Functional forms for technical change functions

The Journal of Productivity Analysis, 2, 143-152 (1991) © 1991 Kluwer Academic Publishers, Boston. Manufactured in the Netherlands.

Functional Forms for Technical Change Functions*

THOMAS MITCHELL AND DANIEL PRIMONT Department of Economics, Southern Illinois University at Carbondale, Carbondale, Illinois 62901-4515

Abstract

In this paper we examine the four properties of technical change functions introduced by Sato [1980, 1981], pay- ing particular attention to the composition property. We show by way of examples that two of the properties are independent, but that a third is a necessary consequence of two others. A theorem providing a necessary condition for the composition property is stated and proved. Finally, we identify the general continuous functional form of technical change functions for which the efficiency of factor i is independent of the quantities of all factors j ;~ i. Noting an unappealing characteristic of a member of this class of transformations, we propose an additional property for technical change functions and note that it is independent of the others.

1. Introduction

A great deal of attention has been given to finding desirable functional forms for production functions while rather less attention has been given to the same question for technical change functions. In this paper, we address the latter question. The main objectives are to examine several reasonable properties of technical change functions and to investigate their implications for functional form.

The starting point for the analysis is the set of four properties introduced by Sato [1980, 1981]. They are called the identity, inverse, composition, and associative properties and are formally defined in the next section. 1 Sato's interest in these properties stems from their usefulness in characterizing the class of production functions for which the effect of technical change can be separated from the effect of scale economies. Moreover, they form a reasonable set of axioms for technical change functions; they are satisfied by all of the forms of technical change functions used in the literature.

As it turns out, the associative property is always satisfied whenever technical change can be represented by technical change functions. Moreover, the identity property is implied by the inverse and composition properties. Thus the inverse and composition properties alone provide all of the restrictions on the functional form of the teclmical change functions. Of these two, the latter property has the most profound effect.

Production with technical change is modelled by a production function, F, which gives the amount of output y for each vector of effective inputs x ; i.e. y = F ( x ' ) , and technical change functions, 4~ = (~b~, ~b2 . . . . . ~bn), which give the amounts of effective inputs for each vector of nominal inputs x and each value of a technical change parameter t, i.e. x ' = ¢(x; t). The technical change parameter is traditionally taken to be time, however other interpretations are possible. 2

*The refereeing process of this paper was handled through S.T. Hackman.

144 T. M I T C H E L L A N D D. P R I M O N T

When we consider the special and interesting case in which the effective amount of each input depends only on the amount of that nominal input and on t, then the composition property alone implies that

x" = ¢i(xi; t) = f / - l ( f / ( x i ) + k ( t ) ) , (1)

where f/and k are increasing functions. Letting f / (xi) = In x i and k(t) = at , for example, we get the popular form:

X i ' : ~i(Xi'~ t) : Xi eat.

In practice, one's choice o f f / and k will depend on the choice of functional form for the production function F. Better yet, one can make these choices simultaneously. Note that the form of tbi in (1) implies that f /(xi ') = f i(xi) + k(t) . Thus, it is quite convenient to specify a function H such that

F ( x ' ) = H [ f l ( x ; ) . . . . . f~(x~)]

= H [ f ~ ( x , ) + k( t ) , . . . , fn(Xn) -Jr k(t)].

To illustrate this application, suppose H is a quadratic function. Then

, , ,

F ( x ' ) = olifi(xi ) -t- ~ijfi(xi ) fj(xj ) i=1 i=1 j = l

= ~ oLi[f/(x,) + k(t)] + ~ ~ flij[f/(x,) + k(t)][fj(xj) + k(t)]. i = l i=1 j = l

Advocates of translog production functions will want to choose f i(xi) = ( 1 / 6 i ) In xi and k(t) = In t, while advocates of the generalized Leontief functional form will want to choose f i(xi) = n/~xi/6i and k(t) = ~ However, many other choices are possible.

In the next section we define and describe the four properties and show that the associative property is satisfied for all technical change functions. In Section 3 we show that the composition property is independent of the others and present a necessary condition for the composition property to hold. In Section 4 we demonstrate that the identity property is implied by the inverse and composition properties. In Section 5 we derive the result given in (1). Noting an unappealing feature of the solution, we propose an additional property and show that it is independent of the others.

