Lucy and Melvin share an apartment. They spend some of their income on private goods like food and clothing that they consume separately and some of their income on public goods like the refrigerator, the household heating, and the rent, which they share. Lucy’s utility function is 2XL + G and Melvin’s utility function is XMG, where XL and XM are the amounts of money spent on private goods for Lucy and for Melvin and where G is the amount of money that they spend on public goods. Lucy and Melvin have a total of $8,000 per year between them to spend on private goods for each of them and on public goods.
(a) What is the absolute value of Lucy’s marginal rate of substitution between public and private goods? ___________ What is the absolute value of Melvin’s? __________
(b) Write an equation that expresses the condition for provision of the Pareto efficient quantity of the public good.
c) Suppose that Melvin and Lucy each spend $2,000 on private goods for themselves and they spend the remaining $4,000 on public goods. Is this a Pareto efficient outcome?
(d) Give an example of another Pareto optimal outcome in which Melvin gets more than $2,000 and Lucy gets less than $2,000 worth of private goods.
(e) Give an example of another Pareto optimum in which Lucy gets more than $2,000.
(f) Describe the set of Pareto optimal allocations.
(g) The Pareto optima that treat Lucy better and Melvin worse will have (more of, less of, the same amount of) public good as the Pareto optimum that treats them equally.