- Consider an agent with CES utility function
u(x1, x2) = 1
1 − σ
(α
σ
1 x
1−σ
1 + α
σ
2 x
1−σ
2
),
where α1, α2, σ > 0 are parameters. (As usual, σ = 1 corresponds to Cobb-Douglas.) Let w > 0 be
the wealth of the agent and p1, p2 > 0 be the price of goods. Compute the demand. - Consider an economy with two agents and two goods with Cobb-Douglas utilities
u1(x1, x2) = α log x1 + (1 − α) log x2,
u2(x1, x2) = β log x1 + (1 − β) log x2,
where 0 < α, β < 1. Suppose that the aggregate endowment is e = (e1, e2). Find all Pareto efficient
allocations such that each agent consumes a positive amount of each good. - Read Section 3.2 of the lecture note before answering this question. Let E = {I, J,(ei),(ui),(Yj ),(θij )}
be an Arrow-Debreu economy with production.
(a) Define (mathematically) the concept of Pareto efficiency.
(b) Show that if ui
’s are locally nonsatiated and {p,(xi),(yj )} is a competitive equilibrium, then
{(xi),(yj )} is Pareto efficient.
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