application of duality theory

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Frontiers of Qumti tatiue Economics. Volume IT, ed . M. D. Infriligaforand D. A. Kerldrlck.0 1974 North-Holland Publishing C ompany.

CHAPTER 3APPLICATIONS OF DUALITY THEORY*

W.E. DIEWERTDepartment of Manpower and Immigration, Ottawa

Department of Economics, University of British Columbia

0. IntroductionThere appear to be two principal practical applications of duality theory ineconomics.

The first principal application of duality theory is that it enables us toderive systems of demand equations which are consistent with maximizing orminimizing behavior on the part of an economic agent, consumer or pro-ducer, simply by differentiat ing, a function, as opposed to solving explicitlya constrained maximization or minimization problem. Perhaps the firstperson to appreciate the econometric implications of the above statement inthe context of consumer demand theory was Rent Roy (1942):

Cette conception conduit k I'emploi de coordonndes tangentielles pour la dkfinition dessurfaces d'indifference e t elle offre ainsi la possibilitk d'obtenir pour I'optimum duconsommateur, de nouvelles relations d'kquilibre, homologues des relations obtenues

* This research was partially supported by a Canada Council grant. In the five yearsthat 1 have been studying duality theory, many people have assisted me: S.N. Afriat,E.R. B erndt, L.R. Christensen, M. Denny, L. Epstein, M. Fuss, Z. Griliches, R.E. Hall,G. Hanoch, R. Harrls, D.W. Jorgenson, L.J. Lau, M. Nerlove, D. McFadden, P.A.Samuelson, R. W. Shephard, R. M. van Slyke,E. Wiens and A. D. Woodland. The edito -rial assistance of M. lntriligator is also greatly appreciated. My thanks also to Lise Bldoofor typing a difficult manuscript.Jhe views expressed are solely those of the author anddo not necessarily correspond to any policy or position of the Department of Manpowerar~dmmigration.

Applications of duality theory, en coordonnker ponctuelles, cer nouvelles relations pr6sentant l'avantag e d e !.-:.A '!e-,,,!.xplicitementh uantitks en fonction du revcnu et des prix.: ..it Ainsi parait s'ouvrir un champ fkcond d'investigations pour les Ccono' ;v$! :$$, (Roy 1947, p. 225).. - ; -, . ~ . ..,?.,,.;&'-,

k?-8 I. ...,. One of the first authors to appreciate the usefulness of duality the: deriving systems of factor demand functions in the context of pro. , theory was McFadden (1966, p. 13):. ,..

The introduction of cost, revenue and profit functions into production theory ytheoretical return and several practical advantages. Most of the qualitative reproduction theory follow from properties of these functions, without resassumptions on the divisibility of commodities and convexity and smoothproduction possibilities.

The principal practical advantage lies in the simple relation between thestions and the corresponding demand and supply functions. F or example, difftion of the profit function yields net output supply functions, and summaprices times net output supply functions yields the profit function.The second principal advantage of duality theory is that it enables

derive in an effortless way the "comparative statics" theorems origdeduced from maximizing behavior.' This second principal advantaits origins in the work of Hotell ing (1932, pp. 594, 597). McFadden 1970) has extensively pursued this second principal advantage.

In sections 1,2 and 3 of the present paper, we provide some exampthe first principal advantage of duality theory, while sections 4, 5 provide some examples of the second principal advantage of duality t

In section 1, we study the duality between unit cost funct ions and coreturns to scale production functions. The duality which is developsection 1 was known to SamuelsonZ 1953) and can be obtained by sizing Shephard's (1970) more general results. However, our present mof developing the duality theorem is somewhat more direct than Shepmethod, since we proceed directly from the cost function to the prodfunction via definition (8). In the appendix to this paper (section loutline a proof of a theorein which appears in the main text of the pa(i) the theorem contains a "new" result3 or (ii) the proof is new (and

That the assumption of maximizing behavior implies certain restrictions sumer dema nd functions was known to Antonelli (1968, pp. 33-39) (originally p1886) and Pareto (1971, pp. 417-426) (originally published 1906), but the fir st comparative statics result was obtained by Slutsky (1915). Hicks (1946) and Sa(1947) systematically used the assumption of maximizing behavior in ord er tocomparative statics theorems.

