Beliefs-Based Preferences (Part I) April 14, 2009
Where are we? Prospect Theory Modeling reference-dependent preferences RD-VNM model w/ loss-aversion Easy to use, but r is taken as given What s the problem?
Example Suppose Finn has preferences over money u(c r) that, fixing r, meet assumptions A0-A4 for v(c r) u(c r) u(r r). We know nothing else about Finn, except that he is facing a choice between $0 for sure, or a 50/50 gamble of lose $Y or gain $X, with X, Y > 0. Q: If Finn s reference point is r = 0, for what values do we know for sure that he will choose 0?
Example Suppose Finn has preferences over money u(c r) that, fixing r, meet assumptions A0-A4 for v(c r) u(c r) u(r r). We know nothing else about Finn, except that he is facing a choice between $0 for sure, or a 50/50 gamble of lose $Y or gain $X, with X, Y > 0. Q: If Finn s reference point is r = 0, for what values do we know for sure that he will choose 0? A: We know for sure that Finn will choose $0 if and only if X Y.
Example Suppose Finn has preferences over money u(c r) that, fixing r, meet assumptions A0-A4 for v(c r) u(c r) u(r r). We know nothing else about Finn, except that he is facing a choice between $0 for sure, or a 50/50 gamble of lose $Y or gain $X, with X, Y > 0. Q: If Finn s reference point is r = 0, for what values do we know for sure that he will choose 0? A: We know for sure that Finn will choose $0 if and only if X Y. Q: Suppose we don t know what Finn s reference point is. When do we know for sure that he ll choose $0?
Example Suppose Finn has preferences over money u(c r) that, fixing r, meet assumptions A0-A4 for v(c r) u(c r) u(r r). We know nothing else about Finn, except that he is facing a choice between $0 for sure, or a 50/50 gamble of lose $Y or gain $X, with X, Y > 0. Q: If Finn s reference point is r = 0, for what values do we know for sure that he will choose 0? A: We know for sure that Finn will choose $0 if and only if X Y. Q: Suppose we don t know what Finn s reference point is. When do we know for sure that he ll choose $0? A: We can t be sure for any values of X, Y > 0. For r < X, it will be consistent to reject the game; for r > X it will be consistent to take the gamble.
Example Suppose Finn has preferences over money u(c r) that, fixing r, meet assumptions A0-A4 for v(c r) u(c r) u(r r). We know nothing else about Finn, except that he is facing a choice between $0 for sure, or a 50/50 gamble of lose $Y or gain $X, with X, Y > 0. Q: If Finn s reference point is r = 0, for what values do we know for sure that he will choose 0? A: We know for sure that Finn will choose $0 if and only if X Y. Q: Suppose we don t know what Finn s reference point is. When do we know for sure that he ll choose $0? A: We can t be sure for any values of X, Y > 0. For r < X, it will be consistent to reject the game; for r > X it will be consistent to take the gamble. Q: What s the point?
Example Suppose Finn has preferences over money u(c r) that, fixing r, meet assumptions A0-A4 for v(c r) u(c r) u(r r). We know nothing else about Finn, except that he is facing a choice between $0 for sure, or a 50/50 gamble of lose $Y or gain $X, with X, Y > 0. Q: If Finn s reference point is r = 0, for what values do we know for sure that he will choose 0? A: We know for sure that Finn will choose $0 if and only if X Y. Q: Suppose we don t know what Finn s reference point is. When do we know for sure that he ll choose $0? A: We can t be sure for any values of X, Y > 0. For r < X, it will be consistent to reject the game; for r > X it will be consistent to take the gamble. Q: What s the point? Specifying the reference point is crucial to making predictions about risk-aversion.
