Handbook of Econometrics Vols1-5 _ Chapter 26 pps

It would be ideal if the extrapolative domain of applicability consists of all nonnegative or positive prices in the case of a unit cost function or of all nonnegative or positive prices

FUNCTIONAL FORMS IN ECONOMETRIC

3 Compatibility of the criteria for the selection of functional forms

3.1 Incompatibility of a global domain of applicability and flexibility

3.2 Incompatibility of computational facility and factual conformity

3.3 Incompatibility of a global domain of applicability, flexibility and

Handbook of Econometrics, Volume III, Edited by 2 Griliches and M.D Intriligator

0 Elsevier Science Publishers BV, 1986

1516 L J Lml

1 Introduction

Econometrics is concerned with the estimation of relationships among observable (and sometimes even unobservable) variables Any relationship to be estimated is almost always assumed to be stochastic However, the relationship is often specified in such a way that it can be decomposed into a deterministic and a stochastic part The deterministic part is often represented as a known algebraic function of observable variables and unknown parameters A typical economic relationship to be estimated may take the form:

where y is the observed value of the dependent variable, X is the observed value

of the vector of independent variables, cu is a finite vector of unknown constant parameters and E is a stochastic disturbance term The deterministic part, f( X, a), is supposed to be a known function The functional form problem that

we consider is the ex ante choice of the algebraic form of the function f( X; CX)

prior to the actual estimation We ask: What considerations are relevant in the selection of one algebraic functional form over another, using only a priori information not specific to the particular data set?

This problem of ex ante choice of functional forms is to be carefully dis- tinguished from that of ex post choice, that is, the selection of one functional form from among several that have been estimated from the same actual data set

on the bases of the estimated results and/or post-sample predictive tests The

ex post choice problem belongs properly to the realm of specification analysis and hypothesis testing, including the testing of nonnested hypotheses

We do not consider here the choice of functional forms in quanta1 choice analysis as the topic has been brilliantly covered by McFadden (1984) elsewhere

in this Handbook In our discussion of functional forms, we draw our examples largely from the empirical analyses of production and consumer demand because the restrictions implied by the respective theories on functional forms are richer But the principles that we use are applicable more generally

Historically, the first algebraic functional forms were chosen because of their ease of estimation Almost always a functional form chosen is linear in parame- ters, after a transformation of the dependent variable if necessary Thus, one specializes from

to

Y= Cfi(x)ai9

where g(a) is a known monotonic transformation of a single variable Moreover,

it is often desirable to be able to identify the effect of each independent variable

on the dependent variable separately Thus, one specializes further to:

In addition, functional forms of the type in eqs (1.1) and (1.2) may be interpreted as first-order approximations to any arbitrary function in a neighbor- hood of some X= X,, and that is one reason why they have such wide currency However, linear functions, while they may approximate whatever underlying function reasonably well for small changes in the independent variables, fre-

1518 L J L.uu

quently do not work very well for many others purposes For example, as a production function, it implies perfect substitution among the different inputs and cons&~ marginal products It cannot represent the phenomenon of di- minishing marginal returns Moreover, the perfect substitution property of the linear production function has the unacceptable implication that almost always only a single input will be employed and an ever so slight change in the relative prices of inputs will cause a complete shift from one input to another

Another linear-in-parameters functional form that was used is that of the Leontief or fixed-coefficients production function in its derived demand functions representation:

Xi=ffiY, i=l , , m,

where Xi is the quantity of the ith input, i =l, , m and Y is the quantity of output However, this production function implies zero substitution among the different inputs No matter what the relative prices of inputs may be, the relative proportions of the inputs remain the same This is obviously not a good functional form to use if one is interested in the study of substitution possibilities among inputs

The first widely used production function that allows substitution is the Cobb-Douglas (1928) production function, which may be regarded as a special case of eq (1.2):

However, it should be noted that the Cobb-Douglas production function was discovered not from a priori reasoning but through a process of induction from the empirical data Cobb and Douglas observed that labor’s share of national income had been approximately constant over time and independent of the relative prices of capital and labor They deduced, under the assumptions of constant returns to scale, perfect competition in the output and input markets, and profit maximization by the firms in the economy that the production function must take the form:

y = AjpL(1-a)

where K and L are the quantities of capital and labor respectively Fq (1.4) reduces to the form of eq (1.3) by taking natural logarithms of both sides The Cobb-Douglas production function became the principal work horse of empirical analyses of production until the early 1960s and is still widely used today

