Download Government Expenditure and Stochastic Endogenous Growth Model

Government Expenditure and Stochastic Endogenous Growth Model
Xiong Yan
(College of Economics, South-Central University for Nationalities, Wuhan 430073)
Abstract: This paper employs a continuous-time stochastic endogenous growth model with government
expenditure on production and welfare. By the method of dynamic optimization, the macroeconomic
equilibrium is derived. And then, the model is used to analyze the macroeconomic effects of government
expenditure and its volatility on economic growth.
Key words: stochastic endogenous growth; congested government expenditure; volatility
1 Introduction
Government expenditure has been introduced into economic growth model in many papers. In
former literatures, it has only one effect, and a series of conclusions have been drawn from this. But this
isn’t in accordance with reality. Tournovsky [1] talks about the congested government expenditure of
stochastic endogenous growth model, and uses government expenditure, a kind of public consumption
as a utility function, then gets the best financial policy in stochastic condition. Wang [2] analyses
congested government expenditure on consumption in stochastic endogenous growth model.
Considering that government expenditure has effect on both production and welfare, the thesis shows
how the effect works on economic growth.
2
A Stochastically Growing Economy
We consider a closed economy populated by a continuum
[ 0,1] of
identical long-living
individuals. We use k , y, c to denote the individual’s physical capital stock, output and consumption
respectively, while K , Y , C are the aggregate variables corresponding to k , y, c . The flow of output,
dy , produced by the typical individual over the period (t , t + dt ) , is determined by his privately
owned capital stock, k ; and the rate of flow of services, H , derived from his use of a public good, in
accordance with the stochastic production function.
1
dy = z (dt + dy ) = α H β k 1− β (dt + dy ) = (α h β )1−β k (dt + dy ) = A(h)k (dt + dy )
A(h) (α h β )
1
1− β
， 0 < β < 1 ， Where dy is
，（1）
a proportional productivity shock that is
independently and normally distributed over time, having zero mean and variance
σ 2 . Equation
u
(1)
asserts that the individual’s mean rate of output, z , is subject to positive, but diminishing, marginal
physical product, in both the level of services of public expenditure, H , and his individual capital
stock, k , and constant returns to scale in these two factors of production, together. Public expenditure
H = hz , h stands for the ratio of government productive expenditure to output.
The productive expenditure of government over the period (t , t + dt ) is specified by:
，
，
dGP = Hdt + H ' dy = z (hdt + h ' dy )
H = hz , H ' = h ' z
0 < h, h ' < 1
Where H denotes the deterministic (expected) rate of expenditure over the period dt and H '
denotes the current stochastic expenditure flow, as a proportion of the stochastic output shock. We
assume that the government sets H and H ' as fraction of the current deterministic and stochastic
rates of flow of aggregate output: H = hz , H ' = h ' z .
Thus the fraction, h , represents the policy maker’s choice of the (deterministic) size of
government, while h '(0 ≤ h ' ≤ 1) represents the fraction of the aggregate output shock, absorbed by
the government. If h is constant, the government is claiming a fixed share of the growing (mean)
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output, so that an increase in h parameterizes a deterministic expansion in expenditure in a growing
economy.
The welfare expenditure of government over the period (t , t + dt ) is specified by:
dGs = AK (
k 1−δ
) ( gdt + g ' dy )
K
，
0 < g, g ' < 1
（ 2）
Equation (2) characterizes what one may call relative congestion, in that the welfare expenditures
derived by an agent from a given rate of public expenditure depends upon the usage of his individual
capital stock relative to aggregate usage. This is a plausible assumption, a good example of which is the
use of highway, and in general the level of services he does enjoy is adversely affected by the total usage
of cars by others insofar as this causes congested roads. The fact that the benefits from the public good
are tied to the individual’s use of his capital encourages private investment.
The exponent δ parameterizes the relative congestion associated with the public good. The case
δ = 1 corresponds to a non-rival, non-excludable public good that is available equally to each agent,
independent of the size of the economy; there is no congestion. There are few, if any, examples of such
pure public goods, so that this case should be treated largely as a benchmark. 0 < δ < 1 describes
congested level.
The representative agent’s welfare is represented by the intertemporal isoelastic utility function:
u ≡ E0 ∫
∞
[C (Gs )η ]γ
γ
0
e − ρ t dt
Where the exponent
γ
（ ρ > 0 ；η > 0 ； −∞ < γ < 1/(1 + ηδ ) < 1 ；ηδγ < 1 ），
is related to the constant relative risk aversion (CRRA) with γ = 0 being
equivalent to a logarithmic utility function. σ = 1 − γ is the constant relative risk aversion
(CRRA). η measures the influence of government expenditure on private welfare. We assume private
and government consumption have opposite marginal utility, so η > 0 .
We assume only deterministic government welfare expenditures are contained in agent’s utility,
while stochastic government welfare expenditures have no effect on agent’s utility. Therefore, we can
get Gs = AgK (
k 1−δ
) .
K
3. The Centrally Planned Economy
The representative agent’s welfare is represented by the intertemporal isoelastic utility function:
∞
max E0 ∫ u (C , Gs )e − ρ t dt
c
s.t.
0
dK = [ A(h)(1 − h − g ) K − C ]dt + Kdu
dk / k = dK / K , k = K and is constant. The
representative agent’s formal optimization problem is to choose C and his individual rate of capital
In
the macroeconomic equilibrium,
accumulation to satisfy the Bellman equation:
[C (Gs )η ]γ
ρV = max 
c

