Micro-macro relation of production: double scaling law for statistical physics of economy

Abstract

We show that an economic system populated by multiple agents generates an equilibrium distribution in the form of multiple scaling laws of conditional probability density functions, which are sufficient for characterizing the probability distribution. The existence of the double scaling law is demonstrated empirically for the sales and the labor of one million Japanese firms. Theoretical study of the scaling laws suggests lognormal joint distributions of sales and labor and a scaling law for labor productivity, both of which are confirmed empirically. This framework offers 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.

Appendix A: Proof of the consequences of the double scaling laws

In this Appendix, we shall derive complete solutions of Eq. (10) and prove the consequences given in Sect. 3.

Let us first describe how Eq. (10) follows from the double scaling laws, (6) and (7). The joint probability (1) reads

$$\begin{aligned} P_{Y\!L}(Y,L)&= \frac{L}{L_0}^{-\alpha } \varPhi _Y\,(L/L_0)^{-\alpha }Y\,P_L(L) \nonumber \\&= \frac{Y}{Y_0}^{-\beta } \varPhi _L\,(Y/Y_0)^{-\beta }L\,P_Y(Y). \end{aligned}$$

(29)

In the following, we do not need to assume that this law is valid in all of the $(Y,L)$ space, but we only need to assume that this law is valid only in a region containing the reference point, $(Y_0,L_0)$.

Let us express the marginal PDFs, $P_Y$ and $P_L$, by the invariant functions, $\varPhi _Y$ and $\varPhi _L$. By substituting $Y=Y_0$, $L=L_0$ into Eq. (29), we have

$$\begin{aligned} P_{Y\!L}(Y_0,L)&= \frac{L}{L_0}^{-\alpha }\! \varPhi _Y\,(L/L_0)^{-\alpha }Y_0\,P_L(L) =\varPhi _L(L)P_Y(Y_0),\end{aligned}$$

(30)

$$\begin{aligned} P_{Y\!L}(Y,L_0)&= \varPhi _Y(Y)P_L(L_0) =\left( \frac{Y}{Y_0}\right) ^{-\beta }\! \varPhi _L\,(Y/Y_0)^{-\beta }L_0\,P_Y(Y), \end{aligned}$$

(31)

respectively. Also, by substituting $(Y,L)=(Y_0,L_0)$ into Eq. (29), we have

$$\begin{aligned} P_{Y\!L}(Y_0,L_0) =\varPhi _Y(Y_0)P_L(L_0) =\varPhi _L(L_0)P_Y(Y_0). \end{aligned}$$

(32)

Eq. (8) follows from Eqs. (30) and (32). Similarly, Eq. (9) follows from Eqs. (31) and (32). Thus we have expressed the marginal PDFs in terms of the invariant functions. Then, by substituting Eqs. (8) and (9) into Eq. (29) and by using the identity (32), we can arrive at Eq. (10). The equation (10) yields interesting consequences for the functions of $\varPhi _Y(Y)$ and $\varPhi _L(L)$, which we shall prove in what follows.

Let us define

$$\begin{aligned} \phi _y(y)&= \ln \varPhi _Y(Y), \end{aligned}$$

(33)

$$\begin{aligned} \phi _\ell (\ell )&= \ln \varPhi _L(L). \end{aligned}$$

(34)

Then Eq. (10) can be rewritten as

$$\begin{aligned}&\phi _\ell (-\beta y +\ell )-\phi _\ell (-\beta y) +\phi _\ell (0)- \phi _\ell (\ell ) \nonumber \\&\quad =\phi _y(-\alpha \ell +y)-\phi _y(-\alpha \ell )+\phi _y(0)-\phi _y(y). \end{aligned}$$

(35)

Let us expand Equation (35) with respect to $y$ around $y=0$. At $O(y^0)$, it is satisfied trivially. At the next order, $O(y^1)$, we have

$$\begin{aligned} -\beta y\phi '_\ell (\ell ) +\beta y \phi '_\ell (0) =y\phi '_y(-\alpha \ell )-y \phi _y'(0), \end{aligned}$$

(36)

where we note that the prime means the derivative with respect to the variable of that function, i.e.,

$$\begin{aligned} \phi '_y(y):=\frac{d}{dy}\phi _y(y), \quad \phi '_\ell (\ell ):=\frac{d}{d\ell }\phi _\ell (\ell ). \end{aligned}$$

