The demand function for a certain brand of mattress is given by p = 450e^−0.04x where p is the unit price in dollars and x (in units of a hundred) is the quantity demanded per month.

Question

(a) Find the number of mattresses demanded per month if the unit price is set at $200/mattress. (b)Use the results of part (a) to find the consumers' surplus if the selling price is set at $200 per mattress.

Answer 1

For part (a):

You want to start by just plugging in the value that they give you for p. If the unit price is set at $200/mattress and p is the unit price, then we see that p = $200/mattress. Plugging that in we get

200 = 450e^(-0.04x) to solve this we need to first get the exponential alone (by dividing both sides by 450)

4/9 = e^(-0.04x) now we can take the natural log of both sides to get rid of the e

ln(4/9) = -0.04x and remember we want to solve for x so we divide both sides by -0.04

-25 ln(4/9) = x

20.27 ≈ x

So you would need -25 ln(4/9) ≈ 2,027 mattresses (since x units are in the hundreds)

For part (b):

The area below the demand function p = 450e^(-0.04x) and above the equilibrium price p = $200/mattress is the consumer surplus (CS).

To get the area above $200/mattress and below the demand function, we want to integrate the demand function minus the equilibrium price ($200/mattress) from x = 0 to x = 20.27 (the x value that gives us the price of $200/mattress) to get the consumer surplus.

Integral (from x =0 to x= 20.27) 450e^(-0.04x) -200 = 450/(-0.04) e^(-0.04x) -200x evaluated from x =0 to x= 20.27

= 450/(-0.04) e^(-0.04*20.27) - 450/(-0.04) e^(-0.04*0) -200(20.27)-0 ≈ 2195.35