Question

A company produces and sells a consumer product and is able to control the demand for the product by varying the selling price. The approximate relationship between price and demand is

p = $38 +2,700/D −5,000/D2, for D > 1,

where p is the price per unit in dollars and D is the demand per month.

The company is seeking to maximize its profit. The fixed cost is $1,000 per month and the variable cost (Cv) is $40 per unit.

a. What is the number of units that should be produced and sold each month to maximize profit?

b. Show that your answer to Part (a) maximizes profit

Answer 1

a. Profit = Revenue – Cost

Cost = fixed cost + (variable cost)D = 1000 + 40D

Revenue = PD = (38 + $\frac{2700}{D}$ – $\frac{5000}{D^2}$ )D

Profit = 38D + 2700 - $\frac{500}{D}$ – 1000 – 40D

P = 1700 – 2D - $\frac{5000}{D}$

dP/dD = -2 + $\frac{5000}{D^2}$

For maximize profit

dP/dD = 0

D² = 2500

D = 50

Demand to maximize to profit is 50 units.

b. At D = 49

Profit = 1700 – 2D - $\frac{5000}{49}$ = 1499.95

At D = 51

Profit = 1700 – 100 - $\frac{5000}{51}$ = 1499.96

At D = 50

Profit = 1700 – 100 - $\frac{5000}{50}$ = 1500

So, D = 50 maximize profit.