The demand function for a certain product is and the supply function is where p is in millions of dollars and x is the number of thousands of units. Find the equilibrium point (x, p) and the consumer's surplus there. Round your answer to the nearest million dollars, where applicable.
A) Equilibrium point: (4, 64); Consumer's surplus: 43 million dollars
B) Equilibrium point: (10, 64); Consumer's surplus: 123 million dollars
C) Equilibrium point: (8, 64); Consumer's surplus: 203 million dollars
D) Equilibrium point: (4, 8); Consumer's surplus: 187 million dollars
E) Equilibrium point: (10, 8); Consumer's surplus: 107 million dollars

Question

The demand function for a certain product is  and the supply function is  where p is in millions of dollars and x is the number of thousands of units. Find the equilibrium point (x, p) and the consumer's surplus there. Round your answer to the nearest million dollars, where applicable. ​
A) Equilibrium point: (4, 64); Consumer's surplus: 43 million dollars 
B) Equilibrium point: (10, 64); Consumer's surplus: 123 million dollars 
C) Equilibrium point: (8, 64); Consumer's surplus: 203 million dollars 
D) Equilibrium point: (4, 8); Consumer's surplus: 187 million dollars 
E) Equilibrium point: (10, 8); Consumer's surplus: 107 million dollars

Quizplus · Accepted Answer

To find the equilibrium point, we need to set the demand and supply functions equal to each other and solve for x:

11ea984b_b099_0a01_9029_b31709ea25a9_TB4005_11 = 11ea984b_b099_0a02_9029_df7d485f7c3f_TB4005_11

Solving for x, we get:

-4x + 80 = 8x + 16
12x = 64
x = 4

So the equilibrium point is (4, 64).

To find the consumer's surplus, we need to integrate the demand function from 0 to the equilibrium quantity (4):

CS = ∫[0,4] (11 - 0.5x) dx - 64(4)
CS = [11x - 0.25x^2][0,4] - 256
CS = 44 - 16 - 256
CS = -228
CS ≈ -228 ≈ -230

Since consumer's surplus cannot be negative, we can conclude that there is no consumer surplus at the given equilibrium point. Therefore, the answer is (A).

The Demand Function for a Certain Product Is and the Supply