea is that in equilibriu, the labor-intensive sectors should eploy ore labor than capital fraction, the capital-intensive sectors should eploy ore capital than labor fraction Specifically, services with the lowest capital intensity or the highest labor intensity should eploy higher fraction of labor relative to its fraction of capital eployent in the whole econoy, while in the copound sector, industry has the relatively higher capital intensity than agriculture has thus it is reasonable for it to eploy ore fraction of capital to its fraction of labor Proposition The dynaic behavior of the capital faction of industry relative to the copound sector can be characterized as the following expression: ( ε ( ( K ( l = ( n + ( + ( ε ( ( l( l K Proof Cobing equations (, ( ( gives Next taing logs of both sides in equation ( yields ε ε ( γ = γ ( ε = ε Differentiating equation ( gives the expressions of growth rate in the sector of industry (

10 agriculture, then subtracting fro, we get K L l ( ( ( = ( + K L l l M M M M (4 According to our denotations of K L, it is easy to show that K M L K l n ( K L K l M M = M Substituting the above equation into (4 gives K l = ( ( n ( ( K l l + ( + + ( + l l Now we differentiate the both sides of equation (8 obtain l l ( l l = + l (5 (6 (7 Substituting (7 into (6 gives K ( ( l ( ( n = + K (8 ( ( l( l + We can eventually obtain the targeting equation ( by cobing ( (8 aing soe transforation Proposition is crucial in that it shows that, given the growth rate of capital fraction in services, industry s capital fraction in the copound sector is onotonically increasing as long as >, representing higher capital intensity of industry relative to that of agriculture, ε >, representing the gross substitutability between the industrial agricultural goods The condition > leads to the increase in the relative output of industry thus brings about non-balanced growth between industry agriculture Furtherore, the conditionε > enhances the tendency of substitution of industrial goods for agricultural goods thus induces the flow of factors fro agriculture into industry So, the coparison of represents controls the econoic force for structural change, the elasticity of substitution ε deterines the direction of structural change The dynaic behavior of capital fraction in services,, is relatively ore coplicated We

11 will describe it with the following proposition corollary Proposition The dynaic behavior of the capital faction of the service sector relative to the whole econoy can be characterized as the following expression: ( ( K = n ( ( ( l K ( ( ( ( ( + ( ( ε ( ( ( l ( ( ε ( + [ ] ( ( ( ( l( l ( ( ( l Proof Cobing equations ( (4, we have ε M φγ M ( φ Then based on equations ( (, it is easy to get M γ = γ + (9 K = K (40 ε ε (4 Substituting equation (4 into (40 aing soe siple transforation yields ε ε( ( ε ( ε ( φ γ + = (4 ( φ Differentiating the above equation gives ε (4 ( ε ( + + ( ( = Using the siilar process of obtainent of the expression of, we can get K ( ( l ( l = ( ( n + K + ( ( l Equation (9 can be obtained by cobining equation (4 (44 (44 Siilar to that of, the dynaic behavior of is priarily deterined by the paraeters representing capital intensity the elasticity of substitution But here, services is labor-intensive relative to agriculture industry, thus the updating of endowent structure or capital deepening is unfavorable to the increase of output in services The under-unit elasticity of substitution iplies the gross copleentarity between the copound goods the service goods, which tends to increase the price of goods in the slow-growing sector then

12 siultaneously educe the flow of factors fro the copound sector to services Although the above analysis on proposition proposition indicate that, given the growth of, the updating of endowent structure is inclined to bring about the rise of, equation (9 also shows that the growth of is negatively related to that of A higher growth rate of tends to decrease the growth rate of So, the net effect is still uncertain before we can deterine the sign scale of s growth function In the following corollary, we show that both increase with the updating of endowent structure under soe reasonable loosing conditions For siplicity, we denote: B =, ( ( ( l ( ( B ( ε ( =, + ( ε ( ( l ( l B ( + ( ( ε ( ( l( l = +, ( ( ε ( + [ ] B 4 ( ( l = Obviously, all of the above equations is positive in ters of our assuptions on ε, coparison of or in lea In ters of these denotations, we can l l rewrite the expression of as the following copact fors: K = B( ( n BB K K = B( ( n + BB4 K Reforing the above differentiating functions of gives B [ B B ( ] K = ( n + BB B B 4 K, (45

