Article non publié
May 11, 2007
A Simple Version of the Lucas Model
Mazamba Tédie
Abstract
This discrete-time version of the Lucas model dot not include the physical capital. We intregrate in the utility function the leisure time. We examine the social
planer and the competitive equilibrium. The main conclusions are that the consumer
always chooses to train, the human capital growth rate increases with the externality
and the quality of training, and that the equilibrium defined by Lucas (1988) is a
competitive equilibrium under some conditions.
1. Introduction
This model is a discrete-time version of the model of Lucas without physical capital.
The consumer devotes the fraction θ of his non-leisure time to current production and the
remaining (1−θ) to human capital accumulation. We consider that the utility of consumer
increases with his leisure time. This assumption implies that the utility increases with the
human capital accumulation that is with the training. Following Lucas (1988), the human
capital has : 1- an external effect through the externality. 2- an internal effect which
increases the productivity through the medium of training.
This paper is organized into seven sections. Section 2 introduces assumptions and examines the social planer problem. After defining the equilibrium (according to Lucas and
Romer) and competitive equilibrium, section 3 shows that an equilibrium is a competitive
equilibrium. The following sections conclude and give some proofs.
2. Social Planer
The utility function is concave (0 < µ < 1 and 0 < ζ < 1) :
max
+∞
X
β t cµt (1 − θt )ζ
t=0
Subject to,
The author thanks Gourdel, P. (University of Paris 1, CERMSEM) & Le Van, C. (University of Paris 1, CNRS-CERMSEM) for helpful
comments during the course of this presentation. He takes responsability for any remaining errors
Keywords : Human capital, Economic Growth, Competitive Equilibrium, Equilibrium
1
∀t ≥ 0, 0 ≤ ct ≤ hγt (θt ht )α
ht+1 = ht (1 + λφ(1 − θt ))
0 < α < 1, γ ≥ 0, 0 ≤ θt ≤ 1, h0 > 0 given
We make the following assumptions :
H1 : φ is concave, increasing and twice continuously differentiable. φ(0) = 0, φ(1) = 1,
λ > 0 and φ0 (0) > 1.
H2 : 0 < β(1 + λ)(α+γ)µ < 1.
The parameter λ balanced the technology of training.
Let us define the function ψ :
−1 1
(x − 1) . Where φ−1 denotes the inverse func[1, 1 + λ] → [0, 1] by ψ(x) = 1 − φ
λ
tion of φ. ψ is clearly decreasing. It is easy to verify that : ψ(1) = 1 and ψ(1 + λ) = 0.
This function gives the working time when the human capital grows by factor x. ψ is
continuously differentiable, decreasing, with ψ 0 (1) = − λφ10 (0) , ψ 0 (1 + λ) = − λφ10 (1) and
concave.
The problem becomes :
max
+∞
X
t=0
(α+γ)µ
β t ht
αµ ζ
ht+1
ht+1
)
1 − ψ(
)
ψ(
ht
ht
Subject to :
∀t ≥ 0, ht ≤ ht+1 ≤ ht (1 + λ) and h0 > 0 given.
Proposition 1 Under H1-H2, there exists a solution.
Proof. See the appendix 1.
Proposition 2 Each optimal path of human capital h = (h0 , h1 , ..., ht , ...) verifies h0 <
h1 < · · · < ht < · · · .
Proof. See the appendix 2.
Proposition 3 Under assumptions H1,H2 and H3 : (α + γ)µ − 1 < 0 :
(a) The optimal path of human capital has a constant growth rate, strictly positive and
which increases with parameter γ.
(b) The optimal path of human capital is an increasing function of λ.
Proof. We prove (a) in several stages.
1. Let V be the Value Function of our problem of optimal growth :
αµ ζ
+∞
X
ht+1
ht+1
t (α+γ)µ
V (h0 ) = max
)
1 − ψ(
)
β ht
ψ(
ht
ht
t=0
Under the constraints : ∀t ≥ 0, ht ≤ ht+1 ≤ ht (1 + λ), and h0 > 0 given.
This value function satisfied (see Le Van & Morhaim 2002) :
2
(α+γ)µ
V (h0 ) = Ah0
Let us consider h0 , the optimal value h1 of the human capital at date 1 is the solution of
the following equation :
αµ ζ
(α+γ)µ (α+γ)µ
y
y
h0
maxy∈[h0 ,(1+λ)h0 ]
ψ( h0 )
1 − ψ( h0 ) + βA hy0
We can
h∗1 = νh0 where ν is the solution
of the equation :
see that
αµ
ζ
(α+γ)µ
. Since the problem is stationary, if {ht } is
max (ψ(z)) (1 − ψ(z)) + βA(γ)z
the optimal path, then : ht = ν t h0 , ∀t.
