- Email: [email protected]
Value and quantity data in economic and technical efficiency measurement
Economics Letters 124 (2014) 108–112
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Value and quantity data in economic and technical efficiency measurement Maria Conceição A. Silva Portela ∗ CEGE - Centro de Estudos em Gestão e Economia da Universidade Católica Portuguesa, Centro Regional do Porto, Rua Diogo Botelho, 1327, 4169-005 Porto, Portugal
highlights • • • • •
The use of value data in efficiency models versus quantity data is addressed. Value data issues are analysed for equal and differing prices across DMUs. Guidelines are proposed regarding the type of models to be used under both cases. Cost efficiency models should be preferred when data are measured in value. Cost efficiency under value data implicitly assumes that prices are discretionary.
article
info
Article history: Received 11 November 2013 Received in revised form 22 April 2014 Accepted 23 April 2014 Available online 10 May 2014
abstract This paper calls attention to the implications of using value data in efficiency measurement through Data Envelopment Analysis (DEA). The main contributions are twofold: (i) it provides a reconciliation of the previous literature on analysing issues of quantity and value data in efficiency measurement, (ii) it provides some guidelines on what to do, when these issues arise in a data set. © 2014 Elsevier B.V. All rights reserved.
MSC: 00-01 99-00 Keywords: Data Envelopment Analysis Technical efficiency Cost efficiency
1. Introduction The main motivation to this paper stems from the fact that insufficient attention has been paid for in the efficiency literature to the implications of using cost or revenue data in technical and economic efficiency assessments. Technical or productive efficiency of a firm ‘‘means its success in producing as large as possible an output from a given set of inputs’’ Farrell (1957, p. 254). Clearly one can also define productive efficiency as the success of firms in using as small as possible inputs to produce a given set of outputs, and this distinguishes a perspective of output expansion from a perspective of input contraction. When factor prices are available, and taken into account in the measurement
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Tel.: +351 226196200. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.econlet.2014.04.023 0165-1765/© 2014 Elsevier B.V. All rights reserved.
of efficiency, the first perspective gives rise to revenue efficiency measurement, and the second perspective gives rise to cost efficiency measurement. In many empirical settings input and/or output quantities and prices are not available as separate variables, and only value measures are available (Cross and Fare, 2009). A typical example found in many empirical applications is the use of capital value to measure the input capital. Since capital is in fact constituted by many sub-items, like buildings, vehicles, equipments, etc., for which real prices are not available or meaningful, it is usual to use some accounting procedure to measure the value of capital of a firm. When value data are used in efficiency models it is questionable what sort of efficiency measure is being computed, since it cannot be a productive efficiency measure (that takes into account only quantities of inputs and outputs) and it cannot be a traditional cost or revenue efficiency measure (that considers that both prices and quantities of all inputs and outputs are available). As stated in
M.C.A. Silva Portela / Economics Letters 124 (2014) 108–112
Banker et al. (2007) ‘‘since aggregate cost or revenue data reflect both quantities and prices, it is not apparent what is measured by the DEA technical efficiency model when such data are used’’ (p. 115). Or, put it another way, ‘‘how does the value-based DEA model relate to the quantity based DEA models? . . . if they do not coincide, then what exactly does the value-based DEA model measure and how do we interpret the difference?’’ Cross and Fare (2008). In this paper we address situations where only value data are available (and distinguish in this case two possible situations: (i) prices are the same across production units, and (ii) prices differ across production units), and situations where for some factors there are quantity data and for others there are value data. The paper contributes to the literature in two ways: (i) by providing a reconciliation of the previous literature on analysing the use of value data in DEA models; and (ii) by providing some guidelines on what to do and not to do, when value data arise in a data set. Some of these guidelines are not new to the literature, but two of them are presented here for the first time.
109
model (3), where the quantities of each input i are replaced by the costs of each input i (Cij = wi xij ).
n min θ λj Cij ≤ θ Cio , i = 1, . . . , m, λj ,θ j =1
n
λj yrj ≥ yro , r = 1, . . . , s, λj ≥ 0 .
The equivalence between models (2) and (3) easily proved, as is n the constraints for each input (i = 1, . . . , m) j=1 λj Cij ≤ θ Cio ⇔
j=1 λj xij ≤ θ xio . Similarly model (4) is equivalent to (1), when prices for each input i are the same across DMUs, being the proof similar to the above.
n
j=1
λj wi xij ≤ θ wi xio , which in turn equals
Consider for each Decision Making Unit (DMU) j (j = 1, . . . , n) a vector xj = (x1j , . . . , xmj ) reflecting m inputs consumed for producing a vector of s outputs yj = (y1j , . . . , ysj ), where prices of inputs are given by a vector wj = (w1j , . . . , wmj ). A cost setting will be analysed in this paper, but the extension of the concepts to a revenue setting is straightforward. The traditional cost efficiency model for DMUo is the solution of the linear program (1), where input quantities, xi , and the intensity variables, λj are the decision variables.
m
min λj ,xi
n
i=1
n wio xi λj xij ≤ xi , i = 1, . . . , m, j =1
λj yrj ≥ yro , r = 1, . . . , s, λj , xi ≥ 0 .