2. The Technical Change Functions

Following Sato [1980, 1981] we assume technical change occurs exogenously, and is represented by changes in a single real-valued parameter, t, representing the "state of

FUNCTIONAL FORMS FOR TECHNICAL CHANGE FUNCTIONS 145

technology." The effects of technical change appear as changes in the efficiency, or productivity, of the factors of production. Let x = (x~, x2, • • . , xn) be the nonnegative real vector of inputs to a production process. If we measure the inputs in "efficiency units" to account for ongoing technical change, then we represent the vector of "effective inputs" by x ' -- (x; , x 2 . . . . . x ' ) ? As is the convention (see, e.g., Chambers [1988]), we suppose that there are n functions, epi(x; t), (i = 1, 2 . . . . . n), which describe the transformation of "raw" or "nominal" inputs into "effective" inputs,

X' : ~(X; t) : [¢~1(/; t), . . . , ~n(X; t)] ~> O. (2)

The convention is to make the ~i'S nonnegative, real-valued, and continuously differentiable (i.e., analytic) functions of the vector x and the parameter t. In addition, the functions are assumed to be functionally independent with respect to the xi's, so that the n-th order Jacobian determinant, [O¢/Oxl , is nonvanishing. Additional properties for the ,~/'s have been introduced by Sato [1980, 1981]. We briefly review them here.

PROPERTY 1 (Existence of an identity transformation). There exists a real number to such that no technical change occurs, i.e.,

x ' : ~(x; to) = x, for all x >__ O.

That is, when t = to, all nominal input vectors are transformed, or mapped, into themselves. While Sato uses to = 0 without loss of generality, this is not the only possibility. (See Mitchell [1984].)

PROPERTY 2 (Existence of an inverse transformation). For every real value of the parameter t, there exists a real value s such that

x " = ~ ( x ' ; s) = ~,[~(x; t); s] = x , for a l l x >_ O.

Stated differently, any amount of technical change can be "undone," and the effective inputs can always be returned to their original "identity" values.

PROPERTY 3 (Composition property). For any pair of real values, tl and t2, which define x ' and x" by x ' = ,~(x; tO and x " = ~b(x'; t2), there exists a real value t3 such that

X" = O(X'; t2) = ~[O(X; tl); t2] = O(X; t3).

Moreover, t3 depends only on t~ and rE, t3 = K ( h , t2), so that

¢,[qi(x; tl); t2] = ,~[x; K(tl, t2)] (3)

is an identity in the Xi'S, tl, and t2. Hence, any two operations of the transformation scheme, (2), are equivalent to some

single operation of the transformation scheme. (See Sato [1981], Appendix, Sec. II.)

146 T. MITCHELL AND D. PRIMONT

Before stating the next property, we define a function • to represent the application of two operations of the transformation scheme, i.e.,

~I~(x; I i , 12) ~ ~[~(X; tl); /2]-

PROPERTY 4 (Associative law). Two operations followed by a single operation of the transformation scheme are equivalent to a single operation followed by two operations that have the same values of x, tl, t2, and t3. Specifically,

q~[¢(X; tl , /2); t3] = ¢ [~ (X; t , ) ; t2, t3].

Sato [1981, p. 26] noted that "the associative law . . . holds trivially" when the transformation scheme has a functional representation such as in (2).

The composition property, restated using ~, says that ~(x; tl, tz) = qi[x; K(q, tz)]. Thus

qi{qi[x; K(t,, t2)]; t3} = ~[¢(x; t l , t2); t3l,

= ¢[~(X; tl); t2, t3] ,

= qi[O(X; t,); K(t2, t3)],

by the composition property

by the associative law

by the composition property.

Therefore, when the composition property is satisfied, the associative law takes the form

O{~[x; K(t,, t2)]; t3} = O[q~(x; t,); K(t2, t3)], (4)

in terms of 0, 4 or

'l~[x; K(q, t2) , t3] = ¢[X; tl, g( t2 , t3)], (5)

in terms of ~ . Sato [1981; 27] has observed that "all of the known types [of technical change functions]

discussed in the economic literature thus far do in fact satisfy the foregoing assumptions." He allows, however, that "it may be possible to construct a type of technical progress which is meaningful, yet does not satisfy the group properties" [1981; 27]. In demonstrating the independence of the composition property, the next section presents an example of such a transformation.

