28
~ Pergamon Socio-Econ. Plann. Sci. Vol.29, No. 2, pp. 85 112, 1995 Copyright © 1995 Elsevier Science Ltd 0038-0121(95)00002-X Printed in Great Britain. All rights reserved 0038-0121/95 $9.50 + 0.00 DEA and Stochastic Frontier Analyses of the 1978 Chinese Economic Reforms W. W. COOPEW, SUBAL KUMBHAKAR 2, ROBERT M. THRALL 3 and XUELIN YU 4 ~Graduate School of Business, University of Texas at Austin, Austin, TX 78712-1174, U.S.A. 2Department of Economics, University of Texas at Austin, Austin, TX 78714, U.S.A. 3professor Emeritus, Jones Graduate School of Administration, Rice University, Houston, TX 77251, U.S.A. 4Tianjin Institute of Scientific and Technical Information, Tianjin, People's Republic of China Abstract--Using data obtained from Chinese sources for the period 1966-88, this paper reports results from a study of the impact of the 1978economic reforms for the period 1966-88on the Textiles,Chemicals and Metallurgical Industries. In all three, the effects were found to be dramatic and manifested almost immediately in: (1) drastic changes in capital-to-labor ratios; (2) large increases in output; and (3) significant increases in efficiency. DEA (Data Envelopment Analysis) and SF (Stochastic Frontier Analysis) provide analytic frameworks which are interpreted as follows. Via its single optimization over all observations, SF estimates are addressed to behavior across all periods while n optimizations used in DEA (one for each observation) are addressed to performances in each of the periods that are applicable for effectingits evaluations. Although based on different principles and treating the data in different ways, these two different approaches to performance evaluation, as used here, provide confirmation of each other's findings. The adjustments to the data obtained from efficiencyevaluations in these two approaches are also used to show that trends underlying the observed behavior would have accelerated under fully efficient production. The joint use made here of DEA and SF contrasts with points of view that have regarded these two approaches as mutually exclusivealternatives. A new ratio measure of efficiencyis also introduced in furtherance of DEA as a body of concepts in its own right. INTRODUCTION Using data from Chinese sources, this paper reports results from a study of the impact of the 1978 economic reforms for the period 1966-88 on the Textiles, Chemicals and Metallurgical Industries. In all three, the effects were found to be dramatic and immediate as manifested in: (1) large increases in output, as well as the rate of increase in output; (2) drastic changes in labor and capital output ratios accompanied by large increases in both inputs; and (3) dramatic increases in efficiency resulting from removal of the mix inefficiencies that characterized production in the pre-reform periods. Our approach, as well as our results, differ markedly from other parts of the literature dealing with the economic reforms in China that began in 1978. Possibly because of data difficulties, especially in earlier years, much of this literature is qualitative, reportorial or anecdotal in character. The parts with quantitative components rely on central tendency estimates in the form of averages, index numbers or statistical regressions with inferences synthesized from unidentified mixtures of efficient and inefficient performances.t Our study, however, utilizes frontier estimation methodologies in the form of DEA (Data Envelopment Analysis) and SF (Stochastic Frontier) regressions, Hence, one reason for contrasting results would appear to arise from differences in the methodologies employed. Partly because these frontier oriented methodologies are still undergoing rapid development, this study does not use the full panoply of possibilities that are now available. The DEA used here is deterministic. It is not stochasticized along lines that have recently become possible as described tRecent examples include the article by Yusuf [36] and the Asian Development Bank's Asian Development Outlook 1993 [5]. Differences between our results and what is reported in the literature were found to be even more striking at the time (ca 1991) when our study was completed. See Yanrui [35] for a survey of more recent studies, some of which apply techniques like those used in the present paper. 85

DEA and stochastic frontier analyses of the 1978 Chinese economic reforms

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Page 1: DEA and stochastic frontier analyses of the 1978 Chinese economic reforms

~ Pergamon Socio-Econ. Plann. Sci. Vol. 29, No. 2, pp. 85 112, 1995 Copyright © 1995 Elsevier Science Ltd

0038-0121(95)00002-X Printed in Great Britain. All rights reserved 0038-0121/95 $9.50 + 0.00

DEA and Stochastic Frontier Analyses of the 1978 Chinese Economic Reforms

W. W. C O O P E W , SUBAL K U M B H A K A R 2, R O B E R T M. T H R A L L 3 and X U E L I N Y U 4

~Graduate School of Business, University of Texas at Austin, Austin, TX 78712-1174, U.S.A. 2Department of Economics, University of Texas at Austin, Austin, TX 78714, U.S.A.

3professor Emeritus, Jones Graduate School of Administration, Rice University, Houston, TX 77251, U.S.A.

4Tianjin Institute of Scientific and Technical Information, Tianjin, People's Republic of China

Abstract--Using data obtained from Chinese sources for the period 1966-88, this paper reports results from a study of the impact of the 1978 economic reforms for the period 1966-88 on the Textiles, Chemicals and Metallurgical Industries. In all three, the effects were found to be dramatic and manifested almost immediately in: (1) drastic changes in capital-to-labor ratios; (2) large increases in output; and (3) significant increases in efficiency. DEA (Data Envelopment Analysis) and SF (Stochastic Frontier Analysis) provide analytic frameworks which are interpreted as follows. Via its single optimization over all observations, SF estimates are addressed to behavior across all periods while n optimizations used in DEA (one for each observation) are addressed to performances in each of the periods that are applicable for effecting its evaluations. Although based on different principles and treating the data in different ways, these two different approaches to performance evaluation, as used here, provide confirmation of each other's findings. The adjustments to the data obtained from efficiency evaluations in these two approaches are also used to show that trends underlying the observed behavior would have accelerated under fully efficient production. The joint use made here of DEA and SF contrasts with points of view that have regarded these two approaches as mutually exclusive alternatives. A new ratio measure of efficiency is also introduced in furtherance of DEA as a body of concepts in its own right.

I N T R O D U C T I O N

Using data from Chinese sources, this paper reports results from a study of the impact of the 1978 economic reforms for the period 1966-88 on the Textiles, Chemicals and Metallurgical Industries. In all three, the effects were found to be dramatic and immediate as manifested in: (1) large increases in output , as well as the rate of increase in output; (2) drastic changes in labor and capital output ratios accompanied by large increases in both inputs; and (3) dramatic increases in efficiency resulting from removal of the mix inefficiencies that characterized production in the pre-reform periods.

Our approach, as well as our results, differ markedly from other parts of the li terature dealing with the economic reforms in China that began in 1978. Possibly because of data difficulties, especially in earlier years, much of this literature is qualitative, reportorial or anecdotal in character. The parts with quant i ta t ive components rely on central tendency estimates in the form of averages, index numbers or statistical regressions with inferences synthesized from unidentified mixtures of efficient and inefficient per formances . t Our study, however, utilizes frontier est imation methodologies in the form of D E A (Data Enve lopment Analysis) and SF (Stochastic Frontier) regressions, Hence, one reason for contras t ing results would appear to arise from differences in the methodologies employed.

Partly because these frontier oriented methodologies are still undergoing rapid development, this study does not use the full panoply of possibilities that are now available. The D E A used here is deterministic. It is not stochasticized along lines that have recently become possible as described

tRecent examples include the article by Yusuf [36] and the Asian Development Bank's Asian Development Outlook 1993 [5]. Differences between our results and what is reported in the literature were found to be even more striking at the time (ca 1991) when our study was completed. See Yanrui [35] for a survey of more recent studies, some of which apply techniques like those used in the present paper.

85

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86 W.W. Cooper et al.

in Banker [6] and Banker and Cooper [7]. However, we do apply statistical concepts and methodologies in the form of SF regressions, and this provides an added advantage in that we are able to use different methodologies (and concepts) to help ensure that our results are not simply reflections of a single methodology. +

We provide brief introductions to our DEA and SF approaches in the sections that follow. In the next section, we begin with an overview of DEA and the definitions of efficiency that we use while allocating further details to an Appendix--where a new algorithm with accompanying geometric portrayals is provided for situations, like ours, that involve only one output and two inputs. Data considerations are discussed next, after which they are joined with DEA to obtain the changes that occurred in the capital and labor ratios. Output increases are then exhibited in the section that follows after a brief discussion of the SF regressions--which are used here in a confirmatory manner to examine those output increases found to have occurred after 1978 in our DEA analyses. Next, a study of efficiency increases is examined, as determined from DEA. This topic is developed in a way that also allows us to examine the sensitivity of our results to variations in the periods covered. We find that our results are not changed appreciably by restricting the data to sub-periods during the interval 1966-88. A final section summarizes what has been accom- plished, identifies sources of the resources available for utilization in China, and suggests some future research possibilities.

DEA (DATA E N V E L O P M E N T ANALYSIS) MEASURES OF E F F I C I E N C Y

Following Charnes et al. [15, p. 201], we characterize DEA as a body of concepts and methods for investigating efficiencies and inefficiencies and evaluating the observed performances of each of a collection of Decision Making Units (DMUs)- -where each D M U is regarded as an entity or organization that converts inputs to outputs. Suppose, therefore, that a set of common input and output values has been collected for each member of a set of DMUs. We then want to determine whether the input-output values for any D M U provide evidence of inefficient performance. In empirical social science, efficiency evaluations will generally have to take the form of relative measures since access to theoretical norms (e.g. as in chemical thermodynamics) are not available. The measures used will thus only reflect the performance of each D M U compared to other DMUs.

To emphasize this feature, we occasionally refer to a D M U as being "DEA efficient," by which we mean that its efficiency has been evaluated from its observed input-output values relative to what has been recorded for other DMUs in these same inputs and outputs. In a tradition in economics going back to Alfred Marshall, an "average" or "representative D M U " is sometimes used for such evaluations; but, DEA proceeds in a different manner. First, its evaluations are based on a "best comparison set" consisting of a subset from a larger collection of DMUs. Second, each member of this subset entering actively (i.e. with non-zero coefficients) in the evaluation is determined to be "100% efficient by DEA," rather than "representative" or "average". Third, this "best comparison set" will usually change as DEA evaluations of different members of the entire collection of D M U s are undertaken.

The measures of efficiency used in DEA can be conceptualized in the following form:

hk(u, v) O* = m a x - - , (1)

u,,, ho(u, v)

where

ho(u ,v ) = max hj (u ,v) . j = 1 . . . . . n .

Here (u, v) are vectors with all components non-negative and associated with outputs and inputs, respectively. For instance, we can represent each hi(u, v) as a ratio of linear forms, viz.

hi(u, v) = uTy//vTxj = u,y,j ViX~, (2) r = l i 1

++Further development of the importance of this topic may be found in the exchange between Charnes et al. [14].

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DEA 87

where yrj, x~ >1 0 are the observed values for each of r = 1 . . . . . s outputs and i = 1 . . . . . m inputs for every DMUj, j = 1 . . . . . n, so that ur is associated with Yrj, and vi is associated with x~.

We refer to this measure, which is new, as the " T D T measure" of relative efficiencyt and note that, by construction,

0 ~ 02' ~< I. (3)

Full (100%) efficiency is attained by DMUk if and only if 0* = 1 because, then, no other D M U has a higher efficiency ratio than DMUk. See (1).

To place this in perspective, we turn to the " C C R ratio" model of DEA as given in Charnes et al. [13], which we can represent as:

~ UrYrk max Ok --r= 1 , (4)

m

E Ui Xlk i = 1

subject to

~. Ur Y~i 1 ~> r=l j = 1 . . . . . n ;

Vi X U i = 1

0~<ur,v/; r = 1 . . . . . s ; i = l . . . . . m.

Here, DMUk, the D M U to be evaluated, is positioned in the functional where it is to be maximized. Its output-to-input ratio is also included in one of the constraints, so the conditions specified in (3) follow, as before, with max Ok = 0* = l if and only if DMUk is fully (100%) efficient, and 0* < 1 otherwise.

