*Corresponding author.
E-mail address: [email protected](A. Picazo-Tadeo).
The calculation of shadow prices for industrial
wastes using distance functions: An analysis for
Spanish ceramic pavements
"
rms
Ernest Reig-Mart
m
H
nez
!
,
"
, Andre
H
s Picazo-Tadeo
!
,
*, Francesc Herna
H
ndez-Sancho
!
!Department of Applied Economics II, University of Valencia, Avda dels Tarongers s/n, 46022 Valencia, Spain"Valencian Institute of Economic Research (IVIE), Guardia Civil 22, 46020 Valencia, Spain
Received 10 August 1999; accepted 27 January 2000
Abstract
This paper deals with the calculation of shadow prices for two industrial wastes generated on their production processes by 18"rms belonging to the Spanish ceramic pavements industry. These prices are then used to calculate an extended productivity index which takes into consideration wastes going with the production of marketable goods. We follow the methodological approach"rst proposed by FaKre et al. (The Review of Economics and Statistics 75 (1993)). A negative correlation is found between absolute shadow prices and wastes production intensity, re#ecting a greater marginal cost of eliminating wastes for those"rms using less contaminant production processes. Di!erences between a conventional labour productivity index and an extended productivity index are also statistically related to"rms characteristics such as size, previous investments in cleaner technologies and a$liation to a Technological Insti-tute. ( 2001 Elsevier Science B.V. All rights reserved.
Keywords: Shadow prices; Distance functions; Ceramic pavements industry; Environment; Productivity
1. Introduction
The growing recognition of the environ-ment as a public good has unleashed a debate with regard to the convenience of breaking the tradition of assessing the value of industrial production by implicitly assuming that all goods produced are socially desirable. If it is accepted that a part of industrial production is undesirable,
and public authorities establish regulations to limit the emissions of polluting wastes, the cost that"rms face to ful"l legal environmental restric-tions, should be evaluated. Therefore, shadow pri-ces for undesirable outputs have to be computed in order to measure in terms of opportunity costs the impact on"rms performance of environmental re-strictions preventing free disposal of industrial wastes.
Shadow prices of undesirable outputs are under-stood in this context as the marginal cost due to a marginal reduction in the possibility of freely disposing of wastes generated in the production
1This can be veri"ed by de"ning a cost function and setting up a maximising pro"t problem. Given a vector of (desirable and undesirable) outputs u, with pricesr, and being xand pthe quantity and price input vectors, respectively, the cost function isc(u,p)"minMpx:xcan produceuN, while the pro"t function can be set up asn(r,p)"maxMru!c(u,p)N. From"rst-order condition for undesirable outputjwe get
r j"
Lc(u,p) Lu
j , wherer
jis the shadow price of wastej.
2This expression is equivalent to the reciprocal of the output oriented e$ciency measure of Farrell [6,7]. Also note that
u3P(x) if and only ifD
0(x,u))1.
process.1From the point of view of public policies for environmental protection, the availability of these shadow prices reports several important be-ne"ts; among them, the possibility of comparing the marginal bene"ts of environment policies, with the cost they generate for private"rms; the chance of checking if all"rms face the same shadow prices; and,"nally, the feasibility to adapt traditional pro-ductivity indexes to allow for the consideration of di!erent intensity of waste production among "rms, sectors or even countries.
This paper deals with the calculation of shadow prices for undesirable outputs that areby-products
of the industrial production of ceramic pavements, with data coming from a sample of Spanish "rms located at Castellon, on the Valencian region. We follow the distance function approach suggested by FaKre et al. [1] (FGLY henceforth), recently applied by Coggins and Swinton [2]. This method uses output distance functions to derive shadow prices for all outputs (desirable and undesirable) gener-ated by"rms in their productive processes. In par-ticular, it makes it feasible to obtain shadow prices for undesirable outputs without having to use exo-genous information on wastes elimination costs coming from other studies (as Pittman [3] does in a paper aimed to adapt the multilateral productiv-ity indexes pioneered by Caves et al. [4] for taking stock of polluting emissions). Secondly, this paper aims to propose anextendedmeasure of productiv-ity that takes into account residuals emerging as aby-productof current industrial production pro-cesses. Availability of residuals output data for each of the "rms in our sample allows us to undertake this correction.