Samuelson does not provide a proof of the theorem.Of course, many "new" results are straig htfonva rd modifications of known


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.* W.E. Diewert Applications of duality theoryf d y simpler than existing proofs). In any case, the reader who is primarily

in applying the existing body of duality theory can read t he paper

l to peruse the proofs.In section 7, we briefly survey the no nparametric methods of obtaining

In section 8, we note how duality theory can m ake the variable coefficients

From a mathematical point of view, duality theory rests on a theorem dueMink owski (191 1): every closed4 convex5 set in R~ can be characterizedhalf space^.^ We shall see how this theo-

Notation: x 2 0, means each compone nt of the vector x is nonnegative;is an N-dimensional vector of zeroes; x > ON means x 2 ON, but x # ON;D ON means each component of x is positive; E means "belongs to";

y s Zxi yi, the inner product of the vectors x and y; Vf(x) is the gradientn matrix of second order partial derivatives o f f evaluated a t x;

uT means the union of the sets S an d T; an d Sn T is the intersection ofan d T.

between unit cost and production functionsf, where y = (x , ,

2 , ...,xN )= (x) means y is the maximal amount of output which can begiven period of time using xi units of input i fo r i =1 , 2 , ...,N , where x = (x , , x , , ...,xN) s the vector of input levels. If the

A set S in RN Euclidean N space) is closed if 2' S for n = 1,2, . , lim x" x0x0 E S.A set S in RN s convex if for every x E S, y S and scalar 1 uch that 051 < 1,

have k+ ( l- 1) Y E S.A halfspace in RN s a set of'the form { x: a T x 5 k } , where aT x= a, x l is thethe vectors a and x . See Fenchel(1953, pp. 48-50) or Rocka fellar (1970,99) for a treatment of Minkowski's theorem.

1g production function satisfies certain regularity conditions, then we maycalculate the producer's total minim um cost function C( y ; l , 2 , . N )=

2;*, =C(y ; p), where p = (p , , ,, ...,p,) is the vector of input prices, as thej@4 +- solution to the following constrained minimization problem:C ( y ; p ) = minx {pTx: f (x ) 2 y ) .

i In other w ords, the producer takes prices as given and attem pts to minimizethe cost of producing a specified output level, y. In general, total cost Cdepends on y, the chosen outp ut level; p, the given vector of input pricesan d f, he given production function.

A production function f determines a cost function C through definition(1). Wh at is not as well known, is the converse, tha t a cost function satisfyingcertain regularity conditions determines a produ ction fun ction; tha t is, thereis a duality between cost and production functions. Given one of these func-tions, under certain regularity conditions, the other can be uniquely deter-mined, a result originally due to Shephard (1953) and Samuelson (1953-4)The production function can in general be obtained from a cost functionsatisfying the appropriate regularity conditions as the solution to the following constrained maximization problem:

f * ( ~ ) m ax, { y : C ( y ; p ) _< p T ~or every p 2 ON), ( 2I where R = Z,, Z,, ...,ZN) s a given vector of inputs and C is the given cosfunction. Note tha t problem (2) has an infinite number of constraints.A geometric interpretation of the maximization problem (2) can be ob-

tained as follows: for every vector of factor prices p >ON ,we can graph theI set of input combinations x such that pTx = C( y; p) for some fixed outpu

level y. Since C(y; p) is supposed to represent the m inimum cost of producingoutput y given prices p, it is reasonable to assume that the set L(y) == {x: (x) 2 ) lies within the halfspace H (y ; p) = { X : ~ ~ XC(y ; p)) . Infact, we may define the envelope produ ction possibilities set L4 (y) as theintersection over all price vectors p > 0, of the halfspaces H( y; p) ; i.e.L* y) = " p > ON H( y; p). Now choose j so that Z belongs to the boundaryof L* ( j ) , nd we have determined j = (Z) using only the given cost functionC. Moreover, we will have C(7; p) = pTZ for some p > ON ,wh ere C( j ;p ) 5S p T 2 o r ev ery p 2 ON . Notice that we have defined the set L"(y) as thintersection of a fam ily of halfspaces, where each halfspace was defined by anisocost surface. In order for the envelope production possibilities set L* y) tocoincide with the "true" production possibilities set L(y) = {x: (x) 2 y)


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>is,-W.E. Diewert i,&-[g?Mangasarian, O.L., 1969, Nonlinear Programming, New York, McGraw-Hill. 4 25.Rockafellar, R.T.. 1970, Convex Analysis, Princeton, New Jersey, Princeton University { {zPress. b -


