Define weak pareto efficiency and strong pareto efficiency

Question

Economics 711, Part II SECOND MIDTERM EXAM Fall 1997-

Part 1-

Q1. Define Weak Pareto Efficiency and Strong Pareto Efficiency. Add the assumptions and show that an allocation is Weakly Pareto Efficient if and only if it is also Strongly Pareto Efficient.

Q2. Prove the following:

Proposition: Suppose x* is Pareto efficient and that preferences are non-satiated. Then there exists a p' such that (p', x*) is a Walrasian equilibrium.

Q3. Suppose (p, x*) is a Walrasian equilibrium in pure exchange. Assume utility functions are concave. Then show that there exist welfare weights θ_i, i = 1, . . . , n, and a linear social welfare function W = ∑θ_iu_i(x_i) for which x* is a solution. Find the θ's you need. What interpretation can you put on θ's in this case?

Q4. In the proof of the Second Theorem of Welfare Economics we had to assume that all agents hold positive amounts of all goods and that preferences are strictly convex. Explain why we need to do this.

Q5. Bill and Al consume two goods, speeches x and votes y, and they each have identical Cobb-Douglas utility functions U_i = x_i^1/4y_i^1/4. Bill and Al are each endowed with one unit of labor, l, which they supply perfectly inelastically. Speeches are produced using only labor: x = 4l_x. However, votes require the use of some speeches as well as labor: y = x_y^1/2l­_y^4/2. Find the Competitive Equilibrium in Bill and Al's economy.

Part 2-

Q1. There are two possible states of the world tomorrow, s = 1 and s = 2. Today, n consumers trade contingent claims on m ≥ 1 physical goods. Consumers have preferences

U_i(x₁ⁱ,x₂ⁱ) = π₁u_i(x₁ⁱ) + π₂u_i(x₂ⁱ),

where (π₁, π₂) is the probability distribution on states 1 and 2, and xⁱ_s is the consumption of consumer i in state s. Finally, let be i's endowment of good kin state s. Then assume that the endowments of the physical goods satisfy for all k goods.

(a) Show that if all the functions u(.) are strictly concave, then any Arrow-Debreu equilibrium allocation x^{^} satisfies x^{^i}₁ = x^{^i}₂ for all i. Interpret this result.

(b) For the case of m = 1 (i.e. one physical good), what are the equilibrium prices (assuming at least one consumer's utility is differentiable).

(c) Consider the case where m = n = 2, π₁ = π₂ = 1/2, and

Find the Arrow-Debreu equilibrium. Interpret the result.

Q2. An economy has a private good x and a public good G. The private good can be converted to the public good on a 1-for-1 basis. There are N consumers. They each have preferences U_i = x_i +In G, and are endowed with money m_i for i = 1, . . . , N. Finally, assume m_i > 1 for all i.

(a) Find the Pareto Efficient allocation in this economy. Is it unique?

(b) Find the Lindahl Solution for the economy. Is it unique?

(c) Give the definition of a core allocation.

(d) Suppose that an improving coalition cannot be excluded from consuming the public good G that the rest of the economy produces. Identify the core.

(e) Now suppose that an improving coalition can be excluded from consuming the public good G that the rest of the economy produces. Identify the core.