On differentiability of cost functions

On differentiability of cost functions

JOURNAL OF ECONOMIC THEORY 38, 233-237 (1986) On Differentiability of Cost Functions* ROLF FARE AND DANIEL Department of Economics, Received ...

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JOURNAL

OF ECONOMIC

THEORY

38, 233-237

(1986)

On Differentiability

of Cost Functions*

ROLF FARE AND DANIEL Department

of Economics, Received

Southern May

Illinois

PRIMONT

University,

21, 1984; revised

Carbondale,

September

Illinois

62901

17, 1985

It was shown by T. Saijo that, under the assumption of strict monotonicity of the production function, differentiability of the cost function in prices is equivalent to strict quasiconcavity of the production function. In this paper we merely assume that the production function is nondecreasing thus weakening Saijo’s maintained hypothesis. Differentiability of the cost function is characterized in terms of properties of efficient subsets of inputs. The result is then extended to characterize continuous differentiability and to the case of multi-output technologies. Journal qf 1’ 1986 Academx Press. Inc. Economic Literature Classification Numbers: 022. 213.

1.

INTRODUCTION

Shephard’s lemma is one of the most useful and important propositions in the economic theory of production. The lemma states that under certain regularity conditions, the demand function for input i is obtained as the partial derivative of the cost function with respect to the ith input price. In his proof of the lemma (see Shephard [S, 9]), Shephard assumed that the cost function is differentiable with respect to input prices. This assumption has prompted Diewert (see [a]) to ask: “Under what conditions on the production function F will the cost function C be differentiable with respect to input prices?” He continues: “It is fairly easy to show that a sufficient condition for C to be once differentiable with respect to input prices is that F be strictly quasiconcave” (Diewert [2, p. 5211)’ From a Leontief production function, however, he shows that strict quasiconcavity of the production function is not necessary for a differentiable cost function. In a subsequent paper (Saijo [6]), it is shown under the assumption of strict monotonicity of the production function, i.e., if x, x’ E R;, .x 3 x’, is equivalent to strict x # x’, then F(x) > F(x’), that differentiability quasiconcavity. * This paper has benefited from useful comments by two anonymous referees and from the encouragement of one of the Associate Editors. ’ Under strict concavity of the production function, Sakai [7] shows that the cost function is differentiable in input prices.

233 OO22-0531/86 641 ix’?-4

$3.00

Copynghr !’ 1986 by Acadenuc Press, Inc. All rights of reproduchon m any form reserved

234

FARE

AND

PRIMONT

The main purpose of this paper is to give a necessary and sufficient condition on the production function, weaker than strict quasiconcavity and allowing for Leontief production functions, for differentiability of the corresponding cost function in positive input prices. Our condition is a restriction on the set of efficient input vectors (for each producible output). This set is called the efficient subset and, loosely stated, the condition requires that it mainly consists of extreme points of its upper level set. We rely on two previously established results. The first is a characterization of efficient subsets given by Arrow, Barankin, and Blackwell [ 11. The second result states that differentiability of the cost function is equivalent to uniqueness of the cost-minimizing input vector at each positive price vector and output level. This second result appears in various forms in Rockafellar [S], McFadden [4], and Saijo [6]. Thus our condition can be viewed as a characterization of the uniqueness of the costminimizing input demand. In interpreting our results the following points may be noted. First, the proof of our main theorem does not require that the production function satisfy any monotonicity assumptions. In the statement of the theorem, we assume that the production function is nondecreasing only to ensure the usual duality relationship between the production and the cost function. Second, as explained later, our condition is also necessary and sufficient for continuous differentiability of the cost function. And finally, none of our arguments depends on the number of outputs produced; thus our results are valid for multi-output production technologies.

2. DEFINITIONS The production technology y E Iw+ is here modelled by a by an input correspondence and the input correspondence

L(y):= @F(x)>

y}

AND

NOTATIONS

transforming inputs x E iwz into net output production function F: IR; --f Iw+ or inversely L: R, + 9’( EP+ ). The production function F L are inversely related by and

F(x) :=max{ y: XE L(v)).

In this paper we assume that the following properties hold for the production function:2 F.l. F(O)=O, F.2. F(x)>F(x') if xbx’, F.3. F is quasiconcave, F.4. F is upper semicontinuous. and

2 For discussions FLre [lo].

of the axioms

on the production

function

see Diewert

[3]

and Shephard

ON DIFFERENTIABILITY

Clearly, a Leontief production WI,

235

OF COST FUNCTIONS

function,

x2) :=min{x,h,

x2/a,),

a,>o, i= 1, 2,

satisfies properties F. 1-F.4. The efficient subsets related to a production function or an input correspondence play an important role in this paper. For y E Im F (Im F is the image of F), the efficient subset is defined by3 E(y) := (x:xG5(y),

x’
y > 0; E(O) = 0.

Denote the input price vector by p, p = ( p1, p*,..., p,). We assume that p is strictly positive in the sense that pi > 0, i = 1, 2,..., n, and denote it p $0. Given such an input price vector, the cost function is defined for y E Im F as C(y,p):= min{px:F(x)>y). 1 Given the cost function C and the production demand correspondence is given by

function

F, the input

K( y, p) := (x: px = C( y, p), F(x) 3 y}.

Finally, define the correspondence K(y)=

3. MAIN

IJ WY> P). pa0

RESULT AND EXTENSIONS

We will make use of the following two results. PROPOSITION

3.1 (Arrow, Barankin,

and Blackwell [ 11).

