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A NOTE ON THE MEASUREMENT OF TOTAL FACTORPRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL CHANGEUSING DATA ENVELOPMENT ANALYSIS

著者Author(s) Umetsu, Chieko

掲載誌・巻号・ページCitat ion 神戸大学農学部学術報告,25:9-28

刊行日Issue date 2001-02-25

資源タイプResource Type Departmental Bullet in Paper / 紀要論文

版区分Resource Version publisher

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DOI

JaLCDOI 10.24546/00038947

URL http://www.lib.kobe-u.ac.jp/handle_kernel/00038947

PDF issue: 2020-04-03

~*.~fi! (Sci. Rept. Fac. Agr. Kobe Univ.) 25: 9-28. 2001

A NOTE ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL

CHANGE USING DATA ENVELOPMENT ANALYSIS

Chieko UMETSU·

Abstract

9

This paper reviews approaches to the measurement of total factor productivity, efficiency and technological change. N onfrontier analysis assumes that the production technology is efficient and technological change is equivalent to total factor productivity change. On the other hand, the frontier approach explicitly considers inefficiency of production. In the presence of inefficiency, total factor productivity consists of efficiency and technological change. The traditional parametric approach to technological change may give results with technological bias that are sensitive to a specific parametric form applied in the analysis. The advantage of the nonparametric frontier approach using Data Envelopment Analysis in productivity analysis has been recently recognized. The Malmquist productivity index (Fare, Grosskopf, Kindgren and Ross, 1989) as a nonparametric frontier approach, is not constrained by specific functional forms and has shown a relative advantage in productivity analysis in developing countries where price information is limited and only quantity data is available. The Malmquist productivity index can be decomposed into efficiency change and technological change, which can provide useful information for policy analysis.

1. INTRODUCTION

The purpose of the paper is to explore methods of total factor productivity, efficiency and technological

change measurement. Emphasis is given to the nonparametric frontier approach which literature is recently

exploring in the field of productivity analysis. Section 2 overviews the conventional methods to total factor

productivity by clarifying frontier/nonfrontier and parametric/nonparametric distinctions. The following two

sections then concentrate on nonparametric approaches. Section 3 explains the nonparametric nonfrontier

approach such as growth accounting and index number studies. Section 4 introduces the nonparametric frontier

approach that includes the theoretical construct of the Malmquist total factor productivity index.

2. ANAL YSIS OF TOTAL FACTOR PRODUCTIVITY

In order to understand the relative advantages and disadvantages of the Malmquist total factor

productivity index which was adopted by Thirtle, Hadley and Townsend (1994) in their cross-country study of

sub-Saharan Africa, an overview of various productivity measurement is fIrst presented with particular reference

to nonfrontier vs. frontier and nonparametric vs. parametric approaches to productivity analysis.

2.1 Nonfrontier vs. frontier approaches

Productivity measures such as labor productivity and land productivity involve single-factor

productivity which is a ratio of output (an index) to a particular input, for example, labor or land. Total factor

• The Graduate School of Science and Technology, Kobe University.

~*.~fi! (Sci. Rept. Fac. Agr. Kobe Univ.) 25: 9-28. 2001

A NOTE ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL

CHANGE USING DATA ENVELOPMENT ANALYSIS

Chieko UMETSU·

Abstract

9

This paper reviews approaches to the measurement of total factor productivity, efficiency and technological change. N onfrontier analysis assumes that the production technology is efficient and technological change is equivalent to total factor productivity change. On the other hand, the frontier approach explicitly considers inefficiency of production. In the presence of inefficiency, total factor productivity consists of efficiency and technological change. The traditional parametric approach to technological change may give results with technological bias that are sensitive to a specific parametric form applied in the analysis. The advantage of the nonparametric frontier approach using Data Envelopment Analysis in productivity analysis has been recently recognized. The Malmquist productivity index (Fare, Grosskopf, Kindgren and Ross, 1989) as a nonparametric frontier approach, is not constrained by specific functional forms and has shown a relative advantage in productivity analysis in developing countries where price information is limited and only quantity data is available. The Malmquist productivity index can be decomposed into efficiency change and technological change, which can provide useful information for policy analysis.

1. INTRODUCTION

The purpose of the paper is to explore methods of total factor productivity, efficiency and technological

change measurement. Emphasis is given to the nonparametric frontier approach which literature is recently

exploring in the field of productivity analysis. Section 2 overviews the conventional methods to total factor

productivity by clarifying frontier/nonfrontier and parametric/nonparametric distinctions. The following two

sections then concentrate on nonparametric approaches. Section 3 explains the nonparametric nonfrontier

approach such as growth accounting and index number studies. Section 4 introduces the nonparametric frontier

approach that includes the theoretical construct of the Malmquist total factor productivity index.

2. ANAL YSIS OF TOTAL FACTOR PRODUCTIVITY

In order to understand the relative advantages and disadvantages of the Malmquist total factor

productivity index which was adopted by Thirtle, Hadley and Townsend (1994) in their cross-country study of

sub-Saharan Africa, an overview of various productivity measurement is fIrst presented with particular reference

to nonfrontier vs. frontier and nonparametric vs. parametric approaches to productivity analysis.

2.1 Nonfrontier vs. frontier approaches

Productivity measures such as labor productivity and land productivity involve single-factor

productivity which is a ratio of output (an index) to a particular input, for example, labor or land. Total factor

• The Graduate School of Science and Technology, Kobe University.

10 Chieko UMETSU

productivity, on the other hand, is a ratio of an index of output to an index of total inputs. The growth of

productivity refers to productivity change from one period to the next. The defmition of productivity followed

here is by Grosskopf (1993) which states that productivity growth is the net change in output due to "efficiency

change" and "technological change". Efficiency change shows a change of relative distance from the observed

production activity and the production frontier (the best practice frontier), and technological change means a

shift of the production frontier over time. This defmition makes clear distinction between frontier and

nonfrontier approaches to total factor productivity.

The methods to measure productivityl are divided into two groups, namely nonfrontier and frontier

approaches (See Figure 1). The nonfrontier approach, which includes growth accounting (Solow, 1957) and

index number approaches (Diewert, 1980; Caves et aI., 1982) as well as econometric estimation of production

functions, assumes that production is always efficient and there is no technical inefficiency in production. In the

nonfrontier approach, technical change is equivalent to total factor productivity change over time. However, if

there is allocative inefficiency2, i.e., the input share is not equal to the cost-minimizing share, this approach may

result in a biased productivity measurement even though technical inefficiency does not exist (Grosskopf, 1993).

Figure 1. Method of Total Factor Productivity Measurement

Nonfrontier

(no technical inefficiency)

Frontier

(technical inefficiency)

Nonparametric (no parameters to be

estimated)

• growth accounting • index number approach

- Divisia index - Tornqvist index - Fisher ideal index

* econometric/mathematical programming • Farrell measure • Distance function • Malmquist index

Parametric (parameters to be estimated)

* econometric method • average practice function

- production function - cost function - revenue function - profit function

* econometric/mathematical programming • deterministic frontier • probabilistic frontier • stochastic frontier

On the other hand, the frontier approach, including the Farrell measure of technical efficiency (Farrell,

1957), distance function (Shephard, 1953, 1970), and Malmquist index (Malmquist, 1953; Caves et aI., 1982)

explicitly considers inefficiency of production. In the presence of inefficiency, productivity change is no longer

equal to technical change but includes additional factor, i.e., efficiency 9hange to account for productivity.

Compared to the nonfrontier approach, the frontier approach is not constrained by a behavioral assumption of

profit-maximizing producers. Both the nonfrontier and frontier approaches include parametric (econometric and

1 Diewert (1980) provides a succinct review of various methods of productivity measurement. 2 The definition of allocative efficiency is given by Farrell (1957). See 4.2 for a detailed explanation.

10 Chieko UMETSU

productivity, on the other hand, is a ratio of an index of output to an index of total inputs. The growth of

productivity refers to productivity change from one period to the next. The defmition of productivity followed

here is by Grosskopf (1993) which states that productivity growth is the net change in output due to "efficiency

change" and "technological change". Efficiency change shows a change of relative distance from the observed

production activity and the production frontier (the best practice frontier), and technological change means a

shift of the production frontier over time. This defmition makes clear distinction between frontier and

nonfrontier approaches to total factor productivity.

The methods to measure productivityl are divided into two groups, namely nonfrontier and frontier

approaches (See Figure 1). The nonfrontier approach, which includes growth accounting (Solow, 1957) and

index number approaches (Diewert, 1980; Caves et aI., 1982) as well as econometric estimation of production

functions, assumes that production is always efficient and there is no technical inefficiency in production. In the

nonfrontier approach, technical change is equivalent to total factor productivity change over time. However, if

there is allocative inefficiency2, i.e., the input share is not equal to the cost-minimizing share, this approach may

result in a biased productivity measurement even though technical inefficiency does not exist (Grosskopf, 1993).

Figure 1. Method of Total Factor Productivity Measurement

Nonfrontier

(no technical inefficiency)

Frontier

(technical inefficiency)

Nonparametric (no parameters to be

estimated)

• growth accounting • index number approach

- Divisia index - Tornqvist index - Fisher ideal index

* econometric/mathematical programming • Farrell measure • Distance function • Malmquist index

Parametric (parameters to be estimated)

* econometric method • average practice function

- production function - cost function - revenue function - profit function

* econometric/mathematical programming • deterministic frontier • probabilistic frontier • stochastic frontier

On the other hand, the frontier approach, including the Farrell measure of technical efficiency (Farrell,

1957), distance function (Shephard, 1953, 1970), and Malmquist index (Malmquist, 1953; Caves et aI., 1982)

explicitly considers inefficiency of production. In the presence of inefficiency, productivity change is no longer

equal to technical change but includes additional factor, i.e., efficiency 9hange to account for productivity.

Compared to the nonfrontier approach, the frontier approach is not constrained by a behavioral assumption of

profit-maximizing producers. Both the nonfrontier and frontier approaches include parametric (econometric and

1 Diewert (1980) provides a succinct review of various methods of productivity measurement. 2 The definition of allocative efficiency is given by Farrell (1957). See 4.2 for a detailed explanation.

A NOTE ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL CHANGE USING DATA ENVELOPMENT ANALYSIS

11

mathematical programming) and nonparametric (linear programming) approaches. The Malmquist index is a

frontier nonparametric approach to total factor productivity measurement as shown in Figure 1.

