Exponential Income Distribution and Evolution of Unemployment Compensation in the United Kingdom

We show that an exponential income distribution will emerge spontaneously in a peerto-peer economic network that shares the publicly available technology. Based on this finding, we identify the exponential income distribution as the benchmark structure of the well-functioning market economy. However, a real market economy may deviate from the well-functioning market economy. We show that the deviation is partly reflected as the invalidity of exponential distribution in describing the super-low income class that involves unemployment. In this regard, we find, theoretically, that the lower-bound μ of exponential income distribution has a linear relationship with (per capita) unemployment compensation. In this paper, we test this relationship for the United Kingdom from 2001 to 2015. Our empirical investigation confirms that the income structure of low and middle classes (about 90% of populations) in the United Kingdom exactly obeys exponential distribution, in which the lower-bound μ is exactly in line with the evolution of unemployment compensation.


Introduction
Income inequality has become a hot subject of both scholarly investigation and public discussion (Fuchs-Schündeln et al., 2010;Piketty and Saez, 2014;Autor, 2014;Ravallion, 2014;Dawid et al., 2018;Song et al., 2019;Aghion et al., 2019;Cowell et al., 2019). Over the past two decades, it has been commonly acknowledged that income distribution consists of two parts: the low-and middle-income class (larger than 90% of populations) and the top income class (less than 5% of populations); each of the two classes follows a different empirical law. For a long time, there was great interest in investigating the top income class. It has been observed widely that the top income class follows a Pareto distribution (Mandelbrot, 1960;Dragulescu and Yakovenko, 2001;Nirei and Souma, 2007;Atkinson et al., 2011;Tao et al., 2019). A large body of literature has attributed the cause of the Pareto distribution to the Matthew effect of income accumulation (Champernowne, 1953;Wold and Whittle, 1957;Dutta and Michel, 1998;Lux and Marchesi, 1999;Reed, 2001;Nirei and Souma, 2007;Benhabib et al., 2011;Malevergne et al., 2013). However, the singular focus on the top income class of households overlooks the component of earnings' inequality that is arguably most consequential for the low and middle income classes of citizens (Autor, 2014).
For example, unemployed population is almost always located in the low and middle classes rather than the top class. In this paper, we focus instead on the income structure of the low and middle classes. The existing literature has shown that earnings' inequality in the low and middle classes follows an exponential distribution (Dragulescu and Yakovenko, 2001;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). Using a large sample, Tao et al. (2019) have analyzed datasets of household income from 66 countries and Hong Kong SAR, ranging from Europe to Latin America, North America, and Asia: For all the countries, the income distribution for the low and middle classes of populations uniformly follows an exponential law.
Despite these empirical advances, there is scant understanding of the underlying mechanism driving the exponential income distribution. By contrast, the underlying mechanism driving the Pareto distribution is recognized commonly as the Matthew effect of income accumulation.
In recent years, many scholars, such as Jones (2015), and Nirei and Aoki (2016), have argued that it is important to analyze the income distribution in a general equilibrium model, because many variables that influence the distribution are determined endogenously. For the top income class, Aoki and Nirei (2017) constructed a tractable neoclassical growth model to generate Pareto's law of income distribution by using the dynamic general equilibrium theory. In this paper, we use a long-run Arrow-Debreu economy (ADE) in which there are many households, each of whom independently operates a firm, to simulate a peer-to-peer economic network. In particular, to eliminate opportunity inequality of market competition, we assume that this network shares the publicly available technology. Then, we show that an exponential income distribution will emerge spontaneously in such a peer-to-peer economic network. We use this finding to demonstrate that the underlying mechanism for driving the exponential income distribution is due to the equal opportunity of market competition.
Here, we outline briefly the basic idea of deriving exponential income distribution.
As is well known, in the long-run competition, each firm's revenue (size) is indeterminate (Mas-Colell et al., 1995). This indicates multiple equilibria of firms' revenue allocations. In this case, we show that, if each equilibrium revenue allocation is allowed to occur with an equal opportunity, then the exponential distribution of revenue is most probable, indicating a spontaneous order (Tao, 2016). Furthermore, by introducing the property right of a firm, we demonstrate that each firm's revenue in the peer-to-peer economic network described by the long-run ADE is equivalent to a household's income. This implies that an exponential income distribution will emerge spontaneously in such a peer-to-peer economy. There is a large body of literature relating a firm's revenue to a household's income (Lucas, 1978;Rosen, 1982;Luttmer, 2007;Gabaix and Landier, 2008;Jones and Kim, 2014). By contrast, the novelty of our model is in introducing the property right of a firm into an ADE, to rigidly describe the income structure of a peer-to-peer economy. The other literature is not based on the peer-to-peer economy. This paper has two contributions. The first is to show that the exponential income distribution emerges spontaneously in a peer-to-peer economic network that shares the publicly available technology. Because this peer-to-peer economic network is an ideal type of competitive market, we identify the exponential income distribution as the benchmark structure of the well-functioning market economy. The second contribution is to relate endogenous growth theory and unemployment theory to the ADE by using exponential income distribution. Here, we explain the second contribution simply. On the one hand, we prove theoretically that the lower-bound of exponential income distribution has a linear relationship with the unemployment compensation . This implies that, by fitting household income data to exponential income distribution, one should predict realistic unemployment compensation. In this paper, by fitting household income data in the United Kingdom to exponential income distribution, we indeed predict exactly the evolution of unemployment compensation from 2001 to 2015. This is strong evidence for the validity of the exponential income distribution. In the literature documenting search models, the unemployment compensation is a wellknown central variable. There is a rich strand of literature dealing with this topic (Moen, 1997;Smith, 1999;Giuseppe and Postel-Vinay, 2013;Kaas and Kircher, 2015;Jerez, 2017;Wright et al., 2019). As our model provides a linear function relationship between and , this potentially bridges the search model and the ADE. On the other hand, we show theoretically that, by using the exponential income distribution, the income summation over all households leads to a neoclassical aggregate production function (or GDP), in which the technology factor emerges naturally. In our model, to guarantee that general equilibrium will occur at each time point, the emerging technology factor must be determined endogenously by the existing labor and capital stock. This bridges the endogenous growth model and the ADE. There is a huge body of literature that assumes the technology factor to be a function of either labor or capital (Arrow, 1962;Uzawa, 1965;Lucas, 1988;Romer 1994). By contrast, in our model, the relationship between the technology factor, capital, and labor arises because a standard general equilibrium mechanism has been taken into account.
The rest of the paper is organized as follows. Section 2 shows that an exponential income distribution emerges spontaneously in a peer-to-peer economic network that shares the publicly available technology. Section 3 shows that the lower-bound of exponential income distribution has a linear relationship with the unemployment compensation. Section 4 fits the household income data in the United Kingdom to the exponential income distribution and shows that this leads to an exact prediction for the evolution of unemployment compensation from 2001 to 2015. Section 5 shows that, to guarantee that general equilibria occur at each time point, the technology factor must be determined endogenously by the existing labor and capital stock. This indicates a path of general equilibrium growth. Section 6 introduces an index for measuring the potential deviation from general equilibrium and estimates empirically the deviation values of the United Kingdom from 2001 to 2015. Section 7 concludes. All formal proofs are collected in the Appendices.