2. We know that the human capital path verifies ht+1 > ht , ∀t ≥ 0. The Euler
equation is given by :
h i
(α+γ)µ−1 0 ht+1
ht+1
ht+1
)
αµ
1
−
ψ(
)
−
ζψ(
)
ht
ψ ( ht )Ψ( hht+1
ht
ht
t
(α+γ)µ−1
ht+2
ht+2
ht+2
= −β(α + γ)µht+1
Ψ( ht+1 ) ψ( ht+1 ) 1 − ψ( ht+1 )
h i
(α+γ)µ−1 ht+2 0 ht+2
ht+2
ht+2
ht+2
+βht+1
ψ
(
)Ψ(
)
αµ
1
−
ψ(
)
−
ζψ(
)
ht+1
ht+1
ht+1
ht+1
ht+1
Ψ( hht+1
)
t
αµ−1 ψ( hht+1
)
t
ζ−1
ψ( hht+1
)
t
With
=
1−
.
This equation gives the human capital growth rate that is constant (ν) :
β
ζ ψ(ν)
(α+γ)µ
(α+γ)µ
1 = αµ
1
−
βν
− α (α + γ)ν (α+γ)µ−1 ψψ(ν)
0 (ν) + βν
1−ψ(ν)
ζ ψ(ν)
(α+γ)µ
(α+γ)µ
Let F (ν) = αµ
1
−
βν
with G(ν) = − αβ (α + γ)ν (α+γ)µ ψψ(ν)
.
0 (ν) + βν
1−ψ(ν)
Functions F and G are decreasing
since
:
ζ ψ0 (x)
ψ(x)
− β(α+γ)
x(α+γ)µ−1 < 0, G0 (x) = − αβ (α +
F 0 (x) = 1 − βx(α+γ)µ αµ
(1−ψ(x))2
α 1−ψ(x)
h
i
ψ 00 (x)
1
x
+
x
−
µ
< 0. Moreover, F (1+λ) =
γ)x(α+γ)µ−2 ψψ(x)
((α
+
γ)µ
−
1)
−
0 (x)
ψ(x)
α
0, limx→1 F (x) = +∞, G(1) = β(α+γ)
λφ0 (0) and G(1 + λ) = β(1 + λ)(α+γ)µ < 1 acα
cording to H2. Hence, there exists a unique solution ν ∈]1, 1 + λ[.
3. We know that the value function verifies the Bellman equation :
V (h) = h(α+γ)µ maxν∈[1,1+λ] (ψ(ν))αµ (1 − ψ(ν))ζ + βA(γ)ν (α+γ)µ
The derivate of function (ψ(ν))αµ (1 − ψ(ν))ζ + βA(γ)ν (α+γ)µ is cancelled :
−αµ ψ 0 (ν ∗ )(ψ(ν ∗ ))ζ + ζ (ν ∗ )(ψ(ν ∗ ))αµ (1 − ψ(ν ∗ ))ζ−1
= βA(γ)(α + γ)µν ∗(α+γ)µ−1
When γ increases, the graph of the function βA(γ)(α + γ)µν ∗(α+γ)µ−1 moves to the top
while the left-hand side remains constant. Consequently, the growth rate increases with
the parameter of the externality. This ends the proof of the claim (a).
4. Let us rewrite the Euler equation : 1 = Fλ (x) + Gλ (x). Note that λ < λ0 ⇒ ψλ <
ψλ0 and −ψλ0 < −ψλ0 0 . Hence, F and G are increasing with λ. Moreover, F and G are
decreasing with ν, then :
dν
∂F
∂G
∂G ∂F
=−
+
/
+
>0
dλ
∂λ
∂λ
∂ν
∂ν
3
3. Equilibrium and Competitive Equilibrium
We introduce the concepts of equilibrium (according to Lucas and Romer) and competitive equilibrium. Take a human capital path h̄ = (h̄1 , ..., h̄t , ...) to be given. Given h̄,
consider the problem :
max
ct
+∞
X
β t u(ct , θt )
t=0
Under the constraints,
∀t, 0 ≤ ct ≤ G(h̄)f (θt ht )
ht+1 = ht (1 + λφ(1 − θt ))
0 ≤ θt ≤ 1, h0 > 0 given
The solution h = (h0 , h1 , ..., ht , ...) of this model depends on h̄. In others words, h =
Φ(h̄). A equilibrium is a human capital path h∗ = (h0 , ..., h∗t , ...) such that h∗ = Φ(h∗ ).
