A statistical approach to the problem of negatives in input-output analysis

A statistical approach to the problem of negatives in input-output analysis Thijs ten Raa and Rick van der Ploeg

The construction

of input

is complicated Sectors also

product output

dividing

available (ten

Scncta The

away.

products

and Vict

purest

and

output,

and [

postulates

the consequent

while

account

of methods

for

equation. for

superior

method

is

analysis;

inputs. vector

can be solved simultaneously solution

in view

to us.

The

WC have a commodity The

model.

Icast

is simple:

authors

are

5COO LE Tilburg,

with

Tilburg

University,

Postbox

90153.

The Netherlands.

We thank Anton Markink for the cxccllcnt programming of the statistical adjustment procedure. Thijs ten Raa’s rcscarch has been made possible by a Senior Fellowship of the Royal Netherlands Academy of Arts and Sciences and a grant (R 46-177) of the Ncthcrlands Organization for the Advancement of Pure Research (ZWO). Final manuscript

2

received I5 February

0X4-9993/89/010002-

smnllncss

organized

iIs

how

may

it

the negatives The

fourth

gcncratc

third

section

for UK

section

applies

in

established

practice ofdcaling

them

model

that

zero.

fifth

in

i&.scssm~nt

appenl The

technology negative prcscnts

section.

at

of the

of the negatives.

follows.

results

but must,

underlies

as the last section

;I

of ncgativcs

second

model and

input-output a diagnosis

of

some intuition.

il rc-estimation

discussed setting

is

is surprising,

data to provide

the negatives: the

which prcscnts

a statistical

conclusion

shows

The

paper

of the theoretical

the commodity

procedure

are presented They

confirm

and the

with ncgutivcs by simply none the Icss,

the construction

reject the

of coefficients,

concludes.

The commodity technology model The

1988.

This

reviavs

coctlicicnts.

ncgativc.

bc Icd to rcjcct the commodity

section

to eliminate The

is

corn--

shortcoming.

This

the problem

and the empirical paper

one

out

it allows

We will

technology model

of

meaningful.

the

and has

based on the

have

turn

to deal with

model

for the outputs

of them

methodology

calculates

model. This

model

not

input -output

rcquircmcnts

coclficients.

Some

is basically matrix,

invariance.

cocllicicnts

economically

of the problem.

and

1I]).

These equations

the technical

arc

such as scaling

technology

Fukui

input ~-output coctlicicnts,

each sector

modity

matrix

by the output

output

coctticicnts

[7].

each sector and equates the sum to the observed Thus,

input

howcvcr.

secondary

wc must

nice propcrtics. The

cocllicicnts

divided

of technical

theoretically

direct

by

matrix

Small

given by the cort~r~dir~~ rc~4r~loy~~ simply

arc dctcrmincd

and a number

Chakraborty

[2]

input

In reality

for the construction

Raa.

the input --output

outputs.

cocllicicnts

by primary

is assumed

for secondary

matrices

owfl or prirrrtrrj*output, but sccwrrtl~wy outputs. In textbook

or

analysis

inputs

output

cocllicicnts

of secondary

not only

each others’

input

output

by the prcscncc

system

I9 SO3.00

of national

>c 1989

accounts

Butterworth

(UN

[S])

includes

& Co (Publishers)

Ltd

The prohlrm an input table

or ‘use’

Cl Entry

consumed of

product

k.

The

of input

i equals

latter

the

industry

Hence

model

tion

sectors

wrong.

for

all

summed

over

-

* inversion.

commute.

where

’

the

the example

sector,

secondary

and negatives

nevertheless

of input -output

of the commodity This

technology

incompatibility

outcome

arise

in the

tables, the basic assumpmodel

between

is the subject

must

theory

of this

be and

study.

Diagnosis of negative coefficients

input-output

compositions

confusion.)

one

if the use and make data are measured

error

empirical

R. ran der Ploeg

outputs: (Since

their

7: ten RLIU and

unul_rsis:

construction

technology

of input

and

without

(I,~

CJ = .4tV’ or A = liV-‘*

with

some

amount

j requires

to consider

economy

i

industry

-’ without

It is instructive

j’s

in input-output Alternatively.

‘make’

k. Its consumption

requirements

operations

producing

is

coefficients

transposition

two

rlt

or

of commodity

[ IO] ). In particular,

may be denoted sector

j.

i for output

I(,~ = I:tc~,trjk. denotes

amount

commodity

technical

(van Rijckeghem

L; and an output the

by industry

postulates

llle~jk

table

rdij is

ofnryotires

say

of a two-

the

first

one,

output:

The

data

national

used

this

and Weale

are square

tables:

is million matrix,

1975

are

in

the

of the UK

[ I]).

