Problem 4. [23 TOTAL POINTS] Consider a consumer faced with a vice good such as vodka which he is tempted to consume rapidly. In period 0, the consumer can buy a large 2-serving bottle or a small 1-serving bottle of vodka and plan consumption over the next periods. There are 3 options:

Plan A: Buy the small bottle in period 0 and consume 1 serving in period 1. Plan B: Buy the large bottle in period 0, consume 1 serving in period 1, and consume

1 serving in period 2.

Plan C: Buy the large bottle in period 0 and consume 2 servings in period 1.

Vodka gives immediate utility when consumed but reduces utility next period. The large bottle has a lower per-serving price so consuming from it gives more utility. The table below shows the utility of each of the consumption plans:

2

Plan Period 1 Utility Period 2 Utility Period 3 Utility A 5 -4 B 6 2 -4 C 12 -13

Assume that consumers have quasi-hyperbolic preferences and = 0.9. Analyze the optimal consumption plan of an exponential agent with = 1 and a quasi-hyperbolic agent with = 0.5.

a. Assuming it is Period 0, calculate the discounted utility of each plan for the exponen- tial agent. What size of vodka bottle would he purchase? [3 points]

b. Assuming it is Period 0, calculate the discounted utility of each plan for the quasi- hyperbolic agent. What size of vodka bottle does a naive quasi-hyperbolic agent purchase? [5 points]

c. Assume it is Period 1 and the large vodka bottle was purchased in Period 0. Calculate the discounted utility of plans B and C for the exponentially discounting agent. Will he consume one or both servings? [5 points]

d. Assume it is Period 1 and the large vodka bottle was purchased in Period 0. Calculate the discounted utility of plans B and C for the quasi-hyperbolically discounting agent. Will he consume one or both servings? [5 points]

e. Assume it is period 0 and there is a sophisticated quasi-hyperbolically discounting agent. The sophisticated quasi-hyperbolic agent can foresee that he wont be able to resist consuming two servings in period 1 if he buys the large bottle in Period 0. Therefore, based on this insight, he reasons that his choices in Period 0 are really Plan A and Plan C. Which size of bottle would he purchase? [5 points]