By definition of Fg (∙), Fg [x, y, ug (z, x, y)] = z, which means the above expression
can be written as
Fg [x, y, ug(z, x, y)] = F
x,y,u z+ + У* μ(χ)dχ,χ,y
') - μ μ(χ)dχ
J J gyg
(36)
From (14)
Fg [x, y, ug (z, x, y)] = Fg
Therefore
χ,y,u^z + У μ(x)dχ,χ,y^
ug (z,χ,y) = u(z +[ μ(χ)dχ,χ,y
∖ ygμ .
Proof of lemma 1.
uh g
Expression I uh ) can be written as
vh yh
-2hh - h + μ(χ)dχ,xh
-1 J<j∙
xh,yh - yh).
(37)
(38)
(39)

looks similar but has the extra term μ(yh)).
For these preferences, it can be shown that the Marshallian demands for
commodity x is of the form
dbh(q) - χh
χh(q, mh) = χh + b(qd)qx- m. p [mh - mh(p)]. (40)
where b(q ) is minimal expenditure necessary to generate one unit of utility and
mh(p) =f zh + qxχh + qyyh denotes survival income. Similar expressions hold for
the other two commodities.
Under the lemma’s assumptions, expression (39) reduces for every agent h to
-i (
-1
∂b(q)
∂ qz
χ + μ[
∂b(q)
d qy
- y],
∂ b(q)
∂ qχ
x,
∂b(q)
∂qy
- y),
(41)
since the expression is homogenous of degree 0 in its arguments and the term
mq)-jm(p) thus drops out. Every agent will then have the same vector of (first best)
13
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