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The Lucas-Uzawa Models:
Closed-Form Solutions(-- Preliminary Version--)
9th of May 2011
Solomon M. Antoniou
SKEMSYSS cientific Knowledge Engineering
and Management Systems
37 oliatsou Street, Corinthos 20100, [email protected]
Abstract
We present explicit solutions in a number of Lucas-Uzawa Models. The models
are solved without dimensional reduction, using two different methods. The first
method uses a procedure similar to the dimensional reduction. However in ourmethod we do not consider solutions along the balanced growth path. The second
method has recently appeared in the literature. However, in our own calculations
the intermediate steps appear explicitly in an easy to follow algorithm. Some of
the results we derive exhibit differences compared to the results already found.
The solution procedure of models with externalities uses quite different techniques
to those known so far. The closed-form solutions of the models with externalities
appear for the first time in the literature.
Keywords: Economic Dynamics, Lucas-Uzawa model, Closed-Form Solutions,
Special Functions, Hypergeometric Functions.
-------------------------------------
The paper is available from: www.docstoc.com/profile/solomonantoniou
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1. Introduction
Modern Economic Growth Theory has received considerable attention after its
formulation by Romer [26] and [27], and Lucas [17]. A central role is played by
the Uzawa-Lucas Model (Uzawa [33] and Lucas [17]). The reader may consult
any of the well-known text books on the subject, like Acemoglu [2], Aghion and
Howitt [3], Barro and Sala-i-Martin [5], Greiner, Semmler and Gong [13] and
Romer [28]. We should also mention Greiner and Semmler [12], Uzawa [34] and
Xepapadeas [36] who consider a number of environmental issues examined within
the Lucas-Uzawa models.
There is a number of papers which claim either they have found closed-form
solutions or they have simplified considerably the equations of motion, like
Benhabib and Perli [6], Bethmann [7], Bucekkine and Ruiz-Tamarit [8], Caballè
and Santos [9], Hiraguchi [14], Mattana [19] and [20], Moro [21] and Mulligan
and Sala-i-Martin [23] among others. Most of the solution methods first use
dimensional reduction and then the model is solved along the balanced growth
path (usually known as BGP). The only exception is the paper by Bucekkine and
Ruiz-Tamarit [8]. In this paper the authors use a method which does not use
dimensional reduction.
The mathematical tool in analysing the dynamics of the Lucas-Uzawa models is
the Pontryagin optimization method . The reader may consult any of the known
references of this technique, like Malliaris and Brock [18], Pontryagin et. al. [24]
or Seierstad and Sydsaeter [30].
We use two different methods in obtaining closed-form solution to the models weconsider. The first method uses a procedure similar to the dimensional reduction.
However in our method we do not consider solutions along the balanced growth
path. The second method uses the procedure introduced by Bucekkine and Ruiz-
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Tamarit [8]. However our line of reasoning is more transparent and will offer a
simple method of solving this kind of models.
The solution procedure of models with externalities uses quite different techniques
to those known so far. The closed-form solutions of the models with externalitiesappear for the first time in the literature.
The closed-form solutions of the models considered is based on the evaluation of
the two integrals
∫ νµω −t
0
0ss ds)Ce(e and ∫ µ−ϕ η+η ν−
t
0
t012
t dt)eC;1;,(Fe
These two integrals are evaluated in Appendices A and B respectively, expressed
in terms of hypergeometric functions. The first one is expressed in terms of the
hypergeometric function ()F12 (Appendix A, equation (A.16)), while the second is
expressed in terms of the generalized hypergeometric function []F23 (Appendix
B, equation (B.11)). The evaluation of the first integral is based on the integral
representation of the hypergeometric function, while the second on an integral
formula of the generalized hypergeometric function, appearing in the existing
literature.We should stress the fact that the paper does not examine in general growth
models. It only serves to introduce some standard methods in solving these
models. Especially the relation between BGP and exact solutions will be examined
in an expanded version of the paper (under preparation).
The paper is organized as follows:
In Section 2 we consider a simple model which is solved explicitly following two
methods. The first method of solution uses a type of dimensional reduction at the
beginning and then introducing two auxiliary functions, we find that these
functions satisfy a system of ordinary differential equations. After solving the
system, we can evaluate the function of the physical capital, the control variables
and then the human capital function. The second method uses the same steps
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followed by Bucekkine and Ruiz-Tamarit. However all the intermediate steps are
made explicit in an easy to follow way. There is however a difference to the
expression for the human capital. According to our own calculations, the human
capital is expressed in terms of the generalized hypergeometric function []F23 .
In Section 3 we consider a simple model with externalities. The solution
procedures we follow are quite different compared to the methods considered in
the model of Section 2 and appear for the first time in the literature.
In Section 4 we solve the model considered by Bucekkine and Ruiz-Tamarit which
is again solved using two methods.
In section 5 we consider the model of Bucekkine and Ruiz-Tamarit equipped with
externalities. The methods used in Section 3 are used again in this model. We thus
conclude that the methods introduced in Section 3 can be applied to more
complicated models.
In Section 6 we consider and solve explicitly a model introduced by Ruiz-Tamarit
and Sánchez-Moreno [29], on optimal regulation in a natural-resource-based
economy.
Finally in section 7 we consider some issues on optimal fiscal policy along the
lines of reasoning by Gómez [11].
A word of caution: The various parameters introduced in the text, like ϕ νµ ,, etc.,
are defined in different ways in the various Sections (or even in the various
subsections).
2. A simple Lucas-Uzawa model
2.0. The model.
We work with a Cobb-Douglas production function
α−α= 1)Hu(KAY
where
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A: represents the technological level of the economy, considered to be
constant
K: represents the stock of the physical capital
H: represents the stock of the human capitalu: is the portion of the human capital devoted to the production of output Y
The portion of the human capital devoted to the production of more human capital
will then be u1− . This leads to the equation of motion for the human capital:
H)u1()t(H −γ =
where γ is a constant coefficient.
The equation of motion of the physical capital is derived by the equation
CYK −= where C is the consumption, or using the expression for Y,
C)Hu(KAK 1 −= α−α
We consider the dynamic optimization problem
dte1
1)t(Cmax t
0
1ρ−
∞ σ−
∫ σ−−
(2.1)
subject to
)t(C))t(H)t(u()t(K)t(K 1 −= α−α (2.2)
)t(H))t(u1()t(H −γ = (2.3)
with initial conditions
0K)0(K = , 0H)0(H = (2.4)
where
0)t(C ≥ , ]1,0[)t(u ∈ , 0)t(K ≥ , 0)t(H ≥ (2.5)
In equation (2.1) we work with an isoelastic utility function and maximize the
integral, where σ is the instantaneous elasticity of substitution and ρ is the
instantaneous discount rate. In equation (2.2) we have considered the case 1A =
for simplicity.
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2.1. The equations of motion.
The current value Hamiltonian is given by
=)K,H,u,C(Hc
+−λ+σ−−
= α−ασ−
])t(C))t(H)t(u()t(K[1
1)t(C 1K
1
])t(H))t(u1([H −γ λ+ (2.6)
where Kλ and Hλ are the costate variables corresponding to K and H
respectively.
We can write down the dynamic equations of the model, using Pontryagin’s
optimization method . The reader may consult any of the known references of this
technique, like Malliaris and Brock [18], Pontryagin et. al. [24] or Seierstad and
Sydsaeter [30].
The first order conditions read
0eC0C
HK
tc
=λ−⇔=∂∂ ρ−σ− (2.7)
0]H)Hu(K)1([H0u
HKH
c
=α−λ+γ λ−⇔=∂∂ α−α (2.8)
We also have the two Euler equations
0)Hu(K0K
HK
11KK
c
=λ+λα⇔=λ+∂∂ α−−α (2.9)
⇔=λ+
∂
∂0
H
HH
c
0H)Hu(K)1()u1( H11
KH =λ+α−λ+−γ λ⇔ −α−α (2.10)
the dynamic constraints
C)Hu(KK 1 −= α−α , 0K)0(K = (2.11)
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H)u1(H −γ = , 0H)0(H = (2.12)
and the transversality conditions
0eKlim tK
t=λ ρ−
∞→(2.13)
0eHlim tH
t=λ ρ−
∞→(2.14)
2.2. First method of solution.
This method uses a procedure similar to the dimensional reduction. The derived
system however is not solved along the balanced growth path (BGP). Considering
two auxiliary functions, we get a system of decoupled first order ordinary
differential equations, which, when solved, determine the physical capital, the
control variable u and finally the human capital.
