MEASURING TECHNICAL CHANGE IN INPUT-OUTPUT MODELS BY MEANS OF DATA ENVELOPMENT ANALYSIS

ABSTRACT

The goal of the present research is to introduce a model to evaluate potential technical

change in an input-output framework by means of Data Envelopment Analysis, DEA. This

mathematical programming technique allows researchers to assess productivity trends in the

form of technical coefficients −input requirements− variation. By constructing envelopment

unitary isoquants within compatible technologies −their production functions have the same

positive (but different) technical coefficients−, DEA identifies as a benchmark those productive

sectors which use the lowest amounts of inputs to produce one unit of output. Once these

reference frontiers have been defined in a given period it is possible to compare previous years’

technologies with the benchmark and to assess how technical coefficients have reduced over

time in the presence of technical progress. These calculations allow us to compare potential

productivity gains to those actually observed in the economy and to simulate what would have

been the benefits of innovations from an economy-wide perspective if they had been available

in previous years. From an equivalent perspective, these simulations identify the necessary

changes that each industry needs to undertake in order to reach the productivity levels of the

most successful technologies in successive years since it may experience productivity losses, i.e.

innovations −understood as technological changes− may not lead to technical improvements but

rather higher technical coefficients. The process is empirically illustrated making use of the

1985-90 input-output tables of the Spanish region of Castile and Leon, Castilla y León.

1. INTRODUCTION In basic textbook input-output analysis it is generally agreed that input-output tables

reflect a general equilibrium model of the economy where inputs are allocated according to

2

technological availability. In this scheme, according to the European Union Statistical Office,

this representation of the economy for a given period is characterized by a complete account of

the production activities, supply and demand of goods and services, interindustry transactions,

primary inputs and foreign trade, Eurostat (1992). Therefore, they provide a clear view of the

interdependencies between these economic variables. In fact, as an integral part of the European

System of Integrated Economic Accounts, ESA, their basic qualities −generality, coherence and

interpretability− ensure their adequacy when analyzing a wide range of issues such as input

allocation −including energy requirements− as well as output distribution −including

environmental disposals. Of great importance for this research is the analysis of sectoral

production technologies and their change over time, i.e. technical change.

Within the input-output framework, the industrial transactions matrix constitutes the

core of the analysis from a technological perspective. This matrix determines the technical

coefficients collected in the direct industry-by-industry requirements tables which depict the

sectoral technologies. However, these tables provide a static picture of the different production

technologies for a given year, although they do eventually change over time as new goods and

services are produced −due to many factors such as substitution effects and relative price

variations, see Vaccara (1970)− and innovations that replace obsolete technologies with

advanced ones take place. The relevant issue is how to characterize and measure such

technological changes. One may compare two consecutive input-output tables and analyze

actual technical change as is customary in the literature −e.g. Blair and Wickoff (1989), Fontela

and Pulido (1991)− or potential technical change, which compares those sectoral technical

trends which incorporate higher productivity gains −technical coefficient reductions− with

actual ones.

In this scheme, the goal of the present research is to develop the latter approach by

introducing a model to evaluate best practice technical change in an input-output framework

using a mathematical programming technique known as Data Envelopment Analysis, DEA.

3

This optimizing technique enables researchers to assess potential productivity change by

identifying those technical coefficient transformations that imply the highest productivity gains,

i.e. lower input requirements. The process searches for those productive sectors which

experience the largest technical coefficients reductions within compatible technologies, i.e. they

employ the same positive inputs, and then defines a benchmark unitary isoquant or frontier, i.e.

the lowest amounts of inputs required to produce one unit of output, which envelops those

sectors which do no meet this condition. When doing so, DEA assesses whether a given sector

behaves according to the highest productivity gains −potential and actual technical changes

coincide− or it lags behind without profiting fully from the available innovations.

2. THE TECHNOLOGY

2.1. The Leontief Production Function in the Input-Output Model

From a technological perspective, input-output analysis yields a system of equations

which distribute the output produced by the n = 1,...,i,j,...,N sectors among themselves −i.e.

interindustry sales− and toward final demand. In this scheme, it is customary to represent sector

i’s purchases from sector j in period t by ztji, z

tji ≥ 0. These transactions are shown in Table 1 for

the simple case of N=3 productive sectors. Clearly, sector j’s production does not only reach the

ztji intermediate consumptions of the existing n = 1,2,3 sectors −rows−, but it must also satisfy

the domestic demand from household consumption (Ctj), investments from private companies

(Itj) and government purchases (Gt

j), as well as foreing demand, i.e. exports, (Etj). Domestic and

foreign demand represent sectors j’s final demand, Ytj, which along with all intermediate

consumptions ztji, constitute total output, Xt

j.

Please insert Table 1

Input−Output Table for Three Sector Economy

Once sector j’s output has been allocated to intermediate and final demand, it is possible

to completely represent its production function by looking at the amount of inputs it employs

when generating that output. Table 1 shows how sector j consumes the summation of all

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intermediate inputs, ztij −domestic goods and services from the economy’s N existing sectors−

as well as foreign imports, Mtj −columns−, while yielding a value added amount represented by

employee compensation, (Ltj) and other items including gross operating surplus (Nt

j). Regarding

value added, Wtj = Lt

j + Ntj, it is worth while to recall that employee compensation is a proxy to

labor services while gross operating surplus –less gross fixed capital consumption, which is a

proxy to capital services when available− represents capital profits. Hence, it is possible to

consider all intermediate consumptions, employee compensation and additional primary inputs

as the productive factors needed by sector j in order to produce total output Xtj while yielding a

capital profit.

Given these relationships, one may introduce sector j’s production function which

relates produced output with inputs amounts in a given moment in time:

Xtj = f ( zt

1j, ..., ztij, z

tjj,..., z

tNj, L

tj, N

tj, M

tj). (1)

In the input-ouput model, the functional relationship between inputs and outputs can be

expressed in unitary terms, i.e. showing intermediate inputs as well as labor and imports

amounts per unit of output. To do so, it is necessary to divide (1) by total output amount, Xtj,

leaving:

1tj = f ( at

1j, ..., atij, a

tjj,..., a

tNj, b

tj, c

tj, d

tj ), (2)

where atij = zt

ij/Xtj, i =1,....,N, represents a generic technical coefficient which shows the direct

input requirements from the N sectors necessary to produce a unit of output. In a similar way,

btj = Lt

j/Xtj, ct

j = Ntj/X

tj and dt

j = Mtj/X

tj respectively represent labor, other primary inputs and

imports coefficients.

The relationship corresponding to Equation (2) represents the production function

inherent to the input-output model which is known in production economics as the Leontief

function. According to this technology characterization, output is produced according to the

inputs amounts represented by the observed technical coefficients. Following the three sector

economy introduced in Table 1 and assuming for all sectors equal intermediate consumption

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from the third industry, tja3 , it is possible to illustrate the production functions of t

1S , t2S and

t3S through their respective unitary isoquants in the technical coefficient space, at

ij, i =1,2

−excluding labor, primary inputs and imports coefficients for simplicity.

Please insert figure 1

Technologies’ Unitary Isoquants, tja3

The Leontief production function, which characterizes all sectors’ technologies in the

input-output model, exhibits two important properties. First, it presents constant returns to scale,

i.e. the technology is homogeneous of degree one, and any proportional increase in intermediate

consumptions increases production in the exact same amount, αXtj = αf (zt

1j, ..., ztij, z

tjj,..., z

tNj, L

tj,

Ntj, Mt

j) = f(αzt1j, ..., αzt

ij, αztjj,..., αzt

Nj, αLtj, αNt

j, αMtj), α > 0, thus leaving the technical

coefficients as well as the unitary isoquants unchanged. Secondly, in order to produce sectoral

output, inputs are to be used in the fixed proportions shown by its technical coefficients, i.e.

input substitution is impossible. Therefore, in the case of sector t1S , output can only be

produced if intermediate production from the three sectors are combined according to the

observed proportion, e.g. tj12,P = zt

11/zt21 = at

11/at21.

So far the case of zero input requirements, i.e. sector j does not consume any input

amount from some of the existing sectors, ztij = 0, has not been considered in equations (1) and

(2). In such case, the particular specification of the production function in (2) yielding:

,MNL

XN

N

1

1

tj

tj

tj

tj

tj

tj

tj

tj

tjj

tjj

tij

tij

tj

tj

j dcba

z

a

z

a

z

a

z========= LL (3)

would be meaningless since ztij / a

tij = ∞. In this circumstance, Miller and Blair (1985) propose

the following customary definition of the input-output production functions:

=========

tj

tj

tj

tj

tj

tj

tj

tj

tjj

tjj

tij

tij

tj

tj

jdcba

z

a

z

a

z

a

z MNLminX

N

N

1

1LL (4)

6

This notation shows how sector j’s technology can be defined in a quantitative manner,

i.e. as a function of positive intermediate consumptions and remaining inputs, e.g. ztij > 0, since

those zero valued coefficients are overlooked in the process of searching for the smallest among

the ratios −note that ztij / at

ij is infinitely large if atij = 0. Given a particular technology, it is

considered that a set of productive sectors share the same technology if they use the same

inputs, i.e. in terms of the input-output model they present the same positive technical

coefficients as well as those associated to the remaining inputs: Ltj, N

tj and Mt

j. This relationship

is quite important since it is the condition on which the technological compatibility among

sectors is based and it allows intertemporal productivity comparisons across sectors in order to

establish technical change.

2.2. Data envelopment analysis and technical change measurement

In the Input-Output framework, several authors have focused their attention on the

effect of technical change over productivity as a way to determine input requirement evolution

over time. Among these, it is possible to mention the studies by Wolff (1985), Blair and

Wickoff (1989) and Fontela and Pulido (1991), who propose several ways to model technical

coefficient evolution for each sector and its consequences in the input-output system, i.e. how

such productivity change in the atij, t = 1,..,T, technical coefficients is transferred through the

economy allowing for larger production or income. In this same scheme, Carter (1990) suggests

a way to analyze the effects of innovation −as technical progress− on the whole economy: from

intermediate consumption reduction to higher production levels, i.e. the upstream and

downstream benefits of innovation.

The analysis proposed in this study fits into this research field by stating what would be

the benefits of using today’s more advanced technology if it had been available in the past, i.e.

defining a base year for the comparison, what would have been the benefits if firms had been

able to produce their goods and services making use of today’s most advanced technologies

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−improved thanks to technical progress? In terms of the input-output model this analysis

requires fulfilling these two steps:

1) To establish the benchmark technology against which technical progress in successive

years is measured. Given the technical coefficients which define the technology of a

generic sector j in period t according to the production function (4), the question to answer

is the following: what would have been their value if this sector’s firms could produce in t

according to the highest productivity levels existing in t+1?, i.e. potential technical change.

Equivalently, if there were comparable technologies in different periods of time −i.e. a set

of positive atij plus additional primary inputs −bt

j and ctj− and intermediate inputs, dt

j,

would it be possible to reduce the amount of such consumptions in t according to the

existing technologies in t+1? If so, is there a difference between potential and actual

technical changes?

2) To determine the economy-wide benefits of such time reversal productivity projections in

order to identify (1), how large is the aggregate economic benefit of potential and actual

technical progress? (2) how are the benefits and possible divergences between both

technical trends distributed in the system?

This section is concerned with the first step regarding technical progress quantification

while the following one deals with the analysis of the economy-wide benefits of the potential

and actual technical changes of such innovations. Assuming the existence of data presenting the

economy in a similar way to that portrayed in Table 1 in two consecutive periods, t y t+1, figure

2 illustrates these questions. Focusing the analysis on the third sector, t3S , and assuming that

intermediate consumption from this industry is the same across sectors and remains unchanged,

tja3 = 1

3+tja , its technical coefficients )2313 ,( tt aa are shown along with those observed for the two

compatible sectors in t+1, 11S +t and 1

2S +t −being tija > 0 for j=1,2 in both periods. Clearly, in the

transition from t to t+1 both sectors experience productivity gains as their technical coefficients

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representing intermediate consumption are reduced, 1+tija < t

ija , j=1,2, −i.e. technical progress is

observed. However, this is not the case for the third sector as its intermediate consumption from

the second sector increases, 123+ta > 1

23+ta . In this situation, actual technical change reflects

productivity decline, which could be compared to the potential productivity gain that could had

been possible if t3S had followed the productivity trends of the other two compatible industries.

However, before a measure of productivity loss is given, it is necessary to establish the

benchmark technology for t3S by optimizing its technical coefficients so this sector could

produce according to the t+1 highest productivity levels.

Such evaluation is done through a comparative technique known as Data Envelopment

Analysis, DEA, which enables the researcher to determine the benchmark productivity levels as

a linear combination of the t+1 compatible technologies. By minimizing the distance from the

observed technical coefficients of the third sector in t )2313 ,( tt aa toward the reference unitary

isoquant, 11,2S +t −produced as linear combination of the technical coefficients of sectors 1

1S +t and

12S +t , )( 1

211

11 , ++ tt aa and )122

112 ,( ++ tt aa , it is possible to identify the potential technical change of the

industry corresponding to the reference values )ˆ,ˆ( 2111tt aa . Thus, it is necessary to determine the

projected technical coefficients of t3S onto the t+1 reference technology given by 1

1,2S +t . This

process identifies the exact coefficient vector which serves as a reference for t3S and allows

quantification of technical change as the difference between the observed and projected

coefficient vectors in t+1.

Please insert figure 2

Technical Change from t to t+1, tja3 = 1

3+tja .

The DEA process allows the generic tjS technology to be optimized according to the

productivity levels of compatible t+1 technologies. This procedure, which is generally described

9

in a producer context by Cooper et al. (2000:7), identifies the benchmark technologies for tjS

according to the following criterion: they use less inputs to get one unit of output, thus creating

a envelopment surface or frontier characterized by the fact that no sector can improve one of its

technical coefficients values without worsening the other. In the general n=1,...,i,j,...N case, it is

likely that the evaluated sector j in t+1, 1S +tj , represents lower productivity levels that those

observed in the base period. In such case 1S +tj would not show up as reference peer for itself in

the period t+1 frontier and the reference values will be given by the remaining compatible

technologies.

Therefore, by identifying the benchmark technical coefficients belonging to peer

compatible sectors, DEA signals how the evaluated sector should have restructured its

production process so as to meet the most productive technologies in t+1. The following DEA

models provide solutions which assess input requirement reductions in the form of lower

technical coefficients. Once the optimizing programs are solved, the reference unitary isoquants

for each sector show the benchmark technical coefficient vector which constitutes a linear

combination of the reference sectors.

In order to introduce the linear programs which solve for the economy’s technical

progress, i.e. the above mentioned technical coefficients reductions, it is first necessary to

introduce the following DEA additive program: 1

dr-minN

1

+ ∑

=ii (5)

s.t.

1 The program presented in (5) has been simplified for a clearer exposition by excluding

employee compensation, Ltj −as well as any additional primary inputs. However, their inclusion, as in the

empirical application presented in the next section using the input−output tables of Castile and Leon

−employee compensation−, requires inserting its associated restriction: tji in

tn LlL

N

1

1 −=−− ∑ =+ λ .

10

N.,...,1,0d

,0being,0ë

N,,....,1,0ë

N,,...,1,d

,XrX

N

1

1

N

1

1

=≥

=≠≠∀=

=≥

=−=−−

=−

∑

∑

=

+

=

+

i

zzzjn

n

izz

i

ijijinn

n

tij

nin

tin

tj

nn

tn

λ

λ

where r and di represent the amounts of output increases and input reductions of sector j in

period t necessary to reach the t+1 productivity levels observed in the benchmark sectors. The

program presented in equation (5) corresponds to a standard constant returns to scale additive

DEA formulation −see Ali and Seiford (1993:130)− where r and di are the output and input

slacks. The only exception is the set of restrictions imposed on the vector of lambda multipliers,

λn, which guarantee that the intertemporal productivity comparison is carried out with

compatible technologies. Thus, if the evaluated sector j does not share the same non-negative

intermediate consumptions with some of the n sectors, these latter ones are removed from the

optimizing program and do not define the reference frontier. The above program can be

expressed in units of output, i.e. unitary isoquant, dividing outputs and inputs by the observed

production value Xtj. Thus the following program is obtained:

es-minN

1

+ ∑

=ii (6)

s.t.

N.,...,1,0e

,0being,0ë

N,,....,1,0ë

N,,...,1,e

,1s1

N

1

1

N

1

1

=≥

=≠≠∀=

=≥

=−=−−

=−

∑

∑

=

+

=

+

i

aaajn

n

iaa

i

ijijinn

n

tij

nin

tin

tj

nn

tn

λ

λ

11

where the unitary output and technical coefficients are now explicitly considered in the

optimizing program. The solution to program (6) quantifies the technical coefficient change

from period t to t+1 through the ei slack variables, which can be equally interpreted as the

necessary technological change which takes sector j’s productivity levels to those observed in

the most productive industries. The reference benchmark sectors which define the unitary

isoquant frontier correspond to those whose associated multipliers are positive, i.e. λn > 0,

rendering it possible to check if a given industry in period t identifies itself as reference

benchmark in t+1 thus matching actual and potential technical change. Once again the above

formulation slightly differs from standard DEA additive formulations in the set of constraints

which take into consideration compatible technologies while excluding those sectors which do

not employ the same inputs, i.e. 0being,0ë =≠≠∀= ijijinn aaajn −. In figure 2, the solution

for t3S when solving (6) shows the amount of technological progress for compatible

technologies from period t to t+1 in the form of lower input requirements or, from this sector’s

perspective, how it needs to reduce its technical coefficient corresponding to the second sector’s

intermediate consumption by e2 −as well as the first sector’s coefficient by e1, if it is to reach the

productivity levels observed in period t+1. Therefore, if sector t3S were to produce according to

the highest t+1 productivity levels it should employ the technology represented by the projected

technical coefficients vector 13 23 )ˆ ˆ( ,t ta a = )21 2313 ,( eaea tt −− . Thus, DEA allows the

determination of the optimal production technology for sector t3S if it could have reached the

following period’s technological levels, i.e. the magnitude of potential technical progress

−productivity gains− experienced by the economy’s industries from period t to t+1 in the form

of lower technical coefficients.

