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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.  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 .
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.
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