Georgetown University - Macroeconomics II

Spring 2010 Th 5:40 - 8:30 ICC 115
Office hours:
  • every Th before/after class
  • every other Friday (I will  be around in the dept all day) starting from Feb 12th
  • by appointment
Some useful stuff in reviewing your final exam:

1) Victor Rios Rull class notes

2) Optimal Taxation in models of Endogenous Growth;
Jones, Manuelli, Rossi, Journal of Political Economy, Vol 101 N. 3, p. 485-517, 1993

January 19th 2010

Today we introduced the types of questions that we want to  be able to answer with the model/s that we develop, especially in the first part of the course. Then we defined an Arrow-Debreu equilibrium and discussed each agent's decision problem. 
We also develop a quite general argument to show under some assumptions that feasibility constraints, and budget constraints are always binding at a solution to agents' maximization problems.

January 26th 2010

Today we proved 2 results that we are going to use for the proof of the second welfare theorem:
  1. constraints on investment in the households problem "don't matter" in equilibrium
  2. we gave a social welfare characterization of Pareto Optimal allocations

January 28th 2010

Today we proved the second welfare theorem: we made some assumptions on differentiability of utility and production functions in order to tackle a slightly different version of the proof from the standard one you may have seen in your micro class: the idea is to compare the equations that characterize the Pareto optimal allocation with the equations that characterize a competitive equilibrium and therefore construct prices and a redistribution of endowments accordingly. The right prices are going to take care of agents wanting to choose what we want them to optimally choose (the Pareto Optimal allocation) in their maximization problems, and the redistribution of endowments is going to take care of agents being able to afford what we want them to afford.

February 2nd 2010

Today we defined a sequential market environment for a simple 2 period, 2 agents, no production 1 good economy. We defined a sequential market equilibrium, discussing the role for a constraint on borrowing in the households' problem. We argued that provided that the borrowing limit is large enough an allocation that is part of an Arrow Debreu equilibrium is also part of a Sequential Market Equilibrium (appropriately defining prices and loan amounts). We argued that the converse is also true.

February 4th 2010

Today we discussed aggregation: we talked about two methods 
  1. everyone is identical 
  2. we allow for hh endowment to differ
and showed how we can reconcile the competitive equilibrium framework we have been dealing with so far with the one sector neoclassical growth model.
Then we introduced the language and notation we are going to use to study dynamic programming for macro, along the lines of SLP (Stokey, Lucas, Prescott). We set up the sequential problem and the functional equation problem and discussed how we can rewrite several problems in canonical form. We also stated the fundamental theorems of dynamic programming (the principle of optimality) and explained what it means for a function to solve the sequential problem and what it means for a function to solve the functional equation problem. Next time we are going to prove these theorems.

February 18th 2010

Today we proved in details theorems 4.2 and 4.3 in Stokey, Lucas, Prescott. We discussed the relationship between the sequential problem and the functional equation problem and we talked about what goes wrong in a finite horizon. Could we write the equivalent of theorems 4.2 and 4.3 for a finite horizon economy?

February 25th 2010

Today we discussed some details of the proof of theorem 4.3 in SLP, in particular we thought about the difference between the cases when the value function v(x) = - or + infinity. Then we proved in details theorems 4.4 and 4.5 in Stokey, Lucas, Prescott. And we setp up the logic behind theorem 4.6 in SLP, explaining the routine that allows us to find the value function.

March 04th 2010

Today we discussed in depth the proof of theorem 4.6 in SLP, then we turned to proving in detail theorems 4.7, 4.8 in SLP.
We stressed how crucial it is to make assumptions in those theorems that preserve the completeness of the metric space we are working with (both in the case of weakly vs strictly increasing continuous and bounded functions and in the case of  weakly vs strictly concave continuous and bounded functions). 
We also started proving theorem 4.9, and for that we introduce the theorem and the lemma in chapter 3 of SLP that the proof relies on. We proved the lemma, next time we'll continue with the theorem in chapter 3 and then with theorem 4.9.

March 18th 2010

Today, after the midterm exam, we finished the proofs of theorem 4.9, and of the theorem it uses from chapter 3.
We pointed out what arguments we have learned from these proofs that we could repeat in applications.

March 25th 2010

Today, ...

April 8th 2010

Today we studied some results about the Ramsey approach to the theory of optimal taxation: we first derived results about:
  • redundancy of some taxes (relative to allocations)
  • steady state characterization  
Then we derived the implementability constraint and talked about a characterization proposition (between allocations that satisfy resource and implementability constraints and a TDCE) and why it is relevant.

Then we discussed the optimal taxation problem as in the original Ramsey  set up, and moved to study in detail the primal approach to the Ramsey problem. 
We derived results on long term optimal capital and labor income taxes

April 15th 2010

Today we studied some basic concepts of measure theory and introduced some basic tools useful for stochastic dynamic programming. Then we defined a complete markets environment, with uncertainty on endowment streams of households, and showed equivalence between an Arrow Debreu and a sequential market equilibrium. 
We also started to set up a recursive formulation for a planner's problem and introduced the idea of big K and little k.

April 26th 2010

Today we finished discussing the equivalence between Arrow Debreu and sequential market environments with uncertainty. Then we introuced economies with capital and studied in detail a planning problem (a stochastic one sector growth model) and its decentralization.
We stressed the concept of recursive competitive equilibrium and how/when consistency conditions are imposed to solve for an equilibrium.
We also spent some time explaining how to use the policy functions, that solve the agents' problem together with value functions, when checking consistency conditions and when characterizing the equilibrium (looking at marginal rate of substitutions for instance).

April29 th 2010

Today ...

some useful stuff in studying for the comps:

-link to UMN prelims archive

syllabus: see below

Problems set 1: see below

Problems set 2: see below

Problems set 3: see below

Problem set 4: see below

Problem set 5: see below

Problem set 6: see below

Problem set 7: see below

Problem set 8: see below

Problem set 9: see below

Problem set 10: see below

francesca carapella,
Jan 20, 2010, 2:47 PM
francesca carapella,
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francesca carapella,
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francesca carapella,
Jan 27, 2010, 9:50 AM
francesca carapella,
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francesca carapella,
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