Exponential income distribution in a peer-to-peer economy
In the economic literature, the relationship between exponential income distribution and equal opportunity has been implied by the generalized Pareto distribution, which is defined as (Cowell, 2000;Jenkins, 2016;Blanchet et al., 2017): where > 0 , denotes the income level, and ( ≥ ) denotes the fraction of population with the income higher than . Here, * and * are two undetermined parameters.
It has been known that the generalized Pareto distribution (2.1) is a fairly general family which includes the Pareto distribution and the exponential distribution as two special cases (Singh and Maddala, 1976;Cowell, 2000). Recently, Blanchet et al (2017Blanchet et al ( , 2018 proposed to use the generalized Pareto distribution (2.1) to describe the income structure of the total population that includes the low-and middle-income class and the top income class. It is easy to check that, when * = * ⁄ , the generalized Pareto distribution (2.1) becomes the Pareto distribution (Jenkins, 2016;Blanchet et al., 2017): where the Pareto exponent is denoted by 1⁄ , which measures the degree of income inequality (Jones and Kim, 2018), that is, a larger Pareto exponent is associated with lower income inequality. Intuitively, a higher income inequality hints a larger decline of equal opportunity among households. Thus, we anticipate that, as the Pareto exponent 1⁄ → ∞ , equation (2.1) yields a distribution being close to equal opportunity; that is, which is an exponential income distribution.
Here, we show that the exponential income distribution can be generated in a peerto-peer economy that underlines equal opportunity of market competition. A peer-topeer economy is a decentralized model whereby two individuals interact to buy or sell goods and services directly with each other or produce goods and service, without an intermediary third-party (Einav et al., 2016;Davidson et al., 2018). In this paper, we use a long-run ADE in which there are many households, each of whom independently operates a firm, to simulate a peer-to-peer economic network. Long-run competition implies that the production function of each firm should take the constant returns form, which is the only sensible long-run production technology (Mas-Colell et al., 1995;Tao, 2016). This economic network can work in a Blockchain system (Davidson et al., 2018).
In particular, to eliminate opportunity inequality of market competition, we assume that this economic network shares the publicly available technology. The property rights for the peer-to-peer economy are arranged as follows: The household can use a linear production function, which is a Cobb-Douglas form, to produce a unit of labor by inputting a unit of labor. In this case, the labor can be regarded as a type of product. Therefore, households can exchange their labors with each other in the peer-to-peer economy. 2 Here, ≫ 1 is necessary for describing long-run competition, as described by Marshall; see Mas-Colell et al. (1995).

