Some correspondences between Index Number Theory in economy and the General Theory of Relativity in physics

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by  Ali Hosseiny

Some correspondences between Index Number Theory in economy and the General
Theory of Relativity in physics
Ali Hosseiny1,2
1Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839, Iran
2
School of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM)
P.O.Box 19395-5531, Tehran, Iran∗
(Dated: January 19, 2017)
GDPofChinaisabout11trilliondollarsandGDPoftheUnitedStatesisabout18trilliondollars.
Suppose that we know for the coming years, economy of the US will experience a real growth rate
7 equal to %3 and economy of China will experience a real growth as of %6. Now, the question is
1 how long does it take for economy of China to catch the economy of the United States. The early
0 impressionisthatthedesiredtimeistheansweroftheequation11×1.06X =18×1.03X. Thecorrect
2 answer however is quitedifferent. GDP is not a simple numberand the gap between two countries
can not be addressed simply through their sizes. It is rather a geometrical object. Countries pass
n
differentpathsinthespaceofproduction. ThegapsbetweenGDPof differentcountriesdependon
a
thepaththateachcountrypassesthroughandlocalmetric. Toaddressdistancebetweeneconomies
J
ofChinaandoftheUSweneedtoknowtheirutilitypreferencesandthepaththatChinapassesto
7
reach the US size. The true gap then can be found if we calculate local metric along this path. It
1
resembles impressions about measurements in the General Theory of Relativity. Path dependency

of aggregate indexes is widely discussed in the Index Number Theory. Our aim is to stick to the
]
N geometricalviewpresentedintheGeneralRelativitytoprovideavisualunderstandingofthematter.
Weshowthatdifferentelementsinthegeneral relativityhavetheirowncounterpartsin economics.
G
Weclaimthatnationalagencieswhoprovideaggregatedataresemblefallingobserversintoacurved
. space time. It is while the World Bank or international organizations are outside observers. The
n
visionprovidedhere,leavesreaderswithaclearconclusion. IfChinakeepsitsgrowthrate,thenthe
i
f economy of China should catch theeconomy of the UnitedStates sooner than what we expect.
-
q
[
I. INTRODUCTION Pa isitsprice. Thoughtheoutputoftheaggregationisa
1
singlenumber,thisnumberisnotaregularnumber. The
v
fact is that prices are our meter to perform aggregation.
4 Suppose that you want to extrapolate the time that
Elements of this meter however vary with heterogenous
1 economyofChinaneedstobecomeasbigastheeconomy
1 of the United States. The current size of economy of rates over time.
5 China is about 11 trillion dollars and the current size of Difficulty nowarisesifweaimtocompareGDP oftwo
0 economy of the US is 18 trillion dollars. Now, suppose different countries. Even for a single country we have
.
1 that we are sure that economy of China in coming years difficultywhenweaimtocompareGDPofdifferentyears.
0 will experience a real growth as of %6 and economy of Problem is as follows. Nominal GDP in the year i is
7
the USexperiencesa%3rate. Then, the questionishow
1 GDPi =Σ PiYi, (3)
many years does it take for economy of China to catch a a a
:
v economyoftheUS.Ourimpressionisthatwecansimply and nominal GDP in the consecutive year is
i
X find the answer through solving the equation
GDPi+1 =Σ Pi+1Yi+1. (4)
r a a a
a 11 1.06X =18 1.03X
× × IfwedecidetocomparebothGDPs andfindgrowthrate
ln(18/11) (1)
⇒X = ln(1.06/1.03) ≈17years. g = GDPi+1 1= ΣaPai+1Yai+1 1, (5)
nominal GDPi − Σ PiYi −
a a a
This answer is however untrue. GDP is not a simple
number. To find GDP we aggregatea set of productions we face problem. The problem is that GDPs of different
through their prices. In other words we have years have been measured with different set of prices.
As a result, inflation may result in an overestimation of
GDP =Σ P Y , (2) the measurement. So, we should find which part of the
a a a
nominalgrowthisrealgrowthandwhichpartisinflation.
in which a ranges for any form of final goods or services
The usual method is to compare the GDP of the con-
providedinacountry,Y isthequantityofeachgoodand
a secutive years with the same set of prices and define the
real growth as
Σ PiYi+1
∗ g = a a a 1. (6)
[email protected],[email protected] Σ PiYi −
a a a
2
Now, it seems that we have overcome the problem with inconsistency (see Feenstra, Ma and Prasada Rao 2009
inflation. We however have created another problem. [12],Feenstra,InklaarandTimmer2013[13],andOulton
The definition of the real growth in Eq. (6) fails the 2015 [14]) where by space people mean measurements in
circularity test. different countries.