This introduction is followed by a description of the methodology. Section 3 describes the sample
and establishes the main results, while Section 4 concludes.
2. The output distance function and the derivation of shadow prices
In order to illustrate the basic aspects of the methodological approach proposed by FGLY to derive output shadow prices from distance func-tions, let us assume that we have a set of"rms using a vector of inputs x3RN
` to produce a vector of
outputsu3RM
`, some of which can be considered
undesirable orbadoutputs. Thetechnology of refer-ence is represented by an output correspondence which is a mapping P:RN
`PP(x)-RM`, where
theoutput setP(x) represents the set of all feasible vectors of outputs given a vector of inputsx. It is also assumed that thetechnologysatis"es the usual axioms initially proposed by Shephard [5], which allows to de"ne thedistance function in outputs as the inverse of the maximum radial expansion of a given output vector, in such a way that the result-ing output vector remains withinP(x). Thedistance functioncan be de"ned on the output set as2
D
0(x,u)"infMh: (u/h)3P(x)N. (1)
The assumptions made on the disposability properties of the technology are a key issue in order to derive output shadow prices. In particular, it is assumed that"rms cannot freely eliminate (without any cost) the industrial wastes (undesirable out-puts) that they generate in their production pro-cesses, either because it would require a greater use of inputs or because resources would have to be diverted from marketable production in order to eliminate undesirable outputs. This condition can be incorporated to the characterisation of the tech-nology by means of the axiom of weak disposal of outputs, in the sense that ifu3P(x), it will also hold
that hu3P(x), being in this case 0)h)1. This
produc-tion, that"rms would have to incur to ful"l envir-onmental regulations imposing a reduction of in-dustrial wastes, and allows for these undesirable outputs to have nonpositive shadow prices.
Following FGLY's approach, under certain assumptions, the shadow prices we seek can be obtained from the gradient vector of partial deriva-tives of an output distance function. The formal reasoning starts recognising the existence of a dual-ity between the revenue function R(x,r) and the output distance functionD
0(x,u). Denoting byrthe
output price vector, some of which components can be negative, the revenue function can be expressed as
R(x,r)"sup
u
Mru:D0(x,u))1N, (2)
and the dual distance function in outputs is given by
D
0(x,u)"sup
r
Mru:R(x,r))1N, (3)
beingruthe inner product of the output prices and quantity vectors.
Then, assuming that the revenue and distance functions are both di!erentiable, aLagrange prob-lem can be set up to maximise revenue, and "rst-order conditions yield the relationship [8]:
r"R(x,r)+
uD0(x,u), (4)
where+is the gradient operator.
Expression (3) can be developed as a relationship between the distance function and the shadow prices, so that
D0(x,u)"rH(x,u)u, (5)
where rH(x,u) represents the output price vector that maximises revenue. ApplyingShephard's dual lemmato expression (5), yields
+
uD0(x,u)"rH(x,u), (6)
which combined with (4), leads to
r"R(x,r)rH(x,u). (7)
In expression (7),rH(x,u) are obtained from the gradient of the distance function, and denote rev-enue-de#ated output prices. The main di$culty to obtain absolute shadow prices from expression (7) is the dependence of the revenue functionR(x,r) on r, that is precisely the vector of shadow prices we seek. If we assume that the observed price of an output m, r0
m, equals its absolute shadow price,
represented byr
m, then the maximum revenue is
R(x,r0
m)"r0m/rHm(x,u), (8)
which can be used to compute the absolute shadow prices of the remaining outputs from its de#ated shadow prices. Denoting by r
m{ the absolute
shadow prices for outputs other thanm, we get
r
An alternative way to handle the problem above is to suppose that a zero-pro"t and revenue-maxi-mising"rm would incur in an observed cost equiv-alent to its observed revenueR(x,r) [8], and simply use expression (7) to obtain absolute shadow prices. Finally, the distance function has to be para-meterised and estimated to proceed with the calcu-lation of the absolute shadow prices along the lines we have showed.