2.2.1. Simplification of the dynamical equations.
Taking logarithms and differentiation with respect to time, we obtain from
equation (2.7) that
K
K
C
C
λλ=ρ−σ−
(2.15)
Similarly, taking logarithms and differentiation with respect to time, we obtain
from equation (2.8) that
H
H
K
K
H
H
u
u
K
K
λλ=α−α−α+
λλ
(2.16)
From equation (2.9) we obtain
α−−αα−=λλ 11
K
K )Hu(K
(2.17)
From equation (2.10) we obtain
H
1)Hu(K)1()u1(
H
K1
H
H ⋅λλ
α−−−γ −=λλ α−α (2.18)
From equation (2.11) we obtain
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K
C)Hu(K
K
K 11 −= α−−α (2.19)
From equation (2.12) we obtain
)u1(HH −γ = (2.20)
We introduce a function named Y defined by
α−α= 1)Hu(KY (2.21)
Equations (2.17)-(2.19) can be expressed in terms of the function Y as follows:
K
Y
K
K α−=λλ
(2.22)
H
Y)1()u1(
H
K
H
H ⋅λλα−−−γ −=
λλ (2.23)
K
C
K
Y
K
K−= (2.24)
From equation (2.8) we find that
u1H
Y
H
K
α−γ
=⋅λλ
(2.25)
Because of the previous relation, we can simplify further equation (2.23):
γ −=λλ
H
H (2.26)
Combining equations (2.15) and (2.22) we find an expression for the ratio
σρ−⋅
σα=
K
Y
C
C(2.27)
Equation (2.16), upon substituting the expressions forK
K
λλ
, K
K, H
Hand
H
H
λλ
given
by (2.22), (2.24), (2.20) and (2.26) respectively, we obtain the equation
K
Cu
)1(
u
u−γ +
αα−γ
= (2.28)
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2.2.2. Auxiliary Functions and their Differential Equations.
We now introduce two more functions U and Z defined by
K
YU = (2.29)
and
K
CZ = (2.30)
respectively.
We shall establish a system of two ordinary differential equations satisfied by
these two functions.
Taking logarithms and differentiation of the defining equations (2.29) and (2.30),we obtain the equations
K
K
Y
Y
U
U −= (2.31)
and
K
K
C
C
Z
Z −= (2.32)
respectively.We can express the right hand sides of the two previous equations in terms of the
functions U and Z. From equation (2.24) we obtain
ZUK
K −= (2.33)
From equation (2.27) we obtain
σ
ρ−⋅
σ
α= U
C
C(2.34)
We now have to express the ratioY
Yin terms of U and Z. Taking logarithms and
differentiation of (2.21), we obtain the equation
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H
H)1(
u
u)1(
K
K
Y
Y α−+α−+α= (2.35)
Substituting the ratiosK
K,
u
uand
H
Hgiven by (2.33), (2.28) and (2.20)
respectively into the previous equation, we obtain
αα−γ
+−α=)1(
ZUY
Y(2.36)
Therefore we obtain from (2.31) and (2.32), using (2.36), (2.33) and (2.34), the
system of two equations
U)1()1(
U
Uα−−
αα−γ
= (2.37)
and
σρ−
σσ−α
+= UZZ
Z(2.38)
Equation (2.37) can be solved by separation of variables. The solution is given by
0t
t
Ce
eU
−⋅αγ = ν
ν(2.39)
where
αα−γ
= ν)1(
and)0(U
1C0 αγ
−= (2.40)
Using the expression (2.39) for the function U, equation (2.38) is converted into
the equation
2
0t
t
ZZCe
e
)1(Z =
σρ−
− ν⋅
α−σσ−α
− ν
ν (2.41)
This is a Bernoulli differential equation, solved under the substitution
1ZX −= (2.42)
Equation (2.41) is converted in terms of X into the linear differential equation
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1XCe
e
)1(X
0t
t
−=
σρ−
− ν⋅
α−σσ−α
+ ν
ν (2.43)
The integrating factor of the above equation is
ζ νω −= )Ce(e)t(I 0tt (2.44)
where
σρ−=ω and
)1( α−σσ−α
=ζ (2.45)
Multiplying (2.43) by the integrating factor, we obtain the equation
ζ νω −−= )Ce(e)X)t(I(dt
d0
tt
which, upon integration in the interval ]t,0[ , gives
ds)Ce(e)0(X)0(IX)t(It
0
0ss∫ ζ νω −−=− (2.46)
The integral on the right hand side of the previous equation is calculated in
Appendix A. We have, using the notation of this section
)t(1
ds)Ce(et
0
0ss Ω
ην−=−
∫
ζ νω (2.47)
where
)C;1;,(F)eC;1;,(Fe)t( 0t
0t η+ηζ−−η+ηζ−=Ω ν−ην− (2.48)
and
νζ ν+ω
−=η (2.49)
Therefore we obtain from (2.46) the following expression for the function X:
ζ νω
ζ
−
Ωην+−
=)Ce(e
)t(1
)0(X)C1(
X
0tt
0
(2.50)
We thus have that the function Z related to X by (2.42) is given by
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)t(1
)0(X)C1(
)Ce(eZ
0
0tt
Ωην+−
−=
ζ
ζ νω(2.51)
Since from (2.47) we have upon differentiation with respect to t the relation
Ω
ην=−− ζ νω )t(
1
dt
d)Ce(e 0
tt (2.52)
equation (2.51) is written as
W
WZ
−= (2.53)
where
)t(1)0(X)C1(W 0 Ωην+−= ζ (2.54)
On the other hand, equation (2.39) can be expressed as
V
V
1
1U
⋅α−
= (2.55)
where
0t
CeV −= ν (2.56)
2.2.3. Equation for the physical capital.
Equation (2.33) can be written, because of (2.55) and (2.53), as
W
W
V
V
1
1
K
K +⋅
α−= (2.57)
The above equation can be integrated once, providing us with the following
expression for the physical capital
WVDK 1
1
0 α−=
or
Ωην+−−= ζα− ν
)t(1
)0(X)C1()Ce(D)t(K 01
1
0t
0 (2.58)
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where
Ω
ην
+−−
==ζα−α− )0(
1)0(X)C1()C1(
K
)0(W)0(V
KD
01
1
0
0
1
10
0 (2.59)
2.2.4. Equation for the control variable u.
From equation (2.28), using (2.30) and (2.53), we derive the equation
W
Wu
)1(
u
u +γ +
αα−γ
=
which is equivalent to the equation
2uuW
W)1(u γ =
+
α
α−γ −
(2.60)
This is a Bernoulli differential equation, which under the substitution
1uy −= (2.61)
becomes a linear first order differential equation
γ −=
+αα−γ
+ yW
W)1(y
(2.62)
The previous equation admits the integrating factor
We)t(Jt ν= (2.63)
Multiplying equation (2.62) by the integrating factor, we obtain the equation
We)y)t(J(dt
d t νγ −=
which, upon integration in the interval ]t,0[ , gives us
∫ ν
γ −=−
t
0
t
ds)s(We)0(y)0(Jy)t(J (2.64)
Let
∫ ν=t
0
t ds)s(We)t(X (2.65)
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Using the expression (2.54), we find that
== ∫ νt
0
s ds)s(We)t(X
+
η+ηζ−
ην−−= ∫ νζ t
0
s00 dse)C;1;,(F
1)0(X)C1(
∫ ν−η− ν η+ηζ−ην+
t
0
s0
s)1( ds)eC;1;,(Fe1
(2.66)
The last integral is evaluated in Appendix B. We get from (B.16), using the
notation of this section,
=η+ηζ−∫ ν−ϕt
0
t0
t dt)eC;1;,(Fe
+η+ηζ−
−
+η+ηζ−
ν−= ν−ϕ
023t
023t C
1,1p
,,pFeC
1,1p
,,pFe
p
1(2.67)
where
)1( η− ν=ϕ and νϕ−=p (2.68)
Therefore we have the following expression for the function )t(X :
== ∫ νt
0
s ds)s(We)t(X
−
ν−
η+ηζ−ην−−=
νζ 1e
)C;1;,(F1
)0(X)C1(t
00
+η+
ηζ−−
+η+
ηζ−
ην− ν−ϕ
023t
023t
2
C
1,1p
,,pFeC
1,1p
,,pFe
p
1(2.69)
We then obtain from (2.64)
)t(J
)t(X)0(y)0(Jy
γ −= (2.70)
and then
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)t(X)0(y)0(J
)t(J)t(u
γ −= (2.71a)
or, comparing (2.63) and (2.65),
)t(X)0(y)0(J
)t(X)t(uγ −=
(2.71b)
2.2.5. Equation for the human capital.
Equation (2.20) for the human capital, because of the previous expression (2.71),
takes on the form
)t(X)0(y)0(J
)t(X
H
H
γ −γ −
+γ =
or even
S
S
H
H +γ = (2.72)
where
)t(X)0(y)0(J)t(S γ −= (2.73)
Equation (2.72) gives upon integration
t0
e)t(S)0(S
H
)t(Hγ
= (2.74)
2.3. Second method of solution.
The second method of solution follows the procedure introduced by Bucekkine
and Ruiz-Tamarit [8]. The intermediate steps however are more transparent and
the overall method provides a standard algorithmic procedure. On the other hand
there is a difference in the human capital evaluation. In our calculation the human
capital is expressed in terms of generalized hypergeometric functions.
