Harrod's Revival of Growth Theory and His Contribution to Keynesian Macroeconomics

Harrod was intimately involved in the origins and development of Keynesian economics. As the galley proofs of The General Theory emerged from the print­ers from June 1935 onwards, copies were sent to Harrod, Kahn and Joan Robinson, and with their assistance, Keynes rewrote extensively for final pub­lication.

Your view, as I understand it is broadly this:

Volume of investment determined by:

Marginal efficiency of capital schedule

and

Rate of interest.

Rate of investment determined by:

Liquidity preference schedule

and

Quantity of money.

Volume of employment determined by:

Volume of investment

and

Multiplier.

Value of multiplier determined by:

Propensity to save (Harrod to Keynes, 30 August 1935, in Keynes 1973: 553; italics in original).

Keynes responded: ‘I absolve you completely of misunderstanding my the­ory. It could not be stated better than on the first page of your letter’ (Keynes to Harrod, 10 September 1935, in ibid.).

Almost immediately after the appearance of The General Theory, Harrod published The Trade Cycle (Harrod 1936a) which contained for the first time in the Keynesian literature the concept of an economy growing at a steady rate. Keynes wrote of it to Robinson on 25 March 1937: ‘I think he has got hold of some good and important ideas. But, if I am right, there is one fatal mistake’ (Keynes 1987: 149), and to Harrod himself on 31 March: ‘I think that your theory in the form in which you finally enunciate it is not correct, being fatally affected by a logical slip in the argument' (ibid.: 151).

Harrod convinced Keynes, who on 12 April congratulated him for ‘having invented so interesting a theory' (ibid.: 170), but with the reservation that, ‘I should doubt whether any reader who has not talked or corresponded with you could be aware that the whole of the last half of the book was intended to be in relation to a moving base of steady progress' (ibid.). Keynes added that it was vital that Harrod carry his ideas further and restate them more comprehensibly.

Harrod made important progress in the next 15 months, and on 3 August 1938 he sent Keynes a preliminary draft of the article, “An Essay in Dynamic Theory”, and wrote in his accompanying letter that:

My re-statement of the “dynamic” theory...is, I think, a great improvement on my book. I have been throwing out hints in a number of places of the possi­bility of formulating a simple law of growth and I want to substantiate the claim. It is largely based on the ideas of the general theory of employment; but I think it gets us a step forward (ibid.: 301).

A lengthy correspondence then developed between Harrod and Keynes in which the two most original elements in Harrod's contribution which later excited much interest and controversy in the economics profession were extensively discussed. Harrod's principal innovation was the invention of a moving equilibrium growth path for the economy, and he described this as the “warranted” line of growth. He had perceived before he wrote The Trade Cycle that there was a fundamental contradiction between the assumptions preva­lent in the microeconomic theory of the firm and industry, to which he had made notable contributions, and the new Keynesian macroeconomics.

Harrod's moving equilibrium or warranted growth path required that saving (of s % of the national income) be continually absorbed into investment, so he asked the question: At what rate of growth will firms all the time choose to invest the s % of the national income, which equilibrium growth requires? To answer this question, he made use of the acceleration principle or “the rela­tion”, as he called it, that firms need say, C_r units of additional capital to pro­duce an extra unit of output. It follows from these premises that the warranted rate of growth of output will be s/ C_r% per annum. Since each rise in output by 1 unit entails that C_r extra units be invested, a rise in output by s/ C_r% of the national income will call for an equilibrium investment of C_r times this, which is precisely s % of the national income, the ratio of ex ante saving in the national income. In Harrod's examples, he suggested a typical s of 10% of the national income and a C_r of 4, to produce a warranted rate of growth of 2.5%.

This idea that if there is continual saving, then equilibrium entails a con­tinual geometric growth in production, came as a considerable surprise to Keynes and the other members of the Cambridge “Circus”.

But Harrod's second discovery had equally radical implications. Suppose the actual growth of output is marginally above the equilibrium or warranted rate of growth. In Harrod's numerical example with s at 10% and C_r at 4, it can be supposed that output actually grows 0.1 of a percentage point faster than the warranted rate, that is by 2.6% instead of 2.5%. Then with 2.6% output growth, the acceleration principle or relation will entail that 4 times 2.6% be added to the capital stock, so that ex ante investment is 10.4% of the national income. With ex ante saving limited to 10.0%, the 0.1 of a percent­age point excess of actual growth over warranted growth then produces an excess in ex ante investment over ex ante saving of 0.4% of the national income. Any excess in ex ante investment over ex ante saving will be associ­ated with extra expansion of the national income according to the economics of The General Theory. Thus, if the actual rate of growth exceeds the warranted rate of s∕C_r%, the tendency will be for actual growth to rise and rise, for as soon as actual growth rises from 2.6% to say, 3.0%, required investment will rise further to 4 times 3.0% which equals 12% and so exceed the 10.0% savings ratio by a still greater margin. Conversely, when actual growth comes out at a rate just short of the warranted 2.5%, ex ante investment will be below the 10.0% savings ratio, which will cause the rate of growth to decline. This second discovery, which became known as Harrod's “knife-edge”, meant therefore that any rate of growth in excess of the equilibrium or warranted path he had discovered would set off a continual acceleration of growth, while any shortfall would set off a deceleration.

If in static theory producers produce too little, they will be well satisfied with the price they get and feel happy; but this is not taken to be the right amount of output; they will be stimulated to produce more. The equilibrium output is taken to be that which just satisfies them and induces them to go on as before. Similarly, the warranted rate [of growth] is that which just satisfies them and leaves them going on as before. The difference between the warranted rate and the old equilibrium (i.e. the difference between dynamic and static theory) is, in my view, that if they produce above the warranted rate, they will be more than satisfied and be stimulated, and conversely, while in the case of equilibrium in static conditions the opposite happens. The “field” round the [static] equilib­rium contains centripetal, that round the warranted centrifugal forces (ibid.: 336—337; italics in original).

It took Keynes time to absorb Harrod's startling discovery. On 19 September, he proposed a counter-example in which C_r was merely one-tenth, while s was also one-tenth. With this counter-example, a deviation of output by a small amount from the warranted path, say by δx, which would raise planned investment above the level at which it would otherwise be by C_rδx would merely raise this by 0.10δx, which would equal the rise in planned sav­ing of sδx, which would also come to 0.10δx, so there would be no tendency towards an explosive growth in effective demand. This would grow explosively if C_r was one-ninth (in which case planned investment would rise by 0.11 δx and saving by only 0.10δx) but the further growth of output would be damped if C_r was merely one-eleventh, so, Keynes insisted, ‘neutral, stable or unstable equilibrium' (ibid.: 341).

Harrod protested on 22 September that ‘it is absurd to suppose extra capi­tal required [C_r] only 1/10 of annual output, when the capital required in association with the pre-existent level of incomes in England today is 4 or 5 times annual output' (ibid.: 344).

1. The relevant marginal capital coefficient (C_r) which determines how much planned investment will rise is the net new requirement of induced invest­ment. In so far as investment decisions are autonomous of short-term fluc­tuations in output, the relevant C_r will be lower than the economy's overall capital-output ratio.

2. The relevant coefficient which determines the increase in planned saving is the marginal and not the average propensity to save. Planned saving will rise more where output deviates upward from the warranted rate, the greater is the marginal propensity to save in relation to the average propensity.

The circumstances that could produce a stable upward deviation of growth from the warranted rate and the avoidance of Harrod's knife-edge are there­fore a very high marginal propensity to save in combination with a situation where most investment is autonomous so that the induced investment coef­ficient, C_r, is considerably less than 1. In “An Essay in Dynamic Theory”, Harrod covered this possibility with the caveat that, ‘when long-range capital outlay is taken into account...the attainment of a neutral or stable equilib­rium of advance may not be altogether improbable in certain phases of the cycle' (Harrod 1939a: 26). The possibility he had in mind here is that in the early stages of a cyclical recovery there may be so much excess industrial capac­ity that C_r will be quite low for a time, and therefore quite possibly lower than the marginal propensity to save. But, in general, any deviation of growth from the warranted line of advance would raise ex ante investment by a greater margin than ex ante saving with the result that the rate of growth would devi­ate further.

In addition to establishing the existence of the warranted line of advance and its instability, Harrod had to define the equilibrium investment behav­iour by businesses which would actually lead to expansion at the requisite rate. In his 1939 article, he omitted to offer any behavioural rule but simply asserted that the warranted rate was ‘that rate of growth which, if it occurs, will leave all parties satisfied that they have produced neither more nor less than the right amount' (ibid.: 16). That is no more than a description of equi­librium growth, and much the same can be said of his definition of the war­ranted rate in Towards a Dynamic Economics (Harrod 1948a) as ‘that overall rate of advance which, if executed, will leave entrepreneurs in a state of mind in which they are prepared to carry on a similar advance' (ibid.: 82). It was

only in the article “Notes on Trade Cycle Theory” (Harrod 1951b) that Harrod arrived at a behavioural assumption that matched his algebraic for­mulation of the warranted rate:

Let the representative entrepreneur on each occasion of giving an order repeat the amount contained in his order for the last equivalent period, adding thereto an order for an amount by which he judges his existing stock to be deficient, if he judges it to be deficient, or subtracting therefrom the amount by which he judges his stock to be redundant, if he does so judge it (ibid.: 274).

With this assumption, an economy which once achieves growth at the war­ranted rate will sustain it, while any upward or downward deviations will lead to still greater deviations wherever C_r exceeds the marginal propensity to save. However, it emerged by 1964, when Harrod published “Are Monetary and Fiscal Policies Enough?”, that even that assumption fails to define growth at the warranted rate, for it must also be assumed that the representative entre­preneur will expand at a rate of precisely s∕C_r when he judges his capital to be neither deficient nor redundant. This requires an expectation by the represen­tative entrepreneur that his market will grow at a rate of precisely s∕C_r Hence the full requirement for growth along Harrod's warranted equilibrium path is that entrepreneurs expect growth at this rate and expand and continue to expand at that rate so long as their capital stock continues to grow in line with their market so that it is neither deficient nor redundant. They will of course increase their rate of expansion if their capital should prove deficient, and curtail it if part of their stock becomes redundant.

