Optimal consumption under an uncertain inter-temporal budget: stochastic dynamic Slutsky equations Текст научной статьи по специальности «Математика»

УДК 517.929 Вестник СПбГУ. Сер. 10, 2013, вып. 3

D. W. K. Yeung

OPTIMAL CONSUMPTION UNDER AN UNCERTAIN INTER-TEMPORAL BUDGET: STOCHASTIC DYNAMIC SLUTSKY EQUATIONS

1. Introduction. In a ground-breaking analysis by Slutsky [1], the foundation for rigorous analysis of optimal consumption decision was laid. This masterpiece which brought mathematical rigor to demand analysis is undisputedly an integral part of contemporary mainstream economics. It allows the problem of the consumer to be analyzed in terms of a utility maximization problem subject to a budget constraint. A dual problem to the utility maximization problem is the minimization of the budget (income) subject to maintaining the utility level achieved before. In particular, the effect of a price change on the demand of goods can be decomposed into tractable terms from the primal and dual problems yielding significant economic implications. This prominent contribution in consumer theory, known as the Slutsky equation, was christened by John Hicks as the "Fundamental Equation of Value Theory". In particular, the Slutsky equation states that

dfh(M, p) дф\ум,р) d^h(M,p) .

dpi = ---(1)

where <ph (M,p) is the ordinary demand for good h, фн (vM,p) is the income compensated demand for good h, p = {p1,p2,... ,pn} is the price vector, M is the consumer's income, and vM is the consumer's utility level given a budget of income M.

Ordinary demands are obtained from the consumer's utility maximization problem and compensated demands are obtained from the dual problem of minimizing consumption expenditures subject to achieving the utility level vM. An important economic implication of the eponymous equation (1) is the now famous Hicksian decomposition which separate the effect of a change in price on demand d^h(M,p)/dpj into a pure substitution effect d^h(vM,p)/dpj and an income effect -[dv>h(M,p)/dM]v>j(M,p). The papers by Allen [2, 3], Hicks and Allen [4] and Schultz [5] propagated Slutsky's classic work.

Though Slutsky was aware of the notion of time in his analysis as he wrote x1,x2,...,xn being the quantities of the various goods bought by the subject in a given interval of time, and U the utility gained by the subject by means of that combination of goods his work is an analysis in a single time interval. Fischer [6] and Epps [7] investigated wealth effects in a broad class of portfolio-choice models and Slutsky equations for assets. Yet a direct counterpart of the Slutsky result in pure consumption under a stochastic intertemporal framework still has to be derived. This paper extends Slutsky's work to a dynamic framework in which the consumer has a planning horizon of T periods or time intervals. Moreover, uncertainty in future incomes is introduced to reflect another facet of reality in consumer choice. In particular, utility maximization for the consumer with an uncertain inter-temporal budget is examined. This represents the first time that inter-temporal wealth-dependent ordinary demand functions and wealth compensated demand functions

David W. K. Yeung — professor, 199034, St. Petersburg State University, Russian Federation; and SRS Consortium for Advanced Study in Cooperative Dynamic Games, Shue Yan University, Hong Kong, China; e-mail: [email protected]. © D. W. K. Yeung, 2013

are obtained. Last but not least, the Slutsky equations under an uncertain inter-temporal budget are formulated.

The paper is organized as follows. We first present a model of consumer utility maximization under an uncertain inter-temporal budget in section 2. In section 3, a set of wealth-dependent ordinary demands and characterizes the solution of the utility maximization problem. The dual problem is set section 4 and wealth compensated demand functions are obtained. Stochastic dynamic Slutsky equations for the consumer with an uncertain inter-temporal budget are formulated in the section 5. An illustration with explicit utility functions is given in the section 6. Finally, novel diagrammatic economic analysis emanated from the inter-temporal model are provided in section 7.

2. Dynamic utility maximization under uncertainty. Consider the case of a consumer whose planning horizon involves T periods. We use xk = (x\,xk,...,xkk) to denote the quantities of goods consumed and pk = (pk,p\,... ,pr{,k) the corresponding prices in period k G {1,2,...,T}. The consumer maximizes his expected intertemporal utility

e91,92,...,9t ^ £ V'k (xk ,xl,...,xk' ^ = e9i,02,-,9t ^ £ uk(Xk ^ (2)

subject to the budget constraint characterized by the wealth dynamics

nk nk

Wk+1 = Wk — £Phkxhk + r(Wk — £Phkxhk) + dk = (3)

h=1 h=l

= (1 + r)(Wk - Pkxk) + 0k, Wi = W0,

where r is the interest rate; 6k is the random income that the consumer will receive in period k +1; and 0k, for k G {1, 2,..., T}, is a set of statistically independent random variables, and Eq1q2. qt is the expectation operation with respect to the statistics of d2,d3,..., 0T. The random variable 0k has a non-negative range {dlk,d'1v,---,^''kk} with corresponding probabilities {Ak, X2k, ..., Akk}. The time preference factor is embodied in the utility function. Since the consumer will receive no income in period T + 1, the random variable Qt has a value of zero with probability 1. The function uk(xlk,xk,... ,xkk) is continuously differentiable and quasi-concave yielding convex level (indifference) curves. Moreover, under the axiom of non-satiation, the consumer will spend all his wealth in his planning horizon and therefore WT +1 = 0. Note that the amount of unconsumed wealth Wk — pkxk in period k will generate an interest income r(Wk — pkxk) in period k +1.

To characterize the solution of the stochastic dynamic consumer problem (2)-(3) we first denote the set of optimal consumption strategies by yk (W) = {<pk(W),y2k(W),...,ynk(W)}, for k G {1, 2,...,T} and a set of value functions V(t,W) such that:

V (t,W )= max Egt ,et+i,...,ST { ¿) uk (xk)\Wt = w1 =

Xt,Xt + l,...,XT I J

T

= Ee^e+i^eJ £uk[yk(Wk)]\Wt = wl (4)

^ k=t >

for t G{1, 2,...,T}.

