Question

A firm is planning to manufacture a new product. The sales department estimates that the quantity that can be sold depends on the selling price. As the selling price is increased, the quantity that can be sold decreases. On the other hand, the management estimates that the average cost of manufacturing and selling the product will decrease as the quantity sold increases. Numerically they estimate:

P = $35.00–0.02Q

C = $4.00Q + $8000

P = selling price per unit

Q = quantity sold per year

C = cost to produce and sell Q per year

The firm’s management wishes to produce and sell the product at the rate that will maximize profit, that is, where total revenue (P*Q) minus total cost (C) will be a maximum. What quantity should they plan to produce and sell each year? Be sure to include a graphical representation of total revenue and total cost. Within the graphical representation indicate where the breakeven (total revenue = total cost) points are, where the profit region is, and the point at which maximum profit is achieved.

Answer 1

Answer - The firm has estimated a inverse demand function, P = 35.00 - 0.02Q and total cost function, C = 4.00 + 8000.

We know that profit is calculated by the following way,

Profit () = TR(Q) - TC(Q)

Total revenue is price time quantity, TR(Q) = P * Q

TR = (35.00 - 0.02Q) * Q

TR = 35.00Q - 0.02Q² ..............(1)

TC(Q) = 4.00Q + 8000

= 35.00Q - 0.02Q² - (4.00Q + 8000)

= 31.00Q - 0.02Q² -8000

Now we need to maximize the profit function to know the level of production which will given maximum profit to the firm. The second derivative must be negative. Now find first and second derivative of the profit function.

/Q = 31.00 - 0.04Q - 0

² / Q² = -0.04

We can see that second derivative is negative (² / Q² < 0), thus this profit function will have maximum value where,

/Q = 0

31.00 - 0.04Q = 0

Q = 31 / 0.04

Q = 775 units

The firm will earn maximum profit if it produces 775 units.