The next advance in functional forms for production functions came when Arrow, Chenery, Minhas and Solow (1961) introduced the Constant-Elasticity- of-Substitution (C.E.S.) production function:

where y, 6 and p are parameters This function is not itself linear in parameters However, it gives rise to average productivity relations which are linear in parameters after a monotonic transformation:

0 3

where p, r, and w are the prices of output, capital and labor respectively and (Y, p and u are parameters The C.E.S production function was discovered, again through a process of induction, when the estimated u from eq (1.6) turned out to

be different from one as one would have expected if the production function were actually of the Cobb-Douglas form

Unfortunately, although the C.E.S production function is more general than the Cobb-Douglas production function (which is itself a limiting case of the C.E.S production function), and is perfectly adequate in the two-input case, its generalizations to the three or more-input case impose unreasonably severe restrictions on the substitution possibilities [See, for example, Uzawa (1962) and McFadden (1963)] In the meantime, interest in gross output technologies dis- tinguishing such additional inputs as energy and raw materials continued to grow Almost simultaneously advances in the computing technology lifted any con- straint on the number of parameters that could reasonably be estimated This led

to the growth of the so-called “flexible” functional forms, including the gener- alized Leontief functional form introduced by Diewert (1971) and the tran- scendental logarithmic functional form introduced by Christensen, Jorgenson and Lau (1973) These functional forms share the common characteristics of linearity- in-parameters and the ability of providing second-order approximations to any arbitrary function In essence they allow, in addition to the usual linear terms, as

in eqs (1.1) and (1.2), quadratic and interaction terms in the independent variables

Here we study the problem of the ex ante choice of functional form when the true functional form is unknown (Obviously, if the true functional form is known, we should use it.) We shall approach this problem by considering the relevant criteria for the selection of functional forms

1520

2 Criteria for the selection of functional forms

What are some of the criteria that can be used to guide the ex ante selection of an algebraic functional form for a particular economic relationship? Neither eco- nomic theory nor available empirical knowledge provide, in general, a sufficiently complete specification of the economic functional relationship so as to determine its precise algebraic form Consequently the econometrician has wide latitude in deciding which one of many possible algebraic functional forms to use in building

an econometric model Through practice over the years, however, a set of criteria has evolved and developed These criteria can be broadly classified into five categories:

a neighborhood of the prices of commodities and income of interest

Obviously, not all functional forms can meet these theoretical requirements, not even in a small neighborhood of the values of the independent variables of

‘Summability means that the sum of expenditures on all commodities must be equal to income or total expenditure

interest However, a sufficiently large number of functional forms will satisfy the test of theoretical consistency, at least locally, that other criteria must be used to select one from among them Moreover, many functional forms, while they may satisfy the theoretical consistency requirement, are in fact readily seen to be rather poor choices For example, the cost function

c(P>y)=y

i=l

where pi is the price of the ith input and Y is the quantity of output and (Y~ > 0,

i=l >* , m, satisfies all the theoretical requirements of a cost function It is homogeneous of degree one, nondecreasing and concave in the prices of inputs and nondecreasing in the quantity of output However, it is not regarded as a good functional form in general because it allows no substitution among the inputs The cost-minimizing demand functions corresponding to this cost function are given by Hotelling (1932)-Shephard (1953) Lemma as:

First, we consider the system of derived demand functions of a cost-minimiz- ing, price and output-taking firm with the constant-elasticity property:

ln Xi = ai + e &ln pj + &ln Y, i=1,2 ,*.-, m (2.1)

j-l

where Xi is the quantity demanded of the ith input, pj is the price of the jth input, and Y is the quantity of output The elasticities of demand with respect to

X (necessarily positive), h’X’/ap and aX/aY at any specified positive values of

p=I_i and Y=Y through a suitable choice of the parameters &,‘s and /Iiv’s However, if it were required, in addition, that the system of derived demand functions in eq (2.1) be consistent with cost-minimizing behavior on the part of the producer, at least in a neighborhood of the prices of input and the quantity of output, then certain restrictions must be satisfied by the parameters pij’s and piv’s Specifically, the function:

C(p,Y)= 2 exp clli+ 5 (/3ij+6ij)lnp,+&lnY ,

(Bki + aki)exP

i ak + 5 j=l (bkj + skj> In Pj + &dn Y

i=l , ,m, (2.3) must be identically equal to the original system of derived demand functions in

