Lucas Uzawa

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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]

[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)


H

1)Hu(K)1()u1(

H

K1

H

H ⋅λλ

α−−−γ −=λλ α−α (2.18)




8

K

C)Hu(K

K

K 11 −= α−−α (2.19)


)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)


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)



13

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



17

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 :



18

α−α

−µα−α

−×λ

αγ

α−γ

=λ 10

tH

1K )Ce()0(

1(2.86)


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



19

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)



20


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



21

−=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



22

+

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.



23

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 =



24

where γ is a constant coefficient.


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.


=)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)


respectively.


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].



25


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)


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)


CH)Hu(KAK1 −= βα−α , 0K)0(K = (3.11)

H)u1(H −γ = , 0H)0(H = (3.12)


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.




26


equation (3.7) that

K

K

C

C

λ

λ=σ−

(3.15)


equation (3.8) that

H

H

K

K

H

H)(

u

u

K

K

λλ=α−β+α−α+

λλ

(3.16)


βα−−αα−ρ=λ

λH)Hu(AK 11

K

K (3.17)


H

K11

H

H H)Hu(AK)1()u1(λλ

β+α−−−γ −ρ=λλ −βα−α (3.18)


K

CH)Hu(AK

K

K 11 −= βα−−α (3.19)


)u1(H

H−γ = (3.20)


βα−α= H)Hu(AKY1 (3.21)


KY

K

K α−ρ=λλ (3.22)

H

Y)1()u1(

H

K

H

H ⋅λλ

β+α−−−γ −ρ=λλ

(3.23)



27

K

C

K

Y

K

K−= (3.24)


u1H

Y

HK α−γ =⋅λλ (3.25)


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.


K

YU = (3.29)

and

K

CZ = (3.30)

respectively.



Taking logarithms and differentiation of the defining equations (3.29) and (3.30),

we obtain the equations



28

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


ZUK

K−= (3.33)


σρ−⋅

σα= U

C

C(3.34)




H

H)1(

u

u)1(

K

K

Y

Y β+α−+α−+α= (3.35)


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



29

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



30

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)



31

On the other hand, equation (3.39) can be expressed as

V

V

1

1U

⋅α−

= (3.55)

where

0t

CeV −= ν (3.56)


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 =


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



32

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


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)




0

t0

t dt)eC;1;,(Fe



33

+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)


== ∫ νt

0

s ds)s(We)t(X

−

ν−


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)


)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)


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



34

)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.



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)





36

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



37

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



38

=−

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



39

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







42

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.



43

(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.


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.


=)K,N,H,u,c(Hc



44

+−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)


respectively.


optimization method .


K

c

c0c

Hλ=⇔=

∂∂ σ− (4.7)

0]N)HNu(KA)1([0u

HHK

c

=δλ−β−λ⇔=∂∂ β−β (4.8)


⇔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)


NcK)HNu(KAK1 −π−= β−β , 0K)0(K = (4.11)

HH)u1(H ϑ−−δ= , 0H)0(H = (4.12)


0eKlim tK

t=λ ρ−

∞→(4.13)

0eHlim tH

t=λ ρ−

∞→(4.14)



45

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.



equation (4.7) that

K

K

C

C

λλ=σ−

(4.16)


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)


β−−ββ−π+ρ=λλ 11

K

K )HNu(KA

(4.18)


H

1)HNu(AK)1()u1(

H

K1

H

H ⋅λλ

β−−−δ−ϑ+ρ=λλ β−β (4.19)


K

CN)HNu(KA

K

K 11 −π−= β−−β (4.20)


ϑ−−δ= )u1(H

H(4.21)


nN

N= (4.22)



46


β−β= 1)NHu(KAY (4.23)


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)


u1H

Y

H

K

β−δ=⋅

λλ

(4.27)


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.


K

YU = (4.31)

and



47

K

CNZ = (4.32)

respectively.



