Exchange economies with indivisible goods

Consider the following housing market - an exchange economy with indivisible goods. The set of agents isN={1, 2, 3, 4}and the set of houses isX={H₁, H₂, H₃, H₄}. Each agentihas as initial endowment houseH_i. Each agent wishes to consume exactly one house.

Agents have strict preferences over houses described as follows:

Agent 1:H11H41H21H3

Agent 2:H12H22H32H4

Agent 3:H23H33H43H1

Agent 4:H34H44H14H2

For instance, in the description of preferences above,H32H4 reads "agent 2 strictly prefersH₃toH₄".

An allocation is denotedH. At allocationHif agentireceives houseH_j, this is denotedH_ji. An example of an allocation isH= (H₁₁, H₂₂, H₃₃, H₄₄). At this allocation, each agent keeps the house he is initially endowed with.

Afeasible allocationisHsuch that each agent receives exactlyone house, andno two agents are assigned the same house. Given the definition of a feasible allocation, there are 4! = 24 feasible allocations in this economy -i.e. there are 24 different ways of assigning the four houses to the four agents.

Finally a price vector isp= (p₁, p₂, p₃, p₄)R+4 , wherep_kis the price of houseH_k.

(i) Find all the Pareto efficient allocations -Note: there are 5 efficient allocations.

(ii) Consider allocationH= (H₁₁, H₂₂, H₃₃, H₄₄) -i.e. the allocation at which no trade takes place- and the price vectorp= (p₁, p₂, p₃, p₄) = (1,1,1,1). Show that (p, H) is not a Walrasian equilibrium. Explain.

(iii) Consider againH= (H₁₁, H₂₂, H₃₃, H₄₄). Find the price vectorpsuch that (p, H) is a Walrasian equilibrium.

(iv) Does the second Welfare theorem hold?

(v) Does the first Welfare theorem hold?