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# Behavioral Economics: Problem Set 4 Exercise 4.1 Consider the ...

Behavioral Economics: Problem Set 4 Exercise 4.1 Consider the ...

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<strong>Behavioral</strong> <strong>Economics</strong>: <strong>Problem</strong> <strong>Set</strong> 4<br />

<strong>Exercise</strong> <strong>4.1</strong> <strong>Consider</strong> <strong>the</strong> following simple three–period addiction model. The three periods<br />

are “youth”, “middle age” and “old age”. In each period, <strong>the</strong> person decides whe<strong>the</strong>r or not to<br />

consume an addictive good. The preferences in each of <strong>the</strong> three periods can be represented as<br />

follows:<br />

utility from consuming<br />

utility from abstention<br />

when unhooked 10 0<br />

when hooked -8 -25<br />

A person is hooked in period t, if and only if she has consumed <strong>the</strong> addictive good in period<br />

t − 1. This implies that a single period of restraint gets a person completely unhooked. Suppose<br />

that <strong>the</strong> person is unhooked initially.<br />

a) Determine <strong>the</strong> behavior of a time–consistent person for δ = 1.<br />

b) <strong>Consider</strong> a naive hyperbolic discounter with δ = 1, β = 1/2. When does <strong>the</strong> person choose<br />

to consume <strong>the</strong> addictive good?<br />

c) <strong>Consider</strong> a sophisticated hyperbolic discounter with δ = 1, β = 1/2. When does this<br />

person choose to consume <strong>the</strong> addictive good?<br />

d) Compare <strong>the</strong> behavior of naive and sophisticated persons? Why can <strong>the</strong> awareness of<br />

self–control problems hurt a person?<br />

1<br />

p.t.o.

<strong>Exercise</strong> 4. 2 A consumer lives for T + 1 periods 1, 2, 3, . . . , T + 1. In each period t ∈<br />

{1, 2, . . . , T }, she spends her income y t on apples and potato chips. (Thus, <strong>the</strong>re is no borrowing<br />

or saving.) Denote <strong>the</strong> period-t consumption of apples and potato chips by a t ∈ R + and<br />

c t ∈ R + , respectively. The consumer’s instantaneous utility in period t ∈ {2, . . . , T } is<br />

a t + u(c t ) − ηc t−1 ;<br />

with c 0 = 0 and her utility in period T + 1 is<br />

−ηc T .<br />

The function u(·) is twice continuously differentiable, increasing and concave with lim c→0 u ′ (c) =<br />

∞ and lim c→∞ u ′ (c) = 0, and η > 0. The consumer is a hyperbolic discounter with discount<br />

parameter δ ≤ 1 and present bias of β < 1. That is, consumption of potato chips is “harmful”<br />

because it causes disutility for <strong>the</strong> consumer in <strong>the</strong> period after consumption. The price of<br />

apples is 1, and <strong>the</strong> price of potato chips is p.<br />

a) Find <strong>the</strong> first-order condition for a naive consumer’s consumption of potato chips in<br />

period t. (Assume that y t is large enough to ensure an interior solution.) Prove that a<br />

sophisticated consumer behaves in exactly <strong>the</strong> same way. Explain why behavior does not<br />

depend on <strong>the</strong> degree of sophistication in this model.<br />

b) Show verbally that <strong>the</strong>re is a way to change <strong>the</strong> consumer’s consumption profile to make<br />

all selves better off than with her actual choices. Explain intuitively why <strong>the</strong> decisionmaker<br />

voluntarily chooses <strong>the</strong> inferior consumption path.<br />

c) Suppose self 1 can choose a per-unit potato chips tax τ to be instituted starting in period<br />

2. The proceeds of <strong>the</strong> tax are going to be lump-sum redistributed in each period. What<br />

tax level would she choose? Explain intuitively. Is it possible for a competitive market<br />

instead of a government to provide <strong>the</strong> welfare-improving effects of a tax to <strong>the</strong> consumer?<br />

2<br />

p.t.o.

<strong>Exercise</strong> 4.3 <strong>Consider</strong> a person who can work on two unpleasant tasks, A and B, during<br />

<strong>the</strong> next three days (t = 1, 2, 3). Each task needs two days of work to be completed, a starting<br />

period and a finishing period. When <strong>the</strong> person has completed a task within <strong>the</strong> first three<br />

days he will receive a reward at day four. On each day t = 1, 2, 3 <strong>the</strong> person can take one of<br />

<strong>the</strong> following three actions: working on task A, working on task B or not working. Not working<br />

costs zero. The immediate costs for working on a project j ∈ {A, B} in period t = 1, 2, 3, c jt ,<br />

and <strong>the</strong> delayed rewards, r j , are given in <strong>the</strong> following table.<br />

project c j1 c j2 c j3 r j<br />

A 4 22 45 40<br />

B 8 4 15 25<br />

It is assumed that people do not discount future utilities, i.e. δ = 1. Fur<strong>the</strong>rmore, assume that<br />

when <strong>the</strong> person is a hyperbolic discounter his degree of present biasedness is β = 1/2.<br />

a) Calculate <strong>the</strong> behavior of a time-consistent person.<br />

b) <strong>Consider</strong> a sophisticated hyperbolic discounter. When does this person work and on which<br />

c) Calculate <strong>the</strong> behavior of a naive hyperbolic discounter.<br />

d) Compare <strong>the</strong> behavior of sophisticates and naives. Give a brief intuitive explanation for<br />

<strong>the</strong> different behavior of <strong>the</strong>se two types of hyperbolic discounters.<br />

3