The demand function for a certain brand of CD is given by
p = −0.01x^2 − 0.2x + 10
where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Determine the consumers' surplus if the market price is set at $2/disc. (Round your answer to two decimal places.)
To determine the consumers' surplus, we first need to find the quantity demanded at the market price of $2/disc.
The demand function for the CD is given by:
p = -0.01x^2 - 0.2x + 10
Substituting p = 2 into the demand function and rearranging the equation, we get:
2 = -0.01x^2 - 0.2x + 10
Now, we can solve this quadratic equation to find the value(s) of x:
0 = -0.01x^2 - 0.2x + 8 (by subtracting 2 from both sides)
To solve this equation, we can either factor it or use the quadratic formula. In this case, let's use the quadratic formula:
x = (-(-0.2) ± sqrt((-0.2)^2 - 4(-0.01)(8))) / (2 * -0.01)
Simplifying further:
x = (0.2 ± sqrt(0.04 + 0.32)) / (-0.02)
x = (0.2 ± sqrt(0.36)) / (-0.02)
x = (0.2 ± 0.6) / (-0.02)
We have two possible solutions for x:
1. x = (0.2 + 0.6) / (-0.02) = 0.8 / (-0.02) = -40
2. x = (0.2 - 0.6) / (-0.02) = -0.4 / (-0.02) = 20
Since x represents the quantity demanded each week in units of a thousand, a negative value (-40) doesn't make sense. Therefore, the quantity demanded at a price of $2/disc is 20,000 units.
Next, we need to find the consumers' surplus. It represents the difference between the maximum price consumers are willing to pay and the market price, multiplied by the quantity demanded.
To find the maximum price consumers are willing to pay, we take the derivative of the demand function with respect to x, which gives us the marginal revenue:
MR = dp/dx = -0.02x - 0.2
Setting MR equal to zero, we can find the quantity where marginal revenue is maximized:
-0.02x - 0.2 = 0
-0.02x = 0.2
x = 0.2 / -0.02
x = -10
Since we want a positive quantity, we ignore the negative value of x. Therefore, the quantity where marginal revenue is maximized is 10,000 units.
Now, we can calculate the maximum price consumers are willing to pay by substituting x = 10 into the demand function:
p = -0.01(10)^2 - 0.2(10) + 10
p = -0.01(100) - 2 + 10
p = -1 - 2 + 10
p = 7
The maximum price consumers are willing to pay is $7/disc.
Finally, we can calculate the consumers' surplus:
Consumers' surplus = (maximum price consumers are willing to pay - market price) * quantity demanded
Consumers' surplus = ($7 - $2) * 20,000
Consumers' surplus = $5 * 20,000
Consumers' surplus = $100,000
Therefore, the consumers' surplus is $100,000.
To determine the consumer surplus, we need to find the area between the demand curve and the market price line.
Step 1: Set up the equation for the market price line:
The market price is set at $2/disc. We can represent this line by the equation p = 2.
Step 2: Set the demand function equal to the market price line:
-0.01x^2 - 0.2x + 10 = 2.
Step 3: Solve for x:
-0.01x^2 - 0.2x + 8 = 0.
Using the quadratic formula, x = [-b ± √(b^2 - 4ac)] / (2a),
where a = -0.01, b = -0.2, and c = 8.
x = [-(-0.2) ± √((-0.2)^2 - 4(-0.01)(8))] / (2(-0.01))
x = [0.2 ± √(0.04 + 0.32)] / (-0.02)
x = [0.2 ± √0.36] / (-0.02)
x = [0.2 ± 0.6] / (-0.02)
x1 = (0.2 + 0.6) / (-0.02) = -0.8 / (-0.02) = 40
x2 = (0.2 - 0.6) / (-0.02) = -0.4 / (-0.02) = 20
Since the quantity demanded cannot be negative, the valid solution is x = 20.
Step 4: Calculate the consumer surplus:
To find the consumer surplus, we need to calculate the area between the demand curve and the market price line up to x = 20.
Substituting x = 20 into the demand function:
p = -0.01(20)^2 - 0.2(20) + 10
p = -0.01(400) - 4 + 10
p = -4 - 4 + 10
p = 2
The price at x = 20 is $2/disc.
To calculate the consumer surplus, we first find the area under the demand curve up to x = 20. This is the integral of the demand function from 0 to 20.
The integral of -0.01x^2 - 0.2x + 10 with respect to x is:
∫ (-0.01x^2 - 0.2x + 10) dx = [-0.01 * (x^3/3) - 0.2 * (x^2/2) + 10x] evaluated from 0 to 20.
Plugging in the values, we get:
[-0.01 * (20^3/3) - 0.2 * (20^2/2) + 10 * 20] - [-0.01 * (0^3/3) - 0.2 * (0^2/2) + 10 * 0]
= [-0.01 * (8000/3) - 0.2 * (400/2) + 200] - [0]
= [-800/3 - 400 + 200]
= -800/3 - 200
= -800/3 - 600/3
= -1400/3
≈ -466.67
The area under the demand curve up to x = 20 is approximately -466.67.
Now, we need to find the area between the demand curve and the market price line, which is a rectangle.
The width of the rectangle is x = 20, and the height is the difference between the market price and the price at x = 20. The market price is $2/disc, and the price at x = 20 is also $2/disc, so the height of the rectangle is 2 - 2 = 0.
The area of the rectangle is 0 * 20 = 0.
To find the consumer surplus, we subtract the area under the demand curve from the area of the rectangle:
Consumer Surplus = Area of Rectangle - Area under Demand Curve
Consumer Surplus = 0 - (-466.67)
Consumer Surplus ≈ 466.67
Therefore, the consumer surplus is approximately $466.67.