THE EXISTENCE THEOREM OF OPTIMAL GROWTH MODEL

2005,25B(1):30-40

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1~4mJl~m THE EXISTENCE THEOREM OF OPTIMAL GROWTH MODEL 1 Gong Liutang ( Jt.*~ ) Guanghua School of Management, Peking University, Beijing 100871, China Institute for Advanced Study, Wuhan University, Wuhan 430072, China E-mail: [email protected] Peng Xianze ( jj 'ht 11'1 ) China University of Geosciences, Beijing 100083, China Abstract

This paper proves a general existence theorem of optimal growth theory. This

theorem is neither restricted to the case of a constant technology progress, nor stated in terms of mathematical conditions which have no direct economic interpretation and moreover, are difficult to apply. Key words

Optimal growth, existence, piecewise continuity

2000 MR Subject Classification

1

90A16, 49L20, 44AIO

Introduction

The Ramsey-Cass-Koopmans model is widely used in growth theory (Barro and Sala-iMartin, 1995; Sargent, 2000; and Gong 2000 et al). Since Ramsey (1928) investigated the optimal saving and optimal consumption for the specific utility function and production function, more and more economists worked on this model. Cass (1965) and Koopmans (1961) studied the convergent properties for the optimal capital accumulation path and consumption level without technology progress. They also studied the existence for the optimal capital accumulation path and consumption level in their framework, but the existence theorem under technology progress has not been studied yet. This paper is to present a general existence theorem, it is neither restricted to the case of a constant technology progress nor stated in terms of mathematical conditions which have no direct economic interpretation. This paper is organized as follows. In Section 2, we state the main results of this paper; we prove the theorem in Section 3 and Section 4.

2

The Model and the Main Result

Following the framework of Ramsey-Cass-Koopmans model (Cass, 1965, Ramsey, 1928, and Koopmans, 1962), the representative agent is assumed to have infinite horizon, to face 1 Received

September 13, 2001; revised December 17,2003.

Natural Science Foundation of China

Project 70271063 supported by the National

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Gong & Peng: THE EXISTENCE THEOREM OF OPTIMAL GROWTH MODEL

31

perfect capital market, and to have perfect foresight. In this environment, he is postulated to choose his private consumption, c(t), capital stock, k(t), to maximize his discounted utility, namely, (P)

subject to

dk(t)

cit

max

c(t),k(t) i o

roo

u(c(t))e-13tdt

= ext f(k(t)) - nk(t) - c(t)

(1.1)

with the initial capital stock k(O) given. Where (3, n, and X are positive constants, they represent the discounted rate, population growth rate, and technology progress, respectively. Equation (1.1) asserts that the output is used to consume and investment. The instantaneous utility function u(c(t)) has the following properties: Al For any Clb(t) < c(t) < 00, u"'(c(t)) is continuous, u'(c(t)) > O,u"(c(t)) < O,u"'(c(t)) > 0, and a'(c(t)) 2: 0, where a(c(t)) = C(~f(~gS)) and 0 ~ Clb(t) =const. Assumption Al states that the representative agent is assumed to derive positive, but diminishing marginal utility from consumption goods. u"'(c(t)) > 0 shows that the consumer is prudent (Kimball 1990). The condition Clb(t) < c(t) confirms that the consumption level must be larger than a given level, and we assume this level is increasing with time. a(c(t)) is the coefficient of relative risk aversion, and it also represents the substituting elasticity of intertemporal consumption; a' (c(t)) 2: 0 requires that the coefficient of relative risk aversion is an increasing function of consumption, this condition is also applied by Gollier (2001). Per-capita output y(t) is produced by a neoclassical production function exhibiting positive, but diminishing marginal productivity in the factor of production; that is y(t) = ext f(k( t)), where k(t) is the capital stock at time t, and it satisfies A2 X 2: 0, f(k(t)) is twice continuous differentiable, and satisfies f(O) = 0, f'(k(t)) > 0, f"(k(t)) < O. Furthermore, we suppose d(1n x)

(1.2)

c(x) = d(ln f(x)/ f'(x) _ x) < 1.

