Problem $1.$ Bargaining in a supply chain.

There are two firms in a supply chain: an Upstream firm sells inputs to a Downstream firm, and the Downstream

firm uses the inputs to produce a final good that it sells to consumers. Both firms are monopolists in their

respective market. The Upstream firm chooses a unit price w $0$ for the inputs it sells, and it has a per$-$unit

production cost of $4.$ The Downstream firm produces a quantity q $0$ of the final good, and it faces the inverse

demand curve p$($q$)=100$ q$.$ The Downstream firm has a production technology that turns $5$ units of input

into $1$ unit of output, i$.$e$.,$ its production cost of q units of output is $5$wq$.$ The firms$$ profits are, therefore,

u$1($w$,$ q$)=($w $4)5$q$,$ u$2($w$,$ q$)=(100$ q $5$w$)$q$.$

In absence of any agreement between the two firms, we predict the subgame$-$perfect equilibrium of the

dynamic game in which the Upstream firm chooses w and the Downstream firm observes w and chooses q$.$

The goal of this problem is to see if the firms can improve their profits through Nash bargaining.

a$.$ Find the unique subgame$-$perfect equilibrium and the firms$$ equilibrium profits.

b$.$ What is the set of alternatives the two firms$$ bargain over? Characterize the efficient frontier.

Hint: The sum of the firms$$ profits does not depend on w if the Upstream firm produces to meet the demand

of the Downstream firm. Thus, you can find the efficient frontier by maximizing total profits over q $0.$

c$.$ Find the Nash bargaining solution if the disagreement point d are the firms$$ SPE profits from $($a$).$ What

price w of inputs and quantity q of outputs should the firms agree on$?$

d$.$ Which firm has higher profits in the Nash bargaining solution? Explain intuitively.

e$.$ Are consumers better off with or without the Nash bargaining agreement of $($c$)?$ Explain intuitivel