Luiss - MACROECONOMIS II (MOSEC)
Today we talked about the plan for the course, then we started defining an Arrow Debreu equilibrium with households, firms that produce consumption goods and firms that produce investment goods. We derived an expression for the households' budget constraint in equilibrium, that does not depend on the allocation of investment good: this expression is convenient, for us especially in the proof by construction of the 2nd welfare thm.
We defined the set of feasible allocations and utility possibilities set, we defined the pareto frontier (these are utility levels) and pareto optima (these are allocations).
We talked about the social welfare characterization of optimal allocations: we mentioned that using the supporting hyperplane thm (which is a version of one of the separating hyperplane thms -see Takayama Mathematical Economcs ) we can prove that a Pareto optimal allocation solves a maximization problem where the objective function is the weighted average of each household's utility and where the constraint set is just given by all of the feasibility constraints. And we mention how to prove the reverse: any allocation that solves the above problem is Pareto optimal.
Then we talked about the 1st welfare thm -you will be asked in the problem set to do a proof of it for the economy that we described today- and the 2nd welfare thm. We only mentioned how to prove it by constructing a system of prices and initial endowments that decentralizes the Pareto optimal allocation as an Arrow Debreu equilibarium. We will do this more formally next time.
Today we cleared up some algebra on the derivation of the budget constraint of the households' problem in equilibrium, and then we did the proof of the 2nd welfare thm in a constructive fashion:
we started from a Pareto optimal allocation and we compared the first order conditions from the Pareto optimal maximization problem with the first order conditions from the maximization problems in a competitive equilibrium (assuming interior solutions, so FOC hold at equality). Doing so we constructed prices for consumption and investment goods, interest rate and wages such that the FOC of the households' problem in a CE are consistent with the PO allocation that we started out with.
This gave us the right choices at the margin: that is to say the right choices to implement as a CE the PO allocation we are given, but these up to now are only marginal choices because they are based only on marginal rates of substitution comparisons. We still need to get the right levels in order to be sure that the PO allocation we are given get implemented as a CE. We do so by constructing new endowments of initial capital and labor (that is to say redistributing initial resources) so that all the constraints in the households' problem ad the firms' problem are satisfied. In particular we want to make sure that we give the households enough wealth to be able to afford the allocation we want to implement. So we want to redistribute initial endowments so that the allocation we want to implement satisfies the households' budget constraint in a CE. This is the idea we should keep in mind when we reverse engineer from the result we want (that the PO allocation satisfies all the necessary and sufficient -under strict concavity of the utility fcn- of a CE) back to the assumption we need (both in terms of prices and initial endowments) to obtain the result.
Then we also noted that in order to check that everything in the definition of a CE is satisfied with the construction of prices and the redistribution of initial endowments, we still need to make sure that firms make zero profits given the prices we constructed.
Today we defined a TDCE (Tax Distorted Competitive Equiliibrium), that is a CE where there is another agent, the government, who must raise some revenue to finance an exogenous stream of government expenditure. In the definition of TDCE the sequence of taxes (on commodities, on income - both labor and capital income- and lump sum) and government expenditures is exogenous, it is given. That is to say fixing a specific fiscal policy (exogenously given to us), a TDCE is basically a CE with the additional requirement that government budget must be balanced.
We discussed two different versions of a government budget balance condition:
In what we did so far we used the government budget balance condition in every period.
Then we gave an example of a specific TDCE: one in which the only taxes are income taxes (on labor and capital income) and the value of the proportional tax is the same. We showed that under this specific fiscal policy, we can rewrite the TDCE as a CE (therefore without government and without the gov't budget balance condition) with a modified production function and modified price system, therefore as a CE of an alternative economy. Then we noticed that in this alternative economy the first welfare theorem applies: this means that the CE of the alternative economy is PO for the modified production function, i.e. it solves a social planner' s problem associated with the modified production function.
Therefore combining all of the steps we concluded that the TDCE -which is unique under the assumptions of strict concavity of U, and convexity of the constraint sets, so concavity of F- solves the CE of the alternative economy -which under the same assumptions on U and F is unique as well- and in turn it solves the social planner's problem with the modified production function. Since the solutions to these problems are unique, then the unique TDCE is also the unique solution to the CE of the alternative economy and the unique solution to the social planner's problem with the modified production function.
