Applications of

to Economic Utility Theory

Benjamin DanielsMath 101Pomona College

Ordinality and Cardinality

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?

Ordering Preferences

Basic Rules:

Ordered

Transitive

Positive

Convex

Guns

Butter

A

B

C

Indifference Curves

Equivalence classes in

utility space.

Making Decisions

Constrained

Optimization:

Some Lagrangian stuff you’ll be thrilled

to be reminded

of.

How Happy Are “We”?

Would something that makes any of us happier

make all of us

happier?

A Social Utility Function

Familiar Rules:

Ordered

Transitive

Positive

Convex

Bread

Circuses

A

B

C

Distributional Equity

Familiar Rules:

Ordered

Transitive

Positive

Convex

Bread

Circuses

A

B

C

Me

You

Cardinalizing Utility Space

For each person, we

can arbitrarily

assign useful

cardinal units,

based on their own subjective perception

s.1

2

3

4

Efficient Social Decisions

Me You

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?

Measuring Happiness

To make the best

decisions, we need to measure

how happy people are.

Time Period Surveys Sum of Momentary Happiness Intensity Weighting

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

First-Order Approaches

Remember, maximization problems are solved using first-

order conditions.

If we can cardinalize

the derivative,

we can avoid the problems of measuring

value directly.

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”

CBA as Cardinal Values

Can we estimate the value

of a human life?

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.

QALYs as Cardinal Values

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

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)

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.

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.

The Space of Social Utility Functions

Rawls Benthamite

Formalizing Fairness

The theories of Bentham and Rawls are upper and lower bounds for this notion of fairness. Rawls: α -∞

Benthamite: α 1

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.