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 = 102 − 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 determine the consumers' surplus and the producers' surplus, we first need to find the equilibrium price and quantity.
In this scenario, the equilibrium price and quantity occur when the quantity demanded is equal to the quantity supplied. Therefore, we can set the two equations equal to each other and solve for x:
102 - x^2 = 48 + (1/2)x^2
Combining like terms:
(1/2)x^2 + x^2 = 102 - 48
(3/2)x^2 = 54
Dividing both sides by (3/2):
x^2 = 36
Taking the square root of both sides:
x = ±6
Since we're dealing with quantities, the negative value doesn't make sense in this context, so we take the positive value.
Therefore, when the market unit price is set at the equilibrium price, the quantity demanded and supplied is x = 6 thousand units.
Now, let's calculate the equilibrium price by substituting x = 6 into either of the initial price equations:
p = 102 - x^2
p = 102 - (6^2)
p = 102 - 36
p = 66
Therefore, the equilibrium price is $66.
To calculate the consumers' surplus, we need to find the area between the demand curve and the equilibrium price line. We can calculate this area using the formula for the area of a triangle:
Consumers' Surplus = (1/2) * (base) * (height)
The base of the triangle is the quantity of tires (x) and the height is the difference between the equilibrium price and the price at which x units are made available.
Consumers' Surplus = (1/2) * x * (p - (48 + (1/2)x^2))
Plugging in the values x = 6 and p = 66:
Consumers' Surplus = (1/2) * 6 * (66 - (48 + (1/2) * 6^2))
Consumers' Surplus = (1/2) * 6 * (66 - (48 + 18))
Consumers' Surplus = (1/2) * 6 * (66 - 66)
Consumers' Surplus = (1/2) * 6 * 0
Consumers' Surplus = 0
Therefore, the consumers' surplus is $0.
To calculate the producers' surplus, we need to find the area between the supply curve and the equilibrium price line. The calculation is similar to the consumers' surplus:
Producers' Surplus = (1/2) * x * ((48 + (1/2)x^2) - p)
Plugging in the values x = 6 and p = 66:
Producers' Surplus = (1/2) * 6 * ((48 + (1/2) * 6^2) - 66)
Producers' Surplus = (1/2) * 6 * ((48 + 18) - 66)
Producers' Surplus = (1/2) * 6 * (66 - 66)
Producers' Surplus = (1/2) * 6 * 0
Producers' Surplus = 0
Therefore, the producers' surplus is also $0.
In conclusion, both the consumers' surplus and the producers' surplus are $0 when the market unit price is set at the equilibrium price.