Condition (1.2) is to eliminate the possibility of super-exponential growth, i.e., of growth paths satisfying lim k(t)e- gt = +00 for any finite constant g > O. Under assumptions Al and t-+oo A2, we have some basic properties: Lemma 1 Under assumption AI, condition (1.2) implies a'(x) ~ 0, where a'(x) = xJ;~)). Lemma 2 Under assumptions Al and A2, an indefinitely constant and positive saving · . (t) = nk(t)+k(t) po IICY, i.e., S y(t) = -S E (0 , 1]' , Imp I'ies t h at lim k(t) = lim iJ(t) = lim c(t) = _X_ 1-+00

where a =

lim

k(t)-+oo

k(t)

1-+00

y(t)

1-+00

c(t)

1-

a

a(k(t)).

In order to derive the existence theorem for the optimization problem, we transfer the optimization problem (P) to a variational problem

maxI(k(t))'g" g k(t)

ioroo u(e

xt f(k(t)) - nk(t) - k(t))e- i3tdt

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The admissible and feasible paths for the above equation can be defined as Definition 1

stants

Cs

> 0,

~

Given a growth model satisfying assumptions Al and A2, and given con-

2: 0, ko 2: km > 0, a path k(t) is said to be indefinitely admissible if

1. k(t) is absolutely continuous on 0 ::::: t ::::: 00; 2. k(O)

= ko, k(t) 2: km , 0 < t < 00;

3. For any t2 > tl 2: 0, k(t) satisfies -~

12 1

k(t)dt ::::: k(t2) - k(td :::::

112

I,

(y(t) - nk(t) - cs)dt

1,

The set of all indefinitely admissible k(t) is denoted by [k(t)O'J. Furthermore, k(t) is said to be indefinitely feasible (we denote the set of all indefinitely feasible k(t) as [k(t)O'J*) if the condition 1 is replaced by 1'. .k( t) is piece-wise continuous on 0 ~ t ~ 00; The existence of optimal path for the optimal problem (1.1) is to find a path in [k(t)O'J* to maximize the above optimization problem, we present the main result of this paper: Theorem 0 1.

There exists an optimal path k(t) E [k(t)O'J* if

fJ > max{O; (1 -

V)(glb - n)}, where v = -

lim

c(t)--+oo

O"(c(t));

> cuTheorem 0 presents the existence of optimal capital accumulation path and consumption path under the discount rate constraint (condition 1) and the initial consumption level constraint (condition 2). Condition 1 of Theorem 0 ensures the convergence of I(k(t))O' in [k(t)O'J*. Condition 2 of Theorem 0 ensures that [k(t)O'J* is not empty. From Theorem 0, we also know that the capital accumulation path and consumption are piece-wise continuous. We will prove Theorem 0 by two steps, first, we prove the optimal path is absolutely continuous (i.e., k(t) E [k(t)O'D in Section 3; second, we prove the path is piece-wise continuous (i.e., 2. j(k o) - nk o 2: c,

k(t) E [k(t)O'J*) in Section 4.

3

Existence of an Absolutely Continuous Optimal Path

In this section, we prove the existence properties of an absolutely continuous optimal path for the optimal problem (1.1), i.e., Theorem 1 Under the same conditions with Theorem 0, there exists an optimal growth path k(t) E [k(t)O'J. Proof It is sufficient that the conditions 1-7 of Theorem A in the appendix are fulfilled by [k(t)O'J and I(k(t))O'. 1. [k(t)O'J is not empty since it obviously contains k(t) = ko; 2. The equi-continuity of the path k(t) on every finite interval 0 ::::: t ::::: T follows from the Lipschitz condition Ik(t 2) - k(tdl ::::: max{~ks(T),

eAT j(ks(T))}T

(2.1)

where ks(t) is the solution of the following differential equation

dk(t) - = e' I j(k ( t)) dt

- nk(t)

(2.2)

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Gong & Peng: THE EXISTENCE THEOREM OF OPTIMAL GROWTH MODEL

33

with the initial condition k(O) = ko; 3. The Lipschitz condition ensures that every limit curve k1(t) of [k(t)O"J is continuous. It follows that there is an 1] for every interval

h StS

t 2 which satisfies

(2.3) Now, if 1] £5(2

> 0, this leads, for any k(t), which lies in the neighborhood £5 of k1(t), 1] >

+ ~(t2 - td),

to the absurd conclusion that 1]

S k(t 2 )

-

k(td - (k[(t 2 )

k[(td)

-

+~

1

t2

(k(t) - k[(t))dt

1,

S £5(2 + ~(h - h)) < 1]. Hence, Condition 3 must hold true.

