Problem $3.$ A coffee shop manager realizes that demand for coffee is more elastic among students, and is trying to determine the optimal pricing schedule. Specifically, she estimates the following average demands:

$-$ Student: $\backslash ($ q$^\{$S$\}=18-5$ p $\backslash )$;

$-$ Non$-$student: $\backslash ($ q$^\{$N$\}=10-2$ p $\backslash ).$

The two groups visit the coffee shop in equal numbers on average. Assume that the marginal cost for coffee is $\backslash (\backslash$$ $2\backslash )$ and there is no fixed cost.

$1.$ If the coffee shop cannot distinguish whether customers are students or not, what is the optimal uniform price $\backslash ($ p$^\{$U$\}\backslash )?(5$ points$)$

$2.$ Assume that students can buy a cup of coffee with a discounted price $\backslash ($ p$^\{$S$\}\backslash )$ by showing their student ID card and non$-$students can buy it with a normal price $\backslash ($ p$^\{$N$\}\backslash ).$ What price should be set for each group? $(5$ points$)$

$3.$ How has price discrimination affected the coffee shop's profit? $(5$ points$)$

$4.$ Calculate consumer surplus under uniform pricing and with price discrimination. How does price discrimination affect consumer surplus? $(5$ points$)$