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Consider a representative household in the static consumption-leisure model with prefer- ences given by u(c,l) = Ac1/2 +11/2, where A is an exogenous parameter that determines the relative amount of utility gained from consumption over leisure. The household has a unit time endowment given by 1 =1+n, faces a price of P on consumption, and earns nominal wages at rate W on their labor supply, n. In addition, the household faces a proportional tax on wage income of T. (a) Using a Lagrangian, derive the first order conditions for c and 1. (b) Use the first order conditions to derive the consumption-leisure optimality condition. (c) Solve for the consumption demand function and the labor supply function. (d) Congress often proposes lowering the tax rate on wages as an incentive for house- holds to increase their supply of labor. Use comparative static analysis to determine whether the labor supply function you have derived in part (c) lends support to this.
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