In particular, V(t, W) reflects the expected inter-temporal utility that the consumer will obtain from period t to period T.

Now one can set up a theorem to characterize the solution of the stochastic intertemporal consumer problem (2)-(3) as

Theorem 1. A set of consumption strategies {yk (W)}, for k G {1, 2,..., T}, provides an optimal solutions to the consumer problem (2) — (3) if there exist functions V(t, W), for t G {1, 2,..., T}, such that the following recursive relations are satisfied:

V(t, W) = Eet{ut(xt) + V[t + 1, (1 + r)(W — ptxt) + Qt]} =

Xt

mt

= m&xY Ajt {ut(xt) + V[t +1, (1 + r)(W — ptxt) + Qj]},

Xt

jt=i

for t G {1, 2,...,T}, and

V(T +1,W) = 0 and Wt +i = 0. (5)

Proof 1. The conditions in (5) are standard results from the technique of stochastic dynamic programming (see Bertsekas and Shreve [8], Puterman [9] and Yeung-Petrosyan [10]) which characterize an optimal solution to problem (2)-(3). Hence theorem 1 follows.

eji ej2 ej'-i

We use W* to denote the set which contains all possible values of wealth W^ 2 1-1 at stage l, for all Q3k G {Qk, Qk,..., Qkmk} and k g{1, 2,...,l — 1} along the optimal trajectory generated by theorem 1. Crucial to solving the stochastic dynamic programming problem (2) — (3) with theorem 1 is the derivation of the value functions V (t, W), for t G {1,2,..., T}, and the corresponding consumption strategies. This will be done in the following section.

3. Primal problem and 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 utility subject to an uncertain intertemporal budget. With wealth-dependent ordinary demand functions derived at various periods the value functions in theorem 1 can be explicitly characterized and used to solve the consumer problem (2)-(3).

Invoking the backward induction procedure of Bellman's [11] dynamic programming we first consider the last period of consumption decision, that is period T. Let W 0 G WT* denote the consumer's wealth at period T. Given that V(T + 1, W) = 0 and WT +k = 0, to exhaust all the wealth in this period, W0 — pTxT = 0. Hence the consumer faces the problem

max uT (xT) (6)

XT

subject to

W0 = ptxt. (7)

Problem (6)-(7) 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 [12]) 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 W0 and pT, that is

xT = yh (W0 ,pt ), for h G{1, 2,...,ut }. (8)

One can readily observe that ^(W0,pT) corresponds to the optimal consumption strategies ^T (W0) in theorem 1.

Substituting (8) into (6) yields the indirect utility function in period T as

vT (W0 ,pt ) = uT \pr (W0 ,pt )]. (9)

Invoking the definition in (4), vT(W0,pT) in (9) equals the function V(T, W0) in theorem 1.

Now consider the problem in the second last period T — 1. If wealth equals W0_1 e 1 in this period, the problem in concern becomes

max EgT_ 1 < uk (xk)>

XT-1,XT Z-'

Kk=T-1 J

subject to the inter-temporal budget

Wt = (1+ T)(Wt-i — Pt-IXt-I ) + 0t-I, Wt-1 = W0-1. (10)

Let WTT-1 = (1 + r)(W0_

-1 — Pt-1xt-1) + 1 denote the wealth at period T if 93T_ 1 e

{00 _ 1, 0T-1, ''' , 0T—1 1 } has occurred. Using (9) and (10), the problem facing the consumer

in period T — 1 can be expressed as a single-period problem:

{TT-1

ut-1(xt-1)+ y, k-1VT[(1 + r)(W0-1 — pt-1Xt-1) + 03T-1,pt]\ . (11) First order condition for a maximizing solution yields

mT-1

(xT-1) — E XT-1VWT X T 1 j=1

x[(1 + r)(W0-i — pT-iXT-i) + e°T-1,PT ]pT-1(1 + r)=0, i e {1, 2,...,nT-i}.

(12)

Again, with the implicit function holding, (12) can be solved to yield the ordinary demands in period T — 1 as

xT-1 = vT-i(W0-i,PT-i,PT ) = vT-i(W0-i ), for h e{l, 2,...,nT-i}. (13)

Substituting vT-1 (W0_1 ,pT-1,pT) in (13) into (11) yields the inter-temporal indirect utility function vT-1(W0_1, pT-1, pT). Invoking (4) and theorem 1, vT-1 (W0_1, pT-1, pT) corresponds to V(T — l, W0).

Repeating the analysis for periods T — 2 to 1 yields the consumer problem at period l e{l, 2,...,T — 2} as

{ m; |

ul(xi) + J^ Xjvl+1 [(l + r)(W0 — pf xf ) + ej ;pi+i,pi+2,... ,pt] j> , (14) where W0 e W*.

First order condition for a maximizing solution to the problems in (14) can be obtained as

mi

uXi —J2 Xj V+i+1 [(l + r)(W0 — pixi ) + ej ; pi+i,pi+2,.. .,pt m + r)=0, (15) j=i

for i e{l, 2,...,ni} and l e{l, 2,...,T — 2}. 124

Note also the condition that period l, good i will be consumed up the point where marginal utility of consumption ulxi (xi) equals (1 + r)p\ times the expected marginal utility

of wealth £k=i H(W0 — p^xx^ + Qj;pi+i,pi+k,... ,pt). Solving (15) yields the ordinary demands in period l as

xh = yh(W0,pi ,pi+i,... ,pt ), for h G{1, 2,..., ni}. (16)

Substituting (16) into (14) yields the inter-temporal indirect utility function vi(W0,pi, p i +i,.. .,pt ), for l G{1, 2,...,T — 2}.

Hence, a solution to theorem 1 can be obtained with the set of optimal consumption strategies y1h(W) being the set of wealth-dependent ordinary demand

y1h(W,pk ,pk+i,... ,pt ), for h G {1, 2,...,nk} and k G {1, 2,...,T};

and the value functions V(k, W) being the inter-temporal indirect utility function

vk(W,pk,pk+i,. ..,pt ), for k G{1, 2,...,T}.

