Determine the producers' surplus if the market price is set at the equilibrium price. (Round your answer to the nearest dollar.)

Question

Producers' Surplus, The demand function for a certain brand of CD is given by

p = −0.01x² − 0.2x + 15

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 = 0.01x² + 0.8x + 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

Denote Pd = -0.01x² - 0.2x +15 as the demand function and denote Ps = 0.01x² + 0.8x + 3 as the supply function. In equilibrium, Pd = Ps, thus -0.01x² - 0.2x +15 = 0.01x² + 0.8x + 3. Moving all the terms to one side, we have 0.02x² + x - 12 = 0, or equivalently 0.02(x² + 50x - 600) = 0. So 0.02(x+60)(x-10) = 0. As x represents quantity supplied, x > 0 and thus x = 10 as x = -60 < 0. Plugging in x=10 into either Ps or Pd should give the same result, so plugging in x=10 into Ps we get Ps = 0.01(10)² + 0.8(10) + 3 = 1+8+3 = 12. So, the equilibrium price is $12. Producer surplus is the area below the market price and above the supply curve, which in this case is equivalent to the integral from 3 to 12 of (12 - (0.01x² + 0.8x + 3)) dx. When computed we get producer surplus is 21.33. 12

3 12 − (0.01x2 + 0.8x + 3) d12

3 12 − (0.01x2 + 0.8x + 3) d