After absolute shadow prices have been com-puted, we make use of them to formulate an
extended productivity indexallowing for the consid-eration of di!erent waste production intensity among"rms. Let us make a partition of the quanti-ty and output price vectors, so that u"(u
a,ub)
M) are the quantity vectors of
de-sirable and undede-sirable outputs, respectively, while r
a"(r1,2,r
J) and rb"(rJ`1,2,r
M) are the
price vectors also for good and bad outputs. Considering only the production of desirable outputs, the partial productivity index for input x
Table 1
Sample description
Variable Description Units Mean Standard deviation Maximum Minimum
u
1 Ceramic pavements m2 2,031,077 1,168,311 4,500,000 200,000
u
2 Watery muds t 2,908 4,108 15,648 14
u
3 Used oil kg 1,822 3,135 12,000 100
x
1 Clay, kaolin, felspar and
limestones
t 50,192 40,762 144,000 3,300
x
2 Labour Number of workers 128 121 428 25
x
3 Capital kw/h 4,573 4,705 20,000 500
r0u1 Observed price Euros/m2 6.76 3.74 15.91 2.80
can be denoted as
PI"raua
x
j
. (10)
However, the estimation of shadow prices for undesirable outputs allows to de"ne an extended index of partial productivity for input x
j:
Given that shadow prices for bad outputs are nonpositive, the extended productivity index pro-posed by expression (11) will always take values equal or smaller than the traditional productivity index of expression (10); this allows for calculating theproductivity deviation indexas
PDI"PI
EPI"
r
aua
ru . (12)
Expression (12) will be equal or greater than one and measures the degree of overvaluation in quan-tifyinginputx
j productivity levels when only
mar-ketable output (good output) is being considered with disregard of those wastes that arise as a by-productand have potential harmful e!ects on the environment.
3. Data and empirical5ndings
The sample used in this paper comes from a cross-section data set of 18 Spanishceramic pav e-mentsproducers located at Castellon, on the
Span-ish Mediterranean coast. The source of the data is theValencian Region Inventory of Industrial Resid-ualselaborated in 1995 by theDepartment of Env i-ronmentof the Valencian Regional Government. All "rms face the same productive process, which is characterised by the production of one desirable output,ceramic pavements(u
1), and two wastes or
undesirable outputs,watery muds(u
2) andused oil
(u
3). Intermediate input isclay, kaolin, felspar and
limestones(x1), whilelabour(x2) andcapital(x3) are primary inputs. Labour input is measured as the number of workers, and capital is proxied by en-ergy consumption in kilowatts/hour. Table 1 pres-ents some descriptive statistics of the data.
As in FGLY, the distance function in output is parameterised as atranslogfunction, which is given by the expression
Table 2
Estimated parameters of the translog distance function. Expres-sion (13)
Thetranslogfunction is a#exible functional form that does not impose strong disposability of out-puts; however, given the large number of para-meters in relation to the size of our sample it is not possible to estimate using econometric methods. Alternatively, the parameters in expression (13) are obtained using mathematical programming tech-niques [9], by solving the following optimisation program (see again FGLY):
Max18+
wherekdenotes"rms.
The set of restrictions in (14a) imply that each observation is located either on or below the tech-nological frontier; the restrictions contained in (14b) guarantee that the desirable output will have a nonnegative shadow price for all"rms, while (14c) assures that undesirable outputs will have nonposi-tive shadow prices, also for all"rms. The assump-tion of weak disposal of outputs is introduced by restriction (14d) that imposes homogeneity of degree#1 in outputs;"nally (14e) and (14f) impose symmetry.
