The demand function for a certain brand of CD is given by
p = −0.01x2 − 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 equil
To determine the producer's surplus at the equilibrium price, we first need to find the equilibrium price and quantity. The equilibrium occurs when the quantity demanded equals the quantity supplied. We can find this by equating the demand and supply functions:
-0.01x^2 - 0.2x + 12 = 0.01x^2 + 0.5x + 3
Let's solve this equation to find the equilibrium quantity, x:
-0.01x^2 - 0.2x + 12 - 0.01x^2 - 0.5x - 3 = 0
Combining like terms, we have:
-0.02x^2 - 0.7x + 9 = 0
To solve this quadratic equation, we can either use factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the equation -0.02x^2 - 0.7x + 9 = 0, we have:
a = -0.02
b = -0.7
c = 9
Substituting these values into the 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 yields two possible values for x:
1) x = (0.7 + 1.1) / (-0.04) ≈ -0.1 / (-0.04) ≈ 2.5 (non-negative, so valid)
2) x = (0.7 - 1.1) / (-0.04) ≈ -0.4 / (-0.04) ≈ 10 (not realistic, so ignore)
Therefore, the equilibrium quantity is x = 2.5.
To find the equilibrium price, substitute this quantity back into either the demand or supply function. Let's use the demand function:
p = -0.01x^2 - 0.2x + 12
p = -0.01(2.5)^2 - 0.2(2.5) + 12
p = -0.01(6.25) - 0.5 + 12
p = -0.0625 - 0.5 + 12
p ≈ 11.44
Thus, the equilibrium price is approximately $11.44 per unit.
Now that we have the equilibrium price and quantity, we can calculate the producer's surplus. The producer's surplus represents the difference between the market price and the minimum price the producers are willing to accept.
Let's find the minimum price the producers are willing to accept. This can be found by evaluating the supply function at the equilibrium quantity:
p = 0.01x^2 + 0.5x + 3
p = 0.01(2.5)^2 + 0.5(2.5) + 3
p = 0.01(6.25) + 1.25 + 3
p = 0.0625 + 1.25 + 3
p ≈ 4.31
Therefore, the minimum price the producers are willing to accept is approximately $4.31 per unit.
To calculate the producer's surplus, subtract the minimum price from the equilibrium price and multiply by the equilibrium quantity:
Producer's Surplus = (Equilibrium Price - Minimum Price) * Equilibrium Quantity
= ($11.44 - $4.31) * 2.5
= $7.13 * 2.5
= $17.83
Hence, the producer's surplus in this scenario is approximately $17.83.