Let us consider a time series of prices Pi and produc- In this paper we borrow the concept of measurement
a
tions Yi over a period of m years. For each year we find and metric in the General Relativity to understand the
a
the realgrowththroughEq. (6). If atthe beginning and problem. Though path-dependency of measurements for
theendoftheperiodproductionandpricesarethesame the growth rate is an old known fact in economy and
thenitmeansthatGDPhavenotgrownovertheperiod. has being discussed in the Index Number Theory, it has
So, we expect beenbarelyknownoutsideofeconomicsliterature. Even
in the world of economy it is not discussed in regular
Σi+mg 0. (7) textbooks. The geometrical vision provided in this pa-
j=i j ≈
perhelpstoclearlyunderstandwhyGDPisnotasimple
This trivial requirement howeveris dismissed in our def- numberandwhymeasurementforthegrowthrateispath
inition. What is the consequence? The consequence is dependent. Itthencanbeeasilyvisionedandunderstood
pathdependency ofmeasurements. I havesupplemented by researches in different areas of sciences. Previously,
a couple of csv files. In one of them I have produced a interestingeffortshavebeendevotedto findsomeinvari-
time series of prices and production for a bi-sector econ- ant measures for aggregate indexes through an analysis
omy. Two countries start with exactly initial conditions. ofthe gaugeinvarianceofeconomicmodels,seeMalaney
Over a century, one of the countries experiences a real and Weinstien1996 [15], and Smolin 2009[16]. Our work
growth annually equal to 3.7% and another country ex- however relies on geometrical imaging of the problem
periences a realgrowthequal to 2.2%. Despite this huge andmainlynotifiessomesimilaritiesbetweenobjectsand
differences, both countries end with exactly the same conceptsinthegeneraltheoryofrelativityandtheindex
conditions and exactly the same set of production and number theory.
prices. Guess what! Despite their equal performance for Metric in the general relativity is addressed by Ein-
the whole period, the governing party of the first coun- stein’s action. In economics, equilibrium constraints
try is proud and celebrates a century of annual growth and production functions address the relation of prices.
of3.7%andthe governingparty ofthe secondcountry is While in the general relativity objects move along
ashamed. geodesics, in economics, utility preferences address the
Dependency of measurements to the paths first at- path of growth in the space of productions. We discuss
tractedattentionofFisherinpriceindexproblemin1927 thatnationalagencieswhomeasureGDPresemblefalling
[1]. He noticed if we aim to measure inflation then our observersinacurvedspace-timeandhavetheirownmet-
measurement can be path dependent. Finding a proper ric to calculate their growth rate. It is while the World
methodtomeasureinflationthenwasunderattentionfor Bankisanoutsideobserverwhichusesastandardmetric
decades. Some influential movements in indexing were for all of these countries. Besides, we discuss that to ad-
addressedbyLaspeyres,Paasche,Törnqvist,andFisher. dressthegapofeconomyofChinaandoftheUSweneed
For recent works see Balk 1995 [2], Diewert 1976 [3] and to know production functions and utility preferences to
1993[4],FeenstraandReinsdorf2000[5]andOulton2008 findtheirpathsinthespaceofproduction. Thenmoving
[6]. alongthispathwecanfindoutthetimeitneedsforecon-
In recent years paradoxical results concerning evalua- omy of China to catch economy of the US. The overall
tionofGDPreportedbyinternationalorganizationssuch impression is that China may catch economy of the US
as the World Bank have reraised attentions. In Interna- sooner than what we expect.
tional Comparison Program (ICP) and Penn World Ta- Besides problem with measurement we shortly review
ble,GDPofeachcountryisevaluatedthroughpurchasing the cost disease phenomenon and its role to address a
power parity (PPP) exchange rate. When we calculate major feature of heterogeneity of dynamics of long run
GDPofdifferentcountriesthroughPPPexchangerateit prices. In econophysics we usually have been interested
means that we have used a standard price for each good in short run prices and its volatility. A vast range of
worldwide. In other words we have used the same me- workshavebeendedicatedtothe price inthe stockmar-
ter to perform aggregation in Eq. (2) for all countries ket,seeforexample[17]-[20]. Besidestockmarket,physi-
and measure the size of GDP of them. We then can find cistsalongwithacommunityofeconomistshavedevoted
out growth of GDP of each country for a time interval. big efforts to the heterogeneous agent based models and
Surprisinglywhatweearnis differentfromthe onesthat emergent macroeconomics, see for example [21]-[28]. In
are reported in the system of national accounts (SNA). majorpartsoftheseworksshortrunbehaviorofthemar-
For works concerning paradoxical results see [7]-[11]. In ket is of interest. In our work however we mainly focus
these works both dependency of measurements on the on the long run behavior.
pathsofgrowthandaswellotherproblemssuchasmiss- While in the general theory of relativity, metric is
measurements have been discussed. Paradoxical results foundthroughthevariationofEinstein’s action,inecon-
in this direction sometimes have been called space time
3
omy the long run prices are addressed through the in- Same scenario goes for capital. Renting new capital is
teraction of different sectors and the hypothesis of equi- reasonableifextraproductioncompensatesfortheinter-
librium conditions. We shortly review production func- est rate and depreciation of capital.
tionandnotifythesourceofheterogeneityindynamicsof
pricesinthecontextoftheBaumol’scostdisease,see[29]- Pa∆Ya ∆Ka(r+δa) (12)
≥
[32].
in which r indicates the interest rate and δ indicates
a
This paper is organized as follows. In the following
depreciation of capital. In equilibrium we expect for all
sections we shortly review production function and the
sectors
Baumol’s cost disease phenomenon. Here we notify the
source of heterogeneity of dynamics of prices in the long ∂Ya
P =R +δ , (13)
run. In Sec. III we discuss path dependency of mea- a∂K c a
a
surements in economics. Besides, we introduce our toy
in which R is the net rate of return on capital. Classi-
model to produce a time series of prices for a bisector c
cally it was supposed that given the price of production
economy. Our time series is supplemented so that en-
in each sector, Eq. (11) indicates the level of wage of
thusiastic reader can perform his/her own analysis on
laborsin that sector. Itwas howevernotified by Baumol
measurements of the growthrate. In Sec. IV we provide
and Bowen in their influential paper that W should be
the main core of our work. In this section we borrow
the same for all sectors and it is the price of production
geometrical vision in the General Relativity to visualize
that should be adjusted in Eq. (11). The consequence
pathdependencyofmeasurementsinmacroeconomy. We
of the statement of Baumol and Bowen was important.
as well notify correspondence of concepts in the general
Since wage and rate of return on capital should be the
relativity with the mentioned problem. In Sec. V on the
sameforallsectors,thenequations(11)and(13)leaveus
basis of geometrical vision provided we explain how the
with a conclusion. Sectors which have relatively higher
gap between economy of China and the United States
ratesofgrowthinproductivityshouldloosetheirrelative
can be addressed and conclude our paper.
prices. It is because Y is an ever increasing function of
a
L orT . Let’scomparetwosectorswhichwecallsectors
a a
A and B. At a given time Eq. (11) states that
II. DYNAMICS OF LONG RUN PRICES IN A
HETEROGENOUS WORLD
∂Y (T L ,K ) ∂Y (T L ,K )
A A A A B B B B
P =P (14)
A B
∂L ∂L
A B
In neoclassical framework of economy, long run prices
and its relation to the wages and interest rate can be Now, if technology or actually the value of TA substan-
addressed through studying production functions which tially grows in sector A, then to keep the equity we
we shortly review in this section. Production function is need to relatively increase PB. So, in a heterogeneous
a function that indicates output of a firm in sector a as world, prices grow with heterogeneous rates. Baumol
andBowentherebyanticipatedthatpricesofproductions
Qa =Ya(TaLa,Ka) (8) or services in the stagnant sectors of the market should
grow comparing to the sectors which have high rate of
in which L indicates the number of employed labors,
a growth in technology. In 1900 each labor in agriculture
K states the value of invested capital, T stands for
a a sector could annually harvest say 1000 kg of carrot. At
productivity, and Q indicates the physical quantity of
a the same time a teachercouldteach25 students. So,if a
productioninthefirm. Thefunctionisaneverincreasing
farmer compensated 40 kg of carrot for teaching his sun
function of its variables. Besides, it is supposed to have
then we had a peace. In 2015 each labor may annually
diminishing return to capital. In other words we expect
harvest 50000 kg of carrot. Despite growth of produc-
ittobe a convexupfunction respecttocapital. Further,
tivity in agriculture sector,in education sector a teacher
we require the function to have constant return to scale
still canteachonly 25 students. So to keep balance,cost
of education for a farmer’s sun should be equal to 2000
Y (zT L ,zK )=zY (T L ,K ). (9)
a a a a a a a a
kgofcarrot. So,from1900to2015inflationineducation
For a manager it is reasonable to hire more labor if sector should have been much higher than agriculture
the value of addedproduction compensatesfor the extra sector.
wages or equivalently Since productivity in service sectors regularly do not
growasfastasmanufacturingsectorsthenitmeansthat
Pa∆Ya W∆La (10) therateofgrowthofpricesinservicesideshouldbemuch
≥
higher than industrial side. This fact is called the Bau-
in which P is the price of unit of production and W is
a mol’s cost disease phenomenon. The disease has been
the wage of each labor. In equilibrium in a competitive
observed in services in recent decades. In Baumol et al.
market we expect
2012[31]manyevidenceshavebeencited. Forexampleit
∂Y mentionsthataccordingtothereportofNationalCenter
a
P =W. (11)
a∂L forPublicPolicyandHigherEducationin2008[34],since
a
4
theearly80s,whilepricesinaveragehadgrownby110%, Solving equations (16) to (19) theoretically we can find
expenses for education had faced a growth as of 440%. L andtherebyY andperformaggregation. Thisiswhat
a a
Besides, expenses for health care had grown as of 250%. we will do in our toy model and measure GDP growth
If we notice that service sector consists a big portion of rates for different conditions.
sectors in economy,then an averageof 110%means that Before closing the section I should notify that our dis-
inflationforindustrialsideshouldhavebeenmuchbelow cussion is rough and wont work for the short run prices
this rate. Beside education and health care,higher rates at all. We ignored fluctuation of prices in the market.
of inflation has been observed in many other sectors in We as well ignored accumulation of capital which itself
service side. According to the report of the U.S. Bureau can be addressed in the context of endogenous growth
ofLabourStatistics2009athecostoffuneralserviceshas models. We simply supposed that R is the same for all
c
roughly doubled since 1987 to 2008[35]. As another ex- sectors which is not true in the short run. Despite being
amplewhileoverallinflationhasbeenaround3%,lawyers roughintheshortrun,thediscussedframeworkprovides
fee has grown around 4.5% between 1986 and 2008 [36]. acceptable approximationfor the long run relations over
Back to our discussion over production function we the prices and distribution of capital.
