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 consumer's surplus, we need to find the area under the demand curve up to the price of $21 per tape. The consumer's surplus represents the difference between what consumers are willing to pay for a product and what they actually pay.
The demand function equation provided is:
p = -0.02x^2 - 0.1x + 24
To find the quantity demanded at a wholesale unit price of $21, we substitute this price into the equation and solve for x:
21 = -0.02x^2 - 0.1x + 24
We now have a quadratic equation. Rearranging the equation to standard form:
0.02x^2 + 0.1x - 3 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the values:
a = 0.02, b = 0.1, c = -3
x = (-0.1 ± √(0.1^2 - 4(0.02)(-3))) / (2 * 0.02)
Simplifying further, we get two possible solutions:
x = (-0.1 ± √(0.01 + 0.24)) / 0.04
x = (-0.1 ± √0.25) / 0.04
Now calculate both solutions:
x₁ = (-0.1 + 0.5) / 0.04 = 11.25
x₂ = (-0.1 - 0.5) / 0.04 = -15
Since x represents the quantity demanded, a negative value (-15) does not make sense in this context. Thus, we consider x = 11.25 as the quantity demanded when the wholesale unit price is $21.
To calculate the consumer's surplus, we need to integrate the demand function from 0 to 11.25 (as we are considering quantity measured in units of a thousand):
Consumer's surplus = ∫[0,11.25] (p - 21) dx
Integrating the demand function:
∫[-0.02x^2 - 0.1x + 24 - 21] dx
∫[-0.02x^2 - 0.1x + 3] dx
The integral of -0.02x^2 with respect to x gives you -0.01x^3, the integral of -0.1x is -0.05x and the integral of 3 is 3x.
Plugging these values back into the consumer's surplus equation:
-0.01x^3 - 0.05x + 3x | [0,11.25]
Evaluating the integral with the upper and lower limits:
[-0.01(11.25)^3 - 0.05(11.25) + 3(11.25)] - [-0.01(0)^3 - 0.05(0) + 3(0)]
Simplifying further:
[-0.01(1419.14) - 0.5625 + 33.75] - [0]
[-14.1914 - 0.5625 + 33.75]
= 18.9961
Therefore, the consumer's surplus when the wholesale unit price is $21 per tape is approximately $18.9961.