INEFFICIENCY, UNCERTAINTY AND THE

STRUCTURE OF COST, COST-SHARE

AND INPUT-DEMAND FUNCTIONS

C.J. O~Donnell

No. 89- August 1996

ISSN 0 157 0188

ISBN 1 86389 368 7

INEFFICIENCY, UNCERTAINTY AND THE STRUCTUREOF COST, COST-SHARE AND INPUT-DEMAND FUNC~ONS

by

C.J. O’Donnell*

Department of EconometricsUniversity of New England

Armidale NSW 2350AUSTRALIA

Email: codonnel @metz.une.edu.au

July 1996

Abstract

We introduce random variables representing technical inefficiency and uncertainty into theproblem of maximising the expected utility of profits. We then use the results of Pope andChavas (1994) to identify cost minimisation problems which are not only consistent withexpected utility maximisation but are also independent of risk preferences.

The first-order conditions for our cost minimisation problems are solved for a set of

behavioural equations which are functions of deterministic unobserved ’planned outputs’ or

’aggregator functions’. In empirical work we can replace these deterministic arguments with

observed quantities and, in the process, endow the behavioural equations with a mutuallyconsistent stochastic structure representing technical inefficiency and/or uncertainty. In this

way, it is possible to overcome the ’Greene Problem’ encountered in the frontier literature.

* Not to be quoted without the author’s permission. This research was undertaken while the author wasvisiting Texas A&M University and the University of Guelph.

1. Introduction

Many econometric models of technically inefficient firm behaviour use as their starting point a set ofbehavioural equations (eg. cost and cost-share equations) which have been derived from a deterministic produceroptimisation problem. Technical inefficiency is then introduced by appending to these equations a carefullychosen set of error terms, some of which may be truncated in order to capture certain stylised economic facts (eg.that technical inefficiency increases costs). Examples include the cost and cost-share equations of Greene (1980)and Ferrier and Lovell (1990). A well-recognised problem with this approach is that the error terms appended todifferent behavioural equations may not provide internally consistent representations of inefficient behaviour.Greene (1980), for example, assumes that the errors in the cost-share equations are independent of the error in thecost equation, even though incorrect shares are known to increase cost. Ferrier and Lovell (1990) attempt tocapture the cost and cost-share effects of technical inefficiency by appending an error term to the cost equationonly.

One method of overcoming this so-called ’Greene Problem’ is to introduce parameters representing technicalinefficiency directly into the optimisation problem, and to carry these parameters into the behavioural equationsby way of the first-order optimising conditions. In this way, all behavioural equations are endowed with amutually consistent representation of inefficiency, albeit a parametric one. Examples of this approach includethe cost and cost-share equations of Atldnson and Comwell (1993, 1994a). A distinctive feature of this approachis that the optimisation problem is still deterministic, so the optimising conditions yield a set of deterministicbehavioural equations which must still be ’embedded in a stochastic framework’. Atkinson and Cornwell (1993,1994a), for example, introduce statistical noise.

A natural extension of this approach is to use ’producer-observable variables’ (rather than parameters) to representtechnical inefficiency in the optimisation problem. By ’producer-observable variables’ we mean variables whosevalues are known to the producer at the time input decisions are made. To use the terminology of Scttmidt(1985-86), technical inefficiencies are ’foreseen’. Examples of this approach include the cost equation and first-order-optimising conditions of Sclmaidt and Lovell (1979), and the additive general error models of McElroy(1987) and Kumbhakar (1989)1. Of course the optimisation problem, the first-order optimising conditions andthe behavioural equations are all still deterministic. Most authors, including McElroy (1987), Schmidt andLovell (1979) and Kumbhakar (1989), ’embed’ their behavioural equations in a ’stochastic framework’ byassuming (sometimes implicitly) that the ’producer-observable variables’ are unobserved by the researcher2. Inthis way, the optimisation problem is used to endow the behavioural equations with a mutually consistentrepresentation of inefficiency, this time a stochastic one.

