2. Labor Supply Practice - Quasi-Linear Utility (25 pts) Suppose that we have a standard model of consumption and leisure tradeoff model. Households have an endowment of one unit of time which they can divide between labor, N;, and leisure, l;. Households choose their labor supply for which they are paid an exogenous wage, w; for each unit of labor. Households enjoy consumption and dislike work but need to work to be able to consume. In addition, households must pay a proportional labor tax T" on all earnings. This means that they take home (1 T")w; for each unit of labor. They have zero savings and no income from other sources. Suppose that households have the following utility function U(C, lt) =24/Cr + s (a) (5 pts) Write out the household budget constrain (b) (5 pts) Write out the Lagrangian and take first order conditions. Combine the first order conditions to create the intratemporal optimality condition. What are our system of two equations that we can use to solve for labor supply (c) (5 pts) Explain intuitively how the tax impacts the intratemporal condition (i.e. explain how an increase impacts the marginal benefit or cost or the price) (d) (5 pts) Now suppose instead of a proportional income tax the government instead institutes a lump sum tax, Ts. This means they will just subtract a fixed amount from the household regardless of labor activity. Rewrite the budget constraint now with no income tax and instead using the lump sum tax (e) (5 pts) Finally rewrite the lagrangian and take first order conditions to create the intratemporal opti- mality condition. How has this changed relative to part (b), why