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Exogenously Determined, Lifetime Utility, Techniques Best, Completely Rational, Projection Bias, Current Preferences, Simple Projection Bias, Consumption Pattern, Same Preferences, Maximizes Intertemporal Preferences are some points from this exam paper.
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There are 3 questions on the exam. Please answer all to the best of your ability. Questions 1 and 3 count somewhat more than Question 2. 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 if you do not get all parts of all problems.
Question # Alec will live forever. Every night t = 1, 2 , ... of his infinite life he chooses between two free meals: he can eat either fish (et = f ) or prawns (et = p). His utility in each period is fully given as follows: u(et = f|et− 1 = p) = 8 u(et = p|et− 1 = p) = 2 u(et = f|et− 1 = f) = 5 u(et = p|et− 1 = f ) = 4. Alec was exogenously determined to have eaten fish in period 0. Alec each period seeks to maximize his expected lifetime utility given by Ut^ ≡
P∞ τ =t δ
τ (^) u τ , with δ < 1. Assume in all parts that δ is very close to 1; Alec is very patient. (Your answer should therefore correspond to what would happen in the limit as δ → 1 , and it is part of the problem to figure what techniques best let you solve for this.) a) If Alec is completely rational, what will be his dinner choice each period of his life from period 1 on? b) If Alec suffers from complete “projection bias”, not realizing in each period that his utility function in all future periods will differ from his current preferences (that is, assume the usual definition of projection bias, where the ‘state’ is the previous period’s consumption choice), what pattern of consumption will we observe? Very briefly give an intuition, and discuss how it does or doesn’t differ from your answer to part (a). What would be the answer if instead of having exogenously eaten fish in period 0 Alec instead exogenously ate prawn? c) Again assuming that Alec exogenously ate fish in period 0, now suppose that Alec suffers from simple projection bias that may not be complete: each period t, Alec thinks that in each future period τ > t he will have instantaneous utility euτ = (1 − α)u(eτ |eτ − 1 ) + αu(eτ |et− 1 ), where α ∈ [0, 1] is a measure of his degree of projection bias. Without worrying about what Alec might do for knife-edge values of α that might generate indifference, state what Alec’s observed consumption pattern will be as a function of α, showing briefly how you reached that conclusion. Pete has the same preferences as Alec each period, and likewise exogenously ate fish in period 0. But Pete does not suffer from projection bias – he is completely aware of what his instantaneous utility function will be each period as a function of contemporaneous and previous consumption. But Pete has self-control problems resulting from present-biased pref- erences, and each period maximizes intertemporal preferences given by Ut^ ≡ ut +β
P∞ τ =t δ τ (^) uτ ,
with β ∈ (0, 1], and δ < 1. As with Alec, Pete is very patient, with δ close to 1 in all parts below. Pete’s degree of sophistication, defined as usual, is given by βb ∈ [β, 1]. d) Suppose Pete is completely naive: βb = 1. As a function of β, what will Pete consume each period of his life? (Per always, don’t worry about behavior in knife-edge cases.) Explain briefly how you reached your conclusion, preferably with clear statements about what options of payoff streams Pete chooses among in period 1 and beyond. e) (Harder) Suppose Pete is completely sophisticated: βb = β. For what values of β does Pete do the fully rational behavior he would do if β = 1 (that is, the behavior you found for Alec in part (a)) in all perception-perfect equilibria? For what values of β does there exist a perception-perfect equilibrium in which Pete does the fully rational behavior he would do if β = 1? How would your answers change if, instead of living forever, Pete merely lives for a very, very long time. I.e., what is the answer if Pete lives for T finite but T → ∞? Explain your answers briefly.
