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Dynamics of Investment in Human Capital and Infrastructure in the Process of Urbanization and Economic Growth LIU Anguo ZHANG Wei Beijing University of Chemical TechnologyBeijing, China, 100029 University of Jinan Jinan,China,250022 [email protected] : Abstract Standard two-sector endogenous growth model with human and physical capitals as inputs does not involve allocation of public expenditure between human capital accumulation and investment in infrastructure. However, public expenditure takes a predominant role in human capital accumulation and investment in infrastructure. By introducing public expenditure into a model of urbanization and endogenous growth, the allocation of public expenditure between human capital accumulation and investment in infrastructure is endogenized. The condition of inter-temporal inter-sector transfer of human capital lends itself to the analysis of the transitional dynamics of the model. Results from numerical simulation indicate that there are complicated transitional dynamics in the model of urbanization and endogenous growth with public input. When there are diminishing returns to physical capital, the size and allocation of public expenditure are endogenously determined and stabilized. If there is somewhat degree of increasing returns to physical capital, the inter-sector allocation of public expenditure is endogenously unstable. In the latter case, government policy in public expenditure plays an important role in stabilizing the economy. In addition, an analysis of the comparative statics of the model suggests that there exists an optimal tax rate and consequently an optimal size of public expenditure, implying that public expenditure supported by an income tax is pro-growth in a certain range, beyond which an increase in public spending will exert negative effect on growth. This is in line with the findings in Barro (1990). :Public Expenditure, Investment, Human Capital, Urbanization, Growth Key words 1. Introduction To effectively allocate public expenditure between infrastructure building and investment in human capital so as to facilitate the process of urbanization and long-run economic growth has been a common challenge faced by China and many other developing countries. Since1990s, Barro, Sala-I-Martin, Corsetti and Roubini et. al have carried out a series of research on the role of public expenditure in economic growth. By introducing tax-financed government services into a model of endogenous growth, Barro (1990) finds out that “[g]rowth and saving rates fall with an increase in utility-type expenditures; the two rates rise initially with productive government expenditures but subsequently decline”. Corsetti and Roubini (1996) emphasize the productive nature of public spending and assert that “[i]f higher distortionary taxes are used to finance a higher productive public spending, the larger scale of government activities may actually enhance the growth rate, despite the negative effect on growth of taxation.” By building a one-sector model of endogenous growth with public input and examining its transitional dynamics, Palivos, Yip and Zhang (2003) show that “a continuum of equilibria and global indeterminacy can arise for reasonable parameter values, simply due to the presence of endogenous public policy.” Alvarez-Albelo (2000) has examined the long-run effect of public expenditure on education and shown that “when the government modifies the proportion of resources assigned to education, the time spent in school may increase, fall or remain constant while the growth rate increases.” This conclusion goes along interestingly with that of Bond, Wang and Yip (1996) on the effect of taxes, who developed an inter-temporal non-arbitrage condition for physical and human capital and generalized following two observations: (1) A decrease in the time preference rate raises the balanced growth rate, but leaves the BGP values of wage, interest rate, and prices unaffected. The level of consumption per effective unit is higher, and the physical to human capital ratio rises if and only if 710 the final goods sector is more intensive in physical capital than education sector; (2) When factor taxes are too distortionary, intensity rankings need not be consistent, which leads to the possibility of either an unstable node or indeterminacy. The current study attempts to