University of A Coruna
From the SelectedWorks of Manuel A. Gómez
April, 2004
Optimality of the competitive equilibrium in the
Uzawa-Lucas model with sector-specific
externalities
Manuel A. Gómez
Available at: http://works.bepress.com/manuel_gomez/4/
Optimality of the competitive equilibrium in the Uzawa-Lucas
model with sector-specific externalities*
Manuel A. Gómez
Faculty of Economics, University of A Coruña, Campus de Elviña, 15071 A Coruña,
Spain (e-mail: [email protected])
Summary. In this paper, we show that the competitive equilibrium is optimal in the
Uzawa-Lucas model with sector-specific externalities associated to human capital in the
goods sector. Thus, these external effects do not provoke a market failure and do not
provide a rationale for government intervention.
Keywords and Phrases: Endogenous growth; Externalities; Efficiency
JEL Classification *umbers: E62; H21; O41
* I wish to thank Sandra López Calvo and an anonymous referee for their valuable
comments. Financial support from the Spanish Ministry of Science and Technology and
FEDER through Plan Nacional de Investigación Científica, Desarrollo e Innovación
Tecnológica (I+D+I) Grant SEC2002-03663 is gratefully acknowledged.
1
1. Introduction
In the past decade, the Uzawa [13] and Lucas’ [9] model of endogenous growth
driven by human capital accumulation has been the subject of active research. The
possibility that the technology at the level of the aggregate economy may differ from the
technology faced by an individual firm because of the presence of external effects has
been considered by a number of authors, and is one of the rationales for government
intervention. Lucas [9] considers the case where the average human capital has an
external effect on the sector producing goods. Chamley [6] and Benhabib and Perli [5]
consider the case where the sector producing human capital exhibits a positive external
effect arising from the average learning time. Such externalities cause the fraction of
time devoted to human capital accumulation be inferior to the optimal. This market
failure may call for public intervention to correct it. In fact, García-Castrillo and Sanso
[7] and Gómez [8] derive fiscal policies that are capable to make the competitive
equilibrium be optimal when there in an externality à la Lucas.
Judging from these and other works one may conjecture that optimal growth paths
and competitive equilibrium paths do not coincide when there are external effects in the
Uzawa-Lucas model. In this paper we shall show that this conjecture is not entirely
correct. To this end, we shall consider the case where there is a sector-specific
externality associated with human capital in the goods sector, i.e., the human capital
employed in the production of goods has an external effect in this sector. Lucas [9]
assumed instead the presence of an externality associated to the average stock of human
capital. However, a sector-specific externality is also plausible, since it can be argued
that having smart colleagues in the department is not very useful to one unless they
spend time in the office. This kind of externality has been recently considered, for
example, by Benhabib and Farmer [2], Benhabib et al. [3], Benhabib and Nishimura [4]
and Mino [11,12]. We shall show that the competitive equilibrium is Pareto optimal in
the Uzawa-Lucas model with such a sector-specific externality. Thus, this external
effect does not provoke a market failure and does not require that the government
intervene to correct it.
2
The intuition for this result is simple. In the Uzawa-Lucas model, human capital is
produced using effective time as the only input and, therefore, the sole cost of investing
in human capital is foregone earnings. Since the individual agent does not take into
account the effect of the sector-specific externality, the private return to effective time
(the wage rate) is lower than its social return. However, as the return to and the
opportunity cost of human capital accumulation augment in the same proportion, the
time allocation decision of the individual agent is the same as that of the central planner.
The remainder of this paper is organized as follows. Section 2 describes the
decentralized economy. Section 3 describes the centrally planned economy and shows
that the competitive equilibrium is optimal. Section 4 concludes.
2. The decentralized economy
In this section we introduce sector-specific externalities associated with human
capital into the Uzawa-Lucas model. Consider an economy populated by a large number
of identical infinitely-lived representative agents who derive utility from the
consumption of a private consumption good, c. For simplicity, we assume that
population is constant and normalized to one. The intertemporal utility derived by the
agent is represented by the isoelastic utility function
∫
∞
0
e − ρt
c 1−σ − 1
dt
1−σ
ρ > 0, σ > 0 .
(1)
The endowment of time is normalized as a constant flow of one unit per period. A
fraction u of time is allocated to work, and a fraction 1−u to learning. Human capital, h,
is accumulated according to the dynamic equation
h& = δ (1 − u )h
δ > 0.
(2)
Output, y, is produced with the Cobb-Douglas technology
y = A k α (uh) β (uh)1−α − β
A > 0, 0 < β < 1, 0 < α < 1, α + β < 1 ,
(3)
where, k is the stock of physical capital, h is the stock of human capital, u is the time
individuals supply as labor, and uh is the average human capital devoted to the
production of goods. This specification assumes that the production of goods exhibits
constant returns-to-scale at the social level but decreasing returns-to-scale at the private
3
level. The term uh expresses sector-specific externalities associated with human capital
employed by the sector producing goods. Note that Lucas [9] assumed instead aggregate
externalities associated with average human capital, h . In the market solution, the
atomistic agents treat uh as given. By symmetry, the value of uh is equal to uh in
equilibrium.
