Abstract
Harrod’s dynamic theory faced two major criticisms. The first was Harrod’s failure to provide a basis for the existence of a unique warranted line of advance. Harrod responded by making a more general assumption about entrepreneurial behavior, and by introducing the concept of the representative entrepreneur, which did not satisfy his critics. A second line of attack championed by post-Keynesians and neoclassical economists alike argued that cumulative deviations around a warranted line of advance resulted from the assumption of constant parameters which led them to highlight the “knife-edge” properties of Harrod’s dynamics. Harrod never made such assumptions and vehemently opposed the term knife-edge to describe the workings of his dynamic theory. Harrod reacted by giving a more prominent role to the rate of interest which led him to develop his second fundamental equation and to redefine the natural rate of growth (Gn) as a welfare optimum, which eventually Harrod did not consider important.
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Notes
- 1.
- 2.
According to Kregel (1980) the misrepresentation of Harrod’s dynamic theory is not only due to this knife-edge view but also to the assumption that Harrod’s dynamics is a long-run growth theory based on Keynes’s short-period analysis of the GT.
- 3.
However, other elements introduced by Alexander misrepresented Harrod’s view. One such element is the specification of savings as a lagged function of output. This was one specifically to translate Harrod’s model into a first order differential equation without regard to the conceptual coherence of the model. As explained by Miconi (1967, p. 461), this leads to the view that “the amount of investment of one period depends on the amount of savings that the income of the previous period can generate, so that savings governs investment rather than the other way around.” Baumol (1949) also uses the same specification. Miconi provides a taxonomy of the misrepresentations of Harrod’s model mainly by Alexander, Baumol (1948, 1949), Encarnación (1965, 1966), Hahn and Mathews (1964), Jorgenson (1960), and Neville (1960). Harrod (1971a, p. 78) called Niconi’s taxonomy “an admirable catalogue of some of the misrepresentations.” See Annex to this chapter for a summary of Niconi’s taxonomy.
- 4.
In his “Notes on the Trade Cycle” (1951, p. 272), Harrod defined the representative entrepreneur as: “…taken to be representative in those attitudes of courage and restraint, of optimism and pessimism…which together govern a man’s reaction to the current outturn of business. He may be defined more precisely as one whose orders in response to a given out-turn are such that that the sum of the excesses of all entrepreneurs in the economy who would order more in a precisely similar situation over what he would order is equal to the sum of the shortfalls of all those who would order less. The formula that correctly describes the state of mind of this representative entrepreneur may be applied to the macro-economy.”
- 5.
If \(s_{w} = 0\) then \(S = s_{c} P \Rightarrow I = S \Leftrightarrow I = s_{c} P\) and \(P = \frac{I}{{S_{c} }}\). Dividing both sides by K and considering that the rate of growth of capital \(\left( {G_{K} } \right)\) in steady state is equal to the rate of growth of output \(\left( {G_{Y} } \right)\), the Cambridge equation obtains, \(\frac{I}{K} = s_{c} \frac{P}{K} \Leftrightarrow G_{K} = G_{Y} = s_{c} r\;\left( {r = {\text{rate}}\,{\text{of}}\,{\text{profit}}} \right)\). Robinson (1978, p. xvi) argued that Harrod did not have a profit rate in his model.
- 6.
