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 $
consumer surplus is 341,333 and producer surplus is 170,667.
To determine the consumers' surplus and producers' surplus when the market unit price is set at the equilibrium price, we need to first find the equilibrium price and quantity.
At equilibrium, the quantity demanded (x) and the quantity supplied (also x) are equal. Therefore, we need to set the demand function and supply function equal to each other and solve for x.
Demand function:
p = 144 - x^2
Supply function:
p = 48 + (1/2)x^2
Setting them equal to each other:
144 - x^2 = 48 + (1/2)x^2
Combining like terms and rearranging the equation:
(3/2)x^2 = 96
x^2 = (2/3)*96
x^2 = 128
x = √128
x ≈ 11.31
So, at equilibrium, the quantity demanded and supplied is approximately 11.31 thousand units of tires.
To find the equilibrium price, substitute this value of x into either the demand or supply function:
p = 48 + (1/2)(11.31)^2
p ≈ 103.34
Therefore, at the equilibrium price, the market unit price is approximately $103.34.
To calculate the consumers' surplus and producers' surplus, we need to integrate the demand and supply functions, respectively, between their respective quantities and the equilibrium price.
Consumers' Surplus:
To find the consumers' surplus, we integrate the demand function from 0 to the equilibrium quantity (11.31):
Consumers' surplus = ∫[0, 11.31](144 - x^2)dx
Using integral calculus, we get:
Consumers' surplus = [144x - (1/3)x^3] evaluated from 0 to 11.31
Consumers' surplus ≈ $1599.4
Producers' Surplus:
To find the producers' surplus, we integrate the supply function from 0 to the equilibrium quantity (11.31):
Producers' surplus = ∫[0, 11.31](48 + (1/2)x^2)dx
Using integral calculus, we get:
Producers' surplus = [48x + (1/6)x^3] evaluated from 0 to 11.31
Producers' surplus ≈ $960.4
Therefore, the consumers' surplus is approximately $1599 and the producers' surplus is approximately $960 when the market unit price is set at the equilibrium price.
To find the consumers' surplus and producers' surplus at the equilibrium price, we first need to determine the equilibrium price by setting the demand and supply equations equal to each other:
144 - x^2 = 48 + 1/2x^2
Rearranging the equation, we have:
2x^2 + x = 144 - 48
2x^2 + x = 96
2x^2 + x - 96 = 0
To solve this quadratic equation, we can factorize it or use the quadratic formula. Factoring is not possible in this case, so we'll use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 2, b = 1, and c = -96. Plugging in these values, we get:
x = (-(1) ± √((1)^2 - 4(2)(-96))) / (2(2))
x = (-1 ± √(1 + 768)) / 4
x = (-1 ± √769) / 4
Since we're dealing with a quantity, we can ignore the negative value:
x = (√769 - 1) / 4
Now, we can substitute this value back into the demand equation to find the equilibrium price:
p = 144 - x^2
p = 144 - ((√769 - 1) / 4)^2
Calculating this, we find:
p ≈ 120.315
Now that we have the equilibrium price, we can calculate the consumers' surplus and producers' surplus.
Consumers' Surplus:
To find the consumers' surplus, we need to integrate the demand equation from 0 to the quantity at the equilibrium price:
Consumers' Surplus = ∫[0, (√769 - 1) / 4] (144 - x^2) dx
Using integration techniques, we find:
Consumers' Surplus ≈ $873.16
Producers' Surplus:
To find the producers' surplus, we need to calculate the area between the equilibrium price and the supply equation curve from 0 to the quantity at the equilibrium price:
Producers' Surplus = ∫[(√769 - 1) / 4, (√769 - 1) / 4] (48 + 1/2x^2 - 120.315) dx
Simplifying and integrating the equation, we find:
Producers' Surplus ≈ $384.42
Therefore, the consumers' surplus is approximately $873 and the producers' surplus is approximately $384.