(37)

Therefore, $\phi '_y(-\alpha \ell )=-(1/\alpha )(d/d\ell ) \phi _y(-\alpha \ell )$. As a result, we obtain the following differential equation for $\phi _\ell (\ell )$:

$$\begin{aligned} \frac{d}{d\ell } \phi _\ell (\ell )=\frac{1}{\alpha \beta }\frac{d}{d\ell }\phi _y(-\alpha \ell ) +\frac{1}{\beta }\phi '_y(0)+\phi '_\ell (0), \end{aligned}$$

(38)

whose general solution is,

$$\begin{aligned} \phi _\ell (\ell )=\frac{1}{\alpha \beta }\phi _y(-\alpha \ell ) +\left( \frac{1}{\beta }\phi '_y(0)+\phi '_\ell (0)\right) \ell +c, \end{aligned}$$

(39)

where $c$ is an arbitrary constant. By substituting $\ell =0$ in the above, we fix $c$:

$$\begin{aligned} c=\phi _\ell (0)-\frac{1}{\alpha \beta }\phi _y(0). \end{aligned}$$

(40)

By substituting Equation (39) to Equation (35), we obtain the following equation for $\phi _y$:

$$\begin{aligned}&\phi _y(\alpha \beta y-\alpha \ell )-\phi _y(\alpha \beta y)-\phi _y(-\alpha \ell ) \nonumber \\&-\alpha \beta \left[ \phi _y(-\alpha \ell +y)-\phi _y(-\alpha \ell )-\phi _y(y)\right] +(1-\alpha \beta )\phi _y(0)=0. \end{aligned}$$

(41)

Before proceeding to $O(y^2)$, we note that this equation is satisfied by an arbitrary function $\phi _y(y)$ for $\alpha \beta =1$. Therefore, we study cases $\alpha \beta =1$ and $\alpha \beta \ne 1$ separately in the following.

Case: $\varvec{\alpha \beta =1}$

In this case, Equations (39) leads to,

$$\begin{aligned} \phi _\ell (\ell )=\phi _y\left( -\alpha \ell \right) + a\ell + c, \end{aligned}$$

(42)

where we denoted $a:=\alpha \phi _y'(0)+\phi _\ell '(0)$. In terms of $\varPhi$s, we can see that this equation is equivalent to Eq. (11) by using (40). Then, substituting (11) into (8) and (9), we obtain the following:

$$\begin{aligned} P_L(L)&= \frac{L}{L_0}^{a+\alpha }\! P_L(L_0),\end{aligned}$$

(43)

$$\begin{aligned} P_Y(Y)&= \frac{Y}{Y_0}^{(1+a)/\alpha } \! P_Y(Y_0). \end{aligned}$$

(44)

We can express these equations as Eqs. (12) and (13), where the exponents are mutually related as shown in Eq. (14). These were what to be proved for this case.

Case: $\varvec{\alpha \beta \ne 1}$

In this case, the expansion of Equation (41) to the next order, $O(y^2)$, leads to following:

$$\begin{aligned} \frac{1}{2} \alpha \beta (\alpha \beta -1) \phi ^{\prime \prime }_y(-\alpha \ell )-\phi ^{\prime \prime }_y(0)=0, \end{aligned}$$

(45)

which shows that $\phi _y(y)$ is a quadratic function of $y$.

Let us parametrize the general solution $\phi _y(y)$ as follows;

$$\begin{aligned} \phi _y(y)=-\beta p y^2 +q y +r \end{aligned}$$

(46)

By substituting this into Equation (39) and using Equation (40), we obtain the following:

$$\begin{aligned} \phi _\ell (\ell )=-\alpha p \ell ^2 +\phi '_\ell (0) \ell + \phi _\ell (0). \end{aligned}$$

(47)

In other words, the coefficient of the linear and the constant terms are not constrained by $\phi _y(y)$.

By denoting $\phi '_\ell (0)=s$ and $\phi _\ell (0)=t$, and by substituting (46) and (47) back into the original expressions, we can prove what to be shown, namely Eqs. (16), (17), (18) and (19). These were what to be proved for this case.