Functional forms for profit and transformation functions

JOURNAL

OF ECONOMIC

Functional

THEORY

6, 284316

(1973)

Forms for Profit and Transformation

Functions*

W. E. DIEWERT’ Department of Economics, University of British Columbia, Vancouver 8, Canada Received September 8, 1971

1.

INTRODUCTION

AND

Sum.4w

In contrast to the volumnious literature on single output, multiple input production functions, the problem of obtaining functional forms for multiple output production functions, has received attention from only a handful of auth0rs.l In order to obtain economically meaningful functional forms for multiple output, multiple input production functions (or transformation functions as they are more commonly called), it is first necessary to determine what restrictions we wish to impose on the underlying production possibilities set, since these restrictions will determine the properties of the transformation function. Also, it is well known2 that given a production possibilities set and fixed prices of inputs and outputs, a profit function may be calculated along with the associated profit maximizing output supply functions and input demand functions.3 What is not so well known is that McFadden [16] [ 171 and Gorman [9] have shown that a profit function may be used to determine the corresponding production possibilities set, and thus we obtain a duality between transformation functions, production possibilities sets, and profit functions, which is similar to the Shephard [29], Uzawa [31], and McFadden [IS] duality theorems between production and cost functions. The main purpose of this paper is twofold: first, we wish to show how different a priori assumptions on the production possibilities set generate different restrictions on functional forms for transformation and profit * This paper is a substantial revision of Report 6925, Center for Mathematical Studies in Business and Economics, University of Chicago, August 1969. t The author is with the University of British Columbia and the Department of Manpower and Immigration, Ottawa. 1 See for example McFadden [16], Samuelson [28], or Powell and Gruen [23]. 2 See Debreu [3, pp. 43-51 and Hicks [12, pp. 78-881. 3 These derived demand and supply functions are sometimes set valued, in which case they are called correspondences. See Debreu 13, pp. 441)

284 Copyright All rights

0 1973 by Academic Press, Inc. of reproduction in any form reserved.

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FORMS FOR PROFIT

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functions, and secondly, we wish to exhibit some functional forms which should prove useful in empirical applications. In Section 2, we state a number of theorems (essentially due to McFadden and Gorman) which develop a set of equivalent conditions on production possibilities sets, transformation functions, and profit functions. We also state some other theorems which establish an equivalence between a production possibilities set which has the property of constant returns to scale in all factors, a transformation function and a variable prqfit jimction, which gives the maximum profits to the firm holding some inputs fixed. Proofs of the theorems are given in Section 7. In Section 3-5, we apply the theorems of Section 2 to the problem of obtaining functional forms for transformation functions and profit functions. In general, we try to obtain functional forms which satisfy the following conditions: (i) they are linear in the unknown parameters so that linear regression techniques may readily be applied to the problem of estimating the unknown parameters, (ii) the functional form contains precisely the number of parameters needed in order to provide a second order approximation to an arbitrary twice differentiable transformation (or profit) function satisfying the appropriate regularity conditions, and (iii) a simple set of inequality restrictions on the (unknown) parameters of the functional form are sujicient for the functional form to satisfy the appropriate regularity conditions over a range of values for the independent variables. Finally, in Section 6, we attempt to relate the contents of this paper to the recent work of others. We make the following notational conventions: x’ > x” means that each component of the vector x’ is equal to or greater than the corresponding component of x”; x’ > x” means x’ 3 x” and x’ # x”; x’ > x” means that each component of x’ is greater than the corresponding component of x”; xT denotes the transpose of the vector x; ONdenotes a vector of zeroes; E means “belongs to”; p*x denotes the inner product of the vectors p and x. 2. DUALITYTHEOREMS Let us assume that a firm can produce various combinations of M nonnegative outputs denoted by the vector y = (vl, yZ ,..., yM) during a given period of time using the services of various combinations of N

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nonnegative inputs, x = (x1 , zc2,..., xN). The firm’s production possibilities set T is the set of all input and output combinations which the firm can produce. Let us assumethat T satisfiesthe following regularity conditions. 2.1. CONDITIONS1 ON THE PRODUCTIONPOSSIBILITIES SET T: (i) T is a non-empty subset of the nonnegative orthant in M + N dimensional Euclidean space, (ii) T is a closed4set, (iii) T is a convex5 set, (iv) if ( y; x) E T, x’ 3 X, then ( y; x’) E T, (v) if (y; x) E T, OM < y’ < y, then ( y’; x) E T, and (vi) for every (finite) input vector x 3 0, , the set of producible outputs {y = (,v; x) E T} is bounded from above. Condition (ii) above is a weak mathematical regularity property which cannot be contradicted by empirical data. Condition (iii) above is a generalization of the Hicksian [12, pp. 86-871 regularity conditions that the technology exhibit diminishing marginal rates of transformation of outputs for inputs (i.e., decreasing returns to scale), increasing marginal rates of substitution of outputs for outputs and diminishing marginal rates of substitution of inputs for inputs. Properties (iv) and (v) above are free disposal assumptions while condition (vi) assertsthat a bounded vector of inputs can produce only a bounded vector of outputs. We note that conditions (iii)-(v) can be contradicted by a finite body of data whereas the other conditions cannot. Rather than describe technology by means of a set T, many authors use the concept of a transformation function in order to describe the set of efficient input-output combinations. The set of efficient input-output combinations may be described symmetrically as the set of (y; x) which satisfy the equation t*( y; x) = 0 where t* is the transformation function, or one output can be singled out (say yr) and the efficient set may be described by y1 = t(y, ,..., y,, ; x) where the transformation function t tells us what the maximum production of y, is given the vector of inputs x and the vector of other outputs y = (vZ , y, ,..., y,J to produce. Some authors, e.g., Hicks [12, pp. 78-88, 319-3201, define transformation functions in the symmetric manner and some do not, e.g., Samuelson [28]. We will use the unsymmetric definition of a transformation function in what follows, since it is easier to characterize the properties of such a function. 4 T is a closed set if z” E T, n = I,2 ,..., lim,,, 2”’ = 9 implies 9’ E T. 5 T is a convex set if z’ E T, z* E T, 0 < h Q I implies AZ’ + (1 - X)z” E T.

FUNCTIONAL

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The first thing to note about unsymmetric transformation functions of the form y, = t(~+ ,..., y, ; x) is that the function need not be well defined and finite for all nonnegative vectors of other outputs y = (y, )..., y,)= and nonnegative input vectors x = (x1, x2 ,..., x,,,)=, for if we choose the components of 5 to be large while the components of x remain small, then it may be impossible to produce any nonnegative amount of y, _If this is the case,we shall define t($; x) = -cc. Thus if we are given a production possibilities set T which satisfies conditions I above, we may define the (unsymmetric) transformation function t which corresponds to T as follows:

2.2. DEFINITION: max,,{)‘, : (n .9; x) E Tj t(j; x) = 1 --co

if there exists a y, such that (Y, ,A 4 E T,

otherwise

for all vectors $ 2 Ofircrl, x 2 0, .

2.3 THEOREM: If T satisfies 2.1 and the ,function t is defined b-y 2.2, then t satisfies Conditions II given by 2.4 below. 2.4. CONDITIONS II ON THE TRANSFORMATION FUNCTION t: (a) t(y;x) is an extended real valued function (i.e., it can take on the value --co for finite ( j; x)) defined for each (9; x) 3 (O,-, , 0,) and is nonnegative on the set where it is finite, t is a continuous from above function6 (c) t is a (proper) concave function,’ (d) t is nondecreasingin the components of x, (e) t is nonincreasing in the components of j, and (b)

(f) for every x > 0, , there exists a number 01large enough such that if any component of the vector? is greater than CL,then t(i; x) = -co. Since a concave function is always continuous from below over the interior of the domain of definition where it is finite,8 it can be seenthat 6 The function t(z) is continuous from above if for it = 1,2,..., we have r(zn) > i(z”), lim,-, zn = 9 then limn-mi(~n) = r(z”). ’ The function f(z) is a proper concave function over a convex set S if (i) for every Z’ ES, z” E S and 1 < h < 1 we have t(hz’ + (1 - h)z”) > xt(z’) + (1 - X) t(z”), (ii) r(z) : + io for every z E 7’, and (iii) t(z) > - w for at least one z E T. See Rockafellar [25, pp. 23-41 for more on proper concave and convex functions. 8 See Rockafellar [25, pp. 51, 841 for material on the continuity of convex functions. Note that a functionfis concave iff -f is a convex function.