See also Kaldor (1961, pp. 196–197): “It is the most significant feature of Keynes’s theory to have shown that equilibrium between savings (ex ante) and investment (ex ante) is secured through forces operating in the commodity markets. When investment exceeds savings, the demand for commodities will exceed the supply. This will lead either to an expansion of supply (assuming the prevalence of ‘Keynesian’ unemployment…) or to a rise in process relatively to costs (assuming ‘full employment’ in the Keynesian sense…). In both cases an increase in the demand for commodities will lead to an increase in savings: in the first case because savings are an increasing function of real income at any given relationship of…profits to wages; in the second case, because the rise in prices relative to costs implies a rise in profits and a fall in wages…which increases savings.” In the case of Kaldor, he assumes initially that investment (I) is exogenous (Kaldor 1955–1956, p. 229), and thus independent of profits. However, as his growth model developed in “A Model of Economic Growth” (1957), “Capital Accumulation and Growth” (1961), and a “New Model of Economic Growth” (1962) investment is endogenized as a function of output and profits. Investment behavior assumes an increasingly greater part in guaranteeing stability and given the specification of the savings function. The investment function jointly with a technical progress function, through which output is made dependent on investment with the assumption that an increase in investment produces a less than proportionate rise in output, provide the basic working mechanism for Kaldor’s growth model and the basis for a steady-state solution. In the model profits not only act as a reinforcing mechanism for growth but they also play a crucial role in maintaining the stability of the steady-state solution. Kaldor made the explicit point that full employment was the “normal” state of affairs, if not of all market economies, at least of the “successful” ones (Kaldor 1955–1956, p. 99).
- 7.
The conditions for a well-behaved production function were established by Inada (1961) and are as follows. Let a production be specified as \(Y = F\left( {K, L} \right),\) where Y, K, L are output, capital, and labor. The assumption of constant returns to scale allows the production function to be formulated in per capita terms as \(y = f\left( k \right);\,y = \frac{Y}{L}, \,k = \frac{K}{L} \cdot \,f\left( {..} \right)\) is a continuously differentiable function with decreasing marginal returns. A well-behaved production function satisfies the following conditions:
\(f\left( 0 \right) = 0;\,f\left( \infty \right) = \infty\) (zero input zero product; infinite input, infinite product).
\(f^{{\prime }} \left( k \right) > 0;\,f^{{{\prime \prime }}} \left( k \right) < 0\) (marginal decreasing returns).
\(\mathop {\lim }\limits_{k \to 0} \,f^{{\prime }} \left( k \right) = \infty ;\;\mathop {\lim }\limits_{k \to \infty } \,f^{{\prime }} \left( k \right) = 0\).
Given that factor prices equal their marginal productivities, payments to the factors of production exhaust output.
See Harris (1980, p. 43).
- 8.
Steady state refers to an equilibrium position where every variable considered is constant or growing at the same rate which excluding technical change is the rate of growth of labor (and as a result the rate of growth of output is determined by technological progress which is exogenous). Growth Theory, at least prior to endogenous growth or what we call neoclassical growth theory, centers around the notion of a steady-state growth and its properties (“Most of the modern theory of economic growth is devoted to analyzing the properties of steady states and to finding out whether an economy not initially in a steady state will evolve into one…” Solow 1988, p. 4). Neoclassical theory views the steady state as a starting point of growth and dynamic analysis since it describes the most-simple form of growth path (Dixit 1976; Haache 1979). At the same time, it is consistent with the central organizing concept of neoclassical economics, equilibrium, in its intertemporal version, characterizing a market economy in terms of “short period equilibrium positions in sequence over time.” The fundamental steady-state growth equation provides a characterization of the dynamics of capital through a first order linear differential equation. Formally, (1) \(\dot{k} = sf\left( k \right) - \delta k,\)
where \(k, \,\dot{k}, \,{\text{and}}\, s\) refer to the capital–output ratio, its growth rate, and the savings rate. And δ is the depreciation rate of the capital stock. With technological change, the steady-state growth equation is (2) \(\dot{k} = sf\left( k \right) - \left( {\delta + \tau } \right)k,\) where τ is the rate of technical progress (which in line with the logic of the model grows at a constant rate). In order to include technological progress or technical change, steady-state growth requires that it take a particular form. One such form is labor-augmenting technical change. Labor-augmenting technical change is equivalent to Harrod neutral technical progress (that is “technical change that leaves the capital-to-output or capital share unchanged when the rate if the rate of interest is constant” Jones and Scrimgeour 2008). The proof is attributed to Uzawa (1961). The other alternative types of technical progress include Hicks and Solow neutral technical progress. Hicks’ neutral technical progress maintains the constancy of the labor-to-capital ratio and Solow’s version maintains constant the ratio of labor-to-output or the labor share. These three types of technical progress are formalized as follows: \(Y_{t} = F\left( {K_{t} , \,A_{t} L_{t} } \right)\) is Harrod neutral; \(Y_{t} = A_{t} F\left( {K_{t} ,\,L_{t} } \right)\) is Hicks neutral; \(Y_{t} = F\left( {A_{t} K_{t} , \,L_{t} } \right)\) is Solow neutral. If the production function is of the Cobb-Douglas form (i.e., the elasticity of substitution between labor and capital is unity) technical progress is Harrod-Hicks-Solow neutral (Haache, ibid.).
