$($i$)$ The Dominant Firm Model explains the functioning of the crude oil market in the presence of a cartel such as OPEC.$$ What are the assumptions underlying this model?

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$($ii$)$ Using a diagram, show the equilibrium price and quantity that would be obtained under the dominant firm model.$$ Clearly indicate the global demand curve for oil, as well as the demand curve facing the dominant firm.$$ How much would be supplied by the dominant firm and the competitive fringe, respectively?

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$($iii$)$ With the aid of two separate diagrams, show how this model could yield a global oil market that is

a$.$ a monopoly $($i$.$e$.,$ only the dominant firm operates in the oil market$),$ and

b$.$ perfect competition $($i$.$e$.,$ only the competitive fringe operates in the oil market$).$

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If you think that either case $($a$)$ or case $($b$)$ is not possible under the Dominant Firm Model, explain why you think so$.$