After solving the primal consumer problem which maximizes expected utility subject to an uncertain inter-temporal budget, we proceed to consider the dual problem in the next section.

4. 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 by considering the dual problem of minimizing expenditure covered by the current wealth subject to maintaining the level of utility achieved in the primal problem. Again we first consider the last period in which W0 G W0 is the consumer's wealth. Since wealth equals income in this period, to derive the compensated demand we follow the standard single period consumer problem of

min pT xT (17)

XT

subject to achieving the level of utility

uT (xt )= vWT = vT (W0 ,pt ).

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

xhT = № (vWT ,pt ), for h G{1, 2,..., nT}. (18)

Substituting (18) into (17) yields the wealth-expenditure function

£t (v0wt ,pt )= pt ^t (v0wt ,pt ) = W0. (19)

Now we proceed to period T — 1 and let wealth in this period be W0_i G W*_i. To

obtain the wealth compensated demand function in period T— 1 we consider the problem of

minimizing expenditure covered by current wealth in the period to bring about the expected w t

inter-temporal utility Vy^Z-1 = vT-i{W0_i,pT-i,pT) from the primal problem. However,

wealth W0_ _i in period T — 1 does not only cover consumption expenditure pt-ixt-i in the period T — 1 but also part of the consumption expenditure in period T. To delineate expenditures attributed to wealth in period T — 1 we first invoke the dynamical equation (3) and express WT-i as

Wt -i = pt-ixt-i + (1+r)- (Wt — Qt-i)- (20)

Using (19) and taking expectation over the random variable QT-i in (20) one can obtain a crucial identity wealth to current and expected future expenditures attributable to wealth as

W0-i = pt-ixt-i + (1+ r)- x

(21)

x £ j-AZt(VWt ,pt) — Qj-i], j=i

ej

W °T -1 pj pj

where VWT = vT(WTT-l p) and WTT-l = (1 + r)(W0-i — pt-ixt-i) + Qj-i. Using (21) the consumer's dual problem in period T — 1 can be formulated as minimizing wealth expenditure

mT-i ejr-l

pT-ixT-I + (1 + r)-iY, j-ilfr(VW ,pT) — Qj-i], (22)

j=i

with respect to xT-i subject to the constraint

mT-i j -l T

uT-i(xT-i)+yl j-iuT(VWT ,pT)]= VW--1. (23)

j=i

W -i

Since ^T (VTT ,pT) is a set of wealth compensated demands that leads to the level of

utility VWt , so uT x T (VWt , pT)] equals VWt . Invoking VWt = vT [(1 + r)

70

-i pT-ixT-i ) + QT-i;

mT-l

i(xt-i)+ j2 XT-ivT[(1 + r)(W0-i — pt-ixt-i)+QjT-1;pt]= VT-i . j=i

The corresponding Lagrange function of the problem in period T - 1 becomes

(W0_i — pT-ixT-i) + Qj_^pT] the constraint (23) can be expressed as

uT-i(xT-1)+ V x^ VT [(1 + r)(W°_i- VT-ixt-i)+Qi_1; pt] = VW-

L(xt -i,£T-l )= pt -ixt -i + (1 + r) i

mt-l

X E XjT-i[zt(VW ,pt) — QT-i] +

j=i

WT-i WT

+ £vt-1 X [V-1 — uT-i(xT—

mT-l

— Z XjT-iVT[(1 + r)(W0-i — pT-ixT-i)+QjT-i;pt]]

(24)

j=i hwT -i

where £ t-1 is the Lagrange multiplier.

w

X

Perform the relevant minimization on the Lagrange function in (24) one obtains

pt-i— Ê X0

8:

WT

T-1

d£.T(vTT ,pT)

-1 Z^ ^T-1 ei

j=1 ,,WTT

T-1

dvlp T . <W0-1 t-i (25)

x —2—,-p^-e^-i [u1, (xT_i) -

8T-1 UjT-1

T

- + l(i + r)] = 0>

j=1 dwTT -1

for i e{l, 2,..., nT-1}; and

x

v

~w--1 — uT-1(xT-i) X0-iVT[(l+ r)(W0-i — pt-ixt-i) + e0-i;pt] = 0. (26) j=1

v wf

Condition (25), (26) yields a system of equations with variables xT-1 and £Vt-1 and

wT

VT -1

compensated demand functions can be obtained as

T 0

parameters pT-1, pT and V0T-1. With the implicit function theorem holding, the wealth

w o

X0-1 = ^T -AVt-i1 , pt -1, pt ), for h e{1, 2,...,nT-1}. (27)

Substituting the wealth compensated demand functions in (27) into (22) yields the wealth-expenditure function in period T — 1:

wT 1 wT 1

£t-i(vtt1 , pt -1, pt )= pt -iWt -1 (v0 t- ,pt-i,pt ) +

T-1

+ (1+ r)-1 53 X0-1 fr(T \pt)■ (28)

T=1

Now we proceed to period T — 2 and let wealth be W0_2 e 2 in the period. Again using (3) we can express wealth in period T — 2 as WT-2 = pT-2xT-2 + (1 + r)-1(WT-1 — 0T-2). Taking expectations over the random variable 0T-2 and invoking (28), one can obtain the identity

pt-2Xt-2 + (1 + r)-1 X

tt-2 eT-2

X 53 XT-2 £T-1(V^T-1 ,pt-1,pt) — 0T-2] = W0-2, (29)

T=1

eT — 2 Tj Tj

where Vwt-1 = vT-1(WTT--12,pt-1,pt) and WTr--12 = (1 + r)(W0-2 — pt-2Xt-2) + 00-2.

