An alternative way to model merit good arguments



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
m
h(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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