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 = ?
To find the consumer's surplus and the producer's surplus, we first need to find the equilibrium price and quantity.
Equilibrium occurs when the quantity demanded is equal to the quantity supplied. Therefore, we will equate the two equations for p:
-0.2x^2 + 220 = 0.1x^2 + 8x + 110.
By rearranging and simplifying this equation, we get:
0.3x^2 + 8x - 110 = 0.
Now, we can solve this quadratic equation for x using factoring, completing the square, or the quadratic formula. For simplicity, we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a),
where a = 0.3, b = 8, and c = -110.
Plugging in these values, we get:
x = (-8 ± √(8^2 - 4*0.3*(-110))) / (2*0.3).
Simplifying further, we have:
x = (-8 ± √(64 + 132)) / 0.6.
x = (-8 ± √196) / 0.6.
x = (-8 ± 14) / 0.6.
This gives us two possible values for x:
x = (6/0.6) or x = (-22/0.6).
Therefore, x = 10 or x = -36.67 (rounded to two decimal places).
Since the quantity cannot be negative, we discard the negative value, and we have x = 10.
Now, we can find the equilibrium price by substituting this value of x back into either of the equations:
p = 0.1x^2 + 8x +110.
p = 0.1(10^2) + 8(10) + 110.
p = 1 + 80 + 110.
p = 191.
The equilibrium price is $191.
To find the consumer's surplus, we need to calculate the area of the triangle above the equilibrium price and below the demand curve.
The equation for the demand curve is p = -0.2x^2 + 220.
Substituting the equilibrium price into this equation, we get:
191 = -0.2x^2 + 220.
0.2x^2 = 29.
x^2 = 145.
x ≈ 12.04.
Since x represents the quantity in hundreds, the approximate quantity demanded is 1200 units.
To find the consumer's surplus, we calculate the area of the triangle:
Consumer's surplus = (1/2) * (p_max - p_eq) * (x_eq - x_min)
= (1/2) * (220 - 191) * (12.04 - 10)
= (1/2) * (29) * (2.04)
≈ $29.82.
Therefore, the consumer's surplus is approximately $29.82.
To find the producer's surplus, we need to calculate the area of the triangle below the equilibrium price and above the supply curve.
The equation for the supply curve is p = 0.1x^2 + 8x + 110.
Substituting the equilibrium price into this equation, we get:
191 = 0.1x^2 + 8x + 110.
0.1x^2 + 8x = 81.
x^2 + 80x = 810.
Solving this quadratic equation, we find:
x ≈ -24.69 or x ≈ 14.69.
Since the quantity cannot be negative, we discard the negative value, and we have x ≈ 14.69.
The approximate quantity supplied is 1469 units.
To find the producer's surplus, we calculate the area of the triangle:
Producer's surplus = (1/2) * (p_eq - p_min) * (x_max - x_eq)
= (1/2) * (191 - 110) * (14.69 - 10)
≈ (1/2) * (81) * (4.69)
≈ $190.24.
Therefore, the producer's surplus is approximately $190.24.
To find the consumer's surplus and the producer's surplus, we first need to find the equilibrium price and quantity.
Equilibrium occurs when the quantity demanded is equal to the quantity supplied:
-0.2x^2 + 220 = 0.1x^2 + 8x + 110
Combining like terms:
0.3x^2 - 8x + 110 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
where a = 0.3, b = -8, and c = 110.
Calculating the discriminant (b^2 - 4ac):
(8^2) - 4(0.3)(110) = 64 - 132 = -68
Since the discriminant is negative, we have no real solutions for x. This means there is no equilibrium price and quantity for this market, and thus, we cannot calculate the consumer's surplus and the producer's surplus.
Therefore, the consumer's surplus and producer's surplus cannot be determined.