$4.(25$ points$)$ Vertical and horizontal differentiation. Consider a Hotelling duopoly

with two firms located at the extremes of a segment of length $1.$ Specifically, firm

$1$ is located at point $0$ and firm $2$ is located at point $1.$ A mass $1$ of consumers are

uniformly distributed over the line. A consumer located at point x in $[0,1]$ derives

utility u$1=$ v$1$ p$1$ tx if s$/$he buys from firm $1$ and u$2=$ v$2$ p$2$ t$(1$ x$)$ is

s$/$he buys from firm $2,$ where p$1$ and p$2$ are the prices charged by firm $1$ and firm

$2,$ respectively, v$1$ and v$2$ are the qualities of the goods provided by firm $1$ and firm

$2,$ respectively, and t $0$ is the per$-$unit transport cost. Firms produce with zero

marginal costs. For each firm i $=1,2,$ the quality of the good can just take two

values, vi in $\{$vL$,$ vH$\},$ with vH $>$ vL $>0,$ and let $$v $=$ vH $$ vL in $(0,$ t$).$ Each

firm initially invests in the quality of its good. Investment is binary and is denoted

Ii in $\{0,1\}$ for each firm i $=1,2.$ Investment costs firm i cIi

$,$ with c in $(0,$v$)$ and

vi $=$ vH only if Ii $=1.$ That is$,$ producing a high$-$quality good requires a costly

investment, whereas producing a low$-$quality good does not. The timing of the game

is as follows.

$1.$ Firms simultaneously and non$-$cooperatively invest in the quality of their goods.

$2.$ Firms observe the quality of the goods and choose their prices simultaneously

and non$-$cooperatively.

$3.$ Consumers observe $($p$1,$ v$1)$ and $($p$2.$v$2)$ and choose where to buy the good. Payoffs

realize.

Assume that the market is covered irrespective of the quality provided by the firms.

$($a$)$ Find the demand that each firm faces as a function of the quality of the goods

and prices.

$3$

$($b$)$ For given qualities $($q$1,$ q$2),$ what is the equilibrium in prices of the subgame

starting at stage $2?$

$($c$)$ Determine under what condition neither firm invests in quality and under what

condition both firms invest.