The consumer attempts to minimize wealth expenditure represented by the left-hand-side of identity in (29) subject to the constraint

TT-2 0

uT-2(xt-2)+Y: x0-2vT-1[(1+ r)(W0_2 — pt-2Xt-2) +0T_2\pt-1,pt]= v7t-2

/ , "T-2V )\"T-2 — LJ0 -2^ 0 -2)^ UT-2ilJ0 -hF 0 J UT-2

j=1

Following the above analysis for period k G {T — 3,T — 2,...,!}, the consumer's wealth expenditure minimization problem can be expressed as

mk ,,.ejk

I 1 V W k

min ^Pk Xk + (1+r)" (ik+1 (v + ,Pk+1 ,Pk+2,...,PT ) — dik)

jk=1

subject to

uk(xk) + £ Af vk+1 [(1 + r)(W0 - pkxk) + 0jk; pk+upk+2, ...,PT ] = vW , jk=1

for k G {1, 2,...,T - 3} and W0 G W*k.

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

xk = ^k(v W ,P k,P k+1, . ..,PT ),

for k G {1, 2,...,T - 2}, h G {1,2,...,nk} and W0 G Wkk.

Similarly, the wealth-expenditure function can be obtained as

£k (v W ,Pk,Pk+1, . . . ,PT ) = Pk^k (v W ,Pk,Pk+1,. ..,Pt ) + mk ■ wekk

+ (1 + r)-1Y, Ajkk (Sk+1 (vk+11 ,Pk+1,Pk+2,..., Pt) - of). jk=1

The wealth compensation demand functions and wealth-expenditure functions derived represent the dual results of the primal problem.

5. Stochastic dynamic slutsky equations. In this section, we derive the Slutsky equations under an uncertain inter-temporal budget. Invoking the duality results in the third and fourth sections we have

^k(v Wk ,Pk ,Pk+l,... ,Pt ) = yk(Wk,Pk,Pk+1,..., Pt ), (30)

and00

£k (vk k ,Pk ,Pk+1,...,PT) = W0 and vk (W0,Pk ,Pk+1,...,PT) = vwk,

for Wk0 G Wl and k g{1, 2,...,T} and h e{1, 2,..., n}.

Substituting W0 by £k(vWk ,Pk,Pk+1,...,PT) into the wealth-dependent ordinary demand function in (30) yields the identity

^k(v W ,Pk,Pk+l, ...,Pt ) = [£k(v W ,Pk,Pk+l,..., Pt ),Pk,Pk+l,..., Pt L (31)

for h G {1, 2, ...,nk}. 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 an uncertain inter-temporal framework as follows.

Theorem 2. In an uncertain inter-temporal framework, the effect of a price change on the demand of a commodity can be formulated into a stochastic dynamic version of the Slutsky equation as

d(Pk(Wk>Pk,Pk+i,---,Pt) = dip^(vyk ,Pk,Pk+i, ■ ■ ■ ,Pt)

dPj dPj

, dy>k(Wk'Pk,Pk+i, • • • ,Pt) dvk(W°,pk,pk+1,... ,pT)

+ dw° dpi ■ {i2)

^ dvk(W%,pk,pk+1,...,pT) dwk

for j = {1, 2,. ..,ni} and l e {k, k +1, ...,T}.

Proof 2. Differentiating the identity (31) with respect to pj yields

dpj dpj drim*r>,pk,pk+u .. .,pt),pk,pk+u ,pt] d^vt\pk,pk+u ■ ■ ■ ,pt);

d£k (v ,.k ,pk ,pk+1,...,pT) dpj

for j e {1, 2,...,ni} and l e {k,k + 1,...,T}.

Involving £k(vWk ,pk,pk+1,... ,pT) = W° one can express (33) as

d<A(W0,pk ,pk+1, ...,pt ) _ di^l (vTk ,pk,pk+1,... ,pt )

(33)

dpj dpj

dfUWk>Pk,Pk+1, ...,Pt) d£k(i>Yk ,Pk,Pk+1, • • • ,Pt) 9W°k dpi

(34)

To derive the term —'Pk'Pk+1'in a more readily computable form we first note that

dpl

vW = vk (W0,pk ,pk+1 ,...,pt ). (35)

To derive the effect on (vWk ,pk,pk+1,... ,pT) brought about by a change in p:j, with v^ being held constant, we totally differentiate (34) to obtain

,o »„krvSfO . T A.,kfT/f/0

W o

With dv t k =0 and dpT = 0 for all jh e {1, 2,..., nh} and h e {k, k + 1,...,T} except

dpj, equation (35) becomes

0 0

9W° dWk + ^ dPl'

which yields

dW°k dvk(W^pk,pk+u...,pT) . dvk(W^pk,pk+u...,pT) _ 8W°k

dpj dpj ' dW0 dpj vkWk

W o

Involving £k(vt k ,pk,pk+1,... ,pT) = W° and using (35) one can readily obtain

<17 ~~ <1 7 \~wk

dpj dpj vkk

0

dvk(W°,pk,pk+1,...,pT) ^ dvk(W°,pk,pk+1,...,pT)

dp] ' aw° ■ [6i)

Substituting (37) into (34), one obtains (322). Hence theorem 2 follows.

The stochastic dynamics Slutsky equation (32) in theorem 2 generalizes the classic Slutsky equation to a multi-period framework with uncertainty. 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 (32) represents how the demand for good h at period k changes in response to a change in price pj, and the first term on the right hand side of the equation gives the change in demand caused by a change in price pj

W 0

holding utility fixed at vkk. The second term on the right hand side of equation (32) is the product of the change in demand when wealth changes and the required change in wealth

jW0

brought about by a change in pj with utility kept fixed at Vkk. 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. Moreover, the price change can come from the change in a price in the current period or that in any other future periods.

In the last period, that is when k = T, wealth W0 e WT* becomes a single period income and in this single period problem,

dvT(W°,PT) . dvT(WZ,pT) 0 , --^ dw.° - -<pt(wt,pt)

because of the Roy's identity [13]. Therefore equation (32) collapses into the standard Slutsky equation (1).

