Question: Consider an economy with the representative consumer, representative firm, and government. Suppose that G = 12. The consumer's preferences over the consumption good and leisure are given by the utility function U(C, l) = Cl. Let h = 24. The firm's production function is Y = zN, where z = 2. Suppose that the government imposes a lump-sum tax on the representative consumer.
(a) Write down the social planner's maximization problem and the social planner's optimality condition. Derive the Pareto optimal allocation for consumption, output, leisure, and labor.
(b) In this economy the welfare theorems apply, and hence the Pareto optimal allocation can be supported as a competitive equilibrium. Write the definition of competitive equilibrium. Find the real wage w that supports the Pareto Optimal allocation you found in part (b) as a competitive equilibrium.
(c) Suppose that the total factor productivity z increases from 2 to 3. How do equilibrium consumption, labor, output, and the wage rate change? What does your answer imply regarding whether the income or the substitution effect dominates for labor?