A geometrical imaging of the real gap between economies of China and the United States
aa r X i v : . [ q -f i n . GN ] A p r A geometrical imaging of the real gap between economies of China and the UnitedStates
Ali Hosseiny
1, 2 Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839, Iran School of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM)P.O.Box 19395-5531, Tehran, Iran ∗ (Dated: April 25, 2019)GDP of China is about 11 trillion dollars and GDP of the United States is about 18 trillion dollars.Suppose that we know for the coming years, economy of the US will experience a real growth rateequal to %3 and economy of China will experience a real growth as of %6. Now, the question ishow long does it take for economy of China to catch the economy of the United States. The earlyimpression is that the desired time is the answer of the equation × . X = 18 × . X . The correctanswer however is quite different. GDP is not a simple number and the gap between two countriescan not be addressed simply through their sizes. It is rather a geometrical object. Countries passdifferent paths in the space of production. The gaps between GDP of different countries depend onthe path that each country passes through and local metric. To address distance between economiesof China and of the US we need to know their utility preferences and the path that China passes toreach the US size. The true gap then can be found if we calculate local metric along this path. Itresembles impressions about measurements in the General Theory of Relativity. Path dependencyof aggregate indexes is widely discussed in the Index Number Theory. Our aim is to stick to thegeometrical view presented in the General Relativity to provide a visiual understanding of thematter. We show that different elements in the general relativity have their own counterparts ineconomics. We claim that national agencies who provide aggregate data resemble falling observersinto a curved space time. It is while the World Bank or international organizations are outsideobservers. The vision provided here, leaves readers with a clear conclusion. If China keeps itsgrowth rate, then the economy of China should catch the economy of the United States sooner thanwhat we expect. I. INTRODUCTION
Suppose that you want to extrapolate the time thateconomy of China needs to become as big as the economyof the United States. The current size of economy ofChina is about 11 trillion dollars and the current size ofeconomy of the US is 18 trillion dollars. Now, supposethat we are sure that economy of China in coming yearswill experience a real growth as of %6 and economy ofthe US experiences a %3 rate. Then, the question is howmany years does it take for economy of China to catcheconomy of the US. Our impression is that we can simplyfind the answer through solving the equation × . X = 18 × . X ⇒ X = ln (18 / ln (1 . / . ≈ years. (1)This answer is however untrue. GDP is not a simplenumber. To find GDP we aggregate a set of productionsthrough their prices. In other words we have GDP = Σ a P a Y a , (2)in which a ranges for any form of final goods or servicesprovided in a country, Y a is the quantity of each good and ∗ [email protected], [email protected] P a is its price. Though the output of the aggregation is asingle number, this number is not a regular number. Thefact is that prices are our meter to perform aggregation.Elements of this meter however vary with heterogenousrates over time.Difficulty now arises if we aim to compare GDP of twodifferent countries. Even for a single country we havedifficulty when we aim to compare GDP of different years.Problem is as follows. Nominal GDP in the year i is GDP i = Σ a P ia Y ia , (3)and nominal GDP in the consecutive year is GDP i +1 = Σ a P i +1 a Y i +1 a . (4)If we decide to compare both GDPs and find growth rate g nominal = GDP i +1 GDP i − a P i +1 a Y i +1 a Σ a P ia Y ia − , (5)we face problem. The problem is that GDPs of differentyears have been measured with different set of prices.As a result, inflation may result in an overestimation ofthe measurement. So, we should find which part of thenominal growth is real growth and which part is inflation.The usual method is to compare the GDP of the con-secutive years with the same set of prices and define thereal growth as g = Σ a P ia Y i +1 a Σ a P ia Y ia − . (6)Now, it seems that we have overcome the problem withinflation. We however have created another problem.The definition of the real growth in Eq. (6) fails thecircularity test.Let us consider a time series of prices P ia and produc-tions Y ia over a period of m years. For each year we findthe real growth through Eq. (6). If at the beginning andthe end of the period production and prices are the samethen it means that GDP have not grown over the period.So, we expect Σ i + mj = i g j ≈ . (7)This trivial requirement however is dismissed in our def-inition. What is the consequence? The consequence ispath dependency of measurements. I have supplementeda couple of csv files. In one of them I have produced atime series of prices and production for a bi-sector econ-omy. Two countries start with exactly initial conditions.Over a century, one of the countries experiences a realgrowth annually equal to . and another country ex-periences a real growth equal to . . Despite this hugedifferences, both countries end with exactly the sameconditions and exactly the same set of production andprices. Guess what! Despite their equal performance forthe whole period, the governing party of the first coun-try is proud and celebrates a century of annual growthof . and the governing party of the second country isashamed.Dependency of measurements to the paths first at-tracted attention of Fisher in price index problem in 1927[1]. He noticed if we aim to measure inflation then ourmeasurement can be path dependent. Finding a propermethod to measure inflation then was under attention fordecades. Some influential movements in indexing wereaddressed by Laspeyres, Paasche, Törnqvist, and Fisher.For recent works see Balk 1995 [2], Diewert 1976 [3] and1993 [4], Feenstra and Reinsdorf 2000 [5] and Oulton 2008[6].In recent years paradoxical results concerning evalua-tion of GDP reported by international organizations suchas the World Bank have reraised attentions. In Interna-tional Comparison Program (ICP) and Penn World Ta-ble, GDP of each country is evaluated through purchasingpower parity (PPP) exchange rate. When we calculateGDP of different countries through PPP exchange rate itmeans that we have used a standard price for each goodworldwide. In other words we have