(1)
j =1
From the optimal solution to (1) cost efficiency is computed as the ratio between optimal cost and observed cost (C ∗ /Co ). The technical efficiency for DMUo is obtained from the solution of model (2) (see e.g. Charnes et al., 1978), where constant returns to scale (CRS) are assumed.
n min θ λj xij ≤ θ xio , i = 1, . . . , m, λj ,θ
n
m n min Ci λj Cij ≤ Ci , i = 1, . . . , m, λj ,Ci i=1
j =1
λj yrj ≥ yro , r = 1, . . . , s, λj , Ci ≥ 0 .
(4)
j =1
The above leads us to our first proposition: Proposition 1. When prices are equal across DMUs, cost efficiency and technical efficiency can be computed using cost data rather than quantity and price data. This means that, when prices are equal across units, in fact price information is not required to compute cost efficiency and decompose it into technical and allocative components. This proposition is not new to the literature. In particular Fare et al. (1990) addressed the equivalence between technical efficiency models based on quantity and value data in a profit setting, and Banker et al. (2007) addressed this equivalence in a revenue setting (see also Cross and Fare, 2008). Not addressed in the literature, to the author’s knowledge, is the equivalence between model (4) and a model mwhere input costs are aggregated into a single input (Cj = i=1 Cij ), where Cij > 0 ∀i and j. That is, (4) is equivalent to (5) with a single aggregated input cost (where no component of total cost can be zero, as otherwise total costs are not comparable across DMUs):
n n min C λ j Cj ≤ C , λj yrj ≥ yro , r = 1, . . . , s, λj , C ≥ 0 . λj ,C j =1
j=1
(5) Proof. The optimal solution to model (5) yields variable C > 0, and the optimal solution to model (4) yields Ci > 0, ∀i (otherwise it would be possible to produce positive output with some zero costs). Consider now the dual of model (4) in (6).
j =1
λj yrj ≥ yro , i = r , . . . , s, λj ≥ 0 .
n
n
2. Using value data under equal and different prices across DMUs
(3)
j =1
(2)
j =1
The product of technical efficiency and allocative efficiency is equal to the measure of cost efficiency, and therefore allocative efficiency can be retrieved residually, after solving models (1) and (2). The above models assume that factor prices are known for each input and that these may be different across DMUs, which are considered price takers. In what follows we will assume that prices are not known, and therefore disaggregated price and quantity data are not available, but just value data on inputs are available. 2.1. Equal prices across DMUs When prices are unknown there is the special case where prices are the same across all units (i.e. wij = wi ). Under this circumstance the technical efficiency model (2) is equivalent to
max v i ,u r
s r =1
m
ur yro −
vi Cij +
i=1
s
ur yrj ≤ 0,
r =1
j = 1, . . . , n, vi ≤ 1, i = 1, . . . , m .
(6)
Through complementary slackness conditions of linear programming the constraints vi ≤ 1 corresponding to the basic variables Ci are binding. Similarly the dual of model (5) is shown in (7), where through complementary slackness conditions of linear programming the constraint v ≤ 1 corresponding to the basic variable C is binding. max v,ur
s r =1
s ur yro −v Cj + ur yrj ≤ 0, j = 1, . . . , n, v ≤ 1 . (7) r =1
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M.C.A. Silva Portela / Economics Letters 124 (2014) 108–112
Since m the input weights vi in (6) are m1 at its optimal solution, the i=1 vi Cij in (6) can be written as i=1 Cij , which is equal to Cj , and since v Cj in (7) is equal to Cj (as v = 1 at the optimal solution of (7)) the duals of the two models are exactly the same. Note that this result is also valid for variable returns to scale. Model (5) has an important implication in terms of aggregation issues addressed e.g. in Fare and Zelenyuk (2002) and Fare and Grosskopf (2004). These authors were concerned with the use of the technical efficiency model (2) and were interested in ‘‘learning when the input-oriented measure of technical efficiency based on quantity data yields the same score when based on the cost aggregated data’’ (Fare and Grosskopf, 2004, p. 121). The authors conclude that aggregation is unbiased only when units are allocatively efficient and prices are the same across all units. To see this, note that if all inputs have been converted into a single aggregate input cost, then a technical efficiency score could be obtained from model (5), as C can be replaced by θ Co without changing the optimal solution to this model. However, model (5) is a cost model and not a technical efficiency model, which means that it would only be equal to a technical efficiency model when there is no allocative inefficiency. Therefore we provide a new result to the aggregation issue: Proposition 1.1. Aggregation of input costs into a single cost value, provides unbiased cost efficiency results, but no technical efficiency can be computed through a single aggregate cost value. That is, to measure technical efficiency in the presence of value data one needs to use disaggregate cost data in a radial model like (3), and assume that prices are the same across DMUs. Cost efficiency, on the other hand, can be computed through aggregate cost data, and the assumption of equal prices does not need to be maintained. This issue will be further addressed in the next sections. 2.2. Different prices across DMUs In this section we assume that prices are different across DMUs and unknown, and only value data are available for each input. Little attention has been paid for in the literature to this situation, particularly to the fact that technical efficiency lacks a solid interpretation under such circumstances. Note, however, that technical efficiency can be computed directly from value data when firms are weakly rational and relative prices are equal across firms, as shown by Cross and Fare (2009). Cross and Fare (2008) also investigated value based efficiency measurement and put forward an identity stating that the value-based technical efficiency (or cost efficiency, as we call it) can be decomposed, multiplicatively, into the quantity-based technical efficiency score (or simply technical efficiency), a technology effect, and a firm effect. Following from this identity, and Proposition 1 we put forward Proposition 2: Proposition 2. When prices are unknown and different across DMUs, cost efficiency can be computed using value cost data, but technical efficiency cannot be computed from cost data. When this is the case cost efficiency can be measured through model (4) (or indeed model (5)), but model (3) should not be used to compute technical efficiency because it implicitly assumes that trade-offs between costs exist, which is an argument difficult to sustain. We explain the reasons for this argument through the illustrative example shown in Fig. 1, where 3 units producing the same level of output use different combinations of costs of two inputs (capital and labour).
