random horizon stochastic dynamic slutsky equation under ... · center of game theory, st...

Applied Mathematical Sciences, vol. 8, 2014, no. 147, 7311 - 7340

HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.4112

Random Horizon Stochastic Dynamic Slutsky

Equation under Preference Uncertainty

David W. K. Yeung

Center of Game Theory, St Petersburg State University, Russia and

SRS Consortium for Advanced Study in Cooperative Dynamic Games

Shue Yan University, Hong Kong

Copyright © 2014 David W.K. Yeung. This is an open access article distributed under the

Creative Commons Attribution License, which permits unrestricted use, distribution, and

reproduction in any medium, provided the original work is properly cited.

Abstract

This paper extends Slutsky’s classic work on consumer theory to a random

horizon stochastic dynamic framework in which the consumer has an

inter-temporal planning horizon with uncertainties in future incomes, preferences

and life-span. Utility maximization leading to a set of ordinary wealth-dependent

demand functions is performed. A dual problem is set up to derive the wealth

compensated demand functions. This represents the first time that

wealth-dependent ordinary demand functions and wealth compensated demand

functions are obtained under these uncertainties. The corresponding Roy’s identity

relationships and Slutsky equations in a random horizon stochastic dynamic

framework with uncertain preferences are derived. The analysis incorporates

realistic characteristics in consumer theory and advances the conventional

microeconomic study on consumption to a more realistic optimal control

framework.

Keywords: Optimal consumption, Stochastic dynamic programming, Roy’s

identity, Slutsky equation

1 Introduction

The ground-breaking work by Slutsky (1915) laid the foundation for rigorous

analysis of optimal consumption decision in microeconomics. The analysis, which

7312 David W. K.Yeung

showed that the effect of a price change on the demand of a good can be

decomposed into substitution effects and income effect, yields significant

economic implications (see Varian (1992)). This prominent contribution in

consumer theory, known as the Slutsky equation, was christened by John Hicks as

the ‘Fundamental Equation of Value Theory’. It becomes an integral part of

mainstream economics and consumer theory. The papers by Allen (1936 and

1950), Hicks and Allen (1934), Schultz (1935), Dooley (1983) and Epps (1975)

propagated Slutsky’s classic work. Another milestone in consumer theory is

Roy’s identity (1947) which provides an often invoked mathematical result in

consumer theory. In addition, the identity is also instrumental in proving the

Slutsky equation. Yeung (2013) extended Slutsky’s work to a stochastic dynamic

framework in which the consumer has a T -period life-span with future incomes

being uncertain and derived the stochastic dynamic Slutsky equations. Yeung

(2014) further extended Slutsky’s consumer problem into a dynamic framework in

which the consumer has a random life-span and with uncertain future incomes.

In this paper, uncertainties in the consumer’s future preferences is

incorporated to Yeung’s (2014) extension of the Slutsky framework to reflect an

often observed reality in consumer choice. Changes in conditions that could not

be perfectly foreseen like health, taste, habits, technology, style, culture and

family composition contribute to uncertainty in future preferences. In particular,

optimal consumption choice under a dynamic framework with uncertainties in the

consumer’s life-span, future incomes and future preferences is examined.

Inter-temporal wealth-dependent ordinary demand functions and wealth

compensated demand functions are obtained. Two of the most crucial foundations

in consumer theory – Roy’s identity and Slutsky equation – are derived in a

random horizon stochastic dynamic framework with uncertainty in the consumer’s

future preferences.

The paper is organized as follows. We first present a dynamic model of

utility maximization by a consumer with an uncertain life-span, uncertain future

incomes and uncertain future preferences in Section 2. In Section 3, a set of

wealth-dependent ordinary demands is characterized. The Roy’s identity result in

a random horizon stochastic dynamic framework with uncertain future

preferences is derived in Section 4. The dual problem is formulated in Section 5

and the corresponding wealth compensated demand functions are obtained.

Stochastic dynamic Slutsky equations for the consumer with uncertainties in

life-span, future incomes and preferences are formulated in Section 6. Section 7

concludes the paper.

2 Dynamic Utility Maximization under Uncertainties in Incomes,

Preferences and Life-span

Consider the case of a consumer whose life-span involves T̂ periods where T̂

Random horizon stochastic dynamic Slutsky equation 7313

is a random variable with range },,2,1{ T and corresponding probabilities

},,,{ 21 T . Conditional upon the reaching of period , the probability of the

consumer’s life-span would last up to periods ,1, T, becomes respectively

(see Yeung and Petrosyan (2011)):

T

,

T

1

T

T

,, . (2.1)

We use ),,,( 21 kn

kkkk xxxx to denote the quantities of goods consumed and

),,,( 21 kn

kkkk pppp the corresponding prices in period },,2,1{ Tk . The

preference or utility function of the consumer in period 1 is known to be )( 1

)1(1 xu .

His future preferences are not known with certainty. In particular, his utility

function in period },,3,2{ Tk is known to be )()(

k

kxu k with probability

k

k

for },,2,1{ kk m if he survives in period k . We use k~ to denote the

random variable with range },,2,1{ kk m and corresponding probabilities

},,,{ 21 km

kkk .

The consumer maximizes his expected inter-temporal utility

TE ,,, 32

T

TT

1ˆˆ

)()(

11

ˆ

1

k

kk

c

c

m

k

T

k

xu k

k

k

k

T

E ,,, 32 )( 1

)1(1 xu

T

TT

2ˆˆ

)()(

11

ˆ

2

k

kk

c

c

m

k

T

k

xu k

k

k

k

, (2.2)

subject to the budget constraint characterized by the wealth dynamics

1

11

1 )(

k

n

h

h

k

h

kk

n

h

h

k

h

kkk

kk

xpWrxpWW , 0

11 WW . (2.3)

where r is the interest rate, is the consumer’s subjective one-period

discount factor for the duration from period 1 to period , the initial period

discount factor 11 , 1k is the random income that the consumer will

receive in period 1k ; and k , for },,3,2{ Tk , is a set of statistically

independent random variables, and 122 ,,, T

E is the expectation operation with


respect to the statistics of 2 , 3 , 1, T . The random variable k has a

non-negative range },,,{ 21 km

kkk with corresponding probabilities

},,,{ 21 km

kkk . The random variable 1T has a value of zero with probability 1

because the consumer will receive no income in period 1T . Moreover, under

the axiom of non-satiation, the consumer will spend all his wealth in the last

period of his expected life-span and therefore 01 TW . The consumer’s discount

factor c may or may not coincide with the market discount factor r1

1 and

may also be equal to 1. For notational simplicity we adopt the notation

k

c

c

k

1

1 and

k

c

c

k

.

The problem (2.2)-(2.3) is a random horizon discrete-time stochastic control

problem with uncertain payoffs (see Yeung and Petrosyan (2013 and 2014)).

