Appendix 1

Mathematical note to Garegnani, pp. 13–74

Michele Tucci

Note: Equation numbering follows original.

THE MODEL

Let us take into consideration the following model (F):

	(5.1f)
	(5.2f)
	(5.3f)

If in the first relation of (5.3f) the inequality sign holds, then P_a0 = 0; if it holds in the second one, then P_b0 = 0. However, as is specified in Paragraph [3] of Appendix I, Assumption (ii), it should be noted that such cases will not be taken into consideration in the economic discussion in the text. The inclusion in the present demonstration is due to the need for clarifying the mathematical passages.

If the inequality sign holds in the third relation of (5.3f), then W = 0.

	(5.4f)
	(5.5f)
	(5.6f)
	(5.7f)

where P_a0, P_b0, P_a1 and P_b1 refer to prices, W indicates wages, S and I represent savings and investments, respectively, A₁ and B₁ correspond to quantities of produced

Figure 5.4 An Example of case 1.

commodities, A₀^D and B₀^D specify demands, and A₀^S, B₀^S and L refer to endowments. The single-valued mappings D_a0, D_b0, D_a1 and D_b1 designate standard Walrasian demand functions, which are characterised by the following assumptions:

1 the set of independent variables in the functions are those indicated by the first five among the symbols specified above;

2 in the non-negative orthant of the independent variables, each demand function is positive and continuous.

For the sake of simplicity, we suppose that the technical coefficients are all strictly positive. Moreover, the usual vitality conditions will hold. Finally, r_b specifies the own interest rate of commodity b over period t = 0. The first eleven among the symbols listed above constitute the unknowns of the model, while r_b is exogenously defined.

Let us substitute equations (5.6f) and (5.2f) into the second equation of (5.1f), thus obtaining:

(5.8)

i.e. equation (5.1a) of Paragraph [2] of Appendix I.

Easy passages allow the following propositions to be derived from equations (5.1f), (5.2f), (5.6f) and (5.8):

(a) The quantities W and P_a1 can be defined as functions of the single variable P_a0.

(b) Consider the interval H_rb = {−1 ≤ r_b ≤ max_rb}, with max_rb = (1 − b_b)/b_b. Define β_rb = [1 − b_b(r_b + 1)′ /a_b. For every r_b ∈ H_rb, we can determine an interval H_Pa0 = {0 ≤ P_a0 ≤ β_rb} such that, for every P_a0 ∈ H_Pa0, W ≥ 0, P_a1 ≥ 0.

(c) In the interval H_Pa0, the functions W(P_a0) and P_a1(P_a0) are continuous.

(d) If −1 < r_b ≤ max_rb, then P_b0 > 0; if r = −1, then P_bo = 0.

Due to the income correction, which is specified in Paragraph 9 of the text, the third of (5.3f) is always satisfied with the equality sign, except in border solutions, which are examined in connection with Assumption (iii), Paragraph [5] of Appendix I, and in Paragraphs [6], [10], [13] and [14] of the same Appendix, where there will be a continuous set of solutions characterised by W = 0.

Let us assume that the equality sign holds in the first two relations of (5.3f). Substituting the last two equations of (5.3f) into the expressions on the right-hand side of the equality sign in the first two relations of (5.3f) and in equation (5.4f), we are able to define the variables A₀^D, B₀^D and I. Moreover, equation (5.5f) defines the variable S. In the interval H_Pa0, the four above-quoted expressions are continuous functions of the unique variable P_a0.

Define

	(5.9)
	(5.10)

In the interval H_Pa0, the function δ(P_ao) is continuous.

Taking into consideration equation (5.7f), for every r_b ∈ H_rb one, and only one, of the following three sentences is necessarily true:

1 There exists P_a0* ∈ H_Pa0 such that δ(P_ao*) = γ.

2 For every P_a0 ∈ H_Pa0, δ(P_a0) < γ.

3 For every P_a0 ∈ H_Pa0, δ(P_a0) > γ.

Figure 5.4 shows an example of case (1). Here, P_a0 ≥ 0 and the first two relations of (5.3f) are satisfied with the equality sign. Assumption (ii) in Paragraph [3] of Appendix I confines the main argument there to case (1) above. Therefore, the remaining two cases will be examined only for the sake of completeness.

In case (2), at the given level of r_b, commodity a₀, taken in the sum of both its consumption and investment uses, cannot be employed in as high a proportion to b₀ as the ratio γ in which it is found in the endowment. As a result, (F) can only admit solution if we allow the quantity A₀^D ‘demanded’ of equation (5.7f), to exceed the quantity used expressed by the R.H.S. of the first part of equation (5.3f), provided P_a0 = 0 (cf. n. 5 to Appendix I).

In case (3) we have the case symmetrical to II, where b₀ cannot be used in as high a proportion to a₀ (i.e. a₀ in as low a proportion to b₀), as the ratio in which the two commodities are found in the endowment. The inequality sign of the second relation in (5.3f) will allow for a solution of (F) provided P_bo = 0, i.e. if r_b is set at r_{b min} = −1. But no solution would exist if we set r_b, our independent variable, at a level r_b > −1 − a case, however, which is of no economic importance since (F) could never, then, provide a solution of (E), where r_b > −1 implies P_b0 > 0, and is therefore incompatible with an inequality sign in the second of relations (5.3e).