Fig. 1. Illustrative example.
Technical efficiency computed through model (3) identifies the production possibilities frontier as the line joining units A and B and its extensions to the axes. The two frontier units would be ‘technically’ efficient despite the fact that they use different strategies (labour intensive for B and capital intensive for A) to produce the same level of output. However, since the production factors are expressed in costs, we are implying that producing at a point with high costs in capital is equally efficient as producing at a point with high costs in labour. This is clearly not a sensible assumption when the DMU with high capital costs incurs in almost the double total costs than the DMU with high labour costs. As a result, trade-offs between the costs of production factors do not exist when prices differ across DMUs (if prices are the same across DMUs, cost trade-offs indeed reflect trade-offs between quantities). To further elaborate our argument consider that the disaggregate data for this example are known and shown in Table 1. In this table we can see that units A and B use exactly the same amount of capital, and B is technically inefficient when compared to the technically efficient units A and C. Therefore the tradeoff observed in Fig. 1 between the costs of A and B is not really an effect of quantities, but an effect of prices. As a result, in the absence of disaggregated information, the efficiency of these units should be computed through models (4) or (5), which yield a cost minimising frontier at Min(C1 +C2 )—the straight black line in Fig. 1. Results from model (4) show unit B as 100% cost efficient and units A and C as cost inefficient (cost efficiency of 57.04% and 58.31%, respectively). Cost efficiency shall be interpreted as the proportion of the costs of a DMU in relation to the minimum cost observed to produce a certain level of outputs. Note that in addition to the above, the example in Table 1 also shows that the cost model (assumed as the best model to be applied in this case) yields cost efficiency scores that may be quite far from the ‘true cost efficiency’ (that obtained when disaggregated data are available). The cost model, based on value data, implicitly assumes that minimum observed cost for a certain level of produced output is achievable irrespective of technical efficiency, which cannot be known when quantities are not known (e.g. in the illustrative example unit B is the cost minimising unit, but it is not technically efficient). However, in reality the minimum cost may be unachievable if prices cannot be changed. Take the example of DMU C, which is technically and cost efficient (for prevailing prices) when one uses the disaggregated data on quantities and prices in models (1) and (2). This means that the cost efficiency of 58.31% identified for DMU C is a result of excessive prices paid for inputs. Therefore the minimum cost target for DMU C is only achievable if this unit can to some extent influence the prices it pays for labour and capital. This result has lead us to Proposition 2.1:
M.C.A. Silva Portela / Economics Letters 124 (2014) 108–112
111
Table 1 Hypothetical disaggregate data for illustrative example. DMU
Capital
Labour
Price of capital
Price of labour
Cost of capital
Cost of labour
Total cost
A B C
40 40 30
10 16 15
500 200 500
300 320 500
20 000 8 000 15 000
3000 5120 7500
23 000 13 120 22 500
Proposition 2.1. In the absence of price information, and differing prices across DMUs, minimum cost is only achievable when prices can be controlled/changed to some extent, and therefore DMUs cannot be considered price takers. Note this result is addressed here for the first time, but the relaxation of the price-taking assumption of DMUs has been addressed by other authors. (see e.g Portela and Thanassoulis, 2014). 3. Mixing value and quantity data in DEA models Often in empirical applications we encounter some inputs defined on value terms and others defined on quantity terms (e.g. N. of workers and value of capital). We consider a subset of inputs i = 1, . . . , k expressed in quantity, and a subset of i = k + 1, . . . , m expressed in value. 3.1. Equal prices across DMUs for value data As seen in the previous section, when prices are the same across units it is equivalent to use value or quantity data, and therefore the mix of both types of data does not pose a problem in the estimation of efficiencies. However, when only some input costs are known, one cannot compute a cost efficiency model for the overall set of inputs, but just for a sub-set. Such models are known as sub-vector cost efficiency and shown in (8) (see also Fare and Grosskopf, 2004).
min λj ,Ci
m
i=k+1
n λj xij ≤ xio , i = 1, . . . , k, Ci j =1
n
λj Cij ≤ Ci , i = k + 1, . . . , m,
j =1 n
λj yrj ≥ yro , r = 1, . . . , s, λj ≥ 0 .
