3 General Comparative Statics for a Two-Person
Household
In this section we perform comparative statics with respect to the balance of bargaining
power within a two-person household denoted by h. We allow for an arbitrary number
of commodities and we consider the general case of intra-household externalities. The
entire population consists of an arbitrary number, n of households.
Negative intra-household externalities allow for the possibility that a household has
a bliss point despite the fact that each household member has monotonic preferences
with respect to her individual consumption (see Haller (2000) for examples). If this
happens, the corresponding notion of competitive equilibrium among households has
to be less demanding. The social feasibility or market clearing condition (ii) has to be
replaced by the free disposal condition
(iii) Pixi ≤ ω.
If in fact an equilibrium with free disposal prevails and the household does not
exhaust its budget, then after a small shift of intra-household bargaining power, the
resulting equilibrium will most likely be one with free disposal again and the household
will still not exhaust its budget. As a consequence, the household’s budget constraint
remains non-binding. This means that the household is not exposed to any price effect.
In the sequel, we treat first the simpler case of non-binding budget constraint and,
hence, zero price effect. We then proceed to the case of a binding budget constraint
and typically non-zero price effect. This general comparative statics helps identify two
relevant effects, a pure bargaining effect and a price effect.
3.1 Preliminaries
We shall perform comparative statics with respect to the bargaining weights within a
select two-person household h, with members h1 and h2. Whenever convenient and
unambiguous, we shall drop the household name and simply refer to consumers 1 and
2. Without restriction, we may also assume that our select household has the lowest
number, i.e. h = 1 and the other households are labelled k = 2, . . . , n. For the sake
of convenience, we shall further adopt the notation α = αh1 and 1 - α = αh2 so that
comparative statics can be performed with respect to the parameter α ∈ (0, 1). Finally,