Bargaining Power and Equilibrium Consumption

denote F ≡ ln S_h . Explicitly, we obtain

F = F (U1(xh), U2(xh); α) = αlnU1(xh) + (1 - α)lnU2(xh).           (2)

While α is treated as variable, the other characteristics of household h as well as all
the characteristics of the rest of the households remain fixed. Each household k 6= h is
assumed to choose an efficient consumption plan, x_k ∈ EB(p). It may, but need not,
maximize a Nash product.

We assume sufficient regularity in the sense that for each α ∈ (0, 1) the economy
has an equilibrium (p(α); x(α)) satisfying:

(iv) local uniqueness and

(v) continuous differentiability in α.

For each α, at the given price system p(α), household h solves the problem

max F (U1(xh), U2(xh); α) s.t. G(xh; α) ≤ 0                     (3)

where G(x_h; α) = p(α)[(x₁ + x₂) - ω_h].  The corresponding solution is x_h(α) =

(x₁ (α), x₂ (α)). The budget constraint G(x_h; α) ≤ 0 can be rewritten x_h ∈ B_h(p(α)).
In turn the household budget set Bh(p(α)) defines a set V(α) of feasible utility al-
locations for household h, given the price system p(α):

V(α) ≡ {(V1, V2) ∈ IR² : (V1, V2) = (U1(xh), U2(xh)) for some xh ∈ Bh(p(α))}

In the sequel, the term Pareto frontier refers to the Pareto frontier of V (α) in the
space of utility allocations for the household. In particular, (U1 (xh(α), U2 (xh(α)) lies
on the Pareto frontier and solves the problem

max F (V1, V2; α) s.t. (V1, V2) ∈ V (α).                          (4)

Finally, for the household under consideration and a given α, the term α-indifference
curve refers to a locus in IR² given by an identity F (V1 , V2; α) ≡ const.

It is instructive to look first at the case ` = 1 of a single good. Assuming that
the equilibrium price is positive, the household’s budget set and, therefore, its Pareto
frontier is price-independent and the household’s consumption decision is reduced to

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