(8)
j =1
Note that model (8) could equivalently be solved through a model where the i constraints (i = k + 1, . . . , m) have been replaced by a single aggregate cost constraint (see Section 2.1). For decomposing the sub-vector cost efficiency into its technical and allocative components, the sub-vector technical efficiency model shown in (9) should be solved, where only a subset of inputs is contracted towards the efficient frontier, in consistency with (8).
n min θ λj xij ≤ xio , i = 1, . . . , k, λj ,θ
j =1
n
λj Cij ≤ θ Cio , i = k + 1, . . . , m,
from the ratio between minimum cost from (8) and observed cost) and the sub-vector technical efficiency (obtained from (9)). It is important to note that model (8) is in many respects different from model (4) (applicable when all inputs are expressed in value), as minimum cost in (8) depends not only on produced outputs, but also on the input levels of the inputs expressed in quantity. This means that the unit showing the lowest quantity on a certain input (i = 1, . . . , k) is sub-vector-cost efficient irrespective of the cost observed for the inputs expressed in value. This is a sensible assumption, since lowest input quantities would also imply lowest costs (if prices were to be known) for those inputs (under similar prices across DMUs). 3.2. Different prices across DMUs for value data When prices are different across DMUs one can invoke the same arguments as in Section 2.2 to disregard the computation of sub-vector technical efficiency through (9). Indeed, such model assumes not only trade-offs between costs but also between input costs and input quantities. Such trade-offs are only sensible in the situation where production units face the same prices and costs are a good surrogate for quantities. Regarding the computation of sub-vector cost efficiency the arguments of Section 2.2 are still valid. That is, one can compute sub-vector cost efficiency, but needs to be aware that sub-vector cost minimising units are not necessarily sub-vector technically efficient, or that sub-vector technically efficient units may be subvector cost inefficient (note however that one is assuming subvector technical efficiency cannot be known). Therefore, the use of model (8) implicitly assumes that DMUs are not price takers and minimum cost may be obtained both from changing quantities and prices. In addition, there is a further assumption invoked in this case: the assumption that cost minimising units are those producing the highest level of outputs and using the lowest level of input quantities at the minimum costs. Lower level of input quantities may not be associated with lower cost levels, when prices are not constant across DMUs, and therefore this assumption may be problematic. However, if the input quantities cannot in any way be expressed in value form due to the complete absence of prices (e.g. input quantities may be related to environmental or contextual variables such as density of population, and number of years of education, which cannot be expressed in value) then the computation of sub-vector cost efficiency is valid. These arguments, and the results in Fare and Grosskopf (2004) led us to our final proposition. Proposition 3. When there is cost data for some inputs and quantity data for others, sub-vector technical and sub-vector cost efficiency should be computed when prices are assumed the same across DMUs, and no measure of sub-vector technical efficiency should be computed when prices differ across DMUs. Sub-vector cost efficiency can be computed under this setting but the assumptions behind this model should be verified.
j =1 n
4. Conclusion
λj yrj ≥ yro , r = 1, . . . , s, λj ≥ 0 .
(9)
j =1
Sub-vector allocative efficiency can be computed residually through the ratio between the sub-vector cost efficiency (obtained
The aim of this paper is to call the attention for some underlying assumptions that are being invoked when value data are used instead of quantity data in efficiency assessments, particularly those undertaken through DEA models. We analyse situations
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M.C.A. Silva Portela / Economics Letters 124 (2014) 108–112
where all inputs or all outputs are taken in value form and situations where some inputs are expressed in value, whereas others are expressed in quantity. For the former situation, all cost efficiency models can be applied and decomposed into technical and allocative efficiency provided that prices are assumed equal across DMUs. When this is not the case, only cost efficiency can be computed and no decomposition is possible, as there is not a way to compute technical efficiency. For the latter situation we conclude that when prices are assumed equal across DMUs sub-vector cost and technical efficiency measures can be used, and when this is not the case sub-vector technical efficiency should not be used. Acknowledgement The author acknowledges the financial support of the Portuguese Foundation for Science and Technology (FCT) through project PTDC/EGE-GES/112232/2009.
References Banker, R., Chang, H., Natarajan, R., 2007. Estimating dea technical and allocative inefficiency using aggregate cost or revenue data. J. Productivity Anal. 27, 115–121. Charnes, A., Cooper, W.W., Rhodes, E., 1978. Measuring efficiency of decision making units. European J. Oper. Res. 2, 429–444. Cross, R., Fare, R., 2008. Farrell efficiency under value and quantity data. J. Productivity Anal. 29, 193–199. Cross, R., Fare, R., 2009. Value data and the bennet price and quantity indicators. Econom. Lett. 102, 19–21. Fare, R., Grosskopf, S., 2004. New Directions: Efficiency and Productivity. Kluwer Academic Publishers, USA. Fare, R., Grosskopf, S., Lee, H., 1990. A nonparametric approach to expenditure constrained profit maximization. Am. J. Agric. Econ. 12 (3), 574–581. Fare, R., Zelenyuk, V., 2002. Input aggregation and technical efficiency. Appl. Econ. Lett. 9, 635–636. Farrell, M.J., 1957. The measurement of productive efficiency. J. R. Stat. Soc., Ser. A (Gen.) 120 (Part 3), 253–281. Portela, M., Thanassoulis, E., 2014. Economic efficiency when prices are not fixed: disentangling quantity and price efficiency. Omega 47, 36–44.
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Value and quantity data in economic and technical efficiency measurement Maria Conceição A. Silva Portela ∗ CEGE - Centro de Estudos em Gestão e Economia da Universidade Católica Portuguesa, Centro Regional do Porto, Rua Diogo Botelho, 1327, 4169-005 Porto, Portugal
highlights • • • • •
The use of value data in efficiency models versus quantity data is addressed. Value data issues are analysed for equal and differing prices across DMUs. Guidelines are proposed regarding the type of models to be used under both cases. Cost efficiency models should be preferred when data are measured in value. Cost efficiency under value data implicitly assumes that prices are discretionary.
article
info
Article history: Received 11 November 2013 Received in revised form 22 April 2014 Accepted 23 April 2014 Available online 10 May 2014
abstract This paper calls attention to the implications of using value data in efficiency measurement through Data Envelopment Analysis (DEA). The main contributions are twofold: (i) it provides a reconciliation of the previous literature on analysing issues of quantity and value data in efficiency measurement, (ii) it provides some guidelines on what to do, when these issues arise in a data set. © 2014 Elsevier B.V. All rights reserved.
MSC: 00-01 99-00 Keywords: Data Envelopment Analysis Technical efficiency Cost efficiency
1. Introduction The main motivation to this paper stems from the fact that insufficient attention has been paid for in the efficiency literature to the implications of using cost or revenue data in technical and economic efficiency assessments. Technical or productive efficiency of a firm ‘‘means its success in producing as large as possible an output from a given set of inputs’’ Farrell (1957, p. 254). Clearly one can also define productive efficiency as the success of firms in using as small as possible inputs to produce a given set of outputs, and this distinguishes a perspective of output expansion from a perspective of input contraction. When factor prices are available, and taken into account in the measurement
∗
Tel.: +351 226196200. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.econlet.2014.04.023 0165-1765/© 2014 Elsevier B.V. All rights reserved.
of efficiency, the first perspective gives rise to revenue efficiency measurement, and the second perspective gives rise to cost efficiency measurement. In many empirical settings input and/or output quantities and prices are not available as separate variables, and only value measures are available (Cross and Fare, 2009). A typical example found in many empirical applications is the use of capital value to measure the input capital. Since capital is in fact constituted by many sub-items, like buildings, vehicles, equipments, etc., for which real prices are not available or meaningful, it is usual to use some accounting procedure to measure the value of capital of a firm. When value data are used in efficiency models it is questionable what sort of efficiency measure is being computed, since it cannot be a productive efficiency measure (that takes into account only quantities of inputs and outputs) and it cannot be a traditional cost or revenue efficiency measure (that considers that both prices and quantities of all inputs and outputs are available). As stated in
M.C.A. Silva Portela / Economics Letters 124 (2014) 108–112
Banker et al. (2007) ‘‘since aggregate cost or revenue data reflect both quantities and prices, it is not apparent what is measured by the DEA technical efficiency model when such data are used’’ (p. 115). Or, put it another way, ‘‘how does the value-based DEA model relate to the quantity based DEA models? . . . if they do not coincide, then what exactly does the value-based DEA model measure and how do we interpret the difference?’’ Cross and Fare (2008). In this paper we address situations where only value data are available (and distinguish in this case two possible situations: (i) prices are the same across production units, and (ii) prices differ across production units), and situations where for some factors there are quantity data and for others there are value data. The paper contributes to the literature in two ways: (i) by providing a reconciliation of the previous literature on analysing the use of value data in DEA models; and (ii) by providing some guidelines on what to do and not to do, when value data arise in a data set. Some of these guidelines are not new to the literature, but two of them are presented here for the first time.
109
model (3), where the quantities of each input i are replaced by the costs of each input i (Cij = wi xij ).
n min θ λj Cij ≤ θ Cio , i = 1, . . . , m, λj ,θ j =1
n
λj yrj ≥ yro , r = 1, . . . , s, λj ≥ 0 .