The

sector

in Tables pounds.

use and make

tables

of sectors.

is omitted.)

U

and

derived

is in Table

technical 3. They

tables

39.

V are

coetficients

arc multiplied

by a factor of 100, so that the unit is pcnnics All

of van

I and 2. The unit of measurement The

A = UV-‘.

system

(Barker,

the size is the number

‘Unallocated’

reproduced

study

for

der Ploeg (The

Then

in

accounts

per pound.

arc in the appendix.

Thcrc arc three ncgativcs on digit lcvcl two. namely - - 0.007, t~~~..~, = - 0.015 and (12H..11 = -0.(X)5.

;?i:&-;trc multiplied The

biggest

when

indirect

rcquircmcnts

(1, z =

II, z /r12

is well

known

by

own

whcro

structures,

why

input

it

is

sectors

subtraction.

secondary reported

structure,

arc purilicd Ncgativs

over

output whcrc net output

IIL’I

of secondary the

the technical products

associated

net

the input

cannot

cxcced

cocflicicnts negative. pure input

However, or with

input

technical

ECONOMIC

table,

MODELLING

be

in its

tables

Thcrcfore

products

of the sector. Then coefficient,

the

make

errors.

of secondary net input

so

A, cannot

may not be valid

the use and

In

products

of the sector,

measurcmcnt

a negative

negative

input

the theory

at least

rcquircmcnts

the obscrvcd yields

the total

net

net of

requirements.

of secondary

of the input -output

form,

obscrvcd

product

requirements

output

is input

o,,

January

in this

1989

natural

gas

pctrolcum

output

and natural output

sector produce petroleum In

theory,

the chemical petroleum

value

there

integration,

case.

arc

created

if

that,

in

components

of the sector

at hand, as

Equation

above a single be taken

a sizable 4)

up

which,

petroleum

products

arc three

sector

possible

it could

and

namely the

less,

gas. The

amounts

to

division

by

precisely

for

can the chemical without

petroleum?

answers:

or altcrnativo

wcrc vertically then

after

accounts

How

in turn.

produces

requirement,

gas rcquiremcnt

of No.,,,.

throughput sector,

( I )).

secondary

IO (chemicals)

c’,~.,~ = 6 928.0,

the

a

with

(commodity

c , o.c)= 0.62 * 78.9 = 48.6

primary

input activities

products). None c LO.9 -- 78.9 (pctroloum sector IO itself uses no pctrolcum and natural (I~.‘,

of the

identify

inputs

Each one will

the ncgativc

the subtraction

or a?,,

one sccdndary

is assumed

for the negative value of the input -

coellicient.

are

may cxcccd

and hence we observe

accounts

listed

olhcr

technology

of secondary

First, (I~ ,,, = - 0.007. Sector

arc trer input

is total

and net input

secondary

theory

coetlicicnts

To

by the use table, U (recall

product output

No

irrcspcctivc

have input

In each of the casts

words.

account

‘.

Each commodity

fabricated.

products

A)-

the commodity

negatives.

sum, exceed the actual inputs

In other

into

on digit lcvcl two arc crcatcd in the invcrsc.

sector

I we obtain

taken

(I -

3.)

persists

ncgativcs

to have its

For sector 2 WC have the usual cocllicicnts,

arc

invcrsc

one that

(ho

It

Leonticf

of 100 in Table

is the only

through

model products

and (I 2z = II~~/L.~~, but for sector

by a factor

one, (12n.J,,

vertical

technology.

integrated produce

into

If the

petroleum

3

The problem

of‘nryutices

in input-output

unulpis:

I: tm Rua and R. cun der Plory

products from petroleum and natural gas inputs. The latter inputs are not well represented in the chemical sector. so that vertical integration is not the answer in this case. The second possibility, throughput, turns out to be the right answer. The chemical sector produces petroleum products out of petroleum products. It has a sizable petroleum products output. r10,9 = 78.9, us &l as input, uy, Lo= 494.9. Thus, the first negative. in the chemical sector. is due to the problem associated with products having much orcn input (ten Raa, Chakraborty and Small [ 71. p 93). It can be considered as an alternative technology instance, namely one with own input coefficient one and all others zero. (It will not be so extreme in practice. but one petroleum product may be turned into another, which is essentially an aggregation problem.) Next take the second negative, (I,~.~, = - 0.015. Sector 31 (water) produces one secondary output with a sizable construction (commodity 28) requirement, The rcquircmcnt namely LI~,.~~ = 73.3 (construction). amounts to (I~~.~~I’_,,,~~- 0. I8 * 73.3 = 13.3 which, after division by primary output I’~,,>, = 654.5, accounts prcciscly for fhc reduction of (12H,J, to its ncgativc value. How can the water department produce construction with rclativcly littlc construction? This is the mirror image of the first cast. Now WC have the problem of products with much O~LVI input, not in the sector at hand (3l), but in the sector of rclfiwncc of Ihe secondary input structure (38). So the answer is that construction use of construction in its own sector. Lf1H.2n = 2836.3, is big. The third and last negative, (‘ZH.32= - 0.005, is similar. The construction secondary output, L’,~,,,~, is again the source of the problem: its commodity 28 (construction) requirement accounts for the reduction of ~~~~~~~to its negative value. Our diagnosis of negative input-output coetficicnts can now be summarized. The source of the trouble is the presence of much throughput of secondary products, either in the sector under consideration (u~.,~ -t I’,~.~ which causes negativity of Us,,,,). or in the sector of reference of the secondary product ~~~~~~~~which causes negativity of u28.3, and tt2B,,2). Throughput typically remains within a firm and its statistics are considered worthless relative to interindustry data for reasons of definition of transactions as well as conlidentiality. Thus, our diagnosis of the problem of negatives directs attention to the reliability of the data (the use and make tables).

The re-estimation procedure The negatives gencratcd in the process of constructing an input-output coeficients matrix arc clearly a nuisance. Something must be wrong. Either the model underlying the construction is misspecified or the data

4

must be erroneous because of measurement error and so on. We begin by exploring the latter case. Our null hypothesis is that the model is correct. Data (U, V) fail to observe non-negativity of input-output coefficients, u/v-‘20

(2)

but this constraint may hold for the true values of the inputs and the outputs. The wedge between data and true values is error. The question is if, given our null hypothesis. the errors take probable values. If not. we must reject the commodity technology model. The situation is reminiscent of accounting theory. This is easily explained by incorporating the valueadded vector of the system of national accounts, x, in our presentation. For each sector, the value of input and value-added must add to the value of output

whcrc cl is the vector with all cntries equal to enc. Data (U, V.y) typically fail to meet this balance constraint. Accountants proceed to adjust the data until constraint (3) is obscrvcd. For this purpose a rc-estimation procctlurc has been dcsigncd by Stone, Champcrnownc and Mcadc [6] and cxtcndcd by van dcr Plocg [S]. WC adopt the idea and will rc-cstimatc U and V so that constraint (2) instead of (3) is observed. We need more precise notation. From now on, I(,~ an d L’,k rcfcr to trtrr values of inputs and outputs of sector j. Attached to them are error terms Si, and I:,~. Errors can bc positive due to over-reporting and negative in the cast of under-reporting. True value plus error makes the datum: observed dufu are indexed by a superscript ‘I: 11;;and uyk. It follows that the data equal

and liTk= Vjk + [Zjk Thcsc data arc scctoral statistics which arc obtained by adding establishment figures. Assume that establishmcnts report with errors which arc indcpendcnt and identically distributed. Then. by the central limit theorem. scctoral errors Oij and i:jk are distributed normally. We also assume that these errors arc indcpendcnt, across cells (i, j. k = I,. . . ,39). The first assumption is natural, the second less so. However, the presence of correlations (for example between inputs and outputs within sectors) would modify the re-estimation procedure in a straightforward way (van der Ploeg [9]) without affecting our conclusions. ECONOMIC

RIODELLING

January

1989

The problrm ofnryuricrs

In mainstream econometrics we need many observations uj and c;~ for each i, j and k to infer the mean and variance of bij and cjL. In input-output analysis. on the contrary, we typically have only one observation. This hampers the application of sound statistical analysis. None the less. we have used subjective information on the accuracy of the data as furnished by the statisticians who gather them. We believe that this direct method of estimating errors in measurement is a good substitute for inference. As regards the mean of the errors, we assume that in the absence of accounting or economic constraints, statisticians have compiled data without systematic bias. Hence the means are zero. With the variances the specification is more delicate. Sir Richard Stone has pushed for revelation of such error information. All that we know is available are the standard deviations reported as percentages of the sectoral statistics underlying Barker, van dcr Ploeg and Wcalc [ I]. For self-containcdness we publish the sectors and the pcrcentagcs in Table 4 (in the appendix). So the variance of the first datum, I(‘;,, isa:, = (5% of I 42O.2)r = 5 042.4201. The second one is similar, but the third is mom complicated. since rr;, is not confined to scotors of the same reliability. Its accuracy will bc neither 5% nor IO%, but some avcragc. The reporting of errors as pcrccntagcs suggests that mixccl data have gcomctric mean accuracy. Hcncc, it is natural to set the variance of u;, equal to 05, = (JO.O5*O.lor~,,)’ =0.05*0.10*3.5’ = 0.06125. The vari-, anccs of all other data arc dctcrmined in the same way. We arc now in a position to write down the likelihood of real values (U, V). Its logarithm is