In order to define a competitive equilibrium, we need before to define the space of the
prices which supports this equilibrium. Observe that all feasible paths of consumption c
verify for all t : 0 ≤ ct ≤ hα+γ
with ht ≤ h0 (1 + λ)t . In others words, c belongs to :
t
|ct |
∞
< +∞
` = c : sup
(α+γ)t
t=0,..,+∞ (1 + λ)
∞
Let `∞
+ be the set of non negative
P+∞sequences of ` . The price sequence pt is such as all
consumption paths ct verify t=0 pt ct < +∞. Likewise, the wage path wt is such as
P
∞
t=0 wt ht < +∞. In order to satisfy these two conditions, we must take the prices space
and the wages space as follows :
(
)
(
)
+∞
+∞
X
X
`1p = p :
|pt |(1 + λ)(α+γ)t < +∞ ; `1w = w :
|wt |(1 + λ)t < +∞
t=0
t=0
Let us denote `1+ , the set of non-negative sequences of `1 .
We define a competitive equilibrium for the model of Lucas.
A collection of sequences (h∗ , c∗ , θ ∗ , p∗ , w∗ ) is a competitive equilibrium if :
1. (c∗ , θ ∗ ) is a solution of the consumer program :
max
+∞
X
ct ,θt
β t u(cc , θt )
t=0
Under the constraints,
+∞
X
p∗t ct ≤
+∞
X
t=0
t=0
wt∗ θt ht + Π∗
∀t ≥ 0, θt = ψ( hht+1
), h0 > 0 given
t
2. θ ∗ is a solution of the firm program :
4
Π∗ = max
θ
( +∞
X
p∗t (h∗t )γ (θt h∗t )α −
+∞
X
)
wt∗ θt h∗t
t=0
t=0
3. Equilibrium on the goods and services market :
∀t ≥ 0, c∗t = (h∗t )γ (θt∗ h∗t )α
Proposition 4 h∗ is a equilibrium from h0 > 0 if and only if it verifies the three following
conditions :
1. Interiority : ∀t ≥ 0, h∗t < h∗t+1 < (1 + λ)h∗t , h∗0 = h0 > 0
2. Euler equation (∀t ≥ 0),
αµ−1 ζ
h∗
h∗t+1
ψ( ht+1
)
1
−
ψ(
)
∗
h∗t
t
αµ ζ−1
∗
∗
∗
h
h
h
∗(α+γ)µ−1 0 t+1
t+1
)
)
1
−
ψ(
−ζ ht
ψ ( h∗ ) ψ( ht+1
∗
h∗
t
t ∗ αµ
ζt
∗
h
h
∗(α+γ)µ−1
ψ( ht+2
)
1 − ψ( ht+2
)
+βαµ ht+1
∗
∗
t+1
t+1
ζ
αµ−1 h∗t+2
h∗t+2
∗(α+γ)µ−1 h∗t+2 0 h∗t+2
−βαµ ht+1
ψ
(
)
ψ(
)
1
−
ψ(
)
h∗t+1
h∗t+1
h∗t+1
h∗t+1
αµ ζ−1
∗
∗
∗
∗
h
h
h
h
∗(α+γ)µ−1 t+2 0 t+2
t+2
t+2
+βζ ht+1
ψ
(
)
ψ(
)
1
−
ψ(
)
=0
h∗
h∗
h∗
h∗
∗(α+γ)µ−1
αµ ht
ψ0(
h∗t+1
)
h∗t
t+1
t+1
t+1
t+1
3. Transversality condition,
h∗t+1 αµ−1
h∗t+1 ζ−1
ψ( ∗ )
1 − ψ( ∗ )
ht
h iht
h∗t+1
h∗t+1
αµ 1 − ψ( h∗ ) − ζψ( h∗ ) h∗t+1 = 0
∗
∗(α+γ)µ−1 0 ht+1
lim β t ht
ψ( ∗ )
t→+∞
ht
t
t
Proof. See the appendix 3.
