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 + 0.7x + 1
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 find the producers' surplus, we first need to find the equilibrium price. The equilibrium price occurs at the intersection of the demand and supply curves, where the quantity demanded equals the quantity supplied.
1. Set the demand and supply functions equal to each other:
-0.01x^2 - 0.2x + 12 = 0.01x^2 + 0.7x + 1
2. Simplify the equation:
-0.01x^2 - 0.2x + 12 - 0.01x^2 - 0.7x - 1 = 0
3. Combine like terms:
-0.02x^2 - 0.9x + 11 = 0
4. Solve the quadratic equation for x. We can either factor or use the quadratic formula to find the solutions. In this case, let's use the quadratic formula:
x = (-(-0.9) ± √((-0.9)^2 - 4(-0.02)(11))) / (2(-0.02))
5. Simplify and solve for x:
x = (0.9 ± √(0.81 + 0.88)) / (-0.04)
There are two possible solutions, x₁ and x₂, but only one will be within the range of realistic values for quantity.
6. Calculate x and find the equilibrium quantity:
x ≈ 27.57 or x ≈ -612.43
Since the quantity cannot be negative, the equilibrium quantity is approximately 27.57 thousand units.
7. Substitute the equilibrium quantity back into either the demand or supply function to find the equilibrium price. Let's use the demand function:
p = -0.01(27.57)^2 - 0.2(27.57) + 12
8. Calculate p:
p ≈ $4.96
The equilibrium price is approximately $4.96.
9. Calculate the producers' surplus. The producers' surplus is the area between the supply curve and the equilibrium price line, up to the equilibrium quantity.
To find this area, we need to calculate the integral of the supply function from 0 to the equilibrium quantity (in this case, 27.57). The integral represents the total revenue received by the suppliers.
The integral of the supply function p = 0.01x^2 + 0.7x + 1 can be found using the power rule of integration:
Integral(p dx) = 0.01 * (x^3/3) + 0.7 * (x^2/2) + x + C
10. Evaluate the integral at the upper limit (27.57):
Integral(p dx) = 0.01 * (27.57^3/3) + 0.7 * (27.57^2/2) + 27.57 + C
11. Evaluate the integral at the lower limit (0):
Integral(p dx) = 0.01 * (0^3/3) + 0.7 * (0^2/2) + 0 + C
Since the integral of a constant is zero, we can ignore the last term.
12. Calculate the producer's surplus:
Producer's Surplus = Revenue at equilibrium quantity - Revenue at lower limit
Producer's Surplus = (0.01 * (27.57^3/3) + 0.7 * (27.57^2/2) + 27.57) - (0.01 * (0^3/3) + 0.7 * (0^2/2) + 0)
Once you plug the values into the equation, the producer's surplus will be approximately $934.57, rounded to the nearest dollar.