Spontaneous economic order

Abstract

This paper provides attempts to formalize Hayek’s notion of spontaneous order within the framework of an Arrow-Debreu economy. Our study shows that, if a competitive economy is sufficiently fair and free, a spontaneous economic order will emerge in long-run competitive equilibria so that social members spontaneously occupy an unplanned distribution of income. Despite this, the spontaneous order may degenerate in the form of economic crises whenever an equilibrium economy approaches the extreme competition. Remarkably, such a theoretical framework of spontaneous order provides a bridge linking Austrian economics and neoclassical economics, where a truth begins to emerge: “Freedom promotes technological progress”.

1 Introduction

Spontaneous order in economic interactions presented by Hayek (1948) is an important notion for economics. This notion originates from the interactions of members of society and is something to which everyone contributes, from which everyone benefits, which everyone normally takes for granted, but which individuals rarely understand (Witt 1997). Hayek believed that, if the degree of freedom within a society, under the constraint of limited resources, achieves a maximum, a spontaneous order would emerge so that social members occupy an optimal allocation of resources. Nevertheless, no one clarifies what the spontaneous order is; for instance, is it a natural law? We have known that, in his early years, Hayek focused his research on the theory of business cycles and later turned to the theory of a spontaneous economic order (Witt 1997). Regarding Hayek’s change, one naturally wonders if there were certain relations between economic crisis and spontaneous order. Unfortunately, Hayek never reconsidered the business cycle theory, given his later beliefs.

We are all now witnesses to the huge financial crisis that began in 2008. Regarding this crisis, many people attribute the origin of it to the laissez faire policies that support free markets (Bouchaud 2008). Therefore, a question may arise: “Does the spontaneous order always benefit free economies?” Regrettably, we cannot answer this question because thus far there is no free market model that is truly based on the principle of spontaneous order.¹ For example, the mainstream model of modern economics is called the Dynamic Stochastic General Equilibrium (hereafter DSGE) in which there is not an interesting variable corresponding to spontaneous order or to a degree of freedom of economic systems. In particular, we often do not find a strict solution to DSGE; however, we can prove the existence of the equilibrium solution, which has certain fine properties such that economic crises are eliminated. This means that we cannot clarify the origin of economic crises solely via the general equilibrium theory. Consequently, certain scholars appealed to abandon DSGE and neoclassical economics to develop alternative economic theories for the early warning of economic crises, refer to Hodgson (2009), Leijonhufvud (2009) and Farmer and Foley (2009). However, it may be illogical to abandon neoclassical economics because it had attained great success in the past. This paper aims to formalize the notion of spontaneous order within the framework of neoclassical economics. Later, we will observe that introducing spontaneous order is inevitable if multiple competitive equilibria arise and that economic crisis is an unstable state of spontaneous order.

We first demonstrate why multiple competitive equilibria will occur in the neoclassical economics. As is well known, to guarantee that a competitive economy has a unique equilibrium outcome, we must assume that each consumer’s preferences are strictly convex (Jehle and Reny 2001; Page 188) and that each firm’s production possibility sets are strongly convex (Jehle and Reny 2001; Page 206). Indeed, strict convexity of preferences is necessary because it exhibits the principle of diminishing marginal rate of substitution in consumption (Jehle and Reny 2001; Page 12). However, strong convexity will eliminate constant returns to scale in production, and the latter is considerably important in the neoclassical production theory. If, instead, mere convexity of production possibility sets is assumed, the existence of an equilibrium outcome can still be guaranteed (Debreu 1971; Page 84). It is crucial that the convexity (rather than strong convexity) of production sets allows the possibility of constant returns to scale for firms (Jehle and Reny 2001; Page 216). Thus, the convexity of production possibility sets actually ensures the existence of long-run equilibrium outcome because constant returns technology is a (only sensible) long-run production technology (Varian 1992; Page 356). More importantly, the long-run level of profits for a competitive firm that has constant returns to scale is a zero level of profits (Varian 2003; Page 340); therefore, one must confront a curious equilibrium: each firm always gains zero economic profit regardless of how it behaves. This equilibrium strongly implies that the long-run competitive economy may have multiple (or indeterminate) equilibrium outcomes.

To strictly clarify whether a long-run competitive economy produces multiple equilibria, we need to introduce an exact definition for such an economy. In this paper, we will specify a long-run competitive economy using an Arrow-Debreu economy with additivity and publicly available technology. Traditional literature (Mas-Collel et al. 1995; Page 334) demonstrated that the long-run competitive economy is a situation of competitive economies when free entry is permitted. Generally, additivity means that there is free entry for firms into a possible industry (Debreu 1971; Page 41). Hence, additivity and publicly available technology combined would guarantee that there is free entry for firms into any (technology) industry. Moreover, publicly available technology (Mas-Collel et al. 1995; Page 653) implies that firms produce their products with a similar or an identical technology; therefore, monopolistic competition and perfect competition are allowed for as well. On the basis of these reasons above, we believe that the Arrow-Debreu economy with additivity and publicly available technology exactly describes the long-run competitive economy. Later, we will prove that such an economy indeed has many (infinitely many) equilibrium outcomes. In addition, according to the first fundamental theorem of welfare economics, these equilibrium outcomes should be all Pareto optimal. Because each equilibrium outcome is associated with a different social state, multiple equilibrium outcomes actually imply an uncertain economic world. To eliminate the uncertainty, welfare economists attempt to search the best outcome through an imaginary social welfare function. Unfortunately, Arrow’s Impossibility Theorem refuted the existence of such a welfare function (Jehle and Reny 2001; Page 243).

In this paper, we propose a scheme for “eliminating” the uncertainty: we introduce the paradigm of natural selection into neoclassical economics. In our setting, the natural selection will obey “survival of the likeliest”. The intuition behind our approach is as follows. Now, a long-run competitive economy produces multiple equilibrium outcomes, each of which is Pareto optimal. We may assume that all of these outcomes are equally likely to occur (or equivalently, to be selected with equal opportunities as collective decisions). Equal opportunities among equilibrium outcomes essentially imply an absolutely fair world in which there is no any difference between all of the outcomes. Thus, if there exists an economic order (or a convention) that contains the most equilibrium outcomes, it does occur with the highest probability (compared to other economic orders). We shall define such an economic order with the highest probability as the spontaneous economic order. Our definition is based on Hayek’s core idea (Sugden 1989): Spontaneous order is a convention that is most likely to evolve and survive.

To formalize the intuition above, we may consider a long-run competitive economy that produces four equilibrium outcomes {B ₁, B ₂, B ₃, B ₄}, and each outcome is Pareto optimal. We have provided such an example in section 4 (refer to Example 4.1). If the competitive economy is sufficiently fair,² we can consider it as a fair procedure that would translate its fairness to the outcomes (Rawls 1999; Page 75) so that all of the social members would be indifferent between these equilibrium outcomes; that is,

$$ {B}_1\sim {B}_2\sim {B}_3\sim {B}_4. $$

(1.1)

By the indifference relation (1.1), every social member will have no desire to oppose or prefer a certain outcome. This means that, under the democratic circumstance, every equilibrium outcome should be selected with an equal opportunity as collective decisions (or equivalently, every equilibrium outcome should occur with an equal probability). From this meaning, (1.1) implies that every equilibrium outcome B _i should occur with the probability $\frac{1}{4}$, where i = 1, 2, 3, 4. Assume now that these four equilibrium outcomes could be divided into the following three economic orders³ (or three conventions): a ₁ = {B ₁}, a ₂ = {B ₂, B ₃} and a ₃ = {B ₄}. By such a division, we immediately understand that a ₁ occurs with the probability $\frac{1}{4}$, a ₂ occurs with the probability $\frac{1}{2}$, and a ₃ occurs with the probability $\frac{1}{4}$. Therefore, a ₂ will occur with the highest probability, and by our definition (i.e., survival of the likeliest) a ₂ will be a spontaneous economic order. In addition, by Sen’s argument (Sen 1993), more equilibrium outcomes imply more choice opportunities or greater opportunity-freedom. From this meaning, the spontaneous economic order a ₂ not only obeys fairness but also owns the greatest opportunity-freedom.

Summarizing the analyses above, we are able to develop three steps for seeking the spontaneous economic order. First, we attempt to find all possible equilibrium outcomes of a competitive economy. Second, we divide all of these equilibrium outcomes into different economic orders. Finally, we find the economic order that contains the most equilibrium outcomes.

The main purpose of this paper is to seek the spontaneous order of a long-run competitive economy using the three steps above. To achieve this purpose, with each economic order we shall associate a possible individuals’ revenue distribution. With this setting, we later show that the spontaneous order of a monopolistic-competitive economy will obey a stable rule: exponential distribution; in addition, the spontaneous order of a perfectly competitive economy will obey an unstable rule: Bose-Einstein distribution. Specifically, the instability of the latter may cause economic crises (Tao 2010). It is worth emphasizing that recent empirical investigations have supported that the individuals’ revenue distribution of free economies (e.g., USA) during a stable economic period obeys exponential distribution, refer to Yakovenko and Rosser (2009), Clementi et al. (2012); and obeys Bose-Einstein distribution in the run-up to an economic crisis, refer to Kürten and Kusmartsev (2011), Kusmartsev (2011).

Because our finding may appear moderately surprising, we attempt to convey an intuition for the result. As is well known, in microeconomics, there are four types of markets: perfectly competitive market, monopolistic-competitive market, oligopoly market and perfectly monopoly market. It is acknowledged, among these four types of markets, that the perfectly competitive market is most efficient. It is worth noting that, before every extremely serious economic crisis occurred, without exception, there appeared extremely prosperous economies, particularly in a financial market.⁴ Therefore, a natural question arises: which type of the above four markets shall, most likely, cause the extremely prosperous economy?

Logically, the answer should be the most efficient market: a perfectly competitive market.

