Custom Writing Service http://StudyHub.vip/Approaches-To-The-Solution-Of-Intertemp 👈

1. APPROACHES zyxwvu
TO THE SOLUTION OF INTERTEMPORAL
CONSUMER DEMAND MODELS
RUSSEL J. COOPER zyxwvu
University of Western Sydney, Nepean
KEITH R. McLAREN
Monash University
I. INTRODUCTION
The derivation of systemsof demand equations based on the maximisation of a static utility
function subject to a budget constraint is now well established with many applications available
in the literature. But this paradigm has at least two (related) restrictions. First, it is essentially
an allocation model, treating total expenditure as exogenously given. Second, by its nature
it cannot adequately handle decisionswhich are intrinsically intertemporal in nature. However,
many issuesof current economic policy interest involve analysisof intertemporal choiceswhich
then impinge upon (endogenise) the conditioning variable in demand systems. Examples here
are the consumption/savings and work/leisure choices. The endogenisation of the
consumption/savings choice was first handled explicitly by Lluch zyxw
(1973) in his derivation of
the ExtendedLinear Expenditure System(ELES) (see King, 1985, for later developments),while
the modern approaches to labour supply theory within a life cyclecontext (e.g.Heckman, 1976;
Heckman and MaCurdy, 1980; MaCurdy, 1981; Browning, Deaton and Irish, 1985; Blundell,
1986) are based on intertemporal models. In all of the above-mentioned models, the objective
function consistsof an intertemporally additive function of an underlying instantaneous utility
function. Such a specification allowsthe allocational and intertemporal aspects of the problem
to be separated out optimally in a ’two-stage budgeting’ framework.
Given the importance of policy issues such as the relationship between microeconomic
structure, tax policy and savings, it could still be argued that there are remarkably few
applicationsof intertemporal optimisation to the development of models capable of addressing
these concerns. One reason for this may be the commonly held view that solutions of such
models are intractable in all but the simplest of cases. Another reason may be the view that,
sincethe specificationof an intertemporallyadditiveutility function admits two-stage budgeting
as optimal, the natural separability of the allocational and intertemporal optimisation tasks
allows a separation of study of the aspects of interest. However, this last view ignores the fact
that, since total expenditure is really endogenous, a study of the effect of fundamentally
exogenous factors(such as initial wealth, taxation rules and interest rates) on consumer demands
requires a composition of the allocational and intertemporal optimal decision rules. While
the allocationalcomponent of the compositesolution is invariant to monotonic transformations
of the instantaneous utility function, this is not the case for the intertemporal component of
the rule. Thus the apparently harmless choice of a suitable monotonic transformation of the
functional form of instantaneous utility (chosen to meet the ‘otherwise intertemporally
intractable’view) impinges on the ability of the model to realistically address questions of the
responsiveness of consumer demands to exogenous factors, even though two-stage budgeting
is permissible. zyxwvuts
20

2. 1993 zyxwvutsrqponm
APPROACHES TO THE SOLUTIONOF INTERTEMPORAL CONSUMER DEMAND MODELS 21 zyx
In this paper we consider a general classof consumer demand models set within the context
of an intertemporal optimisation problem which allows two-stage budgeting. We discuss
approaches to obtaining a composite solution to the full optimisation problem in order to
relate individual consumer demands to fundamentally exogenous factors within the context
of reasonably general functional form specifications.
11. THE
MODEL
The issues to be discussed may be characterised in terms of the following general model
of household behaviour:
subject to
W
(
s
)zyxwvu
= zyxwvut
r w(s)-p' zyxwvut
q(s),
w(t) = zyxwv
w,
w(T)2 zyxwvu
0
where w defines net worth!,
The aim is to derive solutions of the form
where Q'denotes the intertemporal demand system which expresses commodity demands as
functions of current wealth and externally determined state variables zyxw
x, where x' = (r,p').
System (3) implies the intertemporal consumption function
c(s) = p'q(s) = C'(w(s),x). (4)
I Notation. Goods are represented by the n-vector q, with corresponding prices p. Cost (expenditure or
consumption) is defined by c = p'q. Generally, values of variableswill be represented by lower-caseletters,
whiletheir corresponding functions will be representedby upper-case letters. For example,the direct utility
function is written u = U(q).Partial differentiation is generally represented by a subscript so that, for
example, U,,
indicates marginal utility of the ith good. Where a variable may be determined by functions
of different arguments, these will be distinguished by superscripts. Thus q = @(c, p ) will represent
Marshallian demands, and q = Q"(u,p ) Hicksian demands. Intertemporal solutions are identified by
an I superscript, e.g. Q'(w, p ) indicates intertemporal demands depending upon net worth w.
*We make two concessions to full generality in order to facilitate a discussion which ignores the
peculiarities of corners and boundaries. Firstly, the direct utility function U(q)is assumed to be concave
in q, rather than quasi-concave, so that necessary conditions for optimality are also sufficient. Secondly,
analysis will be confined to interior solutions.

