Optimal growth and uncertainty: The borrowing models

JOURNAL

OF ECONOMIC

THEORY

24, 168-186

(1981)

Optimal Growth and Uncertainty: The Borrowing Models* L. C. MACLEAN Faculty

of Business Administration, Halifax, Nova Scotia B3H

Dalhousie University, 4H8, Canada

AND

C. A. FIELD AND W. R. S. SUTHERLAND Mathematics Halifax, Received

Department, Nova Scotia

September

Dalhousie B3H 4H8,

21. 1978;

revised

University, Canada

September

4, 1979

1. INTRODUCTION AND SUMMARY

The theory of optimal economic growth is concerned with the problem of how to best allocate some current stock of capital between capital for immediate consumption and capital for future investment, This allocation is normally required to satisfy a strong type of budget constraint at each point in time; deficits are not allowed and surpluses cannot be retained for future use. In this paper we examine the same allocation question under a weaker type of constraint. Deficits and surpluses will now be carried forward in time, but only the total deficit over the planning horizon will be constrained. A decision to incur a deficit (surplus) will be called a borrowing (lending) activity. A similar concept of borrowing appears in the planning models which arise in the theory of the consumer. There an individual must decide between consumption and investment at each point of time in his planning horizon or lifetime. Borrowing, as well as being an economic acitivity, also serves as a device for introducing a concept of life insurance. Yaari [lo] discusses the qualitative differences between consumer models with and without insurance. Hakansson [ 141 gives explicit optimal policies for a specific class of utility functions. * This Canada.

research

was

supported

in part

by grants

168 0022.0531/81/020168-19$02.00/O Copynghf ‘c_ 1981 by Academic Press. Inc. All rights of reproduction in any form reserved.

from

the

National

Research

Council

of

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Returning to the economic growth literature, our starting point is the onesector nonlinear deterministic production model of Koopmans [5], Cass [2] and Gale [3]. The existence of optimal investment and consumption policies and their qualitative properties are well known. The turnpike (or golden-rule) phenomena of long-run investment being the most celebrated results. Brock and Mirman [l] extended this theory to the case of a stochastic production process and showed that there exists a stationary distribution (over the set of turnpike states) which is approached by long-run optimal investment policies. In Section 2, the borrowing activity is introduced in a discrete-time linitehorizon economic growth model with deterministic production process. The production process is then extended to the case of random or uncertain outputs. The activity of intertemporal borrowing (or lending) of capital involves a fixed rate of interest but is otherwise unrestricted and costless. Capital is borrowed from (or lent to) some unspecified exogeneous source. Two versions of the model are considered. If only the average (or expected) net borrowing over the planning horizon must be repaid, we have the optimistic model (of Section 3). However, if the actual net borrowing must be repaid, we have the pessimistic model (of Section 4). Roughly speaking, these two versions of the borrowing model correspond to versions of Yaari’s consumer model depending on whether life insurance is available or not. However, unlike these consumer models, we do not postulate a specific mechanism for insurance. Our exogeneous source is unspecified. It faces no risks in a pessimistic model and acts as the holder of any final deficits or surpluses in an optimistic model. The latter will be in balance if it were to serve as an exchange bank for a large number of identical economies. An alternative justification for using such an exogeneous source can be based on the idea of “rolling plans.” Here an economy revises its policies after each time period while advancing its planning horizon ahead by one period. Using an optimistic policy, the economy always expects to repay its debts, the exogeneous source always expects to have a zero net balance, and the day of final reckoning is ever deferred. Sections 3 and 4 describe the optimal investment and consumption policies for the optimistic and pessimistic models. Not surprizingly, the optimistic model is the simpler version. Its decisions on investment versus consumption are deterministic with borrowing absorbing all the stochastic behaviour. Section 5 presents explicit formulas for the consumption policies in the two versions for the special case of a quadratic utility function.

