Worked Problems
(See related pages)



Problem 7.1 - Profit maximization: TR - TC approach
Problem:
A competitive firm's short-run cost information is shown in the table below.
Output / Fixed Cost / Variable Cost / Total Cost
0 / R 9.00 / R 0.00 / R 9.00
1 / 9.00 / 8.00 / 17.00
2 / 9.00 / 15.00 / 24.00
3 / 9.00 / 21.00 / 30.00
4 / 9.00 / 26.00 / 35.00
5 / 9.00 / 32.00 / 41.00
6 / 9.00 / 39.00 / 48.00
7 / 9.00 / 47.00 / 56.00
8 / 9.00 / 56.00 / 65.00
9 / 9.00 / 66.00 / 75.00
10 / 9.00 / 77.00 / 86.00
  1. Suppose the firm can sell all the output it desires at the market price of R9.10. Compute the firm's total revenue and its total profit (loss) for the potential output choices shown in the table. What output level maximizes the firm's profits (or minimizes its losses)?
  2. Repeat part a. assuming the price has fallen to R7.10.
/ Answer:
  1. The table showing revenue, cost, and profit is completed below. Total revenue is found as price times output level. For example, total revenue at an output level of 3 is R27.30. R9.10 x 3 = R27.30. Total profit is equal to total revenue minus total cost. At 3 units of output, profit = -R2.70 = R27.30 - R30.00
Output / Fixed Cost / Variable Cost / Total Cost
0 / R 9.00 / R 0.00 / -R 9.00
1 / 17.00 / 9.10 / -7.90
2 / 24.00 / 18.20 / -5.80
3 / 30.00 / 27.30 / -2.70
4 / 35.00 / 36.40 / 1.40
5 / 41.00 / 45.50 / 4.50
6 / 48.00 / 54.60 / 6.60
7 / 56.00 / 63.70 / 7.70
8 / 65.00 / 72.80 / 7.80
9 / 75.00 / 81.90 / 6.90
10 / 86.00 / 91.00 / 5.00
  1. Total profit is maximized at 8 units of output.
  2. The new table is presented below:
Output / Fixed Cost / Variable Cost / Total Cost
0 / R 9.00 / R 0.00 / -R 9.00
1 / 17.00 / 7.10 / -9.90
2 / 24.00 / 14.20 / -9.80
3 / 30.00 / 21.30 / -8.70
4 / 35.00 / 28.40 / -6.60
5 / 41.00 / 35.50 / -5.50
6 / 48.00 / 42.60 / -5.40
7 / 56.00 / 49.70 / -6.30
8 / 65.00 / 56.80 / -8.20
9 / 75.00 / 63.90 / -11.10
10 / 86.00 / 71.00 / -15.00
  1. The loss is minimized at an output of 6 units.

Problem 7.2 - Profit maximization: MR = MC approach
Problem:
Suppose a competitive firm's cost information is as shown in the table below. Its total fixed cost is R9.00.
Output / Marginal Cost / Average Variable Cost / Average Total Cost
0
1 / R 8.00 / R 8.00 / R 17.00
2 / 7.00 / 7.50 / 12.00
3 / 6.00 / 7.00 / 10.00
4 / 5.00 / 6.50 / 8.75
5 / 6.00 / 6.40 / 8.20
6 / 7.00 / 6.50 / 8.00
7 / 8.00 / 6.71 / 8.00
8 / 9.00 / 7.00 / 8.13
9 / 10.00 / 7.33 / 8.33
10 / 11.00 / 7.70 / 8.60
  1. Suppose the firm sells its output for R9.10. What is the firm's marginal revenue (MR)?
  2. Compare MR to marginal cost (MC) to determine the firm's profit maximizing (loss-minimizing) output level. Be sure to check whether or not the firm should shut down.
  3. What is the firm's per-unit profit (loss) at this output level?
  4. What is the firm's total profit (loss) at this output level?
  5. Repeat parts a. through d. assuming the price has fallen to R7.10.
  6. Repeat again assuming the price has fallen to R6.10
/ Answer:
  1. Marginal revenue is equal to price, or R9.10 in this instance.
  2. The firm will expand production as long as MR exceeds MC. It produces 8 units.
  3. Per unit profit is equal to average revenue, or price, minus average total cost. Per unit profit = R.97 = R9.10 - R8.13.
  4. Total profit is equal to per unit profit (R.97) times the number sold (8). Profit = R7.76.
  5. MR = price = R7.10. Comparing to MC, the firm produces 6 units. The firm’s per unit loss is R7.10 - R8.00 = -R.90. Since this is negative, check to see if price exceeds average variable cost. At 6 units of output, AVC = R6.50, which is indeed less than price, so the firm should produce 6 rather than shut down. The firm's total loss is R.90 x 6 = R5.40. The firm would lose its fixed cost (R9.00) if it were to shut down.
  6. Marginal revenue is R6.10. This is lower than the lowest possible value of average variable cost, so the firm should shut down, losing an amount equal to its fixed cost, or R9.00.