2.2 Parametric vs. nonparametric approaches

The parametric approach refers to the methods that involve the estimation of parameters either by

statistical (econometric) techniques or mathematical estimation techniques such as linear programming.3 The

parametric nonfrontier approach to productivity is a departure from the nonparametric nonfrontier approach

such as growth accounting which has no parameters to be estimated and thus. does not allow measurement or

sampling errors in general. This approach also has the characteristic that it does not take inefficiency into

account. The estimated parameters are used to derive technical change and productivity change. However, the

parametric approach restricts the number of parameters to be estimated depending on the degrees of freedom and

is also subject to specification error.4 There is a vast literature on parametric methods. Here, I briefly review

parametric approaches to productivity measurement.

The parametric nonfrontier approach involves specifying functional forms for the production, cost,

revenue or profit function and estimating parameters using statistical techniques or mathematical linear

programming from observed data. The estimated functions yield "average" practice functions. For example, the

production function and cost function for time period t = 1, 2, ... , T can be specified as follows:

yt=/(xt,t) t=I,2, ..... ,T. (1)

C t = C(yl, Wi ,t) t = 1, 2, ..... ,T.

where yt is the output and xt is the vector of inputs for the production function. For a cost function, yt is the

vector of outputs, and w t is the vector of inputs. If there is no inefficiency, then the growth of production over

time, which is given by a parameter, 8In/(xt ,t)/8t, or -[8InC()I,w,t)/8t] is equivalent to the growth of total

factor productivity. If input price data is readily available, the cost function as well as indirect production

function approach provide practical tools for analyzing productivity and technological change and bias

(Kawagoe et aI., 1986; Kim, 1988).

The parametric frontier approach tries to account for inefficiency and its deviation from the frontier

production function. These are categorized into deterministic, probabilistic, and stochastic. The earlier models of

the parametric frontier approach are considered "deterministic" because they use programming methods to

estimate the frontier and did not allow measurement error except when due to inefficiency (Grosskopf, 1993). In

other words, all observation is constrained to lie either on or below the frontier. By estimating this deterministic

frontier, Nishimizu and Page (1988) decomposed productivity growth into frontier technical change and

efficiency change.

3 Also, it is possible to distinguish the parametric approach from nonparametric approach purely by existence of statistical inference of parameters in question.

A NOTE ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL CHANGE USING DATA ENVELOPMENT ANALYSIS

11

mathematical programming) and nonparametric (linear programming) approaches. The Malmquist index is a

frontier nonparametric approach to total factor productivity measurement as shown in Figure 1.

2.2 Parametric vs. nonparametric approaches

The parametric approach refers to the methods that involve the estimation of parameters either by

statistical (econometric) techniques or mathematical estimation techniques such as linear programming.3 The

parametric nonfrontier approach to productivity is a departure from the nonparametric nonfrontier approach

such as growth accounting which has no parameters to be estimated and thus. does not allow measurement or

sampling errors in general. This approach also has the characteristic that it does not take inefficiency into

account. The estimated parameters are used to derive technical change and productivity change. However, the

parametric approach restricts the number of parameters to be estimated depending on the degrees of freedom and

is also subject to specification error.4 There is a vast literature on parametric methods. Here, I briefly review

parametric approaches to productivity measurement.

The parametric nonfrontier approach involves specifying functional forms for the production, cost,

revenue or profit function and estimating parameters using statistical techniques or mathematical linear

programming from observed data. The estimated functions yield "average" practice functions. For example, the

production function and cost function for time period t = 1, 2, ... , T can be specified as follows:

yt=/(xt,t) t=I,2, ..... ,T. (1)

C t = C(yl, Wi ,t) t = 1, 2, ..... ,T.

where yt is the output and xt is the vector of inputs for the production function. For a cost function, yt is the

vector of outputs, and w t is the vector of inputs. If there is no inefficiency, then the growth of production over

time, which is given by a parameter, 8In/(xt ,t)/8t, or -[8InC()I,w,t)/8t] is equivalent to the growth of total

factor productivity. If input price data is readily available, the cost function as well as indirect production

function approach provide practical tools for analyzing productivity and technological change and bias

(Kawagoe et aI., 1986; Kim, 1988).

The parametric frontier approach tries to account for inefficiency and its deviation from the frontier

production function. These are categorized into deterministic, probabilistic, and stochastic. The earlier models of

the parametric frontier approach are considered "deterministic" because they use programming methods to

estimate the frontier and did not allow measurement error except when due to inefficiency (Grosskopf, 1993). In

other words, all observation is constrained to lie either on or below the frontier. By estimating this deterministic

frontier, Nishimizu and Page (1988) decomposed productivity growth into frontier technical change and

efficiency change.

3 Also, it is possible to distinguish the parametric approach from nonparametric approach purely by existence of statistical inference of parameters in question.

12 Chieko UMETSU

The probabilistic and stochastic frontier, on the other hand, relaxes the restrictive assumption of

detenninistic models and allows random errors of the estimated frontier. Probabilistic models accomplish this

objective by specifying that a certain percentage of most efficient observations lie above the estimated

production frontier (Timmer, 1971). The stochastic frontier also overcomes the limitations of deterministic

models by incorporating deviation from the frontier using a composed error term (Aigner, 1977). This error term

allows deviation by variables such as weather.

Diamond et aI. (1978) suggested earlier that the traditional parametric approach to technological change

may provide results in bias of technology that are sensitive to a specific parametric form applied in the analysis.

This identification problem asserts that the use of flexible functional forms such as translog or generalized

Leontief for a production, cost, and profit function, is not helpful in solving the identification of a technical

change problem unless we have "a priori information on technology or the nature of technical change"

(Diamond et aI., 1978). A series of Chavas and Cox (1988, 1990) and Cox and Chavas (1990) papers on

productivity and technological change use the nonparametric approach primarily because of this identification

problem in parametric analysis. Also the advantage of the nonparametric frontier approach using Data

Envelopment Analysis in productivity analysis has recently been recognized (Diewert and Mendoza, 1995).

3. NONP ARAMETRIC NONFRONTIER ANALYSIS TO TOTAL FACTOR PRODUCTIVITY

The nonparametric nonfrontier approach to total factor productivity is represented by the Solow (1957)

growth accounting study and various index number approaches. Solow's intention was to show simple way of

separating changes in output per labor due to technical change (viz. shifts of the production function) from those

due to changes in the availability of capital per labor (viz. movements along the production function). He

assumed continuous time, constant returns to scale, no technical and allocative inefficiency (i.e., producers

maximize profits and factors are paid their marginal product), and Hicks neutral technological change. Solow's

formulation is similar to index number approaches except he assumed a single output and two inputs, capital and

labor. For explanatory purposes, a single output and n-input case is considered, drawing from the notation in

Grosskopf (1993).

Let l E 9t + denote a single output in period t and x t E 9t ~ denote the vector of inputs in period t.

From the assumption of Hicks neutral (output augmented) technological change, the production function at time

t is specified as:

l = A(t)f(x' ) (2)

4 Varian (1984) states that "this procedure suffers from the defect that the maintained hypothesis of the parametric form can never be directly tested: it must be taken by faith."

12 Chieko UMETSU

The probabilistic and stochastic frontier, on the other hand, relaxes the restrictive assumption of

detenninistic models and allows random errors of the estimated frontier. Probabilistic models accomplish this

objective by specifying that a certain percentage of most efficient observations lie above the estimated

production frontier (Timmer, 1971). The stochastic frontier also overcomes the limitations of deterministic

models by incorporating deviation from the frontier using a composed error term (Aigner, 1977). This error term

allows deviation by variables such as weather.

Diamond et aI. (1978) suggested earlier that the traditional parametric approach to technological change

may provide results in bias of technology that are sensitive to a specific parametric form applied in the analysis.

This identification problem asserts that the use of flexible functional forms such as translog or generalized

Leontief for a production, cost, and profit function, is not helpful in solving the identification of a technical

change problem unless we have "a priori information on technology or the nature of technical change"

(Diamond et aI., 1978). A series of Chavas and Cox (1988, 1990) and Cox and Chavas (1990) papers on

productivity and technological change use the nonparametric approach primarily because of this identification

problem in parametric analysis. Also the advantage of the nonparametric frontier approach using Data

Envelopment Analysis in productivity analysis has recently been recognized (Diewert and Mendoza, 1995).

3. NONP ARAMETRIC NONFRONTIER ANALYSIS TO TOTAL FACTOR PRODUCTIVITY

The nonparametric nonfrontier approach to total factor productivity is represented by the Solow (1957)

growth accounting study and various index number approaches. Solow's intention was to show simple way of

separating changes in output per labor due to technical change (viz. shifts of the production function) from those

due to changes in the availability of capital per labor (viz. movements along the production function). He

assumed continuous time, constant returns to scale, no technical and allocative inefficiency (i.e., producers

maximize profits and factors are paid their marginal product), and Hicks neutral technological change. Solow's

formulation is similar to index number approaches except he assumed a single output and two inputs, capital and

labor. For explanatory purposes, a single output and n-input case is considered, drawing from the notation in

Grosskopf (1993).

Let l E 9t + denote a single output in period t and x t E 9t ~ denote the vector of inputs in period t.

From the assumption of Hicks neutral (output augmented) technological change, the production function at time

t is specified as:

l = A(t)f(x' ) (2)

4 Varian (1984) states that "this procedure suffers from the defect that the maintained hypothesis of the parametric form can never be directly tested: it must be taken by faith."