Now consider the case when the consumer has lived to period t and his

wealth is W and his preference is )()(

t

txu t . The consumer problem can be

formulized as the maximization of the (discounted) payoff:

121 ,,, TttE )(

)(

1 t

tt xu t

T

t

TT

tT

ˆ

1ˆ

)()(

1

1

ˆ

1

k

kkm

k

T

tk

xu k

k

k

k

, (2.4)

subject to the budget constraint characterized by the wealth dynamics

11 ))(1( kkkkk xpWrW , WWt , (2.5)

for },,1,{ Tttk .

In a stochastic dynamic framework, strategy space with state-dependent

property has to be considered. In particular, given that the preference of the

consumer is )()(

k

kxu k if he survives to period k , a pre-specified class of

mapping )()(k

k

XW : with the property

},,,{ 21 kn

kkk xxx kx )()(Wk

k

)}(,),(),({)(2)(1)(

WWW kkkk n

kkk

, for

},,2,1{ Tk , is the strategy space and each of its elements is a admissible

strategy. We define the value function ),()(

WtV t and the set of strategies

)(*)(* Wx k

kk

, for },,1,{ Tttk which provides an optimal

consumption solution as follows:


),()(

WtV t Ttt xxx ,,, 1

m a x

121 ,,, TttE )(

)(

1 t

tt xu t

T

t

TT

tT

ˆ

1ˆ

)()(

1

1

ˆ

1

k

kkm

k

T

tk

xu k

k

k

k

WWt

121 ,,, Ttt

E )]([)()(

1 Wu tt

t

tt

T

t

TT

tT

ˆ

1ˆ

)]([)()(

1

1

ˆ

1

kk

kkm

k

T

tk

Wu kk

k

k

k

WWt ,

for },,2,1{ Tt . (2.6)

The value function ),()(

WtV t reflects the expected inter-temporal utility (in

present value terms) that the consumer will obtain from period t to the end of his

life-span. Following the analysis of Yeung and Petrosyan (2013 and 2014) one

can derive an optimal solution to the random-horizon consumer problem

(2.2)-(2.3) as follows:

Theorem 2.1. A set of consumption strategies )({*)(

Wk

k

, for

},,2,1{ tt m and }},,2,1{ Tk provides an optimal solution to the

random horizon consumer problem (2.2)-(2.3) if there exist functions

),()(

WtV t , for },,2,1{ Tt and },,2,1{ tt m , such that the following

recursive relations are satisfied:

),1()(

WTV T 0 and 01 TW ,

),()(

WTV T

)])(1(,1[)(max)()(

1 TTT

TT

xxpWrTVxu TT

T

,

),()(

WtV t

txmax

1t

E

)()(

1 t

tt xu t

T

t

T

t

1

1

1

1

1

1

t

t

t

m

t

]))(1(,1[ 1

)( 1

ttt xpWrtV t

,

for t }1,,2,1{ T and },,2,1{ tt m . (2.7)


Proof. Following Bellman’s (1957) technique of dynamic programming we begin

with the last stage/period. By definition, the utility of the consumer at period

1T is therefore

0),1()(

xTV T .

We first consider the case when the consumer survives in the period T , his

preference is )()(

T

Txu T , and the state WWT . The problem then becomes

]))(1(,1[)(max 1

)()(

11

TTTT

TT

xxpWrTVxuE TT

TT

(2.8)

subject to

11 ))(1( TTTT xpWrW . (2.9)

Since 01 T with probability 1, ),1()(

WTV T 0 and 01 TW , the problem

in (2.8)-(2.9) can be expressed as

),()(

WTV T

)])(1(,1[)(max)()(

1 TTT

TT

xxpWrTVxu TT

T

, (2.10)

subject to

))(1(1 TTT xpWrW , WWT and 01 TW . (2.11)

Now consider the problem when the consumer survives to period 1T and his

preference is governed by )( 1

)(1 1

T

Txu T .The problem in period 1T can be

expressed as maximizing

1, TTE

1

1

T )( 1

)(1 1

T

Txu T

T

T

T

1

T

T

T

m

T

1

T

1 )()(

T

Txu T

(2.12)

subject to

11 ))(1( kkkk xpWrW , for },1{ TTk and WWT 1 . (2.13)

If the value function ),()(

WTV T in (2.10) exists, the problem (2.12)-(2.13)

can be expressed as a single stage problem:


),1()1( WTV T

1

1

,maxTT

T

Ex

1

1

T )( 1

)(1 1

T

Txu T

T

T

T

1

T

T

T

m

T

1

]))(1(,[ 11

)(

TTT xpWrTV T

. (2.14)

Now consider the problem that the consumer survives in period t }2,,2,1{ T

and his preference is )()(

t

txu t . Following the analysis above, the problem in

period t becomes the maximization of the expected payoff

121 ,,, TttE

t

1 )()(

t

txu t

T

t

T

tTT

1ˆ

ˆ

)()(

1

1

ˆ

1

k

kkm

k

T

tk

xu k

k

k

k

121 ,,, Ttt

E t

1 )()(

t

txu t

T

t

T

tTT

T

t

T

t

1

1ˆˆ

1

)()(

1

1

ˆ

1

k

kkm

k

T

tk

xu k

k

k

k

(2.15)

subject to

11 ))(1( kkkk xpWrW , for },,1,{ Tttk and WWt . (2.16)

Note that the term

132 ,,, TttE

T

t

T

tTT

1

1ˆˆ

)()(

1

1

ˆ

1

k

kkm

k

T

tk

xu k

k

k

k

can be expressed as:

132 ,,, TttE

T

t

T

tTT

1

1ˆˆ

)()()(

1

1

ˆ

2

1

)(11

1

1

11

1

1

1

k

kkm

k

T

tk

t

ttm

t xuxu k

k

k

kt

t

t

t

.


In (2.17) the term (2.17)

132 ,,, TttE

)()()(

1

1

ˆ

2

1

)(11

11

k

kkm

k

T

tk

t

tt xuxu k

k

k

kt

gives the expected inter-temporal utility to be maximized in period 1t if the

consumer’s preference in period 1t is )( 1

)(1 1

t

txu t . If the value function

),1()( 1 WtV t

exists, we have:

),1()( 1 WtV t

Ttt xxx ,,, 11

max

132 ,,, Ttt

E

T

t

T

tTT

1

1ˆˆ

)( 1

)(11

11

t

tt xu t

)()(

1

1

ˆ

2

k

kkm

k

T

tk

xu k

k

k

k

. (2.18)

Using (2.18) the problem (2.15)-(2.16) can be formulated as a single stage

problem which maximizes the expected payoff

1tE )(

)(

1 t

tt xu t

T

t

T

t

1

1

1

1

1

1

t

t

t

m

t

]))(1(,1[ 1

)( 1

ttt xpWrtV t

. (2.19)

If ),()(

WtV t exists, we have

),()(

WtV t

1

maxt

t

Ex

)()(

1 t

tt xu t

T

t

T

t

1

1

1

1

1

)(

1

t

t

t

m

t

]))(1(,1[ 1

)( 1

ttt xpWrtV t

,

for t }2,,2,1{ T and },,2,1{ tt m . (2.20)

