The demand function for a certain brand of CD is given by the following equation where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand.
p-0.01x^2-0.1x+6
Determine the consumers' surplus if the wholesale market price is set at $4/disc. (Round your answer to two decimal places.)
$
113,333.33
To determine the consumer's surplus, we first need to find the quantity demanded at a wholesale market price of $4/disc. To do this, we need to solve the demand function equation when p = 4.
p - 0.01x^2 - 0.1x + 6 = 4
Subtracting 4 from both sides:
-0.01x^2 - 0.1x + 2 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Where a = -0.01, b = -0.1, and c = 2.
x = (-(-0.1) ± √((-0.1)^2 - 4(-0.01)(2))) / (2(-0.01))
x = (0.1 ± √(0.01 + 0.08)) / (-0.02)
x = (0.1 ± √0.09) / (-0.02)
Simplifying further:
x = (0.1 ± 0.3) / -0.02
This gives us two possible values for x:
x1 = (0.1 + 0.3) / -0.02
x1 = 0.4 / -0.02
x1 = -20
x2 = (0.1 - 0.3) / -0.02
x2 = -0.2 / -0.02
x2 = 10
Since we're talking about quantity demanded, x cannot be negative, so we discard the negative value (-20). Therefore, the quantity demanded at a wholesale market price of $4/disc is 10 thousand units.
Now, we can calculate the consumer's surplus using the following formula:
Consumer's Surplus = ∫[a, b] p(x) dx
Where a = 0 and b = 10.
Consumer's Surplus = ∫[0, 10] (4 - 0.01x^2 - 0.1x + 6) dx
Consumer's Surplus = ∫[0, 10] (10 - 0.01x^2 - 0.1x) dx
Integrating term by term:
Consumer's Surplus = 10x - 0.01 * (x^3) / 3 - 0.1 * (x^2) / 2 | [0, 10]
Consumer's Surplus = 10 * 10 - 0.01 * (10^3) / 3 - 0.1 * (10^2) / 2 - (10 * 0 - 0.01 * (0^3) / 3 - 0.1 * (0^2) / 2)
Simplifying further:
Consumer's Surplus = 100 - 1000/3 - 50 - (0 - 0 - 0)
Consumer's Surplus = 100 - (1000/3) - 50
Consumer's Surplus ≈ 49.33
Therefore, the consumer's surplus at a wholesale market price of $4/disc is approximately $49.33.
To determine the consumer surplus, we first need to find the equilibrium quantity at the given wholesale market price. The equilibrium quantity is the quantity at which the quantity demanded equals the quantity supplied.
Step 1: Set the demand function equal to the wholesale market price and solve for x:
p - 0.01x^2 - 0.1x + 6 = 4
Step 2: Rearrange the equation:
-0.01x^2 - 0.1x + 6 - 4 = 0
-0.01x^2 - 0.1x + 2 = 0
Step 3: Solve the quadratic equation using the quadratic formula:
x = (-(-0.1) ± √((-0.1)^2 - 4(-0.01)(2))) / (2(-0.01))
Simplifying further:
x = (0.1 ± √(0.01 + 0.08)) / (-0.02)
x = (0.1 ± √(0.09)) / (-0.02)
x = (0.1 ± 0.3) / (-0.02)
Since we are looking for a quantity, we take the positive value:
x = (0.1 + 0.3) / (-0.02)
x = 0.4 / (-0.02)
x = -20
So, the equilibrium quantity is -20 (which doesn't make sense in this context).
Given that the equilibrium quantity doesn't make sense, we cannot calculate the consumer surplus based on the given information.