A NOTE ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL CHANGE USING DATA ENVELOPMENT ANALYSIS

13

where A(t) indicates output augmented technological change viz. shifts in the production function. Totally

differentiating with respect to time and dividing by yt yields the following equation:

(3)

. . where y = dy / dt thus y/ y is the growth ofy; and sn is a factor share of the n-th input. Since factors are paid

the marginal product, it follows that a f / a x~ = w~ / p where w~ e 91:+ is the input price vector and p' E 9t++

is the output price in period t.

s = (t3 J . ~J = t3 y . x" = w"x" = w"x" " t3x~ J(x' ) t3x~ y py ~wx

L...J " "

(4)

,,=\

Thus the change in technology over time is obtained by rearranging equation (3) as follows:

A ; N (;J -=-- LS" -A Y ,,=1 X"

(5)

This is equivalent to the growth of total factor productivity that is not accounted for by the growth of

output and input. The index number approach such as the Divisia index has the same formulation as above

assuming continuous time. The continuous time derivative is, however, not practical for empirical study where

only discrete data is available. As an alternative, growth rates in the continuous form can be approximated by the

discrete change of logarithms from time t and t+ 1, ;/ y = lnyt+\ -lny' , and the input shares from period t to

t+ 1 can be estimated by taking an average of the input shares of the two periods. This index is called the

Tornqvist index or discrete Divisia index of total factor productivity (TFP) growth and is as follows:

(6)

where TFpt = A(t) , which accounts for technological change or shifts in the production function. Equation (6)

for a single output can be generalized as the Tornqvist index for m-output and n-input case as:

A NOTE ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL CHANGE USING DATA ENVELOPMENT ANALYSIS

13

where A(t) indicates output augmented technological change viz. shifts in the production function. Totally

differentiating with respect to time and dividing by yt yields the following equation:

(3)

. . where y = dy / dt thus y/ y is the growth ofy; and sn is a factor share of the n-th input. Since factors are paid

the marginal product, it follows that a f / a x~ = w~ / p where w~ e 91:+ is the input price vector and p' E 9t++

is the output price in period t.

s = (t3 J . ~J = t3 y . x" = w"x" = w"x" " t3x~ J(x' ) t3x~ y py ~wx

L...J " "

(4)

,,=\

Thus the change in technology over time is obtained by rearranging equation (3) as follows:

A ; N (;J -=-- LS" -A Y ,,=1 X"

(5)

This is equivalent to the growth of total factor productivity that is not accounted for by the growth of

output and input. The index number approach such as the Divisia index has the same formulation as above

assuming continuous time. The continuous time derivative is, however, not practical for empirical study where

only discrete data is available. As an alternative, growth rates in the continuous form can be approximated by the

discrete change of logarithms from time t and t+ 1, ;/ y = lnyt+\ -lny' , and the input shares from period t to

t+ 1 can be estimated by taking an average of the input shares of the two periods. This index is called the

Tornqvist index or discrete Divisia index of total factor productivity (TFP) growth and is as follows:

(6)

where TFpt = A(t) , which accounts for technological change or shifts in the production function. Equation (6)

for a single output can be generalized as the Tornqvist index for m-output and n-input case as:

14 Chieko UMETSU

(7)

M

where rm(t) = (PmYm) / (LPmYm) is the revenue share of the m-th output. m=1

The recent development of index number theories has shown that a production function or a cost

function with desirable properties can be used to derive a quantity or price index as an aggregator function.

These index numbers are labeled "exact" for this aggregator function. Also, Diewert (1976) termed the index

number "superlative" if the aggregator function is flexible, such as a translog or generalized Leontief function,

and is a second-order approximation of any arbitrary, twice differentiable, and linearly homogeneous function.

The advantage of this nonparametric nonfrontier approach is the ease of computation if complete

quantity and price information is available. Since variables in equation (4) and (5) are all observable, AI A can

be directly obtained. Also, using a flexible functional form, such as translog, made it possible to easily specify

production technology in order to derive the total factor productivity index.

Although the simplicity of computation is well accepted, this approach is subject to at least two types of

bias. One bias is due to technical inefficiency. Since the index number approach assumes production is efficient,

inefficient production is not considered. The other bias comes from input price and input shares. The index

number approach assumes that input price and resulting input shares are cost-minimizing. However, if allocative

inefficiency exists and the input share does not represent the cost-minimizing share, the index is biased even

without technical inefficiency. Grosskopf (1993) suggested that if allocative inefficiency is relevant, then one

should use shadow prices to obtain input shares or other productivity measure, such as the Malmquist

productivity index, which requires quantity information only and not price information.

Otsuka (1988) defmed three problems associated with applying this index approach to firm level data.

First, the, discrete Divisia index requires time series data which may not be available in developing countries.

Second, the assumption of Hicks neutral technological change is too restrictive since non-neutral technological

change could affect input shares. The last problem concerns the zero input case. If there is any input which is not

. used at the particular observation, then in the equation (5) growth of input x / x is not defmed because of

division by zero. In order to overcome this problem, Otsuka suggested using the Tornqvist input price index,

which allows zero input.

4. NONP ARAMETRIC FRONTIER ANALYSIS TO TOTAL FACTOR PRODUCTIVITY

As mentioned in the previous section, the index number approach is restricted to constant returns to

scale and Hicks neutral technological change. Moreover, if price information is not reliable,and production is

inefficient, the resulting total factor productivity index is biased. The alternative method is a productivity

measurement, such as the Malmquist productivity index, which does not rely on price data and is not constrained

by constant returns to scale, Hicks neutral technological change and parametric specification of production

14 Chieko UMETSU

(7)

M

where rm(t) = (PmYm) / (LPmYm) is the revenue share of the m-th output. m=1

The recent development of index number theories has shown that a production function or a cost

function with desirable properties can be used to derive a quantity or price index as an aggregator function.

These index numbers are labeled "exact" for this aggregator function. Also, Diewert (1976) termed the index

number "superlative" if the aggregator function is flexible, such as a translog or generalized Leontief function,

and is a second-order approximation of any arbitrary, twice differentiable, and linearly homogeneous function.

The advantage of this nonparametric nonfrontier approach is the ease of computation if complete

quantity and price information is available. Since variables in equation (4) and (5) are all observable, AI A can

be directly obtained. Also, using a flexible functional form, such as translog, made it possible to easily specify

production technology in order to derive the total factor productivity index.

Although the simplicity of computation is well accepted, this approach is subject to at least two types of

bias. One bias is due to technical inefficiency. Since the index number approach assumes production is efficient,

inefficient production is not considered. The other bias comes from input price and input shares. The index

number approach assumes that input price and resulting input shares are cost-minimizing. However, if allocative

inefficiency exists and the input share does not represent the cost-minimizing share, the index is biased even

without technical inefficiency. Grosskopf (1993) suggested that if allocative inefficiency is relevant, then one

should use shadow prices to obtain input shares or other productivity measure, such as the Malmquist

productivity index, which requires quantity information only and not price information.

Otsuka (1988) defmed three problems associated with applying this index approach to firm level data.

First, the, discrete Divisia index requires time series data which may not be available in developing countries.

Second, the assumption of Hicks neutral technological change is too restrictive since non-neutral technological

change could affect input shares. The last problem concerns the zero input case. If there is any input which is not

. used at the particular observation, then in the equation (5) growth of input x / x is not defmed because of

division by zero. In order to overcome this problem, Otsuka suggested using the Tornqvist input price index,

which allows zero input.

4. NONP ARAMETRIC FRONTIER ANALYSIS TO TOTAL FACTOR PRODUCTIVITY

As mentioned in the previous section, the index number approach is restricted to constant returns to

scale and Hicks neutral technological change. Moreover, if price information is not reliable,and production is

inefficient, the resulting total factor productivity index is biased. The alternative method is a productivity

measurement, such as the Malmquist productivity index, which does not rely on price data and is not constrained

by constant returns to scale, Hicks neutral technological change and parametric specification of production

A NOTE ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL CHANGE USING DATA ENVELOPMENT ANALYSIS 15

technology. This group of non-parametric frontier approaches which uses linear programming methods to derive efficiency, technology and productivity is called Data Envelopment Analysis (DEA S

).

This section deals with the nonparametric frontier approach to total factor productivity. The recent development of the Malmquist productivity index using Farrell measures of technical efficiency is introduced. First, the specification of production technology is defmed with different returns to scale and disposability assumptions. The second section explains the concept of the Farrell measure of technical efficiency as well as a distance function approach. This is followed by the defmition of input scale efficiency and input congestion. Then the formulation of the Malmquist total factor productivity index is defmed including its decomposition to efficiency change and technological change. The last section shows how the Malmquist productivity index can explain scale efficiency and congestion of inputs. Mathematical notations and figures are drawn largely from Shephard (1970) and Fare et al. (1994).

4.1 Technology

(i) Technology

The production technology is characterized by the set of all feasible input and output vectors. Let X' = (x: ,x~, ... ,x~) denote an input vector at period t and l = (y: ,y~, ... ,y~) denote an output vector at period

t (n inputs and m outputs) where x' E!Jl~={X:XE!Jl~,X~O}, and yIE91~={y:YE91~,y~0}. The

technology is represented by the output set, p' (x') , the input set, Lt (y'), or the collection of all feasible input

and output vectors, GR' (x' ,y') as follows (Figure 2; Figure 3):

p' (x') = {V' : (x' ,y') E Sf ~ t = 1, ... , T.

z: (y') = {Xl: (x' ,y') E S'}, t = 1, ... , T. (8) GRf (x' ,y') = {(xt ,yf) E !Jl~+M : (x' ,y') Est}, t = 1, ... , T.

where Sf = {(xt,yt):x'canvroducey'} is the set of technology at period t. The output set, pl(XI), shows all

possible output vectors, y' E 91 ~ , that are attainable from the input vector, Xl E 9l~. The input set, LI (yl), on the other hand, provides all feasible input vectors, x' E 91:, that can produce output vector, yt E!Jl~. Both the output set and input set represent output substitution and input substitution respectively and involve no price information or behavioral assumptions.6

(ii) Returns to scale

Earlier models of non-parametric frontier approaches which use linear programming such as Farrell

5 Charnes, Cooper and Rhodes first introduced the term DEA (Data Envelopment Analysis) in their report "A Data Envelopment Analysis Approach to Evaluation of the Program Follow Through Experiment in U.S. Public School Education," (1978). 6 If price information is available, price-dependent specification of technology is also possible. See Fare et al. (1994) for the further discussion.

A NOTE ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL CHANGE USING DATA ENVELOPMENT ANALYSIS 15

technology. This group of non-parametric frontier approaches which uses linear programming methods to derive efficiency, technology and productivity is called Data Envelopment Analysis (DEA S

).

This section deals with the nonparametric frontier approach to total factor productivity. The recent development of the Malmquist productivity index using Farrell measures of technical efficiency is introduced. First, the specification of production technology is defmed with different returns to scale and disposability assumptions. The second section explains the concept of the Farrell measure of technical efficiency as well as a distance function approach. This is followed by the defmition of input scale efficiency and input congestion. Then the formulation of the Malmquist total factor productivity index is defmed including its decomposition to efficiency change and technological change. The last section shows how the Malmquist productivity index can explain scale efficiency and congestion of inputs. Mathematical notations and figures are drawn largely from Shephard (1970) and Fare et al. (1994).

