We consider a pure exchange economy with a finite number of goods and households. The only differences with respect to the standard model are what follows:
1. each household utility function depends not only on her own consumption but also on other households’ welfare, measured by wealth;
2. households are allowed to promise transfers to other households (and promised are bound to be honored).
After having presented, in Section 2, we present the set up of the model as it was introduced by Kranich (1988) and in Section 3, we show existence of equilibria taking care of some minor details which are not addressed by that paper.
In Section 4, we discuss two main modelling assumptions of Kranich’s model. The first one amounts to perform a normalization of other households’ nominal wealths; as showed in Section 10.3, different normalizations lead to different allocation equilibria.
The second modelling assumption consists in imposing an ad hoc upper bound on promises of transfers.
That bound allows to use a standard proof of existence which requires compactness of the choice set of each
To tackle the first problem, we follow a simple, widely accepted viewpoint and we assume that households may in general care about other households’ “relative wealth”. That choice seems to be more realistic and allows to make households’ maximization problem homogeneous of degree zero in prices, a quite reasonable and convenient assumption.
The nonexistence problem seems harder to be dealt with. First of all in Section 5, we present a simple Cobb-Douglas, two household, one good version of the model and we do show that there is indeed a “large” set of economies for which equilibria do not exists if upper bounds on promises of transfers are not imposed.
We verify that if an upper bound is added, in that large set of economies, at least one of the households chooses the highest allowed level of transfer; some intuition on the nonexistence results are presented. The simple Cobb-Douglas economies allows to get some further results on the equilibria structure; those results are used to get connjectures in the more general model: under some assumptions, such conjectures are proved to be true in a relatively general version of the model.
In Section 6, we then move to further analysis of equilibria. Since it is quite hard to describe properties of equilibria under the general assumptions under which existence was proved, then we make some stronger assumptions about households’ preferences and transfers possibilities. Indeed, we assume that i. households’
utility functions are sufficiently differentiable, ii. they take an additive form with respect to selfish preferences
and other-regarding preferences and iii. promises of transfers can take place only in terms of a numeraire commodity. In this new set up, we cannot apply the existence result used in Section 3: if each household’s utility is the sum of a selfish utility function and an other-regarding function, then the needed assumption of (quasi)-concavity of the overall function is lost if the given household dislikes “too much” some other household.
An equivalence result between true equilibria and some fictitious equilibria allows to overcome the problem. Given all the above preliminary work, we can then proceed as follows. In Section 7, an indispensable, so- called generic regularity result is proved: typically, in the space of economies, equilibria are finite and depend smoothly upon parameters defining an economy. Two further assumptions are needed to get that result: their economic meaning is explained in detail.
Moreover, extending the result obtained in the Cobb-Douglas economy case, we show that a. there exists a set of economies contained in a closed measure set, for which the number of equilibria is infinite, and in which
all household choose a strictly positive transfer; b. typically, for any pair of households, at most one of them provides a transfer to the other one.
In Section 8, after having discussed the notion of Pareto Optimal, we show there exists an open, non-empty set of economies for which at least one associated equilibrium allocation is not Pareto Optimal. Finally, in Section 9, we prove that there exists a open, nonempty set of economies, for which there exists an equilibrium it is possible to Pareto improve upon, through a redistribution of wealth and a change in other regarding attitudes of a very small number of households.
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