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 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
p = 48 + 1\2x^2
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 $
To find the consumers' and producers' surplus, we need to first determine the equilibrium price and quantity. The equilibrium occurs when the quantity demanded (Qd) is equal to the quantity supplied (Qs).
Given the demand function:
p = 144 - x^2
And the supply function:
p = 48 + 1/2x^2
To find the equilibrium price and quantity, we set Qd equal to Qs and solve for x:
144 - x^2 = 48 + 1/2x^2
Combine like terms:
x^2 + 1/2x^2 = 144 - 48
x^2 + 1/2x^2 = 96
Multiply both sides by 2 to eliminate the fraction:
2x^2 + x^2 = 192
Combine like terms:
3x^2 = 192
Divide both sides by 3:
x^2 = 64
Take the square root of both sides:
x = ±8
Since x represents quantity, we take the positive value:
x = 8
Now, substitute x = 8 into the demand or supply function to find the equilibrium price:
For demand:
p = 144 - (8^2) = 144 - 64 = 80
For supply:
p = 48 + 1/2(8^2) = 48 + 32 = 80
The equilibrium price is $80 and the equilibrium quantity is 8 (thousand units).
Consumer's Surplus:
To find the consumer's surplus, we need to calculate the area under the demand curve up to the equilibrium quantity and price. The formula for consumer's surplus is:
Consumer's Surplus = (1/2) * (Equilibrium Quantity) * (Equilibrium Price)
Consumer's Surplus = (1/2) * 8 * 80
Consumer's Surplus = 320 dollars
Producer's Surplus:
For producer's surplus, we need to calculate the area above the supply curve up to the equilibrium quantity and price. The formula for producer's surplus is:
Producer's Surplus = (1/2) * (Equilibrium Quantity) * (Difference between Equilibrium Price and Minimum Supply Price)
The minimum supply price can be found by substituting x = 8 into the supply function:
Minimum Supply Price = 48 + 1/2(8^2) = 48 + 32 = 80
Producer's Surplus = (1/2) * 8 * (80 - 80)
Producer's Surplus = 0 dollars
Therefore, the consumer's surplus is $320 and the producer's surplus is $0.