4.1 Technology

(i) Technology

The production technology is characterized by the set of all feasible input and output vectors. Let X' = (x: ,x~, ... ,x~) denote an input vector at period t and l = (y: ,y~, ... ,y~) denote an output vector at period

t (n inputs and m outputs) where x' E!Jl~={X:XE!Jl~,X~O}, and yIE91~={y:YE91~,y~0}. The

technology is represented by the output set, p' (x') , the input set, Lt (y'), or the collection of all feasible input

and output vectors, GR' (x' ,y') as follows (Figure 2; Figure 3):

p' (x') = {V' : (x' ,y') E Sf ~ t = 1, ... , T.

z: (y') = {Xl: (x' ,y') E S'}, t = 1, ... , T. (8) GRf (x' ,y') = {(xt ,yf) E !Jl~+M : (x' ,y') Est}, t = 1, ... , T.

where Sf = {(xt,yt):x'canvroducey'} is the set of technology at period t. The output set, pl(XI), shows all

possible output vectors, y' E 91 ~ , that are attainable from the input vector, Xl E 9l~. The input set, LI (yl), on the other hand, provides all feasible input vectors, x' E 91:, that can produce output vector, yt E!Jl~. Both the output set and input set represent output substitution and input substitution respectively and involve no price information or behavioral assumptions.6

(ii) Returns to scale

Earlier models of non-parametric frontier approaches which use linear programming such as Farrell

5 Charnes, Cooper and Rhodes first introduced the term DEA (Data Envelopment Analysis) in their report "A Data Envelopment Analysis Approach to Evaluation of the Program Follow Through Experiment in U.S. Public School Education," (1978). 6 If price information is available, price-dependent specification of technology is also possible. See Fare et al. (1994) for the further discussion.

16

Output

Yt

o

Input

o

P(X~

Figure 2. Output Set

~---Figure 3. Input Set

Chieko UMETSU

Output

Input

Output

1

o

,

,." ,.

,. /+ +

, , ,

" + +

a c d ~~-------.

~

,.~ + +

+ +

+

, f{

e

+

+

Constant Returns to Scale (CRS)

Nonincreasing Rettnns to Scale (NIRS)

Variable Returns to Scale (VRS)

Input

+ Observed input -output combination

Figure 4. Returns to Scale of Technology

16

Output

Yt

o

Input

o

P(X~

Figure 2. Output Set

~---Figure 3. Input Set

Chieko UMETSU

Output

Input

Output

1

o

,

,." ,.

,. /+ +

, , ,

" + +

a c d ~~-------.

~

,.~ + +

+ +

+

, f{

e

+

+

Constant Returns to Scale (CRS)

Nonincreasing Rettnns to Scale (NIRS)

Variable Returns to Scale (VRS)

Input

+ Observed input -output combination

Figure 4. Returns to Scale of Technology

A NOTE ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL CHANGE USING DATA ENVELOPMENT ANALYSIS

17

(1957) had a restricted assumption of constant returns to scale and strong disposability (see the following section

for the defmition of disposability). The recent development of DEA analysis has made it possible to relax the

assumption of constant returns to scale.

Figure 4 shows constant returns to scale (CRS), nonincreasing returns to scale (NIRS), and variable

returns to scale (VRS) in input-output space, GR' (x' ,y'). If the technology exhibits constant returns to scale,

technology is expressed as a straight line from the origin, oba. Nonincreasing returns to scale technology, obed,

involves the same line segment as constant returns to scale, ob, and decreasing returns to scale portion, be.

Variable returns to scale technology, ejbed, is a combination of increasing returns to scale, jb, and decreasing

returns to scale, be. The three types of technology and the observed input-output combination demonstrate how

the variable returns to scale technology envelops observed data more tightly compared to the other two types of

technologies.

(iii) Disposability

Another important characteristic of a technology set is "disposability". Disposability assumes the

existence of inefficiency and thus non-boundary production. This assumption differs from the standard

neoclassical production theory where all marginal conditions for profit maximization or cost minimization are

satisfied at the production function.

Figure 5 illustrates strong and weak disposability of outputs. Output y: and y~ are produced from

inputs x t • The different isoquants represent different input vectors, i.e., the isoquant away from the origin shows

a greater input level. Strong disposability of output is given by the straight line segment, ab, where output y: can be disposed of at no cost to the producer without reducing the level of input. However, at the line segment

cd, reducing output y~ is not possible without reducing the output y: ' or in order to reduce output y ~ and to

keep the same level of output y: at point e, the input level needs to increase to the point e. This shows weak

disposability of outputs.

Figure 6 shows strong and weak disposability of inputs. The input requirement set away from the origin

represents a higher level of output. Inputs x: and x~ are transformed into outputs y , . Similar to the previous

case, the line segment ab shows strong disposability of inputs. Decreasing input x: is possible without

decreasing the level of output. On the other hand, the line segment cd shows weak disposability of inputs where

x: cannot be increased without increasing x~ to maintain the same level of output, or a reduction of output is

required to maintain the same level of input x: (point e to e).

While strong disposability of inputs yields zero marginal productivity of inputs, weak disposability of

inputs results in a negative marginal productivity of inputs. Also, it is possible to have different disposability for

inputs and outputs in the same technology set (see Fare et al., 1994).

A NOTE ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL CHANGE USING DATA ENVELOPMENT ANALYSIS

17

(1957) had a restricted assumption of constant returns to scale and strong disposability (see the following section

for the defmition of disposability). The recent development of DEA analysis has made it possible to relax the

assumption of constant returns to scale.

Figure 4 shows constant returns to scale (CRS), nonincreasing returns to scale (NIRS), and variable

returns to scale (VRS) in input-output space, GR' (x' ,y'). If the technology exhibits constant returns to scale,

technology is expressed as a straight line from the origin, oba. Nonincreasing returns to scale technology, obed,

involves the same line segment as constant returns to scale, ob, and decreasing returns to scale portion, be.

Variable returns to scale technology, ejbed, is a combination of increasing returns to scale, jb, and decreasing

returns to scale, be. The three types of technology and the observed input-output combination demonstrate how

the variable returns to scale technology envelops observed data more tightly compared to the other two types of

technologies.

(iii) Disposability

Another important characteristic of a technology set is "disposability". Disposability assumes the

existence of inefficiency and thus non-boundary production. This assumption differs from the standard

neoclassical production theory where all marginal conditions for profit maximization or cost minimization are

satisfied at the production function.

Figure 5 illustrates strong and weak disposability of outputs. Output y: and y~ are produced from

inputs x t • The different isoquants represent different input vectors, i.e., the isoquant away from the origin shows

a greater input level. Strong disposability of output is given by the straight line segment, ab, where output y: can be disposed of at no cost to the producer without reducing the level of input. However, at the line segment

cd, reducing output y~ is not possible without reducing the output y: ' or in order to reduce output y ~ and to

keep the same level of output y: at point e, the input level needs to increase to the point e. This shows weak

disposability of outputs.

Figure 6 shows strong and weak disposability of inputs. The input requirement set away from the origin

represents a higher level of output. Inputs x: and x~ are transformed into outputs y , . Similar to the previous

case, the line segment ab shows strong disposability of inputs. Decreasing input x: is possible without

decreasing the level of output. On the other hand, the line segment cd shows weak disposability of inputs where

x: cannot be increased without increasing x~ to maintain the same level of output, or a reduction of output is

required to maintain the same level of input x: (point e to e).

While strong disposability of inputs yields zero marginal productivity of inputs, weak disposability of

inputs results in a negative marginal productivity of inputs. Also, it is possible to have different disposability for

inputs and outputs in the same technology set (see Fare et al., 1994).

18 Chieko UMETSU

Input

q'

QJtput o a ~ 0

Figure 5. Strong and Weak Disposability of Outputs

Input

xl! a

b

o Input

Figure 6. Strong and Weak Disposability of Inputs

Figure 7. Farrell Measure of Technical Efficiency

Input

18 Chieko UMETSU

Input

q'

QJtput o a ~ 0

Figure 5. Strong and Weak Disposability of Outputs

Input

xl! a

b

o Input

Figure 6. Strong and Weak Disposability of Inputs

Figure 7. Farrell Measure of Technical Efficiency

Input

A NOTE ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL CHANGE USING DATA ENVELOPMENT ANALYSIS

4.2 The Farrell measure of technical efficiency

19

Technology presented in the previous section is called piecewise linear technology. A frontier

technology is constructed using linear programming by estimating the best practice frontier, as piecewise linear

segments relative to all observations in the reference set. Piecewise linear technology satisfies very general

axioms of production theory. Also, as the number of observations in a reference set increases, piecewise linear

technology converges to a differentiable smooth neoclassical production function.

Farrell (1957) defmed a deterministic, nonparametric radial measure of technical efficiency in which

efficiency is obtained by radially reducing the level of inputs relative to the frontier technology. Figure 7

illustrates Farrell's input-oriented measure of technical efficiency. A firm produces a single outputy using inputs

Xl and X2. The isoquant qq' represents combinations of the two inputs which produce the same amount of output.

If production is technically efficient, the firm will produce on qq', i.e., the frontier technology. Point A is off the

frontier technology qq' and its input combination can be reduced radially along the ray, OA, to the input

combination on the frontier B where production is technically efficient. Farrell dermed OBIOA as the technical

efficiency of the firm A, which varies between zero and unity.

The Farrell measure of technical efficiency is constructed using input and output quantity information

and is independent of input prices. If input prices are available, allocative efficiency (or price efficiency) can be

defined. Let pp' be the ratio of input prices. Then allocative efficiency is shown by OCIOB. Farrell defmed

overall efficiency as OCIOA. Farrell's efficiency measure is similar to Debreu' s (1951) "coefficient of resource

utilization" as well as the inverse of a distance function adopted by Shephard (1953). In Figure 7, the distance

function is given by OAIOB which is just the inverse of Farrell measure of technical efficiency, OBIOA.

In order to explain how to estimate the Farrell measure of technical efficiency using linear

programming, the following section uses only input-oriented Farrell efficiency measures; however, an output­

oriented version is also possible. Suppose there are k = 1, ... ,[(1 number of firms which produce M output, m =

1, ... ,M using N inputs, n = 1, ... ,N, at each time period t = 1, ... , T. Let us first defme the input requirement set,

Lt (yt), at period t as follows:

K

Lt(yt)={xt:y~ ~LZk.,y!/ m= 1, ... , M, k=1

K

X I > ~Zk,IXk.1 n = 1 N n -~ n' , ... , ,

(9)

k=!

