Assessment Task:

A consumer lives for T + 1 periods, t = 1 through t = T + 1, where T ≥4. She decides in each of the periods 2 through T whether to hit. Hitting in a period t gives an instantaneous pleasure of 4 in period t, and generates an instantaneous cost of 3 in period t + 1. The price of a hit in the market is 2, which has to be paid immediately in any period in which the consumer hits. (Note that in the model considered in class, we did not have such a price for hitting.) The consumer is a hyperbolic discounter with β =1/2 and δ = 1.

(a) (12p) Show that both if the consumer is sophisticated and if she is naive, she hits in every period 2 through T. Show that all selves would be better off with a lifetime of no hitting.
(b) (12p) Now consider a system of advance orders. In each period t = 1, ... , T −1, the consumer can submit an order for a hit in period t + 1. If she submits an order in period t, a hit is delivered to her in period t + 1, and she must pay for the hit on delivery. If she does not submit an order in period t, she cannot hit in period t + 1.
What does a naive consumer do? What does a sophisticated con-sumer do? Is the system of advance orders effective in preventing harmful overconsumption? If not, can you think of an improvement to the system? Explain the intuition for your answers.
(c) (10p) Suppose that in period 1, the consumer has to decide (e.g. through voting) whether to set up a system of advance orders as in part (b) or have a spot market as in part (a). The cost of setting up the system of advance orders is 1. Does a naive consumer choose to set up advance orders? How about a sophisticated consumer? Explain the intuition.

d) (8p) Now suppose that the pleasure from hitting is uncertain. In each period 2 through T, the pleasure is 3 with probability one-half and 8 with probability one-half, and the pleasures in different periods are independent from each other. The consumer learns the pleasure from hitting in period t at the beginning of period t, and knows only the above two possibilities and their probabilities beforehand. The price of a hit is still 2, and the utility cost in the next period is still 3. Without a system of advance orders, what does each type of consumer do?
(e) (10p) Now consider advance orders in the setting in part (d) with uncertainty in the pleasure from hitting. What does each type of consumer do with advance orders? Do advance orders prevent harmful overconsumption? If not, can you think of an improvement to the system? Explain the intuition for your answers

 

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  • Posted on : November 08th, 2018

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