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2. THE DETERMINISTIC

SUTHERLAND

BORROWING

MODEL

Gale [3] considers a discrete-time one-sector economic growth model. There is a single good (capital) which can either be invested or consumed. An investment x, in time period t produces an output f(x,) in period t + 1. Consumption c1 yields utility u(c,) during period t. The initial capital stock is y,, the planning horizon T periods, the final stock is JJ~. An optimal policy is a sequence (xt, c,) of (non-negative) investments and consumptions which satisfy the budget constraints and yield a maximum for the sum of the discounted utilities. We now introduce a third activity, z~, which represents borrowed capital (if negative) or repayment of capital (if positive) with respect to some unspecified exogeneous stock. Interest on such activities is represented by a factor p > 0; discounting by 6 > 0. This modification yields the following borrowing model (which reduces to Gale’s model if all zt = 0): Find x, > 0, c, > 0 and z1 which maximize Ci: r P’u(c,) XI

+

x1 +

subject to the budget constraints

Cl +

ZI GYOY

c, +

J’T +’

Tt

z, a-(x,-

I>,

t = 2,..., T,

1
and the repayment constraint rt1 \‘ z,

P f- ‘Zf > 0.

The properties of this and related deterministic borrowing models are given elsewhere (see [8]). Our present purpose is to consider the case of a stochastic production process. The single production function f is now replaced by a family f,, where o E 0 and (L!, 9,s) is a probability space. Thus the output Y(w) = F(x) =f,,( x ) is a random variable, for each fixed input x. If the input is a random variable X, then Y = F(X) is a random variable on the probability space (a x Q, 9 x 9, 9 X 9) with Y(% 7%> =f,*P3~J)~ The stochastic production process is assumed to be stationary in that the same probability space is used in all time periods. We also assume that, for each w E Q, f,(x) is an increasing concave function such that f,(O) = 0 and pf,(x) -x 0 for some bound M. The utility function u(c) is also assumed to be an increasing, strictly concave function for c > 0. For notational convenience the same wT+ I E Q is used for both the final

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period of the production process and the final stocks Y,. As usual E(W) represents the expected value of a random variable W.

3. THE OPTIMISTIC

BORROWING

MODEL

When the production process is stochastic the activities x,, c,, z1 become random variables with respect to the product probability space (Q’, 9’, 9’). Letting 13’ = (w, ,..., wJ, these variables are denoted by X,(w,,..., OJ = X,(w’), C,(w, ,..., cu,) = C,(w’), Z,(u, ,..., ot) = Z,(d) or simply as X,, C,, Z,. The production process is denoted by F’(X) =f,(X). The optimistic borrowing model is defined by the following problem: Find random variables X((w’) > 0, C,(o’) 2 0 and Z,(w’) which maximize CT=, 6’-‘E(u(C,)) subject to the budget constraints

x1+c,+z,
for

t = 2,..., T,

Y, + z r+ 1 G W-Th

and the repayment constraint Ffl

x p’-‘E(Z,) t=1

> 0.

All constraints are required to hold pointwise. That is, the typical budget constraint is

X,(4 + wJ> + ztw ct,,(x,-l(~‘-lN for all wt E 0’. Note that the initial and final stocks are considered to be random variables, YO(u’) and Y,(w” I). A sequence (X,, C,, Z,) which satisfies all of the constraints (for all w*+’ E QF+l) is said to be a feasible T-period policy from Y, to Y7. Since Z,(o,) > 0 is interpreted as a repayment (Z,(o,) < 0 being borrowing), then the repayment constraint requires only that the expected total borrowing be repaid. The pessimistic model (discussed in Section 4) differs from this model only in requiring that the actual total borrowing be repaid. The effect of introducing a borrowing activity in a stochastic economic growth model is to relax the budget constraints and hence yield optimal policies which are more or less insensitive to the randomness inherent in the production process.