Zk.t ~ 0, k = 1, ... , K},

where zk,t indicates intensity levels, which allows the activity of each individual frrm to expand or contract in

order to construct a piecewise linear frontier technology (Fare etal., 1994). Let F/(y',X' / C,S) denote the

input-oriented Farrell measure of technical efficiency, and D: (yl ,x' / C,S) denote Shephard's input-oriented

distance function at period t with the assumption of constant returns to scale (C) and strong disposability of

inputs and outputs (S) as follows:

A NOTE ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL CHANGE USING DATA ENVELOPMENT ANALYSIS

4.2 The Farrell measure of technical efficiency

19

Technology presented in the previous section is called piecewise linear technology. A frontier

technology is constructed using linear programming by estimating the best practice frontier, as piecewise linear

segments relative to all observations in the reference set. Piecewise linear technology satisfies very general

axioms of production theory. Also, as the number of observations in a reference set increases, piecewise linear

technology converges to a differentiable smooth neoclassical production function.

Farrell (1957) defmed a deterministic, nonparametric radial measure of technical efficiency in which

efficiency is obtained by radially reducing the level of inputs relative to the frontier technology. Figure 7

illustrates Farrell's input-oriented measure of technical efficiency. A firm produces a single outputy using inputs

Xl and X2. The isoquant qq' represents combinations of the two inputs which produce the same amount of output.

If production is technically efficient, the firm will produce on qq', i.e., the frontier technology. Point A is off the

frontier technology qq' and its input combination can be reduced radially along the ray, OA, to the input

combination on the frontier B where production is technically efficient. Farrell dermed OBIOA as the technical

efficiency of the firm A, which varies between zero and unity.

The Farrell measure of technical efficiency is constructed using input and output quantity information

and is independent of input prices. If input prices are available, allocative efficiency (or price efficiency) can be

defined. Let pp' be the ratio of input prices. Then allocative efficiency is shown by OCIOB. Farrell defmed

overall efficiency as OCIOA. Farrell's efficiency measure is similar to Debreu' s (1951) "coefficient of resource

utilization" as well as the inverse of a distance function adopted by Shephard (1953). In Figure 7, the distance

function is given by OAIOB which is just the inverse of Farrell measure of technical efficiency, OBIOA.

In order to explain how to estimate the Farrell measure of technical efficiency using linear

programming, the following section uses only input-oriented Farrell efficiency measures; however, an output­

oriented version is also possible. Suppose there are k = 1, ... ,[(1 number of firms which produce M output, m =

1, ... ,M using N inputs, n = 1, ... ,N, at each time period t = 1, ... , T. Let us first defme the input requirement set,

Lt (yt), at period t as follows:

K

Lt(yt)={xt:y~ ~LZk.,y!/ m= 1, ... , M, k=1

K

X I > ~Zk,IXk.1 n = 1 N n -~ n' , ... , ,

(9)

k=!

Zk.t ~ 0, k = 1, ... , K},

where zk,t indicates intensity levels, which allows the activity of each individual frrm to expand or contract in

order to construct a piecewise linear frontier technology (Fare etal., 1994). Let F/(y',X' / C,S) denote the

input-oriented Farrell measure of technical efficiency, and D: (yl ,x' / C,S) denote Shephard's input-oriented

distance function at period t with the assumption of constant returns to scale (C) and strong disposability of

inputs and outputs (S) as follows:

20 Chieko UMETSU

F/ (yl ,Xl / C,S) = min{A: Axl ELI (yl / C,S)},

D: (yl ,Xl / C,S) = max{A: (Xl/A-) ELI (yl / C,S)}. (10)

where F/ (l ,Xl / C, S) estimates the minimum possible expansion of X t and D: (yl ,Xl / C, S) estimates the

maximum possible contraction of X t •

The linear programming problem of (9) is given as:

F/ (yk',t xk',t / C S) = [D~ (yk"1 xk',t / C S) ]-1 = min A-I , , I" A,X ' (11)

subject to

K I' ~ k,1 k,t 1 M Ym ::; L..Jz Ym' m = , ... , ,

k=1

K '] k',t ~ k,t k,t 1 N

/L xn ~.L...J Z xn ,n = , ... , , (12)

k=1

zk,t ~ 0, k =1, ... , K.

Since the Farrell measure of technical efficiency varies between zero and unity, the distance function,

or the inverse of the Farrell measure, takes values greater than unity. Both the Farrell measure and the distance

function are equal to unity when the input vector belongs to the input requirement set, LI (yt ), which means

technology is efficient.

The earlier Farrell measure was restricted to constant returns to scale and strong disposability

assumptions. By controlling the intensity variables with additional constraints to linear programming,

nonincreasing returns to scale and variable returns to scale can be imposed as below:

no restriction => constant returns to scale K

LZk,1 :::;;1 k=1

K

LZk'l = 1 k=1

=> nonincreasing returns to scale

=> variable returns to scale.

(13)

Strong disposability of inputs and outputs can be relaxed to weak disposability (W) by an additional

parameter cr in a nonlinear programming problem. An example is provided with the variable returns to scale

assumption.

F/ (yk',t xk',t / V W) = [D~ (yk',t xk',t / V W) }-1 = min A-I , , I" A.,x,O' ' (14)

subject to

20 Chieko UMETSU

F/ (yl ,Xl / C,S) = min{A: Axl ELI (yl / C,S)},

D: (yl ,Xl / C,S) = max{A: (Xl/A-) ELI (yl / C,S)}. (10)

where F/ (l ,Xl / C, S) estimates the minimum possible expansion of X t and D: (yl ,Xl / C, S) estimates the

maximum possible contraction of X t •

The linear programming problem of (9) is given as:

F/ (yk',t xk',t / C S) = [D~ (yk"1 xk',t / C S) ]-1 = min A-I , , I" A,X ' (11)

subject to

K I' ~ k,1 k,t 1 M Ym ::; L..Jz Ym' m = , ... , ,

k=1

K '] k',t ~ k,t k,t 1 N

/L xn ~.L...J Z xn ,n = , ... , , (12)

k=1

zk,t ~ 0, k =1, ... , K.

Since the Farrell measure of technical efficiency varies between zero and unity, the distance function,

or the inverse of the Farrell measure, takes values greater than unity. Both the Farrell measure and the distance

function are equal to unity when the input vector belongs to the input requirement set, LI (yt ), which means

technology is efficient.

The earlier Farrell measure was restricted to constant returns to scale and strong disposability

assumptions. By controlling the intensity variables with additional constraints to linear programming,

nonincreasing returns to scale and variable returns to scale can be imposed as below:

no restriction => constant returns to scale K

LZk,1 :::;;1 k=1

K

LZk'l = 1 k=1

=> nonincreasing returns to scale

=> variable returns to scale.

(13)

Strong disposability of inputs and outputs can be relaxed to weak disposability (W) by an additional

parameter cr in a nonlinear programming problem. An example is provided with the variable returns to scale

assumption.

F/ (yk',t xk',t / V W) = [D~ (yk',t xk',t / V W) }-1 = min A-I , , I" A.,x,O' ' (14)

subject to

A NOTE ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL CHANGE USING DATA ENVELOPMENT ANALYSIS

K.

Y k'" < ~ Zk"yk" m = 1 M . m - L..." m , , ••• , ' k=1

K.

A. 0' X!'·' = L Zk.1 X!", n = 1, ... , N, k=1

0<0' ~ 1,

zk. t ~ 0, k = 1, ... , K, K. L Zk.1 = 1.

k=1

21

(15)

This parameter cr makes it possible for the convex combination of the observed inputs and outputs to scale either

radially or proportionately (Hire et aI., 1985).

Because constant returns to scale and the strong disposability assumption are the most restrictive, the

Farrell measure of technical efficiency is lowest in this case. On the other hand, variable returns to scale and

weak disposability are the least restrictive, thus yielding the highest efficiency. It is obvious from Figure 4 that

variable returns to scale technology envelops data more tightly so that it brings data close to the frontier. In

summary:

(16)

4.3 Input scale efficiency and input congestion

In this section, the concept of input scale efficiency and input congestion are introduced. The input­

oriented Farrell measure of technical efficiency with constant returns to scale and strong disposability is

composed of input scale efficiency, input congestion, and the input-oriented Farrell measure with variable

returns to scale and weak disposability.

(i) Input scale efficiency

Input scale efficiency is defmed as the efficiency of production activity with variable returns to scale

relative to the one with constant returns to scale as follows:

i i _ F;(yi ,xi / C,S) . _ 2 S; (y ,x ) - . i ' ] - 1, , ... , J.

F;(yJ ,x / V,S) (17)

A production activity is input scale efficient if the two input-oriented Farrell measures of technical

efficiency with strong and weak disposability are equivalent, F;(yi ,xl / C,S) = F;(yl ,xl/V,S), which yields

Sj (yi ,xi) = 1. Scale inefficient production activities result in input scale efficiency less than one. Figure 8

illustrates scale efficiency of inputs in input-output space. The production activity at point b is scale efficient

because it is technically efficient in both constant returns to scale (CRS) and variable returns to scale (VRS)

A NOTE ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL CHANGE USING DATA ENVELOPMENT ANALYSIS

K.

Y k'" < ~ Zk"yk" m = 1 M . m - L..." m , , ••• , ' k=1

K.

A. 0' X!'·' = L Zk.1 X!", n = 1, ... , N, k=1

0<0' ~ 1,

zk. t ~ 0, k = 1, ... , K, K. L Zk.1 = 1.

k=1

21

(15)

This parameter cr makes it possible for the convex combination of the observed inputs and outputs to scale either

radially or proportionately (Hire et aI., 1985).

Because constant returns to scale and the strong disposability assumption are the most restrictive, the

Farrell measure of technical efficiency is lowest in this case. On the other hand, variable returns to scale and

weak disposability are the least restrictive, thus yielding the highest efficiency. It is obvious from Figure 4 that

variable returns to scale technology envelops data more tightly so that it brings data close to the frontier. In

summary:

(16)

4.3 Input scale efficiency and input congestion

In this section, the concept of input scale efficiency and input congestion are introduced. The input­

oriented Farrell measure of technical efficiency with constant returns to scale and strong disposability is

composed of input scale efficiency, input congestion, and the input-oriented Farrell measure with variable

returns to scale and weak disposability.

(i) Input scale efficiency

Input scale efficiency is defmed as the efficiency of production activity with variable returns to scale

relative to the one with constant returns to scale as follows:

i i _ F;(yi ,xi / C,S) . _ 2 S; (y ,x ) - . i ' ] - 1, , ... , J.

