The management of the Titan Tire Company has determined that the quantity demand
ID: 2861867 • Letter: T
Question
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 = 176 -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 = 80 + 1/2 X^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 $Explanation / Answer
the equilibrium point is where the two prices are equal.
176 - x² = 80 + x²/2 gives
x = ±8.
we will take only positive value so, x=8, we get p=112 at equilibrium.
By definition, the consumer surplus is the area beneath the demand curve but above and to the left of the equilibrium point . ie
CS = ? [ (176 - x²) - 112] dx, taken from x=0 to x=8. . after integral solving we get
CS = 1024/3=341.333 i.e. $341333 (because x was in thousands of units).
Similarly, the producer surplus is the area above the supply curve but below and to the left of the equilibrium point
PS = ? [ 112 - (80 + x²/2) ] dx, again taken from x=0 to x=6. After integral solving we get
PS = 512/3=170.666= $170,666.
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