Once these benchmark values have been found, it is possible to define a measure of

potential technical change for input i in sector j as the difference between the projected and

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previous period’s technical coefficients, i.e. tij

tij

ttij aaa −= ˆˆ , , as well as a measure of productivity

loss as the difference between potential and actual technical coefficients, i.e. 11, ˆ ++ −= tij

tij

ttij aaa

3. EMPIRICAL APPLICATION

3.1 The Data.

In this section, the proposed methodology to quantify productivity change is illustrated

using the available input-output tables for the Spanish region of Castile and Leon. Since 1985

the Castile and Leon −federal− government, Junta de Castilla y León, has compiled regional

input−output tables every five years. These tables have been designed to be compatible with the

regional and national accountancy framework following the 1970 methodology laid out in the

European System of Integrated Economic Accounts (EAS-70). However the change in

methodology which followed the adoption of the current European Accounts System in 1995

(EAS-95) −see Eurostat (1996), eases the comparison between the 1985 and 1990 input-output

tables, i.e. they present a higher degree of compatibility, see Junta de Castilla y León (1990,

1992, 2000). The 1985 and 1990 tables adopt the NACE R56 sector classification, see Eurostat

(1992); however for a simpler and clearer discussion of the results, they have been reworked to

a 16 sector levels of aggregation −see annex 1 for a detailed account of the R56 sectors

aggregation codes.

To elaborate the tables in constant terms both of them have been calculated according to a

double deflation method, whose setbacks and particularities can be consulted in Cassing (1996).

Schematically, the following process has been applied2: intermediate consumptions for each

sector have been deflated using the corresponding product price index; then, production for each

sector has been deflated using the price index corresponding to the main product. Once Gross

Value Added in constant terms for each sector is obtained, it is then possible to calculate its

2 In the Spanish case, the absence of regional deflators for each region makes it necessary for researches to apply national deflators.

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deflator and, finally, this deflator is used to deflate the GVA components of employee

compensation and gross operating surplus.

3.2 Potential productivity change in the Castile and Leon region from 1985 to 1990.

The first section of the empirical application deals with the measurement of potential

productivity change from 1985 to 1990. Introducing the employee compensation restriction

tji in

tn bb −=−− ∑ =

+N

1

1 lλ , it is possible to optimize the 1985 technical coefficients according to

the 1990 reference technologies by means of equation (6), thus quantifying the magnitude of the

1985 technical coefficients improvements if each one of the N=16 different industries were to

operate according to 1990 benchmark productivity levels. The analysis of the economy’s

productivity change can be undertaken from a double but equivalent perspective: through the

individual technical coefficients −intermediate consumption and additional primary inputs− and

by way of the Leontief inverse matrix.

From the technical coefficients perspective, Table 2 shows the subtraction of the

observed 1985 industry-by-industry coefficients from the optimized 1985 values, i.e.

198519851985,1985 ˆˆ ijijij aaa −= for intermediate consumption and 198519851985,1985 ˆˆjjj bbb −= for employee

compensation −where 1985ˆija and 1985ˆjb are the optimized 1985 technical coefficients with respect

to the 1990 production technologies and 1985ija and 1985

jb correspond to the observed ones, as

well as the proportion of possible technical savings or increases in terms of the 1985

coefficients, i.e. [ ] [ ] 100·/ˆˆN

1

19851985N

1

1985,19851985,19851985,1985ˆ ∑∑ ==++=

i jiji jijj babaS .

Please insert Table 2

Technical change in Castilla y León, 198519851985,1985ˆˆ ijijij aaa −= −intermediate consumption− and

198519851985,1985 ˆˆjjj bbb −= −employee compensation−

Looking at the summation of these differences one finds potential technical progress in

ten out of the sixteen sectors, being specially important in the transport equipment industries (7)

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with an overall 16.8% reduction in the technical coefficients in the five years period (12.1%

corresponding to intermediate consumption and the remaining 4.7% to employee

compensation), followed by chemical products (5), -10.3%, and transport and communication

services (14), -8.0%. On the other hand, productivity losses reach similar levels. Building and

construction (12) employs an additional 14.9% amount of inputs in 1990 (8.4% regarding

intermediate consumption and 6.5% for employee compensation).

3.3. Comparison between actual and potential productivity change.

Table 2 shows potential technical change as the difference between the observed 1985

technical coefficients and their optimized projection according to benchmark productivity levels

of compatible technologies in 1990. Particularly, the solution for the DEA programs provides a

reference benchmark −optimized vector of technical coefficients− which constitutes the most

productive projection of a given sector’s technology from a past period −base year− into the

present −if this projection matches the actual path followed by the industry, then the evaluated

sector constitutes the reference benchmark for itself in the past, and potential and actual

technical change are the same. Therefore, it is significant to compare potential productivity gains

to what has actually happened in the economy, i.e. do industries technological behavior match

that one of the most productive sectors?. The differences between potential technical change:

198519851985,1985 ˆˆ ijijij aaa −= for intermediate consumption and 198519851985,1985 ˆˆjjj bbb −= for employee

compensation, and actual technical change, which corresponds to the observed 1990 technical

coefficients: 198519901990,1985ijijij aaa −= and 198519901990,1985

jjj bbb −= , are presented in Table 3.

Please insert Table 3 Difference between actual and potential technical change

1990198519901985 ˆˆ ijijij aaa −=− and 1990198519901985, ˆˆjjj bbb −=

Clearly, if a given sector in period t identifies itself as reference benchmark in t+1 when

solving Equation (6), there is no difference between potential −optimized− and actual technical

change and this industry has fully exploited the available technological innovations, i.e.

