¹ “A Model of Economic Growth,” in Kaldor, Nicolas, Essays on Economic Stability and Growth (London, 1960), 259–300 Google Scholar; “Capital Accumulation and Economic Growth,” in Lutz, F. A. and Hague, D. C., eds., The Theory of Capital (London, 1961), 177–222.CrossRefGoogle Scholar

² Cf. the savings relation used by Kaldor, in Lutz, and Hague, , eds., Theory of Capital, 194–5.Google Scholar

³ For a short, clear introduction to the concept of functional see Baumol, William J., Economic Dynamics (2nd ed., New York, 1959), 142–8.Google Scholar We express the determinants of output in the unmanageable form of a functional because we are expressing a view of the nature of growth for which it is appropriate, rather than presenting a function we hope to manipulate. For the same reason, we feel no qualms about including the un-quantifiable variable R.

⁴ By capital, K, we shall mean the cumulation of all past investment less the cumulated values of capital goods written off whether as a result of depreciation or as a result of obsolescence or some other type of impairment of their usefulness which is not covered by depreciation charges.

⁵ Although strictly speaking the current values of I_t and R_t are included in the expressions for these variables over time we here separate them to emphasize the importance which will be placed on them subsequendy.

⁶ For instance, mistaken appreciation of the results of investment on output would cause the private to diverge from the social marginal efficiency of investment. Another reason would arise if those making investments did not expect to be able to retain the full increase in output for themselves because they anticipated an increase in wage rates.

⁷ With the assumptions about the connection between private expectations and social possibilities, the more usual relationship of the marginal efficiency of investment could be defined in this context as the increase in output (as a time rate) resulting from an increase in investment expressed as a function of investment:. The relation here used differs from this only in that all terms are divided by Y_t which in essence only has the effect of changing the units in which the X axis is measured. In consequence, assumptions about the connections between increases in the level of investment and increases in the rate of output apply to the effects of increases in the share of investment in output on the relative rate of increase in output and so can be used to give content to the latter function.

⁸ The downward slope of LL′ could also be the result of increases in the prices of investment goods relative to others caused by increased production of these goods.

⁹ Investment in research could be included in this approach. The LL′ curve could show, among other forces, the possibilities arising from research and complementary investment, higher levels of investment in research having the same sort of effect on LL′ and as ordinary investment. But in a fuller treatment it might be better to make such expenditure a separate factor raising the level of LL′ with the extent of such elevation depending on the levels of such expenditure over a considerable period of time as well as on its immediate level. The height of LL′ depends, of course, on other factors besides technological progress such as increases in the labour force, but we shall not consider them directly.

¹⁰ Cf. Kaldor, , “Capital Accumulation and Economic Growth,” 210–14Google Scholar; and Champernowne, D. G., “Capital Accumulation and the Maintenance of Full Employment,” Economic Journal, LXVIII, 06, 1958, 221.Google Scholar

¹¹ In so far as this is true, a depreciation convention which makes real depreciation charges large would be associated with greater willingness to invest than would one making for smaller real depreciation charges.

¹² This, unfortunately, would not leave the willingness to invest independent of investment opportunities, a consideration which we shall ignore.

¹³ The willingness to invest may decrease the higher is the capital-to-output ratio because a higher ratio is accompanied by less liquidity, an increased proportion of fixed to circulating capital, and greater difficulty in obtaining finance as well as because a feeling of deeper initial commitment and a greater burden of risk already borne, accompanying a higher ratio, may make for unwillingness to invest. In so far as the willingness to invest is thus affected, VV′ will be higher the higher the capital-to-output ratio and increases in that ratio will shift VV′ upwards. Here depreciation conventions, because they affect the estimated value of capital, could influence the model.

¹⁴ The economy will approach the intersection point as long as the slope of VV′ is greater than that of LL′.

¹⁵ If VV′ is dependent on K_t /Y_t , then as investment proceeds and possibly alters the capital to output ratio, VV′ may shift up or down, complicating the path to equilibrium and possibly altering the value of .

¹⁶ The growth rate of capital would be tending towards that of output and investment and the capital-to-output ratio would be approaching a definite level. In so far as VV′ shifted with changes in the capital-to-output ratio, a position of equilibrium in the other variables would only be achieved when the ratio assumed its long-run equilibrium value.

¹⁷ The level to which the capital-to-output ratio is tending will rise or fall as the rise in the rate of growth of output is less or greater than the increase in the ratio of net investment to income. In so far as VV′ shifts downward with decreases in K/Y, this will tend to increase M _{t + m} and or decrease them as K/Y decreases or increases. This would tend to make changes in the equilibrium value of K/Y smaller and changes in the share of investment in output more nearly proportional to the change in the rate of growth of output. It is to be noted that only if the share of net investment in output varies in the same proportion as the rate of growth of output could the capital-to-output ratio remain unchanged.

¹⁸ Cf. Meade, J. E., A Neo-classical Theory of Economic Growth (New York, 1961), chap. 5.Google Scholar See also Fellner, William, “Appraisal of the Labour-Saving and Capital-Saving Character of Innovations,” in Lutz, and Hague, , eds., Theory of Capital, 58–72.Google Scholar

¹⁹ The term r does not show the proportion of technological progress in the second as compared to the first case since even with no current technological development there might still be unexploited opportunities arising from previous developments to give a nonzero area under the LL′ curve. Furthermore, the adjustment of g₁ (M) holds only where—as assumed here—the rate of growth of the labour force and other factors making for growth are held constant. Otherwise the curves would need first to be adjusted to allow for other growth factors before they could be compared with regard to the effects of technological progress alone.

²⁰ This way of separating changes in the nature of technological progress from changes in its amount is admittedly arbitrary and any other range might have been chosen for the integration.

²¹ Changes in the L ₃ curve which might arise if the new level of M_t differed from would be described here as changes in the rate of technological progress. They might offset or reinforce the increase in the rate of growth arising simply from the bias in technological progress. It may be remarked that a situation where the ideas forthcoming in technological progress require greater levels of investment to achieve a given rate of growth than those arising in another situation would be described here mainly as a decrease in the rate of technological progress and so would lead to a fall in the rate of growth.

²² With the increase in the rate of growth of output, the capital-to-output ratio would fall while the distribution of income would not change.

²³ This would not be even approximately true if the willingness to invest depended greatly on P_t/Y_t or P_t/K_t .