It follows from this theorem that so long as each agent keeps its budget constraint, a resulting asset market equilibrium is generically Pareto inefficient regardless of the agent’s optimization behavior.
(see chapter 3 of this dissertation). Considering these facts, it may be conjectured that the consequence obtained here depends on whether the asset prices are endogenous or exogenous. The difference does not alter the intrinsic claim described above but slightly change a condition concerning the number of agents, assets and states of nature in the theorem. Actually, it is easily verified that the map F considered in the proof of the theorem is transversal to zero whether the asset prices are endogenous or exogenous. I have considered them to be endogenous in the text, whereas if they are exogenous, then the hypothesis in the theorem should be altered to (S−J)(I −1)> S.
Chapter 5
Inefficiency of Equilibria with Incomplete Markets II
This chapter is devoted to the mathematical erabolation of the arguments in the previous chapter, which is motivated by the improvement of the treatment for function spaces.
It also turns out that this sophistication improves the result of the previous chapter in some respects, e.g., the condition concerning the number of agents, goods and states of nature, which is necessary for the desired outcome. Specifically, I replace the compact open topology with the Whitney topology, which is more appropriate for the space of utility functions. However, such a refinement necessitates abandonment of the foregoing approach since the Whitney topology does not permit the use of the perturbation technique. Thus, I develop a very different way of proving the generic inefficiency of equilibria. That is to say, in order to reach the goal, I make use of the geometrical relation of two specific sets: one is the set dependent on initial endowments which contains all Pareto efficient allocations, and the other is the set dependent on utility functions which contains all equilibrium allocations. Then, possibility of efficiency of financial equilibria depends on whether these two sets intersect or not. I shall show that through two kinds of transversality theorem their intersection is generically empty, which implies the desired result, i.e., the generic inefficiency of equilibria. In the process of the argument, another mathematical tool, i.e., fiber bundle, is effectively used particularly for a nominal asset case.
5.1 Introduction
In the previous chapter, I demonstrated the generic inefficiency of equilibria with incomplete markets by checking compatibility of Pareto efficiency with budget constraints. Here in this chapter, I approach the same issue, i.e., Pareto inefficiency of equilibria with incomplete markets, from a different angle, which is caused by a mathematical refinement of the treatment for function spaces. Specifically, I consider the Whitney topology for the space of utility functions, though I endowed the space with the compact open topology in the previous chapter. It is true that the compact open topology has nice features: for instance it has a complete metric and a countable base, moreover, it allows the perturbation technique described in the previous chapter so that one has only to look at a finite dimensional subset of the space. But, noting that the domain (RS+1++) of a utility function is not compact, that topology is not adequate since it does not control the behavior of a function “at infinity”
very well. In other words, the generic outcome in chapter 4 only holds under the condition that nothing but a local behavior of a function is covered. On the contrary, if one wishes to cover a global behavior of a function for the genericity analysis, one cannot but resort to the Whitney topology. Thus, I consider here the Whitney topology for the space of utility functions.
However, the cost of substituting the Whitney topology for the compact open topology is not small. Above all, I have to abondan the finite dimensional transformation through the perturbation technique, which implies that the approach previously provided is not valid any more. Therefore, in this chapter, I adopt another method which is similar in spirit to the one described in chapter 3. The basic idea is as follows. First, separate two types of parameters: namely, utility functions (u) and initial endowments (ω). Then, it turns out that on the one hand there exists an appropriate set (A(u)) of allocations which depends only on utility functions and includes all Pareto efficient allocations, on the other hand there exists an appropriate set (F(ω)) of allocations which depends only on initial endowments and includes all financial equilibrium allocations. It follows from this formalization that I have only to check the intersection of these two sets in order to examine possibility of efficient financial equilibrium with incomplete markets.
There are some merits in using this formula. First, this formula is independent on choice of a utility function. Indeed, I still consider a utility function free from convexity.
Second, following this procedure, I can make use of the Thom transversality theorem which makes it possible to perform the genercity analysis for the the space of utility functions with the Whitney topology. Third, this formula admits both real and nominal assets, though the latter assets need an extra mathematical treatment through a fiber bundle. Lastly, what may be the most important in economical sense, I can improve the result previously obtained in some respects by means of this formula. In particular, it is shown through this formula that the condition to assure the generic inefficiency of equilibria is relaxed.
In section 2, I describe the model to be considered, which is substantially the same as the one in the previous chapter. Namely, I have chosen the one-good two-period model,
based on a pure excange economy, where only monotonicity is assumed on each agent’s utility function. In section 3, after presenting a specific method for examining inefficiency of equilibria with incomplete markets, I apply it to both a real asset model and a nominal asset model. The analytical key in the method consists in using two kinds of transversality theorem, the Thom transversality theorem and the standard transversality theorem, alter-nately to attain generic impossibility of efficient equilibrium allocations with incomplete markets. As a result, for both real asset and nominal asset models, I show that equilibrium allocations with incomplete markets are generically inefficient with regard to utility func-tions and intial endowments for agents. Finally, in section 4, I address some economical implications derived from the arguments provided here.