class: center, middle, inverse, title-slide .title[ # Choice under Uncertainty (Lecture 1e) ] .subtitle[ ## EC404; Spring 2024 ] .author[ ### Prof. Ben Bushong ] .date[ ### Last updated February 01, 2024 ] --- layout: true <div class="msu-header"></div> <div style = "position:fixed; visibility: hidden"> `$$\require{color}\definecolor{yellow}{rgb}{1, 0.8, 0.16078431372549}$$` `$$\require{color}\definecolor{orange}{rgb}{0.96078431372549, 0.525490196078431, 0.203921568627451}$$` `$$\require{color}\definecolor{MSUgreen}{rgb}{0.0784313725490196, 0.52156862745098, 0.231372549019608}$$` </div> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ TeX: { Macros: { yellow: ["{\\color{yellow}{#1}}", 1], orange: ["{\\color{orange}{#1}}", 1], MSUgreen: ["{\\color{MSUgreen}{#1}}", 1] }, loader: {load: ['[tex]/color']}, tex: {packages: {'[+]': ['color']}} } }); </script> <style> .yellow {color: #FFCC29;} .orange {color: #F58634;} .MSUGreen {color: #14853B;} </style> --- class: inverseMSU name: Overview # Today ### **Our Outline:** (1) [Introduction to Modern Prospect Theory](#intro) (2) [Köszegi and Rabin's Model with Two Goods](#sec1) (3) [A (Not Easy) Exercise](#example) (4) [Köszegi and Rabin's Model with Risky Choice](#sec2) (5) [Application: Labor Supply of Taxicab Drivers](#app1) (6) [Application: Detecting Loss Aversion with Bunching](#app2) (7) [Concluding Thoughts](#conclusion) --- class: MSU name: intro # "Reinventing" Prospect Theory In the next (two? one and a half?) lectures we will "discover" some problems with vanilla Prospect Theory. -- We'll also introduce a bunch of "new" value functions. `\(\dots\)` except the value functions aren't new `\(\dots\)` and the discoveries were lurking in the back of your mind this whole time. -- And we'll do a million exercises. This material is tricky. --- class: MSU # Köszegi & Rabin (2006) We'll follow the KR theory of reference-dependent utility with loss aversion. (Except it's really just the **right** way to do Prospect Theory.) Their innovations address two major issues ("loopholes"): 1. What determines the reference point? 2. When do people experience loss aversion, and what is the magnitude of this experience? -- They address these issues by incorporating two novel features: A person's reference point is her recent beliefs or expectations about outcomes. Gain-loss utility is directly tied to the intrinsic utility from consumption --- so that a person experiences more gain-loss utility for goods that involve more consumption utility. --- class: MSU name: sec1 # Köszegi & Rabin (2006) ### Model Suppose there are `\(2\)` goods: - Person chooses a vector `\((x_{A},x_{B})\)`. - Reference point is a vector `\((r_{A},r_{B})\)`. -- **Preferences:** `$$\text{Total Utility } \equiv \text{ }\left[ \text{ }m_{A}(x_{A})\text{ }+ \text{ }n_{A}(x_{A}|r_{A})\text{ }\right]$$` `$$\qquad \qquad \qquad \qquad +\text{ }\left[ \text{ }m_{B}(x_{B})\text{ }+\text{ }n_{B}(x_{B}|r_{B})\text{ }\right]$$` -- - `\(m_{A}(x_{A})\)` is intrinsic utility for good `\(A\)`, and `\(m_{B}(x_{B})\)` is intrinsic utility for good `\(B\)`. - `\(n_{A}(x_{A}|r_{A})\)` is gain-loss utility for good `\(A\)`, and `\(n_{B}(x_{B}|r_{B})\)` is gain-loss utility for good `\(B\)`. --- class: MSU # Köszegi & Rabin (2006) How to formalize that gain-loss utility is directly tied to intrinsic utility: Assume there exists a *universal gain-loss function* `\(\mu (z)\)` such that the gain-loss utilities are: `$$v_{A}(x_{A}|r_{A}) =\mu \left( \text{ }m_{A}(x_{A})-m_{A}(r_{A})\text{ }\right)$$` `$$v_{B}(x_{B}|r_{B}) =\mu \left( \text{ }m_{B}(x_{B})-m_{B}(r_{B})\text{ }\right)$$` -- In general, `\(\mu (z)\)` takes form of the Kahneman-Tversky value function. But we'll focus on the "easy" case: `$$\mu (z)=\{ \quad \eta*z \quad \text{if} \quad z \geq 0$$` `$$\qquad \qquad \eta*\lambda*z \quad \text{if} \quad z\leq 0$$` --- class: MSU # Köszegi & Rabin (2006) Example: Two goods, shoes ( `\(c\)` ) and money ( `\(w\)` ), with intrinsic utilities: `$$m(c) = \theta * c$$` `$$m(w) = w$$` -- As with mugs, we can represent shoe utility in a 2x2 grid (see board) --- class: MSU # Köszegi & Rabin (2006) Consider the following choice problem: - Suppose Bogi starts with 0 shoes and wealth `\(w\)`, and has the option to purchase a shoe for price `\(p\)`. How does Bogi behave as a function of expectations? **Case 1:** Suppose you expect to buy a pair of shoes `\(\Longrightarrow\)` reference point is `\((r_{c}=1,r_{m}=w-p)\)` : `$$\text{Utility(Buy)} = \left[ \theta +\eta 0\right] + \left[ (w-p)+\eta 0\right]$$` `$$\text{Utility(Not)} = \left[ 0-\eta \lambda \theta \right] + \left[w+\eta p\right]$$` -- `$$\text{Buy when Utility(Buy)} \geq \text{ Utility(Not)}\Longleftrightarrow p\leq \frac{1+\eta \lambda }{1+\eta }\theta$$` --- class: MSU # Köszegi & Rabin (2006) Consider the following choice problem: - Suppose Bogi starts with 0 shoes and wealth `\(w\)`, and has the option to purchase a shoe for price `\(p\)`. How does Bogi behave as a function of expectations? **Case 2:** Suppose you expect not to buy any shoes `\(\Longrightarrow\)` reference point is `\((r_{c}=0,r_{m}=w)\)` : `$$\text{Utility(Buy)} = \left[ \theta +\eta \theta \right] + \left[ (w-p)-\eta \lambda p\right]$$` `$$\text{Utility(Not)} = \left[ 0+\eta 0\right] + \left[ w+\eta 0 \right]$$` `$$\text{Buy when Utility(Buy)}\geq \text{ Utility(Not)}\Longleftrightarrow p\leq \frac{1+\eta }{1+\eta \lambda }\theta$$`. --- class: MSU # Köszegi & Rabin (2006) Because `\(\lambda >1\)` implies `\(\frac{1+\eta \lambda }{1+\eta }>\frac{1+\eta }{1+\eta \lambda}\)`, there are three cases: 1. If `\(p>\frac{1+\eta \lambda }{1+\eta }\theta\)`, don't buy no matter your beliefs. -- 2. If `\(p<\frac{1+\eta }{1+\eta \lambda }\theta\)`, buy no matter your beliefs. -- 3. If `\(\frac{1+\eta }{1+\eta \lambda }\theta <p<\frac{1+\eta \lambda }{1+\eta }\theta\)`, buy if you expect to buy, and don't buy if you expect not to buy. **Point:** If the reference point depends on expectations, then, even in the same situation, a person might exhibit different outcomes depending on which set of self-fulfilling expectations he happens to have. --- class: inverseMSU name: exercise # An Exercise Suppose there are two goods, candy bars ($c$) and money ($m$). Paige has initial income `\(I\)`, and she is deciding whether to buy 0, 1, or 2 candy bars at a price of `\(p\)` per candy bar. Paige's total utility is the sum of her candy-bar utility and her money utility, and her intrinsic