Define u(x) = sup[v(y)/y 2 0, (x, y) 5 l] f or a x in the positive orthant then. 11 u satisfies the following properties: (u) u is finite, continuous and quasi-convex on the ...
Journal
of Mathematical
DUALITY
Economics
12 (1983) 1499165. North-Holland
BETWEEN
DIRECT AND FUNCTIONS
Differentiability
INDIRECT
UTILITY
Properties
J.-P. CROUZEIX Universitt! de Clermont II. 63170 AubiPre, France Received Duality in consumer theory decades. This paper responds be ensured the differentiability
May 1982, tinal version
accepted
May 1983
and production theory has been actively investigated in the last to the question: ‘On what conditions on the primal function can of the dual function?’
1. Introduction A common duality framework in economics can be described as follows: given a real-valued, non-decreasing function u on the positive (non-negative) orthant of R” a dual function u is constructed by the relation
44 = sup CrW(x,
Y> 5
11,
(1)
where (x, y) denotes the usual scalar product of R”. Clearly, u is nonincreasing and turns out to be quasiconvex. If v is quasi-concave and under some additional assumptions which will be described in detail in section 2, a complete duality exists between the primal function r and the dual function u in the sense that u can be obtained from u through the relation 4~) =
infC~X)/5 11.
(2)
x
Let us now describe some applications of this framework. Assume that we have an economy where the behaviour of a consumer can be described through his utility function v, i.e., the consumer determines his choice y (where y denotes a n-dimensional consumption vector) by maximizing u(y) subject to the budget constraint (p, y) sr where p is the vector of commodity prices and I is the maximal amount of money that the consumer can spend. Assume that I is positive and set x =p/r, then the consumer’s utility problem may be written as
4-4 = sup C4Y)l 5 11 0304-4068/83/%3.00
0
1983, Elsevier
Science Publishers
B.V. (North-Holland)
150
J.-P. Crouzeix,
Differentiability
of the dual,finction
Thus u(x) gives the maximum utility level that the consumer can attain when he faces the vector x of normalized prices. This function u is known as the indirect utility function and expresses the utility as a function of prices. The duality says that under certain regularity conditions the behaviour of the consumer can be equivalently described through the indirect utility function, a function of prices. Another situation where this framework can be used concerns the production theory. Consider an economy where only one output is produced using n inputs and the technology can be described by a production function u, i.e., v(Y) is the maximal amount of output that can be produced during a period given the vector of inputs y. Then the producer’s cost function c is defined as
44 P)=infC(p, Y>/~Y) 2 4,
(3)
where J E R and p is the positive vector of input prices. Define a new function r(t,p) by
44 P)= sup Ca(y)lGJ, Y> 5 tl. Y
(4)
Let p be a fixed positive vector, then roughly speaking the function i+c(l,p) can be considered as the inverse function of the function t-+r(t,p). Clearly r(t, P) = 4PlQ The duality expresses that under certain regularity conditions the technology can be completely described by a function of prices. In this case the producer is said to be competitively minimizing costs [Diewert (1982a, b)]. Thus the duality has mainly two applications, one in the producer context and the other one in the consumer context. A considerable literature concerns the duality between utility and indirect utility functions or cost and production functions and we do not intend to give here an exhaustive bibliography. However, we would like to mention Roy (1947), Shephard (1953,1970), Lau (1969) Diewert (1971,1974,1982b) and more specially Diewert (1982a) which provides an excellent and comprehensive study on the question including historical notes and a very precise bibliography. For the sake of simplicity, we shall adopt throughout the following the consumer context. Assume that regularity conditions on u are fulfilled so that relation (2) holds and that for a given positive vector X, j is the optimal solution of the maximization problem (1) i.e., j is the consumer’s demand for the vector of normalized prices X. Assume, in addition, that v has a non-zero
J.-P. Crouzeix, Differentiability of the dual function
derivative
151
at j then
43 = m
(5)
2 = ~‘(Y)l(~‘(Y), j>.