We highlighted that this conclusion DOES NOT mean that the TDCE is PO (for the original production function): in fact in general it is not! That's why we call it Tax DISTORTED CE.
We started discussing again the logic of the example we studied last time, going from the TDCE to the CE of an alternative economy without taxes and to a social planner's problem whose resource constraint is modified according to how we rewrote the technologies of the CE in the alternative economy.
Then we did another example of a fiscal policy made only of consumption taxes: we showed that in order to rewrite this TDCE as the CE of an alternative economy we needed to have two sectors of production: one for consumption goods, one for investment goods.
Then we proceeded according to the same logic, emphasizing the constructive argument.
Today we talked about:
In particular as far as 1. is concerned we discussed how taxes affect steady state level of capital and output and compared the outcome of a tax distorted economy with the one from the one sector growth model. We concluded that the economy with taxes (assuming there exists a steady state) has a lower level of steady state capital ad output with respect to an economy without taxes (and we can do the same argument if we are comparing two different economies with different fiscal policies -one with high taxes one with low taxes-).
As far as 2. is concerned we derived all the conditions that a TDCE must satisfy and in particular we focused on two of them: an Euler equation (between t and t+1) and an intratemporal marginal rate of substitution between consumption and leisure. Then we evaluated them at steady state (still assuming a steady state exists). Since in each of these equations we see that the ratio between consumption and labor income taxes matter and the ratio between investment taxes and capital income taxes matter, then we concluded that our general tax problem is really telling us something about wedges rather than taxes themselves. So we rewrote the general tax problem including only labor and capital income taxes.
As far as 3. is concerned we combined all the conditions that a TDCE must satisfy and we noticed that we can rewrite the household's budget constraint substituting out prices from FOC so that the household's budget constraint is just expressed in terms of allocation! No prices or taxes show up: the only tax that shows up is the tax on the stock of capital at t=0, but we discussed how this is equivalent to lump sum taxation since capital at t=0 is not a choice, but it is given to the household's problem. We called this modified budget constraint the IMPLEMENTABILITY constraint, because it describes, together with the resource constraint, the set of allocations that are implementable as a competitive equilibrium (Arrow-Debreu eqm).
Today we set up the Ramsey problem and discussed it in detail. We know how to derive the implementability from the budget constraint of the households where we plug in the FOC of households and firms to substitute for prices.
We can relabel the new objective function of the Ramsey planner as the sum of U and the LHS of the implementability constraint, with its Lagrange multiplier. Therefore the Ramsey problem looks like a one sector growth model, just with a slightly different objective function.
We looked at the steady state solution to this problem (assuming it exists): we combined the steady state solution to this problem with the FOC from households and firms in a TDCE to learn something about what taxes should be in order to implemente the solution to the Ramsey problem as a competitive equilibrium (this is THE logic that you will always use to say something about taxes once you know that you want to implement a certain allocation -that from your point of view, the one of the Ramsey planner, is optimal).
We discussed the Chamley Judd result on zero capital income taxes in the long run, and we found out that labor income taxes are instead positive.
We also mentioned an issue related to multiplicity of competitive equilibria given a tax schedule. We introduced this issue defining a Ramsey equilibrium (as in Atkeson, Chari and Kehoe or Chari, Kehoe, ch.26 of Handbook of Macroeconomics) and we discussed how we would need to make a selection among the set of competitive equilibria associated with a certain tax system, in case of multiplicity. We select a CE that results in a certain Ramsey equilibrium. Let Ramsey allocation denote the allocation that is part of a Ramsey equilibrium. Since our selection in general is arbitrary, then it won't necessarily be the case that Ramsey allocation and the solution to the Ramsey problem coincide. If the CE for a given tax system is unique then they do, otherwise the Ramsey allocation always solves the Ramsey problem, but the reverse in not necessarily true.
Today we discussed the tax results induced by the solution to the Ramsey problem more in details:
We started talking about two examples from Jones, Manuelli and Rossi (JET 1997) which we will look at more in detail next time:
We also talked about an important issue concerning the Ramsey problem: the constraint set may not be convex (and in general is not convex) because of the implementability constraint. The literature that focuses on relying on the solution to the Ramsey problem to learn something about optimal wedges acknowledges this issue (see for instance Atkeson, Chari and Kehoe) but since their contribution is purely theoretical, they can assume that the stream of gov't expenditure that needs to be financed is arbitrarily small, so that the distortions induced by the implementability constraint into the new objective function of the Ramsey planner (what we called W in class) are small enough to preserve strict concavity of U. Therefore the Ramsey planner's problem would still be a well defined concave problem.
Today we talked about some examples where the long run tax on capital income is different from zero, pointing out that each of these examples is related to a specific feature of the economy that we want to think about. In this exercises the consequences of economy's specific features affect the tax on capital income, but the same logic/procedure shall be followed if we are looking at other taxes. So, there is nothing really special about capital itself, but what matters are the features of the economic environment at hand.
A reference for the examples we did today is Jones, Manuelli and Rossi (JET 1997): we proved and discussed throughly the example of pure rents, and we interpreted our results in terms of economic mechanisms at work for the Ramsey planner to find it optimal to distort capital accumulation. Please always think about the economics of the results you get.
We also talked about another example where we have restrictions on the tax system, and we pointed out what changes we should make in the Ramsey planner's problem. One of this examples is in your problem set 4.
Today we summarized the logic/procedure we should follow to set up the Ramsey problem and to add constraints to it that involve the tax system. Since the Ramsey problem does not have taxes or prices (recall we choose ALLOCATIONS) then we need to translate any constraint on taxes/prices in terms of allocations. We do this via optimality conditions of a TDCE.
Then we started talking about monetary economies, and we studied a CE in an environment with cash and credit goods (reference is Lucas and Stokey, Econometrica, May 1987, Vol. 55, N.3, pg. 491-513). We described the environment carefully and characterized a CE (in particular a steady state monetary equilibrium).
We also characterized the solution to the Pareto Optimal problem associated with this economy, and pointed out that any monetary equilibirum is inefficient.
Next time we will keep working on this allowing for the stock of cash in circulation in the economy to change (be increased or decreased) and see if and when this gets us any closer to PO. Next time we will also look at stochastic versions of the economies we have been studying and we will describe environments with incomplete mkts. In these environments we will mention what is known in the optimal taxation literature.
Today we:
Today we formalized the results on optimal monetary policy that we discussed last time just by comparing the competitive equilibrium with the pareto optimal solution. So we set up the Ramsey problem, where we have an additional constraint with respect to the resource constraint and the implementability constraint. This additional constraint tells us that the interest rate on bonds cannot be smaller than one.
Then we discussed the relationship between this result and both the uniform commodity taxation result and the no intermediate goods taxation result.
Then in the stochastic environment that we introduced last time, we discussed incomplete markets: we focused on what an incomplete mkt environment implies in terms of budget constraints, and therefore in a Ramsey problem, on the implementability constraints (which we normally derive from the time 0 budget constraint of the households). We discussed what difficulties this adds to the analytical characterization of optimal taxes, and we discussed the intuition behind two results:
Today we reviewed the cash-credit goods environment and set up and the characterization of the solution to the Ramsey problem.
Then we talked about the issue of time inconsistency is an 2 period environment in which the government taxes labor income and capital income. We distinguished between 2 cases: one where the revenue raised through the capital income tax set in the first period is sufficient to cover the amount of government expenditure and the other one when it is not. We discussed why time inconsistency is an issue only in the second case, and we showed that, when this happens, welfare is lower than in an environment with commitment. In fact households do not save (anticipating that the government will not keep its promise on the tax rate on capital income) and government expenditure must be ENTIRELY financed through distortions induced by labor income taxation.
Then we discussed a little deeper issue related to time inconsistency: we compared the competitive 2 period environment that we set up with an environment where agents can form a TEAM: we found out that if collusion is feasible, then agents would be better off by colluding and responding by saving, even though the government will raise the tax on capital income in the second period, because in this fashion the equilibrium would feature smaller distortions on the labor margin -because smaller taxation of labor income would be necessary to finance the stream of government expenditure-. In a competitive environment agents free ride, they take prices as given and do not internalize the effect that their actions have on aggregate variables, like prices.
Syllabus:
SyllabusMacroeconomicsII.pdf
Useful links/teaching notes:
Prof. Dirk Krueger and many other notes
Useful textbooks/reference books
Akira Takayama, Mathematical Economics
Stokey, Lucas, with Prescott, Recursive Methods in Economic Dynamics
Ljungqvist and Sargent, Recursive Macroeconomic Theory
Problem sets
Problem set1
Problem set2
Problem set3
Problem set4