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1990,19851985,1985ˆ ijij aa − = ( ) ( )1985199019851985ˆ ijijijij aaaa −−− = 19901985ˆ ijij aa − = 0 regarding intermediate

consumption and 1990,19851985,1985ˆjj bb − = ( ) ( )1985199019851985ˆ

jjjj bbbb −−− = 19901985ˆjj bb − = 0

regarding employee compensation. This is case for all industries marked by “*” in Table 2 and

excluded from Table 3. For the rest of the industries presented in Table 3 it is clear that potential

technical change exceeds actual technical change except for the transport equipment sector (7).

This means that the technical fixed proportions result of the optimizing processes when solving

(6) show higher potential productivity gains than those actually experienced by these industries.

As a matter of fact, in terms of the 1990 technical coefficients summation,

[ ] [ ] 100·/ˆˆN

1

19901990N

1

1985,19901985,19901990,1985ˆ ∑∑ ==++=

i jiji jijj babaS , the other services sector could

have reduced its aggregate input requirements by 3.1% −distributed in a 1.1% aggregate

increase in intermediate consumptions and a 4.2% reduction in employee compensation. This

exact same pattern is observed in the transport and communications services sector (14) with a

−2.1% aggregate reduction −a 7.3% increase and 9.4% reduction respectively. The remaining

industries with potential productivity losses experience reductions in intermediate consumptions

and increases in employee compensations −sectors (3) and (7)− or reductions in both sets of

input requirements: sectors (10) and (11). Finally, the transport equipment sector (7) is the only

one to perform better that the potential productivity gains shown by the DEA process3.

3.4. Comparison between actual and potential benefits of innovations

The final section of this paper is concerned with the aggregate economic benefits and

differentials of the potential and actual technical changes shown in the previous section. If the

projected benchmark differs from the actual technological path followed by the sector, then it is

quite likely that the observed technical coefficient change does not coincide with the highest

available reduction. In this case, it is possible to compare and simulate what would have been

3 For this particular sector, the linear combination of the compatible most productive technologies that define the projected technical coefficients vector when solving (6) represents higher technical coefficients than those actually observed in 1990. This may happen when the set of benchmark sectors which define the isoquant frontier includes the evaluated sector along with other compatible technologies.

16

the benefits of innovations from an economy-wide perspective if all sectors’ technological

changes were those projected by the optimizing process. Clearly, an economy that does not

profit from the most productive technologies incurs in a production loss, i.e. lower aggregate

and sectoral GDP. This cost can be assessed through standard input-output practice by

determining the difference between potential and actual productivity gains.

Following the standard input-output practice to determine the consequences of

innovations set out by Carter (1990), it is possible to check the effects of the changes in the

coefficient matrix on the economy. In order to determine the productivity gains associated with

potential and actual technical change it is necessary to establish a base solution that corresponds

to the 1985 input−output tables. It is customary to present such solution in terms of the

following equation:

1985)1·16(

1)16·16(

19851985)1·16( )( YAIX −−= (7)

where 1985)1·16(X represents sectoral output while 1

)16·16(1985 )( −− AI and 1985

)1·16(Y correspond

respectively to the Leontief inverse and final demand matrixes. Departing from this solution it is

possible to calculate the productivity effects of the potential technical change on the economy

replacing the 1985 technical coefficients matrix by the optimized ones, 1985)16·16(A . Table 4 shows

how the sectoral output necessary to satisfy the final 1985 demand reduces by an aggregated

2.6% −from 22,772.3 million € to 22,181.5 million €. It is significant how potential technical

change distributes throughout the system, being the other manufacturing products sector (11)

that one with the highest output requirements reduction −from 716.8 million € to 577.8 million

€, i.e. a 19.4% reduction. Additional sectors where important reductions values are observed are

the agricultural, forestry and fishery products (1) experiencing a 9.5% reduction and services of

credit and insurance institutions (15) with a 8.1% reduction. However some sectors such as the

17

ferrous and non−ferrous ores and metals (3) and chemical products (5) increase output

requirements by 17.9% and 7.5% respectively4.

Please insert Table 4 Summary of system solutions for potential and actual technical change, million €

The optimized sectoral output vector 1985)1·16(

1)16·16(

19851985)1·16( )ˆ(ˆ YAIX −−= can be related to

its intermediate consumption requirements, )1·16(1985

)16·16('ˆ iZ , employee compensation, 1985)1·16(L and

gross operating surplus, 1985)1·16(N , by means of the following equations:

)1·16(1985

)16·16(1985

)16·16()1·16(1985

)16·16( 'ˆ'ˆ·'ˆ iAXiZ ⟩⟨= (8a)

1985)1·16(

1985)16·16(

1985)1·16(

ˆˆˆ lXL ⟩⟨= (9a)

1985)1·16(

1985)16·16(

1985)1·16(

ˆˆ nXN ⟩⟨= (10a)

where 1985)16·16(

ˆ ⟩⟨ X stands for the diagonalized form of the optimized output vector 1985)1·16(X . Once

this decomposition has been accomplished, one may determine particular productivity gains

calculating the difference between the optimized output less all input requirements including

gross operating surplus, i.e. 1985,1985)1·16(Ä = 1985

)1·16(1985

)1·16()1·16(1985

)16·16(1985

)1·16(ˆˆ·'ˆˆ NLiZX −−− , as well as the

percentage proportion it represents in the optimized output, i.e. for the generic j sector

1985,1985 1985,1985 1985ˆ ˆ ˆ(%) /j j j = Ä Ä X . Table 4 shows how potential aggregate productivity gains

in the Castile and Leon reaches 1.2% (259.4 million €). Among the different productive sectors

ten industries experience productivity gains, being led by the transport equipment (7) and

chemical products (5) sectors with 13.6% (225.3 million €) and 7,0% (47.7 million €) gains. On

the opposite side the building and construction sector (12) reflects the highest productivity

4 The Castile and Leon input-output system summarized in equation (7) as well as the effects of potential and actual technical change on aggregate employment, output levels and final deliveries hinge on several macroeconomic assumptions which may affect the attained results, e.g. whether one deals with an open or

18

losses with a 12.7% increase in input requirements in terms of optimized production. Clearly,

these data are in accordance with the potential technical change trends shown in Table 2.