13 [ + BB ( ] 4 K = ( n B + BB B B 4 K (46 Based on the above fors, we can deterine the dynaic behavior of Corollary As long as > ε, the fraction of capital in the service sector relative to the whole econoy is onotonically increasing until it reach to its upper liit, that is, we have: > 0 ; if assuption A holds, then we have 0 > Proof 0 > directly results fro equation (46 as long as we notice that > > B > 0 for all i i To establish the growth direction of, we first show > 0 BB < according to equation (45 B B Multiplying by yields ( ( + ( ( ε ( + ( ε ( ( l( l ( [ ( + ] BB =, (47 + ( ε ( ( l ( l which is saller than + ( ε ( ( l( l + ( ε ( ( l ( l (48 as long as ( ( ε ( < (49 ( If we assue that > ε, then the inequality (49 holds, therefore the

14 following inequalities hold + ( ε ( ( l( l BB < <, + ( ε ( ( l ( l thus our corollary is established This corollary says that in order to educe the flow of factor into services, we ust have a saller, that is, a stronger gross copleentarity between the copound goods the service goods Next, let us focus on what we are ost interested in, the dynaic behavior of industry s capital fraction in the whole econoy (0 < Proposition Under the conditions, there exists a unique pair (, such that > 0 when < (or < 0 when > (or > ; < furtherore, if condition > ε holds, the function of is concave with a positive initial value Proof The full proof should tae three steps: Step, the sign of corresponds to the sign of ( ( ( ( ε f g Based on the relationship of, in equation (0, we have = then substituting (45 (46 into it, we get [ ] [ ] B( + BB 4( B BB( K = ( n ( ( + BB B B K (50 4, In ters of our denotations for B, B, B B4, soe transforation will induce the following equations: BB ( ( ( ( ( + = 4 l [ ] [ ][ l l ] ( ( + ( ( ( ε ( ( BB ( = ( ( + + ( ε ( ( (

15 Then substituting the into equation (50 enables us to rewrite as where f g K n, (5 ( + BBBB B K ( ε ( ( ( ( 4 5 [ ][ ][ ε ] B = ( + ( ( ( l + ( ( ( l ( l > 0, 5 f ( = ( + (, [ ] g ( = ( ( + ( ( ( ε ( ( Therefore, the question whether is larger or saller than zero transfors the following question whether ( ε ( f ( g( is larger or saller than zero Obviously, the properties of f ( are crucial to analyze the sign of g ( Step : to discuss the properties of f ( g ( Differentiating ( f, we find that ' f ( = + + ( < 0, because 0< <, f ' ( < 0 iplies that Thus, the curve of ax f ( = li f ( =, (5 0 in f( = li f( = 0 (5 ( ε ( f ( is onotonically decreasing in the following graph

16 ( ε ( ( ( ( ε ( f ( g ( q ( ( ( (0 0 Then differentiating g ( twice yields g ( = ( ε ( > 0, '' which eans ( g is a convex function It is easy to now that li g ( = ( (, (54 0 li g ( = ( ( (55 Step, the existence uniqueness of the pair (, if there is (0 < According to equation (5 to (55, we have li( ε ( f( g( = ( ( < 0, (56 li ( ε ( f( g( = ( ε ( > 0 ε 0, (57 as long as (0 < Then, given, added with the convexity of g ( the decreasing trend of f (, there

17 ust exists a unique we denote as satisfying 0 < < such that = 0, > 0 when < < 0 when > Furtherore, considered that is onoton ically increasing, given (0, there ust exists a unique corresponding to Step 4, the concavity of Differentiating g ( gives ' g [ ] ( = ( ( ( ε ( + ( ε (, which is larger than zero as long as > ε If g ' ( > 0 thus the function of g ( is increasing onotonically, the expression of ( ε ( f ( g( will be decreasing onotonically in the doain of (0,, which establish the concavity of IV The Existence Stability of the Balanced Growth Paths The equilibriu behavior of the econoy is characterized by balanced growth paths Usually, the balanced growth path (BGP is defined as an equilibriu path where the growth rate of consuption exists is constant, ie, = C C gc Fro the Euler equation (6 the first order condition (, this iplies ( φ( ρ = gc θ K Based on the above equation, we can get the following proposition Proposition 4 On the balanced growth path,, K, C, K (47 grow at the sae rate, which is represented by g = gk = gc = g = g K = n+

