Ben and Jerry consumes a private good, X$,$ with a price of $$1($Px $=1),$ and a public good,

fireworks, with a price of $$1($PF $=1).$ They each have an income of $$100.$ Because fireworks are

a public good, the total amount provided is the sum of the amount provided by each individual:

F $=$ FB $+$ FJ$.$ Each individual $($i$)$ has a utility function of the form: U $=2\backslash$times log$($Xi$)+$ log$($FB $+$ FJ$),$

which he maximizes subject to the budget constraint: Xi $+$ Fi $=100.$

$($a$)$ Determine the private market equilibrium.

$($b$)$ What is the socially optimal? Explain your answer.

$($c$)$ If Ben has $$125$ income and Jerry earns $$75,$ then calculate the quantities for each individuals

and explain the outcome changes $($if any$)$ with the help of your answer.

$($d$)$ If government decides to solve private sector under provision by fixing supply at $26\mathrm{.}6,$ then

will this solve the problem? Explain with the help of your answer.