Question 1. Altruism and the prodigal son:

Consider the prodigal son problem. Suppose that U(x₁, x₂) = u(x₁) /βu(x₂), where u(.) is a well behaved function of one variable and β is a discounting factor, β ≤ 1. All other notation is as in the notes. Parts 1-3 are worth 15% each and part 4 is worth 25%.

(a) Why do we expect that ∂g/∂x₁ > 0. Explain in words and using the parents' FOC from the notes.

(b) Show (by writing down and solving the problem) that the planner's solution to the intergenerational problem for the allocation of x₁ and x₂ is:

ux₁/ux₂ = β/p₂

Will the same allocation arise from the family's three stage game?

(c) Derive the planner's efficient solution to the allocation of x₂ and x₃. Is the allocation the same as the one that results from the family's three stage game?

(d) Now suppose the parents' utility function is V(x₂, x₃, U) with V_x2 > 0 and V_x2,_x2 < 0 (so V is increasing and concave in x₂ as well as in x₃ and U): x₂ is a merit good to parents. Now derive the planner's solution for the allocation of x₁ and x₂ and compare it with the solution from the three stage game. Is the family's allocation in the three stage game closer or further from the planner's efficient allocation than in part (b)? How can you tell?

Question 2: Utility maximization and the single woman

(a) Jane lives alone and has a preference function U(Z₁, Z₂, Z₃) = Z₁^α, Z₂^β, Z₃^1-β where α ∈ (0, 1), β ∈ (0, 1), Z₁ is eating, Z₂ is hiking in the woods, and Z₃ is going to Raptors games. Each commodity Z has a production function Z_i = f_i(X_i, t_i) = X_i^γit_i^1-γi where X_i are market inputs into commodity i, ti is the time input into commodity γ_i ∈(0,1) and i ∈ {1,2,3} indexes the three commodities Z. What are some reasonable guesses at the values of γ₁, γ₂ and γ₃?

(b) Jane works as a lawyer in big law firm. After years of work as an associate, she is finally promoted to partner and her wage goes up 50%. Jane subsequently changes her Match.com profile from "Seeking a confident, financially secure, responsible man" to "Seeking a man who loves basketball and knows how to cook!". Explain why Jane's dating priorities have changed.