(6)
Next assume
that u has also a non-zero
j = u’(X)/(u’(X), 2).
derivative
at X, then (7)
Hence, it appears that the demand correspondence of the consumer is closely related to the differentiability of the indirect utility function. A mathematician will recognize through relations (5) to (7) a functional transform very similar to the famous Legendre’s one [see, for instance, Lau (1969)]. Unfortunately, the differentiability of v does not necessarily imply the differentiability of u and conversely. A motivation of this paper is to study what conditions on one function imply the differentiability of the other one. It would be noticed that a partial response has been already given by Blackorby and Diewert (1979) in the twice differentiable case. As already said, section 2 is devoted to conditions which imply a complete duality between primal and dual functions. In section 3, differentiability properties of quasi-convex functions are analysed. The convex and convexifiable cases are investigated in section 4, whereas the general case is considered in the next section. In section 6, a duality theorem is reformulated for differentiable functions. Let us recall that a function f defined on a convex Cc R” is said to be quasi-convex on C if
f(x + L(Y- 4) 5 ~42%4, f(y)l, for all x, y E C, 05 t 5 1. If (-f) is quasi-convex, then f is said to be quasiconcave. The function f is said to be strictly quasi-convex on C if
.0x+
L(Y-4)
O for i=l,2,...,n}. Given x,y~R”, xsy means xiSyi for i=l , . . . , n. A function f on R” is said to be increasing if x 5 y and x # y imply f(x) 0
fjlYf',
Yi 2
n. An easy calculation
for i=1,2,...,
u(x)=ifJl
O,
(cli/axi)ai3
xi
gives
>o,
where a=cc,+cc,...+cc,. The Cobb-Douglas functions furnish an example where the indirect utility function is not finite on the non-negative orthant. Other examples given by Diewert (1974, p. 122) shows conditions (i) and (ii) are not sufficient to ensure the continuity of u on the non-negative orthant, and condition (i) cannot be weakened by requiring the continuity of v only on the positive orthant. The following result given by Crouzeix (1977, pp. 222) establishes symmetric duality between direct and indirect utility functions:
a quite
Assume that u is a real valued function Theorem 2.2. K of R” satisfying the following conditions:
orthant
(a)
u is finite,
continuous
and quasi-convex
on K.
on the positive
J.-P. Crouzeix, Dzxferentiabilitv qf the dual function
153
(b) u is decreasing on K. (c) For all YE K the minimization problem inf[u(x)/ (x,y) 5 1, XEK] has at least one optimal solution in K. (d) For all y E K the function of one real variable 0(t) =inf[u(x) / (x, y) 5 t] is continuous at t for all t > 0. For all ~EK, define a function v by u(y) = inf[u(x) / (x, y) 5 1, x E K] then v has the following properties: (a’) v is finite, continuous and quasi-concave on K. (b’) w is increasing on K. (c’) For all XE K the maximization problem. sup[u(y)/ (x, y) 5 1, YE K] has at least one optimal solution in K. (d’) For all x E K, the function of one real variable p(t) = sup [o(y) / (x, y) 5 t, y E K] is continuous in t for all t > 0. Furthermore, the following property holds:
44 = SUPC~Y)/ 6, Y> 5 1,Y E Kl. It would be noticed that conditions (a’), (b’), (c’) and (d’) are exactly symmetric to conditions (a), (b), (c) and (d). I n counterpart the monotonicity assumption in Diewert’s theorem is replaced by a strict monotonicity assumption, but this is a very usual assumption in economics. More restrictive are conditions (c) and (c’). In terms of utility theory condition (c’) means that when faced with a positive vector price a consumer will choose a positive consumption vector. In other terms, even if the price of one unit of goods number i is very large, the consumer will take may be a very small amount of goods i but positive. We shall give later conditions which imply condition (d). To simplify notation, we shall denote by M(y) the set of the optimal solutions of the minimization problem u(y) =inf[u(x) / (x, y) 5 1, x E K] and by N(x) the set of the optimal solutions of the maximization problem u(x)= supC4~) / (x, Y> i 1, Y E Kl. One extends the function u to R” by setting u(x) = + co if x $ K and one denotes by U the greatest lower semi-continuous function which is majorized by u. If u is quasi-convex then ii is also quasi-convex and coincides with u over K [see, for instance, Crouzeix (1977 or 1982)]. Clearly U is nonincreasing. Proposition 2.3.