Model
Using the formula (2.4) of exponential income distribution, the aggregate production function (or GDP) can be written in the form: . (3.1) The total number of households is: Using equations (3.1) and (3.2), we can obtain: The derivation for equation (3.3) can be found in Appendix C.
As is well-known, the aggregate production function in the neoclassical economics can be written as: where, denotes the labor, denotes the capital, and denotes the technology factor.
Since households are the owners of labor and capital, we rewrite equation ( Thus, the complete differential of equation (3.5) yields: In fact, we have the following proposition. only if the following partial differential equation (3.10) holds and is solvable: Proof. The proof can be found in Appendix D.
Proposition 3.2: The partial differential equation (3.10) has the general solution: where Φ( , ) is a smooth function of and .
Proof. The proof can be found in Appendix D.
Propositions 3.1 and 3.2 together guarantee that equations (3.7)-(3.9) hold when equation (3.10) holds. In this paper, we always assume that equation (3.10) holds. In this regard, Tao (2021) has strictly proved that, if equation (3.10) holds, then by using equation (3.9) one has: where ( ) denotes the probability of households taking the collective equilibrum decisions = ( 1 , 2 , … , ) = ( 1 * , … , * ; 1 * , … , * ) . Here, = ( * , * ) denotes the equilibrium strategy of the household and = 1, … , in which * and * denote equilibrium consumption vector and equilibrium production vector of the household , respectively, as described in Examples A.1 and A.2 in Appendix A. Based on Shannon's explanation for amount of information (Shannon, 1948), equation (3.12) indicates that the technology factor can be interpreted as society's information stock 4 (Tao, 2021). From this sense, the technology factor has a broader significance, e.g., interpreting Hayek's theory of knowledge (Tao, 2021). Section 5 will show further that the aggregate production function in the peer-to-peer economy is determined endogenously by equation (3.10), and it includes the Cobb-Douglas form as a special case. By equations (3.7) and (3.8), the economic meanings of the parameters and are determined by and .
Substituting equations (3.7) and (3.8) into equation (2.4) we obtain: The constraint ≥ is considered as the Rational Agent Hypothesis (Tao et al., 2019) in neoclassical economics, which states that firms (or households) enter the market if and only if they can gain the marginal labor-capital return at least to pay for the cost; otherwise they will make a loss. Here, we show further that can be written as a linear function of (per capita) unemployment compensation and interest rate .
The complete differential of equation (3.5) can be written in the form: (3.14) where, = and = denote marginal labor return and marginal capital return, respectively. By the law of diminishing marginal returns, the marginal labor return indicates the lowest wage for a worker that, like (per capita) unemployment compensation 5 , is a critical wage level at which laborers would like to either enter or exit markets. Therefore, we might as well denote by the (per capita) unemployment 4 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 ( ). Therefore, by equation (3.12), the technology factor denotes the information content contained in the event of agents taking the collective decisions . 5 Unemployment compensation is paid by the state to unemployed workers who have lost their jobs due to layoffs or retrenchment. It is meant to provide a source of income for jobless workers until they can find employment. compensation. Furthermore, as we assume that the capital markets are perfectly competitive, the marginal capital return denotes the interest rate.
As Proposition 3.1 holds, by comparing equations (3.6) and (3.14) we have: which can be rewritten as where, = denotes the marginal employment level 6 and = − denotes the marginal rate of technical substitution of labor and capital. Equation (3.16) shows that is a linear function of (per capita) unemployment compensation and interest rate . It implies that indeed includes both contributions from labor wage ( ) and capital revenue ( ).
Consequently, by using equation (3.16), equation (3.13) can be summarized as: Equation (3.17) is the basic form of the exponential income distribution in the peerto-peer economy that shares the publicly available technology. It is the central model of this paper. Because this peer-to-peer economy is an ideal type of competitive market, we identify the exponential income distribution as the benchmark structure of the wellfunctioning market economy. Next, we investigate if the exponential income distribution (3.17) really describes the income structure of a typical market-economy country, i.e., the United Kingdom. Before doing this, we need to clarify that, although equation (3.17) is derived within the ADE, the values of and are unknown. In section 5, we will prove that general equilibrium occurs only when = 0. As a real economy always deviates somewhat from general equilibrium, we should expect that the observed value of is non-zero. By equation (3.17), we realize that exponential income distribution is invalid in describing the income interval [0, ] , in which 6 The marginal employment level stands for the increment of labor once a firm enters markets. unemployed population is located so long as > 0. From this sense, unemployment is a disequilibrium phenomenon in our model. In Figure 1, we use the household income data of the United Kingdom in 2010 to show this invalidity, where, due to the disequilibrium of economy ( = 7204 > 0), the income structure conforms to a rightskewed distribution as described by a Log-Normal function or a Gamma function. Let us denote the population located in [0, ] by the super-low income class. If we remove the super-low income class, by observing Figure 1, the remaining income structure conforms to an exponential distribution. In next section, we confirm this observation by using empirical investigation.