6. An illustration with explicit utility function. In this section, we illustrate the derivation of the results in section 3 to 5 with an explicit form of the consumer's utility function. We consider a consumer with a 2 period horizon and his utility function in period k e {1, 2} is given by uk(x\,x?k) = olk\n(x\) + a2k\n(x?k). His initial wealth in period 1 is W0. In period 2 he expects to receive an income d\ with probability \\ for j e {1, 2}. The consumer maximizes the expected inter-temporal utility

E9l{ £[«k ln(xk ) + ak ln(xk)]} (38)

,.k=l

subject to the budget dynamics

Wk+i = Wk - £phkxhk + r(Wk - £ph4) + 6k, W1 = W0.

h=1 h=1

Note that the utility function in each period in (38) follows the generalized Cobb-Douglas utility preference (see Varian [14]). We first derive the wealth-dependent ordinary demands and obtain the intertemporal indirect utility function. Then we derive the wealth compensated demands and the wealth expenditure function. Finally, we compute the relevant partial derivatives and derive the stochastic dynamic Slutsky equations.

Wealth-Dependent Ordinary Demand. Following the analysis in the third section, the wealth-dependent ordinary demand functions in period 2 can be obtained as

4 = ^(Wlplri) = fori G {1, 2}.

(a2 + a2 )P2

,W0

The indirect utility function in period 2 becomes

1 2

v2(WIpIpD = (o4 + al)HW2°) + a'M^^) + aj\n(-

a.2 + a2 a2 + a2

- a2ln(p2) - al\n(pl).

Now we proceed to period 1. Following previous analysis we obtain the wealth-dependent ordinary demand functions in period 1 as

Proposition 1. The wealth-dependent ordinary demand functions in period 1 are

i ctirO -, — v —

=--, (39)

2Aip1

where Ai = [(«} + aj) + (o^ + a22)}(^±),

ai

Bi = ([W0 + e2(1 + r)-1 ][(a1 + a2) + XKal + a2)] +

+ W0 + e\(1 + r)-1][(a\ + a2) + (a1 + a2)A2j) , Ci = a\[W0 + dl(1+ r)-1][W0 + d2(1+ r)-1 ], fori e {1, 2}. (40)

Proof 3. See Appendix A.

Using (40), the inter-temporal indirect utility function in period 1 can be obtained as

wo ^ i, ,B1-y/B?-4A1C1. , 2. ,B2 - ^B2 - 4A2C2. v\W0,pl,p2) = a\H--) + a?In(-^-) +

2

^ E X2{ (a2 + al)H(1+ r)(W0 - pixi) + ej] + j=2

12 \

+ o4ln( 1°;2 2) + a2^n( T~~ 2 ) - a\Hp\) - a2Mp22) \ , a1 +a2 a2 + a2 J

where x2 = [y1(W0,p1,p2),y2(W0,p1,p2)] is as given in proposition 1.

Wealth Compensated Demand. Following the analysis in the fourth section we obtain the wealth compensated demands in period 2 as

i l

'^(vY^pIpD = [exp(^0)]^?(%)^f, fori, / e {1, 2} and i ± I.

a2 p2

The expenditure in period 2 becomes

6(v Y2 ,p2,p2)

I 1 a£ 2 a1 a£ a1

ti/0 Cv C\ 2 2

= [exp(uf2)]^i [(2)^? + ( =Wo

a22 a12

Now we proceed to period 1. Following the analysis from equation (20) to equation (28) the wealth compensated demand function in period 1 can be obtained as

as

Proposition 2. The wealth compensated demand functions in period 1 can be obtained

Wi (v 1 ,pi,p2) =

= exp

(a1 + a2)

V W° —Y, Xi {(a2 + a2)ln[(l+ r)(W0 — pv) +j ] +

j=1

+ «2In( 1°;2 +a%ln( 1"2 ) - alln(pl) - az2ln(pz2)

2 a2 + a22 2 a2 + a22 2 2 2 2

' «1 P\

for i, l e {l, 2} and i = l.

(42)

Proof 4. See Appendix B.

Using (42) the wealth expenditure function in period 1 can be expressed as

2 ( 1

a^ a2+ai

+ -i

a2\ a2+ai

(p\)ai+ai (p\)ai+ai - e{

where if and I2 are given in (42).

Stochastic Dynamic Slutsky Equation. To establish the stochastic dynamic Slutsky equations as in (32) we have to obtain

d(fi\ d'ipl d(fi\ dv1 dv1 .

(43)

In the sequel, we obtain all the partial derivatives in (44). We first partially differentiate

Vi(W0,p1,p2) in (39) with respect to W0 and obtain

dcp\ _ dBi/dW0 - [Bf(dBi/dW0) - 2A^dCi/dW0)} dW0 ~

2AiP\vŒ2^IX~Ci where dBi/dW0 = 2(a 1 + a2) + (a2 + a2), and

dCi/dW0 = ai [2W0 + e 1(1 + r)-1 + e 2(1 + r)-1 ].

, for i e {l, 2},

(44)

Moreover, dv1

dW0

ai dvi a2 dv2 f lp[ dW0 + tf dW0

+ Xj1

(a2 + a2 )(l+ r)

j=1

(1 + r)(W0 — £ 1=1 pk vk ) + ej

H 1 dv\ 2 dv2 n

(45)

i

x

X

2

a

a

2

2

X

= -ai Bi - y/B'f - AAiC,

dp\ yi1

2Ai(p\)2

--EA i

3=1

12 a2 + a2

(1 + r)(W0 -£k=ipkyk) + 3

(1+ r)y\ - (1+ r)p\

4 Bi - y/B'i - 4AiCi

2Ai(p\)2

for i e {1, 2}

and

dv1

-E=iA>2fc-L fori €{1,2}.