2.3.1. Equations for the control variables.
Solving equation (2.7) with respect to C, we obtain
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σ−
λ
σρ−=
1
K )(texpC (2.75)
Solving equation (2.8) with respect to u, we obtain
H
K
1u
1
H
K
1αα
−
λλ
α−γ
= (2.76)
We also get from the previous relation that
α−α
α−
αα−
−α−
λλ
α−γ
= 1
1
H
K
1
1 K1
)Hu( (2.77)
we shall need later on.
2.3.2. Equations for the costate variables.
Equation (2.10), because of (2.76) and (2.77), takes on the form
0HH =λγ +λ (2.78)
Equation (2.78) admits the general solution
tHH e)0(
γ −×λ=λ (2.79)
Equation (2.9), because of (2.77), becomes
01
K
1
H
K
1
K =λ
λλ
α−γ
α+λαα−
αα−− (2.80)
This equation, written as (because of (2.79))
ααα−
αα−
−αα−
−λ
αα−γ
λ
α−γ
α−=λ1
K
11
H
1
K )(t)1(
exp))0((1
is an equation with separable variables with general solution
Ct)1(
exp)0(1
)(
1
H
1
K +
αα−γ
λα−γ
γ α=λ α
α−−
αα−
−(2.81)
where C is a constant to be determined from the initial conditions. For 0t = we
obtain from the above equation
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C)0(1
))0((
1
H
1
K +
λα−γ
γ α=λ α
α−−
αα−
−
from which we determine the constant C. Therefore equation (2.81) takes the final
form
+
αα−γ
λα−γ
γ α=λ α
α−−
αα−
−t
)1(exp)0(
1)(
1
H
1
K
αα−
−αα−
−
λα−γ
γ α−λ+
1
H
1
K )0(1
))0(( (2.82)
The previous equation can also be written as
×
λα−γ
γ α=λ α
α−−
αα−
−1
H
1
K )0(1
)(
−
λλ⋅
γ α−
αγ +
αα−γ
×αα−
−
1)0(
)0(1t
)1(exp
1
H
K
or
)Ce()0(1
)( 0t
1
H
1
K −×
λα−γ
γ α=λ µα
α−−αα−−
(2.83)
where we have introduced the notation
αα−γ
=µ)1(
(2.84)
and
α
α−
−
λλ⋅
γ α−
αγ −=
1
H
K0
)0(
)0(11C (2.85)
We thus obtain from (2.83) the following expression for the costate variable Kλ :
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α−α
−µα−α
−×λ
αγ
α−γ
=λ 10
tH
1K )Ce()0(
1(2.86)
2.3.3. Equation for the physical capital.
We now consider equation (2.11). This equation, because of (2.77) and (2.75),
takes the form
σρ−λ−
λλ
α−γ
= σ−α
α−
αα−
−texp)(K
1K
1
K
1
H
K
1
(2.87)
In order to simplify the above expression, we need to find the ratio of the costate
variables. Using the expressions (2.86) and (2.79), we obtain
0t
t11
H
K
Ce
e
1 −×
α−γ
αγ =
λλ
µ
µαα−
αα−
(2.88)
Equation (2.87), taking into account (2.88) and (2.86), takes on the form
σρ−−−=
−µ
α−− α−σ
αµ
µ
µtexp)Ce(AK
Ce
e
1
1K )1(
0t
0t
t (2.89)
where A is a constant defined by
σ−
α−σα
−
λα−γ
⋅
αγ
=
1
H)1(
)0(1
A (2.90)
We come now to the solution of (2.89), which is a linear first order differential
equation, with integrating factor
α−−µ
µ
µ−=
−
µα−
−≡ ∫ 1
1
0t
0t
t
)Ce(dtCe
e
1
1exp)t(I (2.91)
Multiplying equation (2.89) by the integrating factor )t(I , we find
σρ−−−= α−
−α−σα
µ texp)Ce(A))t(I)t(K(dt
d 1
1
)1(0
t
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and integrating in the interval ]t,0[ , we arrive at the equation
∫ α−−
α−σα
µ −
σρ−−=−
t
0
1
1
)1(0
s0 ds)Ce(sexpA)0(IK)t(I)t(K (2.92)
Introducing the notation
α−
−α−σα
= ν1
1
)1(and
σρ−=ω (2.93)
equation (2.92) becomes
∫ νµωα−−
α−−µ −−=−−−
t
0
0ss1
1
001
1
0t ds)Ce(eA)C1(K)Ce()t(K (2.94)
The integral appearing in (2.94) is not elementary. It can be evaluated using the
hypergeometric function, known from the theory of differential equations in the
complex domain. All the relevant calculations appear in Appendix A. We have
found (see equation (A.16))
=−∫ νµωt
0
0ss ds)Ce(e
])eC;1;,(F)eC;1;,(Fe[
1 t
0
t
0
t µ−µ−µη−
η+η ν−−η+η ν−ηµ−= (2.95)
We are now in a position to write down the final expression for the function )t(K
using (2.94), (2.91) and (2.95):
)t(A
)C1(K)Ce()t(K 1
1
001
1
0t Ψ
ηµ=−−− α−
−α−
−µ (2.96)
where
)C;1;,(F)eC;1;,(Fe)t( 0t0t η+η ν−−η+η ν−=Ψ µ−µη− (2.97)
From equation (2.96) we obtain the final expression for the physical capital:
α−µα−−
−×
Ψηµ
+−= 1
1
0t1
1
00 )Ce()t(A
)C1(K)t(K (2.98)
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2.3.4. Equation for the human capital.
We come now in finding an explicit solution of the differential equation (2.12) for
the human capital. Substituting into (2.12) the expression for u given by (2.76),
we obtain the following differential equation
K1
HH
1
H
K
1
αα−
λλ
α−γ
γ −=γ − (2.99)
From (2.88) we obtain
α−−µαα−α
−×
α−µ
α−γ
αγ
=
λλ
1
1
0t
1
1
11
H
K )Ce(t1
exp1
Using the previous expression and also (2.98) for K, we get the following
differential equation for H:
=γ − HH
Ψηµ
+−
αγ
αγ
γ −= α−−α−
)t(A
)C1(Ktexp 1
1
001
1
(2.100)
This is a linear first order differential equation with integration factor
te)t(J γ −=
Therefore it can be written as
Ψηµ
+−
γ −αγ
αγ
γ −= α−−α−γ − )t(
A)C1(Ktexp)He(
dt
d 1
1
001
1
t (2.101)
Integrating the previous formula in the interval ]t,0[ , we obtain
∫
Ψ
µη+−
αγ
γ −=− α−−µα−γ −
t
0
1
1
00t1
1
0t dt)t(
A)C1(KeH)t(He (2.102)
Using the expression for the function )t(Ψ , given in (2.97), equation (2.102) can
be transformed into the equation
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−=γ −0
t H)t(He
−
η+η ν−
ηµ−−
αγ
γ − ∫ µα−−α−
t
0
t0
1
1
001
1
dte)C;1;,(FA
)C1(K
η+η ν−µη+ ∫ µ−η−µ
t
0
t0
t)1( dt)eC;1;,(FeA
(2.103)
The last integral is not elementary and needs to be evaluated using some results
from the theory of Generalized Hypergeometric Functions. We put the integral
into the form
∫ µ−ϕ η+η ν−
t
0
t0t dt)eC;1;,(Fe (2.104)
where
)1( η−µ=ϕ (2.105)
The integral has been evaluated in Appendix B. We have found (see equation
(B.11))
=η+η ν−∫ µ−ϕ
t
0
t
0
t dt)eC;1;,(Fe
+η+η ν−
µ+
+η+η ν−
µ−= µ−ϕ
023t
023t C
1,1p
,,pF
p
1eC
1,1p
,,pFe
p
1(2.106)
where the parameter p is defined byµϕ−=p .
Therefore from (2.103), using (2.104) and (2.106), we arrive at the following final
expression for the function )t(H :
αγ
γ −= α−γ )t(XHe)t(H1
1
0t (2.107)
where )t(X is given by
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+
η+η ν−
ηµ−−≡ ∫ µα−
− t
0
t0
1
1
00 dte)C;1;,(FA
)C1(K)t(X
=η+η ν−µη+ ∫ µ−η−µ
t
0
t0
t)1(dt)eC;1;,(Fe
A
−
µ−
η+η ν−
ηµ−−=
µα−
− 1e)C;1;,(F
A)C1(K
t
01
1
00
+η+η ν−
−
+η+η ν−
ηµ− µ−ϕ
023t
023t
2C
1,1p
,,pFeC
1,1p
,,pFe
p
A(2.108)
2.3.5. Expressions for the control variables.
It is now easy to find the expressions of the control variables )t(u and )t(C
using the functions we have already found.