Additional Worked Problems



Monopoly price and output
Problem:
Suppose a monopoly's demand schedule is given by the first two columns of the following table. Its total cost of production is given in the next column.
Output / Price / Total Cost / Total Revenue / MC / MR
0 / R 24 / R 10
1 / 21 / 14
2 / 18 / 20
3 / 15 / 28
4 / 12 / 38
5 / 9 / 50
  1. Fill in the next column by computing the firm's total revenue associated with each output level.
  2. By comparing total cost and total revenue, find the output level that maximizes the firm's profit.
  3. What price should the firm set to achieve maximum profit?
  4. Complete the final two columns to verify that the same conclusions are reached using the MR = MC rule.
/ Answer:
  1. Total revenue is the product of output and price. For example, if the firm wishes to sell two units, it sets a price of R18 and its total revenue is R36 = 2 x R18. The completed table is shown below.
Output / Price / Total Cost / Total Revenue / MC / MR
0 / R 24 / R 10 / R 0
1 / 21 / 14 / 21 / R 4 / R 21
2 / 18 / 20 / 36 / 6 / 15
3 / 15 / 28 / 45 / 8 / 9
4 / 12 / 38 / 48 / 10 / 3
5 / 9 / 50 / 45 / 12 / -3
  1. At one unit of output, profit is R7 = R21 - R14, the difference between total revenue and total cost. As output is increased from 2 to 5, profit becomes R16, R17, R10, and -R5. Profit is greatest at an output level of 3 units.
  2. According to the demand schedule, price must be set at R15 to sell three units.
  3. Comparing MR to MC, output should be expanded to produce the third unit, but not the fourth. The marginal revenue of the fourth unit is R7 less than its marginal cost, and will cause profit to decrease.

Price discrimination
Problem:
Suppose a price discriminating monopoly has segregated its market into two submarkets, and can prevent resale between the two. Assume that its marginal cost is constant and equal to its average total cost of R8. The firm's demand schedule for the first group is given by the first two columns of the following table.
Output / Price / Total Revenue / MR
0 / R 24
1 / 22
2 / 20
3 / 18
4 / 16
5 / 14
6 / 12
7 / 10
8 / 8
  1. Find the firm's total revenue schedule for this submarket, entering the data into the table where indicated. Use these data to determine the marginal revenue schedule in this submarket.
  2. What output level and price will maximize the firm's profit in this submarket?
  3. The firm's demand schedule for the second group is given by the first two columns of the following table.
Output / Price / Total Revenue / MR
0 / R 33
1 / 30
2 / 27
3 / 24
4 / 21
5 / 18
6 / 15
7 / 12
8 / 9
  1. Find the firm's total and marginal revenue schedules in this second submarket. What output level and price will maximize the firm's profit in this submarket?
  2. Based on these prices, which submarket has the more elastic demand?
  3. What is this firm's total economic profit?
/ Answer:
  1. The completed table is shown below.
Output / Price / Total Revenue / MR
0 / R 24 / R 0
1 / 22 / 22 / R 22
2 / 20 / 40 / 18
3 / 18 / 54 / 14
4 / 16 / 64 / 10
5 / 14 / 70 / 6
6 / 12 / 72 / 2
7 / 10 / 70 / -2
8 / 8 / 64 / -6
  1. Using the MR = MC rule, profit is maximized at 4 units of output, implying a price of R16.
  2. The completed table is shown below.
Output / Price / Total Revenue / MR
0 / R 33 / R 0
1 / 30 / 30 / R 30
2 / 27 / 54 / 24
3 / 24 / 72 / 18
4 / 21 / 84 / 12
5 / 18 / 90 / 6
6 / 15 / 90 / 0
7 / 12 / 84 / -6
8 / 9 / 72 / -12
  1. Comparing MR and MC, maximum profits are achieved by selling 4 units in this submarket as well, but at a price of R21 as shown in the demand schedule.
  2. Since the price is higher in the second submarket, demand is more elastic in the first submarket.
  3. The firm earns revenue of 4xR16 = R64 in the first submarket and revenue of 4xR21 = R84 in the second. Its total revenue is then R148 = R64 + R84. Its total cost is 8xR8 = R64, so its total economic profit is R148 - R64 = R84