M100

Matthew Elliott
Cambridge and Caltech

October, 2016
Some Logistics

These slides borrow heavily from teaching notes by Clayton

Featherstone, Jonathan Levin, Paul Milgrom, Antonio Rangel,
Ilya Segal and especially Luke Stein.

My office hour is after class on Thursdays (Office 42).

I’ll be available to chat after class most of the time.

You are lucky you have an outstanding TA, River. Please use
him!

Handouts are intended to allow you to concentrate on the

material in lectures.
Who said this?

“Nobody ever saw a dog make a fair and deliberate exchange of

one bone for another with another dog. Nobody ever saw one
animal by its gestures and natural cries signify to another, this is
mine, that yours; I am willing to give this for that....But man has
almost constant occasion for the help of his brethren, and it is in
vain for him to expect it from their benevolence only. He will be
more likely to prevail if he can interest their self-love in his favour,
and show them that it is for their own advantage to do for him
what he requires of them. Whoever offers to another a bargain of
any kind, proposes to do this. Give me that which I want, and you
shall have this which you want, is the meaning of every such offer;
and it is in this manner that we obtain from one another the far
greater part of those good offices which we stand in need of.”
Or, more famously

“It is not from the benevolence of the butcher, the brewer, or the
baker that we expect our dinner, but from their regard to their own
interest.”
Or, more famously

“It is not from the benevolence of the butcher, the brewer, or the
baker that we expect our dinner, but from their regard to their own
interest.”

Adam Smith, The Wealth of Nations (1776).

Who said this?

“The greatest happiness of the greatest number is the foundation

of morals and legislation.”
Who said this?

“The greatest happiness of the greatest number is the foundation

of morals and legislation.”
“It is vain to talk of the interest of the community, without
understanding what is the interest of the individual.”
Who said this?

“The greatest happiness of the greatest number is the foundation

of morals and legislation.”
“It is vain to talk of the interest of the community, without
understanding what is the interest of the individual.”

Jeremy Bentham (1748-1832).

Economic thinking in early 19th Century

Want to refine Adam Smith’s ideas about an economic system

based on self-interest.

Perhaps the project could be advanced by an index of

self-interest?

How beneficial are different outcomes to any individual?

Utilitarian philosophers in early 19th Century

Wanted an objective criterion for a science of government.

But requires a way of measuring how beneficial different

policies are to different people.
Utilitarian philosophers in early 19th Century

Wanted an objective criterion for a science of government.

But requires a way of measuring how beneficial different

policies are to different people.

Utility-maximization theory of decision making is born.

Individual Decision-Making Under Certainty

Our study begins with individual decision-making under certainty

Individual Decision-Making Under Certainty

Our study begins with individual decision-making under certainty

We start with a very general problem that only includes:

Feasible set (what can you choose from)
Choice correspondence (what you choose)
Individual Decision-Making Under Certainty

Our study begins with individual decision-making under certainty

We start with a very general problem that only includes:

Feasible set (what can you choose from)
Choice correspondence (what you choose)

A fairly innocent assumption will then allow us convert this

problem into an optimization problem (more later).
Individual Decision-Making Under Certainty

Our study begins with individual decision-making under certainty

We start with a very general problem that only includes:

Feasible set (what can you choose from)
Choice correspondence (what you choose)

A fairly innocent assumption will then allow us convert this

problem into an optimization problem (more later).

We’ll generalize to decision making under uncertainty later.

What will the model deliver?

Useful:
Can often recover preferences from choices.
Aggregating into a theory of government it is aligned with
democratic values.
But. . . interpersonal comparisons prove difficult
Accurate (somewhat): many predictions are empirically
verified.
Broad
Consumption and production
Lots of other things
Compact
Extremely compact formulation
Ignores an array of other important “behavioral” factors
How we will use it: A road map

(i) Choice under certainty.

(ii) Choice under uncertainty.
Our theory of decision making can be applied to study:
(i) Consumers’ purchasing decisions.
(ii) Producers’ supply decisions.
(iii) General Equilibrium
Ultimately, this will allow us to model markets and the
economy.
We’ll be able to rigorously flesh out Adam Smith’s ideas
building a model in which the “invisible hand” delivers.
(iv) Limitations
Our general equilibrium analysis will rely on (strong)
assumptions. We’ll start to relax these.
Outline

1 Choice Theory
Preferences
Utility functions
Properties of preferences
Behavioral critiques
Comparative statics
An application: Consumer Theory

2 Choice under Uncertainty

3 General Equilibrium

4 Limitations of the Welfare Theorems: Asymmetric information

Preference relations

Definition (weak preference relation)

 is a binary relation on a set of possible choices X such that
x  y iff “x is at least as good as y.”

How might we use this relation to define strict preferences and

indifference?
Preference relations

Definition (weak preference relation)

 is a binary relation on a set of possible choices X such that
x  y iff “x is at least as good as y.”

How might we use this relation to define strict preferences and

indifference?
Definition (strict preference relation)
 is a binary relation on X such that x  y (“x is strictly
preferred to y”) iff x  y but y 6 x.

Definition (indifference)
∼ is a binary relation on X such that x ∼ y (“the agent is
indifferent between x and y”) iff x  y and y  x.
Properties of preference relations

What properties might we expect  to satisfy?

Properties of preference relations

What properties might we expect  to satisfy?

Definition (completeness)
 on X is complete iff ∀x, y ∈ X, either x  y or y  x.

Completeness implies that x  x

Definition (transitivity)
 on X is transitive iff whenever x  y and y  z, we have x  z.

Rules out preference cycles except in the case of indifference

Properties of preference relations

What properties might we expect  to satisfy?

Definition (completeness)
 on X is complete iff ∀x, y ∈ X, either x  y or y  x.

Completeness implies that x  x

Definition (transitivity)
 on X is transitive iff whenever x  y and y  z, we have x  z.

Rules out preference cycles except in the case of indifference

Definition (rationality)
 on X is rational iff it is both complete and transitive.
Example from Kahneman and Tversky (84)

Suppose you are about to buy a stereo for $125 and a

calculator for $15.

There is a $5 discount on the calculator at a store 10 minutes

away. Do you make the trip? (Yes)

There is a $5 discount on the stereo at a store 10 minutes

away. Do you make the trip? (No)
Example from Kahneman and Tversky (84)

You find out both items are out of stock and need to go to
the other branch 10 minutes away. You get a $5 dollar
discount as compensation. Which item would you rather have
the discount on? (Indifferent).
Example from Kahneman and Tversky (84)

You find out both items are out of stock and need to go to
the other branch 10 minutes away. You get a $5 dollar
discount as compensation. Which item would you rather have
the discount on? (Indifferent).

Which property of rational choice has been violated?

Example from Kahneman and Tversky (84)

Suppose you are about to buy a stereo for $125 and a

calculator for $15.

There is a $5 discount on the calculator at a store 10 minutes

away. Do people make the trip? (Yes)

There is a $5 discount on the stereo at a store 10 minutes

away. Do people make the trip? (No)
Example from Kahneman and Tversky (84)

Suppose you are about to buy a stereo for $125 and a

calculator for $15.

There is a $5 discount on the calculator at a store 10 minutes

away. Do people make the trip? (Yes)

There is a $5 discount on the stereo at a store 10 minutes

away. Do people make the trip? (No)
Consider the following three options:
(i) staying at the store, no discount.
(ii) Traveling 10 minutes for the calculator discount.
(iii) Traveling 10 minutes for the stereo discount.

From the above choices what can we infer: (ii)  (i) and
(i)  (iii).
Example from Kahneman and Tversky (84)

You find out both items are out of stock and need to go to
the other branch 10 minutes away. You get a $5 dollar
discount as compensation. Which item would you rather have
the discount on? (Indifferent).
Recall the following three options:
(i) staying at the store, no discount.
(ii) Traveling 10 minutes for the calculator discount.
(iii) Traveling 10 minutes for the stereo discount.
From the above choices what can we infer: (ii) ∼ (iii).
Example from Kahneman and Tversky (84)

You find out both items are out of stock and need to go to
the other branch 10 minutes away. You get a $5 dollar
discount as compensation. Which item would you rather have
the discount on? (Indifferent).
Recall the following three options:
(i) staying at the store, no discount.
(ii) Traveling 10 minutes for the calculator discount.
(iii) Traveling 10 minutes for the stereo discount.
From the above choices what can we infer: (ii) ∼ (iii).
From before (ii)  (i) and (i)  (iii).
Example from Kahneman and Tversky (84)

You find out both items are out of stock and need to go to
the other branch 10 minutes away. You get a $5 dollar
discount as compensation. Which item would you rather have
the discount on? (Indifferent).
Recall the following three options:
(i) staying at the store, no discount.
(ii) Traveling 10 minutes for the calculator discount.
(iii) Traveling 10 minutes for the stereo discount.
From the above choices what can we infer: (ii) ∼ (iii).
From before (ii)  (i) and (i)  (iii).
(ii)  (i)  (iii)  (ii), so by transitivity (ii) ∼ (i) ∼ (iii).
Example from Kahneman and Tversky (84)

You find out both items are out of stock and need to go to
the other branch 10 minutes away. You get a $5 dollar
discount as compensation. Which item would you rather have
the discount on? (Indifferent).
Recall the following three options:
(i) staying at the store, no discount.
(ii) Traveling 10 minutes for the calculator discount.
(iii) Traveling 10 minutes for the stereo discount.
From the above choices what can we infer: (ii) ∼ (iii).
From before (ii)  (i) and (i)  (iii).
(ii)  (i)  (iii)  (ii), so by transitivity (ii) ∼ (i) ∼ (iii).
Contradicts (ii)  (i) (and (i)  (iii)), transitivity violated.
Summary of preference notation

yx y 6 x
xy x∼y xy
x 6 y yx Ruled out by completeness
Summary of preference notation

yx y 6 x
xy x∼y xy
x 6 y yx Ruled out by completeness

Can think of (complete) preferences as inducing a function

p : X × X → {, ∼, ≺}
Other properties of rational preference relations

Assume  is rational. Then for all x, y, z ∈ X:

Weak preference is reflexive: x  x
Indifference is
Reflexive: x ∼ x
Transitive: (x ∼ y) ∧ (y ∼ z) =⇒ x ∼ z
Symmetric: x ∼ y ⇐⇒ y ∼ x
Strict preference is
Irreflexive: x 6 x
Transitive: (x  y) ∧ (y  z) =⇒ x  z
(x  y) ∧ (y  z) =⇒ x  z, and
(x  y) ∧ (y  z) =⇒ x  z
Approaches to modelling individual decision-making

1 Conventional approach
Start from preferences, ask what choices are compatible
Approaches to modelling individual decision-making

1 Conventional approach
Start from preferences, ask what choices are compatible

2 Revealed-preference approach
Start from observed choices, ask what preferences are
compatible
What can we learn about underlying preferences from
observing choices?
Can we test rational choice theory? How?
Are all decisions consistent with some rational preferences?
Choice rules

Definition (Choice rule)

Given preferences  over X, and choice set B ⊆ X, the choice
rule is a correspondence giving the set of all “best” elements in B:

C(B, ) ≡ {x ∈ B : x  y for all y ∈ B}.

Choice rules

Definition (Choice rule)

Given preferences  over X, and choice set B ⊆ X, the choice
rule is a correspondence giving the set of all “best” elements in B:

C(B, ) ≡ {x ∈ B : x  y for all y ∈ B}.

Theorem
Suppose  is complete and transitive and B finite and non-empty.
Then C(B, ) 6= ∅.
Discussion of theorem

Conclusion of the Theorem says that the choice rule never

selects nothing.
Does this make sense?
What if I don’t like any of the things I am choosing between?
Why can’t my most preferred choice be choosing nothing?
Discussion of theorem

Conclusion of the Theorem says that the choice rule never

selects nothing.
Does this make sense?
What if I don’t like any of the things I am choosing between?
Why can’t my most preferred choice be choosing nothing?
Choice rule is about choosing among items in B only.
The action of choosing nothing could be included in B.
Theorem is a “consistency check” for the choice rule.
Outline of the proof for non-emptiness of choice
correspondence

Proof is by mathematical induction. This is a very useful

technique. Here is an outline of the argument.
With one item B = x, by completeness x  x, so it would be
selected and x ∈ C(A, ).
Outline of the proof for non-emptiness of choice
correspondence

Proof is by mathematical induction. This is a very useful

technique. Here is an outline of the argument.
With one item B = x, by completeness x  x, so it would be
selected and x ∈ C(A, ).
Suppose now there are n items and the choice set is not empty.
Outline of the proof for non-emptiness of choice
correspondence

Proof is by mathematical induction. This is a very useful

technique. Here is an outline of the argument.
With one item B = x, by completeness x  x, so it would be
selected and x ∈ C(A, ).
Suppose now there are n items and the choice set is not empty.
We will show that with n + 1 items the choice set is not empty.
Outline of the proof for non-emptiness of choice
correspondence

Proof is by mathematical induction. This is a very useful

technique. Here is an outline of the argument.
With one item B = x, by completeness x  x, so it would be
selected and x ∈ C(A, ).
Suppose now there are n items and the choice set is not empty.
We will show that with n + 1 items the choice set is not empty.
Without loss of generality, suppose x ∈ C(B, ). Let
A = B ∪ {y}. By completeness either x  y, in which case
x ∈ C(A, ), or y  x. By transitivity y must then be preferred to
all elements of A, so y ∈ C(A, ).
Outline of the proof for non-emptiness of choice
correspondence

Proof is by mathematical induction. This is a very useful

technique. Here is an outline of the argument.
With one item B = x, by completeness x  x, so it would be
selected and x ∈ C(A, ).
Suppose now there are n items and the choice set is not empty.
We will show that with n + 1 items the choice set is not empty.
Without loss of generality, suppose x ∈ C(B, ). Let
A = B ∪ {y}. By completeness either x  y, in which case
x ∈ C(A, ), or y  x. By transitivity y must then be preferred to
all elements of A, so y ∈ C(A, ).
Letting n = 1, we have proved it for n = 2. Letting n = 2 . . . , and
so on.
Revealed preference

Before: used known preference relation  to generate choice

rule C(·, )
Now: suppose agent reveals her preferences through her
choices, which we observe; can we deduce a rational
preference relation that could have generated them?
Revealed preference

Before: used known preference relation  to generate choice

rule C(·, )
Now: suppose agent reveals her preferences through her
choices, which we observe; can we deduce a rational
preference relation that could have generated them?

Definition (revealed preference choice rule)

Any CR : 2X → 2X (where 2X means the set of subsets of X)
such that for all A ⊆ X, we have CR (A) ⊆ A.

If CR (·) could be generated by a rational preference relation (i.e.,

there exists some complete, transitive  such that
CR (A) = C(A, ) for all A), we say it is rationalizable
Examples of revealed preference choice rules

Suppose we know CR (·) for

A ≡ {a, b}
B ≡ {a, b, c}

CR ({a, b}) CR ({a, b, c}) Possibly rationalizable?

{a} {c}
{a} {a}
{a, b} {c}
Examples of revealed preference choice rules

Suppose we know CR (·) for

A ≡ {a, b}
B ≡ {a, b, c}

CR ({a, b}) CR ({a, b, c}) Possibly rationalizable?

{a} {c} X (c  a  b)
{a} {a}
{a, b} {c}
Examples of revealed preference choice rules

Suppose we know CR (·) for

A ≡ {a, b}
B ≡ {a, b, c}

CR ({a, b}) CR ({a, b, c}) Possibly rationalizable?

{a} {c} X (c  a  b)
{a} {a} X (a  b, a  c, b?c)
{a, b} {c}
Examples of revealed preference choice rules

Suppose we know CR (·) for

A ≡ {a, b}
B ≡ {a, b, c}

CR ({a, b}) CR ({a, b, c}) Possibly rationalizable?

{a} {c} X (c  a  b)
{a} {a} X (a  b, a  c, b?c)
{a, b} {c} X (c  a ∼ b)
Examples of revealed preference choice rules

Suppose we know CR (·) for

A ≡ {a, b}
B ≡ {a, b, c}

CR ({a, b}) CR ({a, b, c}) Possibly rationalizable?

{a} {c} X (c  a  b)
{a} {a} X (a  b, a  c, b?c)
{a, b} {c} X (c  a ∼ b)
{c} {c}
Examples of revealed preference choice rules

Suppose we know CR (·) for

A ≡ {a, b}
B ≡ {a, b, c}

CR ({a, b}) CR ({a, b, c}) Possibly rationalizable?

{a} {c} X (c  a  b)
{a} {a} X (a  b, a  c, b?c)
{a, b} {c} X (c  a ∼ b)
{c} {c} X (c 6∈ {a, b})
Examples of revealed preference choice rules

Suppose we know CR (·) for

A ≡ {a, b}
B ≡ {a, b, c}

CR ({a, b}) CR ({a, b, c}) Possibly rationalizable?

{a} {c} X (c  a  b)
{a} {a} X (a  b, a  c, b?c)
{a, b} {c} X (c  a ∼ b)
{c} {c} X (c 6∈ {a, b})
∅ {c}
Examples of revealed preference choice rules

Suppose we know CR (·) for

A ≡ {a, b}
B ≡ {a, b, c}

CR ({a, b}) CR ({a, b, c}) Possibly rationalizable?

{a} {c} X (c  a  b)
{a} {a} X (a  b, a  c, b?c)
{a, b} {c} X (c  a ∼ b)
{c} {c} X (c 6∈ {a, b})
∅ {c} X (Not possible)
Examples of revealed preference choice rules

Suppose we know CR (·) for

A ≡ {a, b}
B ≡ {a, b, c}

CR ({a, b}) CR ({a, b, c}) Possibly rationalizable?

{a} {c} X (c  a  b)
{a} {a} X (a  b, a  c, b?c)
{a, b} {c} X (c  a ∼ b)
{c} {c} X (c 6∈ {a, b})
∅ {c} X (Not possible)
{b} {a}
Examples of revealed preference choice rules

Suppose we know CR (·) for

A ≡ {a, b}
B ≡ {a, b, c}

CR ({a, b}) CR ({a, b, c}) Possibly rationalizable?

{a} {c} X (c  a  b)
{a} {a} X (a  b, a  c, b?c)
{a, b} {c} X (c  a ∼ b)
{c} {c} X (c 6∈ {a, b})
∅ {c} X (Not possible)
{b} {a} X (Not possible)
Examples of revealed preference choice rules

Suppose we know CR (·) for

A ≡ {a, b}
B ≡ {a, b, c}

CR ({a, b}) CR ({a, b, c}) Possibly rationalizable?

{a} {c} X (c  a  b)
{a} {a} X (a  b, a  c, b?c)
{a, b} {c} X (c  a ∼ b)
{c} {c} X (c 6∈ {a, b})
∅ {c} X (Not possible)
{b} {a} X (Not possible)
{a} {a, b}
Examples of revealed preference choice rules

Suppose we know CR (·) for

A ≡ {a, b}
B ≡ {a, b, c}

CR ({a, b}) CR ({a, b, c}) Possibly rationalizable?