288

DIEWERT

2.4(b) and (c) imply that the transformation function t is actually continuous over this region. A formal proof of Theorem 2.3 is given in Section 7 below. On the other hand, suppose that we are given a transformation function t satisfying Conditions II above. Then the production possibilities set T which corresponds to t may be defined as follows: 2.5 DEFINITION:

2.6 THEOREM: If t satisjes 2.4 and the set T is defined by 2.5, then T satisfies Conditions I given by 2.1 above. The proof of Theorem 2.6 is very straightforward if one notes that the set T defined by 2.5 is the epigraph [25, pp. 231 of the concave function t intersected with the half spacey1 3 0. Theorems 2.3 and 2.6 above show that we have two equivalent ways of describing the technology of an M outputs, N inputs firm. We shall now show that another equivalent parameterization of the firm’s technology can be obtained by means of the firm’s projit function. 2.7 DEFINITION: Given a vector of fixed output prices p (pl , p2 ,..., pM)T and a vector of fixed input prices w = (wi , wZ,..., wN)r with all prices positive, i.e., (p; w) > (0, ; ON), and a production possibilities set T, then the producer’s profit function r is defined by “(p; w) = sup,,X{p’y - w=x: (y; x) E r>. We think of the producer as a competitive profit maximizer; i.e., he takes input and output prices as fixed and does not attempt to monopolistically or monopsonistically exploit any demand or supply curves which he may face. Then for a given vector of prices (p, w), the producer is assumed to choose a feasible production plan ( y; X) E T which maximizes his profits.g The resulting maximum profits are a function of the price vector (p; w) which we denote by ~(p; w). However, we should explain why the profit function is defined as a supremum rather than as a maximum. Consider the transformation function given by y1 = t(xl) = x1 + cx,/(l + x1) where y, is output I, x1 , is input 1 and c is a positive constant. When p1 = w1 = 1, the profit function is well defined as a supremum (i.e., ~(1, 1) = lim,,, x + (cx/(l + x)) - x = c but would 0 We may think of all inputs and outputs as being variable or we may think of the set T as being a function of a vector of fixed inputs, and the vector x represents the variable inputs.

FUNCTIONAL

FORMS FOR PROFIT

289

not be well defined as a maximum since the maximum would never be attained for a finite amount of input. 2.8 THEOREM: If T satisjies Conditions I given by 2.1 above,‘O and the .function r is defined by 2.7, then rr satisfies Conditions III below. 2.9. CONDITIONS III ON THE PROFIT FUNCTION n-Q; w): (1) n(p; W) is an extended real valued function (i.e., it can take on the value + cc for finite (p, w)) defined for all (p, w) > (0, , 0,) and x(p, w) 3 pTa - wTb for a fixed vector (a; b) > (0, , O,), (2) rr is nonincreasing in the components of w, (3)

T is nondecreasing in the components of p,

(4) for every p > 0, and w > 0, , we have lim.,, where c 3 O,,, is a vector of fixed constants, (5)

rr(p, nw) < pTc

T is a (proper) convex function,ll and

(6) T is homogeneous of degree 1; i.e., for every scalar h > 0 and vector of prices (p; w) > (0; 0), we have r(hp; hw) = hr(p; w). If the vector of zero outputs and zero inputs belonged to T, then the vector (a: b) in 2.9(l) could be replaced by (0; 0) and the profit function would be nonnegative. However, our present formulation allows us to consider the case of quasifixed factors; i.e., inputs such that a minimum positive amount has to be employed during the production period. With respect to (4) of 2.9, supposethat T is such that the vector of zero inputs can only produce the vector of zero outputs. If T satisfied this condition, (as well as a certain subgradient condition), then the c 3 OM in (4) could be replaced by c = OM. However, our present formulation allows us to consider the case where fixed inputs may be combined with variable inputs x and positive outputs are producible by fixed inputs alone. Whether this is a realistic assumption will depend on the empirical application at hand and how fixed and variable inputs are defined. Now supposewe are given a profit function r(p, w) satisfying CondiloActually, (iii)-(v) of 2.1 may be omitted and the profit function w will still satisfy Conditions III. This point has been emphasized by McFadden [17, 7-81 in the context of cost functions and by Samuelson [27, p. 3591 in the context of consumer theory. However, if UN parts of 2.1 hold and n is constructed from Tvia definition 2.7 and then this function r is used to construct a set T* by means of definition 2.10, then T = T*; i.e., if all parts of 2.1 hold, then 7 completely characterizes T. I1 See Rockafellar 125, pp. 2341 for a definition of a proper convex function. Note that = can be uniquely extended to the entire nonnegative orthant by continuity [25, p. 521.

290

DIEWERT

tions III. Then we may use the profit function possibilities set by means of the following:

to generate a production

2.10 DEFINITION: T = {(y; x) : pr.v - wTx < rr(p; w) for every (p; w) > (0; 0) and (v; x) 3 (0; 0)). 2.11 THEOREM: If rr satisfies Conditions III given by 2.9 above, and the set T is defined by 2.10, then T satisfies Conditions 1 given by 2.1. The above theorems establish a duality between transformation functions and profit functions satisfying Conditions II and 111, respectively. Theorem 2.11 enables us to prove the following lemma. 2.12 HOTELLING'S LEMMAS If a profit function rr(p; w) satisfies Conditions III giuen by 2.9 above and is, in addition, dzjkrentiable with respect to output and input prices at the point (p*; w*) > (0; 0), then we have &(p*; w*)/apm = y&*; w*) for m = 1,2,..., M and ar(p*; w*)/aw, = -x,(p*; w*) for n = 1,2,..., N, where y&p*; w*) is the profit maximizing amount of output m given positive output prices p* and input prices w* and x,(p*; w*) is the projit maximizing amount of input n given prices (p*; w*). Hotelling’s lemma is extremely useful from an econometric point of view since it enables us to obtain functional forms for demand and supply functions consistent with profit maximization simply by choosing a functional form for ?Tand differentiating it with respect to input and output prices. We now turn our attention to the problem of modeling technology in the case where some inputs are fixed in the short run and thus we can expect diminishing returns to scale in the variable inputs and outputs. However, in what follows, we will assume constant returns to scale in all inputs (both fixed and variable) and outputs. I2 The statementof the lemma is due to Hotelling [13, p. 5941. Similar lemmas have been stated or proved by Hicks [12, p. 3311, Samuelson [26, p. 341, Shephard 129, p. 1 I], and McKenzie [19; 20, p. 541 in the context of cost functions or utility theory. McFadden [16] [17] has proven versions of 2.12 and 2.24 in the context of profit functions. We note that if the profit function is not differentiable at the point (p; w) but is finite in a neighbourhood of (p; w), then the convexity of the profit function will imply that the profit function has a nonempty set of supporting hyperplanes (y; -x) at (p; w) and thus the profit maximizing derived input demand and output supply functions become set valued functions or correspondences. See Rockafellar [25, p. 2151.

FUNCTIONAL

FORMS FOR PROFIT

291

The following remarks may help to explain why economists often assume that the production possibilities set T exhibits the property of constant returns to scale. If there is instantaneous free entry into an industry; i.e., no institutional or other barriers to entry, then T will exhibit the property of additivity,13 i.e., T + T is a subset of T. If in addition, T is a convex set containing the vector of zero inputs and outputs (so that there are no indivisibilities), then T will be a cone; i.e., Twill exhibit the property of constant returns to scale. In what follows, we will denote the variable inputs and outputs by the I dimensional vector u 3 (ul, t12 ,..., ul) and the fixed inputs by the J dimensional vector ~1= (c.~, v2 ,..., vJ), and z = (21,u) will denote an I + J dimensional vector of all inputs and outputs. The duality theorems at the beginning of this section assumedthat the role of inputs and outputs did not change and thus we could denote both inputs and outputs by nonnegative vectors. However now we wish to be more general and allow for the possibility of a technology which could either use a good as an input or produce the good as an output. (For example, an automobile plant could produce a part but if its internal labor costs increased sufficiently, it could be more profitable to have the part produced by a private subcontractor). Thus we shall follow the example of Debreu [3, pp. 381 and index inputs with a negative number and outputs with a positive number, so that in particular, the vector of fixed inputs v will be nonpositive. With the above conventions in mind, we can now list another seriesof conditions, definitions, and theorems which are similar to 2.1 to 2.12. 2.13. CONDITIONS I. ON THE PRODUCTION POSSIBILITIES SETT: (i) T is a closed, nonempty subset of I + J dimensional space, (ii) (iii) tion),

if (u; 2))E T, then v < 0, (last J goods are always inputs), T is a convex set (nonincreasing marginal rates of transforma-

(iv)

T is a cone; i.e., z E T, h 3 0 implies Xz E T (constant returns),

(v)

if z’ E T, z” < z’ then z” E T (free disposal), and

(vi) if (u; v) E T, then the components of u are bounded from above (for finite fixed inputs, the set of producible outputs is also finite). Again, we may define an unsymmetric transformation function t which corresponds to T as follows: IS See Debreu [3, p. 411.