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Let f(x) = E(F(x)) denote the expected (or average) production function. Our first result shows that the optimal investment and consumption policies are deterministic in the optimistic model. Moreover, they depend only on the average production function & the expected initial and final stocks E(Y,), E( YT) and the factors 6 and p. THEOREM 1. rf (x,, c,, ZJ is an optimal T-period policy from y, = E( Y,) to yT = E( Y,) in the deterministic borrowing model with production function f = E(F), then (q, c,, 2,) is an optimal T-period policy from Y, to Y, for the optimistic model, where

Z,(d) = Y&o’) - x, - c,, zt (4 = f&*z,+l(WT+l)

=f,,+,(x,>

1)- x, - c, - Y&J+‘).

for

t = 2,..., T, .

Prooj Let (xl, c,, zt) be any feasible policy for the deterministic model and Z, as defined, then it is easily verified that (xt, c,, Z,) is feasible for the optimistic model. Conversely, if (X,, C,, Z,) is feasible for the optimistic model from Y, to Y, then x, = E(X,), c1= E(C,), zt = E(Z,) is feasible for the deterministic model from y, to y,. This follows from the concavity of F and Jensen’s inequality as E(X,) + E(C,) + E(Z,) < E(F(X,- ,)) <.f(E(X, _ ,)); also WY,)

+ EV,,

J < WV,))


Now let (X,, C,, Z,> be any

feasible policy from Y, to Y,, then applying Jensen’s inequality concave function u, we obtain \T 6’-‘E(u(C,)) z-1

< 2 a’-‘u(E(C,)) I=1

since (xt, c,, z,) is the optimal proof. I

deterministic

to the

,< f Plu(c,) I=1

policy, which completes the

Since the optimistic model has optimal policies which are obtainable from a deterministic model, then the qualitative behaviour of these policies can be derived by well-known arguments. Some of the main results are stated below without proof. The interested reader can refer to [7] for details or observe that if the budget constraints are used to eliminate the borrowing variables, then the deterministic model can be reduced to a liner production model so that these results follow from those of McFadden [9]. Next let 2 maximize (pE(F(x)) -x) subject to x > 0. Assume X is unique and let jj = E(F(2)). The special notation p(r) = C;=, p’-‘, where p is a scalar, will be useful here and again in Section 5.

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UNCERTAINTY

THEOREM 2. The optimistic model has an optimal policy if and only if E(K) > 0, where

K(wr+‘)=

Y&J,)+

5 p’-l[pf,,+,(.f)-.q

-prY,(oT+l)

f=l

so that

THEOREM 3. Assume E(K) > 0. A feasible policy (X,, C,, Z,) is optimal for the optimistic model if and only if there exist (nonrandom) competitive prices pt > 0 (for t = I,..., T+ 1) andq>O such that

I

c, maximizes (6’-‘E(u(C))

- ptE(C)}

over all C(o’) > 0 for t =

1 ,..., T; (‘I

(ii)

X, maximizes ( pt+ , E(F(X)) - ptE(X)}

over all X(w’) > 0 for t =

1... .. T: maximizes 1 ,..., T (iii)+

{@‘-‘q - pt) E(Z)}

over

all

Z(d)

for

t=

1.zt

THEOREM 4. Let (X,, C,, Z,) be an optimal policy for the optimistic model. Then investment is stationary, X, = R Moreover, consumption and borrowing are monotone;

E(Ct)>E(Ct+,) WJ


for

t = I,..., T -

1,

for

t = 2,..., T-

1,

when p > 6 (the opposite inequalities holding when p < 6 and equality holding when p = 6). COROLLARY.

if p = 6 then the optimal poiicy has the explicit form C, = F z,=

for

t=

Y&o’)-2-F

l,...,

T,

for

t=

=f”,(+GE

for

t = 2,..., T,

= J-

for

t=T+

Yr(d)

where c=(y-.Q+((E(Y,,)-.?)+p’(Y-E(Y,>))/p(T).