The objective function in (14) minimises the sum of the deviations of individual observations from
the frontier. However, we are in fact maximising because the distance function takes positive values lower or equal than one, and therefore its log can take negative or zero values; in consequence, to maximise the deviations of the distances expressed in logs from zero (represented by the log of one) is equivalent to minimise the sum of the absolute deviations of the individual observations.
Table 2 shows the parameter estimates of the
translog function given by expression (13). Using those "gures and the available information, the output distance function for each individual"rm is computed, with an average value of 0.927. It is well known that the output distance function is the reciprocal of Farrell's measure of output e$ciency whose average value is then 1.079. This means that making an e$cient use of their available resources, the"rms of the sample would be able to increase their production of ceramic pavements almost by 8% on average.
The estimation of the distance function allows us to obtain output shadow prices for each"rm in our sample, as we explained above. In order to get the
shadow income of each productive unit, we use expression (13) under the hypothesis that the shadow price ofgoodoutput (ceramic pavements) is equal to its observed market price; this is equivalent to assume thatr0
Table 3
Output shadow prices!
r0u1 r
u2 ru3
Mean 6.76 !9,830.6 !1,043.7
Standard deviation 3.74 23,345.9 2,512.8
Weighted mean 6.84 !336.6 !125.5
Maximum 15.91 !79,893.5 !9,998.6
Minimum 2.81 0.0 0.0
!Euros.
3Weights are the quantity of wastes produced by each"rm; they are chosen to obtain a weighted mean of shadow prices for bad outputs that represents the cost of reducing in one unit its production in the sample.
of pavement market price, a"gure which is di!erent for each company. Table 3 shows the computations for output shadow prices; simple means and stan-dard deviations, as well as weighted means3 are reported.
The shadow prices of the industrial wastes could be interpreted as the marginal loss of revenue corre-sponding to the volume of resources that a "rm would need to reduce its emission by a marginal unit. Therefore, in average terms, a reduction of one ton of watery mud production, residual u
2, would
require the use of resources valued in 336.6 euros, which in terms of marketable output would imply an implicit loss of 49.2 square meters of pavements, considering that the market price for square meter is 6.84 euros on average. Similarly, and in order to reduce the production of used oil in a kilogram, residualu
3, the opportunity cost would be valued
in 125.5 euros, equivalent to a reduction in good output production of 18.3 square meters.
Shadow price estimates di!er signi"cantly among "rms, as the high standard deviation of those prices reveals; this result is consistent with FGLY, and it is also related to important di!er-ences among"rms in terms of quantities of resid-uals discharged per unit of desirable output produced. After shadow prices at "rm level have been obtained, the correlation between absolute values of these prices and waste emission for square meter of ceramic pavements has been computed,
getting a negative value close to 30% in the case of
watery mud and around 25% for used oil. The correlation coe$cient has the expected sign since companies that produce a greater volume of resid-uals per unit of marketable production are very likely to be those that rely on technical equipment less adapted to recycling those residuals or minim-ising their delivery; for these "rms, investments intended to cut the volume of e%uents would have a relatively small cost in comparison with their prospective yields in terms of reduction of emis-sions. On the other hand, those "rms that have already reached a high performance according to their ability to control the environmental impact of their production processes would face a higher marginal cost in case of stepping-up their e!orts to reduce waste discharges. This would be re#ected in a higher shadow price for bads in these companies.
Table 4
Labour productivity measures!
Firm Labour
1 57.81 44.33 1.304
2 202.81 179.74 1.128
3 120.65 109.12 1.106
4 79.30 65.71 1.207
5 96.66 91.72 1.054
6 91.07 80.38 1.133
7 64.96 62.49 1.040
8 97.02 84.59 1.147
9 127.83 124.01 1.031
10 77.38 73.64 1.051
11 127.91 114.24 1.120
12 194.59 167.23 1.164
13 127.83 118.42 1.079
14 121.29 105.28 1.152
15 115.48 94.10 1.227
16 105.18 92.58 1.136
17 126.67 108.74 1.165
18 166.61 159.20 1.047
Mean 116.73 104.20 1.127
Standard deviation 39.97 36.41 0.072
!Euros per worker. Finally, it seems right to remark that knowledge
on shadow prices allows the researcher to approach a valuation of industrial production not con"ned to marketable production and market prices, but able to include an estimate of the (negative) value of a likely increase in industrial refuses, linked to the expansion of good output levels. The type of shadow prices computed in this paper are derived strictly from technical relations between inputs and outputs and should be considered as estimates arising from a producer approach. Externalities on consumers or other producers are not taken into account, but could be incorporated in a straightfor-ward way, provided adequate data are made avail-able, within the same framework of analysis, as shown in FaKre and Grosskopf [10]. Despite their shortcomings, the shadow prices we have used illustrate well enough the e!ects on production valuation of including in the price vector several components with a negative sign, each correspond-ing to a di!erent undesirable output.
We now proceed to compute two di!erent measures of labour productivity, one along the conventional lines, as a quotient between the mon-etary value of each"rm marketable output (ceram-ics pavements) and labour sta!, and an extended
one, de"ned by expression (11). The outcome of using both types of productivity indexes appear in Table 4. Both productivity measures di!er by values that go from 3% to 30% in the sample, with an averagegapclose to 12%. It means that di!er-ences are important enough to make an impact on "rms comparisons. As an example, "rm 8 is more productive in conventional terms than"rm 5, but it changes when an environmental-friendly approach is taken and waste production from both"rms are included in the numerator of the productivity quo-tient, using shadow prices to translate residuals quantities to monetary"gures. Now"rm 5 is more
productive than "rm 8. Ranking between "rms 4 and 10 is also inverted when moving from a pro-ductivity measure to the other, and two "rms (11 and 13), that have quite similar labour produc-tivity levels when only good output is compared, show marked di!erences when badsare also con-templated.
From society's comprehensive view it is clearly a serious matter of concern. It is not possible to
remain aloof to the fact that a given productivity level can be achieved with very di!erent volumes of real or potential noxious wastes, and a virtue of the very simple indicator we propose is to ease the transition to a new way of analysing entrepreneur-ial performance, apt enough to accueil a growing public interest in environmental values. Firms that have undertook costly investments in new equip-ments that allow cleaner production processes should have their e!orts (that consume productive resources), recognised and not merely dismissed as
less productivein conventional terms.
The next step we have followed is to disclose if there exists a relationship between our deviation productivity index and some "rm's characteristics, making use of variance analysis. We have broken our sample in two groups;groupA includes those "rms with a deviation smaller than the average deviation for the whole sample (8 "rms), while
remain-4The Region of Valencia has a network of Technological Institutes specialised by branches of industrial production, AICE being the one concerned with the ceramics industry. They are non-pro"t entities under the form of business associations sponsored and mostly funded by the public authorities and oriented to promote technological innovation, best practice dif-fusion and quality tests. They provide external services (R&D, among others) at low cost to their a$liates, that are mostly small and medium sized"rms. A$liation is normally considered as an index of innovative behaviour on the part of"rm's management.
5The ceramics pavements industry is highly concentrated in a small number of contiguous municipalities in the CastelloHn Province, where"rms are supposed to enjoy important external economies that could be broadly de"ning a Marshallian type industrial district in the sense coined by the economic geography literature. We hypothesize that close proximity improves the chances of"rms being more sensible to good practice in terms of waste management, trough imitation and easier technical in-formation di!usion.
ing"rms). Then we have tried to ascertain if there is a statistically signi"cant di!erence between both subsamples concerning to"ve variables:size,az li-ation to a Technological Institute,4spatial location,5
a record of past investments in cleaner technologies, anduse of external services for waste management.