- 9.
As explained by Yeager (op. cit., 1954, pp. 56–57): “Harrod’s analysis appears to lack event his sort of empirical basis. Everything flows from mere definitions and assumptions…it strikes me as wholly illegitimate to suppose that the economic system gets into trouble if it does not satisfy the precise conditions implied by one’s own overly precise assumptions.”
- 10.
In the warranted equation, \(C\) is an equilibrium term expressing the requirement for new capital. The warranted rate equates the desire savings to the required increase in capital. In the actual growth equation, \(C\) is an ex-post term (the amount of capital goods actually produced per period). Harrod (1948, p. 82; 1973, pp. 15–19, 167).
- 11.
He argues that both concepts were confused in the concept of time preference or discounting the future. See Robinson (1949, p. 73).
- 12.
Pigou (1920 [1912], pp. 25–26) explains the preference of present over future enjoyment as follows: “…this preference for present pleasures does not…imply that a present pleasure of given magnitude is any greater than a future pleasure of the same magnitude. It implies only that our telescopic faculty is defective, and that we, therefore, see future pleasures, as it were, on a diminished scale.”
- 13.
The principle of diminishing utility of income or money was originally formulated by Jeremy Bentham and was given its modern formulation by William Stanley Jevons (1835–1882): “We can now conceive, in an accurate manner, the utility of money, or of that supply of commodity which forms a person’s income. Its final degree of utility is measured by that of any of the other commodities which he consumes” (Jevons 1965 [1871], p. 140). The principle of diminishing marginal utility entails the assumption of comparability (Howey 1989, p. 54). In this sense, other marginalist authors such as Walras (1954 [1900], pp. 445, 175) and Marshall (1982 [1920], p. 16) shared the same notion (although the notion of comparability has been questioned by other authors including Edgeworth 1967 [1881], pp. 77–78). The principle was part of standard economic theory at the time as illustrated by H. D. Henderson’s Supply and Demand (1922) which was the first Cambridge Economics Handbook (p. 51). Besides making a reference to Henderson’s book, Harrod cites A. C. Pigou, I. Fisher, and Frank Ramsey (1903–1930) (Harrod 1948, pp. 38, 40–41; 1973, p. 59).
- 14.
In his article, Frank Ramsey (1903–1930) provided an answer to the question of how much should a nation save.
- 15.