Proposition 5 Under the assumptions of proposition 3 and H4 : λ ≤ β1 − 1, there exists
an equilibrium h∗ which increases at constant rate ν. The equilibrium growth rate h∗
is weaker than that of the centralized rate. We can associate with this equilibrium the
stationary sequence θ ∗ = (ψ(ν)), a consumption sequence c∗ , a price system p∗ , wage
w∗ such as the collection of sequences (h∗ , c∗ , θ ∗ , p∗ , w∗ ) is a competitive equilibrium.
Proof. 1. We know that if h∗ is an equilibrium then it verifies interiority, the Euler
equation and the transversality condition. In addition, let us show that exists a human
capital sequence that increases at constant rate and satisfies the Euler equation. Indeed,
according to Euler equation, this rate ν must satisfy :
1=
ζ ψ(ν)
αµ 1−ψ(ν)
(α+γ)µ
1 − βν (α+γ)µ − βν (α+γ)µ−1 ψψ(ν)
≡ V (ν)
0 (ν) + βν
5
ζ ψ(ν)
(α+γ)µ
(α+γ)µ
Let F (ν) = αµ
1
−
βν
and H(ν) = −βν (α+γ)µ−1 ψψ(ν)
. We
0 (ν) + βν
1−ψ(ν)
know that F is decreasing, limx→1 F (x) = +∞ and that F (1 + λ) = 0. We show that H
is also decreasing :
h
i
ψ(ν)
ψ(ν)ψ 00 (ν)
0
(α+γ)µ−2
((α + γ)µ − 1) ψ0 (ν) − ν − ν (ψ0 (ν))2 < 0. One has V (x) =
H (ν) = −βν
0
F (x) + H(x), V (x) = F 0 (x) + H 0 (x), limx→1 V (x) = limx→1 F (x) + limx→1 H(x) =
+∞ and V (1 + λ) = F (1 + λ) + H(1 + λ) = β(1 + λ)(α+γ)µ < 1 according to H2.
Consequently, there exists a unique solution ν which belongs to ]1, 1 + λ[. It’s easy to
show that this rate is weaker than the rate of social planer program which is the solution
∗
of the equation : 1 = F (ν) + G(ν), since G(ν) = H(ν) − βγ
ν (α+γ)µ−1 ψψ(ν)
0 (ν) . Let h be
α
the trajectory defined by : h∗0 = h0 , h∗t+1 = νh∗t , ∀t. Obviously, it satisfies the interiority
and Euler equation. We must show than it verifies the transversality condition to conclude
that h∗ is an equilibirum. Now,
αµ−1 ζ−1
h∗t+1
h∗t+1
h∗t+1
∗(α+γ)µ h∗t+1 0 h∗t+1
β t ht
ψ
(
)
ψ(
)
1
−
ψ(
)
)
A(
∗
∗
∗
∗
h
h
h
h
h∗
t
t
t
t
t
(α+γ)µ
= β t h0
ν (α+γ)µt ψ 0 (ν)ν(ψ(ν))αµ−1 (1 − ψ(ν))ζ−1 A(ν)
t
(α+γ)µ
≤ h0
(ψ(ν))αµ−1 νψ 0 (ν)(1 − ψ(ν))ζ−1 A0 β(1 + λ)(α+γ)µ
h∗
h∗t+1
0
Where A = αµ 1 − ψ( h∗ ) − ζψ( ht+1
∗ ). Assumption H2 implies :
t
t
∗
h∗t+1 αµ−1
h∗t+1 ζ−1 0
t ∗(α+γ)µ−1 0 ht+1
lim β ht
ψ ( ∗ ) ψ( ∗ )
1 − ψ( ∗ )
A =0
t→+∞
ht
ht
ht
This is the transversality condition.
2. We show that this trajectory is a competitive equilibrium. Let us define the price
path and the wage path, p∗ , w∗ by :
∗(α+γ)(µ−1)
t ,θt )
= µβ t ht
(ψ(ν))α(µ−1) (1 − ψ(ν))ζ
p∗t = β t ∂u(c
∂ct
∗(α+γ)µ−1
(ψ(ν))αµ−1 (1 − ψ(ν))ζ−1 [αµ(1 − ψ(ν)) − ζψ(ν)]
wt∗ = β t ht
Where h∗t = ν t h0 .
a) It is easy to see that the sequence θ ∗ defined by θt∗ = ψ(ν), for all t, maximizes the
profit of the enterprise according to p∗ and w∗ .
