Product-Specific Economies of Scale based on a multiproduct environment
Suppose that instead of a C(q) cost function, we have a C(q1,q2) cost function. This new function represents the cost of a firm that produces q1 units of product 1 and q2 units of product 2.
The incremental costs of increasing product 2 from 0 to q2 holding product 1 constant is IC2 = C(q1,q2) – C(q1,0)
The average incremental costs of increasing product 2 from 0 to q2 holding product 1 constant is AIC2 = IC2/q2
So the product-specific economies of scale (PS2) of q2 holding the other output, q1, constant is: PS2=AIC2/MC2 , where MC2 is the marginal cost of product 2, which is the partial derivative of C(q1,q2) with respect to q2.
Note that in case with more than two products, we have PSi = AICi/MCi as before, but all outputs except qi must be held fixed (not only q1 as in the example above).