# Which options Dan cannot choose if he wants to be

## Which options Dan cannot choose if he wants to be a benevolent dictator?

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Exercise 3.1. Benevolent Dictators

Two friends, Bob and Ann, cannot agree on how to spend the evening. To solve this problem, the ask a friend, Dan, to act as a benevolent dictator. There are four things to   do.

{ Cinema   Football   Television   Theatre }

Ann and Bob`s preferences are given  by

UA(Cinema)1U            B(Cinema)4

UA(Football)3U            B(Football)1

UA(Television)2.5U      B(Television)0.5

UA(Theatre)1.5U          B(Theatre)2 Answer the following questions. (Hint: draw the set of feasible   utilities)

◼Which options Dan cannot choose if he wants to be a benevolent dictator?

◼Which options should he choose to maximize a utilitarian Social Welfare  Function?

◼Which options should he choose to maximize a Rawlsian Social  Welfare  Function?

◼Which option he cannot choose to maximize a Bergsonian  social  welfare  function?

◼Among the options you just excluded, is the one which is Pareto  efficient?

◼Change either Ann or Bob utility function so that you obtain different results. For example, can you change their utilities so that all SWF choose   Theater?

#### Exercise 3.3. Contract Curves

Robinson and Friday consume coconuts and fish. They have 10 units of fish and 5units   of coconuts. They have the same preferences that are represented by the following utility function

 f

U(c,f)c+

◼Draw the Edgeworth box and the corresponding contract curve. Be careful: the contract curve goes through the points in which Robinson or Friday consume the entire endowment of the two goods. The contract curve is also   continuous...

◼Would it be correct to say that, within limits, there is a distribution of fish which      is optimal, independently from the way we divide coconuts? What  are  these  limits?

Exercise 3.4. Social Welfare Functions. Solve Kreps, prob. 2 Chapter 5. Page 182,183 a&b.

(Hint: before you do any calculation, sketch the level curve for the SWF of the exercise. You will see that these curves have a kink on the line in which the utilities of the two individuals are the same.)