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A detailed analysis of consumer behavior in shopping and consumption, focusing on the impact of price variation, transportation and storage costs, and inattention. It introduces a model of inattention and its implications for revenue maximization in auctions, and discusses the perception of round numbers and its effect on pricing. It also examines a case study of used car auctions and the role of inattention in buyers' decision-making.
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There are two (long) questions on the exam. Please answer both to the best of your ability. Do not spend too much time on any one part of any problem (especially if it is not crucial to answering the rest of that problem), and don’t stress too much if you do not get all parts of all problems.
Question 1
This question is long, and takes a bit of notational set up and investment in comprehension of the setting and all the variables. But it concerns good old-fashioned present bias. The basic issue is shopping and consumption behavior in the face of known price variation of a good, and possible transportation and storage costs. The potential for overconsumption is, for simplicity, solely because spending now reduces future consumption (rather than health costs).
Soo Hong can shop during the day on one or both of two days, for consumption of candied widgets on one or both of those nights. He can shop or not during the 12-hour day, but cannot consume. He can consume or not during the 12-hour night, but cannot shop.
Soo Hong has present bias (^ ≡ +
P =+1 − ) with ∈ [0 1] and degree of sophis-
tication b ∈ [ 1] and no long-term discounting: = 1 Importantly: the periods are 12 hours long, not 24-hour days. So: whenever Soo Hong is shopping (during the day) consumption is in the future, NOT the present. When choosing consumption at night, he can only consume if he has candied widgets home with him. And as formalized below, whenever Soo Hong is choosing consumption at night all costs are either in the past or future.
On each night, Soo Hong can consume 0, 1, or 2 candied widgets. (He gets no utility from consuming more than 2.) His utility on each night is ≥ 0 from consuming one unit, and + from 2 units, where 0 ≤ ≤ (So Soo Hong never gets more utility from the second widget on a night than from the first.)
The cost of going to the store each day is ≥ 0 independent of how much Soo Hong buys. So, if 0 Soo Hong might have a motive for storing goods. But there might be a linear storage cost between Night 1 and Night 2: the cost of storing a candied widget is ≥ 0 per widget. So if Soo Hong buys 3 units on day 1, and eats 2 of them at night, his storage costs will be If he buys 4, and stores 2, his storage costs will be 2 Clearly, if is high, Soo Hong might choose to shop on each of two days
Soo Hong faces an obvious additional cost: the price of candied widgets. These prices can vary across the two days; this is a major point of the question. The prices are 1 0 per unit on day 1, and 2 0 per unit on day 2. These are standard linear prices, so buying two units on day 1 costs 2 1 etc. The prices might be the same, or either can be bigger than the other.
We describe Soo Hong’s behavior by the amount he purchases each of the two days, 1 and 2 and how much he consumes each night, 1 and 2 and whether he goes to the store each day, 1 ∈ { 0 1 } and 2 ∈ { 0 1 } and the storage between day 1 and day 2, (Note that, in practical terms, and 2 are determined by the other values.)
For simplicity, we assume all monetary, storage, and travel costs are in the future. So the instantaneous utilities for Soo Hong on each of the 5 relevant days are:
1 = 0
Again: in what is below we don’t directly observe parameters or b nor behavior ( 1 2 ) Only prices and purchase behavior.
e) Suppose that we observe both of the following: when ( 1 2 ) = ($4 $4) then ( 1 2 ) = (1 1) and that when ( 1 2 ) = ($4 $7) then ( 1 2 ) = (2 1)
In all parts below, your answer should be either we cannot say which of the two cases is true, or–if we can say–say (without explanation) which of the two cases holds. Whenever asked whether we can say whether a parameter is infinite, we of course mean to ask about whether it could be very, very large; assume other parameters are not arbitrarily large so that we are really asking whether we can put any limit on the size of this parameter. And since these are all yes-no questions and you have limited time, we urge you to think carefully about the logic of the problem rather than develop a lot of algebra.
e1) Can we say whether 1 or = 1? e2) If your answer above is that we know 1 can we distinguish b = 1 vs. b = ? e3) Can we say whether = or ? e4) Can we say whether = 0 or 0? e5) Can we say whether ∞ or = ∞? e6) Can we say whether = 0 or 0? e7) Can we say whether ∞ or = ∞? e8) Give an intuition (tightly argued!) for the set of restrictions if any we can conclude in parts (1) to (7) above.