292

DIEWERT

2.14

DEFINITION:

max,,{u, : (ul , i?; 0) E T} t(&; v) = --co

if there exists a u1 such that (ur , zi; ZJ)E T

otherwise,

for all vectors li 3 (uz , uQ,..., u,) and u < 0, . 2.15 THEOREM: If T satisfies Conditions I’ given by 2.13 and the jiinction t is dejined by 2.14, then t satisfies Conditions II’ below. 2.16. CONDITIONS 11’ ON THE TRANSFORMATION FUNCTION t: (a) t extended real valued function defined and bounded from above for I - 1 + J dimensional vector (a; u) such that Y < OJ and t(O,-, ; 0,) (b) t is a continuous from above function, (c) t is a (proper) concave function, (d) t(ALi; Xv) = Xt(fi; v) for every scalar h > 0, (e) t is nonincreasing in the components of (a; v), and (f) for every 0 < OJ , the set ((u, zi) : U, < t(ti; u)} is bounded above.

is an each = 0,

from

Note that it is natural that t be a nonincreasing function in the components of v, for if v’ < 0” < 0,) then in view of our sign conventions on inputs, we see that the amount of fixed inputs available to the firm has decreased going from v’ to 21” and thus it is reasonable that t(li; ZY) < t(ir; v’). On the other hand, suppose that we are given a transformation function t satisfying Conditions II’ above. Then the production possibilities set T which corresponds to t may be defined as follows: 2.17

DEFINITION:

T = {(ul , 6; v) : q < t@; v); v < 0,).

2.18 THEOREM: If t satisjies Conditions II’ given by 2.16 and the set T is defined by 2.17, then T satisfies Conditions I’ given by 2.13. Let us now suppose that the producer’s fixed inputs are fixed at v and that he can buy or sell variable inputs or outputs at the fixed positive prices (PI , P2 ,..., pr) = p > 0, . Then the producer’s variable profit function may be defined as follows; 2.19 DEFINITION: v < OJ .

n-(p; v) = max,{pru

: (u; u) E T} where p > OI ;

FUNCTIONAL

FORMS FOR PROFlT

293

Thus the variable profit function depends not only on the vector of variable input and output prices p but also on the vector of fixed inputs 0. We note that the variable profit function can be defined as a maximum (rather than as a supremum as in Definition 2.7) if T satisfies2.13. Note also if pollutants are taxed, they can be introduced into the ZIvector and should be indexed negatively. The corresponding price of a pollutant is the positive tax rate; i.e., pollutants can be treated formally in the same manner as inputs. 2.20 THEOREM: If the production possibilities set T satisfies Conditions I’ given by 2.13 and the variable profit function rr is defined by 2.19, then rr satisfies Conditions iI1’ below. 2.21. CONDITIONS III’ ON THE VARIABLE PROFIT FUNCTION w (1) is nonnegative and bounded above by pTb for a fixed vector b if p > OI , v < 0 and v is bounded from below, n(p; v) (2) r is linear homogeneous in p; i.e., for every h > 0, GP; v> = WP; 4, (3) rr is convex and continuous in p for every fixed v < OJ, (4) 7 is linear homogeneous in v; i.e., for every A > 0, “(Pi xv> = MP; v), (5) n is nonincreasing in v for every fixed p (recall that v < 0), and (6)

7r is concave and continuous in v for every fixed p.

For a proof of Theorem 2.20, the reader is referred to Section 7. We note that the variable profit function does not attain the value +cc for finite p and v (whereas the profit function could attain the value +cc for finite p and w). We also note that ~(p; v) can be increasing or decreasing with respect to a component of the vector p depending on whether the corresponding good is an output or input. Finally, if we are given a variable profit function 7r satisfying Conditions III’, then we may define the production possibilities set which corresponds to rr as follows: 2.22 DEFINITION: 2’< 0,).

T = {(u; v) : pTu < n( p; v) for every p > 0, and

2.23 THEOREM: If rr satisfies Conditions III’ given by 2.21 above and the set T is dejined by 2.22, then T satisfies Conditions I’ given by 2.13. The above theorems establish equivalent setsof conditions on a production possibilities set T satisfying 2.13, the corresponding transformation

294

DIEWER-I

function t and the corresponding variable profit function ~(p; u). Theorem 2.23 enables us to prove the following version of hotelling’s lemma which we will apply in Section 5. 2.24 given prices &(p*; proj?t prices

LEMMA: If a variable prqfit function n(p; v) satisfies Conditions 111’ by 2.21 above and is in addition diflerentiable with respect to the of variable outputs and inputs at p” > 01, v* < OJ , then we have v*)/~P~ = ui(p*; P*) for i = 1, 2,..., I, where u&p*; v*) is the maximizing amount qf output i (qf input i if ui(p*; o*) < 0) ghen p* andjxed inputs cl*.

3.

A

GENERALIZED

LINEAR

TRANSFORMATION

FUNCTroN

In choosing a functional form for a transformation or profit function which could provide a second order (local) approximation to an arbitrary twice differentiable transformation function satisfying Conditions II or to a twice differentiable transformation function satisfying Conditions III above, one is tempted to choose a quadratic function of the form f(z) = a, + arz + zrAz where z is a vector of quantities or prices as the case may be, a, is a constant, aT is a vector of constants, and A is a symmetric matrix of constants. The main problem with such a strategy is that in many cases, we want a second order approximation to a linear homogeneousfunction, in which casea, and the elements of the matrix A must be taken to be equal to zero and the resulting function aTz can provide only a first order approximation to an arbitrary linear homogeneousfunction. Thus, we must search for other functional forms. Aside from linear homogeneity, the conditions in the previous section of this paper indicate that we will want functional forms which satisfy certain nonnegativity, monotonicity, and convexity and/or concavity properties. Ideally, we would like these last properties to be satisfied by the functional form over the entire domain of definition (such as the nonnegative orthant in the case of a transformation function satisfying Conditions II). However, in general, it is not possible to guarantee that a given functional form satisfy the various regularity conditions over the entire domain of definition for all possible values of the (unknown) parameters in the functional form. Thus, in what follows, we will present various inequalities which will determine an “economic region” over which the given functional form is well behaved. Let M > 1, N > 1, (y, ,..., yhf) be an M - 1 vector of nonnegative outputs and (x1 , x2 ,..., xN) be an N vector of nonnegative inputs. Define the Kdimensional vector z = (z, ,..., zK) 3 (vZ , ,rJ ,..., yM ; x1 , x2 ,..., xN)

FUNCTIONAL

FORMS

where K = M - 1 $ N. Consider function:

FOR

295

PROFIT

the following

generalized

linear

tmns-

a,j

= aji .

formatiorz

y1

=

t(

j;

X)

=

t(z)

=

au0

+

2

t

aoi,$!*

+

f i=l

i=l

f

aijzf’*,-:‘*;

j=l

(3.1) The above functional form is essentially a quadratic form in the square roots of “variable” inputs and outputs which are represented in the vector z and in a “fixed” input which always takes the value one. If aoo= 0 and a,i = 0 for all i, then the transformation function given by 3.1 exhibits constant returns to scale in the vector of variable inputs and outputs z14and if in addition, aij = 0 for all i #j, then 3.1 reduces to the well known linear transformation function. Assuming that the aij in 3.1 are given numbers, we now determine a z region where the functional form given by 3.1 is consistent with Conditions II given by 2.4. Condition (a) of 2.4 implies that z should satisfy the following nonnegatiuity restriction: aOo -+ 2 f

a,,z:”

(3.2)

i=l

Conditions (d) and (e) of 2.4 imply that z > 0 should satisfy the following monotonicity conditions: Zt(z)/az,,,

= aO,,,z,l”

8t(Z)/aZfF = aobzi;“’

$

+ f

L7pj(Zj/ZJc)l’*

3 0;

j;=l

k = M, M + I ,..., M - 1 + N = K. Let [tii(z)] denote the K by K matrix of second order partial derivatives of the function t defined by 3.1, and Iet K(z; il, i, ,..., iJ denote the determinant of the submatrix of [tij(z)] which consists of rows and columns i, , is ,..., i,< . It is well known15that a twice continuously differentiable function t defined over an open convex set S is concave iff its Hessian matrix [t&z)] is negative semidefinite for every z ES. Thus, lil Note that if we started with the constant returns to scale case and then fixed one of the z’s, we would obtain a functional form on the remaining z’s similar to 3.1. I5 Fenchel [7, pp. 87-881 or Rockafellar [25, p. 271.

296

DlEWERT

(c) of 2.4 implies that z p 0 should satisfy the following concavity conditions: (--I)” D(z; i1 , i, )...) ik) 3 0

for

k = 1, 2,..., M - 1 + N

(3.4)

and for all indices such that 1 < il < iz < ... < ik < M - 1 + N. Condition 3.4 is a necessary and sufficient condition for the matrix [tij(z)] to be negative semidefinite.l’j For a given set of aij , let us define S as the closure of the set of z > O,_,+, which satisfy 3.2-3.4, and let us supposethat S is nonempty. Let S’ be a closed, convex subset of S. Then the transformation function/ defined by 3.1 is well behaved over the set S’. We indicate how t defined by 3.1 for z ES’ can be extended to the entire nonnegative M - 1 + N dimensional orthant. We first define the free disposal hull T’ (which corresponds to t defined by 3.1 restricted to S’) as follows: T’ ~2 {(y,, , 9; x) : vn < t(j*;

x*); OMwl< 5 < $*; x* < x; ($*; x*) ES’}. (3.5)

Now we project T’into an M - 1 -+ N dimensional set S”: S” E {($; x) : there exists y, such that (y, , 9; x) E rl>.

