Determine the consumers' surplus (in dollars) if the market price is set at $9/disc. (Round your answer to two decimal places.)

Question

Consumers' Surplus, The demand function for a certain brand of CD is given by

p = −0.01x² − 0.1x + 39

where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand.

Answer 1

Note that if one sets the price at $9/disc, the amount sold can be found by solving

9 = -0.01x²-0.1x+39

This has two solutions: x = -60 which we must discard as it is a negative quantity and x=50 which is the solution we require.

The consumer surplus stems from the fact that while consumers are willing to pay $9 for acquiring 50 units, they would have paid more to acquire the first unit, more to acquire the second unit, ... , more to acquire the 49th unit. Instead they got to purchase each of these units for $9. For instance, how much would consumer pay for just one unit?

p(1)= -0.01*1^2-0.1*1+39 =38.89

For 2, 3, ... ,49 units:

p(2) = -0.01*3^2-0.1*2+39 =38.76

p(3) = -0.01*3^2-0.1*3+39 =38.61

..............

p(49) = -0.01*49^2-0.1*49+39 =10.09

Now, you can find the surplus by taking the difference between each of these numbers and 9 and summing up these numbers. Technically, this is the right way to do it because we cannot really have fractional units. Doing it by hand is a chore, but if you use any software you could find

Consumers' surplus = (38.89-9)+(38.76-9)+(38.61-9) + .... + (10.09-9) = 943.25.

The other way is to use Calculus, although this assumes that we can buy any amount, even fractional amounts.

∫₀⁵⁰ ( -0.01x²-0.1x+39 -9) dx ≈958.33

This integral is a bit higher than the sum above for we are considering the problem via continuous quantities. Personally, I believe the first (discrete) solution is the correct one. The second one is an approximation. Needless to say, all these numbers are in $ units.

I hope this helps.