¹ Here is a detailed list, grouped by authors according to the date of the first item against each author's name. References to these items will henceforward be given, where possible, simply by author's name. (1) Ramsey, F. P., “A Mathematical Theory of Savings,” Economic Journal, XXXVIII (12 1928)Google Scholar; (2) Harrod, R. F., “An Essay in Dynamic Theory,” Economic Journal, XLIX (March 1939)Google Scholar; (3) Harrod, R. F., Towards a Dynamic Economics (London, 1948)Google Scholar; (4) Harrod, R. F., “Second Essay in Dynamic Theory,” Economic Journal, LXX (06 1960)Google Scholar; (5) von Neumann, J., “A Model of General Economic Equilibrium,” Review of Economic Studies, XIII (1945–1946)Google Scholar, a translation of a paper first published in 1938 in German; (6) Domar, E. D., “Capital Expansion, Rate of Growth and Employment,” Econometrica, XIV (04 1946)Google Scholar reprinted along with item (7) that follows in Essays in the Theory of Economic Growth (Oxford, 1957)Google Scholar; (7) Domar, E. D., “Expansion and Employment,” American Economic Review, XXXVII (03 1947)Google Scholar; (8) Tobin, J., “A Dynamic Aggregative Model,” Journal of Political Economy, LXIII (04 1955)Google Scholar; (9) Solow, R. M., “A Contribution to the Theory of Economic Growth,” Quarterly Journal of Economics, LXX (02 1956)Google Scholar; (10) Robinson, Joan, The Accumulation of Capital (London, 1956)Google Scholar; (11) Swan, T. W., “Economic Growth and Capital Accumulation,” Economic Record (11 1956)Google Scholar; (12) Kaldor, N., “Capital Accumulation and Economic Growth,” chapter 10 of The Theory of Capital, proceedings of a conference of the International Economic Association, edited by Lutz, F. A. and Hague, D. C. (London, 1961).Google Scholar

² “It may be well to emphasize at this point that no distinction is drawn in this theory between capital goods and consumption goods.” Harrod, “Essay,” 18.

³ It is this interpretation that is consistent with the original statement in the “Essay,” 30: “… the natural rate of growth. This is the maximum rate of growth allowed by the increase of population, accumulation of capital, technological improvement, and the work-leisure preference schedule, supposing that there is always full employment in some sense.” In Towards a Dynamic Economics, 87, there is a similar statement.

⁴ In the “Second Essay,” 282, he writes: “We may now go to the ‘natural’ (or welfare optimum) rate of growth. … If we are considering the welfare optimum, we should write, T = 1. There ought not to be a time preference! Any authority concerned with planning should, presumably, disregard it.” On p. 286 he suggests why the fraction of income actually saved would not tend to approximate the fraction required for welfare optimum growth: “T [ = 1 for rational time preference] may be less, indeed much less than 1, especially in developing societies … individuals have their own youth-maturity-old age patterns. … companies save importantly.” Harrod, like Ramsey before him, means by time preference the unequal (and as he sees it, irrational) valuation of present and future utilities and not simply, as we use the term, the unequal valuation of present and future goods (which with diminishing marginal utility might be perfectly rational).

⁵ In “Domar and Dynamic Economics,” Economic Journal, LXIX (1959), 455 Google Scholar, Harrod, after remarking that “it is the natural rate that is to be regarded as the welfare optimum,” proposes that “the target of policy should be to bring the warranted rate as near as possible to the natural rate.” In the earlier and the later Harrod alike the natural rate serves both as an equilibrating force and a guide to policy—see for example pp. 89 et seqq. of Towards a Dynamic Economics and p. 286 of the “Second Essay.” But it is the natural rate as it was first defined (with, for example, actual time preferences) that interacts with the real economy, and the later natural rate (with rational time preferences only) that prescribes policy.

⁶ This sorts out situations where although there is no involuntary unemployment in the Keynesian sense some labour has become unemployable (as it might, for example, in a system with fixed coefficients of production where the growth of capital was made constant at a lower rate than the growth of the labour supply).

⁷ In Tobin, 107, for example, balanced growth is identified with “proportional growth of capital, income, and employment.” But balanced growth in “Tobin” is simply a convenient possibility.

