Question

The demand function for a certain brand of CD is given by
p = −0.01x^2 − 0.2x + 12
where p is the unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. The supply function is given by
p = 0.01x^2 + 0.5x + 3
where p is the unit price in dollars and x stands for the quantity that will be made available in the market by the supplier, measured in units of a thousand. Determine the producers' surplus if the market price is set at the equilibrium price. (Round your answer to the nearest dollar.)

Answer 1

0.01x^2 + 0.5x + 3 = −0.01x^2 − 0.2x + 12
0.02^2 + 7x -9 = 0

2x^2 + 70x -900 =0

2(x^2 + 35x -450) = 0

2(x -10)(x + 45)=0

x = 10
x = -45

Solving gives x=10
This is the equilibrium quantity
Plug 10 into either demand or supply function to get equilibrium price = 9

Consumer surplus:
ʃ (−0.01x^2 − 0.2x + 12 )dx on [0,10] - 10*9

(-.01/3x^3/3 -.2x^2/3 + 12x on [0,10] - 90

-10/3 - 20/3 + 120- 90 = 20

Producer surplus:
10*9 -ʃ (0.01x^2 + 0.5x + 3)dx on [0,10]

90 - ((.01/3)x^3 + .5x^2/2 + 3x)) on [0,10]

90 - ( 10/3 + 25+ 30)

90 -175/3 = 31.67