Suppose that x (x 1, x 2, , x n) is a Pareto efficient allocation in an exchange economy with l commodities and n consumers (example 1 117) Assume that cent Individual preferences are convex, continuous and strongly monotonic cent x Show that 1 The set is the set of all aggregate commodity bundles ...

1 Suppose x x Then there exists an allocation x 1 x 2 x such that where x x for every 1 2 Conversely if x 1 x 2 x is an allocation with x x for every ...

Suppose that x* = (x1, x2,..., x*n) is a Pareto efficient allocation in an exchange economy with

Question:

Suppose that x* = (x*1, x*2,..., x*n) is a Pareto efficient allocation in an exchange economy with l commodities and n consumers (example 1.117). Assume that
€¢ Individual preferences are convex, continuous and strongly monotonic.
€¢ x*

Show that
1. The set

is the set of all aggregate commodity bundles that can be distributed so as to make all the consumers at least as well off as at the allocation x*.
2. S = ‰¿(x*) - x* is nonempty, convex and contains no interior points of the nonpositive orthant „œl-.
3. There exist prices p* ˆˆ „œl+ such that (p*)Tx ‰¥ (p*)Tx* for every x ˆˆ ‰¿(x*).
4. For every consumer i, (p*)Txi ‰¥ (p*)Tx*i for every xi ˆˆ ‰»i(x*i).
5. (p*, x*) is a competitive equilibrium with endowments wi = x*i