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 market equilibrium quantity at the given price of $2/disc. The market equilibrium quantity occurs when the quantity demanded equals the quantity supplied.
Given the demand function: p = -0.01x^2 - 0.2x + 10
Set p = 2 and solve for x:
2 = -0.01x^2 - 0.2x + 10
Rearrange the equation to bring it to quadratic form:
0.01x^2 + 0.2x - 8 = 0
Now we can solve this quadratic equation. You can use the quadratic formula or factorization method.
Using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values:
a = 0.01, b = 0.2, c = -8
x = (-0.2 ± √(0.2^2 - 4 * 0.01 * -8)) / (2 * 0.01)
Simplifying the equation, we get:
x = (-0.2 ± √(0.04 + 0.32)) / 0.02
x = (-0.2 ± √0.36) / 0.02
x = (-0.2 ± 0.6) / 0.02
x = (-0.2 + 0.6) / 0.02 or x = (-0.2 - 0.6) / 0.02
Simplifying further, we get:
x = 0.4 / 0.02 or x = -0.8 / 0.02
x = 20 or x = -40
Since we are measuring x in units of a thousand, the market equilibrium quantity is x = 20 thousand units.
Next, we need to find the area under the demand curve from 0 to the equilibrium quantity (x = 20).
The consumers' surplus can be calculated using the following formula:
Consumers' Surplus = ∫[0,x] p dx
In this case, we need to integrate the given demand function from 0 to x = 20:
Consumers' Surplus = ∫[0,20] (-0.01x^2 - 0.2x + 10) dx
Evaluating this definite integral will give us the consumers' surplus.
Let's solve this integral step by step:
∫ (-0.01x^2 - 0.2x + 10) dx
To integrate each term:
∫ (-0.01x^2) dx = -0.01 * (x^3/3) + C1
∫ (-0.2x) dx = -0.2 * (x^2/2) + C2
∫ (10) dx = 10x + C3
Where C1, C2, and C3 are constants of integration.
Now evaluate the integral with the limits from 0 to 20:
Consumers' Surplus = (-0.01 * (20^3/3)) - 0.2 * (20^2/2) + (10 * 20) - [(0.01 * (0^3/3)) - 0.2 * (0^2/2) + (10 * 0)]
Calculating this expression will yield the consumers' surplus.