Firms can often vary the proportions in which joint products are created. Even the classic example of fixed proportions in the joint production of beef and hides holds only over short periods: Leaner or heavier cattle can be bred to provide differing proportions of these two products. When the proportions of joint output can be varied, it is possible to construct separate marginal cost relations for each product.
The marginal cost of either joint product produced in variable proportions equals the increase in total costs associated with a one-unit increase in that product, holding constant the quantity of the other joint product produced. Optimal price/output determination for joint products in this case requires a simultaneous solution of marginal cost and marginal revenue relations. The firm maximizes profit by operating at the output level where the marginal cost of producing each joint product just equals the marginal revenue it generates. The profit-maximizing combination of joint products A and B, for example, occurs at the output level where MRA = MCB and MRB = MCB.
It is important to note, however, that although it is possible to determine the separate marginal costs of goods produced in variable proportions, it is impossible to determine their indi common costs
Expenses that are necessary for manufacture of a joint product vidual average costs. This is because common costs are expenses necessary for manufacture of a joint product. Common costs of production—raw material and equipment costs, management expenses, and other overhead—cannot be allocated to each individual by-product on any economically sound basis. Only costs that can be separately identified with a specific by-product can be allocated. For example, tanning costs for hides and refrigeration costs for beef are separate identifiable costs of each by-product. Feed costs are common and cannot be allocated between hide and beef production. Any allocation of common costs is wrong and arbitrary.
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