《MicroMacro Relation of ProductionThe Double Scaling Law for Statistical Physics of Economy.pdf》由会员分享，可在线阅读，更多相关《MicroMacro Relation of ProductionThe Double Scaling Law for Statistical Physics of Economy.pdf（5页珍藏版）》请在三一文库上搜索。
1、Micro-Macro Relation of Production The Double Scaling Law for Statistical Physics of Economy Hideaki Aoyama Department of Physics, Kyoto University, Kyoto 606-8501, Japan Yoshi Fujiwara ATR Laboratories, Kyoto 619-0288, Japan Mauro Gallegati Dipartimento di Economia, Facolta di Economia “Giorgio Fua
2、”, Universit a Politecnica delle Marche, Piazzale Martelli 8, 60121 Ancona, Italy (Dated: March 12, 2010) We show that an economic system populated by multiple agents generates an equilibrium distribu- tion in the form of multiple scaling laws of conditional PDFs, which are suffi cient for character
3、izing the probability distribution. The existence of the double scaling law is demonstrated empirically for the sales and the labor of one million Japanese fi rms. Theoretical study of the scaling laws suggests lognormal joint distributions of sales and labor and a scaling law for labor productivity
4、, both of which are confi rmed empirically. This framework off ers characterization of the equilibrium distribution with a small number of scaling indices, which determine macroscopic quantities, thus setting the stage for an equivalence with statistical physics, bridging micro- and macro-economics.
5、 PACS numbers:02.50.-r, 05.70.-a, 89.65.-s, 89.75.Da Economics is in crisis. Although there exists a main- stream approach 1, 2, its internal coherence and ability in explaining empirical evidences are increasingly ques- tioned. The causes of the present state of aff airs go back to the mid of the X
6、VIII century, when new fi gures of so- cial scientist (economists) borrowed the method (math- ematics) of the most successful hard science (physics) allowing for the mutation of political economy into eco- nomics. It was, and still is, the Newtonian mechanical physics of the XVII century, which rule
7、 economics. From then on, economics lived its own evolution based on the classical physics assumptions (reductionism, de- terminism and mechanicism). Quite remarkably, Keynes adopted the approach of statistical physics, which deeply aff ected physical science at the turn of the XIX century by emphas
8、izing the diff erence between micro and macro, around the mid 1930s 3. However, after decades of ex- traordinary success it was rejected by the neoclassical school around the mid 1970s, which framed the discipline into the old approach and ignored, by defi nition, any interdependence among economic
9、agents (fi rms, banks, households) and diff erence between microscopic individ- ual behavior and macroscopic aggregate behavior. The ideas of natural laws and equilibrium were transplanted into economics sic et simpliciter.As a consequence of the adoption of the classical mechan- ics paradigm, behav
10、ior of macro-economic systems are treated as a scaled-up version of one individual agent, who is called Representative Agent (RA), and complex- ity that emerges from aggregation was lost. Any learned physicist knows that this is entirely wrong for physi- cal systems with many constituents: macroscop
11、ic behav- ior of gas is qualitatively diff erent from that of a sin- gle molecule. Likewise, economy of a country is not ex- plained by analysing a single RA as if he is on a deserted island all by himself, like Robinson Crusoe without even Friday, and multiplying the number of population to the res
12、ults. Another quite a dramatic example is the concept of equilibrium.In many economic models equilibrium is described as a state in which (individual and aggregate) demand equals supply.The notion of statistical equi- librium, in which the aggregate equilibrium is compati- ble with stochastic behavi
13、or of the constituents, is out- side the box of tools of mainstream economists. Again, physics teaches us that the equilibrium of a system does not require that every single element be in equilibrium by itself, but rather that the statistical distributions de- scribing macroscopic aggregate phenomen
14、a be stable. What modern physics can do for economics is, then, to open a way to a proper treatise of macro economy as an aggregation of individual economic agents, which is one main theme of econophysics on real economy 4, 5. Such a thought is not totally unfamiliar to open-minded economists, whose
15、 keyword is Heterogeneous Interact- ing Agents (HIA) 6, where heterogeneity implies that each has diff erent characteristics; diff erent fi nancial pro- fi le like diff erent energy and momentum, and interaction is trade with exchange of money, goods, workers, infor- mation, etc. just as physical pa
16、rticles interact with each other. In this letter we will show that a system populated by many HIA generates equilibrium distribution in the form of scaling laws. In particular, economic literature arXiv:1003.2321v1 q-fin.GN 11 Mar 2010 2 has shown the existence of large and persistent diff er- ences
17、 in labour productivity across industries and coun- tries 79. Productivity is often measured in terms of the ratio between fi rms revenues and the number of employ- ees. It can be expected to be a unique value only within a very straightjacket hypothesis, such as the Represen- tative Agent. If agent
18、s are heterogeneous and interact, then scaling laws emerge, and dispersion is nothing but a consequence of it. Using Japanese data we empirically demonstrate it. The conclusive remarks points out that a thermodynamical approach (see also 10, 11) may be what economics needs if HIA are the actors of t
19、he drama. For the purpose to study properties in probability dis- tributions, it is essential to observe a large portion of the entire population of fi rms and workers. A database of only listed fi rms, for example, is insuffi cient to ana- lyze properties of distributions. We employ the largest dat
20、abase of Credit Risk Database (CRD) in Japan (years 1995 to 2009), which includes a million fi rms and fi fteen million workers in the year 2006, covering the large por- tion of the whole domestic population. Below we give our analysis for the year 2006, but note that the qualitative results are val
21、id for other years as well. We measure the value added Y and the labor L of each fi rm to have the information of output and input in the production at the individual level. We use simply the business sales/profi ts as a proximity to the value added, and the end-of-year number of workers (excluding
22、man- agers) as the labor in order to calculate distributions for Y and L and to uncover their properties. To understand how workers are distributed among dif- ferent levels of output and productivity, we shall study the distributions of Y and L using the following prob- ability density functions (PD
23、Fs). The joint PDF, PYL, the conditional PDFs, PY |Land PL|Y, and the marginal PDFs, PLand PY , are defi ned by PYL(Y,L) = PY |L(Y |L)PL(L) = PL|Y(L|Y )PY(Y ). (1) The conditional average of f(Y ) is defi ned by E(f(Y )|L) := Z 0 f(Y )PY |L(Y |L)dY,(2) foranarbitraryfunctionf(),andsimilarlyfor E(f(L
24、)|Y ). As we shall see, since the PDFs are heavy-tailed for Y and L, it is convenient for the purpose of statistical analysis to take the logarithms of variables: y := ln Y Y0 , := ln L L0 ,(3) where Y0and L0are arbitrary scales. Fig. 1 shows the scatter plot for (,y). To reveal sta- tistical struct
