Remarks to R.F.H.'s "Essay in Dynamic Theory" [1]


See the accompanying letter


(A) In the chart below I have tried to show what seems to be the time-shape of R.F.H.'s four rates of growth. The [fig. 1 [2] ] full-drawn line is the actual rate of growth of output (G). The dotted line is the "warranted rate" (G w ). The double line is the "ceiling" or the natural rate which, of course, need not be horizontal; finally the line represents the "normal" rate which lies between the extremes attained by G w , and above zero.

The definitions of the actual rate and the natural rate are easily understood; not so the definition of the warranted rate [b] .

If the actual, statistically recorded output is x, .

As to the natural rate, it is the highest value which can reach under conditions of full employment and given technological conditions.

The warranted rate, G w , also depends on technological conditions, but the condition of full employment is removed and replaced by the condition that the rate "will leave all parties satisfied" (p. 4) which is alternatively alluded to as "certain fundamental conditions namely the propensity to save etc." [3]

(B) It seems to me that G w can be easiest understood in the light of the para 2. page 1 of the Essay where the axiomatic basis of the theory is stated. [4] I hope I assume rightly that the magnitudes involved are ex-ante (or warranted) magnitudes. Attaching to them, therefore, the subscript w, we may write:

(a) (Supply function of ex-ante savings)

(b) (Demand function for ex-ante savings)

(g) (Equilibrium equation)

the last equation determining the "equilibrium" or "warranted" values of output (x w ), rate of growth , or relative rate of growth , as functions of time. According to the shape of the demand and supply functions we shall obtain a definite time-shape for G w --so long as the functions f and remain unchanged.

In the simplest case, the demand function may be a simple proportionality law, thus:

( ) ,

where the constant C is identical with the C used in your equation (1) (section 4 of the Essay [5] ): the latter is obtained from ( ) by dividing by Cx w :

( G w )

In this simple case the equilibrium equation g assumes the form

( ) ,

or, dividing by x w C,

This expresses the fact that G w depends on 1) the propensity-to-save-schedule (namely, the function of output, ) and 2) the value of the constant C. I can't help thinking that C itself expresses expected technological conditions and nothing else: it doesn't [c] depend, e.g., on the propensity to save but is an investment factor which, together with the propensity-to-save-schedule, determines the warranted rate G w .

(C) The differential equation (g) or, in the simplest case, ( ), does not yield, in general, a periodical solution of suitable form; though special cases may be constructed {NB. with a simple (linear) form for f we get a non-periodical "explosive" function}. Nor is it attempted to derive the periodicity directly from the supply-and-demand set-up.

(D) To explain the behaviour of G additional postulates are made. (The equation is not such a postulate, it is a definition of C p ). 1 These postulates are allowed to: 1) on p. 15 para 2. [7] where it is postulated that the divergence G - G w induces the individuals to certain actions; 2) on p. 21 [8] where the "ceiling" is introduced; 3) on p. 23 where negative values of both growth rates are discussed. [9]

(E) The reaction to the divergence G - G w as postulated in the Essay (the "cumulative process") may be written, I suggest, as


where l is some increasing function. Am I right in understanding this as an empirical assumption of the boom (and depression) rather than a conclusion following from the other equations. In fact, one could imagine the opposite case: a shortage of capital equipment being met by reduction of output rather than leading to an increase. It seems that the definition of the "unstable equilibrium" implies certain empirical postulates regarding the psychology of entrepreneurs' reactions to current profits made in capital good production. Could this be stated explicitly?

Further, what postulate do you make to conclude that G increases faster than G w ?

(F) A further relationship is the "ceiling relationship"

G  G naturalis .

If it can be postulated that G w continues to rise after intersecting the ceiling (some sort of "inertia"), the turning of G does follow from the postulate of cumulative process {see under (E)}. This is, I think, one of the most interesting points in the Essay and I think you are right that the "lag" involved can be assumed infinitely small [10] {more about this below, section (I)}. But what about G w ? As stated in my section (C) the supply-and-demand-functions do not, in general, produce a periodicity; I presume that you postulate a change in the form of these functions themselves; in fact, you did discuss changes in the parameter C; [11] could the postulates as to the behaviour of C (viz., its reactions to changes in G) be stated explicitly? Similarly as with regard to the "cumulative process" of G, some sort of reaction of entrepreneur's expectations to their realizations is probably assumed: (e.g. a change of expected technological productivity?). [12]

To sum up the preceding sections, it seems to me that the following set-up is implied:

1) ex-ante supply function of savings (= propensity to save).