= u(c(t))e- f3t = u(ex t f(k(t))

4. The proof of W(t, k(t), k(t))

(2.4)

- nk(t) - k(t))e- f3t being

L-integrable along k(t) E [k(t)O"J is trivial. As for the proof of condition 5 and condition 6 in Theorem A, we only need to consider . the case

u(c(t)) = +00.

(2.5)

100(c(t))1 = v S 1,

(2.6)

lim

c(t)--+oo

First, the condition (2.5) implies that lim

c(t)--+oo

Otherwise, if v

= 1 + 21] > 1, then we have (c(t)l+1J u'(c(t)))'

= c(t)1Ju'(c(t)) (O'(c(t)) + 1 + 1]),

(2.7)

where lim (O'(c(t))

c(t)--+oo

+ 1 + 1])

(2.8)

= -1].

Hence, we get

(c(t)l+1J u'(c(t)))' < 0, for some

CE

> 0.

c(t)::::

(2.9)

CE

This is contrary to equation (2.5) by Weierstrass' test for the convergence of

improper integrals. Now v = 1, assumption A2 implies for c;

(c(t)u'(c(t)))'

S c(t) < 00,

= u'(c)(O' + 1) SO

(2.10)

v S 1 implies that for some c* :::: c" c* S c(t) < 00,

(c(t)u'(c(t)))' = u'(c(t))(O'

+ 1) > 0.

(2.11)

Hence, we have lim

C(I)--+DO

c(t)u'(c(t))

= const

or

lim

c(t.)--+oo

c(t)u'(c(t))

= +00.

(2.12)

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ACTA MATHEMATICA SCIENTIA

So we have lim c(t)u'(c(t)) C(t)4oo u(c(t))

=0

or

lim c(t)u'(c(t)) C(t)4oo u(c(t))

=1-

v.

(2.13)

+ ~ks(t))e-l3tdt

(2.14)

It follows that for every k(t) E [k(t)oJ, we have

1

00

u(ext f(k(t)) - nk(t) - k(t))e-l1tdt <

1

00

u(ext f(ks(t))

where the second integral converges because lim i.ln[u(e xt f(ks(t)) dt

t400

+ ~ks(t))e-l3teJ

= (1 - V)(glb - n) - (3

<0

(2.15)

where we have used equation (2.13) and the condition 1 of Theorem 0, I(k(t)o) is therefore upper-bounded on [k(t)oJ. Moreover, I(k(t)o) is uniformly convergent on [k(t)oJ. Indeed, whatever k(t) E [k(t)oJ, we have (2.16)

Now the right side of the above equation can be made smaller than any pre-assigned positive number by choosing the time horizon large enough. Finally, we prove the condition 7 of Theorem A is satisfied, namely, we must prove the upper-semicontinuity of I(k(t)6) on [k(t)ifJ for every T > O. Let k*(t) E [k(t)ifJ and 7r(t) be a polygonal curve inscribed in [k(t)ifJ (end-points coinciding with the initial condition and the end point, vertices on [k(t)if]). So we have a. ji- is continuous on 0 ::; t ::; T; b. Given an E E (0, c*(t) - cs), one has 17r(t) - k*(t)1 and over an interval of measure less than

li(t) - k*(t)1 <

< E for all 0::; t ::; T,

(2.17)

E,

E

for any t,

(2.18)

0::; t ::; T.

It follows that, except possibly over an interval of measure less than max{ks(T), -~ks(T) + E}, for all 0 ::; t ::; T, and

E,

both

Iii <

M

=

E(t, k*(t); i(t), k*(t))

~ W(t, k*(t), k*(t)) - W(t, k*(t), i(t)) - (k*(t) - i(t))Wj,(t, k*(t), i(t))

> {H'dt, k*(t), k*(t)) - Wk,(t, k*(t), i(t))}(k*(t) - i(t)) > -2EU'(C s )

(2.19)

Moreover, wherever it exists, the Weierstrass function E is negative since u( c( t)) is strictly concave. Hence

1 T

(E(t, k(t); i(t), k(t)) - E(t, k*(t); i(t), k*(t))dt < -2EU'(C s)(T

+ 2111).

