Question

A Bacon Factory is located in a small town. Also in the town is a Water Park. The smell of the Bacon factory has adversely affected the Water Park such that it has put in air cleaning equipment to eradicate the odor created by the factory.

The cost function of the Bacon Factory is:

CBF= B2 + 4B1/2 + (1 − x)2

where B denotes the quantity of bacon produced annually and x denotes the quantity of pollutants that A creates in a given year.

Thus, the Bacon Factory can limit production costs by eliminating its air scrubbers. However, the air pollution increases the costs for the water park W, whose cost function is:

CWP = W2 + 2x,

where W denotes the number of visitors to the Water Park on an annual basis. Suppose that the unit price of admission to the water park is $3 and that the unit price of bacon is $32.5 per unit.

Compute the profit maximizing quantity of the Bacon Factory (B) and pollutant (x) produced by Bacon Factory B (assuming B behaves competitively in the output market, i.e., taking the price of Bacon as $32.50). Also, compute the Bacon Factory’s (Firm B) profits. (8 Points)

Compute the profit maximizing visits (represented by W) created by Firm W (assuming W behaves competitively in the output market, i.e., taking the price of visits as given). Notice that W does not choose x. Also, compute W’s profits. (9 Points)

Suppose now that the two firms B and W merge, creating B&W. The management of B&Wnow maximizes B&W’s profits by appropriately choosing x, B, and W. Find the quantities of Bacon, Water Park Visits, and pollutants that the new firm produces. Also, find the profits of B&W. (3 Points)

Answer 1

(a) Calculate the profit maximizing quantity of Bacon for the Bacon factory: MC; = MR 28 +(-){) 84 -32.5 2B+2B 2 = 32.50 B+B 2 = 16.25 B=16 The profit maximizing quantity of bacon is 16 units. Calculate the optimal quantity of x for the Bacon factory. OC 20 2(1-x)(-1) = 0 2(x - 1) = 0 x-1=0 x = 1 The profit maximizing quantity of pollutant is 1 unit.2 Bacon factory's profit = TR3 - TC; = (16)(32.5)-[162 +4v16+(11)] = (16)(32.5)-(162 +16) = (16)(32.5) – 16(16+1) = (16)(32.5) – 16 (17) = (16)(32.5 -17) = (16)(15.5) = 248 (b) Calculate the profit maximizing visits for the waterpark. MC, = MR, 2W = 3 W = 1.5 The profit maximizing quantity of visits is 1.5.

Profits for W=TR - TC = (1.5) (3) -[1.52 + 2(1)] = 4.50 -4.25 = 0.25 (C) Since marginal costs MC, & MC, are independent of x, the optimal quantity of Bacon and waterpark visits will remain the same, at 16 and 1.5 respectively. However, the company will now choose x such that the combined cost is minimized. Combined cost] = 0 B? + 4B 2 + (1 - x)+ W2 +2x = 0 2(1-x)(-1) + 2 = 0 2(x - 1) +2 = 0 2x – 2+2 = 0 2.x = 0 x = 0 Therefore, now the optimal quantity of x is 0. The cost functions for the Bacon Factory and Waterpark tell us that as x falls from 1 to 0, the total cost rises at Bacon Factory by 1 and it falls at the Waterpark by 2. Therefore, the net impact is a fall of 1 (= 2 – 1) in cost, implying an increase of 1 in profit. Therefore, the profit amount for the combined firm is: Profits of B&W = Previous profits + 1 = 248 +0.25 +1 = 249.25