notify that constant return to scale (9) means that pro-
duction function can be written as
III. GDP GROWTH RATE; A PATH
K
Y (T L ,K )=L Y (T , a)=L y (T ,k ) (15) DEPENDENT MEASURE
a a a a a a a a a a a
L
a
inwhichy isproductionperlaborandk = Ka iscapital Ithasbeenknownsince longagothatmeasurementof
per labor.aNow, Eq. 13 reduces to a La thegrowthrateispathdependent. Actuallystudieshave
been more concerned with price index which is almost
∂ya the same problem. There is no way to define a practical
=R +δ , (16)
∂k c a measure for price index which passes a couple of tests
a
suchascircularitytest. Circularitytestmeansthatwhen
and Eq.(11) reduces to wemeasureachainedindexovertimeifatthebeginning
and the end of a period conditions are the same then we
P y =W +k (R +δ ). (17)
a a a c a expect our chained index be equal to one.
Same difficulty goes for measurements for the growth
As it can be seen L is eliminated from equations.
a rate as well. Suppose that we want to compare GDP of
Though prices and capital per labor are addressed in
two different years. Nominal GDP of the i year can be
equations (16) and (17), neither K nor Y can be iden- th
a a written as
tified. OnlygivenR wecanidentifyK /L ,Y /L ,and
c a a a a
Pa. For a more precise discussion over the matter in a GDP =Σ PiYi (20)
heterogeneous world see [33].To address aggregate pro- i a a a
duction we need to know La. Distribution of labors are in which Yai is the physical quantity of production in
givenfromdemandside. Indemandsideweneedtoknow the sector a and Pi is the price for units of production.
a
the utility preference of customers. They optimize their Aggregation should be performed over all sectors. Now,
satisfaction or equivalently maximize a utility function if we aim to compare this level of GDP to the one from
such as anotheryearsayj year,wecanwriteGDP =Σ PjYj.
th j a a a
∞ We then face a problem. Prices are subject to inflation
U = e−ρtu(t)dt (18) and comparing nominal GDP we may overestimate the
Z
t=t0 growth of the size of GDP. If all prices grew with the
same rate then we could simply divide GDP with the
in which u(t) is instant utility. If we forget dynamics j
size of inflation.
and accumulation of capital, we may write utility as
As it was discussed in the previous section, prices are
u(Y δ K ). This is a function that differs in differ-
a a a
− subject to heterogeneous growth rates. As a result ag-
ent countries and cultures. People in southern Europe
gregationof the realgrowthin a invariantmethod is im-
prefer to enjoy sitting in cafe or restaurantwhile Ameri-
possible. In 1939 Konüs defined a measure to be path
cansmayenjoymorehavingbiggercarsorbiggerhouses.
independent [37]. He supposed that all consumers do
So,evenifallproductionfunctions andvariablessuchas
have a utility preferences which sustains its functional
rate of return on capital is the same in these countries,
form over the time and space. Even if we accept such
the relative size of sectors is different. If we know the
hypothesis,it seems thatestimating suchfunction seems
utility function then we can maximize it through varia-
not to be an easy task. There has been efforts to opti-
tion over L . Actually we have Σ L = L . Lagrange
a a a t
mize some methods that may practically evaluate Konüs
method leads to equations which address L
a
index,seeOulton2008[6]. Inanycircumstance,national
∂u(Y δ K ) agencies usually use chained Paasche, Laspayres, Fisher
a a a
− =Constant. (19)
∂L orTörnqvistindexesnowadayswhicharepathdependent
a
5
measures and will be considered in this paper. growsfasterinthissectorandinthelateyearsproductiv-
To find GDP growthrate, usuallyGDP ofthe sequen- ity grows faster in sector B. Quantitatively we consider
tial years are compared with the same price base the following pattern for growth in productivities
g = ΣaPaiYΣai+a1Pa−iYΣaiaPaiYai, (21) TA1900+i =(1+0.06∗ 10099−i)∗TA1900+i−1 (23)
i+1
in which Yam is the production level in sector a in the TB1900+i =(1+0.06∗ 99 )∗TB1900+i−1.
m year and Pm is the price of unit of production in
th a
this sector. Inthe Middle Islandwesupposethatproductivitygrows
with a sustainable rate in both sectors
Now,weintroduceatoymodelandutilizingachained
growth rate we show that dependency of measurements Ti =(1.0305) Ti−1
when structural changes happens can be significant. We A ∗ A (24)
will see that for three countries with exactly initial and TBi =(1.0305)∗TBi−1 .
final conditions, perceptions for the real growth rate by
IntheSouthIslandwesupposethatdespiteNorthIsland
national agencies range from 2% to 3.5%. A reader who
inearlyyearsmosteffortsaredevotedtothesectorBand
isnotwillingtogetengagedintechnicaldiscussionsmay
productivity grows in this sector faster in early years
escapetherestofthecurrentsectionandjumptothefol-
lowingsectionwherewehaveprovidedgeometricalimag-
i+1
ing of the discussion. T1900+i =(1+0.06 ) T1900+i−1
A ∗ 99 ∗ A
Let’s supposethatwe havethreeIslandswhichwecall (25)
100 i
South, Middle, and North Island. Each Island has a T1900+i =(1+0.06 − ) T1900+i−1.
