Consider the following interaction between a single long lived firm, and a sequence

$of$ short$-$lived consumer. $In$ each period $t=1,2$dotsB and $NH$ and $L.$ The payoffs $to$ the two parties are given $by$ the following table, where

$v_H>p>v_L$ and $p>c_H>c_L.$

$a,$ where Each consumer $at$

date $t$ observes the actions $of$ the firm and $of$ previous consumers $in$ all previous periods,

$1,2$dots,$t-1.$ Show that there exists a subgame perfect equilibrium where the firm provides

high quality, $ifis$ large enough. $(Hint$: construct a trigger strategy equilibrium$).$