(2.20)

No.1

Gong & Peng: THE EXISTENCE THEOREM OF OPTIMAL GROWTH MODEL

It follows that

II(k(t)6) - I(k*(t)6)1 < 2w'(c s)(T + 2M)

+

l

35

T

(W(t, k(t), ir(t))

+(k(t) - ir(t))Wk(t, k(t), ir(t)))dt

-I

T

(H! (t , k* (t ), ir(t )) + (k*(t) - ir(t))Wk(t,k*(t),ir(t)))dt.

The continuity of JoT (H!(t, k*(t), ir(t)) + (k*(t) - ir(t))Wk(t, k*(t), ir(t)))dt is obvious. So, the conditions 1-7 of Theorem A are satisfied, thus, there exists an optimal path

k(t) E [k(t)O'].

4

Piece-Wise Continuous Differentiability of the Optimal Path

In this section, we will prove the optimal conditions for the optimal path derived from Theorem 1, that is, Theorem 2 Under the assumptions of Theorem 0, let k(t) be optimal in a given set [k(t)O'J, then, for every given w > there exists <5 > such that one of the following conditions holds over w ::::: t ::::: w + <5,

°

°

~~:~ O"(c(t)) + ext f'(k(t)) c(t) = cs ,

(3.2)

k(t) = -~k(t),

(3.3)

k(t) Moreover, if w w - <5*

> 0, then there exists <5* >

< t < w.

(3.1)

(3 - n = 0,

°

= k rn .

(3.4)

such that one of the above conditions holds over

In order to prove Theorem 2, it will be convenient to write ks(t; w, a) as the solution of the following equation dk(t) = ext f(k(t)) - nk(t) - Cs ; k(w) = a, dt

w::::: t

< 00.

and kd(t; w, a) as the solution of the following equation

dk(t)

~

= -~k(t); k(w) = a,

w::::: t <

00

over its interval of existence beyond w. Let us further denote by k(t; w, a, b) the solution of the Euler equation

2(t) (~()) +e \tf'(~()) c(t)O"ct kt - (3 -n-O,

k(w)

= a, k(w) = b

over its interval of existence in the open region

R

= {(t, R)IO < t < 00,0 < k < 00, -00 < k(t) < ext f(k(t))

- nk(t) - Clb(t)}.

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We have the following Lemmas (we will prove them in Appendix B): Lemma 3 Let k(t; w, a, b) be defined as above, then it is uniquely determined over its interval of existence in the region R. Moreover, both k(t; w, a, b) and k(t; w, a, b) are continuous. Lemma 4

If k*(t) E [k(t)go] is absolutely continuous over 0

< w ::; t ::; 0, then

e- Bw {I[k(t)~] - I[k*(t)~]} 11 d d < w (k(t) - k*(t))(dk - dt ak)(u'(c*(t)) - (k(t) - k*(t))u'(c*(t))e-6tl~·

a

1

Lemma 5 If k1(t) is continuous over w < t < w + 6 and k(t;w,k1(w),bd < k1(t) < k(t;w,k1(w),b2 ) holds over w < t < w + 61 for some b1 and b2 , then there exists b* E (b1,b2 ) and w* E (w,w + 6) such that k(w*; w, k1(w), b*) = k1(w*). Lemma 6 Let k(t) E [k(t)go] be the optimal solution, then there is no w 2: 0 and 6 > 0 such that k(t; w, k(w), ks(w;w, k(w))) < k(t) < ks(t;w, k(w))

or

kd(t; w, k(w)) < k(t) < k(t; w, k(w), kd(w; W, k(w))) for w < t < w + 6. Lemma 7 Let k(t) E [k(t)go] be the optimal solution. If

k(t;w,k(w),ks(w;w,k(w))) > ks(t;w,k(w)) > k(t) > k(t;w,k(w),ks(w;w,k(w)) -E) for the given w 2: 0 and any

E

> O,W < t < w + 6, then k(w) = ks(w;w, k(w)).