3. 22 AUSTRALIANECONOMIC PAPERS JUNE zyx
The intertemporaldemand system (3) is equivalentto the compositionof the staticMarshallian
demand system
with the intertemporal consumption function zyxwv
(
4
)
.That is
Solutions of the form (3) and (4)will be referred to as closed loop or synthesised solutions
sincethey represent decision rules which generate values of the control variables as functions
of the state zyxwvut
of the system at the time of the decision.
There are a variety of approaches to the derivation of solutions to problem (l), each of which
has certain advantages in particular situations. The approaches to be discussed are the control
theoretic approach applied firstly to direct and then to indirect preference representations,
variants of the dynamic programming approach, and the intertemporal duality approach.
Not all of our terminology is standard and some justification is in order. We attribute the
observation that optimisation problems are best solved by considering carefully the 'natural'
independent (or conditioning) variable to Gorman (1976). Since marginal utility turns out to
be the natural conditioning variable in intertemporal problems so structured as to admit two-
stage budgeting, we introduce a zyxwvut
'G' superscript for functions conditioned on marginal utility
and refer to these as 'Gorman' functions in order to distinguish them from Marshallian and
Hicksian functions. Gorman functions are therefore closely related to Frisch functions, where
the latter are understood to be conditioned on the 'price of utility' (the reciprocal of marginal
utility). We develop these inter-relationships more fully as necessary in the sequel. zyx
111. CONTROL
THEORETIC
APPROACHES
a) The basic approach: direct representation of preferences
The standard procedure (see, for example, Takayama, 1985; Kamien and Schwartz, 1991;
or Leonard and Long, 1992) would be to apply the control theoretic approach to the
representation as set out in (1). The current value Hamiltonian is
H zyxwvutsr
= U(q)+X(rw-p'q)
with first order conditions
Hq = Uq(q)-Xp = 0
H, = r
X = AX-X
Hx = rw-p'q = zyxwv
W
w(0 = w, X(T) zyxwvutsr
2 0, w(T) 2 0, X(T)w(T) = 0.
(7)

4. 1993 zyxwvutsrqpon
APPROACHESTO THE SOLUTION OF INTERTEMPORALCONSUMER DEMAND MODELS zyx
23 zyx
The standard solution method may be described in five steps:
(i) Invert (8a) for zyxwvuts
q. This gives the 'Gorman' (marginal-utility-conditioned) demands
(ii) Theseconditional demands are substituted into (8c) to give, with (8b), a pair of differential
equations in zyxwvuts
X and zyxwvut
w.It should be noted that qualitative analyses of intertemporal problems
typically stop at this point and simply analyse the qualitative characteristics of the state (w)
and costate (A) differential equations in a phase diagram. However, when the interest is in
determiningconsumerresponsesto fundamentallyexogenousfactors, it is necessaryto synthesise
the solution. This requires:
(iii) The differential equations in X and w,given conditions (8d), are solved to give the open
loop solutions zyxwvutsr
w(s) = zyxwv
W(s;t, w,x),
X(S) = A0(s; t, w,x).
(iv) Application of (lob) allowsX to be eliminated from (9)to give the open loop control paths
(v) The final step involves the manipulation of (loa) and (11) in order to obtain the closed
loop intertemporal Marshallian demand system (3) (i.e. to synthesise the solution).
In this five-stepstyliseddescription of the solution process, steps two and four simply involve
substitution or compositionof functions. However, difficultiesin the application of this solution
procedure typically arise at steps one, three and five. Step one involves the solution of a set
of nonlinear simultaneousequations which, it may be noted, is equivalentto the problem which
arisesin staticdemand models specifiedin terms of the direct utility function. A useful approach
to the problems at step one would therefore be to make use of results in static duality theory.
The remainder of Section 111 discusses ways in which these results may help at step one while
remaining within the control theoretic approach.
The control theoretic approach continues to present problems at steps three and five. In
Section IV we outline approaches to dealing with the problems at step three by making use
of resultsin dynamicprogramming. These results obviatethe need to solvedifferentialequations.
While some progress can be made using these techniques, to some extent these approaches
simply concentrate the problems towards step five, where the difficult issue of synthesis must
be tackled. Dynamic programming approaches rely for their successful implementation on a
restricted set of functional representations. In section V, therefore, we turn to an approach
based on intertemporal duality theory which acts directly to eliminate problems at step 5.
b) Indirect representation of preferences: use of the indirect utility function
The indirect utility function is defined by
UM(c,
p ) = max { U(q):p'q 5 c }

5. 24 AUSTRALIANECONOMICPAPERS JUNE zyx
Properties of zyxwvut
UMand other functions are noted in Appendix zyxw
11.
If an explicit representation for UMis specified then the allocational aspect of problem (1)
is immediately resolved by using Roy's Identity to give the Marshallian demand system
conditional on zyxwvuts
c
The intertemporal problem may now be expressed as
subject to zyxwvutsrq
W(s) zyxwvuts
= zyxwvut
r(s)w(s)-c(s), w(t) = w,
The current value Hamiltonian now becomes
HM = UM(c,
p )+X(rw-c)
with first order conditions
HY = UY(c,p)-X = 0
HY = r
X = SX-X
HF = rw-c = zyxwv
w
w(T) 2 0.
The solution of (14) by the control theoretic approach follows the same series of steps as
outlined for problem (1). However, inspection of (17a) compared with (8a) indicates that the
advantage of the use of the indirect utility function lies in the reduction of the dimensionality
of the problem at step one. The problem at step one is reduced to that of the inversion of
a scalar monotonic function.
c) Indirect representation of preferences: use of the profit function
It is interestingto note that problemsassociatedwith stepone could have been avoidedentirely
by explicit specification of a Gorman cost function
c = CqX, p).
Since the Gorman cost function is defined as the inverse of the UY(c,p ) function, (17a)
would simply be replaced by (18) in a control theoretic approach, completely eliminating step
one. This simplification is achieved firstly at the cost of the need to specify a functional form
for which the regularity conditions are not immediately obvious, and secondly at the cost of
a lack of a suitable variant of Shephard's Lemma which would allow derivation of the
allocationalresponses. Theseapparent problems are resolved by specificationof a profit function

