# Return to Example 5.2, where we explored treating the third pool as a period or a product cost…

Return to Example 5.2, where we explored treating the third pool as a period or a product cost pool. Can you find an LLA and allocation scheme for the third pool that makes the resulting unit cost for the first product as close as possible to that product’s marginal cost? What happens to the ability of the second product’s unit cost to approximate that product’s marginal cost? Explain.

((Example 5.2 For a second illustration, we return to Example 4.1, where three factors are used to produce two products, subject to an upper bound on the third factor. Denote the output by q = [q1, q2]. We assume output is sufficiently high that the third factor is at its upper bound, implying a specific region of the long-run cost curve or a specific version of the shortrun cost curve with the third factor fixed at its upper bound.

Either way, the three factors are naturally catalogued in three distinct cost pools, and this leads to the following cost pool expressions18

which reflects the economic cost curve in this region of C(q; P) = 75 + 4 3 q 2 1 + q 2 2 , along with respective marginal costs of MC1(q; P) = 8 3 q1 and MC2(q; P) = 2q2. The accounting library will treat the first two cost pools as direct product cost pools, while the third will be treated as a period or a product cost pool. Now suppose, as in the original example, that q1 = 15 and q2 = 25 are produced and we have cost pool totals of cost1 = 300, cost2 = 625 and cost3 = 75. This gives us an LLA of cost1 = 20q1 (as cost1/q1 = 300/15 = 20) for the first pool, and cost2 = 25q2 (as cost2/q2 = 625/25 = 25). Treating the third pool as a period cost thus implies respective unit costs of 20 and 25 for the two products. In turn, treating the third pool as an indirect product cost pool requires we specify some synthetic variable. We’ll use q= q1 + q2; and this implies an LLA for the third pool of cost3 = 1.875q (as cost3/q= 75/40 = 1.875).

This leads to respective unit costs of 20 + 1.875 = 21.875 and 25 + 1.875 = 26.875.

The respective marginal costs at this point are MC1(q; P) = 8 3 (15) = 40 and MC2(q; P) = 2(25) = 50. With output sufficiently large that the third factor is at its upper bound, the firm is producing in a region where its marginal costs are increasing in output. This leads, as flagged in Table 4.5, to the accountant’s systematic assignment of actual costs to products leading to unit costs that are systematically below marginal costs. Now glance back at Table 4.5. All we have done here is reconstruct that display, but now using the explicit costing mix of aggregation, LLAs and cost allocation.

Stepping back, any application of the accountant’s product costing art leads to a product cost statistic. Suppose we decide to treat some product cost statistic as an approximation of marginal cost. Is this best thought of as an approximation to long-run or to some short-run marginal cost? No general answer is possible. The accountant’s art produces a number. This number may be close to or far removed from the portion of the long-run marginal cost curve we had in mind. Similarly, this number may be close to or far removed from the portion of some short-run marginal cost curve we had in mind.

This may appear curious or even cynical. Yet it is the natural manifestation of treating accounting as a library. Various choices go into design of the library. The resulting choices may produce something that is close to what we want or not so close. We and our immediate curiosity are just one of many users of that library. How we use the library depends on how the library was constructed and on our context. Rules and recipes stand in the way of professional quality interrogation of the accounting library. Professional judgment is an essential ingredient in the use of the accounting library.))

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