Christopher A. Sims

English

Research Papers in Economics

This PDF is a selection from an out-of-print volume from the National Bureau 
of Economic Research
Volume Title: Annals of Economic and Social Measurement, Volume 6, number 1
Volume Author/Editor: NBER
Volume Publisher:
Volume URL: http://www.nber.org/books/aesm77-1
Publication Date: 1977
Chapter Title: Remarks on Real Value Added
Chapter Author: Christopher A. Sims
Chapter URL: http://www.nber.org/chapters/c10506
Chapter pages in book: (p. 127 - 131)A ,i,,als of Leonn,n,eand .coc,/ %iea.urt',nenI 6 / //977
REMARKS ON REAL.VALUE ADDED
B' CI!RIST0PIIPR A.SIMS
In a recent article in this journal, Stefano I-enoaltea
[1976] discussedmea- surement of real value added. The article concludesthat a cellainclass of measures is superior to all others: "TheCurrentprjce valuesadded of dif- ferent industries are. . .to be deflated by thesame price index"' Aspart o an extensive discussion of the criteria for a goodmeasure of real value added on which the article bases its choice ofa class of indices,the article asserts that the assumption of separability of the
production functionon which earlier discussions by' Arrow [1974Jand myself 11969]has been based, is reasonable only "on a literal butunusual definition of'real value added' as a thing in its own right; and it has littleto do with themeaning of real value added' in the context which coinedthe phrase in thefirst place."2
Though I did once write (1969] thatseparability is required inorder for "the notion of real value added" to "makeany sense", I agree with
Fenoaltea that it may sometimes be useful toconipule "real value added"
for industries with non-separable productionfunctions justas it may
sometimes he useful to compute "real income"for groups ofconsumers with different utility functions. Nonetheless, itmust be expected thatany
good measure of real value added will sometimesmisbehave accordingto
some intuitively natural criteria, when industry productionfunctions are
non-separable.
Likewise 1 agree with Fenoaltea that histype of index is a reasonable
one in some applications--- I have used an index of thistype in sonic em-
pirical work of my own [1968]. Nonetheless heis wrong to claim that his
class of indexes is better than the other typesof index to which hecorn-
pares his own. In any situation where there ismore than one primary in-
put, Fenoaltea's index is capable of producing anomalousresults, even in
some situations where the alternatives to which hecompares his index
behave reasonably. The simulations reported inFenoaltea's article never
make his index misbehave, because they all dealonly with the case of a
single primary input.
SectionIbe1ow presents examples of undesirable behaviorby
Fenoaltea's type of index. Section 2argues that Fenoaltea's own criteria
for a good real value addedmeasure point to the central importance of a
separability assumption. Section 3 summarizesconclusions.
'Fenoaltea 19761.p.21. emphasis in ori'inaI.
2FenoaILea 11976j.pp. 19 120.
127
aI.ANo\1,xi.lis ARIStN(lRu,\l Assi
A11)rvHAS ONIV ONIPRiti
For an industry which produces its output dirthhni
lactors, without anpurchases oF intermediate inputs
wou Id agree that measurement oF rca I "a Inc added presentno i
real value added should in this case he identicalith iealtitpti1In fact all of the indexes Fenoaltea considers, except hisov ri agree in1Itic, in measuring real value added as real outputhe indces1enoaltcaCO. siders are: gross output: 1)avid 's index: the standardmdcx ofactjvjtand the double-deflated index. For the ease ofOne Output, t\ 0primtr
(K and L) and one intermediate input (41 ), thesemeasure the
change in real value added as, respectively,
Gross output:
(I) (/Q
Q
E)avid index:
((/Q.V,(/I%f\
MF(-'W_TY
Sr .sjJ\!'T)
Activity:
.VAdA' S(IL--+
L
DoubIedeflatjon:
1Q.s,,dM
Al
where Q is output, Kcapital, L labor, A!materiils V value added. P the price of output, Ttheprice of materials, andsubscripted c's are shares of output. Fenoaltea'sown indexes measure thepercentage change in real value addedas
Fenoaltei indexes:
II
4141(/1
p -
1)'I
3ThiscornpIicite1 forrnurnaioi make apparelu thes:or11 puiatio;ial aiPed f David\ mdcc: Thcmndcis currem value addeddeflated b thej)1iCe of output
128
awhere Pv is aprice index br value added. Clearly if thereare 110 material
inputs, = I, indexes (I), (2), and (4) all reduceto
dQ/Q as does3) under competitive assumptions withQ =F(K, I.)I"
oinogCne0uS of degree one. (5)s (IQ/Q ithis case only ifdP/P1
dPI P.I hus for two ntdustueiii which idative price haschanged (5) (j5
not give thenatural answer, regardless of how P1; iS measured
For example, if demand br the output of each of two industries is
unit_elastic and neither uses intermediate inputs, l-enoaltea's indexes will
never showchange inthe industries' relative real valueadded,
though (l)(4) could all agree on large changes in relative real value
added.
Another class ofexaniples, in which intermediate inputs are allowed
arises when we consider twoindustries with lixed-cociliciciits technologies
and inelastic demands. All inputs and outputs in these industries remain
fixed while relative prices change. Most economists would agree that a rea-
sonable detinitiOrl of real value added would not vary as prices change in
this situation. Obviously indexes (I). (3), (4) behave reasonably here,
while (2) and (5) do not.
Fenoaltea claims, "A proper value measure of industrial production
measures both the valueof activity and the value of itsresults.''4
Though he does not give a precise explanation of what he means by "re-