These results deal with the economy−wide implications of the calculated potential

technical change. However, it is possible to compare potential to actual technical change by

calculating equivalent measures of productivity gains −losses− to those already introduced. In

this case the productivity effects of actual technical change on the economy can be found by

replacing the 1985 technical coefficients matrix in (7) by the 1990 ones, 1990)16·16(A . Table 4 shows

how the aggregate sectoral output necessary to satisfy the 1985 final demand reduces by 3.6%

−from 22,772.3 million € to 22,032.7 million €. In this case, actual technical change distributes

in a similar way to the optimized solution, e.g. the other manufacturing products sector (11)

experiences the highest output requirements reduction −from 716.6 million € to 594.4 million €,

i.e. a 17.1% reduction. Once again, sectors with important reduction values are the agricultural,

forestry and fishery products (1) experiencing a 9.7% reduction and services of credit and

insurance institutions (15) with a 8.2% reduction. The difference between potential and actual

sectoral production reduction when satisfying the final 1985 demand levels can be found in the

fuel and power products sector (2) −in the former case, it experiences a 7.3% reduction while in

the latter it reduces by 13.6%, and, more importantly, in the ferrous and non ferrous ores and

metals sector (3). For this sector, in the case of potential technical change, the solution to the

input-output system shows the above mentioned 17.9% production increase while regarding

actual technical change, it shows a 2.2% reduction.

However, even if aggregate production reduction is larger when taking into account

actual technical change than potential technical change −by 148.8 million €, productivity gains

are larger in the latter case. In order to obtain real technical change in the Castile and Leon

economy it is necessary to follow the steps already introduced for the optimized solution. Once

closed model where final demand is held constant (exogenous) or varies according to the new income values −for simplicity we have assumed in this case an open model regarding final demand values.

19

the output vector necessary to satisfy 1985 final demand according to the 1990 technical

coefficients has been calculated, i.e. 1985)1·16(

1)16·16(

19901990)1·16( )(~ YAIX −−= , it is possible to find its

intermediate consumption requirements, )1·16(1990

)16·16('~ iZ , employee compensation, 1990

)1·16(~L and

gross operating surplus, 1990)1·16(

~N , through the following equations:

)1·16(1990

)16·16(1990

)16·16()1·16(1990

)16·16( ''~·'~ iAXiZ ⟩⟨= (8b)

