Bargaining Power and Equilibrium Consumption

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, x_h(α) ∈
Bh (p ( α + e)) and

[Uι(xh(α + e))]^α+) ∙ [U2(x_h(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(x_h(α)),U₂(x_h(α))) = (U₁(x_h(α + e)),U₂(x_h(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, x_h(α + e) ∈
Bh(p(α)), x_h(α) ∈ Bh(p(α + e)), and x_h(α) and x_h(α + e) are close enough so that
Ui(Xi(α) ≥ Ui(Xi(α + e)) and Ui(X_i(α + e)) ≥ Ui(Xi(α)), hence Ui(x_h(α)) = Ui(Xi(α)) =
Ui(Xi(α + e)) = Ui(xh(α + e)) for i = 1,2.

The second possibility is (U₁(x_h(α)), U₂(x_h(α)) = (U₁(x_h(α + e)), U₂(x_h(α + e))).
Again an increase of α makes the household’s α-indifference curves steeper. Hence, as
long as x_h(α + e) ∈ Bh(p(α)) and x_h(α) ∈ Bh(p(α + e)), the revised utility allocation
(U₁(x_h(α + e)), U₂(x_h(α + e))) must lie to the southeast of (U₁(x_h(α)), U₂(x_h(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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