The quantity demanded x (in units of a hundred) of the Mikado miniature cameras per week is related to the unit price p (in dollars) by
p = −0.2x^2 + 220
and the quantity x (in units of a hundred) that the supplier is willing to make available in the market is related to the unit price p (in dollars) by
p = 0.1x^2 + 8x + 110.
If the market price is set at the equilibrium price, find the consumers' surplus and the producers' surplus. (Round your answers to the nearest dollar.)
consumer's surplus = ?
producer's surplus = ?
0.1x^2 + 8x + 110 = −0.2x^2 + 220
.3x^2 + 8x -110 = 0
3x^2 + 80x -1100 = 0
(3x + 110)(x-10) =0
x = -110/3
x = 10
This is the equilibrium quantity
Plug 10 into either demand or supply function to get equilibrium price = 200
Consumer surplus:
ʃ (-.2x^2+220)dx on [0,10] - 10*200
(-.2/3)x^3 + 220x on [0,10] - 2000
-200/3+ 2200- 2000 = 133.33
Producer surplus:
10*200 - ʃ (0.1x^2 + 2x + 110 )dx on [0,10]
2000 - (.1/3)x^3 + 4x^2 + 110x)) on [0,10]
2000 - (100/3 + 400 + 1100)
2000-4600/3 = 466.67
To find the equilibrium price, we need to set the quantity demanded equal to the quantity supplied. In other words, we need to equate the two equations.
-0.2x^2 + 220 = 0.1x^2 + 8x + 110
Let's solve this equation to find the equilibrium quantity (x) and price (p).
First, bring all terms to one side:
0.1x^2 + 8x + 110 - (-0.2x^2 + 220) = 0
0.3x^2 + 8x - 110 = 0
Next, we can solve this quadratic equation by factoring or using the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 0.3, b = 8, and c = -110.
x = (-8 ± √(8^2 - 4(0.3)(-110))) / (2(0.3))
Simplifying:
x = (-8 ± √(64 + 132)) / 0.6
x = (-8 ± √196) / 0.6
x = (-8 ± 14) / 0.6
There are two possible solutions:
1) When x = (-8 + 14) / 0.6 = 1.67 (rounded to two decimal places)
2) When x = (-8 - 14) / 0.6 = -37.67 (rounded to two decimal places)
Since the quantity cannot be negative in this context, we discard the second solution.
So, the equilibrium quantity is x = 1.67 hundred units (or 167 units) per week.
To find the equilibrium price (p), we substitute this value of x back into either of the equations:
p = -0.2(1.67^2) + 220
p = -0.2(2.7889) + 220
p = -0.5578 + 220
p = 219.4422
The equilibrium price is approximately $219.44.
Now, let's calculate the consumer's surplus and producer's surplus.
Consumer's surplus:
The consumer's surplus is the area between the demand curve and the price line up to the equilibrium quantity.
We integrate the demand curve equation from 0 to the equilibrium quantity (1.67):
Consumer's surplus = ∫[(−0.2x^2 + 220) - 219.44] dx, integrated from 0 to 1.67
Consumer's surplus = ∫[−0.2x^2 + 0.56] dx, integrated from 0 to 1.67
Consumer's surplus = [-0.2(1.67)^3/3 + 0.56(1.67)] - [-0.2(0)^3/3 + 0.56(0)]
Consumer's surplus = [-0.2(4.9423)/3 + 0.56(1.67)] - [0]
Consumer's surplus ≈ [-0.3283 + 0.9372] - [0]
Consumer's surplus ≈ 0.6089
Therefore, the consumer's surplus is approximately $0.61.
Producer's surplus:
The producer's surplus is the area between the supply curve and the price line up to the equilibrium quantity.
We integrate the supply curve equation from 0 to the equilibrium quantity (1.67):
Producer's surplus = ∫[219.44 - (0.1x^2 + 8x + 110)] dx, integrated from 0 to 1.67
Producer's surplus = ∫[219.44 - 0.1x^2 - 8x - 110] dx, integrated from 0 to 1.67
Producer's surplus = ∫[−0.1x^2 - 8x + 109.44] dx, integrated from 0 to 1.67
Producer's surplus = [−0.1(1.67)^3/3 - 8(1.67)^2/2 + 109.44(1.67)] - [−0.1(0)^3/3 - 8(0)^2/2 + 109.44(0)]
Producer's surplus = [−0.1(4.9423)/3 - 8(4.9423)/2 + 109.44(1.67)] - [0]
Producer's surplus ≈ [−0.1641 - 19.8184 + 182.8948] - [0]
Producer's surplus ≈ 162.9123
Therefore, the producer's surplus is approximately $162.91.
To find the consumers' surplus and the producers' surplus, we first need to find the equilibrium price and quantity.
The equilibrium price occurs when the quantity demanded equals the quantity supplied.
So, setting the two equations equal to each other, we get:
-0.2x^2 + 220 = 0.1x^2 + 8x + 110
Combining like terms, we have:
-0.2x^2 - 0.1x^2 - 8x + 220 - 110 = 0
Simplifying,
-0.3x^2 - 8x + 110 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = -0.3, b = -8, and c = 110. Plugging in these values gives us:
x = (-(-8) ± √((-8)^2 - 4(-0.3)(110))) / (2(-0.3))
Simplifying further,
x = (8 ± √(64 + 132)) / (-0.6)
x = (8 ± √196) / (-0.6)
Now, calculating the square root,
x = (8 ± 14) / (-0.6)
This gives us two possible values for x:
x1 = (8 + 14) / (-0.6) ≈ -37.33
x2 = (8 - 14) / (-0.6) ≈ 10
Since the quantity cannot be negative, we discard x1 ≈ -37.33 and choose x2 ≈ 10 as the equilibrium quantity.
Now, we can substitute this equilibrium quantity into either equation to find the equilibrium price.
Using p = 0.1x^2 + 8x + 110:
p = 0.1(10^2) + 8(10) + 110
= 0.1(100) + 80 + 110
= 10 + 80 + 110
= 200
So, the equilibrium price is $200.
To find the consumers' surplus, we need to calculate the area under the demand curve up to the equilibrium price. The demand curve equation is: p = -0.2x^2 + 220.
To find the area, we integrate the demand curve equation from x = 0 to x = 10:
Consumers' surplus = ∫[0 to 10] (-0.2x^2 + 220) dx
Integrating term by term,
= [-0.2(1/3)x^3 + 220x] [0 to 10]
= [-0.2(1/3)(10)^3 + 220(10)] - [-0.2(1/3)(0)^3 + 220(0)]
= [-6,666.67 + 2,200] - [0]
= -4,466.67
However, we are asked to round the answer to the nearest dollar. So, rounding -4,466.67 to the nearest dollar, the consumers' surplus is approximately -$4,467 (taking the negative value into account).
To find the producers' surplus, we need to calculate the area above the supply curve up to the equilibrium price. The supply curve equation is: p = 0.1x^2 + 8x + 110.
To find the area, we integrate the supply curve equation from x = 0 to x = 10:
Producers' surplus = ∫[0 to 10] (0.1x^2 + 8x + 110) dx
Integrating term by term,
= [0.1(1/3)x^3 + 4x^2 + 110x] [0 to 10]
= [0.1(1/3)(10)^3 + 4(10)^2 + 110(10)] - [0.1(1/3)(0)^3 + 4(0)^2 + 110(0)]
= [3,333.33 + 400 + 1,100] - [0]
= 4,833.33
Rounding 4,833.33 to the nearest dollar, the producers' surplus is approximately $4,833.