{a} {c} X (c  a  b)
{a} {a} X (a  b, a  c, b?c)
{a, b} {c} X (c  a ∼ b)
{c} {c} X (c 6∈ {a, b})
∅ {c} X (Not possible)
{b} {a} X (Not possible)
{a} {a, b} X (Not possible)
A necessary condition for rationalizability

Suppose that CR (·) is rationalizable (in particular, it is generated

by ), and we observe CR (A) for some A ⊆ X such that
a ∈ CR (A) (a was chosen).
So: a  z for all z ∈ A
A necessary condition for rationalizability

Suppose that CR (·) is rationalizable (in particular, it is generated

by ), and we observe CR (A) for some A ⊆ X such that
a ∈ CR (A) (a was chosen).
So: a  z for all z ∈ A
b ∈ A (b could have been chosen)
A necessary condition for rationalizability

Suppose that CR (·) is rationalizable (in particular, it is generated

by ), and we observe CR (A) for some A ⊆ X such that
a ∈ CR (A) (a was chosen).
So: a  z for all z ∈ A
b ∈ A (b could have been chosen)

We can infer that a  b

A necessary condition for rationalizability

Suppose that CR (·) is rationalizable (in particular, it is generated

by ), and we observe CR (A) for some A ⊆ X such that
a ∈ CR (A) (a was chosen).
So: a  z for all z ∈ A
b ∈ A (b could have been chosen)

We can infer that a  b

Now consider some B ⊆ X such that

a∈B
b ∈ CR (B) (b was chosen)
So: b  z for all z ∈ B
A necessary condition for rationalizability

Suppose that CR (·) is rationalizable (in particular, it is generated

by ), and we observe CR (A) for some A ⊆ X such that
a ∈ CR (A) (a was chosen).
So: a  z for all z ∈ A
b ∈ A (b could have been chosen)

We can infer that a  b

Now consider some B ⊆ X such that

a∈B
b ∈ CR (B) (b was chosen)
So: b  z for all z ∈ B

We can infer that b  a

A necessary condition for rationalizability

Suppose that CR (·) is rationalizable (in particular, it is generated

by ), and we observe CR (A) for some A ⊆ X such that
a ∈ CR (A) (a was chosen).
So: a  z for all z ∈ A
b ∈ A (b could have been chosen)

We can infer that a  b

Now consider some B ⊆ X such that

a∈B
b ∈ CR (B) (b was chosen)
So: b  z for all z ∈ B

We can infer that b  a

Thus a ∼ b
A necessary condition for rationalizability

Suppose that CR (·) is rationalizable (in particular, it is generated

by ), and we observe CR (A) for some A ⊆ X such that
a ∈ CR (A) (a was chosen).
So: a  z for all z ∈ A
b ∈ A (b could have been chosen)

We can infer that a  b

Now consider some B ⊆ X such that

a∈B
b ∈ CR (B) (b was chosen)
So: b  z for all z ∈ B

We can infer that b  a

Thus a ∼ b, hence a ∈ CR (B) and b ∈ CR (A) by transitivity

Houthaker’s Axiom of Revealed Preferences

A rationalizable choice rule CR (·) must therefore satisfy “HARP”:

Definition (Houthaker’s Axiom of Revealed Preferences)
Revealed preferences CR : 2X → 2X satisfies HARP iff ∀a, b ∈ X
and ∀A, B ⊆ X such that
{a, b} ⊆ A and a ∈ CR (A); and
{a, b} ⊆ B and b ∈ CR (B),
we have that a ∈ CR (B) (and b ∈ CR (A)).
Houthaker’s Axiom of Revealed Preferences

A rationalizable choice rule CR (·) must therefore satisfy “HARP”:

Definition (Houthaker’s Axiom of Revealed Preferences)
Revealed preferences CR : 2X → 2X satisfies HARP iff ∀a, b ∈ X
and ∀A, B ⊆ X such that
{a, b} ⊆ A and a ∈ CR (A); and
{a, b} ⊆ B and b ∈ CR (B),
we have that a ∈ CR (B) (and b ∈ CR (A)).

It turns out HARP is not only necessary, but sufficient for

rationalizability
Illustrating HARP

A violation of HARP:
Example of HARP

Suppose
1 Revealed preferences CR (·) satisfy HARP, and that
2 CR (·) is nonempty-valued (except for CR (∅))

If CR ({a, b}) = {b}, what can we conclude about

CR ({a, b, c})?

If CR ({a, b, c}) = {b}, what can we conclude about

CR ({a, b})?
Example of HARP

Suppose
1 Revealed preferences CR (·) satisfy HARP, and that
2 CR (·) is nonempty-valued (except for CR (∅))

If CR ({a, b}) = {b}, what can we conclude about

CR ({a, b, c})?

CR ({a, b, c}) ∈ {{b}, {c}, {b, c}}

If CR ({a, b, c}) = {b}, what can we conclude about

CR ({a, b})?
Example of HARP

Suppose
1 Revealed preferences CR (·) satisfy HARP, and that
2 CR (·) is nonempty-valued (except for CR (∅))

If CR ({a, b}) = {b}, what can we conclude about

CR ({a, b, c})?

CR ({a, b, c}) ∈ {{b}, {c}, {b, c}}

If CR ({a, b, c}) = {b}, what can we conclude about

CR ({a, b})?
CR ({a, b}) = {b}
HARP is necessary and sufficient for rationalizability

Theorem
Suppose revealed preference choice rule CR : 2X → 2X is
nonempty-valued (except for CR (∅)). Then CR (·) satisfies HARP
iff there exists a rational preference relation  such that
CR (·) = C(·, ).
Proof Logic

Already argued that rationality implies HARP.

To see HARP implies rationality we construct a preference
relation.
For any x and y, let x c y iff there exists some A ⊆ X such
that y ∈ A and x ∈ CR (A).

Can consider all pairs of x, y, so constructed relation is

complete.
HARP then implies that transitivity holds, so relation is
rational.
Revealed preference and limited data

Our discussion relies on all preferences being observed

All elements of CR (A)

CR (A) for every A ⊆ X

Revealed preference and limited data

Our discussion relies on all preferences being observed. . . real data

is typically more limited
All elements of CR (A). . . we may only see one element of A
eR (A) ∈ CR (A)
i.e., C
CR (A) for every A ⊆ X
Revealed preference and limited data

Our discussion relies on all preferences being observed. . . real data

is typically more limited
All elements of CR (A). . . we may only see one element of A
eR (A) ∈ CR (A)
i.e., C
CR (A) for every A ⊆ X. . . we may only observe choices for
certain choice sets
bR (A) : B → 2X for B ⊂ 2X with C
i.e., C bR (A) = CR (A)
Revealed preference and limited data

Our discussion relies on all preferences being observed. . . real data

is typically more limited
All elements of CR (A). . . we may only see one element of A
eR (A) ∈ CR (A)
i.e., C
CR (A) for every A ⊆ X. . . we may only observe choices for
certain choice sets
bR (A) : B → 2X for B ⊂ 2X with C
i.e., C bR (A) = CR (A)

Other “axioms of revealed preference” hold in these environments

Weak Axiom of Revealed Preference (WARP)
Generalized Axiom of Revealed Preference (GARP)—necessary
and sufficient condition for rationalizability
Weak Axiom of Revealed Preferences

Definition (Weak Axiom of Revealed Preferences)

bR : B → 2X defined only for choice sets
Revealed preferences C
X
B ⊆ 2 satisfies WARP iff ∀a, b ∈ X and ∀A, B ∈ B such that
{a, b} ⊆ A and a ∈ CbR (A); and
{a, b} ⊆ B and b ∈ C
bR (B),
we have that a ∈ C
bR (B) (and b ∈ C
bR (A)).
Weak Axiom of Revealed Preferences

Definition (Weak Axiom of Revealed Preferences)

bR : B → 2X defined only for choice sets
Revealed preferences C
X
B ⊆ 2 satisfies WARP iff ∀a, b ∈ X and ∀A, B ∈ B such that
{a, b} ⊆ A and a ∈ CbR (A); and
{a, b} ⊆ B and b ∈ C
bR (B),
we have that a ∈ C
bR (B) (and b ∈ C
bR (A)).

HARP is WARP with all possible choice sets (i.e,. B = 2X )

WARP is necessary but not sufficient for rationalizability

Weak Axiom of Revealed Preferences cont.

Example
bR : B → 2{a,b,c} defined for choice sets
Consider C
B ≡ {{a, b}, {b, c}, {c, a}} ⊆ 2{a,b,c} with:
CbR ({a, b}) = {a},
bR ({b, c}) = {b}, and
C
bR ({c, a}) = {c}.
C

C
bR (·) satisfies WARP, but is not rationalizable.

Think of CbR (·) as a restriction of some CR : 2{a,b,c} → 2{a,b,c} ;

there is no CR ({a, b, c}) consistent with HARP.
Outline

1 Choice Theory
Preferences
Utility functions
Properties of preferences
Behavioral critiques
Comparative statics
An application: Consumer Theory

2 Choice under Uncertainty

3 General Equilibrium

4 Limitations of the Welfare Theorems: Asymmetric information

From abstract preferences to maximization

Our model of choice so far is entirely abstract

Utility assigns a numerical ranking to each possible choice
By assigning a utility to each element of X, we turn the
choice problem into an optimization problem

Definition (utility function)

Utility function u : X → R represents  on X iff for all x, y ∈ X,

x  y ⇐⇒ u(x) ≥ u(y).

Then the choice rule is

C(B, ) ≡ {x ∈ B : x  y for all y ∈ B} = argmax u(x)

x∈B
Utility representation implies rationality

Theorem
If utility function u : X → R represents  on X, then  is rational.

Can show  is complete. How?

Can show  is transitive. How?

Ordinality of utility and interpersonal comparisons

Note that  is represented by any function satisfying

x  y ⇐⇒ u(x) ≥ u(y)

for all x, y ∈ X

Thus any increasing monotone transformation of u(·) also

represents 
The property of representing  is ordinal

There is no such thing as a “util”

Failure of interpersonal comparisons

Interpersonal comparisons are impossible using this theory

1 Disappointing to original utilitarian agenda

2 Rawls (following Kant, following. . . ) attempts to solve this by

asking us to consider only a single chooser
Can we find a utility function representing ?

Theorem
Any complete and transitive preference relation  on a finite set X
can be represented by some utility function u : X → {1, . . . , n}
where n ≡ |X|.

Intuitive argument—a constructive algorithm:

1 Assign the “top” elements of X utility n = |X|
2 Discard them; we are left with a set X 0
3 If X 0 = ∅, we are done; otherwise return to step 1 with the set
X0
What if |X| = ∞?

If X is infinite, our proof doesn’t go through, but we still may be

able to represent  by a utility function

Example
Preferences over R+ with x1  x2 iff x1 ≥ x2 .
 can be represented by u(x) = x. (It can also be represented by
other utility functions.)

However, if X is infinite we can’t necessarily represent  by a

utility function
Example (lexicographic preferences)
Preferences over [0, 1]2 ⊆ R2 with (x1 , y1 )  (x2 , y2 ) iff
x1 > x2 , or
x1 = x2 and y1 ≥ y2 .
Continuous Preferences

Definition (continuous preference relation)

A preference relation  on X is continuous iff for any sequence
{(xn , yn )}∞
n=1 with xn  yn for all n,

lim xn  lim yn .
n→∞ n→∞
Continuous Preferences

Definition (continuous preference relation)

A preference relation  on X is continuous iff for any sequence
{(xn , yn )}∞
n=1 with xn  yn for all n,

lim xn  lim yn .
n→∞ n→∞

Theorem
A continuous, rational preference relation  on X ⊆ Rn can be
represented by a continuous utility function u : X → R.
Outline

1 Choice Theory
Preferences
Utility functions
Properties of preferences
Behavioral critiques
Comparative statics
An application: Consumer Theory

2 Choice under Uncertainty

3 General Equilibrium

4 Limitations of the Welfare Theorems: Asymmetric information

Reasons for restricting preferences

Analytical tractability often demands restricting “allowable”

preferences
Some restrictions are mathematical conveniences and cannot
be empirically falsified (e.g., continuity)
Some hold broadly (e.g., more is good–monotonicity.)
Some require situational justification
Reasons for restricting preferences

Analytical tractability often demands restricting “allowable”

preferences
Some restrictions are mathematical conveniences and cannot
be empirically falsified (e.g., continuity)
Some hold broadly (e.g., more is good–monotonicity.)
Some require situational justification

Restrictions on preferences imply restrictions on utility functions.

Reasons for restricting preferences

Analytical tractability often demands restricting “allowable”

preferences
Some restrictions are mathematical conveniences and cannot
be empirically falsified (e.g., continuity)
Some hold broadly (e.g., more is good–monotonicity.)
Some require situational justification

Restrictions on preferences imply restrictions on utility functions.

And perhaps more relevant, restrictions on utility functions imply
restrictions on preferences.
Reasons for restricting preferences

Analytical tractability often demands restricting “allowable”

preferences
Some restrictions are mathematical conveniences and cannot
be empirically falsified (e.g., continuity)
Some hold broadly (e.g., more is good–monotonicity.)
Some require situational justification

Restrictions on preferences imply restrictions on utility functions.

And perhaps more relevant, restrictions on utility functions imply
restrictions on preferences.
Assumptions for the rest of this section

1  is rational (i.e., complete and transitive).

2 For simplicity, we assume preferences over X ⊆ Rn .
Interpretation and notation

We are considering a choice from the set X ⊆ Rn

How can we interpret this?

Interpretation and notation

We are considering a choice from the set X ⊆ Rn

How can we interpret this?

What then is a element x ∈ X?

Interpretation and notation

We are considering a choice from the set X ⊆ Rn

How can we interpret this?

What then is a element x ∈ X?

What might we mean when x, y ∈ X and we say x > y?

Interpretation and notation

We are considering a choice from the set X ⊆ Rn

How can we interpret this?

What then is a element x ∈ X?

What might we mean when x, y ∈ X and we say x > y?

There are different conventions: For us, x is weakly greater than y
in every coordinate, and strictly greater in at least one coordinate.
Interpretation notation

Letting x, y ∈ Rn , we write

x ≥ y if and only if xi ≥ yi for all i.

x > y if and only if xi ≥ yi for all i and xi > yi for some i.

x >> y if and only if xi > yi for all i.

Properties of rational : Monotonicity

Definition (monotonicity)
 is monotone iff x > y =⇒ x  y. (N.B. MWG differs)
 is strictly monotone iff x > y =⇒ x  y.

In words: If you give me more of everything, I weakly (strictly)

prefer it.
Properties of rational : Nonsatiation

Definition (local non-satiation)

 is locally non-satiated iff for any y and ε > 0, there exists x
such that ||x − y|| ≤ ε and x  y.

In words: For every bundle y ∈ Rn there exists a close by bundle x

that is better. For example, a little bit more of everything might be
a bit better.

 is locally non-satiated iff u(·) has no local maxima in X

Properties of rational : Convexity I

Convex preferences capture the idea that agents like diversity

1 Satisfying in some ways: rather alternate between juice and
soda than have either one every day
2 Unsatisfying in others: rather have a glass of either one than a
mixture
3 Key question is granularity of goods being aggregated
Over time? What period?
Over what “bite size”?
Properties of rational : Convexity II

Definition (convexity)
 is convex iff x  y and x0  y together imply that

tx + (1 − t)x0  y for all t ∈ (0, 1).

Equivalently,  is convex iff the upper contour set of any y (i.e.,

{x ∈ X : x  y}) is a convex set.
 is strictly convex iff x  y and x0  y (with x 6= x0 ) together
imply that
tx + (1 − t)x0  y for all t ∈ (0, 1).

i.e., one never gets worse off by mixing goods

Examples illustrating convexity definition I

Indifference Curve

(y1,y2)

Good 2
Examples illustrating convexity definition II

(x1,x2)

(y1,y2) (x’1,x’2)

Good 2
Examples illustrating convexity definition III

Upper Contour Set

(y1,y2)

Good 2
Examples illustrating convexity definition IV

(x1,x2)

(y1,y2) (x’1,x’2)

Good 2
Examples illustrating convexity definition V

(x1,x2)
(y1,y2)

(x’1,x’2)

Good 2
Quasiconcavity

Definition (quasiconcavity)
f : X → R is quasiconcave iff for all x ∈ X, the upper contour set
of x 
UCS(x) ≡ ξ ∈ X : f (ξ) ≥ f (x)

if f (ξ1 ) ≥ f (x) and f (ξ2 ) ≥ f (x), then

is a convex sets; i.e.,
f λξ1 + (1 − λ)ξ2 ≥ f (x) for all λ ∈ [0, 1].
f (·) is strictly quasiconcave iff for all x ∈ X,
f (ξ1 ) ≥ f (x) and
f (ξ2 ) ≥ f (x) implies that f λξ1 + (1 − λ)ξ2 > f (x) for all
λ ∈ (0, 1).
Concavity and Quasiconcavity

A function f (x) that is concave is quasi-concave.

A function f (x) that is quasiconcave function need not be concave.

Examples: Concavity and Quasiconcavity I

x
Examples: Concavity and Quasiconcavity II

x
Examples: Concavity and Quasiconcavity III

Convex set

x
Examples: Concavity and Quasiconcavity IV

x
Examples: Concavity and Quasiconcavity V

x
Examples: Concavity and Quasiconcavity VI

x
Examples: Concavity and Quasiconcavity VII

x
Examples: Concavity and Quasiconcavity VIII

x
Examples: Concavity and Quasiconcavity IX

x
Convex preferences and quasiconcave utility functions

A preference relation  is (strictly) convex if and only if u is

(strictly) quasiconcave.
Example 1: Preferences

Suppose: (x1 , y1 )  (x2 , y2 ) if and only if

log(x1 ) + log(y1 ) ≥ log(x2 ) + log(y2 ).
10

0
0 2 4 6 8 10
Example 1: Preferences

Suppose: (x1 , y1 )  (x2 , y2 ) if and only if

log(x1 ) + log(y1 ) ≥ log(x2 ) + log(y2 ).
10

0
0 2 4 6 8 10
Example 1: Utility Function

Represent these preferences by u(x, y) = log(x) + log(y).

Example 1: Utility Function

Represent these preferences by u(x, y) = log(x) + log(y).

Example 1: Utility Function

Represent these preferences by u(x, y) = log(x) + log(y).

Example 2: Preferences

Suppose (x1 , y1 )  (x2 , y2 ) if and only if

log(x1 ) + log(y1 ) + max(x1 , y1 ) ≥ log(x2 ) + log(y2 ) + max(x2 , y2 ).
10

2 4 6 8 10
Example 2: Preferences

Suppose (x1 , y1 )  (x2 , y2 ) if and only if

log(x1 ) + log(y1 ) + max(x1 , y1 ) ≥ log(x2 ) + log(y2 ) + max(x2 , y2 ).
10

2 4 6 8 10
Example 2: Utility function

Represent these preferences by

u(x, y) = log(x) + log(y) + max(x, y).
Example 2: Utility function

Represent these preferences by

u(x, y) = log(x) + log(y) + max(x, y).
Example 2: Utility function

Represent these preferences by

u(x, y) = log(x) + log(y) + max(x, y).
Properties of rational : Homotheticity

Definition (homotheticity)
 is homothetic iff for all x, y, and all λ > 0,

x  y ⇐⇒ λx  λy.

Definition (Homogeneity of degree k)

A utility function u : Rn → R is homogeneous of degree k if and
only if for all x ∈ Rn and all λ > 0, u(λx) = λk u(x).
Properties of rational : Homotheticity

Definition (homotheticity)
 is homothetic iff for all x, y, and all λ > 0,

x  y ⇐⇒ λx  λy.

Definition (Homogeneity of degree k)

A utility function u : Rn → R is homogeneous of degree k if and
only if for all x ∈ Rn and all λ > 0, u(λx) = λk u(x).

