RAC and a measure of scale economy based on a multiproduct environment

The Ray Average Costs (RAC) are defined as total costs divided by q. That is, RAC(q) = C(rq,(1-r)q)/q where q1=rq, q2=(1-r)q and r is the proportion in which product 1 is produced.

We know that RAC(q) falls iff dRAC(q)/dq <0.

But dRAC(q)/dq = (1/q)[r(dC/dq1)+(1-r)(dC/dq2)]-(1/q)(1/q)C(q)
= (1/q)(1/q) [rq(dC/dq1)+(1-r)q(dC/dq2)-C(q)]
= (1/q)(1/q) [q1(dC/dq1)+q2(dC/dq2)-C(q)]

So we have economics of scale if q1(dC/dq1)+q2(dC/dq2)-C(q) is negative or q1(dC/dq1)+q2(dC/dq2) is less than C(q)

The measure of scale economy is the following: S= C(q)/ [q1(dC/dq1)+q2(dC/dq2)].
We can see that q1(dC/dq1)+q2(dC/dq2) is less than C(q) only if S is greater than 1.

Consequently, RAC(q) falls, rises or is constant as q increases depending on whether S is above, below or equal to 1. Note that this S is the multiproduct analogue of the ratio of average to marginal cost (See Tuesday, May 2, 2006).