Neoclassical economics on the edge: Fisher, Knight, and the theory of interest in the 1930s

Abstract

Irving Fisher and Frank Knight had a brief exchange on the former’s “The Theory of Interest”, published in 1930, which culminated in a full book review by Knight in 1931. We assess the episode by recasting Fisher’s interest theory as a maximisation under constraint calculation. After that, Knight’s objections to subjective time preference are examined, as well as his ensuing attack on the Austrian School’s definition of capital. Fisher’s and Knight’s reflections on business cycles, grounded on expectations, delays, and speculation, are assessed in sequence. We conclude by indicating some limitations of the theories of interest formulated by both economists.

Keywords:

• Time-preference
• productivity
• capital
• interest rate
• business cycles

JEL:

• B13
• B31
• B25

Acknowledgement

We thank the helpful comments by two anonymous reviewers who greatly helped improve the overall result. The usual disclaimer applies.

Disclosure statement

No potential conflict of interest was reported by the authors.

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Notes

1 For a compilation of the critical reactions to ROI and TOI, along with Fisher’s responses, see Dimand (2007, I: caps. 19–30, and III: caps. 81–87). Modern assessments of this literature are available in Barber (1997a, 473–480) and Dimand (2019, 99–104).

2 Frank Albert Fetter, contemporary to the debate, had earned his doctorate in economics at the University of Halle in Germany and afterwards became professor at Cornell and Princeton. Fetter sought inspiration in Carl Menger’s work and became the leader of the “American psychological school”. He critically intervened on the Clark-Böhm-Bawerk polemic in several occasions. All his papers on the issue were later compiled by Murray Rothbard in Capital, Interest, and Rent (Fetter 1977, see especially 33–74, 119–142, and 226–255).

3 Fisher’s 1891 famous doctoral thesis Mathematical Investigations in the Theory of Value and Prices developed a formal system of general equilibrium in prices without direct knowledge of Léon Walras’ contribution (Fisher (1892) 1926: Parts I and II). Knight, for his part, as we shall see below, displayed solid command of the mathematics of compound interest in his essays on the subject (see particularly Knight 1936a, 441–450).

4 On Fisher’s general contribution to economics, see Dimand (2019, all chapters). For his monetary ideas, see Boianovsky (2013, 206–237), Gomez Betancourt and Boyer des Roches (2013, 167–173), and Laidler (2013, 174–205). Knight’s economic theory and social conception, especially in comparison with J. Maynard Keynes’, is examined in Dimand (2021, 570–584), Brooke and Cheung (2021, 901–918), O’Donnell (2021, 1099–1125), as well as in Packard, Bylund, and Clark (2021, 1099–1125).

5 The British astronomer and mathematician Edmund Halley, in the article Of Compound Interest, published in 1861, was the first one to correctly calculate the present value $z$ of a fixed annuity $a$ received for $t$ years at the interest rate $r (=1+i)$ as $z=\frac{a}{r-1}-\frac{a}{r^t\times r-1}$ (Halley 1861, 265).

6 John Rae, Böhm-Bawerk, Alfred Marshall, and Adolphe Landry had based the depreciation of future goods on the uncertainty of life and the human difficulty of visualizing needs far off in time as vividly as the more pressing ones whose means of satisfaction are readily available to the senses. Böhm-Bawerk also mentioned the people’s lack of will to withstand present privations (Böhm-Bawerk (1891) 1930, 253–259; Landry 1904, 57–62; Marshall (1890) 1920, 117–123; Rae (1834) 1965, 118–123). For a brief history of the idea of time-preference in the works of pioneers such as Ferdinando Galiani, Anne Robert Jacques Turgot, and others, see Dimand (2019, 85–99) and Rothbard (2008, 281–284).

7 Fisher was the first one to show that longer periods of production do not always materialize in higher present values since two alternative income-streams could turn out to be the best choice depending on the interest rate, a situation that anticipated the phenomenon of reswitching in capital theory (Fisher (1907) 1997, 83–104, 381–385; see also Dorfman 1995, 21–34; Samuelson 1997, 16–28; Velupillai 1975, 679–680).