Hence Theorem 2.1 follows. ■

The stochastic optimal state trajectory derived from Theorem 2.1 is

characterized by:


0

11 WW ,

12

2 ;

2

j

W 21

2

0

1

)(

11

0

1 )]()[1(j

WpWr ,

213

32

2 ;

3

jj

W )]()[1( 12

2212

2 ;

2

)(

22

;

2

jj

WpWr 3

3

j , if preference is )( 2

)(2 2 xu

in

period 2;

3214

43

32

2 ;

4

jjj

W )]()[1( 213

32

23213

32

2 ;

3

)(

33

;

3

jjjj

WpWr 4

4

j ,

if preference is )( 3

)(3 3 xu

in period 3;

121

33

22 ; T

TjT

jj

TW

)]()[1( 2211

13

32

212211

13

32

2 ;

1

)(

11

;

1

TTj

Tjj

TTTj

Tjj

TTTT WpWr

Tj

T ,


)(1 1

T

Txu T in period 1T ;

TTj

Tjj

TW 21

11

33

22 ;

1

)]()[1( 1213

32

21213

32

2 ;)(; TTj

Tjj

TTTj

Tjj

TTTT WpWr

1

1

Tj

T

0 .

We use *

W to denote the set of possible values of wealth 1213

32

2 ;

jjj

W

at period along the optimal trajectory generated by Theorem 2.1.

3 Wealth-Dependent Ordinary Demand

In this section, we consider the primal problem of deriving wealth-dependent

ordinary demand functions in which the consumer maximizes his inter-temporal

expected utility subject to uncertain inter-temporal budget, life-span and future

preferences. Following the analysis in Yeung (2014) we first consider the case

when the consumer survives in the last period and his preference is )()(

T

Txu T .

Let *0

TT WW denote the consumer’s wealth in period T . Given that

),1()(

WTV T 0 and 01 TW , to exhaust all the wealth in this period,

TTT xpW 0 0 . Hence the consumer faces the problem

)(max)(

T

T

x

xu T

T

subject to TTT xpW 0 0 . (3.1)

Problem (3.1) is a standard single period utility maximization problem.

Setting up the corresponding Lagrange problem and performing the relevant

maximization one obtains a set of first order conditions. It is well-known (see

Cheung and Yeung (1995)) that if the set of first order conditions satisfies the


implicit function theorem, one can obtain the ordinary demand as explicit

functions of the parameters 0

TW and Tp , that is:

h

TTx

)(),( 0)(

TT

h

T pWT , for },,2,1{ Tnh and },,2,1{ tt m . (3.2)

Note that ),( 0)(

TT

h

T pWT corresponds to the optimal consumption strategies

)( 0)(

T

h

T WT in Theorem 2.1. To show this we note that the problem in period T

in Theorem 5.1 is

]))(1(,1[)(max 1

)()(

11

TTTT

TT

xxpWrTVxuE TT

TT

)(max

)(

1 T

TT

xxu T

T

subject to ))(1(1 TTT xpWrW 0 .

Since the problem )(max)(

T

T

x

xu T

T

subject to TTT xpW 0 0 and the

problem and the problem )(max)(

T

TT

xxu T

T

subject to the same constraint yields

the same controls, therefore ),( 0)(

TT

h

T pWT = )( 0)(

T

h

T WT .

Substituting (3.2) into (3.1) yields the indirect utility function in period T

as ),( 0)(

TT

TpWv T )],([ 0)()(

TTT

TpWu TT . Invoking the definition in (2.6),

T

1 ),( 0)(

TT

TpWv T equals the function ),( 0)(

TWTV T in Theorem 2.1.

Now consider the case when the consumer survives in period 1T , and his

preference is )( 1

)(1 1

T

Txu T . If wealth equals *

1

0

1 TT WW in this period, the

problem in concern becomes

TT xx ,1

max

1, TTE )( 1

)(1 1

T

Txu T

T

T

T

1

T

T

T

m

T

1

T )()(

T

Txu T

subject to the inter-temporal budget

TTTTT xpWrW ))(1( 111 ,

11 ))(1( TTTTT xpWrW 0 , 0

11 TT WW *

TW . (3.3)

Let j

TTTTT xpWrW Tj

T

))(1( 11

0

1

; 1 denote the wealth at period T if j

T },,,{ 21 Tm

TTT occurs and the consumer’s preference is )( 1

)(1 1

T

Txu T in

period 1T . Using the indirect utility function ),( 0)(

TT

TpWv T and (3.3), the

problem facing the consumer in period 1T can be expressed as a single-period

problem:


1

maxTx

)( 1

)(1 1

T

Txu T

T

T

T

1

j

T

m

j

T

1

T

T

T

m

T

1

T

),( 1;)(

TT

TpWv T

jTT

. (3.4)

First order condition for a maximizing solution yields

)( 1

)(1 1

1

T

T

xxu T

iT

T

T

T

1

j

T

m

j

T

1

T

T

T

m

T

1

),( 1;)(

TT

T

WT pWv Tj

TT

T

i

Tp 1 )1( r 0 ,

for },,2,1{ 1 Tni ; (3.5)

Again, with the implicit function holding, (3.5) can be solved to yield the ordinary

demands in period 1T as

h

TTx

)(

11

),,( 1

0

1

)(

11

TTT

h

T ppWT

, for

},,2,1{ 1 Tnh .

Note that ),,( 1

0

1

)(

11

TTT

h

T ppWT

corresponds to the optimal consumption

strategies )( 0

1

)(

11

T

h

T WT in Theorem 2.1. To show this we transform (3.4) into a

similar problem by multiplying the maximand by 1

1

T and obtain

)(max 1

)(11

11

1

T

TT

xxu T

T

T

T

T

1

j

T

m

j

T

1

T

T

T

m

T

1

T

1

),( 1;)(

TT

TpWv T

jTT

.

(3.6)

Moreover, recalling that T

1 ),( 0)(

TT

TpWv T equals the ),( 0)(

TWTV T in

Theorem 2.1 we can express (3.6) as

1

maxTx

T

E

)( 1

)(11

1

T

TT xu t

T

T

T

1

T

T

T

m

T

1

]))(1(,[ 11

)(

TTT xpWrTV T

,

which is the period 1T condition to be maximized in Theorem 2.1. Since the

controls for problem (3.4) and the transformed problem (3.6) are the same, we

have ),,( 1

0

1

)(

11

TTT

h

T ppWT

)( 0

1

)(

11

T

h

T WT .

Substituting ),,( 1

0

1

)(

11

TTTT ppWT

into (3.4) yields the inter-temporal


indirect utility function ),,( 1

0

1

)(1 1

TTT

TppWv T

. Invoking (2.6) and Theorem 2.1,

the term 1

1

T ),,( 1

0

1

)(1 1

TTT

TppWv T

corresponds to ),1( 0)( 1

TWTV T in

Theorem 2.1.

