Consider the plight of two equally skilled economists, Ann and Bill. Ann lives in Ohio, where she pays a monthly rent of $1000 for a 1000 square foot apartment. Bill lives in Oakland in an 800 square foot apartment that is otherwise identical to Ann's but which costs him $2000 per month. Assume that the only other good that Ann and Bill consume is food, which costs $10 per unit in Ohio and only $5 per unit in Oakland.

a) Suppose that Ann and Bill both have ordinal utility functions U(h, z) = α log(h) + (1 - α) log(z), with 0 ≤ α < 1, where h is housing services (measured in square feet occupied per month) and z is food. Derive Ann and Bill's Marshallian Demands for housing and food. Give an interpretation of the parameter α ; how does it affect choices between housing services and food?

1. Suppose preferences are as in (a) but with α = 0.25. What are the locally prevailing monthly salaries for economists in Ohio and Oakland?
2. Now suppose that we don't know what Ann and Bill's incomes are equal to (though they each know, and also know each others'), and that further we don't know what the value of α is, and Ann's α may be different from Bill's α. Either Ann or Bill could choose to move to the other's location and get a job as an economist there at the prevailing local salary, yet they choose not to. What can you say about the value of Ann's α relative to Bill's α? (Hint: Ann's indirect utility in Ohio must be greater than her indirect utility would be in Oakland; similarly, Bill must be happier in Oakland than in Ohio.)
3. A pollster surveys a random sample of Ohio residents and Oakland residents, and asks each respondent, "Would you say that you're (1) Very happy; (2) Pretty happy; or (3) Not so happy?" After tabulating the results, the pollster finds that on average people in Ohio are just as happy as people in California. Noting that the cost of living in Ohio is lower than in California, she argues that many Californians should move to Ohio, where they could be just as happy but spend less money. Comment on the quality of this argument. Are you tempted to move?