If

k(t;w, k(w), kd(w; W, k(w))) < kd(t;w, k(w)) < k(t) < k(t; w, k(w), kd(WiW, k(w)) - E) for the given w 2: 0 and any E > O,W < t < w + 6, then k(w) = kd(w;w,k(w)). Proof of Theorem 2 Now, we begin to prove Theorem 2. We consider two cases,

k(w) > km and k(w) = kmCase 1 k(w) > k m. In this case, there must be a 61 > 0 such that k(t) satisfies either one of the following three conditions over w ::; t ::; w + 61:

= ks(t,w,k(w)),

(3.8)

k(t) = kd(t,w,k(w)),

(3.9)

kd(t,w,k(w)) < k(t) < ks(t,w,k(w)).

(3.10)

k(t)

Indeed, by virtue of the definition of[k(t)ifL k(Wl) > kd(Wl, W, k(w)) and k(wd < ks(Wl'W, k(w)), > w, imply respectively k(t) > kd(t,w, k(w)) and k(t) < ks(t,w, k(w)), t> WI· To prove Theorem 2 for k(w) > h,« over w ::; t ::; w + 6, it is therefore sufficient to show that equation (3.10) implies (3.1). For this purpose, we have to consider five different alternatives separately. Alternative 1 is that there is an k(t;w,k(w),bd and an k(t;w,k(w),b 2 ) , kd(w, w, k(w)) ::; bj ::; b2 < ks(w, w, k(w)), which are both feasible over some open interval starting at w and satisfy

Wj

(3.11)

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Gong & Peng: THE EXISTENCE THEOREM OF OPTIMAL GROWTH MODEL

No.1

for some 62 > O. Now, by Lemma 4 and Lemma 5, we know this implies (3.1) for some 6. If alternative 1 does not hold, then either

k(t) < k(t;w,k(w),b);

w < t::; w + 63

(3.12)

k(t) > k(t;w,k(w),b);

w<

t::; w + 63

(3.13)

or for b satisfies (3.14) and some 63 , which may depend on b. Indeed, Lemma 3 and Lemma 4 imply that

k( t) cannot be

both optimal and cross k(t; w,k(w), b) at some W2 > w unless k(t) = k(t; w, k(w), b); w ::; t ::; W2. Now, each of the two alternatives (3.12) and (3.13) gives rise to two further alternatives. We are thus left four alternatives, which, together with alternative 1, constitute the five alternatives we have to consider. Let us first consider the two alternatives, say, alternative 2 and alternative 3, associated with (3.13). Alternative 2 is that, over some open interval starting at w,

k(t; w, k(w), ks(w;w, k(w))) < k(t) < ks(w; w, k(w))

(3.15)

Now by Lemma 6, (3.15) is contrary to the optimality of k(t). Alternative 3 is that

k(t,w; k(w), ks(w;w,k(w))) > ks(w; w, k(w))

(3.16)

over some interval starting at w. From equation (3.13) and Lemma 3, equation (3.16) implies the existence of a ()

<

ks(w;w, k(w)) and we> w such that k(t;w,k(w),()) < ks(w;w,k(w)),w < t < we

(3.17)

k(we,w;k(w),()) > ks(we;w,k(w)).

(3.18)

and These, in turn, imply that there is an

Wb

for every b satisfying (3.19)

such that both (3.17) and (3.18) hold if f3 is everywhere replaced by b. It follows that k(t;w,k(w),b), where b satisfies (3.19), crosses k(t) at some point of time

wI;, w < wI; <

Now if there were an k(t;w,k(w),b) admissible over w < t < Wb, where b satisfies (3.19), then k(t) could not be optimal unless k(t) = k(t; w, k(w), b). This would, however, be contrary to our assumption that alternative 1 does not hold. Hence, there is no Wb.

feasible k(t; w, k(w), b) over w

< t < Wb. By Lemma 7, this implies that k(w+)

= ks(w+;w,k(w))

(3.20)

It follows from Lemma 3 that over some interval starting at w,

- ~!-) {v.(c)e- Bt } < 0 (~ dk dt ok

(3.21)

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c

where is defined by ks(t) = ks(t;w,k(w)). This implies that there exists an e" > 0 such that (3.1) remains true over some interval w < t < Ws' If ks(t) is replaced by a path, say ksElwhich satisfies

dk(t)

.

