A large company in the communication and publishing industry

A large company in the communication and publishing industry has quantified the relationship between the price of one of its products and the demand for this product as Price=150-0.01*Demand for an annual printing of this particular product. The fixed costs per year (i.e., per printing)=$49,000 and the variable cost per unit=$40. What is the maximum profit that can be achieved? What is the unit price at this point of optimal demand? Demand is not expected to be more than 6,000 units per year. Please help and explain step by step!

Solution

Demand function is as follows -

Price = 150 - 0.01D

Calculate the total revenue -

TR = Price * Demand

TR = (150 - 0.01D) * D = 150D - 0.01D²

Calculate the marginal revenue -

MR = dTR/dD = d(150D - 0.01D²)/dD = 150 - 0.02D

Variable cost is constant at $40 per unit. When variable cost is constant then VC equals MC.

So, MC will also be $40 per unit.

Profit is maximized when that level of output is produced correspondinig to which MR equals MC.

MR = MC

150 - 0.02D = 40

0.02D = 110

D = 5,500

The optimal demand is 5,500 units.

P = 150 - 0.01D = 150 - 0.01(5,500) = 150 - 55 = 95

Profit = TR - TC

Profit = (Price * demand) - (FC + VC)

Profit = (95 * 5,500) - [49,000 + (40 * 5,500)]

Profit = 522,500 - [49,000 + 220,000]

Profit = $253,500

So,

The maximum profit that can be achieved is $253,500.

The unit price at this point of optimal demand is $95 per unit.