08.07.2022 - 01:29

# The management of the Titan Tire Company has determined that the quantity demanded x of their Super Titan tires/week is related to the unit price p by the relation p = 144 – x^2 where p is measured in

Question:

The management of the Titan Tire Company has determined that the quantity demanded x of their Super Titan tires/week is related to the unit price p by the relation {eq}p = 144 – x^2 {/eq} where p is measured in dollars and x is measured in units of a thousand. Titan will make x units of the tires available in the market if the unit price is {eq}p = 48 + 1/2x^2 {/eq} dollars.

Determine the consumers’ surplus and the producers’ surplus when the market unit price is set at the equilibrium price. (Round your answers to the nearest dollar.) consumers’ surplus $producers’ surplus$

• The equilibrium price occurs when the quantity demanded equals the quantity supplied. Setting the two price equations equal to each other and solving for x, we have: 144 - x^2 = 48 + 1/2x^2 3/2x^2 = 96 x^2 = 64 x = 8 So the equilibrium price is p = 144 - 8^2 = 80 dollars per thousand units. To find the consumers' surplus, we need to find the area beneath the demand curve and above the equilibrium price, up to the quantity of 8 thousand units. This is a triangle with base 8 and height (144 - 80) = 64, so the consumers' surplus is (1/2)(8)(64) = 256 dollars. To find the producers' surplus, we need to find the area above the supply curve and below the equilibrium price, up to the quantity of 8 thousand units. This is also a triangle, but with base 8 and height (80 - 48) = 32, so the producers' surplus is (1/2)(8)(32) = 128 dollars. Therefore, the consumers' surplus is $256 and the producers' surplus is$128.