In this paper we further extend this approach by using ’random variables’ (rather than parameters or producer-observable variables) to represent technical inefficiency in the producer optimisation problem. By ’randomvariables’ we mean variables whose values are not known to the producer at the time input decisions are made.Thus, technical inefficiency is ’unforeseen’ (Schmidt 1985-86) and the decision-making environment is said to becharacterised by ’uncertainty’. The plausibility of unforeseen technical inefficiency has been only brieflydiscussed in the frontier literature by, for example, Schmidt (1985-6). In contrast, decision-making underuncertainty has been widely discussed in the agricultural economics literature, with important recentcontributions by Daughtey (1982), Pope (1980) and Pope and Chavas (1994). By dealing with technical

1 Although Schmidt and Lovell (1979) and Kumbhakar (1989) do not make their assumptions regarding

(producer) observability of technical inefficiencies explicit, this is implied by the assumption that producersminimise cost. McElroy (1987) does, however, make this assumption explicit, and it is clear that

Kumbhakar (1989) relies heavily on this earlier work.2 Schmidt and Lovell (1979) and Kumbhakar (1989) also introduce statistical noise.

inefficiency in an uncertainty framework we hope to provide an overview of, and perhaps build something of abridge between, the frontier and uncertainty/agricultural economics literatures.

We take the classical economic theory approach to decision-making under uncertainty by assuming that producers

maximise the expected utility of profits. In the past, this assumption has severely limited the empiricalconvenience and usefulness of duality theory, since cost functions have usually depended on risk preferences.Recently, however, Pope and Chavas (1994) have shown how to remove this dependency, by imposingrestrictions on the form of the stochastic revenue function. In this paper we consider two such restrictions, bothof which guarantee the existence of a tractable cost function which does not depend on risk preferences, and oneof which ensures that random variables representing technical inefficiency are carried into at least some of ourbehavioural equations. Thus, at least some of our behavioural equations inherit a stochastic structure directlyfrom the producer optimisation problem, and can be estimated without being further ’embedded in a stochasticframework’. When it occurs, this ’stochastic inheritance’ has a great deal of theoretical appeal (see McElroy1987) and at least one important implication for econometric work: full-information estimators (eg. FIML) willbe identical for all representations of the technology.

One of the restrictions we consider in this paper may actually prevent this ’stochastic inheritance’. Therestriction which does this is one which constrains the variances of all but one random variable to zero. If thevariances of all random variables are equal to zero, technical inefficiencies are once again represented byparameters (specifically, the means of the distributions of our random variables) and our behavioural equationscollapse to the deterministic behavioural equations found in the frontier literature, such as those derived byAtkinson and Cornwell (1993, 1994a, 1994b). Thus, our paper is not as narrowly focused as it might ftrstappear: our model of producer behaviour under uncertainty is general enough to accommodate much of thefrontier literature as a special case.

The outline of the paper is as follows. In Section 2 we provide a brief description of the production technology,which is described in terms of the production possibilities set. Arguably the most important simplifyingassumption we make in this Section is that of output separability, which allows us to represent our multiple-output technology in single-output terms. In Section 3 we make specific assumptions concerning uncertaintyand inefficiency, a task which is made easier by our single-output representation of the technology. The produceroptimisation problem is the subject matter of Section 4, where we specify two restrictions which guarantee thatcost minimisation is consistent with maximisation of the expected utility of profits, and which also guaranteethat the cost function is not an explicit function of risk preferences. The cost, input-demand and cost-shareequations derived under each restriction are presented separately in Sections 5 and 6. In Section 7 we use shadowinput prices as a vehicle for introducing allocative inefficiencies into our model, which is then complete. Thepaper is concluded in Section 8, where we summarise our results and offer suggestions for further research.

2. The Production Technology

We begin by assuming that the firm employs an (Nxl) non-negative real-valued input vector x = (x1 ..... xN)’ toproduce an (Mxl) non-negative real-valued output vector y = (Yl ..... YM)’. We choose to describe thetechnology using the production possibilities set T = {(x, y): x can produce y} and we follow Chambers (1988,

p.252) by assuming that:

A.1A.2A.3A.4

T is a non-empty, closed, convex setIf (x, y) ~ T and xl>x then (x1, y) ~ T (free disposability of x)

If (x, y) ~ T and yl<y then (x, yl) ~ T (free disposability of y)(x, 0M) ~ T

A.5 If y>0, (ON, y) ~ T (weak essentiality)A.6 For every f’mite x, T is bounded from above

These reasonably straightforward assumptions are a subset1 of the minimal properties due to Shephard (1970).Assumption A.1 ensures that some feasible input-output combinations exist, that there are no ’holes’ in theboundary of T, and that the average of any two technically feasible input-output combinations is also feasible.A.2 and A.3 ensure that extra inputs do not hamper production, A.4 says that producing zero output is alwaysfeasible, while A.5 says it is not possible to produce outputs without inputs. A.6 is a regularity conditionwhich guarantees the existence of a transformation function and the existence of well-defined exlrema for some ofthe producer optimisation problems below.

