Technology factor as society’s information stock: An elegant proof

: It has been empirically observed that the income structure of the vast majority of populations in market-economy countries follows an exponential distribution. The empirical evidence has covered more than 66 countries, ranging from Europe to Latin America, North America, and Asia. Here, to further support exponential income distribution as a signature of the well-functioning market economy, we empirically show how the income structure of China evolved towards an exponential distribution after the market-oriented economic reformation. In particular, we strictly prove that, if the income structure of an economy obeys an exponential distribution, the income summation over all households leads to a neoclassical aggregate production function, in which the technology factor is exactly equal to society’s information stock. This finding provides an insight into understanding the underlying implication of technological progress.


Introduction
The technology factor in aggregate production function is a mysterious variable.
Distinguishing from labor and capital, Solow (1956) has used his famous model to show that the technology factor is the most powerful driver of economic growth. However, in his model, the technology factor is exogenously given. As a result, for the given technology, Solow's model predicts that, due to the diminishing marginal return of capital, the growth rate of per-capita income will decline to zero. This implies that, in the economic world described by Solow's model, the growth of per-capita income is unsustainable. To overcome the dilemma of Solow's model, motivated by Arrow's learning by doing (Arrow 1962), Lucas (1988), Romer (1986Romer ( , 1990, and Barro (1990) endogenized the technology factor. Following their models, the technology factor will persistently rise with knowledge accumulation so that the growth rate of per-capita income is always higher than zero. These models are called the "endogenous growth models", in which the technology factor has been proposed as "knowledge" or "information stock". In the 1990s, the conception of technology factor as "knowledge" or "information stock" was further used to explain the success of China's gradual reform of economic development 1 (Murrell, 1992). Although the technology factor has been commonly interpreted as "knowledge" or "information stock", to date there is no formal proof for the validity of this interpretation. Our paper fills this gap. Here, we strictly prove that if the income structure of an economy obeys an exponential distribution, the technology factor in aggregate production function is exactly equal to society's information stock.
In the economic literature, it has been observed empirically that, for marketeconomy countries, the low-and middle-income class (about 95% of populations) follows an exponential distribution (Nirei and Souma, 2007;Newby et al., 2011;1 Recently, Lin (2011) attempted to explain the success of China's gradual reform of economic development from the perspective of matching technological choice and factor endowment. This is in accordance with the conception of technology factor as "knowledge" or "information stock". In this regard, Murrell (1992) has pointed out that a society's stock of personal knowledge is acquired through a long historical process and is shaped by the institutions and organizations of that particular society that has the given factor endowment. Clementi et al., 2012;Prinz, 2016;Irwin and Irwin, 2017;Rosser, 2019;Tao et al., 2019;Ma and Ruzic, 2020) and the top income class (less than 5% of populations) follows a Pareto distribution (Mandelbrot, 1960;Nirei and Souma, 2007;Atkinson et al., 2011;Aoki and Nirei, 2017;Tao et al., 2019). In this regard, Blanchet et al. (2017Blanchet et al. ( , 2018 have used the generalized Pareto distribution to describe the income structure of the total population. The generalized Pareto distribution is a fairly general family that includes the Pareto distribution and the exponential distribution as two special cases (Singh and Maddala, 1976;Cowell, 2000;Jenkins, 2016). In this paper, we further demonstrate that when the equal opportunity is guaranteed, the generalized Pareto distribution yields an exponential distribution, and when the equal opportunity is undermined, the generalized Pareto distribution turns to a Pareto distribution. This finding implies that the exponential distribution is expected to describe the income structure of an equalopportunity market economy, just as described by a peer-to-peer economy (Tao, 2016), which indicates a well-functioning market economy. Using a large sample, Tao et al.
(2019) analyzed datasets of household income from 66 countries, ranging from Europe to Latin America, North America, and Asia, and found that, for all the countries, the income structure for the low and middle classes (about 95% of populations) uniformly follows an exponential distribution. Because the empirical evidence has shown that exponential income distribution dominates the vast majority of the populations, we assume that the income structure of the total population in a market-economy country is approximately described by an exponential distribution. Using this assumption, we strictly prove that the income summation over all households leads to a neoclassical aggregate production function, in which the technological factor is exactly equal to society's information stock.
The remainder of the paper is organized as follows. In section 2, we empirically show the validity of the exponential income distribution in describing the income structure of the well-functioning market economy. In section 3, we show that if the income structure of an economy obeys the exponential distribution, then the income summation over all households leads to a neoclassical aggregate production function with Hicks-neutral-like technical progress. In section 4, we further prove that the technology factor in this aggregate production function is exactly equal to society's information stock. Section 5 concludes the paper.