Repeating the analysis for periods 2T to 1 yields the consumer problem

at period }2,,2,1{ T with preference )()(

xu

as:

)(m a x)(

xux

T

T

1 1

1

1

1

1

j

m

j

1

1

1

1

1

m

1 );(;

1

)(1 111 pWv

j

,

(3.7)

where 0

W *

W , 11

1

1

0;

1 ))(1(

j

xpWrWj

, and );(;

1

11 pW

j

is the

short form for the vector ),,,,( 21

;

1

11

TpppWj

.

First order condition for a maximizing solution to the problems in (3.7) can

be obtained as:

)()(

xu ix

T

T

1 1

1

1

1

1

j

m

j

1

1

1

1

1

m

1 );(;

1

)(1 111

1pWv

j

W

0)1( rpi

,

(3.8)

for },,2,1{ ni and }2,,2,1{ T .

Note also the condition that in period , good i will be consumed up the

point where marginal utility of consumption )()(

xu ix

equals )1( r ip times

the expected marginal utility of wealth

T

T

1 1

1

11

1

1

1

W

jm

j

v

1

1

1

1

1

m

);(;

1

)(1

1

111

1pWv

j

W

.

In particular, the expected marginal utility of wealth takes into consideration

the random future income and preferences plus the probability of the consumer

surviving in period 1 . Solving (3.8) yields the ordinary demands in period

as:

h

x)(

),( 0)(pW

h

, for },,2,1{ nh and },,2,1{ hh m . (3.9)


After solving the primal consumer problem which maximizes expected

utility subject to uncertainties in future income, future preferences and life-span,

we proceed to derive the Roy’s identity result in a dynamic framework with

uncertainties in future life-span, preferences and incomes.

4 Random Horizon Stochastic Dynamic Roy’s Identity under

Uncertain Preferences

In this section we derive the stochastic dynamic version of Roy’s identity with

uncertainties in future life-span, preferences and incomes. Invoking (3.7) we

obtain the identity

),( 0)(pWv

)],([ 0)()(pWu

T

T

1 j

m

j

1

1

1

1

1

1

1

1

1

m

];)),()(1[( 1

0)(0)(1 1 ppWpWrv j

.

(4.1)

Differentiating the inter-temporal indirect utility function in (4.1) with

respect to jp :

jp

pWv

),( 0)(

)],([ 0)()(

1

)( pWu k

n

k

T

T

1 j

m

j

1

1

1

1

1

1

1

1

m

;

1

;

1

)(1

11

11 ),(j

j

W

pWv

k

j

W)(

;

11

j

k

p

pW

),( 0)(

T

T

1 j

m

j

1

1

1

1

1

1

1

1

m

;

1

;

1

)(1

11

11 ),(j

j

W

pWv

jp

Wj

;

11

, (4.2)

where jpWpWrWj

1

0)(0;

1 )),()(1(1

, k

t

j

W)(

;

11

kpr )1( and

jp

Wj

;

11

)1( r ),( 0)(pW

j

.


Invoking the first order conditions in (3.8) the term inside the curly brackets

vanishes, condition (4.2) then becomes:

jp

pWv

),( 0)(

T

T

1 j

m

j

1

1

1

1

1

1

1

1

m

;

1

;

1

)(1

11

11 ),(j

j

W

pWv

)1( r ),( 0)(

pWj

.

(4.3)

The effect of a change in initial wealth on the maximized utility can be

obtained by differentiating ),( 0)(pWv

in (4.1) with respect to 0

W :

0

0)(),(

W

pWv

)],([ 0)()(

1

pWu k

n

k

T

T

1 j

m

j

1

1

1

1

1

1

1

1

m

;

1

;

1

)(1

11

11 ),(j

j

W

pWv

k

j

W)(

;

11

0

0)(),(

W

pWk

T

T

1 j

m

j

1

1

1

1

1

1

1

1

m

;

1

;

1

)(1

11

11 ),(j

j

W

pWv

)1( r .

(4.4)

Again, invoking the first order conditions in (3.8), the term inside the curly

brackets vanishes, condition (4.4) then becomes:

0

0)(),(

W

pWv

T

T

1 j

m

j

1

1

1

1

1

1

1

1

m

;

1

;

1

)(1

11

11 ),(j

j

W

pWv

)1( r .

(4.5)

Dividing the right-hand-side of equation (4.3) by the right-hand-side of

equation (4.5) and the left-hand-side of (4.3) by the left-hand-side of (4.5) yields:

jp

pWv

),( 0)(

0

0)(),(

W

pWv

),( 0)(pW

j

, for },,2,1{ nj . (4.6)


Condition (4.6) provides a random horizon stochastic dynamic version of Roy’s

identity involving a change in current prices under uncertain future preferences.

Then we consider deriving the random horizon stochastic dynamic Roy’s

identity under uncertain future preferences for a change in prices in current and

future periods.

Theorem 4.1. Random Horizon Stochastic Dynamic Roy’s Identity under

Uncertain Preferences

jp

pWv

),( 0)(

0

0)(),(

W

pWv

),( 0)(pW

j

, for },,2,1{ nj ; (4.7)

k

hp

pWv

),( 0)(

0

0)(),(

W

pWv

h

h

h

m

j

j

h

m

j

jm

j

j

11

2

1

1

2

2

2

1

1

1

2

2

1

1

1

1

2

1

1

mm

h

h

h

h

m

h

1

h

111

22

11

112

21

1

;

;)(),(

hhj

h

jj

hhj

h

jj

h

h

h

h

W

pWv

),( 112

21

1 ;)(pW h

hj

h

jj

h

h

k

h

)()1( hr

h

h

h

m

w

w

h

m

w

wm

w

w

11

2

1

1

2

2

2

1

1

1

2

2

1

1

1

1

2

1

1

mm

h

h

h

h

m

h

1

h

1

112

21

1

112

21

1

;

;)(),(

hhw

hww

hhw

hww

h

h

h

h

W

pWv

, (4.8)

for },,2,1{ T , },,2,1{ Th , },,2,1{ hnk and

},,2,1{ m ,

where 0

WW ,

;

1

11

j

W 1

1

0)(0 )],()[1(

j

pWpWr ,

12

21

1 ;

2

jj

W )],()[1(;

1

)(

11

;

1

111

11 pWpWr

jj

2

2

j ,


)(1 1

xu

in period 1 ;


213

32

21

1 ;

3

jjj

W )],()[1( 12

21

1112

21

1 ;

2

)(

22

;

2 pWpWrjjjj

3

3

j ,


)(2 2

xu

in period 2 ;

112

21

1 ,;

TTj

Tjj

TW

)]()[1( 211

12

21

11211

12

21

1 ;

1

)(

11

;

1

TTj

T

jj

TTTj

T

jj

TTTT WpWr

Tj

T ;


)(1 2

T

Txu in period 1T ;

TTj

Tjj

TW

1

11

22

11 ,;

1

)]()[1( 112

21

1112

21

1 ;)(;

TTj

Tjj

TTj

Tjj

TTTT WpWr

01

1

Tj

T .