Let us recall that there was a common feature in the past three serious economic crises⁵; that is, before these crises occurred, without exception, extremely prosperous economies had appeared. Of course, for every past economic crisis, one always could find an explanation that appears appropriate for the origin of that crisis, e.g., asymmetric information, currency mismatch between assets and liabilities of firms (Deesomsak et al. 2009), or greed (selfishness), to explain the origin of economic crisis in 2008. However, we need to remember that the “selfishness” is one of several axioms of economics. Perfect competition is regarded as the extreme case of competitive economies. Logically, as the competition in a free economy increases, the economy shall naturally evolve toward extreme competition (i.e., perfect competition); however, according to our argument, perfect competition is unstable and may cause economic crisis. That is, a thing turns into its opposite if pushed too far.

The remainder of our paper is organized as follows. Section 2 introduces the definition of long-run competitive equilibrium within the framework of an Arrow-Debreu economy. Subsections 3.1 and 3.2 prove that a long-run competitive economy has at least an equilibrium outcome. Subsection 3.3 and section 4 show that, by a long-run equilibrium outcome, one can produce infinitely many long-run equilibrium outcomes. Section 5 introduces the concepts of economic order and spontaneous economic order, and we further show that all of the long-run equilibrium outcomes can be appropriately (non-repeated) divided into different economic orders. Section 6 shows that one can seek the spontaneous economic order through normative criteria regarding fairness and freedom. Section 7 investigates the possible link between spontaneous economic order and neoclassical macroeconomics, and later introduces certain empirical evidence that supports our results. Section 8 explores the relation between technological progress and social freedom. In section 9, our conclusion follows.

2 Preliminaries

We begin by describing a competitive economy that is composed of a vast number of agents (consumers and firms) and diverse industries. In accordance with the standard framework of the neoclassical economics (Mas-Collel et al. 1995; Page 579), we assume that there are M consumers, N firms and L commodities. Every consumer i = 1, …, M is specified by a consumption set X _i  ⊂ R ^L, a preference relation ^≻_∼ i on X _i , an initial endowment vector ω _i  ∈ R ^L, and an ownership share θ _ij  ≥ 0 of each firm j = 1, …, N (where ${\displaystyle \sum_{i=1}^M{\theta}_{ij}}=1$). Each firm j is characterized by a production set Y _j  ⊂ R ^L. All allocations for such an economy are a collection of consumption and production vectors:

$$ \left((x),(y)\right)=\left({x}_1,\dots, {x}_M,{y}_1,\dots, {y}_N\right)\in {X}_1\times \dots \times {X}_M\times {Y}_1\times \dots \times {Y}_N, $$

where x _i  = (x _1i , …, x _Li ) and y _j  = (y _1j , …, y _Lj ).

2.1 Arrow-Debreu economy and competitive equilibrium

A well-known definition for competitive equilibrium is introduced as follows.

Definition 2.1 (Mas-Collel et al. 1995; Page 579): An allocation ((x ^c), (y ^c)) and a price vector p = (p ₁, …, p _L ) constitute a competitive (or a Walrasian) equilibrium if the following three conditions are satisfied:

• (1). Profit maximization: For every firm j, y ^c _j  ∈ Y _j maximizes profits in Y _j ; that is,

$$ p\cdot {y}_j\le p\cdot {y}_j^c\ \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{all}\;{y}_j\in {Y}_j $$

• (2). Utility maximization: For every consumer i, x ^c _i  ∈ X _i is maximal for $\underset{\sim i}{\succ }$ in the budget set:

$$ \left\{{x}_i\in {X}_i:p\cdot {x}_i\le p\cdot {\omega}_i+{\displaystyle \sum_{j=1}^N{\theta}_{ij}p\cdot {y}_j^c}\right\}. $$

• (3). Market clearing: ${\displaystyle \sum_{i=1}^M{x}_i^c}={\displaystyle \sum_{i=1}^M{\omega}_i}+{\displaystyle \sum_{j=1}^N{y}_j^c}$.

One can verify that the equilibrium allocation ((x ^c), (y ^c)) does exist if the following nine conditions are satisfied (Debreu 1971; page 84).

For every consumer i:

• (a) Each consumer’s consumption set X _i is closed, convex, and bounded below;

• (b) There is no satiation consumption bundle for any consumer;

• (c) For each consumer i = 1, …, M, the sets $\left\{{x}_i\in {X}_i\left|{x}_i\underset{\sim i}{\succ}\right.{x}_i^{\prime}\right\}$ and $\left\{{x}_i\in {X}_i\left|{x}_i^{\prime}\underset{\sim i}{\succ}\right.{x}_i\right\}$ are closed;

• (d) If x ¹ _i and x ² _i are two points of X _i and if t is a real number in (0, 1), then ${x}_i^2\underset{\sim i}{\succ }{x}_i^1$ implies $t{x}_i^2+\left(1-t\right){x}_i^1\underset{i}{\succ }{x}_i^1$;

• (e) There is x ⁰ _i in X _i , such that x ⁰ _i  < < ω _i ;

For every firm j:

• (f) 0 ∈ Y _j ;

• (g) $Y={\displaystyle \sum_{j=1}^N{Y}_j}$ is closed and convex;

• (h) − Y ∩ Y = {0};

• (i) − R ^L₊  ⊂ Y.

If a competitive economy satisfies (a)-(i), it is called an Arrow-Debreu economy (Arrow and Debreu 1954). In particular, (d) will guarantee that each consumer’s preferences are strictly convex. It should be noted that Debreu did not restrict his discussion on the strict convexity of preferences. Instead, mere convexity of preferences was assumed in his famous book (Debreu, 1971; Page 84). However, strict convexity of preferences is necessary for neoclassical economics because it exhibits the principle of diminishing marginal rate of substitution in consumption (Jehle and Reny 2001; Page 12). Technically, strict convexity of preferences will guarantee that (x ^c₁ , …, x ^c _M ) is a unique equilibrium consumption allocation.

Here, we do not assume the strong convexity of production possibility sets because it will eliminate constant returns to scale in production (Jehle and Reny 2001; Page 206). In section 3, we will observe that constant returns technology is inevitable when long-run competition is considered and that (f)-(i) allow the possibility of constant returns to scale for firms.

2.2 Long-run competitive equilibrium

In accordance with Mas-Collel et al. (1995; Page 334), we consider the case in which “an infinite number of firms can potentially be formed”; that is, N → ∞. Moreover, each firm has access to the publicly available technology; thus, it may enter and exit an industry in response to profit opportunities. “This scenario, known as a situation of free entry, is often a reasonable approximation when we think of long-run outcomes” in an industry (or a market). Under such a scenario, Mas-Collel et al. deduced (1995; Page 335): “A firm will enter the market if it can earn positive profits at the going market price and will exit if it can make only negative profits at any positive production level given this price. If all firms, active and potential, take prices as unaffected by their own actions, this implies that active firms must earn exactly zero profits in any long-run competitive equilibrium; otherwise, we would have either no firms willing to be active in the market (if profits were negative) or an infinite number of firms entering the market (if profits were positive)”. Thus, if all of the industries (or markets) remain at long-run competitive equilibria, we have:

$$ p\cdot {y}_j^c=0, $$

(2.1)

$$ j=1,\dots, N $$

By substituting (2.1) into Definition 2.1, we can present a natural definition for long-run competitive equilibrium as follows.

Definition 2.2: An allocation ((x*), (y*)) and a price vector p = (p ₁, …, p _L ) constitute a long-run competitive equilibrium if the following three conditions are satisfied:

• (1). For every firm j, there exists y ^* _j  ∈ Y _j such that p ⋅ y _j  ≤ p ⋅ y ^* _j  = 0 for all y _j  ∈ Y _j .

• (2). For every consumer i, x ^* _i  ∈ X _i is maximal for $\underset{\sim i}{\succ }$ in the budget set:

$$ \left\{{x}_i\in {X}_i:\kern0.1em p\cdot {x}_i\le p\cdot {\omega}_i\right\}. $$

• (3). ${\displaystyle \sum_{i=1}^M{x}_i^{*}}={\displaystyle \sum_{i=1}^M{\omega}_i}+{\displaystyle \sum_{j=1}^N{y}_j^{*}}$.

In contrast to Definition 2.1, Definition 2.2 has a stronger constraint; that is, the maximum profit of every firm j, p ⋅ y ^* _j , is restricted to be null. Because of this constraint, we cannot guarantee that ((x*), (y*)) does exist although (a)-(i) are satisfied. In the next section, our focus will be on the existence of the long-run equilibrium allocation ((x*), (y*)). To avoid confusion, when we note an equilibrium allocation in the remainder of this paper, we always mean that it denotes a long-run equilibrium allocation.

3 Long-run competitive economy

In this section, we will verify the existence of long-run competitive equilibrium. Before proceeding, let us first explore what conditions will restrict the maximum profit of every firm j to be null within the framework of the Arrow-Debreu economy.

3.1 Assumptions

Assumption 3.1 (additivity): Y _j  + Y _j  ⊂ Y _j for every j.

If the production set of the j th firm, Y _j , can be interpreted as an industry, the Assumption 3.1 means that there is free entry for firms into that industry (Debreu 1971; Page 41). More importantly, we have the result as below:

Theorem 3.1: If Assumption 3.1 and (f) are satisfied, the maximum profit of every firm j is zero; that is, p ⋅ y ^* _j  = 0 for j = 1, …, N.

Proof. Refer to page 45 in Debreu (1971). □

Theorem 3.1 demonstrates that the Arrow-Debreu economy under Assumption 3.1 will restrict the maximum profit of every firm to be null (if the maximum profit exists). Unfortunately, Assumption 3.1 does not really imply free entry because Y _j represents a private production set (of the j th firm) rather than a public industry. However, if the following assumption is satisfied, Assumption 3.1 will imply free entry.

Assumption 3.2 (publicly available technology⁶): Y ₁ = Y ₂ = … = Y _N .

Assumption 3.2 implies that every firm has free access to one another’s technology. Then, every Y _j represents a public production set, and thus can be interpreted as a public (open) industry. Therefore, Assumptions 3.1 and 3.2 combined imply free entry. Furthermore, we have the following proposition:

Proposition 3.1: Assumptions 3.1 and 3.2 combined guarantee that Y _j  = Y for j = 1, 2, …, N and Y + Y ⊂ Y, where $Y={\displaystyle \sum_{j=1}^N{Y}_j}$.

Proof. To verify this proposition, we need to prove that, for any y ∈ Y there must be y ∈ Y _j . Because y ∈ Y, we have $y={\displaystyle \sum_{k=1}^N{y}_k}$, where y _k  ∈ Y _k . Then, by Assumption 3.2, we immediately obtain y _k  ∈ Y _j for k = 1, 2, … N, where j = 1, 2, …, N. Finally, by Assumption 3.1, we have $y={\displaystyle \sum_{k=1}^N{y}_k}\in {Y}_j$. □

3.2 Existence of long-run competitive equilibrium

Because the Arrow-Debreu economy under Assumption 3.1 will restrict the maximum profit of every firm to be null (if the maximum profit exists), and because Assumptions 3.1 and 3.2 combined imply free entry, we note that, if the Arrow-Debreu economy under Assumptions 3.1 and 3.2 has an equilibrium allocation ((x′), (y′)), ((x′), (y′)) does satisfy the Definition 2.2. Therefore, we can introduce an exact definition for long-run competitive economy as follows.

Definition 3.1: A competitive economy is a long-run competitive economy (hereafter LRCE) if and only if:

• (1). (a)-(i) are satisfied;

• (2). Assumptions 3.1 and 3.2 hold.

To verify that a LRCE has at least an equilibrium allocation, we solely need to prove that (a)-(i) are compatible with Assumptions 3.1 and 3.2. This is because (a)-(i) would ensure the existence of an equilibrium allocation (Debreu 1971; page 84). Before proceeding, let us introduce two lemmas.

Lemma 3.1: If (f)-(g) are satisfied, and if Assumptions 3.1 and 3.2 hold, Y is a cone with vertex 0, i.e., y ∈ Y implies ty ∈ Y for any scalar t ≥ 0.

Proof. First, by (f), one has 0 ∈ Y, and by (g), Y satisfies convexity; therefore, for any y ∈ Y and any c ∈ [0, 1], one has cy = cy + (1 − c) ⋅ 0 ∈ Y. Second, by proposition 3.1 (because Assumptions 3.1 and 3.2 hold) Y satisfies additivity; that is, for any non-negative integer k, one has ky ∈ Y. Let t be any non-negative number satisfying t ≤ k, then the two results above imply $ty=\frac{t}{k}\cdot ky\in Y$. □

Because a cone with vertex 0 implies constant returns to scale (Debreu 1971; page 46), we immediately have two corollaries:

Corollary 3.1: The production set Y exhibits constant returns to scale.

Corollary 3.2: The production set of each firm, Y _j , exhibits constant returns to scale.

Lemma 3.2: If Y is a cone with vertex 0, and if Y is closed and convex, Y must be a closed, convex cone with vertex 0.

Proof. Refer to page 42 in Debreu (1971) □

Combining Lemmas 3.1 and 3.2, we have the following theorem:

Theorem 3.2: A LRCE has at least an equilibrium allocation ((x*), (y*)).

Proof. We start to prove that (a)-(i) are compatible with Assumptions 3.1 and 3.2. By Lemma 3.1, (f)-(g) together with Assumptions 3.1 and 3.2 guarantee that Y is a cone with vertex 0. Thus, by (g) and Lemma 3.2, Y is further a closed, convex cone with vertex 0. Such a result does not contradict (f)-(g). This means that (a)-(i) still hold although Assumptions 3.1 and 3.2 arise. □

Now that the LRCE satisfies (a)-(i) (refer to the proof above), we immediately obtain two corollaries as below:

Corollary 3.3: The LRCE is an Arrow-Debreu economy.

Corollary 3.4: Any long-run equilibrium allocation ((x*), (y*)) is a competitive (or a Walrasian) equilibrium.

3.3 Multiplicity of long-run competitive equilibria

In subsection 3.2, we have proved that the LRCE has at least an equilibrium allocation ((x*), (y*)) = (x ^*₁ , …, x ^* _M , y ^*₁ , …, y ^* _N ). Next, we show that by ((x*), (y*)) one can produce infinitely many equilibrium allocations.

Let $z(p)={\displaystyle \sum_{j=1}^N{y}_j^{*}}$ denote the aggregate production vector, then we have three results as below:

Proposition 3.2: z(p) ∈ Y _j for j = 1, …, N.

Proof. By Proposition 3.1, one has z(p) ∈ Y = Y _j for j = 1, … N. □

Lemma 3.3: tz(p) ∈ Y _j for j = 1, …, N, where t ≥ 0.

Proof. By Lemma 3.1 and Proposition 3.2, one concludes tz(p) ∈ Y _j . □

Lemma 3.4: p ⋅ z(p) = 0.

Proof. By Definition 2.2, one has p ⋅ y ^* _j  = 0 for j = 1, …, N. □

Let us consider a sequence of numbers, {t _j } ^N _j = 1 , satisfying:

$$ \left\{\begin{array}{c}\hfill \begin{array}{ccc}\hfill {t}_j\ge 0\hfill & \hfill for\hfill & \hfill j=1,2,\dots, N\hfill \end{array}\hfill \\ {}\hfill {\displaystyle \sum_{j=1}^N{t}_j}=1\hfill \end{array}\right.. $$

(3.1)

Then, by Lemmas 3.3 and 3.4, we can prove the following proposition.

Proposition 3.3: Let

$$ {y}_j^e\left({t}_j\right)={t}_jz(p) $$

(3.2)

for j = 1, 2, …, N, then (x ^*₁ , …, x ^* _M , y ^e₁ (t ₁), …, y ^e _N (t _N )) constitutes a long-run equilibrium allocation.

Proof. Because ${\displaystyle \sum_{j=1}^N{y}_j^e\left({t}_j\right)}=z(p)={\displaystyle \sum_{j=1}^N{y}_j^{*}}$, we only need to verify that each y ^e _j (t _j ) satisfies the condition (1) in Definition 2.2. Thus, by Lemma 3.3 y ^e _j (t _j ) ∈ Y _j and by Lemma 3.4 p ⋅ y ^e _j (t _j ) = 0, where j = 1, …, N. □

The proof above implies two corollaries as below:

Corollary 3.5: ${\displaystyle \sum_{j=1}^N{y}_j^e\left({t}_j\right)}=z(p)$.

Corollary 3.6: p ⋅ y ^e _j (t _j ) = 0 for j = 1, …, N.

Proposition 3.3 indicates that each sequence {t _j } ^N _j = 1 satisfying (3.1) will correspondingly produce a different long-run equilibrium allocation. Undoubtedly, there are infinitely many possible sequences {t _j } ^N _j = 1 satisfying (3.1); therefore, there will be infinitely many possible long-run equilibrium allocations, too.

Lemma 3.5 (First fundamental theorem of welfare economics): Any Walrasian equilibrium allocation is Pareto optimal.

Proof. Refer to page 549 in Mas-Collel et al. (1995). □

Using Corollary 3.4 and Lemma 3.5, we can prove an important proposition.

Proposition 3.4: Any equilibrium allocation (x ^*₁ , …, x ^* _M , y ^e₁ (t ₁), …, y ^e _N (t _N )) obeying (3.1) is Pareto optimal.

Proof. By Proposition 3.3 and Corollary 3.4 any (x ^*₁ , …, x ^* _M , y ^e₁ (t ₁), …, y ^e _N (t _N )) obeying (3.1) is Walrasian equilibrium. Then, by Lemma 3.5, we complete this proof. □

4 Uncertainty of social choice

Propositions 3.3 and 3.4 combined demonstrate that the LRCE will produce uncertain equilibrium outcomes (x ^*₁ , …, x ^* _M , y ^e₁ (t ₁), …, y ^e _N (t _N )), each of which is Pareto optimal. Additionally, the following proposition will further reveal that the uncertainty of equilibrium outcomes is due to production rather than consumption.

Proposition 4.1: (x ^*₁ , x ^*₂ , …, x ^* _M ) is a unique equilibrium consumption allocation.

Proof. If there were another equilibrium consumption allocation (x ^′₁ , x ^′₂ , …, x ^′ _M ) satisfying ${x}_i^{\prime}\underset{\sim i}{\succ }{x}_i^{*}$ for i = 1, …, M, by the condition (d) we do have x ^″ _i  = tx ^′ _i  + (1 − t)x ^* _i  ∈ X _i so that ${x}_i^{{\prime\prime}}\underset{i}{\succ }{x}_i^{*}$ contradicting the condition (2) of Definition 2.2, where 0 < t < 1. □

Using the Proposition 4.1 and the condition (3) of Definition 2.2 we immediately arrive at two corollaries:

Corollary 4.1: (x ^*₁ , …, x ^* _M , y ^e₁ (t ₁), …, y ^e _N (t _N )) can be reduced to (y ^e ₁ (t ₁ ) …, y ^e _N (t _N )).

Corollary 4.2: z(p) is a fixed vector.

Proposition 4.1 and Corollary 4.1 imply that all equilibria involve the same consumption vector; hence, all consumers are indifferent between all equilibria, and the multiplicity arises solely from the distribution of production. Then, a doubt may occur; because the publicly available technology is assumed (refer to Assumption 3.2), the multiplicity is perhaps a meaningless (or spurious) multiplicity. However, the multiplicity of equilibria must be admitted because the economic crises are hidden in such a multiplicity. To observe this, we consider a possible long-run equilibrium outcome (x ^*₁ , …, x ^* _M , z(p), 0, …, 0), where y ^e₁ (t ₁) = z(p) and y ^e _j (t _j ) = 0 for j = 2, …, N. This equilibrium strongly indicates an economic crisis: only one firm survives (wins), and others all go bankrupt because they cease production for a long time. Later, we shall observe that such an equilibrium involving economic crisis will not occur in a monopolistic-competitive economy.⁷ Unfortunately, we cannot eliminate it in a perfectly competitive economy.⁸

Moreover, the Corollary 4.1 reminds us: to describe the uncertainty of equilibrium outcomes, we only need to consider the equilibrium production allocation (y ^e₁ (t ₁), …, y ^e _N (t _N )). Thus, for convenience, we may denote by (y ^e₁ (t ₁), …, y ^e _N (t _N )) the long-run equilibrium allocation.

4.1 Equilibrium revenue allocation

Without loss of generality, we assume that z(p) has at most one positive component (namely, a single output⁹).

Assumption 4.1:

$$ \left\{\begin{array}{c}\hfill z(p)=\left({z}_1(p),\dots, {z}_m(p),\dots, {z}_L(p)\right)\hfill \\ {}\hfill {z}_m(p)\ge 0\hfill \\ {}\hfill \begin{array}{cc}\hfill {z}_l(p)\le 0\hfill & \hfill l=1,\dots, m-1,m+1,\dots, L\hfill \end{array}\hfill \end{array}\right. $$

(4.1)

where, z _m (p) represents the outputs’ amount, and z _l (p) represents the inputs’ amount.

Substituting (4.1) into (3.2), we understand that the equilibrium outputs’ amount of the j th firm is specified by t _j z _m (p). Because the equilibrium price of the m th commodity (i.e., output) is denoted by p _m , the j th firm will obtain t _j p _m z _m (p) units of revenue.

If we refer to ε _j (t _j ) as the equilibrium revenue of the j th firm, and refer to Π as the equilibrium total revenue of all firms, we have:

$$ {\varepsilon}_j\left({t}_j\right)={t}_j{p}_m{z}_m(p), $$

(4.2)

$$ {\displaystyle \sum_{j=1}^N{\varepsilon}_j\left({t}_j\right)}=\varPi . $$

(4.3)

Substituting (4.2) into (4.3) we obtain:

$$ \varPi ={p}_m{z}_m(p). $$

(4.4)

Using (4.4), the formula (4.2) can be rewritten as:

$$ {\varepsilon}_j\left({t}_j\right)={t}_j\varPi . $$

(4.5)

Definition 4.1: An equilibrium revenue allocation is a collection of firms’ revenue scalars:

$$ \left({\varepsilon}_1\left({t}_1\right),\dots, {\varepsilon}_N\left({t}_N\right)\right). $$

(4.6)

The equilibrium revenue allocation (ε ₁(t ₁), …, ε _N (t _N )) shows revenue allocations among N firms when the economy achieves long-run competitive equilibria; therefore, there is no essential distinction¹⁰ in denoting the long-run equilibrium allocation either by (ε ₁(t ₁), …, ε _N (t _N )) or by (y ^e₁ (t ₁), …, y ^e _N (t _N )). However, compared to a production allocation (y ^e₁ (t ₁), …, y ^e _N (t _N )), a revenue allocation (ε ₁(t ₁), …, ε _N (t _N )) is often easier to be empirically tested. Therefore, we are more interested in exploring the revenue allocation of an economy. In the remainder of this paper, we always denote the long-run equilibrium allocation (or equilibrium outcome) by (ε ₁(t ₁), …, ε _N (t _N )).

4.2 Uncertain equilibrium outcomes

Combining (3.1) and (4.5), one easily notes that any revenue allocation (ε ₁(t ₁), …, ε _N (t _N )) satisfying the following requirements

$$ \left\{\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\varepsilon}_j\left({t}_j\right)\ge 0\hfill & \hfill for\hfill & \hfill j=1,2,\dots, N\hfill \end{array}\hfill \\ {}\hfill {\displaystyle \sum_{j=1}^N{\varepsilon}_j\left({t}_j\right)}=\varPi \hfill \end{array}\right. $$

(4.7)

is a long-run equilibrium allocation.

Therefore, by Corollary 3.4 and Lemma 3.5, we immediately arrive at¹¹:

Corollary 4.3: Any revenue allocation (ε ₁(t ₁), …, ε _N (t _N )) satisfying (4.7) is Pareto optimal.

Undoubtedly, (4.7) implies that there are infinitely many possible equilibrium outcomes. We soon show that these equilibrium outcomes (ε ₁(t ₁), …, ε _N (t _N )) can be simply depicted as different figures. To help readers follow our idea more easily, the constraint ${\displaystyle \sum_{j=1}^N{\varepsilon}_j\left({t}_j\right)}=\varPi$ in (4.7) will be temporarily abandoned (specifically, we abandon this constraint in subsection 4.2, sections 5 and subsections 6.1-6.2). However, we will resume the constraint ${\displaystyle \sum_{j=1}^N{\varepsilon}_j\left({t}_j\right)}=\varPi$ starting from subsection 6.3.

We now consider a simple LRCE with two firms as below:

Example 4.1: Assume that there exists an LRCE in which there are a total of two firms and two industries. Moreover, assume that, if a firm enters industry 1, it will obtain ε ₁ units of revenue; if a firm enters industry 2, it will obtain ε ₂ units of revenue. In addition, assume that ε ₁ < ε ₂.

Let us first explore how many equilibrium outcomes Example 4.1 has. Because there are a total of two firms, we need to count all possible revenue allocations (ε ₁(t ₁), ε ₂(t ₂)) that satisfy (4.7). However, because the constraint ${\displaystyle \sum_{j=1}^2{\varepsilon}_j\left({t}_j\right)}=\varPi$ has been temporarily abandoned, we only need to count all possible revenue allocations (ε ₁(t ₁), ε ₂(t ₂)) satisfying ε _j (t _j ) ≥ 0 for j = 1, 2. Consequently, there are a total of four equilibrium outcomes, which are, respectively, as follows:

$$ {A}_1=\left({\varepsilon}_2,{\varepsilon}_2\right),{A}_2=\left({\varepsilon}_1,{\varepsilon}_2\right),{A}_3=\left({\varepsilon}_2,{\varepsilon}_1\right),{A}_4=\left({\varepsilon}_1,{\varepsilon}_1\right). $$

If we denote a firm by a ball and denote an industry by a box, each equilibrium outcome A _i  (i = 1, 2, 3, 4) can be depicted as a different figure, refer to Figs. 1, 2 and 3. For example, Fig. 1 depicts the equilibrium outcome A ₁ in which firms 1 and 2 both occupy industry 2, and each thereby obtains ε ₂ units of revenue, where ball 1 represents firm 1 and box 1 represents industry 1, and so forth.

Fig. 1

In the equilibrium outcome A ₁, firms 1 and 2 both occupy industry 2, and each obtains ε ₂ units of revenue

Fig. 2

In the equilibrium outcome A ₂, firm 1 occupies industry 1 and obtains ε ₁ units of revenue; firm 2 occupies industry 2 and obtains ε ₂ units of revenue. In the equilibrium outcome A ₃, firm 1 occupies industry 2 and obtains ε ₂ units of revenue; firm 2 occupies industry 1 and obtains ε ₁ units of revenue

Fig. 3

In the equilibrium outcome A ₄, firms 1 and 2 both occupy industry 1, and each obtains ε ₁ units of revenue

Although Example 4.1 merely describes a simple situation of (4.7) when N = 2 and ε _j (t _j ) takes two possible values: ε ₁ or ε ₂, four equilibrium outcomes remain. Additionally, because each equilibrium outcome is Pareto optimal, we are not able to clarify which equilibrium outcome is best for society so that all of the social members want to select it. In accordance with the conventional economic analysis, welfare economists believe that one can find the best equilibrium outcome by taking advantage of an imaginary social welfare function. Unfortunately, Arrow’s Impossibility Theorem has refuted the existence of such a social welfare function in the framework of ordinal utility (Jehle and Reny 2001; Page 243). Consequently, one must confront an uncertain economic world that exhibits four possible social states.¹² This fact is well known as the “dilemma of social choice”.

In fact, evolutionary economists have already been aware that neoclassical economics lacks a body of economic analysis that could address choice under uncertainty. These economists further argued that what is missing in conventional economic analysis is a treatment of “economic emergence” (Foster and Metcalfe 2012). Therefore, it is considerably significant to materialize “economic emergence” in neoclassical economics. Our plan is to introduce Hayek’s principle of spontaneous order into neoclassical economics (From an evolutionary economic perspective, order and emergence are inseparable). We hope that such an attempt will facilitate eliminating the “dilemma of social choice”. We next introduce the concept of economic order.

5 Economic order

To simplify the analysis, we begin to introduce the concept of economic order by investigating the four equilibrium outcomes of the Example 4.1.

5.1 Definition

As noted in subsection 4.2, the LRCE described by Example 4.1 has four possible equilibrium outcomes: A ₁, A ₂, A ₃ and A ₄; each of these can be associated with a figure. If one observes Figs. 1, 2 and 3 carefully, one may find that these four outcomes can be divided into three different groups. To observe this, we consider an ordered pair {a ₁, a ₂}, where a ₁ represents that there are a ₁ firms, each of which obtains ε ₁ units of revenue; a ₂ represents that there are a ₂ firms, each of which obtains ε ₂ units of revenue. Adopting this notion, one easily finds that Figs. 1, 2 and 3 can be denoted by {a ₁ = 0, a ₂ = 2}, {a ₁ = 1, a ₂ = 1} and {a ₁ = 2, a ₂ = 0}, respectively.

It is here worth noting that, although Fig. 2 depicts two equilibrium outcomes (thereby two figures): A ₂ and A ₃, we can still use a unique ordered pair {a ₁ = 1, a ₂ = 1} to denote it. This is because A ₂ and A ₃ obey a unified rule (or convention): one firm obtains ε ₁ units of revenue, and another obtains ε ₂ units of revenue. Thus, the ordered pair {a ₁, a ₂} can be considered as a ‘set’ whose elements are equilibrium outcomes. For example, A ₂ and A ₃ obey the rule {a ₁ = 1, a ₂ = 1}; therefore, we obtain:

$$ \left\{{a}_1=1,{a}_2=1\right\}=\left\{{A}_2,{A}_3\right\}. $$

(5.1)

Similarly, we have:

$$ \left\{{a}_1=0,{a}_2=2\right\}=\left\{{A}_1\right\}. $$

(5.2)

$$ \left\{{a}_1=2,{a}_2=0\right\}=\left\{{A}_4\right\}. $$

(5.3)

If we extend the analysis regarding the two firms above to N firms, we have the following definition regarding economic order.

Definition 5.1: Let W denote the set of all possible equilibrium outcomes satisfying (4.7). A sequence of non-negative numbers, {a _k } ⁿ _k = 1  = {a ₁, a ₂, …, a _n }, is called an economic order if and only if it denotes a subset of W, obeying the following four conventions:

• (1). There are a total of n possible revenue levels¹³: ε ₁ < ε ₂ < … < ε _n ;

• (2). There are a _k firms each of which obtains ε _k units of revenue, and k runs from 1 to n;

• (3). These a _k firms are distributed among g _k industries¹⁴;

• (4). ${\displaystyle \sum_{k=1}^n{a}_k}=N$.

It is easy to observe that, with every economic order {a _k } ⁿ _k = 1 , one can associate a different revenue distribution as follows: There are a ₁ firms each of which obtains ε ₁ units of revenue; there are a ₂ firms each of which obtains ε ₂ units of revenue, and so on. From this meaning, an economic order actually denotes an ordered distribution rule of society’s wealth. In general, any distribution rule is always due to certain social institutions or conventions. Thus, we are eager to clarify what distribution rule (or economic order) a free economy would obey. Hayek believed that, if a competitive economy is sufficiently free, a spontaneous economic order will arise. According to Hayek’s belief, a striking feature of the spontaneous economic order is that it is more likely to emerge or more able to survive than other economic orders (Sugden 1989). With this belief, we can present a concrete definition for spontaneous economic order as below:

Definition 5.2: Among all possible economic orders {a _k } ⁿ _k = 1 satisfying Definition 5.1, if there exists an economic order {a ^* _k } ⁿ _k = 1 which would occur with the highest probability, {a ^* _k } ⁿ _k = 1 is called a spontaneous economic order.

To understand Definition 5.2, we can informally adopt the following statistical notion: One considers W as a sample space in which each equilibrium outcome is regarded as a sample outcome (or an outcome of ‘experiment’), and one considers an economic order as a random event that is identified with a collection of sample outcomes. Adopting such a notion, the spontaneous economic order is of course the most likely event (this is why it can spontaneously arise). In section 6, we will formalize this notion and further show how to seek the spontaneous economic order from among all possible economic orders.

Before proceeding, we are particularly interested in counting how many equilibrium outcomes a given economic order would contain. Let us next attempt to accomplish this task.

5.2 Monopolistic competition and perfect competition

For convenience, we denote by¹⁵ Ω({a _k } ⁿ _k = 1 ) the number of elements in a given economic order {a _k } ⁿ _k = 1 . In other words, the economic order {a _k } ⁿ _k = 1 contains Ω({a _k } ⁿ _k = 1 ) equilibrium outcomes. In microeconomics, we have clarified that there are two types of competitive structures, that is, perfect competition and monopolistic competition. Hence, we need to find Ω({a _k } ⁿ _k = 1 ) in terms of these two types of economic structures, respectively.

Definition 5.3: A monopolistic-competitive economy is an Arrow-Debreu economy in which firms are completely distinguishable¹⁶ (or heterogeneous).

In fact, targeting the LRCE described by Example 4.1, Figs. 1, 2 and 3 have exhibited the situation of monopolistic competition where two balls (firms) are marked by serial numbers so that we can distinguish which is firm 1 and which is firm 2. For monopolistic-competitive LRCE, Tao (2010) has computed the number of elements in a given economic order {a _k } ⁿ _k = 1 in the form:

$$ \varOmega {\left({\left\{{a}_k\right\}}_{k=1}^n\right)}_{mon}=\frac{N!}{{\displaystyle \prod_{k=1}^n{a}_k!}}{\displaystyle \prod_{k=1}^n{g}_k^{a_k}}. $$

(5.4)

For instance, the LRCE described by Example 4.1 requires that N = 2, n = 2 and g ₁ = g ₂ = 1. Thus, using the formula (5.4) we can compute the number of elements in each economic order as follows:

$$ \varOmega {\left(\left\{{a}_1=0,{a}_2=2\right\}\right)}_{mon}=\frac{2!}{0!\times 2!}\times {1}^0\times {1}^2=1, $$

(5.5)

$$ \varOmega {\left(\left\{{a}_1=1,{a}_2=1\right\}\right)}_{mon}=\frac{2!}{1!\times 1!}\times {1}^1\times {1}^1=2, $$

(5.6)

$$ \varOmega {\left(\left\{{a}_1=2,{a}_2=0\right\}\right)}_{mon}=\frac{2!}{2!\times 0!}\times {1}^2\times {1}^0=1. $$

(5.7)

Clearly, the results (5.5)–(5.7) are consistent with the numbers of equilibrium outcomes listed by Figs. 1, 2 and 3, respectively.

Definition 5.4: A perfectly competitive economy is an Arrow-Debreu economy in which firms are indistinguishable¹⁷ (or identical).

To observe the difference between perfect competition and monopolistic competition, we continue to concentrate on the LRCE described by Example 4.1. Because firms are indistinguishable in a perfectly competitive economy, we use the following three figures (refer to Figs. 4, 5 and 6) to list all possible equilibrium outcomes and the corresponding economic orders.

Fig. 4

The economic order {a ₁ = 0, a ₂ = 2} allows a single equilibrium outcome in which two indistinguishable firms occupy industry 2, and each obtains ε ₂ units of revenue

Fig. 5

The economic order {a ₁ = 1, a ₂ = 1} allows a single equilibrium outcome in which one firm occupies industry 1 (hence, obtains ε ₁ units of revenue), and another firm occupies industry 2 (hence, obtains ε ₂ units of revenue)

Fig. 6

The economic order {a ₁ = 2, a ₂ = 0} allows a single equilibrium outcome in which two indistinguishable firms occupy industry 1, and each obtains ε ₁ units of revenue

Typically, for the case of perfect competition, two balls (firms) are not marked by serial number (refer to Figs. 4, 5 and 6); thus, we cannot distinguish which is firm 1 or which is firm 2. Therefore, the economic order obeying perfect competition, {a ₁ = 1, a ₂ = 1}, contains only one equilibrium outcome, refer to Fig. 5. This is clearly different from the case of monopolistic competition (comparing Figs 2 and 5).

For a perfectly competitive LRCE, Tao (2010) has computed the number of elements in a given economic order {a _k } ⁿ _k = 1 in the form:

$$ \varOmega {\left({\left\{{a}_k\right\}}_{k=1}^n\right)}_{per}={\displaystyle \prod_{k=1}^n\frac{\left({a}_k+{g}_k-1\right)!}{a_k!\left({g}_k-1\right)!}}. $$

(5.8)

Similarly, targeting the LRCE described by Example 4.1, using the formula (5.8) we can compute the number of elements in each economic order as follows:

$$ \varOmega {\left(\left\{{a}_1=0,{a}_2=2\right\}\right)}_{per}=\frac{\left(0+1-1\right)!}{0!\times \left(1-1\right)!}\times \frac{\left(2+1-1\right)!}{2!\times \left(1-1\right)!}=1, $$

(5.9)

$$ \varOmega {\left(\left\{{a}_1=1,{a}_2=1\right\}\right)}_{per}=\frac{\left(1+1-1\right)!}{1!\times \left(1-1\right)!}\times \frac{\left(1+1-1\right)!}{1!\times \left(1-1\right)!}=1, $$

(5.10)

$$ \varOmega {\left(\left\{{a}_1=2,{a}_2=0\right\}\right)}_{per}=\frac{\left(2+1-1\right)!}{2!\times \left(1-1\right)!}\times \frac{\left(0+1-1\right)!}{0!\times \left(1-1\right)!}=1. $$

(5.11)

The results (5.9)–(5.11) are consistent with the numbers of equilibrium outcomes listed by Figs. 4, 5 and 6, respectively.

In summary, we have:

$$ \varOmega \left({\left\{{a}_k\right\}}_{k=1}^n\right)=\left\{\begin{array}{cc}\hfill {\displaystyle \prod_{k=1}^n\frac{\left({a}_k+{g}_k-1\right)!}{a_k!\left({g}_k-1\right)!}}\hfill & \hfill \left( perfect\kern0.6em competition\right)\hfill \\ {}\hfill \frac{N!}{{\displaystyle \prod_{k=1}^n{a}_k!}}{\displaystyle \prod_{k=1}^n{g_k}^{a_k}}\hfill & \hfill \left( monopolistic\kern0.6em competition\right)\hfill \end{array}\right.. $$

(5.12)

6 Fairness, freedom and spontaneous economic order

In this section, we show how to seek the spontaneous economic order from among all possible economic orders by using the normative criteria of fairness and freedom.

6.1 Social fairness

Undoubtedly, an ideal framework of considering social fairness or equity is the theory of social choice. To keep the following analysis simple, we proceed to investigate the Example 4.1. As noted in subsection 4.2, the Example 4.1 has four possible equilibrium outcomes: A ₁, A ₂, A ₃ and A ₄; each of which is Pareto optimal. In this case, the task of the theory of social choice is to answer: Which of these four equilibrium outcomes is best for society. In accordance with the welfare economists’ convention, one may denote the set of equilibrium outcomes by A = {A ₁, A ₂, A ₃, A ₄}. Then, if one can find a ranking of the equilibrium outcomes in A that reflects ‘society’s’ preferences, one would capture the best social choice. Unfortunately, Arrow’s Impossibility Theorem has refuted the existence of such ‘society’s’ preferences (Jehle and Reny 2001; Page 243). Hence, one is not able to compare any two alternatives in A from a perspective that is individually and socially consistent; otherwise, the social choice will be unfair. To ensure fairness, a wise treatment is to abandon comparing any two equilibrium outcomes in A, and furthermore, to admit equality between all of these equilibrium outcomes; that is,

$$ {A}_1\sim {A}_2\sim {A}_3\sim {A}_4, $$

(6.1)

where, the symbol ~ represents the indifference relation.

Such a treatment exhibits Leibniz’s principle of the identity of indiscernibles (Arrow 1963; Page 109). Now that social members are indifferent between all equilibrium outcomes, we cannot ensure which outcome will be selected as a collective decision. This means that collective choices should be completely random.

Because of the randomness of the collective choices, we are very interested in exploring the probability that a certain equilibrium outcome will be selected as a collective decision in a just society. Thus, let us concentrate on Rawls’ pure procedural justice (Rawls 1999; Page 74) which aims to design the social system (or economic institutions) so that the outcome is just whatever it happens to be, at least so long as it is within a certain range. With this idea, a just economy can be regarded as a fair procedure that will translate its fairness to the (equilibrium) outcomes; thus, every social member would have no desire to oppose or prefer a certain outcome. That is, (6.1) holds. Technically, to ensure that the economy is one of pure procedural justice, Rawls suggested considering the principle of fair equality of opportunity (Rawls 1999; Page 76). In accordance with this principle, a fair economy implies that each outcome should be selected with equal opportunities¹⁸; in other words, each outcome will then occur with an equal probability. Based on the analyses above, we can present the following axiom for social fairness.

Axiom 6.1: If a competitive economy produces ω equilibrium outcomes, and if this economy is absolutely fair, each equilibrium outcome will occur with an equal probability $\frac{1}{\omega }$.

For instance, if the LRCE described by Example 4.1 is absolutely fair, Axiom 6.1 implies:

$$ P\left[{A}_1\right]=P\left[{A}_2\right]=P\left[{A}_3\right]=P\left[{A}_4\right]=\frac{1}{4}, $$

(6.2)

where, we denote by P[X] the probability that an equilibrium outcome X occurs.

Due to Axiom 6.1, we may apply the concept of classical probability (refer to page 21 in Larsen and Marx (2001)) to the set of equilibrium outcomes of the LRCE. Specifically, we adopt the following three conventions:

• (i). The set of all possible equilibrium outcomes satisfying (4.7), W, is referred to as the sample space.

• (ii). Each element (or equilibrium outcome) of the sample space W is referred to as a sample outcome.

• (iii). Each economic order that is identified with a collection of sample outcomes is referred to as a random event.

Adopting the conventions (i)–(iii), we are able to compute the probability that any economic order occurs, provided that all possible equilibrium outcomes had been found.

For example, by (6.2) and (5.1) one has

$$ P\left[\left\{{a}_1=1,{a}_2=1\right\}\right]=P\left[\left\{{A}_2,{A}_3\right\}\right]=P\left[{A}_2\right]+P\left[{A}_3\right]=\frac{1}{2}, $$

and by (6.2) and (5.2) one has $P\left[\left\{{a}_1=0,{a}_2=2\right\}\right]=P\left[\left\{{A}_1\right\}\right]=P\left[{A}_1\right]=\frac{1}{4}$.

6.2 Social freedom

The concept of freedom is very complex, and every attempt to formalize it must neglect important aspects (Puppe 1996). As most authors have done (Sen 1993) (Pattanaik and Xu 1998), this paper concentrates on the opportunity aspect of freedom. In this case, if social members are indifferent between alternatives, the extent of freedom offered to the social members is entirely determined by the size of the set of alternatives (i.e., opportunity set), refer to Sen (1993).

In subsection 5.2, we have known that a given economic order {a _k } ⁿ _k = 1 contains Ω({a _k } ⁿ _k = 1 ) equilibrium outcomes. Therefore, if an economy obeys the economic order {a _k } ⁿ _k = 1 , the social members will encounter Ω({a _k } ⁿ _k = 1 ) possible choices. In the spirit of the opportunity-freedom (Sen 1993), one can refer to an economic order as an opportunity set. Because social members are indifferent between equilibrium outcomes (refer to subsection 6.1), the degree of freedom of an economic order may be denoted by its size (i.e., number of elements in it). Thus, we have the following axiom:

Axiom 6.2: If a competitive economy obeys an economic order {a _k } ⁿ _k = 1 , which contains Ω({a _k } ⁿ _k = 1 ) equilibrium outcomes, the degree of freedom of this economy is denoted by Ω({a _k } ⁿ _k = 1 ).

Because valuing freedom of choice may involve psychology (Verme 2009), we will not discuss the relation between freedom and preference.¹⁹ In addition, we must emphasize that²⁰ the degree of freedom defined by the Axiom 6.2 has no ethical standard regarding ‘good’ or ‘bad’. The larger degree of freedom merely implies more possible choices. For example, by (5.1), the degree of freedom of {a ₁ = 1, a ₂ = 1} equals 2; by (5.2), the degree of freedom of {a ₁ = 0, a ₂ = 2} equals 1. Then, we do not mean that {a ₁ = 1, a ₂ = 1} is better than {a ₁ = 0, a ₂ = 2}.

In subsection 5.1, we have defined the spontaneous economic order {a ^∗ _k } ⁿ _k = 1 as an economic order with the highest probability, refer to Definition 5.2. In accordance with this definition, we will now show that, if a competitive economy is not only absolutely fair but also has the largest degree of freedom, it would obey the spontaneous economic order {a ^∗ _k } ⁿ _k = 1 .

Lemma 6.1: If the competitive economy is absolutely fair, the probability of the economic order {a _k } ⁿ _k = 1 occurring is given by:

$$ P\left[{\left\{{a}_k\right\}}_{k=1}^n\right]=\frac{\varOmega \left({\left\{{a}_k\right\}}_{k=1}^n\right)}{{\displaystyle \sum_{{\left\{{a}_k^{\hbox{'}}\right\}}_{k=1}^n}\varOmega \left({\left\{{a}_k^{\hbox{'}}\right\}}_{k=1}^n\right)}}, $$

(6.3)

where, ${\displaystyle \sum_{{\left\{{a}_k^{\hbox{'}}\right\}}_{k=1}^n}\varOmega \left({\left\{{a}_k^{\prime}\right\}}_{k=1}^n\right)}$ denotes the sum of the numbers of equilibrium outcomes over all possible economic orders.

Proof. Because an economic order {a _k } ⁿ _k = 1 contains Ω({a _k } ⁿ _k = 1 ) equilibrium outcomes, one easily counts that the competitive economy totally produces ${\displaystyle \sum_{{\left\{{a}_k^{\hbox{'}}\right\}}_{k=1}^n}\varOmega \left({\left\{{a}_k^{\prime}\right\}}_{k=1}^n\right)}$ equilibrium outcomes. According to Axiom 6.1, if the competitive economy is absolutely fair, each equilibrium outcome will occur with an equal probability $\frac{1}{{\displaystyle \sum_{{\left\{{a}_k^{\hbox{'}}\right\}}_{k=1}^n}\varOmega \left({\left\{{a}_k^{\prime}\right\}}_{k=1}^n\right)}}$. In accordance with the conventions (i)-(iii), the probability of the economic order {a _k } ⁿ _k = 1 occurring is denoted by (6.3). □

Using the Lemma 6.1, we can prove the following important result.

Proposition 6.1: If a competitive economy is absolutely fair, and if it obeys the spontaneous economic order {a ^∗ _k } ⁿ _k = 1 , it would have the largest degree of freedom; that is,

$$ \varOmega \left({\left\{{a}_k^{*}\right\}}_{k=1}^n\right)=\underset{{\left\{{a}_k\right\}}_{k=1}^n}{ \max}\varOmega \left({\left\{{a}_k\right\}}_{k=1}^n\right). $$

(6.4)

Proof. According to Definition 5.2, the spontaneous economic order {a ^∗ _k } ⁿ _k = 1 is the most likely economic order; therefore, we have:

$$ P\left[{\left\{{a}_k^{*}\right\}}_{k=1}^n\right]=\underset{{\left\{{a}_k\right\}}_{k=1}^n}{\operatorname{ma}}\operatorname{x}P\left[{\left\{{a}_k\right\}}_{k=1}^n\right]. $$

(6.5)

Substituting (6.3) into (6.5) yields (6.4). □

The proof above implies a corollary as below:

Corollary 6.1: If a competitive economy is not only absolutely fair but also has the largest degree of freedom, it would obey a spontaneous economic order.

Proposition 6.1 is a central result of this paper because it not only tells us that a spontaneous economic order is completely determined by fairness, freedom and competition but also implies a method of seeking the spontaneous economic order. By (6.4), seeking the spontaneous economic order {a ^∗ _k } ⁿ _k = 1 is equivalent to solving an extremum problem $\underset{{\left\{{a}_k\right\}}_{k=1}^n}{ \max}\varOmega \left({\left\{{a}_k\right\}}_{k=1}^n\right)$. This is what we will accomplish in the next subsection.

6.3 Spontaneous economic order

When we introduced the concept of economic order in section 5, we abandoned the constraint ${\displaystyle \sum_{j=1}^N{\varepsilon}_j\left({t}_j\right)}=\varPi$ in (4.7). Such a treatment is incorrect. To observe this, let us now return to Example 4.1. It is easy to observe that, if one assumes ε ₁ + ε ₂ = Π, one does have ε ₁ + ε ₁ < Π and ε ₂ + ε ₂ > Π; therefore, the economic orders {a ₁ = 2, a ₂ = 0} and {a ₁ = 0, a ₂ = 2} do not satisfy (4.7). To eliminate the economic orders transgressing (4.7), we now resume the constraint ${\displaystyle \sum_{j=1}^N{\varepsilon}_j\left({t}_j\right)}=\varPi$.

Without loss of generality, all of the economic orders obeying (4.7) must satisfy the following two constraints:

$$ {\displaystyle \sum_{k=1}^n{a}_k}=N, $$

(6.6)

$$ {\displaystyle \sum_{k=1}^n{a}_k{\varepsilon}_k}=\varPi . $$

(6.7)

It is easy to observe that (6.7) is the constraint ${\displaystyle \sum_{j=1}^N{\varepsilon}_j\left({t}_j\right)}=\varPi$. Thus, seeking the spontaneous economic order {a ^∗ _k } ⁿ _k = 1 from among all of the possible economic orders, obeying (4.7) is equivalent to solving the extremum problem as below:

$$ \left\{\begin{array}{c}\hfill \underset{{\left\{{a}_k\right\}}_{k=1}^n}{ \max}\varOmega \left({\left\{{a}_k\right\}}_{k=1}^n\right)\hfill \\ {}\hfill \begin{array}{cc}\hfill s.t.\hfill & \hfill N={\displaystyle \sum_{k=1}^n{a}_k}\hfill \end{array}\hfill \\ {}\hfill \varPi ={\displaystyle \sum_{k=1}^n{a}_k{\varepsilon}_k}\hfill \end{array}\right. $$

(6.8)

where, Ω({a _k } ⁿ _k = 1 ) is denoted by (5.12).

To solve the extremum problem (6.8), we need to introduce a lemma.

Lemma 6.2: Let U[Ω] = ln Ω({a _k } ⁿ _k = 1 ). If U[Ω] achieves the maximum value at {a ^∗ _k } ⁿ _k = 1 , then Ω({a _k } ⁿ _k = 1 ) achieves the maximum value at {a ^∗ _k } ⁿ _k = 1 as well.

Proof. If one observes that U[Ω] is a monotonically increasing function of Ω({a _k } ⁿ _k = 1 ), one easily completes the proof. □

Using the Lemma 6.2, the extremum problem (6.8) is equivalent to the following extremum problem:

$$ \left\{\begin{array}{c}\hfill \underset{{\left\{{a}_k\right\}}_{k=1}^n}{ \max } \ln \varOmega \left({\left\{{a}_k\right\}}_{k=1}^n\right)\hfill \\ {}\hfill \begin{array}{cc}\hfill s.t.\hfill & \hfill N={\displaystyle \sum_{k=1}^n{a}_k}\hfill \end{array}\hfill \\ {}\hfill \varPi ={\displaystyle \sum_{k=1}^n{a}_k{\varepsilon}_k}\hfill \end{array}\right.. $$

(6.9)

Substituting (5.12) into (6.9), we obtain the spontaneous economic order of the LRCE in the form:

$$ {a}_k^{\ast }(I)=\frac{g_k-I}{e^{\alpha +\beta {\varepsilon}_k}-I}\left\{\begin{array}{cc}\hfill I=1\hfill & \hfill \left( perfect\kern0.6em competition\right)\hfill \\ {}\hfill I=0\hfill & \hfill \left( monopolistic\kern0.6em competition\right)\hfill \end{array}\right., $$

(6.10)

$$ k=1,2,\dots, n $$

where, α and β are Lagrange multipliers. For detailed calculations, refer to Appendix A and B.

The spontaneous economic order (6.10) is a central result of this paper; it is called the Bose-Einstein distribution whenever I = 1 and is called the Boltzmann distribution whenever I = 0 (Carter 2001). Such an economic order determines the following rule of revenue distribution:

There are a ^∗ _k (I) firms each of which obtains ε _k units of revenue, and k runs from 1 to n.

It is easy to observe that a ^∗ _k (I) will decrease as k grows. This result strongly implies revenue inequality. However, revenue inequality does not contradict our definition for social fairness (refer to Axiom 6.1). In fact, Axiom 6.1 merely indicates that each firm has an equal chance of occupying any possible revenue level. In this case, to obtain a higher revenue, luck and effort are similarly important (Alesina and Angeletos 2005) (Alesina et al. 2012).

It must be noticed that α and β in (6.10) are two indeterminate multipliers, and the principle of spontaneous order cannot determine them.

7 Empirical investigation to spontaneous economic order

To clarify the economic meanings of α and β, we need to introduce the neoclassical macroeconomics (Romer 2000; Page 120) in which the aggregate revenue²¹ Π is completely determined by labor L, capital K and technological progress T; that is,

$$ \varPi ={L}^x{K}^y{T}^z. $$

(7.1)

As is well known, an important role of a firm is to collect labor and capital. Naturally, a firm can be considered as composed of labor and capital (Williamson and Winter 1993), e.g., a unit of a firm corresponds to a unit of labor and capital. Hence, the total number of firms, N, can be written as a function with respect to labor L and capital K; that is,

$$ N=N\left({L}^x{K}^y\right). $$

(7.2)

Using (7.2), the function (7.1) can be rewritten in the form:

$$ \varPi =\varPi \left(N,T\right). $$

(7.3)

The complete differential of (7.3) yields:

$$ d\varPi \left(N,T\right)=\mu dN+\theta dT, $$

(7.4)

where, $\mu =\frac{\partial \varPi }{\partial N}$ and $\theta =\frac{\partial \varPi }{\partial T}$.

μ and θ denote the marginal labor-capital return and the marginal technology return of an economy, respectively.

In addition, using (6.6), (6.7) and (6.10), Tao (2010) obtained:

$$ d\varPi =-\frac{\alpha }{\beta }dN+\frac{1}{\beta }d\left( \ln W-\alpha \frac{\partial \ln W}{\partial \alpha }-\beta \frac{\partial \ln W}{\partial \beta}\right), $$

(7.5)

where $W=W\left(\alpha, \beta \right)={\displaystyle \prod_{k=1}^n{\left(1-I{e}^{-\alpha -\beta {\varepsilon}_k}\right)}^{-\frac{g_k-I}{I}}}$.

Remarkably, the differential aggregate revenue (7.4) (from neoclassical economics) and the differential aggregate revenue (7.5) (from spontaneous order theory) share the same functional form. This means that neoclassical economics and Austrian economics will be compatible with each other as long as (7.4) equals (7.5).

In accordance with the belief of unifying neoclassical and Austrian theories, by comparing (7.4) and (7.5), we obtain:

$$ \alpha =-\frac{\mu }{\lambda \theta }, $$

(7.6)

$$ \beta =\frac{1}{\lambda \theta }, $$

(7.7)

$$ T=\lambda \left( \ln W-\alpha \frac{\partial \ln W}{\partial \alpha }-\beta \frac{\partial \ln W}{\partial \beta}\right), $$

(7.8)

where λ is a positive constant.

Substituting (7.6) and (7.7) into (6.10) yields a definite form:

$$ {a}_k^{\ast }(I)=\frac{g_k-I}{e^{\frac{\varepsilon_k-\mu }{\lambda \theta }}-I}\left\{\begin{array}{cc}\hfill I=1\hfill & \hfill \left( perfect\kern0.6em competition\right)\hfill \\ {}\hfill I=0\hfill & \hfill \left( monopolistic\kern0.6em competition\right)\hfill \end{array}\right., $$

(7.9)

$$ k=1,2,\dots, n. $$

The formula (7.9) has earlier been obtained by Tao (2010). The formula describes the firms’ revenue distribution in an economy. If we associate each firm with a different agent, the formula (7.9) may describe the income distribution of the society. An attractive idea is to test (7.9) by collecting firms’ revenue data or individuals’ income data. It is worth emphasizing that there had been empirical evidence supporting (7.9), refer to Yakovenko and Rosser (2009), Kürten and Kusmartsev (2011), Kusmartsev (2011), Clementi et al. (2012). Let us next demonstrate how these empirical investigations support (7.9).

It is easy to observe that {a ^∗ _k (I = 0)} ⁿ _k = 1 is an exponential distribution. Such a distribution is associated with the monopolistic-competitive economy. As noted in microeconomics, monopolistic competition is a common competitive mode, and most real economies usually obey this mode. Yakovenko and Rosser (2009) have confirmed that the income distribution in the USA from 1983 to 2000 obeys the exponential distribution well; refer to Fig. 7. Later, Clementi et al. (2012) also confirmed this point. More specifically, Yakovenko and Rosser (2009) show that approximately 3 % of the population obey Pareto distribution (i.e., power-law distribution), and 97 % obey exponential distribution. This fact that income distribution consists of two distinct parts reveals the two-class structure of the American society.²²

Fig. 7

Reprinted from Yakovenko and Rosser (2009). Points represent the Internal Revenue Service data, and solid lines are fits to Boltzmann (exponential) and Pareto distributions

In contrast, {a ^∗ _k (I = 1)} ⁿ _k = 1 is an unstable distribution. To observe this, let us observe that there may be a singularity ε _k  = μ so that the denominator of (7.9) corresponding to I = 1 equals zero. Therefore, there may be a great many firms (or agents) occupying a very low revenue level (or income level) μ, for details observe the section IV in Tao (2010). Such an unstable distribution is associated with the case of extreme competition, i.e., perfect competition. Recently, Kürten and Kusmartsev (2011) have confirmed that the income distribution in the USA from 1996–2008 obeys the Bose-Einstein distribution well; in addition, the financial crisis in 2008 is due to the instability of the Bose-Einstein distribution. Refer to Fig. 8.

Fig. 8

Reprinted from Kusmartsev (2011). Solid squares represent the Internal Revenue Service data in 2008. The red curve and the black straight line are fits to Bose-Einstein (BE) and Pareto distributions, respectively

8 Technological progress and freedom

From a neoclassical economic perspective, the technological progress T is mysterious, and no one clarifies what the origin of it is. Of course, there had been certain excellent economic models, e.g., Romer (1990), in which the technological progress is interpreted as an endogenous variable, whereas it is artificially taken into these models. Remarkably, soon we shall observe that, if one treats neoclassical economics and Austrian economics in a unified perspective, one will decipher the profound origin of technological progress.

Using (5.12) and (6.10), Tao (2010) obtained:

$$ \ln \varOmega \left(\left\{{a}_k^{*}\right\}\right)= \ln W-\alpha \frac{\partial \ln W}{\partial \alpha }-\beta \frac{\partial \ln W}{\partial \beta }. $$

(8.1)

Substituting (8.1) into (7.8) yields a refined form²³:

$$ T=\lambda \ln \varOmega . $$

(8.2)

From (8.2), we surprisingly find that the technological progress T is exactly proportional to ln Ω. In addition, by Axiom 6.2, the variable Ω (or equivalently ln Ω) represents the degree of freedom of an economy; from this, we conclude: The more freedom, the higher the technological progress.

We attempt to convey an intuition for the result above. To simplify things, we examine Figs 1 and 2. Figure 2 depicts the economic order {a ₁ = 1, a ₂ = 1} whose degree of freedom is denoted by 2, then firm 1 not only may enter industry 1 but also may enter industry 2. In contrast, the degree of freedom of {a ₁ = 0, a ₂ = 2} is denoted by 1, then firm 1 is confined within industry 2 (refer to Fig. 1). Logically, if a firm has the chance of entering two industries, the probability of causing innovation should increase relative to being solely confined within one industry because the genius distribution among agents is heterogeneous. That is, the more freedom (namely, the larger Ω), the greater probability of causing innovation.

As is well known, Schumpeter (1934) emphasized that innovation is a main driving force of promoting economic development. In contrast, Hayek (1948) believed truly that freedom will induce a spontaneous economic order in which economic development is most efficient. Interestingly, (8.2) undoubtedly indicates that the innovation emphasized by Schumpeter is essentially equivalent to the freedom highlighted by Hayek. In other words, our theory may unify the ideas of Schumpeter and Hayek, which appear independent of each other. Thus, when an economy favors a state with more freedom, it essentially favors a state (or evolutionary direction) with higher technology level as well. By (5.12) the “freedom” Ω is an endogenous variable in our theory.²⁴ Substituting (8.2) into (7.3) we obtain:

$$ \varPi =\overset{\wedge }{\varPi}\left(N,\varOmega \right). $$

(8.3)

From (8.3), we observe that the freedom (Ω), as an equivalent replacement of technological progress (T), will become an important driving force of promoting economic growth. Actually, certain empirical studies have found a non-linear relation between economic freedom and growth, refer to Barro (1996).

Thus far, we have presented a complete theoretical framework for spontaneous economic order. At the moment, it is the proper time to summarize the logical establishments of this theoretical framework. First, we prove that a LRCE will produce infinitely many equilibrium outcomes. Second, we show that all of these equilibrium outcomes can be appropriately (non-repeated) divided into different economic orders. Third, according to normative criteria of fairness and freedom, one can determine an economic order that will occur with the highest probability, and we dub such an economic order the spontaneous economic order. Finally, we verify that Austrian economics and neoclassical economics will become compatible with each other within the framework of spontaneous economic order, provided that the technological progress T is proportional to the freedom variable ln Ω. Because of these, we believe that our spontaneous order theory presents a possible link between Austrian economics and neoclassical economics.

9 Conclusion

This paper presents a theoretical framework for spontaneous economic order in which the union of Austrian economics and neoclassical economics lies at the heart of our attempt. Our study shows that, if a competitive economy is sufficiently fair and free, an unplanned economic order will spontaneously emerge. It must be noticed that such an economic order is not the result of any process of collective choice (in contrast to that expected by many welfare economists), but is an unplanned and spontaneous consequence (as expected by Hayek).

In general, we cannot guarantee that an Arrow-Debreu economy has one sole equilibrium outcome. If an Arrow-Debreu economy (e.g., long-run competitive situation) produces multiple equilibrium outcomes, according to the first fundamental theorem of welfare economics, each outcome will be Pareto optimal. Consequently, social members must confront a problem of social choice: They must choose an equilibrium outcome that is best for society. For this problem, the proposal of welfare economists is to seek the best outcome through an imaginary social welfare function. Unfortunately, Arrow’s Impossibility Theorem warns us: There will be no such social welfare function if we insist on the perspective of ordinal utility. This means, a plan of seeking the best equilibrium outcome through a social welfare function is doomed to failure.

Our proposal here is to abandon seeking the best equilibrium outcome and furthermore, admit the equality between all possible equilibrium outcomes. In this case, we attempt to divide all of these equilibrium outcomes into different groups, where each group exhibits different convention. These conventions are called the economic orders by us. Based on these preparations, we further show that, if one adds normative criteria regarding fairness and freedom into the Arrow-Debreu economy, one will find that there is an economic order with the highest probability, which we call the spontaneous economic order. In accordance with the Darwinian spirit of natural selection, an economic order with the highest probability would most likely occur; this is why we call it the spontaneous order. Thus, we can say that the economic world does change in the manner it does because it seeks an economic order of higher probability. Our attempt has very strong theoretical and practical significance. The goal of human society should be to insist on the criteria regarding fairness and freedom (similar to Axioms 6.1–6.2). In accordance with these criteria, the competitive society will spontaneously obey an economic order. Specifically, we conclude that the spontaneous order of a monopolistic-competitive economy will obey a stable rule: exponential distribution; in addition, the spontaneous order of a perfectly competitive economy will obey an unstable rule: Bose-Einstein distribution. Additionally, the instability of the latter may cause economic crises. These conclusions have been supported by recent empirical investigations.

Obviously, our spontaneous order theory is, in principle, based on the theoretical framework of an Arrow-Debreu economy; therefore, it may present a bridge linking Austrian economics and neoclassical economics. Then, an interesting conjecture arises: Could Austrian economics and neoclassical economics yield to a unified framework? Our spontaneous order theory strongly supports this conjecture. As a possible result of unifying these two types of economics, we should comprehend a truth: “Freedom promotes technological progress”. This is because, within the framework of our spontaneous order theory, “technological progress” and “freedom” will become equivalent to each other.

Appendices

A. Spontaneous economic order of perfectly competitive economy

Allowing for the number of firms N → ∞ in a long-run competitive economy, we assume that every a _k is a sufficiently large number.

If one considers the perfect competition, using (5.12), the function U[Ω] can be written in the form:

$$ U\left[{\varOmega}_{per}\right]={\displaystyle \sum_{k=1}^n \ln \left({a}_k+{g}_k-1\right)!}-{\displaystyle \sum_{k=1}^n \ln {a}_k!}-{\displaystyle \sum_{k=1}^n \ln \left({g}_k-1\right)!}. $$

(A.1)

Because the value of a _k is sufficiently large, using the Stirling’s formula (Carter 2001; Page 218)

$$ \ln m!=m\left( \ln m-1\right),\left(m>>1\right) $$

(A.2)

(A.1) can be rewritten in the form:

$$ U\left[{\varOmega}_{per}\right]={\displaystyle \sum_{k=1}^n\left[\left({a}_k+{g}_k-1\right) \ln \left({a}_k+{g}_k-1\right)-{a}_k \ln {a}_k-{g}_k- \ln \left({g}_k-1\right)!+1\right].} $$

(A.3)

The method of Lagrange multiplier for the optimal problem (6.9) gives

$$ \frac{\partial \left\{U\left[\varOmega \right]\right\}}{\partial {a}_k}-\alpha \frac{\partial N}{\partial {a}_k}-\beta \frac{\partial \prod }{\partial {a}_k}=0,k=1,2,\dots n $$

(A.4)

where, α and β are Lagrange multipliers.

Substituting (6.6), (6.7) and (A.3) into (A.4) yields

$$ \ln \left(\frac{a_k+{g}_k-1}{a_k}-\alpha -\beta {\varepsilon}_k\right)=0, $$

(A.5)

$$ k=1,2,\dots, n. $$

which is the spontaneous economic order of a perfectly competitive economy:

$$ {a}_k=\frac{g_k-1}{e^{\alpha +\beta {\varepsilon}_k}-1}, $$

(A.6)

$$ k=1,2,\dots, n. $$

B. Spontaneous economic order of monopolistic-competitive economy

If one considers the monopolistic competition, using (5.12) the function U[Ω] can be written in the form:

$$ U\left[{\varOmega}_{mon}\right]= \ln N!+{\displaystyle \sum_{k=1}^n{a}_k \ln {g}_k}-{\displaystyle \sum_{k=1}^n \ln {a}_k!}. $$

(B.1)

Using the Stirling’s formula (A.2), (B.1) can be rewritten in the form:

$$ U\left[{\varOmega}_{mon}\right]= \ln N!+{\displaystyle \sum_{k=1}^n{a}_k \ln {g}_k}-{\displaystyle \sum_{k=1}^n{a}_k \ln {a}_k}+{\displaystyle \sum_{k=1}^n{a}_k}. $$

(B.2)

Substituting (6.6), (6.7) and (B.2) into (A.4) gives the spontaneous economic order of a monopolistic-competitive economy:

$$ {a}_k=\frac{g_k}{e^{\alpha +\beta {\varepsilon}_k}}, $$

(B.3)

$$ k=1,2,\dots, n. $$