To align this with (1), we note that at least one constraint in (4) must be at its upper bound of unity if the (u, v) choice is to be maximal. The optimal solutions of (4) therefore coincide with (1) in this special case where the numerator has a value of unity. It is this set of constraints that limit the value of Ok. Further, the D M U s associated with the constraints that are critical in limiting the value of Ok provide the best comparison set since they yield those (u, v) choices that render Ok maximal.

The components of the (u, v) vector are determined anew for each D M U i so that a new "best" set of values is also determined each time a different DMUk is to be evaluated. No a priori specification of weights is needed and the resulting 0* values are invariant with respect to the units of measure used. (See Charnes and Cooper [9].) Moreover, the resulting optimal ratios may be regarded as corresponding to "tangent planes" (or hyperplanes) or, more generally, as "supports" for points generated from underlying functions. The production function concept used in much of the econometric-regression literature is thus replaced by "tangency relations" generated from observations without requiring specification of the underlying functions. (See Charnes et al. [10].)

From this latter property, we see that DEA can offer considerable advantages for use in studies where the underlying technologies are largely unknown, complex--and possibly changing, during the periods studied. Various efficiencies are identified and available for use from the DEA literature. These include "allocative" ( = price or cost) and "scale" efficiencies. However, price or cost data are either not available or are of questionable value over many of the periods covered in our study. Also, we do not wish to allow substitutions between inputs and outputs or even to consider the exchanges between inputs and outputs needed to implement findings relative to changing scales of operations. Hence, we confine attention to "technical efficiency" defined as follows:

DMUk is technically efficient if and only if it is not possible to improve any of its inputs or outputs without worsening some of its other inputs or outputs.

Once the inputs, outputs and D M U s are specified, the technical efficiency evaluations are relatively value free and this permits the data to be determinative in effecting the evaluations in

tSee Ref [26].

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88 W.W. Cooper et al.

DEA. Indeed, technical inefficiency is sometimes simply referred to as "waste" in that elimination of such inefficiencies results in improved performance. So long as the inputs and outputs have "some" positive value (which need not be specified), the elimination of such inefficiencies can generally be regarded as resulting in a "betterment" since no other inputs or outputs are sacrificed to secure this improvement.

TWO INPUTS AND ONE OUTPUT: A GRAPHIC PORTRAYAL

The formulation (4) is a nonlinear-nonconvex programming problem and hence best used for conceptual clarification. Use of a fractional programming transformation, as described in Cooper [18], provides access to a dual pair of linear programming problems that can be used in place of (4) for computation. We need not do this here for, returning to (1), we see that any method for securing these maxima may be used and, indeed, one need not be confined to a use of mathematical programming formulations or algorithms for such purposes. We take advantage of these possibilities and provide a special algorithm for use with DEA analyses involving only one output and two inputs--the case of interest here. (See Fig. A1 and Table A1 in the Appendix.)

Figure 1 below, can provide help in understanding the algorithm we supply in the Appendix. It can also provide insight into the portrayals we subsequently use to analyze the effects of the 1978 economic reforms. It is constructed from the following "data matrix"

P = - 2 - 3 - 4 - 6 - 2 . 5 - 8 . 4 - 7 = P1, P2 . . . . , P8 • (5)

- 5 - 3 - 2 - 1 - 7 . 5 - 1 . 2 - 5 - 1 . 5

In this representation, we associate points (or vectors) P~, P2 . . . . . P8 with the data for each of eight DMUs, which produced one unit of a single output (as listed in row one) and used varying amounts of two inputs (as listed in the next two rows). See Ref. [15] for conventions on the use of positive and negative signs in such "data matrices."

The solid line in Fig. 1 constitutes a frontier in the form of an envelope generated from the supports to this set of points. All points on or inside this frontier represent production possibilities. The frontier is formed in a piecewise linear fashion, as follows. Bounded on the left by the vertical ray extending upward from 1, the efficient portion of the frontier is initiated by the segment from 1 to 2. It continues from 2 to 3 and then from 3 to 4, after which the horizontal ray extending to the right from 4 continues to bound the set of production possibilities. It is from this envelopment of the data that DEA derives the name "Data Envelopment Analysis." The "efficiency frontier," which is of special interest in evaluating the performance of all DMUs, consists of the three segments that connect 1-4 in a continuous (piecewise linear) manner. The "extended frontier" includes the vertical and horizontal rays as well. Restated mathematically, the efficiency and extended frontiers portrayed in Fig. 1 are 2-dimensional cones generated by their respective intersections with the plane y = !.

We can now interpret the definition of efficiency given earlier in a different manner. Here, we decompose our measures into two parts, consisting of: (i) a 0* value that represents the distance from the origin to Pk, the point associated with the DMU to be evaluated, relative to the distance to the point on the frontier used in the evaluation; and (ii) any non-zero slack that remains. This allows us to identify different types and amounts of inefficiency as follows. Using 0* to represent the ratio of the distance from the origin to the evaluating point relative to the point being evaluated, we see that a value of 0* < 1, as in part (i) of the proposed efficiency evaluation, is associated with a reduction of all inputs in the same proportionin amounts determined by applying 0* to every input. We refer to this as a measure of "radial efficiency." It is sometimes referred to as "Farrell efficiency," after Farrell [21], and also called "weak efficiency" since it does not necessarily identify all sources of inefficiency in terms of the definition we gave above for "technical efficiency." For the latter purpose, account must also be taken of the possible presence of non-zero slack even when 0 " = !. Correction for non-zero slack, as in (ii), changes the input proportions and therefore

Page 5: DEA and stochastic frontier analyses of the 1978 Chinese economic reforms

DEA

x 2

9--

8--

7- -

6--

5 -

4--

3--

2--

1--

"g, i !

~ 7 /

/ /

/

3~ .46 4 6

i i i i i I I I I 1 2 3 4 5 6 7 8 9 x l

Fig. 1. An example of a unit ized f ront ier wi th y = 1, m = 2, s = 1 and n = 8.

89

represents a correction for what we will refer to as "mix inefficiencies." Full ( = 100%) " D E A efficiency" is thus attained if and only if both of the following conditions are satisfied:

(i) 0 " = 1;

(ii) all slacks are zero. (6)

Attention should be called to the fact that the value of 0 is minimized preemptively in DEA after which the slack values are maximized. (See Refs [13, 9, 12, 20].) This way of proceeding insures that inefficiencies are all identified and may be removed for each D M U without changing the proportions in which the inputs are used. Removal of any non-zero slack that remains must therefore change these input proportions. Since "mix" is a matter of using inputs in correct proportions, we identify these non-zero slacks as originating in "mix inefficiencies." As will subsequently be seen, labor is always the source of such mix inefficiencies in the pre-reform years. Moreover, non-zero labor slack occurs with 0* < 1 in every case. Since this means that all resources were used in excess, the evidence suggests that this mix inefficiency is best removed by reducing labor rather than by increasing capital.

Returning to Fig. 1 we illustrate how these different sources of inefficiency may be located and their amounts estimated. To start, we note that points such as 5 and 6 can be improved in one input without worsening any other input (or output). Thus 3 has 0* = 1 but 5 is not technically efficient because, without augmenting x~, this same one unit output is achievable at the point labeled 1. Thus 3 is not efficient even though located on the frontier. The same situation is evident for ?), where movement to 4 may be effected by reducing the slack associated with the different values of x~ for points 4 and 6. The inefficiencies identified at 5 and 6 are both mix inefficiencies, and the adjustment amounts needed to eliminate these inefficiencies are represented in the non-zero slack values with which removal of these mix inefficiencies is associated.

Turning to7 , we have 0* < 1 so that x, and x2 can both be decreased in the proportion (1 - 0") to attain 7 = 7--where one unit of output continues to be possible under the efficient production segment connecting 2 and 3. This is a situation with 0* < 1 and no slack, and, hence, represents a radial inefficiency that permits a reduction in both inputs. For 6, however, both 0 " < 1 and non-zero slack occur, in which case both inputs are first decreased in the same proportion to attain 6. This eliminates "radial inefficiency" after which slack in x, may be removed by going from to 4 to remove "mix inefficiency." A similar situation occurs for 5, where 0* < 1 means that both inputs can be reduced. But this is to be followed by the further slack reduction in x2 attainable by going from 5 to 1.

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90 W.W. Cooper et al.

For the moment, this is as far as we carry our discussion of these technical efficiency concepts. The efficiency frontiers in Fig. 1 may be interpreted as an isoquant under the usual assumption of efficient production in micro-economics. The graphs used to guide our discussion of the Chinese economic reforms will, however, depart from the customary isoquant concept since we will need to deal with different levels of output for each observation. To retain access to 2-dimensional representations, we divide each input by its corresponding output. This means that we will be dealing with ra t e s of input per unit output in diagrams that portray what we shall refer to "unitized front iers"--for which the isoquant in Fig. 1 provides a special case with all of the outputs represented by a divisor y = 1.

There is no need for concern in this departure from the standard isoquant concept, however, because the algorithm we supply in the Appendix makes it possible to obtain estimates of the a m o u n t s of inefficiency in each input and output in accordance with the following formulas:

fci~ = O * x i ~ - s + * ; i = 1 . . . . . m , (7)

f ~ r k = Y r k + S r * ; r = 1 . . . . . s,

where 0 <~ 0* ~< 1 and s +* , s T * represent optimal values for 0 and slack, respectively, as obtained from our algorithm.

The expressions in (7) are called "CCR projection formulas"--as introduced in Ref. [13J--since they enable us to project the original xik, Yrk values into points ~ik, Yrk which are on the efficiency frontier so that ~k ~< X~k for all i and Yrk t> Yrk for all r. These ~ and Yrk values are, in fact, the coordinates of the point on the efficiency frontier used to evaluate the performance of DMUk. Differences in inputs and output values can thus be associated with the values of 0", s +* and sT*. Unlike situations with larger numbers of outputs and inputs, the results for the one output- two input case considered here are always unique so there is no need to examine possibilities that might arise from the presence of alternate optima. Note, for instance, that 8 in Fig. 1 may be expressed in terms of 3 and 4. It may also be expressed in terms of non-negative combinations of other points on this segment of the frontier, but, in every case, the solutions give 0* = 1 and all slacks are zero.

In the single output case we can also ignore the possibility of non-zero o u t p u t slack because by virtue of a theorem in Cooper and Gallegos [19]. Thus, at least one output slack and one input slack will always equal zero and this allows us to use the observed outputs to convert the inefficiency ra t e s to corresponding amounts. (A precise statement and proof are given in the Appendix.) Here we only note that the output projection in (7) has been included, despite its redundancy. This is done since we will also seek to replace the above "input oriented" projections with one which is "output oriented". The orientation is thus one which maximizes output--wherein we replace (7) with:

.~/k = Xik-- ~,-*; i = 1 . . . . . m,

)~rk=q~*yrk+g+*; r = 1 . . . . . s, (8)

where ¢* = 1/0" so that, always, ~b*/> 1. [See (3), above.]

DATA AND DMU CHOICES

Tables A2-A4 contain data on our three industries: Textiles, Chemicals, and Metallurgy. The units in which the data are stated conform to the source from which they were s e c u r e d - - v i z ,

Y e a r b o o k o f C h i n a ' s 4 0 Y e a r s as issued by the Chinese Statistical Bureau [17]. Output and capital are measured in 10,000 Yuan adjusted to 1980 prices--based on the official index of prices. Labor is stated in number of employees and requires no such adjustment. The resulting data are listed in the three columns captioned "Observations" in all three tables.

Having no way to check the validity of these data, we simply call attention to the following remark of Jefferson and Rawski [22, p. 47]: "We refrain from detailed comment on the accuracy of Chinese economic statistics, except to note that officially compiled data on industrial inputs and outputs typically convey (correctly) the order of magnitude of relevant levels and rates of change."

We selected Textiles to represent an industry which is labor intensive, Chemicals to represent an industry which is capital intensive, and Metallurgy to be in between. We think that choice of

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DEA 91

an industry provides a meaningful basis for analyzing behavior and evaluating efficiency both before and after the 1978 reforms. Even though decision making was moved to a more decentralized basis after the reforms, planning and decision making organizations remain in place at industry levels.

