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, we first need to find the equilibrium price and quantity.
The equilibrium occurs when the quantity demanded is equal to the quantity supplied. So, we can set the demand equation equal to the supply equation:
160 − x^2 = 64 + 1/2x^2
Simplifying this equation will give us the equilibrium quantity (x value):
160 − x^2 = 64 + 1/2x^2
160 - 64 = x^2 + 1/2x^2
96 = 1.5x^2
64 = x^2
x = √64 = 8
Now, substitute the equilibrium quantity (x = 8) into either the demand or supply equation to find the equilibrium price (p value):
p = 160 − x^2
p = 160 − (8^2)
p = 160 − 64
p = $96
So, the equilibrium price is $96 and the equilibrium quantity is 8 thousand units.
To calculate the consumer's surplus, we need to find the area under the demand curve but above the equilibrium price.
The equation for the demand curve is p = 160 − x^2. We substitute the equilibrium price (p = 96) into this equation:
96 = 160 − x^2
x^2 = 160 - 96
x^2 = 64
x = √64 = 8
The consumer's surplus is the area of the triangle formed by the equilibrium quantity (8) and the equilibrium price (96). Using the formula for the area of a triangle (A = 1/2 x base x height), we get:
Consumer's Surplus = 1/2 x base x height
= 1/2 x 8 x (160 - 96)
= 1/2 x 8 x 64
= 256 dollars
So, the consumer's surplus is $256.
To calculate the producer's surplus, we need to find the area under the supply curve but below the equilibrium price.
The equation for the supply curve is p = 64 + 1/2x^2. We substitute the equilibrium price (p = 96) into this equation:
96 = 64 + 1/2x^2
1/2x^2 = 96 - 64
1/2x^2 = 32
x^2 = 64
x = √64 = 8
The producer's surplus is the area of the triangle formed by the equilibrium quantity (8) and the equilibrium price (96). Again using the formula for the area of a triangle, we get:
Producer's Surplus = 1/2 x base x height
= 1/2 x 8 x (96 - 64)
= 1/2 x 8 x 32
= 128 dollars
So, the producer's surplus is $128.
Therefore, the consumer's surplus is $256 and the producer's surplus is $128 when the market unit price is set at the equilibrium price.