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.)

Question

Consumers' and Producers' Surplus, 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 − x²

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+1/2x^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 $___

Answer 1

176-x^2 = 80 + (1/2)x^2

96 = (3/2)x^2

x^2 = (2/3)(96) = 2(32) = 64

x =sqr64

x = 8 thousand sold at the equilibrium price of $112

equilibrium price = 176-8^2 = 176-64 = $112

= 80+(8^2)/2 = 80+64/2 = 80+32 = $112

consumer surplus = the integral of 176-x^2 -112 evaluated from 0 to 8

= 341 1/3 = $341,333 rounded to nearest dollar

producer surplus = integral of 112 -80 -x^2 evaluated from 0 to 8

= 170 2/3 thousand= $170,667 rounded to nearest dollar

the basic method to a solution is

CS = the area below the demand curve and above the line P=112

PS = area below P=112 and the supply curve

total surplus = total gains from trade = CS +PS = area under the demand curve and above the supply curve = 341 1/3 + 170 1/3 = 511 2/3 = integral of 96 -3x^2/2 evaluated from 0 to 8 = 512 = $512,000 for 8,000 tires sold/purchased 341 1/3 + 170 2/3 = 512 = $512,000

Answer 2

First, we'll need to determine the equilibrium price p and quantity x. We do this by finding the point where the demand and supply curves intersect (i.e. set them equal to each other).

176 − x² = 80+1/2 x²

96 = 3/2 x²

x² = 64, so x = 8 (remember, this means eight THOUSAND units).

We can plug that back into either equation to find the equilibrium price p:

p = 176 - x² = 176 - 8² = 176 - 64 = 112.

Now that we have the intersecting point of the supply and demand curve, (8, 112), we can find consumer and producer surplus by using the curves and the horizontal equilibrium price line p=112.

Consumer surplus = area under demand curve but above the horizontal equilibrium line, up until the equilibrium point

= ∫₀⁸ 176 - x² - 112(8-0)

= 1237.3 - 896

= 341.3. But remember, x was in thousands of units, so it's not just $341.30, it's $341,300.

Producer surplus = area below the horizontal equilibrium line but above the supply curve, up until the equilibrium point

= 112(8-0) - ∫₀⁸ 80 + 1/2 x²

= 896 - 725.333

= 170.667. Same as above, remember that x was in thousands of units, so it's not just $170.667, it's $170,667.