In order to more easily accomodate uncertainty and technical inefficiency within our model, we also assumeoutput separability2:

A.7 There exists a set T* and a finite, non negative, real-valued index q(y) such that (x, q(y)) ~ T*~ (x, y) ~ T

This assumption is popular in the applied agricultural economics literature (eg. Ball and Chambers 1982) andmeans we can represent our multiple-output technology in single-output terms (see Chambers 1988 p.285):

(1) f(x) = max{q(y): (x, q(y)) ~ T*}

= max{q: (x, q) ~ T*}

where q -- q(y). Assumptions A.1 to A.6 imply that the input requirement set V(q) -- {x: f(x)>q} is closed andnon-empty for all q>0, and that f(x) satisfies monotonicity, quasi-concavity and weak essentiality (Chambers1988 p.258-9). Moreover, since q(y) is fmite, non-negative and real-valued, f(x) is also t-mite, non-negative andreal-valued for all finite and non-negative x.

Finally, to permit the use of differential calculus, and to allow us to establish consistency between costminimisation and maximisation of the expected utility of profits, we assume:

A.8A.9

f(x) is everywhere continuous and everywhere twice-continuously differentiable.f(x) is nonrandom and does not depend on risk preferences

3. Technical Inefficiency and Uncertain _ty

W~An input-output combination (~, ~1) is technically efficient if and only if there is no (x, q) ~ such that x _< ~and q > ~1; that is, if there is no way to produce more output with the same inputs or to produce the same outputwith fewer inputs (Varian 1992, p.4). Clearly, for a technically efficient input-output combination (~, ~1) we can

12

We do not assume, for example, that if x>0 there exists ~. such that (kx, y) ~ T for all finite y>0.Another assumption which would have allowed us to accomodate uncertainty and inefficiency is theassumption of input non-jointness: for every (x, y) ~ T there exist vectors xm->0 such that ym<fm(XTM) andExm < x, where fro(’) is a production function satisfying weak monotonicity, weak essentality,nonemptiness, and closedness of the input requirement set (see Chambers 1988, p.287). The implications ofinput non-jointness for the stochastic structure of transformation, input-demand, cost-share and cost functionsare addressed in another paper.

In the remainder of this paper we shall use overbar notation to denote input-output combinations which aretechnically efficient. Moreover, we shall make explicit the relationships between technically efficient andinefficient input-output combinations by assuming that for all (x, q) E T*:

A.10 xn=~n~)n a)n>l n = 1 .....NA.11 q=i10 0<0<1

where 0 and ~n (n = 1 ..... N) are finite and real-valued. Clearly an input-output combination (x, q) E T* istechnically efficient if 0 = ~n = 1 for all n.

Various parameterisations of assumptions A.10 and A.11 can be found in the frontier literature, whereequivalents of 0 and ~n (n = 1 ..... N) are (usually implicitly) assumed to be ’foreseen’. Assumption A.10 hasbeen maintained by, for example, Kumbhakar (1988) who uses an equivalent of ~n to reflect ’input-specifictechnical inefficiency’ (relative to some fixed factor). Atkinson and Cornwell (1993, 1994a, 1994b) alsomaintain A.10, but impose the restriction that ~n = ~ for all n (ie. technical inefficiency is not input-specific).Assumption A.11 has been maintained by Sclmaidt and Lovell (1979) and Atkinson and Comwell (1993, 1994a,1994b), with their equivalents of 0 usually being referred to as ’output technical inefficiency’.

Slightly different interpretations have emerged in the uncertainty literature, where the random variables 0 and a)n(n = 1 ..... N) are assumed to be ’unforeseen’. Waiters (1960), for example, maintains A.10 but uses hisequivalent of a)n to represent ’random variations in flows of factor services’. O’Donnell and Woodland (1995)maintain an unrestricted1 version of A.11 and use 0 to represent ’the random effects of industrial disputes,weather and disease’. Some of these interpretations are consistent with those from the frontier literature (eg.reductions in flows of factor services can be viewed as a form of input technical inefficiency) but some are not(eg. variations in weather would not generally be regarded as a source of output technical inefficiency). In mostsections of this paper we shall attempt to accommodate a multiplicity of plausible interpretations by referring to0 and x)n (n = 1 ..... N) as an ’output scaling factor’ and ’input scaling factors’ respectively.