_ _

Qpk — j=L ±

Differentiating the wealth compensated demand in (42) with respect to price one obtains the following substitution effects brought about by price changes

di> dp\

1 - I Z1——2 I

a11 + a21

2 1 i 2 (pir' + JZxX

j=1

a11 + a21

X

1+r

[(1 + r)(W0 -p\y\ -p2y2) + d\]

dW0

2 dlPi PldW0

for i, l e {1, 2} and i = l,

(46)

M _

p)-1 -

2

j=1

12 Ai —i—■—n

1 + r

a11 + a2\ [(1+ r)(W0 - p\y\ - p2y2)+ei]

for i, l e {1, 2} and i = l;

dW0

dp2

1 l d(Pi 1 -Pl

dW0

M

8W0

(47)

d^i dpf

= ri

(a11 + a2)

5>*

j=1

p2

'2 1 , 2

x

j=1

1 9

a11 + a21

(48)

1+r

[(1 + r)(W0 - p\v\ - p21y21)+031 ]

dW0

dp%

1 - p11

M

dW0

2 diA

PldW0

for i,w e {1, 2},

Differentiating the wealth-dependent ordinary demand in (40) with respect to price yields

a-A B - y/Bf-AAiQ and d<p\

dy1 dy1

dpi1

2Ai(p\)2

dpk

dp\

dp2

0.

(49)

Substituting the partial derivatives obtained in (45)-(49) the set of stochastic dynamic Slutsky equation in (44) are specified explicitly.

x

x

x

1

a

2

x

Numerical Verification of the Slustky Equation. To verify the duality results and Slutsky equations numerically we consider the illustration in section 6 with the following parameter values: a \ = 0.4, a\ = 0.6, a2 = 0.35, a\ = 0.65, A \ = 0.6, X\ = 0.4, d\ = 45, d2 = 65, p\ = 3, p2 = 2, p2 =2, =4, r = 0.004, W0 = 120. Using these parameters we first show the equality of the wealth-dependent ordinary demand and the wealth compensated demand. We also derive the indirect utility and wealth expenditure. Then we compute the partial derivatives (45)-(49) and verify the stochastic dynamic Slutsky equations.

In table 1, the results showing that the ordinary demand y>\(W0,p\,p2) equals the

W0

wealth-compensated demand (£x 1 ,p\,p2), for i G {1, 2}, are given in the first two rows. The indirect utility and wealth expenditure are given in the third row. Note that wealth expenditure is indeed equal to initial wealth W0 = 120.

Table 1. Numerical depiction of weaith-dependent ordinary demands, wealth compensated demands, indirect utility and wealth expenditure

Ordinary Demands & Indirect Utility Wealth Compensated Demands & Expenditure

<p11(W°,p1,p2) 11.33155866 iH-wï ^ VlK ,P1,P2> 11.33155866

<P21(W°,p1,p2) 25.49600698 i2/~wi \ VlK ,Pl,P2) 25.49600698

v1(W°,p1,p2) 5.610421304 1-1 C to 120.000000

Table 2. Numerical depiction of the partial derivatives in (45)—(49) and stochastic dynamic Slutsky equations

Stochastic Dynamics Slutsky Equation 1 Stochastic Dynamics Slutsky Equation 2

d<p\/dW^ 0.067031375

—dv1/dp\4- 11.33155866 —dv1/dpf-\r 25.49600698

dip\/dp\ -3.777186219 dip\!dp\ 0.000000

d'tt>l/dp\ -3.017616267 d'ipl/dpf 1.709032393

Bvi ev1 . . dv1 • -0.759565731 6<p[ Qv1 . . dv1 • ëwf -1.709032393

dip\/dp\ = = di>\ldp\ + , dv1 ^ dp{ ew\> -3.777186219 = -3.777186219 dip\/dp'[ = = di>\ldp\ + dv\ gvi . + aw" e^f ' . dv1 • dW\} 0.000 = 0.000

Stochastic Dynamics Slutsky Equation 3 Stochastic Dynamics Slutsky Equation 4

—dv1/dp2 -r 14.87267074 —dvi/dp2^r 13.81033711

dipj /dp2 0.000000 dipj /dp^ 0.000000

di/>| /dp2 0.996935562 dip [/dpi. 0.925725879

dv1 . 8W° dp\ ' . dv1 • ëwf -0.996935562 dv1 . aw" dpi ' . dv1 • ëw^ -0.925725879

dip^/dp2 = = dfl/dpl + 1 Qv 1 . ^ ew° dp\ • . dv1 ' dW\> 0.000 = 0.000 dip[/dp g = = di>\/dpl + 1 Qv 1 . ^ Wf ' . dv1 • ew,0 0.000 = 0.000

End to Table 2

Stochastic Dynamics Slutsky Equation 5 Stochastic Dynamics Slutsky Equation 6

difl /dW^ 0.150820593

dip{ /dpf -12.74800349 Off /dpf 0.000000

d'ipf /dp\ -8.902680606 dipf /dp] 1.709032393

Bv 1 dv1 . ÖW" dpi • . dv1 ' ÖW" -3.845322892 Bvi dv1 . ÖW" dp\ • . ÖU1 • ÖW" -1.709032393

dip'f /dpf = = dtßf/&pf + difl Qv1 . t8W1° dpi ' . dv1 ' ÖW" -12.74800349 = = -12.74800349 dipf/dpf = = dff/dpf + dipf Qv1 . ^ ÖW" dp\ ' . dv1 ' Sfff 0.000 = 0.000

Stochastic Dynamics Slutsky Equation 7 Stochastic Dynamics Slutsky Equation 8

dipf /dp 2 0.000000 difl /dpi 0.000000

dipf /dp£ 2.243105015 dipf /dpi 2.082883229

6<fi'i dv1 ■ ÖW" dp\ • . dv1 ' ÖWf -2.243105015 6<fi'i dv1 ■ ÖW" dpi ' . dv1 ' ÖW^ -2.082883229

dipf/dpi = = d'4>21/dp12 + i Bifl gv 1 . t8W1° dp\ ' . dv1 ' dW\> 0.000 = 0.000 dipf /dpi = = d^l/dpl + 1 Bifl ötJl . ^ ÖW" dpi • . ÖU1 • dW\> 0.000 = 0.000

In table 2, the results for the eight stochastic dynamic Slutsky equation

d(fi\ drip\ d(fi\ dv1 dv1

^,/€{1,2},

are given as stochastic dynamic Slutsky equations 1 to 8 in table 1. The numerical values of partial derivatives (45)-(49) are derived and the Slutsky results are shown in the last row of each equation block.