The control variable )t(C is given by (2.75). Substituting into this expression the
Kλ given by (2.86), we obtain
σ
ρ−×−×
α
γ
λ
α−
γ = α−σ
αµα−σ
α−
σ−
texp)Ce()0(
1
)t(C )1(0
t)1(
1
H (2.109)
Using (2.76) for the control variable )t(u , we first find the ratioH
K
λλ
from (2.88)
and then substitute the expressions for K and H from (2.98) and (2.107)
respectively. We have
)t(XH
)t(A
)C1(K
t1
exp)t(u
11
0
1
1
001
1
α−
α−−
α−
αγ
γ −
Ψηµ
+−
×
γ −α−
µ
×
α
γ
=(2.110)
where all the functions and parameters are defined in the text.
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3. A Lucas-Uzawa model with externalities.
3.0. The model.
We work with a Cobb-Douglas production function
α−α= 1)LHu(KAY
where
A: represents the technological level of the economy, considered to be
constant
K: represents the stock of the physical capital
L: represents the labour, considered to be constant
H: represents the stock of the human capital
u: is the portion of the human capital devoted to the production of output Y
Therefore uHL represents the amount of labour used to produce output.
Externality is introduced to the model in the form of average schooling of the
human capital. We thus introduce the production function
βα−α= a1 H)LHu(KAY
where aH is the measure of the externality.
The physical capital, ignoring depreciation, is expressed by
CH)HLu(KAK a1 −= βα−α
where C is the consumption. Considering L constant, and dividing through by L,
we can write the previous equation in per capita terms (retaining the notation)
CH)Hu(KAK a1 −= βα−α
In equilibrium condition, we have HHa = .
The portion of the human capital devoted to the production of more human capital
will then be u1− . This leads to the equation of motion for the human capital:
H)u1()t(H −γ =
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where γ is a constant coefficient.
We consider the dynamic optimization problem
dte
1
1)t(Cmax t
0
1ρ−
∞ σ−
∫ σ−
−(3.1)
subject to
)t(C)t(H))t(H)t(u()t(KA)t(K1 −= βα−α (3.2)
)t(H))t(u1()t(H −γ = (3.3)
with initial conditions
0K)0(K = , 0H)0(H = (3.4)
where
0)t(C ≥ , ]1,0[)t(u ∈ , 0)t(K ≥ , 0)t(H ≥ (3.5)
In equation (3.1) we work again with an isoelastic utility function and maximize
the integral, where σ is the instantaneous elasticity of substitution and ρ is the
instantaneous discount rate.
3.1. The equations of motion.
The current value Hamiltonian is given by
=)K,H,u,C(Hc
+−λ+σ−−
= βα−ασ−
])t(C)t(H))t(H)t(u()t(KA[1
1)t(C 1K
1
])t(H))t(u1([H −γ λ+ (3.6)
where Kλ and Hλ are the costate variables corresponding to K and H
respectively.
We can write down the dynamic equations of the model, using Pontryagin’s
optimization method . The reader may consult any of the known references of this
technique, like Malliaris and Brock [18], Pontryagin et. al. [24] or Seierstad and
Sydsaeter [30].
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The first order conditions read
0C0C
HK
c
=λ−⇔=∂∂ σ− (3.7)
0H]H)Hu(KA)1([0u
HH
1K
c =γ λ−α−λ⇔=∂∂ β+α−α (3.8)
We also have the two Euler equations
KK11
KKK
c
H)Hu(AKK
Hλρ=λ+λα⇔λρ=λ+
∂∂ βα−−α (3.9)
⇔λρ=λ+∂∂
HH
c
H
H
HHH11
K )u1(H)Hu(KA)1( λρ=λ+−γ λ+β+α−λ⇔ −βα−α (3.10)
the dynamic constraints
CH)Hu(KAK1 −= βα−α , 0K)0(K = (3.11)
H)u1(H −γ = , 0H)0(H = (3.12)
and the transversality conditions
0eKlim
t
Kt =λ
ρ−
∞→ (3.13)
0eHlim tH
t=λ ρ−
∞→(3.14)
3.2. First method of solution to the model.
This method uses a procedure similar to the dimensional reduction. The derived
system however is not solved along the balanced growth path (BGP). Considering
two auxiliary functions, we get a system of decoupled ordinary first order
differential equations, which, when solved, determine the physical capital, the
control variable u and finally the human capital.
3.2.1. Simplification of the dynamical equations.
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Taking logarithms and differentiation with respect to time, we obtain from
equation (3.7) that
K
K
C
C
λ
λ=σ−
(3.15)
Taking logarithms and differentiation with respect to time, we obtain from
equation (3.8) that
H
H
K
K
H
H)(
u
u
K
K
λλ=α−β+α−α+
λλ
(3.16)
From equation (3.9) we obtain
βα−−αα−ρ=λ
λH)Hu(AK 11
K
K (3.17)
From equation (3.10) we obtain
H
K11
H
H H)Hu(AK)1()u1(λλ
β+α−−−γ −ρ=λλ −βα−α (3.18)
From equation (3.11) we obtain
K
CH)Hu(AK
K
K 11 −= βα−−α (3.19)
From equation (3.12) we obtain
)u1(H
H−γ = (3.20)
We introduce a function named Y defined by
βα−α= H)Hu(AKY1 (3.21)
Equations (3.17)-(3.19) can be expressed in terms of the function Y as follows:
KY
K
K α−ρ=λλ (3.22)
H
Y)1()u1(
H
K
H
H ⋅λλ
β+α−−−γ −ρ=λλ
(3.23)
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K
C
K
Y
K
K−= (3.24)
From equation (3.8) we find that
u1H
Y
HK α−γ =⋅λλ (3.25)
Because of the previous relation, we can simplify further equation (3.23):
u1
)1()u1(
H
H ⋅α−β+α−γ
−−γ −ρ=λλ
(3.26)
Combining equations (3.15) and (3.22) we find an expression for the ratio
σ
ρ−⋅
σ
α=
K
Y
C
C(3.27)
Equation (3.16), upon substituting the expressions forK
K
λλ
,K
K,
H
Hand
H
H
λλ
given
by (3.22), (3.24), (3.20) and (3.26) respectively, we obtain the equation
K
Cu
1
)1()1(
u
u−
α−β+α−γ
+αβ+α−γ
= (3.28)
3.2.2. Auxiliary functions and their differential equations.
We now introduce two more functions U and Z defined by
K
YU = (3.29)
and
K
CZ = (3.30)
respectively.
We shall establish a system of two ordinary differential equations satisfied by
these two functions.
Taking logarithms and differentiation of the defining equations (3.29) and (3.30),
we obtain the equations
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K
K
Y
Y
U
U −= (3.31)
and
KK
CC
ZZ
−= (3.32)
respectively.
We can express the right hand sides of the two previous equations in terms of the
functions U and Z. From equation (3.24) we obtain
ZUK
K−= (3.33)
From equation (3.27) we obtain
σρ−⋅
σα= U
C
C(3.34)
We now have to express the ratioY
Yin terms of U and Z. Taking logarithms and
differentiation of (3.21), we obtain the equation
H
H)1(
u
u)1(
K
K
Y
Y β+α−+α−+α= (3.35)
Substituting the ratiosK
K,
u
uand
H
Hgiven by (3.33), (3.28) and (3.20)
respectively into the previous equation, we obtain
αβ+α−γ
+−α=)1(
ZUY
Y(3.36)
Therefore we obtain from (3.31) and (3.32), using (3.36), (3.33) and (3.34), the
system of two equations
U)1()1(
U
Uα−−
αβ+α−γ
= (3.37)
and
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σρ−
σσ−α
+= UZZ
Z(3.38)
Equation (3.37) can be solved by separation of variables. The solution is given by
0t
t
Cee
11U
− ν⋅α−= ν
ν(3.39)
where
αβ+α−γ
= ν)1(
and)0(U)1(
)1(1C0 α−α
β+α−γ −= (3.40)
Using the expression (3.39) for the function U, equation (3.38) is converted into
the equation
2
0t
t
ZZCe
e
)1(Z =
−
ν⋅α−σσ−α−
σρ+ ν
ν (3.41)
This is a Bernoulli differential equation, solved under the substitution
1ZX −= (3.42)
Equation (3.41) is converted in terms of X into the linear differential equation
1X
Ce
e
)1(
X
0
t
t
−=
σ
ρ−
−
ν⋅
α−σ
σ−α+ ν
ν (3.43)
The integrating factor of the above equation is
ζ νω −= )Ce(e)t(I 0tt (3.44)
where
σρ−=ω and
)1( α−σσ−α
=ζ (3.45)
Multiplying (3.43) by the integrating factor, we obtain the equation
ζ νω −−= )Ce(e)X)t(I(dt
d0
tt
which, upon integration in the interval ]t,0[ , gives
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ds)Ce(e)0(X)0(IX)t(It
0
0ss∫ ζ νω −−=− (3.46)
The integral on the right hand side of the previous equation is calculated in
Appendix A. We have, using the notation of this section
)t(1
ds)Ce(et
0
0ss Ω
ην−=−∫ ζ νω (3.47)
where
)C;1;,(F)eC;1;,(Fe)t( 0t
0t η+ηζ−−η+ηζ−=Ω ν−ην− (3.48)
and
ν ζ ν+ω−=η
(3.49)
Therefore we obtain from (3.46) the following expression for the function X:
ζ νω
ζ
−
Ωην+−
=)Ce(e
)t(1
)0(X)C1(
X
0tt
0
(3.50)
We thus have that the function Z, related to X by (3.42), is given by
)t(1
)0(X)C1(
)Ce(eZ
0
0
tt
Ωην+−−=
ζ
ζ νω(3.51)
Since from (3.47) we have upon differentiation with respect to t the relation
Ω
ην=−− ζ νω )t(
1
dt
d)Ce(e 0
tt (3.52)
equation (3.51) is written as
WWZ
−= (3.53)
where
)t(1
)0(X)C1(W 0 Ωην+−= ζ (3.54)
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On the other hand, equation (3.39) can be expressed as
V
V
1
1U
⋅α−
= (3.55)
where
0t
CeV −= ν (3.56)