(3.6)

The extension t* of t defined by 3.1 restricted to S’ can now be defined as follows, where (j; x) > (O,-, ; 0,): t*(j;

x) =

max,,{ Y, : (Y, , A .Y>E T’) otherwise. I -03

if

($; x) ES”

It may be verified that t coincides with t* if z E S’ and that the extension t* satisfiesConditions II given by 2.4. Figure 1 illustrates how the above extension may be accomplished for some simple cases of 3.1. In the one output, one input cases, the economic region s’ is the set of x1 which satisfy the three inequalities t(xl) 3 0, t/(x1) 3 0 and t”(x& < 0 where t(xl) = a,, + ~u,~x~/~+ allxl and t’ denotes the derivative of the function t. These three inequalities correspond to 3.2, 3.3, and 3.4, respectively. In the two output, no variable input cases,the economic region S’ is the set of y2 which satisfy the three inequalities t(vz) > 0, f’(Y2) < 0 and t”(y2) < 0, where t(Y*) = %I + 2%J9'2 + %Y2 - The broken lines in Fig. 1 indicate how the function t restricted to the domain of definition S’ can be extended to the entire nonnegative orthant. I6 Debreu (2), except that we have reduced the number of determinantal conditions by noting that an interchange of two rows and the same two columns does not change the value of a determinant.

FUNCTIONAL ONE v,

TWO

FORMS

FOR

OUTPUT,

ONE

INPUT

= a00 + 230,

x, =+a,,

a,

OUTPUTS. NO VARIABLE Y, = aoo+ 2a0,Q +a,,Y2

aoO >o, aO, = 0, a,, < 0 Y, = aoO’ v2 = - aoO/a,,

297

PROFIT CASES

INPUT

CASES

“1, ( 0. “0, ) 0. a00 > a07 /a,, 5 = a00 - a01=/a1, v2 = IQ, + I+ aop,,i :2 / /a,:

FIG. 1. Generalized linear transformation function. As it is difficult to check the concavity conditions 3.4 in general, we note that a suficient condition for the transformation function t defined by 3.1 to be concave over the entire nonnegative orthant is that aolc > 0 for k = 1, 2,..., K and aii>O for all ij such that 1
298

DIEWERT

3.8 LEMMA: Given an arbitrary transformation function t* (consistent with Conditions II) which is twice continuously d@erentiable at the point z* = (yZ* ,..., y,,,*; x1* ,..., xN*) > (O,,,-, ; 0,) where yl* ES t*(z*), at*(z*)jaq s tie for i = 1, 2,..., M - 1 + N and a2t*(z*)/azL azj = tz for 1 < i < j < M - 1 + N = K, then there exists a generalized linear transformation function which provides a second order approximation to t” at the point z*. Proof. Take t(z) to be given by 3.1, partially differentiate t with respect to zi at the point z* and upon setting the resulting partial derivative equal to ti*, we obtain:

aoizi*-112+ f aij(q*/zi*)li2 = ti*; j,l

i = 1, 2,.. ., K.

(3.9)

If we differentiate &(z*)/az, with respect to zi and then set the resulting partial derivative equal to t$ , we obtain the following system of equations:

- +aoizt?-3i2-

i=

1,2 ,..., K.

(3.10)

i#i

If we differentiate &(z*)/azi with respect to zj* for j # i and then set the resulting partial derivative equal to tii , we obtain: &ii(zj*zi*)-1'2

= t: ;

1
(3.11)

Since the vector z* is fixed and the ti*, tij are given numbers, we may solve the system of Eqs. 3.9-3.11 for values of the unknown parameters a,,, and aij . The aii for i # j may be determined by solving 3.11, then the a,, may be determined from 3.10, and finally, the aii may be determined from 3.9. The parameter aoo which occurs in 3.1 can now be determined once all of the other a’s are known by solving the equation yl* = t(z*) where t is given by 3.1. Q.E.D. It is also easy to show that the homogeneous version of 3.1 (i.e., aoO= 0 and aoi = 0) can provide a second order (local) approximation to an arbitrary twice differentiable transformation function which exhibits constant returns to scale. Lemma 3.8 is important because it is well known [21] that the first and second order partial derivatives of a production or transformation function completely determine the function’s elasticity of substitution properties. Thus if a given functional form has only a few unknown

FUNCTIONAL

FORMS

FOR

PROFIT

299

parameters occurring in its Hessian matrix, then the flexibility of the functional form in modeling substitution possibilities will be severely 1imited.l’ Of course, in situation’s where the number of products and factors is large and the number of degrees of freedom is small, it may not be possible to estimate all of the parameters occurring in 3.1. In this last case, we could place a priori linear restrictions on the elements of the Hessian matrix (tii) of the functional form t* that we are attempting to approximate and these linear restrictions on the tij could be translated into linear restrictions on the aij parameters (at a given point z*) by means of 3.10 and 3.11. Thus the econometrician on the basis of his a priori knowledge, could determine the restrictions on the flexibility of the functional form. Given data on outputs and inputs for a number of firms who are suspected of having the same technical coefficients on the average, an appropriate econometric specification for 3.1 could be obtained by assuming that the a’s differed from firm to firm but that they had a common mean. Thus linear regression techniques could be applied to this random coefJicients model 124, p. 1921 in order to estimate the mean a’s. We now turn to the problems involved in estimating technological coefficients when price data are available in addition to quantity data.

4. A

GENERALIZED

LEONTIEF

PROFIT

FUNCTION

If wt assume that an entrepreneur maximizes profits subject to the constraints of his production possibilities set, taking all input prices w and output prices p as fixed, then given a production possibilities set T which satisfies Conditions I of Section 2, the profit maximizing input and output quantities will be functions of the given prices. This is often a convenient framework for econometric estimation, with quantities as dependent variables and prices as independent variables. However, rather than postulating a functional form for the transformation function and then working through the algebra of the profit maximization procedure in order to derive profit maximizing quantities as functions of prices, it is much simpler to postulate a functional form for the profit function satisfying Conditions III of Section 2 and then differentiate the profit function to obtain profit maximizing quantities as functions of prices (see 2.12). I7 For example, the many factor version of the constant elasticity of substitution production function, which is a mean of order r < 1 [l I], imposes the restriction that all elasticities of substitution between any two factors be identical.

300

DIEWERT

Consider the following generalized Leontief projit function: M+N

77(q)=

c i=l

M+h’

c

bjjq;‘2qy;

hij = bii )

(4.1)

j=l

where qm = pm = price of output m for 1~1= 1, 2,..., M and qM+n

=

wn

=

price of input n for n = 1,2 ,..., N.

We will use Conditions III in order to determine an economic region of q’s where the profit function defined by 4.1 is well behaved. From (2) of Conditions III, we want n(q) = ~(p; w) to be a nonincreasing function in the components of w; i.e., we want M+N

&-(q)/3qi =

c bij(qi/qi)l’z < 0 J==l

for

i=

Mf

l,M+2

,..., MfN. (4.2)

From (3) of Conditions III, we want ~(p; w) to be a nondecreasing function in the components of p; i.e., we want M+N

Wq)/+,

=

C bij(qj/qP2

3 0

for

i = I, 2,..., M.

(4.3)

j=l

From (4) of Conditions III, we want n(q) to be a convex function of q and thus if we define A4 + N by A4 + N matrix of second order partial derivatives of T defined by 4.1 as [rrij(q)] and define D(q; il, iz ,..., iK) as the determinant of the matrix which consists of rows and columns . ‘1 3l2 ,..., ik of [nij(q)], we see that we want q to satisfy the following determinantal inequalities? D(q; il , iZ ,..., i3 3 0

for

k = 1, 2,..., M + N

(4.4)

and all indices such that I < il -C iZ < ... < il, < A4 + N. Note that a sufficient condition for the functional form given by 4.1 to be a convex function over the positive orthant is that bij < 0 for all i #j. For a given set of coefficients by bii occurring 4.1, define S as the set of which satisfy 4.2-4.4, and let us suppose that S is not empty. 4 > OMM+N I* The determinantal inequalities given by 4.4 are necessary and sufficient for the matrix [nij(q)] to be positive semidefinite (2). The matrix [rij(q)] is positive semidefinite for all q E S where S is an open convex set iff the twice continuously differentiable function n is a convex function over S [7, pp. 87-881 [25, p. 271. Note that the determinant of the entire matrix [TJq)] will be identically equal to zero since n(q) is homogeneous of degree one.

FUNCTIONAL

301

FORMS FOR PROFIT

Let s’ be a convex subset of S. Then the profit function 7~defined by 4.1 is well behaved over the set S’. We now indicate how rr defined for q ES can be extended to the entire positive M + N dimensional orthant. Define the production possibilities set T’ which is consistent with n defined by 4.1 for q ES’ as follows: T’ = {(y; x) :p’y - wTx < T(p; WY) for every (p; w) ES’; (Jo;x) > (0; 0)). (4.5) It is easy to seethat 7” is the intersection of a set of closed half spaces and thus is a closed convex set and in fact satisfiesConditions I. It can also be verified that the profit function n* which corresponds to T’ by means of definition 2.7 coincides with the function rr defined by 4.1 for (p; M’)E S’ and thus YT* is the appropriate extension of 7~(restricted to the economic region S’) to the entire positive orthant. ONE II

INPUT.

