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”.

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)

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