6. 1993 zyxwvutsrqpon
APPROACHESTO THE SOLUTION OF INTERTEMPORALCONSUMER DEMAND MODELS zyx
25 zyx
based on the interpretation of zyxwvut
1/X as the price of utility. Following Gorman (1976) (see also
Browning, Deaton and Irish, 1985) the profit function is defined by
?r = nF(p, zyxwv
P) = mq,bu-C"(u, PI}, (19)
where CH(u, zyxwvutsrq
p ) is the Hicksian cost function, p is the 'price' of utility and the superscript F
denotes p-conditioned (Frisch) functions.
By Hotelling's Lemma the Frisch demands are
Moreover, Hotelling's Lemma also provides an expression for optimal u (which might be
termed a Frisch utility function)
and hence the Frisch cost function may be recovered from the profit function on rearrangement
of (19) to give
c = zyxwvu
CFb,P ) = P)-nFb,
PI. (22)
It may be noted in passing, that the Frisch cost function is also recoverable from (20) as
(23)
c = CFb,P) = P'Pb,P) = -P'n;cp, PI.
The equivalence of (22) and (23) reflects the fact that the profit function is homogeneous
of degree one in p and p.
The Frisch cost function (23) is translated very simply to the Gorman cost function (18)
since p = C," = l / U y = 1/X at the optimum. Thus:
CG(X, p ) icqm,p ) . (24)
In summary, to apply a control theoretic solution procedure the most natural specification
of preferencesis in terms of the consumer's instantaneous profit function. This eliminates step
one. This procedure is illustrated in the next sub-section by a number of examples. To resolve
the problems associated with steps three and five, one may turn to alternative solution
procedures, which are introduced in following sections.
d) Examples
Example 1: Cobb-Douglas
As usual, the Cobb-Douglas function provides the simplest illustration. While allocational
aspects will be unaffected by monotonic transformations, this is not so for the intertemporal
solution. Thus for this example choose the additive specification
U(q) = EP)nqj, Pj 2 zyxwv
0,
which is clearly concave. (Appendix I provides a brief summary of useful results on convexity

7. 26 AUSTRALIAN zyxwvut
ECONOMIC PAPERS JUNE zyx
and quasi-convexityto which reference will often be made.) The pi may be normalised as zy
Cpj zy
= 1. Define the price index zyxwvuts
PIby PI = ll@j/@j)pj. PIis a concave, increasing, homogeneous
of degree 1 (HD1)function of positive prices. Then the corresponding atemporal functions
can be defined as
CH = PI@
UM= ln(c/P,)
TIF = p[ln(p/P,)- 11
CF = p zyxwvutsr
C G = 1/X.
Use of Cc = 1/X gives zyxwvut
c(s) = l/X(s),allowing open loop solutions (for r > 6)
with synthesis X(s) = 1/(6w(s))and c(s) = C' = zyxw
6w(s).Finally, Roy's Identity gives q,(s) =
@ = (fii/pJGw(s).
Example 1 could be immediately generalised by allowing PIto range over
the class of concave,positive, increasing,HD1 functions. Intertemporal aspects are unaffected;
only allocation aspects are modified.
Example 2: Generalised ELES
Consider the generalisation of ELES introduced by Cooper and McLaren (1983)
1-(ll(q,-y
,
)
p
,
]
a
U(q) =
where zyxwvutsr
fi, > 0 V j , Cfij = 1 and -1 < a c 00. For this range of a,(1 -k-u)/a is increasing
and concave in k > 0, so by property C7(d) of Appendix I, U(q)is concave and increasing
for qj > yj. Define PI as before, and let Po= p'y. Then the corresponding dual functions
are found to be
C
" = (1 -au)-"wP1 +Po
1 -[(c-P,)/P,]--u
UM = , c > P ,
a
C F = p(p/P,) - f +Po
CG = (l/A)(l/APl)-f+Po
P
O
where E = m/(1 +a),i.e. a = d(l-E), Note that the admissiblerange -1 < a < 00 corresponds
to -00 < E < 1. That these are legitimate functions follows by duality, but for example a
direct proof that UMis quasi-convex in prices follows from Q9(c) and Q7(a) of Appendix I,

8. 1993 zyxwvutsrqpon
APPROACHES TO THE SOLUTIONOF INTERTEMPORAL CONSUMER DEMAND MODELS zyx
21 zyx
and properties C7(b) and C4 demonstrate that zyxwvu
llFis convex in p. Application of the control
theoretic method allows solution similar to Example 1, giving the synthesised consumption
function zyxwvut
C
'
zyxwvuts
= Po+[~r+(l
-c)6](w-Po/r).
In this example, both Poand PImay be allowed to range over the class of concave, positive,
increasing, HDl functions, to give commodity demand equations with marginal budget shares
which are functions of r, 6 and p.
Example 3: Extended almost ideal demand system (EAIDS)
Specify
The corresponding atemporal functions can be derived as
UM = [ln(c/P,)]/P,
llF = p(lnp-lnp, -Inp, -l)/P2
C F = p/P2
C G = l/wz.
Provided PIand P2are twice continuously differentiable, Slutsky symmetry will be assured.
Adding up is assured by linear homogeneity of the cost function in prices, and this can be
met by the conditions that PIis HD1 in p and Pzis HDO in p. These conditions imply that
the Engel curves corresponding to this model are of the Working-Leser form. Consideration
of the indirectutility function or of the profit function shows more clearlythan the cost function
that a reasonable set of sufficient conditions for satisfaction of the remaining regularity
conditions would be that both PI and P2be positive, concave and increasing in prices.
However, it is impossible for P2to satisfy these conditions and also be HDO in p, and hence
global regularity does not apply in Example 3. It is on the basis of considerations other than
regularity that Deaton and Muellbauer (1980) take PIto be translog and PzCobb-Douglas
(with sum of exponents equal to zero).
These regularity issues are reconsidered in a generalisation of extended AIDS developed in
Section IV. Compared with Example 1, the role of P2is only in allocation, and hence the
intertemporal consumption function is again C
' = 6w.
IV. DYNAMIC
PROGRAMMING
APPROACHES
a) Introduction
While the previous examplesillustrate the advantages of indirect specification of preferences
for the control theoretic approach, they also make clear that difficulties will continue to arise
at stepthree of that approach and that this limitsthe range of generalfunctional representations
which are amenable to solution. One way of looking at the dynamic programming approach