f) Suppose that we observe both of the following: when ( 1 2 ) = ($4 $5) then ( 1 2 ) = (3 0) and that when ( 1 2 ) = ($4 $9) then ( 1 2 ) = (2 0)
f1) Can we say whether 1 or = 1? f2) If your answer above is that we know 1 can we distinguish b = 1 vs. b = ? f3) Can we say whether = or ? f4) Can we say whether = 0 or 0? f5) Can we say whether ∞ or = ∞? f6) Can we say whether = 0 or 0? f7) Can we say whether ∞ or = ∞? f8) Give an intuition for the set of restrictions if any in parts (1) to (7) above.
g) Suppose that we observe both of the following: when ( 1 2 ) = ($4 $4) then ( 1 2 ) = (1 1) and that when ( 1 2 ) = ($4 $6) then ( 1 2 ) = (3 0)
g1) Can we say whether 1 or = 1? g2) If your answer above is that we know 1 can we distinguish b = 1 vs. b = ? g3) Can we say whether = or ? g4) Can we say whether = 0 or 0? g5) Can we say whether ∞ or = ∞? g6) Can we say whether = 0 or 0?
g7) Can we say whether ∞ or = ∞? g8) Give an intuition for the set of restrictions if any in parts (1) to (7) above.
h) Suppose that we observe both of the following: when ( 1 2 ) = ($5 $5) then ( 1 2 ) = (0 0) and that when ( 1 2 ) = ($3 $7) then ( 1 2 ) = (4 0)
h1) Can we say whether 1 or = 1? h2) If your answer above is that we know 1 can we distinguish b = 1 vs. b = ? h3) Can we say whether = or ? h4) Can we say whether = 0 or 0? h5) Can we say whether ∞ or = ∞? h6) Can we say whether = 0 or 0? h7) Can we say whether ∞ or = ∞? h8) Give an intuition for the set of restrictions if any in parts (1) to (7) above.
i) Suppose that we observe both of the following: when ( 1 2 ) = ($5 $5) then ( 1 2 ) = (2 2) and that when ( 1 2 ) = ($5 $4) then ( 1 2 ) = (1 1)
i1) Can we say whether 1 or = 1? i2) If your answer above is that we know 1 can we distinguish b = 1 vs. b = ? i3) Can we say whether = or ? i4) Can we say whether = 0 or 0? i5) Can we say whether ∞ or = ∞? i6) Can we say whether = 0 or 0? i7) Can we say whether ∞ or = ∞? i8) Give an intuition for the set of restrictions if any in parts (1) to (7) above.
f) Table 4 in Hossain and Morgan (2007) presents the revenue raised for treatment C (shipping cost = 2) and treatment D (high shipping cost. = 6). Using the information on the average revenue raised, provide an estimate for ˆ.(exclude the unsold item). Provide an explanation for why ˆ may be lower in this case (other than because of sampling error)
g) Consider now the Chetty et al. (AER) paper on taxation and inattention. As you recall, the setting considered there is one in which consumers may be inattentive with respect to the tax, so = and = which is the price of an item pre-tax. Without going into details, explain the strategy to identify the inattention parameter in that paper, and compare it to the strategy used to identify in the Hossain and Morgan (2007) paper.
h) Finally, consider a related model of inattention which refers to the perception of round numbers. A seller sells one unit of a good produced at zero marginal cost. The sale price is which we can write as the sum of $ and cents, that is, = + 01 , with ∈ { 0 1 99 }. The buyers are attentive to the dollars, but inattentive to the cents. That is, plays the role of above and plays the role of . Briefly discuss the plausibility of this assumption. How does an inattentive consumer perceive a price = + 01 ? Plot the perceived price ˆ as a function of the true price .
i) A monopolist seller maximizes profits from the sale of the one unit knowing that the consumer will only buy if ˆ ≤ ¯ = ¯ + 01¯ a reservation price. What pattern of pricing will we observe with full inattention, that is, = 1? Will the pattern persist for intermediate inattention (0 1)? Characterize the solution.
j) Lacetera, Sydnor, and Pope (2009) consider a related case in which buyers of used cars pay full attention to the first digit of kilometers from the left in the odometer, but are inattentive with respect to the other digits. The attached Figure plots the distribution of auction prices for used cars on a wholesale auction sites (where the buyers are car dealers
who will then re-sell the car to used car buyers) as a function of the number of miles on the odometer. Explain in as much detail as you can to what extent these findings refer to inattention, and approximately the amount of inattention that this graph suggests.