Results are shown in Table 5 and are clearly conclusive for all the variables under consideration. Firms of greater size show a smaller deviation and the same happens for "rms that have recorded investments conducive to clean production pro-cesses. External management of wastes is mainly associated to the"rms with higher bias; this was an expected outcome, given that those"rms are more intensive in terms of production of industrial wastes, what forces them into a greater dependence of external suppliers of services (transport, storage) for waste disposal. A$liation to a Technological Institutespecialised in the ceramics industry (AICE) helps to reduce the deviation, as variance analysis shows. This"nding is probably linked to the servi-ces provided by theInstitute in terms of technolo-gical consultancy, easing the access to R&D in di!erent"elds, including the development and ap-plication of waste reducing techniques, and favour-ing di!usion of industry's best practice. The variablespatial locationis employed to distinguish the"rms that are located in one area of very dense concentration of this type of industry (the Plana Baixa zone) and the others. We presume that the
"rst enjoy an advantage in terms of sharing the external economies generated by a dynamic indus-trial district, that gives them rapid access to in-formation released by suppliers of new industrial equipments and competitors and facilitates the acquisition of better practices in dealing with industrial residuals. Evidence, after performing variance analysis, conforms to it.
4. Concluding remarks
This paper has been devoted to estimate shadow prices for two di!erent types of undesirable outputs or industrial wastes generated in their production processes by eighteen Spanish"rms in the ceramic pavements industry. The methodology follows the approach suggested by FaKre et al. [1]. Accordingly, central importance has been given to the duality between output distance and revenue func-tions, using the former to derive output shadow prices.
The shadow prices obtained forwatery mudsand
used oil (industrial wastes that areby-products in the production of marketable ceramic pavements) have made it feasible to measure in terms of loss of marketable output production (square meters of ceramic pavements) or its equivalent in cash rev-enue, the opportunity costs "rms would have to incur in order to achieve a marginal decrease in the production of those refuses. A negative correlation has been observed between waste production inten-sity by"rms (as a proportion of their own output of ceramic pavements) and absolute shadow prices computed for thesebadsoutputs. This feature prob-ably re#ects a greater marginal cost of polluting residues elimination for those "rms that had pre-viously invested in cleaner technologies directed to get rid of undesirable outputs, and are currently generating less units of wastes for square meter of ceramic pavements produced.
Table 5
Variance analysis
Firm's size# A$liation to Technological Institute$
Spatial location%
Investments on cleaner technologies&
External management of wasteu
2'
Mean group A! 2,966.2 0.63 0.75 1.00 0.13
Mean group B" 1,283.0 0.10 0.30 0.60 0.90
StatisticF 18.983 7.063 4.000 4.740 24.063
p-value 0.000 0.017 0.063 0.045 0.000
!Deviation smaller than average deviation for the whole sample; 8"rms.
"Deviation greater or equal average deviation for the whole sample; 10"rms.
#Thousand of square meters of ceramic pavements.
$Dummy variable that equals to 1 if the"rm is a$liated to Technological Institute AICE and 0 otherwise.
%Dummy equal to 1 if the"rm is located at the industrial district of Plana Baixa and 0 otherwise.
&Dummy equal to 1 if the"rm has invested on clean technologies and 0 otherwise.
'Dummy that equals to 1 if external management of wasteu
2is carried out and 0 otherwise.
since for a similar marginal social bene"t obtained from a cut in current pollution levels across sample "rms, the marginal costs that those "rms would have to face would be quite di!erent.
Finally we wish to use shadow prices to emphasise on the need to improve current productivity indexes in order to take industrial waste into consideration. Our results show that clear di!erences arise within the sample when com-puting conventional and extended measures of la-bour productivity at "rm level. These di!erences are statistically associated to "rms characteristics, such assize, record offormer investments in cleaner technologies, andazliation to a Technological Insti-tute for the Ceramics Industry.
Acknowledgements
We thank the Bureau of Environmental Quality of the Valencian Regional Government for statist-ical help, as well as two anonymous referees for useful comments. We are also grateful tothe Minis-try of Science and Technology of the Spanish Gov-ernment for the Financial aid received (Project SEC 2000-0803).
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