9. 2szyxwvutsrqpon
AUSTRALIANECONOMIC PAPERS JUNE zyx
is as a procedure to avoid the need to solve differential equations as is required at step three
of the control theoretic approach. Intuitively, it is as if the problem has been moved to step
five, where the need arises to obtain synthesis of the control as a function of the state.
The key to the various approaches based on dynamic programming is the concept of the
optimal value function. The optimal value function V(w,x, t) may be defined as the maximised
value of problem (14). The optimal value function satisfies the Hamilton-Jacobi-Bellman
equation zyxwvuts
0 zyxwvutsrq
= maxc{UM(c,p)+(rw-c)V,(w, x, t)-6V(w, x, t)+V,(w, x, zyxw
t)}, (25)
yielding the first order condition zyxwvu
uy = v, (26)
and, for optimal c, the Hamilton-Jacobi equation
(rw-c)V,+V, zyxwvu
1 6V-U'. (27)
A standard reference for these results is Dreyfus (1965).
The alternative approaches to be discussed combine these relationships to varying degrees
with equations (17). The unifying variable here is X, since (17a) and (26) imply
uy = v, = X. (26')
b) Dynamic programming: zyxwvuts
the standard approach
This approach firstly solves (26) for c to obtain the Gorman cost function by inversion as:
c = C"(V,, p). (28)
Secondly, (28) is substituted into (27) to give a nonlinear partial differential equation in V.
Solution of this for V and substitution of V, into (28) gives the closed loop solution for c.
To illustrate, consider Example 1. In this case (28) gives c = l/Vw, allowing (27) to take
the form
rwVw-1 = 6V+InVW+InP,.
It can be verified that a solution is given by
V = [In6+Inw-InP, +(r-6)/61/6
so that V
, = 1/6w and the closed loop solution for c becomes
c = C'(W, x, t) = 6w.

10. 1993 zyxwvutsrqpon
APPROACHES TO THE SOLUTION OF INTERTEMPORALCONSUMER DEMAND MODELS 29zyx
Clearlythe difficulty with this approach arises from the need to derivean analytical solution
to a nonlinear partial differential equation. It may be noted in passing that some scope exists
to broaden the applicability of this approach by noting that the solution for zyxw
V is used only
to determine V,. Thus a variant of this approach would be to differentiate (27) with respect
to w and reapply (26) to give the partial differential equation zyxw
(rw-c)Vww+ V,,zyxwvutsr
= zyxwv
(6-r)vw (29)
which, on use of (28), leads to a relationship to be solved for V,. Under some specifications
of (28) this may be more tractable than the solution of (27) for K
c) The matched Gorman functions approach
This approach uses (26') to observe that Uy(c,p ) = A = V,(w, x, t), so that zyx
c = CG(A,p ) (30a)
and zyxwvuts
w = V ( X , X , t)
where C
? is the inverse function of U y for c (the Gorman cost function) and WG is the inverse
function of V, for w (hereafter termed the Gorman wealth function). In principle, upon
specification of p,
the function CGmay be obtained from a rule linking CGto p.
This
rule, derived in the context of consumer decision making under uncertainty by Cooper, Madan
and McLaren(1989), followsfrom (29) upon rewriting partial derivativesof Vin terms of partial
derivatives of using the identity wG(V,(w, x, t),x, t) I w. Specifically, since
V,, = l / q , V,, = zyxwv
-e/q,
and Vw = X (31)
then (29) may be rewritten as
c = f v - ( 6 - f ) x q - q . (32)
This implies that the functions CGand are related by
CG = D V (33)
where the operator D is defined by
a a
ax at
D = rZ-@-r)A - - -
In practice this methodology may be implemented by specifying a profit function ITF@,p),
applying (22) to derive CF@,
p), and re-expressing this as CG@,
p). The matching functional
form of may then be derived 6y the method of undetermined coefficients.
To illustrate this approach, consider Example 2, for which
CG(A,p ) = P,,+X'-'P;.