The warranted rate of growth and its instability were Harrod's great innova­tions. From 1939 onwards, he contrasted this equilibrium rate with the natu­ral rate of growth, ‘the rate of advance which the increase of population and technological improvements allow' (Harrod 1948a: 87), which was entirely independent of the warranted rate. Harrod defined the rate of technical prog­ress more precisely as the increase in labour productivity ‘which, at a constant rate of interest, does not disturb the value of the capital coefficient' (ibid.: 23). This then entered the language of economics as Harrod-neutral technical progress, which, together with growth in the labour force, determines the natural rate of growth, that is, the rate at which output can actually be increased in the long run. This raised few theoretical problems, and there was nothing novel in the proposition that long-term growth must depend on the rate of increase of the labour force and technical progress. Keynes himself had said as much several years earlier in “Economic Possibilities for Our Grandchildren” (Keynes 1930 [1972]). But the contrast between this natural rate and Harrod's innovatory warranted rate offered entirely new insights.

If the warranted rate exceeds the feasible natural rate, the achievement of equilibrium growth must be impractical because the economy cannot con­tinue to grow faster than the natural rate. It must deviate downwards from the warranted rate towards the natural rate far more than it deviates upwards with the result that ‘we must expect the economy to be prevailingly depressed' (Harrod 1948a: 88). If the natural rate is greater, output will tend to deviate upwards towards the natural rate with the result that the economy should enjoy ‘a recurrent tendency to develop boom conditions' (ibid.).

Keynes's own reaction to the dichotomy between the warranted and natural rates was characteristically that the warranted rate always exceeded the natural:

In actual conditions...[ suspect the difficulty is, not that a rate in excess of the warranted is unstable, but that the warranted rate itself is so high that with pri­vate risk-taking no one dares to attain it. I doubt if, in fact, the warranted rate—let alone an unstable excess beyond the warranted—has ever been reached in USA and UK since the war, except perhaps in 1920 in UK and 1928 in USA. With a stationary population, peace and unequal incomes, the warranted rate sets a pace which a private risk-taking economy cannot normally reach and can never maintain (Keynes to Harrod, 26 September 1938 in Keynes 1987: 349—350; italics in original).

This is characteristic Keynes, but Harrod had persuaded him to express his familiar analysis in the language of his new theory of growth. In the immedi­ate post-war decades when full employment and creeping inflation prevailed, it was widely argued that the natural rate had come to exceed the warranted. The richness of Harrod's model is demonstrated by its ability to illuminate both kinds of situation.

Evsey Domar's growth model, which has a good deal in common with Harrod's, was published seven years after “An Essay in Dynamic Theory”, and a considerable literature emerged in the next 15 years on the stability condi­tions and other important features of what came to be known as the Harrod- Domar growth model (see Domar 1946, 1947). This is elegantly summarised by Frank Hahn and Robin Matthews in their celebrated 1964 survey article.

The development of neoclassical growth theory in the 1950s led to an increasing realisation that the warranted and natural growth rates could be equated by an appropriate rate of interest. If the warranted rate was excessive so that oversaving led to slump conditions, a lower interest rate which raised C_r sufficiently would bring it down to the natural rate. Conversely, the infla­tionary pressures that resulted from an insufficient warranted rate would be eliminated if higher interest rates reduced C_r sufficiently. If the real rate of interest and C_r responded in this helpful way, s∕C_r, the warranted rate could always be brought into equality with the natural rate.

Harrod's response included his “Second Essay in Dynamic Theory” (Harrod 1960a), a title which underlines its significance. He proposed that there was an optimum real rate of interest r_n which would maximise utility, with a value of G_p∕e, G_p being the economy's long-term rate of growth of labour productiv­ity and e the elasticity of the total utility derived from real per capita incomes with respect to increases in these. If a 1.0% increase in real per capita incomes raises per capita utility by 0.5%, e will be 0.5, and r_n the optimum rate of interest which maximises utility will be Gp/0.5, namely twice the rate of growth of labour productivity. If the marginal utility of income does not fall at all as real per capita incomes rise, per capita utility will grow by 1.0% when incomes rise by 1.0% so that e is unity, and r_n equals G_p. The more steeply the marginal utility of incomes fall, the more e will fall below unity, and the more the optimum real rate of interest, G_p∕e, will exceed the rate of growth of labour productivity.

If a society actually seeks to establish the optimum rate of interest deter­mined in this kind of way, the value of C_r will depend upon this optimum rate of interest, so it will not also be possible to use the rate of interest to equate the natural and warranted rates of growth in the manner that the neoclassical growth models of, for instance, Robert Solow (1956) and Trevor Swan (1956) propose. There will therefore still be difficulties because the warranted rate of growth with real interest rates at their optimum level will not in general be equal to the natural rate. Therefore, as Harrod suggested in the final articles he published in 1960 and 1964, governments will have to run persistent bud­get deficits or surpluses if they are to avoid the difficulties inherent in discrep­ancies between the natural and the warranted rates of growth.

3