IP,:

w,,

‘11 (I---0 b,,

=

ONE b,,P,

+

x,

OUTPUT 1

b22w,

b,\ /‘0

-b22 >O,

CASES

I

2b,2p,‘w,‘+

b12

= 0.

TWO

b22<0

OUTPUT. TI~P,.P~)=

NO bllP,+2b12~,

“I

t,

b,,

>O.

b12

‘

0.

X,

= -

b22

+

b,$/b,,

VARIABLE i

INPUT ~2

t

+

b22

bl;‘bll

C

CASES

bz2p2

“I

b~~~::----j ~2 ?‘I 0 b,,>O.

b22 b,2=0,b22’0

0

v2

b,,>0.b12<0. 7,

FIG. 2. Generalized

v2

= b,,

bz2>0. -

b 2/b 12

22

b,,b22>b,s Vz

Leontief profit function.

=

b22

-

b,;/b,,

302

DIEWERT

Figure 2 illustrates what the sets T’ look like for some simple casesof 4.1. In the one input, one output cases,the economic region S’ is the set of positive (pl ; WJ such that &T(p, ; w,)/+, 3 0 and &r(p, ; w,)/LJw, < 0. (In order that rr(~ 1 ; wl) = b,,p, + 2b,,p:“~:~’ + b22wl be a convex function for any (pl ; wl) > (0; 0), it is necessary and sufficient that blz < 0 in which case, rr is a convex function over the entire nonnegative orthant). Applying Hotelling’s lemma 2.12 to the one input, one output case gives us the two equations y1 = bll + b12(w,/p,)1/2 and --x1 = b,, + b12(p1/w,)1/2for (pl ; wl) ES’. The locus of points (ul ; xi) satisfying the last two equations is given by a solid line in Fig. 2. In the two output cases,the economic region S’ is the set of positive (pl, p2) such that &(p, , p2)/ap1 3 0 and &(p, , p2)/ap2 > 0 where r(pl , p2) = blIpI + 2b12p:‘2pi’2+ b,,p, . (Again we must have b,, < 0.) Again, Hotelling’s lemma yields the two equations y1 = b,, + b12(p2/p1)l12and y, = b,, + b12(p1/p2)1/2and the locus of points (JJ~; y2) satisfying the last two equations for (pl , p2) E S’ is denoted by a solid line in the last two casesof Fig. 2. Assuming that the economic region s’ is not empty, it can be seenfrom definition 4.5 (seealso Fig. 2) that in generaP the production possibilities set T’ which is generated by 4.1 will exhibit diminishing returns to scale. However, we may generalize the functional form given by 4.1 to cases where there are constant returns to scale or even increasing returns to scale by assuming that there is a fixed output which must be produced, or a single fixed input which cannot be expanded during the production period. (We will consider the caseof several fixed inputs in the following section.) Call this fixed input or output z, . Then we may generalize the profit function given by 4.1 to

44;

zo)

=

h(z,)

M+N

M+N

1

C j=l

i=l

biid’2d’2,

(4.6)

where bii = bji and h is a nonnegative and nondecreasing real valued function. If h(z,) = z, , then we are back in the classical case of constant returns to scale in all quantities, including z, . If h is differentiable and the derivative of h is an increasing function, then we will have increasing returns to scale in all quantities, but constant or decreasing returns to scale will still prevail in the variable quantities. (Note that if the profit IDHowever, if a(g; w) = for all (p; w) E s’, then T’ defined by 4.5 will be a cone and hence wll exhibit constant returns to scale. In this case, ~(p; w) will be a linear function over s’ and thus it will not provide a very interesting approximation from an econometric point of view,

FUNCTIONAL

FORMS

FOR

303

PROFIT

function can be written as rr(q; z,) = h(P)f(q), then the technology is homothetic with respect to the fixed input z,). To show that the generalized Leontief profit function defined by 4.6 is a generalization of the generalized Leontief production function defined in [6], let q = W, an N dimensional vector of input prices; that is, assume that there are no variable outputs. Assume that z, = y, a fixed output. Then our present profit maximization problem is equivalent to a cost minimization problem, and we see that our present profit function becomes minus the generalized Leontief cost function defined in [6]. We now indicate how the unknown parameters bij occurring in 4.6 could be estimated econometrically. If we had data on profits rr, prices q = (p; w) and on the fixed quantity z, , then we could apply linear regression techniques to Eq. 4.6, (if we had also assumed a known functional form for h). If in addition, we had data on profit maximizing output levels y 3 O,+r and on profit maximizing input levels x > ON, then we could use Hotelling’s lemma 2.12 and estimate the parameters of the following system of equations: M+N pi

=

%)

C

Mqi/W”,

i = 1, 2,..., M

j=l

and

(4.7) M+N -xfi

=

hkJ

C

bcMqJ1’2,

k = M + 1, M + 2 ,..., M + N.

j=l

The main advantage of the functional form given by 4.6 over other possible functional forms is that the unknown bii parameters occur in Eqs. 4.6 and 4.7 in a linear fashion and thus modifications of linear regression techniques may be used in order to estimate the parameters. Note that in the system of equations given by 4.7, the validity of the symmetry20 restrictions bii = b,i could be tested. It is also easy to show that the generalized Leontief profit function can provide a good local approximation to an arbitrary twice differentiable profit function. 4.8 LEMMA: Given an arbitrary profit function T* (consistent with Conditions III) which is twice continuously d@erentiabIe at the point q* = (ql*, q2*,..., q&+N) > OMMCNwhere &r*(q*)/aqi = TV* for i = 1, 2,..., M + N and a2n*(q*)/aq, aqi = m$ for 1 < i
304

DIEWERT

Proof. Since ++ satisfies Conditions III, it is homogeneous of degree one and thus by Euler’s identity, rr*(q*) = xz” vri*qi* and for i = 1, 2,..., M+N, dfqj* = 0 and thus rr*(q*) and rrZGfor Cz” i = 1, 2,..., A4 + N cannot b”d determined independently of the other rri*, r&. Also, since Z-* is twice continuously differentiable at q*, we have by Young’s theorem that rrz = rr$ . Now take z-(q) to be given by 4.1, partially differentiate rr with respect to qi at the point q* and upon setting the resulting partial derivative equal to 7ri*, we obtain:

If we differentiate &(q*)/aqi with respect to qj for i # j and set the resulting partial derivative equal to Z$ , we obtain: ~~ii(qi*qf*)-1’2 = Tii* .3

1
(4.10)

Thus, the parameters bij are determined by 4.10 and then the parameters bii may be determined from 4.9. Q.E.D. If there are many outputs and inputs and only a relatively small number of observations, it may not be possible to estimate a complete set of the parameters bij occurring in 4.1 or 4.6, in which casevarious linear restrictions on the bii may be imposed by the econometrician. For example, if the profit maximizing amount of output (or input) i is not expected on a priori grounds to respond to a change in the price of output (or input) j, then bij may be set equal to zero.21 We now turn to the problem of obtaining functional forms for profit functions when more than one input is fixed in the short run.

5. FUNCTIONAL

FORMS FOR VARIABLE

PROFIT FUNCTIONS

Let p = (pl ,..., p,) denote a vector of positive prices for variable outputs and inputs, let u = (ul ,..., u,) denote a vector of variable outputs and inputs (recall that outputs are now indexed positively, inputs negatively), let 27= (ur ,..., vJ) denote a vector of (nonpositive) fixed inputs and p1The Allen [I, p. 5041 and Uzawa [31] partial elasticity of transformation between goods c and j may be defined as utj(q)- -m(q)nij(q)/ni(q)nj(q) which in view of Hotelling’s lemma is a normalization of the response of the profit maximizing output (or input) i to a change in the price of output (or input) j. It is easy to translate restrictions on these elasticities of transformation at a given price quantity point q*. z* into restrictions on the parameters bij by means of 4.10 and Hotelling’s lemma, which states that n<(q*)= zi*, where (zI* ,..., &,) - (,vl* ,..., yM*; --x~* .. ... -xN*).

FUNCTIONAL

let Z > 1, J > 1. Consider function:

FORMS

FOR

the following

305

PROFIT

candidate

for a variable profit

Qih =al,,;

bi, = b,i ; aii = 0 ; bji . (5.1)

ix1 j-1

The restrictions aii = 0, bjj = 0 can be replaced by other normalizations on the aih , b,i and cij .22 Note that rr defined by 5.1 is homogeneous of degree one in both p and v. Note also that the function (ipi + +ph2)l12 is a mean of order 2 with weights (&, 4) and is a convex function of (pi ;ph) for (pi ; p,,) > (0; O).23 The function (-Uj)“2(-Uk)1’2 is a mean of order 0 with weights (4, 4) and is a concave function in (-uj ; --LIP) for (Vi , v3 < (0, 0). Given a set of parameters aih , bi, , and cii , we now determine an economic region of (p; v)‘s where the function defined by 5.1 is consistent with Conditions III’ given by 2.21. Condition (I) of 2.21 implies that we want n(p; v) to be nonnegative for p > OJz4and v < 0,. More formally: “(Pi v) 3 0,

where 7~(p; v) is defined by 5. I.