Empirical investigation and falsifiability
Equation ( To conduct empirical analysis for exponential income distribution (3.17), we consider the cumulative distribution of equation (4.1); that is, where denotes the size of sample.
As the peer-to-peer economy describes a competitive economy with equal opportunity, the exponential law (4.5) should be invalid in describing top income households who benefit from the Matthew effect of income accumulation. Therefore, when we fit equation (4.5) to the empirical data, the top income part should be removed.
Furthermore, due to the constraint ≥ , we should also remove the super-low income part that is less than . When the top and super-low income parts are both removed, we denote the remaining income part by the low and middle income classes. We expect that the exponential income distribution (3.17) describes the low and middle income classes.
In this paper, we conduct a rigid investigation for the United Kingdom, since it has a large size sample that involves 99 quantiles (see Data resource in Fig. 2), that is, To eliminate the government's intervention in markets 9 , we use the data of income 8 The data was released by the GOV.UK, see Data resource in Fig. 1 [ ] Second, we investigate if the predicted value of the parameter in equation (4.2) agrees with actual data. By fitting equation (4.5) to household income data, one can obtain the predicted value of . Equation (4.4) predicts a linear relation between and . If the exponential law (4.2) indeed describes actual societies, by equation (4.4) we should expect that the predicted value of is in line with the evolution of (per capita) unemployment compensation announced by the United Kingdom Government. This is the most crucial testing in the falsifiability of the exponential income distribution (3.16), which distinguishes our model from other distribution models, such as Log-Normal distribution and Gamma distribution.
As equation (

Endogenous technological change
In section 3, we mentioned that, when the exponential income distribution (2.4) emerges, the aggregate production function is determined endogenously by the partial differential equation (3.10), which has a general solution (3.11). Now, we show further that, if the economy stays at the general equilibrium, the aggregate production function includes the Cobb-Douglas form as a special case, where the technology factor is determined endogenously by labor and capital. Equation (3.11) can be rewritten as: where χ( ) is a smooth function of .
To find the concrete form of the function χ( ), we need to introduce Proposition 5.1. Before doing this, let us first introduce Lemma 5.1 and Assumption 5.1.
Lemma 5.1: If the economy arrives at general equilibrium, then one has = 0.
Proof. The proof can be found in Appendix E.
By equation (3.16), we know that includes contributions from both labor wage and capital revenue. = 0 implies that the two contributions cancel each other out, so there is no profit on marginal return. This is in accordance with the existing conclusion based on general equilibrium analysis.
Proof. The proof can be found in Appendix E.
To guarantee that the economy stays at general equilibrium, by equations (5.1) and (5.2), the aggregate production function should be written as: Now, we explore the relationship between and when the economy stays at general equilibrium. By equation ( To determine the function form of ( , ), we have the following proposition. (5.7) Proof. The proof can be found in Appendix E.
Here, we simply take: where ≥ 0, ≥ 0, and > 0. By Proposition 5.2, equation (5.8) implies (5.9) By equations (5.8) and (5.9), equation (5.6) can be rewritten as: 11 In section 3, we have interpreted the technology factor as society's information stock. In this sense, labor and capital play a role of a "container" that stores the information or knowledge. should match the existing labor and capital stock. This implies that an efficient growth path for the market-economy countries that pursue the catch-up strategy should be a gradual matching process, in which industrial developments cannot be divorced from the existing labor and capital stock (Lin, 2011;Ju et al., 2015).

