Cap and trade policies set a cap on greenhouse gas emissions within a certain

industry $($say$,$ petrol$)$ and give companies in this industry the option to trade

allowances for greenhouse gasses with each other. When designing these policies,

regulators often find it useful to try to anticipate how companies will respond. In

this question, we'll walk through such an exercise.

Assume a group of six petrol companies has a collective cap of $30$ allowances. In the

network in Figure $2,$ the nodes are these six companies and an edge signifies that

two companies have a strong enough relationship to consider trading allowances.

The companies have mutually agreed to $1)$ follow the one$-$exchange rule $($i$.$e$.,$ each

node can be involved in an exchange with only one of its network neighbors$)$ and $2)$

allow each exchanging pair to split $10$ emission allowances. So far, the companies

have arrived at the following outcome.

a$.$ Show that the outcome in Figure $2$ is stable and balanced.

b$.$ Sunoco is unhappy with only having $4$ allowances. In order to negotiate a

better deal from Pemex, Sunoco wants to try developing a new relationship

with exactly one of the four other companies it's not already connected to$.$ If

successful, this would add one new non$-$exchanging edge to the network in

Figure $2($without adding any new nodes$).$ Intuitively $($without doing any

calculations$),$ what is one company Sunoco can form a relationship with that

would destabilize the current arrangement and give it the opportunity to

maximize its allowances? Explain.

c$.$ Assume that Sunoco is able to form the relationship you described in part b

and tries to use this to renegotiate its deal with Pemex. Draw the resulting

network and find a balanced outcome where only the split between Sunoco

and Pemex changes. All other exchanging pairs should have the same split as

they do in Figure $2.$ Explain why this new exchange is balanced.

d$.$ Does Sunoco get more in the outcome from part $c$ than in the original

arrangement?

e$.$ After what happened in part c$,$ Sunoco has gotten greedy and wants even

more allowances. Is it better for Sunoco to try to $($i$)$ create new relationships

with other companies $($i$.$e$.,$ add a new edge to the network in part c$,$ between

itself and one of three nodes it's not already connected to$)$ or $($ii$)$ disrupt

existing relationships between other companies $($i$.$e$.,$ remove an edge

between two other nodes, from the network in part c$)?$ If you said $($i$),$ what is

an example of a company Sunoco would want to create a new relationship

with? If you said $($ii$),$ what is a pair of companies whose relationship Sunoco

would want to disrupt? Offer an intuitive explanation of both answers in $1-3$

sentences.