Harrod provided a formal explanation for savings as follows starting front the condition of equilibrium the marginal utility of consumption in t. Consider one individual and two time periods (year 0 and year t). Let \(T\) be time preference (the present equivalent of a unit of utility in one year’s time). This is a negative function of the rate of discount. The greater is the rate of discount the less is the present equivalent of one future unit of utility. And let R be the amount of money that accumulates at the end of the year at the current rate of interest. The higher is the rate of interest the greater is the amount of money that accumulated after one year. The condition of equilibrium requires that the marginal utility of a sum of money today \(\left( {MU_{t} } \right)\) times \(R^{t}\) and \(T^{t}\) be equal to the marginal utility of a sum money in year 1 \(\left( {MU_{0} } \right)\). That is,
$$MU\left( {C_{0} } \right) * R^{t} * T^{t} = MU\left( {C_{t} } \right)$$(8.2)Defining the average elasticity of the income utility curve \(\left( \varepsilon \right)\) as the ratio of the percent change in marginal utility of income to the percent change in consumption
$$\varepsilon = - \frac{{\frac{{C_{t} - C_{0} }}{{C_{0} }}}}{{\frac{{MU\left( {C_{t} } \right) - MU\left( {C_{0} } \right)}}{{MU\left( {C_{0} } \right)}}}} = - \frac{{C_{t} - C_{0} }}{{C_{0} }} * \frac{{MU\left( {C_{0} } \right)}}{{MU\left( {C_{t} } \right) - MU\left( {C_{0} } \right)}}$$(8.3)where \(C_{0} ,\,C_{t} = \,{\text{consumption at time 0 and time }}t .\)
Substituting (8.1) into (8.2) yields,
$$C_{t} = C_{1} \left\{ {1 + \varepsilon \left( {1 - \frac{1}{{R^{t} *T^{t} }}} \right)} \right\} \Leftrightarrow \frac{{C_{t} }}{{C_{1} }} = \left\{ {1 + \varepsilon \left( {1 - \frac{1}{{R^{t} *T^{t} }}} \right)} \right\}$$(8.4)According to Eq. (8.3), the greater the elasticity of the income utility curve the greater the consumption (the less the savings) in time t relative to time 1. Similarly, the higher the interest accumulated the lower the consumption (the greater the savings) in time t relative to time 1. Finally, the less important is time preference the higher the consumption in time t (the lower the savings in time t) relative to time 1.
- 16.
- 17.
See Chapter 4 and the discussion of Keynes’s comments to the draft Essay.
- 18.
Ralph Hawtrey (1879–1975) defined a market economy as “an organization for clearing supply against demand” where dealers (traders, merchants) played a pivotal role (Hawtrey 1925, p. 17). As he put it: “The wholesale merchants …judge demand and regulate supply…set the machinery of production at work by giving orders to the producers, an incidentally to start the machinery of credit. This is especially true of manufactures and the production of raw materials” (Hawtrey 1978 [1919], p. 8). Dealers clear supply and demand by holding stocks and transport commodities from excess demand to excess supply areas. Dealers’ actions are affected by the future evolution of demand and supply, prices and interest rates. As he asserted: “…a high rate of interest acts in the first instance on the wholesale dealer or merchant who restricts his orders to the manufacturer or producer” (Hawtrey, ibid., p. 108. See also Hawtrey 1925, p. 22). Hawtrey’s views had attracted criticism from different economists including Walsh (1934), Tinbergen (1939, 1951), Tinbergen and Polak (1950), and also Harrod. Tinbergen (1939) found that short-term rates of interest had a negligible effect on investment and later on in 1950 pointed out that he could not corroborate any influence from short-term rates of interest to mew investment in stocks or the total magnitude of stocks. According to Deutscher (1990, p. 226): “When Harrod brought Tinbergen’s results to his attention, Hawtrey was unimpressed.”
- 19.
Harrod argued: “…there is no reason why interest rates, supposing them to have some magical property of restoring equilibrium, should cause \((C_{r} )\) to rise (or fall) proportionately to any increase (or decrease) in the savings ratio, thereby holding the velocity of the growth rate (\(G_{w} )\) constant.”
- 20.
On this point, see also Harrod (1963, p. 404).
- 21.
See Keynes (1936, Chapter 14).
- 22.
As Harrod formulated his dynamics analogous with statics, Keynes argument about the indeterminacy of the equilibrium rate of interest must also have a counterpart in dynamic theory.
- 23.
- 24.
