# 237. W. M. Allen to Harrod , 28 April 1932 [a]

[Balliol College, Oxford] # [b]

28 April 1932

Dear Harrod,

I hope that this will be a somewhat more coherent version of the problem I thrust on you after lunch:  Assume the existence of a unit in which the physical volume of goods can be measured, and of a unit by which the amount of the "original factors" can be expressed.

(i) [c] Let there be n units of the original factors.

(ii) Let a unit of the original factors, when initially applied, produce k units of goods.

(iii) Let the times elapsing between the application of the original factors, and their product reaching the consumer, be .

(iv) Let i be the interest rate per unit of time, so that one unit of present goods may be exchanged for (1 + i) units of goods obtainable after a unit of time. In effect, k units of goods become k(1 + i) units after unit time.

The unit of the original factors first applied,--then yielding k units of goods,--will have matured units by the time it reaches the consumer, afterwards. If the original factor is applied continually, then at any moment there will be (unfinished) goods in process. = . {=0, if = 0}. Similarly the second unit of the original factor yields after time mp and implies the existence of units of goods in process.

The total of goods in process is, therefore, .

Now may be constant without being constant. When , then is a minimum. When , and the constant to which is equated, then is maximised.

So it would appear that, given the amount of the original factors, n; the coefficient of production of labour, and the interest rate, i; together with the value of , (which I have taken to be the index of the average period of production), yet is the quantity of goods in process not uniquely determined.

No doubt there are other limitations that apply to these formulae. For one, output must be maximised given the amount of goods in process,--if that is taken as the index of the amount of capital, the productivity of labour being constant--, and for another, it is probable that along with any one rate of interest there are only a few organisations of the system (i.e. distributions of the t's) possible. But it seems to me that it might still be possible for an increase in --which I would hold to be the "amount of capital"--to occur, while decreases.

I hope I have made myself clear. I must confess to a feeling of faintness whenever I try to work in this field, and I expect you'll tell me I have good reasons [d] to mistrust myself. But I can't see the way out of this problem of definition.

Yours sincerely,

Maurice Allen.

1. 1. This seems to be the only extant document concerning Allen's problem
1. a. ALS, four pages on a single folded sheet, in HP IV-14-15c.

b. The letterhead consists of the College emblem only.

c. The numbering of the assumptions was added afterwards in the margin--it is difficult to judge whether this was done by the recipient or the sender.

d. Ms: possibly "reason".

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