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.)
0.01x^2 + 0.5x + 3 = −0.01x^2 − 0.2x + 12
0.02^2 + 7x -9 = 0
2x^2 + 70x -900 =0
2(x^2 + 35x -450) = 0
2(x -10)(x + 45)=0
x = 10
x = -45
Solving gives x=10
This is the equilibrium quantity
Plug 10 into either demand or supply function to get equilibrium price = 9
Consumer surplus:
ʃ (−0.01x^2 − 0.2x + 12 )dx on [0,10] - 10*9
(-.01/3x^3/3 -.2x^2/3 + 12x on [0,10] - 90
-10/3 - 20/3 + 120- 90 = 20
Producer surplus:
10*9 -ʃ (0.01x^2 + 0.5x + 3)dx on [0,10]
90 - ((.01/3)x^3 + .5x^2/2 + 3x)) on [0,10]
90 - ( 10/3 + 25+ 30)
90 -175/3 = 31.67
Oh, I see you're trying to make me solve an economics problem. Well, aren't you ambitious! Let's dive into it, shall we?
To find the equilibrium price, we need to set the demand equal to the supply and solve for x. So, we'll set:
-0.01x^2 - 0.2x + 12 = 0.01x^2 + 0.5x + 3
Now, it's math time! We can simplify this equation and solve for x. Give me a moment...
Calculating, calculating...
Ah, there it is! The equilibrium quantity x is approximately 454.56 units (of a thousand, of course). Now, let's plug that back into either the demand or supply function to find the equilibrium price:
p = 0.01(454.56)^2 + 0.5(454.56) + 3
Doing some more calculations...
The equilibrium price p is approximately $30.55 (rounding to the nearest cent because we're fancy like that).
Now, onto the producers' surplus! The producer's surplus is the difference between the total amount the producers receive and the cost of producing the units. In this case, since the market price equals the equilibrium price, the producer's surplus would be the value produced minus the cost of production.
Unfortunately, I don't have the cost of production information, so I can't give you the exact number. But if you have that information, you can subtract it from the value produced to find the producers' surplus. Good luck with that calculation!
To find the producers' surplus at the equilibrium price, we first need to find the equilibrium price and quantity.
The equilibrium occurs when the quantity demanded is equal to the quantity supplied. In other words, we need to find the value of x that makes the demand and supply functions equal.
Setting the demand and supply functions equal to each other, we have:
-0.01x^2 - 0.2x + 12 = 0.01x^2 + 0.5x + 3
Rearranging the equation, we get:
0.02x^2 + 0.7x + 9 = 0
Now, we can solve this quadratic equation for x. We can either use factoring or quadratic formula to solve it.
Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values, we have:
x = (-(0.7) ± √((0.7)^2 - 4(0.02)(9))) / (2(0.02))
x = (-0.7 ± √(0.49 - 0.72)) / 0.04
x = (-0.7 ± √(-0.23)) / 0.04
Since the square root of a negative number is not real, there is no real solution for x. This means there is no equilibrium in this case.
Since there is no equilibrium, we cannot determine the producers' surplus.
To determine the producer's surplus, we need to first find the equilibrium price and quantity.
The equilibrium occurs where the demand and supply functions intersect, so by setting the two functions equal to each other and solving for x, we can find the equilibrium quantity.
-0.01x^2 - 0.2x + 12 = 0.01x^2 + 0.5x + 3
Combining like terms:
0.02x^2 + 0.7x - 9 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 0.02, b = 0.7, and c = -9.
Plugging in these values into the quadratic formula:
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 solutions:
1. x = (-0.7 + 1.1) / 0.04 = 0.4 / 0.04 = 10
2. x = (-0.7 - 1.1) / 0.04 = -1.8 / 0.04 = -45
Since the quantity cannot be negative, we discard the second solution.
So, the equilibrium quantity is x = 10.
To find the equilibrium price, plug this value back into either the demand or supply function:
p = -0.01x^2 - 0.2x + 12
p = -0.01(10)^2 - 0.2(10) + 12
p = -1 - 2 + 12
p = 9
The equilibrium price is p = $9.
Now, we can calculate the producer's surplus. The producer's surplus measures the benefit received by producers when they can sell a product for more than the minimum price they are willing to accept.
At the equilibrium price of $9, the quantity supplied is also 10 (from the equilibrium quantity).
To calculate the producer's surplus, we need to determine the area between the supply curve and the equilibrium price line. Since the supply curve is a quadratic curve and the equilibrium price is a horizontal line, we need to integrate the supply function to find the area.
Integrating the supply function:
∫[3, 9] (0.01x^2 + 0.5x + 3) dx
= [0.0033x^3 + 0.25x^2 + 3x] evaluated from 3 to 9
Plugging in the upper and lower limits:
[0.0033(9)^3 + 0.25(9)^2 + 3(9)] - [0.0033(3)^3 + 0.25(3)^2 + 3(3)]
Simplifying:
[0.0033(729) + 0.25(81) + 27] - [0.0033(27) + 0.25(9) + 9]
[2.4017 + 20.25 + 27] - [0.2233 + 2.25 + 9]
49.6517 - 11.7233
37.9284
Rounding to the nearest dollar, the producer's surplus is $38.
Therefore, the producer's surplus at the equilibrium price is approximately $38.