MECANISMOS DE EXIGIBILIDAD JURÍDICA

There are two reasons why the constrained planner’s feasibility correspondences will not have compact image sets. The first concerns the structure of the recursive problem and the second concerns the behaviour of interest rates around regions where capital is zero.

Structure of recursive problem

In the recursive problem, to showΛhas compact image sets, we need to place some further restrictions on the spaceYand equipYwith a suitable topology. In the sup- norm topology, the Arzela-Arscoli Theorem (see example6.2, Mas-colell and Zame

(1991)) states uniformly bounded, equicontinuous family of functions on a compact

interval will be compact. However, A will not be bounded and policy functions may not be bounded. As such,Λ(µ)cannot be compact valued as it is defined here.

A possible approach could be restrictingMto measures on a compact support. For

each µ ∈ M, we can then restrict policy functions in Λ(µ) to be defined on the

bounded support of µ. If the mean of µ is positive, then policy functions in Λ(µ)

will also be bounded. Notwithstanding the pathologies (see below) as interest rates diverge, to now use the Arzela-Arscoli Theorem, we also need to restrict feasible policy functions in each period to an equicontinuous family of functions. Similarly, to use Helly’s Selection Theorem (see Cao (2016), Lemma11) to verify compactness

in the product topology onY, we need to restrictYto the space of monotone policy functions.

While restricting the space of equilibrium policy functions to monotone functions is a common approach in proofs for existence of general equilibrium (Cao, 2016),

applying the approach here will be difficult. The reason is subtle; in proofs of existence of general equilibrium, we already know a policy function exists as a solution to the consumers’ finite dimensional problem given a sequence of prices. We are then interested in searching for a policy function consistent with the price sequence it generates. In the proof, we can restrict the space of policy functions by using properties of the policy functions, which, once again, arealready known to exist. Now, in the search for constrained optima, we are searching for an optimal policy function for the planner. We cannot use information about the policy function conditional on existence of the policy function, such as necessary conditions, to restrict the search for an optimizer. This is because an optimizer may not exist. In particular, there may be pathological sequences of policy functions outside the equicontinuous or monotone family converging to the supremum. The problem is

confounded by the fact that a proof for the necessity of functional Euler equations has not been completed, as I argue in section B.3of the appendix.10

The appendix for this chapter gives further detail of pathologies in the weak topol- ogy if we letYbe the space of square integrable functions onS, whereΦfails to be defined.

Note the pathologies in the recursive problem also prevent the application of the ideas in the non-compact existence result of section 2.3 to the recursive problem.

This is because we cannot place restrictions such as monotonicity or equicontinuity on policy functions in the upper contour sets of Assumption2.3.1.

Non-compactness near zero capital

Consider the setting and notation of section 2.4. Once we move to L2 space with

the weak topology, the feasibility correspondence will be compact valued and have a closed graph. The sequential problem will also be well-defined. However, there will exist a compact setC⊂St such thatΓt(C)is not compact. (Recall from section

A.1.1 in the appendix for this chapter that the image of a compact set under a

compact-valued and upper hemicontinuous is compact.)

For the following claim, assume F(K,L) = α−1Kα and α ∈ (0,1). Furthermore, assume x0 ∈ S0, the initial assets for the economy are a uniform random variable

on the interval [0,1]. Assume the random variable e0 is large enough to satisfy

˜

w(x0)e0 > 1.

Claim2.5.1. There exists a compact set C, satisfyingC⊂S₁, such that the image set Γ1(C)is not compact.

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One way to proceed could be to construct a proof showing policy functions that are monotone dominate all other policy functions, however, this line of attack has not yielded any results so far.

The proof is in the appendix for this chapter. Roughly, we can construct a sequence of asset distributions inΓ1(C)whose means converge but variances diverge. This is

because a smaller and smaller measure of agents can accumulate assets that go to infinity due to higher and higher interest rates as aggregate assets converge to zero.

How can Assumption2.3.1guarantee existence even whenΓ does not have compact

image sets? Intuitively, the role of compactness ofΠ∞_t=0Γt(x0)is to guarantee feasible

sequences, such as the ones in the above example that diverge in norm to infinity, do not "escape" the feasible space and take the infinite sum of pay-offs associated with them to the supremum of the problem. To rule out that such sequences cause a pathology, a natural way to show existence in a problem with non-compact feasible spaces is to show that any sequence that converges to the supremum must belong to a compact space. This is exactly what Assumption 2.3.1 achieves; any sequence

in the infinite product space converging to the supremum must eventually have at-least some per-period pay-offs always greater than the infimum of the problem, Assumption 2.3.1 then requires the heads of the sequences belong to a compact

space. And the proof for Theorem 2.3.1 constructs a compact space in the product

topology form such successive compact spaces.

We can verify 2.3.1 for the constrained planner’s problem since sequences of the

type identified in the above claim are ones that converge in mean of assets to zero and must lead to pay-offs in the future that converge to zero. Hence if a pay-off must be at-least some value strictly above the infimum of the problem at a given time, the mean and norm of assets in the head of the sequence leading to that time must be finite and strictly positive.