Question

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!

Answer 1

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.