Why does production function take the Cobb–Douglas form?

Abstract

We directly observed a Cobb–Douglas symmetric plane using the index of surface openness, which is used in geography, and successfully identified it. Based on this observation, we measured the capital shares (capital elasticity) and labor shares (labor elasticity) and compared them with the results of multiple regression analysis used in economics. We confirmed consistent agreement in seven countries: Japan, Germany, France, Spain, Italy, the UK, and the Netherlands. Thus, we show that the Cobb–Douglas production function can be clearly captured in empirical data as a geometric entity with a quasi-inverse symmetry of variables. Based on the above discussion, we theoretically clarified why the Cobb–Douglas production function is better fitted to empirical data in economics, because it uniquely derives the fact that its variable follows a power-law distribution.

Appendices

Appendix 1

Here, we show that under the inverse symmetry (4), according to Fujiwara et al. (2003, 2004), the power laws of $x _ 1$ and $x _ 2$ with the same Pareto index are derived from Gibrat’s law (5). Using growth rate $R=x_2/x_1$, (4) is rewritten by variables $x_1, R$ as follows:

$$\begin{aligned} P_{12}(x_{1},R)=R^{-1}~P_{12}(R x_{1}, R^{-1}). \end{aligned}$$

(33)

Using conditional PDF $Q (R | x_1) = P_{12} (x_1, R) / P (x_1)$ and Gibrat’s law (5), this can be reduced:

$$\begin{aligned} \frac{P(x_1)}{P(R x_{1})} = \frac{1}{R} \frac{Q(R^{-1}|R x_{1})}{Q(R|x_1)} = \frac{1}{R} \frac{Q(R^{-1})}{Q(R)}. \end{aligned}$$

(34)

Since the system has inverse symmetry, Gibrat’s law (5) is also established under the inverse transformation of $x_1 \leftrightarrow R x_1 ~ (= x_ {2})$. The right side of Eq. (34) is only a function of R, and so, we signify it by G(R) and expand Eq. (34) by $R = 1 + \epsilon ~(\epsilon \ll 1)$ in $\epsilon$. The zeroth order term of $\epsilon$ is trivial, and the first-order term yields the following differential equation:

$$\begin{aligned} G'(1) P(x_1) + x_1 \frac{d}{dx_1} P(x_1) = 0. \end{aligned}$$

(35)

Here, $G '(\cdot )$ means the R differentiation of $G (\cdot )$. No more useful information can be obtained from the second-order and higher order terms of $\epsilon$. The solution to this equation is uniquely given:

$$\begin{aligned} P(x_1) \propto {x_1}^{- G'(1)}. \end{aligned}$$

(36)

This solution satisfies Eq. (34) even if R is not near $R = 1$, when $Q (R) = R ^ {-G '(1) -1} ~Q (R)$ holds (this is called a reflection law). Reflection law has been confirmed with various actual data (Fujiwara et al. 2004). Finally, if we set $G '(1) = \mu + 1$, Eq. (36) matches the power laws (1), (2), and (3) described by variables at 1 year $x_1$. Since this system has symmetry $x_1 \leftrightarrow x_ {2}$, the power laws holds for the same Pareto index $\mu$ even at time 2.

Appendix 2

Here, we show that according to Ishikawa (2006, 2009), Mizuno et al. (2012), and Ishikawa et al. (2013, 2014), under the quasi-inverse symmetry (6), the power laws of $x _ 1$ and $x _ 2$ with different Pareto indices are derived from Gibrat’s law (5). Using extended growth rate $R = x_{T+1}/{a {x_T}^{\theta }}$, (6) is rewritten by variables $x_1, R$ as follows:

$$\begin{aligned} P_{12}(x_{1},R)=R^{1/{\theta }-2}~P_{12}(R^{1/\theta } x_1, R^{-1}). \end{aligned}$$

(37)

Equation (37) is reduced to Eq. (33) at $\theta = 1$. Using the conditional PDF $Q (R | x_T)$ and Gibrat’s law (5), this is reduced to:

$$\begin{aligned} \frac{P(x_1)}{P(R^{1/\theta } x_{1})} = R^{1/{\theta }-2} \frac{Q(R^{-1}|R^{1/\theta } x_{1})}{Q(R|x_1)} = R^{1/{\theta }-2} \frac{Q(R^{-1})}{Q(R)}. \end{aligned}$$

(38)

Here, we assume that Gibrat’s law (5) holds under a transformation: $x_1 \leftrightarrow R^{1/\theta } x_1 ~(=(x_{2}/a)^{1/\theta })$. This is valid in a system that has quasi-inverse symmetry. Since the last term in Eq. (38) is only a function of R, we signify it by $G _ {\theta } (R)$ and expand Eq. (38) to R near 1 as $R = 1 + \epsilon ~(\epsilon \ll 1)$. The zeroth order of $\epsilon$ is trivial, and the first-order term yields the following differential equation:

$$\begin{aligned} {G_{\theta }}'(1) P(x_1) + \frac{x_1}{\theta } \frac{{\text {d}}}{{\text {d}}x_1} P(x_1) = 0. \end{aligned}$$

(39)

Here, ${G_{\theta }}'(\cdot )$ denotes the R differentiation of $G_{\theta }(\cdot )$. No more useful information can be obtained from the second-order and higher order terms of $\epsilon$. The solution to this equation is uniquely given:

$$\begin{aligned} P(x_1) \propto {x_1}^{- \theta {G_{\theta }}'(1)}. \end{aligned}$$

(40)

Similar to Appendix 1, this solution satisfies Eq. (38) even if R is not near $R = 1$, when $Q(R) = R^{-{G_{\theta }}'(1)-1}~Q(R)$ holds.

Next, in quasi-static system $(x_1, x_ {2})$, we identify distribution $P (x_ {2})$. Actually, we should write $P_ {x_1} (x_1)$, $P_ {x_ {2}} (x_ {2})$; however, because function forms are complicated, they are collectively written as P. From Eq. (40) and $P(x_1) dx_1 = P(x_{2}) dx_{2}$, $P (x_ {2})$ can be expressed:

$$\begin{aligned} P(x_{2}) = P(x_1) \frac{{\text {d}} x_1}{{\text {d}} x_{2}} \propto {x_{2}}^{- {G_{\theta }}'(1) + 1/\theta - 1}. \end{aligned}$$

(41)

Here, we signify Pareto indices at year 1, 2 by $\mu _{1}$, $\mu _{2}$ and represent $P(x_1)$, $P(x_{2})$ as follows:

$$\begin{aligned} P(x_1) \propto {x_1}^{- \mu _1 - 1} ,\quad P(x_{2}) \propto {x_{2}}^{- \mu _{2} - 1}. \end{aligned}$$

(42)

Comparing Eqs. (40) and (41) to Eq. (42), we obtain $\theta {G_{\theta }}'(1) = \mu _1 + 1$, ${G_{\theta }}'(1) - 1/\theta + 1 = \mu _{2} + 1$ and conclude the relation among $\mu _1$, $\mu _{2}$, and $\theta$ as follows:

$$\begin{aligned} \theta = \frac{\mu _{1}}{\mu _{2}}. \end{aligned}$$

(43)