A Note on Reswitching and Intertemporal Prices

Abstract

Bliss (Capital Theory and Distribution of Income, Amsterdam/New York: North-Holland/Elsevier) claims that reswitching is nothing but an ‘optical illusion’ due to the exclusion of non-stationary price sequences from the analysis. This note develops this point. The standard case for choice of techniques and reswitching is reformulated in terms of Arrow-Debreu intertemporal prices and the conditions making these prices stationary are highlighted separately. It is then shown that the analysis of the choice of techniques in terms of ‘switch points’ requires stationary conditions.

Keywords:

• prices
• capital
• reswitching

JEL Codes:

• B51
• D21
• D46

Acknowledgments

Sincere thanks to Christian Gehrke for comments and suggestions. The suggestions of two anonymous referees are also gratefully acknowledged. The usual disclaimer applies for any remaining errors.

Notes

¹This point is also presented by Bliss (1990, p. 152) in his entry on ‘equal rates of profit’ in the New Palgrave Dictionary of Economics, but the argument there, rapidly sketched, is less clear than in the above quoted passage.

²This model corresponds to the one considered by Garegnani (1966, pp. 566–567) in the appendix of his article for the 1966 symposium in the Quarterly Journal of Economics.

³The wage-interest curve associated with technique α is the graph of the function wα = wα(r), where wα solves the equations (1) and (2) for a given r. In the same way, a function wβ = wβ(r) can be defined by the equations (1) and (3).

⁴By definition, the own-rates of interest of the two commodities are: r _x = p _{x , t} /p _{x , t +1} – 1 and r _y = p _{y , t} /p _{y , t +1} – 1. It is therefore immediately evident that r _x = r _y = r if and only if p _{y , t} /p x_{, t} = p _{y , t +1}/p _{x , t +1} = π _y . Therefore, in a stationary price vector, ‘all present-value price vectors [are] proportional to each other and all own-rates of interest equal’ (Bliss, 1975, p. 69). See also Bliss (1990, p. 151).

⁵Given a solution of System 1–3, Equalities 4 and 5 make it possible to determine the corresponding prices and univocally. In particular, we have: ; and . Therefore, since solves the System 1–3, then satisfies Equations 6–8.

⁶This clearly means that and ; and similarly and . See also the previous footnote.

⁷The proof of the possibility of non-negative prices satisfying the System 6–8 with a null wage rate can be briefly outlined as follows. With w = 0, Equation 6 implies p_{y, t} = 1/a₁₂ – (a₁₁/a₁₂) p_{x, t}, which is the equation of a decreasing straight line crossing the vertical axis at point 1/a₁₂. Equations 7 and 8 instead bring about p_{y, t} = –(a₂₁ – a₃₁)/(a₂₂ – a₃₂) p_{x, t}, which is the equation of a straight line starting from the origin of the axes. If, therefore, in the production of y, one of the two alternative methods employs more of commodity x per unit of output and the other more of commodity y, then the straight line derived from Equations 7 and 8 is increasing and consequently intersects the other line for a pair of strictly positive prices (p_{x, t}, p_{y, t}). And this also implies p_{y, t+1} > 0. If instead the method of production of y that employs more of commodity x per unit of output employs more of commodity y too, then there is a minimum strictly positive level of w compatible with the coexistence of techniques. There is a very simple reason for this, namely that the method employing both less of x and less of y per unit of output is unquestionably the most advantageous when w = 0. This appears, however, to be a particular case as it allows us to say which technique is more capital-intensive than the other through direct reference to physical capital without knowing the prices.

⁸It is clear that P_α ∪ P_β ⊂ P, and also that the set P \ {Pα ∪ P_β} is non-empty.

⁹The case in which the wage-interest curves of the two techniques coincide on a certain stretch is ignored here because it is associated with a non-generic choice of technical coefficients.

¹⁰It can be observed that in the case of stationary prices, wage rate levels above W are regarded as economically inadmissible because they bring about negative interest rates r. When non-stationary price vectors are instead admitted, it seems possible to have vectors of non-negative prices associated with wage rates higher than W. This possibility appears, however, to play no role in the argument presented in this note.

¹¹To see this it is sufficient to observe that, once w is taken as given, there are three unknown prices in the System 11–13 and therefore its degrees of freedom depend on the number of conditions satisfied with the inequality sign.

¹²The idea for this kind of graphic representation is taken from Figure 17.D.6 in Mas-Colell, Winston and Green (1995, p. 598), where points on the vertical axis are assumed to be vectors of L – 1 prices.

¹³This refers here to the 1966 symposium on the Quarterly Journal of Economics.

¹⁴This very peculiar idea of aggregating a vector of quantities of different commodities into a scalar quantity of a factor of production appears to derive from confusion over two different concepts, namely ‘capital’ and ‘capital goods’, which are often erroneously seen as two sides of the same coin. Capital is a scalar quantity but not an input, being an amount of purchasing power used to anticipate the costs of production before revenues are obtained. For example, wages paid ex ante are ‘advanced’ from capital, whereas post factum wages are paid from revenue (Sraffa 1960, p. 10). Capital goods—whose costs, beside the others, are anticipated by capital—are inputs but cannot be aggregated without the risk of arriving at paradoxical results. In order to avoid any confusion, it would be better to use ‘means of production’ instead of ‘capital goods’ and drop the terms ‘aggregate’ and ‘disaggregate’ capital, as they do not correspond to rigorously scientific objects.

¹⁵As Marshall (1920, p. 534) wrote: ‘interest, being the price paid for the use of capital in any market, tends towards an equilibrium level such that the aggregate demand for capital in that market, at that rate of interest, is equal to the aggregate stock forthcoming there at that rate.’

¹⁶An instability argument grounded on the increasing shape of the demand for capital schedule due to reverse capital deepening was already present in Garegnani (1970). Other relevant examples are Garegnani (1990) and Kurz and Salvadori (1995).

¹⁷For a discussion of the reasons why Wicksell was essentially forced to take the value of the existing stock of capital as given, see Garegnani (1990), Kurz (2000) and Fratini (2013b).

¹⁸See also Potestio (1999) and Kurz and Salvadori (2001).

¹⁹While the most relevant contributions of this second phase are Bliss (1975) and Hahn (1975, 1982), its beginning can perhaps be dated earlier and associated with the publication of Bliss's (1970) comment on Garegnani's article (1970) in the Review of Economic Studies.

²⁰Stiglitz (1974) and Dixit (1977) can also be mentioned. They expressed views on the debate although they did not actually take part in it.

²¹The point is complex and cannot be given the attention it deserves here. Readers are therefore referred in particular to Garegnani (1976) and Petri (1978).

²²This seems to be recognised even by neo-Walrasian authors when they write that the Arrow-Debreu equilibrium model is just a benchmark against which the real economy can be measured and not a state toward which it tends. See Hahn (1985) and Geanakoplos (1989).

²³See Mandler (2002, 2005) for a neoclassical reply to the arguments of Garegnani and Schefold.

²⁴On the one hand, if outputs can be sold in the very moment when inputs have to be paid for, no investment of capital appears necessary because costs can be met directly out of revenues. On the other hand, as saving has the function of moving purchasing power from the present to a subsequent market day, it appears to have no role in a model where markets, for present and future delivery, are open for a single instant, so that the entire purchasing capacity of each household exists and is spent in that moment.