Intermediate Macroeconomics Problem set-
1. Properties of Utility and Production Functions
Consider a representative consumer with the following utility function:
U(C, l) = log C + γ log l
1. Find the marginal utility of consumption and leisure.
2. Show that they are positive. Give an intuitive explanation for your result in no more than 3 lines.
3. Show that they are decreasing. Give an intuitive explanation for your result in no more than 3 lines.
Now consider a firm with the following production function:
F(K, Nd) = zKα(Nd)β
Where 0 < α, β < 1.
1. Let α + β = 1. Show that the production function has constant returns to scale.
2. If α + β > 1, what kind of returns to scale does the production function present? And if α + β < 1? Prove your answers.
2. Competitive Equilibrium with Distortive Taxes
Consider a simple one-period, closed economy model where the representative consumer has utility function U(C, l) and h available hours to divide between work and leisure. The representative firm has technology given by Y = zF(K, Nd). There is a government that imposes a proportional tax τ on total income received by the consumer. The revenue of this tax finances the provision of public goods G.
1. Define a Competitive Equilibrium for this economy.
2. Find the optimal conditions for the CE, that is, characterize the CE.
3. Write down the Social Planner's problem associated with this economy.
4. Find the optimal conditions for the Social Planner's problem, that is, characterize the SPP.
5. Is the Competitive Equilibrium Pareto Efficient? Show it mathematically and explain in no more than 5 lines.
3. Competitive Equilibrium with Multiple Taxes
Consider a simple one-period, closed economy model where the representative consumer has utility function U(C, l) and h available hours to divide between work and leisure. The representative firm has technology given by zF(K, Nd). There is a government that imposes a proportional tax τN on labor income, a proportional tax τc on consumption, and a proportional tax τw on wages paid by the firm for each hour of work hired. Notice that the labor income and consumption taxes are paid by the consumer, and the tax on labor hired is paid by the firm. The revenue of this taxes finances the provision of public goods G.
1. Define a Competitive Equilibrium for this economy.
2. In words, explain if the Competitive Equilibrium of this economy is Pareto Optimal.