Download Dynamics of Investment in Human Capital and Infrastructure in the

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
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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
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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 :
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 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 
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 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).
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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.
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