Fact 1 The value function (8) increases (decreases) in α, if U1 > U2 (U1 < U2).
However, this result alone does not allow the further conclusion that the utility of at
least one household member increases (decreases). A look at a more elementary proof
of the fact proves instructive. Namely, let without loss of generality U1 > U2 > 0 and
consider α and e with 0 < α < α + e < 1. Then for sufficiently small e, xh(α) ∈
Bh (p ( α + e)) and
[Uι(xh(α + e))]α+) ∙ [U2(xh(α + e))]1 -(α+))
≥ [Uι(xh(α))]α+) ∙ [U2(xh(α))]1 -(α+))
= [Uι(xh(α))]α ∙ [U2(xh(α))]1 -α ∙ (U1 /U2))
> [U1(xh(α))]α ∙ [U2(xh(α))]1-α.
The last inequality shows that the shift in bargaining power has a “nominal effect”
on the household’s Nash product even before reoptimization takes place. For this
reason, we cannot conclude from a surge of the household’s maximum value of F per
se that the utility of at least one household member has increased. The impact of a
shift of bargaining power has to be assessed for each household member individually.
When we take a closer look at individual welfare, we encounter the same dichotomy
as in the case ` = 1:
One possibility is (U1(xh(α)),U2(xh(α))) = (U1(xh(α + e)),U2(xh(α + e))). For
instance, assume (E2), the absence of externalities. Then a non-binding budget con-
straint for the household requires that both household members be individually locally
satiated at their equilibrium consumption. Then for sufficiently small e, xh(α + e) ∈
Bh(p(α)), xh(α) ∈ Bh(p(α + e)), and xh(α) and xh(α + e) are close enough so that
Ui(Xi(α) ≥ Ui(Xi(α + e)) and Ui(Xi(α + e)) ≥ Ui(Xi(α)), hence Ui(xh(α)) = Ui(Xi(α)) =
Ui(Xi(α + e)) = Ui(xh(α + e)) for i = 1,2.
The second possibility is (U1(xh(α)), U2(xh(α)) = (U1(xh(α + e)), U2(xh(α + e))).
Again an increase of α makes the household’s α-indifference curves steeper. Hence, as
long as xh(α + e) ∈ Bh(p(α)) and xh(α) ∈ Bh(p(α + e)), the revised utility allocation
(U1(xh(α + e)), U2(xh(α + e))) must lie to the southeast of (U1(xh(α)), U2(xh(α)). Thus
consumer 1 benefits from a small increase of her bargaining power to the detriment of
consumer 2.
The foregoing local comparative statics can be easily globalized.
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