used the same me-ter to perform aggregation in Eq. (2) for all countriesand measure the size of GDP of them. We then can findout growth of GDP of each country for a time interval.Surprisingly what we earn is different from the ones thatare reported in the system of national accounts (SNA).For works concerning paradoxical results see [7]-[11]. Inthese works both dependency of measurements on thepaths of growth and as well other problems such as miss-measurements have been discussed. Paradoxical resultsin this direction sometimes have been called space time inconsistency (see Feenstra, Ma and Prasada Rao 2009[12], Feenstra, Inklaar and Timmer 2013 [13], and Oulton2015 [14]) where by space people mean measurements indifferent countries.In this paper we borrow the concept of measurementand metric in the General Relativity to understand theproblem. Though path-dependency of measurements forthe growth rate is an old known fact in economy andhas being discussed in the Index Number Theory, it hasbeen barely known outside of economics literature. Evenin the world of economy it is not discussed in regulartextbooks. The geometrical vision provided in this pa-per helps to clearly understand why GDP is not a simplenumber and why measurement for the growth rate is pathdependent. It then can be easily visioned and understoodby researches in different areas of sciences. Previously,interesting efforts have been devoted to find some invari-ant measures for aggregate indexes through an analysisof the gauge invariance of economic models, see Malaneyand Weinstien1996 [15], and Smolin 2009[16]. Our workhowever relies on geometrical imaging of the problemand mainly notifies some similarities between objects andconcepts in the general theory of relativity and the indexnumber theory.Metric in the general relativity is addressed by Ein-stein’s action. In economics, equilibrium constraintsand production functions address the relation of prices.While in the general relativity objects move alonggeodesics, in economics, utility preferences address thepath of growth in the space of productions. We discussthat national agencies who measure GDP resemble fallingobservers in a curved space-time and have their own met-ric to calculate their growth rate. It is while the WorldBank is an outside observer which uses a standard metricfor all of these countries. Besides, we discuss that to ad-dress the gap of economy of China and of the US we needto know production functions and utility preferences tofind their paths in the space of production. Then movingalong this path we can find out the time it needs for econ-omy of China to catch economy of the US. The overallimpression is that China may catch economy of the USsooner than what we expect.Besides problem with measurement we shortly reviewthe cost disease phenomenon and its role to address amajor feature of heterogeneity of dynamics of long runprices. In econophysics we usually have been interestedin short run prices and its volatility. A vast range ofworks have been dedicated to the price in the stock mar-ket, see for example [17]-[21]. Beside stock market, physi-cists along with a community of economists have devotedbig efforts to the heterogeneous agent based models andemergent macroeconomics, see for example [22]-[30]. Inmajor parts of these works short run behavior of the mar-ket is of interest. In our work however we mainly focuson the long run behavior.While in the general theory of relativity, metric isfound through the variation of Einstein’s action, in econ-omy the long run prices are addressed through the in-teraction of different sectors and the hypothesis of equi-librium conditions. We shortly review production func-tion and notify the source of heterogeneity in dynamics ofprices in the context of the Baumol’s cost disease, see[31]-[34].This paper is organized as follows. In the followingsections we shortly review production function and theBaumol’s cost disease phenomenon. Here we notify thesource of heterogeneity of dynamics of prices in the longrun. In Sec. III we discuss path dependency of mea-surements in economics. Besides, we introduce our toymodel to produce a time series of prices for a bisectoreconomy. Our time series is supplemented so that en-thusiastic reader can perform his/her own analysis onmeasurements of the growth rate. In Sec. IV we providethe main core of our work. In this section we borrowgeometrical vision in the General Relativity to visualizepath dependency of measurements in macroeconomy. Weas well notify correspondence of concepts in the generalrelativity with the mentioned problem. In Sec. V on thebasis of geometrical vision provided we explain how thegap between economy of China and the United Statescan be addressed and conclude our paper. II. DYNAMICS OF LONG RUN PRICES IN AHETEROGENOUS WORLD
In neoclassical framework of economy, long run pricesand its relation to the wages and interest rate can beaddressed through studying production functions whichwe shortly review in this section. Production function isa function that indicates output of a firm in sector a as Q a = Y a ( T a L a , K a ) (8)in which L a indicates the number of employed labors, K a states the value of invested capital, T a stands forproductivity, and Q a indicates the physical quantity ofproduction in the firm. The function is an ever increasingfunction of its variables. Besides, it is supposed to havediminishing return to capital. In other words we expectit to be a convex up function respect to capital. Further,we require the function to have constant return to scale Y a ( zT a L a , zK a ) = zY a ( T a L a , K a ) . (9)For a manager it is reasonable to hire more labor ifthe value of added production compensates for the extrawages or equivalently P a ∆ Y a ≥ W ∆ L a (10)in which P a is the price of unit of production and W isthe wage of each labor. In equilibrium in a competitivemarket we expect P a ∂Y a ∂L a = W. (11) Same scenario goes for capital. Renting new capital isreasonable if extra production compensates for the inter-est rate and depreciation of capital. P a ∆ Y a ≥ ∆ K a ( r + δ a ) (12)in which r indicates the interest rate and δ a indicatesdepreciation of capital. In equilibrium we expect for allsectors P a ∂Y a ∂K a = R c + δ a , (13)in which R c is the net rate of return on capital. Classi-cally it was supposed that given the price of productionin each sector, Eq. (11) indicates the level of wage oflabors in that sector. It was however notified by Baumoland Bowen in their influential paper that W should bethe same for all sectors and it is the price of productionthat should