dP
dx
—a(p-ρ) £aS1 + (1 — α)S2 — (1 — α)x2xmm — xp — αxim(xs — xr)]
α[Sr — xmm(xr — x)] + (1 — α)(S2 — x2xmm) — xsp
< 0.
Proposition 6
For a type-1 household, using Roy’s identity,
dv1 1 dp 1
=V = vPF + vm
dp p dp
($ — 1)
dp 1
— (x — x) + x
dp
Hence, substituting for dp from (13), we have
d1
dpx
1
m1 xx
αx1m (x1 — xx)
∆
+1
We have seen in the text
that
ifx1 <x2 ,then∆ < 0.
mm
Using this with the
assumption in case 1, that
x1
x ≤ 0, it is seen here that dV1 < 0.
— ’ dp
Similarly,
d1
=v = v
dx
α(x1
(p — rx) χ +--
r∖ rp1
x)xm
and so dV1 > 0.
dx
Likewise, for a type-2 household, as Proposition 3 holds, that is dp < о < dp,
dv2
dpx
22
—m2 x2
dp
dpx
dv2
о;
dx
2 2 dp
vm x 5=
dx
< 0.
30