B ∗ 99 ∗ B
closed economy and in this closed economy they pro-
duce only one good to eat which we denote as good A Given these rates, if we maximize utility we find distri-
and provide only one service as B. Production functions bution of labors as
have Cobb-Douglas form (Y = TλLλN1−λ) and we set
fλoArm=aλsB = 2/3. We supposae thatautailitya function has a LA =LtλA+ΩλBλN0/+TΩA(λR1c−+λδAA)1−λAλA , (26)
A B
u=(YA N0)(YB)Ω, (22) LB =Lt−LA.
L − L
t t
For depreciation we set δ = %5.5. We kept rate of re-
a
in which we suppose N0 =1.6711 and Ω=5. Such util- turn on capital sustainable and equal to %5.5. We then
ity function means that people need a minimum value ran a simple code to carry out calculation concerning
of food, otherwise utility is negative. Once the mini- GDP real growth rates for each Island. Five CSV files
mum food which has been indicated by N0 is fulfilled have been supplemented named by gdpnorth.csv, gdp-
thenpeoplelookforserviceB. Whenproductivityislow middle.csv,andgdpsouth.csvwhichrepresentourresults.
the major portion of labors work in sector A. As long Eachfileisa99*4matrixwhereeachrowrepresents,YA,
as economy grows then the major of labors move to the PA, YB, and PB for of the related year from 1900 to
sector B since Ω = 5. It resembles structural changes 1998. As it can be seen in the file, initial and final level
in economy. In the early decades of the twenties century ofproductionandpricesarethesameforthreecountries.
we had a movement of labors from the agriculture sec- Now,wecanmeasuretherealgrowthrateforthesedata.
tors to the manufacturing sectors. In the recent decades When we use Eq. (21) then for the growth rate of each
as well we have observed substantial structural changes island we come to the results depicted in Fig. 1a. The
where economy has moved from an industrial base to a averageof the growth rate from 1900 to the desired year
service base. Actually structural changes happens in all has been depicted in Fig. 1b. As it be can bee seen
sectors in economy over time where each sector experi- as we approach 1998 all three countries approach to the
ences an Engel’s consumption cycle. Based on this cycle same final condition and we expect the average of the
eachgoodorserviceatthe beginning appearasa luxury real growth rate for the whole period merge to a unique
good with high income elasticity and in the end appears value. ItiswhileinFig. 1bweobservethatintheendof
as necessity with a low elasticity. the period, the average of the real growth for the whole
In our toy model we suppose that total number of periodis%3.5inNorthIsland,%3inMiddleIslandand
labors in each Island is 100000. Concerning productiv- %2.1inSouthIsland. Whatasurprise! Whileinitialand
ity we suppose that in 1900 T = T = 1 and in 1998 final conditions are the same for all three countries, the
A B
T = T = 18.93. Though initial and final conditions governing parties in the North Island are proud of their
A B
arethesameforallIslands,wesupposethatintheinter- performanceforthepastcenturywhilethegoverningpar-
mediate yearseachislandfollowsits ownR&Dprogram. ties in South Island should be ashamed for the actually
WesupposethatinNorthIslandinearlyyearsabigdeal thesameresults. Tomakesurethatdifferentindexescan
ofattentions is devotedto the sector Aand productivity not influence the paradoxicalresult in our toy model we
6
Measurments for GDP growth rate in three islands Average of real growth rate in all three islands since 1900 to the desired year
0.06 0.06
North Island
North Island
Middle Island
Middle Island
South Island
South Island
0.05 0.05
0.04 0.04
Real growth rate0.03 Real growth rate0.03
0.02 0.02
0.01 0.01
0 0
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
Year Year
(a) (b)
Figure 1: a) Real growth rate in each Island. For each year the previous year has been considered as the base year.
b) The averageof growthof eachisland since 1900to the desired yearhas been calculated. Each country has its own
pattern of growth and thereby the average of growth should vary in middle years. The initial and final size of
economies are however the same. So, we expect that the averageof real growth in the end of the period to be the
same for all countries. As it can be seen the average of the growth for three islands do not merge as their final
conditions merge in 1998.
Besides the mentioned csv files, two other files have
Comparing Laspayres VS Paasche Indexed Measurement for the Real Growth
0.06 been submitted as "northconstant.csv" and "southcon-
Laspayres
Paasche stant.csv". Time series in these files are even more in-
0.055
teresting than the former files. Two islands start with
exactly initial conditions. One of them annually experi-
0.05
encesarealgrowtharound3.7%andofthemexperiences
Rate0.045 anannualgrowtharound2.2%(Imeantheannualgrowth
wth for every year within the period and not the average of
Real Gro 0.04 tfehreenacnensu,atlhgerioswlatnhd.s).fiDnaelslpyiteendthuispawnintuhaelxsaucbtslytatnhteiasladmife-
0.035 economy.