Our interest is in the effects of these reforms before, as well as after, 1978 and, hence, it was deemed best to focus on the industry level of decision making (and planning) across the two periods since, prior to 1978, this, rather than the plant level, was critical. We then treat each year as a separate D M U and take advantage of the fact that DEA optimizes on each observation to obtain a "best fit" to the pertinent years for evaluating its performance. In this regard, we portray our results (below) graphically in a manner that will enable us to relate our analysis to what we have already covered with Fig. 1, above. In addition, Fig. A1 and Table A1 in the Appendix are used to show how this geometric portrayal can be used with our data.

In terms of the periods utilized in this study, it seems clear that 1978 should be used and, in fact, we accord it a central position. Going back to 1966 gives us 12 yr of pre-reform behavior. Counting 1978 as the start of reform provides 11 yr of post-reform behavior. This is as far as we could extend the analysis since our data terminate in 1988. (See Ref. [17].)

By 1966, most of the effects of the 1958-59 "Great Leap Forward" as a policy directed to overcoming China's economic deficiencies had pretty well disappeared. We thus believe that few serious effects remain from this attempt at economic reform. There is little doubt that the "Cultural Revolution of 1966-9" affected economic performances. t However, we believe it can be argued that such political-social activity continued to be a part of the environment in which economic activities occurred up to (and possibly beyond) 1988. Hence, we consider such political and social activity to have been a normal backdrop for economic activity over the period of our study.

As discussed below, we conducted limited checks on data effects by omitting some of the years with no large changes in our results. Hence, at least with respect to these changes, we can express some confidence that our results reflect underlying economic behavior rather than vagaries in the data. In any case, this is how we proceed in the sections that follow using industry-level behavior partly for reasons already given and partly because we did not have access to plant level data. Similarly, we were unable to go below the level of one output and two inputs (labor and capital) for each industry since we did not have access to more refined data on their components.

C H A N G E S IN LABOR A N D CAPITAL TO O U T P U T RATIOS

Figures 2~, provide plots of actual behavior obtained from Tables A2-A4 after reduction to normalized form by dividing each input by its output value.$ The solid diamonds show fully efficient production and the dates of occurrence. Evidently, the occurrence of fully efficient behavior was relatively rare, and departures from efficiency were much larger for the pre-reform era in all three industries.

This is corroborated by Tables A2-A4, where the normalizations have been removed. Of special interest is evidence of overuse of labor in the pre-reform era as indicated not only by 0* < 1 values, but also by mix inefficiencies indicated by the non-zero labor slacks which all occur in the pre-reform era. No non-zero SLACAP values occur in this period, further emphasizing the validity of referring to these slack values in labor as "mix inefficiencies," represented by inappropriate capital and labor proportions. The only occurrence of non-zero slack for capital occurs in Textiles during the post-reform years of 1987 and 1988. Note that these occurrences are in only relatively minor amounts, and may thus represent anticipatory actions as well as overshoots in the capital used. (See the left-most points in Fig. 2.)

There have been numerous reports of the overuse of labor. See, for instance, Tidrick and Byrd [30] or Walder [31, 32]. Our methods go further, however, and enable us to supply estimates of these

tThis revolution continued more or less actively until the death of Mao Zedong in 1976. See Perkins [24]. However, as will be seen in Figs 10-12, below, the bottom in output occurred in the period 1967-69, with recovery under way by 1970 in each of our three industries.

++The origin has been eliminated for visual convenience so allowance should be made for this when interpreting these figures. See Fig. A1 and its discussion in the Appendix.

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92 W . W . Cooper et al.

.7

.6

.5

.4

.3

.E

Capital to output ratio

A A

A

8 5 - A

A A A

A A • A A A A AA A A 81 • A A

8O • 70

I i I I I I i i i • 48 .45 .58 .55 .68 .65 .78 .75 .88 .85

Labor to output ratio

Fig. 2. Plot of cap/Y vs lab/Y (unadjusted data). Textile Industry, A = 1 obs, B = 2 obs, etc.

inefficiencies, relat ive to efficient usage, via the non-zero slack and 0-values recorded in Tables A 2 - A 4 . We are also able to es t imate capi ta l inefficiencies that were present (but not no ted by others) via the values o f 0* < 1. In this regard, we can identify their occurrence relative to wha t was possible under efficient p roduc t ion even in the pre-re form years.

We can, addi t iona l ly , br ing out under ly ing tendencies more sharply by focusing on the improvements in pe r fo rmance that p resumably mot iva ted the 1978 economic reforms a n d / o r o ther

Capital to output ratio

.65

. 60

.55

. 5 0 -

.45

.40

.35

8 8 A

A

8 6 & A

A A A A A

A A

A AAB A

• •

78 70

A

I I I I I I I .35 .48 .45 .58 .68 .65 .78

Labor to output ratio

Fig. 3. Plot of cap/Y vs lab/Y (unadjusted data). Chemical Industry, A = I obs, B = 2 obs, etc.

Page 9: DEA and stochastic frontier analyses of the 1978 Chinese economic reforms

DEA 93

Capital to output ratio

.65

.60

.55

.50

.45

.40

.35

.30

.25

8 8 • A

87 • A

A A

A

79 • - A

I I .15 .28

AA

A A

74

A

A A

I I I I I I I I .25 .38 .35 .48 .45 .55 .68 .65

Labor to output ratio

Fig. 4. Plot of cap/Y vs lab/Y (unadjusted data). Metallurgical Industry, A = 1 obs, B = 2 obs, etc.

prods and incentives intended to promote efficiency.t To bring these efficiency considerat ions into prominence, we apply the CCR Projection Operator as given in (7) to obta in the results shown

in the columns headed " D E A adjusted values" in Tables A2-A4. Then, normal iz ing on the ou tput values in each year we obta in the efficiency frontiers shown in Figs 5-7. Not ing that the coordinate

tlncluded here is the increasing exposure to market forces. See the discussion of additional reforms in 1984 and other subsequent years in Perkins [24].

DEA adjusted capital to output ratio

.65

.60

.55

.50

.45

.40

.35

.30

.25

- 85 C

A

-*- A A la A H 80 70..

I I I I I .45 .58 .55 .68 .65

DEA adjusted labor to output ratio

Fig. 5. Plot of cap*/Y vs lab*/Y (DEA adjusted data). Textile Industry, A = 1 obs, B = 2 obs.

Page 10: DEA and stochastic frontier analyses of the 1978 Chinese economic reforms

94 W . W . C o o p e r et al.

DEA adjusted capital to output ratio

.65 -

.60 -

.55 -

.50 -

.45 "

.40

.35

.34

88

8 A A

D . . . . . 7 F

I I I I I I I I • 36 .38 .40 .42 .44 .46 .48 .50

DEA adjusted labor to output ratio

Fig. 6. Plot of cap*/Y vs lab*/Y (DEA adjusted data). Chemical Industry, A = 1 obs, B = 2 obs.

values of each point can be used to form a capital-to-labor ratio, we observe that a sharp change in this ratio occurs in all three industries at (or shortly after) 1978. Indeed, the change occurs immediately in the Chemical Industry and shortly thereafter (in 1979) in Metallurgy.

DEA adjusted capital to output ratio

.55(

. 5 2 -

. 5 0 0 -

.47 -

.45q -

.421 "

.40~ "

.37!

.351 -

.32! l .14

8 8 A \

87

I I I I I I 7 " .16 .18 .20 .22 .24 .26

DEA adjusted labor to output ratio

Fig. 7. Plot of cap*/Y vs lab*/Y (DEA adjusted data). Metallurgical Industry, A = 1 obs, B = 2 obs.

Page 11: DEA and stochastic frontier analyses of the 1978 Chinese economic reforms

DEA 95

The years 1970, 1980, 1981 and 1985 are when efficient production actually occurred in Textiles. The points labeled A, B . . . . . H in Fig. 5, with H representing 8 points, are obtained from the Projection Operator formula (7). The remaining parts of the solid line are obtained by using continuity and interpolating, segment by segment, to connect the endpoints identified with observed efficient production (as in our earlier discussion for Fig. 1).

Chemicals and Metallurgy experience transitions as will be clear when we discuss Table 1. However, since their graphic portrayals are less sharp, we focus on Textiles, where 1980 and 1981 mark the endpoints of the transition. These points are portrayed as efficient in Fig. 2 and, as can be confirmed in Table 1, the year 1982 has zero slack and projects (approximately) into 1980 with a value of 0* = 0.82429; hence, it is included as one of the two points in B.

Turning to the endpoints at 1970 and 1985 in Fig. 5 for Textiles, we can see that the capital-to-labor ratios change dramatically from a value of 0.4 ( - 4 9 , 2 1 8 + 122,428) in 1970 to a value of 1.45 (-311,977+214,961) in 1985. (See the values under Xz/X~ in Table A1 for a more precise derivation.) These are the efficient ratios, as determined by DEA, for the beginning and ending points of the Textiles efficiency frontier.

By returning to (1) and (4) we can examine more closely how these evaluations and projections are determined, where, as noted earlier, the evaluation of each D MU is effected from comparisons with efficient DMUs. The objective in (4) is to maximise the efficiency score for each DMU, where the efficient subset of DMUs used to effect the comparison is chosen accordingly. Thus, in the Textile example, which we are presently considering, the pre-reform performances are all evaluated by DEA relative to the segment stretching from 1970 to 1980. Consider, for instance, the H (=8) values identified with 1970 in Fig. 5. Reference to the last column of Table A2 shows that 7 of these 8 values are associated with non-zero labor slacks that occur in the years 1966-77. In other words, the performance in these years was evaluated by DEA relative to the observed performances in 1970 in the manner described for 6 in Fig. 1. (As noted below, a use of post-reform segments would have resulted in much lower efficiency ratings for these DMUs.)

Thus, as this example shows, the optimizations used in DEA orient the evaluations and projections for each DMU relative to the evidence generated from efficient production occurring in the periods most like the DMU (=per iod) being evaluated. 1982 is thus evaluated in the transition occurring from 1980 to 1981, while the performances through 1979 are evaluated relative to points on the segment from 1970 to 1980, as indicated by the positioning of the letters on these segments. Similarly, the performances for 1982 to 1988 are evaluated relative to the segment stretching from 1981 to 1985. The "overshoots" of capital represented by non-zero slacks in 1987 and 1988 are evaluated by reference to the efficient performance exhibited in 1985. It is of interest to note that this is the only case where non-zero slack occurs for capital. We would suggest that this kind of mix inefficiency may have resulted from the government pushing forward especially

Table 1. Segments and related points on unitized efficiency frontiers

Segment DMUs in the graph (from right to left)

(A) Graph of textiles 1970-80

198(~81

1981 88

(B) Graph of chemicals 1970-78

1978 86

198688

(C) Graph of metals 1974-79

1979-87

1987-88

70, 67, 68, 71, 72, 73, 74, 77 (in point H); 76; 69 and 75 (in point B), 66, 79, 78, 80

80, 82 (in point B); 81

81, 83, 84, and 86; 87; 88, 85 (in point C)

70, 68, 66, 67, 72 and 77 (in point F); 71 and 73 (in point B); 74, 69, 75, 78 (in point D)

78, 80, 79, 81, 82, 84, 83, 85, 86;

86, 87, 88

74, 68, 66, 67, 73, 70, 72 (in point G); 67, 77, 69, 71, 75, 78, 79 (in point B)

79; 80; 82, 83 (in point B) 83, 84, 85, 86, 87

87, 88

Page 12: DEA and stochastic frontier analyses of the 1978 Chinese economic reforms

96 W.W. Cooper et al.

hard with textiles at this time in attempting to create an immediate earner of hard currency. (See Cheung et al. [16].)

Table 1 records the efficiency frontier segments by dates and the projections to each segment for all three industries. As can be seen, the patterns are very similar although the transition segments are much less evident--indeed, are not visible on our g r a p h s i f o r the Metallurgical and Chemical Industries.