Exponential income distribution
For market-economy countries, it has been empirically observed that the income structure of top income class (less than 5% of populations) obeys a Pareto distribution (Mandelbrot, 1960;Nirei and Souma, 2007;Atkinson et al., 2011;Aoki and Nirei, 2017;Tao et al., 2019), while the low-and middle-income class (about 95% of populations) obeys an exponential distribution (Nirei and Souma, 2007;Newby et al., 2011;Clementi et al., 2012;Prinz, 2016;Irwin and Irwin, 2017;Rosser, 2019;Tao et al., 2019;Ma and Ruzic, 2020). Theoretically, it has been known that the generalized Pareto distribution is a fairly general family that includes the Pareto distribution and the exponential distribution as two special cases (Balkema and Haan, 1974;Pickands, 1975;Singh and Maddala, 1976;Cowell, 2000). In this regard, Blanchet et al. (2017Blanchet et al. ( , 2018 used the generalized Pareto distribution to describe the income structure of the total population, where the generalized Pareto distribution is defined as follows: where > 0, denotes the income level, and ( ≥ ) denotes the fraction of population with the income being higher than .
It can be seen that equation (1) yields the Pareto distribution ( ≥ ) = ( ⁄ ) −1⁄ when = ⁄ (Jenkins, 2016;Blanchet et al., 2017), where the Pareto exponent is denoted by 1⁄ , which measures the degree of income inequality, that is, a larger Pareto exponent is associated with lower income inequality (Jones and Kim, 2018). Intuitively, a lower income inequality should correspond to a more equalopportunity society. Thus, we anticipate that, as the Pareto exponent 1⁄ → ∞ (or → 0), equation (1) yields a distribution being close to equal opportunity; that is, Equation (2) is an exponential distribution, which is expected to describe the income structure of an equal-opportunity economy. In the economic literature, a peer-to-peer economy without an intermediary third-party is recognized as an equal-opportunity market economy (Einav et al., 2016;Davidson et al., 2018;Filippi et al., 2020). In this regard, Tao (2016) has theoretically shown that the exponential income distribution emerges spontaneously in a peer-to-peer economic network described by an Arrow-Debreu economy 2 : where 1 < 2 < ⋯ < , and denotes that there are households, each of which obtains units of income. We will clarify the economic implication of and in section 3.
Because exponential income distributions (2) and (3) are derived by assuming an equal-opportunity market economy, we propose to identify the exponential income distribution as a signature of the well-functioning market economy. Next, we employ the empirical evidence to support this proposal.
Before doing this, we first demonstrate the relationship between equations (2) and (3). Equation (2) is the cumulative distribution of income, where the income level takes continuous value. By contrast, equation (3) is the density distribution of income, where the income level takes discrete values to keep up with reality. Tao et al (2019) have shown that the cumulative form of equation (3) is equivalent to equation (2) when the income level takes continuous value. From this sense, the discrete form (3) has included the continuous form (2) as a special example. In this paper, we will use the discrete form (3) of exponential income distribution to perform a theoretical analysis. By contrast, it is more convenient to carry out empirical investigation by using the continuous form (2).
Because there has been a large body of empirical literature to support that the exponential distribution (2) dominates an extremely large proportion of populations (Nirei and Souma, 2007;Newby et al., 2011;Clementi et al., 2012;Prinz, 2016;Irwin and Irwin, 2017;Rosser, 2019;Tao et al., 2019;Ma and Ruzic, 2020), here we simply use the household income data from two representative market-economy countries to support the validity of the exponential distribution (2) in describing the income structure of the well-functioning market economy. The two countries include one typical developed economy (the United Kingdom) and one typical developing economy (China). Figure 1 shows that the low-and middle-income classes (about 97% of populations) in China and the United Kingdom uniformly obey the exponential distribution (2).