1,

1,

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4. THE PESSIMISTIC BORROWING

MODEL

We now turn to the second version of the stochastic economic growth model with borrowing. Here the total borrowing must be repaid in every realization of the stochastic process. This change in the model results in a more complicated behaviour for the optimal policies. The pessimistic borrowing model is defined by the following problem: Find random variables X,(w’) > 0, C,(o’) > 0 and Z,(o’) which maximize CT= r 6’-‘E(u(C,)) subject to the budget constraints x, t c, -I- z, < y,, for

X,tC,+Z,
t = 2,..., T,

YT + ZT, 1 G WT), and the repayment constraint T+l ‘-12,

x:p f=l

>

0.

Again the constraints must hold pointwise (with respect to O’ E fz’). If for each gT+’ the corresponding deterministic borrowing model is feasible, then there exists a feasible policy for the pessimistic model. The existence of optimal policies follows from familiar arguments in dynamic programming. We next turn to the duality theory for the pessimistic model. DEFINITION. A feasible T-period policy (X,, C,, Z,) from Y, to Y, in the pessimistic model is competitive at prices P,(o’) > 0 (for t = l,..., T t 1) and T+l) > 0 if Q<@

(i) C, maximizes {8’-‘E(u(C))‘E(P,C)} over all C(0’) > 0, for t = 1‘..., T; (ii) X, maximizes (E(P,+,F(X)) -E(P,X)} over all X(w’) > 0, for t = l,..., T; (iii) Z, maximizes {E(E,-,+,@‘-‘Q - P,)Z)} over all Z(w’), for t = 1,..., T t 1; where the partial expectation ET-,+, E T--t+,(q

=I

w(wI,...,

is defined (for t = l,..., T + 1) by

WT+l)dgT-‘+‘(WI+,,“‘,

WT+l)’

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ANDUNCERTAINTY

These competitive conditions are similar to those for the optimistic model except that the prices are now random variables. Note that the partial expectation is an averaging over the randomness of a variable W(mr+ ‘) for only the remaining periods after time t. In particular, if t = T + 1 then E,(W) = W and if we set t = 0, then E,, ,(I+‘) = E(W). The competitive conditions have a simple economic interpretation. Consumers maximize their utility subject to a budget constraint, producers maximize their profits and borrowers minimize their anticipated costs on their debts. In order to circumvent a technical detail, we will assume that the repayment constraint is satisfied with equality in the optimal solutions. This can be achieved by increasing the final stock Y, and avoids a separate treatment for the dual variable P,, 1. THEOREM 5. If (XT, Cr, ZF) is a competitive policy which satisfies the repayment constraint with equality in the pessimistic model then it is an optimal policy.

Proof: Let (XI, C,, Z,) be any feasible policy from Y, to Y,, then by feasibility and the competitive conditions, i a’-‘E@(Q) (=I

- c a’- ‘E(u(C,*)) t=1

< x Wt(Ct - CT)>

by 0)

t=1

< 1 L(E(Pt+,F(Xt))-E(PtXt))-(E(Pt+,(~(XT))-E(PtXT))l t=1 Ttl

I=, Wt(Zt

-

-\’

-

ZT>)

Ttl

by (ii) and (iii)

EG%t+,@t-‘Q)V1*- 2,))


\‘

I=1 p’-‘Q(d+‘)

=

d9T-t+‘(~t+,,...,

Q
-Z,(d))]

co=+, >1 dP’G-0 d9T+‘(WT+‘)

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SUTHERLAND

since Q

Ttl F‘

pt-‘zT=O
t=,

holds for all uTt’

E QTt ‘.

Ttl -f-

,o-lZ,

t=,

I

Hence the existence of competitive prices is suffkient for optimality. To show necessity a Slater-type constraint qualification is imposed on the model. Specifically it is assumed that it is always possible to over-repay debts (perhaps by decreasing consumption). THEOREM 6. Let (XT, CT, ZT) be an optimal T-period policy from Y,, to Y, for the pessimistic model. If there exists some feasible T-period policy from Y, to YT satisfying CTztlpf-‘Z,(o’) > 0 for all wTt’ E QTfl, then the policy (XT, C:, ZT) is competitive.

