This document discusses various approaches to quantifying and measuring economic utility and social welfare. It examines how ordinal preferences can be translated into cardinal utility metrics through techniques like indifference curves, willingness to pay analysis, quality-adjusted life years (QALYs), and happiness indexes. These cardinal representations allow welfare and decisions to be analyzed using tools from calculus and constrained optimization. However, quantifying utility also raises issues regarding interpersonal comparisons and distributional fairness.

1. Applications of
to Economic Utility Theory
Benjamin Daniels
Math 101
Pomona College
Ordinality and Cardinality

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. Ordering Preferences
Basic Rules:
Ordered
Transitive
Positive
Convex
Guns
Butter
A
B
C

4. Indifference Curves
Equivalence
classes in
utility space.

5. Making Decisions
Constrained
Optimization
:
Some
Lagrangian
stuff you’ll
be thrilled to
be reminded
of.

6. How Happy Are “We”?
Would
something
that makes
any of us
happier
make all of
us happier?

7. A Social Utility Function
Familiar
Rules:
Ordered
Transitive
Positive
Convex
Bread
Circuses
A
B
C

8. Distributional Equity
Familiar
Rules:
Ordered
Transitive
Positive
Convex
Bread
Circuses
A
B
C
Me
You

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. Efficient Social Decisions
Me You

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. 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. 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. 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. 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. CBA as Cardinal Values
Can we
estimate the
value of a
human life?

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. QALYs as Cardinal Values
The QALY
conversion
gives each
procedure a
cardinal
value so the
most
efficient
choices can
be made.

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. 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. 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. The Space of Social Utility
Functions
Rawls Benthamite

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. 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.