Continuous, strictly monotone  is homothetic iff it can be

represented by a utility function that is
Properties of rational : Homotheticity

Definition (homotheticity)
 is homothetic iff for all x, y, and all λ > 0,

x  y ⇐⇒ λx  λy.

Definition (Homogeneity of degree k)

A utility function u : Rn → R is homogeneous of degree k if and
only if for all x ∈ Rn and all λ > 0, u(λx) = λk u(x).

Continuous, strictly monotone  is homothetic iff it can be

represented by a utility function that is homogeneous of degree one
(note it can also be represented by utility functions that aren’t)
Homogeneity of degree 1 and linearity

Suppose x ∈ R and a function f (x) is homogeneous of degree 1,

then f (x) = bx for some constant b. The function is linear in
terms of the calculus notion of linearity.
Homogeneity of degree 1 and linearity

Suppose x ∈ R and a function f (x) is homogeneous of degree 1,

then f (x) = bx for some constant b. The function is linear in
terms of the calculus notion of linearity.

In linear algebra, we use a different notion of linearity. A function

f (x) is defined to be linear (sometimes called a linear map) if:

f (a + b) = f (a) + f (b).

Thus any linear function is homogeneous of degree 1.

If x ∈ Rn for n > 1, a function f (x) that is homogeneous of

degree 1 need not be linear.
e.g. f (x1 , x2 ) = min(x1 , x2 )
Implications for utility representation

Property of  Property of u(·)

Monotone Nondecreasing
Strictly monotone Increasing
Locally non-satiated Has no local maxima in X
Convex Quasiconcave
Strictly convex Strictly quasiconcave
Implications for utility representation

Property of  Property of u(·)

Monotone Nondecreasing
Strictly monotone Increasing
Locally non-satiated Has no local maxima in X
Convex Quasiconcave
Strictly convex Strictly quasiconcave
Homothetic, continuous, can be represented by a
strictly monotone utility function that is
Homogeneous of degree 1

Why should you care?

Properties of rational : Separability I

Suppose rational  over X × Y ⊆ Rp+q

First p goods form some “group” x ∈ X ⊆ Rp
Other goods y ∈ Y ⊆ Rq

Separable preferences
“Preferences over X do not depend on y” means that

(x0 , y1 )  (x, y1 ) ⇐⇒ (x0 , y2 )  (x, y2 )

for all x, x0 ∈ X and all y1 , y2 ∈ Y .

Note the definition is not symmetric in X and Y .

The critical assumption for empirical analysis of preferences

Properties of rational : Separability II

Example
X = {wine, beer} and Y = {cheese, crisps} with strict preference
ranking
1 (wine, cheese) 
2 (wine, crisps) 
3 (beer, crisps) 
4 (beer, cheese).
Utility representation of separable preferences: theorem

Theorem
Suppose  on X × Y is represented by u(x, y). Then preferences
over X do not depend on y iff there exist functions v : X → R and
U : R × Y → R such that
1 U (·, ·) is increasing in its first argument, and
2 u(x, y) = U (v(x), y) for all (x, y).
Utility representation of separable preferences: example

Example
Preferences over beverages do not depend on your snack, and are
represented by u(·, ·), where

u(wine, cheese) =4
u(wine, crisps) =3
u(beer, crisps) =2
u(beer, cheese) = 1.
Utility representation of separable preferences: example

Example
Preferences over beverages do not depend on your snack, and are
represented by u(·, ·), where

u(wine, cheese) =4
u(wine, crisps) =3
u(beer, crisps) =2
u(beer, cheese) = 1.

Let v(wine) ≡ 3 and v(beer) ≡ 2.

Utility representation of separable preferences: example

Example
Preferences over beverages do not depend on your snack, and are
represented by u(·, ·), where

u(wine, cheese) =4 And let U (3, cheese) ≡4

u(wine, crisps) =3 U (3, crisps) ≡3
u(beer, crisps) =2 U (2, crisps) ≡2
u(beer, cheese) = 1. U (2, cheese) ≡ 1.

Let v(wine) ≡ 3 and v(beer) ≡ 2.

Utility representation of separable preferences: example

Example
Preferences over beverages do not depend on your snack, and are
represented by u(·, ·), where

u(wine, cheese) =4 And let U (3, cheese) ≡4

u(wine, crisps) =3 U (3, crisps) ≡3
u(beer, crisps) =2 U (2, crisps) ≡2
u(beer, cheese) = 1. U (2, cheese) ≡ 1.

Let v(wine) ≡ 3 and v(beer) ≡ 2.

Thus
1 U (·, ·) is increasing in its first argument, and
2 u(x, y) = U (v(x), y) for all (x, y).
Properties of rational : Quasilinearity

Suppose rational  over X ≡ R × Y

First good is the numeraire (a.k.a. “good zero” or “good
one,” confusingly): think money
Other goods general; need not be in Rn
Properties of rational : Quasilinearity

Theorem
Suppose rational  on X ≡ R × Y satisfies the “numeraire
properties”:
1 Good 1 is valuable: (t, y)  (t0 , y) ⇐⇒ t ≥ t0 for all y;
2 Compensation is possible: For every y, y 0 ∈ Y , there exists
some t ∈ R such that (0, y) ∼ (t, y 0 );
3 No wealth effects: If (t, y)  (t0 , y 0 ), then for all d ∈ R,
(t + d, y)  (t0 + d, y 0 ).
Then there exists a utility function representing  of the form
u(t, y) = t + v(y) for some v : Y → R. (Note it can also be
represented by utility functions that aren’t of this form.)
Conversely, any  on X = R × Y represented by a utility function
of the form u(t, y) = t + v(y) satisfies the above properties.
Outline

1 Choice Theory
Preferences
Utility functions
Properties of preferences
Behavioral critiques
Comparative statics
An application: Consumer Theory

2 Choice under Uncertainty

3 General Equilibrium

4 Limitations of the Welfare Theorems: Asymmetric information

Problems with rational choice

Rational choice theory plays a central role in most tools of

economic analysis

But. . . significant research calls into question underlying

assumptions, identifying and explaining deviations using
Psychology
Sociology
Cognitive neuroscience (“neuroeconomics”)
Context-dependent choice

Choices appear to be highly situational, depending on

1 Other available options
2 Way that options are “framed”
3 Social context/emotional state

Numerous research projects consider these effects in real-world and

laboratory settings
e.g., Ariely (2003, QJE); Madrian and Shea (2001, QJE).
Non-considered choice

Rational choice theory depends on a considered comparison of

options
Pairwise comparison
Utility maximization

Many actual choices appear to be made using

1 Intuitive reasoning
2 Heuristics
3 Instinctive desire
Policy

Sharp predictions based on rational agents.

Behavioral critiques call these predictions into question.

Policy

Sharp predictions based on rational agents.

Behavioral critiques call these predictions into question.

Could an alternative model/approach do better?

Outline

1 Choice Theory
Preferences
Utility functions
Properties of preferences
Behavioral critiques
Comparative statics
An application: Consumer Theory

2 Choice under Uncertainty

3 General Equilibrium

4 Limitations of the Welfare Theorems: Asymmetric information

Comparative statics

Comparative statics is the study of how endogenous variables

respond to changes in exogenous variables
Endogenous variables are typically set by
1 Maximization, or
2 Equilibrium
Comparative statics

Comparative statics is the study of how endogenous variables

respond to changes in exogenous variables
Endogenous variables are typically set by
1 Maximization, or
2 Equilibrium
Often we can characterize a maximization problem as a system of
equations (like an equilibrium)
Typically we do this using FOCs
Key comparative statics tool is the Implicit Function Theorem
Runs into lots of problems with continuity, smoothness,
nonconvexity, . . .
Comparative statics tools

We will discuss (and use):

1 Envelope Theorem

2 Implicit Function Theorem

Need different tools without continuity, smoothness, . . .

(but we won’t cover this):
1 Topkis’ Theorem

2 Monotone Selection Theorem

Envelope Theorem

The ET and IFT tell us about the derivatives of different objects

with respect to the parameters of the problem (i.e., exogenous
variables):
Envelope Theorems consider value function

Implicit Function Theorem considers choice function

Envelope Theorem

A simple Envelope Theorem:

v(q) = max f (x, q)

x
= f x∗ (q), q

d ∂ ∂
∂ ∗
f x∗ (q), q + f x∗ (q), q


v(q) = x (q)
dq ∂q |∂x {z } ∂q
=0 by FOC
∂
f x∗ (q), q


=
∂q

Think of the ET as an application of the chain rule and then FOCs

A more complete Envelope Theorem

Theorem (Envelope Theorem)

Consider a constrained optimization problem v(θ) = maxx f (x, θ)
such that g1 (x, θ) ≥ 0, . . . , gK (x, θ) ≥ 0.
A more complete Envelope Theorem

Theorem (Envelope Theorem)

Consider a constrained optimization problem v(θ) = maxx f (x, θ)
such that g1 (x, θ) ≥ 0, . . . , gK (x, θ) ≥ 0.
Comparative statics on the value function are given by:
K
∂v ∂f X ∂gk ∂L
= + λk =
∂θi ∂θi x∗ ∂θi x∗ ∂θi x∗
k=1
P
(for Lagrangian L(x, θ, λ) ≡ f (x, θ) + k λk gk (x, θ)) for all θ
such that the set of binding constraints does not change in an
open neighborhood.

Roughly, the derivative of the value function is the derivative of

the Lagrangian
Example: Cost Minimization Problem

Single-output cost minimization problem

min w · z such that f (z) ≥ q.

z∈Rm
+

L(q, w, λ, µ) ≡ −w · z + λ f (z) − q + µ · z

Applying Kuhn-Tucker here gives

∂f (z ∗ )
λ ≤ wi with equality if zi∗ > 0
∂zi
The ET applied to c(q, w) ≡ minz∈Rm
+ , f (z)≥q
w · z gives
Example: Cost Minimization Problem

Single-output cost minimization problem

min w · z such that f (z) ≥ q.

z∈Rm
+

L(q, w, λ, µ) ≡ −w · z + λ f (z) − q + µ · z

Applying Kuhn-Tucker here gives

∂f (z ∗ )
λ ≤ wi with equality if zi∗ > 0
∂zi
The ET applied to c(q, w) ≡ minz∈Rm
+ , f (z)≥q
w · z gives
∂c(q, w)
=λ
∂q
Example: Cost Minimization Problem

−c(q, w) = L(q, w, λ, µ)|z=z ∗ = −w · z ∗ + λ f (z ∗ ) − q + µ · z ∗

m
d − c(q, w) X ∂zi∗ ∂z ∗ ∂z ∗
= −λ + −wi + λfzi∗ (z ∗ ) i + µi i
dq ∂q ∂q ∂q
i=1
m
X
∂zi∗
= −λ + −wi + λfzi∗ (z ∗ ) + µi
∂q
i=1
Lets look at the first order condition of the Lagrangian in zi :

−wi + λfzi (z) + µi = 0

Substituting this into the above equation:

dc(q, w)
=λ
dq
The Implicit Function Theorem I

A simple, general maximization problem

X ∗ (t) = argmax F (x, t)

x∈X

where F : X × T → R and X × T ⊆ R2 .
Suppose:
1 Smoothness: F is twice continuously differentiable
2 Convex choice set: X is convex
3 00 < 0
Strictly concave objective (in choice variable): Fxx
(together with convexity of X, this ensures a unique
maximizer)
4 Interiority: X ∗ (t) is in the interior of X for all t (which means
the standard FOC must hold)
The Implicit Function Theorem II

Under these assumptions there is a unique maximizer: |X ∗ (t)| = 1.

The first-order condition says the unique maximizer satisfies

Fx0 x(t), t = 0

Taking the derivative in t:

dFx0 x(t), t ∂Fx0 x(t), t ∂x(t) ∂Fx0 x(t), t

= + = 0.
dt ∂x(t) ∂t ∂t
So,
00 x(t), t


0 Fxt
x (t) = − 00

Fxx x(t), t
Note by strict concavity, the
denominator is negative, so x0 (t) and
00 x(t), t have the same sign
the cross-partial Fxt
Illustrating the Implicit Function Theorem

FOC: Fx0 x(t), t = 0

00 > 0 Thus t
Suppose Fxt high > tlow =⇒
Illustrating the Implicit Function Theorem

FOC: Fx0 x(t), t = 0

00 > 0 Thus t
Suppose Fxt 0 0
high > tlow =⇒ Fx (x, thigh ) > Fx (x, tlow )
Illustrating the Implicit Function Theorem

FOC: Fx0 x(t), t = 0

00 > 0 Thus t
Suppose Fxt 0 0
high > tlow =⇒ Fx (x, thigh ) > Fx (x, tlow )

x
Fx0 (·, tlow )

x(tlow )

Fx0 (·, thigh )

x

x(thigh )
Illustrating the Implicit Function Theorem

FOC: Fx0 x(t), t = 0

00 > 0 Thus t
Suppose Fxt 0 0
high > tlow =⇒ Fx (x, thigh ) > Fx (x, tlow )

F (·, tlow )
x
Fx0 (·, tlow )
x
x(tlow ) x(tlow )
F (·, thigh )
Fx0 (·, thigh )
x

x
x(thigh ) x(thigh )
Intuition for the Implicit Function Theorem

00 ≥ 0, an increase in x is more valuable when the

When Fxt
parameter t is higher
In a sense, x and t are complements; we therefore expect that an
increase in t results in an increase in the optimal choice of x
Intuition for the Implicit Function Theorem

00 ≥ 0, an increase in x is more valuable when the

When Fxt
parameter t is higher
In a sense, x and t are complements; we therefore expect that an
increase in t results in an increase in the optimal choice of x

This intuition should carry through without all our assumptions

Monotone comparative statics lead to the same conclusion
without smoothness of F or strict concavity of F in x.
Outline

1 Choice Theory
Preferences
Utility functions
Properties of preferences
Behavioral critiques
Comparative statics
An application: Consumer Theory

2 Choice under Uncertainty

3 General Equilibrium

4 Limitations of the Welfare Theorems: Asymmetric information

The consumer problem

Utility Maximization Problem

max u(x) such that p·x ≤w

x∈Rn
+
|{z}
Expenses

where p are the prices of goods and w is the consumer’s “wealth.”

This type of choice set is a budget set

B(p, w) ≡ {x ∈ Rn+ : p · x ≤ w}

Can then write our problem as:

max u(x)
x∈B(p,w)
Illustrating the Utility Maximization Problem

Suppose there are two goods (X = R2 ), what are the best choices
of x given B(p, w)? When is there a unique best choice?
Assumptions underlying the UMP

Note that
Utility function is general (but assumed to exist—a restriction
of preferences)
Choice set defined by linear budget constraint
Consumers are price takers
Prices are linear
Perfect information: prices are all known
Finite number of goods
Goods are described by quantity and price
Goods are divisible
Goods may be time- or situation-dependent
Perfect information: goods are all well understood
Aside: Marshall

Cambridge mathematician who became Professor of Political

Economy at Cambridge
Utility maximization problem

The consumer’s Marshallian demand is given by correspondence

x : Rn × R ⇒ Rn+

x(p, w) ≡ argmax u(x) ≡ argmax u(x)

x∈Rn
+ : p·x≤w x∈B(p,w)

Rn+ :

= x∈ p · x ≤ w and u(x) = v(p, w)

Resulting indirect utility function is given by

v(p, w) ≡ sup u(x) ≡ sup u(x)

x∈Rn
+ : p·x≤w x∈B(p,w)
Properties of Marshallian demand and indirect utility

Theorem
v(p, w) and x(p, w) are homogeneous of degree zero. That is, for
all p, w, and λ > 0,

v(λp, λw) = v(p, w) and x(λp, λw) = x(p, w).

These are “no money illusion” conditions

Proof.
B(λp, λw) = B(p, w), so consumers are solving the same
problem.
Implications of restrictions on preferences: convexity I

Theorem
If preferences are convex, then x(p, w) is a convex set for every
p  0 and w ≥ 0.

Theorem
If preferences are strictly convex, then x(p, w) is single-valued for
every p  0 and w ≥ 0.
Implications of restrictions on preferences: convexity II
Implications of restrictions on preferences: non-satiation I

Definition (Walras’ Law)

p · x = w for every p  0, w ≥ 0, and x ∈ x(p, w).

Theorem
If preferences are locally non-satiated, then Walras’ Law holds.

This allows us to replace the inequality constraint in the UMP with

an equality constraint
Implications of restrictions on preferences: non-satiation II

Proof.
Suppose that p · x < w for some
x ∈ x(p, w). Then there exists
some x0 sufficiently close to x
with x0  x and p · x0 < w,
which contradicts the fact that
x ∈ x(p, w). Thus p · x = w.
Solving for Marshallian demand I

Suppose the utility function is differentiable

This is an ungrounded assumption
However, differentiability can not be falsified by any finite
data set
Also, utility functions are robust to monotone transformations
We may be able to use Kuhn-Tucker to “solve” the UMP:
Utility Maximization Problem

max u(x) such that p · x ≤ w

x∈Rn
+

gives the Lagrangian

L(x, λ, µ, p, w) ≡ u(x) + λ(w − p · x) + µ · x.

Solving for Marshallian demand II

1 First order conditions:

u0i (x∗ ) = λpi − µi for all i

2 Complementary slackness:

λ(w − p · x∗ ) = 0
µi x∗i = 0 for all i

3 Non-negativity:

λ ≥ 0 and µi ≥ 0 for all i

4 Original constraints p · x∗ ≤ w and x∗i ≥ 0 for all i

We can solve this system of equations for certain functional forms
of u(·)
The power (and limitations) of Kuhn-Tucker

Kuhn-Tucker provides conditions on (x, λ, µ) given (p, w):

1 First order conditions
2 Complementary slackness
3 Non-negativity
4 (Original constraints)

Kuhn-Tucker tells us that if x∗ is a solution to the UMP, there

exist some (λ, µ) such that these conditions hold
The power (and limitations) of Kuhn-Tucker

Kuhn-Tucker provides conditions on (x, λ, µ) given (p, w):

1 First order conditions
2 Complementary slackness
3 Non-negativity
4 (Original constraints)

Kuhn-Tucker tells us that if x∗ is a solution to the UMP, there

exist some (λ, µ) such that these conditions hold; however:
These are only necessary conditions; there may be (x, λ, µ)
that satisfy Kuhn-Tucker conditions but do not solve UMP
If u(·) is concave, conditions are necessary and sufficient
When are Kuhn-Tucker conditions sufficient?

