Now continue to think of log utility and that betaj it has

Question

Labor Economics - Problem Set 2

1. Workers are all equally productive with preferences

U_i(C, A) =C + α_i A.

Assume that α_i is uniform [0,1].

Profits of firms can be written as

f - β_jA - W_A

where β_j is the cost of providing the amenity and WA is the wage they pay depending on the amenity. Suppose β_j 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 β_j is uniform assume that log (β_j) it has a logistic distribution:

Pr (log(β_j) ≤ b) = e^b/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(α_i) =a + ε_i

where ε_i is logistic.

4. Go back to the conditions in Problem 1 but assume further that β_j is uniform [0, 1] Wages now depend on both θ_i and and A, call that W_A(θ_i). Assume that εi is independent of θ_i. What do W_A(θ_i) and the fraction of each θ_i doing the job with amenity A look like? In the cross section what will be the relationship between W_A and W_B on average?

5. Now let's change utility to log utility

U_i(C, A) = log(C) + α_iA

and assume that θ_i = 0.5 with probability ½ and θ_i = 1 with probability ½. Now f = 1. Continue to assume that α_j and β_j are uniform [0, 1]. Now what are W₀(θ_i), W₁(θ_i) 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 β_j it has a uniform distribution, but now get rid of heterogeneity across individual in terms of preferences and ability and assume α_i = 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.