~

~ = eX,t f(k(t)) - nk(t) - c, - e; kSE(w) ~ k(w), 0


:::; f

*

,w :::; t

< 00

then we may choose an E E (0, f*] such that kSf(t) crosses k(t) for the first time after w at a time WE < WE" By Lemma 3 and (3.20), we know that kSf(t) is better than k(t) over w < t < WE' Next, we consider two alternatives, say, alternative 4 and alternative 5, which are associated with (3.12). Alternative 4 is the counterpart of alternative 2 and, as the latter, is excluded by Lemma 6. It is (3.22) ks(t;w,k(w)) < k(t) < k(t,w;k(w),kd(w,w,k(w))) over some open interval starting at w. Alternative 5 is the counterpart of alternative 3. It is

< kd(w;w, k(w))

k(t; w, k(W),kd(W;W, k(w)))

(3.23)

over some open interval starting at w. This alternative is dealt with in a perfectly similar way to alternative 3. Case 2 k(w) = km. In this case, either there is, for every J > 0, t 1 , t z, and w < t 1 < tz < w + J such that (3.24) or one of the following three conditions holds over R:

km

k(t) = ks(t; w, k(w)),

(3.25)

k(t) = k m

(3.26)

,

< k(t) < ks(t;w,k(w)).

(3.27)

In order to prove Theorem 2, it is sufficient to exclude the alternative associated with (3.24). Indeed, the three alternatives (3.25), (3.26), (3.27) can be formally considered as respectively identical to (2.8), (2.9), and (2.10) with ~ = 0 and k(t) = km. Hence, the same reason applies here as in the case ofk(t) > k m , provided the alternative associated with (3.24) can be excluded. Now, in this last alternative, there must be a Jo such that (3.28) (3.29) Otherwise, one would have (3.30) along k(t) = k for some J 4 > O. By Lemma 4, this would be contrary to the optimality of k(t). From here on, we proceed formally as in the case of alternative 5 by setting ~ = 0 and k(w) = km. The proof of w - J* :::; t :::; w is perfectly similar. Theorem 0 is obvious from Theorem 1 and Theorem 2.

No.1

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Gong & Peng: THE EXISTENCE THEOREM OF OPTIMAL GROWTH MODEL

39

Conclusion

This paper presents an existence theorem for the optimal growth model, which is not restricted to the case of a constant technology progress and can be easily to interpret and apply. The two conditions are the discount rate constraint and consumption level constraint, respectively. Further researches would extend this existence theorem to the optimal growth model with the endogenous discounting function, such as: the Uzawa's endogenous time preference, Laibson's hyperbolic discounting function (Laibson 1997); Becker-Mulligan's endogenous time preference (Becker and Mulligan 1997) et al. References 1 Barro R J, Sala-I-Martin X. Economic growth. New York: McGraw-Hill Inc, 1995 2 Becker G S, Mulligan K C. The endogenous determination of time preference. Quarterly Journal of Economics, 1997, August: 729-758 3 Cass D. Optimum growth in an aggregative model of capital accumulation. Review of Economic Studies, 1965, 31: 233-240 4 Fleming W H, Rishel R W. Deterministic and stochastic optimal control. New York: Springer-Verlag Inc, 1975 5 Gollier C. The economics of risk and time. MIT Press, 2001 6 Gong Liutang, Fei Pusheng. Local an global stability for the infinite-horizon variation problems. Acta Mathematica Scientia, 1998, 18(3): 278-284 7 Gong Liutang. Applications of Hamiltonian and Laplace transform in an economic growth model. Acta Mathematica Scientia, 2000, 20B(4): 442-450 8 Kemien M I, Schwartz N L. Dynamic optimization. Elsevier Science Publishing Co Inc, 1991 9 Kimball M S. Precautionary savings in the small and in the large. Econometrica, 1990, 58: 53-73 10 Koopmans T C. On the concept of optimal economic growth. In: The Econometric Approach to Development Planning. Amsterdam: North Holland, 1965 11 Laibson D. Golden eggs and hyperbolic discounting. Quarterly Journal of Economics, 1997, 112: 443-77 12 Pontryagin L S. Ordinary differential equations. Reading, Mass: Addison-Wesley, 1962 13 Ramsey F. A contribution to the theory of taxation. Economic Journal, 1928,31: 47-61 14 Sargent M T. Recursive macroeconomic theory. MIT Press, 2000