Together with equation (2), assumptions A.10 and A.11 imply that for all (x, q) ~ T* the production functioncan be written:

(3) q = f(xl/v1 ..... XN/X)N)0

= m(n(xl/v1 ..... XN/VN))0

= m(n(x)/v)0

= f(x)0/vk

if f(.) is homothetic.

if Vn=V V n and f(.) is homothetic.

if Vn=V V n and f(.) is homogeneous of degree k

0 is not restricted to lie on the unit interval.There is no loss of generality in our restriction that n(.) be homogenous of degree one: a monotonictransform of an homogenous function of degree It0 can always be expressed as a monotonic transform of anhomogenous function of degree one (Shephard 1970 p.31).

12

where m(.) is a monotonic function and n(.) is a function which is homogenous of degree one2. Thus, theproduction function for the (inefficient) fLrm (in the presence of uncertainty) can be viewed as a standard (ie.deterministic and technically efficient) production function evaluated at inputs (Xl/a~1 .....XN/~N) and post-

multiplied by 0.

The empirical implications of equation (3) are twofold. First, both the output scaling factor (0) and the inputscaling factors (~n) should appear in the production function, and in a form which is determined by the

homotheticity and homogeneity properties of the technology. Second, if ~n=~ V n and f(.) is homogeneous ofdegree k then the output and input scaling factors will be underidentified (they may also be underidentified if~n=~ V n and f(.) is homothetic). A large number of frontier equations with an underlying functional structuregiven by (3) have been estimated under the assumptions that ~n=~ V n and f(.) is homogeneous. Examplesinclude the production frontiers of Aigner, Lovell and Schmidt (1977), Pitt and Lee (1981), Schmidt and Sickles(1984), Battese and Coelli (1988), Kumbhakar (1990) and Dawson, Lingard and Woodford (1991).

4. Op_ timising Behaviour

Although the theoretical economics literature contains a number of alternative descriptive models of choice underuncertainty (eg. Quiggin 1982) empirical researchers usually maintain the assumption of maximisation ofexpected utility, partly because it provides for easy pammeterisation of preferences. Recent examples from theapplied agricultural economics literature include the studies by Chavas and Holt (1990) and Duffy, Shalishali andKinnucan (1994). This paper continues in the tradition estabilished in the empirical literature by assuming thatproducers maximise the expected utility of profits:

A.12 U = maXx,q E{U(w0 + pq - w’x: (x, q) ~ T*}

= maxx>0 E{U(w0 + pf(xl/~l, x2/~o2 ..... XN/I~N)0 - w’x}

where U(.) is avon Neuman-Morgenstern utility function representing risk preferences, w0 is initial wealth, p isa nonnegative index of random output prices1, and w is an (Nxl) real-valued vector of non-negative non-randominput prices. Prices and initial wealth are regarded as exogenous, and we follow Pope and Chavas (1994) byassuming that:

A.13A.14

U(W) is differentiable and increasing in W, andE{U(W)} is concave in W.

Result 3 of Pope and Chavas (1994) can be used to show that the expected utility maximisation problem A.12implies cost minimisation if either or both of the following restrictions hold:

R.1

R.2~ = (~1 ..... X~N)’ is nonstochastic �~ E{~’} -- 0, andf(xl/’O1 ..... XN/~N) can be written f(xl/l~1 ..... XN/I~N) -- h(g(x), ~), where g(x) is non-negative, finite,real-valued and differentiable.

These restrictions not only ensure the existence of a deterministic cost minimisation problem, they implyrestrictions on its structure. Specifically, if R.1 is satisfied the cost minimisation problem is independent ofrisk preferences and takes the form:

(4) C(WlX)1 ..... WNX)N, q) = minx>0 {w’x: q = f(xl]~)1 ..... XN/X)N)}

which implies the expected utility maximisation problem A. 12 can be rewritten:

(5) U = max~l>_O E{U(w0 + p~10 - C(WlV1 .....WNVN, ~1)}

1 In this draft paper we have not considered the conditions required for the existence of the price aggregate p.

6

If R.2 is satisfied the cost minimisation problem is again independent of risk preferences, and is defined over thenonstochastic aggregator function ~ -- g(x):

(6) ~(w, ~) = minx>0 {w’x: ~ = g(x)}

which implies the expected utility maximisation problem A.12 can be written:

(7) U = maxg_>0 E{U(w0 + p0h($, v) - ~(w, $)}

Clearly, the Pope and Chavas result implies that the producer optimisation problem can be broken into twoparts: first, the producer chooses a so-called ’planned output’ (~1 and ~ in the case of R.1 and R.2 respectively) tomaximise the expected utility of profits (as described by (5) and (7)); and second, the producer chooses inputs tominimise the cost of producing planned ouput (as described by (4) and (6)). This separation of the expectedutility maximisation problem from the deterministic cost minimisation problem has two important implicationsfor empirical work: first, we can estimate a cost function and/or derivatives of it without having to makeassttmptions about the functional form of U(.) (ie. without having to make assumptions about risk preferences);and, second, it allows us to treat planned and realised outputs as exogenous, which means, for example, that wemay be able to estimate some of our cost, input-demand and cost-share equations in a Seemingly UnrelatedRegressions (SUR) framework, rather than in a simultaneous equations framework.

5. Cost. Inr)ut-Demand And Cost-Share Equations Under R.I: Nonstochastic a)

If R.1 is satisfied the cost function takes the form C(WlV1 ..... WNVN, ~]) and specifies the minimum cost ofproducing planned output level ~1 at prices w and input scaling factors ~) > tN. The increase in cost due to inputuncertainty/inefficiency is Ac = C(WlX)1 ..... WNX)N, ~1) - c(w, ~1), and Euler’s Theorem can be used to show that

this reaches a minimum of zero when a) = tN (details are provided in a Technical Appendix available on reques0.Under our earlier assumptions on the technology it is also possible to show that the cost function has a standardset of properties (Chambers 1988 p.51-52): nonnegative for all w, ~1 > 0; nondecreasing in w, ~) and ~1; concaveand continuous in w; positively linearly homogenous in w and ~); and no fixed costs (ie. C(Wla)1 ..... wNa)N, 0)= 0). Thus, C(Wla)1 ..... WNX)N, ~1) may be viewed as a standard (ie. deterministic and technically efficient) costfunction which is simply evaluated at planned output level ~1 and shadow prices (WlX)1 ..... WNX)N). Of courseplanned output coincides with realised output when 0 = 1, and shadow prices equal observed prices when ~) = tN.

Differentiability of f(x) means the first order conditions for an interior solution (ie. x>0) to the costminimisation problem (4) can be obtained as:

(8)wn - ~Lfn(Xl/X)1 .....XN/X)N)/X)n = 0

~= f(xl/v1 .....XN/~)N)

n=l .....N.

where fn(X) denotes the partial derivative of f(x) with respect to xn. Moreover, closedness and convexity of V(~I)implies that a unique solution exists (see Varian 1992 p.53-4), with conditional factor demand equations givenby (n = 1 ..... N):

(9) xn = Xn(Wl~01 .....WNa3N, ~l)X)n

= Xn(WlV1 ..... WNVN, 1)m-l(~l)X)n if f(.) is homothetic

-- Xn(Wl~1 ..... WNDN, 1)~ll/k~n if f(.) is homogenous of degree k

= Xn(W, 1)m’l(~)x) ff ~n---~ V n and f(.) is homothetic

=Xn(W, 1)~i/k~ if X)n--’~ ~’ n and f(.) is homogenous of degree k

These equations have standard properties (Chambers 1988, p.64): homogeneity of degree zero in w and a), and thesubstitution matrix, with elements ~Xn(Wl’O1 ..... WNX~N, ~l)/~wk = ~xk(wlX~l ..... WN~N, ~l)/~gwn, is negativesemi-defmite. Thus, the nth input demand function for the (inefficien0 firm (in the presence of uncertainty) canbe viewed as a standard input demand function evaluated at planned out~t level ~t and shadow prices (Wl~)1 .....

WNX)N), and post-multiplied by the nonstochastic scalar ~n"

In empirical work we replace the unobserved planned output ~ with q/0 (=~1), thereby introducing the one-sided 0into the empirical factor demand equations. Thus, like the production function, both the output scaling factor (0)and the input scaling factors (Un) should appear in the input-demand functions, and in a manner which isdetermined by the homotheticity and homogeneity properties of the technology. For example, if ~)n=~ V n andf(.) is homogeneous of degree k, then the output and input scaling factors are underidentified, and should appearin the unit input demand functions multiplicatively.