The enthusiasm for cycle analysis that was present in economics from the 1920s to the 1950s faded away during the 1960s and interest and research shifted toward growth theory. The basics for the Neo-classical growth model were laid out by Solow (1956), Swan (1956), Meade (1961), and Tobin (1965). The change in emphasis can be explained by: (i) the maintained growth period in post-World War II era with minor fluctuations; (ii) absence of corroboration of the “periodic regularity of the cycle”; (iii) an inherent contradiction between the deterministic character and recurrence of the cycle models and the assumption of rationality. Rationality implies that agents incorporate the former information in their decisions and utility functions leading to a change in the behavioral parameters thus contradicting the deterministic character of cycle models (see Medio 2008).
- 25.
See Boianovsky (2017b).
- 26.
- 27.
Notice in general the elasticity is expressed as the obverse of expression (8.2) \(\varepsilon = - \frac{{{{\Delta MU\left( C \right)} \mathord{\left/ {\vphantom {{\Delta MU\left( C \right)} {MU\left( C \right)}}} \right. \kern-0pt} {MU\left( C \right)}}}}{{{{\Delta C} \mathord{\left/ {\vphantom {{\Delta C} C}} \right. \kern-0pt} C}}} = - \frac{\Delta MU\left( C \right)}{MU\left( C \right)} * \frac{C}{\Delta C}\)
Harrod attributes the formulation of the elasticity (\(\varepsilon\)) in (8.2) to Marshall (Harrod 1973, p. 78).
- 28.
Harrod indicates (1960, p. 279) this is a purely Keynesian notion. It is indeed at the core of the theory of effective demand, that the level of investment undertaken in a capitalist market economy does not correspond to the level full-employment savings.
- 29.
Harrod (1969, p. 197) argued that \(G_{n} > G_{w}\) prevails in developing countries while \(G_{n} < G_{w}\) in the “highly advanced countries.”
- 30.
- 31.
These cases are presented in terms of growth curves so that the comparison is between the gradients of the different curves.
- 32.
Also, since the warranted rate is above the natural rate of growth and the actual rate cannot be, except temporarily, above the natural rate, eventually the actual rate of growth will fall again below the warranted rate generating a downward cumulative movement unless the warranted rate can be brought down. This however does not create a conflict since contracyclical policies can be, once again, implemented.
- 33.
Harrod makes a reference to John Kenneth Galbraith’s endorsement of public works.
- 34.
Chapter 4 of this book.
- 35.
If savings are increased but there is no investment, the government savings will be offset by private sector savings due to fall in profits. Here again Harrod links the increases/decrease in savings with that of corporate profits.
- 36.
Harrod addressed using the rate of interest as a policy tool. He opined that since the effects of the interest rate are slow working, the rate of interest was a better-suited instrument for long-run objectives (Harrod 1939, p. 32; 1964, p. 908). For its part, the short-run rate of interest could be used a supplementary to the use of fiscal policy when the warranted rate and natural rates were aligned (p. 32).
- 37.
- 38.
See, for example, Blaug (1995, p. 349). Most of the textbooks on growth do not distinguish between Harrod and Domar and simply refer to the Harrod–Domar growth model. Two exceptions are Jones (1975) and Haache (1979). Domar was introduced to Keynes GT in 1939, at the start of his graduate studies, by Arthur Smithies at Michigan University. Later on, in the 1940s, he studied under Alvin Hansen at Harvard University where he obtained his Ph.D. Domar’s most important writings on growth theory include “Capital Expansion, Rate of Growth and Employment” (1946b) and “Expansion and Employment” (1946a). After finishing his B.A. in economics at the University of California in Los Angeles (1939), Domar went to Michigan University, where he took a course in macroeconomics taught by Arthur Smithies. Smithies taught Keynes GT, and Domar found it a revelation (“At last, economics began to make sense to me”). At the insistence of Smithies, Domar went to Harvard University to study for his Ph.D. in economics which he finished in 1947 (at Michigan he earned a Master’s degree in Mathematical Statistics, 1941). At Harvard, he took Hansen’s Fiscal Policy seminar (the highlight of his stay at Harvard) and was his research assistant at the Federal Reserve in 1943. Domar’s well-known writings on debt (“The Burden of the Debt and the National Income,” 1944) were influenced by Hansen’s Fiscal Policy and the Business Cycle. See Colander and Landreth (1996, pp. 179–191). Most of Domar’s writings on growth are included in Essays in the Theory of Economic Growth (1957). See Boianovsky (2017a) for an investigation of the role of Domar in introducing growth modeling in economics.