sults" and "activity", he does give synonymsfor activity (industry, in-
puts) and for results (industrialproduction, net output).5 I agree that it is
essential to most interpl'etatioilS of real value added statistics that they
measure both "real primaryinputs'' and "real net output''It seems to me
reasonable to require that when a given industry uses exactly the same
vector of primary inputs at two diflerent times orin two difl'erent places,
then any measure of "real primary inputs" should he the samein those
two times or two places. This impliesthat "real primary inputs" is SOfl1C
function a of the vector of primary inputs L.Correspondingly, "real net
output" ought to be a function r of the vector rof industry outputs and
intermediate goods inputs. If there is to he anysingle number which al-
ways measures both "realprimary input" and "real net output", then we
must always have r( y) = a(L).for any v and L consistent with the in-
dustry's technology.
Now we have that if a point (i', I.) istechnologically efficient it must
satisfy r(y) = a(L). If conversely any point( v. L) such that r( y) =
[I97bl,p. 118.
p. 112.
2.IsSIPARABII.IrYIRRiI.rvANT'?
129a(1. ) is efficient, then r() = a(/. ) defines the technology,and the tech
nology is separable in r and 1.
When the tech nology is smoothly di lirent ableand conv,ofle
expect that the set oleflicieiit points will he in NI diinensjo1icmi..
where N is the total number of inputs andoutputs, and the set tiPOints r( r) = a(L ) will also he such a surficeSince the lattersurface COntains
the former, and both have the same di mensioti,tl1ey will htt least loall
identical. This completes a rough sketch ofa proof that a 5iflOth
tech- nology which admits an ti nambigtiotisnieasu re of' real value addedifiust he separable.7
3. C'tnsioss
If one adnuts that the real value addedmakes most Sense when"net
Output" is sonic function of' primary inputs, thenseparability of the tech-
nology is important to the notion ofreal value added. tinderperfect
corflpefition, the technology is linear honiogeneotisand separablein- dexes (3) and (4)the Divisia index of' primary inputsand the "douhle_
deflated'' index of' net outputare both locally exact. Thegross output
indexI) is locally exact only when intermediateinputs arc absentor more
!n proportion to total output. The David index islocally exact only iiin-
termediate inputs are absent or the priceof intermediate inputmoves in
proportion to the output price. The Fenoalteaindex is exact onlyif (c/F'/P - s,di'/T)/s1 is the same in all industriesand dJ.//is chosen equal to this quantity.
Now it may sonietimes happenthat even though theseparable-tech-
nology assumption is reasonable,the data necessary tocOmpute (3) or (4)
are unavailable, the technology isnot homogeneous, or competitiveas-
sumptions are not satisfIed. Thenone of the other indexes may he pref-
erable, and it would thereforehe worthwhile to explore inmore detail exactly what assumptio,isbeyond separabilityarc required to justify each
of the other indexes.
i\ tecIinoiuis separable in r and I.iii can heSpressed as g l,(v). h2U 0. where ii; a id h2 arc eachu ne-d i men nirnal
readily tdniit that thisgeolilerric arQ(iiiit is so sketch5ato he irritating to thc skeçn ic. A more completeirgumerii might (tesciop in iCrtjiiiirisi'litilto special cases where real value addedexists iii non-separable tech nologies('tearI hieii coL'ihicient tech- nologies insstiiehtheetticierilsetisthe intersection or r(iall.ssiih some add. tiumi,il ICsiriciiuris pros mdcuric class 01 suChSliecual Cise, Note thai ss lienthere us tints uricprm.Ir>inputasIiillicuioaitea.s etann ptstheieciniiuitogsistilt!ii,iticahlS SepFhte
lire t)avid index norkss%hefliii prices nose iiiproportionss mile the I cluojhtea uindmu principle ssurks under slightlyless restrjctivcasstulniiihBittitjni;t% heihat mu a ctusd econtniiwith scp.ir.mhte techiniotogsiiicactimiLiulsirsaliseth pm ice ol value added across industries implies hicdrei,u(ivprices
130
I,lso, it is not too hard to imagine Situations where "netOutput'' or
"total input" is in some sense meaningful, even though the twoflotL011s
CaflOOt be treated as identical. For example, it may he reasonable tocom-
pute a net output measure as "value added at international prices'' inan
econofflY with tartfl' hai rters. Ihe fact that there inot in general anyway
to identify this "netoutput'' with any one-dimensionalmeasure ot "total
input" need not, in some applications, affect the Usetulness of thenet out-
put meastlre. However, in such applications, where Separability isriot
claimed, a distinction between net output and total input is unts'oidtble
one cannotpretend that "real value added" is both thingsat once,9
(Juiverc itt' of vIw,wvoja
REFER F N('ES
Arross, K. J. [1974), "The Measurement of Real Value Added''fl PA. I)aid md M
Rcder. and Iloun'nolds in Economic (Jronih Ncn York: Academic Press).
FcnooltCa, S. 19761, ''Real Va!ue Added and the Measurement 0)ndustrial Production'',
,wals a! Economic and Social Mea'uremt'ni. 5. II I 35.
Sato, Ka?uo 119761, "1 he Meaning and Measuremeiit of the Real Value Added lndc",
Rec'iew of Economics and Statistics. 55, 434 442.
Sinis. C. A. [1968). "The Dvnam,cs ol ProduciivitChanue'', unpublished Ph.D djssrtj,
lion, Department of Economics, Harvard U nivcrsil.
[19691, "Theoretical baSiS for a L)ouhle-l)eilatcd Index ot Real Value Added" Ri'-
new ofEconomics and Statistics, .5 I. 470 471.
9An importantnew paper on this topic which appeared after this note sas In proofs
is Saw 1l976[.
131

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