Question

The quantity demanded x (in units of a hundred) of the Sportsman 5 ✕ 7 tents, per week, is related to the unit price p (in dollars) by the relation p = −0.1x² − x + 30.

The quantity x (in units of a hundred) that the supplier is willing to make available in the market is related to the unit price by the relation p = 0.1x² + 2x + 10.
If the market price is set at the equilibrium price, find the consumers' surplus and the producers' surplus.

Consumer's surplus_________
Producers surplus_______

Answer 1

Consumer's surplus: $2.4

Producers' surplus: $17.6

To find the consumer's surplus and producer's surplus at equilibrium, we need to first determine the equilibrium quantity (x) and price (p) by setting the demand and supply equations equal to each other:

−0.1x^2−x+30=0.1x^2+2x+10

Solving this quadratic equation, we find x=2 as the equilibrium quantity.

Substituting x=2 into either the demand or supply equation, we get the equilibrium price (p):

p=−0.1(2)^2−2+30=27.6

Now, we can calculate the consumer's surplus (CS) and producer's surplus (PS).

Consumer's Surplus:

CS= 1/2×(30−27.6)×2=2.4

Producer's Surplus:

PS= 1/2×(27.6−10)×2=17.6

Therefore, the consumer's surplus is $2.4, and the producer's surplus is $17.6 at the equilibrium price.