3.2.3. Equation for the physical capital.
Equation (3.33) can be written, because of (3.55) and (3.53), as
W
W
V
V
1
1
K
K +⋅
α−= (3.57)
The above equation can be integrated once, providing us with the following
expression for the physical capital
WVDK 1
1
0α−=
or
Ωην+−−= ζα− ν )t(
1)0(X)C1()Ce(D)t(K 0
1
1
0t
0 (3.58)
where
Ωην+−−
==ζα−α− )0(
1)0(X)C1()C1(
K
)0(W)0(V
KD
01
1
0
0
1
10
0 (3.59)
3.2.4. Equation for the control variable u.
From equation (3.28), using (3.30) and (3.53), we derive the equation
W
Wu
1
)1()1(
u
u +
α−
β+α−γ +
α
β+α−γ =
which is equivalent to the equation
2u1
)1(u
W
W)1(u
α−β+α−γ
=
+αβ+α−γ
− (3.60)
This is a Bernoulli differential equation, which under the substitution
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1uy −= (3.61)
becomes a linear first order differential equation
α−
β+α−γ −=
+
α
β+α−γ +
1
)1(y
W
W)1(y
(3.62)
The previous equation admits the integrating factor
We)t(Jt ν= (3.63)
Multiplying equation (3.62) by the integrating factor, we obtain the equation
We1
)1()y)t(J(
dt
d t ν
α−β+α−γ
−=
which, upon integration in the interval ]t,0[ , gives us
∫ να−β+α−γ −=−
t
0
t ds)s(We1
)1()0(y)0(Jy)t(J (3.64)
Let
∫ ν=t
0
t ds)s(We)t(X (3.65)
Using the expression (3.54), we find that
== ∫ νt
0
s ds)s(We)t(X
+
η+ηζ−ην−−= ∫ νζ
t
0
s00 dse)C;1;,(F
1)0(X)C1(
∫ ν−η− ν η+ηζ−ην+
t
0
s0
s)1( ds)eC;1;,(Fe1
(3.66)
The last integral is evaluated in Appendix B. We get from (B.16), using the
notation of this section,
=η+ηζ−∫ ν−ϕt
0
t0
t dt)eC;1;,(Fe
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+η+ηζ−
−
+η+ηζ−
ν−= ν−ϕ
023t
023t C
1,1p
,,pFeC
1,1p
,,pFe
p
1(3.67)
where
)1( η− ν=ϕ and νϕ−=p (3.68)
Therefore we have the following expression for the function )t(X :
== ∫ νt
0
s ds)s(We)t(X
−
ν−
η+ηζ−ην−−=
νζ 1e
)C;1;,(F1
)0(X)C1(t
00
+η+ηζ−
−
+η+ηζ−
ην− ν−ϕ
023t
023t
2C
1,1p
,,pFeC
1,1p
,,pFe
p
1(3.69)
We then obtain from (3.64)
)t(J
)t(X)0(y)0(Jy
µ−= (3.70)
and then
)t(X)0(y)0(J)t(J)t(uµ−= (3.71)
where we have put
α−β+α−γ
=µ1
)1((3.72)
3.2.5. Equation for the human capital.
Equation (3.71) can also be written, comparing (3.63) and (3.65), as
)t(X)0(y)0(J
)t(Xu
µ−= (3.73)
Equation (3.20) for the human capital, because of the previous expression, takes
on the form
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)t(X)0(y)0(J
)t(X
H
H
µ−γ −
+γ =
or even
SS
HH
⋅µγ +γ = (3.74)
where
)t(X)0(y)0(JS µ−= (3.75)
Equation (3.74) gives upon integration
t0 eS
)0(S
HH γ µ
γ
µ
γ =
or
t1
1
1
10 e)t(S
)0(S
H)t(H γ β+α−
α−
β+α−α−= (3.76)
3.3. Second method of solution to the model.
3.3.1. Equations for the control variables.
Solving equation (3.7) with respect to C, we obtain
σ−
λ=1
K )(C (3.77)
Solving equation (3.8) with respect to u, we obtain
αα−β
αα
λλ
γ α−
= HKA)1(
u
1
H
K
1
(3.78)
We also get from the previous relation that
αα−β
α−αα−
αα−
α−
λλ
γ α−
=)1(
1
1
H
K
1
1 HKA)1(
)Hu( (3.79)
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0H
HH
1
)(
H
HH =γ −
⋅α−β
−λ
λρ−γ +λ
or equivalently
µ=⋅α−β−
λλ
HH
1H
H
(3.85)
where
γ −ρ+α−γ β
−=µ1
(3.86)
Integrating equation (3.85), we get
t1
0H e)t(HC)t(
µα−β
=λ (3.87)where
α−β
λ=
1
H0
)0(H
)0(C (3.88)
3.3.5.2. Expression for the costate variable Kλ .
Using the above expression for Hλ into the equation (3.80) for Kλ , we get the
equation
αλ
αα−µ−−=λρ−λ
1
K0KK )()1(
expB (3.89)
where
αα−
γ α−
α=
1
00
C
A)1(AB (3.90)
Equation (3.89) is a Bernoulli differential equation. Multiplying both members of
this equation by α−
λ1
K )( and introducing a new function λ by α−
λ=λ1
1
K )( , we
obtain the linear first order differential equation
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αα−µ
−
αα−
=λαα−ρ
+λ)1(
expB1)1(
0 (3.91)
The above differential equation is found to admit a solution given by
αα−ρ−×
λµ−ρ−−
αα−µ−ρ
µ−ρ=λ t)1(exp)0(
B)(1t)1()(expB
0
0
Therefore
×
λ
µ−ρ−−
αα−µ−ρ
µ−ρ=λ α
α−−
α−
1
K0
0
11
K ))0((B
)(1t
)1()(exp
B)(
α
α−ρ−× t
)1(exp
from which we obtain the following expression for the costate variable Kλ :
t10
t10K e)De(
B ρα−α− να−
α−
×−
µ−ρ
=λ (3.92)
where
α
α−µ−ρ= ν
)1()((3.93)
and
αα−−
λµ−ρ
−=1
K0
0 ))0((B
)(1D (3.94)
3.3.5.3. Equation for the physical capital.
Equation (3.82) for the physical capital, because of the (3.87), gets simplified into
the equation
σ−
αα−
αα−
λ−
αα−µ
−λ
γ α−
=1
K
1
K
1
0
)(K)1(
exp)(C
A)1(AK
and because of (3.92), into
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=−
µ−ρ
γ α−
− ν
ναα−
KDe
e
BC
A)1(AK
0t
t
0
1
0
σρ−−
µ−ρ
−= α−σ
α
να−σ
α
texp)De(B )1(0
t)1(0 (3.95)
We obtain that
α− ν
=αµ−ρ
=
µ−ρ
γ α− α
α−
1BC
A)1(A
0
1
0
(3.96)
We also introduce the notation
)1(00
BF
α−σα
µ−ρ
= (3.97)
and
σρ−=ω (3.98)
Equation (3.95) then becomes
t)1(0t0
0t
t
e)De(FKDe
e
1
1K ωα−σ
α
ν ν
ν
−−=− ν⋅α−−(3.99)
Equation (3.99) is a linear first order differential equation with integrating factor
given by
α−− ν
ν
ν−=
−
να−
−≡ ∫ 1
1
0t
0t
t
)De(dtDe
e
1
1exp)t(I (3.100)
Therefore equation (3.99), after multiplying by the integrating factor, becomes
α−−
α−σα
νω −−≡ 1
1
)1(0
tt0 )De(eF)K)t(I(
dt
d
Integrating the above equation we arrive at
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dt)De(eFK)0(IK)t(It
0
0tt
00 ∫ ζ νω −−≡− (3.101)
where
)1(1
1)1( α−σ
σ−α=α−
−α−σα=ζ (3.102)
The integral appearing in (3.101) is evaluated to be (Appendix A)
=−∫ ζ νω dt)De(et
0
0tt
−−−ην−= ∫ ∫ ζ−ηζ ν−−η νη−
1
0
01
1
0
t0
1t du)uD1(udx)xeD1(xe1
)t(1Ψ
ην−= (3.103)
where
)D;1;,(F)eD;1;,(Fe)t( 0t
0t η+ηζ−−η+ηζ−=Ψ ν− νη− (3.104)
νζ ν+ω
−=η (3.105)
Using (3.101), (3.103) and the explicit expression for the integrating factor )t(I
given by (3.100), we obtain the following expression for the human capital:
α− να−−
−
Ψ
ην+−= 1
1
0t0
01
1
0 )De()t(F
K)D1()t(K (3.106)
