Benefit Cost in a Benevolent Society Ted Bergstrom, UCSB

A Love Story? Alice and Bob live together. They are fond of each other, but maintain separate budgets. They are offered a new apartment, with two extra rooms. A study for Alice. A lounge for Bob.

Benefits and costs Alice never uses lounge, Bob never uses study. Study is worth $100 a month to Alice and lounge is worth $100 a month to Bob. Alice is willing to pay $50/month for Bob to have lounge. Bob is willing to pay $50 per month for Alice to have study. Extra rent on new apartment is $250/month.

Should they take the apartment? Suppose they take it and split the rent. Alice pays $125, gets a study worth $100 to her. Is there consolation for her in Bob’s benefits? Bob pays $125, gets lounge worth $100 to him.

Altruism in Carolina Viscusi interview: Greensboro, N.C. willingness to pay for product that reduces health risk. Value X if it benefits self only. What if it benefits all N. Carolinans? 5X All Americans? 6X Georgians only? Nothing. They should pay for it themselves.

What can benefit cost do? Without explicit instructions about how to compare one person’s benefits with those to another, benefit cost cannot tell us whether a project should be adopted. Best we can expect is to learn whether a project is potentially Pareto improving. But this is very useful.

Utility Possibilities and Benefit Cost

A Justification for B C With selfish individuals, a project is potentially Pareto improving if and only if the sum of individual willingnesses to pay for the project exceeds its cost. Simple powerful result, but requires a new BC test for every project.

Samuelson condition With selfish preferences, if preferences are convex, then a necessary condition for an increase in the amount of a public good to be potentially Pareto improving is that sum of mrs’s is at least mc.

Benevolence and BC Utility functions U i (v 1 (x 1,y),…,v n (x n,y)) where v j represents j’s private preferences and the U’s are nondecreasing in v’s. Trivial (but useful) result: Pareto improvement in v’s implies Pareto improvement in U’s. With benevolence, “private Pareto improvement” is a sufficient condition for a Pareto improvement.

Sufficient but not necessary

Distributional efficiency An allocation (x,y) is distributionally efficient if no Pareto improvement is possible by simple redistribution. Figure 3—Even for distributional efficient Allocations, passing private BC test is not necessary for Pareto improvement.

Distributional Efficiency?

A marginal necessary condition Private values Samuelson test Does sum of private mrs for a public good exceed mc? If yes, then passes. If no, then fails.

Assumptions for Theorem A.1 Altruistic preferences of form U i (v 1 (x 1,y)…v n (x n,y) where U is quasi- concave, differentiable in the v’s. A.2 Private utility functions are concave, differentiable, increasing in private goods. A.3 Cost function C(y) is convex.

Theorem Given A.1-A.3, if initial allocation is distributionally efficient, then passing Samuelson condtion is a necessary condition for an increase in the amount of a public good to be potentially Pareto improving.

Intuition Not every Pareto improvement is “privately Pareto improving” but under the assumptions of the theorem, if a Pareto improvement can be achieved with an increase in y, then it must also be true that a “private Pareto improvement” can be achieved with this increase.

Applications and Discussion Contingent valuation studies Risk to health and life and limb studies Animals and wilderness studies

Conclusion That’s all