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 = 160 − 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 = 64 + 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.)
consumer's surplus = ?
producer's surplus = ?
To determine the consumer's surplus and producer's surplus at the equilibrium price, we first need to find the equilibrium price. At equilibrium, the quantity demanded (x) will be equal to the quantity supplied. We can set the demand equation equal to the supply equation and solve for x.
Demand equation: p = 160 − x^2
Supply equation: p = 64 + 1/2x^2
Setting them equal: 160 − x^2 = 64 + 1/2x^2
Combining like terms: 3/2x^2 = 96
Simplifying: x^2 = 64
Taking the square root of both sides (since x can't be negative): x = 8
Now we know that at equilibrium, x = 8. We can substitute this value back into either the demand or supply equation to find the equilibrium price (p).
Using the demand equation: p = 160 − x^2
p = 160 − (8)^2
p = 160 − 64
p = 96
Therefore, the equilibrium price is $96 per unit.
Now that we have the equilibrium price, we can find the consumer's surplus and producer's surplus. Consumer's surplus represents the difference between what the consumers are willing to pay for a product and what they actually pay, while producer's surplus represents the difference between the price the producers receive and the minimum price they are willing to accept.
To calculate the consumer's surplus, we need to find the area between the demand curve and the price line up to the equilibrium quantity.
Consumer's surplus:
Consumer's surplus = 0.5 * (Equilibrium quantity) * (Maximum price consumers are willing to pay - Equilibrium price)
In this case, the equilibrium quantity is x = 8, and the maximum price consumers are willing to pay is the price at which x units are available in the market, which is given by p = 64 + 1/2x^2.
Substituting the values: Consumer's surplus = 0.5 * 8 * (64 + 1/2(8)^2 - 96)
Calculating: Consumer's surplus = 0.5 * 8 * (64 + 32 - 96) = 0.5 * 8 * 0 = 0
Therefore, the consumer's surplus at the equilibrium price is $0.
To calculate the producer's surplus, we need to find the area between the supply curve and the price line up to the equilibrium quantity.
Producer's surplus:
Producer's surplus = 0.5 * (Equilibrium quantity) * (Equilibrium price - Minimum price producers are willing to accept)
In this case, the equilibrium quantity is again x = 8, and the minimum price producers are willing to accept is also given by the supply equation p = 64 + 1/2x^2.
Substituting the values: Producer's surplus = 0.5 * 8 * (96 - (64 + 1/2(8)^2))
Calculating: Producer's surplus = 0.5 * 8 * (96 - (64 + 32)) = 0.5 * 8 * 0 = 0
Therefore, the producer's surplus at the equilibrium price is $0.
In summary:
Consumer's surplus = $0
Producer's surplus = $0
To find the consumer's surplus and the producer's surplus, we first need to find the equilibrium price.
Equilibrium price occurs when the quantity demanded equals the quantity supplied. Therefore, we can equate the two equations for p:
160 - x^2 = 64 + (1/2)x^2
Simplifying the equation:
160 - 64 = (1/2)x^2 + x^2
96 = (3/2)x^2
Multiplying both sides by 2/3:
64 = x^2
Taking the square root of both sides:
x = ± 8
Since x represents the quantity of tires, a negative value does not make sense in this context. Therefore, x = 8.
Substituting x = 8 back into the demand equation, we can find the equilibrium price:
p = 160 - x^2
p = 160 - 8^2
p = 160 - 64
p = 96
So, the equilibrium price is $96.
Now, we can find the consumer's surplus and the producer's surplus.
Consumer's surplus is the difference between what consumers are willing to pay (based on their demand) and what they actually pay (equilibrium price). It can be calculated using the formula:
Consumer's surplus = (1/2)(equilibrium quantity – quantity demanded)(equilibrium price)
In this case, the equilibrium quantity is x = 8 units. Substituting the values into the formula:
Consumer's surplus = (1/2)(8 - x)(96)
= (1/2)(8 - 8)(96)
= 0
Therefore, the consumer's surplus is $0.
Producer's surplus is the difference between the equilibrium price and the cost of production for the producer. In this case, the cost of production is not given. As a result, we cannot calculate the producer's surplus without additional information.