In equation (13), p0(α) reflects the price effect. If the price effect is negligible, i.e.
p0 (α) ≈ 0, then Φ0(α) can be signed and the conclusion of Fact 1 holds again. As before,
this alone does not allow to sign individual utility changes. To achieve the latter, let us
assume momentarily that the price effect is negligible and focus on the pure bargaining
effect. Equation (11) is the key to the pure bargaining effect. It differs from equation
(6) by the right-hand term λ(α)p(α). In analogy to (7), let us rewrite (11) as
α ■ DUi = - 1-α ■ DU2 + λ(a)(p(α),p(a)). (14)
U1 U2
Now consider a change ∆xh away from xh(a) while maintaining the budget identity,
i.e. p(a) * (xh(a) + ∆xh) = p(a) * xh(a) = p(a)ωh. Then (p(a),p(a)) ■ ∆xh =
p(a) * ∆xh = 0, hence with (14),
■ ∆xh = -
-a
U2
■ DU2
■ ∆xh.
(15)
Thus (7) essentially holds again. Running through the earlier geometric and topo-
logical arguments yields
Proposition 2 Suppose that the household’s budget constraint is always binding while
the price effect is negligible. If 0 < a* < a* < 1, then one of the following two
assertions holds:
(i) U1(xh(a*)) = U1(xh(a*)), U2(xh(a*)) = U2(xh(a*)).
(ii) U1(xh(a*)) < U1(xh(a*)), U2(xh(a*)) > U2(xh(a*)).
Obviously, Propositions 1 and 2 could be combined into one, assuming zero or
negligible price effects. If, on the contrary, the price effect is drastic, both utilities
may move in the same direction. The magnitude of the price effect — whether it is
negligible or drastic or somewhere in between — depends on the size of the household
relative to the economy. It also depends on preferences, including the preferences of
consumers not belonging to the household, as a comparison of Examples 1 to 3 shows.
The focus on a particular household h amid many might suggest that shifts of
bargaining power are sporadic and therefore price effects are likely to be negligible.
Our general analysis provides valuable insights in case the change of bargaining power
is a sporadic event, indeed. It helps identify the relevant effects. Drastic price effects
13