In Example 7.1, two points on the demand curve were given (see Figure 7.10).
a. Suppose three additional points are estimated by Madison: (1) demand of 460 when price is $65, (2) demand of 355 when price is $75, and (3) demand of 275 when price is $85. With these new points and the original two points, estimate and interpret the best-fitting linear demand curve; do the same for the best-fitting constant elasticity demand curve.
b. Calculate the mean absolute percentage error (MAPE) for each of the two fits, linear and constant elasticity, where each MAPE is the average of the absolute percentage errors for the five points. On the basis of MAPE, which curve provides the better fit?
EXAMPLE 7.1 PRICING DECISIONS AT MADISON
The Madison Company manufactures and retails a certain product. The company wants to determine the price that maximizes its profit from this product. The unit cost of producing and marketing the product is $50. Madison will certainly charge at least $50 for the product to ensure that it makes some profit. However, there is a very competitive market for this product, so that Madison’s demand falls sharply when it increases its price. How should the company proceed?4
Objective To use a demand function in a nonlinear model to find the price that maximizes the company’s profit.
WHERE DO THE NUMBERS COME FROM?
Cost accountants should be able to supply the unit cost. Historical data on demands and prices of the product are needed to estimate the demand function, as discussed next.