8 Fisher’s full mathematical system with $n$ individuals, $m$ goods, $m$ periods, and $m$ intertemporal interest rates is developed in chapter XIII of TOI (Fisher (1907) 1997, 332–345; see also Hirshleifer 1970, 31–40).

9 The first-order conditions for each step, given an initial interest rate $i,$ are respectively (1) $\frac{F_x}{F_y}=(1+i)$ along with $F\left(x,y,c\right)=0,$ and (2) $\frac{U_x}{U_y}=(1+i)$ along with $\left[{\mathit{\text{pv}}}^{*}-x-\frac{y}{(1+i)}\right]=0.$ If the second-order conditions for $P^{*}$ and $Q\prime$ maximum obtain and the resulting $K_{d }$ and $K_s$ differ, then the interest rate will change according to the excess demand for capital, so that $\mathit{\text{di}}/\mathit{\text{dt}}=K_d-K_s,$ and therefore a new round of calculations is performed by all parties until the equilibrium rate $i^{*}$ is eventually achieved.

10 “I made the mistake of youth in not blowing my own horn, expecting others to discover my merits and fall all over themselves in admiring them. Instead they have not even read the book. … The ‘impatience’ side is not original except in form of presentation. The productivity side is original” (Fisher to Brown, May 10, 1928).

11 For Clark’s concept of capital, see Cohen (2008, 151–171); Dewey (1987, 428–431); Henry (1995, 71–88), and Clark ((1899) 1908, 116–205, 1888, 9–69); for Marshall’s, see Bliss (1990, 223–241); Bridel (1987, 7–24); Reisman (1986, 246–282), and Marshall ((1890) 1920, 71–82, 220–236, 351–380, 785–790).

12 Knight 1925b); see William I. Thomas’s and Florian Znaniecki’s methodological reflections on social action as stemming from personal attitude and built-in group values (1918, 1–86), approvingly cited in RUP ((1921) 1964, 333 f.1).

13 Fisher may be guilty just of lack of emphasis, for he had indeed accepted that differences between actual cost and the present value of capital goods would move both variables over time toward a common denominator (Fisher (1907) 1997, 61–63; (1930) 1997, 498–505).

14 “A realistic discussion of the motives involved in decisions as to saving would run rather in terms of interest in security and power, of living standards, of forms of social emulation and of similar facts of social psychology and culture history” (Knight 1934, 137, 1937, 107–110).

15 Knight was not very keen on avoiding inconsistencies, as shown by his remarks about the short-term fixity of savings when he had admitted earlier on that an expansion in credit induced investment could occasion immediate changes in real income by the mobilization of dormant resources. “It is arguable, however, that new purchasing power may serve to bring into use existing wealth or productive power previously idle and so to increase the total output and circulation of goods instead of raising prices” (Knight 1932, 142, 1936b, 627).

16 Knight represents the series of capitalized costs and the infinite discounted series of net revenues through the notation $\beta =S,\pi =R ($both constants), $a=c$ and $A=\left(1+i\right),$ with the respective equation expressed as $\frac{S\left(A^c-1\right)}{i}=\frac{R}{i},$ which is the discrete time version of (5) (Knight 1936a, 445, 1934, 264–267).

17 For our limited purposes here, the Austrian School encompasses all economists that in one form or another accepted the plausibility of such proposition. A more accurate description of the original Austrian way of thinking though would take into account Carl Menger’s judgement of Böhm-Bawerk’s concern with the conditions of production as his “greatest error” due to the implicit downplaying of the subjective aspects involved in the determination of interest (Schumpeter (1954) 2006, 814, f. 8; on this, see Endres 1987; Garrison 1990).