We have also ordered the dates in Table 1 so that their sequence conforms to the order of their appearance in the segments to which they have been projected. These within-segment sequences tend to move in the same order as the periods over which changes in these capital-labor ratios were being effected, where the consistency of the changes appears to improve in later-period segments. In this regard, note that even the less sharp, but longer lasting, transitions in Chemicals and Metals conform to what we are suggesting here.

Returning to Figs 5-7, we see a tendency for efficient production to increase the ratio of capital-to-labor in all three cases. The substitution rate prior to 1978 is readily visible only in the case of metallurgy, but this rate of substitution increases markedly in the post-reform periods for all three industries. It is noteworthy that these changes in technological structure are far from "Hicks" or "Harrod-neutral ." Moreover, they appear to be ongoing as evidenced by the highest capital-to-labor ratio arising in 1988, the concluding year of our study, and, again, for all three industries. (See the items marked ** in Table A1 of the Appendix.)

This brings us to a remark made by one of the referees, who observed that the efficiency portrayal would be quite different if evaluations were effected relative to the post-reform segments where the most efficient production occurs. With suitable constraints such evaluations could be arranged, but then the questions addressed would also differ. Here, we are concerned with changes in structure and the tendencies to efficiency under the conditions when the performance occurred. Introducing constraints--such as might now be done with the "assurance region" or "cone-ratio envelopments" in DEAr- -wou ld move the analysis to questions such as: "what efficiency losses occurred because

tSee Thompson et al. [28] and Charnes et al.[11].

Output Hundred Million (1980) Yuan

60

50

40

30

20

10

slope = 3.2

O o// 0 °

0 o slope = 0.4

I I I I I I I I I I I I I | I I I I I I I I I

66 70 75 77 80 85 88

Fig. 8. Textile Industry output 1966-77; 1977-88.

Year

Page 13: DEA and stochastic frontier analyses of the 1978 Chinese economic reforms

DEA 97

the reforms were not instituted at an earlier date?" Introducing the constraints needed to address such questions would also have required use of value judgements while introducing more a priori

structure than we sought to employ. We thus avoid undertaking this task in the present paper.

O U T P U T EFFECTS

Since the normalizations used in the preceding figures obscure output behavior, we now turn to this topic. First, we continue with the example of Textiles, obtaining the very simple mathematical representation in Fig. 8. Here, the emphasis is on actual data (which is not efficiency adjusted) and visual fits. We next consider formal statistical developments in which stochastic frontier regressions are utilized and compared with output-oriented projections for DEA obtained from (8), above. These DEA and SF comparisons refer to efficiency-adjusted estimates and are preceded by a brief introductory discussion of how the stochastic frontier estimates are obtained. Original (unadjusted) output data are also included for added perspective in these comparisons, as given in Figs 9-11.

Figure 8 is arranged so that 1977 serves as the terminal year for pre-reform output values in the periods beginning with 1966. The second linear segment covers the behavior of output from 1978

O u ~ u t 600000~

550000

500000

450000

4000O0

350000

3OO000

250000

200000

150000

100000

J

J j °

YD: DEA adjusted output noted as x--x YC: Stochastic Frontier output noted as

Y: Original output noted as ........ Units used are 1(I.000 Yuan adjusted to 1980 prices

Fig. 9. Comparison of YC, YD and Y for the Textiles Industry.

1 Year

SEP$ 29/2--B

Page 14: DEA and stochastic frontier analyses of the 1978 Chinese economic reforms

98 W.W. Cooper et al.

600000-

500000

400000

500000.

200000

100000.

1960

~ut'put

I

/

/ /

' / /

J 1

~..../ , , ° , , ~ 1 , | , , , i , 0 , , ,

1970 1978 1980

YD: DEA adjusted output noted as ×--×

YC: Stochastic froniter output noted as o---o

Y: Original output noted as ........ Units of 10,000 Yuan ajusted to 1980 prices

1990 YEAR

Fig. 10. Comparison of YC, YD and Y for the Chemical Industry.

to 1988. The change in slope for the two lines is dramatic. The observations for the period 1978-88 all refer to a line showing an 8-fold increase in slope over the line representing output levels for the period 1966-77. Note that in both periods, behavior of the actual (unadjusted) data conforms closely to the representations by their respective linear segments.

We next turn to statistical confirmation by using SF regressions. These are output-oriented in that a//losses in efficiency are assumed to occur only in the outputs and, accordingly, corrections for inefficiency are effected only in the outputs. Because we also wish to use these regressions as a confirmatory check on DEA, the output oriented adjustments of (8) are called in here, where 0* is replaced with its reciprocal ~b*.

Unlike DEA, which is non-parametric, we need to consider the parametric forms that might be used for our SF regressions. Since our objective is merely to observe if confirmation occurs, we

Page 15: DEA and stochastic frontier analyses of the 1978 Chinese economic reforms

D E A 99

O u ~ u t

4OOOOO

I000{~

Is •/a s.

B - - . ~ o t

I I I | I , i , i , , , , i i i i i i i i i i i i i i i i / Y c ~ . r

YD: DEA adjusted output noted as x m x

YC: Stochastic Frontier output noted as

Y: Original output noted as ........

Units used are 10,000 Yaan adjusted to 1980 prices

Fig. 11 Comparison of YC, Y D and Y for the Metallurgy Industry.

confine our SF approach to the very simple one embodied in the following (static) Cobb-Douglas production function:t

where

Yi = a c i l i e , (9)

Yt = observed output (in 10,000 yuan), ci = amount of capital (10,000 yuan), Ii = amount of labor (total number of employees),

for each of i = 1 . . . . . m observations. We should perhaps emphasize that our use of this simple regression allows us to bring the

behavior of the data into view without obscuring those effects that might otherwise be attributed

tOther approaches have now been developed that make joint use of DEA and statistical regressions. While such at tempts seek to improve performance, we here confine our attention to confirmatory uses in which complete independence of each approach is preserved. See Arnold et al. [4] and Bardhan et a/.[8].

Page 16: DEA and stochastic frontier analyses of the 1978 Chinese economic reforms

100 W.W. Cooper et al.

to additional structures imposed on the regression relation. As noted below, the results are striking. Importantly, we are able to more easily compare results from these regressions fitted across a// periods with the per iod-by-per iod "best fits" utilized in DEA.

In SF estimation, the error term Ei is assumed to be composed of two elements, viz.

E i ~ 1.3 i - - U i

with

ui>~O,i = 1 . . . . . m, (10)

where v is unrestricted in sign. This additive expression has led to its being named a "composed error analysis." Following usual practice, we assume v~ is noise, which is statistically represented by the normal distribution, N(0, 0.~). Further, the u~/> 0 represent inefficiencies, which are assumed to behave in accordance with the half-normal distribution, N(0, 0.2u). (See Aigner et al. [3] and Aigner et al. [2].)

Following Aigner et al. [3], the density function for E is formulated as the sum of a symmetric normal random variable and a truncated normal variable:

with

where

and

- ~ < E ~<oo,

0.2 = 2 0.2 0. u "1- ~,

2 = 0.u/0.,., (11)

f* ~ standard normal density,

F*,,~ cumulative normal distribution function.

The values? in Table 2 are maximum likelihood estimates for these parameters obtained from our data by maximizing the following log-likelihood function:

- i 1 E~. (12) l n L ( y L ~ ' f l ' ~ ' 0 - ) = c ° n s t ' + n l n 0 . - I + ~ = l l n 1--F*(Ei20. ) - -~-~2i

f T h e s e p a r a m e t e r values were o b t a i n e d f r o m the " F r o n t i e r P r o d u c t i o n M o d e l " c o d e descr ibed o n pp. 72 a n d 79 o f T S P Vers ion 4.1 (1983: T S P In t e rna t iona l ) .

Table 2. Maximum likelihood function estimates

Textiles Chemicals Metallurgicals

Parameter Estimate t-statistic Estimate t-statistic Estimate t-statistic

a (constant) 1.2154 1.7797 1.1412 0.1248 -0 .5158 -0 .7817

ct 0.3613"** 32.0319 0.4379"t" 2.3726 0.5932,1, 2.4546

fl 0.6796*** 36.1484 0.6207* 2.7491 0.8064 1.7126

a 9.06* 2.8446 10.1218"t" 2.2717 3.4645 ** * 4.1697

2 6.90* 2.7303 1.80 0.5393 9.4206 0.3139

***statistical significance P < 0.002. **statistical significance P < 0.01. *statistical significance P < 0.05. ?statistical significance P < 0.1.

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DEA 101

Following Jondrow et al. [23], our estimates of inefficiencies are obtained from the mode of u given E, i.e.

u * = M ( u J ~ ) = ~ - - E _ (s2u/a2) ifE~<0,

t o if~ >0 ,

where

,j 20-2 0"2 ~--- .~20"~ = ~2 (0 "2 - - 0"2u) - - ( 1 3 )

(1 + 22) '

We next turn to output behavior or, more precisely, "efficiency adjusted output behavior" for both the DEA and composed error (stochastic frontier) approaches. This is done in Tables A5-A7 in the Appendix, where Yc = Y eu" gives the efficiency-adjusted outputs for the composed error approach and Yo = Y~*, with ~* = 1/0", gives the efficiency-adjusted outputs for DEA. In both cases, Y represents the original output values in units of 10,000 Yuan adjusted to 1980 prices transferred from Tables A2-A4 to Tables A5-A7.

Figures 9-11 allow us to graphically compare the results obtained from the DEA and SF approaches both with each other and with the observed outputs for the indicated years. The actual observations (represented as -), are of course, never above either the DEA adjusted values represented as x - - x ), or the SF adjusted values (represented as • - - 0 ) . Although there are discrepancies, the results from DEA and SF generally tend toward confirmation. For instance, both show large output inefficiencies for Metallurgicals, especially in the earlier years, with the estimates of inefficiency appearing to be larger for SF than for DEA. Turning to Textiles, however, the behavior reverses, with DEA showing larger inefficiency estimates than SF. Finally, the case of Chemicals may be characterized as showing the SF and DEA estimates to be close over the entire period 1966-88. This consistency in results increases our confidence that such behavior is "in the observations" and not simply a reflection of our models and pre-conceived hypotheses or beliefs.

The SF and DEA efficiency-adjusted outputs, as well as the actual data, show strikingly large increases in output starting in 1978, and continuing thereafter, in all three industries. Since this finding is consistent with Fig. 8, we conclude that the changes in capital and output ratios previously described are reflected in large output increases. This, in turn, further reinforces our previous observations that the reforms initiated in 1978 significantly improved economic perform- ance.

SENSITIVITY CHECKS AND EFFICIENCY CHANGES

In referring to studies by others, we learn that our results generally differ from previous findings. For instance, Cheung et al. [16, p. 24], observe that "most studies on the industrial sector conclude that the 1978 reform had no effect on productivity." In fact, using a Cobb-Douglas function to synthesize a total factor productivity index, Cheung et al. conclude (p. 25) "that the 1978 reform reduced output variability but did not improve total factor productivity much." However, the methods used by Cheung et al., as well as those they cite, are not designed to distinguish between efficient and inefficient behavior. Their conclusions therefore rest on mixtures of efficient and inefficient performance and provide little benchmark against which performance can be checked. Indeed, Cheung et al. [16] even fail to explain those improvements in total factor productivity that are apparent in their own Fig. 4. Finally, the regressions used by Cheung et al. [16] produce parameter estimates from a single optimization. Thus, in contrast to the multiple optimizations used in DEA, their methods fail to distinguish sharply between the opportunities for improved performance that should be possible in different periods.

In addition to these differences in methods, allowance must be made for other variances between our study and the one by Cheung et al. [16]. For instance, they covered only cotton yarn production. Differences in periods covered may also help to explain discrepancies--although this does not seem to loom very large since their study covers the periods we consider (and more) except that their data terminate in 1986 while ours end with 1988.

More generally, we must attend to the possibility that variations in the periods covered in our own DEA analyses might also differ since our measures of relative efficiency can change if some

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102 W. W . C o o p e r et al.