3.3.5.4. Equation for the human capital.
We now turn to the evaluation of the human capital using equation (3.83). This
equation can be written, using (3.87) and (3.92), as
×−
µ−ρ
γ α−
γ −=γ − α−− να−
−α
1
1
0t1
1
0
1
0
)De(B
C
A)1(HH
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η ν−ϕ=θ and νθ−=p (3.118)
Therefore (3.115), because of (3.116) and (3.117), takes the form
α−β+α−γ
× α−β+α−
−= t1
)1(exp)t(XG1
1UU 00 (3.119)
where
=
Ψ
ην+−≡ ∫ α−
−ϕt
0
00
1
1
0t dt)t(
FK)D1(e)t(X
−ϕ−
+ηηζ−
ην−−=
ϕα−
− 1e)D;1;,(F
FK)D1(
t
00
01
1
0
+η+ηζ−
−
+η+ηζ−
ην− ν−θ
023t
023t
20 D
1,1p
,,pFeD
1,1p
,,pFe
p
F(3.120)
We thus obtain the following formula for the human capital
t1
1
01
1
e)t(XG1
1)0(HH γ
β+α−α−
α−β+α−
×
α−β+α−
−= (3.121)
taking into account the relation (3.111) between H and U.
3.3.5.5. Expression for the costate variable Hλ .
Using equations (3.87) and (3.121), we are able to find the following expression
for the costate variable Hλ :
t)()1(
1
01
1
0H e)t(XG1
1)0(HC γ +µ
β+α−βα−β+α−
α−β+α−−=λ (3.122)
where 0C is given by (3.88).
4. Application I.
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(The model considered by Bucekkine and Ruiz-Tamarit).
In this Section we solve again the model considered by Bucekkine and Ruiz-
Tamarit [8]. This model is solved using the two methods we already used in
solving the simple model considered in Section 2. Despite the fact that in the
second method we use the same technique to that introduced in reference [8], we
present explicitly all the intermediate steps, in easy to follow calculations. Apart
from that, we have found that the human capital function is expressed in terms of
the generalized hypergeometric function []F23 .
4.0. The model.
We consider the dynamic optimization problem
t
0
1
e)t(N1
1)t(cmax ρ−
∞ σ−
∫ σ−−
(4.1)
subject to
)t(N)t(c)t(K))t(H)t(N)t(u()t(KA)t(K1 −π−= β−β (4.2)
)t(H)t(H))t(u1()t(H ϑ−−δ= (4.3)
with initial conditions 0K)0(K = , 0H)0(H = , 0N)0(N = (4.4)
where
0)t(c ≥ , ]1,0[)t(u ∈ , 0)t(K ≥ , 0)t(H ≥ (4.5)
The notation in this model is as in previous models. In this model however has
been considered depreciation of physical capital (π is the coefficient of
depreciation) and depreciation of human capital (ϑ is the coefficient of
depreciation). The function )t(N represents the population growth.
4.1. The dynamical equations.
The current value Hamiltonian is given by
=)K,N,H,u,c(Hc
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+−π−λ+σ−−
= β−βσ−
])t(N)t(c)t(K))t(H)t(N)t(u()t(KA[)t(N1
1)t(c 1K
1
])t(H)t(H))t(u1([H ϑ−−δλ+ (4.6)
where Kλ and Hλ are the costate variables corresponding to K and H
respectively.
We can write down the dynamic equations of the model, using Pontryagin’s
optimization method .
The first order conditions read
K
c
c0c
Hλ=⇔=
∂∂ σ− (4.7)
0]N)HNu(KA)1([0u
HHK
c
=δλ−β−λ⇔=∂∂ β−β (4.8)
We also have the two Euler equations
⇔λρ=λ+∂∂
KK
c
K
H β−−ββλ−λπ+ρ=λ⇔ 11
KKK )HNu(KA)( (4.9)
⇔λρ=λ+∂∂
HH
c
H
H )u1(H)Nu(KA)1()( H
1KHH −δλ−β−λ−λϑ+ρ=λ β−β−β (4.10)
the dynamic constraints
NcK)HNu(KAK1 −π−= β−β , 0K)0(K = (4.11)
HH)u1(H ϑ−−δ= , 0H)0(H = (4.12)
and the transversality conditions
0eKlim tK
t=λ ρ−
∞→(4.13)
0eHlim tH
t=λ ρ−
∞→(4.14)
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We also suppose that we have the exponential population growth (exogenous)
given by
tn0 eN)t(N = (4.15)
4.2. First method of solution.
4.2.1. Simplification of the dynamical equations.
Taking logarithms and differentiation with respect to time, we obtain from
equation (4.7) that
K
K
C
C
λλ=σ−
(4.16)
Taking logarithms and differentiation with respect to time, we obtain from
equation (4.8) that
H
H
K
K
H
H
N
N)1(
u
u
K
K
λλ=β−β−+β−β+
λλ
(4.17)
From equation (4.9) we obtain
β−−ββ−π+ρ=λλ 11
K
K )HNu(KA
(4.18)
From equation (4.10) we obtain
H
1)HNu(AK)1()u1(
H
K1
H
H ⋅λλ
β−−−δ−ϑ+ρ=λλ β−β (4.19)
From equation (4.11) we obtain
K
CN)HNu(KA
K
K 11 −π−= β−−β (4.20)
From equation (4.12) we obtain
ϑ−−δ= )u1(H
H(4.21)
From equation (4.15) we obtain
nN
N= (4.22)
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We introduce a function named Y defined by
β−β= 1)NHu(KAY (4.23)
Equations (4.18)-(4.20) can be expressed in terms of the function Y as follows:
K
Y
K
K β−π+ρ=λλ (4.24)
H
Y)1()u1(
H
K
H
H ⋅λλ
β−−−δ−ϑ+ρ=λλ
(4.25)
K
CN
K
Y
K
K−π−= (4.26)
From equation (4.8) we find that
u1H
Y
H
K
β−δ=⋅
λλ
(4.27)
Because of the previous relation, we can simplify further equation (4.25):
δ−ϑ+ρ=λλ
H
H (4.28)
Combining equations (4.16) and (4.24), we find an expression for the ratio
σπ+ρ
−⋅σβ= K
Y
C
C(4.29)
From equation (4.17), upon substituting the expressions given by (4.24), (4.26),
(4.22) and (4.21), we obtain the equation
K
CNu
)n()1(
u
u−δ+
β+ϑ−δ+πβ−
= (4.30)
4.2.2. Auxiliary functions and their differential equations.
We now introduce two more functions U and Z defined by
K
YU = (4.31)
and
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K
CNZ = (4.32)
respectively.
We shall establish a system of two ordinary differential equations satisfied by
these two functions.
Taking logarithms and differentiation of the defining equations (4.31) and (4.32),
we obtain the equations
K
K
Y
Y
U
U −= (4.33)
and
KK
CC
NN
ZZ
−+= (4.34)
respectively.
