Pierre André Chiappori (Columbia) "Family Economics" - Cemmap

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6. Uncertainty and Dynamics in the Collective model 247
W a = u a + δ a u b
= u (x)+δ a u (1 − x) (6.2)
W b = δ b u a + u b
= δ b u (x)+u (1 − x) (6.3)
where δ s ≥ 0 is person s’s caring for the other person, with δ a δ b < 1 (see
chapter 3). We assume that the caring parameters are constant and outside
the control of either partner. Rather than choosing an explicit game form
to choose x, we simply assume that there is some (collective) procedure
that leads the household to behave as though it maximizes the function:
W = W a + μW b
³
= 1+μδ b´
u (x)+(δ a + μ) u (1 − x) (6.4)
As discussed in chapter 4, caringmodifies the Pareto weight for b to an
effective value of (δ a ³
+ μ) / 1+μδ b´
.
Now suppose there is a (moving) decision that costlessly increases household
income from unity to y>1. Ifthisistheonlyeffect then, of course,
both partners would agree to move. However, we also assume that the decision
increases b’s Pareto weight to μ (1 + m) where m ≥ 0. Inthiscase
there is a reservation income y ∗ (m) such that person a will veto the move
if and only if y