the division of a given pie. Consider an increase from α to α + e. Then there are only
two possibilities. It can happen that
(Uι(xh(α)), U2(xh(α))) = (Uι(xh(α + e)), U2(xh(α + e)))
because of a kinked Pareto frontier or a corner solution. But whenever
(Uι(xh(α)), U2(xh(α))) = (Uι(xh(α + e)), U2(xh(α + e))),
consumer 1 benefits from her increased bargaining power to the detriment of consumer
2. This follows from the fact that an increase in 1’s bargaining power, that is, in α,
renders the household’s α-indifference curves steeper.
3.2 Non-Binding Budget Constraint
If the household’s budget constraint is not binding, we have a case of equilibrium with
free disposal and the household’s problem can be locally described as
max F (U1 (xh), U2(xh); α). (5)
At the solution xh(α) = (x1(α), x2(α)), the equation
∂F ∂F
(6)
∂U1 ∙ DXi U1 + U ∙ Dxi U2 = 0
holds for i = 1, 2. With DUj = (Dx1 Uj, Dx2Uj) for j = 1, 2, equation (6) amounts to
1-α
U2
• DU2,
(7)
i.e. in general a small utility gain for one household member is accompanied by a small
loss for the other member. For the value function
Φ(α) ≡ F (U1(xh(α)), U2(xh(α)); α), (8)
we obtain
0 ∂F
φ<α)=Σ ∂U-. •
∂U1
∂F
Dxi U1 + W2 • Dxi U2
0 ∂F
xi (α) + ɑ
(9)
which by (6) implies a simple
case
of the envelope theorem:
Φ0(α) =
∂F
— = ln Uι(xh(α)) - ln U2(xh(α)).
(10)
One is tempted to exploit the following immediate consequence of (10):
10
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