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The conjugate convex function of F(y', z), considered a function of y',denoted F * (y'*, z), is given by

where the y'* may again be interpreted as normalized prices. It may bereferred to as the normalized restricted profit function: because it can beidentified with the restricted profit function if the price of the (n + 1)stcommodity is set equal to unity. Other terminology for the restricted profitfunction include gross profit function, ' partial profit f ~ n c t i o n , ~nd variableprofit function.' We note that in the case that z is a vector of outputs, a nd y'is a vector of inputs, F* yl *, z) is the negative of the normalized cost func-tion; in the case that z is a vector of inputs, and y' is a vector of outputs,F* (y l* , z ) is the normalized revenue function; in the case that the quantitiesof all commodities are variable, the normalized restricted profit function isthe normalized profit function.

Seventh, the subgradient of a convex function F(y) at y, denoted y*, isdefined by the system of inequalities

F ( x ) 2 F ( y ) + ( y * , x - y ) V x .The set of all subgradients at y denoted aF(y) is referred to as the subdiffer-ential of F(y ) at y. If the subdifferential at y consists of only on e element, it isequal to the gradient of FQ at y, denoted VF(y). Given th e subdifferential,a closed, proper and convex function is determined up to an additiveconstant.

1. I. Alternative approaches to duality theoryThere are as many alternative approaches to duality theory as there areindividuals working in the field of duality theory. These different approac hesmay be approximately classified, at the risk of gross simplification, into threegroups.The&~f,gfrup 2f approaches is based on the conjugacy correspondencedeveloped by ~e nch el'( l94 9, 953) and extended by R ockafellar (1970a). In

This terminology follows McFadden (1973a).This terminology is used by Gorrnan (1968).This terminology is used by Lau (1969).' his terminology is used by Diewert (1973).

Applications o duality theory : its modern form, this theory states that given F(y), a closed, proper

f $ onvex function, its conjugate dual, defined as. F* Y* )= sup {(Y,y* ) - F(y )}

E r YI is also a closed proper convex fu~ ~c ti on . oreover, the dual of the ! defined as

is equal to F(y):F * * ( ~ ) F ( y ) .

Hence there is a one-to-one correspoi~dencebetween F(y) and F*Economically, this implies a one-to-one correspondence betweenproduction function a nd the normnlized profit function under the assumof closure, properness, and convexity. The precursor of this conjucorrespondence is of course the classical Legendre transformation. transformation was implicit in the pioneering work of Hotelling (1932the normalized profit function. This line of reasoning was taken uSamuelson (1953) and Lau (1969) for the differentiable case, and by Joson and L au (1973) for the general case. One may add that with the additassumption of nonproducibility of the (n + 1)st con~modit y, hat is, it be used as an input, the properties of F(y) and F*(y*) are complidentical - hey are both nonnegative and are equal to zero at the ori

g Any function th at can be used as a produc tion function can also be usednormalized profit function, and vice versa, and the duality is complsymmetric.

The second gr oup of appro aches is based on the symmetric dubetween gauge functions', or distance functions, or polar cones of cosets. Shephard (1953) was the pioneer of this group. He proved the dutheorem between production and cost functions under the assumptiodifferentiability employing the concept of distance functions. Other schwho have used similar approaches include Gorrnan (1968), McFa(1973a), Hanoch (1973), and Jacobsen (1 970, 1972), among others.

The third gr oup of approaches is based 011 he duality between the sprod"&i6n po%sibilitiesand its sup port function. The w ork of Uzawa (1McF adden (1966) and Diewert (1971, 1973) may be classified amongsgroup.

Unskilled labor appears to be the natural choice for the ( ? I + ] ) s t commodity.


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Let y be an arbitrary point in e. Since is (relatively) open, there is ap > 1 such that By E c.Given E > 0, et 6 > 0 e such that llxll < 6 impliesIf (x)l < E. Then for llz - ll < (1 - 8-')6, we have

z = y + (1- 8-')x = p-'(py) + (1- -')xfor some x E c with llxll < 6. Thus z E C andThus f is bounded above in the sphere llz - ll < (1 - 8-')a. It followsthat for sufficiently large r the point (r, y) is an interior point of [f,C];hence, by Proposition 1,f is continuous at y. I

The proof of the following important corollary is left to the reader.Corollary 1. A convex functional defined on a finite-dimensional convex set Cis continuous throughout c.

