The quantity demanded x (in units of a hundred) of the Mikado miniature cameras/week is related to the unit price p (in dollars) by
p = −0.2x2 + 80
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 = 0.1x2 + 2.5x + 60
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.)
consumers' surplus $ ?
producers' surplus $ ?
To find the consumers' surplus and producers' surplus, we first need to find the equilibrium price and quantity. At equilibrium, the quantity demanded by consumers equals the quantity supplied by producers.
Step 1: Set the quantity demanded equal to the quantity supplied:
-0.2x^2 + 80 = 0.1x^2 + 2.5x + 60
Step 2: Rearrange the equation to solve for x:
-0.2x^2 - 0.1x^2 - 2.5x + 80 - 60 = 0
-0.3x^2 - 2.5x + 20 = 0
Step 3: Solve the quadratic equation. You can use the quadratic formula or factorization methods:
Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)
For this equation: a = -0.3, b = -2.5, c = 20
x = (-(-2.5) ± √((-2.5)^2 - 4(-0.3)(20))) / (2(-0.3))
Simplifying further, x = (2.5 ± √(6.25 + 24)) / (-0.6)
x = (2.5 ± √(30.25)) / (-0.6)
x = (2.5 ± 5.5) / (-0.6)
x = -13 / 3 or x = 6.33
Since the quantity cannot be negative, the feasible solution is x ≈ 6.33.
Step 4: Calculate the equilibrium price by substituting the value of x into either of the equations:
Using the first equation, p = -0.2(6.33)^2 + 80 ≈ 68.448
Therefore, at the equilibrium price of approximately $68.45, the quantity demanded and supplied is approximately 6.33 hundred units of Mikado miniature cameras.
Step 5: Calculate the consumers' surplus:
To find the consumers' surplus, we need to calculate the area under the demand curve but above the market price (p = $68.45). The formula for consumers' surplus is:
Consumers' Surplus = 0.5 * (Quantity Demanded at Equilibrium - Quantity Demanded at Market Price) * (Equilibrium Price - Market Price)
Consumers' Surplus = 0.5 * (6.33 - 6.33) * (68.45 - 0) = 0
Therefore, the consumers' surplus is $0.
Step 6: Calculate the producers' surplus:
To find the producers' surplus, we need to calculate the area above the market price but under the supply curve. The formula for producers' surplus is:
Producers' Surplus = 0.5 * (Equilibrium Price - Market Price) * (Quantity Supplied at Equilibrium - Quantity Supplied at Market Price)
First, we need to find the quantity supplied at the market price. Substitute the market price p = $68.45 into the supply equation:
68.45 = 0.1x^2 + 2.5x + 60
Solve the quadratic equation to find x:
0.1x^2 + 2.5x - 8.45 = 0
Using the quadratic formula or factoring, you can find that x ≈ 3.18.
Producers' Surplus = 0.5 * (68.45 - 0) * (6.33 - 3.18) = 0.5 * 68.45 * 3.15 ≈ $107.80
Therefore, the producers' surplus is approximately $107.80.
To find the equilibrium price, we need to find the price at which the quantity demanded is equal to the quantity supplied.
Equating the quantity demanded and quantity supplied equations gives us:
-0.2x^2 + 80 = 0.1x^2 + 2.5x + 60
Combining like terms and simplifying:
-0.3x^2 - 2.5x + 20 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this equation, a = -0.3, b = -2.5, and c = 20.
Plugging in these values:
x = (-(-2.5) ± sqrt((-2.5)^2 - 4(-0.3)(20))) / (2(-0.3))
Simplifying further:
x = (2.5 ± sqrt(6.25 + 24)) / (-0.6)
x = (2.5 ± sqrt(30.25)) / (-0.6)
Now, we need to find the positive value for x since we are dealing with quantities. Therefore, we will take the positive root:
x = (2.5 + sqrt(30.25)) / (-0.6)
x ≈ 4.23
Now, we can plug this value of x back into either the quantity demanded or supplied equation to find the equilibrium price.
Using the supplier's equation:
p = 0.1x^2 + 2.5x + 60
p = 0.1(4.23)^2 + 2.5(4.23) + 60
p ≈ $72.12
So, the equilibrium price is approximately $72.12.
To find the consumers' surplus, we need to calculate the area between the demand curve and the equilibrium price, while the producers' surplus is the area between the supply curve and the equilibrium price.
Consumers' surplus:
To find the consumers' surplus, we need to integrate the demand curve equation from 0 to the equilibrium quantity (4.23).
Consumers' surplus = ∫[0 to 4.23] (-0.2x^2 + 80) dx
Consumers' surplus = -0.2 * ∫[0 to 4.23] x^2 dx + 80 * ∫[0 to 4.23] dx
Using the power rule of integration and evaluating the limits:
Consumers' surplus = -0.2 * [x^3/3] [0 to 4.23] + 80 * [x] [0 to 4.23]
Consumers' surplus = (-0.2 * (4.23^3 / 3)) - (-0.2 * (0^3 / 3)) + (80 * 4.23) - (80 * 0)
Consumers' surplus ≈ $43.82
Producers' surplus:
To find the producers' surplus, we need to integrate the supply curve equation from 0 to the equilibrium quantity (4.23).
Producers' surplus = ∫[0 to 4.23] (0.1x2 + 2.5x + 60) dx
Producers' surplus = 0.1 * ∫[0 to 4.23] x^2 dx + 2.5 * ∫[0 to 4.23] x dx + 60 * ∫[0 to 4.23] dx
Using the power rule of integration and evaluating the limits:
Producers' surplus = 0.1 * [x^3/3] [0 to 4.23] + 2.5 * [x^2/2] [0 to 4.23] + 60 * [x] [0 to 4.23]
Producers' surplus = (0.1 * (4.23^3 / 3)) - (0.1 * (0^3 / 3)) + (2.5 * (4.23^2 / 2)) - (2.5 * (0^2 / 2)) + (60 * 4.23) - (60 * 0)
Producers' surplus ≈ $156.73
Therefore, the consumers' surplus is approximately $43.82 and the producers' surplus is approximately $156.73.