25、ure in data, which can be easily missed by parametric methods, we employ a kernel-based nonpara- metric methods 12. Fig. 1 depicts a nonparametric re- gression curve with error bars (95% signifi cance level) for FIG. 1.Scatter plot for (L,Y ) (gray dots).The curve with error bars is the nonparametri
26、c estimation for E(y|) := E(lnY |L), while the thin straight line is the linear regression (both axes are in units of ln10.) E(y|) = E(lnY |L). We can observe that there exists a range 100.7 L 102.5for which the relation: E(y|) = + const.,(4) holds where is a constant. In fact, the goodness of fi t
27、for nonparametric regression (R2= 44.04%; see 13 for the defi nition) has a same level as that for linear regression (R2= 44.03%) for the range, the estimation of which gives the estimation, = 1.037(0.003) (shown by a straight line in Fig. 1). Similarly, for the range of 104.5 Y 107.0, we have anoth
28、er relation, namely E(|y) = y + const.,(5) with a constant .The validity for this relation is checked by the nonparametric (R2= 47.18%) and lin- ear (R2= 47.09%) regressions, the latter of which gives the estimation, = 0.655(0.002). We fi nd that these relations are simple consequences from two scal
29、ing relations for the conditional PDFs, PY |L(Y |L) and PL|Y(L|Y ). Fig. 2 (a) depicts the con- ditional PDF, PY |L(Y |L), with the conditioning values of L are chosen at a logarithmically equal interval cor- responding to the range 100.7 L 102.0in terms of histograms. By using the values of estimat
30、ed above, we fi nd that the conditional PDF obeys a scaling relation: PY |L(Y |L) = ? L L0 ? Y(Yscaled),(6) where Yscaled:= (L/L0)Y and Y() is a scaling func- tion, as shown by the fact that the PDFs PY |L(Y |L) 3 FIG. 2. (a) The conditional PDF PY |L(Y |L) for L 5,200, where the conditioning values
31、 of L are chosen at a logarithmically equal interval. (b) The scaled conditional PDF Y(Yscaled ) defi ned by Eq. (6). Dots are the scaled data points and the curve is the lognormal function given by Eq. (16). FIG. 3. The scaled conditional PDF L(Lscaled ) defi ned by Eq. (7). The curve is the lognor
32、mal function given by Eq. (18). for diff erent values of L fall onto a curve depicted in Fig. 2 (b).It is straightforward to show that Eq. (4) follows from Eq. (6). Similarly, we have another scaling relation for PL|Y(L|Y ) = ? Y Y0 ? L(Lscaled),(7) where Lscaled:= (Y/Y0)L and L() is a scaling func-
33、 tion, as shown by Fig. 3. And also Eq. (5) immediately follows from Eq. (7). The two scaling laws, Eqs. (6) and (7), which we col- lectively call the double scaling law (DSL) have strong consequences to the function form of the joint PDF. Let us briefl y describe them in the following. Let us choos
34、e the reference scales Y0and L0to be within the region of the (Y,L)-plane where DSL is valid. Then by substituting Y = Y0, L = L0into Eqs. (1), (6) and (7), We obtain the marginal PDFs, PYand PLin terms of the invariant functions, Yand L: PL(L) = ? L L0 ? L(L) L(L0) Y(Y0) Y(L/L0)Y0)PL(L0), (8) PY(Y
35、) = ? Y Y0 ? Y(Y ) Y(Y0) L(L0) L(Y/Y0)L0)PY (Y0). (9) From Eqs. (6) and (7) and the above, we arrive at the following equation for the s: L ?(Y/Y 0)L ? L(Y/Y0)L0) L(L0) L(L) = Y(L/L0)Y ) Y(L/L0)Y0) Y(Y0) Y(Y ) . (10) This equation puts strong constraints the form of s. We have converted the above to
36、 diff erential equations and have derived complete solutions of Eq. (10).14 De- pending on whether = 1 or not, the solution is quali- tatively diff erent, which we shall explain below. When = 1, we fi nd the following relation between s is necessary and suffi cient for Eq. (10): L(L) = Y ?(L/L 0)Y0
37、? ? L L0 ?a L(L0) Y(Y0). (11) In other words, we have one arbitrary function in the 4 solution. In this case, Eqs. (8) and (9) implies that PL(L) = ? L L0 ?L1 PL(L0),(12) PY(Y ) = ? Y Y0 ?Y1 PY(Y0),(13) with = L Y , = Y L ,a = L + Y+ LY Y .(14) This result is straightforward to understand:Due to = 1