F;(yJ ,x / V,S) (17)

A production activity is input scale efficient if the two input-oriented Farrell measures of technical

efficiency with strong and weak disposability are equivalent, F;(yi ,xl / C,S) = F;(yl ,xl/V,S), which yields

Sj (yi ,xi) = 1. Scale inefficient production activities result in input scale efficiency less than one. Figure 8

illustrates scale efficiency of inputs in input-output space. The production activity at point b is scale efficient

because it is technically efficient in both constant returns to scale (CRS) and variable returns to scale (VRS)

22 Chieko UMETSU

technologies. On the other hand, points a and c are not scale efficient because they are technically efficient with

VRS technology but not efficient in CRS technology.

(ii) Input congestion

Input congestion is defmed as the relative efficiency between production activity with strong

disposability and one with weak disposability when technology exhibits variable returns to scale.

C (yi i) F;(yi ,xi / V,S) . 12 J j ,x = ii' ] = , "'" '

F;(Y ,x / V, W) (18)

Input congestion shows the proportional contraction of the input vector in technology with weak

disposability in order to keep the same level of output compared to technology with strong disposability. Figure

9 illustrates congestion of inputs. Consider the case where true technology is characterized by weak disposability

(V, W); however, only strong disposability (V, S) is considered under VRS technology. Then in Figure 9

production activities on line fa are considered inefficient because they are inside of the isoquant h-f-i with

strong disposability. However, those production activities are congesting relative to the technology with strong

disposability because increasing Xl while maintaining the same level of X2 will reduce the output. Thus, input

vector b congests output vector y\ F;(yi ,xi IV,S) < F;(yi ,xi IV,W), and Cj(yi ,xi) < 1. If there is no

congestion, such as production activity c, then the two Farrell measures are equal, i.e.,

F;(yi ,xi / V,S) = F;(yi ,xi I V, W), and Cj(yi ,xi) = 1.

(iii) Decomposition of the Farrell measure of technical efficiency

The Farrell input measure of technical efficiency with constant returns to scale and strong disposability

(C,S) assumption can be decomposed into three terms, as follows:

F ( i xi / C S) = F ( i xi / V W). F'; (y i ,x ~ / C, S) . F'; (y i ,x ~ / V, S) I Y , , I Y , , F (i ) / V S) F (i ) / V W)

I Y ,x , i Y ,x , (19)

where the frrst term shows the input measure of technical efficiency with variable returns to scale and weak

input disposability (V,W); the second term shows input scale efficiency measurement; and the third term

measures input congestion. The Farrell measure F;(yi ,xi I C,S) equals one only when all three components are

equal to one (Fare et at, 1994).

22 Chieko UMETSU

technologies. On the other hand, points a and c are not scale efficient because they are technically efficient with

VRS technology but not efficient in CRS technology.

(ii) Input congestion

Input congestion is defmed as the relative efficiency between production activity with strong

disposability and one with weak disposability when technology exhibits variable returns to scale.

C (yi i) F;(yi ,xi / V,S) . 12 J j ,x = ii' ] = , "'" '

F;(Y ,x / V, W) (18)

Input congestion shows the proportional contraction of the input vector in technology with weak

disposability in order to keep the same level of output compared to technology with strong disposability. Figure

9 illustrates congestion of inputs. Consider the case where true technology is characterized by weak disposability

(V, W); however, only strong disposability (V, S) is considered under VRS technology. Then in Figure 9

production activities on line fa are considered inefficient because they are inside of the isoquant h-f-i with

strong disposability. However, those production activities are congesting relative to the technology with strong

disposability because increasing Xl while maintaining the same level of X2 will reduce the output. Thus, input

vector b congests output vector y\ F;(yi ,xi IV,S) < F;(yi ,xi IV,W), and Cj(yi ,xi) < 1. If there is no

congestion, such as production activity c, then the two Farrell measures are equal, i.e.,

F;(yi ,xi / V,S) = F;(yi ,xi I V, W), and Cj(yi ,xi) = 1.

(iii) Decomposition of the Farrell measure of technical efficiency

The Farrell input measure of technical efficiency with constant returns to scale and strong disposability

(C,S) assumption can be decomposed into three terms, as follows:

F ( i xi / C S) = F ( i xi / V W). F'; (y i ,x ~ / C, S) . F'; (y i ,x ~ / V, S) I Y , , I Y , , F (i ) / V S) F (i ) / V W)

I Y ,x , i Y ,x , (19)

where the frrst term shows the input measure of technical efficiency with variable returns to scale and weak

input disposability (V,W); the second term shows input scale efficiency measurement; and the third term

measures input congestion. The Farrell measure F;(yi ,xi I C,S) equals one only when all three components are

equal to one (Fare et at, 1994).

A NOTE ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL CHANGE USING DATA ENVELOPMENT ANALYSIS

Output Input

(GRN,S)

Input o xt 0

Figure 8. Scale Efficiency of Inputs

Input h

v (ytN,S)

c

V (ytN,W) b

a

1

~----------------------------·Inpm

o x~

Figure 9. Congestion of Inputs

Lt+I(yt+I/C, S)

Figure 10. Input-oriented Malmquist Productivity Index

23

Input

A NOTE ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL CHANGE USING DATA ENVELOPMENT ANALYSIS

Output Input

(GRN,S)

Input o xt 0

Figure 8. Scale Efficiency of Inputs

Input h

v (ytN,S)

c

V (ytN,W) b

a

1

~----------------------------·Inpm

o x~

Figure 9. Congestion of Inputs

Lt+I(yt+I/C, S)

Figure 10. Input-oriented Malmquist Productivity Index

23

Input

24 Chieko UMETSU

4.4 The input-oriented Malmquist productivity index

The Malmquist productivity index was proposed by Hire, Grosskopf, Lindgren, and Roos (1989)

following the work of Caves, Christensen, and Diewert (1982). They show that under a flexible technology

specification such as trans log, Tornqvist indices are equal to the arithmetic mean of two Malmquist indices. The

input-oriented Malmquist productivity index with constant returns to scale and strong disposability (C, S)

between period t and t + 1 consists of the geometric mean of four input oriented Farrell measures of technical

efficiency as follows:

where

1

1+1 1+1 1+1 I 'IC [F/(yI,XIIC,S) Ft1(yl,X

IIC,S)]2

M j (y ,x ,y,x ,S) = F;'(yl+l ,x1+1 I C,S) F;l+l(y'+1 ,x1+1 I C,S)

F/ (y' ,x'I C,S) = min { A: AX' EL' (y' I C,S)},

F;' (yl+l ,x1+1 I C,S) = min{ A: AX'+1 EL' (yl+l I C,S)},

F/+1(y' ,x'I C,S) = min{A: AX' E L'+I(y' I C,S)},

and

F;l+l(yl+l ,x1+1 I C,S) = min{A: AXl+l E Ll+l(yl+l I C,S)}.

(20)

(21)

The cross period Farrell measure needs special attention since. F/ (yl+l ,x t+1 I C,S) means to evaluate

production activities at period t+ 1 with the reference technology at period t. Similarly, the Farrell measure

F;l+l (y' ,Xl I C, S) estimates the efficiency of production activity at period t relative to the frontier technology at

period t+1.

Alternatively, four input-oriented distance functions which are reciprocals of Farrell measures can be

used to construct the Malmquist index as follows:

where

1

M~+I(yl+l Xl+l I Xl I C S) = i Y ,x , i Y ,x , [DI( /+1 1+1 I C S) Dl+l( 1+1 1+1 Ie S)]2

I "y, , D;(yI,xIIC,S) D,~+I(yI,XIIC,S)

D: (yl ,Xl I C,S) = maX{A: (x'I A) EL' (yl I C,S)},

D: (yl+l ,XI+1 I C,S) = maX{A: (X I+1 I A) ELI (yl+l I C,S)},

D:+l(y' ,x'I C,S) = maX{A: (x'I A) E L'+1(yl I C,S)},

and

D:+l(yt+l,X'+l I C,S) = max{A: (X '+1 I A) E Ll+l(yl+l I C,S)}.

(22)

(23)

Figure 10 illustrates the input-oriented Malmquist productivity index. The input set, L' (y') and

Ll+l (y'+l ), shows frontier technology at period t and t+ 1, respectively. The input vector at period t and t+ 1 is

denoted by x' (vector c) and Xl+l (vector e). The Malmquist productivity index in equation (19) can be

24 Chieko UMETSU

4.4 The input-oriented Malmquist productivity index

The Malmquist productivity index was proposed by Hire, Grosskopf, Lindgren, and Roos (1989)

following the work of Caves, Christensen, and Diewert (1982). They show that under a flexible technology

specification such as trans log, Tornqvist indices are equal to the arithmetic mean of two Malmquist indices. The

input-oriented Malmquist productivity index with constant returns to scale and strong disposability (C, S)

between period t and t + 1 consists of the geometric mean of four input oriented Farrell measures of technical

efficiency as follows:

where

1

1+1 1+1 1+1 I 'IC [F/(yI,XIIC,S) Ft1(yl,X

IIC,S)]2

M j (y ,x ,y,x ,S) = F;'(yl+l ,x1+1 I C,S) F;l+l(y'+1 ,x1+1 I C,S)

F/ (y' ,x'I C,S) = min { A: AX' EL' (y' I C,S)},

F;' (yl+l ,x1+1 I C,S) = min{ A: AX'+1 EL' (yl+l I C,S)},

F/+1(y' ,x'I C,S) = min{A: AX' E L'+I(y' I C,S)},

and

F;l+l(yl+l ,x1+1 I C,S) = min{A: AXl+l E Ll+l(yl+l I C,S)}.

(20)

(21)

The cross period Farrell measure needs special attention since. F/ (yl+l ,x t+1 I C,S) means to evaluate

production activities at period t+ 1 with the reference technology at period t. Similarly, the Farrell measure

F;l+l (y' ,Xl I C, S) estimates the efficiency of production activity at period t relative to the frontier technology at

period t+1.

Alternatively, four input-oriented distance functions which are reciprocals of Farrell measures can be

used to construct the Malmquist index as follows:

where

1

M~+I(yl+l Xl+l I Xl I C S) = i Y ,x , i Y ,x , [DI( /+1 1+1 I C S) Dl+l( 1+1 1+1 Ie S)]2

I "y, , D;(yI,xIIC,S) D,~+I(yI,XIIC,S)

D: (yl ,Xl I C,S) = maX{A: (x'I A) EL' (yl I C,S)},

D: (yl+l ,XI+1 I C,S) = maX{A: (X I+1 I A) ELI (yl+l I C,S)},

D:+l(y' ,x'I C,S) = maX{A: (x'I A) E L'+1(yl I C,S)},

and

D:+l(yt+l,X'+l I C,S) = max{A: (X '+1 I A) E Ll+l(yl+l I C,S)}.