b) In order to prove that the consumption path and the working time path (c∗t , θt∗ ) maximize the consumer utility, consider :
∆T =
T
X
β t u(c∗t , θt∗ ) −
t=0
P+∞
P+∞
T
X
β t u(ct , θt )
t=0
P
P+∞ ∗
t 0 ∗
∗
Since t=0 β u
=
+ Π and +∞
t=0 β u (ct )ct <
t=0 wt θt ht + Π
with θt = ψ(ht+1 /ht ), one has :
T
X
h∗t+1
h∗t+1
∗(α+γ)µ−1
t
∗
∆T ≥
β (ht − ht ) αµht
1 − ψ( ∗ ) Φ( ∗ )
ht
h
t=0
t
αµ ζ−1
∗
∗
∗
∗
h
h
h
∗(α+γ)µ−1 ht+1
+ζht
ψ( ht+1
1 − ψ( ht+1
ψ 0 ( ht+1
∗ )
∗ )
∗ )
h∗t
t
t
t
ζ
αµ−1
∗
h∗t+1
h∗
ht+1
∗(α+γ)µ−1
∗
+(ht+1 − ht+1 ) αµht
1 − ψ( h∗ ) ψ 0 ( ht+1
ψ( h∗ )
∗ )
t
t
t
αµ ζ−1
∗
∗
∗
h
h
h
∗(α+γ)µ−1
−ζht
ψ( ht+1
1 − ψ( ht+1
ψ 0 ( ht+1
∗ )
∗ )
∗ )
t 0
(ct )c∗t
∗ ∗ ∗
t=0 wt θt ht
∗
t
t
6
t
h∗
Φ( ht+1
∗ )
t
αµ−1
h∗t+1
ψ( h∗ )
t
=
1−
Where
Using the Euler equation, we obtain :
∗(α+γ)µ−1
∆T ≥ β hT
h h∗
αµ 1 − ψ( hT∗+1 )
T
T
ζ−1
h∗t+1
ψ( h∗ )
t
h∗
ψ( ht+1
∗ )
t
−
h∗t+1 0 h∗t+1
ψ ( h∗ )
h∗t
t
.
αµ−1 ζ−1
h∗T +1
h∗T +1
ψ( h∗ )
1 − ψ( h∗ )
T
T
i
∗
h∗T +1
h
h∗
− ζψ( h∗ ) ψ 0 ( hT∗+1 ) = β T wT∗ ψ 0 ( hT∗+1 )h∗T +1
T
T
T
By definition of wT∗ . According to the transversality condition, we conclude that limT →+∞ ∆T ≥
0.
c) The goods market is balanced since for all t : c∗t = (h∗t )γ (θt∗ h∗t )α .
d) To complete this proof, let us show that p∗ belongs to `1p and w∗ belongs to `1w . One
has :
+∞
X
(α+γ)t
pt (1 + λ)
<
(α+γ)(µ−1)
µB 0 h0
t=0
+∞
X
t
β(1 + λ)(α+γ)µ < +∞
t=0
According to H2 and with B 0 = (ψ(ν))α(µ−1) (1 − ψ(ν))ζ . Likewise,
+∞
X
(α+γ)µ−1
wt (1 + λ)t = C 0 h0
+∞
X
t
β(1 + λ)ν (α+γ)µ−1 < +∞
t=0
t=0
According to H4, 1 < ν < 1 + λ and where C 0 = (ψ(ν))αµ−1 (1 − ψ(ν))ζ−1 [αµ(1 −
ψ(ν)) − ζψ(ν)]. This ends the proof. A collection of paths (h∗ , c∗ , θ ∗ , p∗ , w∗ ) is a
competitive equilibrium.
4. Conclusion
This dicrete-time version of the Lucas model solves the social planer program and
shows that an equilibrium for this model is a competitive equilibrium. Moreover, the
model concludes that : 1- when the utility depends on consumption and leisure time, the
consumer always prefers to increase his skill level. 2- the quality of training increases
the human capital growth rate. 3- the externality is related positively to the human capital
growth rate through it contribution to the productivity of all factors of production.
5. Appendix 1
It’s easy to verify that if c = (c0 , c1 , ..., ct , ...) is a feasible path of consumption, then
: ∀t, 0 ≤ ct ≤ h0α+γ (1 + λ)(α+γ)t . This shows that all feasible paths of consumption are
compact for this topology. Assumption H2 ensures that function :
U (c) =
+∞
X
β t u(ct , θt )
t=0
is continuous for the product topology. Existence of the solution rise from these results.