Having established the simple relation between continuity and interiorpoints, we conclude this section by noting a property off which holds i j[f,C] happens to be closed. As illustrated in Figure 7.8, closure of[f, ] is related to the continuity properties off on the boundary of C.

Figure 7.8 A nonclosed epigraphProposition 3. If [f,C] is closed, then f is lower semicontinuous on C.

Proof. The set {(a, x) E R x X :x E X) is obviously closed for eacha E R. Hence, if [f, ] is closed, so is

[f, ] n { ( a , ~ )x E X) { ( a , ~ ) : i C;f(x) 5 )

67.10 CONJUGATE CONVEX FUNCTIONALS 195for each a E R. It follows that the set

T , = {x : x E C,f (x) I )is closed.

Now suppose {xi) is a sequence from C converging to x E C Letb = lim inf f (xi). If b = - o, hen x E T, = To for each a E R which isX,+Ximpossible. Thus b > - o and x E Tb+= Tb+for all E > 0. In other

words,f (x) I lim inff (xi) which proves that f is lower semicontinuous.X,+X

Figure 7.9 shows the graph of a convex functionalf defined on a disk Cin E2 that has closed [f, ] but is discontinuous (although lower semi-continuous) at a point x.

Figure 7.9

7.10 Conjugate Convex FunctionalsA purely abstract approach to the theory of convex functionals, includinga study of the convex set Lf, C] as in the previous section, leads quitenaturally to an investigation of the dual representation of this set in termsof closed hyperplanes. The concept of conjugate functionals plays a naturaland fundamental role in such a study. As an important consequence of thisinvestigation, we obtain a very general duality principle for optimizationproblems which extends the earlier duality results for minimum normproblems.Definition. Let f be a convex functional defined on a convex set C in anormed space X. The conjugate set C* is defined as

c * = {x * E x*:up [(x, x* > - f x ) ] < co)x EC


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and the functional f * conjugate tof is defined on C* asf *(x*)= sup [(x, x*) - (~11.x 0 C

Proposition 1. The conjugate set C * and the conjugate functional f* areconvex and [ *, C *] is a closed convex subset of R x X *.

Proof. For any x: x; E X* and any u, 0 < u < 1, we haveSUP {(x, ux: + (1- )x,*)- f (x)) = sup {uC(x, x) - (x)]x 0 C x E C

+ (1- u)C(x, x;) - (xlllOr SUP C(x, xr>- f ( 4 1x E C+ (1 - a) SUP C(x, x;> - f (x)lX E C

from which it follows immediately that C* and f * are convex.Next we prove that [ *, C*] is closed. Let {(st, x*)) be a convergent

sequence from [f *, C*] with (s, , x) + s, x*). We show now that(s, x*) E [ *, C *]. For every i and every x E C, we havesr2 *($) 2 x, x - (x).

Taking the limit as i + a , we obtains 2 x, x*) - (x)

for all x E C. Therefore,s 2 up C(x, x*) -X E C

from which it follows that x* E C* and s 2 *(x*). )We see that the conjugate functional defines a set [f *, C*] which is of

the same type as [f,C] ; herefore we write [f,a* Lf *, C*]. Note thatiff = 0, the conjugate functionalf * becomes the support functional of C.Example 1. Let X= C =En and define, for x = (x,, x, , . ,x,), f (x) =l/p x!=, xllP, 1 < p c a. Then for x* = (5,, 5,, .. c,),

$7.10 CONJUGATE CONVEX FUNCTIONALS 197The supremum on the right is achieved by some x since the problem isfinite dimensional. We h d , by differentiation, the solution

= IxrlP-' sgn xi

where l/p + l/q = 1.Let us investigate the relation of the conjugate functional to separating

hyperplanes. On the space R x X, closed hyperplanes are represented byan equation of the form ( r , X ) E I R x X

(s,x% R X K *((b , ) , (s,fl>>= sr + (x, x*) = kwhere s, k, and x* determine the hyperplane. Recalling that we agreed torefer to the R axis as vertical, we say that a hyperplane is nonvertical if itintersects the R axis at one and only one point. This is equivalent to therequirement that the defining linear functional (s, x*) have s # 0. If atten-tion is restricted to nonvertical hyperplanes, we may, without loss ofgenerality, consider only those linear functionals of the form (- 1, x*).Any nonvertical closed hyperplane can then be obtained by appropriatechoice of x* and k.To develop a geometric interpretation of the conjugate functional, notethat as k varies, the solutions (r, x) of the equation

