The US has an absurdly complicated tax and transfer system, but economists sometimes use the following...

 

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The US has an absurdly complicated tax and transfer system, but economists sometimes use the following setup as an excellent approximation. Consider a household earning pre-tax income y>0. The taxes paid by the household are given by the nonlinear function T(y) = y-oy¹- where >0 and ≤ 1 are parameters. 1. Can T(y) <0 for some values of y? Provide an interpretation for negative taxes. 2. Two measures of the tax burden on an individual are • the average tax rate, total taxes divided by pre-tax income • the marginal tax rate is the rate of change of the tax bill with respect to pre-tax income. A tax system is said to be progressive if the marginal tax rate is increasing in pre-tax income. (a) Calculate the average and marginal tax rates paid by a household with pre-tax income y. (b) Suppose 0<< 1. Is the tax system progressive or regressive? (c) Suppose<0. Is the tax system progressive or regressive? 3. Suppose a household's only source of income is labor income. Let the aggregate price level be normalized to 1. The household solves the problem log C. subject to wN - T(wN) = C N² 2 where the function T(y) was defined above. (a) Rewrite the budget constraint by substituting for pre-tax income wN into the tax function defined above. (b) Set up the Lagrangian for this problem, using the notation A for the Lagrange Multiplier on the budget constraint. (c) What are the first order conditions for the households problem? Combine the first order conditions to eliminate the Lagrange multiplier A and obtain one equation relating C, N,w. (d) What is the elasticity of labor supply N to changes in the wage rate w, holding consumption C fixed? Give some intuition for why it has the sign it does. (e) Suppose 0 < < 1 initially, and a presidential candidate proposes to raise it to > such that < 1. What would happen to the elasticity of labor supply that you calculated in the previous part? The US has an absurdly complicated tax and transfer system, but economists sometimes use the following setup as an excellent approximation. Consider a household earning pre-tax income y>0. The taxes paid by the household are given by the nonlinear function T(y) = y-oy¹- where >0 and ≤ 1 are parameters. 1. Can T(y) <0 for some values of y? Provide an interpretation for negative taxes. 2. Two measures of the tax burden on an individual are • the average tax rate, total taxes divided by pre-tax income • the marginal tax rate is the rate of change of the tax bill with respect to pre-tax income. A tax system is said to be progressive if the marginal tax rate is increasing in pre-tax income. (a) Calculate the average and marginal tax rates paid by a household with pre-tax income y. (b) Suppose 0<< 1. Is the tax system progressive or regressive? (c) Suppose<0. Is the tax system progressive or regressive? 3. Suppose a household's only source of income is labor income. Let the aggregate price level be normalized to 1. The household solves the problem log C. subject to wN - T(wN) = C N² 2 where the function T(y) was defined above. (a) Rewrite the budget constraint by substituting for pre-tax income wN into the tax function defined above. (b) Set up the Lagrangian for this problem, using the notation A for the Lagrange Multiplier on the budget constraint. (c) What are the first order conditions for the households problem? Combine the first order conditions to eliminate the Lagrange multiplier A and obtain one equation relating C, N,w. (d) What is the elasticity of labor supply N to changes in the wage rate w, holding consumption C fixed? Give some intuition for why it has the sign it does. (e) Suppose 0 < < 1 initially, and a presidential candidate proposes to raise it to > such that < 1. What would happen to the elasticity of labor supply that you calculated in the previous part?

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