Tax Laffer Curve $(25$ points$)$

Consider a one$-$period closed economy, i$.$e$.$ the consumer lives for one period. He$/$She supplies labor $0$ N $1$ and demands consumption goods C $0.$ The incomes come from wage $($wage rate is w$)$ and firms$$ dividend p $0.$ We will take p as exogenous and given. The government needs to finance an exogenous spending via distorting labor tax: t$.$ That is to say, for each dollar of wage income the consumer earns, he$/$she needs to pay t fraction to the government. Suppose the preference is : u$($C$,$ N$)=$ Cg$(1$ N$)1$g$.$ Answer the following questions.

a Basedontheabovedescription,writeouttherepresentativeconsumer$$soptimizationprob$-$ lem. $(5$ points$)$

b Solve the consumer$$s consumption$-$labor choice problem, taking prices and government policies as given. Derive the function of decisions: C$($w$,$ t$,$ g$),$ N$($w$,$ t$,$ g$).(5$ points$)$

c Suppose the government needs to increase the tax rate. Based on the above solutions you get, determine how that affects the equilibrium labor supply N$($w$,$ t $",$ g$).$ What does the result depend on$?(5$ points$)$

d Usingappropriategraphstoillustratetheeffectsofthistaxratechangeonconsumption$/$labor choice. Explain it with income and substitution effects. $(5$ points$)$

e As the labor tax rate t increases gradually from $0$ to $1,$ do you always see a Laffer curve of tax revenue? Discuss the issue case$-$by$-$case if you need to$.(5$ points$)$