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=132-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=36+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.)
To determine the consumer's surplus and the producer's surplus when the market unit price is set at the equilibrium price, we first need to find the equilibrium price and quantity.
Step 1: Set the two price equations equal to each other:
132 - x^2 = 36 + (1/2)x^2
Step 2: Combine like terms:
2(132 - x^2) = 72 + x^2
Step 3: Distribute the 2:
264 - 2x^2 = 72 + x^2
Step 4: Move all terms to one side:
3x^2 = 192
Step 5: Divide both sides by 3:
x^2 = 64
Step 6: Take the square root of both sides:
x = ±8
Since quantity can't be negative in this context, we take x = 8.
Step 7: Substitute the value of x into one of the price equations to find the equilibrium price:
p = 36 + (1/2)(8^2)
p = 36 + (1/2)(64)
p = 36 + 32
p = 68
So, the equilibrium price is $68 and the equilibrium quantity is 8 thousand units.
Now, let's calculate the consumer's surplus and the producer's surplus.
Consumer's Surplus:
The consumer's surplus measures the benefit consumers receive from purchasing a good, which is equal to the difference between the price they are willing to pay and the price they actually pay.
To calculate the consumer's surplus, we need to integrate the demand equation (132 - x^2) from x=0 to x=8 and subtract the area of the triangle formed by the equilibrium price and x-axis.
Integral of (132 - x^2) => [132x - (1/3)x^3]
Consumer's surplus = [(132*8 - (1/3)*8^3)] - (1/2)*(8)*(68)
= [1056 - 170.67] - 272
= 885.33 - 272
= $613.33
Producer's Surplus:
The producer's surplus measures the benefit producers receive from selling a good, which is equal to the difference between the price they receive and their marginal cost.
To calculate the producer's surplus, we need to find the area under the supply curve (p = 36 + (1/2)x^2) from x=0 to x=8.
Integral of (36 + (1/2)x^2) => [36x + (1/6)x^3]
Producer's surplus = (1/2)*(8)*(68) - [(36*8 + (1/6)*8^3)]
= 272 - (288 + 170.67)
= 272 - 458.67
= -$186.67
Since the producer's surplus is negative, this means that the producers are not making any profit at the equilibrium price.
So, the consumer's surplus is approximately $613, and the producer's surplus is approximately -$187.