(x, x*) - = kdescribe parallel closed hyperplanes in R x X. The numberf *(x*) is thesupremum of the values of k for which the hyperplane intersects [f,C].Thus the hyperplane (x, x*) - = *(x*) is a support hyperplane ofcf, CI .In the terminology of Section 5.13, f *(x*) is the support functionalh[(- 1, x*)} of the functional (- 1, x*) for the convex set [f,C]. Thespecial feature here is that we only consider functionals of the form( - 1, x*) on R x X and thereby eliminate the need of carrying an extravariable.For the application to optimization problems, the most important geo-metric interpretation of the conjugate functional is that it measures verticaldistance to the support hyperplane. The hyperplane

intersects the vertical axis (i.e., x =8) at (- f *(x*), 8). Thus, - *(x*) isthe vertical height of the hyperplane above the origin. (See Figure 7.10.)


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198 OPTIMIZATION OF FUNCTIONALS 7

Figure 7.10 A conjugate convex functionalAnother interpretation more clearly illuminates the duality between

[f,C] and [f*, C*] in terms of the dual representation of a convex setas a collection of points or as the intersection of half-spaces. Given thepoint (s, x*) E R x X*, let us associate the half-space consisting of all(r, x) E R x X satisfying

(x, x*) - l .Then the set [ *, C *] consists of those (nonvertical) half-spaces that con-tain the set [f,C]. Hence [ *, C*] is the dual representation of [ , C].

Beginning with an arbitrary convex functional cp defined on a convexsubset r of a dual space X*, we may, of course, define the conjugate of cpin X** or, alternatively, following the standard pattern for duality rela-tions (e.g., see Section 5.7), define the set * r n X as

and the convex functional*cp(x) = SUP C(x, x*) - cp(x*)Ix* r

on * r . We then write *[p, r] = [*cp, * r] . With these definitions we havethe following characterization of the duality between a convex functionaland its conjugate.Proposition 2. Letf be a convex functional on the convex set C in a normedspace X. If [f ,C] is closed, then [f,C] = *[[f, CI*].

Proof. We show first that [f, C] c *[f*, C*] = *[[f, CI*]. Let(r, x) E [f,C] ; hen for all x* E C*, f *(x*)2 x, x*) - (x). Hence, wehave r 2 (x) 2 x, x*) - *(x*) for all x* E C*. Thus

r 2 sup [(x, x*)- f *(x*)lx* C*and (r,x) ~* [f *, *].

57.1 1 CONJUGATE CONCAVE FUNCTIONALS 199We prove the converse by contraposition. Let (r,, x,) q [f,C]. Since

[f,C] is closed, there is a hyperplane separating (r,, x,) and [f, ].Thus there exist x* E X*, s, and c such that

for all (r, X)E [f,C]. It can be shown that, without loss of generality, thishyperplane can be assumed to be nonvertical and hence s # 0 (see Prob-lem 16). Furthermore, since r can be made arbitrarily large, we must haves < 0. Thus we take s = - . Now it follows that (x, x*) - (x) I cfor all x E C, which implies that (c, x*) E [f *, C*]. On the otherhand, c < (x, ,x*) - , implies (x, ,x*) - > r, , which implies that(r0, xo) $ *[f*, C*l. I7.11 Conjugate Concave FunctionalsA development similar to that of the last section applies to concave func-tional~. t must be stressed, however, that we do not treat concave func-tion al ~ y merely multiplying by - and then applying the theory forconvex functionals. There is an additional sign change in part of thedefinition. See Problem 15.

Given a concave functional g defined on a convex subset D of a vectorspace, we define the set

[g, Dl = ((r, X):x E D, r I (x)).The set [g, Dl is convex and all of the results on continuity, interior points,etc., of Section 7.9 have direct extensions here.Definition. Let g be a concave functional on the convex set D. The con-jugate set D* is defined as

D* = (x * E X*: inf[(x, x*) - g(x)] > -a),x e D

and the functional g* conjugate to g is defined asg*(x*) = inf [(x, x*) - (x)]:,Srp f - f ~ x ) < X , x c

x e D xcDWe can readily verify that D* is convex and that g* is concave. We

write [g, Dl* = [g*, D*].Since our notation does not completely distinguish between the develop-

ment for convex and concave functionals, it is important to make clearwhich is being employed in any given context. This is particularly true whenthe original function is linear, since either definition of the conjugatefunctional might be employed and, in general, they are not equal.


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application of duality theory

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