Measuring the deviation from general equilibrium
Equation (5.12) describes the dynamic general equilibrium in an ADE. Such a dynamic equilibrium does not occur easily in the real world. Therefore, it is meaningful to seek a way of measuring the potential deviation from dynamic equilibrium. In this section, we attempt to construct an index to measure the potential deviation from general equilibrium that can be applied in a real economy.
By Lemma 5.1, we observe that = 0 indicates general equilibrium. The proof for Lemma 5.1 is based on the equation (3.16) (see Appendix E), which yields: By microeconomics (Mas-Colell et al., 1995), we know that general equilibrium leads to = ; that is = 0. Based on this fact, we can construct an index for measuring the deviation from general equilibrium, as below: Obviously, if general equilibrium occurs, by = , we have = 0. In fact, we have the following proposition. Generally, it is hard to collect the data ( ) because it contains the contributions from all potential capital forms. Therefore, here, we assume approximately that ( ) • ( ) • ( ) does not vary with respect to the time ; that is, Thus, equation (6.4) can be rewritten as: which is exactly the equation (4.6) that has indicated that the approximate assumption [ ]

Conclusion
By introducing the property right of a firm, we show that a long-run ADE can be used to simulate a peer-to-peer economic network that shares the publicly available technology in which there are many households, each of whom independently operates a firm. We have shown that an exponential income distribution will emerge spontaneously in such an economic network. Based on this finding, we identify the exponential income distribution as the benchmark structure of the well-functioning market economy. However, a real market economy may deviate from the wellfunctioning market economy. We show that the deviation is partly reflected as the invalidity of exponential distribution in describing the super-low income class that involves unemployment. In this regard, we find, theoretically, that the lower-bound of exponential income distribution has a linear relationship with (per capita) To explore macroeconomic behaviors of a well-functioning market-economy country, we show that, by using the exponential income distribution, the income summation over all households leads to an aggregate production function with Hicks-   Note: The estimated value = 7204 can be found in Table 1.

Figure 2. Exponential income distribution fitting to household income data in the United Kingdom
Note 13 : Household income data in the United Kingdom are from 2001 to 2015 (except 2008), and they are plotted as the black circles. Exponential income distribution (4.5) is plotted as the blue line for each year. In the data resource, the household income data in 2008 are missing. To eliminate the government's intervention in markets, we use the data of income before tax to fit the exponential income distribution.    and 2 = (2 3 ⁄ , 0), respectively. In such a society, firms produce rice by using labor, and, due to the property right arrangement, the products of firm belong to the household , where = 1,2; meanwhile, households will consume rice. For simplicity, we assume that the utility functions of households 1 and 2 are 1 ( 11 , 21 ) = 21 and 2 ( 12 , 22 ) = 22 , respectively. Because there is not any monopolistic technology in the ADE, we assume that firms 1 and 2 employ the same Cobb-Douglas production technology {( − , )| = }; thus, the production vectors of firms 1 and 2 are 1 = (−ℎ 1 , ℎ 1 ) and 2 = (−ℎ 2 , ℎ 2 ), respectively. We further assume that the price of labor is 1 = 1 and the price of rice is 2 ; thus, the price vector is = ( 1 = 1, 2 ).