(4.9)

Proof. See the Appendix. ■

Theorem 4.1 gives the random horizon stochastic dynamic Roy’s identity under

uncertain future preferences. Invoking (A.9) in the proof of Theorem 4.1 in the

Appendix, an alternative form of the random horizon stochastic dynamic Roy’s

identity can be expressed as:

k

hp

pWv

),( 0)(

T

T

h

h

h

h

m

j

j

h

m

j

jm

j

j

11

2

1

1

2

2

2

1

1

1

1

1

1

1

1

m

2

2

2

1

2

m

1

1 1

h

h

h

m

h

h

111

22

11

112

21

1

;

;)(),(

hhj

hjj

hhj

hjj

h

h

h

h

W

pWv

),( 112

21

1 ;)(pW h

hj

h

jj

h

h

k

h

,

(4.10)

for },,2,1{ T , },,2,1{ Th , },,2,1{ hnk and

},,2,1{ m .

5 Duality and Wealth Compensated Demand

In this section, we invoke the duality principle in consumer theory to construct

wealth compensated demand functions under an uncertain inter-temporal budget,

uncertain preferences and a random life-span. To do this, we consider the dual


problem of minimizing expenditure covered by the current wealth subject to

maintaining the level of utility achieved in the primal problem. Following the

analysis in Yeung (2014) we first examine the case when the consumer survives

in the last period and his preference is )()(

T

Txu T . Let *0

TT WW denote the

consumer’s wealth in period T . Since wealth equals income in this period, to

derive the compensated demand we follow the standard single period consumer

problem of

TTx

xpT

m i n

subject to achieving the level of utility )()(

T

Txu T

0

)(ˆ T

T

W

Tv = ),( 0)(

TT

TpWv T .

(5.1)

Setting the corresponding Lagrange function and performing the minimization

operation yields a set of first order conditions. With the implicit function

theorem holding for the first order conditions one can obtain the wealth (income)

compensated demand functions as

h

Tx ),ˆ(0

)(

)(

T

W

T

h

T pv T

T

T

, for },,2,1{ Tnh . (5.2)

Substituting (5.2) into (5.1) yields the wealth-expenditure function

),ˆ(0

)(

)(

T

W

TT pv T

T

T

),ˆ(0

)(

)(

T

W

TTT pvp T

T

T

0

TW .

Now we proceed to period 1T and let wealth in this period be *

1

0

1 TT WW

and preference be )( 1

)(1 1

T

Txu T . To obtain the wealth compensated demand

function in period 1T we consider the problem of minimizing expenditure

covered by current wealth in the period to bring about the expected inter-temporal

utility 0

1

1 )(1ˆ

T

T

W

Tv ),,( 1

0

1

)(1 1

TTT

TppWv T

from the primal problem. However,

wealth 0

1TW in period 1T does not only cover consumption expenditure

11 TT xp in the period 1T but also part of the consumption expenditure in

period T . To delineate expenditures attributed to wealth in period 1T we first

invoke the dynamical equation (2.3) and express 1TW as:

)()1( 1

111 TTTTT WrxpW

. (5.3)

Using the wealth-expenditure function ),ˆ(0

)(

)(

T

W

TT pv T

T

T

in period T and taking

expectation over the random variables T and T~ one can obtain a crucial

identity relating wealth to current and expected future expenditures attributable to

wealth from (5.3) as:


11

0

1 TTT xpW ]),ˆ([)1())(1(

)(

)(

11

1 110

1 j

TT

xpWr

TT

m

T

m

j

j

T pvrj

TTTT

T

T

T

T

T

T

. (5.4)

Using (5.4) the consumer’s dual problem in period 1T can be formulated as

minimizing wealth expenditure

11 TT xp ]),ˆ([)1())(1(

)(

)(

11

1 110

1 j

TT

xpWr

TT

m

T

m

j

j

T pvrj

TTTT

T

T

T

T

T

T

(5.5)

with respect to 1Tx subject to the constraint

)( 1

1

T

T xu )],ˆ([))(1(

)(

)()(

11

1

110

1

T

xpWr

TT

T

T

m

T

m

j

j

TT

T

T pvuj

TTTT

T

TT

T

T

T

T

0

1

1 )(1ˆ

T

T

W

Tv .

(5.6)

Since ),ˆ())(1(

)(

)( 110

1

T

xpWr

TT pvj

TTTT

T

T

is a set of wealth compensated demands

that leads to the level of utility j

TTTT xpWr

Tv

))(1(

)(11

01ˆ , so

),ˆ([))(1(

)(

)()( 110

1

T

xpWr

TT

Tpvu

jTTTT

T

TT

equals

jTTTT

T

xpWr

Tv

))(1(

)(11

01ˆ . Invoking

jTTTT

T

xpWr

Tv

))(1(

)(11

01ˆ ];))(1[( 11

0

1

)(

T

j

TTTT

TpxpWrv T

the constraint (5.6)

can be expressed as:

)( 1

)(1 1

T

Txu T

Tm

j

j

TT

T

T

1

1

T

T

T

m

T

1

T ];))(1[( 11

0

1

)(

T

j

TTTT

TpxpWrv T

01

1 )(1ˆ

T

T

W

Tv .

Setting the Lagrange function and performing the relevant optimization

operation (similar to the analysis in Yeung (2014)) yields a set of first order

conditions. With the implicit function theorem holding, the wealth compensated

demand functions can be obtained as:

),,ˆ( 1)(1

)(

1

)(

1

01

1

11

TT

W

T

h

T

h

T ppvx T

T

TT

, for },,2,1{ 1 Tnh . (5.7)

Substituting the wealth compensated demand functions in (5.7) into (5.5)

yields the wealth-expenditure function in period 1T :


),,ˆ( 1)(1

)(

1

01

1

1

TT

W

TT ppv T

T

T

1 Tp ),,ˆ( 1)(1

)(

1

01

1

1

TT

W

TT ppv T

T

T

Tm

j

j

Tr1

1)1(

T

T

T

m

T

1

),ˆ(),,ˆ(

)(

)( 1

01

1110

1

T

ppvpW

TT pvj

TTTT

W

TTTT

T

T

. (5.8)

Now we proceed to period }1,,3,2{ TTk and let wealth be 0

kW *

kW

and preference be )()(

k

kxu k in the period. Again using (2.3) we can express

wealth in period k as )()1( 11

1

kkkkk WrxpW . Invoking the

wealth-expenditure functions in period 1k and taking expectations over the

random variables 1k and 1~

k , one can obtain the identity

kk xp

1

1

1

1

1

1)1(k

k

k

m

j

j

kr

1

1

1

1

1

k

k

k

m

k

110

1

1

1

))(1(

)(1

)(

1 ),ˆ(

kj

kkkk

k

k j

k

xpWr

kk pv

0

kW , (5.9)

where ),ˆ( 10

1

))(1(

)(1 pvj

kkkk

k

xpWr

k

is the short form for

),,,,ˆ( 21

))(1(

)(11

0

1 Tkk

xpWr

k pppvj

kkkk

k

.