A further implication of equation (9) is that if X)n#’X) V n (ie. the input scaling factors are input-specific) then~xn(wlX~l ..... WN~)N, O)/~wk = ~Xn(WlX)1 ..... wN~)N, ~l)/~*ok for k,n, and input uncertainty/inefficiency willcause input demands to increase or decrease in line with the complementarity/substitutability relationshipsbetween the inputs. Thus, for example, the inefficient use of labour will reduce the demand for labour-substitutes, such as capital.

Finally, the n input demand functions can be used to recover the cost function:

(11) C = W~X

N=nZ__ l(wn~)n)Xn(W IX) 1 ..... WNX)N, 9)

-= C(WlX)l ..... WNX)N, ~1)

= C(Wl~1 ..... WNVN, 1)m-l(q)

= C(Wl~)1 ..... WNX)N, I)~I/k

= c(w,

= C(WlX)1 ..... WNX)N, 1)m-l(q)~

= C(w, 1)ql/ka)

if f(.) is homothetic

if f(.) is homogenous of degree k

if ~n=~) V n

if x)n=~ V n and f(.) is homothetic

if ~n---’o V n and f(.) is homogenous of degree k )

and the input cost-share equations:

(12)WnXn(Wlt~ 1 ..... WNgN, ~1)

Sn- c forn= 1 .....N

~ Sn(WlV1 ..... WNVN, ~)

= Sn(WlV1 ..... WN~N, 1)

= Sn(W, ~)

= Sn(W, 1)

if f(.) is either homothetic or homogenous of degree k

if ~n=l) V n

if ~n=~ V n and f(.) is either homothetic or homogenous ofdegree k

Not surprisingly, the cost-share equations have standard properties: nonnegative for all w, ~]>0, andhomogeneous of degree zero in w and v. Thus, both the cost and cost-share equations can be viewed as standardequations evaluated at planned output level ~1 and shadow prices (Wll)1 ..... WNa)N).

Equations (11) and (12) reveal that replacing the unobserved ~1 with q/0 (=~1) may have different implications forthe stochastic structures of the cost and cost-share equations. Specifically, this substitution always introduces 0into the cost function, but it fails to introduce 0 into any cost-share equations derived from an homogenous orhomothetic production function. Thus, since 0 is the only possible source of stochastic variation in the cost-share equations (v is nonstochastic by assumption), the share equations derived from an homogenous orhomothetic production function are inestimable without appending some form of error term. Of course, if 0 isnonstochastic then, irrespective of the homogeneity properties of the technology, all our behavioural equationsare deterministic and must be ’embedded in a stochastic framework’. In fact, our cost and cost-share equationscollapse to the parametric frontiers of Atkinson and Comwell (1993, 1994a) who, it will be recalled, introducerandom variables representing statistical noise.

6. Co$t. Input-Demand And Cost-Shzre Equations Under R.2: f(xl/~l..... XN/~N~

If R.2 is satisfied the cost function takes the form ~(w, ~) and specifies the minimum cost of producing plannedoutput level $ at prices w. Note that ~(.) and c(.) are different functions, and that ~(.) is not defined over thestochastic v. Although ~(.) is not defined over a), it is possible to show that the increase in cost due to inputuncertainty/inefficiency, Ac = ~(w, $) - c(w, ~1), reaches a minimum of zero when v = tN (again, details areavailable in a Technical Appendix available on reques0. It is also possible to show that the input requirementset V(~) = {x: g(x)>$} is closed and non-empty for all ~>0, and that g(x) satisfies monotonicity and weakessentiality. Thus, the cost function ~(w, $) will have the standard properties (Chambers 1988 p.51-2; Pope andChavas 1994 p.199): nonnegative for all w, $>0; nondecreasing in w; concave and continuous in w; positivelylinearly homogenous in input prices; nondecreasing in ~; and no fixed costs (~(w, 0) = 0). Thus, ~(w, ~) can bealso regarded as a standard cost function, on this occasion evaluated at the ’planned output’ level ~,.

Differentiability of f(x) implies differentiability of g(x), so the first order conditions for an interior solution tothe cost minimisation problem (6) can be obtained as:

(13)wn - ~gn(X) = 0

$=g(x)

n=l .....N.

where gn(X) denotes the partial derivative of g(x) with respect to Xn. Again, closedness and convexity of theinput requirement set implies these first order conditions can be solved to yield the conditional factor demandequations (n = 1 ..... N):

(14) xn = in(W,

= ~n(W, 1)/~

= Xn(W, l)m’l(fi~.u

= ~n(W’ l)~l/k

= Xn(W, I)~]1

if X)n=X) ~’ n and f(.) is homothetic.

if !.)n-----1) ~’ n and f(.) is homogeneous of degree k.