Kuhn-Tucker conditions are necessary and sufficient for a solution

(assuming differentiability) as long as we have a “convex problem”:

1 The constraint set is convex

If each constraint gives a convex set, the intersection is a
convex set

The set x : gk (x, θ) ≥ 0 is convex as long as gk (·, θ) is a
quasiconcave function of x

2 The objective function is concave

If we only know the objective is quasiconcave, there are other
conditions that ensure Kuhn-Tucker is sufficient
Intuition from Kuhn-Tucker conditions I

Recall (evaluating at the optimum, and for all i):

FOC u0i (x) = λpi − µi
CS λ(w − p · x) = 0 and µi xi = 0
NN λ ≥ 0 and µi ≥ 0
Orig p · x ≤ w and xi ≥ 0
We can summarize as

u0i (x) ≤ λpi with equality if xi > 0

And therefore if xj > 0 and xk > 0,

∂u
pj ∂xj
= ∂u
≡ MRSjk
pk ∂xk
Intuition from Kuhn-Tucker conditions II

The MRS is the (negative) slope of the indifference curve

Price ratio is the (negative) slope of the budget line

x2
6 @
@
@
@
@
@ x∗  Du(x∗ )
q
@
p
@
@
@
@
@
@
@
@ - x1
Intuition from Kuhn-Tucker conditions III

Recall the Envelope Theorem tells us the derivative of the value

function in a parameter is the derivative of the Lagrangian:
Value function (indirect utility)

v(p, w) ≡ sup u(x)

x∈B(p,w)

Lagrangian

L ≡ u(x) + λ(w − p · x) + µ · x
∂v
By the Envelope Theorem, ∂w = λ; i.e., the Lagrange multiplier λ
is the “shadow value of wealth” measured in terms of utility
Intuition from Kuhn-Tucker conditions IV

Given our envelope result, we can interpret our earlier condition

∂u
= λpi if xi > 0
∂xi

as

∂u ∂v
= pi if xi > 0
∂xi ∂w
where each side gives the marginal utility from an extra unit of xi
LHS directly
RHS through the wealth we could get by selling it
MRS and separable utility

Recall that if xj > 0 and xk > 0,

∂u
∂xj
MRSjk ≡ ∂u
∂xk

does not depend on λ; however it typically depends on x1 , . . . , xn

MRS and separable utility

Recall that if xj > 0 and xk > 0,

∂u
∂xj
MRSjk ≡ ∂u
∂xk

does not depend on λ; however it typically depends on x1 , . . . , xn

Suppose choice from X × Y where preferences over X do not
depend on y


Recall that u(x, y) = U v(x), y for some U (·, ·) and v(·)
∂u 0 0

∂v ∂u

∂v
∂xj = U1 v(x), y ∂xj and ∂xk = U1 v(x), y ∂xk
∂v ∂v
MRSjk = ∂xj / ∂xk does not depend on y
Separability allows empirical work without worrying about y
Recap: Utility Maximization Problem

Basic problem: How much of different goods will a consumer buy?

Suppose preferences are rational and satisfy continuity and
local non-satiation.
Can represent the consumer’s problem as a constrained
optimization problem.
There exists a continuous and differentiable utility function
representing these preferences.
Any such utility function
Recap: Utility Maximization Problem

Basic problem: How much of different goods will a consumer buy?

Suppose preferences are rational and satisfy continuity and
local non-satiation.
Can represent the consumer’s problem as a constrained
optimization problem.
There exists a continuous and differentiable utility function
representing these preferences.
Any such utility function has no local maxima.
Recap: Utility Maximization Problem

Basic problem: How much of different goods will a consumer buy?

Suppose preferences are rational and satisfy continuity and
local non-satiation.
Can represent the consumer’s problem as a constrained
optimization problem.
There exists a continuous and differentiable utility function
representing these preferences.
Any such utility function has no local maxima.

The Utility Maximization problem

Choose a vector x ∈ Rn+ describing how much we want to
consume of the n available goods.
To maximize utility u(x).
Selecting from bundles that are affordable: p · x ≤ w
Recap: Utility Maximization Problem I

The Utility Maximization problem

max u(x) such that p · x ≤ w

x∈Rn
+
Recap: Utility Maximization Problem I

The Utility Maximization problem

max u(x) such that p · x ≤ w

x∈Rn
+

And this gives the Lagrangian:

L(x, λ, µ, p, w) ≡ u(x) + λ(w − p · x) + µ · x.

Recap: Utility Maximization Problem II

Now solve the Lagrangian.

First differentiate with respect to xi to get first order
conditions:
Recap: Utility Maximization Problem II

Now solve the Lagrangian.

First differentiate with respect to xi to get first order
conditions:

∂L(x, λ, µ, p, w) ∂u(x)
= − λpi + µi = 0 for all i
∂xi ∂xi
Any solution to the consumer’s problem must satisfy these
conditions.

But which constraints bind?

Recap: Utility Maximization Problem II

Now solve the Lagrangian.

First differentiate with respect to xi to get first order
conditions:

∂L(x, λ, µ, p, w) ∂u(x)
= − λpi + µi = 0 for all i
∂xi ∂xi
Any solution to the consumer’s problem must satisfy these
conditions.

But which constraints bind? Using local non-satiation, apply

Walras’ law.
Recap: Utility Maximization Problem II

Now solve the Lagrangian.

First differentiate with respect to xi to get first order
conditions:

∂L(x, λ, µ, p, w) ∂u(x)
= − λpi + µi = 0 for all i
∂xi ∂xi
Any solution to the consumer’s problem must satisfy these
conditions.

But which constraints bind? Using local non-satiation, apply

Walras’ law.

This tells us that w = p · x.

Recap: Utility Maximization Problem II

Now solve the Lagrangian.

First differentiate with respect to xi to get first order
conditions:

∂L(x, λ, µ, p, w) ∂u(x)
= − λpi + µi = 0 for all i
∂xi ∂xi
Any solution to the consumer’s problem must satisfy these
conditions.

But which constraints bind? Using local non-satiation, apply

Walras’ law.

This tells us that w = p · x. We also know λ > 0. Why?

Recap: Utility Maximization Problem II

Answer 1:
The first order condition in xi is:
∂u(x)
= λpi − µi .
∂xi
If λ = 0 then ∂u(x)/∂xi ≤ 0 for all i.
But then there is a local maxima.
Recap: Utility Maximization Problem II

Answer 1:
The first order condition in xi is:
∂u(x)
= λpi − µi .
∂xi
If λ = 0 then ∂u(x)/∂xi ≤ 0 for all i.
But then there is a local maxima.

Answer 2:
As there are no local maxima utility must be increasing locally
in some xi .
So ∂v/∂w > 0.
But by the Envelope Theorem, ∂v/∂w = ∂L/∂w = λ.
Recap: Utility Maximization Problem III

What about the non-negativity constraints? Do they bind

too?
Recap: Utility Maximization Problem III

What about the non-negativity constraints? Do they bind

too?
Know from complementary slackness that µi xi = 0.
So, if xi > 0 then µi = 0.
Recap: Utility Maximization Problem III

What about the non-negativity constraints? Do they bind

too?
Know from complementary slackness that µi xi = 0.
So, if xi > 0 then µi = 0.
From the first order conditions, if xi > 0 and xj > 0

∂u(x) ∂u(x)
= λpi and = λpj
∂xi ∂xj
Recap: Utility Maximization Problem IV

Combining

∂u(x) ∂u(x)
= λpi and = λpj ,
∂xi ∂xj
we get
∂u(x)
∂xi pi
∂u(x)
= .
pj
∂xj
Recap: Utility Maximization Problem IV

Combining

∂u(x) ∂u(x)
= λpi and = λpj ,
∂xi ∂xj
we get
∂u(x)
∂xi pi
∂u(x)
= .
pj
∂xj

Left hand side is the marginal rate of substitution.

Right hand side is the price ratio.
Why we need another “problem”

We would like to characterize “important” properties of

Marshallian demand x(·, ·) and indirect utility v(·, ·)
For instance, does demand for a good always increase when
its price decreases?
Difficult because in the UMP parameters enter feasible set
rather than objective
Why we need another “problem”

We would like to characterize “important” properties of

Marshallian demand x(·, ·) and indirect utility v(·, ·)
For instance, does demand for a good always increase when
its price decreases?
Difficult because in the UMP parameters enter feasible set
rather than objective

Consider a price increase for one good (apples)

1 Substitution effect: Apples are now relatively more expensive
than bananas, so I buy fewer apples
2 Wealth effect: I feel poorer, so I buy (more? fewer?)
apples
Wealth effect and substitution effects could go in opposite
directions =⇒ can’t easily sign the change in consumption
Isolating the substitution effect

We can isolate the substitution effect by “compensating” the

consumer so that her maximized utility does not change
If maximized utility doesn’t change, the consumer can’t feel richer
or poorer; demand changes can therefore be attributed entirely to
the substitution effect
Isolating the substitution effect

We can isolate the substitution effect by “compensating” the

consumer so that her maximized utility does not change
If maximized utility doesn’t change, the consumer can’t feel richer
or poorer; demand changes can therefore be attributed entirely to
the substitution effect

Expenditure Minimization Problem

min p · x such that u(x) ≥ ū.

x∈Rn
+

i.e., find the cheapest bundle at prices p that yield utility at least ū
Illustrating the Expenditure Minimization Problem
Aside: Hicks

Lecturer at Cambridge, 1935-38.

Expenditure minimization problem

The consumer’s Hicksian demand is given by correspondence

h : Rn × R ⇒ Rn

h(p, ū) ≡ argmin p·x

x∈Rn
+ : u(x)≥ū

= {x ∈ Rn+ : u(x) ≥ ū and p · x = e(p, ū)}

Resulting expenditure function is given by

e(p, ū) ≡ min p·x

x∈Rn
+ : u(x)≥ū
Illustrating Hicksian demand
Recap

x(p, w) bundles that maximize utility given

budget constraint

v(p, w) utility generated by optimal bundles

given budget constraint

h(p, u) bundles than minimize expenditure

for given utility

e(p, u) minimum required expenditure

to obtain given utility
Relating Hicksian and Marshallian demand I

Theorem (“Same problem” identities)

Suppose u(·) is a utility function representing a continuous and
locally non-satiated preference relation  on Rn+ . Then for any
p  0 and w ≥ 0,


1 h p, v(p, w) = x(p, w),


2 e p, v(p, w) = w;

and for any ū ≥ u(0),

3 x p, e(p, ū) = h(p, ū), and


4 v p, e(p, ū) = ū.
Relating Hicksian and Marshallian demand II

These say that UMP and EMP are fundamentally solving the same
problem, so:
If the utility you can get with wealth w is v(p, w). . .
To achieve utility v(p, w) will cost at least w
You will buy the same bundle whether you have w to spend, or
you are trying to achieve utility v(p, w)

If it costs e(p, ū) to achieve utility ū. . .

Given wealth e(p, ū) you will achieve utility at most ū
You will buy the same bundle whether you have e(p, ū) to
spend, or you are trying to achieve utility ū
Relating (changes in) Hicksian and Marshallian demand I

Assuming differentiability and hence single-valuedness, we can

differentiate the ith row of the identity


h(p, ū) = x p, e(p, ū)

in pj to get

∂hi ∂xi ∂xi ∂e

= +
∂pj ∂pj ∂e ∂pj
Note that
e(p, ū) = p · h(p, ū),
so
∂e(p, ū)
= hj (p, ū) = xj
∂pj
| {z }
Shepherd’s Lemma
The Slutsky equation I

Slutsky equation


∂xi (p, w) ∂hi p, u(x(p, w)) ∂xi (p, w)
= − xj (p, w)
∂pj ∂pj ∂w
| {z } | {z } | {z }
total effect substitution effect wealth effect

for all i and j.

Sometimes known as the Hicks decomposition (just to confuse

you, something different is known as the Slutsky decomposition!).
The Slutsky equation II

Setting i = j, we can decompose the effect of an increase in pi

∂xi (p, w) ∂hi p, u(x(p, w)) ∂xi (p, w)
= − xi (p, w)
∂pi ∂pi ∂w

An “own-price” increase. . .
1 Encourages consumer to substitute away from good i
∂hi
∂pi ≤0

2 Makes consumer poorer, which affects consumption of good i

in some indeterminate way
∂xi
Sign of ∂w depends on preferences
Illustrating wealth and substitution effects

Following a decrease in the price of the first good. . .

Substitution effect moves from x to h
Wealth effect moves from h to x0

Z
J
JZZ
J Z
J ZZ
J x Z
J Z
J Z
J Z
Z
J Z
JJ Z
Z -
Illustrating wealth and substitution effects

Following a decrease in the price of the first good. . .

Substitution effect moves from x to h
Wealth effect moves from h to x0

Z
J
JZZ
J Z
Z J ZZ
Z J x Z
ZJ Z
Z
0 Z
Z
JZ h(p , u)
J
Z
Z Z
J Z Z
JJ ZZ Z
Z -
Illustrating wealth and substitution effects

Following a decrease in the price of the first good. . .

Substitution effect moves from x to h
Wealth effect moves from h to x0

Z
J
JZZ
J Z
Z J ZZ x0
Z J x Z
ZJ Z
Z
0 Z
Z
JZ h(p , u)
J
Z
Z Z
J Z Z
JJ ZZ Z
Z -
Marshallian response to changes in wealth

Definition (Normal good)

Good i is a normal good if xi (p, w) is increasing in w.

Definition (Inferior good)

Good i is an inferior good if xi (p, w) is decreasing in w.
Graphing Marshallian response to changes in wealth

Engle curves show how Marshallian demand moves with

wealth (locus of {x, x0 , x00 , . . . } below)
In this example, both goods are normal (xi increases in w)

Z
Z
Z
Z Z
Z x00
Z Z
Z Z
Z Z
x 0 Z
Z Z Z
Z x Z Z
Z Z Z
Z Z Z
Z Z Z
Z
Z Z
Z Z -
Marshallian response to changes in own price

Definition (Regular good)

Good i is a regular good if xi (p, w) is decreasing in pi .

Definition (Giffen good)

Good i is a Giffen good if xi (p, w) is increasing in pi .

Potatoes during the Irish potato famine are the canonical example
(and probably weren’t actually Giffen goods)
Marshallian response to changes in own price

Definition (Regular good)

Good i is a regular good if xi (p, w) is decreasing in pi .

Definition (Giffen good)

Good i is a Giffen good if xi (p, w) is increasing in pi .

Potatoes during the Irish potato famine are the canonical example
(and probably weren’t actually Giffen goods)
∂xi ∂hi ∂xi
By the Slutsky equation (which gives ∂pi = ∂pi − ∂w xi for i = j)
Normal =⇒ regular
Giffen =⇒ inferior
Graphing Marshallian response to changes in own price

Offer curves show how Marshallian demand moves with price

In this example, good 1 is regular and good 2 is a gross
complement for good 1

Q
@
JQ
J@Q
J@QQ
00
J@ Q
J @ xQ 0 x
x
J @
Q
Q
J @ Q
Q
J @ Q
J @ Q
Q
J @ Q
J @ Q
Q -
Marshallian response to changes in other goods’ price

Definition (Gross substitute)

Good i is a gross substitute for good j if xi (p, w) is increasing in
pj .

Definition (Gross complement)

Good i is a gross complement for good j if xi (p, w) is decreasing
in pj .

Gross substitutability/complementarity is not necessarily symmetric

Hicksian response to changes in other goods’ price

Definition (Substitute)
Good i is a substitute for good j if hi (p, ū) is increasing in pj .

Definition (Complement)
Good i is a complement for good j if hi (p, ū) is decreasing in pj .

Substitutability/complementarity is symmetric

In a two-good world, the goods must be substitutes (why? )

Wrapping up: Conceptual

Agents (people, firms, etc) make choices from sets of

alternatives.
Wrapping up: Conceptual

Agents (people, firms, etc) make choices from sets of

alternatives.
Their preferences determine these choices.
Wrapping up: Conceptual

Agents (people, firms, etc) make choices from sets of

alternatives.
Their preferences determine these choices.
Can be represented by a utility function if and only if rational.
Wrapping up: Conceptual

Agents (people, firms, etc) make choices from sets of

alternatives.
Their preferences determine these choices.
Can be represented by a utility function if and only if rational.
Properties of some underlying preferences have equivalent
properties in utility space.
Wrapping up: Conceptual

Agents (people, firms, etc) make choices from sets of

alternatives.
Their preferences determine these choices.
Can be represented by a utility function if and only if rational.
Properties of some underlying preferences have equivalent
properties in utility space.
We work in utility space because constrained maximization
problems are easier than binary comparisons!
Wrapping up: Conceptual

Agents (people, firms, etc) make choices from sets of

alternatives.
Their preferences determine these choices.
Can be represented by a utility function if and only if rational.
Properties of some underlying preferences have equivalent
properties in utility space.
We work in utility space because constrained maximization
problems are easier than binary comparisons!
Seen that it helps us understand consumers’ decisions.
Wrapping up: Mathematical tools we’ve used

Constrained optimization problems.

Envelope theorems.

Implicit function theorem.

Wrapping up: Language I

Preferences
- Completeness
- Transitivity
- Rationality
- Reflexivity
- Symmetry
- Continuity
- Monotonicity
- Local non-satiation
- Convexity
- Separability
- Choice rule
- Numeraire Properties
Wrapping up: Language II

Utility representation
- Choice rule
- Revealed preference choice rule
- HARP / WARP
- Quasiconcavity
- Homogeneity of degree k
Consumer theory
- Budget set
- Marshallian Demand
- Hicksian Demand
- Indirect utility
- Expenditure function
- Substitution effect
Wrapping up: Language III

- Wealth effect
- Normal goods
- Inferior goods
- Regular goods
- Giffen goods
- Substitutes
- Complements
- Gross Substitutes
- Gross Complements
Outline

1 Choice Theory

2 Choice under Uncertainty

Uncertainty setup
Expected utility representation
Lotteries with monetary payoffs
Measuring risk aversion
Comparative Statics
Application: demand for insurance
Expected Utility and Social Choice
Subjective Probability
Behavioral criticisms

3 General Equilibrium

4 Limitations of the Welfare Theorems: Asymmetric information

Why study uncertainty?

So far we have covered individual decision-making under certainty

Goods well understood
Prices well known

In fact, decisions typically made in the face of an uncertain future

Workhorse model: objective risk
Subjective assessments of uncertainty
Behavioral critiques
von Neumann-Morganstern expected utility model

Simplifying assumptions include

Finite number of outcomes (“prizes”)

Objectively known probability distributions over prizes

(“lotteries”)

Complete and transitive preferences over lotteries

Other assumptions on preferences over lotteries (to be
discussed)
Outline

1 Choice Theory

2 Choice under Uncertainty

Uncertainty setup
Expected utility representation
Lotteries with monetary payoffs
Measuring risk aversion
Comparative Statics
Application: demand for insurance
Expected Utility and Social Choice
Subjective Probability
Behavioral criticisms

3 General Equilibrium

4 Limitations of the Welfare Theorems: Asymmetric information

Prizes and lotteries

Let X be the set of possible prizes (a.k.a. outcomes or

consequences)
Assume |X | = n < ∞
Since |X | < ∞, there must be a best outcome and a worst
outcome
Prizes and lotteries

Let X be the set of possible prizes (a.k.a. outcomes or

consequences)
Assume |X | = n < ∞
Since |X | < ∞, there must be a best outcome and a worst
outcome

A lottery is a probability distribution over prizes

p = (p1 , . . . , pn ) ∈ Rn+
The set of all lotteries is
n X o
∆(X ) ≡ p ∈ Rn+ : pi = 1 ,
i

called the n-dimensional simplex

Graphing the simplex ∆(X ) ⊆ R2

Suppose there are two prizes (|X | = 2)

The simplex ∆(X ) is the portion of the line p1 + p2 = 1 that
lies in the positive quadrant

p1

∆(X )

p2
Graphing the simplex ∆(X ) ⊆ R3

Suppose there are three prizes (|X | = 3)

The simplex ∆(X ) is the portion of the plane
p1 + p2 + p3 = 1 that lies in the positive orthant
p2

p3 p1
Graphing the simplex ∆(X ) ⊆ R3

Suppose there are three prizes (|X | = 3)

The simplex ∆(X ) is the portion of the plane
p1 + p2 + p3 = 1 that lies in the positive orthant
p2 p2

p3 p1 p3 p1
Convexity of the simplex I

Note that ∆(X ) is a convex set

If pi ≥ 0 and p0i ≥ 0, then αpi + (1 − α)p0i ≥ 0
If i pi = 1 and i p0i = 1, then i [αpi + (1 − α)p0i ] = 1
P P P

This is not surprising given that the simplex is a “triangle”

p2

p
αp + (1 − α)p0
p0
p3 p1
Convexity of the simplex II

We can view αp + (1 − α)p0 as a compound lottery

1 Choose between lotteries: Lottery p with probability α and
lottery p0 with probability (1 − α)
2 Resolve uncertainty in chosen lottery per p or p0

p
α

1−α
p0
Preferences over lotteries

A rational decision-maker has preferences over outcomes X

We consider preferences over lotteries ∆(X ) (note that from here

on,  refers to preferences over lotteries, not outcomes)

Expected utility theory relies on  satisfying

Completeness
Transitivity
Continuity (in a sense to be defined)
Independence (to be defined)
Continuity axiom

Definition (continuity)
A preference relation  over ∆(X ) is continuous iff for any pH ,
pM , and pL ∈ ∆(X ) such that pH  pM  pL , there exists some
α ∈ [0, 1] such that

αpH + (1 − α)pL ∼ pM .
An example I

Consider the following lotteries:

p $12 for sure.
p0 $15 with probability 1/2 and $10 with probability 1/2.
pm $100 with probability 1/2 and $0 with probability 1/2.
An example I

Consider the following lotteries:

p $12 for sure.
p0 $15 with probability 1/2 and $10 with probability 1/2.
pm $100 with probability 1/2 and $0 with probability 1/2.