We can express the right hand sides of the two previous equations in terms of the
functions U and Z. From equation (4.26) we obtain
ZUK
K−+π−= (4.35)
From equation (4.29) we obtain
σπ+ρ
−⋅σβ= U
C
C(4.36)
We now have to express the ratioY
Yin terms of U and Z. Taking logarithms and
differentiation of (4.23), we obtain the equation
H
H)1(
N
N)1(
u
u)1(
K
K
Y
Y β−+β−+β−+β= (4.37)
Substituting the ratiosK
K,
u
uand
H
Hgiven by (4.26), (4.30), (4.22) and (4.21)
respectively, into the previous equation, we obtain
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∫ µ=t
0
t ds)s(We)t(X (4.69)
Using the expression (4.57), with )t(Ω given by (4.51), we find that
== ∫ µt
0
s ds)s(We)t(X
+
η+ηζ−ην−−= ∫ µζ
t
0
s00 dse)C;1;,(F
1)0(X)C1(
∫ ν− νη−µ η+ηζ−ην+
t
0
s0
s)( ds)eC;1;,(Fe1
(4.70)
The last integral is evaluated in Appendix B. We get from (B.16), using the
notation of this section,
=η+ηζ−∫ ν−ϕt
0
t0
t dt)eC;1;,(Fe
+η+ηζ−
−
+η+ηζ−
ν−= ν−ϕ
023t
023t C
1,1p
,,pFeC
1,1p
,,pFe
p
1(4.71)
where
ην−µ=ϕ and νϕ−=p (4.72)
Therefore we have the following expression for the function )t(X :
== ∫ µt
0
s ds)s(We)t(X
−
µ
−
η+ηζ−ην−−=
µζ 1e
)C;1;,(F
1
)0(X)C1(
t
00
+η+ηζ−
−
+η+ηζ−
ην− ν−ϕ
023t
023t
2C
1,1p
,,pFeC
1,1p
,,pFe
p
1(4.73)
We then obtain from (4.68)
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Equation (4.83), using (4.15) and (4.85), takes the simplified form
−λπ+ρ=λ KK )(
ββ
β−−ββ−
λ
β
ϑ−ρ−δ+β−λ
δβ−
β−
1
K
1
H
1
0
)(t)n)(1(
exp))0((
AN)1(
A (4.86)
Introducing the notation
ββ−
δβ−
β=ε
1
0AN)1(A (4.87)
and
β
ϑ−ρ−δ+β−=µ
)n)(1((4.88)
equation (4.86) can also be written as
βµββ−
−λλε−=λπ+ρ−λ
1
Kt
1
HKK )(e))0(()( (4.89)
Equation (4.89) is a Bernoulli differential equation. Multiplying through by
β−
λ1
K )( , this equation becomes
t
1
H
11
K
1
KK e))0(()()()( µββ−−
β−
β−
λε−=λπ+ρ−λλ (4.90)
Introducing a new function λ by
β−
λ=λ1
1
K )( (4.91)
equation (4.90) takes on the form
t
1
H e))0(()1())(1( µβ
β−−
λβ β−ε=λβ π+ρβ−+λ (4.92)
The previous equation admits the function teα as an integrating factor, where
βπ+ρβ−
=α))(1(
(4.93)
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Therefore the general solution of (4.92) is given by
Ce))0(()(
)1(e t)(
1
Ht +λ
α+µββ−ε
=λ α+µββ−−
α (4.94)
For 0t = , we obtain from the previous equation the relation
C))0(()(
)1()0(
1
H +λα+µββ−ε
=λ ββ−
−(4.95)
from which we determine the constant C.
Therefore equation (4.94) takes the form
t
1
Ht
1
H e))0(()(
)1()0(e))0((
)(
)1( α−ββ−−
µββ−−
λ
α+µβ
β−ε−λ+λ
α+µβ
β−ε=λ
The above equation, taking into account (4.91), can also be written as
×λα+µββ−ε
=λ α−ββ−
−ββ−
−t
1
H
1
K e))0(()(
)1()(
λλ
β−εα+µβ
++−×ββ−
−α+µ
1
H
Kt)(
)0(
)0(
)1(
)(e1
or, in equivalent form,
×
β
π+ρβ−−λ
π+δ+ϑ−ε
=λ ββ−
−ββ−
−t
)()1(exp))0((
n)(
1
H
1
K
λλ
επ+δ+ϑ−
+
β
π+δ+ϑ−β−+−×
ββ−
−1
H
K
)0(
)0(nt
)n()1(exp1
From the previous equation we get the following expression for the costate
variable Kλ :
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])C;1;,m(F)eC;1;,m(Fe[D
0t
0t0 η+η−−η+η−
η ν= ν− νη−
or
Ψη ν
+−×−= β−−π−β− ν )t(D
)C1(Ke)Re()t(K 011
00t1
1t (4.111)
where
)C;1;,m(F)eC;1;,m(Fe)t( 0t
0t η+η−−η+η−=Ψ ν− νη− (4.112)
4.3.4. Equation for the human capital.
We come now in finding an explicit solution of the differential equation (4.12) for
the human capital. Substituting into (4.12) the expression for u given by (4.80),
we obtain the following differential equation
KNA)1(
H)(H
11
H
K
1
ββ−
ββ
λλ
δβ−
δ−ϑ−δ= (4.113)
From (4.99) we obtain
β−
−
νβ−β
−×
β ϑ−δ+π× ε π+δ+ϑ−=
λλ
1
1
0t1
11
HK )Ce(texp
n
Using the previous expression and also (4.15) for N, we get the following
differential equation for H:
K)Ce(eQH)(H 1
1
0tt β−
− νζ −××−ϑ−δ= (4.114)
where
ββ−β−β ×
επ+δ+ϑ−
δβ−δ=
1
0
111
NnA)1(
Q (4.115)
and
ββ−
+βϑ−δ+π
=ζn)1(
(4.116)
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H
KHN
A)1(u
1
H
K
11
αβ
ααα−
α
λλ
γ α−
= (5.82)
We also get from the previous relation the two relations
α−αα−β
αα−
αα−
αα−
α−
λλ
γ α−
= 1
)1(1
H
K
11
1 KHNA)1(
)uNH( (5.83)
and
α−αα−βα−
αα−
αα−
αα−
α−
λλ
γ α−
= 1
))(1(1
H
K
11
1 KHNA)1(
)uN( (5.84)
we shall need later on.5.3.2. Equations for the costate variables.
Equation (5.9), because of (5.83), becomes
K
1
H
K
11
KK HNA)1(
A)( λ
λλ
γ α−
α−=λπ+ρ−λ αβ
αα−
αα−
αα−
(5.85)
Equation (5.10), because of (5.84) and (5.82), takes on the form
=λγ −ϑ+ρ−λ HH )(
KHNA)1(
A K
1
H
K
11
λ
λλ
γ α−
β−= αα−β
αα−
αα−
αα−
(5.86)
5.3.3. Equation for the physical capital.
Equation (5.11), using (5.83) and (5.81), takes the simplified form
=π+ KK
σ−
αβ
αα−
αα−
αα−
λ−
λλ
γ α−
=1
K
1
H
K
11
)(NKHNA)1(
A (5.87)
5.3.4. Equation for the human capital.
Equation (5.12), because of (5.82) takes the form
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KHNA)1(
H)(H
1
H
K
11
αβ
ααα−
α
λλ
γ α−
γ −=ϑ−γ − (5.88)
5.3.5. The solution strategy.
Equations (5.85)-(5.88) constitute a system of four equations which determine
completely the dynamics of the model. The solution strategy we follow is quite
different compared to the one we used without the externalities.
The first observation is that equation (5.86) cannot be solved as it is, since its right
hand side contains unknown quantities. Therefore we have to modify the solution
strategy in the presence of externalities.
Dividing (5.86) by (5.88) we obtain the equation
H1H)(H
)( HHH λα−β
=ϑ−γ −λγ −ϑ+ρ−λ
which is equivalent to the equation
0H
H)(H
1
)(
H
HH =ϑ−γ −
⋅α−β
−λ
λγ −ϑ+ρ−λ (5.89)
The previous equation can be written as
µ=⋅α−β−
λλ
H
H
1H
H (5.90)
where
)(1
)( γ −ϑα−β
+γ −ϑ+ρ=µ (5.91)
Equation (5.90) admits the solution
t10H e)t(HC)t(
µα−β
=λ (5.92)
where
α−β
λ=
1
H0
)0(H
)0(C (5.93)
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Equation (5.92) expresses a relation between Hλ and H. However this relation
simplifies considerably the other equations of motion. We then can obtain closed-
form solutions.
5.3.5.1. Solution of equation for the costate variable Kλ .
Using this equation into the equation (5.85) for Kλ , we obtain the equation
ααα−
αα−
λ
αα−µ
−
γ α−
α−=λπ+ρ−λ1
K
11
0KK )(t
)1(expN
C
A)1(A)( (5.94)
which can be solved, since it is a Bernoulli differential equation.
In fact multiplying this equation through byα−
λ
1
K )( , and taking into account
equation (5.15) for the population, we obtain the equation
t
11
K
1
KK e)()()( να−
α−
ε−=λπ+ρ−λλ (5.95)
where
αα−
γ α−
α=ε
1
0
0
C
AN)1(A (5.96)
and
α
α−µ−= ν
)1()n((5.97)
Introducing a new function λ by
α−
λ=λ1
1
K )( (5.98)
equation (5.95) takes on the form
te)1()()1( ν
αα−ε
=λα
π+ρα−+λ (5.99)
The previous equation admits the function teξ as an integrating factor, where
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λε
µ−+π+ρ+
αµ−+π+ρα−
+−× αα−−1
K ))0((n
t)n()1(
exp1 (5.103)
taking into account the expression for ξ and that
µ−+π+ρ=α−ξ+ να
n1
)(
From equation (5.103) we get the following expression for the costate variable
Kλ :
t)(10
t1
K e)De(n
π+ρα−α−ζα−
α−
−×
µ−+π+ρ
ε=λ (5.104)
where we have introduced the notation
αµ−+π+ρα−
=ζ)n()1(
(5.105)
αα−−
λε
µ−+π+ρ−=
1
K0 ))0((n
1D (5.106)
Equation (5.104) is an explicit expression for the costate variable Kλ .
