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.)
-.01x^2 - 0.1x + 6 = 4
x=10 or -20
not sure what the "consumers' surplus" is defined as, unless it's when x<0, since demand is negative
To determine the consumers' surplus, we first need to find the quantity demanded at the given wholesale market price, and then calculate the area of the triangle formed by the demand curve and the market price.
Step 1: Find the quantity demanded at the wholesale market price of $4/disc.
We have the demand function: p = -0.01x^2 - 0.1x + 6.
Setting p = 4, we can solve the equation to find the corresponding x:
4 = -0.01x^2 - 0.1x + 6.
Simplifying the equation, we get:
0.01x^2 + 0.1x - 2 = 0.
We can solve this quadratic equation by factoring or using the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a).
In our case, a = 0.01, b = 0.1, and c = -2:
x = (-0.1 ± √(0.1^2 - 4 * 0.01 * -2))/(2 * 0.01).
After substituting the values into the formula, we get:
x = (-0.1 ± √(0.01 + 0.08))/(0.02).
Simplifying further, we have:
x = (-0.1 ± √0.09)/(0.02).
Now, we have two possible values for x:
x1 = (-0.1 + √0.09)/(0.02) and x2 = (-0.1 - √0.09)/(0.02).
Calculating these values, we find:
x1 = (0.1 + 0.3)/(0.02) = 20,
x2 = (0.1 - 0.3)/(0.02) = -10.
Since we are dealing with quantity, we ignore the negative value of x. Therefore, the quantity demanded at a price of $4/disc is 20 thousand units.
Step 2: Calculate the consumers' surplus.
To calculate the consumers' surplus, we need to find the area of the triangle formed by the demand curve and the market price of $4/disc.
Consumers' surplus = (0.5) * (base) * (height).
The base of the triangle is the quantity demanded (20 thousand units), and the height is the difference between the wholesale unit price ($4/disc) and the demand curve at that quantity demanded.
Substituting the values into the formula, we get:
Consumers' surplus = (0.5) * (20) * (difference between demand curve and price).
First, we calculate the value of demand at the given quantity:
p = -0.01x^2 - 0.1x + 6.
Substituting x = 20 into the equation:
p = -0.01(20)^2 - 0.1(20) + 6.
Calculating this, we find:
p = -0.01(400) - 0.1(20) + 6 = -4 + (-2) + 6 = 0.
The difference between the demand curve and the price (0 - 4) is -4.
Now, we can calculate the consumers' surplus:
Consumers' surplus = (0.5) * (20) * (-4) = -0.5 * 20 * 4 = -40.
Since consumers' surplus cannot be negative, we round our answer to zero.
Therefore, the consumers' surplus, when the wholesale market price is set at $4/disc, is $0.