be adjusted in Eq. (11). The consequenceof the statement of Baumol and Bowen was important.Since wage and rate of return on capital should be thesame for all sectors, then equations (11) and (13) leave uswith a conclusion. Sectors which have relatively higherrates of growth in productivity should loose their relativeprices. It is because Y a is an ever increasing function of L a or T a . Let’s compare two sectors which we call sectors A and B . At a given time Eq. (11) states that P A ∂Y A ( T A L A , K A ) ∂L A = P B ∂Y B ( T B L B , K B ) ∂L B (14)Now, if technology or actually the value of T A substan-tially grows in sector A , then to keep the equity weneed to relatively increase P B . So, in a heterogeneousworld, prices grow with heterogeneous rates. Baumoland Bowen thereby anticipated that prices of productionsor services in the stagnant sectors of the market shouldgrow comparing to the sectors which have high rate ofgrowth in technology. In 1900 each labor in agriculturesector could annually harvest say 1000 kg of carrot. Atthe same time a teacher could teach 25 students. So, if afarmer compensated 40 kg of carrot for teaching his sunthen we had a peace. In 2015 each labor may annuallyharvest 50000 kg of carrot. Despite growth of produc-tivity in agriculture sector, in education sector a teacherstill can teach only 25 students. So to keep balance, costof education for a farmer’s sun should be equal to 2000kg of carrot. So, from 1900 to 2015 inflation in educationsector should have been much higher than agriculturesector.Since productivity in service sectors regularly do notgrow as fast as manufacturing sectors then it means thatthe rate of growth of prices in service side should be muchhigher than industrial side. This fact is called the Bau-mol’s cost disease phenomenon. The disease has beenobserved in services in recent decades. In Baumol et al.2012 [33] many evidences have been cited. For example itmentions that according to the report of National Centerfor Public Policy and Higher Education in 2008 [36], sincethe early 80s, while prices in average had grown by 110%,expenses for education had faced a growth as of 440%.Besides, expenses for health care had grown as of 250%.If we notice that service sector consists a big portion ofsectors in economy, then an average of 110% means thatinflation for industrial side should have been much belowthis rate. Beside education and health care, higher ratesof inflation has been observed in many other sectors inservice side. According to the report of the U.S. Bureauof Labour Statistics 2009a the cost of funeral services hasroughly doubled since 1987 to 2008[37]. As another ex-ample while overall inflation has been around 3%, lawyersfee has grown around 4.5% between 1986 and 2008 [38].Back to our discussion over production function wenotify that constant return to scale (9) means that pro-duction function can be written as Y a ( T a L a , K a ) = L a Y a ( T a , K a L a ) = L a y a ( T a , k a ) (15)in which y a is production per labor and k a = K a L a is capitalper labor. Now, Eq. 13 reduces to ∂y a ∂k a = R c + δ a , (16)and Eq.(11) reduces to P a y a = W + k a ( R c + δ a ) . (17)As it can be seen L a is eliminated from equations.Though prices and capital per labor are addressed inequations (16) and (17), neither K a nor Y a can be iden-tified. Only given R c we can identify K a /L a , Y a /L a , and P a . For a more precise discussion over the matter in aheterogeneous world see [35].To address aggregate pro-duction we need to know L a . Distribution of labors aregiven from demand side. In demand side we need to knowthe utility preference of customers. They optimize theirsatisfaction or equivalently maximize a utility functionsuch as U = Z ∞ t = t e − ρt u ( t ) dt (18)in which u ( t ) is instant utility. If we forget dynamicsand accumulation of capital, we may write utility as u ( Y a − δ a K a ) . This is a function that differs in differ-ent countries and cultures. People in southern Europeprefer to enjoy sitting in cafe or restaurant while Ameri-cans may enjoy more having bigger cars or bigger houses.So, even if all production functions and variables such asrate of return on capital is the same in these countries,the relative size of sectors is different. If we know theutility function then we can maximize it through varia-tion over L a . Actually we have Σ a L a = L t . Lagrangemethod leads to equations which address L a ∂u ( Y a − δ a K a ) ∂L a = Constant. (19) Solving equations (16) to (19) theoretically we can find L a and thereby Y a and perform aggregation. This is whatwe will do in our toy model and measure GDP growthrates for different conditions.Before closing the section I should notify that our dis-cussion is rough and wont work for the short run pricesat all. We ignored fluctuation of prices in the market.We as well ignored accumulation of capital which itselfcan be addressed in the context of endogenous growthmodels. We simply supposed that R c is the same for allsectors which is not true in the short run. Despite beingrough in the short run, the discussed framework providesacceptable approximation for the long run relations overthe prices and distribution of capital. III. GDP GROWTH RATE; A PATHDEPENDENT MEASURE
It has been known since long ago that measurement ofthe growth rate is path dependent. Actually studies havebeen more concerned with price index which is almostthe same problem. There is no way to define a practicalmeasure for price index which passes a couple of testssuch as circularity test. Circularity test means that whenwe measure a chained index over time if at the beginningand the end of a period conditions are the same then weexpect our chained index be equal to one.Same difficulty goes for measurements for the growthrate as well. Suppose that we want to compare GDP oftwo different years. Nominal GDP of the i th year can bewritten as GDP i = Σ a P ia Y ia (20)in which Y ia is the physical quantity of production inthe sector a and P ia is the price for units of production.Aggregation should be performed over all sectors. Now,if we aim to compare this level of GDP to the one fromanother year say j th year, we can write GDP j = Σ a P ja Y ja .We then face a problem. Prices are subject to inflationand comparing nominal GDP we may overestimate thegrowth of the size of GDP. If all prices grew with thesame rate then we could simply divide GDP j with thesize of inflation.As it was discussed in the previous section, prices aresubject to heterogeneous growth rates. As a result ag-gregation of the real growth in a invariant method is im-possible. In 1939 Konüs defined a measure to be pathindependent [39]. He supposed that all consumers dohave a utility preferences which sustains its functionalform over the time and space. Even if we accept suchhypothesis, it seems that estimating such function seemsnot to be an easy task. There has been efforts to opti-mize some methods that may practically evaluate Konüsindex, see Oulton 2008 [6]. In any circumstance, nationalagencies usually use chained Paasche, Laspayres, Fisheror Törnqvist indexes nowadays which are path dependentmeasures and will be considered in this paper.To find GDP growth rate, usually GDP of the sequen-tial years are compared with the same price base g = Σ a P ia Y i +1 a − Σ a P ia Y ia Σ a P ia Y ia , (21)in which Y ma is the production level in sector a in the m th year and P ma is the price of unit of production inthis sector.Now, we introduce a toy model and utilizing a chainedgrowth rate we show that dependency of measurementswhen structural changes happens can be significant. Wewill see that for three countries with exactly initial andfinal conditions, perceptions for the real growth rate bynational agencies range from 2% to 3.5%. A reader whois not willing to get engaged in technical discussions mayescape the rest of the current section and jump to the fol-lowing section where we have provided geometrical imag-ing of the discussion.Let’s suppose that we have three Islands which we callSouth, Middle, and North Island. Each Island has aclosed economy and in this closed economy they pro-duce only one good to eat which we denote as good A and provide only one service as B . Production functionshave Cobb-Douglas form ( Y a = T λa L λa N − λa ) and we set λ A = λ B = 2 / . We suppose that utility function has aform as u = ( Y A L t − N )( Y B L t ) Ω , (22)in which we suppose N = 1 . and Ω = 5 . Such util-ity function means that people need a minimum valueof food, otherwise utility is negative. Once the mini-mum food which has been indicated by N is fulfilledthen people look for service B . When productivity is lowthe major portion of labors work in sector A . As longas economy grows then the major of labors move to thesector B since Ω = 5 . It resembles structural changesin economy. In the early decades of the twenties centurywe had a movement of labors from the agriculture sec-tors to the manufacturing sectors. In the recent decadesas well we have observed substantial structural changeswhere economy has moved from an industrial base to aservice base. Actually structural changes happens in allsectors in economy over time where each sector experi-ences an Engel’s consumption cycle. Based on this cycleeach good or service at the beginning appear as a luxurygood with high income elasticity and in the end appearsas necessity with a low elasticity.In our toy model we suppose that total number oflabors in each Island is 100000. Concerning productiv-ity we suppose that in 1900 T A = T B = 1 and in 1998 T A = T B = 18 . . Though initial and final conditionsare the same for all Islands, we suppose that in the inter-mediate years each island follows its own R&D program.We suppose that in North Island in early years a big dealof attentions is devoted to the sector A and productivity grows faster in this sector and in the late years productiv-ity grows faster in sector B . Quantitatively we considerthe following pattern for growth in productivities T iA = (1 + 0 . ∗ − i
99 ) ∗ T i − A T iB = (1 + 0 . ∗ i + 199 ) ∗ T i − B . (23)In the Middle Island we suppose that productivity growswith a sustainable rate in both sectors T iA = (1 . ∗ T i − A T iB = (1 . ∗ T i − B . (24)In the South Island we suppose that despite North Islandin early years most efforts are devoted to the sector B andproductivity grows in this sector faster in early years T iA = (1 + 0 . ∗ i + 199 ) ∗ T i − A T iB = (1 + 0 . ∗ − i
99 ) ∗ T i − B . (25)Given these rates, if we maximize utility we find distri-bution of labors as L A = L t λ A + Ω λ B N /T A ( − λ A R c + δ A ) − λAλA λ A + Ω λ B ,L B = L t − L A . (26)For depreciation we set δ a = %5 . . We kept rate of re-turn on capital sustainable and equal to %5 . . We thenran a simple code to carry out calculation concerningGDP real growth rates for each Island. Five CSV fileshave been supplemented named by gdpnorth.csv, gdp-middle.csv, and gdpsouth.csv which represent our results.Each file is a 99*4 matrix where each row represents, YA,PA, YB, and PB for of the related year from 1900 to1998. As it can be seen in the file, initial and final levelof production and prices are the same for three countries.Now, we can measure the real growth rate for these data.When we use Eq. (21) then for the growth rate of eachisland we come to the results depicted in Fig. 1a. Theaverage of the growth rate from 1900 to the desired yearhas been depicted in Fig. 1b. As it be can bee seenas we approach 1998 all three countries approach to thesame final condition and we expect the average of thereal growth rate for the whole period merge to a uniquevalue. It is while in Fig. 1b we observe that in the end ofthe period, the average of the real growth for the wholeperiod is % 3.5 in North Island, %3 in Middle Island and%2.1 in South Island. What a surprise! While initial andfinal conditions are the same for all three countries, thegoverning parties in the North Island are proud of theirperformance for the past century while the governing par-ties in South Island should be ashamed for the actuallythe same results. To make sure that different indexes cannot influence the paradoxical result in our toy model we R ea l g r o w t h r a t e Measurments for GDP growth rate in three islands North IslandMiddle IslandSouth Island (a) R ea l g r o w t h r a t e Average of real growth rate in all three islands since 1900 to the desired year North IslandMiddle IslandSouth Island (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 average of growth of each island since 1900 to the desired year has been calculated. Each country has its ownpattern of growth and thereby the average of growth should vary in middle years. The initial and final size ofeconomies are however the same. So, we expect that the average of real growth in the end of the period to be thesame for all countries. As it can be seen the average of the growth for three islands do not merge as their finalconditions merge in 1998. R ea l G r o w t h R a t e Comparing Laspayres VS Paasche Indexed Measurement for the Real Growth LaspayresPaasche (a)
Figure 2: Measurement for the real growth rate basedon different inflation indexes. As it can be seen a resultfrom a Laspayres index is very close to the Paascheindex in the North Island. So, choosing differentindexes can not relax the problem with measurements.considered both Laspayres and Paasche index and recal-culated the real growth rate. As it can be seen in Fig 2the difference is negligible. Since Fisher and Törnqvistindexs are between Laspayres and Paasche indexes thenit means that none of these indexes can not influence theresult noticeably. Besides the mentioned csv files, two other files havebeen submitted as "northconstant.csv" and "southcon-stant.csv". Time series in these files are even more in-teresting than the former files. Two islands start withexactly initial conditions. One of them annually experi-ences a real growth around . and of them experiencesan annual growth around . (I mean the annual growthfor every year within the period and not the average ofthe annual growth.). Despite this annual substantial dif-ferences, the islands finally end up with exactly the sameeconomy. IV. GEOMETRICAL UNDERSTANDING OFTHE GROWTH RATE
We obtained surprising results concerning measure-ments for the growth rate in our toy model. Now, let’srecall a similar problem in physics. We suppose thatwe have two astronauts in two spacecrafts. They starta trip from one point in one side of a star to the otherside as depicted in Fig. 3. In Fig. 3 they start theirtrips simultaneously from point A. One of the spacecraftsmoves along the diameter and the other spacecraft movealong the circumference from A to D. They have differ-ent speeds and manage to end their trip simultaneouslyat point D. During their trip they measure both distanceand time of their trip. When they share their measure-ments they observe that surprisingly the relation betweenradius and circumference is no longer S = 2 Π R . Besideof the distance they disagree on the duration of the trip.Figure 3: Near a star metric for people who are atcircumference is different from who are close to the starin diameter. One astronaut moves along thecircumference from A to D and measure it. Anotherastronaut moves along diameter from A to D tomeasure distance. If he is not cooked in the middle ofthe star and meet the other astronaut at point Dsurprisingly finds that for radius and circumference wehave S < Π R Neither of the astronauts has made mistake. The point isthat space becomes curved by gravity. In Fig. 3 distancebetween each line in the grid shows the same value sayone kilometer.The story about measurements for GDP growth rate isexactly similar to the problems with astronauts. Whenwe measure the real growth rate g = d log GDP = Σ a P a Σ b P b Y b dY a , (27)for any changes in production we use prices as our meters.Let’s look at it from a geometrical point of view.In New York city price for a McDonald’s "TripleCheeseburger" is 3$. Let’s suppose that the average of ahaircut for men is about 21$. Now we can draw a frame-work as of Fig. 4a. In this framework distance from eachaxis shows production level. Red grid however representsdollar values. Distance between parallel lines in the redgrid represents 10$. If you move along haircut directionyour influence in GDP is bigger from moving in burgerdirection. One step along haircut direction passes morethan two lines and has an influence more than 20$. Alongburger direction however you need to have more than 7steps to have such effect in the growth of GDP. In Iran price for both burger and men haircut is around3 $. In this case, in spite of the US moving along haircutdirection has not a bigger influence on GDP size. Whilebased on the meter of the US Bureau of Economic Anal-ysis (BEA) distance between A to C is much longer thanA 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 itsown technology and productivity level. Relative pricesare different in each country. Though each of three is-lands have the same initial and final conditions, sincetheir paths are different in the middle ages their measurefor the growth during the period of 99 years is quite dif-ferent, see Fig. 4c. Metric to aggregate production andmeasure the real growth is different in each country.It should be noted that in this model we supposed thatall Islands have the same utility preference and therebywhenever their technologies were the same their produc-tions were the same as well. So, they ended up in thesame point in the space of production. In real world how-ever in each country people have their own utility prefer-ences. As a result even if they start from the same initialconditions they would follow different paths in the spaceof productions and they barely may meet each other inthis space.If we aim to list the counterparts of elements of thegeneral relativity to the current discussions we can countas follows. - In the general relativity metric, denoted by g µν helps to measure distances in the space time along acurve c through the relation d ( x i , x f ) = Z c p g µν dx µ dx ν . (28)In economics prices are our meters and the gap betweeneconomies can be addressed via ∆ GDP = Z c Σ a P a dY a . (29) - In the general relativity metric is obtained through vari-ation of the Einstein-Hilbert action S = 116 π Z d d xdt √− g ( R + matter term ) . (30)In economics, prices are addressed through interactionsof sectors and actually equations (16) and (17). Thoughrate of return or real interest rate is addressed from theutility preference. - In the general relativity objects move along geodesicsand the extremums of distances in Eq. (28). In eco-nomics, paths of countries in the space of productioncan be addressed through maximizing utility preferencesin equations (19) and (18).
Hair Cut B u r ge r Metric in NY
A B C (a) B u r ge r Metric in Tehran B A C (b) Production in sector A P r odu c t i on i n s e c t o r B Pattern of Growth in Three Islands in Our Toy Model
Middle IslandSouth Island North Island (c)
Figure 4: a) Price for a Mc’Donalds burger in NY is about 3$ and suppose that average price 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 muchgreater 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 of distances in the space of productions, the related data provider of each country has itsown meter in each direction. So, while in Iran moving from A to B is recognized to have a bigger effect on the realGDP, in the US moving from A to C has a bigger effect on the real growth. c)In our problem for three islands, eachisland moves along its own path and national agencies have their own meters to measure distance between initial andfinal points. So what they measure as the real gap between initial and final sizes of GDP is different from each other.
Mathematical precision
Before closing this section I should notify that findingdistance between points in economics and physics areconceptually similar. Though we borrowed the conceptof metric from the general relativity to notify the impactof measurements by different observers, the equationswe reviewed however do not meet the true mathematicaldefinition of metric. Equation (29) resembles a oneform. Mathematically it resembles the work done on anobject in a force field. The work done along each pathhowever depends on the path since P a changes whenlevel of productions changes. Though mathematicallythe problem is closer to the problems with work andenergy, the concepts of measurements and meters foraggregation, dependency of measurement to the path,and dependency of paths to the optimizations and localextremums are conceptually close to the observations inthe general relativity. V. THE GAP BETWEEN ECONOMIES OFCHINA AND THE UNITED STATES
Before paying attention to the gap between economiesof China and the US let’s recall paradoxical results con-cerning Pen World Table and International ComparisonProgram. When you use purchasing power parity ex-change rate to measure GDP of different countries, itmeans that for all of them you have used the same met-ric to measure their GDP, see Fig. 5a. The price base inthis method is standard worldwide. In developed coun- tries such as the US since productivity is very high, wagesare relatively high and services are relatively more ex-pensive. This is while in poor countries relative pricesfor services are low comparing to the US. So, there willbe problem when we aim to compare growth rate of dif-ferent countries. Systems of national account tries toprovide a standard basis worldwide. Currently howevermany countries such as the US or China do not followsuch system. These countries use their own price base.From the general relativity point of view, national agen-cies resemble falling observers in a curved space-time whohave their own meters. It is while international agencieshave standard meters and resemble an outside observer.Though system of national accounts and PPP exchangerates aim to provide a better international basis for aggre-gation. Still, space time inconsistency is observed. Formore studies see [7]-[14].Back to our problem with China it is now clear thatif we even know the current size of the GDP of bothChina and the US and besides we know their futuregrowth rates we still are not able to examine the timethat economy of China catches the economy of theUS. To extrapolate the time that China needs to passeconomy of the US besides knowing the future growthrates of both countries we need to know the path thateach country passes through. We need to find this paththrough production functions and utility preferencesand the level of capital in China and the US and thenfind local metric along this path and thereby find outthe desired time. Since utility preferences in China isdifferent from the US for sure the final shape of economyof each country will be different and thereby China
Service Sector M anu f a c t u r i ng S e c t o r Metric of International Agencies vs National Agencies
USA ChinaIran (a)
Non−tradable goods T r adab l e good s US vs China
US (Now) US (Future)China (Future)China (Now) (b)
Non−tradable goods T r adab l e good s Growth or Inflation
China, Now ! China, Future!Y Y (c) Figure 5: a) While each country has its own prices or its own grid to measure different movements in real GDP, aPPP based measurement (red grid) aims to have a standard metric to measure GDP of different countries in thespace of productions. b) The US and China have their own meter to measure GDP. While their meter to measuretradable goods are close, their meter to measure non-tradable goods are different. Each country pass its own path inthe 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 newgirding (the blue one), the nominal value of Y has grown, from China NBS perspective only ( Y − Y ) P is the realgrowth. From international perspective however meter of China to measure non-tradable goods has become closer tothe international meter and the growth in the volume in non-tradable goods looks real. In other words frominternational perspective the volume of GDP of China has grown as ( Y − Y ) P − Y P .should not meet the US in the space of production, seeFig. 5b. If we know the path and local metric howeverwe can extrapolate the time that nominal sizes of theirGDP are the same. How dose it work?
One may argue that we have nominal GDP of Chinaand of the US. We have their reported real and nom-inal growths. How then Eq. (1) does not hold? Forsure the equation holds for nominal rates. The problemarises when we aim to extract real growth from the nom-inal growth. Each country reports its growth rate andthen some day their nominal GDP sizes should be thesame. The geometrical figure however is different. Trad-able goods have roughly the same prices at border ofboth countries, see Fig. 5b. Non-tradable goods do nothave the same price however. So, while girding of bothcountries along vertical direction roughly has the samelength, their girding along horizontal axis is different. Ifwe think of Baumol’s cost disease and real observations,we find that prices for services are relatively cheaper indeveloping countries.Manufacturing sectors have good overlap with trad-able goods. It is while service sectors have better overlapwith non-tradable goods. As time goes by, according tothe cost disease and observations, relative prices of non-tradable goods increase. As a result girding of countriesalong horizontal axis will be more dense. Let’s look atthe case of China in Fig. 5c. What will happen then?Before the period, production of both countries in trad- able