Question

a firm is planning to manufacture a new product. the sales department estimates the quanity that can be sold depends on the selling price. as the selling price is increased, the quantity that can be sold decreases. they estimate:
P=$35-0.02Q where P=selling price/unit and Q=quantity sold/unit.

on the other hand, management estimates that the average cost of manufacturing and sellling the product will decrease as the quantity sold increases. they estimate:
C=$4Q+$8000.

where C=cost to produce and sell Q/year.

The firm's management wishes to produce and sell the product at the rate that will maximize profit, that is, income minus cost will be a maximum. what quantity should the decision makers plan to produce and sell each year.

I know the answer is 775 units and i have to take derivatives of something because it's asking for a maximum, but how do i put the two equations together.

Answer 1

This is a classic maximize-profits for a monopolist. Always, always, always, maximize where marginal cost (MC) equals marginal revenue (MR).

OK, Total Revenue is P*Q. Using your demand equation TR=35Q-.02Q^2. Marginal revenu is the first derivitive of total revenue, so MR=35-.04Q

Marginal cost is the first derivitive of total cost, so TC = 4.

Maximized profits occur when MC=MR or 4 = 35-.04Q. Solve for Q. (I get 775)