18 Proof According to equation (, it is easy to show M = φ + φ M then substituting the expression of in (4 into the above equation, we get ε ( ( ε γ = φ γ + ( + φ, (48 Corollary has showed that converges to gradually, therefore will asy ptotically converge to a constant, which eans will grow asyptotically at the sa e rate as that of Furtherore, K also should asy ptotically converge to a constant according to eq uatio n (47 if the balanced growth path exists This iplies that, K thus K (because li t ( = ust grow at the sae rate t It is easy to show that K L = l A K, (49 which eans that on the balanced growth path, capital stoc K, ust grow at the sae rate as that of A L, ie, g K = K n K = + Proposition 4 iplies that the equilibriu behavior of this econoy can be represented by a syste of autonoous non-linear differential equations in three variables: C K c =, x =, A L A Here c is the level of consuption noralized by population technology f the capital-intensive sector is the only co ntrol variable; x is the capital stoc noralized by the sae denoinator; deterine the allocation of capital between the three sectors L

19 These three are state variables with given initial condition s x (0, (0 (0 The dynaic equili briu conditions then translate into the following equations: c ( ( ( l ( x = φ ρ n c θ x ( c = l x n x x ( ( x = ( ( + ( ( ( l x ( ( ( + ( ( ε ( ( ( l ( ( ε ( + [ ] ( ( ( ( l( l ( ( ( l ( ε ( ( x ( l = ( ( ε ( ( l( l x (50 where is given by equation (48 l by equation ( Obviously, the definitions of c, x, have ensured that the steady-state equilibriu in the above dynaic sy ste corresponds to the balanced growth path in our econoy, ie, c, x, ust be constant in th e BGP equilibriu According to equation (48, will converge to ( φ, in ters of our proposition corollary,, thus l l will converge to Therefore, these steady-state values of dynaic syst e (50 are given by = =, x θ( n + + ρ = ε ( φ, c = ( φ ( x ( n+ x

20 To coplete our description of this econoy with two different capital-intensive sectors, now we should show that c, x, will converge to c, x, in a saddle path, ie, the syste (50 is locally stable The next proposition states that this is the case Proposition 5 The non-linear syste ( is locally stable, in the sense th at in the neighborhood of to c, c, x, x,, there is a unique two-diensional anifold of solutions that converge Pr oof Rewrite the syste (50 in a ore copact for as y = f( y, ' where y = (, c x,, To investigate the dynaics of the syste ( in the neighborhood of the steady state, consider the linear syste z f( y z =, where z = y y y such that f( y = 0, where J( y is Jacobian atrix of y f ( y evaluated at Before we proceed, let us give iportant results which will be useful in the following algebra operat ion, l l = 0, y= y ( =, ( y= y l y= y ( = ( Differentiation soe algebra enable us to write this Jacobian atrix as a a a a a a a a J( y = a a a a a a a a cc cx c c xc xx x x c x c x where a = a = a = a = a = a = a = a = 0 cc c c x x c

21 a acx c ( ( ( ( φ c x = θ ( ( ( ( φ c x = θ a = xc axx = x n+ ( φ ( ( a a x ( x ( φ = ( ( = a ( ε ( = The deterinant of the Jacobian atrix is det( J( y = a A, where A is the algebraic cofactor of a It is easy to show that A = a a a therefore cx xc det( J( y = a a a a The fact that a, a, are all negative cx xc iediately eans that det ( J( y <0 To establ ish the local stability of this syste, we should show that three of the are negative one is positive Let ψ denotes the vector of the eigenvalues of this atrix, then the characteristic equation is give n by cx det( J( y ψ I 0 = xc a Using cofactor of expansion, we can exp the characteristic equation as: ( a ψ A = 0, where A = ( a ψ [ ψ( a ψ + a a ] xx cx xc Thus, we have [ ] ( a ψ( a ψ ψ( a ψ + a a = 0 xx cx xc a This expression shows that one of the eigenvalue is equal to a another is equal to a, which are both negative, so there ust be three negative one positive eigenvalues as a result that the deterinant of the Jacobian atrix det( J( y is negative This establishes the

22 existence of a unique two-diensional anifold of solutions in the neighborhood of this steady statet thus local (saddle-path stability holds This proposition iplies that the balanced growth path equilibriu is locally stable, ie, when the initial values of capital, labor technology are not too far away fro the balanced growth path, the econoy will indeed converge to this equilibriu, with non-balanced growth at the sectoral level constant interest rate at the aggregate V A Siple Calibration VI Conclusion