11. 30 zyxwvutsrqpon
AUSTRALIANECONOMICPAPERS JUNEzyx
Postulate a form for zyxwvutsr
WC as
with zyxwvuts
O,, O2 undetermined coefficients. The condition zyxw
CG zyxwvutsrqp
= zyxwvut
r W C - ( s - r ) x q
allows the determination 0, = l/r, O2 = l / [ u + ( l - ~ ) 8 ]to give the solution
C
' = Po+[cr+(l- ~ ) 6 1 ( ~ - P ~ / r )
The simplicity with which this approach allows Example 2 to be solved suggestsits potential
for solving problems whose solution procedures may not be at all transparent via the other
approaches.
d) Examples of the matched Gorman functions approach
In this sub-section, the generality of the matched Gorman functions approach is illustrated
by the solution of a further two examples. The first of these (Example 4) generalises both the
Cobb-Douglasand the AIDS examples in the spirit of AIDS, but preserves regularityconditions
which are sacrificed in the standard specification of AIDS.
Example 4: The extended generalised AIDS model
The analysis in Section 111, indicates that it is not at all obvious how, a priori, to check
the extent of the regular region of the AIDS cost function. However, the nature of the regularity
problem may be exhibited on specification of the AIDS profit function zyx
W = p(hp-InP,-InP, - l)/P2.
The major regularity problems with the AIDS profit function are caused by the requirement
that P2 be HEQ. Consider then the following generalisation:
where 0 < q < 1 and PIand P2are now both concave,positive, increasing and HD1 functions
in p. (P, in (34) has been redefined to include the previous specification of P2.)Now (34) is
of the form lll($)vwhere I
I
l is the profit function of the general form of Example 1. Proof
of regularity conditions will require JI' > 0, ie. In&/P,) > 1, so consider zyx
ITF defined over
the cone {p, p : p > eP,@)}.(That HF is decreasing in p follows by direct evaluation; lTFis
clearly HD1 in (p, p);4is > 0 and concave in p (and hence v), pTI' is convex in (p, p ) and
hence llFis quasiconvex in (p, p) by Q9(a), and hence convex by QlO.)
Although this restriction involves the unobservable p , the derivation of the Gorman cost
function below will allow this region to be mapped into an equivalent region in (c, p ) space.

12. 1993 zyxwvutsrqpon
APPROACHES TO THE SOLUTIONOF INTERTEMPORAL CONSUMER DEMAND MODELS 31 zyx
Application of Hotelling's Lemma yields the Frisch utility function
and the Frisch cost function is
To solve the intertemporalaspect of the problem by the matched Gorman functions approach,
firstly re-expressthe Frisch cost function as a Gorman cost function by the change of variable zy
X = l/p:
Recall that, corresponding to zyxwvuts
P(x, zyxw
p ) is a function p ( x , p) where the relationship between
the functions is given by CG = zyxwv
D p and
This implies that F P takes the form
where 8 = l/[(l +q]S-qr]. The intertemporal Marshallian consumption function C1(w,p) is
defined implicitly by the parametric representation (35)-(36).
To completethe solution of the problem it suffices to obtain the Gorman demand functions.
Firstly, the Frisch demands may be generated on application of Hotelling's Lemma, zy
q = QF(p,p)= -Hi. These may be exhibited in expenditure form as
where E
" = alnPi/alnpj; i = 1, 2; j = 1, ...,n. The Gorman demands are of course simply
these functions with the change of variable X = l/p.
In order to facilitate comparison with AIDS it is convenient to exhibit the share form of
the Gorman demand system together with the parametric representation of the consumption-
wealth relationship: zyxwvuts
c = (1/X)[1 +q(-InX-lnP,-l)](l/XP,)~
w = (e/X)[86 +q( -InX-lnP, -1)](1/XP2)~
where sj = pJ4j/c,and 8 = l/[(l+q)6-qr].

13. 32 AUSTRALIANECONOMICPAPERS JUNE zyx
A little manipulation shows that
For given prices, then if (for example) elj zyxwvut
c zyxw
ezi, the share zyxw
sj is bounded from below byzyx
eIj
and increases monotonically and asymptotically to zyxwv
eY as c increases. This attractive property
may be contrasted with the behaviour implied by Example 1 (which in fact corresponds to
7 zyxwvutsrqpo
= 0 above), in which asj/alnc = 0, and with the behaviour implied by the AIDS
specification (Example 3), in which asj/alnc is constant. Neither of these alternative
specifications is attractive by virtue of the empirical implausibility of the constancy of sj on
the one hand, and the potential for sj to fall outside the (0,l) interval on the other hand. It
should be noted that this attractive property follows from the imposition of the regularity
condition that the profit function be monotonic (as well as HD1) in prices.
A commenton estimationmay be in order. As it stands, system (36) containsthe unobservable
variable A. However, it should be noted that in the relationship (36c) w is monotonic in A and
hence for any given parameter values there is an implicit representation of A as a function
of w andp. Sincethis is a one dimensional relationship it could be easilyhandled in a numerical
optimisation program.
It is easy to show that the term in square brackets in (34) may be generalised in the same
way that Example 2 generalises Example 1, since as demonstrated by the notation adopted
above, the derivation of the solution is not dependent upon the specific functional forms for
the price terms. For example provided 7 > 0, a specification of P2Cobb-Douglas maintains
regularity, while PIcould be chosen regular and flexibleby use of the functional specifications
considered by Diewert and Wales (1987).
Example 5: A general regular intertemporal demand system
The previous example demonstrated that a modification of the AIDS profit function leads
to share equations with properties more consistent with the regularity conditions implied by
the utility maximising framework. Consider now an alternative specification for the profit
function which is also amenable to solution by the matched Gorman functions approach but
for which there exists a simple set of sufficient conditions implying global regularity:
where P, and P2are concave, positive, increasing and HD1 in p, and 7 1 0.
Followingthe steps outlined in Example4, one obtains the consumption-wealth relationship
in parametric form:
c = (l/A)[l+7(l/xp2)q
w = (l/A)[ 1/s +ev(l/Wz)~],

