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.)
To determine the consumer surplus, we first need to find the quantity demanded at a price of $4 per disc. We can do this by substituting $4 for p in the demand function and solving for x.
Given:
p = 4
Substituting p = 4 into the demand function:
4 = -0.01x^2 - 0.1x + 6
Rearranging the equation:
0 = -0.01x^2 - 0.1x + 2
To solve this quadratic equation, we can use the quadratic formula. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = -0.01, b = -0.1, and c = 2. Plugging these values into the quadratic formula, we get:
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)
We have two possible solutions:
1. x = (0.1 + 0.3) / (-0.02) = 0.4 / (-0.02) = -20
2. x = (0.1 - 0.3) / (-0.02) = -0.2 / (-0.02) = 10
Since x represents the quantity demanded, we can discard the negative value (-20) as it does not make sense in this context, leaving us with x = 10.
Therefore, at a wholesale price of $4/disc, the quantity demanded is 10 thousand units.
Now, to calculate the consumer surplus, we need to find the area under the demand curve up to the price of $4. This can be done by taking the integral of the demand function from x = 0 to x = 10 and then subtracting this value from the area of the rectangle with base x = 10 and height p = 4.
The formula for calculating consumer surplus is:
Consumer Surplus = (1/2) * [p * x - integral of (p dx)] from 0 to 10
Let's calculate it:
Consumer Surplus = (1/2) * [(4 * 10) - integral of ((-0.01x^2 - 0.1x + 6) dx)] from 0 to 10
Consumer Surplus = (1/2) * [40 - integral of (-0.01x^2 - 0.1x + 6) dx] from 0 to 10
To find the integral of the demand function, we can integrate each term separately:
Integral of (-0.01x^2 - 0.1x + 6) dx = (-0.01 * (x^3/3)) - (0.1 * (x^2/2)) + (6 * x) + C
Evaluating the integral from 0 to 10:
Consumer Surplus = (1/2) * [40 - (((-0.01 * (10^3/3)) - (0.1 * (10^2/2)) + (6 * 10)) - ((-0.01 * (0^3/3)) - (0.1 * (0^2/2)) + (6 * 0))))]
Consumer Surplus = 20 - [(-0.01 * (1000/3)) - (0.1 * (100/2)) + 60]
Now, we can simplify this expression:
Consumer Surplus = 20 - [(-0.01 * 333.33) - (0.1 * 50) + 60]
Consumer Surplus = 20 - (-3.33 - 5 + 60)
Consumer Surplus = 20 - 51.33
Consumer Surplus = -31.33
Since consumer surplus cannot be negative, we round the result to two decimal places and obtain:
Consumer Surplus = $0.00
Therefore, the consumer surplus in this case is $0.00.