Suppose two firms engage in price competition in a single market over an infinite sequence of periods. The firms produce identical goods that last for $1$ period. In each period t$,$ competition is as follows: The firms simultaneously announce prices for that period. Firm i names a price p t i $,$ where p t i is a whole number in $0,1,2,3,4,5,6.$ The market demand is a quantity Qt $=6$ p t $,$ where p t is the lower of p t $1$ and p t $2.$ If firm i names a strictly lower price than firm j$,$ firm i supplies all of Qt on its own, and firm j supplies nothing. If the firms name the same price, they each supply Qt $2.$ Let q t i denote the quantity that firm i supplies in period t$.$ Firm i incurs a cost of $\mathrm{.}01$ for each unit it produces. In each period, firm i$$s profit is p t i q t i $\mathrm{.}01$q t i $.1$ Suppose each firm has a discount factor $\backslash$delta in $(0,1)$ and seeks to maximize the discounted sum of its profits.

IMPORTANT: Consider that there is a price of $5$ be each firm every period starting in t$=1,$ and a price of $1$ for every subsequent period if the other firm ever names a price other than $5.$

$($a$)$ Back to the infinitely repeated interaction: consider strategies such that each firm names a price p t i $=2$ starting in period $1$ and every subsequent period, provided the other firm has always chosen p t j $=2.$ If ever p t j $6=2,$ then firm i states a price of $1$ starting in period t $+1$ and forever after. Explain how this is a grim trigger strategy.

$($b$)$ For what values of $\backslash$delta is the grim trigger strategy in the previous sub$-$part a SPNE of the repeated game?

$($c$)$ How many units are provided to the market as a whole in each period in the SPNE identified in the previous sub$-$part?