The consumer’s wealth expenditure minimization problem can be expressed as:

kxmin kk xp

1

1

1

1

1

1)1(k

k

k

m

j

j

kr

1

1

1

1

1

k

k

k

m

k

11

10

1

1

1

))(1(

)(1

)(

1 ),ˆ(

kkj

kkkk

k

k j

k

xpWr

kk pv

(5.10)

subject to

)()(

k

kxu k

T

k

T

k

1

1

1

1

1

1

k

k

k

m

j

j

k

1

1

1

1

1

k

k

k

m

k

1k ];))(1[( 11

1

0)(1pxpWrv kk j

kkkk

k

0

)(ˆ k

k

W

kv , (5.11)

for }2,,2,1{ Tk and *0

kk WW .

Setting up the Lagrange function and deriving the first order conditions one

can obtain the wealth compensated demand functions (with the implicit function


theorem holding) as ),ˆ(0

)(

)(pvx k

k

k W

k

h

k

h

k

, for }2,,2,1{ Tk ,

},,2,1{ knh and *0

kk WW .

Similarly, the wealth-expenditure function can be obtained as:

),ˆ(0

)(

)(pv k

k

k W

kk

kp ),ˆ(0

)(

)(pv k

k

k W

kk

1

1

1

1

1

1)1(k

k

k

m

j

j

kr

1

1

1

1

1

k

k

k

m

k

11

10

1

1

1

))(1(

)(1

)(

1 ),ˆ(

kkj

kkkk

k

k j

k

pWr

kk pv

. (5.12)

The wealth compensation demand functions and wealth-expenditure

functions derived in this section represent the dual results of the primal problem in

Section 3.

6 Random Horizon Stochastic Dynamic Slutsky Equations

In this section, we derive the Slutsky equations under a dynamic framework with

uncertainties in the consumer’s future income, future preferences and life-span.

Invoking the duality results in Section 3 and Section 5 we have

),ˆ(0

)(

)(pv

Wh

),( 0)(pW

h

, and ),ˆ(

0

)(

)(pv

W

0

W and

),( 0)(pWv

0

)(ˆ

W

v , for *0

WW and },,2,1{ T , },,2,1{ nh and

},,2,1{ m .

Substituting 0

W by ),ˆ(0

)(

)(pv

W

into the wealth-dependent ordinary

demand function yields the identity:

),ˆ(0

)(

)(pv

Wh

]),,ˆ([0

)(

)(ppv

Wh

(6.1)

for },,2,1{ nh and },,2,1{ m .

One can derive a theorem concerning the relationships between the price

effect of the demand of a commodity and the pure substation effect and the wealth

effect in a random horizon stochastic dynamic framework with uncertain future

preferences as follows.


Theorem 6.1. Random Horizon Stochastic Dynamic Slutsky Equation

under Preference Uncertainty

i

h

p

pW

),( 0)(

i

h

p

pW

),( 0)(0

0)(),(

W

pWh

),( 0 pW

i

,

ki

k

h

p

pW

),( 0)(

ki

k

h

p

pW

),( 0)(

0

0)(),(

W

pWh

m

j

j

k

m

j

jm

j

j k

11

2

1

1

2

2

2

1

1

1

2

2

2

1

1

1

1

2

1

1

mm

k

k

k

m

k

1

k

111

22

11

112

21

1

;

;)(),(

kkj

k

jj

kkj

k

jj

k

k

k

k

W

pWv

),( 112

21

1 ;pW k

kj

k

jj

k

k

i

k

)()1( kr

k

k

k

m

w

w

k

m

w

wm

w

w

11

2

1

1

2

2

2

1

1

1

2

2

2

1

1

1

1

2

1

1

mm

h

h

h

m

h

1

h

1

112

21

1

112

21

1

;

;)(),(

hhw

hww

hhw

hww

h

h

h

h

W

pWv

, (6.2)

for },,2,1{ T , },,2,1{ Tk , },,2,1{ kk ni , },,2,1{, nih

and

},,2,1{ m .

Proof. Differentiating the identity (6.1) with respect to ti

tp yields:

ti

t

Wh

p

ppv

]),,ˆ([0

)(

)(

ti

t

Wh

p

pv

),ˆ(0

)(

)(

),ˆ(

]),,ˆ([0

0

)(

)(

)(

)()(

pv

ppv

W

Wh

ti

t

W

p

pv

),ˆ(0

)(

)(

, (6.3)

for },,2,1{ tt ni , },,1,{ Tt and },,2,1{ m .

Invoking ),ˆ(0

)(

)(pv

W

0

W one can express (6.3) as:

ti

t

h

p

pW

),( 0)(

ti

t

Wh

p

pv

),ˆ(0

)(

)(

0

0

)(

)(),(

W

pWh

ti

t

W

p

pv

),ˆ(0

)(

)(

. (6.4)


To derive the term ti

t

W

p

pv

),ˆ(0

)(

)(

in a more readily computable form we

first note that 0

)(ˆ

W

v ),( 0)(pWv

. To derive the effect on

),ˆ(0

)(

)(pv

W

brought about by a change in ti

tp , with 0

)(ˆ

W

v being held constant,

we totally differentiate 0

)(ˆ

W

v to obtain:

0

)(ˆ

W

vd 0

0

0)(),(

dWW

pWv

i

i

T n

j

dpp

pWv

),( 0)(

1

. (6.5)

With 0ˆ0

)(

W

vd and 0

idp for all },,2,1{ ni and },,1,{ T

except ti

tdp , equation (6.5) becomes

0 0

0

0)(),(

dWW

pWv k

t

t

i

ti

t

dpp

pWv

),( 0)(

,

which yields

ti

tdp

dW 0

t

k

i

tp

pWv

),( 0)(

0

0)(),(

W

pWv

0

)(

0

ˆ

Wi

tvp

Wt

. (6.6)

Invoking ),ˆ(0

)(

)(pv

W

0

W and using (6.6) one can readily obtain

ti

t

W

p

pv

),ˆ(0

)(

)(

0

)(

0

ˆ

Wi

tvp

Wt

t

k

i

tp

pWv

),( 0)(

0

0)(),(

W

pWv

. (6.7)

Substituting (6.7) into (6.4) and invoking the random horizon stochastic dynamic

Roy’s identity in Theorem 4.1, one obtains (6.2). Hence Theorem 6.1 follows.

■

The random horizon stochastic dynamic Slutsky equation under uncertain

preferences in (6.2) generalizes the classic Slutsky equation to a multi-period

framework with uncertainties in future income, the consumer’s life-span and

preferences. In particular, the effect of a price change on the demand of a

commodity can be decomposed into a pure substation effect and a wealth effect.

