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.)
$ ?
Did you provide the all information? Because all you can do with the given information is to plug P=4 into the equation and find x, but x is quantity of CD demanded.
Consumer surplus is the difference between the max price a consumer willing to pay and the actual price they do pay.
So you do have the actual price they pay - $4, and you can find how many thousand of CDs is being sold, but you need to have (I believe) extra information, such as price consumers willing to pay.
To determine the consumers' surplus when the wholesale market price is set at $4/disc, we need to find the area under the demand curve from 0 to the quantity demanded at that price. Consumers' surplus represents the difference between the price consumers are willing to pay and the actual price they pay.
Given the demand function P = -0.01X^2 - 0.1X + 6, where P is the price and X is the quantity demanded, we can substitute P = $4 to find the quantity demanded at that price:
4 = -0.01X^2 - 0.1X + 6
Rearranging the equation, we get:
0.01X^2 + 0.1X - 2 = 0
To solve this quadratic equation, we can use the quadratic formula:
X = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 0.01, b = 0.1, and c = -2. Plugging these values into the formula, we get:
X = (-0.1 ± √(0.1^2 - 4(0.01)(-2))) / (2(0.01))
Simplifying further, we have:
X = (-0.1 ± √(0.01 + 0.08)) / 0.02
X = (-0.1 ± √0.09) / 0.02
X = (-0.1 ± 0.3) / 0.02
This gives us two possible values for X:
X1 = (-0.1 + 0.3) / 0.02 = 0.2 / 0.02 = 10
X2 = (-0.1 - 0.3) / 0.02 = -0.4 / 0.02 = -20
Since we're measuring X in units of a thousand, we take X1 = 10 thousand units as the relevant quantity demanded.
To find the consumers' surplus, we need to calculate the area under the demand curve from 0 to 10 thousand units. Since the demand function is quadratic, we can use calculus to find the area.
The consumers' surplus is given by the integral of the demand function from 0 to 10, divided by 1000 (to convert units from thousands to units):
Consumers' surplus = (1/1000) * ∫[0 to 10] (-0.01X^2 - 0.1X + 6) dX
Evaluating this integral, we get:
Consumers' surplus = (1/1000) * [-0.00333X^3 - 0.05X^2 + 6X] (0 to 10)
Plugging in the limits of integration, we have:
Consumers' surplus = (1/1000) * [-0.00333(10)^3 - 0.05(10)^2 + 6(10) - (-0.00333(0)^3 - 0.05(0)^2 + 6(0))]
Simplifying further, we get:
Consumers' surplus = (1/1000) * [-0.00333(1000) - 0.05(100) + 60 - 0]
Consumers' surplus = (1/1000) * [-3.33 - 5 + 60]
Consumers' surplus = (1/1000) * [51.67]
Finally, calculating the value, we have:
Consumers' surplus ≈ 0.05167
Therefore, the consumers' surplus is approximately $0.05.