T E X T I L E S

M Employees

15o •

140 ,

130 •

120-

110 -

10o-

90-

80

0

(Labor)

+ + +

+

+

66 67 68 69 70 71 72 73 74 75

+

++ // +

"', ' Y e ~ I ' S

76 77 78 79 80

+ observed value; - - efficiency adjusted values

10MM (1980) Yuan

90-

80-

70-

60-

5 0 +

30 •

0 66 67 68 69 70 71

(Capital)

+

+ +

' Y e a r s

72 73 74 75 76 77 78 79 80

+ observed value; - - efficency adjusted value

Fig. 12. Observed and efficiency adjus ted inputs , 1966-80.

of the periods (especially, the more efficient post-reform periods) were eliminated. To check the latter possibility, we asked six teams of students in a course on DEA at Aoyama Gakuin University, Tokyo, to select different 15 yr periods (not necessarily contiguous). The question posed was: did their results differ from results for the 23 yr period (1966-80) used in our study? No substantial differences were obtained from these studies.

Figure 12, reported by one of these teams, allows us to draw forth further points of interest. A surge in input levels occurs for both labor and capital beginning in 1978. The increases, which continue thereafter, were so large that attempts at graphic portrayals lost significant detail. However, these continuing increases can be confirmed from Table A2 in the Appendix.'}" Moving to Tables A3 and A4, this surge is also evident in the Chemical and Metallurgical industries. Hence, we conclude that the substitution possibilities in favor of capital, discussed earlier, were accompanied by large increases in the use of labor as well as capital.

t N u m e r i c a l es t imates may differ f rom those shown in Fig, 12 because of differences in the per iods covered, but they are relat ively small (see Table A2). (We are grateful to I samu K a t e for a l lowing us to use these results.)

Page 19: DEA and stochastic frontier analyses of the 1978 Chinese economic reforms

D E A 103

Table 3. Inefficiency components in textiles

M

Year F M F + M

1966 19,206 0 0 1967 5967 10,369 0.64 1968 11,660 9380 0.45 1969 5334 0 0 1970 0 0 1971 1874 18,651 0.91 1972 3828 20,253 0.85 1973 5743 6507 0.53 1974 8068 1489 0.16 1975 8230 0 0 1976 12,065 0 0 1977 15,980 4436 0.22

Total 97,955 71,085 - -

The differences between observed and efficiency adjusted values in Fig. 12 show that the usage of capital was relatively efficient during the period 1966-88, representing the first (pre-reform) segment for efficiency evaluations in Fig. 5. This is not true for labor, however, where substantial parts of these labor surpluses are represented by non-zero slacks in Table A2. The same is true in Tables A3 and A4. This suggests that mix inefficiencies were present in all three industries, and that these inefficiencies in mix subsequently disappeared except for the capital surpluses, as previously noted for Textiles in 1987 and 1988.

To obtain measures for the relative importance of these mix inefficiencies, we adapt the following formulas from Ref. [7], where they form components in a measure of efficiency covering outputs as well as inputs:

Xio - (O*xio - s ~-*) = (Xio - O*x io) + s~-* = F + M . (14)

Here, F (for Farrell) and M (for mix) represent the corresponding algebraic expressions for the amounts of each of these two inefficiencies. Using these formulas, we can tabulate the following values for inefficiencies in manpower used in the pre-reform years. Thus, referring to Table A2 for Textiles, we find that a total inefficiency of 169,040(=97,955 + 71,085) man years contains a mix component of 71,085, or approx. 42%, of this total (See Table 3).

We can perhaps underscore the importance of mix as a source of inefficiency during these years by noting that this proportion would likely be increased substantially if some meaningful way could be found to introduce price or cost considerations in analyzing allocative and other types of inefficiencies. In any case, this is the first study we have seen that, at the least, seeks to separate mix from other sources of inefficiency, and to then estimate the amounts of each.

Referring, now, to the 0 ( = theta) values, we note an upward trend, in general, with retrogressions tending to occur when output levels fall. In all cases, however, the 0 * values are substantially higher for the ending, compared with the beginning, periods. Putting this together with the elimination of mix inefficiencies resulting from reductions in labor surpluses, we conclude that the structural changes also resulted in improved performance efficiencies in all three industries. We also note that these improvements occurred in association with the higher output levels of the post-reform periods. (See Figs 9-11.)

SUMMARY AND CONCLUSIONS

Such drastic changes in structure, accompanied by improvements in efficiency with accompany- ing increases in output and employment of labor and capital, present a remarkable record of achievement from the reforms initiated in 1978. A population of 1.15 billion with more than 14 million new entrants into the labor force each year, however, means that there is "still a long way to go." (See Refs [24, 32].) It would be of interest, of course, to extend our analysis beyond the three industries studied here. (See Table 1 of Perkins [24, p. 24] for data on the growth of Gross Domestic Product (GDP) per capita in China starting in 1978.) It would also be of interest to proceed in the opposite direction, so as to move to plant, as well as industry, levels. Our guess is that a study along the latter lines would reveal that increments in resources subsequent to 1978

Page 20: DEA and stochastic frontier analyses of the 1978 Chinese economic reforms

104 W.W. Cooper et al.

were deployed from less efficient to more efficient plants, where this added to the improvements found in all plants.

A question as to sources for these resource increments naturally arises when looking to the future. A labor force of nearly 600 million would seem more than adequate, especially when accompanied by spare capacity in the range of 100 million underemployed. (See Yusuf [36, p. 79].) Concerning capital increments, we also turn to Yusuf [36, p. 81], who notes that foreign sources of investment have only recently become significant. They were not important during the period covered by our study. Indeed, the high propensity to save in China, and the accompanying developments in investment and economic activity have begun to attract attention, even in the popular press.t Consider, for example, the following excerpt from an article on page A8 of the 16 March 1992, Wall Street Journal:

"The government wants to get a piece of the country's savings glut into circulation. Chinese people currently save about 35 cents of each $1 earned. At the end of 1991, savings deposits and cash held by individuals were estimated to total more than $205 billion, or more than the total annual wages of every person of working age in China."

Further, Trends in Developing Countries 1993 ( W orm Bank [34, p. 110]) reports that Gross Domestic Savings/GDP reached a level of 40.3% in 1990 having risen from a level of 32.2% in 1980. As Yusuf [36, p. 91] notes, this "high rate of saving has not only helped sustain extraordinary rates of investment but it has also been the key to stability as well as the ability to absorb shocks." Apparently, savings patterns have carried over from a time when there were limited opportunities for individuals to buy consumer goods and limited opportunities to invest. Again, quoting from Yusuf [36, p. 77]:

"As reforms progressed, the multiplication of profitable opportunities and the difficulty of checking investment hunge r . . , pushed gross investment to the level of 35-37 percent [of G D P ] . . . and it seems clear that China's growth from 1978 onward has been investment led and that increased capital investment has probably been responsible for about 40 percent of growth. This is higher than in Japan during 1960-70 (32 percent) but in about the same range as South Korea and Singapore."

Our own analysis suggests that reductions in mix inefficiencies and increases in labor usage were also important. In any event, it is now becoming generally recognized that economic reforms in China have unleashed a remarkable period of growth, with recent studies by the World Bank (among others) indicating that China may well become the biggest of the "Asian Tigers"--with a potential for tremendous impact on world economic performance.

tRecognition by the press of these extraordinary developments has now become much more extensive.

Acknowledgements--This paper has benefited from comments received in presentations at seminars in (1) Canterbury University, Christchurch, New Zealand (June, 1991), (2) The OR Society of Japan and Aoyama Gakuin University in Tokyo (June, 1992), (3) McGill University in Montreal, Canada (May, 1993) and (4) the City Polytechnic University of Hong Kong (June, 1993). Acknowledgments are also due Barnett Parker and two anonymous referees. Support for this research by the IC2 Institute of the University of Texas, by Rice University, and by the Tianjin Institute of Scientific and Technical Information of the People's Republic of China are all gratefully acknowledged.

REFERENCES

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2. D. Aigner, T. Amemiya and D. J. Poirier. On the estimation of production frontiers: maximum likelihood estimation of the parameters of a discontinuous density function. Econ. Rev. 17, 377-397 (1976).

3. D. Aigner, K. Lovell and P. Schmidt. Formulation and estimation of stochastic frontier production models. J. Economet. 6, 21-37 (1977).

4. V. Arnold, I. Bardhan, W. W. Cooper and S. Kumbhakar. New uses of DEA and statistical regressions for efficiency evaluation and estimation--with an illustrative application to public secondary schools in Texas. The Ann. of Opers Res. (to appear), (1995).

5. Asian Development Bank. People's Republic of China. In Asian Development Outlook 1993. Oxford University Press (1993).

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6. R. D. Banker. Maximum likelihood, consistency and DEA: statistical foundation. Mgmt Sci. 39, 1265-1273 (1993). 7. R. D. Banker and W. W. Cooper. Validation and generalization of DEA and its uses with accompanying commentaries

by E. Grifell-Taije, J. T. Pastor, P. W. Wilson, E. Ley and C. A. K. Lovell. TOP 2, 249-309 (1994). 8. I. Bardhan, W. W. Cooper, and S. Kumbhakar. A simulation study of joint uses of DEA and statistical regressions

for production function estimation and use in efficiency evaluations. Working Paper, Graduate School of Business and IC 2 Institute, The University of Texas at Austin (1994).

9. A. Charnes and W. W. Cooper. Preface to topics in data envelopment analysis. Ann. Opers Res. 2, 59-112 (1985). 10. A. Charnes, W. W. Cooper, B. Golany, L. Seiford and J. Stutz. Foundations of data envelopment analysis for

Pareto--Koopmans efficient empirical production functions. J. Economet. 30, 91-107 (1985). 11. A. Charnes, W. W. Cooper, Z. Huang and B. Sun. Polyhedral cone-ratio DEA models with an illustrative application

to large commercial banks. J. Economet. 46, 73-91 (1990). 12. A. Charnes, W. W. Cooper and S. Li. Using data envelopment analysis to evaluate relative efficiencies in the economic

performance of Chinese cities. Socio-Econ. Plann. Sci. 23, 235-244 (1989). 13. A. Charnes, W. W. Cooper and E. Rhodes. Measuring efficiency of decision making units. Eur. J. Operl Res. 6, 429~,44

(1978). 14. A. Charnes, W. W. Cooper and T. Sueyoshi. A goal programming/constrained regression review of the Bell system

breakup. Mgmt Sci. 34, pp. 1-26. See also the rejoinder by D. S. Evans and J. J. Heckman in this same issue (1988). 15. A. Charnes, W. W. Cooper and R. M. Thrall. A structure for classifying and characterizing efficiency and inefficiency

in data envelopment analysis. The J. Product. Anal. 2, 197-237 (1991). 16. K. Cheung, S. Archibald and M. Faig. Impact of central planning on production efficiency: the case of the cotton yarn

industry in China. J. Comparative Econ. 17, 23-42 (1993). 17. Chinese Statistical Publishing House. Yearbook of China's 40 Years. Beijing (In Chinese) (1989). 18. W. W. Cooper. Data envelopment analysis. Encyclopedia of Operations Research and Management Science. Kluwer

Academic (1995). 19. W. W. Cooper and A. Gallegos. A combined DEA-stochastic frontier approach to Latin American airline efficiency

evaluations. Working Paper, University of Texas at Austin, Graduate School of Business (1991). 20. W. W. Cooper, K. Tone, H. Takamori and T. Sueyoshi. Data Envelopment Analysis: survey and interpretations.

Communications of the Operations Research Society of Japan (August, September, and October issues). In Japanese. English versions available on request from any of the authors at, respectively, The University of Texas at Austin, Saitama University, Aoyama Gakiun University and Ohio State University (1994).

