The demand function for a certain brand of CD is given by
p = −0.01x^2 − 0.2x + 12
where p is the unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. The supply function is given by
p = 0.01x^2 + 0.5x + 3
where p is the unit price in dollars and x stands for the quantity that will be made available in the market by the supplier, measured in units of a thousand. Determine the producers' surplus if the market price is set at the equilibrium price. (Round your answer to the nearest dollar.)
To determine the producers' surplus at the equilibrium price, we first need to find the equilibrium price and quantity.
The equilibrium occurs where the quantity demanded equals the quantity supplied, so we can set the demand and supply functions equal to each other:
-0.01x^2 - 0.2x + 12 = 0.01x^2 + 0.5x + 3
Next, we need to solve this equation to find the equilibrium quantity (x). We can start by simplifying the equation:
0.02x^2 + 0.7x - 9 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 0.02, b = 0.7, and c = -9. Substituting these values into the quadratic formula, we get:
x = (-(0.7) ± √((0.7)^2 - 4(0.02)(-9))) / (2(0.02))
Simplifying further:
x = (-0.7 ± √(0.49 + 0.72)) / 0.04
x = (-0.7 ± √1.21) / 0.04
x = (-0.7 ± 1.1) / 0.04
This gives us two possible values for x:
x1 = (-0.7 + 1.1) / 0.04 ≈ 10
x2 = (-0.7 - 1.1) / 0.04 ≈ -45
Since the quantity cannot be negative in this context, we can disregard x2. Therefore, the equilibrium quantity (x) is approximately 10 thousand units.
To find the equilibrium price (p), we can substitute this value back into either the demand or supply function. Let's use the demand function:
p = -0.01(10)^2 - 0.2(10) + 12
p = -1 - 2 + 12
p = 9
Therefore, the equilibrium price is $9.
Now that we have the equilibrium price and quantity, we can calculate the producers' surplus.
Producers' surplus is the area above the supply curve and below the equilibrium price. To find this area, we need to integrate the supply function from 0 to the equilibrium quantity (10) and then subtract it from the equilibrium quantity multiplied by the equilibrium price.
Integrating the supply function:
∫ [0.01x^2 + 0.5x + 3] dx = 0.01 * (x^3/3) + 0.5 * (x^2/2) + 3x + C
Evaluating the integral from 0 to 10:
[0.01 * (10^3/3) + 0.5 * (10^2/2) + 3 * 10] - [0.01 * (0^3/3) + 0.5 * (0^2/2) + 3 * 0]
[10/3 + 50/2 + 30] - [0 + 0 + 0]
[40/3 + 75] - 0
(40/3 + 75)
Now, multiply the equilibrium quantity (10) by the equilibrium price ($9):
10 * 9 = 90
Finally, subtract the integral value from the product:
90 - (40/3 + 75) ≈ 49
Therefore, the producers' surplus at the equilibrium price is approximately $49.