7
6. Appendix 2
It’s enough to show that for any initial conditions, h0 > 0, the stationary path (h0 , h0 , ..., h0 , ...)
is not optimal. Let > 0 be a sufficiently small number such as 1 + λφ() ≤ 1 + λ
and a path h = (h0 , h1 , ..., ht , ...) which verify ht = h0 (1 + λφ()), ∀t ≥ 1. The
consumption path associated with this human capital path is c = (c0 , c1 , ..., ct , ...)
that is : c0 = hα+γ
(1 − )α and ct = hα+γ
(1 + λφ())α+γ , ∀t ≥ 1. Moreover, let
0
0
(h0 , h0 , ..., h0 , ...) be a human capital path and c∗ be a consumption path which satisfy :
c∗t = hα+γ
. Compare the utilities generated by these sequences of consumptions, we have
0
:
ζ X
+∞
+∞
X
ht+1
ζ
t µ
) −
β t c∗µ
∆ =
β ct 1 − ψ(
t (1 − ψ(1))
h
t
t=0
t=0
Since ψ(1) = 1, so ∆ > 0. All optimal paths of human capital are increasing.
7. Appendix 3
We give the proof of the Proposition 4 in several stages.
1. Let h∗ be an equilibrium. One can show that any equilibrium is increasing, that
is h∗t+1 > h∗t , ∀t ≥ 0 (Proceed as in the previous appendix). Moreover, since the utility
function verifies the Inada condition, the optimal consumptions are strictly positive on
each date. Hence, h∗t+1 < (1 + λ)h∗t , for all t. This ends the first part of the claim. It is
easy to show that h∗ verifies the Euler equation (see Le Van & Dana 2003). Let us show
now that the transversality condition is satisfied. Let Vh∗ (h0 ) be the value function of this
program, one has :
Vh∗ (h0 ) = max
+∞
X
β t u(ct , θt )
t=0
Under the constraints (∀t),
0 ≤ ct ≤ G(h∗t )f (θt ht )
ht+1 = ht (1 + λφ(1 − θt ))
0 ≤ θt ≤ 1, h0 > 0 given
One can verify that Vh∗ is concave and differentiable (Beneviste & Scheinkman 1979) and
:
αµ ζ
h∗
h∗
(α+γ)µ−1
1 − ψ( h10 )
Vh0 ∗ (h0 ) = αµh0
ψ( h10 )
αµ−1 ζ
h∗1
h∗1
(α+γ)µ−1 h∗1 0 h∗1
−αµh0
ψ
(
)
ψ(
)
1
−
ψ(
)
h0
h0
h0
h0
αµ ζ−1
∗
∗
∗
∗
h
h
(α+γ)µ−1 h1 0 h1
+ζh0
ψ ( h0 ) ψ( h10 )
1 − ψ( h10 )
h0
Moreover, since h∗ is a equilibrium, it must verify 0 ≤ h∗t ≤ h0 (1 + λ)t for all t. Consequently, c∗t ≤ [h0 (1 + λ)t ]α+γ and
0 ≤ Vh∗ (h0 ) =
+∞
X
β
t
u(c∗t , θt∗ )
≤
(α+γ)µ
h0
+∞
X
t=0
t=0
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[β(1 + λ)(α+γ)µ ]t
Like Vh0 ∗ (0) = 0, we have for all t :
∗(α+γ)µ
ht
≥ Vh∗ − Vh∗ (0) ≥ Vh0 ∗ (h∗t )h∗t
(α+γ)µ
1 − β(1 + λ)
Since,
ζ
h∗
1 − ψ( ht+1
∗ )
αµ−1t ζ
h∗t+1
h∗t+1
∗(α+γ)µ−1 h∗t+1 0 ∗(α+γ)µ−1
ψ
(h
)
ψ(
)
1
−
ψ(
)
−αµ ht
∗
∗
∗
t
ht
ht
h
αµ tζ−1
h∗t+1
h∗t+1
∗(α+γ)µ−1 h∗t+1 0 ∗(α+γ)µ−1
ψ (ht
) ψ( h∗ )
1 − ψ( h∗ )
+ζ ht
h∗
∗(α+γ)µ−1
Vh0 ∗ (h∗t ) = αµ ht
ψ(
αµ
h∗t+1
)
∗
ht
t
t
t
