Question

The demand function for a certain brand of CD is given by the following equation where is the wholesale price in dollars and x is the quantity demanded each week, mesured in units of a thousand p=-0.01 -0.22+19 Determine the consumers' surplus if the wholesale market price is set at 51/disc. Round your answer to two decimal places) $ 1,860,000 00 X Need Help?

Answer 1

Given, demand function for a certain brand of CD is
p = -0.01x^2 - 0.2x + 49
where p is wholesale unit price in dollars
x is quantity in demand each week, in units of thousands

Now if the wholesale market price is set at $1/disc
so, for 1000 disc, wholesale price will be $1000
hence, for x disc (in unit of thousands), wholesale price will be 1000x
hence,

supply function will be P = 1000x

so using p = P
-0.01x^2 - 0.2x + 49 = 1000x
0.01x^2 + 1000.2x - 49 = 0
solving we get
x = 0.0489901779656 (in units of thousand)
at this point,
p = P = $ 48.990

also, for x = 0, p = $ 49
then consumer surplus is CS = integral[p*dx] from x = 0 to x 0.0489901779656 - 0.5*0.0489901779656 *48.990
CS = integral[(-0.01x^2 - 0.2x + 49 )*dx] from x = 0 to x 0.0489901779656 - 1.200014409267372
CS = -0.01(0.0489901779656)^3/3 - 0.1(0.0489901779656)^2 + 49*0.0489901779656 - 1.200014409267372
CS = 3.6002927339
now as x is in units of thousands
CS = $ 3600.23