$1)(26$ points$)$ Suppose Quentin's preferences depends only on consumption $($c$)$ and leisure $(1).$ The market wage rate is w and Quentin's non$-$labor income is V$.$ The maximum time that Quentin can devote to work$/$leisure is T$.$

a$)$ Show the optimal point using a graphical illustration of the above problem. Show how the optimal point changes when the wage rate increases.

b$)$ Derive the labor supply curve for an individual worker, according to part $($a$).$

c$)$ Define labor supply elasticity.

d$)$ Suppose the government is planning to cut income taxes. Discuss how results$/$implications of this policy on total hours of work using the labor supply elasticity concept. When is it more likely to boost labor supply?

e$)$ Suppose you want to estimate the labor supply elasticity using a multiple regression framework. Write down the basic regression specification according to the theoretical model in question $(1).$

f$)$ Is it "safe" to use non$-$labor income as a control variable in this regression? What is the problem?