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# Ordinality and Cardinality in Economic Utility Theory

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### Ordinality and Cardinality in Economic Utility Theory

1. 1. Applications of to Economic Utility Theory Benjamin Daniels Math 101 Pomona College Ordinality and Cardinality
2. 2. How Happy Are You? Okay, this is a tough question to start with.  Could you be happier?  What would make you happier?  How much happier would it make you?
3. 3. Ordering Preferences Basic Rules: Ordered Transitive Positive Convex Guns Butter A B C
4. 4. Indifference Curves Equivalence classes in utility space.
5. 5. Making Decisions Constrained Optimization : Some Lagrangian stuff you’ll be thrilled to be reminded of.
6. 6. How Happy Are “We”? Would something that makes any of us happier make all of us happier?
7. 7. A Social Utility Function Familiar Rules: Ordered Transitive Positive Convex Bread Circuses A B C
8. 8. Distributional Equity Familiar Rules: Ordered Transitive Positive Convex Bread Circuses A B C Me You
9. 9. Cardinalizing Utility Space For each person, we can arbitrarily assign useful cardinal units, based on their own subjective perceptions. 1 2 3 4
10. 10. Efficient Social Decisions Me You
11. 11. Are These Decisions Possible? Are utilities comparable between people, even if we assume that utility space is cardinal for each person? If so, is that fair?
12. 12. Measuring Happiness To make the best decisions, we need to measure how happy people are.  Time Period Surveys  Sum of Momentary Happiness  Intensity Weighting
13. 13. Estimating Social Utility Most practical theories avoid estimating individual utility directly.  Econometric Analyses  Assigning dollar values to outcomes.  Proportional Tradeoffs  “Doubling my happiness is equivalent to doubling yours.”  Transformations from Ordinal Values
14. 14. First-Order Approaches Remember, maximizatio n problems are solved using first- order conditions. If we can cardinalize the derivative, we can avoid the problems of measuring value directly.
15. 15. In Practice: Cost-Benefit Analysis Cost-benefit analysis tries to estimate cardinal values by observing what people are willing to pay for various things.  Difference-in-difference analysis  “Marginal willingness to pay”
16. 16. CBA as Cardinal Values Can we estimate the value of a human life?
17. 17. In Practice: The QALY System “How many years of life in state X are equivalent to one year of healthy life?”  Participants answer this question, and their answers are normalized such that one year of perfect health has value 1.  The answers are averaged to create a social valuation of health states.  Each procedure is empirically observed to improve health by some value and extend longevity by some duration.
18. 18. QALYs as Cardinal Values The QALY conversion gives each procedure a cardinal value so the most efficient choices can be made.
19. 19. QALYs as Derivatives Once we have the marginal value of a procedure, cost- efficiency is trivial. Value = Quality ∙ Longevity The marginal value of a procedure is given by its expected impact on value: ∂Value = (Quality ∙ ∂Longevity) + (∂Quality ∙ Longevity)
20. 20. In Practice: The U-Index The U-Index is specifically designed to output a cardinal measure from ordinal inputs.  At each time period, participants rank various emotions from 0 to 5.  Whenever any negative emotion scores higher than all positive emotions, a person is considered “unhappy.”  The U-Index is then reported as the proportion of “unhappy” outcomes.
21. 21. The U-Index as Cardinal Values The main benefit of U- Indexation is the ease of transforming answers to useful values. But it is very difficult to account for differences in intensity.
22. 22. The Space of Social Utility Functions Rawls Benthamite
23. 23. Formalizing Fairness The theories of Bentham and Rawls are upper and lower bounds for this notion of fairness. Rawls: α  -∞ Benthamite: α  1
24. 24. In Conclusion… Social choice is really hard!  Kenneth Arrow proved that efficient social choice is impossible.  But policymakers need to make decisions somehow.  Even given very weak assumptions about fairness and human happiness, we can construct useful cardinal estimations of benefits.