Let u be a function satisfying conditions (a), (b) and (c) of
154
J.-P. Crouzeix.
Theorem 2.2. conditions:
Assume,
Differentiability
of the dual function
in addition, that u satisfies one of the following
(d,) u has a continuous (but not necessarily finite) extension on the non-negative orthant. or
(d2) For all y E K, the set M(y) = {x E K / u(x) = v(y), (x, y) 5 l> is closed. Then u satisfies condition (d) of Theorem 2.2. Proof It can be easily seen that it is sufficient to prove the continuity of functions 8 at t = 1. Let j be fixed in K. From condition (c), the set M(j) is not empty. Let XE M(j), clearly O(t) 5 u(G) for all t > 0 and Q(1) = u(X). Hence 0 is upper semi-continuous at 1 because u is continuous at X. Assume for contradiction that 0 is not continuous at 1. Then there exist E>O and a sequence {t,,} converging to 1 such that O(t,) 5 e( 1) -E for all n. Note that there exists x, E K such that d(t,) = u(x,) and (x,, j) = t, in consequence of conditions (b) and (c). Since j is a positive vector, x, EK and the sequence {t,,) converges to 1, there exist Reel(K) and a subsequence {x,,,} of {x”) which converges to i. Clearly (a, j) = 1 and ti(.?) 5 e( 1) -8. Hence z? belongs to the boundary of K. Because U is quasi-convex, X+ t(i-2) E M(j) for all t E [0,1). This is in contradiction with condition (d,) or (d,). 0
In particular, if M(y) is reduced to a singleton for all ~EK, then condition (d,) is satisfied. This is the case when u is strictly quasi-convex. The next proposition shows that condition (d,) implies condition (d,): Proposition 2.4. Assume that conditions (a), (b), (c) and (d,) are fulfilled. Then condition (d,) is satisfied. Proof
Let ~EK and set G(y) 3 M(y) and A(y) is closed. opposite case there would exist Define ~EK by ji=yi if Zi#O, y#j. Because U is the continuous
~(y)={x/U(x)=v(y),(x,y)~1}. Clearly It suffices to prove that M(y) = M(y). In the X in the boundary of K such that X EM(~). - ji=2yi if Xi=O. Then (x,y)sl, ygJ and extension of u on cl(K), then
v(y) = inf[G(x) / (x, j) 5 l] 5 U(X)= v(y). This is in contradiction
with consequence (b’) of Theorem 2.2.
Proposition 2.5. Assume that conditions (a), (b), (c) and (d2) are satisfied. Then for all ~EK, M(y) is a convex compact set. Furthermore the correspondence M from K is upper semi-continuous.
155
J.-P. Crouzeix, Differentiability of the dual function
ProoJ: Clearly M(y) is bounded because y is a positive vector, M(y) is The continuity property of M is a convex because u is quasi-convex. consequence of the maximum theorem and the continuity of u and u on K.
Cl
Similar results to the above proposition hold for Diewert’s duality [see Diewert (1982a)]. It could be interesting to know conditions which imply condition (c). The following result gives a sufficient but not necessary condition: Proposition 2.6. Let u be a function satisfying Theorem 2.2. and the additional condition: (cl) If X belongs to the boundary Then u satisfies also conditions
conditions
(a) and (b) of
of the positive orthant, U(x) = supXEK u(x). (c) and (d,).
Proof: Let y be fixed in K and consider the set S,= {x/U(x) sn, (x, y) 5 l} for 1< sup [U(X)/XE K]. Clearly S, is a compact set and Snc K. On the other hand, it can be easily seen that M(y)= r),{S,/SI#8]. Hence, M(y) is not empty and closed. 0 From an economic point of view, the transposed condition (c;) means that any positive consumption vector is preferable to any non-negative consumption vector having a component equal to 0. Let us notice that the CobbDouglas functions verify conditions (a), (b) and (ci). The first results concerning the duality described in Theorems 2.1 and 2.2 have appeared for differentiable functions [see, for instance, Lau (1969)]. Assume, for instance, that the conditions of Theorem 2.2 are fulfilled, X and ~EK and XE M(y), then YE N(Z). If in addition u is differentiable at X and u’(X) #O and u is differentiable at j and v’(j) #O, then it is clear that X = v’(j)/(v’(j$
j)
and
j= u’(X)/(u’(X), x).
(8)
Unfortunately, the differentiability of one of the functions does not necessarily imply the differentiability of the other one as it can be seen in the following example [Crouzeix (1977, p. 226)]: u(xi,%)=
-max[(x4~,/4);,(x,x~/4)~,(x,
it can be easily seen that u is differentiable satisfies the conditions of Theorem 2.2.
+x2)/3]
if
on the positive
x1,x2
EK,
orthant
of R2 and
156
J.-P. Crouzeix.
The dual function
v is defined
v(Y~,h)= -3[vfv21-+ = -+[y,y$]-+ v is not differentiable minimization problem
of the dud fiction
by
if
Y,~Y,>O,
if
y,zy,>O.
at (y,,y,)
has not a single solution
3. Differentiability
Di&erentiahility
when y, =y,.
It should
be noticed
that
the
when yi =y,.
properties of quasi-convex
functions
Since our purpose is to obtain conditions which ensure the differentiability of the dual function and according to the specific structures of these functions, we shall list in this section some properties of quasi-convex/quasiconcave functions we shall use later. The proofs of these properties can be found in several papers by the author. For a survey on differentiability properties of quasi-convex functions, see Crouzeix (1982). Throughout this paper, differentiability is taken in its usual sense. Namely, given a real-valued function f which is finite in a neighbourhood of x0, f’(x,) is the derivative off at x0 if
(llllhII)Cf(x~+h)-f(x~)-(f’(x,),h)l~O where llhll denotes Let {x/f(x) and T closure
the usual norm
when h-4
of h.
f
be a quasi-convex function which is finite at x0. Define S= O}. Then S is a convex set is a convex cone. If f is differentiable at x0 and f’(xJ#O, then the of T is a half-space. Conversely we have:
Proposition 3.1. [Crouzeix (1981)]. Let f, S and T be defined as above. If the closure of T is a half-space and if there exists h in the interior of T such that the function of one real variable,
k(t) = f (x,, + th), is differentiable Of course,
at t = 0, then f is differentiable the analogous
at x0.
result holds for quasi-concave
functions.
J.-P. Crouzeix,
D#erentiubility
157
of the dual function
Now, assume that u and o are functions defined as in Theorem 2.1 or 2.2. Let YE K and consider the set M(j) of the solutions of the maximization problem u(j) = inf[u(x) Theorem
3.2.
vector of R”
function
/ (x, j) 5 11.
The function v is differentiable at YE K and v’(j) is a positive M(j) is reduced to a single element X E K and if the
if’ and only if
of one
real
variable
f3(t)= inf[u(x)
/ (x, j) s t]
has
a
non-zero
derivative at t = 1.
