The determination of a profit-maximizing activity level is only slightly more complex if a downturn in demand for either product A or B causes marginal revenue for one product to be negative when all output produced is sold to the marketplace.
Suppose that an economic recession causes the demand for product B (packaging materials) to fall dramatically, while the demand for product A (newsprint) and marginal cost conditions hold steady. Assume new demand and marginal revenue relations for product B of
P'B = $290 - $0.02Qb MR'b = ATR'b/AQb = $290 - $0.04Qb
A dramatically lower price of $90 per ton [= $290 - $0.02(10,000)] is now required to sell 10,000 units of product B. However, this price and activity level is suboptimal.
To see why, the profit-maximizing activity level must again be calculated, assuming that all output is sold. The new marginal revenue curve for Q is
If all production is sold, the profit-maximizing level for output is found by setting MR = MC and solving for Q:
MR = MC $690 - $0.06Q = $50 + $0.02Q 0.08Q = 640 Q = 8,000
At Q = 8,000, the sum of marginal revenues derived from both by-products and the marginal cost of producing the combined output package each equal $210, because
However, the marginal revenue of product B is no longer positive:
Even though MR = MC = $210, the marginal revenue of product B is negative at the Q = 8,000 activity level. This means that the price reduction necessary to sell the last unit of product B causes Vancouver's total revenue to decline by $30. Rather than sell product B at such unfavorable terms, Vancouver would prefer to withhold some from the marketplace. In contrast, Vancouver would like to produce and sell more than 8,000 units of product A because MRA > MC at the 8,000 unit activity level. It would be profitable for the company to expand production of Q just to increase sales of product A, even if it had to destroy or otherwise withhold from the market the unavoidable added production of product B.
Under these circumstances, set the marginal revenue of product A, the only product sold at the margin, equal to the marginal cost of production to find the profit-maximizing activity level:
MRa = MC $400 - $0.02Q = $50 + $0.02Q $0.04Q = $350
Under these circumstances, Vancouver should produce 8,750 units of Q = QA = QB. Because this activity level is based on the assumption that only product A is sold at the margin and that the marginal revenue of product A covers all marginal production costs, the effective marginal cost of product B is zero. As long as production is sufficient to provide 8,750 units of product A, 8,750 units of product B are also produced without any additional cost.
With an effective marginal cost of zero for product B, its contribution to firm profits is maximized by setting the marginal revenue of product B equal to zero (its effective marginal cost):
Whereas a total of 8,750 units of Q should be produced, only 7,250 units of product B will be sold. The remaining 1,500 units of QB must be destroyed or otherwise withheld from the market. Optimal prices and the maximum total profit for Vancouver are as follows:
No other price/output combination has the potential to generate as large a profit for Vancouver.
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