⁸ In the first paragraph of the “Essay,” Harrod sketches his dynamics as “a classification and system of axioms to meet the situation in which certain forces are operating steadily to increase or decrease certain magnitudes in the system.” In this context it is possible to make the labour supply an exogenous variable that grows at a constant rate, and with this and related assumptions about the givens of the system to show that the natural rate of growth (for example) tends to a constant value. But with one or two noted exceptions such stringent assumptions are not necessary to the working of the models. Except in Kaldor constant rates are not to be seen as historical facts with which the models must be reconciled.

⁹ “My warranted rate is simply the dynamized version of Keynes' excess or deficiency of aggregate effective demand in relation to what is required for full employment.” Harrod, , “Domar and Dynamic Economics,” 456.Google Scholar On p. 455 he observes that the “warranted rate is the path on which the supply and demand for goods and services will remain in equilibrium, given the propensity to save; thus the economy could move along the warranted rate with great and growing ‘involuntary’ unemployment in Keynes' sense. It is the natural rate that implies full employment. The actual rate may be below both.” For the identification of the warranted rate with the equality of intended savings and investment see the “Essay,” particularly the arithmetic of p. 18 and of p. 28, and the extended discussion of pp. 19–21. See also the observation on p. 22: “The line of output traced by the warranted rate of growth is a moving equilibrium, in the sense that it represents the one level of output at which producers will feel in the upshot that they have done the right thing, and which will induce them to continue in the same line of advance.”

¹⁰ A good deal of variation could be allowed without affecting the operation of the model. It does not really matter whether all margins internal to the firm and household are exactly satisfied, or whether the world of the model is completely or only very largely Keynesian. The point is that the operation of the model is entirely consistent with the pure abstractions, that is, completely efficient firms and households and a wholly Keynesian economy. Moreover, without them it would be pointlessly difficult to be specific about some of the laws of the system.

¹¹ Because when the liquidity trap acts as a constraint intended savings and investment are unequal.

¹² On the question of wage and interest rate disequilibrium see particularly pp. 91–100 in the third lecture of Towards a Dynamic Economics.

¹³ In the “Essay,” 18, “1/b” is defined as “the amount of capital per unit increment of output required by technological and other conditions (including the state of confidence, the rate of interest, etc.)”. In Towards a Dynamic Economics, 22, Harrod comments that if “there is no technological advance and the rate of interest does not alter, the ratio of the value of capital in use to income per period … will remain constant.” But this is only certain if the production function is linear and homogeneous. When the rate of interest does alter, the capital intensity of methods used for increments of production is changed, with capital being deepened as the rate falls (96).

¹⁴ See Towards a Dynamic Economics, 26-7.

¹⁵ What is here called “neutral” might be better described as “capital-neutral, labour-saving,” for while the output-capital ratio is constant, the output-labour ratio increases. See our “Classification of Technical Progress in Models of Economic Growth,” Economica, Nov. 1963. It is to be noted that comparisons are made from equilibrium to equilibrium, and so it is not correct, as Harrod claims (Towards a Dynamic Economics, 27), that his rules depend “on the intrinsic character of the invention only.” Indeed, it is rather the Hicksian classification that has this property, since it holds factor combinations fixed. Because of this basic difference “Hicks-neutral” and “Harrod-neutral” are never equivalent, except in the limited sense that a given technical change may have both characteristics.

¹⁶ See the discussion of equation (26) below in the account of the Robinson model. On this point the Harrod and Robinson models are equivalent.

¹⁷ “… ‘s” is likely to vary with a change in the size of income. …” “Essay,” 25.

¹⁸ Harrod accepts that “the discovery that output is excessive or deficient, and the consequent emergence of a depressing or stimulating force, takes some time” and notes that this is “the time required for an undue accretion or depletion of capital goods to exert its influence upon the flow of orders.” Ibid., 25-6.

¹⁹ It is difficult to give a compact reference, but see, inter al., pp. 21–3 of the “Essay” where the effect of a difference between warranted and actual rates is described, e.g. on p. 22: “… suppose that there is a departure from the warranted rate of growth. Suppose an excessive output … the actual increase of capital goods … falls below … that which is desired. There will be, in fact, an undue depletion of stock or shortage of equipment, and the system will be stimulated to further expansion. [The actual rate of growth], instead of returning to [the warranted], will move further from it in an upward direction, and the further it diverges, the greater the stimulus to expansion will be. Similarly, if [the actual rate] falls below [the warranted]. …”

²⁰ Harrod considers extended versions of equation (3) that contain autonomous investment and a foreign sector (cf. “Essay,” 27-8, and lecture 4 of Towards a Dynamic Economics), but these are extraneous to the logic of the model.

²¹ In order to use a uniform notation we identify output-capital rather than capital-output rates throughout, although at occasional cost to familiar expressions.

²² The question has been widely discussed. See, for example, Alexander's, S.S. “Mr. Harrod's Dynamic Model,” Economic Journal, LX (1950), or chap. 3Google Scholar of Allen, R. G. D., Mathematical Economics (2nd ed., London, 1963)Google Scholar and the references therein cited.

²³ In the “Essay,” 22, (just following the passage cited in n. 19), Harrod argues that “in the dynamic field we have a condition opposite to that which holds in the static field. A departure from equilibrium, instead of being self-righting, will be self-aggravating. [The warranted rate] represents a moving equilibrium, but a highly unstable one.” It is important to notice that deviations are to be referred to a warranted path, continuously maintained from initial conditions onwards. Once a deviation has occurred, warranted values themselves change (with the change in actual values) from what they would have been on the original line of advance. Thus, in Towards a Dynamic Economics, 86, Harrod remarks on his “extraordinarily simple and notable demonstration of the instability of an advancing system. Around the line of advance, which, if adhered to, would alone give satis-faction, centrifugal forces are at work, causing the system to depart farther and farther from the required line of advance.” Hence it is proportionately increasing deviations that are foreseen.

²⁴ “… the reader, whenever he finds a reference to the excess or deficiency of [ex post investment] compared with [ex ante investment], may substitute, if he prefers it, a supposed deficiency or excess of ex post saving compared with ex ante saving, without affecting the course of the argument.” “Essay,” 20.

²⁵ As before, once there is a deviation, warranted values themselves change.

²⁶ All the assumptions but the last are designed to hold b constant. Without technical change the system would simply expand with the labour force at the original real wage. Neutral technical change allows the result to be extended, although since real wage rates now rise the assumption that s is fixed becomes less plausible.

²⁷ In Towards a Dynamic Economics, 86, Harrod observes that it would be great good luck if “their collective appraisal caused people to hit precisely upon the value of the warranted rate of growth. But if they do not do so, their collective experience will tend to drive them further and further from it.”

²⁸ "The system cannot advance more quickly than the natural rate allows. If the [warranted rate which would obtain in conditions of full employment] is above this, there will be a chronic tendency to depression; the depressions drag down the [warranted rate below its full employment] level, and so keep its average value over a period of years down to the natural rate. But this reduction of the warranted rate is only achieved by having chronic unemployment.” “Essay,” 30.

²⁹ See “Domar and Dynamic Economics,” 455: “If the warranted rate is above the natural rate, the actual rate must be below the warranted rate for most of the time, and the centrifugal forces pull it further down, causing frequent periods of unemployment. (This is the dynamised version of the stagnation thesis.) If the natural rate is above the warranted rate, full employment may be achieved more frequently owing to the upward pull of the centrifugal forces, but only at the cost of inflation.” And then notice also p. 32 of the “Essay”; “… the ideal policy would be to manipulate the full employment warranted rate so that it should be equal to the natural rate … an anti-cycle policy would still be an indispensable supplement … necessary to combat the run-away forces which come into being as soon as a substantial change occurs in the warranted rate.”

³⁰ Cumulative movements must end, of course, at some stage, but the model does not give a dieory of turning points to show how the boom or bust for which it accounts may become a cycle. There are many references to the instability as cyclical, and note is taken that “the value of the warranted rate depends upon the phase of the trade cycle and the level of activity” (“Essay,” 30), but these point to rather than provide the Hicksian extension into cyclical theory.

³¹ “But the construction of a new factory has a dual effect: it increases productive capacity and it generates income. The emphasis on this dual character of the investment process is the essence of the approach in this paper to the problem of employment. If investment increases productive capacity and also creates income … at what rate should it grow in order to make the increase in income equal to that of productive capacity?” Domar, , Essays, 88–9Google Scholar, “… income and capacity should increase at trie same rate.” Ibid., 91.