Suppose pm  p0  p.

Continuity implies that there exists an α ∈ [0, 1] such that

αpm + (1 − α)p ∼ p0 . Reasonable?
An example I

Consider the following lotteries:

p $12 for sure.
p0 $15 with probability 1/2 and $10 with probability 1/2.
pm $100 with probability 1/2 and $0 with probability 1/2.

Suppose pm  p0  p.

Continuity implies that there exists an α ∈ [0, 1] such that

αpm + (1 − α)p ∼ p0 . Reasonable?

When might continuity not hold?

An example II

Same lotteries:
p $12 for sure.
p0 $15 with probability 1/2 and $10 with probability 1/2.
pm $100 with probability 1/2 and $0 with probability 1/2.

Suppose p  p0 .

For some α ∈ [0, 1] consider the following two compound lotteries:

c1 αp0 + (1 − α)pm .
c2 αp + (1 − α)pm .
An example II

Same lotteries:
p $12 for sure.
p0 $15 with probability 1/2 and $10 with probability 1/2.
pm $100 with probability 1/2 and $0 with probability 1/2.

Suppose p  p0 .

For some α ∈ [0, 1] consider the following two compound lotteries:

c1 αp0 + (1 − α)pm .
c2 αp + (1 − α)pm .

Would you expect c1 be preferred to c2?

An example III

Compound Lottery c1:

1/2 100

α 0
1/2
1/2 15
1-α
1/2 10
Compound Lottery c2:
1/2 100

α
1/2 0

1-α 1 12
Independence axiom

Definition (independence)
A preference relation  over ∆(X ) satisfies independence iff for
any p, p0 , and pm ∈ ∆(X ) and any α ∈ [0, 1], we have

p  p0
m
αp + (1 − α)pm  αp0 + (1 − α)pm .

i.e., if I prefer p to p0 , I also prefer the possibility of p to the

possibility of p0 , as long as the other possibility is the same (a
(1 − α) chance of pm ) in both cases
Independence sensible for choice under uncertainty

There is no counterpart in standard consumer theory; e.g.,

p = (2 cokes, 0 apples) and p0 = (0 cokes, 2 apples)
pm = (2 cokes, 2 apples)
1
α= 2

p p0
z }| { z }| {
(2 cokes, 0 apples)  (0 cokes, 2 apples)
Independence sensible for choice under uncertainty

There is no counterpart in standard consumer theory; e.g.,

p = (2 cokes, 0 apples) and p0 = (0 cokes, 2 apples)
pm = (2 cokes, 2 apples)
1
α= 2
There is no reason to conclude that
p p0
z }| { z }| {
(2 cokes, 0 apples)  (0 cokes, 2 apples)
m
(2 cokes, 1 apples)  (1 cokes, 2 apples)
| {z } | {z }
αp+(1−α)pm αp0 +(1−α)pm
Independence sensible for choice under uncertainty

There is no counterpart in standard consumer theory; e.g.,

p = (2 cokes, 0 apples) and p0 = (0 cokes, 2 apples)
pm = (2 cokes, 2 apples)
1
α= 2
There is no reason to conclude that
p p0
z }| { z }| {
(2 cokes, 0 apples)  (0 cokes, 2 apples)
m
(2 cokes, 1 apples)  (1 cokes, 2 apples)
| {z } | {z }
αp+(1−α)pm αp0 +(1−α)pm

Why the difference?

Example

p = (2 cokes, 0 apples) and p0 = (0 cokes, 2 apples)

pm = (2 cokes, 2 apples)
pc = (2 cokes, 1 apples) and p0c = (1 cokes, 2 apples).

Suppose you are very hungry and thirsty.

(i) First priority is fluids: p  p0
(ii) But after 1 drink, food becomes more desirable: p0c  pc
(iii) Now consider a 50:50 chance of p and pm Vs. a 50:50 chance
of p0 and pm .
Example

p = (2 cokes, 0 apples) and p0 = (0 cokes, 2 apples)

pm = (2 cokes, 2 apples)
pc = (2 cokes, 1 apples) and p0c = (1 cokes, 2 apples).

Suppose you are very hungry and thirsty.

(i) First priority is fluids: p  p0
(ii) But after 1 drink, food becomes more desirable: p0c  pc
(iii) Now consider a 50:50 chance of p and pm Vs. a 50:50 chance
of p0 and pm .
(iv) (ii) and (iii) are not the same. Prefer the 50:50 chance of p
and pm because it guarantees liquids. . .
Independence implies linear indifference curves

Independence implies linear indifference curves

Consider p ∼ p0
Let pm = p0
By the independence axiom,

αp + (1 − α)p0 ∼ αp0 + (1 − α)p0

∼ p0
∼p

p2

p
αp + (1 − α)p0
p0
p3 p1
Independence implies parallel indifference curves

Independence implies parallel indifference curves

Consider p ∼ p0
Let pm be some other point, and α some value in (0, 1)
By independence αp + (1 − α)pm ∼ αp0 + (1 − α)pm
αp + (1 − α)pm and αp0 + (1 − α)pm lie on a line parallel to
the indifference curve containing p and p0

p2

p0
pm
p3 p1
Independence implies parallel indifference curves

Independence implies parallel indifference curves

Consider p ∼ p0
Let pm be some other point, and α some value in (0, 1)
By independence αp + (1 − α)pm ∼ αp0 + (1 − α)pm
αp + (1 − α)pm and αp0 + (1 − α)pm lie on a line parallel to
the indifference curve containing p and p0

p2

p0
pm
p3 p1
Recap: The Simplex

p2

(1, 0, 0)
p3 p1
Recap: The Simplex

p2

(1/2, 1/2, 0)

p3 p1
Recap: The Simplex

p2

p3 p1
Recap: The Simplex

p2

(1/6, 1/6, 2/3)

p3 p1
Linear and parallel indifference curves

Suppose X = {x1 , x2 , x3 } and x1  x2  x3 .

Let p1 = (1, 0, 0), p2 = (0, 1, 0) and p3 = (0, 0, 1).

By continuity, there exists a γ ∈ [0, 1] such that

pc = γp1 + (1 − γ)p3 ∼ p2 .
Linear and parallel indifference curves

p2

p3 p1
Linear and parallel indifference curves

p2

γ 1-γ
p3 p1
Linear and parallel indifference curves

By independence for any q, q 0 , and qm ∈ ∆(X ) and any α ∈ [0, 1],

we have

q  q0
m
αq + (1 − α)qm  αq 0 + (1 − α)qm .

Suppose we then have q  q 0 and q 0  q, then

αq + (1 − α)qm ∼ αq 0 + (1 − α)qm .
Linear and parallel indifference curves

By independence for any q, q 0 , and qm ∈ ∆(X ) and any α ∈ [0, 1],

we have

q  q0
m
αq + (1 − α)qm  αq 0 + (1 − α)qm .

Suppose we then have q  q 0 and q 0  q, then

αq + (1 − α)qm ∼ αq 0 + (1 − α)qm .
Letting q = p2 , q 0 = pc and qm = p2 , as p2  pc and pc  p2 , we
have

αp2 + (1 − α)p2 = p2 ∼ αpc + (1 − α)p2

Linear and parallel indifference curves

p2

γ 1-γ
p3 p1
Linear and parallel indifference curves

Using independence again, as pc ∼ p2 :

αpc + (1 − α)p1 ∼ αp2 + (1 − α)p1

There is thus an indifference curve connecting these points.
Linear and parallel indifference curves

p2

α 1-α
p3 p1
Linear and parallel indifference curves

p2
α

1-α

α 1-α
p3 p1
Linear and parallel indifference curves

p2
α

1-α

α 1-α
p3 p1
Linear and parallel indifference curves

β
p2

1-β

1-β β
p3 p1
Linear and parallel indifference curves

p2

p3 p1
Outline

1 Choice Theory

2 Choice under Uncertainty

Uncertainty setup
Expected utility representation
Lotteries with monetary payoffs
Measuring risk aversion
Comparative Statics
Application: demand for insurance
Expected Utility and Social Choice
Subjective Probability
Behavioral criticisms

3 General Equilibrium

4 Limitations of the Welfare Theorems: Asymmetric information

von Neumann-Morganstern utility functions

Definition (von Neumann-Morganstern utility function)

A utility function U : ∆(X ) → R is a vNM utility function iff there
exist numbers u1 , . . . , un ∈ R such that for every p ∈ ∆(X ),
n
X
U (p) = pi ui = p · ~u.
i=1

Can think of u1 , . . . , un as indexing preference over outcomes

Linearity of vNM utility functions I

Theorem
A utility function U : ∆(X ) → R is a vNM utility function iff it is
linear in probabilities, i.e.,

U αp + (1 − α)p0 = αU (p) + (1 − α)U (p0 )

for all p, p0 ∈ ∆(X ), and α ∈ [0, 1]

Linearity of vNM utility functions II

Proof Outline
vNM =⇒ linearity:
Suppose |X | = 3.
Consider p = (p1 , p2 , p3 ) and p0 = (p01 , p02 , p03 ).
Let ~u = (u1 , u2 , u3 )

U αp + (1 − α)p0 = αp + (1 − α)p0 · ~u

= (αp) · ~u + (1 − α)p0 · ~u

= α(p · ~u) + (1 − α)(p0 · ~u)

= αU (p) + (1 − α)U (p0 )
Linearity of vNM utility functions III

linearity =⇒ vNM:
Continue to let |X | = 3 and p = (p1 , p2 , p3 ).
We can write p as a compound lottery:
p = p1 (1, 0, 0) + p2 (0, 1, 0) + p3 (0, 0, 1) = p1 b1 + p2 b2 + p3 b3

By linearity:

X
U p = p1 U b1 + p2 U b2 + p3 U b3 =




pi ui
i

for ui = U bi .


Expected utility representation and ordinality

If preferences  can be represented by a vNM utility function, we

say it is an “expected utility representation” of 
That is “U (·) is an expected utility representation of ” means
1 U (·) is a vNM utility function, and
2 U (·) represents 
Expected utility representation and ordinality

If preferences  can be represented by a vNM utility function, we

say it is an “expected utility representation” of 
That is “U (·) is an expected utility representation of ” means
1 U (·) is a vNM utility function, and
2 U (·) represents 

Linearity of vNM utility functions mean that expected utility

representation is not ordinal
Utility representation is robust to any increasing monotone
transformation
Expected utility representation is only robust to affine
(increasing linear) transformations
Exp. util. representation robust to affine transformation

Theorem
Suppose U : ∆(X ) → R is an expected utility representation of .
Then V : ∆(X ) → R is also an expected utility representation of 
iff there exist some a ∈ R and b ∈ R++ such that

V (p) = a + bU (p)

for all p ∈ ∆(X ).

Level sets of an expected utility function

A von Neumann-Morganstern utility function satisfies

n
X
U (p) = pi ui = p · ~u
i=1

for some ~u ∈ Rn

Indifference curves are therefore p · ~u = c for various c

Level sets of an expected utility function

A von Neumann-Morganstern utility function satisfies

n
X
U (p) = pi ui = p · ~u
i=1

for some ~u ∈ Rn

Indifference curves are therefore p · ~u = c for various c

Indifference curves are therefore
Straight lines (p · ~u = c is a plane that intercepts the simplex
in a line)
Level sets of an expected utility function

A von Neumann-Morganstern utility function satisfies

n
X
U (p) = pi ui = p · ~u
i=1

for some ~u ∈ Rn

Indifference curves are therefore p · ~u = c for various c

Indifference curves are therefore
Straight lines (p · ~u = c is a plane that intercepts the simplex
in a line)
Parallel (all indifference curves are normal to ~u)
Example

Suppose |X | = 3.

vNM utility function: U (p) = p1 u1 + p2 u2 + p3 u3

Suppose p ∼ p0 so U (p) = U (p0 ).

Example

Suppose |X | = 3.

vNM utility function: U (p) = p1 u1 + p2 u2 + p3 u3

Suppose p ∼ p0 so U (p) = U (p0 ).

By linearity,
U (αp + (1 − α)p0 ) = αU (p) + (1 − α)U (p0 ) = U (p).

So indifference curves are linear.

Example

Suppose |X | = 3.

vNM utility function: U (p) = p1 u1 + p2 u2 + p3 u3

Suppose p ∼ p0 so U (p) = U (p0 ).

By linearity,
U (αp + (1 − α)p0 ) = αU (p) + (1 − α)U (p0 ) = U (p).

So indifference curves are linear.

Suppose p00  p ∼ p0 so U (p00 ) > U (p) = U (p0 ).

Example

Suppose |X | = 3.

vNM utility function: U (p) = p1 u1 + p2 u2 + p3 u3

Suppose p ∼ p0 so U (p) = U (p0 ).

By linearity,
U (αp + (1 − α)p0 ) = αU (p) + (1 − α)U (p0 ) = U (p).

So indifference curves are linear.

Suppose p00  p ∼ p0 so U (p00 ) > U (p) = U (p0 ).

Then U (αp00 + (1 − α)p) = αU (p00 ) + (1 − α)U (p)

And U (αp00 + (1 − α)p0 ) = αU (p00 ) + (1 − α)U (p)

So indifference curves are parallel.

So, loosely, we have argued that:

Expected utility representation

⇐⇒
parallel, straight indifference curves
⇐⇒
preferences satisfy completeness, transitivity, continuity and
independence.
Which preferences have expected utility representations? I

The following is a remarkable result due to Von Neumann and

Morgenstern (1953) building on work by Bernoulli.

Theorem
A complete and transitive preference relation  on ∆(X ) satisfies
continuity and independence iff it has an expected utility
representation U : ∆(X ) → R.

Proof Outline Showing that if U (p) = p · ~u represents , then 

must satisfy continuity and independence is (relatively) easy
Showing the other direction is a bit harder. . .
Which preferences have expected utility representations? II

Roughly:
1 Let p ∈ ∆(X ) be the most preferred lottery and p ∈ ∆(X ) be
the least preferred lottery. For every p ∈ ∆(X ) find λp ∈ [0, 1]
such that
p ∼ λp p̄ + (1 − λp )p
This λp . . .
exists by continuity
is unique by independence
2 Let U (p) = λp
3 Show that U (·) is an expected utility representation of 
U (·) represents 
U (·) is linear, and therefore is a vNM utility function
Outline

1 Choice Theory

2 Choice under Uncertainty

Uncertainty setup
Expected utility representation
Lotteries with monetary payoffs
Measuring risk aversion
Comparative Statics
Application: demand for insurance
Expected Utility and Social Choice
Subjective Probability
Behavioral criticisms

3 General Equilibrium

4 Limitations of the Welfare Theorems: Asymmetric information

Money lotteries

We seek to measure the attitude towards risk embedded in

preferences 

We will now consider monetary payoffs

X ⊆ R (note we give up assumption that prize set is finite)
∆(X ) is now a bit more complicated
Money lotteries

We seek to measure the attitude towards risk embedded in

preferences 

We will now consider monetary payoffs

X ⊆ R (note we give up assumption that prize set is finite)
∆(X ) is now a bit more complicated
A probability distribution with finite support can be described
with a pmf; set of distributions is the simplex
A probability distribution over infinite (ordered) support is
described by a cdf

From here on, we’ll represent a lottery by a cdf F (·), where F (x)
is the probability of receiving less than or equal to an $x payout
Example

What lottery does this CDF represent?

1
Probability get less
than this

0
$0 $100
Example

What lottery does this CDF represent?

1
Probability get less
than this

0
$0 $100

You get an amount drawn uniformly from the interval [0, 100].
Example

What lottery does this CDF represent?

1
Probability get less
than this

0
$0 $100
Example

What lottery does this CDF represent?

1
Probability get less
than this

0
$0 $100

50% chance of $0 and 50% chance of $75.

Example

What lottery does this CDF represent?

1
Probability get less
than this

0
$0 $100
Example

What lottery does this CDF represent?

1
Probability get less
than this

0
$0 $100

Approximately: 50% chance of $0 and 50% chance of $75.

 Probability get less
Example

than this

0
1

$0
$100
 Probability get less
Example

than this

0
1
0.8

$0
$50
$100
What is the set of lotteries?

When |X | = n < ∞, the set of all lotteries is

n X o
∆(X ) ≡ p ∈ Rn+ : pi = 1
i
What is the set of lotteries?

When |X | = n < ∞, the set of all lotteries is

n X o
∆(X ) ≡ p ∈ Rn+ : pi = 1
i

When X = R, the set of all lotteries (we will consider) is the set of
cdfs:
F is the set of all functions F : R → [0, 1] such that
F (·) is nondecreasing
limx→−∞ F (x) = 0
limx→+∞ F (x) = 1
F (·) continuous (This is stronger than we need, but will
suffice for our purposes)
Preferences over money lotteries

Our old vNM utility function (over pmfs) was

X X  
U (p) = pi ui = pi u(xi ) ≡ Ep u(x)
i i
Preferences over money lotteries

Our old vNM utility function (over pmfs) was

X X  
U (p) = pi ui = pi u(xi ) ≡ Ep u(x)
i i

The continuous analogue is a vNM utility function over cdfs:

Z
 
U (F ) = u(x) dF (x) ≡ EF u(x)
R

Where
U : F → R (“von Neumann-Morganstern utility function”)
represents preferences over lotteries
u : R → R (“Bernoulli utility function”) indexes preference
over outcomes
Risk aversion I

Definition (risk aversion)

A decision-maker is risk-averse
R iff for all lotteries F , she prefers a
certain payoff of EF (x) ≡ R x dF (x) to the lottery F .