Proof: If the borrowing variables Z, are eliminated by using the budget constraints, then it follows that (XT, CT) is optimal for the problem:

Find X, > 0, C, > 0 which maximize CT= i 6’- ‘E(u(C,)) subject to the (repayment) constraint T

1 pf-92, < Y, + 5 p’?pF(X,) t=1 I=1

-X,)

- PTY,.

Letting X denote the policy (X,, C,), ST(%) denote CT=, 8t-1u(C,) and Y(X) denote the constraint function Y, + 2 p’-‘@F(X,) f=l

-AC,) -pry,

the objective

-- 5 p’-lc, I=1

then the pessimistic model is of the form: Find 5 > 0 which maximizes EST(X) subject to g(X) where 5(X) E L,(RT+‘).

>, 0

The assumption that there exists some feasible policy ‘which strictly satisfies the repayment constraint for all wTt i E Q” ’ is equivalent to Y(X) > 0 for some 3Y 2 0. Hence by Luenberger [6, p. 2241 there exists some w* E L$ (the dual space of L, with respect to the sup norm) such that w* > 0 and P(X, w*) is maximized at 3”* = (XT, C,*) over all 5 > 0 where the Lagrangian is given by

OPTIMAL

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177

Now proceeding as in Zilcha [ 1 l] since w* E Lf!& there exists a bounded finitely additive measure v such that w*(Y) = J”Y dq for L? E L,. Moreover q has the decomposition q = qC + vs, where qC is absolutely continuous with respect to 0, then we have

Also there exists a sequence (A,} ~&I,,) = 0 for all n, with TTtl(A,) .K > 0 define P(d+

‘) = .x(d+

of subsets in 2”+ ‘, A, c A,, , and --) 1 as n + co. For 2T E L&(L!“‘), ‘)

if

zz&-*(gT+‘)

wrtlEA,

otherwise,

then Y(S-*,

y/*)=EF(X*)+E(Q.

.F(S-*))+jy(%*)&r

holds for all n so that we have

>

F(X)

dLF+'

t

Q . L?(v%'-)dYT+'.

Next letting n + co we obtain

I

F(s*) &F+ l f Q . Y(&-*) dPT+'

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for all 5 E I;p,(Q r+ ‘), 2??> 0. Reverting to the notation at the beginning of this proof, we have shown that i a’-‘E(u(C,)) f=l r,+

2 p’-‘(pF(xt)-Xt)-pTY,I=1

f pt-q t=1

is maximized at the policy (XT, CT) over all X, > 0, C, > 0. Since the random variables are non-anticipative, that is, X,(0” ‘) = Xt(cu’), is the product measure on the space C,(wr+ ‘) = C,(w’) and 9r+’ (0 ‘+l, ~3’~~‘), then upon defining random variables Pt(w’)=E,-t+,@‘-lQ(~T+l)) the previous statement reduces to t (8 - ‘E(u(C,)) - E(P, C,)) t=1

+I=1 x (Wtt ,Wt)

- E(PJt)) + W’, Yd - W’,+ 1Yr))

being maximized at the policy (XF, C,*) over all X, > 0, C, > 0. This defmtion of prices P, immediately yields competitive condition (iii). Conditions (i) and (ii) follow by separability of the variables in the above statement. I As usual, the discussion of the qualitative behaviour of optimal policies is facilitated by the existence of competitive prices. In the case of the pessimistic borrowing model the utility function now plays a more significant role than in the deterministic (and hence optimistic) borrowing models. THEOREM 7. Assume that u’(c) exists. If (XT, C,*, 2:) policy for the pessimistic model and if CT > 0, then

E@‘(W)

G E(u’(CT+ 1))

for

t = l,.:., T-

is an optimal

1

when p > 6 (the opposite inequalities holding when p > 6 and equality holding for p = 6). Proof:

By differentiating

condition (i) we have

Pu’(C~(w’))

- P,(w’) < 0,

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UNCERTAINTY

By condition (iii) holds if C~(o’) > 0. )) = P,(o’) holds for t = l,..., T + 1. Thus if E,(P,+ ,) denotes expectation with respect to w(+ i we have E,(P,+ r(w’+ ‘)) = pP,(w’). Now assuming that Cf(o’) > 0 holds for almost all wTtl and t = I,..., T, these equations combine to yield where

equality

E T-f+lcP--lQ(~T+l

GE,(u’(C,*,,(w’+

‘))) = pu’(c:(w’)).

