3. Lucy and Melvin share an apartment. They

spend part of their income on private goods

such as food and clothing, which they

consume separately, and part of their income

on public goods such as the refrigerator,

heating and rent, which they share. Lucy's

utility function is 2XL+G and Melvin's utility

function is XM6, where XL and XM are the

sums of money spent on private goods for

Lucy and Melvin, and G is the sum of money

spent on public goods. Lucy and Melvin have a

total of $8,000 a year to spend on private

goods for each of them and on public goods.

a. What is the absolute value of the marginal

rate of substitution between private and public

goods for Lucy? What is this value for Melvin?

Write an equation to calculate the Pareto

efficient quantity of public goods.

b. Suppose Melvin and Lucy each spend

$2,000 on private goods and the remaining

$4,000 on public goods. Is this Pareto

efficient?

c. Give an example of another Pareto optimal

outcome in which Melvin receives more than

$2000 and Lucy receives less than $2000 for

their consumption of private goods. Give an

example of another Pareto optimum in which

Lucy receives more than $2000./nd. The Pareto optima that makes Lucy better

off and Melvin worse off will have (more, less,

the same amount) of public goods than the

Pareto optimum that treats them the same.