Definition (strict risk aversion)

A decision-maker is strictly risk-averse iff for all non-degenerate
lotteries F (i.e, all lotteries for which the support of F is not a
singleton),R she strictly prefers a certain payoff of
EF (x) ≡ R x dF (x) to the lottery F .
Risk aversion II

Risk aversion says that for all F ,

 
u EF [x] ≥ EF u(x)

or equivalently
Z  Z
u x dF (x) ≥ u(x) dF (x)
R R

By Jensen’s inequality, this condition holds iff u(·) is concave

Theorem
A decision-maker is (strictly) risk-averse iff her Bernoulli utility
function is (strictly) concave.
Illustrating risk aversion

Consider a risk-averse decision-maker (i.e., one with a concave

Bernoulli utility function) evaluating a lottery F with a two-point
distribution

u(x)
 
u EF (x)
 
EF u(x)

x
EF (x)

 
u EF [x] ≥ EF u(x)
Outline

1 Choice Theory

2 Choice under Uncertainty

Uncertainty setup
Expected utility representation
Lotteries with monetary payoffs
Measuring risk aversion
Comparative Statics
Application: demand for insurance
Expected Utility and Social Choice
Subjective Probability
Behavioral criticisms

3 General Equilibrium

4 Limitations of the Welfare Theorems: Asymmetric information

The certain equivalent

A risk-averse decision-maker prefers a certain payoff of EF (x) to

the lottery F
 
u EF [x] ≥ EF u(x)
How many “certain” dollars is F worth? That is, what is the
certain payoff that gives the same utility as lottery F ?
The certain equivalent

A risk-averse decision-maker prefers a certain payoff of EF (x) to

the lottery F
 
u EF [x] ≥ EF u(x)
How many “certain” dollars is F worth? That is, what is the
certain payoff that gives the same utility as lottery F ?
Definition (certain equivalent)
The certain equivalent is the size of the certain payout such that a
decision-maker is indifferent between the certain payout and the
lottery F :
Z

 
u cu (F ) = EF u(x) ≡ u(x) dF (x).
R
Illustrating the certain equivalent

Consider a risk-averse decision-maker (i.e., one with a concave

Bernoulli utility function) evaluating a lottery F with a two-point
distribution

u(x)
 
u EF (x)
 
EF u(x)

x
cu (F )EF (x)

 

u cu (F ) = EF u(x) ≤ u EF [x]
The certain equivalent as a measure of risk aversion

For a risk-averse decision-maker

 

u cu (F ) = EF u(x) ≤ u EF [x]

so assuming increasing u(·),

cu (F ) ≤ EF [x]

The certain equivalent gives a measure of risk aversion

A consumer u is risk-averse iff cu (F ) ≤ EF [x] for all F
Consumer u is more risk-averse than consumer v iff
cu (F ) ≤ cv (F ) for all F
The Arrow-Pratt coefficient of absolute risk aversion

Definition (Arrow-Pratt coefficient of absolute risk aversion)

Given a twice differentiable Bernoulli utility function u(·),

u00 (x)
Au (x) ≡ − .
u0 (x)

Where does this come from?

Risk-aversion is related to concavity of u(·); a “more concave”
function has a smaller (more negative) second derivative
hence a larger −u00 (x)
The Arrow-Pratt coefficient of absolute risk aversion

Definition (Arrow-Pratt coefficient of absolute risk aversion)

Given a twice differentiable Bernoulli utility function u(·),

u00 (x)
Au (x) ≡ − .
u0 (x)

Where does this come from?

Risk-aversion is related to concavity of u(·); a “more concave”
function has a smaller (more negative) second derivative
hence a larger −u00 (x)
Normalization by u0 (x) takes care of the fact that au(·) + b
represents the same preferences as u(·)
We can also view it as a “probability premium”
The A-P coefficient of ARA as a probability premium

Consider a risk-averse consumer:

She prefers x for certain to a 50-50 gamble between x + ε and
x−ε
The A-P coefficient of ARA as a probability premium

Consider a risk-averse consumer:

She prefers x for certain to a 50-50 gamble between x + ε and
x−ε
If we wanted to convince her to take such a gamble, it
couldn’t be 50-50—we need to make the x + ε payout more
likely
The A-P coefficient of ARA as a probability premium

Consider a risk-averse consumer:

She prefers x for certain to a 50-50 gamble between x + ε and
x−ε
If we wanted to convince her to take such a gamble, it
couldn’t be 50-50—we need to make the x + ε payout more
likely
Consider the gamble G such that she is indifferent between G
and receiving x for certain, where
(
x+ε with probability 12 + π,
G=
x−ε with probability 12 − π
The A-P coefficient of ARA as a probability premium

Consider a risk-averse consumer:

She prefers x for certain to a 50-50 gamble between x + ε and
x−ε
If we wanted to convince her to take such a gamble, it
couldn’t be 50-50—we need to make the x + ε payout more
likely
Consider the gamble G such that she is indifferent between G
and receiving x for certain, where
(
x+ε with probability 12 + π,
G=
x−ε with probability 12 − π

It turns out that Au (x) is proportional to (π/ε) as ε → 0; i.e.,

Au (x) tells us the “premium” measured in probability that the
decision-maker demands per unit of spread ε
Our measures of risk aversion are equivalent

Theorem
The following definitions of u being “more risk-averse” than v are
equivalent:
1 Whenever u prefers F to a certain payout d, then v does as
well; i.e., for all F and d,
   
EF u(x) ≥ u(d) =⇒ EF v(x) ≥ v(d);

2 Certain equivalents cu (F ) ≤ cv (F ) for all F ;

3 u(·) is “more concave” than v(·); i.e., there exists some

increasing concave function g(·) such that u(x) = g v(x) for
all x;
4 Arrow-Pratt coefficients of absolute risk aversion
Au (x) ≥ Av (x) for all x.
How does risk aversion change with “wealth”

Example
Suppose
 
$120 with probability 2/3
 ($110 for certain).
$60 with probability 1/3

We might then reasonably expect that

 
$220 with probability 2/3
($210 for certain).
$160 with probability 1/3
How does risk aversion change with “wealth”

Example
Suppose
 
$120 with probability 2/3
 ($110 for certain).
$60 with probability 1/3

We might then reasonably expect that

 
$220 with probability 2/3
 ($210 for certain).
$160 with probability 1/3

This is the idea of decreasing absolute risk aversion:

decision-makers are less risk-averse when they are “richer”
Decreasing absolute risk aversion

Definition (decreasing absolute risk aversion)

The Bernoulli utility function u(·) has decreasing absolute risk
aversion iff Au (·) is a decreasing function of x.
Decreasing absolute risk aversion

Definition (decreasing absolute risk aversion)

The Bernoulli utility function u(·) has decreasing absolute risk
aversion iff Au (·) is a decreasing function of x.

Definition (increasing absolute risk aversion)

The Bernoulli utility function u(·) has increasing absolute risk
aversion iff Au (·) is an increasing function of x.

Definition (constant absolute risk aversion)

The Bernoulli utility function u(·) has constant absolute risk
aversion iff Au (·) is a constant function of x.
Relative risk aversion

Definition (coefficient of relative risk aversion)

Given a twice differentiable Bernoulli utility function u(·),

u00 (x)
Ru (x) ≡ −x = xAu (x).
u0 (x)
Relative risk aversion

Definition (coefficient of relative risk aversion)

Given a twice differentiable Bernoulli utility function u(·),

u00 (x)
Ru (x) ≡ −x = xAu (x).
u0 (x)

We can define decreasing/increasing/constant relative risk aversion

as above, but using Ru (·) instead of Au (·)
DARA means that if I take a $10 gamble when poor, I will
take a $10 gamble when rich
DRRA means that if I gamble 10% of my wealth when poor, I
will gamble 10% when rich
Outline

1 Choice Theory

2 Choice under Uncertainty

Uncertainty setup
Expected utility representation
Lotteries with monetary payoffs
Measuring risk aversion
Comparative Statics
Application: demand for insurance
Expected Utility and Social Choice
Subjective Probability
Behavioral criticisms

3 General Equilibrium

4 Limitations of the Welfare Theorems: Asymmetric information

When is one lottery “better than” another?

We can compare lotteries given a Bernoulli utility function

representing some preferences u

But when will two lotteries be consistently ranked under a broad

set of preferences? e.g.,
1 All nondecreasing u(·)
2 All nondecreasing, concave (i.e., risk averse) u(·)
Comparing lotteries: examples I

Example
$95 for certain vs. $105 for certain.

Example
 
$90 with probability 1/2
vs. $95 for certain.
$110 with probability 1/2
Comparing lotteries: examples II

Example
 
$90 with probability 1/2
vs. $105 for certain.
$110 with probability 1/2

Example
 
$90 with probability 1/2
vs. $110 for certain.
$110 with probability 1/2
Comparing lotteries: examples III

Example

   
$90 with probability 1/2 $80 with probability 1/2
vs. .
$110 with probability 1/2 $120 with probability 1/2
First-order stochastic dominance

Definition (first-order stochastic dominance)

Distribution G first-order stochastic dominates distribution F iff
lottery G is preferred to F under every nondecreasing Bernoulli
utility function u(·).

That is, for every nondecreasing u : R → R, the following

(equivalent) statements hold:

G u F,
   
EG u(x) ≥ EF u(x) ,
Z Z
u(x) dG(x) ≥ u(x) dF (x).
R R
Characterizing first-order stochastic dominant cdfs

Theorem
Distribution G first-order stochastic dominates distribution F iff
G(x) ≤ F (x) for all x.

That is, lottery G is more likely than F to pay at least x for any
threshold x
 Probability get less
Example

than this

0
1

$0
$100
 Probability get less
Example

than this

0
1
0.8

0.1
$0
$50
$100
Back to our examples. . .

Example

Lottery
x $95 $105 $110 $80 or $120 $90 or $110
< 80 F (x) = 0 0 0 0 0
[80, 90) 0 0 0 1/2 0
[90, 95) 0 0 0 1/2 1/2

[95, 105) 1 0 0 1/2 1/2

[105, 110) 1 1 0 1/2 1/2

[110, 120) 1 1 1 1/2 1

≥ 120 1 1 1 1 1
Back to our examples. . .

Example

Lottery
x $95 $105 $110 $80 or $120 $90 or $110
< 80 F (x) = 0 0 0 0 0
[80, 90) 0 0 0 1/2 0
[90, 95) 0 0 0 1/2 1/2

[95, 105) 1 0 0 1/2 1/2

[105, 110) 1 1 0 1/2 1/2

[110, 120) 1 1 1 1/2 1

≥ 120 1 1 1 1 1

($110) FOSD ($105) FOSD ($95)

($110) FOSD ($90 or $110)
Every other combination is ambiguous in terms of FOSD
Characterizing FOSD with upward shifts

Start with a lottery F and construct compound lottery G

First resolve F
Then if the resolution of F is some x, hold a second lottery
that could potentially increase (but can’t decrease) x

x f (·)

4 1
F (·)

3 1/2

2 1/2

1 1/2

0 0 x
1 2 3 4
Characterizing FOSD with upward shifts

Start with a lottery F and construct compound lottery G

First resolve F
Then if the resolution of F is some x, hold a second lottery
that could potentially increase (but can’t decrease) x

x f (·) g(·)

4 1/4 1
F (·)
G(·)
3 1/2 1/4

2 1/2

1 1/2 1/2

0 0 x
1 2 3 4
Characterizing FOSD with upward shifts

Start with a lottery F and construct compound lottery G

First resolve F
Then if the resolution of F is some x, hold a second lottery
that could potentially increase (but can’t decrease) x

x f (·) g(·)

4 1/4 1
F (·)
G(·)
3 1/2 1/4

2 1/2

1 1/2 1/2

0 0 x
1 2 3 4

G FOSD F iff we can construct G from F using upward shifts

Which of these lotteries can be ranked?

Payoff L1 L2 L3 L4
0 1/2 0 1/4 1/4
1 0 1/4 0 0
2 1/4 1/2 0 1/2
3 1/4 1/4 0 0
4 0 0 3/4 1/4
Which of these lotteries can be ranked?

Payoff L1 L2 L3 L4
0 1/2 0 1/4 1/4
1 0 1/4 0 0
2 1/4 1/2 0 1/2
3 1/4 1/4 0 0
4 0 0 3/4 1/4

L3 FOSD L4 FOSD L1 and L2 FOSD L1.

Second-order stochastic dominance

FOSD said a lottery was preferred by all nondecreasing u(·). . .

Consider whether a lottery is preferred by all risk-averse u(·)

Definition (second-order stochastic dominance)

Consider two lotteries with which pay out according to the
distributions F and G.
Distribution G second-order stochastic dominates distribution F iff
lottery G is preferred to F under every concave, nondecreasing
Bernoulli utility function u(·).
That is, for every concave, nondecreasing u : R → R,
   
EG u(x) ≥ EF u(x) .
Characterizing second-order stochastic dominant cdfs

Theorem
Distribution G second-order stochastic dominates distribution F iff
Z x Z x
G(t) dt ≤ F (t) dt for all x.
−∞ −∞
Example: Which would you prefer

1
Probability get less
than this

0
$0 $100
Example: Which would you prefer

1
Probability get less

0.8
than this

0.2
0
$0 $35 $50 $100
Example: Which would you prefer

1
Probability get less

0.8
than this

0.2
0
$0 $20 $65 $100
 Probability get less
than this

0
1

$0
Example: SOSD

$100
 Probability get less
than this

0
1

$0
Example: SOSD

x
$100
 Probability get less
than this

0
1

$0
Example: SOSD

x
$100
 Probability get less
than this

0
1

$0
Example: SOSD

x
$100
 Probability get less
than this

0
1

$0
Example: SOSD

x
$100
 Probability get less
than this

0
1

$0
Example: SOSD

$100
Characterizing SOSD with mean-preserving spreads I

We can construct a lottery F from G using mean-preserving

spreads if there exists a way of reaching lottery F using the
following procedure:
1. Set lottery H = G
2. Take a possible outcome of H and change it so that instead
of receiving that outcome an additional lottery is held that
has zero-mean.
3. Update lottery H to this new lottery.
4. If H = F stop. Otherwise return to step 2.
Characterizing SOSD with mean-preserving spreads II

x g(·)

4 1
G(·)

3 1/2

2 1/2

1 1/2

0 0 x
1 2 3 4
Characterizing SOSD with mean-preserving spreads III

x g(·) f (·)

4 1/4 1
G(·)
F (·)
3 1/2

2 1/4 1/2

1 1/2 1/2

0 0 x
1 2 3 4

Suppose two lotteries F and G have the same mean. Then G

SOSD F iff we can construct F from G using mean-preserving
spreads
Outline

1 Choice Theory

2 Choice under Uncertainty

Uncertainty setup
Expected utility representation
Lotteries with monetary payoffs
Measuring risk aversion
Comparative Statics
Application: demand for insurance
Expected Utility and Social Choice
Subjective Probability
Behavioral criticisms

3 General Equilibrium

4 Limitations of the Welfare Theorems: Asymmetric information

Demand for insurance: Setup I

Strictly risk-averse agent with wealth w

Risk of loss L with probability p

Insurance available for cost qa pays a in event of loss (agent

chooses a)

She solves
Consumer Problem

max pu[w − qa − L + a] + (1 − p)u[w − qa]

a
Demand for insurance: Setup II

The demand for insurance problem

max pu[w − qa − L + a] + (1 − p)u[w − qa]

a | {z }
≡U (a)

Note strict concavity of u(·) gives strict concavity of U (·):

U 0 (a) = (1 − q)pu0 [w − qa − L + a] − q(1 − p)u0 [w − qa]

U 00 (a) = (1 − q)2 pu00 [w − qa − L + a] + q 2 (1 − p)u00 [w − qa] < 0

Thus FOC is necessary and sufficient:

p(1 − q)u0 [w − qa∗ − L + a∗ ] = q(1 − p)u0 [w − qa∗ ]

Actuarially fair insurance

What if insurance is actuarially fair?

That is, insurer makes zero-profit: q = p
FOC becomes
 0
−
p(1

q)u [w − qa∗ − L + a∗ ] =  −
q(1  0
p)u [w − qa∗ ]
w − qa∗ − L + a∗ = w − qa∗
a∗ = L

Agent fully insures against risk of loss

Non-actuarially fair insurance

What if insurance is not actuarially fair?

Suppose cost of insurance is above expected loss: q > p
FOC is
u0 [w − qa∗ − L + a∗ ] q(1 − p)
0 ∗
=
u [w − qa ] p(1 − q)
>1
u0 [w − qa∗ − L + a∗ ] > u0 [w − qa∗ ]
w − qa∗ − L + a∗ < w − qa∗
a∗ < L

Agent under-insures against risk of loss; it’s costly to transfer

wealth to the loss state, so she transfers less
Understanding the inequality reversal in the imperfect
insurance problem

Increasing concave utility

𝑢(𝑥)

𝑥
Understanding the inequality reversal in the imperfect
insurance problem

If 𝑢(𝑥 ′ ) > 𝑢(𝑥 ′′ ) then 𝑥 ′ > 𝑥 ′′

𝑢(𝑥)

𝑥
Understanding the inequality reversal in the imperfect
insurance problem

Concavity implies decreasing

marginal utility
𝑢′ (𝑥)

𝑥
Understanding the inequality reversal in the imperfect
insurance problem

If 𝑢′ (𝑥 ′ ) > 𝑢′ (𝑥 ′′ ) then 𝑥 ′ < 𝑥 ′′

𝑢′ (𝑥)

𝑥
Outline

1 Choice Theory

2 Choice under Uncertainty

Uncertainty setup
Expected utility representation
Lotteries with monetary payoffs
Measuring risk aversion
Comparative Statics
Application: demand for insurance
Expected Utility and Social Choice
Subjective Probability
Behavioral criticisms

3 General Equilibrium

4 Limitations of the Welfare Theorems: Asymmetric information

Utilitarianism and interpersonal comparisons

Bentham proposed that society should maximize the sum of

individual utilities: X
Ui
i

But this implies interpersonal comparisons.

I could represent j’s preferences with a utility function

Vj = 106 Uj

I’d get different prescriptions if I instead maximized:

X
Ui + Vj ,
j6=i
A weaker criterion

Bergson (38), as a Harvard undergrad, proposed instead

maximizing a function:

W (U1 , . . . , Un ),

under the assumption that W is increasing in all its

arguments.

Does not require interpersonal comparisons.

Embeds a notion of Pareto optimality.

When can social preferences be represented in this way?

A weaker criterion

Bergson (38), as a Harvard undergrad, proposed instead

maximizing a function:

W (U1 , . . . , Un ),

under the assumption that W is increasing in all its

arguments.

Does not require interpersonal comparisons.

Embeds a notion of Pareto optimality.

When can social preferences be represented in this way?

So requires transitivity and completeness of social preferences
Transitivity is a strong assumption (e.g. Condorcet’s Paradox).
An aside: Condorcet’s Paradox

Consider the following social preferences among three

alternatives:
1/3rd of people have preferences A  B  C.
1/3rd of people have preferences C  A  B.
1/3rd of people have preferences B  C  A.

Suppose we say society prefers D to E (D S E) if more

people prefer D than E.

Then we have A S B, B S C and C S A.

Arrow’s impossibility theorem formalizes the difficulty in

generating complete and transitive social preferences.
Harsanyi’s extension

Harsanyi (1955) proposed refining Bergson’s proposal to

include uncertainty.

Showed that if all agents and society satisfy the vN-M axioms,
then society should maximize:
X
θi Ui ,
i

for individual vN-M utilities Ui and some weights θi .

As before, this representation is cardinal but does not make

interpersonal comparisons.

Can rescale the θi terms.

We are back to a (weighted) utilitarian welfare function!

Harsanyi’s argument

Consider decision making behind the veil of ignorance.

