The demand function for a certain brand of DVD is given by
p = −0.01x^2 − 0.2x + 8
where p is the unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Determine the consumers’ surplus if the market price is set at $5 per disc.
$5 = −0.01x^2 − 0.2x + 8
0 = −0.01x^2 − 0.2x +3
or
0.01x^2 + 0.2x - 3 = 0
x = -30 or +10
so at a price of $5, people will buy 10,000 a week
To determine the consumers' surplus, we need to calculate the area under the demand curve up to the market price of $5. Consumers' surplus represents the difference between the total amount consumers are willing to pay for a good and the actual price they pay.
The first step is to find the quantity demanded (x) at a price of $5. We can do this by setting the price (p) to $5 in the demand equation and solving for x:
5 = -0.01x^2 - 0.2x + 8
This is a quadratic equation, so we can solve it by setting it equal to 0:
0 = -0.01x^2 - 0.2x + 3
Next, we can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values from the equation, we get:
x = (-(-0.2) ± √((-0.2)^2 - 4(-0.01)(3))) / (2(-0.01))
Simplifying further:
x = (0.2 ± √(0.04 + 0.12)) / (-0.02)
x = (0.2 ± √0.16) / (-0.02)
Now, we have two possible solutions:
x1 = (0.2 + 0.4) / (-0.02) = -0.6 / (-0.02) = 30
x2 = (0.2 - 0.4) / (-0.02) = -0.2 / (-0.02) = 10
Since we are measuring x in units of a thousand, x1 represents a demand of 30,000 units, and x2 represents a demand of 10,000 units.
To find the area under the demand curve up to a quantity of 10,000 units, we need to calculate the integral of the demand function from 0 to 10:
∫[0,10] (-0.01x^2 - 0.2x + 8) dx
To calculate the integral, we can use the power rule and the definite integral property:
= [-0.01(1/3)x^3 - 0.2(1/2)x^2 + 8x] from 0 to 10
= [-0.01(1/3)(10)^3 - 0.2(1/2)(10)^2 + 8(10)] - [-0.01(1/3)(0)^3 - 0.2(1/2)(0)^2 + 8(0)]
= [-0.01(1/3)(1000) - 0.2(1/2)(100) + 80] - [0]
= -3.33 - 10 + 80
= 66.67
Therefore, the consumers' surplus is $66.67.