(5.2)

Let [a2n-(p; u)/ap, 3pn] denote the Z by Z matrix of second order partial derivatives with respect to p of the function rr defined by 5.1. Then (3) of Conditions III’ implies that we want the following regularity condition to be satisfied: V2dPi

fJ)l3Pi

aPhI

is a positive semidefinite matrix

where rr is defined by 5.1. Condition (5) of 2.21 implies that we want the following be satisfied : iirr(p; vya11, = - i: i: i (a.r,‘ t bjJ($piz i-1 k=l Ah1 for

(5.3) condition

to

+ +ph”)lp (-vr/-uj)1/2

j = 1, 2 ,..., J.

z Note that we can add a constant to each ailc and subtract the same constant from each bjk without changing the value of TT. 63See Hardy, Littlewood, and Polya [l l] for the properties of means of order r. 64We can always uniquely extend a convex function defined over an open convex set to its boundary points by continuity. See Rockafellar [25, p. 521.

306

DIEWERT

Finally, let [a2rr(p; v)/aq av,] denote the J by J matrix of second order partial derivatives with respect to 11of the function ?T defined by 5.1. Then (6) of Conditions III’ implies that we want the following regularity condition to be satisfied: [a’TT(p;

V)/aVj

al’&e]

is a negative semidefinite

matrix

where rr is defined by 5.1.

(5.5)

Now given a set of parameters aih , bjk , and cij , suppose that the of (p; v) which satisfy 5.2-5.5 where p> 0, and v < OJ is nonempty and let S denote the closure of this set. Let S’ be a closed, convex subset of S. The homogeneity of the function r(p; v) defined by 5.1 in p and in v implies that there is no loss of generality in taking S’ to be a set with the following properties: (i)

if (p; v) E S’ then (Ap; U) E S’ for all scalars h > 0, and

(ii) if (p; v) ES’ then (p; Au) ES’ f or all scalars h 3 0. We take S to have these last two properties in what follows. Define the scalar ill where r is given by 5.1 as follows: OL= max,,,{n(p;

v) : (p; v) ES’; Fp = 1; lTv = -l}.

(5.6)

The symbol Zrepresents a vector of l’s of the appropriate dimensionality. The maximum exists since n is a continuous function over the closed and bounded set of (p; v)‘s which is specified in 5.6. We now use the variable profit function rr defined by 5.1 restricted to S’ in order to define the following production possibilities set: T’ = {(u; v) : pTu < rr(p; v) for every (p; v) E S’ and u < I( --otlTv)}. (5.7) It can be verified that T’ is a closed, convex cone which is bounded from above if z) is bounded. Define the free disposal hull of T’ as follows:

T” = ((u*; v*>: (u*; v*)

< (u; v), (u; v) E T’>.

(5.8)

It can be verified that T” satisfies Conditions I’ and that the variable profit function n* defined by r*(p; v) = max,(pTu : (u; v) E T”> for p > 0,) v < 0, satisfies Conditions TIT.’and coincides with z-(p; v) defined by 5.1 for (p; v) E s’. We note that a su@cient condition for rr(p; v) defined by 5.1 to satisfy the convexity condition 5.3 and the concavity condition 5.5 for all p > 0, and all v < OJ is that aih + bjl, > 0 for all indices i, h, j, k.

FUNCTIONAL

FORMS

FOR

307

PROFIT

In the case where there are no variable inputs, i.e., where u is a vector of outputs, then a suficient condition for n(p; v) given by 5.1 to satisfy Conditions III’ for all p > OI and all z! < 0, is that aih >, 0, bjk >, 0 and Cij > 0 for all indices. It can be shown that the variable profit function defined by 5.1 can provide a good local approximation to an arbitrary twice differentiable variable profit function. 5.9 LEMMA. Let rr* be an arbitrary variable projiit function (consistent with Conditions III) which is twice continuously dlrerentiable at the point (p; v) where p > 01, v Q OJ and 3rr*(p; v)/3pi = rITi* for i = 1,2,..., I; &*(p; v)/&Q = &* for j = 1, 2,..., J; a2n2(p; v)/ap, ap, = T& for 1 < i < II < I; a2~*(p; v)/3q 3v, = T@* for 1 < j < k < J; a2n*(p; v)/3pi 3vj = r$ for i = 1, 2,..., Z and j = 1, 2,..., J. Then in general, there exists a rr(p; v) given by 5.1 which provides a second order approximation to r* at the point (p; v). Proof. Differentiate 7~given by 5.1 and set the partial derivatives equal to the corresponding partial derivative of T*, evaluating all derivatives at the point (p; v). We obtain the following system of equations, where we set -v = x > OJ for notational convenience: gl ,‘& gl (aih + b&+p:

+ *ph’)-“’

(xjx#‘2 Pi + i

C& = Ti*;

j=l

i =

il il

g1 (aih + b&Pi2

1, 2 ,..., 2.

+ iPh2)1/2(Xk/Xj)lI” + i=li CijPi = -&t;

j = 1, 2,..., J-& i j=l

i

(5.10)

(aih + bjk)($pt

1. (5.11)

+ +ph2)e312(xjxk)1’2p
k=l

1 < i < h < I. 4 il

jl

(aih

f

b&p:

+ &ph2)‘/’ (XjXk)-l12 1
(5.12)

= T+*;

(5.13)

Ii1 il (aih+ bjk)(+pie+ ~pk2)-1’zpi(xk/xj)1’z + cij = -n:*; i = 1, 2,..., Z-l

and j=

I,2 ,..., J-l.

(5.14)

308

DIEWERT

By Euler’s identity n*(p; v) = Cizlpini* = & x$*. Therefore, not all of the parameters rrITi* and +* can be independent and thus we have only J - 1 equations in 5.11. Furthermore, if we partially differentiate the identity n*(p; Xv) = hrr*(p; u), we find that ni*(p; hv) = hri*(p; zl) and thus ri* = & Ujnr for i = 1, 2,..., I. Similarly &* = xi=, pini for j = 1,2,..., J. Taking into account the identity Cpini* = 2 vi&*, it can be seen that only (I - l)(J - 1) of the /J parameters $ can be independent. Now the system of (I + J - 1) + Z(Z - I)/2 + J(J - 1)/2 + (I- l)(J - 1) linear equations 5.10 to 5.14 in the I(1 - 1)/2 unknown aih , J(J - 1)/2 bi, and IJ+ can be solved if the coefficient matrix is nonsingular. Q.E.D. By the modified Hotelling’s lemma 2.24, the profit maximizing output supply (or variable input demand) functions are given by ui(p; v) = ui(p; -x) = &r(p; --x)/+~ = 7ri* where rri* is defined in 5.10 for i = 1, 2,..., 1. Thus the output supply (and the variable input demand) functions are linear in the unknown parameters aih , bj, , and cii and thus linear regression techniques may be applied to the system of equations given by 5.10. Again, a priori information on the partial derivatives of the variable profit function at a given point (p; v) = (p; -x) may be used in order to impose linear restrictions on the parameters aih , bjl, , and cij . (See Eqs. 5.10-5.14.) Another functional form for the variable profit function which may prove to be useful in empirical applications is the following one, where all variables are defined as in 5.1 with x = -v: n(p;

V)

=

i

i

i

ai,(Spi2

+

+Ph2)li2

Xj

i=l h-1 j=l

+

i

i

i-1 j=l

ciiPixi

+ i i i bjkpix;‘2x;‘2; i=l j=l k=l aii = 0;

aih = ahi ; bik = b,j ; bjj = 0. (5.15)

A sufficient condition for rr given by 5.15 to satisfy Conditions III’ everywhere is for all parameters aih , bjk , and Cij to be nonnegative. (In this case, there will be no variable inputs.) The function r defined by 5.15 can also provide a second order local approximation; that is, we can prove a counterpart to Lemma 5.9. The counterparts to the relations 5.10-5.14 follow. kl gl a&pi2

+

&ph2)-1’2piXy

+ jil

CijXj

+ i

j=l

i bj&‘2xi’2 R=l

= pi*.

(5.16)

FUNCTIONAL

jl

309

FORMS FOR PROFIT

CijPi

+

il

il

bjdxJxjY’”

Pi

=

-

+**

(5.17) -$aih($pi2

+

4.ph3)-313

P i Ph

(]il

xj)

=

TTTi* f

(5.18) I 3bjk(xjxjc)-1’2

i

aih(*pi2 + *ph2)-1/2pi i

R =l

cij + i

@ i=l

pi zzz @i*. ! (5.19)

bik(x,/xj)l12pi = - ,r;-.

,:=1

(5.20) Although the functional form given by 5.15 is not as symmetric as 5.1, we seefrom 5.18 and 5.19 that the a priori restrictions r& = 0 or T@* = 0 can be imposed rather more easily if the functional form for z- is given by 5.15 rather than 5.1.