goods have had close prices. In non-tradable sectorshowever since prices in China have been smaller than theUS, then nominal size of these sectors in China has beensmaller respect to the US one.When economy of China grows along tradable axis wehave no problem. Tradable goods are gridded worldwideroughly with the same price. The problem is with non-tradable goods or in some senses services. For these sec-tors, after a while prices of services in China will be closerto the prices of the US if we believe in the cost disease.What would happen then? For these sectors accordingto the National Bureau of Statistics of China (NBS) wehave a growth in GDP size as: ∆ GDP = Y P f − Y P i . (31)This is nominal share of growth for non-tradable sectors.The NBS however report part of this growth as inflation.According to the NBS the real growth is ∆ GDP = ( Y − Y ) P i . (32)What does it mean? China considers only the part of thegrowth that is real in this direction for the real growthrate. What about international comparisons. In inter-national comparisons the nominal share of non-tradablegoods has grown and from international perspective econ-omy of China has grown. At the end of the period whenGDP of China grows and China faces inflation along non-tradable goods, its price gets closer to the US one, nom-inal volume of the non-tradable goods grows. ThoughChina considers parts of these growth as inflation, but0since prices for these sectors have really become closerto the US one, and nominal GDP of China has becomecloser to the US one, from international comparisons itis a real growth.In international comparisons tradable goods haveworldwide prices. Now that volume of services in Chinagrows, then nominal GDP of China will be closer to theUS one. It does not matter if China percepts it as infla-tion or real growth rate. To clarify it I have provided anumerical example in table I.I have supposed that the US and China produce onlyone good and provide only one service. Prices and pro-ductions are as the level in the table. In this hypotheticalmodel from 2015 to 2016 nothing changes in the US. So,both inflation and growth experience a zero value rates.In China tradable good A has the same value as of theUS. Service B however experiences a relative growth inprices in China since economy in this country is growingfaster than the US. For this case the China NBS reportsa real growth as of 3.9%. According to the NBS partof the growth of the volume in service sector is inflationrather than being a real growth. So, China report an an-nual inflation equal to 1.3%. So, if we consider reports ofthe NBS we should conclude that economy of China hasbecome closer to the economy of the US by 3.9%. If welook from an international perspective things are differ-ent. Growth of prices in service sector in China is not afluctuation. It is permanent and is a result of the growthand the cost disease. Volume of economy of China hasgrown by 5.2% from an international perspective. Meterof the NBS of China has become closer to the meter ofthe US BEA. From an international perspective volumeof GDP of China has become closer to the US one by5.2% despite reports of the national agencies for the realgrowths.Two conclusions can be deduced from our discussions: The first conclusion is that extrapolating the real timethat nominal GDP of China will be equal to the nominalGDP of the US is a hard task even if we know their realgrowths. To find the proper answer we need to knowboth the relative growth of prices as a result of the costdisease and as the utility preferences. The problem holdswith both nominal and PPP values of GDP sizes.We have two forms of variation in prices. One formof variation is the regular fluctuations of prices in themarket as a result of short time interaction in supplyand demand curves. The other form of variation is thegrowth in prices in service side as a result of the costdisease which is permanent as long as we can not find atechnology to grow productivity. Though from nationalperspective growth in prices in service sector is inflation,from international perspective it is a permanent growthof volume of service sector and a permanent growth ofthe volume of GDP.Prices for non-tradable goods are lower in developingcountries. If developing countries experience fast growthand relative prices for their services grow faster thanmanufacturing side, then this growth only make nomi-nal GDP for them closer to the PPP values. CurrentlyPPP value of GDP of China is bigger than the US. WhenGDP of China expands we expect prices in the serviceside in China become closer to the prices in the US. As aresult, nominal size of GDP of China become closer to itsPPP value. So, the second conclusion is that if prices forservices in China become closer to the US ones then itseconomy become closer to the economy of the US soonerthan what an equation such as Eq. (1) suggests. ACKNOLEDGMENTS
