An interesting case of joint production is that of by-products produced in fixed proportions. Products that must be produced in fixed proportions should be considered as a package or bundle of output. When by-products are jointly produced in fixed proportions, all costs are common, and there is no economically sound method of cost allocation. Optimal price/output determination for output produced in fixed proportions requires analysis of the relation between marginal revenue and marginal cost for the combined output package. As long as the sum of marginal revenues obtained from all by-products is greater than the marginal cost of production, the firm gains by expanding output.
Figure 12.4 illustrates the pricing problem for two products produced in fixed proportions. Demand and marginal revenue curves for each by-product and the single marginal cost curve for production of the combined output package are shown. Vertical summation of the two marginal revenue curves indicates the total marginal revenue generated by both by-products. Marginal revenue curves are summed vertically because each unit of output
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