6. CONCLUDING

COMMENTS

Thus far, we have exhibited somefunctional forms for a transformation function (see3. I), for a profit function (see4. l), and for a variable profit function (see 5.1), all of which have the property that they can provide a (different) second order local approximation to an arbitrary technology (satisfying the appropriate regularity conditions). Furthermore, the functional forms are all linear in their unknown parameters, which should facilitate econometric estimation. The question arises as to which functional form should be used in an actual empirical application. If one does not wish to make the assumption of competitive price taking behavior, then the transformation function should be estimated. If one is willing to accept price taking behavior as a first approximation to real world behavior, then profit functions can be estimated. If all inputs and outputs are variable and the technology exhibits strict decreasing returns to scale,25then the parameters of the profit function should be estimated by using the system of derived demand and supply functions which are obtained from the profit function via Hotelling’s lemma. If there is a z6 This last restriction is important, since if we assume constant returns to scale in all factors, then the firm’s true profit function n(q) will have the property that either m(q)= 0 or v(q) = + cc. It is impossible for a functional form such as that given by 4.1 to adequately model such discontinuous behavior. 642/o/3-7

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single fixed input and the technology exhibits strict decreasing returns in all of the other outputs and inputs, then the parameters of the system of variable supply and demand equations (given by 4.7 for example) should be estimated. If there are several fixed inputs but the technology exhibits constant returns in all inputs and outputs, then the parameters of the system of variable supply and demand functions (such as that given by ui(p; -x) = TV* for i = 1, 2 ,..., Z, where rTTi* is defined by 5.10) should be estimated. The reason for preferring the profit function (or the variable profit function) to the transformation function in the last cases described above is that we obtain M + N (or in the case of the variable profit function, I) degrees of freedom for each observation compared to only one degree of freedom per observation using the transformation function. Thus the assumption of price taking behavior buys us degrees of freedom in econometric applications. (Similarly, the imposition of the symmetry conditions bij = bj< in the system of supply and demand equations given by 4.7 also will buy us degrees of freedom.) We conclude with some brief historical remarks. Shephard [2] showed that a cost function could describe technology as validly as a production function. Nerlove [22] econometrically implemented Shephard’s basic idea by estimating the parameters of a Cobb-Douglas production function by using its cost function. Uzawa [32] relaxed Shephard’s differentiability assumptions and showed that the duality between cost and production functions was a consequence of the fact that a closed convex set (such as a production possibilities set) can be represented as the intersection of its supporting (total cost) hyperplanes. Making use of Rockafellar’s [25] concept of a proper, closed convex function f and the relationship of such a function to its epigraph {( y; x) : f(x) 2 v}, the proofs of 2.3, 2.6,2.15 and 2.18 are trivial. Versions of Theorems 2.9 and 2.11 were proven by McFadden [16] and versions of Theorems 2.20 and 2.23 were proven by Gorman [9]. More recently, McFadden [17] and Shephard [30] have used the distance or gauge function (see also Rockafellar [25, p. 1281) in order to characterize transformation functions. Hanoch [lo] has also used the distance function to generate new functional forms for single output production functions. A formal proof of Hotelling’s lemma (see also Rockafellar [25, p. 2181) may be found in Fenchel [7, p. 1041 who proved it in the context of conjugate duality theory, (an approach which has been pursued by Lau [14] in the context of production theory). The profit function was introduced into the economics literature by Debreu [3]. The variable profit function was studied by Gorman [9] who called it a gross profit function.

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The usefulnessof Hotelling’s lemma in generating functional forms for systems of derived demand functions consistent with cost minimization was demonstrated in [6], where Hotelling’s lemma was called Shephard’s lemma. Other interesting functional forms derived via Hotelling’s lemma have been exhibited by Denny [4], Fuss and McFadden [8], Lau [14], and McFadden [18]. Several authors [5] [14] [17] [18] [30, Chapter 93 [25, pp. 112-1501 have studied the implications of various kinds of homotheticity [29], separability and nonjointness in production [28].

7. APPENDIX:

PROOFSOF THEOREMS

Proof of 2.3. (a) (b) (c) and (f) follow directly from definition 2.2 and (i) (ii) (iii) and (vi) of 2.1. See Rockafellar [25; 23; 511.Proof of (d): Let (i; x) 3 (OMM--l , 0,), x’ 2 x and suppose that y, = t(5; x) is finite. Then 2.2 implies that ( y1 , 9; x) E T and (iv) of 2.1 implies that (yl, 9; x’) E T. By 2.2, t(j; x’) 2 y1 = r($; x). Proof of (e): Let (9; x> 3 aw-I, Qv>, 0 G 9” G 9 and supposethat y1 = t($; x) is finite. Then 2.2 implies that (y, , 9; x) E T and (v) of 2.1 implies that (yl, $*; x’) E T. By definition 2.2, t(j*; x) > y1 = t(j; x). Q.E.D. Proof of 2.6. (i) follows from (a) and (c). (ii) and (iii) follow from (b) and (c). Proof of (iii): Let (yl , 5; x) E T and x’ > x. By definition 2.5, 0 < y, < t(j; x) where 9 3 O,-, , x 2 0,. By (d) @; x) < t($; x’) and thus by definition 2.5, (yl , 5; x’) E T. Proof of (iv): Let (yr , 5; x) E T, 0 < y,* < y, and OMel < $* < 9. By 2.5, 0 < y, < t(g; x), 9 > O,-, , X>ON. By (e), t(j*, x) 3 t(B; x) 3 y1 > yl* 3 0 and thus by definition 2.5, (yl*, g*; x) E T. (vi) follows from 2.5 and (f). Q.E.D. Proof of 2.8. For every integer m 3 WI”, define the set T,, = {(y; x) : (y; x)T and x ,< ml}. By (i), (ii), and (vi) of 2.1, the set T, is a nonempty, closed bounded set for m* sufficiently large. Define 7rm(p; W) = max,,,{pTy - wTx : (y; x) E Tm}. Obviously, ~(p; W) = linkrn rr,(p; W)where the limit is well defined since ~~+~(p; W) > rr,(p; W) for every (p; w) > (0, ; 0,) and every integer m Z m*. Parts (2), (3), (5), and (6) of 2.8 will be true if we can show that for every sufficiently large positive integer m, Z-&J; W) satisfies Conditions III given by 2.9. Proof of (I): Since T is nonempty by (i) of 2.1, there exists (a; b) E T. Thus by definition 2.7, ~(p; W) > pTa - ~~6. Proof of (2): Let (p; w) > (0, , O,), w’ 3 w and m > m*. Then n,(p; w’) E max(pTy - w’=x : (y; x) E T,} = pry’ - w’~x’ ,( pTy’ - wTx’ G rr,(p; w). Proof of (3): Let (p; w) > (0; 0), p’ 3 p and m > m*. Then n-&p; W) = max{pTy - wTx : (y; x) E T,} =

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p=y* - W=X* m*. Using (iii) and (iv), one can show thatf(m) is a (proper) concave function for m 3 m*. Therefore f has subgradients and there exist scalars c1 and p such that f(m) < a + /3m for all m. Let (y; x) E T. Then by the definition off, ITy -Oo,. Then V( p; nw) = sup{ pTy - mvTx : ( y; x) E r} < sup,{pET(ol + /?ETX) - nw=x : x 3 0,} < pTh since the term in x can be made nonpositive for n large enough if w > 0, . Proof of (5): Let (p’; MJ’)> (0; 0), (p”; WV”)> (0; O), 0 < h < 1, and m 3 m*. Then 7r,(hp’ + (1 - X)p”; hw’ + (1 - A) w”) = max{(hp’

+ (1 - A) p”)’ y - (XW’ + ( 1 - A) w”)= x : ( y; x) E T.}

= (Ap’ + (1 - A) p”)’ y” - (Xw’ + (1 - A) w”)T x* = h(p’=y* - w’=x*) + (1 - h)(p”%* - w”=x*) < hn,(p’;

w’) + (1 - A) 7Tm( pn; w”).

Proof of (6): Let X > 0, (p; w) > (0; 0), and m 3 m*. Then Xr,(p

: w) = X max{pTy - W=X: ( y; x) E r,) = max{hpTy - hwTx : (y; x) E T,,} = x,,(Ap; Xw).

Q.E.D.