A cost function is homogeneous of degree one in the prices of inputs and the first-order partial derivative of a cost function with respect to the price of an input is therefore homogeneous of degree zero, implying:

Applying eq (2.6) to eq (2.4) yields:

Pi,exP ai + 5 PikIn Pk + Pivln y

Second, pij > 0 and fiji > 0 (they cannot have opposite signs because of the

positivity of the exponential function and the nonnegativity of prices), in which

case the relative expenditures on the two inputs are constants independent of the

prices of inputs and quantity of output, implying the following restrictions on the

proportionality of expenditures implies that pii + 1 - Pki = 0 for all k such that

Pik # 0, k # i Hence all &‘s, k # i, must have the same sign-positive, in this case All Pki’s, k # i, must have the same positive sign and magnitude And PiY= bjY=PkY'

By considering all the i’s it can be shown that the inputs are separable into n,

n I m, mutually exclusive and jointly exhaustive groups such that

(1) Cross-price elasticities are zero between any two commodities belonging to different groups;

(2) Relative expenditures are constant within each group

Such a system of derived demand functions corresponds to a cost function of the form:

where

A,.>O; aii > 0, i; Caji=l; fi,>O, j=l, , n

Third, pij < 0 and pij < 0, in which case the relative expenditures on the two inputs are again constants independent of the prices of inputs and quantity of output, implying the same restrictions on the parameters as those in eq (2.8) However, as derived earlier, all &‘s that are nonzero must have the same

sign-negative, in this case But then cy= i& cannot be zero as required by zero degree homogeneity We conclude that a cost function of the form in eq (2.9) is the only possibility, with rather restrictive implications

From this example we can see that the requirement of theoretical consistency, even locally, may impose very strong restrictions on an otherwise quite flexible functional form

Second, we consider the complete system of demand functions of a utility-max- imizing, budget-constrained consumer with the constant-elasticity property: 3

In Xi = ai + F pijln pj + piMln M, i=l,2 ,.-., m;

j-l

(2.10)

where Xi is the quantity demanded of the i th commodity, pj is the price of the jth commodity, and M is income (or equivalently total expenditure) The elasticities of demand with respect to own and cross prices and to income are all constants:

p = p and M = &? through a suitable choice of the parameters pij’s and PiM’s However, if it were required, in addition, that the system of consumer demand functions in eq (2.10) be consistent with utility-maximizing behavior on the part

of the consumer, at least in a neighborhood of the prices of commodities and

3Such a system was employed by Schultz (1938), Wold with Jureen (1953) and Stone (1953)

Differentiating eq (2.12) with respect to In M twice, we obtain:

5 (&M-1)2exp a,+ f (/?i,+Sij)lnpj+(&,-l)lnM (2.14)

which by a similar argument implies

( PiM -l) = ‘3 i=l , > m

We conclude that (local) summability alone implies that the system of consumer demand functions must take the form:

i=l

which is no longer flexible.4 For this system, the own-price elasticity is minus unity, the cross-price elasticities are zeroes, and the income elasticity is unity for the demand function of each and every commodity

We conclude that theoretical consistency, even if applied only locally, can indeed impose strong restrictions on the admissible range of the values of the parameters of an algebraic functional form It is essential in any empirical application to verify that the algebraic functional form remains reasonably flexible even under all the restrictions imposed by the theory We shall return to the concept of “flexibility” in Section 2.3 below

2.2 Domain of applicability

The domain of applicability of an algebraic functional form can refer to a number

of different concepts The most common usage of the domain of applicability refers to the set of values of the independent variables over which the algebraic functional form satisfies all the requirements for theoretical consistency For example, for an algebraic functional form for a unit cost function C( p; a), where

(Y is a vector of parameters, the domain of applicability of the algebraic functional form, for given (Y, consists of the set

{PIP~O; c(p;+o; v ct p; a) 2 0; v2C( p; a) negative semidefinite}

For an algebraic functional form for a complete system of consumer demand functions, X( p, M; a), the domain of applicability, for given (Y, consists of the set

{P, WP, M20; X(pdwd20;

X(Xp, AM; a) = X(p, M; a); and

the corresponding Slutsky substitution matrix being symmetric

and negative semidefinite}

4This result is well known The proof here follows Jorgenson and Lau (1977) which contains a more general result

