Labor Economics - Problem Set 2
1. Workers are all equally productive with preferences
Ui(C, A) =C + αi A.
Assume that αi is uniform [0,1].
Profits of firms can be written as
f - βjA - WA
where βj is the cost of providing the amenity and WA is the wage they pay depending on the amenity. Suppose βj is uniform [β, β-] figure out the two wages and the fraction of jobs that have the amenity (as a function of f, β, and β-).
2. Same as 1, but rather than assuming βj is uniform assume that log (βj) it has a logistic distribution:
Pr (log(βj) ≤ b) = eb/1 + e = 1/e-b + 1
In this case you don't need to solve it numerically, just write down the formula to solve for the compensating differential ? = W0 - W1.
3. Same as 2 but now let αi have a distribution of
log(αi) =a + εi
where εi is logistic.
4. Go back to the conditions in Problem 1 but assume further that βj is uniform [0, 1] Wages now depend on both θi and and A, call that WA(θi). Assume that εi is independent of θi. What do WA(θi) and the fraction of each θi doing the job with amenity A look like? In the cross section what will be the relationship between WA and WB on average?
5. Now let's change utility to log utility
Ui(C, A) = log(C) + αiA
and assume that θi = 0.5 with probability ½ and θi = 1 with probability ½. Now f = 1. Continue to assume that αj and βj are uniform [0, 1]. Now what are W0(θi), W1(θi) and the fraction of each θi doing the job with amenity A = 1? Which ability type is more likely to work with the job amenity. In this case you are going to have to figure out a numerical way to solve for the wages. Also keep in mind that it is one set of firms that is hiring both types of workers so the marginal firm will end up being indifferent between four things which type of worker × where to offer A.
6. Now continue to think of log utility and that βj it has a uniform distribution, but now get rid of heterogeneity across individual in terms of preferences and ability and assume αi = 0.5 for everyone. In a standard compensating differential model what are the wages and proportion doing the two jobs? Now suppose you are a worker facing those wages. What fair lottery would maximize their expected utility? Calculate the expected value with and without the lottery.