We want to repeat our analysis of optimal menu pricing $($Lecture $11,$ p$\mathrm{.}31)$

when there are three types of a buyer, $\backslash$theta $1<\backslash$theta $2<\backslash$theta $3$ with $\backslash$beta i $=$ Pr$(\backslash$theta $=\backslash$theta i$)$ in $(0,1).$

To simplify the problem, we assume that q$3>=$ q$2>=$ q$1$ and the buyer of type $\backslash$theta $1$

$($resp$.\backslash$theta $3)$ cannot pick $($q$3,$ T$3)($resp$.($q$1,$ T$1)).$

$(1)$ Write down the seller$$s problem.

$(2)$ Show that we can safely ignore the participation constraints for the buyer

of type $\backslash$theta $2$ and $\backslash$theta $3.$

$(3)$ As in the case of two types, the important incentive constraints are that

a higher type i shouldn$$t pick $($qi$1,$ Ti$1),$ i $=2,3.$ Given this, rewrite

the seller$$s problem and express the optimal prices Ti as a function of

quantities qj $$s$.$

$(4)$ Discuss your finding in terms of full surplus extraction, information rent,

or distortion of quantities.