Maximize revenue (linear programming)

Maximize revenue (linear programming)

 

Quantitative Exercise

Scenario

You are the new owner of Drespie Corn Products and Refineries. You are interested in your company’s cost and revenue relationships as well as its future pricing strategies. Accordingly, you have developed the following relationships, which you believe to be accurate on the basis of historical data:

P = $50 – $0.005Q

TC = $73,500 + $6Q + $0.0006Q2

MR = $50 – $0.01Q

MC = $6 – $0.0012Q

where P is the price, Q the quantity, TC the total cost, MC the marginal cost, and MR the marginal revenue.

Tasks:

  1. Using the Microsoft Excel Template given below, complete all the data in the template.
  2. Use the following formula to calculate the profit-maximizing point: MR – MC = 0. Explain your answer.
  3. Using the equation for MR (given below), calculate the revenue-maximizing level of output. Explain your answer.

MR = $50 – $0.01Q = 0

  1. What is the difference between the output level where the total profit is maximized and the output level where the total revenue (TR) is maximized? What is the significance of these two values in the decision-making process?
  2. Using Microsoft Excel, graph the data in the completed Template given below. Using the graph, identify the point where the total profit is maximized and the point where the TR is maximized. Explain your answers.
Quantity Price ($)   Total Revenue ($)  Marginal Revenue ($) Total Cost ($) Marginal Cost ($) Total Profit ($)
0 50 0 50 73,500 6 73,500
500
1,500
2,500
3,500
4,500
5,500
6,500
7,500
8,500
9,500
10,500

 

Solution

Answer 1:

Table 1: Data from Excel Template

Quantity Price ($)   Total Revenue ($)  Marginal Revenue ($) Total Cost ($) Marginal Cost ($) Total Profit ($)
0 50 0 50 73,500 6 73,500
500 47.5 23750 45 76650 5.4 -52900
1,500 42.5 63750 35 83850 4.2 -20100
2,500 37.5 93750 25 92250 3 1500
3,500 32.5 113750 15 101850 1.8 11900
4,500 27.5 123750 5 112650 0.6 11100
5,500 22.5 123750 -5 124650 -0.6 -900
6,500 17.5 113750 -15 137850 -1.8 -24100
7,500 12.5 93750 -25 152250 -3 -58500
8,500 7.5 63750 -35 167850 -4.2 -104100
9,500 2.5 23750 -45 184650 -5.4 -160900
10,500 -2.5 -26250 -55 202650 -6.6 -228900

 

Answer 2:

The profit-maximizing point is where MR-MC=0. Or where MR = MC.

We know that MR is the increase in revenue for each additional unit sold by the company and MC is the increase in cost for each additional unit produced by the company. Therefore as long as MR > MC, we know that the revenue generated by the additional unit is higher than the additional cost to produce it resulting in higher profits. Profits will maximize when the cost of additional unit produce equals the revenue generated by it.

Therefore,

50-0.01Q = 6-0.0012Q

  • 44 = 0.0088Q
  • Q=44/0.0088 = 5000

Thus, the profit is maximum when the company produces 5000 units which mean that when the company produces 5000 units, the cost of additional unit produced by the company is equal to the revenue generated by selling this additional unit.

Answer 3:

Total Revenue is maximum when MR=0

Revenue maximizing point is when an additional unit produced (and sold) does not result in the increased revenue. In other words, marginal revenue for the output, which is increase in revenue from each additional unit sold by the company, becomes zero. That is, MR = 0.

Therefore,

  • 50-0.01Q=0
  • 50=0.01Q
  • Q=50/0.01 = 5000

Thus, the total revenue is maximum when the company produces 5,000 units which mean that when the company produces 5000 units, increase in revenue from each additional unit sold by the company becomes zero.

Answer 4:

Difference in output levels is 5000 – 5000 = 0

Companies who focus on maximizing profits , keep producing more quantities of products as long as there profits increase because MR > MC, till the point MR tapers off and MR becomes equal to MC. At this point, their profit realization is maximum. For this example, the unit price at which profit maximization happens is $25 ($50 – 0.005*5000). After this MC > MR and profit starts to come down with the production of additional quantities.

Some companies focus on revenue maximization. This way they can reduce the price of the product to maximize revenues without incurring losses. This strategy may suit certain types of companies like not-for-profit companies. It may also be a preferred strategy in certain situations for a for-profit companies. For example, a company may want to reduce the prices as long as it is not making a loss in order to hit the rival companies to drive them out of the market.

In this example, the unit price at which revenue-maximization happens is $25 ($50 – 0.005*5000).

Answer 5:

Total profit is maximized, as shown in the graph above, where MR = MC. That is, marginal revenue, revenue increase per unit increase in quantity, is equal to the marginal cost.

Total revenue is maximized, when MR = 0. That is, marginal revenue, revenue increase per unit increase in quantity, is zero.

And in this example, profit-maximization and revenue-maximization happen at the same point i.e. at 5000 units.

Figure 1: Graph showing Profit and Revenue maximization

Quantity Price ($)   Total Revenue ($)  Marginal Revenue ($) Total Cost ($) Marginal Cost ($) Total Profit ($)
0 50 0 50 73,500 6 73,500
500 47.5 23750 45 76650 5.4 -52900
1,500 42.5 63750 35 83850 4.2 -20100
2,500 37.5 93750 25 92250 3 1500
3,500 32.5 113750 15 101850 1.8 11900
4,500 27.5 123750 5 112650 0.6 11100
5,500 22.5 123750 -5 124650 -0.6 -900
6,500 17.5 113750 -15 137850 -1.8 -24100
7,500 12.5 93750 -25 152250 -3 -58500
8,500 7.5 63750 -35 167850 -4.2 -104100
9,500 2.5 23750 -45 184650 -5.4 -160900
10,500 -2.5 -26250 -55 202650 -6.6 -228900
5,000 25 125000 0 118500 0 6500
0.0088
5000 5000