Find the competitive equilibrium prices and allocations

Question

1. Production set and supply correspondance

Let Y be the production set and y(p) the associated supply correspondance. Show that

• If Y is convex, then y(p) is convex, ∀p;

• If Y is strictly convex, then y(p) is single-valued (if not empty).

2. Edgeworth box

Consider a pure-exchange, private-ownership economy, consisting in two consumers, denoted by i = 1, 2, who trade two commodities, denoted by 1 = 1, 2. Each consumer i is characterized by an endowment vector, ω_i  ∈ R²₊, a consumption set, X_i = R²₊, and > i on X_i. Initial endowments are given by ω₁ = (1/2, 1/2) and ω₁ = (3/2, 1/2). Individual utility functions are u₁ (x₁₁, x₂₁) = x₁₁ + x₂₁ and U₂ (X₁₂, x₂₂) = min X₁₂. X₂₂.

1. Draw the Edgeworth Box for this economy, plotting the endowment point w and the indifference curves passing through it for both consumers.

2. Show graphically the set of Pareto efficient allocations, that is the set of allocations for which there is no other allocation indifferent for one consumer and strictly preferred by the other consumer.

3. Find the competitive equilibrium prices and allocations. Draw the equilibrium in the Edgeworth Box.

3. Robinson Crusoe

Consider a "Robinson Crusoe Economy", i.e. a private-ownership, competitive economy with only one consumer (I = 1), one producer (J = 1), and two commodities (L = 2). The consumer is characterized by a consumption set X = { x = (x₁, x₂) ∈ R²₊₊}, where x₁ and x₂ denote the quantities consumed of "leisure time" and "consumers good", respectively, and a Cobb-Douglas utility function u (x₁, x₂) = x₁^1/2 x₂^1/2. The consumer owns the endowments ω' = (L, 0) ∈ R²₊, where L = 24 denotes the time units the consumer has at his disposal (time can be used as "leisure time"or "working time"); the endowment of consumers good is nil. The consumer owns the producer, getting the latter's profits entirely. The producer is characterized by a single-output technology, with production set

Y = {y = (-z, q) ∈ R² | F(y) = F ((- z, q)) = q - f (z) ≤ 0 and z ≥ 0}

where z is the quantity of input ("labor time"), q is the quantity of output ("consumers good"), F(y) is the trans-formation function and f (z) = 2z^1/2 is the production function.

1. Check that the consumer's utility function is strictly quasi-concave, twice continuously differentiable and such that ∇u_i (x_i) >> 0 for all x ∈ X. Check that the production funtion f : R₊ → R₊ is strictly concave, that the transformation function F : R² → R is strictly convex, twice continuously differentiable, and such that F(0) ≤ 0 and ∇F(y) >> 0 for all y ∈ eff (Y) = {y E Y | F(y) = 0}.

2. Compute the (unique) Pareto-efficient allocation of the "Robinson Crusoe Economy" , denoted ( (x₁*, x₂^* , (-z*, q*)) .

3. Let p and ω be respectively the price of the consumers good and the wage rate, i.e., the price of both the "leisure time" and the "working time". By taking the consumers good as the numeraire, thereby setting p ≡ 1, compute the shadow prices implicit in the Pareto-efficent allocation, (w*, 1), and the corresponding profits, Π*. Show that, at those prices and by assigning those profits to the consumer, the Pareto-efficient allocation can be obtained as a Walrasian competitive equilibrium.

4. Illustrate the producer's and the consumer - worker's problems in one and the same diagram.