14. 1993 zyxwvutsrqpon
APPROACHES TO THE SOLUTIONOF INTERTEMPORALCONSUMER DEMAND MODELS 33 zyx
with corresponding share equations:
El .+EaV(1/W2)~ zyxwvutsr
s. zyxwvutsrq
= zyxwvut
J
' 1+7)(l/W2)'1'
A major characteristic of these share equations is that they are globally well-behaved.
This exampleclearly demonstrates the strength of the matched Gorman functions approach.
By choosing to specify the preference structure in terms of the profit function, it is possible
to exhibit an (implicit) closed form solution to the intertemporal optimisation problem for
a wide variety of regular profit functions. The cost of achievinga solution to the intertemporal
problem is that the consumption-wealth relationship may only be defined parametrically.
However the defining relationships of the parametric representation are single dimensional
and assured to be monotonic as a consequence of the theory, and hence the numerical burden
in either estimation or simulation is relatively minor.
V. THEINTERTEMPORAL
DUALITY
APPROACH
The previous examples illustrate the power of the dynamic programming approach and its
variants, giving direct synthesis provided that the functional forms are such that (26) or (26')
are tractable. We now introduce an approach which in principle can be applied to any well-
specifiedproblem. The key to this approach is an explicit specification of a (regular) optimal
value function zyxwvut
V(w, zyxwvuts
x, t).
This approach makes use of a result (initially derived in the context of the theory of the
firm in McLaren and Cooper (1980)) which may be obtained by rearranging (29). This implies zy
w = [(S-r)Vw-Vw,]/Vww (37)
and hence the synthesised solution for c is available immediately upon specification of a
functional form for V as
c = C'(w, x) = rw-[@-r)Vw- Vw,l/Vww. (38)
This approach presupposes that one starts from a specification of the function V rather
than the function U
'
. The duality between the functions UMand Vis investigatedin Cooper
and McLaren (1980). The allocationalaspect of the problem is solved by noting that, in principle,
UMmay be defined in terms of V using (27) together with (37). The implication of R
o
y
'
s
Identity for such a construction is
This approach may be illustrated for Example 1 by noting that the application of (38) and
(39)to the particular functional form for Vgeneratesthe appropriate intertemporal Marshallian
consumption and demand equations. The potential usefulness of this approach lies in the ease
with which such equations may be generated from a suitablevalue function Vwhich is specified
initially rather than being derived from the initial specification of some other function.

15. 34 zyxwvutsrqponm
AUSTRALIAN zyxwvut
ECONOMICPAPERS JUNEzyx
Example 6:
Recall the value function derived for Example 1 zyxwvu
V zyxwvutsr
= [In6+Inw-lnPl+(r-6)/6]/S.
Although this was derived for a particular (Cobb-Douglas) functional form for P,, it is
straightforward that PI could have been any regular function of p. However, this source of
generalisation is available through any of the various approaches. The aim of this section is
to demonstrate a generalisation which is specifically available to the intertemporal duality
approach. To this end, note that for the Cobb-Douglas specification the term (lnw-I*,)
could have been written as cD,ln(@,w/pj). An interestinggeneralisationis to define the addilog
value function by replacin: this term with its addilog equivalent, giving the value function
which will satisfy the regularity properties provided -1 < zyx
a
, < zyx
03, j = 1, ..., n.
Application of (38) and (39) leads (after some manipulation) to
where zyxwvutsr
f;= hi(l +Cgj/Chj-cri)
J J
This is a systemwhich is rich in the specification of the dependence of the marginal budget
shares on prices and wealth. Of course, other generalisations are immediately available. For
example, in (40) w could be replaced by supernumerary wealth, leading to intercept terms in
(41) and (42). It is not at all obviousthat models of this type could be derived by the alternative
approaches.
VI. CONCLUSION
Thispaper has surveyed several approachesto the solutionof intertemporalconsumerdemand
models which have not received much attention in the literature but which allow the solution
of such models under more general representations of preferencesthan is apparently generally
recognised. The approaches which have been discussed all exploit to varying degrees the two-
stage budgeting aspect of the type of model which has been considered. While the very nature
of such a model may itself be consideredsomething of a restriction, the relative predominance

16. 1993 zyxwvutsrqpon
APPROACHESTO THE SOLUTIONOF INTERTEMPORALCONSUMER DEMAND MODELS zyx
35 zyx
of this type of model structure together with the relatively simplistic functional forms which
have been employed to date, combine to make it imperative to consider procedures which allow
the derivation of synthesised solutions to such models under more general functional
specifications, if issues of the responsiveness of consumer demand to fundamentally exogenous
factors are to be realistically examined.
APPENDIX
I zyxwvu
Convexity and quasi-convexity
The results below are selected because of their relevance in the checking of regularity
conditions in the functions which occur in the paper. They are collected from various sources
including Mangasarian (1969),Rockafellar (1970),Greenberg and Pierskalla (1971),Roberts
and Varberg (1973),and Madden (1986).Explicit references are only given for the less common
cases.
Let f(x),g(x),h(x):X - V and zyxwvu
F: V-R' where X and V are convex subsets of R" and R',
respectively.
a) Convex and concave functions
C1. f(x) convex zyxwvutsr
-flhu+(l -A)y] zyxwvu
5 Af(x)+(l-A)f(y) for AE[O, 11, x, y E X .
C2.f(x) convex -f(x)-f(y) h fx, (y)(x-y).
0.
f(x) convex -f
,
, positive semi definite.
C4. f(x) concave ~3 -f ( x ) convex.
C5. f(x) convex ~3 F(A) = flhu+(l -A)y] convex over A E [0, 1J V x, y E X .
C6. f(x) convex 3 {x:f(x) 5 a} a convex set.
C7. Define h(x) = m(x)].
Then
(a) f convex, F convex, F non-decreasing 3 h convex.
(b) zyxwvutsrqp
f concave, F concave, F non-decreasing 3 h concave.
(c) f convex, F concave, F non-increasing 3 h concave.
(d) f concave, F convex, F non-increasing 3 h convex.
C8. f(x) concave and V = Q, 3 l/f(x)convex.
C9. (a) f(x),g(x) convex a, b E Q+ 3 h(x) = af(x)+bg(x)convex.
(b) f(x),g(x) concave a, b E Q+ 3 h(x) = af(x++g(x) concave.
b) Quasi-convex and quasi-concave functions
Q1. f(x) quasi-convex -f[hu+(l-X)yJ smax{f(x),f(y)}for X E [0, 11, x, y E X.
Q2. f(x) quasi-convex -f ( y ) 5 f(x) 3 fx, (x)@-x) 5 0.
Q3. f(x)quasi-convex -lDj(x)l 5 0 forj = 1 , . . . ,n, where D is the bordered Hessian
and Dj is jth principal minor.

