Problem Set

PROBLEM SET 10 – due Thu at 11pm



  1. Consider Anna, who gets a regular transfer from her parents that equals 0 dollars per working da She also works L hours per day, for hourly wage w. Let’s group everything she buys under a “composite good” x, with price .


Anna has the following preferences over leisure (Le) and consumption of other goods (x):


( , )  =   0.75 0.25

Where leisure = 24 −


  1. Write down Anna’s budget constraint: an equation that expresses the tradeoff the consumer must make between, in this case, leisure and other

(Hint: you can read the constraint from the graph)


  1. Solve Anna’s optimization problem to find the demand functions ( , , 0)and ( , , 0): how much good x and how much leisure does she purchase at different possible values of the wage, etc? (Hint: think of this as a typical utility maximization problem: calculate MRS, set the tangency condition, )


  1. Let’s say 0=0 (there is no unearned income). What are the expressions for x and Le? Does leisure depend on wage w?

In what way is relevant?


You can set = 1 from here on.


  1. Now let 0=$100. How do leisure Le and labor L vary with w?


  1. Keeping 0 at $100, let w=$ What values do you get for x and Le, and how much does Anna work? How do you interpret these results?


  1. What does the wage need to be in order for Anna to work 3 hours/day (keeping 0=100)?



  1. Just like Anna, Nancy also has a job that pays her an hourly Her utility over leisure (Le) and other things (x) is:




Assume for simplicity that = 1

= 260 ln( ) +



  1. If the money Nancy earns is all she has to spend, solve her utility maximization problem to derive Le and x as functions of


  1. How many hours does Nancy work if she gets paid an hourly wage w=$20?


  1. Her grandmother Olga is worried that Nancy works too much, so she starts sending her transfers that amount to $100/day. How much will Nancy work now? (You can continue to assume w=$20)


  1. [You don’t need to turn in, but think about it:]

Is there some other (legal) scheme Olga could implement to get Nancy to work less? Assume Olga is able to see how much Nancy is getting paid each day.

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