# Public Economics

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).

Find the Pareto optimal allocations. (Hint: There are 3).

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 = {(xJ,, xT) : xJ ? 0; xT ? 0; xJ + xT ? 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 = {(xJ,, xT, xC): xJ ? 0; xT ? 0 and xC ? 0; xJ + xT + xC ? 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 x1 and 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 ? and ? 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 1 denote the allocation of x1 to individual 1, and let 2 denote the allocation of x2 to individual 1. Then use the fact that the remainder of the economy’s endowment is allocated to individual 2 to denote individual 2’s allocation as (E1 ? 1) and (E2 ? 2) for x1 and x2 respectively.

Derive the contract curve in the form 2 = x2( 1)—i.e. with the allocation of x2 to person 1 as a function of the allocation of x1 to person 1.

(b) Simplify your expression under the assumption that tastes are identical — i.e. ? = ?. What shape 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.

Bread

Ale 0J

0D Ale

Bread

a) Dinah’s utility function is UD = (aD)1/2 +bD

where aD and bD 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 = (aJ )1/2 +bJ

where aJ and bJ 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.

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