Find the pareto optimal allocations - find the best

Question

Question 1

1. The following table describes the set of possible allocations. There are 12 possible allocations.

1/2                         b1                           b2                           b3

a1                           2, 7                         -48, 2                     12, 2

a2                           12, 2                       12, 2                       12, 2

a3                           32, -8                     17, 12                    -3,27

a4                           9, -5                       2, 2                         10,4

For example, x = (a1; b3) is an allocation. The first number reported in the (a1; b3) position represents u1(x), the utility of agent 1 at x (in this case, 12); the second number represents the utility of agent 2 (in this case 2).

(i) Find the Pareto optimal allocations.

(ii) Find the best allocation according to the utilitarian criterion.

Question 2

Recall the 'splitting $100' example discussed in class. In that example there are two individuals (Jack and Tina) and the set of possible allocations is X = {(x^J,,x^T) : x^J≥0; x^T≥0; x^J+ x^T≤100}:

(i) Is a (60; 30) split (Jack gets 60, Tina gets 30) Pareto optimal?

(ii) Is a (20, 80) split Pareto optimal?

(iii) Does (20, 80) Pareto-dominate (60, 30)?

Assume now that $100 must be split among three people Jack, Tina, Chris.

Thus, the set of possible allocations is now

Y = {(x^J,,x^T, x^C): x^J≥0; x^T≥0 and x^C ≥0; x^J+ x^T + x^C ≤100}:

An example of a Pareto optimal allocation is now (20, 30, 50).

(iv) Give two other examples of Pareto optimal allocations in this 'three person divide $100' game.

(v) Give an example of two allocations in set Y (i.e., a three way split of $100) such that the second Pareto dominates the first.

Question 3

Consider a 2-person/2-good exchange economy in which person 1 is endowed with (e , e ) and person 2 is endowed with (e , e ) of the goods x₁and x₂.

Suppose that the tastes for individuals 1 and 2 can be described by the utility functions u¹=x x and u²= x x (where α and β both lie between 0 and 1). Some of the questions below are notationally a little easier to keep track off if you also denote E₁= (e  + e ) as the economy's endowment of x₁ and E₂= (e  + e )as the economy's endowment of x₂.

(a) Let ₁denote the allocation of x₁to individual 1, and let ₂denote the allocation of x₂toindividual 1. Then use the fact that the remainder of the economy's endowment is allocatedto individual 2 to denote individual 2's allocation as (E₁- ₁) and (E₂- ₂) for x₁and x₂respectively.

Derive the contract curve in the form ₂= x₂( ₁)-i.e. with the allocation of x₂to person 1 as a function of the allocation of x₁to person 1.

(b) Simplify your expression under the assumption that tastes are identical - i.e. α = β. Whatshape and location of the contract curve in the Edgeworth Box does this imply?

Question 4

Consider an economy in which two people, Dinah and Joe, exchange two goods, ale (A) and bread (B). Let p be the price of ale measured in bread.

a) Dinah's utility function is U_D = (a_D)^1/2 +b_D wherea_D and b_D are her consumption of ale and bread, respectively. She is endowed with 10 loaves of bread but no ale. Find

i) Dinah's budget constraint, ii) her best attainable (u-maximizing) commodity bundle, and iii) her excess demand for ale.

b) Joe's utility function is UJ = (a_J)^1/2 +b_J wherea_J and b_J are his consumption of ale and bread, respectively. Joe has 18 pints of ale but no bread. Find

i) Joe's budget constraint, ii) his best attainable (U-maximizing) commodity bundle, and iii) his excess demand for ale.

c) Given that p is the price of ale measured in bread, find the market-clearing value of p. Find the quantity of each good traded, and ?nd each person's actual consumption of ale and bread.