The demand function for a certain brand of CD is given by p = -0.01x^2 - 0.2x +

Question

The demand function for a certain brand of CD is given by p = -0.01x^2 - 0.2x + 13 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 + 0.8x + 1 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.)

Explanation / Answer

For equlibrium

-0.01x^2 - 0.2x + 13 = 0.01x^2 + 0.8x + 1
0.02^2 + x -12 = 0 => x^2 + 50x -600 =0 =>(x -10)(x + 60)=0 => x=10 or -60

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

-0.01x^2 - 0.2x + 13 we get equilibrium price = 10

therefore Producer surplus:

10*10 - (0.01x^2 + 0.8x + 1)dx on [0,10]

= 100 - ((.01/3)x^3 + 0.8x^2/2 + x)) on [0,10]

=100 - ( 10/3 + 40+ 10)

=100 -160/3 = 140/3=46.66

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