Taking expected values (with respect to 0’) yields the conclusions of the theorem. I COROLLARY. If u is a quadratic function, then E(CT) > E(C,*, ,) for t = I ,...r T - 1 when p > 6 (the opposite inequalities holding when p < S and equality holding when p = 6).

Proof. Let u’(C) = B - C where B > 0, then as E(u’(CT)) = B - E(CT) the inequalities of the theorem statement are reversed. I

Theorem 7 together with Theorem 4 suggest strong similarities in the expected behaviour of the consumption policies of the two versions of the borrowing model. These similarities will be examined in more detail in Section 5. The investment policy for the optimistic model was stationary (at the “golden-rule” level 2). Investment in the pessimistic model, while not stationary, does- belong to a set analogous to that described by Brock and Mirman [ 11. Let x(o) maximize {pf,(x) - x} for all x > 0 (recall that JW is concave) and let X,, A?~denote the infimum and supremum values of X(w) on R. 8. If (XT, CT, Zf) is an optimal policy for the pessimistic model then 3,,, < e < XM for t = l,..., T. THEOREM

Proof:

By condition (ii), z(w’)

I P,, ,(u’+ ‘)f,,+,(x,(~‘)>

maximizes dy(w,+

1) - P,(w’) X,@‘).

Using conditions (i) and (iii) (as in the proof of the previous theorem) this is equivalent to, *(w’) maximizes

Since P,+,(w ‘+ ’ ) > 0 this then implies the conclusion of the theorem.

1

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Note that if ZM = Xm then the optimal investment policy will be stationary and at the golden-rule level 2 (for f = E(F)).

5.

COMPARISONS

The purpose of this final section is to compare the optimal consumption policies for the optimistic and pessimistic models. Specifically we show that the two policies are very similar during the initial periods of a model with a long planning horizon. Furthermore the pessimistic policy exhibits an adaptive behaviour with respect to the realized path of the production process. Assume that an investment policy X,(0’) has been chosen. Then the total capital accumulation is defined by

K,(WT+‘>= Y,(q)

+

i P’-‘k??~,+,w~fN -&(41 - y,(WT+‘). t=1

Recall that an investment policy X,(0’) is feasible for the optimistic model if E(K,) > 0 or for the pessimistic model if K,(w~+‘) > 0 for all mT+‘. If the investment policies are fixed, then the statements of the two models can be simplified (by elimination of the borrowing variables Z,). A consumption policy C, > 0 is optimal (with respect to the chosen X,) for the optimistic model if it solves: Maximize

2 cvE(u(C,)) t=1

subject to

s pf - sq C,) < E(K,). I=1

Similarly, a consumption policy C, > 0 is optimal (with respect to the chosen XT) for the pessimistic model if it solves: Maximize

i

6’-‘E(u(C,))

I=1

subject to

2 p

C,(d) < K;(d+‘)

I=1

for all

(?J+1

E-p+‘.

But by the previous sections, optimal policies exist and can be assumed to be competitive. Thus there exist prices q and Q for the two models respectively such that the following constraints are satisfied:

OPTIMAL

; p’-lE(C,) I=1

GROWTH

AND

UNCERTAINTY

= c p’-‘u(@/6)‘-‘q)

181

= E(K,),

I=1

T ” *=I

pf-lc,=

i f=l

p’%(@/6)‘-‘E,~,+,(Q))=KgoT),

where u is the inverse function of the marginal utility function U’ and Kg(w’) is the minimum of Kg(oT+i) as a function of ~~+i. In order to carry out our comparison of optimal consumption policies we need to solve these constraints for the competitive prices q and Q. Explicit solutions can be obtained for the case of a quadratic utility function u. In this case v(q) = B -4, where B > 0 is the “bliss point” of U. Since the calculations for the pessimistic model are somewhat complicated, the following notation will be helpful. Let v = p*/6, and define

wT

=

@(tP

-

KT)/V(T)~

q

= @(t)B - K;)/v(T).

