Problem $7$: Collusive Monopsony $(40$ points$).$ Suppose there are two teams in the NBA $($team $1$ and team $2).$ Each team has a monopoly in their product market with a demand curve given by:

$P(q)=270-2q$

Each team's production function is given by:

$q(l)=l$

The aggregate supply of labor is given by:

$w(L)=30+2L$

where $L=l_1+l_2$ is the total number of players in the market.

$(15$ points in this part$)$ First, assume the teams compete by simultaneously choosing the number of players.

a$)$ Find team l$\text{'}$s best response function. $(6$ points$)$

b$)$ Find the number of players $(l)$ hired by each team $(3$ points$).$ Find the wage level $w(3$ points$).$

c$)$ Find each team's profits ${}_n(3$ points$).$

$(15$ points in this part$)$ Next, assume the teams can collude to form a monopsony.

a$)(8$ points$)$ Find the number of players hired in the market $(L).$ Find the wage level $w.$

b$)(3$ points$)$ Find the number of players hired by each team $(l.$ Find each team's profits $({}_c).$

c$)(4$ points$)$ Imagine that team $1$ deviates from the collusive agreement while team $2$ keeps choosing the collusive amount of labor. Given team $2\text{'}$s choice of labor, how many players should team $1$ hire to maximize profits? And what is the corresponding level of profit $({}_d)?$

$(10$ points in this part$)$ Now, consider the infinitely repeated version of this game.

a$)$ Fill in the blanks to complete the grim trigger strategy. You don't need to provide explanations for your answers. $(6$ points; each blank carries $1$ point$)$

$"$In the first period, choose $l=$ In all future periods, choose $l=$ unless any team has chosen any other amount in any previous period, in which case choose $l=$