Find the competitive equilibrium prices and allocations

Assignment Help Managerial Economics
Reference no: EM13688941

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, ωi  ∈ R2+, a consumption set, Xi = R2+, and > i on Xi. 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) = x11/2 x21/2. 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) = 2z1/2 is the production function.

1. Check that the consumer's utility function is strictly quasi-concave, twice continuously differentiable and such that ∇ui (xi) >> 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.

Reference no: EM13688941

Analyze the economic impact of contracting

Propose how differences in demand and elasticity lead managers to develop various pricing strategies and analyze the economic impact of contracting, governance and organizatio

Calculate expected profit and calculate expected utility

Calculate the expected profit and calculate the expected utility of profit and comparing the utility of $3,200 with the expected utility of the risky project, what should the

Determine market structure in which the low-calorie frozen

Determine the market structure in which the low-calorie frozen, microwavable food company operates - You are now aware of recent changes in the selling environment that sugge

What is the expected profit that jonathan earns

Find the equilibrium contract assuming that Cristina is a monopolist and derive expressions for E(Π1) and E(Π2), the expected value of Cristina's profits from loans that finan

Lesson 3 problem set a

Suppose that bondholders converted $500 million worth of bonds to equity. How would this affect Loewen's earnings before income taxes as compared with its economic value added

Describe the present market reality

Describe the present market reality, market structure and/or power, in terms of sales, industry leadership, and profits for selected firm, if one particular firm is selected

Find combined effect of increase in wages of farm workers

Describe when increase in price of what a company sells will lead to its total revenue to increase and determine the combined effect of increase in wages of farm workers and

What is the maximum capital cost of the conversion unit

What is the maximum capital cost of the conversion unit for which a refiner should proceed with the investment - Change the price delta for Jet Fuel & Kerosene from

Reviews

Write a Review

 
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd