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Spontaneous economic order


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”.

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  1. It must be noticed that there have been much literature in which certain authors attempt to connect the principle of spontaneous order and the method of evolutionary game, e.g., refer to Schotter (1981), Sugden (1989) and Young (1993) (1996). Nonetheless, these excellent attempts pay more attentions to the order of social rules (e.g., conventions or institutions) rather than the order of economic rules (e.g., distribution of wealth or income). Obviously, the imbalance of the latter is more likely associated with economic crises. Additionally, the latter, in which we are chiefly concerned, is easier to be empirically tested.

  2. It is worth emphasizing that there may be difficulty concerning the possibility of satisfying fairness and Pareto optimality objectives simultaneously when interpersonal comparisons of utility are allowed (Pazner and Schmeidler 1974). However, one can eliminate this difficulty by insisting on the perspective of ordinal utility (Pazner and Schmeidler 1978).

  3. a 1 = {B 1} represents an economic order or a convention that allows equilibrium outcome B 1 to occur. Similarly, a 2 = {B 2, B 3} allows B 2 and B 3; a 3 = {B 4} allows B 4.

  4. Interestingly, compared to all the other real markets, the financial market is closest to a perfectly competitive market. This is the reason why the Black-Scholes equation of option pricing can be well applied in a financial market. The starting point of the Black-Scholes equation of option pricing is that the change in the price of stock obeys the law of Brownian movement. Only the perfectly competitive market, which is free of monopolization, is closest to such an ideal state. Minsky (1986) continuously claimed that the finance was the cause of the instability of capitalism. Now, according to our theory, it is because the financial market is closest to perfect competition.

  5. These three economic crises are, respectively, the Great Depression in 1929, the Asian financial crises in 1997, and the American subprime crisis in 2008.

  6. Publicly available technology coincides with Rawls’ principle of fair equality of opportunity (Rawls 1999; Page 63)

  7. The formula (7.9) shows that firms’ revenue (or equivalently “output value”) distribution in a monopolistic-competitive economy obeys the exponential law. Then there is no possibility that one firm’s output value is positive, and others’ all are null.

  8. The formula (7.9) shows that firms’ revenue (or equivalently “output value”) distribution in a perfectly competitive economy is unstable, because the denominator of (7.9) corresponding to I = 1 may equal zero. Then, there is indeed a possibility that one firm’s output value is positive, and others’ all are null. For more details, refer to Tao (2010).

  9. The assumption regarding single output appears very restrictive; however, it does not affect our final results. This assumption is made solely to keep our writing to follow succinct.

  10. With each equilibrium revenue allocation one may associate several or many equilibrium production allocations. For example, we cannot eliminate a possibility that there were another equilibrium production allocation (y a1 , …, y a N ) whose every vector y a j has two positive components: y a1j and y a2j , which are defined by ε j (t j ) = p 1 y a1j  + p 2 y a2j for j = 1, …, N, where \( z(p)={\displaystyle \sum_{j=1}^N{y}_j^a} \).

  11. When we here say that an equilibrium revenue allocation is Pareto optimal, we actually mean that the corresponding equilibrium production allocation is Pareto optimal. In this case, (ε 1(t 1), …, ε N (t N )) corresponds to (y e1 (t 1), …, y e N (t N )) at least, refer to (3.2) and (4.2).

  12. From the perspective of empirical observation, there must be one and only one equilibrium outcome (or social state), which would occur (at a given time, although we do not know which equilibrium outcome would occur).

  13. To guarantee that all possible equilibrium outcomes satisfying (4.7) can be, without loss of any outcomes, divided into different economic orders fulfilling Definition 5.1, we may require that n → ∞ and ε l + 1 − ε l  → 0, where l = 1, 2, …, n − 1.

  14. It must be noted that we cannot prevent the possibility that g k  > 1. To observe this, suppose that there were an equilibrium production allocation which contains several different equilibrium production vectors each of which generates a same revenue level. These different equilibrium production vectors (any two vectors must be linearly independent with each other and otherwise should be considered as an industry) can be considered as different industries. However, (3.2) and (4.1) together imply g k  = 1 for k = 1, 2, …, n.

  15. Adopting this notation, Ω({a k } n k = 1 ) should be a function of a k , where k = 1, 2, …, n.

  16. Namely, every firm corresponds to a different brand (Varian 2003; Page 453)

  17. Namely, firms produce homogeneous products (Varian 2003; Page 380); thus, the notion of brand does not exist. Perhaps, certain people may argue that homogeneous products, generally, solely hold in one industry. However, in the long run, if a firm exits an industry, then it can enter an arbitrary industry in which there should not be differentiated products; otherwise there exists monopoly. Consequently, homogeneous products, in the long run, hold in all industries; this case can be understood as products without brands (or equivalently, firms without brands).

  18. In accordance with Rawls (1999; Page 134), fairness here has been modeled as a demand for uncertainty. For more investigations concentrating on the relation between random choice and fairness, refer to Broome (1984).

  19. However, certain authors believe that judgments regarding the degree of freedom offered to an agent by different opportunity sets must consider the agent’s preferences over alternatives, refer to Sen (1993), Kreps (1979) and Koopmans (1964).

  20. Sudgen (1998) also emphasized this point, and he further noted that the problem of measuring opportunity has many similarities with the familiar preference-aggregation problems of welfare economics and social choice theory.

  21. In fact, we should here consider the aggregate production function z m (p) rather than the aggregate revenue function Π. However, (4.4) implies that there is no essential difference between z m (p) and Π (except a constant factor p m ).

  22. It is worth noting that (7.9) is due to the Axiom 6.1, which arises because the society is assumed to be absolutely fair. However, human society cannot be absolutely fair; therefore, this (7.9) may be only suitable for a segment of the population. Therefore, we can conclude that approximately 97 % of the population in the American society obeys fair behavior rules; however, the remaining fraction may involve unfair behaviors.

  23. (8.2) implies that technological progress T appears similar to the entropy in physics (Tao 2010). The latter is often related to “information” (or “knowledge”).

  24. Therefore, technological progress T is also an endogenous variable.


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Correspondence to Yong Tao.

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This article is dedicated to my mother. The work is supported by the Scholarship Award for Excellent Doctoral Student Granted by Ministry of Education of China (Grant No. 0903005109081-019) and the Fundamental Research Funds for the Central Universities (Grant No. SWU1409444).


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:

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

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.1) can be rewritten in the form:

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

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

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

where, α and β are Lagrange multipliers.

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

$$ \ln \left(\frac{a_k+{g}_k-1}{a_k}-\alpha -\beta {\varepsilon}_k\right)=0, $$
$$ k=1,2,\dots, n. $$

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

$$ {a}_k=\frac{g_k-1}{e^{\alpha +\beta {\varepsilon}_k}-1}, $$
$$ k=1,2,\dots, n. $$

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:

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

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

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

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

$$ {a}_k=\frac{g_k}{e^{\alpha +\beta {\varepsilon}_k}}, $$
$$ k=1,2,\dots, n. $$

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Tao, Y. Spontaneous economic order. J Evol Econ 26, 467–500 (2016).

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  • General equilibrium
  • Spontaneous order
  • Rawls’ fairness
  • Freedom
  • Technological progress

JEL classifications

  • D5
  • D63
  • B25