The subscript T will usually be omitted in W, and WC. Some simple identities for v(r) will be needed (their proofs, being straightforward manipulations, are omitted). LEMMA

1. For any v > 0 and integer T > 0,

(i)

v(T) - v(t - 1) = v’-‘v(T-

(ii)

(:, v(t):;;I)=l-v(Tv:i), f-l

(iii)

\‘ ,r

v(T-

t + 1)for t = l,..., T,

vs- ‘v(T) t + s) v(T- t + s + 1) v(t

=v(T-t+

- 1) 1)

The prices for the optimistic expressions

for

t = 2,..., T.

and pessimistic models are defined by the s=E(W)

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MACLEAN,FIELD,AND

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and T-l

Q(m=+I)= I+- - 1 FT &g+ s=l [

,)ww

1

w*.

Note that as V = WYC(W~)then Q is actually independent of Ok+, . Thus we have the equation ET-t+l(Q(oT+l)) = E,-XQ(d)) for t = 1, 2,..., T. Note that E,(A) = A for any random variable A. LEMMA

2. E T-,(Q)=ET-I(Ul?k) andfor t= 2,...,T f-l

ET-~(Q)=

ET-I(P) v(T-

$?

Proof. Applying E,-, collecting common terms

t +

l>

-

1 s=1

T-l

1 Y'-'

s=1

-ET-i@?

v(T) V

+

to the expression for Q(o’)

ET--,=E,-,(W*)-

=m

,,,~:‘,‘,“;,;r”~+‘,

1)

1 *

we have, upon

v(T) v(s) v(s + 1) ET-LES(W*))

I T-l

ET-~(~)

-

sgl

v(sj;;'+

l> E~-t(Es(W*))

1 *

But ET-,(E,(W*)) equals ETet(V) for s = l,..., T - t and equals E,(W) for s = T- t + l,..., T - 1. This second possibility does not arise for the case t = 1. When t = 1, we have s-l

E,,(Q)=$

[I-

y' v ]E,-,(w)=E,-,(~n) ,.I v(s)v(s+ 1)

by identity (ii) of Lemma 1. For t = 2 ,..., T, ET--I(~'+?

-

,,-’

t-1 -Y

L

s=l

vS-lE T--L+&@? v(T-t+s)v(T-t+++

1

1) ’

which simplifies to the required form upon applying the same identity.

m

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UNCERTAINTY

LEMMA 3. The prices q and Q are competitive for the optimistic and pessimistic models.

Proof.

In the optimistic model: T

\’ p’-‘v(@/Lyq) tz

=p(T)B

- v(T)q

as required. In the pessimistic model, applying Lemma 2, T x t=1

v’-‘ET-t(Q(mT)) ‘, ET-t(~) + V(T) 5 tT2 v(T- t +

=ET--I(v) T

1)

t-l \‘

vs- ‘E T-t+su+T t:2 ,YI v(T-t+s)v(T-t+st

-v(T)

\‘

1)’

The first term is combined with the second, the double summation rearranged (using the change of variable r = t-s), to obtain T-l

T-r

vSwlET-J

r?I S?, v(T-r)v(T-rt 7’

tYI v(T-tt = v(T) E&V+)/v(

l)-,!!l

ET+(v)

v(T-rt

1)

1

1) = v(T) w”.

Hence T 1 pt-‘V(@/6)‘-‘E,-t+,(Q(wT+‘)) t= 1

as required.

