$(20$ points$)$ 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 $p_i^t,$ where $p_i^t$ is a whole number in $0,1,2,3,4,5,6.$ The market

demand is a quantity $Q^t=6-p^t,$ where $p^t$ is the lower of $p_1^t$ and $p_2^t.$

If firm i names a strictly lower price than firm $j,$ firm i supplies all

of $Q^t$ on its own, and firm $j$ supplies nothing. If the firms name the

same price, they each supply $\frac{Q^t}{2}.$ Let $q_i^t$ 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 $i\text{'}$s profit is $p_i^t{q}_i^t-\mathrm{.}01q_i^t.{?}^1$ Suppose each firm has a

discount factor $in(0,1)$ and seeks to maximize the discounted sum of

its profits.

$($a$)$ Consider the stage game, such that competition occurs only one

time and then interaction ends. Show that the Nash equilibrium in

the stage game is that each firm names a price $p_i=1.$ How many

units are provided to the market as a whole in this equilibrium?

$($b$)$ Back to the infinitely repeated interaction: consider strategies

such that each firm names a price $p_i^t=2$ starting in period $1$

and every subsequent period, provided the other firm has always

chosen $p_j^t=2.$ If ever $p_j^{t}2,$ then firm i states a price of $1$

starting in period $t+1$ and forever after. Explain how this is a

grim trigger strategy.

$($c$)$ For what values of $$ is the grim trigger strategy in the previous

sub$-$part a SPNE of the repeated game?

$($d$)$ How many units are provided to the market as a whole in each

period in the SPNE identified in the previous sub$-$part? $(20$ points$)$ 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.

$($a$)$ Consider the stage game, such that competition occurs only one

time and then interaction ends. Show that the Nash equilibrium in

the stage game is that each firm names a price pi $=1.$ How many

units are provided to the market as a whole in this equilibrium?

$($b$)$ 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 $=2,$ then firm i states a price of $1$

starting in period t $+1$ and forever after. Explain how this is a

grim trigger strategy.

$($c$)$ For what values of $\backslash$delta is the grim trigger strategy in the previous

sub$-$part a SPNE of the repeated game?

$($d$)$ How many units are provided to the market as a whole in each

period in the SPNE identified in the previous sub$-$part?