(22)

(23)

Figure 10 illustrates the input-oriented Malmquist productivity index. The input set, L' (y') and

Ll+l (y'+l ), shows frontier technology at period t and t+ 1, respectively. The input vector at period t and t+ 1 is

denoted by x' (vector c) and Xl+l (vector e). The Malmquist productivity index in equation (19) can be

A NOTE ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL CHANGE USING DATA ENVELOPMENT ANALYSIS

25

expressed by the defmition of Farrell measures. The production activity e at period t+ 1 is evaluated relative to

the frontier technology at period t along the ray from the origin Of. Also, the production activity c at period t is

evaluated relative to the frontier technology at period t+ 1 along the ray Oa.

1

M~+l(yl+l X I+1 I Xl / C S) = [Ob / Oc . Oa /oc]i I "Y , , Oe / of Od / Oe

(24)

The input-oriented Malmquist productivity index shows progress when the value of an index is less

than one and regress when the value is greater than one. When the Malmquist index is equal to unity, there is no

improvement in productivity between two periods. The output-oriented Malmquist index is the reciprocal of the

input-oriented Malmquist productivity index where an index greater than one shows progress and the value less

than one shows regress of productivity.

The Malmquist productivity index is further decomposed into a change in efficiency, and a change in

technology which shows a shift of frontier technology between two time periods as follows.

I

r:;ol (yl I / C S) [ r;"l+l (y1+1 1+1 / C S) r:;ot+1 (yl I / C S)]2" M~+I(yI+1 XI+1 I Xl / C S) = rj ,x , r i ,X , r i ,x , . I "Y , , pt+1 (y1+1 Xt+1 / C S) pl(y1+1 Xt+1 / C S) pl(yl Xl / C S)

i , , i , , i' ,

(25)

Th fi pI (yl ,Xl / C, S) . di h' ffi' fr . d l·th th e Irst term, I ,m cates a c ange m e ICIency om peno t to t+ ,WI respect to e Ftl (yt+1 ,x l+1 / C,S)

1

[

F;I+I(yt+l'Xt+1 / C,S) F;t+l(yl ,Xl / C,S)]2" , technology prevailing at each point in time. The second term,

F;I (y1+1 ,Xt+1 / C, S) F;I (yl ,Xl / C, S)

indicates technological change, with constant returns to scale and strong disposability of inputs.

In Figure 10, this decomposition is also written as the ratio of distances along the ray from the origin as

follows:

I I

M~+I (yt+1 Xl+l I Xl / C S) = Ob /Oc [Od / Oe . Oa / oc]2" = Ob / Oc [Od . oa]2" I "y, , Od/Oe Of/oe Ob/Oc Od/Oe Of Ob

(26)

This decomposition is particularly useful in policy analysis. Both terms, changes in efficiency and

technology over time, should be intetpreted as progress when the value is less than one and regress when they

are greater than one. In the output-oriented case, the intetpretation is the same as output-oriented Malmquist

index. Hire and Grosskopf (1992) showed that under certain conditions the Fisher ideal index can be derived

from Malmquist indices.

The advantages of the Malmquist index are summarized as follows (Grosskopf, 1994): (a) The

Malmquist index is a useful way of estimating total factor productivity when inefficiency of production exists.

(b) The measurement of the Malmquist index does not require price or cost share data, and requires only

A NOTE ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL CHANGE USING DATA ENVELOPMENT ANALYSIS

25

expressed by the defmition of Farrell measures. The production activity e at period t+ 1 is evaluated relative to

the frontier technology at period t along the ray from the origin Of. Also, the production activity c at period t is

evaluated relative to the frontier technology at period t+ 1 along the ray Oa.

1

M~+l(yl+l X I+1 I Xl / C S) = [Ob / Oc . Oa /oc]i I "Y , , Oe / of Od / Oe

(24)

The input-oriented Malmquist productivity index shows progress when the value of an index is less

than one and regress when the value is greater than one. When the Malmquist index is equal to unity, there is no

improvement in productivity between two periods. The output-oriented Malmquist index is the reciprocal of the

input-oriented Malmquist productivity index where an index greater than one shows progress and the value less

than one shows regress of productivity.

The Malmquist productivity index is further decomposed into a change in efficiency, and a change in

technology which shows a shift of frontier technology between two time periods as follows.

I

r:;ol (yl I / C S) [ r;"l+l (y1+1 1+1 / C S) r:;ot+1 (yl I / C S)]2" M~+I(yI+1 XI+1 I Xl / C S) = rj ,x , r i ,X , r i ,x , . I "Y , , pt+1 (y1+1 Xt+1 / C S) pl(y1+1 Xt+1 / C S) pl(yl Xl / C S)

i , , i , , i' ,

(25)

Th fi pI (yl ,Xl / C, S) . di h' ffi' fr . d l·th th e Irst term, I ,m cates a c ange m e ICIency om peno t to t+ ,WI respect to e Ftl (yt+1 ,x l+1 / C,S)

1

[

F;I+I(yt+l'Xt+1 / C,S) F;t+l(yl ,Xl / C,S)]2" , technology prevailing at each point in time. The second term,

F;I (y1+1 ,Xt+1 / C, S) F;I (yl ,Xl / C, S)

indicates technological change, with constant returns to scale and strong disposability of inputs.

In Figure 10, this decomposition is also written as the ratio of distances along the ray from the origin as

follows:

I I

M~+I (yt+1 Xl+l I Xl / C S) = Ob /Oc [Od / Oe . Oa / oc]2" = Ob / Oc [Od . oa]2" I "y, , Od/Oe Of/oe Ob/Oc Od/Oe Of Ob

(26)

This decomposition is particularly useful in policy analysis. Both terms, changes in efficiency and

technology over time, should be intetpreted as progress when the value is less than one and regress when they

are greater than one. In the output-oriented case, the intetpretation is the same as output-oriented Malmquist

index. Hire and Grosskopf (1992) showed that under certain conditions the Fisher ideal index can be derived

from Malmquist indices.

The advantages of the Malmquist index are summarized as follows (Grosskopf, 1994): (a) The

Malmquist index is a useful way of estimating total factor productivity when inefficiency of production exists.

(b) The measurement of the Malmquist index does not require price or cost share data, and requires only

26 Chieko UMETSU

quantity data. (C) The Malmquist index is a useful tool to analyze public service where inputs and outputs are not

marketable (i.e., schools, hospitals, utilities, etc.). (d) The Malmquist index can be decomposed into the change

in efficiency and the change in frontier technology. For both components, it is possible to progress and regress

over time. (e) Technical change is not constrained as Hicks neutral even under constant returns to scale, since it

is not constrained to a specific functional form. (f) The distance function and thus Malmquist index do not

require behavioral assumptions such as profit maximization, cost minimization, or revenue maximization. (g)

The Malmquist index is useful for applying to cases where data is limited, and the assumption of profit

maximization and efficiency in production is not appropriate.

The disadvantage of this approach includes: (a) there is no statistical inference and the index is sensitive

to measurement errors; (b) computation is more demanding since four linear programming programs are

required to estimate one Malmquist index.

4.5 The input-oriented Malmquist productivity index with scale and congestion efficiency

The fIrst efficiency change term of the input oriented Malmquist index is decomposed into three terms

using the Farrell measure decomposition explained in section 4.3 as follows:

1 (27)

.[F/+1(yt+l,Xt+l / C,S) F/+1(yt ,x' / c,S)]2" F/(yt+l,Xt+1 /C,S) F/(yt,xt IC,S)

h th fi Ft (yt xt / V W) . d' h . I hn' I f~ . th d were e Irst term, I' , , m lcates c anges m pure y tec lca e llclency; e secon term, F;t+l(yt+l,Xt+l / V, W)

shows changes in scale efficiency; and the third term,

congestion between two periods. These three terms indicate improvement of productivity when their values are

less than one. The output oriented version should be interpreted in an opposite direction.

5. CONCLUSION

This paper briefly introduced the method for measuring total factor productivity. The nonfrontier

approach to total factor productivity does not consider technical inefficiency while the frontier approach

explicitly considers technical efficiency. In the former case, total factor productivity is equivalent to

technological change; however, in the latter case productivity is equivalent to efficiency change and

technological change. The nonparametric approach is not constrained by specific functional forms but does not

26 Chieko UMETSU

quantity data. (C) The Malmquist index is a useful tool to analyze public service where inputs and outputs are not

marketable (i.e., schools, hospitals, utilities, etc.). (d) The Malmquist index can be decomposed into the change

in efficiency and the change in frontier technology. For both components, it is possible to progress and regress

over time. (e) Technical change is not constrained as Hicks neutral even under constant returns to scale, since it

is not constrained to a specific functional form. (f) The distance function and thus Malmquist index do not

require behavioral assumptions such as profit maximization, cost minimization, or revenue maximization. (g)

The Malmquist index is useful for applying to cases where data is limited, and the assumption of profit

maximization and efficiency in production is not appropriate.

The disadvantage of this approach includes: (a) there is no statistical inference and the index is sensitive

to measurement errors; (b) computation is more demanding since four linear programming programs are

required to estimate one Malmquist index.

4.5 The input-oriented Malmquist productivity index with scale and congestion efficiency

The fIrst efficiency change term of the input oriented Malmquist index is decomposed into three terms

using the Farrell measure decomposition explained in section 4.3 as follows:

1 (27)

.[F/+1(yt+l,Xt+l / C,S) F/+1(yt ,x' / c,S)]2" F/(yt+l,Xt+1 /C,S) F/(yt,xt IC,S)

h th fi Ft (yt xt / V W) . d' h . I hn' I f~ . th d were e Irst term, I' , , m lcates c anges m pure y tec lca e llclency; e secon term, F;t+l(yt+l,Xt+l / V, W)

shows changes in scale efficiency; and the third term,

congestion between two periods. These three terms indicate improvement of productivity when their values are

less than one. The output oriented version should be interpreted in an opposite direction.

5. CONCLUSION

This paper briefly introduced the method for measuring total factor productivity. The nonfrontier

approach to total factor productivity does not consider technical inefficiency while the frontier approach

explicitly considers technical efficiency. In the former case, total factor productivity is equivalent to

technological change; however, in the latter case productivity is equivalent to efficiency change and

technological change. The nonparametric approach is not constrained by specific functional forms but does not

A NOTE ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL CHANGE USING DATA ENVELOPMENT ANALYSIS

27

account for measurement errors. The parametric approach is subject to specification errors and bias of

technology itself is sensitive to functional forms.