make an extension of the public financing model of Barro (1990) and further introduce public expenditure into sector of human capital production. The incorporation of public expenditure and human capital accumulation into a unified model of endogenous growth not only facilitates the analysis of the optimal allocation of public expenditure between investment in education and that in infrastructure, but also allows us to further examine the transitional dynamics of this optimal allocation as well as some other important parameters in the process of their inter-temporal evolution. .A Model of Urbanization and Endogenous Growth with Public Inputs 2 2.1 Basic Framework Assume that there exists a closed economy that comprises of atomistic individuals (or households) and firms of infinite lives. The population of the economy is held constant. In the economy there are two productive sectors: a goods sector and an education sector. Aggregate utility function u( ) of the economy takes the form C1−θ − 1 (1) u (C ) = 1−θ in which C is aggregate consumption and 1/ θ is a measure of inter-temporal elasticity of substitution. Goods sector uses as inputs physical capital, human capital and infrastructure financed with public expenditure to produce consumption/capital goods. The education sector uses human capital and public expenditure to produce human capital. Both sectors employ a Cobb-Douglas technology of production. The production function of the goods sector takes the form β (2) Y = A( µ H )α ( vG ) K 1−α − β in which Y is the output, A is a measure of the level of technology, K and H are respectively aggregate stocks of physical capital and human capital, and G is government expenditure. µ (0 < µ < 1) and v (0 < v < 1) are respectively fraction of human capital and fraction of public expenditure used in goods sector. α > 0 stands for the share of human capital input in the production function, and β > 0 stands for the share of input of government expenditure in the production function, with α + β < 1 . Similar to the specification used in Barro (1990), government expenditure is supported by an income tax. The government collects an income tax T with a rate of τ and it is assumed that the government always keeps a balanced budget, that means the government can neither finance deficits by issuing debt nor run surpluses by accumulating assets. As T = G = τ Y , we have β 1 α G = (τ A )1− β v1− β ( µ H )1− β K 1− α 1− β (3) and β 1 β α Y = τ 1−β A1−β v1− β ( µ H )1− β K α 1− 1− β (4) With a depreciation rate of δ , the process of physical capital accumulation can be expressed as β • 1 β α K = (1 − τ )τ 1− β A1−β v1− β ( µ H )1− β K 1− α 1− β − C −δ K (5) With g K standing for the growth rate of physical capital, we have • β 1 β g K = K / K = (1 − τ )τ 1− β A1− β v 1− β K / ( µ H ) − α 1− β − C / K −δ (6) The education sector uses a fraction of (1 − µ ) of the human capital and a fraction of (1 − v ) of the government expenditure for the production of human capital. Assume that human capital depreciates with the same rate of δ as physical capital does, the production function of human capital is described 711 as • H = E [ (1 − µ ) H ] [ (1 − v )G ] γ 1−γ (7) −δ H γ in which stands for the share of human capital input in the education sector, productivity of education sector. Inserting (3) into (7), we have 1−γ • 1−γ H = Eτ 1− β A1− β (1 − µ )γ (1 − v)1−γ µ α (1−γ ) β (1−γ ) 1− β 1− β v γ+ H α (1−γ ) 1− β E for a measure of (1−α − β )(1−γ ) 1− β K −δ H (8) With g H standing for the growth rate of human capital, we have 1− γ • 1− γ g H = H / H = Eτ 1− β A1−β (1 − µ ) γ (1 − v)1−γ µ α (1− γ ) 1− β v β (1−γ ) 1− β (K / H ) (1−α − β )(1− γ ) 1− β −δ (9) Taking K (0) , H (0) and G (0) as given and subject to resources constraints of (5) and (8), by selecting a path of { µ(t), v(t), C(t), K(t), H(t)} a social planner maximizes the inter-temporal utility ∞ U = ∫ e− ρ t u (C )dt (10) 0 where ρ is a measure of the consumer’s subjective inter-temporal preference. The above problem can be taken as one of dynamical programming for the social planner. Write J , the Hamiltonian of the discounted value of inter-temporal utility as: β 1 α β α α 1− J = u (C )e − ρ t + λ (1 − τ )τ 1− β A1− β µ 1− β v 1− β H 1− β K 1− β − C − δ K α (1−γ ) β (1−γ ) α (1− γ ) 1−α − β )(1− γ ) ( 1− γ 1− γ γ+ + φ Eτ 1− β A1− β (1 − µ )γ (1 − v)1−γ µ 1− β v 1− β H 1− β K 1− β −δ H (11) Define λ and φ as co-state