Optimal Pricing for Joint Products Produced in Fixed Proportions

For joint products produced in fixed proportions, the optimal activity level occurs at the point where the marginal revenues derived from both products (MRT) equal the marginal cost of production.

Optimal Pricing for Joint Products Produced in Fixed Proportions

For joint products produced in fixed proportions, the optimal activity level occurs at the point where the marginal revenues derived from both products (MRT) equal the marginal cost of production.

provides revenues from the sale of both by-products. The intersection of the total marginal revenue curve MRt with the marginal cost curve identifies the profit-maximizing output level.

The optimal price for each by-product is determined by the intersection of a vertical line at the profit-maximizing output level with each by-product's demand curve. Qj represents the optimal quantity of the output package to be produced, and PA and PB are the prices to be charged for each by-product.

Notice that the MRT curve in Figure 12.4 coincides with the marginal revenue curve for product B at all output quantities greater than Q2. This is because MRA becomes negative at that point, and the firm would not sell more than the quantity of product A represented by output package Q2. The total revenue generated by product A is maximized at output Q2; sales of any larger quantity of product A would reduce revenues and profits.

If the marginal cost curve for the output package intersects the total marginal revenue curve to the right of Q2, profit maximization requires that the firm raise output up to this point of intersection. At that point, product B must be priced as indicated by its demand and marginal revenue curves. Because product B sales offer the sole motivation for production beyond the Q2 level, the marginal revenue generated from product B sales must be sufficient to cover the marginal costs of producing the entire output package. In this instance, profit maximization requires that MRB = MC. Beyond the Q2 level, the marginal cost of product A is zero; product A is the unavoidable by-product of product B production. Beyond the Q2 level, the price of product A is set in order to maximize profits in that MRA = MCA = 0. This pricing situation is illustrated in Figure 12.5, which shows the same demand and marginal revenue curves presented in Figure 12.4, along with a new marginal cost curve. The optimal output quantity is Q3, determined by the intersection of the marginal cost curve and the total marginal revenue

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