38、, we have L scaled Y L Yscaled.(15) Therefore, an arbitrary function of Yscaledis a function of Lscaledas far as dependence on Y and L is concerned. This is why an arbitrary function is left in the solution. Also the marginal PDFs in Eqs. (12) and (13) results from the relation (1). For 6= 1, we obt
39、ain, Y(Y ) = epy 2+qy Y(Y0), (16) L(L) = ep 2+s L(L0), (17) PL(L) = e(1)p 2+(s+(q+1)P L(L0). (18) PY(Y ) = e(1)py 2+(q+(s+1)yP Y(Y0). (19) The joint PDF is given by the following: PYL(Y,L) = ep 2+2pypy2+s+qyP YL(Y0,L0). (20) We fi nd that in the limit 1 we obtain the power laws for PL(L) and PY(Y ),
40、 which is consistent with the results (12) and (13) above. The parameter of p is estimated in two ways: The best fi t of the theoretical expression (16) in Fig. 2 (b) yields p = 0.692(0.027), while Eq. (16) in Fig. 3 yields p = 0.704(0.016). These measured values of p agree with each other very well
41、, with combined value p = 0.698(0.025), assuming equal weight and no corre- lation. The marginal PDF PL(L) and PY(Y ) in Eqs. (18) and (19) agrees with empirical data very well with these values of p. These analysis show that our theoretical re- sults above are in good agreement with data. We stress
42、 that the above results are proven in the lo- cal region of the (Y,L)-plane where DSL is valid.On the other hand, if the lognormal PDF (20) is valid every- where on the (Y,L) plane, one may obtain the marginal PDFs and s as given in Eq. (17)(19), as PL(L) can be obtained by integrating Eq. (20) over
43、 Y 0,), and then obtain L(L) in Eq. (17) from Eqs. (6) and (7), and so forth, which constitute easy checks of the relation between various functions. Let us now study the labor productivity C := Y/L in case of 6= 1, as such is the reality as shown empiri- cally. The joint PDF of (C,L), PCL(C,L) is g
44、iven by the following: PCL(C,L) = LPYL(CL,L).(21) Substituting the expression (20) into the above, we fi nd that is express, we obtain that PCL(C,L) too is of log- normal form like the r.h.s. of (20) with ,p replaced by := 1, := 1 + 2 ,(22) p := + 2 1 p,(23) respectively. By substituting empirical v
45、alues found for , and p (the average of the two central values of p found above), we fi nd that they are = 0.037(0.003), = 0.072(0.007), p = 6.353(0.680). By comparing this with Eq. (19), we fi nd the following marginal PDF for the productivity C: PC(C) e1c 2+2c, (24) 1:= (1 ) + 2 p,(25) where c :=
46、ln(C/C0) with C0:= Y0/L0and the mea- sured values of ,p yield 1= 0.456(0.017). Also, the conditional PDF PL|C (C,L) satisfi es the scaling law; PL|C(C,L) = ? C C0 ? L ? C C0 ? L ! ,(26) L e pc2+ qc, (27) while the average of the labor L for a given productivity C is given by the following; EL|C ? C
47、C0 ? .(28) Fig. 4 checks the result of Eq. (28) with the parameter estimated from the relation in Eq. (27), where the thick straight line is the theoretical calculation. Fig. 5 depicts the scaling function (), where the curve is given by the fi tting in Eq. (27). Both of these results confi rm our r
48、esults in the region where the scaling relations are valid. In this paper, we have shown that the Japanese data for some one million fi rms show that the fi rm distribu- tion in (Y,L) plane satisfy the double scaling law (DSL). We have shown that DSL leads to either power-law for the marginal PDF of
49、 Y and L, or the lognormal PDF for the joint PDF of Y and L. Although we have con- centrated on these specifi c two variables because of their importance for the labour productivity, we believe that 5 FIG. 4.Scatter plot for (C,L) (gray dots).The curve with error bars is the nonparametric estimation for E(|c) := E(lnL|C), while the thick straight line is the theoretical pre- diction (28) with given by Eq. (23) (in units of ln10). FIG. 5. The scaled conditional PDF L(Lscaled ) defi ned by Eq. (27). The curve is the