The equivalence between models (2) and (3) easily proved, as is n the constraints for each input (i = 1, . . . , m) j=1 λj Cij ≤ θ Cio ⇔
j=1 λj xij ≤ θ xio . Similarly model (4) is equivalent to (1), when prices for each input i are the same across DMUs, being the proof similar to the above.
n
j=1
λj wi xij ≤ θ wi xio , which in turn equals
Consider for each Decision Making Unit (DMU) j (j = 1, . . . , n) a vector xj = (x1j , . . . , xmj ) reflecting m inputs consumed for producing a vector of s outputs yj = (y1j , . . . , ysj ), where prices of inputs are given by a vector wj = (w1j , . . . , wmj ). A cost setting will be analysed in this paper, but the extension of the concepts to a revenue setting is straightforward. The traditional cost efficiency model for DMUo is the solution of the linear program (1), where input quantities, xi , and the intensity variables, λj are the decision variables.
m
min λj ,xi
n
i=1
n wio xi λj xij ≤ xi , i = 1, . . . , m, j =1
λj yrj ≥ yro , r = 1, . . . , s, λj , xi ≥ 0 .
(1)
j =1
From the optimal solution to (1) cost efficiency is computed as the ratio between optimal cost and observed cost (C ∗ /Co ). The technical efficiency for DMUo is obtained from the solution of model (2) (see e.g. Charnes et al., 1978), where constant returns to scale (CRS) are assumed.
n min θ λj xij ≤ θ xio , i = 1, . . . , m, λj ,θ
n
m n min Ci λj Cij ≤ Ci , i = 1, . . . , m, λj ,Ci i=1
j =1
λj yrj ≥ yro , r = 1, . . . , s, λj , Ci ≥ 0 .
(4)
j =1
The above leads us to our first proposition: Proposition 1. When prices are equal across DMUs, cost efficiency and technical efficiency can be computed using cost data rather than quantity and price data. This means that, when prices are equal across units, in fact price information is not required to compute cost efficiency and decompose it into technical and allocative components. This proposition is not new to the literature. In particular Fare et al. (1990) addressed the equivalence between technical efficiency models based on quantity and value data in a profit setting, and Banker et al. (2007) addressed this equivalence in a revenue setting (see also Cross and Fare, 2008). Not addressed in the literature, to the author’s knowledge, is the equivalence between model (4) and a model mwhere input costs are aggregated into a single input (Cj = i=1 Cij ), where Cij > 0 ∀i and j. That is, (4) is equivalent to (5) with a single aggregated input cost (where no component of total cost can be zero, as otherwise total costs are not comparable across DMUs):
n n min C λ j Cj ≤ C , λj yrj ≥ yro , r = 1, . . . , s, λj , C ≥ 0 . λj ,C j =1
j=1
(5) Proof. The optimal solution to model (5) yields variable C > 0, and the optimal solution to model (4) yields Ci > 0, ∀i (otherwise it would be possible to produce positive output with some zero costs). Consider now the dual of model (4) in (6).
j =1
λj yrj ≥ yro , i = r , . . . , s, λj ≥ 0 .
n
n
2. Using value data under equal and different prices across DMUs
(3)
j =1
(2)
j =1
The product of technical efficiency and allocative efficiency is equal to the measure of cost efficiency, and therefore allocative efficiency can be retrieved residually, after solving models (1) and (2). The above models assume that factor prices are known for each input and that these may be different across DMUs, which are considered price takers. In what follows we will assume that prices are not known, and therefore disaggregated price and quantity data are not available, but just value data on inputs are available. 2.1. Equal prices across DMUs When prices are unknown there is the special case where prices are the same across all units (i.e. wij = wi ). Under this circumstance the technical efficiency model (2) is equivalent to
max v i ,u r
s r =1
m
ur yro −
vi Cij +
i=1
s
ur yrj ≤ 0,
r =1
j = 1, . . . , n, vi ≤ 1, i = 1, . . . , m .
