Suppose a concave utility function has two peaks g 1

c) You can easily see that the concavity implies single peakedness. Suppose a concave utility
function has two peaks G
1
and G
2
, and G
1
< G
2
. Since G
1
and G
2
are local maxima, the
slop of utility function is zero.
∂U
i
∂G
( G
1
) =
∂U
i
∂G
( G
2
) = 0
However, concavity implies that the slope decreases as G increases:
∂U
i
∂G
( G
1
) >
∂U
i
∂G
( G
2
)

which is a contradiction.
d) With the single-peakness the median voter is the decisive voter. Let Y
Med
be the median
income of the member of society, the outcome is given by median voter’s preferred level:
αY
Med
N
Problem 3: Non-For-Profits (20 percent)
a) If municipal unions control the voting behavior of their members, then they can endorse
a candidate and swing close elections. Moreover, they can provide local manpower to a
political campaign or donate money to a campaign.
b) You can require unions to hold an internal votes before they can endorse a politician.
You can try to pass laws that restrict political contributions to individuals (is that con-
stitutional?). Of course, unions may just serve as counter-balancing weights to employer
organizations and firms that have competing interests.