Consider an economy where the representative consumer has a utility function u(c, ?) over consumption c and leisure ?. Assume preferences satisfy the standard properties we assumed in class. The consumer has an endowment of one unit of time. The representative firm has a production technology given by y = zf(¯k, n), where ¯k is the fixed capital input and n is labor input. Suppose that the government levies a proportional tax on labor income ? , where 0 < ? < 1. The revenues from the tax on labor income are rebated lump-sum to the households. Let d and t denote the lump-sum dividend from firms and transfers from the government, respectively, that the representative consumer receives. So the consumerâs budget constraint is: c (1 ? ? )w(1 ? ?) + d + t
1. Use the Lagrangian to derive an equation that implicitly defines the optimal labor supply n of the household as a function of (w, ?, d, t).
2. Define the competitive equilibrium of this economy.
3. Show that the competitive equilibrium is not Pareto optimal.
Consider an economy where the representative consumer has a utility function u(c, ?) over consumption c and leisure ?. Assume preferences satisfy the standard properties we assumed in class. The consumer has an endowment of one unit of time. The representative firm has a production technology given by y = zf(¯k, n), where ¯k is the fixed capital input and n is labor input. Suppose that the government levies a proportional tax on labor income ? , where 0 < ? < 1. The revenues from the tax on labor income are rebated lump-sum to the households. Let d and t denote the lump-sum dividend from firms and transfers from the government, respectively, that the representative consumer receives. So the consumerâs budget constraint is: c (1 ? ? )w(1 ? ?) + d + t
1. Use the Lagrangian to derive an equation that implicitly defines the optimal labor supply n of the household as a function of (w, ?, d, t).
2. Define the competitive equilibrium of this economy.
3. Show that the competitive equilibrium is not Pareto optimal.