I

=p(T)B

- 4 V’-‘E,-t@(~T)) te,

=dT)B

- v(T) I+-

=p(T)B

- @(T)B -Kg)

= K;

w*)

1)

1

is

184

MACLEAN,FIELD,AND

SUTHERLAND

THEOREM 9. If u is a quadratic utility function (wirh bliss point B), then the optimal consumption policies are

C; = B - (p/6)‘-’

E( W,)

in the optimistic model and CT = B - (p/6)‘-’

E,-,(G)

+ 0;

in the pessimistic model, where the deviation 0: = 0 and Dfzp’-t for

f-1 x ,,S-1 v(T--+s)v(T-t+s+l)-v(T-tt s=1

ET--l(V)

1) I

t = 2,..., T.

Proof:

In the optimistic model C;=

v(@/6)‘-‘q)

= B - (p/6)‘-‘E(W)

as claimed. In the pessimistic model

%,(Q(wT)N

C: = v@/~)‘-’ and applying Lemma 2 C;=

B -ET-,(Q)

= B - ETml(W);

and for t > 1,

C; = B - @/6)‘-’ ET-((Q)

-

t-1

vs- ‘E T-l+s(w*)

Sk1 v(T--ts)v(T-ttst = B - (p/6)‘-’

1 t I--1

ETJV) Vt-1 _

P

1) I

v(T) v(T-tt

t-1 t

v(T)

K'

v"-'Er-t+s(w*)

,kl v(T--ts)v(T-t+s+

1) ET-I(~)

1

1) ’

which reduces to the required expression upon applying identity Lemma 1. I

(i) of

OPTIMALGROWTH COROLLARY.

Proof

E(DT) = ofir

185

AND UNCERTAINTY

t = l,..., T.

This clearly holds for t = 1. If t > 1 then t-1 E(DT)

=

p’-’

y

v(T)

vs-’

S=l

1

-

1) I E(W*)

v(T-t+ t-1

V

1-t =P

1)

v(T-t+s)v(T-t+s+

K s;l v(Tv(T-t+

S-l

v(T-t+s)v(T-t+s+

1) 1) I

1) v(T)

W+T

=o by identity (iii) of Lemma 1. I Although the consumption policies of the theorem have similar forms, the optimistic policy is based upon the expected capital accumulation over the entire horizon whereas the pessimistic policy depends upon the realized past history up until time t and the expected capital accumulation over the remaining horizon. The corollary shows that the deviations in the pessimistic consumption policy do average out to be zero (the deviation in the initial period being exactly zero). For certain choices of p and 6 the repayment constraint becomes less important as the planning horizon T tends towards infinity. Our final results show that for such choices the distinction between the optimistic and pessimistic models disappears. 10. Let u be a quadratic utility (with bliss point B). If p < 1 and v > 1 then lim,+, DT = 0 for each t. THEOREM

ProoJ Since p < 1 and f,(x) - x ( M for all x, all CD,then it follows that KT(o ‘) is uniformly bounded (for reasonable initial and final stocks Y,,, Yr). But as

G = @V’)B-W/v(T) it follows that v(w follows that

‘) is uniformly bounded in mT and in T. Now if v > 1 it 1

v(T-t+

1) ET-tvw

186

MAC LEAN, FIELD, AND SUTHERLAND

and

vu-7 v(T-t+s)v(T-t+s+ tend to zero (for fixed o=o.

1)

E r-f+suG>

t, s) as T+ co so that lim,,,

0: = p’-’ c:L:

v’-’

*

1

COROLLARY. If p < 1 and v > 1 then the optimal consumption policies in both models satisfy lim,, Cf’= B or equivalently lim,, PT = 0 for each finite t.

Proof From Theorems 9 and 10 we have (for the pessimistic model) CT = lim,,(B - (p/B)‘-’ ET+( @) + 0:) = B since v -+ 0 as lb+, T-+ co under the conditions on p and v. P,‘+ 0 follows from PT = 6’-‘u’(CT) = 8-‘(B - Cf). 1

ACKNOWLEDGMENTS The authors would like to express their thanks to several referees for helpful suggestions on an earlier version of this paper.

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