PLEASE EXPLAIN THIS SUM STEP BY STEP ON PAPER AND PEN.... $[$Exclusive Contracts$]$

Consider a market with an incumbent firm, a potential entrant and a buyer with demand

D$($p$)=120-$ p:

Suppose the potential entrant is more efficient than the incumbent. The entrant has a unit

production cost of $$10$ whereas the incumbent has a cost of $$20$ per unit. However, the entrant

has to pay an entry cost of F $=500.$ Before entry, the incumbent can propose the buyer to sign

an exclusive contract in exchange of a payment t to the buyer. An exclusive contract prevents the

buyer to purchase from the entrant. The timing is as follows:

$1.$ Incumbent offers buyer exclusive contract with payment t

$2.$ B accepts or rejects

$3.$ E decides to enter or not

$4.$ active firms name price to B

$5.$ B chooses supplier $($respecting exclusive contract if in place$)$

a$)$ Suppose no exclusive contract has been signed. Derive the equilibrium prices, profits and

consumer surplus if $($i$)$ the entrant enters and $($ii$)$ the entrant does not enter.

b$)$ Give the minimum amount $^$t the incumbent has to pay to induce the buyer to accept the

exclusive contract. Give the maximum amount $-$t the incumbent is willing to pay for an exclusive

contract. Compare $^$t and $\_$t and explain the overall outcome of the game.

Suppose now there are two identical buyers with the same demand as above. At stage $1,$ the

incumbent offers payments t$1$ and t$2$ for buyer $1$ and buyer $2,$ respectively, in exchange for an

exclusive contract. At stage $2,$ the buyers simultaneously decide whether to accept or reject the

offer. Then stages $3-5$ follow as above. Note that at stage $4,$ each active firm is able to name a

different price for each buyer. Finally, suppose that the entry cost of the entrant is now F $=1500.$

c$)$ Derive the equilibrium for stages $3-5$ as a function of whether $($i$)$ both buyers sign exclusive

contracts, $($ii$)$ exactly one buyer signs and $($iii$)$ no buyer signs.

d$)$ Derive the equilibrium of the buyer acceptance game at stage $2$ as a function of the values $($t$1$;

t$2).$

e$)$ In case there are multiple equilibria in your answer to d$),$ assume that buyers can coordinate

on the Pareto dominant equilibrium. Derive the optimal values for $($t$1$; t$2).$ Do the buyers sign

exclusive contracts in equilibrium? Does entry occur or not? Compare your results to b$)$ and

explain briefly