The left hand side of equation (6.2) represents how the demand for good h at

period changes in response to a change in price ti

tp , and the first term on the

right hand side of the equation gives the change in demand caused by a change in

price ti

tp holding utility fixed at 0

)(ˆ

W

v . The second term on the right hand side


of equation (6.2) is the product of the change in demand when wealth changes and

the required change in wealth brought about by a change in ti

tp with utility kept

fixed at 0

)(ˆ

W

v . Thus, the change in the demand of a good caused by a price

change can be decomposed into a pure substation effect and a wealth effect.

7 Concluding Remarks

This paper extends the conventional consumer analysis to a random horizon

stochastic dynamic framework in which there are uncertainties in the consumer’s

future incomes, future preferences and life-span. The extension incorporates

realistic and essential characteristics of the consumer into conventional consumer

theory. The paper derives two of the most crucial foundations in consumer

theory – Roy’s identity and Slutsky equation – in a random horizon stochastic

dynamic framework with uncertain future preferences. The analysis advances the

microeconomic study on optimal consumption decision to a random horizon

stochastic dynamic framework with uncertain consumer preferences. Further

research, development and propagations which explore further economic

implications of the results in this paper are in order.

8 Appendix: Proof of Theorem 4.1.

Invoking (3.7) we obtain the identity

),( 0)(pWv

)],([ 0)()(pWu

T

T

1 1

1

1

1

1

j

m

j

1

1

1

1

1

m

1 )],([;

1

)(

1

)(1 1111 pWu

j

T

T

2

2

2

2

1

1

1

1

2

1

1

m

j

jm

j

j

1

1

1

1

1

m

2

2

2

1

2

m

2

1

)],([ 1

22

1122 ;

2

)(

2

)(2pWu

jj


T

T

h

h

h

h

m

j

j

h

m

j

jm

j

j

11

2

1

1

2

2

2

1

1

1

1

1

1

1

1

m

2

2

2

1

2

m

h

h

h

m

h

1

h

1 )],([ 112

21

1 ;)()(pWu h

hj

h

jj

hh

hh

h

T

T

h

1

1

1

1

2

2

2

1

1

1

1

1

1

2

1

1

h

h

h

m

j

j

h

m

j

jm

j

j

1

1

1

1

1

m

2

2

2

1

2

m

1

1

1

1

1

h

h

h

m

h

1

1

h

)(1 1 hh

v

),( 11

12

21

1 ;

1 pW hhj

h

jj

h

. (A.1)

Differentiating (A.1) with respect to k

hp yields:

k

hp

pWv

),( 0)(

)],([ 0)()(

1

pWu i

n

i

T

T

h

1

1

1

1

2

2

2

1

1

1

1

1

1

2

1

1

h

h

h

m

j

j

h

m

j

jm

j

j

1

1

1

1

1

m

2

2

2

1

2

m

1

1

1

1

1

h

h

h

m

h

1

1

h

h

hjh

jj

hhj

hjj

h

h

h

h

W

pWv

11

12

21

1

11

12

21

11

;

1

;

1

)(1),(

i

hh

hj

h

jj

W)(

;

11

11

22

11

k

h

i

p

pW

),( 0)(

h

1

m

j

jm

j

jm

j

j

11

2

1

1

2

2

2

1

1

1

1

1

1

1

1

m

2

2

2

1

2

m

m

1

n

i 1

T

T

1 )],([ 11

22

11

)(

;)()(pWu

jjj

i


T

T

h

1

1

1

1

2

2

2

1

1

1

1

1

1

2

1

1

h

h

h

m

j

j

h

m

j

jm

j

j

1

1

1

1

1

m

2

2

2

1

2

m

1

1

1

1

1

h

h

h

m

h

1

1

h

h

hjh

jj

hhj

hjj

h

h

h

h

W

pWv

11

12

21

1

11

12

21

11

;

1

;

1

)(1),(

i

hh

hjh

jj

W)(

;

11

11

22

11

k

h

h

i

p

pW hhj

h

jj

),( 11

12

21

1 ;

1

)(

T

T

h

1

1

1

1

2

2

2

1

1

1

1

1

1

2

1

1

h

h

h

m

j

j

h

m

j

jm

j

j

1

1

1

1

1

m

2

2

2

1

2

m

1

1

1

1

1

h

h

h

m

h

1

1

h

h

hjh

jj

hhj

hjj

h

h

h

h

W

pWv

11

12

21

1

11

12

21

11

;

1

;

1

)(1),(

k

h

h

p

W hhj

h

jj

11

12

21

1 ;

1 .

(A.2)

Using (4.8) we have

i

hh

hjh

jj

W)(

;

11

11

22

11

ih pr 1)1( and

k

h

h

p

W hhj

h

jj

11

12

21

1 ;

1 )1( r ),( 112

21

1 ,;)(pW h

hj

h

jj

h

h

k

h

. (A.3)

Substituting (A.3) into (A.2) yields

k

hp

pWv

),( 0)(

)],([ 0)()(

1

pWu i

n

i

T

T

h

1

1

1

1

2

2

2

1

1

1

1

1

1

2

1

1

h

h

h

m

j

j

h

m

j

jm

j

j

1

1

1

1

1

m

2

2

2

1

2

m

1

1

1

1

1

h

h

h

m

h

1

1

h

h

hjh

jj

hhj

hjj

h

h

h

h

W

pWv

11

12

21

1

11

12

21

11

;

1

;

1

)(1),(

ih pr 1)1(


k

h

i

p

pW

),( 0)(

h

1

m

j

jm

j

jm

j

j

11

2

1

1

2

2

2

1

1

1

1

1

1

1

1

m

2

2

2

1

2

m

m

1

n

i 1

T

T

1 )],([ 11

22

11

)(

;)()(pWu

jjj

i

T

T

h

1

1

1

1

2

2

2

1

1

1

1

1

1

2

1

1

h

h

h

m

j

j

h

m

j

jm

j

j

1

1

1

1

1

m

2

2

2

1

2

m

1

1

1

1

1

h

h

h

m

h

1

1

h

h

hjh

jj

hhj

hjj

h

h

h

h

W

pWv

11

12

21

1

11

12

21

11

;

1

;

1

)(1),(

ih pr 1)1(

k

h

h

i

p

pW hhj

h

jj

),( 11

12

21

1 ;

1

)(

T

T

h

1

1

1

1

2

2

2

1

1

1

1

1

1

2

1

1

h

h

h

m

j

j

h

m

j

jm

j

j

1

1

1

1

1

m

2

2

2

1

2

m

1

1

1

1

1

h

h

h

m

h

1

1

h

h

hj

h

jj

hhj

h

jj

h

h

h

h

W

pWv

11

12

21

1

11

12

21

11

;

1

;

1

)(1),(

)1( r ),( 112

21

1 ;)(pW h

hj

h

jj

h

h

k

h

. (A.4)

Invoking (4.5) we obtain:

112

21

1

112

21

1

;

;)(),(

jjj

jjj

W

pWv


T

T

h

1

1

1

1

2

2

2

1

1

1

1

1

1

2

1

1

h

h

h

m

j

j

h

m

j

jm

j

j

1

1

1

1

1

m

2

2

2

1

2

m

1

1

1

1

1

h

h

h

m

h

1

1

h

h

hjh

jj

hhj

hjj

h

h

h

h

W

pWv

11

12

21

1

11

12

21

11

;

1

;

1

)(1),(

1)1( hr . (A.5)

Using (A.5) the terms inside the square brackets in (A.4) can be written as

T

T

1 )],([ 11

22

11

)(

;)()(pWu

jjj

i

T

T

h

1 1

1

1

1

1

j

m

j

1

1

1

1

1

m

1

11

12

21

1

11

12

21

11

;

1

;

1

)(1),(

jjj

jjj

W

pWv)1( r

jp

.

