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.)
Is this a trick question? I thought the equilibrium price is where supply = demand. That is, there is zero surplus.
To find the producers' surplus when the market price is set at the equilibrium price, we need to first find the equilibrium price and quantity. The equilibrium occurs when the quantity demanded equals the quantity supplied.
Step 1: Set the demand function equal to the supply function and solve for x:
-0.01x^2 - 0.2x + 12 = 0.01x^2 + 0.5x + 3
Combine like terms:
0.02x^2 + 0.7x - 9 = 0
Step 2: Solve the quadratic equation. We can do this by factoring, completing the square, or using the quadratic formula. In this case, we'll use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
where a = 0.02, b = 0.7, and c = -9. Plugging in these values, 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
We have two possible values for x:
x1 = (-0.7 + 1.1) / 0.04 = 10
x2 = (-0.7 - 1.1) / 0.04 = -45
Since x represents quantity and cannot be negative in this context, we discard x2 = -45.
Step 3: Calculate the equilibrium price using either the demand or supply function. Let's use the supply function:
p = 0.01x^2 + 0.5x + 3
Substituting x = 10 into the supply function:
p = 0.01(10^2) + 0.5(10) + 3
p = 1 + 5 + 3
p = 9
The equilibrium price is $9.
Step 4: Calculate the producers' surplus. Producers' surplus is the difference between the total amount received by producers and the cost of production.
To calculate the producers' surplus, we need to find the area between the supply curve and the equilibrium price. Since the supply function is a quadratic equation, we can calculate the area using integration:
Producers' surplus = ∫[a, b] (Supply function - Equilibrium price) dx
where [a, b] is the range of x values that corresponds to the quantity supplied.
Substituting the values into the supply function:
Producers' surplus = ∫[0, 10] (0.01x^2 + 0.5x + 3 - 9) dx
Producers' surplus = ∫[0, 10] (0.01x^2 + 0.5x - 6) dx
Evaluating the integral:
Producers' surplus = [0.0033x^3 + 0.25x^2 - 6x]0 to 10
Producers' surplus = (0.0033(10^3) + 0.25(10^2) - 6(10)) - (0.0033(0^3) + 0.25(0^2) - 6(0))
Producers' surplus = (0.0033(1000) + 0.25(100) - 60) - (0)
Producers' surplus = (3.3 + 25 - 60) - 0
Producers' surplus = 3.3 + 25 - 60
Producers' surplus = -31.7
Round the answer to the nearest dollar:
Producers' surplus ≈ -32 dollars
Since the value is negative, it means that the producers are incurring a loss of $32 when the market price is set at the equilibrium price.