Proof of 2.11. By (1) of 2.9, n(p; w) > pTa - wTb for all (p; w) > (0; 0) where (a; b) > (0; 0). Therefore by 2.10, (a; b) E T and T is nonempty. From definition 2.10, T is the intersection of a family of closed halfspaces and hence is a closed, convex set. (See Rockafellar [25, p. IO].) Proof of (iv): Suppose that (y; x) E T, 0 < x ,< x’ and (p; w) > (0; 0). Then pTy - wTx’ < pTy - wTx < nfp; w) for every (p; w) > (0; 0). Thus (y; x’) E T. Proof of (v): Suppose (y; x) E T and 0 < y’ < y. Then pTy’ - wTx < pTy - W=X< ~(p; w) for every (p; w) > (0; 0). Thus ;E x) E T. Proof of (vi): By (4) if p > 0, and w > ON we have ~(p; nw) < pTc where c > 0 is a vector of fixed constants. Let (p*:*G*) > (0; 0) be such that r(p*; w*) < +co. By definition 2.10, (y; x) E T implies p**y - W*=X < n(p*; w*). Thus if the components of x are bounded, we have p*‘y < n-(p*; w*) + W*=X and this implies that the components of y are also bounded, since p* > 0 and y > 0. This completes the proof of 2.11 as stated. Note that we have used only (1) and (4) of 2.9 in order to show that T defined by 2.10 satisfiesConditions I. Define the profit function Z-* which corresponds to T defined by 2.10 by r*(p; W) = sup{pTy - wTx : (y; x) E T} for every (p; w) > (0; 0). Then we have ~*(p; w) < ~(p; w) for every (p; W)> (0; 0). However

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FOR

PROFIT

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using (2)-(6) of 2.9, it can be shown that n*(p; w) = z-(p; w). (See Fenchel [7, p. 1021 or Rockafellar [25, Corollary 13.2.1, p. 1141.) Proof of 2.12. Given rr, define T by 2.10. Assume that 7r is once continuously differentiable in a neighbourhood of (p*; w*) > O,,, and define the vectors of partial derivatives of 7r at (p*; w*) by y* = V, n(p*; w*) and -x* = V, n(p*; w*), by (2) and (3) of 2.9, .J* 3 0, and x* > 0,. We wish to show that ~*~y* - w*~x* = s~p(p*~y - W*~X : (v; X) E T). Since n is a convex function by (6), it lies above its supporting hyperplanes, and thus since rr is differentiable at (p*; MS*),we have (seeRockafellar [25, p. 2421):for every (p; w) > OM+N, QT(p;w) 3 ?T’(p*;w*) + (p -p*)=V,n(p*; w*) + (w - w*)Tvg(p*; w*) = pTV,r(p*; w*) + w’V,r(p*; w*) where the last equality follows from Euler’s identity using (6) of 2.9. Thus we have pTy* - wTx* < n(p; w) for every (p; w) > (0, ; 0,) with an equality if (p; w) = (p*; w*). By the definition of T given by 2.10, the desired result follows.26 Q.E.D. Proof of 2.15. The function t defined by 2.14 is well defined as a maximum in view of (i) and (vi) of 2.13. Proof of (a): By (iv) and 2.14, t(OIPl ; 0,) 3 0. If t(O,_, ; 0,) > 0, then (vi) would be contradicted. The proof of (b) and (c) follows directly from 2.14 and (i) and (iii). Proofs of (d), (e), and (f) follow directly from (iv), (v), and (vi), respectively. Q.E.D. Proof of 2.18. Since by (a), t(O,-, ; 0,) = 0, by definition 2.17, T will be nonempty. Properties (b) and (c) imply that the epigraph of t (which is the set T defined by 2.17) will be a closed, convex setwhich proves (i) and (iii). Proof of (ii): Follows from 2.17. Proof of (iv): Let z = (ul , zi; v) E T and X 3 0. Then by 2.17, u1 < t(ti; v). By (d) Xu, < t(Xti; hv) and thus )Iz E T. Proof of (v): Let z’ = (Us’,a’; 0’) E T and z” = (u”, li”; v”) < z’. Then by 2.17, 0 < t(z.2’;u’) - u,’ (using (e)) < t($‘; v”) - u; since U; < u,‘. Thus z” E T. (vi) follows directly from 2.17 and (f). Q.E.D. Proof of 2.20. (1): Let p > 0, and v* < OJ. By (vi), there exists an upper bounding vector b such that pTu
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linear function over a closed, bounded set is attained. Proof of (2): Let p > 0,) v d OJ 3 X > 0. Then n(Ap; v) E max(hpTu : (u; v) E r} = X max{pTu : (u; u) E ??I = Xr(p; v). Proof of (3): Let p’ > 0,) p”>O,, vfO,, and O O1 now follows since a convex function is continuous over the interior of its domain of definition (see Fenchel [7, p. 751). Proof of (4): Let p > 0,) v < 0,) and h > 0. Then 77(p; Au) = max,{pru : (u; Au) E r} = maxAu{pT(Au) : (Au; hzl) E T] = maxu(Apru : h(u; 21)E T} = h max,{pru : (u; ZJ)E A-lT} = hn(p; v) since A-IT G T since T is cone by (iv). Proof of (5): Let vn < v’ f OJ and p > 0,. Then ~(p; v’) = max,{pru : (u; 0’) E Tj = pTu’. Since u” < u’, by (v) we have (u’; u”) E T. Therefore n(p; v”) = max,{pru : (u; v”) E T} > pTu’ = n(p;u’).Proofof(6):Letp>00,,v’ 0, for v < 0,. Continuity of ~(p; v) with respect to v on the boundary of the negative orthant can be established by using the maximum theorem (see Debreu [3, p. 191). Q.E.D. Proof of 2.23. The set T defined by 2.22 is not empty, since by (I), (0, ; 0,) E T. For every integer n, let (u”; v”) E T and let limlz&Un; 0”) = (u”; u”). Let p > 0, . Since (u”; pn) E T, by definition 2.22, pTun < n-(p; v”). Upon taking limits on both sides of this last inequality as n tends to infinity and making use of (6), we obtainpTUo < ~(p; v”). Thus (no; v”) E T which showsthat Tis a closed set. Definition 2.22 implies (ii). Proof of (iii): Let (u’; u’) E T, (u”; II”) E T and 0 < X < 1. By 2.22 for every p > 0, , we havepTU’ < ~(p; II’) and pTU” < ~(p; v”). Thereforepr(hu’ + (1 - A) u”) < hn(p; v’) + (1 - A) r(p; u”) < ~(p; Au’ + (1 - A) v”) by (6). Therefore by definition 2.22, (Au’ + (1 - A) u”; hv’ + (1 - A) v”) E T. Proof of (iv): Let (u; u) E T. Then by 2.22, pTu < z$p; v) for every p 3 0,. Thus if X > 0, we have hpTu < An-(p; v) or using (4), p=(h) < a(p; hv). Therefore (Au; Au) E T. Proof of (v): Let (u’; v’) E T and (u”; u”) < (u’; v’). Then for every p > 0,) we have pTu” < pTu’ < n-(p; u’) < n-(p; v”) where the last inequality follows from (5). Therefore (u”; v”) E T. Proof of (vi): Let v* < OJand let (a*; v*) E T. By definition 2.22, we have for every p > 0, ,

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315

pTu* < Z-(p; v*) < pTb where the upper bounding vector b is given by (1) and depends on D*. Thus we have u* < b. Q.E.D. Proof of 2.24. Given 7~ satisfying Conditions III’, define T = {(u; 2)): pTu < n(p; v) for every p > 0, ; v ,< 0,). Let p* > 0,) ZI* < 0, and denote the vector of first order partial derivatives of rr with respect to the components of p evaluated at (p*; v*) by u* = V,n(p*; u*). Since ~(p; II*) is a convex function of p and once continuously differentiable with respect to the components of p at p*, we have (see Rockafellar [25, p. 2421): for everyp > 0,) ~(p; v*) > r(p*; v*) + (p -~*)~V,r(p*; v*) = pTV,n(p*; v*), where the last equality follows using the linear homogeneity of 7~ in p. Thus we have pTu* < ~(p; v*) for every p > OI and PERU* = nfp*; v*). Thus by the definition of T, PERU* = max,{p*Tu : (u; o*) E T}. Q.E.D. ACKNOWLEDGMENT My thanks for the assistance of Daniel McFadden, Giora Hanoch, and Robert Engel, but they are absolved from any remaining errors. I am also indebted to D. W. Jorgenson for his initial encouragement. REFERENCES 1. R. G. D. ALLEN, “Mathematical Analysis for Economists,” St. Martin’s Press, New York, 1938. 2. G. DEBREU, Definite and Semidefinite Quadratic Forms, Econometrica, 20 (1952), 295-300. 3. G. DEBREU, “Theory of Value An Axiomatic Analysis of Economic Equilibrium,” J. Wiley and Sons, New York, 1959. 4. M. DENNY, The Relationship between Functional Forms for the Production System, Working Paper 7007, Institute for Quantitative Analysis, Univ. of Toronto, August, 1970. 5. M. DENNY, Notes on the Specification of Technologies, Working Paper 7008, Institute for Quantitative Analysis, Univ. of Toronto, August, 1970. 6. W. E. DIEWERT, An application of the Shephard duality theorem A generalized Leontief production function. J. Political Economy, 79 (1971), 481-507. 7. W. FENCHEL, Convex cones, sets and functions, unpublished lecture notes, Dept. of Mathematics, Princeton Univ., Sept. 1953. 8. M. Fuss AND D. MCFADDEN, Flexibility versus efficiency in ex ante plant design, in “An Econometric Approach to Production Theory,” (D. McFadden, Ed.), forthcoming. 9. W. M. GORMAN, Measuring the quantities of fixed factors, in “Value, Capital and growth: Papers in honour of Sir John Hicks,” (J. N. Wolfe, Ed.), pp. 141-172, Aldine Publishing Co., Chicago, 1968. 10. G. HANOCH, Generation of new production functions through duality, in “An Econometric Approach to Production Theory,” (D. McFadden, Ed.), forthcoming.

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