7. Novel diagrammatic analysis in consumer theory. One of the most profound developments of the Slutsky equation by Hicks and others in consumer theory is a diagrammatic analysis which appears in the almost all microeconomic textbooks. In particular, in a single-period consumption model, suppose a consumer with preferences given by the indifference curve containing point a and his spending constraint is given by the budget line AA in the conventional price-substitution-income effect diagram (figure 1). He will choose the consumption bundle indicated by point a. Consider a decrease in the price of good 1. The budget line will become AC and a new equilibrium at point c will result. The movement from a to b depicts the pure effect. The movement from b to c represents the income effect added to the pure substitution effect in a price decrease.

To illustrate we begin with the simple case in two periods and two goods in each period and future income equals O\ = Of = O1 with certainty. The optimal consumptions in period 1 and period 2 are depicted by the points ai and af in the inter-temporalprice-substitution-wealth effects delineation. The utility level obtained from period 1 consumption along the indifference curve containing a1 is u\; and the utility level obtained from period 2 consumption along the indifference curve containing a2 is ua (figure 2).

Figure 2. Inter-temporal price, substitution and wealth effects delineation

The budget line A1 A1 gives an expenditure or spending in period 1 as Y2 and the budget line A2A2 gives an expenditure or spending in period 2 as Y2°. Moreover, Ya + Y22 =

W0 + r(W0 — Y0) + 91. The consumer's inter-temporal utility is u\ + u2 = vW.

Consider a decrease in the price of the first good in period 1 from p 1 to p 1. The new budget line in period 1 becomes C1C1 and yields a spending of Yf, and the new budget line in period 1 becomes C2C2 and yields a spending of Y2c. Yhe optimal consumption bundles in period 1 and period 2 are respectively c 1 and c2.

Note that there is a wedge between A1 and C1 along the vertical axis which reflects that spending in period 1 after the price decrease is different from the original spending. Whether Yf and Yf would be the same as the original spending Y2 and Y2 depends on the preference of the consumer. In many cases, there exists inter-temporal substitution in spending so that Yf and Yf would not be exactly the same as Ya and Y2°. As a simple

demonstration of intertemporal substitution of consumption spending we consider the case where the consumer's intertemporal utility is

[(a(xi)p + (1 - a)(x2)p)1/p]1/2 + [(a(x12)p + (1 - a)(xf )P)1/P]1/2,

and 0 < a < 1 and 0 < p < 1. We first consider the single-period consumer problem which maximizes

[(a^x1 )p + (1- a)(x2)p)1/p]1/2 subject to the spending constraint

p1x1 + p2x2 = Y. The corresponding indirect utility function is

V(Y,p\p2) = Y1/2[a'J(p1)1-'J + (1 - aYip2)1-*]^.

Making use of the indirect utility function the original inter-temporal utility maximization is analogous to the following problem which chooses spending Y1 in period 1 to maximize inter-temporal utility

(y!)1/VCpi)1-7 + (i - anpl)1-"}^ +

+ [(1 + r)« - YJ + 01]1'VC^)1-"7 + (1 - anp2)1-*}^. Solving the above problem yields an optimal choice of Y1 as

Y =_[(1 + r)W? + e^jpl)1- + (1 - ar(p21)1-'J}^_

[a<J(p}_)1-<J + (1 - a)17(p21)1-'I}TJ^ + (1 +r)[a<J(^)1-<J + (1 - a)<J(^)1-<J]T^

The derivative of spending Y1 with respect to p1 is positive, and therefore Y1 < Y". Analysis and evidence of inter-temporal substitution in consumption (spending) have been provided in [15], [16] and [17]: in the inter-temporal delineation, wealth-compensating budget lines in period 1 is depicted as B1B1 and wealth-compensating budget lines in period 2 is depicted as B2B2. The wealth-compensated consumption in period 1 and period 2 are depicted by the points b and b2 respectively. The utility level obtained from period 1 consumption along the indifference curve containing b 1 is u 1, and the utility level obtained from period 2 consumption along the indifference curve containing b2 is uf. As stated before,

wealth compensated leads to u1 + uf = u" + u" = vW .

The pure substitution effect in period 1 is the movement from a to b and the pure substitution effect in period 2 is the movement from a2 to b2. The wealth effect in period 1 is the movement from b1 to c1 and the wealth effect ion period 2 is the movement from b2 to C2.

The price, substitution and wealth effects in an inter-temporal framework depicted are significantly different from the price, substitution and income effects in the conventional one-period consumer model. Finally, diagrammatic analysis for more than two periods can be obtained and analysis with uncertainty in income can be constructed following the above explication.

8. Concluding remarks. This paper extends Slutsky's analysis to a stochastic dynamic frame work in which the consumer has a planning horizon of T periods and there is uncertainty in future incomes. The extension incorporates realistic and essential characteristics of the consumer into conventional consumer theory. The paper provides three novel features in mathematical and economic analysis. First, the discrete-time dynamic stochastic utility maximization model analyzed is new. For the first time, wealth dependent ordinary demand functions are derived. Secondly, a non-trivial construction of the dual problem of wealth-expenditure minimization in multi-stages is provided. For the first time, wealth compensated demand functions are derived. Thirdly, a set of stochastic dynamic Slutsky equations for the consumer is derived for the first time. A novel diagrammatic analysis in consumer theory can be derived from the stochastic dynamics Slutsky equation (32) r - temporal consumption framework.

Moreover, using the intertemporal consumption function, novel elasticity concepts and measurements which do not exist in the conventional one-period consumer theory could be obtained. These include:

(i) wealth elasticity of spending = ,PT) w?

aw»

9pkVk(Wk ,Pk,Pk+i,--- ,PT)

dp3.

