Answer
(a) The market is in equilibrium when the quantity is 3800 units.
(b) The maximum total surplus is $\$324,900$
Work Step by Step
We can find the quantity $X$ when the market is in equilibrium:
$228.4-18X = 27X+57.4$
$45X = 171$
$X = 3.8$
The market is in equilibrium when the quantity is 3800 units.
(b) We can find the price $P$:
$p(x) = 228.4-18(3.8) = \$160$
We can find the consumer surplus:
$\int_{0}^{X}[p(x)-P]~dx$
$=\int_{0}^{3.8}[(228.4-18x)-160]~dx$
$=\int_{0}^{3.8}(68.4+18x)~dx$
$=68.4x-9x^2~\vert_{0}^{3.8}$
$=68.4(3.8)-9(3.8)^2~-0$
$= 129.96$
We can find the producer surplus:
$\int_{0}^{X}[P-p_s(x)]~dx$
$=\int_{0}^{3.8}[160- (27x+57.4)]~dx$
$=\int_{0}^{3.8}(102.6-27x)~dx$
$=102.6x-13.5x^2~\vert_{0}^{3.8}$
$=102.6(3.8)-13.5(3.8)^2~-0$
$= 194.94$
The total surplus is $129.96+194.94$ which is $324.9$
Since $x$ is in thousands, the maximum total surplus is $\$324,900$
