Question

Bill has received an endowment ω > 0 of a perfectly divisible consumption good from his rich relatives. Bill is Canadian — so we think of ω as a certain quantity of maple syrup, the only thing Canadians need to survive and thrive in this world. Now, Bill cares a great deal about Denise, and if he consumes an amount x of his endowment of maple syrup, Denise receives the remaining ω−x, as long as this last quantity is non-negative. Note we limit the amount of x that Bill can consume to be 0 ≤ x ≤ ω. In other words, Bill has no access to credit markets to expand his (or Denise’) consumption beyond ω. In the language of our rational choice model, Bill’s choice set is the closed interval X = [0,ω] ⊆ R. To capture the fact that Bill cares about Denise, we want to incorporate her wellbeing (expressed here as her consumption of maple syrup) directly into his utility function. This class of preferences are called ‘altruistic’ or ‘other-regarding’ preferences. Assume that Bill’s utility function is u(x) = Axα(ω−x)1−α. Here α ∈ [0,1] measures how much weight Bill gives to his own share of ω. The higher the value of α, the less altruistic Bill feels with respect to Denise, because her consumption of maple syrup is assigned less weight. The parameter A > 0 is just a constant. We include it to emphasize that Bill’s utility function is just an ordinal measure of his preferences, thus any A > 0 will yield the same result (but not the same value for the utility).
(a) Show that when α = 1 Bill’s utility function is strictly increasing in x. Draw a picture with x in the horizontal axis and u(x) in the vertical axis. Remember that X = [0,ω].1
(b) Show that when α = 0 Bill’s utility function is strictly decreasing in x. Draw a picture here too.

(c) Showthatwhen α ∈ (0,1), Bill’sutilityfunctionisstrictlyconcaveon (0,ω). Draw a picture here too. Hint: obtain the second derivative of u(·) and verify that u′′(x) < 0 on (0,ω).

Answer 1