1990)1·16(

1990)16·16(

1990)1·16(

~~~ lXL ⟩⟨= (9b)

1990)1·16(

1990)16·16(

1990)1·16(

~~ nXN ⟩⟨= (10b)

where 1990)16·16(

~⟩⟨ X stands for the diagonalized form of the output vector 1990)1·16(

~X . Matching the

outlay already introduced for the optimized case, it is possible to obtain productivity change

calculating the difference between the projected 1990 output −necessary to satisfy the 1985 final

demand− less all input requirements including gross operating surplus, i.e. 1985,1990)1·16(Ä =

1990 1990 1990 1990(16·1) (16·16) (16·1) (16·1) (16·1)' ·− − −X Z i L N% % % % , as well as the percentage proportion it represents in

the calculated 1990 output, i.e. for the generic j sector [ ]19901985,19901985,1990 ~/(%) jjj XÄÄ = . Results

for these calculations are shown in Table 4. In this case aggregate productivity gains in the

Castile and Leon reaches just 0.8% (169.9 million €) in comparison to the potential productivity

gain that reaches 1.2%. The direction of the different sectoral productivity gains or losses equals

that already discussed for potential productivity change. Nine industries experience productivity

gains, being led by the transport equipment (7) and chemical products (5) sectors with 14.1%

(236.8 million €) and 7,0% (47.7 million €) gains. The remaining industries experience

productivity losses, with the building and construction sector (12) experiencing the highest

productivity decline, 12.7%

A comparison between the economy-wide benefits of potential and actual technical

change, which justifies the productivity gain differential in favor of the former by 0.4%, is

shown in the last two columns of Table 4 −both in million € and percentage terms. In six out of

20

the seven sectors potential and actual technical change differs, productivity gains of the former

being greater than those of the latter. The largest productivity difference can be found in the

other services sector where it accounts for 77.7 million €. Similar comments apply to the

transport and communications services and paper and printing products sectors which exceed a

1.0% difference. These results closely follow the difference between potential and actual

technical change portrayed in Table 3 once such innovations have been introduced in the

input−output system to calculate their economy-wide effects.

The main conclusion from these results is that the Castile and Leon region has not

benefited from innovations up to the amount that potential technical change allowed. In fact, the

economic cost of the difference between the observed 1990 productivity levels and the potential

ones accounts for 89.5 million €. Clearly if actual technical progress had followed that of the

most productive technologies the benefits from such innovations would had been larger and the

potential production and income gains would had reached both capital profits and labor earnings

in a manner and proportions that fall beyond the scope of this research −see Carter (1990) for

this side extension of the analysis.

4. CONCLUSION

When analyzing technical change in an input-output framework it is customary to

compare the technical coefficients corresponding to successive periods and simulate what would

be the benefits if today’s technologies had been available in the past. However the technical

innovations followed by the different industries may not be those which incorporate the highest

productivity gains, thus failing to match and benefit from the largest input requirements

reductions, i.e. potential productivity increases. In this research we introduce a model to

evaluate potential technical change by means of linear programming techniques known as Data

Envelopment Analysis, DEA. These techniques are specially suited for this task given the linear

nature of the input-output framework. In fact, this method, which enables researchers to identify

21

the most productive benchmark technologies, can be implemented with minor changes to basic

DEA additive models. Once potential technical change has been identified in the form of

optimized technical coefficients, it is possible to compare these values to actual ones and check

if there have been productivity losses due to some industries failing to follow the most

productive technological changes. The empirical implications of this process taking into account

the economy-wide effects of these productivity differences are analyzed considering the input-

output tables of the Spanish region of Castile and Leon. Results for 1985 and 1990 show how

this region presents productivity losses in several sectors, which in turn causes aggregate

production and income losses. Therefore, from a base year perspective, the model allows

quantification of potential and actual technological changes comparing the economy’s most

productive and real technological paths.

22

Figure 1. Technologies’ Unitary Isoquants, tja3

at1j

at2j

Unitary Isoquant for St1

•

0

(at11, a

t21)

(at12, a

t22) •

Unitary Isoquant for St2

(at13, a

t23)

• Unitary Isoquant for St

3

23

Figure 2. Technical Change from t to t+1, tja3 = 1

3+tja .

at1j

at2j

•

0

•

• ),( 1

211

11++ tt aa

),( 122

112

++ tt aa

)2313 ,( tt aa

Unitary Isoquant 11,2S +t

• e2

e1

13 23)ˆ ˆ( ,t ta a

)123

113 ,( ++ tt aa

•

24

Table 1. Input−Output Table for Three Sector Economy Productive

Sectors Final Demand

St1 St

2 St3 Yt

i

Total Output

St1 zt

11 zt12 zt

13 Ct1 It

1 Gt1 Et

1 Xt1

St2 zt

21 zt22 zt

23 Ct2 It

2 Gt2 Et

2 Xt2

Productive Sectors

St3 zt

31 zt32 zt

33 Ct3 It

3 Gt3 Et

3 Xt3

Lt1 Lt

2 Lt3 Lt

C LtI Lt

G LtE Lt Value

Added Wt

j Nt

1 Nt2 Nt

3 NtC Nt

I NtG Nt

E Nt Payments Sector Mt

1 Mt2 Mt

3 MtC Mt

I MtG Mt

E Mt

Total Outlays

Xt1 Xt

2 Xt3 Ct It Gt Et Xt

Source: Miller and Blair (1985)

25

Table 2. Technical change in Castilla y León, 198519851985,1985 ˆˆ ijijij aas −= −intermediate consumption− and 198519851985,1985 ˆˆjjj bbb −= −employee compensation−

Sector 1* 2* 3 4* 5* 6* 7 8 9* 10 11 12* 13* 14 15* 16