Appendix A: Existence Theorem for the Variational Problem The following theorem presents the existence properties for the variational problem, which is proved by Tonell-Cinquini methods (Fleming and Rishel (1975)). Theorem A If the following conditions are satisfied 1. [x( t)O"J is a non-empty set offunctions x( t), which satisfy the initial condition x(O) 2. Every x(t) E [x(t)O"J is absolutely equi-continuous over every finite interval;

= Xo;

3. [x(t)O"J is closed in the finite; 4. W(t, x(t), x(t)) is an L-integrable function along all x(t) E [x(t)O"J over every finite interval; oo 5. J(x(t)O") = Jo W(t, x(t), x(t))dt is upper-bounded on [x(t)O"J; 6. J(J:(t)O") is uniformly convergent on [x(t)O"J; 7. J(x(t)'{;) = J;r TV(t, x(t), :i:(t))dt is upper semi-continuous on the set [x(t)O"J of all x(t) E [x(t)O"J truncated at time t = T; TheIL there is an x(t) E [x(t)O"J such that it maximizes J(x(t)O"). Appendix B: Proofs of Lemma 3-Lemma 7

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ACTA MATHEMATICA SCIENTIA

Vol.25 Ser.B

Proof of Lemma 3 The lemma follows from the fact that the Euler equation has the form k(t) = h(t, k(t), k(t)) , where function h(t, k(t), k(t)) is continuous differentiable over R. Proof of Lemma 4 By virtue of the strict concavity of the utility function and produc-

tion function, the continuity of u"(c(t)) and f'(k(t)), the absolute continuity of k(t) and k(t) over 0 < w :::; t :::; 11, we have

i ill _ill ill Il

<

<

(u(c(t)) - u(c*(t)))e- i3tdt <

i

Il

u'(c*(t))(c(t) - c*(t))e- i3tdt

u'(c*(t))(eyt(f(k(t)) - f(k*(t))) - n(k(t) - k*(t)))e- i3tdt u'(c*(t))(k(t) - k*(t))e- i3tdt

u'(c*(t))(e yt f'(k*(t)) - (n + (3) + a(c*) ~~:~ )e- i3tdt

-(k(t) - k*(t))u'(c*(t))e-i3tl~ This is just what we need in Lemma 4. Proof of Lemma 5 is obvious. Proof of Lemma 6 We consider the first part. From Lemma 3 and the fact c(t) ~ Cs > cis, k(t; W, k(w), ks(w; w, k(w)) + E) exists over w < t < w + 0 and is greater than ks(t; w, k(w)), provided only that 0 < E < C s - Clb. Hence, by Lemma 5, the first part of Lemma 6 implies there is an E* E (0, C s - Clb) such that

ks(w + o~, w, k(w)) = k(w + o~; w, k(w), ks(w; w, k(w)) + E) for any

E

E (0, E*) and some

Ow > o~ > O. By Lemma 4, this implies that k(t) can be optimal if

eyt f'(ks(t,w, k(w))) - (n

+ (3) > 0,

w

< t < w*

for some w* < w. It follows that there is an E such that

_ {ks(t,w'k(W))' w:::;t:::;w+o~, k(t) = _ ~. ~ , k(t; w, k(w), ks(w; w, k(w)) + E), W+ of:::; t :::; w + Of is feasible and better than k(t) over w :::; t :::; w + Of. Proof of Lemma 7 We consider the first case, the second case is similar to prove. There is an E such that

· ( ) k(w,) - k(w) ks(w,;w,k(w)) - ks(w;w,k(w)) < ----'----'-----'----'-----'------''-'-kIt, E < w, - w

for 0 <

E :::;

E. Now, as

E

w, - w

tends to zero, we get the conclusion.