17. 36 AUSTRALIANECONOMIC PAPERS JUNE zyx
Q4. f(x) zyxwvutsr
quasi-concave zyxwvuts
~3 -f(x) quasi-convex.
Q5. f(x)quasi-convex ~3 F(X) =flxX+(l -X)y] quasi-convex over X zyxwv
E [0, 11 vX , zyxw
y E X .
Q6. f(x ) quasi-convex zyxwvut
c3 { x :f(x) zyxwvu
5 a } a convex set.
Q7. Define h(x) = Fv(x)].Then
(a) f quasi-convex, F non-decreasing h quasi-convex.
(b) zyxwvuts
f quasi-concave, F non-decreasing =3 h quasi-concave.
(c) f quasi-convex, F non-increasing 3 h quasi-concave.
(d) f quasi-concave, F non-increasing * h quasi-convex.
(a) f ( x ) quasi-convex, V = Q+ 3 l/f(x)quasi-concave.
(b)f ( x ) quasi-concave, V = Q+ * l/f(x)quasi-convex.
Let h(x) = g(x)/f(x).Then h(x)is quasi-convex if any of the following hold:
(a) g convex, f(x) > 0, and f linear.
(b) g convex, f(x) > 0, f convex and g(x) 5 0.
(c) g convex, f ( x ) > 0, f concave and g(x) 2 0.
(d) g concave, Ax) < 0, and f linear.
(e) g concave, f(x) < 0, f convex and g(x) 5 0.
U
, g concave, f(x) < 0, f concave and g(x) 2 0.
(See Greenberg and Pierskalla, 1971, p. 155.)
QS.
Q9.
QlO. Let X be a convex cone, and let f be such that for x E X- (0) and zyx
X 2 0, f(x) > 0,
Thenf is convex (concave) if and only if it is quasi-convex (quasi-concave) (Berge, 1963,
p.208).
Q11. Let f l , f2, .. . ,f,: X - V, and g : Rm - V. Construct h(x) = gV,(x),f2(x),. . . ,
f(W = x f o .
f,Ol. Then
(a) f , , . . . ,f
, convex, g quasi-convex non-decreasing 3 h quasi-convex.
(b) f , , . . . ,f , concave, g quasi-concave non-decreasing 3 h quasi-concave.
(c) f l , . . . ,f , convex, g quasi-concave non-increasing 3 h quasi-concave.
(d) f,, . . . ,f , concave, g quasi-convex non-increasing * h quasi-convex.
(Berge, 1963, p.207)
APPENDIX
I1
Definitions and Properties of Functions
The following functions and their properties are discussed in Diewert (1982)and McFadden
(1978). The properties listed are somewhat stronger than those necessary for duality, but are
convenient in order to focus attention on the intertemporal aspects of consumer demand. Let
Q" and 0; be the non-negative, and strictly positive, orthants of A", respectively.
a) Direct utility function
= U(9) U : Q " -R.

18. 1993 zyxwvutsrqpon
APPROACHESTO THE SOLUTION OF INTERTEMPORAL CONSUMER DEMAND MODELS zyx
31 zyx
Properties of U zyxwvuts
DU1. U concave in zyxwvuts
q. (Assume concavity rather than quasi-concavity for the
intertemporal model.)
DU2. U is increasing in q.
Marshallian demand equations q zyxwvu
= QM(c,
p) are defined as solutions to:
Maximise { U(q):p'q zyxwvu
5 c, q E Q"}.
b) Hicksian cost function
c = P ( u , p) = min{p'q : U(q) z u, q zyxwv
E Q"}
= p'QH(u, PI
where q = QH(u,p) are the Hicksian (utility compensated) demand equations.
CH(u,p) : R x 0; -Q.
Properties of CH(u,p) :
HCl. CHis non-negative.
HC2. CHis increasing in u.
HC3. zyxwvutsrq
CHis convex in u.
HC4. CHis increasinp in p.
HCS. CHis concave in p.
HC6. CHis HD1 in p.
Shephard's Lemma:
PI = q.
Shadow price of utility p = Cf.
c) Indirect utility function
u = V ( c ,p) = ma{u(q): p'q 5 c, q E Q"}
= V(Q% PN.
Note: CH(UM(c,
p), p) I C .
UH(c,p): Q x 0
: - R.
Properties of UM(c,
p):
IU1. UM is increasing in c.
I n . U M is concave in c.
IU3. UMis decreasing in p.
IU4. UMis quasi-convex in p.
1115. UMis H W in (c,p).