(6)
Through complementary slackness conditions of linear programming the constraints vi ≤ 1 corresponding to the basic variables Ci are binding. Similarly the dual of model (5) is shown in (7), where through complementary slackness conditions of linear programming the constraint v ≤ 1 corresponding to the basic variable C is binding. max v,ur
s r =1
s ur yro −v Cj + ur yrj ≤ 0, j = 1, . . . , n, v ≤ 1 . (7) r =1
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Since m the input weights vi in (6) are m1 at its optimal solution, the i=1 vi Cij in (6) can be written as i=1 Cij , which is equal to Cj , and since v Cj in (7) is equal to Cj (as v = 1 at the optimal solution of (7)) the duals of the two models are exactly the same. Note that this result is also valid for variable returns to scale. Model (5) has an important implication in terms of aggregation issues addressed e.g. in Fare and Zelenyuk (2002) and Fare and Grosskopf (2004). These authors were concerned with the use of the technical efficiency model (2) and were interested in ‘‘learning when the input-oriented measure of technical efficiency based on quantity data yields the same score when based on the cost aggregated data’’ (Fare and Grosskopf, 2004, p. 121). The authors conclude that aggregation is unbiased only when units are allocatively efficient and prices are the same across all units. To see this, note that if all inputs have been converted into a single aggregate input cost, then a technical efficiency score could be obtained from model (5), as C can be replaced by θ Co without changing the optimal solution to this model. However, model (5) is a cost model and not a technical efficiency model, which means that it would only be equal to a technical efficiency model when there is no allocative inefficiency. Therefore we provide a new result to the aggregation issue: Proposition 1.1. Aggregation of input costs into a single cost value, provides unbiased cost efficiency results, but no technical efficiency can be computed through a single aggregate cost value. That is, to measure technical efficiency in the presence of value data one needs to use disaggregate cost data in a radial model like (3), and assume that prices are the same across DMUs. Cost efficiency, on the other hand, can be computed through aggregate cost data, and the assumption of equal prices does not need to be maintained. This issue will be further addressed in the next sections. 2.2. Different prices across DMUs In this section we assume that prices are different across DMUs and unknown, and only value data are available for each input. Little attention has been paid for in the literature to this situation, particularly to the fact that technical efficiency lacks a solid interpretation under such circumstances. Note, however, that technical efficiency can be computed directly from value data when firms are weakly rational and relative prices are equal across firms, as shown by Cross and Fare (2009). Cross and Fare (2008) also investigated value based efficiency measurement and put forward an identity stating that the value-based technical efficiency (or cost efficiency, as we call it) can be decomposed, multiplicatively, into the quantity-based technical efficiency score (or simply technical efficiency), a technology effect, and a firm effect. Following from this identity, and Proposition 1 we put forward Proposition 2: Proposition 2. When prices are unknown and different across DMUs, cost efficiency can be computed using value cost data, but technical efficiency cannot be computed from cost data. When this is the case cost efficiency can be measured through model (4) (or indeed model (5)), but model (3) should not be used to compute technical efficiency because it implicitly assumes that trade-offs between costs exist, which is an argument difficult to sustain. We explain the reasons for this argument through the illustrative example shown in Fig. 1, where 3 units producing the same level of output use different combinations of costs of two inputs (capital and labour).
Fig. 1. Illustrative example.
Technical efficiency computed through model (3) identifies the production possibilities frontier as the line joining units A and B and its extensions to the axes. The two frontier units would be ‘technically’ efficient despite the fact that they use different strategies (labour intensive for B and capital intensive for A) to produce the same level of output. However, since the production factors are expressed in costs, we are implying that producing at a point with high costs in capital is equally efficient as producing at a point with high costs in labour. This is clearly not a sensible assumption when the DMU with high capital costs incurs in almost the double total costs than the DMU with high labour costs. As a result, trade-offs between the costs of production factors do not exist when prices differ across DMUs (if prices are the same across DMUs, cost trade-offs indeed reflect trade-offs between quantities). To further elaborate our argument consider that the disaggregate data for this example are known and shown in Table 1. In this table we can see that units A and B use exactly the same amount of capital, and B is technically inefficient when compared to the technically efficient units A and C. Therefore the tradeoff observed in Fig. 1 between the costs of A and B is not really an effect of quantities, but an effect of prices. As a result, in the absence of disaggregated information, the efficiency of these units should be computed through models (4) or (5), which yield a cost minimising frontier at Min(C1 +C2 )—the straight black line in Fig. 1. Results from model (4) show unit B as 100% cost efficient and units A and C as cost inefficient (cost efficiency of 57.04% and 58.31%, respectively). Cost efficiency shall be interpreted as the proportion of the costs of a DMU in relation to the minimum cost observed to produce a certain level of outputs. Note that in addition to the above, the example in Table 1 also shows that the cost model (assumed as the best model to be applied in this case) yields cost efficiency scores that may be quite far from the ‘true cost efficiency’ (that obtained when disaggregated data are available). The cost model, based on value data, implicitly assumes that minimum observed cost for a certain level of produced output is achievable irrespective of technical efficiency, which cannot be known when quantities are not known (e.g. in the illustrative example unit B is the cost minimising unit, but it is not technically efficient). However, in reality the minimum cost may be unachievable if prices cannot be changed. Take the example of DMU C, which is technically and cost efficient (for prevailing prices) when one uses the disaggregated data on quantities and prices in models (1) and (2). This means that the cost efficiency of 58.31% identified for DMU C is a result of excessive prices paid for inputs. Therefore the minimum cost target for DMU C is only achievable if this unit can to some extent influence the prices it pays for labour and capital. This result has lead us to Proposition 2.1:
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Table 1 Hypothetical disaggregate data for illustrative example. DMU
Capital
Labour
Price of capital
Price of labour
Cost of capital
Cost of labour
Total cost
A B C
40 40 30
10 16 15
500 200 500
300 320 500
20 000 8 000 15 000
3000 5120 7500
23 000 13 120 22 500
Proposition 2.1. In the absence of price information, and differing prices across DMUs, minimum cost is only achievable when prices can be controlled/changed to some extent, and therefore DMUs cannot be considered price takers. Note this result is addressed here for the first time, but the relaxation of the price-taking assumption of DMUs has been addressed by other authors. (see e.g Portela and Thanassoulis, 2014). 3. Mixing value and quantity data in DEA models Often in empirical applications we encounter some inputs defined on value terms and others defined on quantity terms (e.g. N. of workers and value of capital). We consider a subset of inputs i = 1, . . . , k expressed in quantity, and a subset of i = k + 1, . . . , m expressed in value. 3.1. Equal prices across DMUs for value data As seen in the previous section, when prices are the same across units it is equivalent to use value or quantity data, and therefore the mix of both types of data does not pose a problem in the estimation of efficiencies. However, when only some input costs are known, one cannot compute a cost efficiency model for the overall set of inputs, but just for a sub-set. Such models are known as sub-vector cost efficiency and shown in (8) (see also Fare and Grosskopf, 2004).
min λj ,Ci
m
i=k+1
n λj xij ≤ xio , i = 1, . . . , k, Ci j =1
n
λj Cij ≤ Ci , i = k + 1, . . . , m,
j =1 n
λj yrj ≥ yro , r = 1, . . . , s, λj ≥ 0 .
(8)
j =1
Note that model (8) could equivalently be solved through a model where the i constraints (i = k + 1, . . . , m) have been replaced by a single aggregate cost constraint (see Section 2.1). For decomposing the sub-vector cost efficiency into its technical and allocative components, the sub-vector technical efficiency model shown in (9) should be solved, where only a subset of inputs is contracted towards the efficient frontier, in consistency with (8).
n min θ λj xij ≤ xio , i = 1, . . . , k, λj ,θ
j =1
n
λj Cij ≤ θ Cio , i = k + 1, . . . , m,
from the ratio between minimum cost from (8) and observed cost) and the sub-vector technical efficiency (obtained from (9)). It is important to note that model (8) is in many respects different from model (4) (applicable when all inputs are expressed in value), as minimum cost in (8) depends not only on produced outputs, but also on the input levels of the inputs expressed in quantity. This means that the unit showing the lowest quantity on a certain input (i = 1, . . . , k) is sub-vector-cost efficient irrespective of the cost observed for the inputs expressed in value. This is a sensible assumption, since lowest input quantities would also imply lowest costs (if prices were to be known) for those inputs (under similar prices across DMUs). 3.2. Different prices across DMUs for value data When prices are different across DMUs one can invoke the same arguments as in Section 2.2 to disregard the computation of sub-vector technical efficiency through (9). Indeed, such model assumes not only trade-offs between costs but also between input costs and input quantities. Such trade-offs are only sensible in the situation where production units face the same prices and costs are a good surrogate for quantities. Regarding the computation of sub-vector cost efficiency the arguments of Section 2.2 are still valid. That is, one can compute sub-vector cost efficiency, but needs to be aware that sub-vector cost minimising units are not necessarily sub-vector technically efficient, or that sub-vector technically efficient units may be subvector cost inefficient (note however that one is assuming subvector technical efficiency cannot be known). Therefore, the use of model (8) implicitly assumes that DMUs are not price takers and minimum cost may be obtained both from changing quantities and prices. In addition, there is a further assumption invoked in this case: the assumption that cost minimising units are those producing the highest level of outputs and using the lowest level of input quantities at the minimum costs. Lower level of input quantities may not be associated with lower cost levels, when prices are not constant across DMUs, and therefore this assumption may be problematic. However, if the input quantities cannot in any way be expressed in value form due to the complete absence of prices (e.g. input quantities may be related to environmental or contextual variables such as density of population, and number of years of education, which cannot be expressed in value) then the computation of sub-vector cost efficiency is valid. These arguments, and the results in Fare and Grosskopf (2004) led us to our final proposition. Proposition 3. When there is cost data for some inputs and quantity data for others, sub-vector technical and sub-vector cost efficiency should be computed when prices are assumed the same across DMUs, and no measure of sub-vector technical efficiency should be computed when prices differ across DMUs. Sub-vector cost efficiency can be computed under this setting but the assumptions behind this model should be verified.
j =1 n
4. Conclusion
λj yrj ≥ yro , r = 1, . . . , s, λj ≥ 0 .
(9)
j =1
Sub-vector allocative efficiency can be computed residually through the ratio between the sub-vector cost efficiency (obtained
The aim of this paper is to call the attention for some underlying assumptions that are being invoked when value data are used instead of quantity data in efficiency assessments, particularly those undertaken through DEA models. We analyse situations
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where all inputs or all outputs are taken in value form and situations where some inputs are expressed in value, whereas others are expressed in quantity. For the former situation, all cost efficiency models can be applied and decomposed into technical and allocative efficiency provided that prices are assumed equal across DMUs. When this is not the case, only cost efficiency can be computed and no decomposition is possible, as there is not a way to compute technical efficiency. For the latter situation we conclude that when prices are assumed equal across DMUs sub-vector cost and technical efficiency measures can be used, and when this is not the case sub-vector technical efficiency should not be used. Acknowledgement The author acknowledges the financial support of the Portuguese Foundation for Science and Technology (FCT) through project PTDC/EGE-GES/112232/2009.
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