and h∗t ≤ h0 (1 + λ)t . Multiply the two previous equations by β t , we obtain the transversality condition :
αµ−1 ζ−1
h∗t+1
h∗t+1
lim
ψ( h∗ )
1 − ψ( h∗ )
t
t→+∞ h
t ∗
∗
ht+1
h∗t+1 0 h∗t+1
ht+1
αµ 1 − ψ( h∗ ) ψ( h∗ ) − h∗ ψ ( h∗ )
t
t i
t
t
h∗t+1
h∗t+1
h∗t+1 0 h∗t+1
∗
+ζ h∗ ψ ( h∗ )ψ( h∗ ) 1 − ψ( h∗ ) ht = 0
t
t
t
t
∗(α+γ)µ−1
β t ht
The Euler condition implies :
lim
t→+∞
∗(α+γ)µ−1 0 h∗t+1
β t ht
ψ ( h∗ )
t
h
αµ 1
αµ−1 ζ−1
h∗t+1
h∗t+1
ψ( h∗ )
1 − ψ( h∗ )
t
i t
∗
h∗t+1
ht+1
− ψ( h∗ ) − ζψ( h∗ ) h∗t+1 = 0
t
t
2. We prove the converse now. Let (c∗ ,h∗ ) and (c,h) be two
sequencesαsets with the
same initial condition h0 . The last verifies ∀t, 0 ≤ ct ≤ (h∗t )γ ht ψ( hht+1
) . Show that
t
P
PT
T
y
t
t
∗ ∗
t=0 β u(ct , θt ) ≥ 0. Observe that u, (x, y) → xψ( x ) and ψ are
t=0 β u(ct , θt ) −
concave functions, hence :
T
X
∗
0
∗ ∗ ∂ct
∗
∗
0
∗ ∗ ∂θt
∗
∗
β (ht − ht ) u1 (ct , θt )
(h , h ) + u2 (ct , θt )
(h , h )
∆T ≥
∂ht t t+1
∂ht t t+1
t=0
i
∗
0
∗ ∗ ∂ct
∗
∗
0
∗
∗ ∗ ∂θt
+(ht+1 − ht+1 ) u1 (ct , θt ) ∂ht+1 (ht , ht+1 ) + u2 (ct , θt ) ∂ht+1 (ht+1 − ht+1 )
t
Where u01 (c∗t , θt∗ ) =
∂u ∗ ∗
(c , θt )
∂ct t
T
∆T ≥ β
h
and u02 (c∗t , θt∗ ) =
By the Euler equation,
αµ−1 ζ−1
h∗T +1
h∗T +1
ψ( h∗ )
1 − ψ( h∗ )
T
T
i
∗
h∗T +1
h∗T +1
0 hT +1
ψ( h∗ ) − ζψ( h∗ ) ψ ( h∗ )h∗T +1
T
T
T
∗(α+γ)µ−1
hT
αµ 1 −
∂u
(c∗ , θt∗ ).
∂θt t
The transversality condition is written limT →+∞ ∆T = 0 since ψ 0 < 0.
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References
[1] Barro, R., & Lee ,J. (1993) "International comparisons of educational attainment"
Journal of Monetary Economics 32, 363-394.
[2] Benhabib, J., & Spiegel, M. (1994) "The role of human capital in economic development : Evidence from aggregate cross-country data" Journal of Monetary Economics 34, 143-173.
[3] Le Van, C. & Morhaim, L. (2002) "Optimal Growth Models with Bounded or Unbounded Returns : a Unifying Approach" Journal of Economic Theory 105, 158187.
[4] Le Van, C., Morhaim, L. & Dimaria, C. (2002) "The discrete time version of the
Romer Model" Economic Theory 20, 133-158.
[5] Lucas, R.Jr (1988) "On the mechanics of economic development" Journal of Monetary Economics 22, 3-42.
[6] Mankiw, G., Romer, D. & Weil, D. (1992) "A Contribution to the Empirics of Economic Growth" The Quaterly Journal of Economics 107 (2), 407-437.
[7] Romer, P. (1990) "Endogenuous Technological Change" Journal of Political Economics 5, S71-S102.
[8] Romer, P. (1986) "Increasing returns and long-run growth" Journal of Political Economy 94, 1002-1037.
[email protected]
http://mazambatedie.free.fr
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