³² “… the productive capacity of the whole economy may increase by a smaller amount [than b′], because the operations of these new projects may involve a transfer of labour (and other factors) from other plants, whose productive capacity is therefore reduced.” ibid., 73.

³³ Here b′ and b″ correspond to Domar's s and a. We have made the change so that all output-capital ratios will be identified by a b and all savings propensities by an s, with affixes distinguishing variants within each family. The definitional phrases are found in ibid., 73-4.

³⁴ We deduce this from ibid., 76, n. 11. Domar incorrectly conjectures that “owing to capital-saving inventions in existing plants” it might even be found that b′ < b″. But the nature of technical change has nothing to do with the comparison: the productivity of the investment must be no smaller without a restriction than with one. That is, b′ is indeed a ceiling to b″. However, since Domar judges that the propensity to save “is sufficiently high in our society to make [b′ < b″] in a continuous state of full employment more an exception that a rule,” we conclude that he expects labour to be fully employed whenever capital is.

³⁵ “Domar and Dynamic Economics,” 457.

³⁶ In the Foreword to the Essays, 7, Domar himself speaks of his attempted “distinction between the productivity of total investment in a specific project … and the productivity of total investment as measured by the increment in the output of the whole economy … the latter might be affected by shift of labour and of other factors from existing to new projects. Some such distinction is needed, but I do not think that my attempt was successful, because the model employed an inadequate production function to show what determined the difference between these two productivities. The difference was taken as given; it involved, I believe, a tautology.…”

³⁷ See the second numbered paragraph, ibid., 80.

³⁸ Domar's view of the insufficiency of the distinction between b′ and b″ is that the “correct alternative lay either in disregarding the difference in the hope that technological progress and the growth of population would provide a sufficient labour force to man the new plants without denuding the old, or in employing a more realistic, and a more complex production function with an active participation of factors other than capital. In the subsequent essays I chose the easier of the two solutions, but with an ever-guilty conscience.” Ibid., 7. Evidently there would always exist a rate of growth of the labour force sufficient to erase the difference between b′ and b″. But again this has not the slightest connection with the difference between Harrod's warranted and natural rates.

³⁹ If a system simply expands in step with its labour force, with real wage rates held constant, then a fixed average propensity is a natural assumption; but it is less so if there is technical change or an increase in capital per head, changes that should be reflected in time preference. Cf. our comments in “Sir Roy Harrod's Equation of Supply,” Oxford Economic Papers (11 1963), 271.Google Scholar

⁴⁰ Domar, , Essays, 75 Google Scholar; Tobin, 104; Solow, 66; Swan, 335.

⁴¹ That it can is the point (at least within the present context) of von Neumann's argument, shown by a fixed point version of the minimax theorem.

⁴² See Ramsey, 543–4 and 551–3. Ramsey's “community” is in effect a single person.

⁴³ To paraphrase the short proof suggested to Ramsey by Keynes: suppose that in the equilibrium approach to bliss the rate of utility enjoyed is plotted against time. Then to reduce the rate of savings now by a dollar must shift the utility path by an interval of 1/S. Utility lost is thus 1/S multiplied by the gap between present utility and bliss. Utility gained is the marginal utility of consumption. See ibid., 547.

⁴⁴ If U (X, Y) is a utility index of two independent goods, then by definition U _XY must equal zero. Any other admissible index must therefore be a linear transformation of U(X, Y).

⁴⁵ See Ramsey, 544. Labour has little more than a descriptive role anywhere in the model, and has in effect to be assumed out of existence from time to time to facilitate the arithmetic.

⁴⁶ “… we must know how much capital our man feels it necessary to leave his heirs. …” Ibid., 552. It is not that the choice of terminal capital reflects an arbitrary preference but that die preference is absolute, so that there is no rate of substitution between this good and other goods.

⁴⁷ In the “Second Essay,” 285, Harrod comments that his “article is concerned with a steady shift of the ‘production function’ outwards from the origin. It is not implied that the successive functions, moving outwards with time, are parallel to each other; the analysis has been consistent with innovations being neutral, labour saving or capital saving. Some students of growth have laid stress on the phenomenon of a movement along the productivity function in consequence, not of innovation, but of a rise in the ratio of capital to other factors. … It is essentially a phenomenon of a falling natural rate of growth.”