K(Y)EE(Y)~K(.v) where K(y) is the closure of K( y). PROPOSITION 3.2 (Saijo [6]). K( y, ~5) is a singleton if and only if C( y, p) is differentiubZe at ( y, $) in p.

The condition

that characterizes differentiability

of the cost function is:

(*) For all x, x’EE( y), x#x’, there exists f, T’EE( y), i #Y, arbitrarily close to x and x’, respectively, such that E + (I- A),?’ 4 E(y). for all I,O
means x’
but x'fx.

236

FARE

AND

PRIMONT

As a way of understanding condition (*), it may be useful to see that (*) is implied by: (**) For all x,x’~E(y), x#x’, Lx+(l-A)x’$E(y) for all %,O<%< 1. Condition (M) says that every point in E(y) must be an extreme point of L(y). (w) is a sufficient condition for differentiability of the cost function but it is not necessary. It is the weaker condition (*) that is both necessary and sufficient. PROPOSITION 3.3. Let production function satisfy properties F.llF.4. Assume y E Im F and y > 0. C( y, p) is differentiable in p, p $0, if and onl-v if condition (Y ) holds.

ProoJ Sufficiency of (*). If E(y) is a singleton, so that (*) holds vacuously, then K( y, p) is a singleton for each ~$0. Thus by Proposition 3.2, C( y, p) is differentiable. To continue, suppose there exists x, x’ E K( y, p), x #x’, for some p b 0, i.e., suppose C( y, p) is not differentiable. Then x, x’ E E(y). Using (*), there exist .?, 2’ E E(y), 1# R’, arbitrarily close to x and x’ such that ;l,i- + (1 - J,).? $ E(y), Since L(y) is convex, 1.? + (1 - L).f’ E L(y). Hence there exists x0 EL(~) such that x0 < 22 + (1 - A)?‘. This implies that px” < Api + (1 - A) p?. Since 1 and .? are arbitrarily close to x and x’, px” < Apx + ( 1 - 1) p-x’ = px = px’. This contradicts the supposition that x, x’ E K( y, p). Necessit-y of (a). If E(y) is a singleton then (*) holds vacuously. To continue, suppose x, x’ E E(y), x # x’. Then x, x’ E K(y). Thus there exist .< and Y arbitrarily close to x and x’, respectively, such that i-, ?-’ E K(y) c E(y),.?#f’.

Supposex”=~~+(1-~)~‘~E(y)forsomeII,0~~~1,i.e.,suppose(*) fails. Then x0 E K(y) and for some i” arbitrarily close to x0, 2” E K(y), i.e., .?’ E K( y, p) for some p % 0. Since K( y, p) is a singleton, p.?’ < pi and pl” < pi’. Since .i? is arbitrarily close to x0, px” < pf and px” < p-2’. Thus px”=Apx”+(l-i)px” < 2p-e + (1 -%)pZ = p(ll+(I -l)x’)=px”, a Q.E.D. contradiction. Thus x0 $ E(y) and (*) holds. As noted earlier, no use of monotonicity was made in the proof of Proposition 3.3. Differentiability of the cost function is intimately related to the strict convexity of the upper level sets, but strict convexity need only hold over the economic region, i.e., for efficient input vectors. As long as input prices are strictly positive the nature of the upper level sets outside the economic region is irrelevant. At no cost, Proposition 3.3 can be extended in two ways. First, a result by Rockafellar [S, p. 2461 says that any finite, convex (or concave)

ON DIFFERENTIABILITY

OF COST FUNCTIONS

231

function which is differentiable on a convex domain must also be continuously differentiable on the same domain. Thus we may replace “differentiable” by “continuously differentiable” in Proposition 3.3. Second, if y is considered to be a vector of outputs nothing in the proof of Proposition 3.3 need be changed. The set of assumptions F.l-F.4 can easily be replaced by an equivalent set of assumptions for the input correspondence L thereby facilitating the extension to multi-output technologies.

REFERENCES E. BARANKIN, AND D. BLACKWFLL, Admissible points of convex sets, in “Contributions to the Theory of Games” (H. W. Kuhn and A. W. Tucker, Eds.), Princeton Univ. Press, Princeton, N. J., 1953. W. E. DIEWERT, Generalized concavity and economics, in “Generalized Concavity in Optimization and Economics” (S. Schaible and W. Ziemba, Eds.), Academic Press, New York, 1981. W. E. DIEWERT, Duality approaches to microeconomic theory, in “Handbook of Mathematical Economics” (K. Arrow and D. Intriligator, Eds.), Vol. II, North-Holland, Amsterdam, 1982. D. MCFADDEN, Cost, revenue, and profit functions and Appendix A.3, in “Production Economics, a Dual Approach to Theory and Applications” (M. Fuss and D. McFadden, Eds.), Vol. I, North-Holland, Amsterdam, 1978. R. T. ROCKAFELLAR, “Convex Analysis,” Princeton Univ. Press, Princeton, N. J., 1970. T. SAIIO, Differentiability of the cost function is equivalent to strict quasiconcavity of the production function, Econ. Letf. 12 (1983), 135-139. Y. SAKAI, An axiomatic approach to input demand theory, hf. Econ. Rev. 14, 735-752. R. W. SHEPHARD, “Cost and Production Functions,” Princeton Univ. Press, Princeton, N. J., 1953. R. W. SHEPHARD, “Theory of Cost and Production Functions,” Princeton Univ. Press, Princeton, N. J., 1970. R. W. SHEIPHARD AND R. FARE, The law of diminishing returns. Z. Nationaliikon. 34, 69-90.

1. K. ARROW,

2. 3. 4. 5. 6. 7. 8. 9. 10.