Solution：
It is easy to list the following equations.
Maximizing firms' profit follows: The assumption of two types of goods (labor and rice) indicates a kind of schizophrenic society, which seems to be unrealistic. However, the assumption is made solely to guarantee that Example A.1 is easily accessible. In this regard, Tao (2016) has considered any types of goods. uniquely belongs to household 2. However, in a general ADE, the profit • 1 (and • 2 ) will be shared by households 1 and 2 (Arrow and Debreu, 1954). Therefore, the peer-to-peer economy is only a special case of the general ADE. where denotes an arbitrary number satisfying 0 ≤ ≤ 1.
Equations (A.6)-(A.10) show that, although * , 1 * , and 2 * are unique, 1 * and 2 * will change as changes. This means that { * ; 1 * , 2 * ; 1 * , 2 * } are multiple equilibria. Because * , 1 * , and 2 * are fixed, the multiplicity of equilibria is due to the uncertainty of 1 * and 2 * . For simplicity, we might as well denote { * ; 1 * , 2 * ; 1 * , 2 * } by { 1 * , 2 * }. According to the first fundamental theorem of welfare economics, general equilibrium is always Pareto optimal. Regarding multiple equilibria, welfare economists propose that the best outcome can be selected through a "social welfare function" (Mas-Colell et al., 1995). Unfortunately, such a social welfare function has been refuted by Arrow's Impossibility Theorem (Arrow, 1963). For multiple equilibria, this leads to a dilemma of social choice. Next, we show that the dilemma can be eliminated if one introduces the maximum likelihood principle.
Before doing this, let us first introduce the definition of household income.

Definition A.1 (Household income):
Based on the property rights for the peer-to-peer economy, the income of a household equals the total sale of the products that its firm has produced.
Definition A.1 indicates that the income of a household is determined by the revenue (or size) of its firm. There has been a large body of literature relating a firm's revenue to a household's income (Lucas, 1978;Rosen, 1982;Luttmer, 2007;Gabaix and Landier, 2008;Jones and Kim, 2014). Here, Definition A.1 differs from the definition for wealth. For instance, the wealth of the household should be defined by * • + * • * , where = 1,2 . However, in this paper, we only investigate the income distribution of households.
Using equations (A.6), (A.9), and (A.10) it is easy to ascertain that the equilibrium revenues of firm 1 and 2 are and 1 − , respectively. Since the revenue of a firm equals the sale of products that it has produced, by Definition A.1, the equilibrium income allocation between households 1 and 2 is as below: ( 1 , 2 ) = ( , 1 − ).
where denotes the income of household and = 1,2. Therefore, the GDP of this society equals = + (1 − ) = 1. Now, we introduce the maximum likelihood principle. It is a basic principle of statistics to infer unknown probability distributions under given constrained conditions (Foley, 1994;Martin et al., 2012). For example, Gauss (1857) first used this principle to determine the normal distribution (Tao et al., 2017), which was also known as the Gauss distribution. In the context of evolutionary game, this principle is referred to as "stochastically stable" (Young, 1998). According to previous discussion, these three allocations are all Pareto optimal, so one cannot choose the best one by using the theory of social choice. To apply the maximum likelihood principle, let us first introduce the notion of income distribution.
If we assume that the number of households whose income is 0 equals 1 , the number of households whose income is 1/2 equals 2 , and the number of households whose income is 1 equals 3 , then { } =1 3 = { 1 2 3 } is called an income distribution. In Example A.2, we will give the general definition for income distribution.
Obviously, according to the definition of income distribution above, the multiple allocations (A.12) only lead to two income distributions, as below: By using the maximum likelihood principle we choose the likeliest one from income distributions (A.13) and (A.14). To this end, we observe that the ADE is a decentralized system with procedural justice, in which there is no criterion for what constitutes a just outcome (equilibrium) other than the procedure itself. Rawls (1999) has proposed that each outcome produced by a system with procedural justice is fair and, hence, can be selected with an equal opportunity. Following Rawls' proposal, we make the axiom of equal opportunity as follows:  According to the maximum likelihood principle, the MLD can be taken as a real occurrence (i.e., spontaneous order). The main purpose of this section is to find the general function form of the MLD in an ADE. To obtain such a general function form, we have to extend a 2-household economy to an -household economy.
Example A.1 can be easily extended to the case of an -household, as below.

Example A.2: -household peer-to-peer economy
We consider an -household agricultural society, in which each household 16 Here, we have considered non-dictatorship as an axiom of defining an Arrow-Debreu economy. (1). There are a total of possible income levels: 1 < 2 < ⋯ < ; (2). There are households, each of which obtains units of income, and runs from 1 to .

Appendix B
To guarantee ≫ 1 , here we assume ≫ 1 for = 1,2,3 . Because and ln have the same monotonicity, the extremum problem (A.26) can be written as: where we have abandoned the constant of integration.

Proof of Proposition 5.2
Proof. By the setting in section 2, the long-run production technology exhibits the constant return, so we have: