The proportion

 

See accompanying letter

 

I do not wish to quibble but you are not quite right, I think, in saying that in the case in which Mc starts from the origin the condition that should be constant is that the second derivative [b] of Mc in respect to x (output) should be zero. [6] It is true that if is zero in these conditions, will be constant; but it is not true necessarily that if is constant, will be zero. Suppose that the elasticity of the average cost curve is constant for all values of x and equals e. Then = constant, or ; integrating we have , where k is a constant, or , or so that , where a and e are constants. This then is the form of a constant elasticity curve. Now we know that , so that in the case of all curves of the form , is constant. With the equation , both Ac and Mc start from the origin, and . In other words for to be constant, e must be constant and in this case Ac and Mc must start from the origin. But [c] will be positive or negative for all values of according as e is < or >1. is only zero in the special case in which , in which case both Mc and Ac are straight lines through the origin.

I agree with your assertion that in the normal case, in which Mc is at first < Ac and then > Ac, will necessarily be rising at the point of equality between Mc and Ac, since will be changing from a negative to a positive value at this point. Moreover I agree that I have exaggerated the probability that will in any normal case be falling. Is not the simplest way of putting it this:--We know that in all circumstances . It is probable that the average cost curve will become less and less elastic for all values of output in which we are interested, i.e. in which Mc > Ac, as output increases. For this reason will be rising. This is a form of statement which is simple and accurate for all cases.

I do not think that it is possible to accept [d] your general proposition that in the normal case to the right of the point of equality between Mc and Ac [fig. 1], the second derivative of Mc must be negative if is to diminish. Let .

.

Now

and .

We know that y > o [and] x > o and that if M is > A (i.e. if we are to the right of the point of equality between M & A), .

But these facts do not involve that if is to be <  , (i. e. if is to be negative [e] ), then must be < 0.

I feel sure that the only simple thing is to talk in terms of increasing or decreasing values of e, the elasticity of the average cost curve.


Welcome page

top of page

Return to index of this section

Go to previous page

Go to next page