utilities for the two goods are: `$$w_{c}(c) \equiv \{\quad 0 \quad \text{if }\quad c=0$$` `$$\qquad \qquad \qquad \theta _{1} \quad \text{if } \quad c=1$$` `$$\qquad \qquad \qquad \theta _{1} + \theta_2 \quad \text{if } \quad c=2$$` Where `\(\theta _{1}>\theta _{2}\)` and `\(w_{m}(m) \equiv m\)` . --- class: inverseMSU # An Exercise **(a)** If Paige were a standard agent who only cares about her intrinsic utilities, how would she behave as a function of the price `\(p\)`? In other words, for what prices would she buy zero candy bars, for what prices would she buy one candy bar, and for what prices would she buy two candy bars? **(b)** Now suppose that Paige behaves according to the Koszegi-Rabin model. In other words, in addition to intrinsic utilities, she also cares about gain-loss utility, where the gain-loss utility for each good is derived from the universal gain-loss function described above. If Paige expects to buy no candy bars, how would she behave as a function of the price `\(p\)`? In other words, for what prices would she buy zero candy bars, for what prices would she buy one candy bar, and for what prices would she buy two candy bars? --- class: MSU name: sec2 # Köszegi & Rabin (2007) In a second paper, Köszegi and Rabin investigate the implications of their approach for basic risk preferences. Assume one good, money ($x$), with intrinsic utility `\(w(x)=x\)`. - Note: `\(w(x)=x\)` implies there is no intrinsic risk aversion --- all risk aversion will derive from gain-loss utility! -- Applying their approach, if consume money `\(x\)` given reference point `\(r\)`, then total utility is `$$u(x|r)= \{ x+ \eta*(x-r) \quad \text{if } \quad x > r$$` `$$\qquad \qquad \qquad x +\eta*\lambda*(x-r) \quad \text{if } \quad x\leq r$$` --- class: MSU # Köszegi & Rabin (2007) How to incorporate uncertainty: - If consume lottery `\(X\equiv (x_{1},p_{1};...;x_{N},p_{N})\)` given reference point `\(r\)`, then "expected" total utility is `$$U(X|r)\text{ }=\text{ }\sum_{i=1}^{N}\text{ }p_{i}\text{ }u(x_{i}|r)\text{.}$$` -- **Example:** If `\(X=(200,\frac{1}{4};0,\frac{3}{4})\)` and `\(r=100\)`, then `$$U(X|r)=\frac{1}{4}u(200|100)+\frac{3}{4}u(0|100).$$` -- But might expect a lottery, in which case **the reference point would be a lottery**. --- class: MSU # Köszegi & Rabin (2007) If consume money `\(x\)` given reference point `\(R\equiv (r_{1},q_{1};...;r_{M},q_{M})\)`, then "expected" total utility is `$$U(x|R)\text{ }=\text{ }\sum_{j=1}^{M}\text{ }q_{j}\text{ }u(x|r_{j})\text{.}$$` --- class: MSU # Köszegi & Rabin (2007) - If consume lottery `\(X\equiv (x_{1},p_{1};...;x_{N},p_{N})\)` given reference point `\(R\equiv (r_{1},q_{1};...;r_{M},q_{M})\)`, then total utility is `$$U(X|R)\text{ } =\text{ }\sum_{i=1}^{N}\text{ }p_{i}\text{ }U(x_{i}|R)$$` `$$=\text{ }\sum_{j=1}^{M}\text{ }q_{j}\text{ }U(X|r_{j})$$` `$$=\text{ }\sum_{i=1}^{N}\text{ }\sum_{j=1}^{M}\text{ }p_{i}\text{ }q_{j}\text{ }u(x_{i}|r_{j})\text{.