Model of Tomes
Tomes (1981) studied a simultaneous relationship between inheritance and the recipient’s income. Although it is not stated explicitly, Tomes’ model can be defined by
.У?/ = УіУ2і + x’ufii + “н> У* = У2Уи + + м2,’
Уи = Уи if J>u>0 (10.9.9)
= 0 if yftSO,
where у fr is the potential inheritance, yu is the actual inheritance, and y2t is the recipient’s income. Note that this model differs from Nelson’s model defined by (10.9.5) and (10.9.6) only in that yu, not yf,, appears in the right-hand side of (10.9.8). Assuming yxy2< 1 for the logical consistency of the model (as in Amemiya, 1974b, mentioned in Section 10.6), we can rewrite (10.9.7) as
У*=(1 ~Уі УаГЧУі (*2 tPi + «2.) + х’і A + «и! (10.9.10)
and (10.9.8) as
У2І = У2,} = (1 – У УіҐ’Ш^иРі + Ml,) + *2iPi + М2,] (10.9.11)
if у * > 0,
= У% = **р2+Щі if yf, S0.
Thus the likelihood function of the model is
L = n Г №> y%) dy* П Луш У*), (10.9.12)
which is the same as (10.9.2).