As a nonparametric frontier approach, the Malmquist productivity index has shown a relative advantage

in productivity analysis in developing countries where price infonnation is limited and only quantity data is

available. The Malmquist productivity index is based on the Farrell measure of technical efficiency which is an

inverse of Shephard's distance function. The Malmquist productivity index can be decomposed into efficiency

change and technological change, which can provide useful information for policy analysis.

REFERENCES

Aigner, D. J., C. A. K. Lovell, and P. Schmidt. (1977). "Formulation and Estimation of Stochastic Frontier Production Function Models." Journal of Econometrics 6(1): 21-37.

Caves, D. W., L. R. Christensen, and W. E. Diewert. (1982). "The Economic Theory of Index Numbers and the Measurement of Input, Output, and Productivity." Econometrica 50(6): 1393-1414.

Charnes, A., W.W. Cooper and E. Rhodes (1978). "A Data Envelopment Analysis Approach to Evaluation of the Program Follow Through Experiment in U.S. Public School Education," Management Science Research Report No. 432, Carnegie-Mellon University, School of Urban and Public Affairs, Pittsburgh, PA.

Chavas, J.-P. and T. L. Cox (1990). "A Non-Parametric Analysis of Productivity: The Case of U.S. and Japanese Manufacturing." American Economic Review 80(3): 450-464.

Cox, T. L. and J.-P. Chavas (1990). "A Nonparametric Analysis of Productivity: The Case of US Agriculture." European Review of Agricultural Economics 17: 449-464.

Debreu, G. (1951). "The Coefficient of Resource Utilization." Econometrica 19(3): 273-292. Diamond, P., D. McFadden, and M. Rodriguez. (1978). Measurement of the Elasticity of Factor Substitution and

Bias of Technical Change. Production Economics: A Dual Approach to Theory and Application. M. Fuss and D. McFadden, Eds. Amsterdam, North-Holland: Ch. IV.2.

Diewert, W. E. (1976). "Exact and Superlative Index Numbers." Journal of Econometrics 4: 115-145. Diewert, W. E. (1980). "Capital and the Theory of Productivity." American Economic Review 70(2): 260-267. Diewert, W. E. and M. N. F. Mendoza (1995). "Data Envelopment Analysis: A Practical Alternative? "

Vancouver, Department of Economics, University of British Columbia. Hire, R., R. Grabowski, and S. Grosskopf. (1985). "Technical Efficiency of Philippine Agriculture." Applied

Economics 17: 205-214. Hire, R. and S. Grosskopf (1992). "Malmquist Productivity Indexes and Fisher Ideal Indexes." The Economic

Joumall02: 158-160. Hire, R., S. Grosskopf, B. Lindgren, and P. Roos. (1994). Productivity Developments in Swedish Hospitals: A

Malmquist Output Index Approach. Data Envelopment Analysis: The Theory, Applications and the Process. Charnes, A., W. W. Cooper, A. Y. Lewin and L. M. Seiford, Eds. Boston, Kluwer Academic.

Farrell, M. J. (1957). "The Measurement of Productive Efficiency." Journal of the Royal Statistical Society 120 (Part III): 11-290.

Grosskopf, S. (1993). Efficiency and Productivity. The Measurement of Productive Efficiency. H. O. Fried, C. A. K. Lovell and S. S. Schmidt, Eds. New York, Oxford University Press: 160-194.

Kawagoe, T., K. Otsuka, and Y. Hayami. (1986). "Induced Bias of Technical Change in Agriculture: The United States and Japan, 1880-1980." Journal of Political Economy 94(3): 523-544.

Kim, H. Y. (1988). "Analyzing the Indirect Production Function for U.S. Manufacturing." Southern Economic Journal 54: 494-504.

Malmquist, S. (1953). "Index Numbers and Indifference Surfaces." Trabajos de Estadistica 4: 209-242. Nishimizu, M. and J. M. Page (1982). "Total Factor Productivity Growth, Technological Progress and Technical

Efficiency Change: Dimensions of Productivity Change in Yugoslavia 1965-78." Economic Journal 92: 920-936.

Shephard, R. W. (1953). Cost and Production Functions. Princeton, N.J., Princeton University Press.

A NOTE ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY, EFFICIENCY AND TECHNOLOGICAL CHANGE USING DATA ENVELOPMENT ANALYSIS

27

account for measurement errors. The parametric approach is subject to specification errors and bias of

technology itself is sensitive to functional forms.

As a nonparametric frontier approach, the Malmquist productivity index has shown a relative advantage

in productivity analysis in developing countries where price infonnation is limited and only quantity data is

available. The Malmquist productivity index is based on the Farrell measure of technical efficiency which is an

inverse of Shephard's distance function. The Malmquist productivity index can be decomposed into efficiency

change and technological change, which can provide useful information for policy analysis.

REFERENCES

Aigner, D. J., C. A. K. Lovell, and P. Schmidt. (1977). "Formulation and Estimation of Stochastic Frontier Production Function Models." Journal of Econometrics 6(1): 21-37.

Caves, D. W., L. R. Christensen, and W. E. Diewert. (1982). "The Economic Theory of Index Numbers and the Measurement of Input, Output, and Productivity." Econometrica 50(6): 1393-1414.

Charnes, A., W.W. Cooper and E. Rhodes (1978). "A Data Envelopment Analysis Approach to Evaluation of the Program Follow Through Experiment in U.S. Public School Education," Management Science Research Report No. 432, Carnegie-Mellon University, School of Urban and Public Affairs, Pittsburgh, PA.

Chavas, J.-P. and T. L. Cox (1990). "A Non-Parametric Analysis of Productivity: The Case of U.S. and Japanese Manufacturing." American Economic Review 80(3): 450-464.

Cox, T. L. and J.-P. Chavas (1990). "A Nonparametric Analysis of Productivity: The Case of US Agriculture." European Review of Agricultural Economics 17: 449-464.

Debreu, G. (1951). "The Coefficient of Resource Utilization." Econometrica 19(3): 273-292. Diamond, P., D. McFadden, and M. Rodriguez. (1978). Measurement of the Elasticity of Factor Substitution and

Bias of Technical Change. Production Economics: A Dual Approach to Theory and Application. M. Fuss and D. McFadden, Eds. Amsterdam, North-Holland: Ch. IV.2.

Diewert, W. E. (1976). "Exact and Superlative Index Numbers." Journal of Econometrics 4: 115-145. Diewert, W. E. (1980). "Capital and the Theory of Productivity." American Economic Review 70(2): 260-267. Diewert, W. E. and M. N. F. Mendoza (1995). "Data Envelopment Analysis: A Practical Alternative? "

Vancouver, Department of Economics, University of British Columbia. Hire, R., R. Grabowski, and S. Grosskopf. (1985). "Technical Efficiency of Philippine Agriculture." Applied

Economics 17: 205-214. Hire, R. and S. Grosskopf (1992). "Malmquist Productivity Indexes and Fisher Ideal Indexes." The Economic

Joumall02: 158-160. Hire, R., S. Grosskopf, B. Lindgren, and P. Roos. (1994). Productivity Developments in Swedish Hospitals: A

Malmquist Output Index Approach. Data Envelopment Analysis: The Theory, Applications and the Process. Charnes, A., W. W. Cooper, A. Y. Lewin and L. M. Seiford, Eds. Boston, Kluwer Academic.

Farrell, M. J. (1957). "The Measurement of Productive Efficiency." Journal of the Royal Statistical Society 120 (Part III): 11-290.

Grosskopf, S. (1993). Efficiency and Productivity. The Measurement of Productive Efficiency. H. O. Fried, C. A. K. Lovell and S. S. Schmidt, Eds. New York, Oxford University Press: 160-194.

Kawagoe, T., K. Otsuka, and Y. Hayami. (1986). "Induced Bias of Technical Change in Agriculture: The United States and Japan, 1880-1980." Journal of Political Economy 94(3): 523-544.

Kim, H. Y. (1988). "Analyzing the Indirect Production Function for U.S. Manufacturing." Southern Economic Journal 54: 494-504.

Malmquist, S. (1953). "Index Numbers and Indifference Surfaces." Trabajos de Estadistica 4: 209-242. Nishimizu, M. and J. M. Page (1982). "Total Factor Productivity Growth, Technological Progress and Technical

Efficiency Change: Dimensions of Productivity Change in Yugoslavia 1965-78." Economic Journal 92: 920-936.

Shephard, R. W. (1953). Cost and Production Functions. Princeton, N.J., Princeton University Press.

28 Chieko UMETSU

Otsuka, K. (1988). A Note on the Measurement of Total Factor Productivity Index Using Farm Level Data. Los Banos, Laguna, International Rice Research Institute.

Shephard, R. W. (1953). Cost and Production Functions. Princeton, N.J., Princeton University Press. Shephard, R. W. (1970). Theory of Cost and Production Functions. Princeton, New Jersey, Princeton University

Press. Solow, R. W. (1957). "Technical Change and the Aggregate Production Function." Review of Economics and

Statistics 34: 312-320. Thirtle, C., D. Haddey, and R. Townsend. (1995). "Policy Induced Innovation in Sub-Saharan African

Agriculture: A Multilateral Malmquist Productivity Index Approach." Development Policy Review. 13: 323-348.

Timmer, C. P. (1971). "Using a Probabilistic Frontier Production Function to Measure Technical Efficiency." Journal of Political Economy 79(4): 776-94.

Varian, H. R. (1984). "The Nonparametric Approach to Production Analysis." Econometrica 52(3): 579-597.

28 Chieko UMETSU

Otsuka, K. (1988). A Note on the Measurement of Total Factor Productivity Index Using Farm Level Data. Los Banos, Laguna, International Rice Research Institute.

Shephard, R. W. (1953). Cost and Production Functions. Princeton, N.J., Princeton University Press. Shephard, R. W. (1970). Theory of Cost and Production Functions. Princeton, New Jersey, Princeton University

Press. Solow, R. W. (1957). "Technical Change and the Aggregate Production Function." Review of Economics and

Statistics 34: 312-320. Thirtle, C., D. Haddey, and R. Townsend. (1995). "Policy Induced Innovation in Sub-Saharan African

Agriculture: A Multilateral Malmquist Productivity Index Approach." Development Policy Review. 13: 323-348.

Timmer, C. P. (1971). "Using a Probabilistic Frontier Production Function to Measure Technical Efficiency." Journal of Political Economy 79(4): 776-94.

Varian, H. R. (1984). "The Nonparametric Approach to Production Analysis." Econometrica 52(3): 579-597.