Besides the prices and measurements there are some
0.03 other interesting issues which can be discussed in the
growth models with the glass of path dependency. For
0.025
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 someinteresting discussionssee[38]. Actually the model
Year
discussed here and the Fig. 4 have been first presented
(a) and discussed with more details there. Though the
mentioned reference is more concerned with structural
Figure 2: Measurement for the real growth rate based changes, the current paper is more concerned with geo-
on different inflation indexes. As it can be seen a result metrical imaging of the growth rate.
from a Laspayres index is very close to the Paasche
index in the North Island. So, choosing different
indexes can not relax the problem with measurements.
IV. GEOMETRICAL UNDERSTANDING OF
THE GROWTH RATE
considered both Laspayresand Paasche index and recal- We obtained surprising results concerning measure-
culated the real growth rate. As it can be seen in Fig 2 ments for the growth rate in our toy model. Now, let’s
the difference is negligible. Since Fisher and Törnqvist recall a similar problem in physics. We suppose that
indexs are between Laspayres and Paasche indexes then we have two astronauts in two spacecrafts. They start
itmeansthatnoneoftheseindexescannotinfluencethe a trip from one point in one side of a star to the other
result noticeably. side as depicted in Fig. 3. In Fig. 3 they start their
7
axisshowsproductionlevel. Redgridhoweverrepresents
dollar values. Distance between parallel lines in the red
grid represents 10$. If you move along haircut direction
your influence in GDP is bigger from moving in burger
direction. One step along haircut direction passes more
thantwolinesandhasaninfluencemorethan20$. Along
burger direction however you need to have more than 7
steps to have such effect in the growth of GDP.
InIranpriceforbothburgerandmenhaircutisaround
3$. Inthis case,inspite ofthe US movingalonghaircut
direction has not a bigger influence on GDP size. While
basedonthe meter ofthe US BureauofEconomic Anal-
ysis(BEA) distancebetweenAto Cis muchlongerthan
A to B, in Iran based on the meter of the central bank,
A to C is shorter than A to B.
For our toy model, each country in each year has its
own technology and productivity level. Relative prices
are different in each country. Though each of three is-
lands have the same initial and final conditions, since
theirpathsaredifferentinthemiddleagestheirmeasure
for the growth during the period of 99 years is quite dif-
ferent, see Fig. 4c. Metric to aggregate production and
Figure 3: Near a star metric for people who are at measure the real growth is different in each country.
circumference is different from who are close to the star
Itshouldbenotedthatinthismodelwesupposedthat
in diameter. One astronaut moves along the
all Islands have the same utility preference and thereby
circumference from A to D and measure it. Another
whenever their technologies were the same their produc-
astronaut moves along diameter from A to D to
tions were the same as well. So, they ended up in the
measure distance. If he is not cooked in the middle of
samepointinthespaceofproduction. Inrealworldhow-
the star and meet the other astronaut at point D
everineachcountrypeoplehavetheirownutilityprefer-
surprisingly finds that for radius and circumference we
ences. Asaresulteveniftheystartfromthesameinitial
have S <2ΠR
conditions they wouldfollow different paths in the space
of productions and they barely may meet each other in
this space.
tripssimultaneouslyfrompointA.Oneofthespacecrafts
If we aim to list the counterparts of elements of the
movesalong the diameter andthe other spacecraftmove
generalrelativitytothecurrentdiscussionswecancount
along the circumference from A to D. They have differ-
as follows.
ent speeds and manage to end their trip simultaneously
atpointD.Duringtheirtriptheymeasurebothdistance
- In the general relativity metric, denoted by g
and time of their trip. When they share their measure- µν
helps to measure distances in the space time along a
mentstheyobservethatsurprisinglytherelationbetween
radius and circumference is no longer S = 2ΠR. Beside curve c through the relation
of the distance they disagree onthe duration of the trip.
Neitheroftheastronautshasmademistake. Thepointis d(x ,x )= g dxµdxν. (28)
i f Z µν
thatspacebecomescurvedbygravity. InFig. 3distance cp
between each line in the grid shows the same value say
In economics prices are our meters and the gap between
one kilometer.
economies can be addressed via
ThestoryaboutmeasurementsforGDPgrowthrateis
exactly similar to the problems with astronauts. When ∆GDP = Σ P dY . (29)
Z a a a
we measure the real growth rate c
P
g =dlogGDP =Σ a dY , (27) -Inthegeneralrelativitymetricisobtainedthroughvari-
a a
Σ P Y
b b b ation of the Einstein-Hilbert action
foranychangesinproductionweusepricesasourmeters.
1
Let’s look at it from a geometrical point of view. S = ddxdt√ g(R+matterterm). (30)
16π Z −
In New York city price for a McDonald’s "Triple
Cheeseburger"is 3$. Let’s suppose that the averageof a In economics, prices are addressed through interactions
haircutfor menis about21$. Now wecandrawaframe- of sectors and actually equations (16) and (17). Though
workasofFig. 4a. Inthisframeworkdistancefromeach rate of return or real interest rate is addressed from the
8
Metric in NY Metric in Tehran Pattern of Growth in Three Islands in Our Toy Model
12 12
1101 B 1101 B
9 9 South Island
Burger45678 C Burger 45678 C Production in sector B Middle Island North Island
3 A 3 A
2 2
1 1
00 1 2 3 4 5Hair6 Cut7 8 9 10 11 12 00 1 2 3 4 5Hair6 Cut7 8 9 10 11 12 Production in sector A
(a) (b) (c)
Figure 4: a) Price for a Mc’Donalds burger in NY is about 3$ and suppose that averageprice for men haircut is 21$.
