You have been hired as a marketing consultant to Big Book Publishing, Inc., and you have been approached to determine the best-selling price for the hit calculus text

Question

You have been hired as a marketing consultant to Big Book Publishing, Inc., and you have been approached to determine the best-selling price for the hit calculus text by Whiner and Istanbul entitled Fun with Derivatives. You decide to make life easy and assume that the demand equation for Fun with Derivatives has the linear form q = mp + b, where p is the price per book, q is the demand in annual sales, and m and b are certain constants you must determine.

(a)

Your market studies reveal the following sales figures: when the price is set at $55.00 per book, the sales amount to 10,000 per year; when the price is set at $85.00 per book, the sales drop to 1,000 per year. Use these data to calculate the demand equation.

q = ____

 

(b)

Now, estimate the unit price that maximizes annual revenue.

$ ____

Predict what Big Book Publishing, Inc.'s annual revenue will be at that price.

$ ___

Answer 1

(10, 55) and (1, 85) are two points on a line with slope = 30/-9 = - 10/3 = (P-55)/(q-10)

q-10 = (-3/10)P - 33/11

q = -.3P - 3 + 10

q = -.3P + 7 where P is measured in dollars and q measured in thousands, m=-.3, b= 7

q=-.3P +7 is the demand equation

Revenue = R = Pq = P(.3P+7) = -.3P^2 + 7P

take the derivative of R and set = 0, then solve for P

R' = -.6P + 7 = 0

.6P = 7

P =7/.6 = 70/6 = $11.67 = revenue maximizing textbook price, rounded off to nearest cent

q=-.3(70/6) + 7 = -7/2 +7 = 7/2 = 3.5 thousand = 3,500 = revenue maximizing output sales level

max R = Pq = 70/6)(7/2) - 245/6 = 40.8333...thousand dollars = $40,833.33

max revenue = -.3(70/6)^2 +7(70/6)

= -(3/10)(70/6)(70/6) + 7(70/6)

= -(7/2)(70/6) + 7(70/6)

= (7/2)(70/6) = 245/6 = 40.888...thousand dollars = $40,833.33

it might help to graph the original two points (10,55) and (1,85)

and the straight line through them.

q = (-3/10)P + 7 has q (or x) intercept = 7, and has P (or y) intercept of 70/3 = 23 1/3

that line segment from one intercept to the other has a midpoint of (3.5, 11 2/3)

which represents q = 3,500 and P = $11.67

the revenue maximizing quantity and price is the midpoint of the demand curve