⁴⁸ Tobin, 104.

⁴⁹ “A given or constant state of knowledge is only capable of being defined implicitly.” Kaldor, 205. For a description of equation (5) that follows, see 208.

⁵⁰ See ibid., 215.

⁵¹ “The first term … relates to the amount of investment induced by the change in output the previous period, and assumes that this investment will be such as to make the growth in output capacity in period (t + θ) equal to the growth in output in period t.” Ibid., 215.

⁵² Ibid., 216. The lag is really a descriptive detail. As the model is developed nothing turns upon its existence.

⁵³ Again see ibid., and also relationships (i), (ix), and (a) of p. 220. Kaldor notes boundary conditions outside of which his model becomes, say, Keynesian, but his attention is centred on a natural rate of growth. For example, on p. 201 he observes that “it is only under conditions of ‘Keynesian’ full employment that the growth-potential of an economy (indicated by its ‘natural’ rate of growth) is exploited in terms of actual growth.”

⁵⁴ “…investment in ‘fixed assets’… is considered to be far more risky or illiquid than either investment in financial assets or in working capital. …” Ibid., 218. “… circulating capital stands always in a linear relationship to output….” Ibid., 219. “ … a higher capital-output ratio … requires for any given rate of interest a higher minimum rate of profit.” Ibid.

⁵⁵ See Ramsey, 544.

⁵⁶ All of these properties can be deduced from the use Ramsey makes of the function. See esp. 544–6. For example, the product is exactly distributed by marginal productivity rules—hence equation (9) must be linear and homogeneous. Once bliss is reached savings comes to a halt—hence Y is net income and the final stages of investment raise K to a level where, combined with L, it maintains Y indefinitely.

⁵⁷ The point is to separate the contributions of capital and labour, a computational device that suggest treating capital and labour quite independently (as in fact Black, J. does in his “Optimum Savings Reconsidered, or Ramsey without Tears,” Economic Journal, LXXII, 1962).Google Scholar See Ramsey, 549, n. 1.

⁵⁸ Ramsey deals with technical change only in passing, but it is interesting to note that he regards it (549) not only as autonomous but foreseen: “… the probability that future inventions … are likely to make income available with less sacrifice than at present is a reason for saving less.” But this view is not taken up, for example, in “Harrod II,” where savings follow “Ramsey” but none the less adapt to technical change only as it comes and not by anticipation.

⁵⁹ Harrod, , Towards a Dynamic Economics, 96.Google Scholar

⁶⁰ Wicksell, Knut, Lectures on Political Economy (London, 1935), 236–7Google Scholar: “We can either adopt Walras’ method of taking a cross-section through social production at a moment of time. … Or else we can refer everything back to the original factors in conjunction with waiting … we make a longitudinal section instead. …”

⁶¹ See Solow, 66 and 76.

⁶² See Swan, 334–5 and 340. When land is fixed the “only possible equilibrium with constant labour growth (and no technical progress) would be one in which output per head and the wage rate were perpetually falling.” But the question might then be put: “What rate of labour growth will maintain constant output per head?” The answer must be, a rate that tends to zero.

⁶³ Solow speaks (65) of “the Harrod-Domar” model in which the “fundamental opposition of warranted and natural rates turns out in the end to flow from the crucial assumption that production takes place under conditions of fixed proportions, … If this assumption is abandoned, the knife-edge notion of unstable balance seems to go with it.”

⁶⁴ Solow (67) observes that as “a result of exogenous population growth the labour force increases at a constant relative rate n. In the absence of technological change n is Harrod's natural rate of growth.” Then on p. 70 he speaks of the “warranted rate of growth, warranted by the appropriate real rate of return to capital. …”

⁶⁵ Swan's account of warranted and natural rates is much like Solow's—see Swan's section on “The Harrod Model,” 342–3. Since in discussing these models we have been giving little attention to special cases, extensions, and the like, we should add that in Swan the appended “Notes on Capital” (343 et seqq.) are perhaps of greater interest than the model proper.

⁶⁶ Kaldor discusses this at some length on 215, n. 1.

⁶⁷ With Y _K and β fixed it is the output-to-labour ratio that rises to show the gains from technical progress.

⁶⁸ “In each process P _i … quantities a _ij , … are used up, and quantities b _ij are produced, of the respective goods G _j. … Each process to be of unit time duration.” Von Neumann, 2.

⁶⁹ The processes that maximize profit at one point of time will (together with their multiples) maximize profit thereafter.

⁷⁰ On p. 3, von Neumann suggests that the assumption “is not very far reaching, although … otherwise [the system] might break up into disconnected parts.” But as Champernowne, D. G. observed in his interpretative article, “A Note on ‘A Model of Economic Equilibrium,’” Review of Economic Studies, XIII (1945–1946), 18 Google Scholar, “the commodities with the lowest rate of expansion may be trivial goods … [yet they may] determine the rate of expansion of the whole system.”

⁷¹ Von Neumann, 2–3 and 8.

⁷² See Tobin, 105. Changes in the total are monetary gifts or tributes.

⁷³ Ibid., 104–6. Given the assumptions about expectations and risk, the properties of equation (15) are straightforward. In strict terms, adjustment affects both sides of the equation, although the direct effect is through p.

⁷⁴ “In this model, unlike those of Harrod, Hicks, and others, failure of the labour supply to grow at the rate necessary for balanced growth does not mean that growth at a slower rate is impossible.” Ibid., 107–8. “An alternative to price deflation is expansion of the supply of currency.” Ibid., 108. The extra flexibility is there, even if the Harrod model is not as rigid as Tobin supposes.

⁷⁵ Is new money distributed in proportion to existing balances? Probably not, since Tobin assumes the own rate of interest to be zero. Is it distributed in proportion to wealth? Does it establish claims on government? If it is introduced systematically is its growth foreseen? It is not obvious what defines a neutral system of making gifts and exacting tribute.

⁷⁶ Robinson, 404.

⁷⁷ Robinson, Joan, “Findlay's Robinsonian Model of Accumulation: A Comment,” Economica, 12 1963, 409–10.Google Scholar

⁷⁸ Robinson, 96. Mrs. Robinson continues: “Even more important than speeding up discoveries is speeding up of the rate at which innovations are diffused. When entrepreneurs find themselves in a situation where potential markets are expanding but labour hard to find, they have every motive to increase productivity; and the experience of wage rates rising with output overcomes the reluctance of the workers to assist them to do so.” Technical progress, then is induced by the difference between the rates of growth of capital and labour, and is induced the more rapidly the greater the difference.

⁷⁹ In “Findlay's Robinsonian Model of Accumulation: A Comment,” 410, Mrs. Robinson points to “the importance of Harrod's antinomy between the ‘natural’ rate of growth of the effective labour supply and the rate of accumulation of capital that satisfies the capitalists.” Her model, she then says, “is intended to show that when the urge to accumulate (‘animal spirits’) is high relatively to the growth of the labour force … near-enough steady growth with near-enough full employment may be realized. … In the converse case, the existence of a growing surplus of labour … can not be relied upon to bring the ‘natural’ rate of growth down to equality with the sluggish rate of accumulation.”

⁸⁰ See our “Classification of Technical Progress in Models of Economic Growth,” Economica, 11 1963, 376 Google Scholar et seqq. The propensity to save out of profits might be regarded as fractional, in which case equation (23) becomes απ = S ( = I), where α is a fraction and where the development of the model is adjusted accordingly.

⁸¹ “When technical progress is neutral, and proceeding steadily, without any change in the time pattern of production, the competitive mechanism working freely, population growing (if at all) at a steady rate and accumulation going on fast enough to supply productive capacity for all available labour, the rate of profit tends to be constant and the level of real wages to rise with output per man. … We may describe these conditions as a golden age.” Robinson, 99.

⁸² The passage continues (ibid., 81): “… it is only too easy for a surplus of labour to grow, relatively to the stock of capital, while investment fails to increase and the economy sinks into stagnation; whereas entrepreneurs will not accumulate and maintain redundant capital, so that when the rate of accumulation is too high (relatively to the labour force) … one way or another it is certain to be cut down.”

⁸³ Cf. ibid., 140–2.

⁸⁴ See our “Classification of Technical Progress,” 381 et seqq., for the two-sector paraphrase and for a detailed development of these concluding remarks. A similar two-sector paraphrase was given by Findlay, R. in “The Robinsonian Model of Accumulation,” Economica, 02 1963, 1–12.Google Scholar