A person imagines themselves being allocated uniformly at

random to any position in society.

That person associates each position in society with some

utility vi .

Now use expected utility theory to maximize that individual’s

expected utility.
P
This implies the person maximizing: i vi .
Outline

1 Choice Theory

2 Choice under Uncertainty

Uncertainty setup
Expected utility representation
Lotteries with monetary payoffs
Measuring risk aversion
Comparative Statics
Application: demand for insurance
Expected Utility and Social Choice
Subjective Probability
Behavioral criticisms

3 General Equilibrium

4 Limitations of the Welfare Theorems: Asymmetric information

Subjective probabilities

We have assumed the decision-maker accurately understands the

likelihood of each outcome

But in some cases subjective probabilities more reasonable?

Subjective probabilities

We have assumed the decision-maker accurately understands the

likelihood of each outcome

But in some cases subjective probabilities more reasonable?

Can probably agree on the probability distribution for a coin
toss.
Much harder to agree on an objective probability for the
existence of intelligent alien life.
Arguably most situations are like the second—for example,
what is the probability Trump will win the US presidential
election.
Choice with subjective probabilties

Suppose there are bookmakers taking bets on who will win

the election.
These bookmakers offer odds on each candidate winning the
election.
Modeling bets:
Set of outcomes X (e.g., dollar payouts)
Set of “states of the world” S
Bets (a.k.a. “acts”) are mappings from states of the world to
outcomes f : S → X
Suppose I am offered every possible bet and each time you
observe whether I take it or not.
What axioms might we expect my choices to satisfy?
Savage (1954)

Suppose that choices in this world satisfy axioms similar in spirit to

the vN-M axioms.
(Completeness, transitivity, something close to continuity, . . . )
Then Savage shows that choice must be must be as if maximizing
a vN-M utility function given:

Some probability distribution p over S

A Bernoulli utility function u : X → R
P P
An act f  g if and only if s∈S u(f (s))p(s) ≥ s∈S u(g(s))p(s).
Savage (1954)

The Savage result is remarkable and a key achievement in the

development of choice theory. Why?
Savage (1954)

The Savage result is remarkable and a key achievement in the

development of choice theory. Why?
It was not assumed that any probability distribution exists!
Savage (1954)

The Savage result is remarkable and a key achievement in the

development of choice theory. Why?
It was not assumed that any probability distribution exists!
We didn’t impose on the world the need to have a probability
in mind for intelligent alien life existing.
Instead, we just considered people making choices about what
bets to accept.
Savage (1954)

The Savage result is remarkable and a key achievement in the

development of choice theory. Why?
It was not assumed that any probability distribution exists!
We didn’t impose on the world the need to have a probability
in mind for intelligent alien life existing.
Instead, we just considered people making choices about what
bets to accept.
As long as those choices satisfy some (reasonable) criteria.
Choices are made as if people have some probability
distribution in mind.
Savage (1954)

The Savage result is remarkable and a key achievement in the

development of choice theory. Why?
It was not assumed that any probability distribution exists!
We didn’t impose on the world the need to have a probability
in mind for intelligent alien life existing.
Instead, we just considered people making choices about what
bets to accept.
As long as those choices satisfy some (reasonable) criteria.
Choices are made as if people have some probability
distribution in mind.
The existence of such a probability distribution is derived!
Justifies expected utility theory even when there is no
objective probability distribution. Any limitations?
Savage (1954)

Assumes you can list all possible states.

Savage (1954)

Assumes you can list all possible states.

Different people may have different probability distributions.

Outline

1 Choice Theory

2 Choice under Uncertainty

Uncertainty setup
Expected utility representation
Lotteries with monetary payoffs
Measuring risk aversion
Comparative Statics
Application: demand for insurance
Expected Utility and Social Choice
Subjective Probability
Behavioral criticisms

3 General Equilibrium

4 Limitations of the Welfare Theorems: Asymmetric information

Problems with the independence axiom

Example
Which lottery would you rather face?

$0 $48 $55
Lottery A 1% 66% 33%
Lottery B 100%
Problems with the independence axiom

Example
Which lottery would you rather face?

$0 $48 $55 Expected payout

Lottery A 1% 66% 33% $50
Lottery B 100% $48
Problems with the independence axiom

Example
Which lottery would you rather face?

$0 $48 $55 Expected payout

Lottery A 1% 66% 33% $50
Lottery B 100% $48
Problems with the independence axiom

Example
Which lottery would you rather face?

$0 $48 $55 Expected payout

Lottery A 1% 66% 33% $50
Lottery B 100% $48

Example
Which lottery would you rather face?

$0 $48 $55 Expected payout

Lottery C 67% 33% $18
Lottery D 66% 34% $16
Problems with the independence axiom

Example
Which lottery would you rather face?

$0 $48 $55 Expected payout

Lottery A 1% 66% 33% $50
Lottery B 100% $48

Example
Which lottery would you rather face?

$0 $48 $55 Expected payout

Lottery C 67% 33% $18
Lottery D 66% 34% $16
Illustrating Allais’ experiment

$0 $48 $55 Expected payout

Lottery A 1% 66% 33% $50
Lottery B 100% $48
Lottery C 67% 33% $18
Lottery D 66% 34% $16

$0

D C

B A
$48 $55
Alternative Models

Experimental violations of the independence axiom have generated

a huge literature looking for alternative models.
These models don’t impose linearity in probabilities.
Some have alternative axiomatic foundations.
Others just consider more general forms. For example:
X
U (p) = ui g(pi ),
i

for some increasing (but not necessarily linear) function g,

Problems with risk aversion

It seems like people are way too risk averse on small-stakes

gambles (Rabin, 2000).
Example
Would you bet on a fair coin toss where you lose $1000 or win
$1050?
Problems with risk aversion

It seems like people are way too risk averse on small-stakes

gambles (Rabin, 2000).
Example
Would you bet on a fair coin toss where you lose $1000 or win
$1050?

If you would always turn down such a bet (at any wealth level),
you would turn down a bet on a fair coin where you lose $20,000
or gain any amount

“Loss aversion” has been suggested as an explanation

Risk aversion implied by small gambles: Intuition

-100 5 110
Risk aversion implied by small gambles: Intuition

-100 5 110
Risk aversion implied by small gambles: Intuition

-1000 100M
Ellsberg Paradox

Two urns with white and black balls:

Urn A 49 black balls, 51 white balls.
Urn B also has 100 balls, each of which is black or white, but
the mix is unknown.

Three lotteries:
(i) $100 if a black ball is drawn from A, nothing otherwise.
(ii) $100 if a black ball is drawn from B, nothing otherwise.
(iii) $100 if a white ball is drawn from B, nothing otherwise.

Lotteries (ii) and (iii) have subjective risk. Lottery (i) has objective
risk.
Ellsberg Paradox

Two urns with white and black balls:

Urn A 49 black balls, 51 white balls.
Urn B also has 100 balls, each of which is black or white, but
the mix is unknown.

Three lotteries:
(i) $100 if a black ball is drawn from A, nothing otherwise.
(ii) $100 if a black ball is drawn from B, nothing otherwise.
(iii) $100 if a white ball is drawn from B, nothing otherwise.

Lotteries (ii) and (iii) have subjective risk. Lottery (i) has objective
risk.
By Savage, should prefer either (ii) or (iii) to (i), but most people
choose (i).
Ellsberg Paradox

Motivates a large behavioural literature.

Can model Urn B as having a number of white selected from some

distributions.

People might then behave as if their choice affects the distribution

selected.

For example, the worst possible distribution might always be

selected.

This allows people to be ambiguity averse.

Framing experiment 1

Example
The U.S. is preparing for an outbreak of an unusual disease, which
is expected to kill 600 people. Two alternative programs to
combat the disease have been proposed. Scientists predict that:
If program A is adopted, 200 people will be saved.
If program B is adopted, there is a 2/3 chance that no one will
be saved, and a 1/3 probability that 600 people will be saved.

Which program would you choose?

Framing experiment 1

Example
The U.S. is preparing for an outbreak of an unusual disease, which
is expected to kill 600 people. Two alternative programs to
combat the disease have been proposed. Scientists predict that:
If program A is adopted, 200 people will be saved. [72%]
If program B is adopted, there is a 2/3 chance that no one will
be saved, and a 1/3 probability that 600 people will be saved.
[28%]
Which program would you choose?
Framing experiment 2

Example
The U.S. is preparing for an outbreak of an unusual disease, which
is expected to kill 600 people. Two alternative programs to
combat the disease have been proposed. Scientists predict that:
If program C is adopted, 400 people will die with certainty.

If program D is adopted, there is a 2/3 probability that 600

people will die, and a 1/3 probability that no one will die.

Which program would you choose?

Framing experiment 2

Example
The U.S. is preparing for an outbreak of an unusual disease, which
is expected to kill 600 people. Two alternative programs to
combat the disease have been proposed. Scientists predict that:
If program C is adopted, 400 people will die with certainty.
[22%]
If program D is adopted, there is a 2/3 probability that 600
people will die, and a 1/3 probability that no one will die.
[78%]
Which program would you choose?

These two examples are exactly the same question stated in

different ways!
Outline

1 Choice Theory

2 Choice under Uncertainty

3 General Equilibrium
Introduction
Exchange economies: the Walrasian Model
The Welfare theorems
Characterizing optimality and equilibrium with first-order conditions
Existence of Walrasian equilibria
Properties of Walrasian equilibria
A useful restriction: the “gross substitutes” property
General equilibrium with production
Limitations of the Welfare Theorems: Externalities
An example of inefficiency
Public goods
Introduction

From the time of Adam Smith’s Wealth of Nations in

1776, one recurrent theme of economic analysis has been
the remarkable degree of coherence among the vast
numbers of individual and seemingly separate decisions
about the buying and selling of commodities. [...]
Would-be buyers ordinarily count correctly on being able
to carry out their intentions, and would-be sellers do not
ordinarily find themselves producing great amounts of
goods that they cannot sell. This experience of balance is
indeed so widespread that it raises no intellectual disquiet
among laymen; they take it so much for granted that
they are not disposed to understand the mechanism by
which it occurs.
Ken Arrow (1973)
Equilibrium

In a single market, we can think about price equalizing demand

and supply.

At an equilibrium price:
Consumers’ maximize their utility generating demand.
Producers’ maximize their profits generating supply.
Market clears so that demand equals supply.
But the story is a bit more complicated. . .

Demand for each good depends on prices of other goods

Marshallian demand ~x(~
p, w)

Supply also depends on the vector of prices: ~y (~

p)

General equilibrium prices satisfy

~y (~
p) = ~x(~
p, w),

Potentially a very complicated system of equations.

Have to clear all markets simultaneously.
Equilibrium Price and Efficiency

Fundamental question: Are equilibrium prices efficient?

Relevant notion of efficiency due to Pareto (1909) and
Bergson (1938)
Equilibrium Price and Efficiency

Fundamental question: Are equilibrium prices efficient?

Relevant notion of efficiency due to Pareto (1909) and
Bergson (1938)

Inquiry culminated in the Welfare Theorems (Arrow (1951)

and Debreu (1951)).
Equilibrium Price and Efficiency

Fundamental question: Are equilibrium prices efficient?

Relevant notion of efficiency due to Pareto (1909) and
Bergson (1938)

Inquiry culminated in the Welfare Theorems (Arrow (1951)

and Debreu (1951)).

Identifies a strong relationship between efficient outcomes and

market equilibria.
Equilibrium Price and Efficiency

Fundamental question: Are equilibrium prices efficient?

Relevant notion of efficiency due to Pareto (1909) and
Bergson (1938)

Inquiry culminated in the Welfare Theorems (Arrow (1951)

and Debreu (1951)).

Identifies a strong relationship between efficient outcomes and

market equilibria.

But, efficient outcomes could also be obtained in other ways.

Key advantage of market equilibria minimal informational
requirements—prices are a sufficient statistic for guiding choices.
General equilibrium: other key questions

Does a general equilibrium exist?

When is it Unique

When is it “Stable”

How does the economy reach general equilibrium prices?

An important simplification

Finding prices that equalize production and demand is hard.

We will ignore production: exchange economy

Finite number of agents
Finite number of goods
Predetermined amount of each commodity (no production)
Goods get traded and consumed

Results will extend to production economies—but this is beyond

the scope of this class.
Other assumptions

None of the following assumptions should surprise at this point,

but should be kept in mind when interpreting our following results:
Markets exist for all goods
Agents can freely participate in markets without cost
“Standard” consumer theory assumptions
Preferences can be represented by a utility function
Preferences are LNS/monotone/strictly monotone (as needed)
All agents are price takers
Finite number of divisible goods
Linear prices
Perfect information about goods and prices
All agents face the same prices
Outline

1 Choice Theory

2 Choice under Uncertainty

3 General Equilibrium
Introduction
Exchange economies: the Walrasian Model
The Welfare theorems
Characterizing optimality and equilibrium with first-order conditions
Existence of Walrasian equilibria
Properties of Walrasian equilibria
A useful restriction: the “gross substitutes” property
General equilibrium with production
Limitations of the Welfare Theorems: Externalities
An example of inefficiency
Public goods
Exchange economies: the Walrasian Model

Primitives of the model

L goods ` ∈ L ≡ {1, . . . , L}
I agents i ∈ I ≡ {1, . . . , I} (for the most part, we’ll set
I = 2)
Endowments ei ∈ RL + ; agents do not have monetary wealth,
but rather an endowment of goods which they can trade or
consume
Preferences represented by utility function ui : RL
+ →R

Endogenous prices p ∈ RL
+ , taken as given by each agent
Exchange economies: the Walrasian Model

Primitives of the model

L goods ` ∈ L ≡ {1, . . . , L}
I agents i ∈ I ≡ {1, . . . , I} (for the most part, we’ll set
I = 2)
Endowments ei ∈ RL + ; agents do not have monetary wealth,
but rather an endowment of goods which they can trade or
consume
Preferences represented by utility function ui : RL
+ →R

Endogenous prices p ∈ RL
+ , taken as given by each agent

Each agent i solves

max ui (xi ) such that p · xi ≤ p · ei ≡ max ui (xi )

xi ∈RL
+
xi ∈B i (p)

where B i (p) ≡ {xi ∈ RL i i

+ : p · x ≤ p · e } is the budget set for i
Walrasian equilibrium

Definition (Walrasian equilibrium)

Prices p and quantities (xi )i∈I are a Walrasian equilibrium iff
1 All agents maximize their utilities; i.e., for all i ∈ I,

xi ∈ argmax ui (x);
x∈B i (p)

2 Markets clear; i.e., for all ` ∈ L,

X X
xi` = ei` .
i∈I i∈I
A graphical example: the Edgeworth box

𝑒21 + 𝑒22

𝑒21 𝑒1

𝑒11 𝑒11 + 𝑒12

Agent 1
A graphical example: the Edgeworth box

𝑒21 + 𝑒22

𝑒22 𝑒2

𝑒12 𝑒11 + 𝑒12

Agent 2
A graphical example: the Edgeworth box
A graphical example: the Edgeworth box
A graphical example: the Edgeworth box

𝑒11 + 𝑒12
𝑒2

𝑒12
Agent 2
+ 𝑒22

𝑒22
1
2
A graphical example: the Edgeworth box
A graphical example: the Edgeworth box
A graphical example: the Edgeworth box

Agent 2
𝑒11 + 𝑒12 𝑒12

𝑒2 𝑒22

𝑒21 + 𝑒22
A graphical example: the Edgeworth box

Agent 2
𝑒11 + 𝑒12 𝑒12

𝑒2 𝑒22

𝑒21 + 𝑒22
A graphical example: the Edgeworth box

𝑒21 + 𝑒22

𝑒21 𝑒1

𝑒11 𝑒11 + 𝑒12

Agent 1
A graphical example: the Edgeworth box

Agent 2
𝑒12

𝑒21 𝑒22

𝑒11
Agent 1
Budget Constraints

p · x1 = p · e1 coincides with p · x2 = p · e2

𝑒21 + 𝑒22

𝑒21 𝑒1

𝑒11 𝑒11 + 𝑒12

Agent 1
Budget Constraints

p · x1 = p · e1 coincides with p · x2 = p · e2

𝑒21 + 𝑒22

𝑒22 𝑒2

𝑒12 𝑒11 + 𝑒12

Agent 2
Budget Constraints

p · x1 = p · e1 coincides with p · x2 = p · e2
Agent 2
𝑒11 + 𝑒12 𝑒12

𝑒2 𝑒22

𝑒21 + 𝑒22
Budget Constraints

p · x1 = p · e1 coincides with p · x2 = p · e2

Agent 2
𝑒11 + 𝑒12 𝑒12

𝑒2 𝑒22

𝑒21 + 𝑒22
Budget Constraints

p · x1 = p · e1 coincides with p · x2 = p · e2

Agent 2
𝑒12

𝑒21 𝑒22

𝑒11
Agent 1
The offer curve

The offer curve traces out Marshallian demand as prices change

𝑒21 𝑒1

𝑒11
Agent 1
The offer curve

The offer curve traces out Marshallian demand as prices change

𝑒21 𝑒1

𝑒11
Agent 1
The offer curve

The offer curve traces out Marshallian demand as prices change

𝑒21 𝑒1

𝑒11
Agent 1
The offer curve

The offer curve traces out Marshallian demand as prices change

𝑒21 𝑒1

𝑒11
Agent 1
The offer curve

The offer curve traces out Marshallian demand as prices change

𝑒21

𝑒11
Agent 1
The offer curve

The offer curve traces out Marshallian demand as prices change

𝑒21

𝑒11
Agent 1
The offer curve

The offer curve traces out Marshallian demand as prices change

𝑒21

𝑒11
Agent 1
The offer curve

The offer curve traces out Marshallian demand as prices change

𝑒21

𝑒11
Agent 1
The offer curve

The offer curve traces out Marshallian demand as prices change

𝑒21

𝑒11
Agent 1
The offer curve

The offer curve traces out Marshallian demand as prices change

Offer Curve

𝑒21

𝑒11
Agent 1
Non-equilibrium prices give total demand 6= supply

Agent 2

Agent 1
Equilibrium prices give total demand = supply

Agent 2

Agent 1
Equilibrium Uniqueness and Multiplicity

Agent 2

Agent 1
Equilibrium Uniqueness and Multiplicity

Agent 2

Agent 1
Equilibrium Uniqueness and Multiplicity

Agent 2

Agent 1
Some Clarifications

1 Offer curves can include infeasible allocations

They are defined just by the optimal bundle of goods for a
given agent as prices change. There is no mention of whether
that bundle is feasible or not.
2 Why are intersections of offer curves in the Edgeworth box
Walrasian equilibria whenever they are not at the endowment
point?
Offer curves with infeasible allocations

𝑒21

𝑒11
Agent 1
Offer curves with infeasible allocations

𝑒21

𝑒11
Agent 1
Offer curves with infeasible allocations

𝑒21

𝑒11
Agent 1
Offer curves with infeasible allocations

𝑒21

𝑒11
Agent 1
Offer curves with infeasible allocations

𝑒21

𝑒11
Agent 1
Offer curves with infeasible allocations

𝑒21

𝑒11
Agent 1
Offer curves with infeasible allocations

𝑒21

𝑒11
Agent 1
Some Clarifications From Last Time

1 Offer curves can include infeasible allocations

They are defined just by the optimal bundle of goods for a
given agent as prices change. There is no mention of whether
that bundle is feasible or not.
2 Why are intersections of offer curves in the Edgeworth
box Walrasian equilibria whenever they are not at the
endowment point?
Intersections of offer curves and equilibria

𝑒21

𝑒11
Agent 1
Intersections of offer curves and equilibria

𝑒21

𝑒11
Agent 1
Intersections of offer curves and equilibria

𝑒21

𝑒11
Agent 1
Intersections of offer curves and equilibria

𝑒12

𝑒22

Agent 1
Intersections of offer curves and equilibria

𝑒12

𝑒22

Agent 1
Intersections of offer curves and equilibria

𝑒12

𝑒22

Agent 1
Intersections of offer curves and equilibria

𝑒21

𝑒11
Agent 1
Intersections of offer curves and equilibria

Agent 2

Agent 1
Intersections of offer curves and equilibria

Agent 2

Agent 1
Intersections of offer curves and equilibria

Agent 2

Agent 1
Intersections of offer curves and equilibria

Agent 2

Agent 1
Intersections of offer curves and equilibria

Agent 2

Agent 1
Intersections of offer curves and equilibria

Agent 2

Agent 1
Pareto optimality

Definition (feasible allocation)

Allocations (xi )i∈I ∈ RI·L
+ are feasible iff for all ` ∈ L,
X X
xi` ≤ ei` .
i∈I i∈I
Pareto optimality

Definition (feasible allocation)

Allocations (xi )i∈I ∈ RI·L
+ are feasible iff for all ` ∈ L,
X X
xi` ≤ ei` .
i∈I i∈I

Definition (Pareto optimality)

Allocations x ≡ (xi )i∈I are Pareto optimal iff
1 x is feasible, and
2 There is no other feasible allocation x̂ such that
ui (x̂i ) ≥ ui (xi ) for all i ∈ I with strict inequality for some i.
Pareto optimality: General case

More generally, Pareto optimality can be defined over outcomes X

for any set of agents I given notions of
1 Feasibility: a mapping X → {infeasible, feasible}
2 Individual preferences: rational preferences i over X for each
i∈I

Definition (Pareto optimality)

Outcome x ∈ X is Pareto optimal iff
1 x is feasible, and
2 There is no other feasible outcome x̂ ∈ X such that x̂ i x
for all i ∈ I with x̂ i x for some i.

