Q.(a) Find the consumer's surplus and producer's surplus for the demand function $P_d=25-3x$ and supply function $P_s=5+2x$.

(a) $x_0=4,\ p_0=13$: CS $=24$, PS $=16$. (b) $P(X\le20)=0.9772,\ P(0\le X\le12)=0.4987$.

(a) Consumer's and producer's surplus. $P_d=25-3x,\ P_s=5+2x$.

Equilibrium (set $P_d=P_s$): $25-3x=5+2x\Rightarrow 20=5x\Rightarrow x_0=4$, and $p_0=25-3(4)=13$.

Consumer's surplus:

$CS=\int_0^{x_0}P_d\,dx-x_0p_0=\int_0^4(25-3x)\,dx-4(13).$

$\int_0^4(25-3x)\,dx=\Big[25x-\frac{3x^{2}}{2}\Big]_0^4=100-24=76.$

$CS=76-52=24.$

Producer's surplus:

$PS=x_0p_0-\int_0^{x_0}P_s\,dx=52-\int_0^4(5+2x)\,dx.$

$\int_0^4(5+2x)\,dx=[5x+x^{2}]_0^4=20+16=36\ \Rightarrow\ PS=52-36=16.$

(b) $X\sim N(\mu=12,\ \sigma=4)$. Standardise with $Z=\dfrac{X-\mu}{\sigma}$. …