I would like to thank GR Jafari for comments on anearlier draft and H Arfaei for discussion. [1] Fisher, I. (1927) [1967], The Making of Index Numbers:A Study of Their Varieties, Tests, and Reliability (3rdedition). Augustus M. Kelley, New York[2] Balk, B.M. 1995, Axiomatic price index theory: a survey.International Statistical Review, 63, 1, 69-93.[3] Diewert, W. E., 1976 "Exact and Superlative Index Num-bers." Journal of Econometrics, 4: 115-46.[4] Diewert, W.E. Chapter 5: "Index Numbers" in Essaysin Index Number Theory. eds W.E. Diewert and A.O.Nakamura. Vol 1. Elsevier Science Publishers: 1993[5] Feenstra, R. C. and M. B. Reinsdorf 2000. "An ExactPrice Index for the Almost Ideal Demand System." Eco-nomics Letters, 66: 159-162.[6] Oulton, N., 2008, "Chain indices of the cost of living andthe path-dependence problem: An empirical solution",Journal of Econometrics 144 (1), 306-324[7] Crawford, I, and J. P. Neary 2008, "Testing for a Refer-ence Consumer in International Comparisons of Living Standards". American Economic Review, 98(4), 1731-1732.[8] Deaton, A. and A. Heston 2010, "Understanding PPPsand PPP-based national accounts". American EconomicJournal: Macroeconomics 2010, 2:4, 1-35.[9] Deaton, A., and Bettina A. 2014, "Trying to understandthe PPPs in ICP2011: why are the results so different?".NBER Working Paper no. 20244.[10] Feenstra, R. C., H. Ma, J. Peter Neary and D. PrasadaRao, 2013 "Who Shrunk China? Puzzles in the Measure-ment of Real GDP," The Economic Journal 123 (573),1100-1129[11] Prasada,R., A. Rambaldi and H. Doran 2010, "Extrapo-lation of Purchasing Power Parities using multiple bench-marks and auxiliary information: a new approach". Re-view of Income and Wealth, 56, Special Issue 1, S59-S98.[12] Feenstra, R. C., Hong Ma, and D.S. Prasada Rao 2009,"Consistent comparisons of real incomes across space and Year Country
Table I: Hypothetical production and prices in the US and China. time". Macroeconomic Dynamics, 13(S2), 169-193.[13] Feenstra, R. C., Robert Inklaar, and Marcel P. Timmer2013b, "The next generation of the Penn World Table",NBER Working Paper 19255[14] Oulton, N. 2015, "Understanding the space-time (in)consistency of the national accounts". Economics Letters,132, 21-23[15] Malaney, P. 1996, "The Index Number Problem: A Dif-ferential Geometric Approach." Ph.D thesis, HarvardUnivesity, 1996; (Chapter 2: Welfare Implications of Di-visia Indices cauthored with Eric Weinstein);[16] Smolin, L. 2009, "Time and symmetry in models of eco-nomic markets", arXiv:0902.4274 [q-fin.GN][17] Mantegna, R. N. and H. E. Stanley, "An Introduction toEconophysics" (Cambridge University Press, Cambridge,2000)[18] Preis, T., Kenett, D. Y., Stanley, H. E., Helbing, D.& Ben-Jacob, E. , 2012, "Quantifying the behaviour ofstock correlations under market stres"s. Sci. Rep. 2, 752[19] Onnela, J.P., A Chakraborti, K Kaski, J Kertesz, AKanto 2015, "Dynamics of market correlations: Taxon-omy and portfolio analysis", Phys. Rev. E 68 (5), 056110[20] Namaki, A., A.H. Shirazi, R. Raei, G.R. Jafari, 2011,"Network analysis of a financial market based on genuinecorrelation and threshold method", Physica A 390, 3835-3841[21] Silva, A., C., V. M. Yakovenko, 2003, "Comparison be-tween the probability distribution of returns in the Hes-ton model and empirical data for stock indexes". PhysicaA, 324, 1-2, 303-310[22] Fujiwara, Y., W. Souma, H. Aoyama, T. Kaizoji, M.Aokie, 2003, "Growth and fluctuations of personal in-come". Physica A, 321, 3-4, 598-604[23] Delli Gatti, D., C. D. Guilmi, E. Gaffeo, G. Giulioni,M. Gallegati, A Palestrini 2005, "A new approach tobusiness fluctuations: heterogeneous interacting agents,scaling laws and financial fragility". Journal of EconomicBehavior & Organization 56, 4, 489-512[24] Lux, T. 2009, "Applications of statistical physics in fi-nance and economics,". in Handbook of Research onComplexity, edited by J. B. Rosser (Edward Elgar, Chel-tenham, UK and Northampton, MA)[25] Schweitzer, F., G. Fagiolo, D. Sornette, F. Vega-Redondo, A. Vespignani, D. R. White 2009, "EconomicNetworks: The New Challenges". Science 325(5939), 422-425[26] J. Doyne Farmer, M. Gallegati, C. Hommes, A. Kirman,P. Ormerod, S. Cincotti, A. Sanchez, D. Helbing 2012, "Acomplex systems approach to constructing better models for managing financial markets and the economy". TheEuropean Physical Journal Special Topics, 214, 1, 295-324[27] Chakraborti, A., D. Challet, A. Chatterjee, M. Marsili,Y.C. Zhang, B. K. Chakrabarti, 2015 "Statistical me-chanics of competitive resource allocation using agent-based models". Physics Reports 552, 1-25[28] H Safdari, A Hosseiny, SV Farahani, GR Jafari, "A pic-ture for the coupling of unemployment and inflation"Physica A: Statistical Mechanics and its Applications444, 744-750[29] A. Hosseiny, M. Bahrami, A. Palestrini, M. Galle-gati, "Metastable Features of Economic Network andResponse to Exogenous Shocks", PloS one 11 (10),e0160363.[30] Mastromatteo, I., B. Toth, J.P. Bouchaud 2014, "Agent-based models for latent liquidity and concave price im-pact". Phys. Rev. E 89, 042805[31] Baumol,W. and W. Bowen 1966 "Performing arts: Theeconomic dilemma." New York:Twentieth Century Fund.[32] Baumol, W., S. Batey Blackman, and E. Wolff. 1989"Productivity and American leadership: The long view."Cambridge, Mass.: MIT Press[33] Baumol WJ, de Ferranti D, Malach M, Pablos MendezA, Tabish H and Gomory Wu L 2012, "The Cost DiseaseWhy Computers Get Cheaper and Health Care Doesn’t."New Haven & London, Yale University Press.[34] Ngai, L. R. and Pissarides, C. 2007 "Structural Changein a Multisector Model of Growth." American EconomicReview, 97(1), 429-443[35] A. Hosseiny and M. Gallegati, "Role of Intensive andExtensive Variables in a Soup of firms in Economy toAddress Long Run Prices and Aggregate Data", PhysicaA: Statistical Mechanics and its Applications, 470, 51-59,[36] National Center for Public Policy and Higher Educa-tion. 2008. "Measuring up 2008: The national reportcard on higher education". San Jose, Calif, p.8, fig. 5.http://measuringup2008.highereducation.org/print/NCP-PHEMUNationalRpt.pdf.[37] U.S. Bureau of Labor Statistics 2009b. Funeral ex-penses consumer price index. All Urban Consumers (Cur-rent Series) Database. http://data.bls.gov/PDQ /out-side.jsp?survey=cu[38] U.S. Bureau of Labor Statistics 2009a. Legal servicesconsumer price index. All Urban Consumers (Cur-rent Series) Database. http://data.bls.gov/PDQ /out-side.jsp?survey=cu.[39] Konüs, A.A. (1939), The problem of the true index of thecost-of-living. Econometrica 7, 10- 29. (English transla-tion; first published in Russian in 1924).