1 -.0094 .0000 .0000 -.0005 -.0015 .0000 .0000 -.0957 .0038 -.0036 .0010 .0003 -.0051 .0000 .0000 .0020 2 -.0134 -.0019 .0003 -.0080 -.0454 -.0050 -.0069 -.0170 -.0019 -.0271 .0037 .0053 -.0023 .0546 -.0016 .0046 3 .0000 .0005 -.0680 .0058 -.0002 .0126 .0071 -.0001 .0000 .0013 .0035 .0152 .0001 .0041 .0000 .0000 4 .0000 .0001 .0000 .0155 -.0039 -.0005 -.0036 .0018 .0000 -.0013 .0001 .0172 -.0001 .0004 .0000 .0002 5 -.0038 .0016 .0026 .0045 .0023 .0016 .0064 .0057 .0052 .0087 .0535 .0045 .0021 .0033 .0001 .0016 6 .0063 .0064 .0025 .0098 -.0001 -.0276 -.0340 .0009 .0016 -.0004 .0032 .0316 .0030 .0245 .0001 .0054

7 .0000 .0001 .0000 .0000 .0000 .0000 -.0305 .0000 .0000 .0000 .0000 .0001 .0055 -.0120 .0000 .0032

8 -.0145 .0000 .0000 .0000 -.0007 .0000 .0000 .0358 .0058 .0011 .0004 .0000 -.0104 .0000 .0000 .0068 9 .0001 .0001 .0010 .0002 -.0012 .0000 -.0047 .0003 .0073 -.0012 .0018 .0003 .0001 -.0010 .0000 .0010 10 .0000 .0000 .0020 -.0006 -.0042 -.0011 .0003 .0033 -.0003 -.0123 .0082 .0000 -.0031 -.0006 -.0008 .0028 11 .0002 .0008 .0000 -.0001 -.0032 -.0028 -.0287 -.0004 .0003 -.0002 .0043 .0021 .0011 -.0096 -.0026 .0014 12 .0069 -.0024 -.0035 -.0002 -.0011 -.0011 -.0021 -.0008 -.0006 -.0006 .0004 .0000 -.0033 .0035 -.0026 -.0190 13 -.0033 -.0005 -.0050 -.0030 -.0036 -.0032 -.0007 -.0034 -.0021 -.0384 -.0008 .0042 -.0071 -.0254 -.0013 .0075 14 -.0028 -.0001 -.0018 .0118 .0006 .0009 -.0026 -.0026 .0006 -.0018 .0015 -.0077 .0010 .0010 .0019 .0041 15 .0000 -.0043 -.0001 .0025 -.0073 -.0001 -.0115 .0000 -.0001 .0004 -.0005 -.0179 -.0008 -.0035 -.0021 .0001 16 .0039 .0064 .0066 .0024 .0017 .0022 .0138 -.0009 .0075 -.0004 .0012 .0164 .0102 .0077 .0043 .0190

1985,1985ˆjb -.0023 .0326 .0408 .0003 -.0026 .0029 -.0382 .0323 .0195 .0526 -.0027 .0555 .0089 -.1102 -.0461 -.0413

∑ =+

N

1

85,8585,85 ˆˆi jij ba -.0321 .0393 -.0225 .0403 -.0703 -.0212 -.1359 -.0409 .0466 -.0232 .0786 .1271 .0000 -.0631 -.0508 -.0007

1985,1985ˆjS -5.90% 5.92% -2.95% 5.91% -10.29% -2.79% -16.81% -4.71% 6.95% -2.84% 10.72% 14.92% 0.00% -8.03% -9.16% -0.11%

* Industries which identify themselves as reference technologies in 1990, i.e. actual and potential technical change are equal. Source: Own, Junta de Castilla y León (1990, 1992)

Table 3. Difference between actual and potential technical change 1990198519901985 ˆˆ ijijij aaa −=− and 1990198519901985, ˆˆ

jjj bbb −= .

Sector 3 7 8 10 11 14 16

1 .0000 .0000 -.0052 .0000 -.0004 .0000 .0016 2 .0001 .0047 .0001 -.0028 -.0003 .0849 .0011 3 -.0007 .0247 .0000 -.0002 -.0004 -.0020 .0000 4 .0000 -.0006 .0000 .0005 .0002 -.0006 .0001 5 .0000 -.0019 .0031 .0106 .0011 .0025 -.0017 6 .0000 -.0087 .0000 -.0002 -.0003 .0148 -.0018

7 .0000 -.0107 .0000 .0000 .0000 -.0115 .0024

8 .0000 .0000 -.0014 .0015 .0004 .0000 .0067 9 .0000 -.0006 .0000 -.0001 -.0001 -.0013 -.0003 10 .0000 .0006 -.0001 -.0146 -.0002 -.0018 -.0004 11 .0000 -.0048 .0001 .0001 -.0021 -.0087 -.0001 12 .0000 .0002 .0000 .0000 .0000 .0031 -.0028 13 .0001 .0012 -.0001 -.0023 -.0005 -.0191 .0008 14 .0000 .0002 .0001 .0009 .0002 -.0046 .0005 15 .0000 .0000 .0001 .0005 -.0002 -.0012 .0004 16 .0000 -.0017 .0000 .0001 .0000 -.0005 .0014 bj .0004 .0024 .0007 -.0058 -.0011 -.0692 -.0288

∑ =+

N

1

90,8585,90 ˆˆi jij ba -.0000 .0051 -.0026 -.0117 -.0036 -.0153 -.0209

1990,1985ˆjS -0.00% 0.76% -0.31% -1.45% -0.44% -2.07% -3.05%

Source: Own, Junta de Castilla y León (1990, 1992)

Table 4. Summary of system solutions for potential and actual technical change, million €.

Potential Technical Change Actual Technical Change

(1) (2) (3) (4) Productivity differences

Sectors 1985jX 1985ˆ

jX 1985ˆjZ 1985ˆ

jL 1985ˆjN 1985,1985ˆ

jÄ (%)1985,1985jÄ 1990~

jX 1990~jZ 1990~

jL 1990~jN 1985,1990

jÄ (%)1985,1990jÄ (1)-(3) (2)-(4)

1 3,389.0 3,065.6 1,446.9 120.5 1,399.8 98.3 3.21 3,060.9 1,444.7 120.4 1,397.7 98.2 3.21 0.2 0.00

2 1,953.8 1,810.4 848.1 423.9 609.4 -71.1 -3.93 1,687.4 790.5 395.1 568.0 -66.3 -3.93 -4.8 0.00

3 300.5 354.4 207.5 54.5 84.4 8.0 2.25 294.0 172.3 45.1 70.0 6.6 2.25 1.4 0.00

4 511.7 529.4 265.5 116.7 168.5 -21.3 -4.03 530.9 266.3 117.1 169.0 -21.4 -4.03 0.1 0.00

5 630.2 677.7 299.3 116.2 214.5 47.7 7.03 677.9 299.4 116.2 214.6 47.7 7.03 0.0 0.00

6 845.8 874.1 430.8 215.1 209.7 18.6 2.12 880.8 434.1 216.7 211.3 18.7 2.12 -0.1 0.00

7 1,698.5 1,658.0 913.1 201.9 317.8 225.3 13.59 1,679.7 920.5 200.5 321.9 236.8 14.10 -11.5 -0.51

8 2,357.4 2,345.8 1,667.0 275.0 307.9 95.9 4.09 2,319.8 1,656.1 270.5 304.5 88.9 3.83 7.0 0.26

9 292.0 289.1 133.9 73.2 95.4 -13.5 -4.66 293.0 135.7 74.2 96.7 -13.6 -4.66 0.2 0.00

10 245.5 252.3 134.9 65.5 46.1 5.9 2.32 259.6 140.3 68.9 47.4 3.0 1.15 2.9 1.17

11 716.8 577.8 335.8 133.4 154.1 -45.4 -7.86 594.4 346.9 137.8 158.5 -48.9 -8.22 3.4 0.36

12 1,522.4 1,454.1 966.8 456.6 215.6 -184.8 -12.71 1,458.7 969.8 458.0 216.3 -185.4 -12.71 0.6 0.00

13 3,036.7 2,940.2 867.9 600.5 1,471.8 0.0 0.00 2,948.9 870.5 602.3 1,476.1 0.0 0.00 0.0 0.00

14 839.6 822.4 327.7 266.5 176.4 51.9 6.31 822.3 283.3 323.3 176.3 39.3 4.78 12.6 1.53

15 888.1 816.3 172.8 238.4 363.7 41.5 5.08 814.9 172.5 238.0 363.1 41.4 5.08 0.1 0.00

16 3,544.6 3,714.0 863.2 1,610.6 1,237.5 2.6 0.07 3,709.4 832.9 1,715.5 1,236.0 -75.0 -2.02 77.7 2.09

Total 22,772.3 22,181.5 9,881.1 4,968.3 7,072.6 259.4 1.17 22,032.7 9,735.8 5,099.5 7,027.5 169.9 0.77 89.5 0.40

Source: Own, Junta de Castilla y León (1990, 1992)

28

Annex 1. Input-output aggregation codes

16 Sector Aggregation Related R56 NACE Codes

1. Agricultural, forestry and fishery products 01,02,03 2. Fuel and power products 04,05,06,07,08,09 3. Ferreous and non-ferrous ores and metals 10,11 4. Non-metallic mineral products 12,13,14,15 5. Chemical Products 16 6. Metal products, except machinery and transport equipment 17,18,19,20 7. Transport equipment 21,22 8. Food, beverage and tobacco 23,24,25,26,27,28 9. Textiles and clothing, leather 29,30 10. Paper and printing products 32,33 11. Other manufacturing products 31,34,35,36 12. Building and construction 37 13. Recover and repair services, wholesale and retail trade 38,39,40 14. Transport and communication services 41,42,43,44,45 15. Services of credit and insurance institutions 46 16. Other services 47 through 56 Source: Own, Junta de Castilla y León (1990, 1992)

29

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