Again, these equations have standard properties: homogeneity of degree zero in w; and the substitution matrix,with elements 0~n(W, ~)/0wk, is negative semi-definite. Thus, the nth input demand function for the(inefficient) ftrm (in the presence of uncertainty) can be viewed as a standard input demand function evaluated at ~rather than q. Of course, in empirical work we replace ~ with h-l(q, ~))0"1 (=~,), so the empirical factor demandequations once more become functions of ~) and 0. Not surprisingly, the conclusions we reached in Section 5concerning homogeneity, homotheticity and the specificity of the input scaling factors are sl~ valid.

Finally, the factor demand equations can be used to recover the cost function:

(15) c = w’x

N=n~__lWn~n(w, ~)

-= C’(w, ~)

= ~(w, 1)~

= c(w, 1)m-l(fi.).v

= ~,(w, 1)~1/k

J= c(w, 1)~1/k~)

if 1)n=X)~’ n and f(.) is homothetic

if ~n=~) V n and f(.) is homogenous of degree k

and the input cost share equations:

Wn~n(W, ~)(16) Sn - C

-- ~n(W, ~) n = 1 ..... N.

I ff X)n = ~ V n and f(.) is either homothetic or homogenous of degree~’n(W, 1)k

= Sn(W, 1)

10

The share equations are nonnegative for all w, ~>0 and are homogenous of degree zero in prices. Thus, onceagain, both the cost and cost-share equations can be viewed as standard equations evaluated at the planned outputlevel ~ (which, in empirical work, we replace with h-l(q, ~)0-1).

It is worth noting from equations (14), (15) and (16) that if ~n = v V n and f(x) is either homothetic orhomogenous of degree k, then the cost, input-demand and cost-share equations obtained under R.2 can beexpressed as functions of the cost, input-demand and cost-share equations obtained under R.1. This is notsurprising since, if ~n = ~ ’q n and f(x) is eilher homothetic or homogenous of degree k, the input and outputscaling factors are underidentified. Furthermore, because output scaling factors play no role in establishingconsistency between cost minimisation and maximisation of the expected utility of profits, it is immaterialwhether R.1, R.2, both or neither are satisfied.

7. Allocative Inefficiency

A cost minimising fLrm is allocatively inefficient if and only if it operates off its least cost expansion path, ie. ifit produces output q (or ~1 or ~) using input combinations which do not minimise the cost of producing q (or/1 or~). Thus, allocative inefficiency implies that the first order conditions are not met exactly.

A convenient way of motivating allocative inefficiencies is to assume that producers maximise the expectedutility of profit subject to S new constraints rs(W, x) = 0 (s = 1 ..... S) (in addition to the usual technologyconstraint (x, q) ~ T*). These new constraints reflect the pursuit of non-expected-utility-maximising behaviour:they may, for example, reflect government regulations, or dynamic considerations in the choice of some inputs.Importantly, under R.1 and R.2, producers still minimise costs, with the first N first-order-conditions for aninterior solution now reflecting the new constraints (n=l ..... N):

(17) wn - ~,fn(Xl/X)l ..... XN/VN)/~)n = 0

in the case of R.1, and

(18) wn - ~,gn(X) = 0

in the case of R.2, where

S(19) wn = wn - s__El~,s3rs(W, x)/~xn

By analogy with equations (8) and (13), it is clear that allocative inefficiencies can be introduced into thetheoretical input-demand equations of Sections 5 and 6 by simply replacing the vector of observed prices w with

the vector of shadow prices w = (w1 ..... WN). In empirical work it is often useful to parameteriseoptimisation (allocation) errors as

(20) Kn = 1 -

ss__ZlksOrs(W, x)/OXn

so that (19) can be rewritten:

(21) wn = Wn~n ~n->O

11

Notice that the condition Kn->0 ensures that the shadow prices w will be non-negative, and that a rum will beallocatively efficient if and only if Kn = K V n. Equation (21) has been used to model the relationship betweenobserved and shadow prices by, for example, Toda (1976), Atldnson and Halvorsen (1984, 1990), Lovell andSickles (1983) and Kumbhakar (1992).