19. 38 AUSTRALIANECONOMICPAPERS JUNE
Roy’s Identity: zyxwvutsr
QM(c,p) zyxwvuts
= zyxwvu
-u;/uy = u;/p’u;.
Marginal utility zyxwvuts
X = U y = l/Cf = 1/p.
d) Profit function
?r = zyxwvu
nF(p,
p ) = max{pu-C”(u, zyxwvu
PI}
U
= pUF(p,P)-CH(UF0I,
PI, P)
where UF@,
p ) is defined by p zyxwvut
I Cf(UF@,p), p).
nF(p,
p ) : Q+ x zyxwvut
i
?
: -R.
Properties of zyxwvutsr
n(p,p):
P1. nFis decreasing in p .
n.
nFis convex in (p, p ) .
P3. ITF is HD1 in (p, p ).
nF(p,
p ) can also be defined as
max{pu-p’q : u = U(q)}
us 4
= pUFb,P)-P‘Q% P);
= PUWFb,PI,P)-CF0L, P)
or max{pW(c, p ) - c }
E
where u = UF(p,
p ) is Frisch utility, q = QF(p, p ) are Frisch demands, and
c = CF@,
p ) = p’@(p, p ) is Frisch cost.
Hotelling’s Lemma:
UFb’P) = n;@,P);
P
O
L
,P) = -~;h
PI.
Frisch cost CFcan be recovered as
c = CF(p,p ) = CH(IIF(p. p), p ) = pn;-rF
or P‘QFOL, P) = -~’nFb,
P)
which are equivalent as, by HD1, nF= pH; + p‘n;.
Define pa,p ) = CF(l/X,
p).
REFERENCES
Berge, C. (1963), Topo/ogicu/ Spaces (Edinburgh: Oliver and Boyd).
Blundell, R. (1986), “Econometric Approaches to the Specification of Life-Cycle Labour Supply and
Commodity Demand Behaviour”, Econometric Reviews, vol. 5.
Browning, M., Deaton, A. and Irish, M. (1985). “A Profitable Approach to Labor Supply and Commodity
Demands over the Life Cycle”, Econometrica, vol. 53.

20. 1993 zyxwvutsrqpon
APPROACHES TO THE SOLUTIONOF INTERTEMPORALCONSUMER DEMAND MODELS 39zyx
Cooper, R.J., Madan, D. and McLaren,K.R. (1989), “AGormanesqueApproach to the Solutionof Stochastic
Intertemporal Consumption Models”, Working Paper No. 7/89, Department of Econometrics, Monash
University, October.
Cooper, R.J. and McLaren, K.R. (1980), “Atemporal, Temporal and Intertemporal Duality in Consumer
Theory”, zyxwvutsrqp
International Economic Review, vol. 21.
Cooper, R.J. and McLaren, K.R. (1983). ‘‘TvlodellingPrice Expectationsin IntertemporalConsumer Demand
Systems: Theory and Application”, Review of Economics and Statistics, vol. 65.
Deaton, A. and Muellbauer, J. (1980),“An Almost Ideal Demand System”,American Economic Review,
vol. 70.
Diewert, W.E. (1982), “Duality Approaches to Microeconomic Theory”, in K. Arrow and M. Intriligator
(eds) Handbook of Mathematical Economics, vol. 2 (Amsterdam: North Holland).
Diewert, W.E. and Wales, T. (1987), “Flexible Functional Forms and Global Curvature Conditions”,
Econornetrica, vol. 55.
Dreyfus, S.E. (1965),Dynamic Programming and the Calculus of Variations(New York: Academic Press).
Gorman, W.M. (1976). ‘Trickswith Utility Functions”, in M. Artis and R. Nobay (eds)Essays in Economic
Analysis (Cambridge: Cambridge University Press).
Greenberg, H.J. and Pierskalla, W.P. (1971). “A Review of Quasi-ConvexFunctions”, Operations Research,
vol. 19.
Heckman, J.J. (1976), “A Life-CycleModel of Earnings, Learning and Consumption”, Journal of Political
Economy, vol. 84.
Heckman, J.J. and MaCurdy, T. (1980), “A Life-CycleModel of FemaleLabor Supply”, Journal of Political
Economy, vol. 47.
Kamien, M.I. and Schwartz,N.L. (1991),Dynamic Optimization: The Calculus of Variations and Optimal
Control in Economics and Management, Second Edition (Amsterdam: North-Holland).
King, M.A. (1985), ‘The Economics of Saving: A Survey of Recent Contributions”, in K.J. Arrow and
S. Honkapohja (eds), Frontiers of Economics (Oxford: Basil Blackwell).
Leonard, D. and Long, N.V. (1992), Optimal Control Theory and Static Optimization in Economics
(Cambridge: Cambridge University Press).
Lluch, C. (1973), ‘The Extended Linear Expenditure System”, European Economic Review, vol. 4.
MaCurdy, T.E. (1981). “An Empirical Model of Labor Supply in a Life-CycleSetting”, Journal of Political
Economy, vol. 89.
Madden, P., Concavity and Optimization in Microeconomics (Oxford: Basil Blackwell, 1986).
Mangasarian, O.L. (1969). Non-Linear Programming (New York: McGraw-Hill).
McFadden,D. (1978). “Cost, Revenue, and Profit Functions”,in M. Fuss and D. McFadden (eds),Production
Economics:A Dual Approach to Theoryand Applications Vol. 1 (Amsterdam: North Holland, 1978).
McLaren, K.R. and Cooper, R.J. (1980), “Intertemporal Duality: Application to the Theory of the Firm’’,
Econornetrica, vol. 48.
Roberts, A.W. and Varberg, D.E. (1973). Convex Functions (New York: Academic Press).
Rockafellar, R.T. (1970), Convex Analysis (Princeton: Princeton University Press).
Takayama, A. (1985). Mathematical Economics (Cambridge: Cambridge University Press).