Profit maximization by competitive firms implies that labor and capital are used up
to the point at which marginal product equates marginal cost:
r =α
y
,
k
(4a)
w=β
y
,
uh
(4b)
where r is the rate of return on physical capital, and w is the wage rate. Since the
production function exhibits decreasing returns-to-scale at the private level, the
competitive firm earns positive profits,
π = (1 − α − β ) y .
(4c)
We assume that these profits are distributed back to households as dividends. As
Benhabib and Nishimura [4] pointed out, positive profits should be combined with a
fixed entry cost to determine the number of firms along the equilibrium, unless the
number of firms is fixed.
The representative household maximizes (1) subject to the flow budget constraint
k& = rk + wuh + π − c ,
(5)
and the constraint on human capital accumulation (2). Let J be the current value
Hamiltonian of the household’s optimization problem, and let λ and θ be the multipliers
for the constraints (5) and (2), respectively:
J=
c 1−σ − 1
+ λ [ rk + wuh + π − c ] + θ [ δ (1 − u )h ] .
1−σ
The necessary conditions for an optimum are
c −σ = λ ,
(6a)
λ wh = θδ h ,
(6b)
λ& = ( ρ − r )λ ,
(6c)
4
θ& = ( ρ − δ (1 − u ))θ − λwu ,
(6d)
plus the usual transversality conditions. These conditions are also sufficient since the
optimal Hamiltonian can be readily shown to be concave in the state variables, k and h,
given the specification of preferences and technology.
Hereafter, let γ z = z& z denote the growth rate of the variable z. In what follows,
the equilibrium condition uh = uh will be imposed, and the expressions (4a)–(4b) for r
and w will be taken into account. From (6a) and (6c) we derive the consumption growth
rate:
γc =
α y k−ρ
.
σ
(7)
From (4) and (5), we can obtain the market equilibrium condition in the goods
sector:
y c
− .
k k
γk =
(8)
Substituting the expression for λ from (6b) into (6d) yields
γθ = ρ −δ .
(9)
Log-differentiating (3) and (6b) with respect to time yields, respectively,
γ y = αγ k + (1 − α ) (γ u + γ h ) ,
(10)
γ λ + γ y − γu = γθ + γ h .
(11)
From (2), (6c), (8), (9), (10) and (11), the growth rate of u is obtained as
γu =
δ (1 − α )
c
+δ u − .
k
α
(12)
The system (2), (7), (8) and (12) characterizes the dynamics of the decentralized
economy in terms of k, h, c and u. This system can be reformulated in terms of variables
that are constant in the steady state, defining x = k h and q = c k :
γq =
(α − σ ) Ax α −1u 1−α
σ
+q−
ρ
,
σ
γ x = Ax α −1u 1−α − δ (1 − u ) − q ,
γu =
δ (1 − α )
+δu − q.
α
5
(13a)
(13b)
(13c)
Note that this is the exactly the solution to the Uzawa-Lucas model without externalities
(see, e.g., Barro and Sala-i-Martín [1, Sect. 5.2]). The steady state of system (13) is
u* =
q* =
δ (σ − 1) + ρ
,
δσ
δ (1 − α )
+ δ u* ,
α
x* = (αA δ ) 1 (1−α ) u * .
The condition 0<u*<1 holds if and only if δ > ρ > δ (1 − σ ) , and the conditions q*>0
and x*>0 are satisfied if 0<u*<1. The transversality conditions can be easily shown to
be fulfilled if 0<u*<1. The saddle-path stability of the steady state has been shown, e.g.,
by Barro and Sala-i-Martín [1, Sect. 5.2].
3. The centrally planned economy
The central planner possesses complete information and chooses all quantities
directly, taking all the relevant information into account. Now, the social planner
maximizes (1) subject to (2) and
k& = Ak α (uh)1−α − c .
(14)
Let J be the current value Hamiltonian of the planner’s maximization problem, and
let φ and ψ be the multipliers for the constraints (14) and (2), respectively:
c 1−σ − 1
J=
+ φ [ Ak α (uh) 1−α − c] + ψ [δ (1 − u )h] .
1−σ
The necessary conditions for an interior solution are
c −σ = φ ,
(15a)
φ (1 − α ) Ak α u −α h 1−α = ψ δ h ,
(15b)
φ& = ( ρ − αAk α −1u 1−α h 1−α )φ ,
(15c)
ψ& = ( ρ − δ (1 − u ))ψ − φ (1 − α ) Ak α u 1−α h −α ,
(15d)
plus the usual transversality conditions. These conditions are also sufficient since the
optimal Hamiltonian can be readily shown to be concave in the state variables, k and h,
given the specification of preferences and technology.
