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

248 6. Uncertainty and Dynamics in the Collective model
her (δ b =0) this will be the case if:
(δ a + μ) ≥ (δ a (1 − σ)+(1+m) μ)
⇔ δ a σ ≥ mμ (6.7)
That is, there will be a move with no betrayal if δ a and σ are sufficiently
large relative to m and μ. For example, a husband who lacks power (and
hence relies on his wife’s caring for resources) or has a small increase in
power (so that mμ is small) will be less likely to betray; and the same holds
if his wife cares a lot for him (δ a ) and she feels the betrayal strongly (σ
close to unity).
Psychological games
Adifferent but related analysis is provided by Dufwenberg (2002), who
uses "psychological games" to discuss commitment in a family context. The
basic idea, due to Geanakoplos et al (1989), is that the utility payoffs of
married partners depend not only on their actions and the consequences in
terms of income or consumption but also on the beliefs that the spouses may
have on these actions and consequences. The basic assumption is that the
stronger is the belief of a spouse that their partner will act in a particular
manner, the more costly it is for that partner to deviate and disappoint their
spouse. This consideration can be interpreted as guilt. A crucial restriction
of the model is that, in equilibrium, beliefs should be consistent with the
actions. Dufwenberg (2002) uses this idea in a context in which one partner
(the wife) extends credit to the other spouse. For instance, the wife may
work when the husband is in school, expecting to be repaid in the form
of a share from the increase in family income (see Chapter 2). But such a
repayment will occur only if the husband stays in the marriage, which may
not be the case if he is unwilling to share the increase in his earning power
with his wife and walks away from the marriage.
Specifically, consider again the two period model discussed in Chapter
2. There is no borrowing or lending and investment in schooling is lumpy.
In the absence of investment in schooling, each spouse has labor income
of 1 each period. There is also a possibility to acquire some education;
if a person does so then their earnings are zero in the first period and 4
in the second period. We assume that preferences are such that in each
period each person requires a consumption of 1
2 for survival and utility
is linear in consumption otherwise. This implies that without borrowing,
no person alone can undertake the investment, while marriage enables the
couple to finance the schooling investment of one partner. We assume that
consumption in each period is divided equally between the two partners
if they are together and that if they are divorced then each receives their
own income. Finally, suppose that each partner receives a non monetary
gain from companionship of θ =0.5 for each period they are together. The
lifetime payoff if neither educate is (2 + 2θ) =3for each of them. Since