(A.6)

Invoking the first order conditions in (3.8) the term inside the square brackets

in (A.6) vanishes and therefore (A.4) becomes:

k

hp

pWv ),( 0)(

T

T

h

1

1

1

1

2

2

2

1

1

1

1

1

1

2

1

1

h

h

h

m

j

j

h

m

j

jm

j

j

1

1

1

1

1

m

2

2

2

1

2

m

1

1

1

1

1

h

h

h

m

h

1

1

h

h

hjh

jj

hhj

hjj

h

h

h

h

W

pWv

11

12

21

1

11

12

21

11

;

1

;

1

)(1),(

)1( r ),( 112

21

1 ;)(pW h

hj

h

jj

h

h

k

h

. (A.7)

Using (A.5), one has

1

1

1

1

1

h

h

h

m

j

j

h

1

1

1

1

1

h

h

h

m

h

1hh

hjh

jj

hhj

hjj

h

h

h

h

W

pWv

11

12

21

1

11

12

21

11

;

1

;

1

)(1),(


T

h

T

h

1

112

21

1

112

21

1

;

;)(),(

hhj

hjj

hhj

hjj

h

h

h

h

W

pWv

1)1( r . (A.8)

Substituting (A.8) into (A.7) yields

k

hp

pWv

),( 0)(

T

T

h

h

h

h

m

j

j

h

m

j

jm

j

j

11

2

1

1

2

2

2

1

1

1

1

1

1

1

1

m

2

2

2

1

2

m

1

1 1

h

h

h

m

h

h

111

22

11

112

21

1

;

;)(),(

hhj

h

jj

hhj

h

jj

h

h

h

h

W

pWv

),( 112

21

1 ;)(pW h

hj

h

jj

h

h

k

h

.

(A.9)

Invoking (A.5) one obtains

0

0)(),(

W

pWv

T

T

h

h

h

h

m

j

j

h

m

j

jm

j

j

11

2

1

1

2

2

2

1

1

1

1

1

1

1

1

m

2

2

2

1

2

m

1

1 1

h

h

h

m

h

h

111

22

11

112

21

1

;

;)(),(

hhj

hjj

hhj

hjj

h

h

h

h

W

pWv

hr)1( .

(A.10)

Dividing (A.9) by (A.10) yields another form of the random horizon Roy’s

identity as:

k

hp

pWv

),( 0)(

0

0)(),(

W

pWv

h

h

h

m

j

j

h

m

j

jm

j

j

11

2

1

1

2

2

2

1

1

1

h

h

h

h

m

h

mm

11

2

1

1

2

2

1

1

1

h

1

11

22

11

112

21

1

;

;)(),(

hhj

hjj

hhj

hjj

h

h

h

h

W

pWv


),( 112

21

1 ;)(pW h

hj

h

jj

h

h

k

h

)()1( hr

h

h

h

m

w

w

h

m

w

wm

w

w

11

2

1

1

2

2

2

1

1

1

2

2

1

1

1

1

2

1

1

mm

h

h

h

h

m

h

1

h

1

hw

h

ww

hhw

h

ww

h

h

h

h

W

pWv

22

11

112

21

1 ),(;)(

, (A.11)

for },,2,1{ T , },,2,1{ Th , },,2,1{ hnk and

},,2,1{ m .

Hence Theorem 4.1 follows. Q.E.D.

Acknowledgements. Financial support by HKSYU is gratefully acknowledged.

References

[1] R.G.D. Allen, Professor Slutsky’s Theory of Consumers’ Choice. Review of

Economic Studies, 3(1936), 120-129.

[2] R.G.D. Allen, the Work of Eugen Slutsky. Econometrica, 18(1950),

209-216.

[3] R. Bellman, Dynamic Programming. Princeton, Princeton University Press,

1957.

[4] M.T. Cheung and D.W.K. Yeung, Microeconomic Analytics, New York,

Prentice Hall, 1995.

[5] P.C. Dooley, Slutsky’s Equation Is Pareto’s Solution. History of Political

Economy, 15(1983), 513-517.

[6] T.W. Epps, Wealth Effects and Slutsky Equations for Assets. Econometrica,

43(1975), 301-303.

[7] J.R. Hicks and R.G.D. Allen, A Reconsideration of The Theory of Value.

Parts 1-2. Economica, New Series, 1(1934), 52-76 & 196-219. (Reprinted In

Hicks 1981, pp. 114-32).


[8] R. Roy, La Distribution Du Revenu Entre Les Divers Biens. Econometrica,

15(1947), 205-225.

[9] H. Schultz, Interrelations of Demand, Price, and Income. Joumal of Political

Economy, 43(1935), 433-81.

[10] E.E. Slutsky, Sulla Teoria Del Bilancio Del Consumatore. Giornale Degli

Economisti, 51(1915), 1–26. Translated as “On the Theory of the Budegt of

the Consumer” in G. Stigler and K. Boulding (eds.), Reading in Price Theory,

Published for the Association by Richard D. Irwin, Inc., Homewood, IL,

1952.

[11] H. Varian, Microeconomic Analysis, 3rd ed., W.W. Norton, New York, 1992.

[12] D.W.K. Yeung, Optimal Consumption under an Uncertain Inter-temporal

Budget: Stochastic Dynamic Slutsky Equations. Vietsnik St Petersburg

University: Mathematics, 10(2013), 121-141.

[13] D.W.K. Yeung, Optimal Consumption under Uncertainties: Random Horizon

Stochastic Dynamic Roy’s Identity and Slutsky Equation, forthcoming in

Applied Mathematics, (2014).

[14] D.W.K. Yeung and L. A. Petrosyan, Subgame Consistent Cooperative

Solution of Dynamic Games with Random Horizon. Journal of Optimization

Theory and Applications, 150(2011), 78-97.

[15] D.W.K. Yeung and L. A. Petrosyan, Subgame-consistent cooperative

solutions in randomly furcating stochastic dynamic games. Mathematical and

Computer Modelling, 57(2013), 976–991.

[16] D.W.K. Yeung and L. A. Petrosyan, Subgame-consistent Cooperative

Solutions for Randomly Furcating Stochastic Dynamic Games with

Uncertain Horizon, forthcoming in International Game Theory Review,

(2014).

Received: August 23, 2014

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