[ ]
In particular, because China is a special sample that has undergone the transition from a planned economy to a market economy, it is more important to examine if there is a transition towards an exponential income distribution after the market-oriented economic reformation. Figure 2 empirically shows that the transition of China's income structure evolving to an exponential income distribution occurred after the marketoriented economic reformation (i.e., 1978). The transition of China's income structure is compelling evidence for identifying the exponential income distribution as a signature of the well-functioning market economy.
[ ] Because the empirical evidence has shown that the exponential distribution (2) dominates an extremely large proportion of populations, we assume that the income structure of the total population in a market economy is approximately described by an exponential distribution. Based on this assumption, we next use the discrete form (3) of exponential income distribution to verify that the technology factor in aggregate production function is exactly equal to society's information stock.

Aggregate production function
According to the economic meaning of exponential income distribution (3), the total number of households, , and the GDP, , of the economy can be written as below: Here, we order: By using equations (6) and (7), equations (4) and (5) can be rewritten as Equations (4) and (5) indicate that and are related via two parameters and . This means that the aggregate production function (or GDP) should be a function of total number of agents, . By using equations (8) and (9), we can obtain The derivation for equation (10) can be found in Appendix A.
Let us order: The following proposition 1 guarantees that if equation (12) is solvable, then is independent of (Tao, 2019).
Proposition 1: Assume that agents obey the exponential income distribution (3) so that equation (10) holds. Then is independent of if and only if the following partial differential equation (12) holds and is solvable: Proof. The proof can be found in Appendix B. □ It is easy to verify that equation (12) is solvable (Tao, 2019). Henceforth, we always assume that equation (12) holds. Following this assumption, is independent of . Therefore, in the context of exponential income distribution (3), we have the aggregate production function as below: Substituting equations (6), (7), and (11) into equation (10) we obtain Since is independent of , by equation (14) we have = ( , ) .
To identify the economic implication of and , we denote by the technology factor. Here, we further assume that households are the owners of labor and capital.
This assumption is natural. If a household has no labor or capital, it cannot survive in a real society. Therefore, the total number of households, , can be regarded as the function of the labor and the capital stock ; that is, we have In the context of neoclassical economics, the aggregate production function can be written as = ( , , ).
Because equation (12) holds, is independent of . Therefore, by equations (17) and (18), the aggregate production function (13) can be written in the form: which implies a Hicks-neutral-like technical progress.

Main results
To prove that stands for society's information stock, we need to make three assumptions.
The economic meaning of Assumption 1 is that the GDP of a country is determined by the economic strategy (or economic decision) of each agent. This assumption is natural. Indeed, the GDP consists of all agents' incomes and an agent's income is determined by her economic strategy. For example, a good strategy may increase income and a bad strategy may reduce income. Furthermore, = ( 1 , 2 , … , ) indicates that the number of agents is discrete (or countable). However, as with Aumann's research of a continuum of traders (Aumann, 1966), Assumption 1 can be applied to the case that the number of agents is continuous (or uncountable). For a continuum of agents, we order = ( ) ∈ with = [0, ].