2) ex-ante demand function of savings, involving a changing parameter C.

3) A postulate describing the changes of C as function of (say) G, or of G - G w , and making for a periodic shape of the G w curve.

4) A postulate stating explicitly the reactions making for the progressive growth of the difference .

(1), (2) and (4) are given in the Essay more explicitly than (3), but perhaps not always explicitly enough.

(G) The stage in which G and G w are both below zero is discussed. It is proved in the Essay [d] that the two rates do get below zero. [13] This is necessary but not sufficient to produce an intersection between the two curves below the zero-line. It would be necessary to show (theoretically or on empirical grounds) that while G rises faster than G w in the upswing, it falls slower than G w in the downswing. E.g., the equation proposed in (E). would be replaced by


so that the rate of increase of the difference between G and G w is positive when the difference is positive and negative, if the latter is negative. But I am not clear about the economic implications of this. But perhaps I have overlooked your proof of the existence of the intersection.

(H) For the empirical verification, the ex-post quantities (G, ex-post savings) would not present any but purely technical difficulties. But with the ex-ante magnitudes (G w , C) there are difficulties of principle, as they are not recorded statistically. I wonder whether they could be replaced by some indices of expectations, based, e.g., on current profits. That would of course, imply the reformulation of certain postulates.

(I) I entirely agree with you that dynamics is not identical with Theory of Lag's. It is sufficient to have velocities, accelerations etc. and no finite lags, to have dynamics. By definition, velocity involves the comparison between two points of time but they may be as near each other as we please. It seems to me therefore that p. 7 para 2 is a little too polemic: "those who define dynamic as having a cross reference to two points of time" [14] do not necessarily require lags and will certainly agree that your equation (1) is dynamic involving as it does a rate of growth (a velocity).

(K) Small alterations suggested:

p. 6 and footnote. [15] The proposition about C being inversely proportionate to the time-unit chosen is rather obvious: since to add A to the annual output is equivalent to adding to the monthly output. The discussion given is not very easy to follow. On line 2 of the footnote, a misprint (1 instead of 12) .

p.7. Misprint: the "s" in the second equation should be the same as in the first (i.e. small). [16]

p. 9 l. 4. after "goods" insert "per unit increment of output". [17]

p. 5 last line: "increment of" can be cancelled. [18]

p. 6 line 3-4 from below: "<it> ... etc.": too truistic since saving has been defined as ex-post investment. [19]

p. 11 <+> from below. This seems to be a slip. The steadiness of advance of two variables does not exclude finite lags; e.g. two trains moving steadily, the one lagging behind the other by 100 yards. {The objection has nothing to do with my attitude to your general definition of dynamic etc.: See above, under (I)}. [20]

p. 15 line 11: insert "per unit of output increment" after "capital goods". [21] Or write s w x w and sx instead of C p and C in this sentence. (The substitution between s w and s is, however, explicitly discarded, for simplicity, on p. 12 ff. [e] , and I don't argue about this).

p. 16, l. 1-2. The marginal propensity to save is . The propensity-to-save-schedule is the function f in the supply equation . The word "expressed" should, therefore, be replaced by some other word. [22]

16, l. 2. I should avoid to use a phrase "the warranted G, G w )"; [23] the reader must remain clear that G and G w are two variables (which may or may not have equal values), so that G w is not a particular value of G (as, e.g., x 1 is a particular value of output x).

21 para 4 (last para of Section 12). [24] I should prefer to read (v. graph) "From this point on the actual rate coincides with the natural rate. When the warranted rate becomes greater than the natural rate, it also becomes greater than the actual rate and so enters the field ... etc." Next sentence: "If on the other hand, the warranted rate remains below the natural rate, an immediate ... etc.": These alterations are suggested because it is the natural rate which is conceived as the independent variable: you wouldn't say: if the ceiling finds itself in a higher position than the ball, then ...

p. 22. line 5. [25] best, but not necessarily!

On p. 22-23 I have not always been clear why you write "normal warranted rate": the cumulative divergency assumed earlier is the one between the actual and the warranted rate tout court. [26]

On p. 24, l. 3 from below: [27] "can". But: will it?

p. 26, l. 5: [28] negative? or diminishing? l. 14: put "numerical" before "value of C p "?

p. 27 para 1 "G w " should be "G" (in two places)? [29]

p. 35 para 1 [30] (same remark as to pp. 22-23).

p. 37, second sentence. [31] Does this mean that people have for a [f] long time expected the impossible? (This remark is not a criticism).

J. M.

  1. 1. Harrod, "An Essay in Dynamic Theory" ( 1939:7 ); the manuscript of the draft Marschak is referring to is reproduced here as essay 19 .

    2. In the preliminary version of this note (reference is given in source note a to this letter), the G w line intersects the G line from below at the point where the latter begins its descent from the "ceiling". The line of the normal rate of growth was not drawn.

    3. Essay 19 , [jump to page] and [jump to page] , respectively.

    4. Essay 19 , [jump to page] .

    5. Essay 19 , [jump to page] .

    6. Essay 19 , sections 4 and 5.

    7. Essay 19 , [jump to page] .

    8. Essay 19 , [jump to page] .

    9. The notion of negative rates of growth is introduced in page 23 of the draft (essay 19 , [jump to page] ); the slump is discussed in more detail in section 15 of the draft (essay 19 , [jump to page] ).

    10. Harrod did not explicitly discuss the length of the lag: he argued instead that, in the event of steady growth, lags are unimportant (essay 19 , [jump to page] ). Marschak inferred that Harrod assumed infinitely small lags from his own interpretation of Harrod's equation as a differential equation, considered as a special case of lagged functional equation with a lag of vanishing length.

    11. Essay 19 , [jump to page] .

    12. Harrod did not follow Marschak's suggestion. This had unfortunate consequences, as after the war commentators--neoclassical critics, in particular--failed to appreciate that the parameter C changes in the course of the cycle (see D. Besomi, "Failing to Win Consent: Harrod's Dynamics in the Eyes of His Readers", in G. Rampa, L. Stella and A. P. Thirlwall, Economic Dynamics, Trade and Growth. Essays on Harrodian Themes ( 1998), in particular pp. 57-61.

    13. Essay 19 , section 15.

    14. Essay 19 , [jump to page] .

    15. Essay 19 , [jump to page] .

    16. Marschak probably refers to equation (1a) (essay 19 , [jump to page] ). Since the Ms is correct, the slip must be due to the typist (the typescript is not extant).

    17. Essay 19 , [jump to page] . Harrod added instead: "in any period".

    18. Essay 19 , [jump to page] (it is not clear, however, to which of the two contiguous sentences containing this expression Marschak was referring to). Harrod did not follow Marschak's suggestion.

    19. Essay 19 , [jump to page] .

    20. Essay 19 , [jump to page] . The passage was not altered in the final version (Harrod 1939:7 , p. 20).

    21. Essay 19 , [jump to page] . Marschak's suggestion was accepted.

    22. Essay 19 , [jump to page] . This passage was altered, following both Marschak's and Keynes's suggestions: see note 33 to essay 19 .

    23. Essay 19 , [jump to page] .

    24. Essay 19 , [jump to page] . From this point on, the text was drastically modified, and Marschak's suggestions were not incorporated.

    25. Essay 19 , [jump to page] .

    26. Essay 19 , sections 13 and 14.

    27. Essay 19 , [jump to page] .

    28. Essay 19 , [jump to page] .

    29. Essay 19 , [jump to page] , paragraph beginning "In these circumstances ...".

    30. Essay 19 , [jump to page] , paragraph beginning "If in the absence ...".

    31. This sentence could not be identified with precision; it is likely, however, to belong to sections 23 or 24.

    1. a. ANI, 12 pages on 12 numbered leaves, in HP V-113a. A preliminary version of this note is preserved in JMP 1275, Box 147, folder H.

      b. Ms: «rates».

      c. Ms: «does'nt».

      d. Ms: «essay».

      e. Ms: «<seg.>».

      f. Ms: «for long time».

1. I felt it difficult to understand the pp. 5-9 of the Essay [6] until it became clear to me that the equation was a definition of C p , while the equation is the equilibrium equation for saving, viz. the equality of demand and supply. I wonder whether you agree. [Marschak's footnote]

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