Here, P is price in dollars, Q is the number of hogs processed (with an average weight of 100 pounds), and QP and QB are pork and render by-product per hog, respectively; both total and marginal costs are in dollars. Total costs include a risk-adjusted normal return of 15% on a $50 million investment in plant and equipment.
Currently, the city allows the company to dump excess by-product into its sewage treatment facility at no charge, viewing the service as an attractive means of keeping a valued employer in the area. However, the sewage treatment facility is quickly approaching peak capacity and must be expanded at an expected operating cost of $3 million per year. This is an impossible burden on an already strained city budget.
A. Calculate the profit-maximizing price/output combination and optimal total profit level for Satriale.
B. How much by-product will the company dump into the Musconetcong sewage treatment facility at the profit-maximizing activity level?
C. Calculate output and total profits if the city imposes a $35 per unit charge on the amount of by-product Satriale dumps.
D. Calculate output and total profits if the city imposes a fixed $3-million-per-year tax on Satriale to pay for the sewage treatment facility expansion.
E. Will either tax alternative permit Satriale to survive in the long run? In your opinion, what should the city of Musconetcong do about its sewage treatment problem?
A. Solution to this problem requires that one look at several production and sales options available to the firm. One option is to produce and sell equal quantities of pork (P) and byproduct (B). In this case, the firm sets relevant MC = MR.
MC = MRP + MRb = MR $60 = $110 - $0.0001Q + $10 - $0.0002Q 0.0003Q = 60
Thus, the profit-maximizing output level for production and sale of equal quantities of P and B would be 200,000 hogs. However, the marginal revenues of both products must be positive at this sales level for this to be an optimal activity level.
Evaluated at 200,000 hogs:
Because the marginal revenue for B is negative, and Satriale can costlessly dump excess production, the sale of 200,000 units of B is suboptimal. This invalidates the entire solution developed previously because output of P is being held down by the negative marginal revenue associated with B. The problem must be set up to recognize that Satriale will stop selling B at the point where its marginal revenue becomes zero because, given production for P, the marginal cost of B is zero.
Qb = 50,000 units
Thus, 50,000 units of B are the maximum that would be sold. Any excess units will be dumped into the city's sewage treatment facility. The price for B at 50,000 units is
To determine the optimal production of P (pork), set the marginal revenue of P equal to the marginal cost of hog processing because pork production is the only motive for processing more than 50,000 units:
MRP = MCP = MCq $110 - $0.0001QP = $60 0.0001QP = 50
QP = 500,000 units
PP = $110 - $0.00005QP = 110 - 0.00005(500,000) = $85
Excess profits at the optimal activity level for Satriale are
= $85(500,000) + $5(50,000) - $10,000,000 - $60(500,000) = $2,750,000
Because total costs include a normal return of 15% on $50 million in investment,
Total profits = Required return + Excess profits = 0.15($50,000,000) + $2,750,000 = $10,250,000
B. With 500,000 hogs being processed, but only 50,000 units of B sold, dumping of B is
Units B dumped = Units produced - Units sold = 500,000 - 50,000 = 450,000 units
C. In part A, it is shown that if all P and B produced is sold, an activity level of Q = 200,000 results in MRB = -$30. A dumping charge of $35 per unit of B will cause Satriale to prefer to sell the last unit of B produced (and lose $30) rather than pay a $35 fine. Therefore, this fine, as does any fine greater than $30, will eliminate dumping and cause Satriale to reduce processing to 200,000 hogs per year. This fine structure would undoubtedly reduce or eliminate the need for a new sewage treatment facility.
Although eliminating dumping is obviously attractive in the sense of reducing sewage treatment costs, the $35 fine has the unfortunate consequence of cutting output substantially. Pork prices rise to PP = $110 - $0.00005(200,000) = $100, and by-product prices fall to PB = $10 - $0.0001(200,000) = -$10. This means Satriale will pay the pet food company $10 per unit to accept all of its by-product sludge. Employment will undoubtedly fall as well. In addition to these obvious short-run effects, long-run implications may be especially serious. At Q = 200,000, Satriale's excess profits are
= $110Q - $0.00005Q2 + $10Q - $0.0001Q2 - $10,000,000 - $60Q = $110(200,000) - $0.00005(200,0002) + $10(200,000) - $0.0001(200,0002) - $10,000,000 - $60(200,000) = -$4,000,000 (a loss)
This means that total profits are
Total profits = Required return + Excess profits = 0.15($50,000,000) + (-$4,000,000) = $3,500,000
This level of profit is insufficient to maintain investment. Although a $35 dumping charge will eliminate dumping, it is likely to cause the firm to close down or move to some other location. The effect on employment in Musconetcong could be disastrous.
D. In the short run, a $3 million tax on Satriale has no effect on dumping, output, or employment. At the Q = 500,000 activity level, a $3 million tax would reduce Satriale's total profits to $7,250,000, or $250,000 below the required return on investment. However, following imposition of a $3 million tax, the firm's survival and total employment would be imperiled in the long run.
E. No. Satriale is not able to bear the burden of either tax alternative. Obviously, there is no single best alternative here. The highest fixed tax the company can bear in the long run is $2.75 million, the full amount of excess profits. If the city places an extremely high priority on maintaining employment, perhaps a $2.75 million tax on Satriale plus $250,000 in general city tax revenues could be used to pay for the new sewage system treatment facility.
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