5.3.5.2. Solution of equation for the physical capital.
We now consider equation (5.87). This equation, because of (5.92), takes the form
=π+ KK
σ−
αα−
αα−
αα−
λ−
αα−µ
−λ
γ α−
=1
K
1
K
11
0
)(NKt)1(
exp)(NC
A)1(A (5.107)
Using the expressions (5.104) for Kλ and (5.15) for the population, we get from
the previous equation
=π+ KK
×−×
εµ−+π+ρ
γ α−
= ζαα−
)De(n
C
A)1(A 0
t
1
0
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−
αα−µ
−α
α−π+ρ+
αα−
× Kt)1()1()()1(n
exp
σ
π+ρ
−×−×
µ−+π+ρ
ε
−α−σα
ζα−σα
tnexp)De(nN
)1(
0
t)1(
0 (5.108)
The previous equation can be simplified considerably. Using (5.96) in place of ε
we find
α−ζ
=α
µ−+π+ρ=
εµ−+π+ρ
γ α− α
α−
1
nn
C
A)1(A
1
0
(5.109)
where we also have used expression (5.105) for ζ .
We also have that
tet)1()1()()1(n
expζ=
αα−µ
−α
α−π+ρ+
αα−
(5.110)
using (5.105) again.
Therefore (5.108) takes on the form
−=
−
ζ⋅α−
−π+ ζ
ζK
De
e
1
1K
0
t
t
σπ+ρ−×−×
µ−+π+ρ
ε− α−σα
ζα−σα
tnexp)De(n
N )1(0
t)1(
0 (5.111)
The above differential equation is a linear first order differential equation with
integrating factor
α−−ζπ
ζ
ζ−×=
−
ζ⋅
α−
−π= ∫ 1
1
0tt
0
t
t
)De(edt
De
e
1
1exp)t(I (5.112)
Therefore multiplying (5.111) by the integrating factor, we obtain the equation
−=))t(I)t(K(dt
d
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σπ+ρ−+π×−×
µ−+π+ρ
ε− α−−
α−σα
ζα−σα
tnexp)De(n
N 1
1
)1(0
t)1(
0
and integrating in the interval ]t,0[ , we arrive at the equation
∫ ηζω −−=−t
0
0tt
00 ds)De(eF)0(IK)t(I)t(K (5.113)
where
)1(
00n
NFα−σα
µ−+π+ρ
ε= (5.114)
σ
π+ρ
−+π=ω n (5.115)
α−
−α−σα
=η1
1
)1((5.116)
The integral appearing in the right-hand-side of (5.113) has been evaluated in
Appendix A. We have, using the notation of this section:
)t(1
ds)De(et
0
0ss Ψ
ζυ−=−∫ ηζω (5.117)
where
ζζη+ω
−=υ (5.118)
and
)D;1;,(F)eD;1;,(Fe)t( 0t
0t +υυη−−+υυη−=Ψ ζ−υζ− (5.119)
We thus get the following expression for the physical capital K:
α−ζα−−
−×
Ψζυ
+−= 1
1
0t01
1
00 )De()t(F
)D1(K)t(K (5.120)
5.3.5.3. Solution of equation for the human capital
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The last integral is not elementary and needs to be evaluated using some results
from the theory of Generalized Hypergeometric Functions. We have, using
Appendix B
=υ+υη−∫ ζ−t
0
t0
ta dt)eD;1;,(Fe
+υ+υη−
−
+υ+υη−
ζ−= ζ−
023t
023ta D
1,1p
,,pFeD
1,1p
,,pFe
p
1(5.131)
where
ζυ−φ=a andζ−=
ap (5.132)
We then obtain from (5.130)
=
ζυ+−≡ ∫ α−
−φt
0
01
1
00t dt)t(F
~F)D1(Ke)t(X
−
φ−
υ+υη−
ζυ−−=
φα−
− 1e)D;1;,(F
F)D1(K
t
001
1
00
+υ+υη−−
+υ+υη−
ζυ− ζ−
023t
023ta
20 D
1,1p,,pFeD
1,1p,,pFe
pF (5.133)
From (5.129) we arrive at the following expression for the function U:
α−γ −ϑβ+α−
−×
α−β+α−
−= t1
))(1(exp)t(XG
1
1)0(UU 0 (5.134)
where )t(X is given by (5.133).
Using the fact that α−
β+
= 1
1
HU , we find the following expression for the human
capital H:
t)(1
1
01
1
0 e)t(XG1
1)H()t(H ϑ−γ
β+α−α−
α−β+α−
×
α−β+α−−= (5.135)
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85
where Kλ and Qλ are the costate variables corresponding to K and Q
respectively.
We can write down the dynamic equations of the model, using Pontryagin’s
optimization method .
The first order conditions read
0C0C
HK
c
=λ−⇔=∂∂ σ− (6.7)
0])Qu(AK)1([)1(0u
HKQ
c
=β−λ+δ+λ−⇔=∂∂ β−β (6.8)
We also have the two Euler equations
β−−ββλ−λρ=λ⇔λρ=λ+∂∂ 11
KKKKK
c
)Qu(AKK
H (6.9)
⇔λρ=λ+∂∂
c
Q
H
)u)1((Q)Qu(K)1( Q11
KQQ δ+−δλ−β−λ−λρ=λ⇔ −β−β (6.10)
the dynamic constraints
)t(CN))t(Q)t(u()t(AK)t(K1 −= β−β , 0K)0(K = (6.11)
)t(Q)t(u)t(Q))t(u1()t(Q −−δ= , 0Q)0(Q = (6.12)
and the transversality conditions
0eKlim tK
t=λ ρ−
∞→(6.13)
0eQlim tQ
t=λ ρ−
∞→(6.14)
6.2. The method of solution.
6.2.1. Equations for the control variables.
Solving equation (6.7) with respect to C, we obtain
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dt)tz1()t1(t)bc()b(
)c()z;c;b,a(F
1
0
a1bc1b∫ −−−− −−−Γ Γ
Γ = (A.11)
when 0)bRe()cRe( >> .
When 01bc =−− , i.e. b1c += , the above integral representation can be written
as
dt)tz1(t)1()b(
)b1()z;b1;b,a(F
1
0
a1b∫ −− −Γ Γ +Γ
=+ (A.12)
and since )b(b)b1( Γ =+Γ and 1)1( =Γ , we have
dt)tz1(tb)z;b1;b,a(F1
0
a1b∫ −− −=+ (A.13)
The above formula is going to be used for expressing (A.10) in terms of the
hypergeometric function. We find, because of (A.13)
)eC;1;,(F1
dx)xeC1(x t0
1
0
t0
1 µ− νµ−−η η+η ν−η=−∫ (A.14)
and
)C;1;,(F1dx)xC1(x 0
1
0
01 η+η ν−η=−∫ ν−η (A.15)
Therefore
=−∫ νµωt
0
0ss ds)Ce(e
])C;1;,(F)eC;1;,(Fe[1
0t
0t η+η ν−−η+η ν−
ηµ−= µ−µη− (A.16)
where η is given by (A.5).
Note. Formula (A.16) is valid within a range of the parameters involved. The
series expansion of the hypergeometric function )z;c;b,a(F is given by
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n
1n n
nn z!n)c(
)b()a(1)z;c;b,a(F ∑
∞
=+=
where
)1n()1()( n −+α+αα=α , 1n ≥1)( 0 =α
is the usual Pochhammer symbol.
The above series converges for 1|z| < . For different values of z, an analytic
continuation needs to be considered.
Appendix B. Evaluation of the integral
∫ µ−ϕ η+η ν−t
0
t0
t dt)eC;1;,(Fe (B.1)
We shall evaluate this integral in two steps:
In the first step we shall convert the above integral into a combination of integrals
having each one of them limits 0 and 1.
Under the transformation
seu µ−= ,u
du1dsµ−= and µ
ϕ−ϕ = ue s (B.2)
and taking into account that the new limits are
=⇒==⇒=µ− teuts
1u0s
the integral in (B.1) transforms into
∫
µ−
+ηη ν−
µ− µ
ϕ−te
1 0 u
du)uC;1;,(Fu
1
which can also be written as
+ηη ν−−+ηη ν−
µ− ∫ ∫ −−
µ− 1
0
01p
e
0
01p du)uC;1;,(Fudu)uC;1;,(Fu
1t
(B.3)
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109
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