Question 4.9: The demand function for a good is given as Q = 65 – 5P. Fixe......
The demand function for a good is given as Q = 65 – 5P. Fixed costs are £30 and each unit produced costs an additional £2.
(a) Write down the equations for total revenue and total costs in terms of Q.
(b) Find the break-even point(s) algebraically.
(c) Graph total revenue and total costs on the same diagram; hence, estimate the break-even point(s).
(a) TR=P×Q. Therefore, if P is written in terms of Q, then TR will also be expressed in terms of Q. The expression for P in terms of Q is obtained from the equation of the demand function
Q = 65 − 5P
5P = 65 − Q
P = \frac{65 − Q}{5}= 13 − 0.2Q
Substitute the expression for P (price per unit) into the equation, TR = P × Q; therefore, TR = (13 – 0.2Q)Q = 13Q – 0.2Q². Total cost is given as TC = FC + VC = 30 + 2Q.
(b) The break-even points occur when TR = TC, therefore,
13Q − 0.2Q² = 30 + 2Q
0 = 0.2Q² − 11Q + 30
The reader is expected to solve the quadratic equation for Q correct to one decimal place. The solutions are Q = 2.91 and Q = 52.1.
(c) A table of values for TR and TC from Q = 0 to Q = 70 is given in Table 4.6. These points are plotted in Figure 4.10. The break-even points occur at the intersection of the two functions. The break-even points on the graph agree with those calculated in (b).
| Table 4.6 Total revenue and total cost | ||
| Q | TR | TC |
| 0 | 0 | 30 |
| 10 | 110 | 50 |
| 20 | 180 | 70 |
| 30 | 210 | 90 |
| 40 | 200 | 110 |
| 50 | 150 | 130 |
| 60 | 60 | 150 |
| 70 | -70 | 170 |
