A firm is planning to manufacture a new product. As the selling price is increased, the quantity that can be sold decreases. Numerically they estimate

P = $35.00 - 0.02Q

(P = selling price per unit, Q = quantity sold per year)

On the other hand, management estimates that the average cost of manufacturing and selling the product will decrease as the quantity sold increases

C = $4.00Q + $8000

where C = cost to produce and sell Q per year

The want to maximize profit. What quantity should the decision makers plan to produce and sell each year?

Profit Maximising Quantity = 775

Explanation:

Price P = 35 - 0.02Q

Total Revenue TR = Price x Quantity = P X Q

= (35 - 0.02Q)(Q)  = 35Q - 0.02Q^2

Total Cost TC = 8000 + 4Q

Profit = TR - TC

[35Q - 0.02Q^2] - [8000+4Q]  =  35Q - 0.02Q^2 - 8000 - 4Q

Profit Function = - 0.02Q^2 + 31Q - 8000

To find out profit maximising Quantity , we will differentiate Profit Function with respect to Q & equate it to 0.

dTR/ dQ = -0.04Q + 31 = 0

Q = 31/0.04 = 775

To verify whether 775 is profit maximising Q, we will do second derivative & check that it is negative.

d^2TR/ dQ^2 = -0.04 i.e < 0 (negative)

So 775 is profit maximising quantity

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