Equation (21) implies that our cost functions can be recovered as:

N(20) c = C(Wl~1 .....

WN~N’ q) n~__lsn(W~X)1 .....WN~N’ q)g~l

in the case of R.1 and

(21) _ , N ,C=C(W , ~) nE__lg’n(W , ~)~¢~1

in the case of R.2. Moreover, the share equations can be recovered as:

(22) Sn(Wl~) 1 ..... WN~N,Sn= N

* * 1.....

in the case of R. 1 and

(23) Sn(W , ~)~1Sn= jj~lgj(w*’ ~)~:jl

in the case of R.2. Thus, the cost and cost-share equations can be regarded as standard cost and cost-shareequations evaluated at shadow prices and different outputs, and multiplied or divided by a share weighted averageof the inverse allocative inefficiencies.

Two final observations are in order. First, the increase in cost due to allocative inefficiencies is non-negative andreaches a minimum of zero when rn=~ V n. Second, equation (21) collapses to equation (20), and (23) collapsesto (22), if ~n--~ V n and fix) is homogenous of degree k. In this case, allocative inefficiencies and the input andoutput scaling factors are all underidentified.

8. Conclusion

Stochastic specification is an important part of applied econometric work. Unfortunately, as McElroy (1987p.738) succinctly points out, "most of this work proceeds as if the static neoclassical model of a .o. firm has noimplications for the specification of the stochastic parts of the implied [behavioural] equations (cost function,input demand, and share functions), and vice versa". In contrast, this paper introduces random variables into theproblem of maximising the expected utility of profits, and carries these random variables into the behaviouralequations by way of the first-order optimising conditions. Thus, most of our cost, cost-share and input-demandfunctions inherit a mutually consistent stochastic smacture directly from our static model of firm behaviour. Inthis way, we overcome the ’Greene Problem’.

The behavioural equations implied by our model of flrm behaviour can be viewed as standard (ie. technicallyefficient and deterministic) behavioural equations evaluated at shadow prices and planned outputs, and sometimes

12

multiplied by (functions of) random variables. Thus, in empirical work we can choose a functional form for ourcost, cost-share and input-demand systems from the full menu of functional forms commonly used in appliedproduction economics. This includes exact (eg. Cobb-Douglas and Constant Elasticity of Substitution) as wellas flexible (eg. Translog, Generalised Leontief and Generalised Quadratic) functional forms. Importantly, ourchoice is not limited to self-dual functional forms (eg. Cobb-Douglas) as Atkinson and Comwell (1994a p. 233)and Bauer (1990 p.44) have suggested. However, our choice may be limited by the fact that our behaviouralequations inherit a stochastic structure from the optimisation problem, making it more difficult to craft acombination of functional form and distributional assumptions which will make our (systems of) equationsempirically tractable. The problem is likely to be greater for multiple-equation rather than single-equationmodels (because we need to craft multiple rather than single combinations of assumptions), for cost plus cost-share systems rather than input-demand systems (because the structure of the cost function will differ from thestructure of the cost-share functions, whereas the structure of each input demand function will be identical), andfor models in which all scaling factors and allocative inefficiencies, rather than just some, are assumed to berandom (because some functional forms will give rise to nonlinearities, and a function with a nonlineardeterministic component is often easier to estimate than a model with a nonlinear error term having momentsthat are difficult or impossible to derive). The problem may also be greater when using flexible functionalforms: in such cases we can only guarantee that our behavioural equations are consistent with maximisation ofthe expected utility of profits by assuming the input scaling factors are non-random or, if the production functionis homogenous, by assuming they are not input-specific (ie. ~n =x~ V n).

A cornerstone of our analysis is the assumption that the inputs and outputs of a technically efficient fLrm can berepresented by the very general production function ~1 = f(~). The generality of this assumption is thendiminished by our assumptions concerning inefficiency and uncertainty, leading us to specify a relationshipbetween observed inputs and outputs of the form q = f(xl/~1 ..... XN/X~N)0. It is, however, quite easy tomotivate and accommodate other inefficiency/uncertainty assumptions within our methodological framework,leading to a wide range of other input-output relationships, including a variant of the production functions usedby Just and Pope (1978) and Griffiths and Anderson (1982): q = f(xl/~1 ..... XN/~N) + h(Xl/91 ..... XN/~ON)0.These alternative uncertainty/inefficiency assumptions provide clear opportunities for further theoretical andempirical research.

13

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15

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16

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