The demand function for a certain brand of blank digital camcorder tapes is given by
p=−0.02x^2−0.1x+24
where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Determine the consumers' surplus if the wholesale unit price is 21 dollars per tape.
To determine the consumers' surplus, we need to compute the area of the triangle below the demand curve and above the price level of $21 per tape.
First, we need to find the quantity demanded (x) when the wholesale unit price (p) is $21. We can substitute the value of p into the demand function and solve for x:
21 = -0.02x^2 - 0.1x + 24
Rearranging the equation, we get:
0.02x^2 + 0.1x - 3 = 0
Next, we can solve this quadratic equation using the quadratic formula:
x = (-0.1 ± sqrt(0.1^2 - 4 * 0.02 * -3)) / (2 * 0.02)
x = (-0.1 ± sqrt(0.01 + 0.24)) / 0.04
x = (-0.1 ± sqrt(0.25)) / 0.04
x = (-0.1 ± 0.5) / 0.04
This gives us two possible values for x:
x1 = (-0.1 + 0.5) / 0.04 = 10 thousand units
x2 = (-0.1 - 0.5) / 0.04 = -15 thousand units
Since we can't have a negative quantity demanded, we discard the negative value and conclude that x = 10 thousand units.
Now that we know the quantity demanded (x), we can calculate the consumers' surplus. The consumers' surplus represents the difference between what consumers are willing to pay for a product and what they actually pay. In this case, it is the area of the triangle formed by the demand curve and the price of $21 per tape.
To calculate the area of the triangle, we use the formula for the area of a triangle: 1/2 * base * height.
The base of the triangle is the quantity demanded (x), which is 10 thousand units. The height of the triangle is the difference between the maximum price consumers are willing to pay (which we can find by evaluating the demand function at x=0) and the actual price of $21 per tape.
We substitute x=0 into the demand function:
p = -0.02(0)^2 - 0.1(0) + 24
p = 24
So, the maximum price consumers are willing to pay is $24 per tape.
The height of the triangle is 24 - 21 = 3 dollars per tape.
Now we can calculate the consumers' surplus:
Consumers' Surplus = 1/2 * base * height
= 1/2 * (10 thousand units) * (3 dollars per tape)
= 15 thousand dollars
Therefore, the consumers' surplus is $15,000.