Exam 32: Welfare

Suppose that Paul and David have utility functions U = 4A P + O P and U = A D + 4O D, respectively, where A P and O P are Paul's consumptions of apples and oranges and A D and O D are David's consumptions of apples and oranges. The total supply of apples and oranges to be divided between them is 20 apples and 12 oranges. The fair allocations consist of all allocations satisfying the following conditions.

Suppose that Romeo has the utility function U = S⁷_RS³_J and Juliet has the utility function U = S³_RS⁷_J, where S_R is Romeo's spaghetti consumption and S_J is Juliet's. They have 60 units of spaghetti to divide between them.

Suppose that Paul and David have utility functions U = 4A P + O P and U = A D + 5O D, respectively, where A P and O P are Paul's consumptions of apples and oranges and A D and O D are David's consumptions of apples and orange. The total supply of apples and oranges to be divided between them is 20 apples and 14 oranges. The "fair" allocations consist of all allocations satisfying the following conditions.

The utility possibilities frontier is the boundary of the production possibilities set.

A parent has two children living in cities with different costs of living. The cost of living in city B is 3 times the cost of living in city A. The child in city A has an income of $2,000 and the child in city B has an income of $6,000. The parent wants to give a total of $2,000 to her two children. Her utility function is U(C A, C B ) = C A C B, where C A and C B are the consumptions of the children living in cities A and B respectively. She will choose to give

An allocation is fair if whenever one person envies another, the envied person does not envy the envier.

A tiny fishing village has 3 residents. Ann has a utility of 10, Bruce has a utility of 6, and Charlie has a utility of 7. If the mayor uses a Rawlsian social welfare function, the social welfare of this tiny village would be

Suppose that Romeo has the utility function U = S⁵_RS¹_J and Juliet has the utility function U = S¹_RS⁵_J, where S_R is Romeo's spaghetti consumption and S_J is Juliet's. They have 36 units of spaghetti to divide between them.

According to Arrow's impossibility theorem, it is impossible to find a social ordering that is complete, reflexive, and transitive.

No one is meaner and uglier than Gladys. Someone is meaner and uglier than Harold. Therefore Gladys is meaner and uglier than Harold. Is this reasoning correct? If so, explain why. If not, explain why not. (Assume that people can be ranked from ugliest to least ugly by a complete transitive ordering and that there are no ties. Likewise assume that people can be ranked from meanest to least mean by a complete transitive ordering and that there are no ties.)

Hatfield and McCoy burn with hatred for each other. They both consume corn whisky. Hatfield's utility function is and McCoy's utility is , where W_H is Hatfield's whisky consumption and W_M is McCoy's whisky consumption, measured in gallons. The sheriff has a total of 58 gallons of confiscated whisky which he could give back to them. For some reason, the sheriff wants them both to be as happy as possible and he wants to treat them equally. The sheriff should give them each

An allocation that is worse for somebody than the initial allocation cannot be Pareto optimal.

If a social welfare function is an increasing function of each person's utility, then every allocation that maximizes this social welfare function must be a Pareto optimum.

In a competitive equilibrium, no matter how different their preferences may be, no two people with the same income will envy each other's consumption bundles.

In a pure exchange economy if the initial allocation is Pareto optimal, then competitive equilibrium is fair.

In a pure exchange economy, if an allocation is Pareto efficient, it is impossible to have two people who prefer each other's consumption bundles to their own.

A parent has two children living in cities with different costs of living. The cost of living in city B is 4 times the cost of living in city A. The child in city A has an income of $3,000 and the child in city B has an income of $12,000. The parent wants to give a total of $1,000 to her two children. Her utility function is U(C A, C B ) = C A C B, where C A and C B are the consumptions of the children living in cities A and B respectively. She will choose to give

A Borda count is used to decide an election between 3 candidates, x, y, and z, where a score of 1 is awarded to a first choice, 2 to a second choice, and 3 to a third choice. There are 29 voters. 10 voters rank the candidates x first, y second, and z third; 3 voters rank the candidates x first, z second, and y third; 8 voters rank the candidates z first, y second, and x third; 8 voters rank the candidates y first, z second, and x third. Which candidate wins?

A Borda count is used to decide an election between 3 candidates, x, y, and z, where a score of 1 is awarded to a first choice, 2 to a second choice, and 3 to a third choice. There are 14 voters. 6 voters rank the candidates x first, y second, and z third; 3 voters rank the candidates x first, z second, and y third; 3 voters rank the candidates z first, y second, and x third; 2 voters rank the candidates y first, z second, and x third. Which candidate wins?

If allocation x is Pareto optimal and allocation y is not, then everyone is at least as well off with x as with y, and someone is better off with x than with y.