Proof. Assume that v is differentiable at j and v’(j) is positive. Then H is differentiable at 1 and Q(l)= - (v’(j),j) v(j)} and V= {k/ (k, h) ~0 for all he U]. Then U is a convex open cone because v is quasi-concave and continuous and V is a convex cone. It follows from the definition of M(j), the non-increasingness of u and the uniqueness of M(j) that (X, j) = 1. Prove that X E - 1/: Indeed,
v(Y)= 4-3 = sup (Y-K-320 V(Y) ’
v(Y)
[v(y) / (Y, 2) 5 = *
=a
11,
u(y)5v(j), (y-j,X)>O.
(9)
Now, prove that the closure of U is a half-space. If not there exists 2~ - V such that f# t2 for all t >O. Since XE - q X is positive and V is a convex cone then Z? can be taken positive and satisfying (i, j) = 1. From the definition of V it follows that (y, a) 2 1 implies u(y) 5 u(j), hence since u can be defined from u, u(a) =v(j). But then i E M(j) and thus is a contradiction. The proof is achieved by noticing that the function p(t) =v(j+ tj) is differentiable at 0 and applying the quasi-concave version of Proposition 3.1 to the function v. 0 Thus, the differentiability of v at J is characterized in terms of uniqueness of the set of solutions M(j) and differentiability of a function of one real variable. Now we intend to express these conditions in terms of the primal function u. It can be easily seen that M(j) is a singleton for all YE K if and only if u is strictly quasi-convex. To express the second condition, we
158
J.-P. Crouzeix,
Differentiability
of the dual function
associate to the already defined function 9
The function of one real variable 8- defined by
Let T= A=R and consider the epigraph of 8 and the hypograph respectively defined by
of 8-,
epi(8) = {(t, A)E T x A/e(t) 5 A}, hypo(e-)={(A,t)eA
x T/8-(il)2t}.
Then the function 8 can be considered as the inverse function of 0- and, conversely, 8- as the inverse function of 0 in the sense that the closure of the epigraph of 8 coincide with the closure of the hypograph of K when these two sets are considered as subsets of the same space T x A [see for instance, Crouzeix (1982)]. Set X= e(l) =v(Y). It follows that 8 is differentiable at 1 and 0’(l) #O if and only if F is differentiable at 1 and (K)‘(I) #O. Besides e’(l)e-r(X)= 1. To have an economic interpretation of the relation between 8 and 8- it is preferable to consider the analogous functions derived from the direct utility function,
B(r)= sup CO(Y) / (Y, P> 5 rl, B-(4 = infC / 4~) L 4, where
p
is a fixed vector of commodity prices.
Then j?(r) is the maximum utility level that the consumer can attain with the income r and p-(A) is the minimal price that he has to pay to have a consumption having a utility level greater than or equal to A. Then functions /I and p- can be considered as the inverse function of each other. In the following two sections we shall try to express the differentiability of 0 or 8in function of u.
4. The convex and convexifiahle
case
One of the easier cases which can be considered is the case where the function u is convex on a compact neighbourhood C of the unique element X
J.-P. Crouzeix,
Differentiability
of the dual function
159
of M(j). It could be easily seen that (3(t)=inf[u(x) /x E C, (x, j) 5 t] in a neighbourhood I of 1 because the correspondence M is upper semicontinuous. Also, 8 is convex on I and since u is non-increasing 19(t)= min [U(X)/XEC, (x, j) =t]. Calculate the subgradient of 8 at 1. By definition t* E %( 1) if e(t)-l3(1)2t*(t-
1).
Replace 0 by its expression in function of u, U(X)-u(X)~t((x,j)-(i,j)),
XEC.
Hence t* E %( 1) if and only if t*j E h(X). Thus if u is differentiable at X and u’(X)#O, then 8 is differentiable at 1, u is differentiable at j, el(l)j=u’(X), 0’(l) = (u’(X),%) and (u’(j),p) = -(u’(X), X) since e(t) = u(y/t). Expressions (8) become u’(j) = - (u’(X), X)X,
- -
(10)
U’G)= - (u’(Y),Y>Y.
