Consider an economy consisting of H households each with a utility function at time t = 0 given by

With β ∈ (0, 1), where ch(t) denotes the consumption of household h at time t. Suppose that u(0) = 0. The economy starts with an endowment of y > 0 units of the final good and has access to no production technology. This endowment can be saved without depreciating or gaining interest rate between periods.

(a) What are the Arrow-Debreu commodities in this economy?

(b) Characterize the set of Pareto optimal allocations of this economy.

(c) Prove that the Second Welfare Theorem (Theorem 5.7) can be applied to this economy.

(d) Consider an allocation of y units to the households, .

Given this allocation, find the unique competitive equilibrium price vector and the corresponding consumption allocations.

(e) Are all competitive equilibria Pareto optimal?

(f ) Derive a redistribution scheme for decentralizing the entire set of Pareto optimal allocations.