Reswitching With Smooth Production Functions

I cite authority:

"Something precious I gained from Robinson's work and that of her colleagues working in the Sraffian tradition. As I have described elsewhere, prior to 1952 when Joan began her last phase of capital research, I operated under an important misapprehension concerning the curvature properties of a general Fisher-von Neumann technology.

What I learned from Joan Robinson was more than she taught. I learned, not that the general differentiable neoclassical model was special and wrong but that a general neoclassical technology does not necessarily involve a higher steady-state output when the interest rate is lower. I had thought that such a property generalized from the simplest one-sector Ramsey-Solow parable to the most general Fisher case. That was a subtle error and, even before the 1960 Sraffa book on input-output, Joan Robinson's 1956 explorations in Accumulation of Capital alerted me to the subtle complexities of general neoclassicism.

These complexities have naught to do with finiteness of the number of alternative activities, and naught to do with the phenomenon in which, to produce a good like steel you need directly or indirectly to use steel itself as an input. In other words, what is wrong and special in the simplest neoclassical or Austrian parables can be completely divorced from the basic critique of marginalism that Sraffa was ultimately aiming at when he began in the 1920s to compose his classic: Sraffa (1960). To drive home this fundamental truth, I shall illustrate with the most general Wicksell-Austrian case that involves time-phasing of labor with no production of any good by means of itself as a raw material.

As in the 1893-1906 works of Knut Wicksell, translated in Wicksell (1934, Volume I), let corn now be producible by combining labor yesterday, labor day-before-yesterday, etc):

Q_t = f(L_t-1, L_t-2, ..., L_t-T) = f(L)                                                     (1)

Q = f(L₁, L₂, ..., L_T) in steady states                                              (2)

    = L₁ f(1, L₂/L₁, ..., L_T/L₁) 1st^o-homogeneous and concave      (3)

    = L₁ (df(L)/dL₁) + ... + L_T (df(L)/dL_T), Euler's theorem            (4)

df/dL_j = f_j(L), d²f/(dL_i dL_j) = f_ij(L) exist for L ≥0                         (5)

f_j > 0, (z₁, ..., z_T)[f_ij(L)](z₁, ..., z_T)' < 0 for z_j ≠ b L_j > 0                (6)

Nothing could be more neoclassical than (1)-(6). If it obtained in the real world, a Sraffian critique could not get off the ground.

Yet it can involve (a) the qualitative phenomena much like 'reswitching', (b) so-called perverse 'Wicksell effects', (c) a locus between steady-state per capita consumption and the interest rate, a(i, c) locus, which is not necessarily monotonically negative once we get away from very low i rates. This cannot happen for the 2-period case where T = 2. But for T ≥ 3, all these 'pathologies' can occur, and there is really nothing pathological about them. No matter how much they occur, the marginal productivity doctrine does directly apply here to the general equilibrium solution of the problem of the distribution of income.

Remarks. What eternal verities do always obtain, even when corners in the technology make derivatives [dQ_j/dL_j, dQ_j/dQ_ij] be somewhat undefined? Always, it remains true:

(a) To go from an initial sub-golden-rule steady state to a maintainable golden-rule steady state of maximal per capita consumption, must involve for society a transient sacrifice of current consumptions ('waiting' or 'abstinence' a la Senior, Böhm, and Fisher!).

(b) For non-joint-product systems, there is a steady-state trade-off frontier between the interest rate and the real-wage (expressed in terms of any good).

This monotone relation between (W/P_j, i) was obscurely glimpsed by Thunen and other classicists and by Wicksell and other neoclassicists. But the factor-price trade-off frontier did not explicitly surface in the modern literature until 1953, as in R. Sheppard (1953), P. Samuelson (1953), and D. Champernowne (1954). One can prove it to be well-behaved for (1)-(3), or any convex-technology case, by modern duality theory. Before Robinson (1956), I wrongly took for granted that a similar monotone-decreasing relation between ( i, Q/(L₁ + ... + L_T) ) must also follow from mere concavity - just as does the relation -d²C_t+1/(dC_t)² = di/dC_t) > 0. But this blythe expectation is simply wrong! I refer readers to my summing up on reswitching: Samuelson (1966).

I realize that there are many economists who tired of Robinson's repeated critiques of capital theory as tedious and sterile naggings. I cannot agree. Beyond the effect of rallying the spirits of economists disliking the market order, these Robinson-Sraffa-Pasinetti-Garegnani contributions deepen our understanding of how a time-phased competitive microsystem works." -- Paul A. Samuelson (1989) "Remembering Joan" in Joan Robinson and Modern Economic Theory (ed. by George R. Feiwel), New York University Press.

(I have changed some of the symbols above.) I've noted before comments from Samuelson in papers that have made claims much the same as above.