18 Formulas for Böhm-Bawerk’s average period of production under continuous input-point output productive processes are developed in Gifford (1933) and Dorfman (2001). As we are dealing here with flows of constant value, this measure can be safely skipped by simply taking the length of construction as $a.$

19 Most of the content of this section about Fisher’s theory of instability is covered in Stabile and Putnam (2002), where a non-formal but compelling case is presented for qualifying the American economist as a forerunner of modern portfolio theory, particularly with respect to the statistical measurement of earnings and risk in stocks. The main approach here though centres on the return of capital goods and its dependence on time-preference and present values, expectational variables which go beyond Bayesian estimates. That is the characteristic which sets Fisher apart from other neoclassical economists like Marshall, Wicksell, or even Knight, since no natural or long-run interest rate is supposed to exist working as an attractor for the short-run monetary interest rate. All economic phenomena in ROI and TOI are ultimately reflexes of mental processes.

20 Since $p^t=e^{\mathit{\text{tln}}(p)},$ we have for the integrand on the left-hand side of (6) $\pi (t)e^{-\left[i-\mathit{\text{ln}}\hspace{0.17em}\left(p\right)\right]t}.$ But as the Taylor series for $\mathit{\text{ln}}\hspace{0.17em}\left(x\right)$ around $x=1$ is $\mathit{\text{ln}}\hspace{0.17em}\left(x\right)=\left(x-1\right)-({(x-1)}^2/2!)+\dots ,$ we can write the integrand, retaining only the first term of the expansion and remembering that $p=1-q,$ as being approximated by the one on the right-hand side of (6) for small values of $q$ (the same result obtains for finite intervals, as shown in Fisher (1906) 1907, 404–405).

21 Fisher understood that the difficulty with business cycles theories resided in the fact that changes in the price level always get mixed up with other disturbances: “Imagine, then, a rocking chair on the deck of a rocking ship, on a rolling sea. The ultimate chair is subjected to so many influences that its motion will not conform with any simple rhythm” (Fisher (1932) 1997, 117; see also Dimand 2003).

22 Investors would thus exhibit herd behaviour: “In whatever direction the leaders of fashion first chance to move, the crowd will follow in mad pursuit until almost the whole social body will be moving in that direction” (Fisher (1930) 1997, 126; (1932) 1997, 93).

23 Conditions can snowball fairly quickly if a demand shock makes the expected rate of deflation overcomes the fall in the loan rate, so that the expected real interest rate rises substantially during an economic contraction. In such case, the Fisher effect may push the economy way off its corridor of stability (Fisher (1932) 1997, 98–99; see also Dimand 2005, 185–199).

24 Such response to changing commodity prices may be intensified by the very short-term interference of professional traders who stand between consumers and producers in most organized markets, a supplementary first layer introduced by Knight in Marshall’s period analysis (Knight (1930) 1955, 170–172).

25 Knight was likely aware of Ragnar Frisch’s well-known pioneering study on impulse and propagation dynamics in mixed difference-differential equations included in Economic Essays in Honour of Gustav Cassel, published in 1933, a book where the American economist had also a chapter on capital and time of production (Frisch (1933) 1967: 171-205; Knight (1933) 1967: 327-342).

26 The price trajectory in (7) depends on the boundary conditions $p\left(0\right)$ and the entire sequence of price values over the initial interval $(-\tau ,0).$ Let now a solution of (7) take the form $p\left(t\right)=e^{\lambda t}.$ Then $\lambda$ must be a root of the characteristic equation of the homogeneous part $\lambda -A-B{e}^{-\lambda \tau }=0,$ which is transcendental and possesses multiple roots. For $A+B>0,p(t)$ will be unstable. If $A+B<0,$ and $B\ge A,$ then the solution is asymptotically stable. But if $B<A,$ there is a ${\tau }^{\prime }$ such that for $\tau <{\tau }^{\prime },$ the trajectory of prices is asymptotically stable, whereas for ${\tau }^{\prime }<\tau ,$ it becomes unstable (Smith 2010, 41–56).

27 Keynes’s theory of a monetary economy was criticized by Knight as overly ambitious, confuse, and quite useless for its sheer neglect of the technical aspects which ultimately determine the long-run interest rate (Knight 1941, 63–65; 1937, 107. f. 9; (1921) 1964, xlii-xlvi).

Additional information

Funding

This work was supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (304783/2022-6).