Problem 1

(20 points total)

Consider a pure exchange economy with

1 < <

consumers and

I

commodities. Let

0

be the vector of resources in the economy, and let the consumers' preferences be specified by

?

(

?

) =

?

?

??

?

for all

i

,

where

?

?

> 0

for all

i

,

j

, and

?

?

= 1

?

,

.

A.

(5 points)

Show that the linear sharing rules, i.e.,

??

=

?

?

?

,

,

, are Pareto optimal.

B.

(5 points)

Will the result in (a) hold if

?

?

> 1

?

for some

j

?

C.

(5 points)

Suppose that the initial assignment of endowment is

??

=

?

,

,

. How

can the Social Planner implement the linear sharing rules?

D.

(5 points)

How will your answer to (c) change if in addition to all above properties,

?

?

= 1

?

,

?