Question # Each year, Americans give over 150 billions of dollars to charities (Andreoni, 2004). Despite the importance of charitable giving, we know little about why people give. One view is that consumers give to charity because they are altruistic and care about worthy causes. A competing view is that giving is mostly the result of professional fund-raisers who put potential givers under social pressure. In this problem you are asked to solve a simple model of fund-raising and discuss, in light of the model, how a field experiment can provide evidence to sort the two explanations (It may be useful to read through the whole problem before working on it) We model the interaction between a giver G and a fund-raiser F. The fund-raiser F visits G’s home. The giver G is at home with probability h 0. If G is at home, she decides a donation g, which can only be either a fixed positive amount ¯g > 0 or zero: g ∈ {¯g, 0 }. If G is absent, g equals 0. The giver G has utility function
u (WG − g) + αv (WF + g) , (1)
that is, she cares about her own consumption, which is WG (the pre-giving wealth) minus the giving g, and about the consumption of people in the charity, which is WF (the wealth of the charity) plus the giving g. The altruism parameter is α ≥ 0. If the maximum donation ¯g is small, assuming u and v smooth, we can use a Taylor approximation around g = 0 and write the utility function (1) as
A + g [αv^0 (WF ) − u^0 (WG)] , (2)
where A ≡ u (WG)+αv (WF ) is a constant that does not affect the giving decision g. [You do not need to derive this, just use expression (2) in the rest of the problem] The final element in the utility function is social pressure. The giver G pays a utility cost S ≥ 0 if she gives zero while the fund-raiser F is present. If the giver G is away when F visits, instead, G pays no social-pressure cost for giving 0. a) Is this a plausible formulation for altruism and social pressure, based on the evidence covered in the Psych and Econ classes? For social preferences, you can refer for example to Mas and Moretti (2007) on the effort of retail workers at the check-out counters. b) Write the maximization problem with respect to g for the case in which G is at home (otherwise the solution is trivially g = 0). Show that G gives if the altruism parameter α is larger than a threshold ¯αS. How does this threshold depend on the social pressure cost S? Provide intuition for how altruism (α) and social pressure (S) affect giving g. Denote the altruism threshold for the case S = 0 as ¯α 0. c) Consider now the case in which the giver G learns one day in advance about the visit of the fund-raiser. The giver can now affect in the first stage the probability of being at home or being out in the second stage, at the time of the visit. In the new game, in the first stage, upon learning of the visit, the giver chooses a probability of staying at home h ∈ [0, 1] at a quadratic cost (h − h 0 )^2 / 2 η. Then, in the second stage if G is at home once the fund-raiser visits, she decides the level of giving g as above (if she is not at home, g = 0 as above). Write down the utility function that G maximizes in the first stage. d) Solve for the level of giving g∗^ and for the optimal probability of being at home h∗. For simplicity, neglect the constraint that h ∈ [0, 1] [Hint: Distinguish the cases α < α¯S , α ¯S ≤ α < α¯ 0 , and α > α¯ 0 ] Provide intuition on the solution.
e) Compare the probability of the giver G being at home h∗^ and the overall level of giving h∗^ ∗ g∗^ in the case without advance notice (point b)) and the case of advance notice (points c) and d)). If you cannot solve this analytically, provide a qualitative answer. Why is the overall giving h ∗ g and not just g? f) Three economists, John, Stefano, and Ulrike, inspired by this model, plan a field experiment as follows. In a first treatment, Fund-Raising with Advance Notice, potential givers are notified one day in advance about the visit of a fund-raiser at their home. In a control group, the Standard Fund-Raising treatment, the visit comes as a surprise (this is the standard procedure). The three economists claim that this field experiment provides a test for the importance of altruism and social pressure in fund-raising. The test consists in comparing giving and the probability that a giver is at home (that is, opens the door) across the two treatments. Discuss in light of your results in point e). g) Consider the market response to the non-standard preferences above (altruism and social pressure). You are not required to do formal derivations here. Assume that the fund- raiser F wants to maximize giving (that is, h ∗ g) and that advance notices are costless. What would the fund-raiser do if giving is driven by pure altruism (α > 0 and S = 0)? What if giving is driven by pure social pressure (α = 0 and S > 0)? What does the absence of advance notices in actual door-to-door fund-raising suggest? h) Comment, again intuitively, on the welfare implications of fund-raising in the cases of pure altruism (α > 0 and S = 0) and pure social pressure (α = 0 and S > 0). i) (Extra credit). The three economists would like to use the results of the above experi- ments to derive quantitative estimates for the altruism parameter α and the social pressure parameter S. They claim that, while the two treatments above are not enough to identify the parameters, a third treatment can in principle allow them to identify the parameters. The third set of treatments, Survey with Advance Notice, is similar to the Fund-Raising with Advance Notice treatment, except that the flyer notifies the resident of a visit asking for collaboration to complete a survey for a varying amount of money ($0 vs. $20 vs. $40). (That is, different givers find flyers that state different amounts) Discuss.