In each axis distance from origin shows the number of burgers and haircuts. Grids however show dollar values.
Distance between parallel red lines in either directions is 10$. In NY money value of distance of A to C is much
greater than distances of A to B. b)In Iran services are much cheaper and distance of A to C is smaller than A to B.
When defining money value ofdistances inthe spaceofproductions,the relateddata providerofeachcountryhas its
own meter in each direction. So, while in Iran moving from A to B is recognized to have a bigger effect on the real
GDP, in the US moving from A to C has a bigger effect on the real growth. c)In our problem for three islands, each
islandmovesalongits ownpath andnationalagencieshavetheir ownmeters to measuredistance between initialand
final points. So what they measureas the realgapbetween initial andfinal sizes ofGDP is different fromeachother.
utility preference. V. THE GAP BETWEEN ECONOMIES OF
CHINA AND THE UNITED STATES
- In the general relativity objects move along geodesics
and the extremums of distances in Eq. (28). In eco-
Before paying attentionto the gap betweeneconomies
nomics, paths of countries in the space of production
of China and the US let’s recall paradoxicalresults con-
can be addressedthroughmaximizing utility preferences
cerning Pen World Table and International Comparison
in equations (19) and (18).
Program. When you use purchasing power parity ex-
change rate to measure GDP of different countries, it
Mathematical precision
means that for all of them you have used the same met-
ric to measure their GDP, see Fig. 5a. The price base in
Before closing this section I should notify that finding
this method is standard worldwide. In developed coun-
distance between points in economics and physics are
triessuchastheUSsinceproductivityisveryhigh,wages
conceptually similar. Though we borrowed the concept
are relatively high and services are relatively more ex-
ofmetricfromthe generalrelativityto notifythe impact
pensive. This is while in poor countries relative prices
of measurements by different observers, the equations
for services are low comparing to the US. So, there will
we reviewedhoweverdo notmeet the true mathematical
be problem when we aim to compare growth rate of dif-
definition of metric. Equation (29) resembles a one
ferent countries. Systems of national account tries to
form. Mathematically it resembles the work done on an
provide a standard basis worldwide. Currently however
object in a force field. The work done along each path
many countries such as the US or China do not follow
however depends on the path since P changes when
a such system. These countries use their own price base.
level of productions changes. Though mathematically
From the generalrelativity point of view, national agen-
the problem is closer to the problems with work and
ciesresemblefallingobserversinacurvedspace-timewho
energy, the concepts of measurements and meters for
have their own meters. It is while international agencies
aggregation, dependency of measurement to the path,
have standard meters and resemble an outside observer.
and dependency of paths to the optimizations and local
Though system of national accounts and PPP exchange
extremums are conceptually close to the observations in
ratesaimtoprovideabetterinternationalbasisforaggre-
the general relativity.
gation. Still, space time inconsistency is observed. For
more studies see [7]-[14].
Back to our problem with China it is now clear that
if we even know the current size of the GDP of both
China and the US and besides we know their future
growth rates we still are not able to examine the time
9
Metric of International Agencies vs National Agencies US vs China Growth or Inflation

China, Future!
US (Future)
USA
US (Now)
Manufacturing Sector China Tradable goods China (Future) Tradable goods China, Now !
China (Now)
Iran
Y Y
0 1

Service Sector Non−tradable goods Non−tradable goods
(a) (b) (c)
Figure 5: a) While each country has its own prices or its own grid to measure different movements in real GDP, a
PPP based measurement (red grid) aims to have a standard metric to measure GDP of different countries in the
space of productions. b) The US and China have their own meter to measure GDP. While their meter to measure
tradable goods are close, their meter to measure non-tradable goods are different. Each country pass its own path in
the space of production and their meter specially along the non-tradable goods changes. c) As GDP in China grows,
its meter along non-tradable goods changes and become closer to the ones for developed countries. While via new
girding (the blue one), the nominal value of Y0 has grown, from China NBS perspective only (Y1 Y0)P0 is the real
−
growth. From international perspective howevermeter of China to measure non-tradable goods has become closer to
the international meter and the growth in the volume in non-tradable goods looks real. In other words from
international perspective the volume of GDP of China has grown as (Y1 Y0)P1 Y0P0.
− −
that economy of China catches the economy of the we think of Baumol’s cost disease and real observations,
US. To extrapolate the time that China needs to pass we find that prices for services are relatively cheaper in
economy of the US besides knowing the future growth developing countries.
rates of both countries we need to know the path that Manufacturing sectors have good overlap with trad-
each country passes through. We need to find this path ablegoods. Itiswhileservicesectorshavebetteroverlap
through production functions and utility preferences with non-tradable goods. As time goes by, according to
and the level of capital in China and the US and then the cost disease and observations, relative prices of non-
find local metric along this path and thereby find out tradable goods increase. As a result girding of countries
the desired time. Since utility preferences in China is along horizontal axis will be more dense. Let’s look at
differentfromthe USforsurethe finalshapeofeconomy the case of China in Fig. 5c. What will happen then?
of each country will be different and thereby China
Beforetheperiod,productionofbothcountriesintrad-
should not meet the US in the space of production, see
ablegoodshavehadcloseprices. Innon-tradablesectors
Fig. 5b. If we know the path and local metric however
howeversincepricesinChinahavebeensmallerthanthe
we can extrapolate the time that nominal sizes of their
US, then nominalsize ofthese sectorsin China has been
GDP are the same.
smaller respect to the US one.
