Suppose we have a model of a consumer choosing between labor and consumption. Their per-period utility...

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Suppose we have a model of a consumer choosing between labor and consumption. Their per-period utility function over consumption and labor is given by U(C, N) = C1/4 N3/2 1/4 3/2 Now when they spend any income on consumption the government taxes that at an additional rate T so that total consumption expenditures are (1+T)C. Agents are paid a wage w for each unit of labor N provided in factor markets. (Note I've made the utility function a function of labor and not leisure, it doesn't actually change anything about the problem so don't panic). They have zero savings and no income from other sources. They have a total endowment of time of 1 (so N cannot be larger than 1) (a) (5 pts) Write out the budget constraint of the household (b) (5 pts) Set up the Langrangian for the household take First Order Conditions (c) (5 pts) Combine your first order conditions for the intratemporal optimality condition for the household. Intuitively based just on the optimality condition how would the tax impact their choices of labor. (Do not solve for labor supply, explain in the context of the optimality condition) (d) (5 pts) Solve for the labor supply function as a function of the wage and tax rate T. (e) (5 pts) For the above labor supply function is the wealth or substitution effect dominant with respect to the wage? How do you know? (Hint: You can take the derivative with respect to the wage) Suppose we have a model of a consumer choosing between labor and consumption. Their per-period utility function over consumption and labor is given by U(C, N) = C1/4 N3/2 1/4 3/2 Now when they spend any income on consumption the government taxes that at an additional rate T so that total consumption expenditures are (1+T)C. Agents are paid a wage w for each unit of labor N provided in factor markets. (Note I've made the utility function a function of labor and not leisure, it doesn't actually change anything about the problem so don't panic). They have zero savings and no income from other sources. They have a total endowment of time of 1 (so N cannot be larger than 1) (a) (5 pts) Write out the budget constraint of the household (b) (5 pts) Set up the Langrangian for the household take First Order Conditions (c) (5 pts) Combine your first order conditions for the intratemporal optimality condition for the household. Intuitively based just on the optimality condition how would the tax impact their choices of labor. (Do not solve for labor supply, explain in the context of the optimality condition) (d) (5 pts) Solve for the labor supply function as a function of the wage and tax rate T. (e) (5 pts) For the above labor supply function is the wealth or substitution effect dominant with respect to the wage? How do you know? (Hint: You can take the derivative with respect to the wage)