21. M. J.. Farrell. The measurement of productive efficiency. J. R. Statist. Soc. Series A 120, 253-290 (1957). 22. G. H. Jefferson and T. G. Rawski. Enterprise reform in Chinese industry. J. Econ. Perspectives 5, 47~0 (1994). 23. J. Jondrow, C. A. Knox Lovell, I. S. Materov and P. Schmidt. On the estimation of technical inefficiency in the

stochastic frontier production function model. J. Economet. 19, 233-238 (1982). 24. D. Perkins. Completing China's move to market. J. Econ. Perspect. Spring, 23-46 (1994). 25. L. M. Seiford and R. M. Thrall. Recent developments in DEA--the mathematical programming approach to frontier

analysis. J. Economet. 46, 7-38 (1990). 26. R. G. Thompson, P. S. Dharmapala and R. M. Thrall. DEA sensitivity analysis of efficiency measures with applications

to Kansas farming and Illinois coal mining. Data Envelopment Analysis: Theory, Methodology and Applications. Kluwer Academic, Norwell, MA (1995).

27. R. G. Thompson, L. N. Langemeier, Ch.-T. Le, E. Lee and R. M. Thrall. The role of multiplier bounds in efficiency analysis with applications to Kansas farming. J. Economet. 46, 93-108 (1990).

28. R. G. Thompson, E. Lee and R. M. Thrall. DEA/AR-efficiency of U. S. independent oil/gas producers over time. Computers Opers Res. 19, 377-391 (1992).

29. R. G. Thompson, R. D. Singleton, R. M. Thrall and B. A. Smith. Comparative site evaluations for locating a high-energy physics laboratory in Texas. Interfaces 16, (1986).

30. G. M. Tidrick and W. A. Byrd. Factor allocation and enterprise incentives. China's Industrial Reform. Oxford University Press (1987).

31. A. G. Walder. Industrial reform in China. The Limits of Reform in China. Westview Press, Boulder, CO (1988). 32. A. G. Walder. Factory and manager in an era of reform. China Q. 118, 242-264 (1989). 33. World Bank. Social Indicators of Development 1993. The John Hopkins University Press, Baltimore, MD (1993). 34. World Bank. Trends in Developing Countries 1993. The World Bank, Washington, DC (1993). 35. W. Yanrui. The measurement of efficiency: a review of the theory and empirical applications to China. Chinese

Economy Research Unit Report No 93/1. University of Adelaide, Australia (1993). 36. S. Yusuf. China's macroeconomics performance and management during transition. J. Econ. Perspect. 8, 71-92 (1994).

A P P E N D I X

S o m e Theorems

T a b l e A1, in the set tha t fo l lows, will be used to i l lus t ra te the a l g o r i t h m we d e v e l o p herein . T h e

n u m e r i c a l va lues in this tab le resul t f r o m n o r m a l i z i n g the d a t a in Tab le s A2, A3 and A 4 on the i r

r espec t ive ou tpu t s . T u r n i n g to 1966 in T a b l e A2 fo r texti les, as an example , we ob ta in :

Xl/Y = 0.73648 = 106,910/145,163,

where Xl is the value o f labor in row 1 and Y is the value of output in that same row. We similarly obtain:

X 2 / Y = 0.33068 = 48,002/145,163

as the ratio of capital-to-output in this same year.

SEPS 29/2~C

Page 22: DEA and stochastic frontier analyses of the 1978 Chinese economic reforms

106 W.W. Cooper et al.

Table A1. The ratios of lab/output = XL/Y, cap/output = X2/Y and cap/lab = X2/X t for testiles, chemicals and metallurgicals

Textiles Chemicals Metallurgicals Observation number Year Xt/Y X2/Y X2/X I XI/Y X2/Y X2/XI XI/Y X:_/Y X2/X I

1 1966 0.73648 0.33068 0.44899 0.58862 0.49438 0.83990 0.48911 0.51908 1.06126 2 1967 0.88427 0.31502 0.35625 0.68716 0.47402 0.68983 0.51117 0.54705 1.07019 3 1968 0.82191 0.29935 0.36422 0.70049 0.45515 0.64976* 0.59297 0.64723 1.09151 4 1969 0.64786 0.28369 0.43789 0.48331 0.40242 0.83262 0.49149 0.64648 1.31535 5 1970 0.66671 ~ 0.40202 0.49840 0.78830 0.57852 0.48831 0.84407* 6 1971 0.79476 0.27203 0.34228 0.48558 0.39640 0.81635 0.38543 0.52206 1.35449 7 1972 0.81550 0.27603 0.33849* 0.50983 0.40080 0.78616 0.34327 0.34193 0.99610 8 1973 0.73395 0.28004 0.38155 0.49246 0.40296 0.81827 0.31221 0.35333 1.13172 9 1974 0.71494 0.28404 0.39730 0.48159 0.40054 0.83170 0.26584 ~ 1.24302

10 1975 0.65744 0.28805 0.43814 0.47549 0.39874 0.83858 0.24175 0.37259 1.54123 11 1976 0.70807 0.29460 0.41606 0.54847 0.39986 0.72904 0.30943 0.39225 1.26766 12 1977 0.77571 0.30115 0.38822 0.58099 0.41016 0.70598 0.30572 0.40078 1.31094 13 1978 0.54785 0.30770 0.56165 0.45919 0.39373 0.85745 0.21131 0.40027 1.89420 14 1979 0.51949 0.28446 0.54758 0.45403 0.42190 0.92923 0.19043 0.42404 2.22679 15 1980 0.47662 0.27788 0.58302 0.50547 0.43855 0.86760 0.18986 0.43911 2.31276 16 1981 0.46183 0.29107 0.63026 0.49465 0.46306 0.93615 0.19800 0.44713 2.25827 17 1982 0.56102 0.35246 0.62826 0.47108 0.47290 1.00387 0.19151 0.45070 2.35343 18 1983 0.54695 0.38714 0.70781 0.45199 0.49021 1.08456 0.20182 0.45946 2.27656 19 1984 0.47059 0.59457 1.26345 0.47539 0.50376 1.05967 0.18105 0.47382 2.63014 20 1985 ~ 0.63403 1.45132 0.41223 0.54109 1.31260 0.17436 0.53084 3.04450 21 1986 0.47065 0.67775 1.44003 0.38895 0.56071 1.44162 0.15689 0.51256 3.26694 22 1987 0.45264 0.71651 1.58297 0.37408 0.61743 1.65054 0.14826 0.52980 3.57348 23 1988 0.44274 0.73785 1.66656"* ~ 0.66788 1.93610"* ~ 0.54009 3.66985*

*The lowest ratio value. **The highest ratio value.

Proceeding in this manner, year by year, we obtain the normalized coordinates used for the figures portrayed for each industry in the text of the article. To locate the unitized frontiers and the 0* and slacks for these figures, we introduce an algorithm based on the following theorems.

We start informally by referring back to Fig. 1 and note that the endpoints of the "efficiency frontier," which occur at the points for 1 and 4, are associated with the lowest input-to-output ratios. Formally, we have the following:

Capital to output ratio

.7 87A / "

"A ,,, 86

8 5 .6 t - A 84

.5

.4

. 3 --

.2 .48

m= .58 / / m / = . 6 3 / ~' m = .40

/ t \ , - . . , -

8 1 ~ " i A , ,A~9 A 78 69 75 A __ . . . " A A . ~ / ~ A A A ~3 8 0 ~

6-~ 7o F~ I I I I I I I I .45 .58 .55 .68 .65 .78 .75 .88

Labor to output ratio

5 Segments F l . . . F$

Fig. A I . C C R pro jec t ions for textiles.

I . 85

Page 23: DEA and stochastic frontier analyses of the 1978 Chinese economic reforms

DEA

Table A2. Observations and DEA results for the Textile Industry

107

Observations DEA adjusted values DEA efficiency scores

Output Inputs Inputs Score Slacks Observation number Year Y Cap Lab CAPADJ LABADJ Theta SLACAP SLALAB

I 1966 145,163 48,002 1 0 6 , 9 1 0 39,378.44 87,703.62 0.82035 0 0 2 1967 121,057 38,135 1 0 7 , 0 4 7 32,446.78 80,710.87 0.85084 0 10,369 3 1968 135,572 40,584 1 1 1 , 4 2 8 36,337.29 90,388.17 0.89536 0 9380 4 1969 177,849 50,454 1 1 5 , 2 2 2 48,118.48 109,888.37 0.95371 0 0 5 1970 183,630 49,218 1 2 2 , 4 2 8 49,218.00 122,428.00 1.00000 0 0 6 1971 160,291 43,604 1 2 7 , 3 9 3 42,962.59 106,868.05 0.98529 0 18,651 7 1972 161,853 44,677 131,991 43,381.37 107,910.26 0.97100 0 20,253 8 1973 181,968 50,958 1 3 3 , 5 5 5 48,772.41 121,319.83 0.95711 0 6507 9 1974 188,066 53,419 1 3 4 , 4 5 6 50,407.24 125,386.37 0.94362 0 1489

10 1975 206,317 59,430 1 3 5 , 6 4 2 55,824.38 127,412.60 0.93933 0 0 11 1976 196,584 57,914 1 3 9 , 1 9 5 52,894.01 127,129.58 0.91332 0 0 12 1977 187,317 56,410 1 4 5 , 3 0 3 50,206.03 124,886.58 0.89002 0 4436 13 1978 229,308 70,558 1 2 5 , 6 2 7 63,522.66 113,100.73 0.90029 0 0 14 1979 258,529 73,542 1 3 4 , 3 0 3 71,462.23 130,504.91 0.97172 0 0 15 1980 313,734 87,180 1 4 9 , 5 3 3 87,180.00 149,533.00 1.00000 0 0 16 1981 3 6 1 , 1 5 5 1 0 5 , 1 2 3 166 ,794 105,123.00 166,794.00 1.00000 0 0 17 1982 3 4 7 , 2 2 9 122,385 t94,801 100,880.73 160,572.52 0.82429 0 0 18 1983 369,631 143,098 202 ,171 120,179.42 169,791.29 0.83984 0 0 19 1984 4 6 5 , 2 3 5 2 7 6 , 6 1 5 2 1 8 , 9 3 7 260,007.04 205,792.02 0.93996 0 0 20 1985 4 9 2 , 0 5 3 3 1 1 , 9 7 7 214 ,961 311,977.00 214,961.00 1.00000 0 0 21 1986 4 9 5 , 8 2 2 3 3 6 , 0 4 2 2 3 3 , 3 5 8 312,152.77 216,768.58 0.92891 0 0 22 1987 5 2 3 , 0 3 6 3 7 4 , 7 6 2 2 3 6 , 7 4 6 331,619.54 228,495.40 0.96515 30,082 0 23 1988 5 3 1 , 0 5 9 3 9 1 , 8 4 2 2 3 5 , 1 2 0 336,709. t8 232,002.31 0.98674 49,937 0

Endpoint theorem For the case o f two inputs and one output, the right and left extreme rays of the efficiency frontier

are generated by, respectively, DMUR and DMUL, where R and L are defined by

X2_..~ R = min x2j, YR J Yj

and

XIL min xjj YL J Yj

These two D M U s are extreme efficient--i.e, they are associated with extreme points which are also efficient.