Where are we? Prospect Theory Modeling reference-dependent preferences RD-VNM model w/ loss-aversion Easy to use, but r is taken as given Next: endogenize the reference point
Where are we? Prospect Theory Modeling reference-dependent preferences RD-VNM model w/ loss-aversion Easy to use, but r is taken as given Next: endogenize the reference point r is recent expectations about consumption This introduces expectations/beliefs into the utility function
Where are we? Prospect Theory Modeling reference-dependent preferences RD-VNM model w/ loss-aversion Easy to use, but r is taken as given Next: endogenize the reference point r is recent expectations about consumption This introduces expectations/beliefs into the utility function Need some machinery/tools for this
Today: Preferences Over Information/Beliefs One may care about Information/news directly (cancer,idiocy) Curvature of information-utility attitude towards information (concavity implies info aversion) Expectations about future outcomes/consumption Anticipatory utility Reference dependence Beliefs (own,others) about unobservable attribute (signaling)
Example Getting a diagnosis: linear beliefs utility state, action s, a {0, 1}, initial belief: Pr(s = 1) =.1 utility = v[1 a s ] + w(1 p), w > 0 Go to doctor? What if new symptom indicates Pr(s = 1) =.5? What if patient can convince self that Pr(s = 1) is still.1, but not after doctor s visit?
Example Getting a diagnosis: concave beliefs-utiilty Suppose health = s 0; equally likely to be s 1 = 49 or s 2 = 64 Choose action t 0, deviation from s is costly U incorporates anticipatory & outcome utility: U = 25 ps 1 + (1 p)s 2 p s 1 t (1 p) s 2 t Visit doctor? What about if s 1 = 0 (condition is more serious)
Example Getting a diagnosis: concave beliefs-utiilty Suppose health = s 0; equally likely to be s 1 = 49 or s 2 = 64 Choose action t 0, deviation from s is costly U incorporates anticipatory & outcome utility: U = 25 ps 1 + (1 p)s 2 p s 1 t (1 p) s 2 t Visit doctor? What about if s 1 = 0 (condition is more serious) s 1 = 0 = information is more useful, but also potentially more harmful. Concave beliefs-utility implies information aversion.
Beliefs-Utility w/ Rational Expectations General Formulation: U(x, φ): utility over outcomes, beliefs (instrumental vs. beliefs-utility) Preferences depend upon beliefs, but beliefs about outcomes depend upon actions determined simultaneously in equilibrium Let σ be a vector of actions. (σ, φ) is a personal equilibrium if Optimization: σ arg max E[U(x, φ)] Consistency: Beliefs about future choices are correct or φ satisfies Bayes Rule
Personal Equilibrium: Example Suppose you get instrumental and anticipatory utility from eating either a muffin or a smoothie. x, e {m, s} U(x, e) is given by e\x m s m 3 2 s 0 1
Personal Equilibrium: Example Suppose you get instrumental and anticipatory utility from eating either a muffin or a smoothie. x, e {m, s} e\x m s U(x, e) is given by m 3 2 s 0 1 Self-fulfilling expectations: ff you expect m, m is the optimal choice; if you expect s, s is optimal
Personal Equilibrium: Example Suppose you get instrumental and anticipatory utility from eating either a muffin or a smoothie. x, e {m, s} e\x m s U(x, e) is given by m 3 2 s 0 1 Self-fulfilling expectations: ff you expect m, m is the optimal choice; if you expect s, s is optimal Multiple equilibria, but (m, m) yields higher utility
Personal Equilibrium: Example Suppose you get instrumental and anticipatory utility from eating either a muffin or a smoothie. x, e {m, s} e\x m s U(x, e) is given by m 3 2 s 0 1 Self-fulfilling expectations: ff you expect m, m is the optimal choice; if you expect s, s is optimal Multiple equilibria, but (m, m) yields higher utility Refinement: preferred personal equilibrium. Based on the assumption that you should be able to make any credible plan for your own behavior, choose the best plan.
Ahead Next: Rational expectations as the reference point, use personal equilibrium (KR model of RD preferences) Later: More beliefs-based preferences, preference-signaling in social-preferences