N ote I
See accompanying letter
If technique is unchanged we can write
where x = output of consumption goods, C = stock of capital in consumption industries and L = amount of "prime" factors called labour in consumption industries.
If there are constant physical returns to both factors L &C, the function is homogeneous of the first degree and
is the marginal physical product elasticity of labour (i.e. the
proportionate increase in output proportionate increase in labour).
In perfect competition 
= the real earnings of labour so that 
will be equal to the proportion of income going to labour.
In what follows I shall call q the proportion of income going to labour and 1 - q the proportion going to profits of capital. But my argument does not at first depend on this, & to extend it to imperfect competition q is simply defined as the marginal product elasticity of labour. In any case 1 > q > 0.
Differentiating the equation 
we have
This is the fundamental relation connecting the proportionate rate of
increase of consumption per unit of time (
) with the proportionate rate of increase of employment per unit of
time (
) and the output of capital goods per unit of time (
).
On
p. 78 at the bottom
[3] you say that you are assuming at first (i) no changes
in technique and (ii) advances or recessions that are neutral as far
as the use of capital is concerned, & this neutrality is defined
as the situation in which 
. If we assume constant returns to all factors as well as no changes
in technique my equation (ii) is applicable and it is clear that if
, then 
.
Your neutral case is therefore essentially the case in which the proportionate rate of increase in consumption is equal to the proportionate rate of increase in employment, in which case the proportionate rate of increase in the stock of capital will have the same value also. In other words, the ratio between the stock of capital and the volume of employment is assumed constant. On this assumption everything you say on page 79 is correct. [4]
On page 84 you suggest two modifying
influences which lessen the force of the "Relation" 
, i.e. that the output of capital goods will vary in direct
proportion to the rate of change in consumption. May I for the moment
leave out your second modifying influence, and still assume tastes
and technique unchanged? I suggest that the proportion of capital to
labour in producing a unit of consumption goods may change not only
because of changes in the rate of interest but also because of
changes in the price of capital goods in relation to the price of the
prime
factors,
i e. of the wage-rate of labour.
[5] C, the amount of capital goods, must be measured in
terms of physical units--called "Machines". The marginal product of
machines is the increase in output of consumption due to having one
more machine. The cost of using an extra machine is the price of a
machine ¥ the rate of interest. The proportion between Machines
and Labour used will remain constant, if the cost of using an extra
machine bears a constant ratio to the cost of using an extra unit of
labour, i.e. if the money price of a machine ¥ the rate of
interest bears a constant proportion to the money wage rate of
labour. Now when there is an increased money demand for commodities,
(i.e. in a boom), we know (i) that the rate of interest rises, (ii)
that the price of machines rises more quickly than the price of
consumption goods and (iii) that the money wage-rate rises less
quickly than the price of consumption goods. All three of these facts
cause the rise in the demand for machines to be less than
proportionate to the rise in the rate of expansion of consumption,
because all give rise to substitution of labour for capital.
Assuming perfect competition we can easily express these facts in my equations. s, the elasticity of substitution between L and C,
i.e. the proportionate change in the ratio between L and C the
proportionate change in the ratio between their marginal physical
products. In perfect competition the price of each factor is equal to
the value of its marginal product so that 
and 
, where w = money wage rate, p = the money price of consumption
goods, P = money price of capital goods and r the rate of interest.
(I have already argued that the money cost of a machine which is
equated to the value of its marginal product is the price of a
machine ¥ the rate of interest.) Then
N.B. s is always negative [c] .
If
we substitute the value of 
from (iv) in (iii) we have
Thus if the proportionate rate of change of wage rates minus
[d] the proportionate rate of change in the price of
capital goods minus the proportionate rate of change in the rate of
interest = zero, 
. But if in a boom wages are rising slowly, and the price of machines
and the rate of interest are rising fast, then the proportionate rate
of increase of consumption will rise more than in proportion to the
output of capital goods; and this phenomenon will be the more
important the greater the possibility of substitution (i.e. the
greater the numerical value of s), and the greater is q, the
proportion of income going to labour.
For interest we can add, though it is not directly relevant, an
interesting relation between the rate of change of consumption prices
and factor prices. We know that 
and 
(since factors are paid their marginal products). Differentiating we
have
From the first equation (i) we know
and substituting the value of 
from equation (iii) we have
and substituting for 
and 
from (v) we have
and dividing both sides by x and substituting for 
and 
from (ii) we get
i.e. the proportionate rate of rise of consumption prices must be equal to q times the proportionate rate of rise of money wage rates + (1 - q) times the proportionate rate of rise of the "price" of machines, i.e. of Pr. This can be interpreted to mean that since factor prices will be bid up to equal the value of their marginal products, then given a certain raise in commodity prices, in wage-rates and in the rate of interest the price of machines will be raised by competition to satisfy the equation.
I am sorry to have bored you with this long notes. For your book I only wish to make two suggestions:--(i) That you should make it clear that your neutral advance or recession from the point of view of capital is one in which the ratio between capital and labour is left constant so that the proportionate rate of increase in consumption = proportionate rate of increase in employment = proportionate rate of increase in capital stock; [6] and (ii) that this will happen if the proportionate rate of change in the money wage rate is equal to the proportionate rate of change in the price of machines while the rate of interest is constant, or in other words that your Relation exaggerates the situation for two reasons apart from changes in technique--both because the rate of interest rises and because the price of machines rises more quickly than the money wage rate, both of which lead to substitution of labour for capital.
I may add that I think that the rise and fall of machine prices out of relation with wage-rates in boom and slump is one of the most important stabilizing elements in the capitalist system, because it is the most important modification of your "Relation."
,
.