γ

1
+ [ A(h)(1 − h − g ) K − C ]VK + σ u2 K 2VKK 
2

(2)
Therefore, deriving partial differential to Bellman equation with respect to C , K , we obtain the
following first-order conditions:
First- order conditions:
C γ −1 (Gs )η γ = Vk
（3）
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ρVK =
ηγ [C (Gs )η ]γ
K
（ 4）
1
+ [ A(h)(1 − h − g )]VK + [ A(h)(1 − h − g ) K − C ]VKK + σ u2 KVKK + σ u2 K 2VKKK
2
Since representative agent now perceives only one state variable, K , we consider a value function
of the form
V ( K ) = xK (t )(1+η )γ
(5)
Where the coefficient x is to be determined.
Equation (3)~(5) yields the ratio of consumption to capital C / K in equilibrium:
µ
ρ − [ A(h)(1 − h − g )][(1 + η )γ ] +
2
(1 + η )(1 − γ )
(1 + η )γ [(1 + η )γ − 1]
（ 6）
k and K have the same stochastic economic growth rate:
A(h)(1 − h − g )
E[(dk / dt ) / k ] = A(h) − C / K =
−S
(7)
1− γ
In
Φ
C/K =
σ u2
the
equilibrium,
ρ+
Where S
σ u2
2
Combined equation
(1 + η )γ [(1 + η )γ − 1]
(1 + η )(1 − γ )
（3）and（5），this yields:
x = [ A(h) g ]ηγ µ γ −1 /[γ (1 + η )]
Proposition 1 When 0 < γ < 1 , The change of government expenditure h
effects on economic growth. When
opposite effects on economic growth.
Proof when 0 < γ < 1 ,
By (7), we have
γ > 1,
、g
has the negative
The change of government expenditure h
、g
has the
∂Φ A '(h)(1 − h − g ) − A
=
<0
∂h
1− γ
A(h)
∂Φ
=−
<0
∂g
1− γ
Namely, a increase in
∂Φ
>0
∂g
、
h g will decrease economic growth. Similarly, when γ > 1 ,
∂Φ
>0
∂h
，
，then a increase in government expenditure will be in favor of economic growth.
Proposition 2 when 0 < γ <
1
2
, an increase in σ u will leads to consumption decrease,
1+η
people invest more capital in production, which will rise economic growth.
When
1
< γ < 1 , an increase in σ u2 will leads to consumption increase, people invest few
1 +η
capital in production, which will decrease economic growth.
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when 0 < γ <
Proof
increase in
σ u2
1
∂Φ
, from (6)\(7), we have
>0
1+η
∂σ u2
， ∂(∂Cσ/ K ) < 0 ，Namely, an
will increase economic growth. Similarly, when
2
u
1
∂Φ
< γ <1 ,
<0
1 +η
∂σ u2
，
∂ (C / K )
> 0 . Then an increase in σ u2 will be in favor of economic growth.
2
∂σ u
4. Stochastic Optimization in the Decentralized Economy
The agent’s stochastic optimization problem is to choose his individual ratio of consumption to
capital and his capital accumulation rate to maximize his utility:
U=
[C ( Ak 1−δ K δ )η ]γ
γ
First-order condition
γ −1
ηγ
：
（）
8
(Gs ) = Vk
(1 − δ )η c
1
ρVk =
Vk + [ A(h)(1 − h − g )]VK + [ A(h)(1 − h − g )k − c ]Vkk + σ u2 kVkk + σ u2 k 2Vkkk
k
2
C
（ 9）
Since the individual now perceives two state variable, k , K ,we consider a value function of the
[(1−δ )η +1]γ
form: V ( k , K ) = bk
From equation 8 ~
consumption to capital
K (t )δηγ
（10）
（ ） （10）and capital accumulation equation we have the equilibrium ratio of
：
µ* =
1
2
ρ − [ A(h)(1 − h − g )][(1 − δ )η + 1]γ + σ u2 [(1 − δ )η + 1]γ − 1]{1 + {[(1 − δ )η + 1]γ − 2}}
（ ）：
(1 − δ )η − {[(1 − δ )η + 1]γ − 1}
From equation 8
γ −1
µ * ( Ag )ηγ
b=
[(1 − δ )η + 1]γ
Proposition 3 when δ → 1 , the ratio of consumption to capital in decentralized economy tends to
that in centrally planned economy.
This thesis talks about government expenditure’s effect on both production and welfare, analyzes
how volatility influences growth rate and Propensity to consume. If without congestion, the ratio of
individual consumption to capital tends to best ratio of consumption to capital.
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