Table A3. Observations and DEA results for the Chemical Industry

Observations DEA adjusted values DEA efficiency scores

Output Inputs Inputs Score Slacks Observation number Year Y Cap Lab CAPADJ LABADJ Theta SLACAP SLALAB

I 1966 116,884 57,785 68,800 45,997.44 54,765.49 0.79601 0 0.0 2 1967 100,354 47,570 68,959 39,427.92 50,016.98 0.82884 0 7139.0 3 1968 102,307 46,565 71,665 40,195.37 50,989.94 0.86321 0 10,872.0 4 1969 160,948 64,768 77,788 63,324.32 76,054.11 0.97771 0 0.0 5 1970 171,774 67,488 85,612 67,488.00 85,612.00 1.00000 0 0.0 6 1971 195,428 77,468 94,896 76,851.35 94,140.63 0.99204 0 0.0 7 1972 197,665 79,225 1 0 0 , 7 7 5 77,660.3l 98,516.09 0.98025 0 268.6 8 1973 209,838 84,557 1 0 3 , 3 3 6 82,523.40 100,850.77 0.97595 0 0.0 9 1974 219,256 87,820 1 0 5 , 5 9 1 86,306.86 103,771.67 0.98277 0 0.0

10 1975 232,676 92,777 1 1 0 , 6 3 6 91,561.62 109,186.67 0.98690 0 0.0 11 1976 209,057 83,593 114,661 82,135.97 104,013,76 0.98257 0 8648.7 12 1977 214,105 87,818 1 2 4 , 3 9 2 84,119.11 106,709.41 0.95788 0 12,443.2 13 1978 2 7 2 , 9 3 5 1 0 7 , 4 6 3 125 ,328 107,463.00 125,328.00 1.00000 0 0.0 14 1979 2 7 8 , 1 4 2 1 1 7 , 3 4 8 126 ,285 116,104.11 124,946.38 0.98940 0 0.0 15 1980 2 8 6 , 4 9 9 1 2 5 , 6 4 4 144 ,818 113,781.95 131,145.73 0.90559 0 0.0 16 1981 3 0 9 , 7 1 7 1 4 3 , 4 1 9 153 ,201 129,974.90 138,839.94 0.90626 0 0.0 17 1982 3 3 8 , 9 8 9 1 6 0 , 3 0 9 159 ,691 149,494.55 148,918.25 0.93254 0 0.0 18 1983 3 5 4 , 5 1 5 1 7 3 , 7 8 8 160 ,238 164,971.73 152,109.13 0.94927 0 0.0 19 1984 373,221 188,013 177 ,426 170,918.86 161,294.43 0.90908 0 0.0 20 1985 4 1 0 , 7 2 0 2 2 2 , 2 3 6 169 ,310 217,017.90 165,334.60 0.97652 0 0.0 21 1986 4 3 2 , 5 1 4 2 4 2 , 5 1 7 168 ,225 242,517.00 168,225.00 1.00000 0 0.0 22 1987 4 6 4 , 5 5 6 286,833 173,781 282,989.44 171,452.33 0.98660 0 0.0 23 1988 5 0 8 , 8 2 9 3 3 9 , 8 3 9 175 ,528 339,839.00 175,528.00 1.00000 0 0.0

Page 24: DEA and stochastic frontier analyses of the 1978 Chinese economic reforms

108 W. W. Cooper et al.

Table A4. Observations and DEA results for the Metallurgy Industry

Observations DEA adjusted values DEA et~ciency scores

Output Inputs Inputs Score Slacks Observation number Year Y Cap Lab CAPADJ LABADJ Theta SLACAP SLALAB

1 1966 51,060 26,504 24,974 16,872.45 13,573.74 0.63660 0 2324.71 2 1967 50,087 27,400 25,603 16,550.97 13,315.11 0.60405 0 2150.38 3 1968 46,259 29,940 27,430 15,285.87 12,297.34 0.51055 0 1707.05 4 1969 59,358 38,374 29,174 20,168.61 15,333.27 0.52558 0 0.00 5 1970 64,959 31,720 37,580 21,465.24 17,268.58 0.67671 0 8162.18 6 1971 70,446 36,777 27,152 24,276.50 17,923.04 0.66010 0 0.00 7 1972 89,592 30,634 30,754 29,605.00 23,816.94 0.96641 0 5904.03 8 1973 112,588 39,781 35,151 37,203.99 29,930.29 0.93522 0 2943.63 9 1974 121,013 39,988 32,170 39,988.00 32,170.00 1.00000 0 0.00

10 1975 132,093 49,216 31,933 48,320.27 31,351.82 0.98180 0 0.00 11 1976 109,393 42,909 33,849 36,502.69 28,795.34 0.85070 0 0.00 12 1977 119,411 47,857 36,506 40,507.12 30,899.41 0.84642 0 0.00 13 1978 171,228 68,538 36,183 68,315.25 36,065.41 0.99675 0 0.00 14 1979 185,703 78,746 35,363 78,746.00 35,363.00 1.00000 0 0.00 15 1980 196,998 86,504 37,403 85,213.36 36,844.95 0.98508 0 0.00 16 1981 199,055 89,003 39,412 85,036.14 37,655.41 0.95543 0 0.00 17 1982 208,897 94,149 40,005 91,157.89 38,734.04 0.96823 0 0.00 18 1983 2 2 6 , 1 8 7 103,925 45,650 97,036.85 42,624.32 0.93372 0 0.00 19 1984 2 2 8 , 1 0 9 108,083 41,094 105,280.41 40,028.43 0.97407 0 0.00 20 1985 2 3 5 , 9 8 1 125,269 41,146 116,663.02 38,319.27 0.93130 0 0.00 21 1986 2 6 8 , 5 4 8 137,646 42,133 136,977.04 41,928.23 0.99514 0 0.00 22 1987 2 8 9 , 8 2 4 153,549 42,969 153,549.00 42,969.00 1.00000 0 0.00 23 1988 3 0 4 , 9 5 5 164,703 44,880 164,703.00 44,880.00 1.00000 0 0.00

Remark

Ties can be readily resolved by comparing the ratio of input 2 to input 1--which, in our case, will be the ratio o f capital to labor.

Proof

Choosing uj = 1, v~ = E, v2 = 1, we can normalize for our one output case and write:

1 hi(u, ) = - - . x2j + Ex~j

The statement o f the theorem and the remark give: either (i) x2R < x2j or (ii) x2R -- Xzj and xlR < xu for the right end-point. If (i) holds then for E sufficiently small we have x2j + cx u > x2R + extR. In case (ii), once again, x2j + Ex~j > x2R + EX~R for E > 0. Hence whether j falls in case (i) or case (ii)

Table A5. Comparison of SF with YC = Ye "° and DEA results with YD = Y~p* for the Textile Industry

Original Stochastic frontier DEA

Observation Cap Lab number Year Y U* e"" YC YD $ slack slack

1 1966 145,163 0 . 0 7 2 0 2 1 . 0 7 4 6 8 156,003.31 176,952.52 1.21899 0 0 2 1967 121,057 0 . 1 6 9 3 0 1 . 1 8 4 4 8 143,389.04 142,279.39 1.17531 0 10,369 3 1968 135,572 0 . 1 0 7 1 2 1 . 1 1 3 0 7 150,900.83 151,416.19 1.11687 0 9380 4 1969 177,849 0.0000 1.00000 177,849.00 186,481.22 1.04854 0 0 5 1970 183,630 0 . 0 0 0 0 0 1 . 0 0 0 0 0 183,630.00 183,630.00 1.00000 0 0 6 1971 160,291 0 . 0 5 7 6 0 1 . 0 5 9 2 9 169,794.84 162,684.08 1.01493 0 18,651 7 1972 161,853 0 . 0 8 0 3 0 1 .08361 175,385.87 166,686.92 1.02987 0 20,253 8 1973 181,968 0 . 0 1 9 9 6 1 . 0 2 0 1 6 185,636.57 190,122.35 1.04481 0 6507 9 1974 188,066 0 . 0 0 8 8 4 1 . 0 0 8 8 8 189,735.87 199,302.69 1.05975 0 1489

10 1975 2 0 6 , 3 1 7 0 . 0 0 0 0 0 1 . 0 0 0 0 0 206,317.00 219,642.72 1.06459 0 0 11 1976 196,584 0 . 0 1 7 1 0 1 . 0 1 7 2 5 199,974.49 215,241.10 1.09491 0 0 12 1977 187,317 0 . 0 8 3 6 7 1 . 0 8 7 2 7 203,664.16 210,463.81 1.12357 0 4436 13 1978 2 2 9 , 3 0 8 0 . 0 0 0 0 0 1 . 0 0 0 0 0 229,308.00 254,704.60 1.11075 0 0 14 1979 2 5 8 , 5 2 9 0 . 0 0 0 0 0 1 . 0 0 0 0 0 258,529.00 266,052.98 1.02910 0 0 15 1980 3 1 3 , 7 3 4 0 . 0 0 0 0 0 1 . 0 0 0 0 0 313,734.00 313,734.00 1.00000 0 0 16 1981 3 6 1 , 1 5 5 0 . 0 0 0 0 0 1 . 0 0 0 0 0 361,155.00 361,155.00 1.00000 0 0 17 1982 3 4 7 , 2 2 9 0 . 0 0 0 0 0 1 . 0 0 0 0 0 347,229.00 421,246.16 1.21317 0 0 18 1983 369,631 0 . 0 0 0 0 0 1 . 0 0 0 0 0 369,631.00 440,120.74 1.19070 0 0 19 1984 4 6 5 , 2 3 5 0 . 0 2 8 1 7 1 . 0 2 8 5 7 478,527.01 494,951.91 1.06388 0 0 20 1985 4 9 2 , 0 5 3 0 . 0 0 3 6 5 1 . 0 0 3 6 6 493,852.28 492,053.00 1.00000 0 0 21 1986 4 9 5 , 8 2 2 0 . 0 7 7 1 3 1 . 0 8 0 1 8 535,578.24 533,767.53 1.07653 0 0 22 1987 5 2 3 , 0 3 6 0.07298 1.07571 562,634.54 541,921.98 1.03611 30,082 0 23 1988 5 3 1 , 0 5 9 0 . 0 6 9 2 5 1 . 0 7 1 7 0 569,138.11 538,195.47 1 . 0 1 3 4 4 49,937 0

Page 25: DEA and stochastic frontier analyses of the 1978 Chinese economic reforms

DEA

Table A6. Comparison of SF with Y C = Ye u~ and DEA results with YD = Y(o* for the Chemical Industry

109

Original Stochastic frontier DEA

Observation Cap Lab number Year Y U* e ~* Y C YD ~b slack slack

1 1966 116,884 0.13622 1.14593 2 1967 100,354 0.18874 1.20773 3 1968 102,307 0.18512 1.20336 4 1969 160,948 0.00000 1.00000 5 1970 171,774 0.00000 1.00000 6 1971 195,428 0.00000 1.00000 7 1972 197,665 0.02137 1.02160 8 1973 209,838 0.00940 1.00944 9 1974 219,256 0.00000 1.00000

10 1975 232,676 0.00000 1.00000 11 1976 290,057 0.05774 1.05944 12 1977 214,105 0.09464 1.09926 13 1978 272,935 0.00000 1.00000 14 1979 278,142 0.00000 1.00000 15 1980 286,499 0.06404 1.06614 16 1981 309,717 0.07546 1.07838 17 1982 338,989 0.06339 1.06544 18 1983 354,515 0.05780 1.05950 19 1984 373,221 0.09317 1.09765 20 1985 410,720 0.05376 1.05523 21 1986 432,514 0.04042 1.04125 22 1987 464,556 0.05738 1.05906 23 1988 508,829 0.04931 1.05055

160,266.44 155,337.19 157,042.88 158,665.25 171,452.06 194,146.38 207 912.38 214 701.81 219 166.56 231 100.25 239 157.50 266 715.13 266 295.50 278 068.19 332 561.19 369 120.50 392 904.19 405 635.56 463 371.94 465 583.94 475 424.19 531 039.94 570 914.38

146,837.35 121.077.65 118,519.25 164,617.32 171,774.00 196,996.09 201 647.54 215 008.97 223 100.01 235 764.52 212 765.50 223 519.65 272 935.00 281 121.89 316 367.23 341 752.92 363 511.48 373 460.66 410 548.03 420 595.58 432 514.00 470 865.60 508 829.00

1.25627 0 0.0 1.20651 0 7139.0 1.15847 0 10,872.0 1.02280 0 0.0 1.00000 0 0.0 1.00802 0 0.0 1.02015 0 268.6 1.02464 0 0.0 1.01753 0 0.0 1.01327 0 0.0 1.01774 0 8648.7 1.04397 0 12,443.2 1.00000 0 0.0 1.01071 0 0.0 1.10425 0 0.0 1.10344 0 0.0 1.07234 0 0.0 1.05344 0 0.0 1.10001 0 0.0 1.02404 0 0.0 1.00000 0 0.0 1.01358 0 0.0 1.O0000 0 0.0

we have hR (u, v) > hj (u, v). Since this strict inequality holds for a n y j ~- R we can conclude that R is extreme efficient. See Charnes et al. [15].

