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using the consumption and leisure model to solve the question below: Consider a representative household in the static consumption-leisure model with a gen- eral utility function uſc,l) that exhibits positive and diminishing marginal utility in each argument. The household faces a price of P on consumption and earns nominal wages at rate W on their labor, n which is the portion of a unitary time endowment not spent on leisure so that 1 = n +1. In addition, the household faces a proportional sales tax on consumption at rate Te and a proportional tax on wage income of Tn. (a) Write down the budget constraint for the representative household. (b) Express the consumption-leisure optimality condition in terms of u(c,l). (c) Using indifference curve analysis, graphically show the optimal choice (c*,7*) given the budget constraint from part (a). Label slopes and intercepts on your graph. (d) Suppose under present law that the federal wage tax rate is Tn = .2, while the federal consumption tax rate is Te = 0. Congress wants to lower the wage income tax rate to Tn = .15, but does not want to alter the aggregate optimal choice of consumption and leisure (c*,7*). Use economic logic based on the households' optimality condition to explain how they can achieve this using the consumption tax. (e) Compute the consumption tax rate that achieves the scenario described in part (d). (f) Suppose Congress only lowers Tn without raising Te. Use your graph from part (c) to show how the optimal choice of aggregate consumption and leisure change, assuming that the substitution effect dominates the income effect on labor supply and consumption.
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