3. The Independence of the Composition Property

Consider the following technical change transformation.

Example 1.

4,i(x; 0 = H(t)xi,

aet + 1 H ( t ) - - - ,

et + a

-) i = 1 , 2 . . . . . n where /

a > 1. (6)


Technical change is of the uniform, or equal, factor augmenting type, although here the "augmenting function," H(t), is different from the usual exponential form, e ~. Interest- ingly, plotting ¢i(x; t) = H(t)xi versus t produces an elongated S-shaped curve: for fixed xi, ¢i increases at an increasing rate until t = In a > 0; for t > In a, 0i increases at a decreasing rate. This is more consistent with general economic theory than t~i(x; t) = eatxi, under which efficiency always increases at an increasing rate (for a > 0). Even more real- istically, Oi in (6) has an upper bound, which is only approached in the limit as t ~ + 0o. With the usual factor augmenting form, eatxi, the efficiency of a fixed amount of factor i, xi, increases without bound as t --" + 0o. In these respects, (6) is a more realistic depic- tion of technical change than the exponential factor augmenting form. 5

The range of the function H is the open interval (1/a, a), hence H(t) > 0 for all real t. Clearly ~i is [infinitely] differentiable and the Jacobian of ~, ]O¢/c3xl = [H(t)] n > 0, is nonvanishing. The identity property is satisfied by to = 0, the inverse property by s = - t , and the associative law holds trivially. The composition property, however, does not hold: it requires that for each q and t2 there exist a t3 for which

H(tl) • H(t2) = H(t3).

Since H(0 is a strictly increasing function that converges to a > 1 as t approaches + 00, the above equality cannot hold for t~ and t2 sufficiently large. This establishes the independence of the composition property.

To see what is happening in this example, define a point-to-set mapping S:

S(x) = {x' E (Rn: x ' = O(x; t); t E ~ } , (7)

and a point-to-set mapping A:

A(x) = {x" E (R": x " = qi[qi(x; tl); t2]; tl, t2 E (R}. (8)

The reason that the composition property fails to hold for the technical change functions in (6) is that there exist q and t2 such that O[O(x; q); t2] = x " ~ S(x) , hence a value t~ satisfying (3) does not exist. This suggests the following theorem.

TnEOREU. If the technical change functions satisfy the composition property, then they satisfy the following contraction property: A(x) c_ S(x), for all x.

The proof of the Theorem is obvious from the discussion above. It would be of interest to know whether the contraction property of the Theorem is also

a sufficient condition for the composition property. The answer to this question is affir- mative in the special case that tbi(x; t) = H(t)xi, i = 1, 2, . . . , n, i.e., when ql i is (multiplicatively) separable in xi and t. In this case, the condition that A(x) c_ S(x) implies that (dropping subscripts)

x " = H(q)H(t2)x E S(x).

148 T. M I T C H E L L A N D D. P R I M O N T

This implies that there exists a t 3 such that

x " = n ( t l ) H ( t 2 ) x = n ( t a ) x .

Thus, as long as H has an inverse, t3 = H - I [ H ( h ) H ( t 2 ) ] , i .e. , t3 depends only on t~ and t2 as required by the composit ion property.

Unfortunately, this result breaks down for more general forms of the technical change function. To see this we need only consider a one-input ("nonseparable") example, viz.

E x a m p l e 2 . th(x; t) = e t x + t, where x is a nonnegative scalar.

To ensure that th(x; t) is nonnegative for all x > 0, restrict t to be nonnegative. Then A ( x ) = S ( x ) = { y : y > x } for all x > 0. Thus, A ( x ) ~ S ( x ) . The composit ion property requires that th[th(x; h); t2] = 0(x; t3) for some t3 that depends only on tx and t2. That is,

et~+t2x + ( t ie t2 + t2) : et~x + t 3.

Since t3 cannot depend on x, t3 must simultaneously equal tl + t2 and tl et2 + t2. This is impossible except for the trivial cases when tl or t2 equals zero.

