Question 4.25: The total cost functions for two soap manufacturers (Plane S......

(a)

TR = P × Q = 115Q − 2Q²

For firm (i), the break-even points are calculated algebraically by solving TC = TR

12 + 2Q = 115Q − 2Q²
2Q² − 113Q + 12 = 0

The solutions to this quadratic are Q = 0.11 and Q = 56.4. Therefore, when plotting the graph, choose 0 ≤ Q ≤ 60.

For firm (ii), the break-even points are calculated algebraically by solving TC = TR:

0.3Q³ − 15Q² + 250Q = 115Q − 2Q²
0.3Q³ − 13Q² + 135Q = 0

In this text, the methods for solving cubic functions are not covered, unless it is possible to factor the cubic function, thereby reducing the problem to the product of linear and quadratic functions. The above cubic is solved as follows:

0.3Q³ − 13Q² + 135Q = 0
Q(0.3Q² − 132Q + 135) = 0       factoring out Q

Therefore, Q = 0 and solving the quadratic 0.3Q² – 13Q + 135 = 0, gives Q = 17.3 and Q = 26.1. Therefore, when plotting the graph, choose 0 ≤ Q ≤ 30.
Set up tables for total revenue with each of the TC functions. Make sure that the tables contain the points of intersection and span intervals where TC < TR and TC > TR (unless, of course, the firm never makes a profit, in which case TC > TR always). (See Figures 4.24 and 4.25.)

(b) (i) Total cost is a linear function, therefore, it increases indefinitely. See Figure 4.24.

TC > TR followed by an interval where TC < TR and by a further interval where TC > TR

(ii) Initially, the rate of increase of TC is decreasing up to a point, after which the rate of increase of TC is increasing. See Figure 4.25.

TC > TR followed by an interval where TC < TR and by a further interval where TC > TR

(c) (i) Break-even at Q = 0.11 and Q = 56.4. Between these levels of output the firm makes a profit.

(ii) Break-even at Q = 17.3 and Q = 26.1. Between these levels of output the firm makes a profit.

Comment: The Plane Soap Co. makes a much larger profit than the Round Soap Co. since the area between the TR and TC curve is substantially greater.