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


ZUK

K−+π−= (4.35)


σπ+ρ

−⋅σβ= U

C

C(4.36)




H

H)1(

N

N)1(

u

u)1(

K

K

Y

Y β−+β−+β−+β= (4.37)


K,

u

uand

H

Hgiven by (4.26), (4.30), (4.22) and (4.21)

respectively, into the previous equation, we obtain











52

∫ µ=t

0


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)




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)


== ∫ µt

0

s ds)s(We)t(X

−

µ

−


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)








55

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)



56

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 :









60

])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)


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)





























74

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)


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)


Equation (5.12), because of (5.82) takes the form



75

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


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)



76

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





78

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



79

−

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



80

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







83

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)


=

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)





85

where Kλ and Qλ are the costate variables corresponding to K and Q

respectively.


optimization method .


0C0C

HK

c

=λ−⇔=∂∂ σ− (6.7)

0])Qu(AK)1([)1(0u

HKQ

c

=β−λ+δ+λ−⇔=∂∂ β−β (6.8)


β−−ββλ−λρ=λ⇔λρ=λ+∂∂ 11

KKKKK

c

)Qu(AKK

H (6.9)

⇔λρ=λ+∂∂

QQ

c

Q

H

)u)1((Q)Qu(K)1( Q11

KQQ δ+−δλ−β−λ−λρ=λ⇔ −β−β (6.10)


)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)


0eKlim tK

t=λ ρ−

∞→(6.13)

0eQlim tQ

t=λ ρ−

∞→(6.14)

6.2. The method of solution.











































105

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



106

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)







109

Dover 1972

[2] D. Acemoglu: “Introduction to Modern Economic Growth”

Princeton University Press 2009

[3] P. Aghion and P. Howitt: “Endogenous Growth Theory”MIT Press 1998

[4] G.E. Andrews, R. Askey and R.Roy: “Special Functions”

Cambridge University Press 1999

[5] R. Barro and X. Sala-i-Martin: “Economic Growth”

MIT Press, Second Edition, 2004

[6] J. Benhabib and R. Perli: “Uniqueness and indeterminacy: On the

Dynamics of Endogenous Growth”

Journal of Economic Theory 63 (1994) 113-142

[7] D. Bethmann: “A Closed-Form Solution of the Uzawa-Lucas Model of

Endogenous Growth”

Journal of Economics 90 (2007) 87-107

[8] R. Bucekkine and J. R. Ruiz-Tamarit: “Special functions for the study

of economic dynamics: The case of the Lucas-Uzawa model”

Journal of Mathematical Economics 44 (2008) 33-54

[9] J. Caballè and M. Santos: “On Endogenous Growth with Physical and

Human Capital”

Journal of Political Economy 101 (1993) 1042-1067

[10] A. Erdélyi (Ed.): “Higher Transcendental Functions”

Volume 1, McGraw Hill 1953

[11] M.A. Gómez: “Optimal Fiscal Policy in the Uzawa-Lucas model with

Externalities”

Economic Theory 22 (2003) 917-925

[12] A. Greiner and W. Semmler: “The Global Environment, Natural

Resources, and Economic Growth”





111

Quarterly Journal of Economics 108 (1993) 739-773

[24] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidge and E.F.

Mishchenko: “The Mathematical Theory of Optimal Processes”

Interscience, 1962[25] E. D. Rainville: “Special Functions”

Chelsea 1971

[26] P.M. Romer: “Increasing Returns and Long-Run Growth”


[27] P.M. Romer: “Endogenous Technological Change”


[28] D. Romer: “Advanced Macroeconomics”

McGraw Hill, N.Y.1996

[29] J. R. Ruiz-Tamarit and M. Sánchez-Moreno: “Optimal Regulation in

a Natural-Resource-Based Economy”

Preprint

[30] A. Seierstad and K. Sydsaeter: “Optimal Control Theory with Economic

Applications”

North-Holland 1987

[31] L.J. Slater: “Generalized Hypergeometric Functions”


[32] V. I. Smirnov: “A Course in Higher Mathematics”

Pergamon Press, Oxford 1964. Volume III, Part 2

[33] H. Uzawa: “Optimum Technical Change in an Aggregate Model of

Economic Growth”

International Economic Review 6 (1965) 18-31

[34] H. Uzawa: “Economic Theory and Global Warming”


[35] E.T. Whittaker and G.N. Watson: “A Course of Modern Analysis”



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