Assumption 2:
is not a function of . In particular, if the agent does not make economic strategy (i.e., inactive), then we order = 0, where, for a discrete number of agents, we have = 1, … , , and, for a continuum of agents, we have ∈ .
( ′ , ) = ′ ; that is, ( ) = ( ′ ). Obviously, these results can be applied to the case of a continuum of agents.
By Proposition 1, we have known that is a function of and ; therefore, by Assumptions 1 and 2 we conclude that is a function of ; that is, For the function = ( ), we further make the continuity assumption as below: Assumption 3 (continuity): If there are two collective strategies and ′ such that ( ) < ( ′ ), then we do find a collective strategy ′′ to satisfy ( ) < ( ′′ ) < Using equations (20) and Assumption 2, equation (14) can be rewritten as ( , ( )) = + ( ).
Based on Assumptions 1-3, we can prove three lemmas as below:

Lemma 1: Assume that agents obey the exponential income distribution (3) and
that equation (12) holds. For any given number 0 , if the probability of each agent acquiring income is independent, the probability of agents taking the collective strategies is denoted by

Lemma 2: Assume that ( , ( )) is a differentiable function with respect to and ( ). For a continuum of agents, if equation (21) holds, then we have
where with 0 being any given number.
Proof. The proof can be found in Appendix D. □ Substituting equations (6) and (7) into equations (22) and (23) yields Lemma 3: For a given constant 0 , if equation (11) holds, by using equations (28) and (29)  Proof. The proof can be found in Appendix E. □ Using Lemmas 1-3, we start to prove the central proposition of this paper.

Proof.
It is easy to verify the following two equations: Substituting equations (30) and (31) into equations (33) and (34) we can obtain where we have used dln = ln • + ln • .
Combining equations (27) and (38) Substituting equation (39) into equation (40) we have which implies that equation (39) can be simplified to that is, (32) is the main result of this paper. By equations (24) and (32), we can provide an economic interpretation for the technology factor ( ) . According to Shannon's information theory (Shannon, 1948), if the probability of an event occurring is denoted by ( ), then the information content contained in the event is equal to −ln ( ) . This means that the lower the probability of an event occurring, the larger the information content contained in the event . By Lemma 1, we have acknowledged that ( ) denotes the probability of agents taking the collective decisions . Therefore, by equation (32), the technology factor ( ) in the aggregate production function (19) denotes the information content contained in the event of agents taking the collective decisions . Because each agent makes decision based on personal information or dispersed knowledge (Hayek, 1945) and

Conclusion
In this paper, we argue that exponential income distribution is a signature of the wellfunctioning market economy. Empirically, we use the household income data from two representative market-economy countries to support the validity of the exponential income distribution in describing the income structure of the well-functioning market economy. The two countries include one typical developed economy (the United Kingdom) and one typical developing economy (China). In particular, because China is a special sample that has undergone the transition from a planned economy to a market economy, we carefully examine whether there is a transition towards an exponential income distribution after the market-oriented economic reformation. The empirical investigation has verified that the transition of China's income structure evolving to an exponential income distribution occurred after the market-oriented economic reformation (i.e., 1978). The transition of China's income structure is compelling evidence for identifying the exponential income distribution as a signature of the well-functioning market economy. Based on the empirical investigation, we assume that the income structure of the total population in a market economy is approximately described by an exponential distribution. Finally, we strictly prove that, if the income structure of an economy obeys an exponential distribution, the income summation over all households leads to a neoclassical aggregate production function, in which the technology factor is exactly equal to society's information stock. This theoretical finding formalizes the validity of interpreting technology factor in the theory of economic growth as "knowledge" or "information stock".

Appendix A
By equations (8) and (9) Next, we verify the necessity. If equation (12) holds and is solvable, then is obviously independent of . Furthermore, Tao (2019) has proved that equation (12)

Proof of Lemma 2
Proof. We consider any two collections of collective strategies and ′ and two numbers and ′ , respectively; that is, agents take collective strategy and ′ agents take collective strategy ′ . Then we order where ∆ ≠ 0 and ∆ ≠ 0. Therefore, we have ′ ≠ .
Because ( , ( )) is a differentiable function with respect to and ( ), by using the mean value theorem of differential, ( ′ , ( ′ )) can be written as Similarly, by using equation (29)