variables associated respectively with K and H. By applying Pontriyagin’s Maximum Principle, we have: (12) 1 α 1−γ v= here ∆ = β γ 1 − ∆ + ∆µ − 1 β 1 (1 − α − β )(1 − τ )τ 1−β A1−β (1 − ∆ + ∆µ −1 ) λ =− λ 1 − β (1 − ∆ + ∆ µ −1 ) • − β 1− β K / ( µ H ) − α 1− β +δ • 1−γ (1−α − β )(1− γ ) 1− γ − φ = −γ E ∆1−γ (τ A )1− β (1 − ∆ + ∆ µ −1 ) 1− β K / ( µ H ) 1− β +δ φ β 1 β α − • − −1 1− β 1− β C 1 (1 − α − β )(1 − τ )τ A (1 − ∆ + ∆µ ) 1− β K / ( µ H ) 1− β = − − δ ρ C θ 1 − β (1 − ∆ + ∆µ −1 ) (13) (14) (15) With p denoting the relative price of human capital in terms of physical capital, we have: φ p = = β∆ γ τ λ β +γ −1 1− β γ 1− β A 1− β −γ −1 (1−α − β )γ 1 − τ (1 − ∆ + ∆µ ) 1− β E K / ( µ H ) 1− β −1 1 − γ 1 − β (1 − ∆ + ∆µ ) −1 (16) Use e as a measure of relative efficiency of public input in terms of the ratio of marginal productivity of public input in the production of goods over that of public input in the accumulation of human capital, we have: β −1 α β α −γ β +γ −1 1−α − β e= K AE µ v H G 1−γ = ∆γ −1τ β +γ −1 1− β (17) γ 1− β −γ (1−α − β )γ β µ −1 1− β A1− β E −1 K / ( µ H ) 1− β (1 − ∆ + ∆µ ) 1− γ 1− µ The relative efficiency of public input e also reflects the opportunity cost of public input in its investment of human capital. From (16) and (17) we can derive a relation between e and p : 712 µ 1 e= 1 − β (1 − ∆ + ∆µ −1 ) p (1 − τ ) ∆ 1− µ From (13), (14) and (16), we obtain the growth rate of relative price: β 1 1− β p (1 − α − β )(1 − τ ) τ A (1 − ∆ + ∆µ ) = p 1 − β (1 − ∆ + ∆µ −1 ) • 1− β −1 1−γ − γ E ∆1−γ (τ A )1− β (1 − ∆ + ∆µ −1 ) − 1−γ 1− β − β 1− β K / ( µ H ) K / ( µ H ) − (18) α 1− β (19) (1−α − β )(1−γ ) 1− β Equation (19) turns out the “inter-temporal no-arbitrage condition” for physical and human capital as in Bond Wang and Yip (1996), which specifies the inter-temporal relative-price adjustment necessary to equalize the net returns on human and physical capitals. If the net return on physical capital (net of depreciation) exceeds that on human capital, there must be a capital gain earned on human capital investments to offset the difference in net returns. By calculating growth rates of variables in (16), we can derive the following “inter-temporal inter-sector transfer condition” for human capital: • • • • (20) (1 − α − β ) γ 1 − β − γ µ (1 − α − β ) γ K H p β∆ ∆ + + = − − K H p 1− β 1 − β (1 − ∆ ) µ + ∆ (1 − β + β ∆ ) µ − β ∆ µ 1− β Equation (20) shows that the difference between the difference of growth rates between physical and human capital after being adjusted by a coefficient representing the difference of input shares of human capital in goods sector as well as in education sector and the growth rate of relative price of human capital, determines the speed of inter-sector transfer of human capital. From (18) an “inter-temporal no-arbitrage condition” for public input analogous to (19) can be further derived: • • • µ p e 1 β∆ = + + e 1 − µ (1 − β + β∆ ) µ − β ∆ µ p (21) Equation (5), (8), (19) and (20) consist of a system of non-linear differential equations involving 2 control variables ( C and µ ) and 2 state variables (K and H) that describe the growth of physical and human capital, consumption and fraction of human capital in goods production. Using “time eliminating method” of Mulligan and Sala-I-Martin (1993), we can proceed with the following analysis with the aid of numerical computation. 2.2 Balanced Growth Path Define ω = K / H and χ = C / K , differential equations (5), (8), (19) and (20) can be rewritten as β • 1 ω (1 − τ )τ 1−β A1− β ω = β ω (1 − ∆ + ∆µ −1 )1−β µ − α 1− β 1−γ 1−γ (1 − µ ) E ∆1−γτ 1− β A1−β ω − 1−γ (1 − ∆ + ∆µ −1 )1−β µ β • 1 (1 − τ ) τ 1−β A1−β ω χ 1 1− α − β = −θ β −1 χ θ 1 − β (1 − ∆ + ∆µ ) (1 − ∆ + ∆µ −1 )1− β µ • (1 − α − β ) γ ω − (1 − α − β ) γ 1− β ω 1− β β = (1−α − β )(1−γ ) 1−β α − 1− β +χ − (22a) −χ (22b) δ (1 − θ ) + ρ θ • + µ β∆ 1− β − γ ∆ + 1 − β (1 − ∆ ) µ + ∆ (1 − β + β∆ ) µ − β∆ µ 1 1 − β (1 − ∆ + ∆µ −1 ) (1 − ∆ + ∆µ 1−γ α (1 − α − β )(1 − τ )τ 1− β A1− β (ω / µ )−1− β β −1 1− β ) − 1−γ γ E ∆1−γτ 1− β A1− β ( ω / µ ) (22c) (1−α − β )(1−γ ) 1− β 1−γ −1 1− β (1 − ∆ + ∆µ ) By setting the growth rates of µ , ω and χ as zero and solving system (22), we arrive at the following equation that determines the stead-state value of µ : (1 − ∆ + ∆µ ) −1 α + β 1 − β (1 − ∆ + ∆µ −1 ) 1−α − β γ − θ (1 − µ ) =η δ (1 − θ ) + ρ 713 γ 1− β +α 1−γ (23) 1−α − β α where η = ∆ατ α + β AE 1−γ (1 − α − β )(1 − τ ) γ . From µ *, the stead-state value of µ , we can get v * , the stead-state value of v , through (12). Steady-state values for physical-human capital ratio( ω * ), consumption-capital ratio ( χ * ) and the average product of physical capital ( Y * / K * ) can be further arrived at γ − θ (1 − µ *) δ (1 − θ ) + ρ ω* = ψµ * where (1 − α − β )(1 − τ ) ψ =τ A γ β α 1 α χ * = 1 − 1− β α : 1− β α β − −1 α (1 − ∆ + ∆µ * ) 1 − β (1 − ∆ + ∆µ *−1 ) − 1− β α (24) . (25) 1−γ α δ (1 − θ ) + ρ µ *+ µ * 1 − α − β γ − θ (1 − µ *) 1 − γ δ (1 − θ ) + ρ Y* 1 α = 1− γ + µ * γ − θ (1 − µ *) K * 1 − τ 1 − α − β (26) 2.3 Steady-state Comparative Statics Due to the structural complexity of the model, it is difficult to carry out an algebraic comparative analysis around the stead state. However, with the aid of numerical computation, the comparative statics of µ , ω and χ around the stead state can still be observed. In Table 1, we report the signs of partial derivatives of the steady-state values of µ , ω and χ as well as g , the growth rate of output with respect to all of the parameters in the model. The baseline parameters used are reported in the notes to the tale. For example, a positive sign “+” in the first column, second row, suggests that an increase in A , the level of technology, in the goods sector leads to an increase in µ *, the steady-state fraction of human capital input in goods sector. μ* Baseline ∆A ∆E ∆α ∆β ∆γ ∆θ ∆τ ∆δ ∆ρ Table 1 Effect of Parametric Changes on Steady-state Values χ* ω* g 0.6302 0.0780 12.0306 0.0235 + + + + + + - + - - - - - - - - - - - - + + - - + + - + - - + - + + - - = = = = Note: The baseline parameters are A 1.1, E 0.2, α 0.3, β 0.1, γ =0.5, θ=2.0, τ=0.2, δ=0.1, ρ=0.02. we also observe that with the increase of τ, the income tax rate, the steady-state value of physical-human capital ratio( ω * ) decreases continuously, but the steady-state growth rate (g), fraction of human capital in goods sector( µ *) and consumption- capital ratio ( χ * ) all rise initially but subsequently decline, exhibiting a rise-fall pattern of “inverted U” (see Figure 1). That means that there exists an optimal tax rate and consequently an optimal size of public expenditure, implying that public expenditure supported by an income tax is pro-growth in a certain range, beyond which an increase in public spending will exert negative effect on growth. This is in line with the findings in Barro(1990). 714 Figure 1. Effect of Change in Income Tax Rate on Steady-state Values .Transitional Dynamics 3 Following the argument of Mulligan and Sala-I-Martin (1991), the transformed dynamical system has a stationary point or stead state that solves the optimal control problem. In the ( µ , χ , ω ) space, the locus of points which, when allowed to evolve according to (22), asymptotically approach the stationary point, form the stable manifold of the stationary point. This stable manifold describes optimal solutions to our optimal control problem. In the absence of externalities, solutions to our model are solutions to a social planning problem that can be represented by a dynamic program solved by a set of policy functions. As the stable manifold and the policy functions describe the same solutions, projections of the stable manifold in the ( µ , χ , ω ) space into the ( µ , ω ) and ( χ , ω ) planes are graphs of policy functions: (27a) µ (t ) = µ (ω (t )) (27b) χ (t ) = χ (ω (t )) Slopes of policy functions are derived by the following computation: • • • • (28a) µ '(t ) = µ / ω = f1 ( µ , χ , ω ) (28b) In 2.2, we have already worked out some boundary conditions for system (27): the stationary point “satisfies” the policy functions µ (t ) and χ (t ) : (29a) µ* = µ (ω*) (29b) χ * = χ (ω*) By applying L’Hopital’s rule or by linearizing the system around its steady state and studying the eigenvectors of the matrix describing the linearized version of the system, the slopes of policy functions can be further specified. Consequently, we arrive at a system of differential equations of an initial value type, the original boundary value problem is transformed into an initial value one that can be easily solved with subroutine ODE45 of Matlab. By working out graphs of the policy functions, we are allowed to visually observe the transitional dynamics of the system. The following two types of behavior of the policy functions are observed: χ '(t ) = χ / ω = f 2 ( µ , χ , ω ) (a) Policy functions µ and v slope downward. In this case, as the physical-human capital ratio ω increases, the growth rate and the relative price of human capital go up, but the growth rate and the average product of physical capital go down, exhibiting a trend of diminishing returns to physical capital. Policy functions µ , v and χ (or C/K) are all 715 downward sloping (see Figure2). The optimal inter-sector allocation and size of public expenditure are endogenously determined and stabilized. This is similar to “the most plausible case” described in the generalized two-sector model in Mulligan and Sala-I-Martin (1993). = = = = Note: The baseline parameters are A 1.1, E 0.2, α 0.3, β 0.1, γ =0.5, θ=2.0, τ=0.4, δ=0.1, ρ=0.02. (b) Policy functions µ and v slope upward. In this case, with an increase of the physical-human capital ratio ω , the growth rate and the average product of physical capital always increase, exhibiting a trend of somewhat increasing returns to physical capital; growth of human capital goes up when ω is lower than ω * and turns to go down when ω is higher than ω * , and the relative price of human capital always goes down. While the policy function χ (or C/K) is still downward sloping, policy functions µ and v are upward sloping (see Figure 3). The equilibrium is unstable. In the case of a disequilibrium, transitional dynamics of the economy as indicated through the condition of inter-temporal inter-sector transfer of human capital (equation (20)) and shown by numerical simulation may even enlarge the disequilibrium. Whether the economy returns to its equilibrium depends on the direction of inter-sector flow of human capital, which depends in turn on the expectation of individuals. A pro-equilibrium expectation may in fact lead to the recovery of economic equilibrium, and a pro-disequilibrium expectation may even amplify an under-way disequilibrium. Similar instability and indeterminacy arising from agents’ expectation about the future are noted in Bond, Wang and Yip (1996) and Palivos, Yip and Zhang (2003). Therefore, government policy in public expenditure plays an important role in stabilizing the economy. 716 = = = = Note: The baseline parameters are A 1.1, E 0.2, α 0.3, β 0.1, γ =0.5, θ=2.0, τ=0.39, δ=0.1, ρ=0.02. Figure 3 Transitional Dynamics with Policy Functions Upward Sloping .Conclusion 4 The introduction of public expenditure into a two-sector endogenous growth model allows us to observe the evolution of behavior of public expenditure, infrastructure building and human capital accumulation within a broader framework in the process of urbanization and economic growth. First, the allocation of public expenditure between human capital accumulation and investment in infrastructure is endogenized. Second, the condition of inter-temporal transfer of human capital derived in the model lends itself to the analysis of the transitional dynamics of the economy. Results from numerical simulation indicate that there are complicated transitional dynamics in the model of urbanization and endogenous growth with public input. When there are diminishing returns to physical capital, the size and allocation of public expenditure are endogenously determined and stabilized. If there is somewhat degree of increasing returns to physical capital, the inter-sector allocation of public expenditure is endogenously unstable. In the latter case, government policy in public expenditure plays an important role in stabilizing the economy. In addition, an analysis of the comparative statics of the model suggests that there exists an optimal tax rate and consequently an optimal size of public expenditure, implying that public expenditure supported by an income tax is pro-growth in a certain range, beyond which an increase in public spending will impart negative effect on growth. References [1] Alvarez-Albelo, Carmen D. A simple growth model of schooling and public expenditure on education. METU Studies in Development, 2000, 27 (1-2): 1-21. 717 [2] Barro, Robert. Government Spending in a Simple Model of Economic Growth. Journal of Political Economy, 1990, 98 : S123–S125. [3] Barro, Robert and Xavier Sala-I-Martin. Public Finance in Models of Economic Growth. 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American Economic Review, 1961, 51 (1): 1-17. [16] Schultz Theodore. Reflections on Investment in Man. Journal of Political Economy, 1962, 70: S1-S8. 718