Example 2 shows that the contraction property is not sufficient for the composition property. The example also fails the inverse property as the reader can easily check. Thus, we provide a second example that satisfies the contraction property, the identity property, and the inverse property, but does not satisfy the composit ion property.

E x a m p l e 3. For a single input, x:

~,(x; t) =

x e t + t t >= 0

x e t + let X + t >= 0, t < 0 .

Once again, A ( x ) = S ( x ) = { y • y > x } . Th,~ identity property is satisfied with to = 0. The inverse property is satisfied with t2 = - t~; we show this in two parts:

1) x ' = th(x; tl) = xe t. + tx i f tl >__ 0. Let t2 = --t~ < 0. Then x ' + t2 = x ' - t~ = xe t' > 0. Thus

x " = x ' e t2 + t2 et2

= ( x e tj + t l ) e -t~ -- t i e - t 1

X.

2) x ' = ~b(x; h) = xe t l + t l et~ if tx <_ 0. Again l e t t 2 - - - t 1 ~_ 0 . T h e n

x " = x 'e tz + t2

= ( x e t~ + t le t~)e - t ' -- t I

X.

To check the composit ion property, consider tl > 0, and proceed as in Example 2 to verify that the property fails.


4. Independence of the Properties

We have shown that the composition property is independent of the others. It is not true, however, that all of the properties are independent. In particular, the identity property may be deduced from the others?

With an example, the inverse property can be shown to be independent. Again, omitting subscripts,

E x a m p l e 4. x ' = th(x; t) = e~t2x, where x is a scalar.

We have differentiability, Ock/c3x ~ 0, the identity property holds (to = 0), and associativity holds (trivially). For the composition property note that

O[O(x; t l ) ; t~] = x e ~<ex+'~)

= ~[x; ~/~ + t~].

Thus the composition property holds since one can take g ( t l , t2) to be ~/tl z + t~. How- ever, the inverse property does not hold, since for each t there must exist an s for which r ~ + s2 = 0. This is impossible (for real s) except for the trivial case t = 0, thus estab- lishing the independence of the inverse property.

We now show that the inverse and composition properties imply the identity property. Fix q. The inverse property implies that a t2 exists for which ~b[q~(x; tl); t2] = x for all x. The composition property implies the existence of a constant K(t l , t2) such that ~b[x; K ( q , t2)] = x for all x. Consequently the identity property holds; the required to is to = K(t l , t2). 7 Of course, to may not be unique. In this case, x = ch(x; to) = ~b(x; t'o), where to ~ t'o. One may rule this out by restricting q~ to be strictly monotonic in t.

5. A General Form Satisfying the Composition Property

We now consider general, continuous functional forms for 4fix; t) and K ( q , tz) when (3) is satisfied under the reasonable assumption that the efficiency of the ith factor is independent of the quantity of the jth factor, for all j ;~ i (see Chambers [1988], p. 210). This assumption allows us to obtain a very sharp result for the implied form of the technical change functions that satisfy the composition property. It is, at the same time, an assumption that is very useful in empirical work and, thus, it is commonly employed?

Given this independence, (3) takes the form

xi" = 4)i[dpi(xi; t0; t2] = 4)i[xi; K( t l , t2)], i = 1, 2, . . . , n. (9)

Equation (9) is a special case of the "generalized associativity [functional] equation" known as the "transformation equation." The general system of solutions of (9), when ~b i and K are strictly monotonic in both arguments, K is continuous, and (~i is continuous in xi, is given by Acz~l [1966]:

150 T. MITCHELL AND D. PRIMONT

X i' = ~ i (X i ; t) = f / - l [ f / ( x i ) -[- k ( t ) ] i = 1, 2 . . . . . n ; - ~

5 t3 = g ( t l , t2) = k-l[k(G) + k ( t 2 ) ] ,

(10)

where f / a n d k are arbitrary, strictly monotonic functions. 9 One immediate consequence of the solutions is that K as given in (10) is associative: K[t~, K(t2, t3)] = K[K(t~, t2), t3], where the common value is k- I lk ( t1 ) q- k(t2) q- k(t3)]. Referring to (4) and note 4, this explicitly shows that technical change is associative: when Oi and K are given by (10), both expressions for xi" in (4) can be reduced, by substitution, to

xi '" = f/-l[f/(xi) + k(G) + k(t2) + k(t3)], i = 1, 2 . . . . . n.

