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. 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. 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. 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. 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. 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. 6.

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

  7. 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. 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. 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. 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. 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. 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. 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. 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. 15.

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

  16. 16.

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

  17. 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. 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. 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. 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. 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. 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. 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. 24.

    Therefore, technological progress T is also an endogenous variable.


  1. Alesina A, Angeletos GM (2005) Fairness and redistribution. Am Econ Rev 95:960–980

    Article  Google Scholar 

  2. Alesina A, Cozzi G, Mantovan N (2012) The evolution of ideology, fairness and redistribution. Econ J 122:1244–1261

    Article  Google Scholar 

  3. Arrow KJ (1963) Social choice and individual values. Wiley, New York

    Google Scholar 

  4. Arrow KJ, Debreu G (1954) Existence of an equilibrium for a competitive economy. Econometrica 22:265–290

    Article  Google Scholar 

  5. Barro RJ (1996) Democracy and growth. J Econ Growth 1:1–27

    Article  Google Scholar 

  6. Bouchaud JP (2008) Economics needs a scientific revolution. Nature 455:1181

    Article  Google Scholar 

  7. Broome J (1984) Uncertainty and fairness. Econ J 94:624–632

    Article  Google Scholar 

  8. Carter AH (2001) Classical and statistical thermodynamics. Pearson Education, Prentice-Hall

  9. Clementi F, Gallegati M, Kaniadakis G (2012) A new model of income distribution: the κ-generalized distribution. J Econ 105:63–91

    Article  Google Scholar 

  10. Debreu G (1971) Theory of value: an axiomatic analysis of economic equilibrium. Yale Univeristy Press, New Haven

    Google Scholar 

  11. Deesomsak R, Paudyal K, Pescetto G (2009) Debt maturity structure and the 1997 Asian financial crisis. J Multinatl Financ Manag 19:26–42

    Article  Google Scholar 

  12. Farmer JD, Foley D (2009) The economy needs agent-based modelling. Nature 460:685–686

    Article  Google Scholar 

  13. Foster J, Metcalfe JS (2012) Economic emergence: an evolutionary economic perspective. J Econ Behav Organ 82:420–432

    Article  Google Scholar 

  14. Hayek FA (1948) Individualism and economic order. The University of Chicago Press

  15. Hodgson GM (2009) The great crash of 2008 and the reform of economics. Camb J Econ 33:1205–1221

    Article  Google Scholar 

  16. Jehle GA, Reny PJ (2001) Advanced microeconomic theory (2nd Edn). ADDISON-WESLEY

  17. Koopmans TC (1964) On the flexibility of future preferences. In: Shelly MW, Bryan GL (eds) Human judgments and optimality. Wiley, New York

    Google Scholar 

  18. Kreps DM (1979) A representation theorem for ‘preference for flexibility’. Econometrica 47:565–577

    Article  Google Scholar 

  19. Kürten KE, Kusmartsev FV (2011) Bose-Einstein distribution of money in a free-market economy. II. Europhys Lett 93:28003

    Article  Google Scholar 

  20. Kusmartsev FV (2011) Statistical mechanics of economics I. Phys Lett A 375:966

    Article  Google Scholar 

  21. Larsen RJ, Marx ML (2001) An introduction to mathematical statistical and its application (3rd Edn). Prentice-Hall, Inc.

  22. Leijonhufvud A (2009) Out of the corridor: Keynes and the crisis. Camb J Econ 33:741–757

    Article  Google Scholar 

  23. Mas-Collel A, Whinston MD, Green JR (1995) Microeconomic theory. Oxford University Press

  24. Minsky HP (1986) Stabilizing an unstable economy. Yale University Press

  25. Pattanaik PK, Xu YS (1998) On preference and freedom. Theor Decis 44:173–198

    Article  Google Scholar 

  26. Pazner EA, Schmeidler D (1974) A difficulty in the concept of fairness. Rev Econ Stud 41:441–443

    Article  Google Scholar 

  27. Pazner EA, Schmeidler D (1978) Egalitarian equivalent allocations: a new concept of economic equity. Q J Econ 92:671–687

    Article  Google Scholar 

  28. Puppe C (1996) An axiomatic approach to ‘preference for freedom of choice’. J Econ Theory 68:174–199

    Article  Google Scholar 

  29. Rawls J (1999) A theory of justice (Revised Edition). Harvard University Press, Cambridge

    Google Scholar 

  30. Romer PM (1990) Endogenous technological change. J Polit Econ 98:S71–S102

    Article  Google Scholar 

  31. Romer D (2000) Advanced macroeconomics (2nd edn). The McGraw-Hill Companies, Inc.

  32. Schotter A (1981) The economic theory of social institutions. Cambridge University Press

  33. Schumpeter J (1934) The theory of economic development. Harvard University Press, Cambrige

    Google Scholar 

  34. Sen A (1993) Markets and freedoms: achievements and limitations of the market mechanism in promoting individual freedoms. Oxf Econ Pap 45:519–541

    Google Scholar 

  35. Sudgen R (1998) The metric of opportunity. Econ Philos 14:307–337

    Article  Google Scholar 

  36. Sugden R (1989) Spontaneous order. J Econ Perspect 3:85–97

    Article  Google Scholar 

  37. Tao Y (2010) Competitive market for multiple firms and economic crisis. Phys Rev E 82:036118

    Article  Google Scholar 

  38. Varian HR (1992) Microeconomic analysis, 3rd edn. Norton & Company, Inc. , New York

    Google Scholar 

  39. Varian HR (2003) Intermediate microeconomics: a modern approach, 6th edn. Norton, New York

    Google Scholar 

  40. Verme P (2009) Happiness, freedom and control. J Econ Behav Organ 71:146–161

    Article  Google Scholar 

  41. Williamson OE, Winter SG (1993) The nature of the firm: origins, evolution, and development. Oxford University Press, Inc.

  42. Witt U (1997) The Hayekian puzzle: spontaneous order and the business cycle. Scott J Polit Econ 44:22–58

    Article  Google Scholar 

  43. Yakovenko VM, Rosser JB (2009) Statistical mechanics of money, wealth, and income. Rev Mod Phys 81:1703–1725

    Article  Google Scholar 

  44. Young HP (1993) The evolution of conventions. Econometrica 61:57–84

    Article  Google Scholar 

  45. Young HP (1996) The economics of convention. J Econ Perspect 10:105–122

    Article  Google Scholar 

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Corresponding author

Correspondence to Yong Tao.

Additional information

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). https://doi.org/10.1007/s00191-015-0432-6

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

JEL classifications

  • D5
  • D63
  • B25