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

452 10. An equilibrium model of marriage, fertility and divorce
10.5.4 Calculations for the example
Properties of θc in the example
We first prove that if the costs of having children are relatively high, that
is if q∗ >c> q∗ +q 0
2 , then an intersection of W1,0(θ) with W1,1(θ) cannot
occur in the region [h1 − a, h0 − a]. The proof is by contradiction. Assume
for some θ ∈ [h1 − a, h0 − a], W1,0(θ) =W1,1(θ). Then this θ must satisfy
7 3 1
Y + θ + p(Y + β)+1 a +(1
2 2 2 2 2 q0 + 1
2 q∗ − c) (10.52)
= 3Y + θ + p(Y + β).
Solving for θ and denoting the solution by θc ,wehave
θc = p(Y + β) − Y − a +2c − (q ∗ + q 0 ). (10.53)
Recalling equation (10.6) forn =0:
we obtain, using c> q∗ +q 0
2 ,
h0 = −Y + p(Y + β),
θc = h0 − a +2c − (q ∗ + q 0 ) >h0 − a. (10.54)
Properties of θm in the example
Proof of (10.20). Consulting Figure 10.2 and allowing V1 to move up or
down, we see that we have to consider three cases for equation 10.42
First, low values of V1 give an intersection with W1,0 (θ) below θ = h0−a.
Equating V1 with W1,0 (θ) this gives:
This requires that:
θm = −Y (10.55)
−Y = θm ≤ h0 − a = −Y + p(Y + β) − a (10.56)
⇒ p(Y + β) ≥ a
For intermediate values of θ ∈ [h0 − a, θc] we equate V1 with W1,0 (θ) evaluated
in the intermediate region of equation (10.17). This gives:
θm = 1
(p(Y + β) − a) − Y. (10.57)
3
Sincewehaveθm≤θc this value and (10.19) requires that:
p(Y + β) ≥ 3(q ∗ − c) − 2a (10.58)