This result can be easily extended to locally convexifiable functions (a function u is said to be locally convexifiable at X if there exists a convex neighbourhood C of X and an increasing and continuous function k defined on u(C) such that ri = k o u is convex on C). Set r?(y)= inf[a(x)/x E C, (x, y) 5 11, &t)=G(j/lt). Then it can be easily seen that in a neighbourhood of 1, e(t)= k[Q(t)]. Assume that u is differentiable at X and u’(X)#O, then since zi is convex k has finite semi-derivatives at x=u(X), k’+(1) = lim nio
4)
- W) 2-2 ’
k’(X)=lim ito
4)
-k(T) 1-I
.
the subgradient of ti at X is given by &i(x)= Furthermore, {tu’(x)/k’(J)~t~k;(;5)}. Th en the subgradient of 0 at 1 can be easily calculated and finally it is seen that 8 has a derivative at t = 1 with e’(1) = (u’(X),%) since o(r)= k[@t)]. Thus we have established the following proposition: Proposition 4.1. If the function u is locally conuexifiable at the unique element X of M(y) and if u has a non-zero derivative at X, then u is differentiable at j and u’(j) = - (u’(X), X)X, u’(X) = -(u’(j), j) y.
Second-order conditions which ensure the local convexifiability can be found in Crouzeix (1977, p. 126), Schaible and Zang (1980) or Zang (1982).
160
J.-P.
For instance,
indicate
Crouzeix,
Differentiability
the following
of the dual function
results:
Let u be a quasi-convex function which is twice continuously differentiable in a neighbourhood of X. Assume in addition that the matrix u”(X) is regular and that (u’(X), [u”(X)] - ‘u’(X)) #O, then u is locally convexifiable at X. Let u be a quasi-convex function which is twice continuously differentiable in a neighbourhood of x. Zf there exists p>O such that the matrix u”(X) + p’(X) [u’(x)]’ is positive definite, then u is locally convexifiable at X. By combining these second-order characterizations for local convexitiability with Proposition 4.1, we obtain similar conditions for the differentiability of the dual fraction to these given by Blackorby and Diewert (1979) [see also Diewert (1982)].
5. The general case Could the result in Proposition 4.1 be extended to functions which are not locally convexiliable? An equivalent formulation of this problem can be expressed as follows: given a quasi-convex function f and the function 8 defined by O(t) =inf[ f (x) / (x, j) 5 t] where j is a fixed non-null vector of R”, is the differentiability off at X sufficient to ensure the differentiability of 6’ at t= (X,7) when it is assumed that X is the unique solution of the minimization problem defining 0(f)? The following example furnishes a negative response to this question: Example.
Consider
the following
sets of R2:
S,={(x,,x,)~R~/~~~2;1-x~~,x~~ S,= {(x1,x2)~ R’/x,s
-x:
-x:+A} +21}
if
250,
if
AlO.
It is clear that Snc S, for all il, p such that ,? 0. In the proof we shall only consider the special case where X =O, f(X) =O, f=O and j= e, basis of R”, the general case where (e1,e2,..., e,) designs the canonical considered in the proposition could be easily deduced thanks to elementary transformations. A vector x E R” will be represented by (a, a) where a is the vector of R”-’ whose components are the first (n- 1) components of x and E
162
J.-P. Crouzeix. Differentiability of the dual function
is the last one. In particular, d will be represented by (46) where A E R”-’ and PER. Set ~,={x/f(x)~~}={(a,~()/f(a,a)~~}. Since f is continuous in a neighbourhood of X=(O,O) and Se is convex there exists W neighbourhood of (0,O) in R” and a concave function g on R”-’ such that S, n W={(a,a)/g(a)~r}. Clearly g(O)=0 and g(a)O. Let EE(O, 1). There exists a neighbourhood I/ of 0 in R”-’ and a,,a, E R such that a1 0,
rcp’(O)(l+e)~cp,(r)~rcp’(O)(l-.s)
if
rO,
if
;1