in input-output

unulysis: 7: ten Raa and R. can drr Ploeg

The use of ( 6) instead of (3) complicates the application of mathematical statistics. not so much by the inequality sign, but by the non-linearity of the constraint in at least one set of variables, namely I! The best linear unbiased estimate property of Stone, Champemowne and Meade’s [6] or van der Ploeg’s [9] re-estimator is lost if some of the constraints are binding Furthermore, if the initial estimates are normally distributed, then the adjusted estimates are not necessary normally distributed. This means that it is difficult to calculate the variances of the re-estimated data. However. it is always possible to use the likelihood ratio test (Silvey [S]. sections 7.1 and 7.2) to investigate whether any binding non-negativity constraints are consistent with the prior covariance matrices of the unadjusted data (see the next section). Since our conclusion will be negative, we do not really need the optimality properties mentioned above. The objective function, /; is exceedingly simple. It has linear first order and constant second order dcrivativcs. The function of constraints. A, is linear in U. but complicated in VI WC can nevertheless write down the sensitivity of the input-output cocfhcicnts with rcspcct to inputs and outputs. Sflij L4,ttitncr 1. .= 0

if

All,,

i #

r.

so .._.L!.

=

wrj.

and

h,,

;y l-1

=-

u~~~L’,~, whcrc wij (i,j = I,. . . ,39) arc the clcmcnts of CV = V-‘. See appendix for proof. We have also been able to calculate dcrivativcs.

the second order

d’c1

Lcttrtt1112. ----‘I = 0, ,Sltk,BIt,, -

I~‘,.~M’,~ and

d z(1 2 j, sr kl

= u~,w~,,w,~ + u,,M’,,M’~~where ,I

rvij (i, j = I,. . . (39) are the elements appendix for proof. $2’39’)

-~,O~(T;k,j.k

1Ol3'7I)

V)”

pJ;2(I~ij1.J

The constraints. A(U,

V)=

ECONOMIC

Ui;)2 + 1 T,;‘( Ujt - L.;k)2 (5) 1.k

rt. are given by

uv-‘20

IMODELLING

(6) January

1989

See

(4)

where 0,; is the variance of llij and ~12 is the variance of rlk. The basic idea is to find the most likely (U, V) that is consistent with non-negativity of input-output cocflicicnts. (2). Since the variances arc assumed to be known. maximizing L is cquivalcnt to minimizing /’ dcfincd by

/(U,

of CV= V-‘.

We turn to a routine for non-linear constrained optimization that exploits analytical knowledge of first order and second order derivatives: EO4WAF of the Numerical Algorithms Group [4]. The computation is complicated by the prohibitive size of the second order derivatives matrix, the non-convexity of the constraint set and the prcsencc of stationary points that do not solve the constrained optimization problem. (5) and (6). globally. To keep it manageable, we aggregate the data. Aggregation usually blurs the analysis, but here it accentuates the problem and the nature of the solution. Aggregation is by the rather traditional scheme, specified in Table 8 of the appendix (p 19). The 5

The problem

ofneyatires

in input-output

unalysis:

7: ten Rua und R. ran der Plorg

constraint set. (6). remains unchanged. The objective function. (5). must be reinterpreted. The coefficients the variances - are now variances of the aggregated

observed. This

Rows. Now,

re-estimation

as the data are independently

normal

distributed, the variances of sums are equal to the sums of variances. In short, the aggregation also applies to the objective function coefficients.

is, of course, very unlikely.

reject unlikely

Statisticians

outcomes. In our context, we shall

be forced to reject the model that procedure. that

the commodity coefficients.

underlies

is constraint

the

(6)

or

technology model for input-output

The raw input-output

coefficients, UC’-’ based on

the aggregated data. as well as the adjusted inputoutput coefficients, U V-‘stemming from the constrained optimization