6
Proceeding in the same manner as in the case of the decentralized economy, from
(15) we can obtain the system that characterizes the dynamics of the centrally planned
economy. From (15a) and (15c) we derive the consumption growth rate:
αAk α −1 (uh) 1−α − ρ
.
γc =
σ
(16)
The growth rate of physical capital can be obtained from (14) as
γ k = Ak α −1 (uh)1−α −
c
.
k
(17)
Substituting the expression for φ from (15b) into (15d) yields
γψ = ρ − δ .
(18)
Log-differentiating (3) and (15b) with respect to time yields, respectively,
γ y = αγ k + (1 − α ) (γ u + γ h ) ,
(19)
γ φ + γ y − γ u = γψ + γ h .
(20)
From (2), (15c), (17), (18), (19) and (20), the growth rate of u is obtained as
γu =
δ (1 − α )
c
+δ u − .
k
α
(21)
The system (2), (16), (17) and (21) characterizes the dynamics of the decentralized
economy in terms of k, h, c and u. This system can be reformulated in terms of variables
that are constant in the steady state, defining x = k h and q = c k :
γq =
(α − σ ) Ax α −1u 1−α
σ
+q−
ρ
,
σ
γ x = Ax α −1u 1−α − δ (1 − u ) − q ,
γu =
δ (1 − α )
+δu − q.
α
This system is identical to system (13) that characterizes the dynamics of the
decentrallized economy. As system (13) describes both the dynamics of the market
economy and the dynamics of the centrally planed economy, the decentralized economy
replicates the first-best optimum attainable by a central planner, not only in the steady
state but also in the transitional path. Thus, we can state the following proposition.
7
Proposition 1 The competitive equilibrium is Pareto optimal in the Uzawa-Lucas model
with sector-specific externalities associated with human capital in the goods sector.
Note that the optimality of the competitive equilibrium in this model can be shown
in a similar manner when the production function exhibits constant private returns-toscale and/or increasing social returns-to-scale, so that this result is not sensitive to the
assumption of decreasing private returns.
Since the competitive equilibrium is optimal despite of the presence of externalities,
there is no room for government intervention. This result can be intuitively explained as
follows. Lucas [10] showed that a constant tax rate on labor income is neutral in the
market economy since the introduction of a wage tax with a constant rate reduces in the
same proportion the returns and the opportunity cost of investment in human capital. A
similar argument can be used to explain the optimality of the competitive equilibrium in
the presence of sector-specific externalities.
Comparing (6c) and (9) with (15c) and (18), we see that competitive and optimal
equilibrium allocations coincide in the valuation of physical and human capital. Thus,
comparison of (6b) and (15b) shows that the difference in perception between the
individual agent in the market economy and the central planner is that the private return
to effective work time, uh, as seen by the individual agent, is the wage rate
w = β y (uh) , whereas its social return, v, as seen by the central planner, is
v = (1 − α ) y (uh) . The private return is lower than the social return as the
representative agent does not take into account the effect of the sector-specific
externality. In this model, human capital is produced using effective time as the only
input and, therefore, the sole cost of investing in human capital is foregone earnings.
Thus, although the agent does not realize that the private return to effective time (the
wage rate) is lower than its social return, as the return to and the opportunity cost of
human capital accumulation augment in the same proportion, the time allocation
decision of the individual agent is the same as that of the central planner.
The former argument can be formally stated by considering the time allocation
margin of choice. As noted by Lucas [10], the allocation of time between working in the
output sector and learning new skills must be such that the value of a unit of time spent
8
producing at each date is equal, on the margin, to the value of spending that unit of time
accumulating new human capital that will enhance production in the future. In the
market economy, this condition can be obtained from (2), (6b), (6c) and (6d) as
∞
w(t ) h(t ) = δ ∫ e − R ( t , s ) w( s ) u ( s ) h( s ) ds ,
t
(22)
where
s
R (t , s ) = ∫ r (τ ) dτ ,
t
is the cumulative rate or return between time t and time s. In the centrally planned
economy, the equivalent condition can be obtained from (2), (15b), (15c) and (15d) as
∞
v(t ) h(t ) = δ ∫ e − R (t , s ) v( s ) u ( s ) h( s ) ds ,
t
(23)
where it has been used that the social return to physical capital and its private return
coincide, r = α y k . The social return to effective labor, v, and the private return, w, are
related by v = (1 − α ) w β . Thus, comparison of (22) and (23) clearly shows that the
time allocation decision of the representative agent is the same as that of the central
planner, since the return and the cost of investing in human capital have changed in the
same proportion.
4. Conclusions
We have shown that the competitive equilibrium is Pareto optimal in the UzawaLucas model when the average human capital employed in the production of goods has
an external effect in this sector. Thus, this externality does not provoke a market failure
and does not provide a rationale for government intervention. Although the UzawaLucas framework is relatively simple, the optimality of the competitive equilibrium in
the presence of a sector-specific externality suggests that not only the presence and
strength but also the type of human capital externalities are important issues in the
debate concerning the subsidy to education on efficiency grounds.
9
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