Equilibrium

x — y — ω must be zero or semi-negative. Taking this reflection into account, we may proceed to define the crucial notion of equilibrium.

10.4.1 Definition

An economic equilibrium is a tuple ((x_i∙), (y_j∙), p, (c,∙)) such that

(1) x 2 4⁽p, ^(c_y⁾⁾;

(2) y 2 Aj⁽p, ^c₇.⁾;

(3) the economy is reproducible; i.e. the net product, after the maintenance of the working class and its additional consumption x has been dis­counted, is at least equal to the original amount of total resources ω; i.e. excess demand z = x — y — ω is zero or semi-negative;

(4) p is a production price; i.e. the profit rate π is uniform with 0 ≤ ^p ≤ ⁽p^yj - pB^y_zVpBy_z;

(5) the financial capital market clears: ₁ c_j∙ = 0;

(6) there exist a reduction w with pB = wy.

In order to prove the existence of an equilibrium, let us introduce some auxil­iary concepts and lemmas. Notice that none of the sets introduced in the defini­tion are empty.

Let Y' be the set {[y, y, y] 2 Y ∣y ≤ l and y ≤ ωg of all production processes that satisfy the labor and material resources restriction.

Let X' be the set {x 2 X∣x ≤ y + ω for some y 2 Y'g of consumptions feasi­ble given the labor and material resources restriction.

Let Y be the set {y — By∣y 2 Y'g of all net products, after basic consumptions are discounted, of processes feasible given the labor and material resources restriction.

Let Z be the set X — Y — {ωg.

Let A:P ? ? C₁ ? ■■ ■ ? C_χ → X ? Y be defined by condition ^A(p; ^(cj⁾⁾ = Ac ⁽p; ^(cj⁾⁾ ? Af ⁽p⁾.

Let ς be the correspondence ς: P x C₁ x... x C_w → Z such that

B⁽p; ^(cj⁾⁾ = ^{x - ^y - ^ω'-^x∙^y) 2 A⁽p; ^(cj⁾⁾}-

In order to prove the existence of an equilibrium, several previous lemmas are needed.

10.4.2 Lemma

For every price vector p and every transfer c, sets B_j(p, c_j∙) and B_w (p) are non­empty, compact, and convex.

172 Classical economics

We have specified these concepts to some extent (with the exception of τ) in order to characterize classical economies. Functions η_k describe the actually observed behavior of the agents, which is supposed to be based on a limited, imperfect knowledge of the actual prevailing equilibrium of the system, or its tendencies of change. The empirical claim is that the agents’ actions, including the occa­sional fixation of local prices, takes place within a band determined by the equi­librium price, which in turn is determined by the social ponderation of the different trades and the structure of the industry.

The Leontief model (in the Tarskian sense of ‘model’) is quite illustrative of the former ideas. It represents a technology in which every kind of good is pro­duced by only one production process (there are no alternative techniques), there is no joint production, and the only primary factor is labor, which is supposed to be homogeneous. The matrix Y of inputs is square of order λ, while the matrix of outputs Y = I is the identity matrix of order λ. If labor is assumed to be

Notes

1 Cf. Debreu (1959: 19), Proposition 4. φ is our Bj and g is our Mj.

2 Quoted by Hamminga (1990: 92).

3 For a precise definition of this concept, cf. Takayama (1985: 370).

4 By theorem 4.D.2 in Takayama (1985: 392).

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