- 39.
The exchange of letters between both took place on May 16 (Weintraub to Harrod) and May 28 (Harrod to Weintraub) 1966.
- 40.
Harrod (1968, p. 173).
- 41.
See also Harrod (1967, p. 499).
- 42.
In his article “Economic Growth: An Econometric Approach” (1952, p. 481), Domar made reference to Harrod’s 1939 famous creation which had to wait for a decade and to be repeated in Towards a Dynamic Economics to receive its “deserved recognition.” Harrod referred to the similarities with Domar in “Domar and Dynamic Economics” (1959). On the similarities between both models, see Schelling (1947), Jones (1975), and Haache (1979).
- 43.
- 44.
Domar also considers the possibility of variations in \(c\) which complicates the relation between investment and income. A high \(c\) implies can prevent the attainment of full employment because it may be difficult for the investment to grow at the same rate as income. This leads to the consideration of the relation between the level of \(c\) and the growth labor, natural resources, and technology. If the latter three factors grow slowly (fast) a high \(c\) is an obstacle to attaining full employment. Domar (1957 [1947], p. 101).
- 45.
I am grateful to Tony Thirlwall for making this point.
- 46.
Another important difference between Domar and Harrod is that the former’s model lacks an investment function (Jones 1975, p. 64).
- 47.
Technology is described by two constants labor and capital requirements per unit of output, that is, by fixed coefficients.
- 48.
Solow (1988, p. x) referred to Harrod’s explanation of instability as a result of, “vague generalization about entrepreneurial behavior.”
- 49.
Jones (1975) identifies two problems in Harrod, the divergence of the warranted from the natural rate and the divergence of the actual from the warranted rate. He argues (p. 89) that Solow simply bypassed the second Harrod problem because Solow’s model assumes that ex ante savings are ex ante investment and thus precludes the existence of an independent investment function dependent on entrepreneurs’ expectations. Similar opinions are expressed, among others, by Sen (1970, p. 23) and Stiglitz and Uzawa (1969, p. 13) and more recently by Halsmayer and Hoover (2015, p. 13). Harrod never made the assumption of the permanent equality between ex ante savings and investment.
- 50.
Halsmayer and Hoover (2015) cite from the correspondence between Solow and Eisner (1956), Solow and Kaldor (1959) and Solow and Harrod (1960) which provide evidence that Solow held on and defended his interpretation of Harrod as a fixed proportions model. As he wrote to Harrod (op. cit., p. 26): “No matter how >I take it, I seem to find your equation to be dependent on the constancy of the social yield or capital” (Letter from Solow to Harrod, September 23, 1960).
- 51.
- 52.
Cited in Besomi, ibid.
- 53.
Cited in Halsmayer and Hoover (op. cit., p. 31). In an earlier article, Solow (1999, p. 640) also recognized the differences between Harrod and Domar: “…the neoclassical growth theory arose as a reaction to the Harrod-Domar models of the 1940s and 1950s…Although their names are always linked, the two versions have significant differences. Harrod is much more concerned with sometimes unclear thoughts about entrepreneurial investment decisions in a growing economy.” In 1988 (op. cit., p. 11), he wrote that he made “some injustice” in referring to Harrod–Domar.
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Pérez Caldentey, E. (2019). Further Developments in Dynamic Economics. In: Roy Harrod. Great Thinkers in Economics. Palgrave Macmillan, London. https://doi.org/10.1057/978-1-349-74085-7_8
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