}$$` --- class: MSU # Köszegi & Rabin (2007) **Example:** If `\(X=(200,\frac{1}{4};0,\frac{3}{4})\)` and `\(R=(150,\frac{1}{3};50,\frac{2}{3})\)`, then `$$U(X|R)=\frac{1}{4}\left[ \frac{1}{3}u(200|150)+\frac{2}{3}u(200|50)\right] + \frac{3}{4}\left[ \frac{1}{3}u(0|150)+\frac{2}{3}u(0|50)\right]$$` or `$$U(X|R)=\frac{1}{3}\left[ \frac{1}{4}u(200|150)+\frac{3}{4}u(0|150)\right] + \frac{2}{3}\left[ \frac{1}{4}u(200|50)+\frac{3}{4}u(0|50)\right]$$` or `$$U(X|R)=\frac{1}{12}u(200|150)+\frac{1}{6}u(200|50)+\frac{1}{4}u(0|150)+\frac{1}{2}u(0|50)$$`. --- class: MSU # Köszegi & Rabin (2007) **Point 1**: Risk aversion when no possible "losses". Consider choice `$$A\equiv (y,1)\ \text{with }y\leq 100 \qquad \text{vs.} \qquad B\equiv (\text{ }200,\frac{1}{2}\text{ };\text{ }0,\frac{1}{2}\text{ })$$` -- **Case 1:** Suppose expect `\(A\Longrightarrow\)` reference point is `\(r=y\)`: `$$U(A|r) = y + \eta*(0) = y$$` `$$U(B|r) = 100 + \left[ \frac{1}{2}\eta (200-y)+\frac{1}{2} \eta \lambda (0-y)\right]$$` -- This implies that you choose `\(A\)` if `\(y\geq \frac{1+\eta }{1+\frac{1}{2}\eta +\frac{1}{2}\eta \lambda }100\equiv \bar{y}_{1}\)` - **Note:** `\(\lambda >1\)` implies `\(\bar{y}_{1}<100\)` --- risk averse! --- class: MSU # Köszegi & Rabin (2007) **Case 2:** Suppose expect lottery `\(B\Longrightarrow\)`; reference point is `\(R=(200,\frac{1}{2};0,\frac{1}{2}\)` `$$U(A|R) = y + \left[ \frac{1}{2}\eta (y-0)+\frac{1}{2}\eta \lambda (y-200)\right]$$` -- `$$U(B|r) = 100 + \frac{1}{2}\left[ \frac{1}{2}\eta (200-0)+ \frac{1}{2}\eta (200-200)\right] +$$` `$$\qquad \qquad \qquad \frac{1}{2}\left[ \frac{1}{2}\eta (0-0)+\frac{1}{2}\eta \lambda (0-200)\right]$$` -- **Result:** Choose `\(A\)` if `\(y\geq 100\equiv \bar{y}_{2}\text{.}\)` - Note: `\(\bar{y}_{2}>\bar{y}_{1}\)`; that is, *expecting risk* makes you less risk averse! - *Intuition:* When expecting risk, even certain outcomes involve gains and losses, and thus they lose part of their advantage relative to risky outcomes. --- class: MSU # Köszegi & Rabin (2007) **Point 2:** Above feature helps explain demand for insurance at actuarially unfair prices. Suppose you have wealth $1000, but there is a 10% chance that you will suffer a loss of $250. Full insurance is available at price `\(\pi > 25\)` . -- - If insure, face lottery `\((1000-\pi ,1)\equiv A\)`. - If don't, face lottery `\((1000,.9;750,.1)\equiv B\)`. -- **Note:** If reference point is `\(r=1000\)`, don't insure! (Prove this.) -- Could it be that you expect to be insured, and still prefer to be insured? In other words, given reference point `\(r=1000-\pi\)`, do you prefer lottery `\(A \equiv (1000-\pi ,1)\)` over `\(B\equiv (1000,.9;750,.1)\)`? --- class: MSU # Köszegi & Rabin (2007) In other words, given reference point `\(r=1000-\pi\)`, do you prefer lottery `\(A\equiv (1000-\pi ,1)\)` over `\(B\equiv (1000,.9;750,.1)\)`? -- `$$U(A|r) = \left[ 1000-\pi \right] + \left[ 0\right]$$` -- `$$U(B|r) = 975 + \left[ .9\eta (\pi )+.1\eta \lambda (\pi-250)\right]$$` -- - **Result:** Insure if `\(\pi \leq \frac{1+\eta \lambda }{1+\eta \lambda -.9\eta (\lambda -1)}25\equiv \bar{\pi}\)` - Note: `\(\lambda >1\)` implies `\(\bar{\pi}>25\)` --- indeed willing to insure at actuarially unfair prices. - *Intuition:* Because expect to pay premium, it's not felt as a loss. --- class: MSU name: app1 # Application: Labor Supply of Taxi Drivers ### Camerer, Babcock, Loewenstein, & Thaler (1997) For many jobs, people choose how to allocate their labor from day-to-day, or from week-to-week, or from