Macroeconomic Theory I: Intertemporal Budget Constraints and Competitive Equilibria - Prof, Assignments of Introduction to Macroeconomics

Download Macroeconomic Theory I: Intertemporal Budget Constraints and Competitive Equilibria - Prof and more Assignments Introduction to Macroeconomics in PDF only on Docsity! ECON7020: MACROECONOMIC THEORY I Martin Boileau Problem Set 3 1. Consider the following closed economy. This economy is populated by a representative consumer that maximizes her utility max c1;c2 u(c1) + ¯u(c2) subject to her intertemporal budget constraint, where u(c) = ¡(1=®) exp(¡®c). She re- ceives a perishable (nonstorable) endowment of y1 in period 1 and y2 in period 2. a) Write the representative agent's intertemporal budget constraints. b) For what values of ® is the period utility increasing and concave? c) De¯ne a competitive equilibrium allocation and price system. d) Solve for a competitive equilibrium. e) What is the impact of an increase in ® on the real rate of interest? How about an increase in ¯? 2. Consider an environment with two consumers. The ¯rst consumer receives a perishable endowment of widget y in each period. She solves the following problems: max c1;c2 u(c1) + ¯u(c2) subject to her intertemporal budget constraint, where u(c) = ln(c). The second consumer receives a perishable endowment of widget y in each period. He solves the following problems: max d1;d2 v(d1) + ¯ ¤v(d2): subject to her intertemporal budget constraint, where v(d) = ln(d). a) Find the market clearing conditions. b) Solve for the competitive equilibrium. c) Show that trade between the two consumers is intertemporally balanced. Why? d) Show that the ¯rst consumer's consumption grows faster than the second consumer when ¯ > ¯¤. Why? 1