(ii) price elasticity of spending =

(iii) interest elasticity of spending

(iv) wealth elasticity of demand = ,pt

Pk<Pk(Wj>,pk,Pk+1,- ,Pt) '

Pk<Pk(Wg,pk,Pk+l,~- ,Pt) '

dpk<fk(Wk ,Pk,Pk+!,••• ,Pt)

dr

PkVk(W£,pk,Pk+1,-" ,Pt) '

W0

f^(W°,pk,Pk+,Pt) '

The analysis advances the conventional static microeconomic study on optimal consumption decision to a stochastic dynamic framework. Further research, development and propagations similar to those by Allen [2, 3], Hicks and Allen [4] and Schultz [5] on Slutsky's work which explore further economic implication of the results in this paper are in order.

Appendix A: Proof of Proposition 1. The problem facing the consumer in period 1 can be expressed as

max ^ \a

xi

[a 1 ln(x 1) + a?ln(x?)j + ^ \{ («2 + «2)ln(1 + r)(W0 - p 1 x 1) + d{ j=1

+

+ o^ln ( 1°;2 + «2In ( 1°;2 2 ) - a\\n{p\) - a22ln(p22)

\a2 + a2/ Va2 + a2

First order condition for a maximizing solution yields

2 («2 + «2)(i + r)pi

ii-VA'

^ + r){w?-Plx1) + e[

= 0, for j = 1, 2.

(50)

Using (50) one obtain p2x2 = —\p\x\, and upon substituting into (50) yields

21 _ xj_(a\+ai)p\

x\ 1 №0 _

0.

Equation (51) can be reduced into a quadratic equation in x1 with roots

r

x

1

Si ± VB'f -AAïC

2Ap\

(52)

where Ai, B i and Ci as given in (39).

One can show that there are two positive real roots and the smaller root yields represents a utility maximizing solution. Following similar analysis, one can obtain ^>2(W0,pi,p2) as that stated in (39). Hence proposition 1 follows. Q.E.D.

Appendix B: Proof of Proposition 2. The consumer's dual problem in period 1 can be formulated as minimizing

[ iX i+ p\x\] + (l + r)-1 £a{ x j=i

exp(41+r)(<^ xi)+ffj )

1 \ i * ,

a2 \ a2+a2

+ I -T

a2

(pl)al2+a2 (pl)al2 + a2 -9{

subject to the constraint

•■ i ln(x i) + a2ln(x2) + ^ Ai (i(i+r)(W2-p2xi)+ej ) = vW.

a i ln(x i ) + a i ln(x

j=i

Setting up the corresponding Lagrange function of the problem and performing the relevant minimization yield the first order conditions

1

p'i

1 ¿Î «2 +«2

2 exp(v2i+r)(W0-PlXl)+0j ) X

i a2 \J ± _Â_

2

a22 a2 2 2

dv.

( i + r)(W0-p2 x1) + 0j

R3

awR1

-p\ +

i

x

1

2

a2+a2

a2+a2

a

x

2

a

2

W0 + £v 1

2

x i j=i

-a — +

' Vi+r)(W0-pix 1 )+R3 \

^-?- (l+r)M

dw2R3 J

= 0, i = {1, 2},

1 - a iln(xi) - a2ln(x2) - Ai (

j=i

,(i +r)(W0-p 1 X1 ) + R 3 ^

(53)

where

2

,( i+r)(W0-p 1 x 1 )+Rj

(a2 + a\)ln(W;R ^

a2 ln

(a2 + a2,)

+ a2ln

(a2 + aR)

—

- a2ln(p2) - a2ln(p2), and WR1 = (1 + r)(W10p ixi) + d{.

i

2

a

a

2

2

2

One can readily obtain

( 1+r)(W0-p1x1)+Ol ^

1+r)(Wi -P1X1)+,11 a2 + a2

ft3 ft3 dw;1 w;1

Note that WR1 = ,pl ,pf). Invoking (41) we can express (54) as

dv

(1+r)(w0-P1X1)+e3 2_

3

W , 2 , (55)

a^ a Q a Q a^

Using (55), one can reduce the first two equations in (53) to

w0 — £Vw 1

1 j=i

' dv(i+r)(w0-p1X1)+e3\

—-¿5- (1 +r)p\

dWR3 J

= 0, i = {1, 2}. (56)

Using (56), one obtains x2 = ^f^-x} and upon substituting into the last equation of (53) yields

2 1

W ,1 ^(aiPii

1 m2 1

1 — a1 ln(x1) — a^n

«1 P1

— 52 j

j=1

a

+

L«2+«2 2

,(1+r)(w0-p1 X1)+e3js

«2 +«2 ^ V «2 Pi)

ln

+

Av.

(1+r)(w0-p1X1)+e3,

1 2 *2a2

«2 + «2 2 which could be expressed alternatively as

_^2P2_\

«2 + a2 111 V «2^2/.

ln

/ 2 1 \ 2

vf - a\Hx\) - allJ^xl) _^(4+r)iW°-PiXi)+el) = 0 \a1p1 / j=1

Solving x1 from (57) yields the wealth compensated demand for good 1 as

(57)

l

a

2

2

2

,P1,P2) = < exp

Invoking the fact that

W0

j=1

,(1+r)(w0-p1X1)+e3j

Q-lPl \ +

2 T a{p1

£+r)(W0-p1X1)+;3 =((a2 + a2) x

X ln[(1 + r)(W0 — pM (W0,p1,p2) — p2M (W0,p1,p2))+ j ] +

l

2

+ a2ln( i"2 + allll( 1°? 2 ) ~ «21п(Р2) - «21п(^2) ч 2 + «2/ Va2 + a2

we have

I U -W0 \

^lK 1 , Pi, P2 ) = exp

(a 1 + a2)

3 W1 -Y Al ((a2 + a2)ln[(1 + r)(W° - p W 1 ) + j ] +

j=1

1

x

x

+ «2In ! , 2 +«2ln ! , 2 -a^ln(^) -a^ln(^)

\a2 + a2/ Va2 + a2

a\p\ \ K +

2 Г a2p1

Following the above analysis, the wealth compensated demand for good 2 can be obtained as that given in Proposition 2. Q.E.D.

References

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Статья рекомендована к печати проф. Л. А. Петросяном. Статья поступила в редакцию 21 марта 2013 г.