Harper is spending a three-day weekend at a beach property. Upon arrival, Harper bought a quart of ice cream and must divide consumption of the quart over each of the three days. Her instantaneous utility of ice cream consumption is given by U(c) = c.5, where c is measured in quarts, so that the instantaneous marginal utility is given by 0.5c-0.5.
a. Suppose Harper discounts future consumption according to the fully additive model, with the daily discount factor δ = 0.8. Solve for the optimal consumption plan over the course of the three days by finding the amounts that equate the discounted marginal utility of consumption for each of the three days, with the amounts summing to 1.
b. Now suppose that Harper discounts future utility according to the quasi-hyperbolic discounting model, with β = 0.05, and δ = 0.8. Describe the optimal consumption plan as of the first day of the weekend. How will the consumption plan change on day two and day three?
c. The model thus far eliminates the possibility that Harper will purchase more ice cream. In reality, if consumption on the last day is too low, Harper might begin to consider another ice cream purchase. Discuss the overall impact of hyperbolic discounting on food consumption or on the consumption of other limited resources.