The argument for the endpoint L is similar and is thus not given. As noted in the text, we use the "CCR projection formulas" of (7) and (8) to ascertain which

parts of the efficiency frontier were used to evaluate the performance of each DMU. Because our D M U s take the form of years, this enables us to determine whether the performance in any year was evaluated by the efficiency that was possible at the time production occurred.

Our use of these projections is assisted by the Cooper-Gallegos [19] theorem which enters into our algorithm. We thus now state and prove it in the following simple manner:

Theorem

(Cooper and Gallegos): an optimal solution for the CCR ratio form of DEA (which we use here) will always have at least one output slack and one input slack equal to zero.

Table A7. Comparison of SF with Y C = Ye u° and DEA results with Y D = Y4a* for the Metallurgical Industry

Original Stochastic frontier DEA

Observation Cap Lab number Year Y U * e u. Y C Y D ~b slack slack

I 1966 51,060 0.39769 1.48838 75,996.81 80,207.35 1.57085 0 2324.71 2 1967 50,087 0.45606 1.57785 79,029.52 82,918.63 1.65549 0 2150.38 3 1968 46,259 0.64165 1.89961 87,874.18 90,606.21 1.95867 0 1707.05 4 1969 59,358 0.58983 1.80368 107,062.94 112,938.09 1.90266 0 0.00 5 1970 64,959 0.59087 1.80556 117,287.28 95,992.37 1.47774 0 8162.18 6 1971 70,446 0.33827 1.40252 98,801.86 106,720.19 1.51492 0 0.00 7 1972 89,592 0.09266 1.09709 98,290.37 92,705.99 1.03476 0 5904.03 8 1973 112,588 0.12657 1.13493 127,779.37 120,386.65 1.06927 0 2943.63 9 1974 121,013 0.00000 1.00000 121,013.00 121,013.00 1.00000 0 0.00

10 1975 132,093 0.01686 1.01700 134,338.97 134,541.66 1.01854 0 0.00 11 1976 109,393 0.16934 1.18452 129,578.50 128,591.75 1.17550 0 0.00 12 1977 119,411 0.20697 1.22995 146,869.04 141,077.72 1.18145 0 0.00 13 1978 171,228 0.05416 1.05565 180,757.44 171,786.31 1.00326 0 0.00 14 1979 185,703 0.03708 1.03778 192,718.12 185,703.00 1.00000 0 0.00 15 1980 196,998 0.07853 1.08170 213,091.91 199,981.73 1.01515 0 0.00 16 1981 199,055 0.12669 1.13507 225,940.38 208,340.75 1.04665 0 0.00 17 1982 208,897 0.12385 1.13185 236,439.25 215,751.42 1,03281 0 0.00 18 1983 226,187 0.20843 1.23174 278,604.19 242,242.86 1,07098 0 0.00 19 1984 228,109 0.13923 1.14939 262,185.85 234,181.32 1.02662 0 0.00 20 1985 235,981 0.19325 1.21319 286,288.86 253,388.81 1.07377 0 0.00 21 1986 268,548 0.13959 1.14980 308,777.10 269,859.52 1.00488 0 0.00 22 1987 289,824 0.14399 1.15487 334,709.78 289,824.00 1.00000 0 0.00 23 1988 304,955 0.16951 1.18472 361,287.57 304,955.00 1.00000 0 0.00

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110 W.W. Cooper et al.

/'roof Cooper and Gallegos use the linear programming equivalent of the CCR ratio form in their

development. Hence, we can simplify their proof by noting that these results follow immediately from the "complementary slackness theorem" of linear programming.

Remark: This theorem allows us to ignore output slack and, without ambiguity, we can always conduct our normalization on the observed outputs. Still further simplication is achieved in the two-input case since, by this theorem, we can have positive slack in at most one of these inputs.

To calculate the slack, we introduce the following

Theorem (non-zero slack)

Part (a): a DMUk will have s*k > 0 if and only if X2k/Xlk < X2R/XIR. Part (b): a DMUk will have s*~ > 0 if and only if X2k/Xlk > X2L/X~L.

Remark

The CCR projection given in (7) consists of two parts: (i) a transformation in which 0* carries k into E. (See Fig. 1); and (ii) a translation in which slack is subtracted to carry E into/¢ on the efficiency frontier. Reference to Fig. 1 shows that non-zero slack can occur when the data have been normalized if and only if/~ is on some part of the horizontal or vertical extension of the unitized frontier.

Proof

To prove part (a) we observe that s* > 0 implies "~lk > XIR as in case (ii) of our endpoint theorem where, also, g2k = x2R. Therefore X2R/X~R > £2~/X1~ = XE~/Xjk the slope of the ray to k which also passes through •. Conversely X2k/Xlk < X2R/XtR together with X2k/X~k = X2k/XI~ implies -Vlk > XIR and s*k = ff~k > 0, as asserted in theorem.

This proves part (a) of the theorem. Proof of part (b) is analogous and so is not given. This theorem (non-zero slack) is applicable without respect to whether the data have been

normalized. However, the remark used in the above proof assumes that the data have been normalized and so we must evaluate how the efficiency adjusted amounts, as well as the rates, may be obtained from our normalizations. To clarify what is involved, we note that the vertical and horizontal extensions in Fig. 1 are, respectively, the planar facets generated by rays joining the origin

to all input/output vectors

F1 --xlj , j = l , 2 . . . . . 8

L -x2J j

in Fig. 1. Formally, we define the "unitized efficiency frontier" in Fig. 1 and the "unitized extended frontier" to be the intersections of these (respective) frontiers with the plane y = 1 in order to align this figure with the other figures in the text.

As already noted, it is assumed that normalization has been previously undertaken. In general, the outputs used for these normalizations will differ from unity and they will differ from each other--sometimes widely, as in Tables A2-A4. However, the ratios of capital-to-labor are independent of the normalization used. We thus extend our results to the non-normalized case and render the relations between the two cases precise by means of the following corollaries:

Corollary 1

The CCR projection formulas map any DMUj with non-zero slack into a multiple, ~, of either DMUR or DMUL.

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DEA 111

Corollary 2

If the data have been normalized then for any D MU i with non-zero slack we have ~ = 1 so the projection is directly onto DMUR or DMUL.

An Algorithm

To initiate the computations, we first locate the right and left endpoints of the unitized efficiency frontier. This is readily done by means of our "endpoint theorem." The results are indicated by the boxes on the values in the first two columns for each industry in Table A1. In Textiles, for instance, we locate the right endpoint R at year 70 since 0.26803 is the smallest value in the )(2/Y (=capital- to-output) column. We similarly locate the left endpoint at L = year 85 since 0.43687 is the smallest value in the Xj/Y (= labor-to-output) column.

Via our endpoint theorem, we know that these points are both efficient and, in fact, they are extreme efficient. For projections to other points via (7), we require the numerical values of 0* and the optimal slack values for each DMU~. Via our "non-zero slack theorem," a DMUk will have non-zero labor slack if and only if its ratio of capital-toqabor is smaller than the ratio for the right endpoint. For example, we observe in Column 3 of Table A1 that the ratio Xz/X~ for DMU67 under Textiles, viz 0.35625, is smaller than the value 0.40202 for DMU70 in this same column. Therefore,

0.26803 0*7 = 0.85084 - 0.3150~'

where the numerator and denominator are the capital-to-output ratios for DMUT0 and DMU67 , respectively. This result, 0.85084, is the value recorded for 0*7 in the theta column of Table A2.

To show how the slack values of DMU67 in the last two columns of Table A2 are obtained, we first determine the slack values for capital. For this, we use 0*7 and apply it to the value

n ~ n x2.67 = 0.31502 to obtain 0 6 7 X 2 , 6 7 ~---£~,67 = 0.26803, where the n superscript denotes a normalized value. Comparing this with x~. 70 = 0.26803, the normalized value identified as the endpoint under X2/Y, we see that there is no difference, i.e. X2,67-n ---- x2.v0, " where x2,70" is the coordinate for capital at this endpoint. Thus, SLACAP (=capital slack) is zero, as noted for 1967 in Table A2.

Turning to labor, for which we have x" = 0.88427, we again apply 0*7 = 0.85084 to obtain: 2, 67

0*7 = (0.85084) (0.88427) = 0.75236 - -" - - X2, 67.

Accordingly, the slack for labor is:

£~.67 - x~.70 = 0.75236 - 0.66671 = 0.08565,

where 0.66671 is the labor-to-output ratio for endpoint 1970. The value, 0.085658, resulting from this subtraction is the normalized slack. We must also

account for the dimension in which the (varying) output values occur. Hence, to obtain this slack in non-normalized units, as required, we multiply by the output value Y67 = 121,057, yielding

0.08565(121,057) = 10,369,

which is the value that is recorded under SLALAB in the second row of the last column in Table A2.

From the remark following the Cooper and Gallegos theorem, we know that non-zero slack in one input implies zero slack in the other input--in the case of one output and two inputs. Hence, we cease further computations for 1967 and proceed to our next illustration.

Referring to L = 85, the other endpoint, we know from our theorem that DMU88, will have positive slack for capital since its capital-to-labor ratio (= 1.66656) exceeds that for DMU85 (= 1.45132). The computations to determine this slack value are analogous to what has just been done. The computations may be applied in any order to the data. Using these computational procedures, we derive all of the DEA efficiency scores and slacks located in the last three columns of Tables A2-A4. This, in turn, enables us to obtain the "DEA adjusted values" which correspond to points which, in normalized form, are on the efficiency frontiers in the figures in the text--as obtained from CCR projections. Everything needed for effecting our DEA evaluations is thus at hand and can thus conclude this discussion by observing that all the results, including the optimal

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112 W.W. Cooper et al.

slacks as well as the 0* values, are unique. Consequently, there is no need to study other possibilities that might be associated with alternate optima.

Having dealt in some detail with the DMUs possessing slacks, we next turn to the remaining inefficient units. The basic geometric ideas for calculating 0* and the optimal multipliers (intensities) can be seen by reference to Fig. 1 and its underlying data matrix in (5). Figure 1 is its counterpart for the Textile Industry, showing that the "cones" associated with DMUs 70, 80, 81 and 85, respectively, play the same roles as the cones associated with 4, 3, 2, 1 in Fig. 1. Both figures display five boundary segments of which the inner three define the efficiency frontier. The extended frontier also contains the cones generated by the two outer (horizontal and vertical) segments, with the slopes of the pictured rays through the four corner points appearing in the corresponding years of the X~/X2 column for the textile portion of Table A1.

We illustrate the algorithm using the year 69 for which the slope of 0.43789-0.44 under ,t" 2/Xl lies between the slope of m = 0.40 fo__5_r 70 and m = 0.58 for 80. The ray from the origin to 69 therefore intersects the boundary at 69--which lies on the segment F2. Hence, the "supporting" year_ss for 69 are 70 and 80. The value for 0*9 is the ratio of the length of the rays from the origin to 69 and 69, respectively, for which the "supporting equations" are:

cap 69 = )̀ 80 cap 80 + )'70 cap 70,

lab 69 = )̀ 80 lab 80 + )`70 lab 70,

1 = )`80 -F )`70,

where, from Table A2, cap 80 = cap80/Y80-- 87,180/313,734. If we substitute, cap 69 = 069 cap 69, lab 69 = 069 lab 69 in (A1) the resulting linear equations

can be solved (uniquely) for the optimal value of 069 , )`80 and )`70. Thus, from Table A1,

0.28369 069 = 0.27788)`80 + 0.26803)`70,

0.64786 069 = 0.47662)`80 + 0.66671)`70,

1 = )`80 + )`70,

we obtain 0~'9=0.95371; )`'0=0.7431 and )`*0=0.2569. Geometrically, )̀ % is t__he ratio of the segment lengths (69, 70) to (80, 70) and )̀ 7"0 is the complementary ratio of (80, 69) to (80, 70) while 0* is the distance from the origin to 69 expressed as the ratio of the distance from the origin to 69.