Results Table 5 (see appendix, p 17) presents the aggregated inputs, the square roots of their variances as percentages (that is standard deviations) and the re-estimates. Table 6 (p 18) presents the same for the outputs. The percentages are basically weighted averages of the disaggregated percentage standard deviations. If the flows are zero so that no weights can be determined. then a blank enters. This

is no problem, since zero

flows remain zero in the maximum likelihood

adjust-

mcnt proccdurc for finitc pcrccntagcstandard deviations. The standard deviations pcrcentagcs arc somctimcs smaller than in the disaggrcgatcd cast, bccausc of the cancclling out of errors. WC wish

to draw the rcador’s attention

to two,

rclatcd results. I:irst, the maximum likelihood estimation involves the setting of some secondary outputs equal to zero. Second, the adjustment sets some data oll’thc ‘true’ values by more than two standard deviations. WC will elaborate on both of thcsc. The solution fcaturcs zero values of some variables. This is easily explained through the example given in the introduction. coellicicnts

Non-negativity

of the input-output

of sector I. (I ), requires

that its inputs

exceed the secondary output requirements.

But, if such

an Input, say pi,,, is zero. then, since standard deviations arc given as percentages so that zeros remain zero, the secondary output requirements, (I, 2u, 2, must be zero. Hence u,? or tlZ must be set zero to meet nonncgalivity of (I,, In short, if an input is zero, then the corresponding

input

requirements

of the secondary

products of that sector must also be zero. The maximum likelihood readjustment brings this about by setting to zero the secondary outputs

interesting to note that. basically. our adjustment procedure sets the negatives equal to zero up to digit level 3. That is the common practice in dealing with the problem. Routine practice is thus given a statistical foundation. Table 6 also confirms that the coefficient adjustments are minor. However, coeflicients are derived constructs. Any change must be conceived as the result of a change in data. Although

the change in

cocllicicnts is small. the underlying change in data must bc large. Large data must bc reduced all the way to zero. This involves many standard deviations and. t hcrcforc, ;I large Icap in terms of likelihood. So although the common practice of ignoring the input output coctticicnts seems justified at first sight. statistical analysis

raises doubts.

One way of obtaining insight

into this question is

the use of the likelihood ratio test (Silvcy [S] sections 7. I and 7.2). Since the vnrianccs in the unadjusted data arc assumed to bc known

from

the Central

Statistical Ollicc. twice times the dill%rence in the log likelihood, (4), equals (minus) the dilrercncc in the ‘sum of squares’, (5). and this is the test statistic

of

the likelihood ratio test. It is distributed as a x’(r) variate, where r is the number ofhintlirtgg non-negativity constraints. In our case r = 9 and 1tiL: test statistic is 1914.2. Since the critical value of x2 (9) at the 5% significance level is 16.92. the non-negativity constraints arc violated at the 5”/;, level. This Icaves no room for other than for an empirical rejection of the commodity technology model.

with such an input

rcquiremenl. In this study, Table 7 shows that sccondury outputs czJ, cZ7, vZn, vZ9 and vsg are set to zero. Clearly thcsc constitute significant adjustment steps. They arc indcpcndcnt of the standard deviations of the variables and may cxccod them by multiples. For cxamplc, if a llow belongs to a sector of which data are accurate up to 5%. then a readjustment towards zero corresponds to 20 standard deviations. This holds for the mining and gas sector, 2. In other words. the data have errors that have much Icss than even I % probability of being

6

problem (5. 6). are reported in Table 7

of the appendix (p 18). They are multiplied by a factor of 100, so that the unit is pennies per pound. It is

Conclusion WC (ind that the magnitude of the adjustments to the USC and make data which are required to ensure the non-negativity of the input -output cocliicicnts, based on the commodity technology model, arc inconsistent with the distribution of the unadjusted data. This means that WC have a statistical basis for rejecting the commodity technology model. This rejection is particularly surprising given the high level of aggregation WC used in our cxcrcisc. At such a high level of

ECONOMIC

I\IODELLING

January

1989

The problem

of negutices

aggregation there are only a few negative input-output coefficients and their magnitude is tiny: but the adjustments required to satisfy non-negativity are nevertheless sweeping and inconsistent with the data. It follows that we must accept that different industries have different technologies for producing the same commodity. This is clear when some industries produce more efficiently than others, but it may hold even in a perfectly competitive world. The A matrix is limited to material inputs, and apparent comparative disadvantages may be offset by lower direct factor costs (fixed capital or labour). Since Kop Jansen and ten Raa [3] reject the alternatives to the commodity technology

model for other

the very

linear

coctficients output

0-I)

tlo~vs

destination

reasons.

framc~vork from

the

we must

technical

unit

black

of

and

(U, V). We must account within

sectors.