When economy of China growsalong tradable axis we
How dose it work?
have no problem. Tradable goods are griddedworldwide
roughly with the same price. The problem is with non-
One may argue that we have nominal GDP of China
tradable goods or in some senses services. For these sec-
and of the US. We have their reported real and nom-
tors,afterawhilepricesofservicesinChinawillbecloser
inal growths. How then Eq. (1) does not hold? For
to the prices of the US if we believe in the cost disease.
sure the equation holds for nominal rates. The problem
What would happen then? For these sectors according
ariseswhenweaimtoextractrealgrowthfromthe nom-
to the National Bureau of Statistics of China (NBS) we
inal growth. Each country reports its growth rate and
have a growth in GDP size as:
then some day their nominal GDP sizes should be the
same. The geometricalfigure howeveris different. Trad- ∆GDP =Y1Pf Y0Pi. (31)
−
able goods have roughly the same prices at border of
both countries, see Fig. 5b. Non-tradable goods do not Thisis nominalshareofgrowthfornon-tradablesectors.
have the same price however. So, while girding of both TheNBShoweverreportpartofthisgrowthasinflation.
countries along vertical direction roughly has the same According to the NBS the real growth is
length, their girding along horizontal axis is different. If
∆GDP =(Y1 Y0)Pi. (32)
−
10
Whatdoesitmean? Chinaconsidersonlythepartofthe the US BEA. From an international perspective volume
growth that is real in this direction for the real growth of GDP of China has become closer to the US one by
rate. What about international comparisons. In inter- 5.2%despite reports of the national agenciesfor the real
national comparisons the nominal share of non-tradable growths.
goodshasgrownandfrominternationalperspectiveecon- Two conclusions can be deduced from our discussions:
omy of China has grown. At the end of the period when The first conclusion is that extrapolating the real time
GDPofChinagrowsandChinafacesinflationalongnon- thatnominalGDPofChinawillbeequalto the nominal
tradable goods, its price gets closer to the US one, nom- GDP of the US is a hard task even if we know their real
inal volume of the non-tradable goods grows. Though growths. To find the proper answer we need to know
China considers parts of these growth as inflation, but both the relative growth of prices as a result of the cost
since prices for these sectors have really become closer diseaseandastheutilitypreferences. Theproblemholds
to the US one, and nominal GDP of China has become with both nominal and PPP values of GDP sizes.
closer to the US one, from international comparisons it
We have two forms of variation in prices. One form
is a real growth.
of variation is the regular fluctuations of prices in the
In international comparisons tradable goods have market as a result of short time interaction in supply
worldwide prices. Now that volume of services in China and demand curves. The other form of variation is the
grows, then nominal GDP of China will be closer to the growth in prices in service side as a result of the cost
US one. It does not matter if China percepts it as infla- disease which is permanent as long as we can not find a
tion or real growth rate. To clarify it I have provided a technology to grow productivity. Though from national
numerical example in table I. perspective growthinprices in servicesectoris inflation,
I have supposed that the US and China produce only from international perspective it is a permanent growth
one good and provide only one service. Prices and pro- of volume of service sector and a permanent growth of
ductionsareasthelevelinthetable. Inthishypothetical the volume of GDP.
model from 2015to 2016 nothing changes in the US. So, Prices for non-tradable goods are lower in developing
both inflation and growth experience a zero value rates. countries. If developing countries experience fast growth
In China tradable good A has the same value as of the and relative prices for their services grow faster than
US. Service B however experiences a relative growth in manufacturing side, then this growth only make nomi-
prices in China since economy in this country is growing nal GDP for them closer to the PPP values. Currently
faster than the US. For this case the China NBS reports PPPvalueofGDPofChinaisbiggerthantheUS.When
a real growth as of 3.9%. According to the NBS part GDP of China expands we expect prices in the service
of the growth of the volume in service sector is inflation sideinChinabecomeclosertothe pricesinthe US.As a
ratherthanbeingarealgrowth. So,Chinareportanan- result,nominalsizeofGDPofChinabecomeclosertoits
nualinflationequalto1.3%. So,ifweconsiderreportsof PPP value. So,the secondconclusionis thatifpricesfor
the NBS we should conclude that economy of China has services in China become closer to the US ones then its
become closer to the economy of the US by 3.9%. If we economybecome closerto the economyofthe US sooner
look from an international perspective things are differ- than what an equation such as Eq. (1) suggests.
ent. Growth of prices in service sector in China is not a
fluctuation. Itispermanentandisaresultofthe growth
and the cost disease. Volume of economy of China has ACKNOLEDGMENTS
grownby 5.2% from an international perspective. Meter
of the NBS of China has become closer to the meter of
I would like to thank GR Jafari for comments on an
earlier draft and H Arfaei for discussion.
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