month-to-month. -- **Benchmark:** The standard life-cycle model of labor supply says that, if your wage varies over time, you should work more when the wage is high than you do when the wage is low. -- - Simple intuition: efficiently allocate your work effort. - Authors test this prediction on NYC cab drivers. --- class: MSU # Labor Supply of Taxi Drivers **First finding:** Their data permits them to calculate an average hourly wage for cab drivers, and they conclude that wages are highly correlated within a day, but not correlated across days. -- Hence, they take their unit of observation to be a day --- in particular, they estimate a daily wage equation: `$$\ln H_{t}=\gamma \ln W_{t}+\beta X_{t}+\varepsilon _{t}$$` -- - `\(H_{t}\equiv\)` hours worked on day `\(t\)` - `\(W_{t}\equiv\)` average wage on day `\(t\)` -- Standard model predicts `\(\gamma >0\)`, but they find `\(\gamma <0\)`. In words, the standard model predicts positive wage elasticities, but they find **negative** wage elasticities. --- class: MSU # Labor Supply of Taxi Drivers Their explanation is income targeting driven by loss aversion: - Drivers have one-day time horizon for decision making. - Their reference point is a daily income target. - They feel losses relative to the target loom larger than gains. --- class: MSU # Labor Supply of Taxi Drivers ### Farber (2005) Provides several critiques of Camerer et al (1997): - There is a "division bias": wages are calculated as earnings divided by hours, but hours are endogenous. - `\(\Rightarrow\)` Negative bias in wage elasticity estimates. -- - After cutting the data in a different way, Farber finds that it is not so clear there is more inter-day variation in the wage than intra-day variation in the wage. -- **Main point:** There is a better approach that gets around these problems: instead of estimating usual wage regressions, estimate a probit optimal-stopping model. --- class: MSU # Farber (JPE 2005)} > [And now for a little econometrics.] -- ### Probit optimal-stopping model: - Stop when `\(R(\tau )\geq 0\)`, where `\(R(\tau )=\gamma _{1}h_{\tau}+\gamma _{2}y_{\tau }+\beta X_{\tau }+\varepsilon _{\tau }\)`. - `\(h_{\tau }\equiv\)` hours worked today after trip `\(\tau\)` . - `\(y_{\tau }\equiv\)` earnings today after trip `\(\tau\)` . -- **Note:** Standard model predicts `\(\gamma _{1}>0\)` and `\(\gamma _{2}=0\)`. Farber indeed finds evidence consistent with `\(\gamma _{1}>0\)` and `\(\gamma_{2}=0\)`, as in the standard model. - BUT it's not clear whether this result is inconsistent with income targeting, since income targeting does not imply `\(\gamma _{1}=0\)`. --- class: MSU # Crawford & Meng (AER 2011) ### Crawford & Meng (AER 2011) They apply the Köszegi-Rabin perspective to this debate: - There should be gain-loss utility over each dimension of consumption. Here, this means over income (as usual) but also over hours worked. - Take the reference point to be people's expectations about outcomes; in particular, take them to be people's average experienced outcomes. `\(H_{t}\equiv\)` hours worked on day `\(t\)` `\(Y_{t}\equiv\)` income on day `\(t\)` `\(W_{t}\equiv Y_{t}/H_{t}\)` `\(H^{e}\equiv\)` average of `\(H_{t}\)` \\ `\(Y^{e}\equiv\)` average of `\(Y_{t}\)` \\ `\(W^{e}\equiv Y^{e}/H^{e}\)` -- Their Hypothesis: Reference point is `\((H^{e},Y^{e})\)`. --- class: MSU # Crawford & Meng (AER 2011) - Working fewer than `\(H^{e}\)` hours generates gain utility, and working more than `\(H^{e}\)` hours generates loss utility. - Earning more than income `\(Y^{e}\)` generates gain utility, and earning less than income `\(Y^{e}\)` generates loss utility. ### Key Idea: - On high-wage days ( `\(W_{t}>W^{e}\)` ), hit `\(Y^{e}\)` first and `\(H^{e}\)` second. - On low-wage days ( `\(W_{t}<W^{e}\)` ), hit `\(H^{e}\)` first and `\(Y^{e}\)` second. -- This suggests splitting the sample into high-wage days vs. low-wage days, because this model predicts that we should see different patterns of behavior. **And** when they do so, they find evidence consistent with their model and strongly reject Farber's analysis. -- Moreover, they show that targets in hours loom larger than targets in wages, which is consistent with the theory. --- class: MSU name: app2 # Other Empirical Work Crawford and Meng (2011) and its predecessors have been influential because of the domain: labor supply. However, this domain can make the analysis more complex than it needs to be. -- An alternative approach (innovated by Saez 2010; and Chetty et. al 2011): search for excess "bunching". - Observed distribution of data exceeds a modeled counterfactual distribution or a normative distribution. - Chetty et al. (2011) application: taxes and kinks in the tax schedule. - Allen et al. (2017) looks for this in marathon runners. --- class: MSU # Time Targets in Runners  --- class: MSU # Time Targets in Runners  --- class: inverseMSU name: conclusion # Reference-Dependent Welfare ### Are people worse off for having made loss-averse decisions? - Samuelson showed us that they are worse of mathematically. - Ultimately, answer to this question depends on modeler's beliefs about whether loss aversion is something that people really *feel*, or merely an artifact of some choice bias or mistake. -- Two camps: (1) Loss aversion is an affective forecasting error; (2) Loss aversion is a real manifestation of preferences. Surprisingly: Kahneman waffles between two; see e.g. Schkade and Kahneman (1998). **Me:** (2.5) Loss aversion is a little bit an affective forecasting error and a little bit "real". --- class: inverseMSU # Epilogue What have we learned in `\(\approx 500\)` years of studying risk preferences? -- **Expected values** matter, but don't wholly determine choice. -- - `\(\dots\)` except they probably should. [Begin rant.] -- - `\(\dots\)` most of the time. [End rant.] -- **Diminishing marginal utility** definitely does not explain most choices over risk. - `\(\dots\)` and I'm suspicious of **all** evidence on diminishing marginal utility of wealth. - Evidence conflated with reference-point effects (e.g. hedonic treadmill). --- class: inverseMSU # Epilogue **Prospect Theory** matters, but you need to apply it correctly. - Misleading conclusions when you fail to account for beliefs. - `\(\dots\)` but when you apply the "correct model", your intuitions are preserved. -- We've also (sneakily) introduced a new category of model: *belief-based utility*. We will return to some other models in this space.