1528 L J Lau

We shall refer to this concept of the domain of applicability as the extrapolative domain since it is defined on the space of the independent variables with respect to

a given value of the vector of parameters (Y

It would be ideal if the extrapolative domain of applicability consists of all nonnegative (or positive) prices in the case of a unit cost function or of all nonnegative (or positive) prices and incomes in the case of a complete system of consumer demand functions for any value of the vector of parameters (Y Unfortunately this is in general not the case

The first question that needs to be examined is thus: for any algebraic functional form f(X; cy), what is the set of (Y such that f( X, a) is theoretically consistent for the whole of the applicable domain? For an algebraic functional form for a unit cost function, the applicable domain is normally taken to be the set of all nonnegative (positive) prices of inputs.5 For an algebraic functional form for a complete system of consumer demand functions, the applicable domain is normally taken to be the set of all nonnegative (positive) prices of commodities and incomes.6 If, for given (Y, the algebraic functional form f( X, CY)

is theoretically consistent over the whole of the applicable domain, it is said to be

globally theoretically consistent or globally valid For many functional forms, however, it may turn out that there is no such (Y, such that f( X; CY) is globally valid, or that the set of such admissible a’s may be quite small relative to the set

of possible a’s Only in very rare circumstances does the set of admissible (Y’S coincide with the set of possible (Y’s

We have already encountered two examples in Section 2.1 in which the set of admissible values of the parameters that satisfy the requirements of theoretical consistency is a significantly reduced subset of the set of possible values of the parameters For the system of constant-elasticity cos&minimizing input demand functions, the number of independent parameters is reduced from m(inputs)X (m + 2)(la,; mpij’s and l&) parameters to at most 2m parameters by the requirements of local theoretical consistency It may be verified, however, that under the stated restrictions on its parameters, the cost function in eq (2.9) as well as the system of constant-elasticity input demand functions that may be derived from it, are globally valid Similarly, for the complete system of constant- elasticity consumer demand functions, the number of independent parameters is reduced from m (commodities)x(m + 2) (la,; m&;‘s and l&) to (m - 1)

parameters by the requirements of local summability It may be verified, however, that under the stated restrictions on its parameters (own-price elasticities of - 1; cross-price elasticities of 0 and income elasticities of l), the complete system of constant-elasticity consumer demand functions is globally valid

51t is possible, and sometimes advisable, to take the applicable domain to be a compact convex subset of the set of all nonnegative prices

61t is possible, and sometimes advisable, to take the applicable domain to be a compact convex subset of the set of all nonnegative prices and incomes

These two examples share an interesting property - for given a, if the algebraic functional form is locally valid, it is globally valid This property, however, does not always hold We shall consider two examples of unit cost functions- the generalized Leontief unit cost function introduced by Diewert (1971) and the transcendental logarithmic unit cost function introduced by Christensen, Jorgenson and Lau (1973)

The generalized Leontief unit cost function for a single-output, two-input technology takes the form:

We note that a change in the units of measurement of the inputs leaves the values

of the cost function and the expenditures unchanged Without loss of generality, the price per unit of any input can be set equal to unity at any specified set of positive prices by a suitable change in the units of measurement The parameters

of the cost function, of course, must be appropriately resealed We therefore assume that the appropriate resealing of the parameters have been done and take (pi, p2) to be (1,l) By direct computation:

C(l,l) = (Ya + (Yi + (Y2,

tive We conclude that for global monotonicity, a0 2 0 and similarly a2 2 0 If (Ye, (pi and a2 are all nonnegative, eq (2.18) will be nonnegative for all nonnegative prices We conclude that the restrictions

are necessary and sufficient for global theoretical consistency of the generalized Leontief unit cost function

The transcendental logarithmic unit cost function for a single-output, two-input technology takes the form:

lnC( ply p2) = a0 + allnpl + Cl- alb p2

Ch 26: Functional Forms in Econometric Model Building 1531

eao is always greater than zero 12 (pi 2 0 is necessary and sufficient for vC(l,l)

to be nonnegative (~r((~r - l)+ j3i1 I 0 is necessary and sufficient for v2C(1, 1) to

be negative semidefinite The set of necessary and sufficient restrictions on the parameters for local theoretical consistency at (1,l) is therefore:

to be arbitrarily large or small, and thus causing the nonnegativity of vC( pl, p2)

to fail Thus, for global monotonicity, &r = 0 If 12 (pi 2 0 and &i = 0, eq (2.27) reduces to:

Having established that functional forms such as the generalized Leontief unit cost function and the transcendental logarithmic unit cost function can be globally valid only under relatively stringent restrictions on the parameters, but that they can be locally valid under relatively less stringent restrictions we turn our attention to a second question, namely, characterizing the domain of theoreti- cal consistency for a functional form when it fails to be global

As our first example, we consider again the generalized Leontief unit cost function We note that (pi 2 0 is a necessary condition for local theoretical consistency Given ai 2 0, eq (2.20) is identically satisfied The set of prices of inputs over which the generalized Leontief unit cost function is theoretically consistent must satisfy:

(2.32)

must also be satisfied

Next we consider the transcendental logarithmic unit cost function We note that 12 (pi 2 0 and al(al -l)+ &i I 0 are necessary conditions for theoretical consistency if (1,l) were required to be in the domain If &i # 0, we have seen that the translog unit cost function cannot be globally theoretically consistent We consider the cases of pii > 0 and pi1 < 0 separately If PI1 > 0, it can be shown that the domain of theoretical consistency is given by:

(2.33) where + 2 (1- a)a 2 pii > 0 If pi1 < 0, it can be shown that the domain of theoretical consistency is given by:

Our analysis shows that both the generalized Leontief and the translog unit cost functions cannot be globally theoretically consistent for all choices of parameters However, even when global theoretical consistency fails, there is still

a set of prices of inputs over which theoretical consistency holds and this set may well be large enough for all practical purposes The question which arises here is that given neither functional form is guaranteed to be globally theoretically consistent, is there any objective criterion for choosing one over the other?

One approach that may provide a basis for comparison is the following: We can imagine each functional form to be attempting to mimic the values of C, VC

and v2C at some arbitrarily chosen set of prices of inputs, say, without loss of generality, (1,l) Once the values of C, VC and v2C are given, the unknown parameters of each functional form is determined We can now investigate, holding C, VC and v2C constant, the domain of theoretical consistency of each functional form If the domain of theoretical consistency of one functional form always contains the domain of theoretical consistency of the other, no matter what the values of C, VC and v2C are, we say that the first functional form dominates the second functional form in terms of extrapolative domain of applicability In general, however, there may not be dominance and one func- tional form may have a larger domain of theoretical consistency for some values

of C, vC and v2C and a smaller domain for other values

We shall apply this approach to a comparison of the generalized Leontief and transcendental logarithmic unit cost functions in the single-output, two-input case

We need to establish the rules that relate the values of the parameters to the values of C, vC, and v2C at (1,l) We shall refer to such rules as the rules of interpolation For the generalized Leontief unit cost function, the rules of interpo- lation are:

Ch 26: Funciional Forms in Econometric Model Building

Leontief unit cost function may be rewritten in terms of k, and k, as:

1535

C( ~1, ~2) = (k, -%)p, +4k3p:‘2p:‘2 + (l- k, -2k,)p, (2.37) For the translog unit cost function, the rules of interpolation are:

1536 L‘.J.Lau

If k, - 2k, 2 0 and (1 - k,)-2k, 2 0, then the domain of theoretical consistency

is the whole of the nonnegative orthant of R* If k, -2k, 2 0 and (l- k,)-2k, -c 0, then the domain of theoretical consistency is given by:

uniquely extended to the whole of the nonnegative orthant of R*) If &i = - k, + k,(l - k2) > 0, then the domain of theoretical consistency is given by:

nonnegative orthant of R2 k, = 0 i mplies that pi1 = k,(l- k2) 2 0 Thus, the domain of theoretical consistency for the translog unit cost function is given by:

exP( (3 + \la- - k,),'k,(l - k,)) 2 E

which is clearly smaller than the whole of the nonnegative orthant of R* We note that the maximum and minimum values of k,(l - k,) over the interval [0, l] is + and 0 respectively Given k, = 0, if k,(l- k,) = 0, pII = 0, which implies that the domain of theoretical consistency is the whole of the nonnegative orthant of

R2 If k,(l - k2) = $, pII = a, and the domain of theoretical consistency reduces

to a single ray through the origin defined by pi = p2 If k,(l- k2) = $, (k2 = )),

the domain of theoretical consistency is given by:

e312=4.48kfi21

P2

Overall, we can say that the domain of theoretical consistency of the translog unit

cost function is not satisfactory for k, = 0

Next suppose k, = k,(l - k,) (which implies that k, I a), then either

will be smaller than the whole of the nonnegative orthant of R2 k, = k,(l - k2)

implies that pii = 0 Thus the domain of theoretical consistency for the translog

unit cost function is the whole of the positive orthant of R2 We conclude that

1538 L .J Lau

neither functional form dominates the other The cases of k, = 0 and k, = k,(l -

k2) correspond approximately to the Leontief and Cobb-Douglas production functions respectively

How do the two functional forms compare at some intermediate values of k, and k,? Observe that the value of the elasticity of substitution at (1,l) is given by:

C(LW,,(L1)

a(lJ) = C,(l,l)C*(l,l) ’

= b’[k,(l- &)I

If we let k, = ), (l- k2) = $, then a(l,l) = a is achieved at k, = i At these

values of k, and k,, the domain of theoretical consistency of the generalized Leontief unit cost function is still the whole of the nonnegative orthant of R* At

these values of k, and k,, pII = -i + 6 = & > 0 The domain of theoretical consistency of the translog unit cost function is given by:

56,233 2 e 2 0.0012,

We see that although it is short of the whole of the nonnegative orthant of R*, for

all practical purposes, the domain is large enough Similarly ~(1, 1) = 3 is achieved

at k, = & At these values of k, and k,, the domain of theoretical consistency of the generalized Leontief unit cost function is given by:

The comparison of the domains of theoretical consistency of different func-

tional forms for given values of k, and k, is a worthwhile enterprise and should

be systematically extended to other functional forms and to the three or more- input cases The lack of space does not permit an exhaustive analysis here It suffices to note that the extrapolative domain of applicability does not often provide a clearcut criterion for the choice of functional forms in the absence of

a priori information Of course, if it is known a priori whether the elasticity of substitution is likely to be closer to zero or one a more appropriate choice can be made

However, it is useful to consider a functional form f( X, a) as in turn a function g( X, k) = f( X, a(k)) where a(k) represents the rules of interpolation

If one can prespecify the set of X’s of interest, over which theoretical consistency must hold, one can then ask the question: What is the set of k’s such that a given functional form f( X, a(k)) = g( X, k) will have a domain of theoretical con- sistency (in X) that contains the prespecified set of X’s We can call this set of

k’s the “interpolative domain’ of the functional form It characterizes the type of underlying behavior of the data for which a given functional form may be expected to perform satisfactorily

2.3 Flexibility

Flexibility means the ability of the algebraic functional form to approximate arbitrary but theoretically consistent behavior through an appropriate choice of the parameters The concept of flexibility, first introduced by Diewert (1973, 1974), is best illustrated with examples First, we consider the cost function:

it cannot be considered “flexible” because it is incapable of approximating any theoretically consistent cost function satisfactorily through an appropriate choice

of the parameters to If we are interested in estimating the price elasticities of the derived demand for say labor or energy, we would not employ the linear cost function as an algebraic functional form because the price elasticities of demands that can be derived from such a cost function are by a priori assumption always zeroes

“There is of course, the question of what satisfactory approximation means, which is addressed

The degree of flexibility required of an algebraic functional form depends on the purpose at hand In the empirical analysis of producer behavior, flexibility is generally taken to mean that the algebraic functional form used, be it a produc- tion function, a profit function, or a cost function, must be capable of generating output supply and input demand functions whose own and cross-price elasticities can assume arbitrary values subject only to the requirements of theoretical consistency at any arbitrarily given set of prices through an appropriate choice of the parameters We can give a working definition of “flexibility” for an algebraic functional form for a unit cost function as follows:

DeJinition

An algebraic functional form for a unit cost function C( p; a) is said to be flexible

if at any given set of nonnegative (positive) prices of inputs the parameters of the cost function, (Y, can be chosen so that the derived unit-output input demand functions and their own and cross-price elasticities are capable of assuming arbitrary values at the given set of prices of inputs subject only to the require- ments of theoretical consistency.”

More formally, let C( p; a) be an algebraic functional form for a unit cost function where (Y is a vector of unknown parameters - Then flexibility implies and

is implied by the existence of a solution a( 3; C, X, S) to the following set of equations:

An example of a flexible algebraic functional form for a unit cost function is the generalized Leontief cost function The generalized Leontief unit cost function

“This definition of flexibility is sometimes referred to as “ second-order” flexibility because it implies that the gradient and the Hessian matrix of the unit cost function with respect to the prices of inputs are capable of assuming arbitrary nonnegative and negative semidetinite values respectively

“Ne g ative semidefiniteness of S follows from homogeneity of degree one and concavity of the unit cost function in the prices of inputs