Mattress Demand Consumers' Surplus Problem

Question

The demand function for a certain brand of mattress is given by p = 300e^(-0.07x) where p is the unit price in dollars and x (in units of a hundred) is the quantity demanded per month.

(a) Find the number of mattresses demanded per month if the unit price is set at $200/mattress. (Round your answer to the nearest integer.)

(b) Use the results of part (a) to find the consumers' surplus if the selling price is set at $200 per mattress.

Answer 1

Given p = 300e^(-0.07x) and letting the price (p) = $200 we have

200 = 300 e ^ (-0.07x) ==> 200 / 300 = e ^ (-0.07x) ==> 2/3 = e ^ (-0.07x) and if we take the ln of both sides we get

ln(2/3) = -0.07 ln e^x and since ln(x) and e^x are inverses

ln(2/3) = -0.07x and finally x = ln(2/3) / -0.07 = 5.79 and since x are in 100's, the answer is 579 Units.

To find Surplus we first find the price where we would not sell any units, so if we substitute x = 0,

we get p = 300 e ^(-0.07 * 0) = 300. No consumer surplus is the area above the price of 200 and the curve 300 e ^(-0.07x) integrated from x 0 to 579 units

300e ^(-0.07x) / 0.07 from x = 0 to 579 units =(300 /.07) [e^0 - e ^(-0.07* 579)] = 4.285.71