(We also observe that according to the solution (10), the function K that determines t3 in (3) is symmetric in its two arguments.)

Mitchell [1984] provides examples illustrating forms of fi(xi) and k( t ) that generate familiar forms of ~b i. We repeat one of those examples here to propose an additional restriction on thi. Let f/(xi) = otixi and k(t) = 7 t with oti, 3' > 0. Then

¢~i(Xi; t) = X i q- ~ i t , w h e r e 13i - 7 /~ i > 0;

this is the "additive" type of technical change. But suppose xj = 0 for some j. Clearly x j = c~j(xj; to) = xj = 0, at the identity value of t. But if tl > to = 0, then since ~j > 0 we have xj' = chj(xj; tO = x j + ~j t l = ~j t l > 0. If the usual interpretation of t as time is taken and xj = 0, then ~j(xj; tl) = ~bj(0; tl) > 0 indicates that we can get something for nothing simply by waiting! This is an economically unappealing property of additive technical change functions. ~° Therefore, we propose an additional restriction for all technical change functions.

PROPERTY 5 (Positive input requirement). For all t, and i = 1, 2 . . . . , n, a technical change transformation must satisfy

(~i(X1, X 2 . . . . . X i _ l , 0 , Xi+ 1 . . . . . Xn; t) = O. (11)

This implies that an additive form, e.g., can never be used if the positive input requirement assumption must hold. 11

The additive form, ~i (X i ; t) = X i -I- 13it ( i ---- 1, 2 . . . . . n), easily establishes the independence of Property 5. (Also, Examples 1 and 4 are unaffected by imposing the positive input requirement; in both cases xi = 0 ~ xi' = 0 for all i.)

Let us return to (10) and see what (11) implies for the representation off/. It implies that f/(O) = f/(O) + k(t) , for all t. I f f/(0) is defined, then k(t) = 0 for all t and the only possible solution for K is trivial. Then meaningful technical change functions ~b i that satisfy the positive input requirement and are independent of xj for all j ~ i, must be such that f/(0) is undefined. An example of such a function is f / (x i ) = (1/ai)ln xi. Then we obtain the technical change function t~i(Xi; t) = A i ( t ) • xi, where A i ( t ) = e aik(t). Note that we can have a nontrivial solution for K, because f/(xi) = (1/ai)ln Xz is undefined at zero. 12 (See also Mitchell [1984].)


Thus, adding the positive input requirement to our list of properties further limits the choice of technical change functions. However, some researchers may be unwilling to ac- cept this additional limitation. In this case it must be admitted that the positive input requirement is less compelling if each xi ~> 0 for the particular data set under study.

6. S u m m a r y

In this paper we have reviewed four properties of technical change functions. With an example of an economically meaningful technical change transformation we demonstrated the independence of the composition property. Similarly we showed the independence of the inverse property. We noted, however, that the identity property is not independent; it is a logical consequence of two other properties.

After restricting technical change functions so that the efficiency of input i is independent of all inputs j ( j ;~ i), we presented the general functional form satisfying the composition property. Consideration of the additive class of technical change functions, which is among the solutions, led to the proposal of the positive input requirement for technical change functions, which the additive class does not satisfy.

The results presented here, particularly the functional representation in (10), will enable empirical researchers to generate functional forms for technical change functions--whether for effective factor-inputs or -prices--and incorporate them into standard models of pro- ducer behavior with productivity change. It is hoped that such applications will yield more rigorous and useful empirical studies of production under technical change.

Acknowledgments

We have benefited from stimulating discussions with Rolf Ffi're and the sharp comments of two anonymous referees.

Notes

1. These four properties applied to technical change functions imply that the technical change functions form a Lie group. Thus Sato [1980, 1981] is able to apply Lie group theory to the study of technical change.

2. For example, labor inputs can be augmented by education, specialized training, or experience. In agriculture, the productivity of seeds can be augmented by improved chemical seed treatments that ward off insects and thereby increase the number of surviving seeds. The productivity of personal computers can be upgraded by adding graphics cards, hard cards, hyper cards, super cards, math co-processors and the like.

