Bargaining Power and Equilibrium Consumption

We will allow for the possibility of consumption externalities. Following Haller
(2000), we shall restrict attention to the case where such consumption externalities, if
any, exist only between members of the same household. This is captured by the notion
of intra-household externalities where utility functions are restricted to the household
consumption x_h, i.e.:

(E1) Intra-Household Externalities: U_i(x) = U_i(x_h) for i = hm, x ∈ X .

A special case is the absence of externalities to which we sometimes pay particular
attention. When there are no externalities, the utility function of an individual i
depends only on his consumption bundle xi , i.e.

(E2) Absence of Externalities: Ui(x) = Ui(xi) for i = hm, x = (xi) ∈ X .

With a fixed household structure, the latter condition is somewhat less restrictive
than it seems. For suppose a consumer i = hm cares about own consumption and
household composition, which could be important for household formation. But if
household membership, i ∈ h, is a fait accompli, one may omit h as an argument of i’s
utility function and work with the reduced form E2.

Budget Constraints: Now consider a household h and a price system p ∈ IR'.
For x_h = (x_h1 , . . . , x_hm(_h)) ∈ X_h, denote total household expenditure

(m ( h )     ∖

xhm .
m=1

Then h’s budget set is defined as B_h(p) = {x_h ∈ X_h : p * x_h ≤ p ■ ω_h}. We define
the efficient budget set EB_h(p) by:

xh = (xh1 , . . . , xhm(h) ) ∈ EBh(p) if and only if xh ∈ Bh(p) and there is no yh ∈
B_h(p) such that

Uhm(yh) ≥ Uhm(xh) for all m = 1, . . . , m(h);

Uhm(yh) > Uhm(xh) for some m = 1, . . . , m(h).

General Equilibrium:

A competitive equilibrium (among households) is a price system p together
with an allocation x = (xi) satisfying

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