information

wc may

continue

pure commodity

technology

such limit

of inputs

its application

vectors

of which

to tinal

as usual. just

dcscribcd

output

or

the

matrix.

or

but

value-oddcd

arc close to the

ones

of the tcohnical

WC can still suppress the ncgativcs is small. but within the

magnitude

class of admissablc

price

to compute

Poutput

projections

scumtrios

will

industrial

bc positive

anyway

the zero setting yields modifications which arc rctluntlant. In short. atljustmcnts. cvcn when based on

and

information trends

about

worst

dctcrminc inpuls

rcliabilitics.

instead

the within

of

make projections

bcttcr.

ssctor

commodity

or limit the applications

along

WC should

cithcr

destination

to scenarios

to the structure

7: ten Raa and R. ran der Ploeg in the years

of the economy

and leave the negatives

of

as they are.

T. Barker, F. van der Ploeg and M. Weale, ‘A balanced system of national accounts for the United Kingdom’. Rwitw oj’ Income and iiklth, Vol 30. No 4. 1984, pp 46 I -4X5. Y. Fukui and E. Seneta. ‘A theoretical approach to the conventional treatment ofjoint product in input-output tables’. Economics Letters. Vol 18. 1985. pp 175 179. P:KoQ’Junsen and T. ten Raa. ‘The choice of model in the construction of input-output coefficients matrices’. workin! paper. Tilburp University. 1987. Numerical Algorithms Group. N.&q FORTRA?; Lihrtrr> ~\lwucd .Md II. Vol 3. Oxford and Downers Grove. IL. IYSI. S. D. Silvey. Sttrtistid /n/kcncr. Chapman and Hall. London. 1975. J. R.N. Stone, D.E. Champernowne and J.E. Mcade,‘The precision of national income estimates’. Rcricw O/ Economic Stutlicv. Vol 9. No 7, 1941, pp I I I - 13. T. ten Haa. D. Chakraborty and J. A. Small. ‘An xltcrnativc tnxtnicnt of secondary products in input outpul analysis’, Rccicw o/‘ Ewmmric3 cod Sftrfi.stks. Vol 66. No I. 10x4. pp XX -Y7. Ilnitcd Nations Statistic~~l Commission. I’ro~~0x1ls /i~r I/I<* Rcpr,l.\ion o/‘ SN.4, Iv.52. Document E,/CN.3; 356. lYh7. I,‘. van dcr I’locg. ‘Rcli;ibility and the ;itljustmsnt of squcnccs of large economic accounting matrices’. Jorrrf~[rl c~/~~hzRo~~trl.Sftrfis/ic.c~/Society A. Vol 145. No 2, IYXZ. pp 16’) IY-1. W. van Rijckcghcm. ‘Ali ou;ict method for dctcrmining the technology matrix in a situ;ition with secondary protluct~‘, Rwic*\c off I:corromic~s trfd Sttrli5fic.s. Vol 4Y, No -l. lYO7. pp 007 00X. V.Q. Vict. ‘Study of input -0ttlpiiI tables: lY70 IYXO’, working paper. Unitcd N;ltions Stxtistical Ollicc. IYX6.

In the absence of

the construction

is based. since their

input

analysis:

References

for the output

demand

the proportions

in the year on which cocllicicnts

input

close

construction

abandon

of deriving box

in input-output

of

proportioned

Appendix Consequently,

the second order derivatives

with respect to

U vanish. The cross partials vanish for i # k by the lirst part L’dII:

To

dctcrminc

clscwhcre and

put

_--“.

put

(ClL:),,

and

= ch,,

’ I’,,

dLt’=

Leros

0 as CV deprnds

To

To dctcrminc we

have

(dlt’)L”+

- t!‘(dC”)Ii: must

put

!z!i,

put

dC’ = 0. Now. difkrentiating Il’dl”=O

or

dIt’=

-

-A(di’)bt:

Hcncc d.4 = - L’IV(dl’r)li.=

evaluate

lemma

I ). Pit (d V),, = SL.,, and zeros clscwhere,

(/,j)th

component

o/ lernrnc~ 2. Lemma

dcrivativcs

ECONOMIC

I shows

that

reads

jw,,=

-w,,6~.,,\v,,

or

then the 2

Then

= ‘r.

- w,,w,,.

We &I,, =

It follows that ;;3[? = - w,,w,,. It remains to LI dctcrminc the second order derivutivcs with rcspcct to V. By lemma

I and

(50 - Gw 3r,,

”

for the ._ Proof

this. note that

= $p by lemma I. II ‘,. ‘,(l dCV = - IV(di;) IV (proof of

L-t’Vr = 1.

Li’(dl”)l“‘=

(d I’),, = cSI.,, and zeros elsewhere.

-

I. If i = k wt‘ have $$

on I’ only. Non-r

rows of dci being zero. it follows that ~~::~~ = 0 for i # r. The ,, rlh row of the equation reads &,, = 61r,,.lr,,, j = I,. ,3Y.

.

-2

of lemma

.-a

the product ,5W

Li.

By lemma

“Or,,

partials

rule,

of

CV with

&=

$(a_w,,)=

I and the above cxprcssion respect

to

V, WC obtain

the first order

with respect to U depend only on LV hence V.

MODELLING

January

1989

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The problem

of nryutires

in input-output

annlgsis:

7: ten Ruu und R. wn der Pkwy

ECONOIMIC MODELLING

January

1989

The problem of negatives in input-output analysis: 7: ten Raa and R. van der Ploeg

ECONOMIC

MODELLING

January

1989

The problem ofnegatices

in input-output

analysis:

‘I: ten Raa and R. van der Ploeg

ccccccoccq~.~.ocpc~qy 2.z _~oooooooooo~o~oro~o

fig g,p

t=

-Y:&

cccacccc3cc3**~****cc*0c0Q~~cr,~~0~~ccccccccccccccccc~

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-

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\o

ECONOMIC

MODELLING

January

1989

Cl

hold

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walqo’d

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The problem of negatices in input-output

analysis:

T. ten Raa and R. van der Pioeg

ECONOMIC

MODELLINC

January

1989

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NOio o-

IO 0

(S'i

L

OOOPI0

io (1

-

IOO-

WC

,".wl"J,\"l

Ilu+lu!lua

;o II-

II.1

,RJ!".U,J

Flu!Jaau!%J~

(0 0

d!qs

au!Pl!"q

IO0

w.vud!nb

ti i

amdwa\

RI'0

cpoo'd

01 0

WI!lWl

WV\

IO0

'Jagma-1

4U!qlOIJ

m

,l00-

Su!qs!lqnd pu' BU!W!l,l

The problem of negatices in input-output

analysis: 7: ten Raa and R. oan der Ploeg

8

‘2

E

‘E

E ~33833333:1a3833~3$33~3~3~3~4~r~3~o~~~~P-~ i)0~,000300000000-r0000d00000300033-~-~0~0ri

16

ECONOMIC

MODELLING

January

1989

The prohlrm

of nryurirrs

Table 6. Aggregated

in input-output

unul_vsis: 7: tm Rua and R. can drr Ploeg

F’. accuracies, and reestimates

FQOd.

Agriculture 5 6 16.90 5.0”” 5617.00 0.00 0.00

\lining

0.00

0.00 0.00 2 61’ JO 2000 2 6”.(H) __

0.00

0.00

O.(X)

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

OSKI

0.00

0.80 5.0”” 0.00

0.00

0.00

0.w

I I844.30 4.I 9, I I 540lx)

O.Oi)

0.00

O.(H)

0.00

O.00

0.00

O.ol)

0.00

0.00

O.(H)

o.oU

0 or)

0.w

ii

o.rH)

I ” I, !!.W

l2413.20 3.5”0 I2 560.00 277.20 4.6” 273.w I.10 5.0” I .OY

YO

0. IO 5 0 I’I, 1)IO

3

0.00

0.00

0.00

0.00

0.w

0.w

I.20

3.60 5.0”0

?.Z”<, 3

I.20

3.60

20 450.70

II

20 7XIwU 457.10

,I

53 110 JJ”,,

5.301

12 S
1,”

2.0”II

I ? 5no.rw)

452.60 23 XI) .3.5 ‘I II 2.l X0

554.80 6.3 “h O.(M)

47.20 3.h”,, 15.73

32.50 5.4 ‘:;, 3 I .Y5

3h3.40 x.0 307.50

” I,

657.70 J..3 ” 0 hSX.40

I,

O.(H)

0.00

0 (H)

O.o(l

ZI.OS

0.00

0.00

O.(W)

O.(H)

O.Or)

0.w

0.00

(I 00

0 (H)

O.(H)

3h.lO 5.0~“,, x1.20

O.(M)

0 (H)

567.30 ?O.h “‘I, IWII00.00

Construction

Sen ices

6I

1lining

27.0x 26. I7

I .YY 2.03

16.6X 16.6X

- 0.04 o.rK)

x50 ?J.YJ

I.02 0.03

- 0 05 0.w

h.YX 6.YX

4.67 4.‘) I

3.5’) 3.x0

30.27 2Y.96

4.35 4.3x

2.x

0.16 0.77

8.73 x.s.3

0.X4 O.YO

0.9’) I .oo

2Y.YX 2Y.55

Hc:1vy m;lnufxrurinp

O.Y7 0.97

I.21 1.30

2.50 I.63

I.X3 1.x3

5.65 5.6 I

Light manufxturing

1.65 1.66

1.35 I .YX

6.2 I 6.51

2.74 2.77

3.65 3.71

5.2 5.43

Conslrucfion

2.21 2.2-l

3.Y3 4.04

0.06 0.10

-0.04 0.00

0.55 0.54

16.65 I7.7Y

15.63 15.61

15.51 I5.H

IO.51 IO.51

507.40 6.2 ‘Ti, 507.w

and thrir re-e%timatcx

- 0.07 O.(M)

18

55.50 5.6?‘u 5 5 .3.s

0 rH)

2.1s 2.33

Services

2 I .JO I..sl’ 21.4,”

O.(H)

0.05 0.06

McrA

‘77.90 IO.700 ?X8.40

0.00

25.2x

pns

56. IO x.2 “f’, 53.87

2 I .YO 27.4 ” II

- O.Gi) 0.01

pcrrolcum

4s.90 ..2”. JS.SY

0 rn1

0.23 O.IX

hlining.

731.00 8.30’0 657.80

0.00

21.X6 5x 50

and

IY.10 5.9 ‘% lY.07

O(H)

-0.03 O.W

drink

5.6u 4.0 “,” 5.60

0 (H)

25.2X

lobxco

69.60 14.80; 0.00

O.(H)

\lining

I-ood.

11.10 8.4?‘0 0.00

0.00

Agriculture

and

36.00 7.1 0” 0 00

.&I).40

.I.‘)I’II

I:ood, drink rnd tobacco

hlming

o.rw

0.20 17.300 0.20

I’) JS2.YO I .Y ” 3) 650 I)0

JO.40

Services

30.20 8.7% 30.20

JO I .SO I .7 “GB JO I .50

2.6”,,

Construction

000

O.(NI

Table 7. Trchnicul coellicirnt\

Agriculluw

Light manufdcturing

_

‘4.80 3.7”a 24.43

O.Gi)

7

Heavy manufacturing

Aletals

0.00

2 I .40 5.0”, ” sg __._

0.00

>lining, gas and petroleum

drink and tobacco

21.7x ‘2.6-l

Kd%

aud petroleum

Xletal5

1IrdVy manulkturing

Light manulxcturing

-0.0

I 0.00

1.58 I.31

0.03 0.02

O.‘X 0.00

-

0.0 I 0.03

0.63 0.62

I.XH

2.13 I.11

0.0’1 0. I2

0.0 0.0

8.54 H.II

2.17 1.77

3x5 2.24

22.87 22.Y3

1.56

I ..I9

9.43 7.6X

1.36 0.7’)

23.36 ‘3.40

I .Y3 I.45

2.60 2.12

I .‘)-I I.13

32.x 30.46

I&X5 12.10

4.15 2.42

0. I3 0.16

0.17 O.IX

IX.10 12.5’)

1.00 0.5x

Il.53 I I .YO

14.7x I426

7.31 5.97

23.30 ! 3.53

- 0.03 0.00

I 2.X’)

ECONOMIC

I.51

I .A0 0.57

I

MODELLING

January

1989

The probkm

Table

vfneyutices

in input-ourpur

anulysis:

7: fen Rau and R. can der Ploeg

8. .Iggregation.

1 Agrtculture

1

etc

? Xftnmg and eas

Z Coal mining 3 Mining 4 Petroleum and natural

3 Food. dnnk

i

and tobacco

Food manufacturing 6 Drink 7 Tobacco

4 Xlinmg and gz+ products

8 Coal products 9 Petroleum products IO Chemtcals Iron and steel Non-ferrous metals Mcchanical engineering fnstrument engineering Electrical engineering

6 Heavy manufacturing

ECONOMlC

16 I7 IX 19 30

MODELLIKG

Ship building Motor vehicles Aerospace cquipmcnt Other vehicles Metal goods

January

1989

Textiles Leather. clothing etc Bricks Timber and furniture Paper and board Printing and publishing Other manufacturing

7 Light manufacturing

Agriculture etc

pas

8 Construction

2!3 Construction

9 Servtces

‘9 30 ?I 33 33 3-t 35 TC XI 37 38 39

GilS

Electricity Water Rail Road Other transport Communication Distrtbutton Business services Professional services Miscellaneous servtccs

19