3. While our analysis views technical change as affecting the quantity of effective inputs, all of our results can be applied to a dual view of technical change as affecting the "effective input prices" paid by a firm. This would be an extension to more general functional forms of the "input-price" or "input-cost-diminishing" interpretation of factor-augmenting technical change presented by Chambers [1988, pp. 222-223, 227-228]; see also Binswanger [1974]. While the effective inputprice interpretation lends itself more readily to empirical implementation, our analysis corresponds to Sato's [1980, 1981] "direct" or "primal" approach and focuses on effective inputs.

152 T. MITCHELL AND D. PR/MONT

4. Note that the composition property applied to (4) implies ~b{x; K[K(tl, t2), hi} = O{x; Kit1, K(h, h)]}. Thus K itself is associative.

5. For example, improved seed treatments increase the survival rate of planted seeds. However, the number of planted seeds is an upper bound on the number of surviving seeds.

6. We will not be concerned with the independence of the associative law; as stated earlier, it will be satisfied whenever technical change can be represented by technical change functions.

7. This relationship was known to Sato [1981]; it can be found in the Appendix where he discussed Lie groups on an abstract level. To further illustrate the relationship between the identity and inverse properties, consider

~b(x; t) = ln(e x + et).

This transformation is associative (trivially), has O4~/Ox -~ O, and satisfies the composition property since q~[O(x; tO; t2] = ~b[x; ln(e t, + et2)]. But since ~b(x; t) ~ x only as t ~ - o o , neither the identity nor the inverse property is satisfied for t E ( - co , + co).

8. While the problem can be solved without this independence assumption--see Mitchell [1990]--the analysis in the general case is much more tedious and it is not at all clear how the results there can be "collapsed" to the results of this paper under the condition of independence. Furthermore, the results in the general case are not, in general, easily applied for empirical use.

9. For this result, Aczdl requires "weak reducibility." Weak reducibility is implied by our assumption of strict monotonicity.

10. That the additive form possesses this unappealing property is not entirely due to the assumption that ~i is independent of xj, j ~ i. Consider the type of technical change that Sato [1981] calls "ratio additive to capital":

K' = dPK(K, L; t) = K + c~Lt L ' = 6L(K, L; t) = L.

In this case, q~K depends on L so the independence assumption has not been made. Yet there is the same "something for nothing" problem as in the example above because even if K = 0, we can nevertheless get a positive effective capital: i f ~ > 0 and we wait (t > 0) with some labor (L > 0), then K' = ~h,(0, L; t) > 0.

11. If a technical change function is defined in a piecewise fashion, it may take an additive form at points away from x i = 0. For simplicity we ignore this possibility.

12. Note that f / i s strictly monotonic in x i even though it is discontinuous at x i = 0. Thus it is (part of) an allowable solution to (9). I f we insist that fi be continuous and thus defined at zero, then there are no such solutions that satisfy both the composition property and the positive input requirement.

References

Acz~l, J.D. (1966). Lectures on Functional Equations and Their Applications. Mathematics in Science and Engi- neering, Vol. 19, New York: Academic Press.

Binswanger, H.P. (1974). "The Measurement of Technical Change Biases with Many Factors of Production." American Economic Review 64, 964-976.

Chambers, R.G. (1988). Applied Production Analysis: A Dual Approach. Cambridge: Cambridge University. Mitchell, T.M. (1984). '~A Functional Equation Approach to the Theory of Production, Technical Change, and

Invariance." Zeitschrift fiJr Nationalbkonomie 44, 177-187. Mitchell, T.M. (1990). "On the Functional Forms of Technical Change Functions." In Conservation Laws and

Symmetry: Applications to Economics and Finance, R. Sato & R. Ramachandran (eds.), Bos~ton: Kluwer Academic. Sato, R. (1980). "The Impact of Technical Change on the Holotheticity of Production Functions?' Review of Economic

Studies 47, 767-776. Sato, R. (1981). Theory of Technical Change and Economic lnvariance: Application of Lie Groups. New York:

Academic Press.

Functional forms for technical change functions

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