This is a very weak notion of optimality, requiring only that there

is nothing “left on the table”
Pareto optimality in the Edgeworth box

If the indifference curves passing through x are not tangent, it is

not Pareto optimal

Agent 2

Agent 1
Pareto optimality in the Edgeworth box

If the indifference curves passing through x are not tangent, it is

not Pareto optimal

Agent 2

Pareto
Improvements

Agent 1
Pareto optimality in the Edgeworth box

If the indifference curves passing through x are not tangent, it is

not Pareto optimal

Agent 2

Agent 1
Pareto optimality in the Edgeworth box

If the indifference curves passing through x are not tangent, it is

not Pareto optimal

Agent 2

Agent 1
Pareto optimality in the Edgeworth box

If the indifference curves passing through x are not tangent, it is

not Pareto optimal

Agent 2

No Pareto
Improvements
𝑥

Agent 1
Pareto optimality in the Edgeworth box

If the indifference curves passing through x are not tangent, it is

not Pareto optimal

Agent 2

Agent 1
Pareto optimality in the Edgeworth box

If the indifference curves passing through x are not tangent, it is

not Pareto optimal

Pareto Set Agent 2

Agent 1
Pareto optimality in the Edgeworth box

If the indifference curves passing through x are not tangent, it is

not Pareto optimal

Contract Curve Agent 2

Agent 1
Outline

1 Choice Theory

2 Choice under Uncertainty

3 General Equilibrium
Introduction
Exchange economies: the Walrasian Model
The Welfare theorems
Characterizing optimality and equilibrium with first-order conditions
Existence of Walrasian equilibria
Properties of Walrasian equilibria
A useful restriction: the “gross substitutes” property
General equilibrium with production
Limitations of the Welfare Theorems: Externalities
An example of inefficiency
Public goods
Relating Walrasian equilibrium and Pareto optimality

Note Walrasian Equilibria and Pareto Optima are very different

concepts
Pareto optimality

1 Allocations
given total endowments and individual preferences.

Walrasian equilibrium

1 Allocations
2 Prices
given individual endowments and preferences.
Walrasian equilibrium allocations are Pareto optimal

Agent 2

Agent 1
Walrasian equilibrium allocations are Pareto optimal

Agent 2

Agent 1
Walrasian equilibrium allocations are Pareto optimal

Agent 2

Agent 1
Walrasian equilibrium allocations are Pareto optimal

Agent 2

Agent 1
Walrasian equilibrium allocations are Pareto optimal

Agent 2

Agent 1
The First Welfare Theorem: WE are PO

Theorem (First Welfare Theorem)

Suppose ui (·) is increasing (i.e., ui (xi0 ) > ui (xi ) for any xi0  xi )
for all i ∈ I.
If p and (xi )i∈I are a Walrasian equilibrium, then the allocations
(xi )i∈I are Pareto optimal.

Proof. Suppose in contradiction that x̂ Pareto dominates x; i.e.,

1 x̂ is feasible,
2 ui (x̂i ) ≥ ui (xi ) for all i ∈ I,
0 0 0
3 ui (x̂i ) > ui (xi ) for some i0 ∈ I.
Proof (continued).

By revealed preference and Walras’ law, p · x̂i ≥ p · xi for all i, and

0 0
p · x̂i > p · xi .
Proof (continued).

By revealed preference and Walras’ law, p · x̂i ≥ p · xi for all i, and

0 0
p · x̂i > p · xi . Thus
X X
p · x̂i > p · xi
i∈I i∈I
XX XX
p` x̂i` > p` xi` .
`∈L i∈I `∈L i∈I
Proof (continued).

By revealed preference and Walras’ law, p · x̂i ≥ p · xi for all i, and

0 0
p · x̂i > p · xi . Thus
X X
p · x̂i > p · xi
i∈I i∈I
XX XX
p` x̂i` > p` xi` .
`∈L i∈I `∈L i∈I

So for some `e it must be that

X X X
x̂i`˜ > xi`˜ = ei`˜,
i∈I i∈I i∈I
Proof (continued).

By revealed preference and Walras’ law, p · x̂i ≥ p · xi for all i, and

0 0
p · x̂i > p · xi . Thus
X X
p · x̂i > p · xi
i∈I i∈I
XX XX
p` x̂i` > p` xi` .
`∈L i∈I `∈L i∈I

So for some `e it must be that

X X X
x̂i`˜ > xi`˜ = ei`˜,
i∈I i∈I i∈I

so x̂ cannot be feasible.
The Second Welfare Theorem: PO endowments are WE

Theorem (Second Welfare Theorem)

Suppose for all i ∈ I,
1 ui (·) is continuous;
2 ui (·) is increasing; i.e., ui (xi0 ) > ui (xi ) for any xi0  xi ;
3 ui (·) is concave; and
4 ei  0; i.e., every agent has at least a little bit of every good.

If (ei )i∈I are Pareto optimal, then there exist prices p ∈ Rl+ such
that p and (ei )i∈I are a Walrasian equilibrium.
Pareto optimal allocations can be supported as a
Walrasian equilibrium

Pareto Set Agent 2

Agent 1
Pareto optimal allocations can be supported as a
Walrasian equilibrium

Agent 2

Agent 1
Pareto optimal allocations can be supported as a
Walrasian equilibrium

Agent 2

Agent 1
Pareto optimal allocations can be supported as a
Walrasian equilibrium

Agent 2

Agent 1
Pareto optimal allocations can be supported as a
Walrasian equilibrium

Agent 2

Agent 1
Pareto optimal allocations can be supported as a
Walrasian equilibrium

Agent 2

Agent 1
The welfare theorems

Theorem (First Welfare Theorem)

Suppose ui (·) is increasing for all i ∈ I.
If p and (xi )i∈I are a Walrasian equilibrium, then the allocations
(xi )i∈I are Pareto optimal.

Theorem (Second Welfare Theorem)

Suppose ui (·) is continuous, increasing, and concave for all i ∈ I.
Further suppose ei  0 for all i ∈ I.
If (ei )i∈I are Pareto optimal, then there exist prices p ∈ Rl+ such
that p and (ei )i∈I are a Walrasian equilibrium.
Thoughts on the second welfare theorem

The assumptions behind the SWT are much stronger than the
FWT—in particular the requirement of convex preferences

ei  0 is required to ensure that each agent has a positive

endowment of some good with a non-zero price—that is,
everyone has non-zero wealth

The SWT is often attributed more importance than it

deserves;
Thoughts on the second welfare theorem

It is tempting to say that FWT shows price equilibria are

sufficient for efficient equilibria.
And the SWT shows price equilibria are necessary for efficient
equilibria.
Thoughts on the second welfare theorem

It is tempting to say that FWT shows price equilibria are

sufficient for efficient equilibria.
And the SWT shows price equilibria are necessary for efficient
equilibria.
But this is wrong.
SWT only shows that we can obtain a given efficient
outcomes via markets.
But if we know an outcome is efficient, why not implement it
directly?
Thoughts on the second welfare theorem

The following claim about the SWT is also commonly made:

Suppose there is a Pareto optimal allocation we want to
implement.
Then this can be done by making lump sum transfers to
redistribute wealth and then allowing the market to clear.
So, redistributional and efficiency concerns can be separated
and redistribution just comes down to lump sum transfers.
Thoughts on the second welfare theorem

But,
in practice can this really work?
Suppose we have an equilibrium in which we there is too
much inequality and we want to help the poor.
So we redistribute wealth to the poor.
But, if people’s consumption level results at least partially
from choices they make such a transfer is not lump sum.
It will distort incentives.
Without perfect information about people’s attributes and
preferences, it is very hard to find ways to make lump sum
transfers.
And people will not want to truthfully reveal this information.
Outline

1 Choice Theory

2 Choice under Uncertainty

3 General Equilibrium
Introduction
Exchange economies: the Walrasian Model
The Welfare theorems
Characterizing optimality and equilibrium with first-order conditions
Existence of Walrasian equilibria
Properties of Walrasian equilibria
A useful restriction: the “gross substitutes” property
General equilibrium with production
Limitations of the Welfare Theorems: Externalities
An example of inefficiency
Public goods
Assumptions enabling a FOC approach

Suppose for all i ∈ I,

1 ui (·) is continuous;
2 ui (·) is strictly increasing; i.e., ui (xi0 ) > ui (xi ) for any
xi0 > xi ;
3 ui (·) is concave;
4 ui (·) is differentiable; and
5 ei  0; i.e., every agent has at least a little bit of every good.
The Pareto Possibility Set

Definition (The Pareto Possibility Set)

The vectors of utility that can be achieved by feasible
consumptions:
( )
X X
1 1 n n i L i i
U = (u (x ), . . . , u (x )) : x ∈ R+ for all i, x ≤ e
i i

We will normalize the utility of consuming no goods to 0:

ui (0) = 0 for all i

The Pareto Possibility Set: An example

𝑢1 (𝑥 1 )

𝑢2 (𝑥 2 )
Finding Pareto optimal allocations

Consider the following algorithm for finding a Pareto optimal

allocation:
1 Decide how much utility to give to each agent i ∈ {2, . . . , I}
2 Maximize agent 1’s utility subject to our decision in step 1
Finding Pareto optimal allocations

Consider the following algorithm for finding a Pareto optimal

allocation:
1 Decide how much utility to give to each agent i ∈ {2, . . . , I}
2 Maximize agent 1’s utility subject to our decision in step 1

That is,
max u1 (x1 )
x∈RIL
+

such that
i i i
P u (xi ) ≥Pū i
for i = 2, . . . , I
i∈I x` ≤ i∈I e` for ` = 1, . . . , L
Graphical Illustration of Pareto Problem

𝑢1 (𝑥 1 )

𝑢2 𝑢2 (𝑥 2 )
Graphical Illustration of Pareto Problem

𝑢1 (𝑥 1 )

𝑢2 𝑢2 (𝑥 2 )
Graphical Illustration of Pareto Problem

𝑢1 (𝑥 1 )

𝑢2 𝑢2 (𝑥 2 )
Graphical Illustration of Pareto Problem

𝑢1 (𝑥 1 )

𝑢2 𝑢2 (𝑥 2 )
Graphical Illustration of Pareto Problem

𝑢1 (𝑥 1 )

𝑢2 (𝑥 2 )
Solving the Pareto problem

The Pareto problem

Maximize u1 (x1 ) such that

x ≥ 0;
ui (xi ) ≥ ūi for i = 2, . . . , I;
i i
P P
i∈I x` ≤ i∈I e` for ` = 1, . . . , L.

Under our assumptions, all the promise-keeping and feasibility

constraints must be binding, thus multipliers > 0:
λi multiplier on ui (xi ) ≥ ūi
µ` multiplier on i xi` ≤ i ei`
P P
Solving the Pareto problem I

I
X L X
I
1 1 i
 i i i
 X
µ` (ei` − xi` ) + γ`i xi`
 
L = u (x ) + λ u (x ) − ū +
i=2 `=1 i=1

Rearranging, and letting λ1 = 1

I
X I
X L X
X I
i i i i i
µ` (ei` − xi` ) + γ`i xi`
 
L= λ u (x ) − λ ū +
i=1 i=2 `=1 i=1

Gives (summarized) FOC

∂ui
λi ≤ µ` with equality if xi` > 0
∂xi`
Solving the Pareto problem II

∂u i
Assuming x  0, the FOC is λi ∂xi = µ` , hence
`

∂ui ∂uj
∂xik µk ∂xjk
MRSik` ≡ ∂ui
= = ∂uj
≡ MRSjk`
µ`
∂xi` ∂xj`
Maximizing a Bergson-Samuelson social welfare function

Consider a planner who simply maximizes a weighted average of

individual utilities:
X
max β i ui (xi )
x∈RIL
+ i∈I

xi` ≤ i
P P
such that i∈I i∈I e` for ` = 1, . . . , L
Graphical Illustration of Bergson-Samuelson Problem

Social Planner’s Indifference Curves

𝑢1 (𝑥 1 )

𝑢2 (𝑥 2 )
Graphical Illustration of Bergson-Samuelson Problem

Changing weights

𝑢1 (𝑥 1 )

𝑢2 (𝑥 2 )
Graphical Illustration of Bergson-Samuelson Problem

𝑢1 (𝑥 1 )

𝑢2 (𝑥 2 )
Graphical Illustration of Bergson-Samuelson Problem

𝑢1 (𝑥 1 )

𝑢2 (𝑥 2 )
Graphical Illustration of Bergson-Samuelson Problem

𝑢1 (𝑥 1 )

𝑢2 (𝑥 2 )
Graphical Illustration of Bergson-Samuelson Problem

𝑢1 (𝑥 1 )

𝑢2 (𝑥 2 )
Maximizing a Bergson-Samuelson social welfare function

Consider a planner who simply maximizes a weighted average of

individual utilities:
X
max β i ui (xi )
x∈RIL
+ i∈I
i i
P P
such that i∈I x` ≤ i∈I e` for ` = 1, . . . , L

X L X
X I
β i ui (xi ) + ψ` (ei` − xi` ) + φi` xi`
 
L=
i∈I `=1 i=1

Gives (summarized) FOC

∂ui
βi i
≤ ψ` with equality if xi` > 0
∂x`
Pareto optimality and Bergson-Samuelson SWFs

Pareto Problem:

I
X L X
X I I
X
λi ui (xi ) + µ` (ei` − xi` ) + γ`i xi` λi ūi
 
L= −
i=1 `=1 i=1 i=2
| {z }
Doesn’t depend on x

Bergson-Samuelson Problem:
I
X L X
X I
β i ui (xi ) + ψ` (ei` − xi` ) + φi` xi`
 
L=
i=1 `=1 i=1
Pareto optimality and Bergson-Samuelson SWFs

Pareto problem

xi ≥ 0
ui (xi ) ≥ ūi
P i P i
i x` ≤ i e`
∂ui
λi ∂xi ≤ µ` with equality if
`
xi` > 0
Pareto optimality and Bergson-Samuelson SWFs

Pareto problem Bergson-Samuelson problem

xi ≥ 0 xi ≥ 0
ui (xi ) ≥ ūi
P i P i
i x` ≤ i e`
P i P i i
i x` ≤
∂u
i e` β i ∂x i ≤ ψ` with equality if
i `
∂u
λi ∂xi ≤ µ` with equality if xi` > 0
`
xi` > 0
Pareto optimality and Bergson-Samuelson SWFs

Pareto problem Bergson-Samuelson problem

xi ≥ 0 xi ≥ 0
ui (xi ) ≥ ūi
P i P i
i x` ≤ i e`
P i P i i
i x` ≤
∂u
i e` β i ∂x i ≤ ψ` with equality if
i `
∂u
λi ∂xi ≤ µ` with equality if xi` > 0
`
xi` > 0

A solution to one is a solution to the other when setting

ūi = ui (xi )
λi = β i
µ` = ψ`
The Walrasian problem

A Walrasian problem
Each individual solves maxxi ∈RL ui (xi ) subject to p · xi ≤ p · ei .
P i ≤
P + i
Markets clear: x
i∈I ` i∈I e` for all ` ∈ L.

This gives Lagrangians (for the individual problems)

L
X
Li = ui (xi ) + ν i p · (ei − xi ) + ζ`i xi`
`=1
The Walrasian problem

A Walrasian problem
Each individual solves maxxi ∈RL ui (xi ) subject to p · xi ≤ p · ei .
P i ≤
P + i
Markets clear: x
i∈I ` i∈I e` for all ` ∈ L.

This gives Lagrangians (for the individual problems)

L
X
Li = ui (xi ) + ν i p · (ei − xi ) + ζ`i xi`
`=1

and (summarized) FOCs

∂ui
≤ ν i p` with equality if xi` > 0
∂xi`
The first welfare theorem

Pareto problem

xi ≥ 0
ui (xi ) ≥ ūi
P i P i
i x` = i e`
∂ui
λi ∂xi ≤ µ` with equality if
`
xi` > 0
The first welfare theorem

Pareto problem Walrasian problem

xi ≥ 0 xi ≥ 0
ui (xi ) ≥ ūi p · xi ≤ p · ei
P i P i P i P i
i x` = i e` i x` = i e`
∂ui ∂ui
λi ∂xi ≤ µ` with equality if ∂xi`
≤ ν i p` with equality if
`
xi` > 0 xi` > 0
The first welfare theorem

Pareto problem Walrasian problem

xi ≥ 0 xi ≥ 0
ui (xi ) ≥ ūi p · xi ≤ p · ei
P i P i P i P i
i x` = i e` i x` = i e`
∂ui ∂ui
λi ∂xi ≤ µ` with equality if ∂xi`
≤ ν i p` with equality if
`
xi` > 0 xi` > 0

If (x, p) is a Walrasian equilibrium, we can get that x is Pareto

optimal by setting
ūi = ui (xi )
λi = 1/ν i
µ` = p`
The second welfare theorem

Pareto problem Walrasian problem

xi ≥ 0 xi ≥ 0
ui (xi ) ≥ ūi p · xi ≤ p · ei
P i P i P i P i
i x` = i e` i x` = i e`
∂ui ∂ui
λi ∂xi ≤ µ` with equality if ∂xi`
≤ ν i p` with equality if
`
xi` > 0 xi` > 0

If x is Pareto optimal, we can get a Walrasian equilibrium (x, p) by

setting
ei = xi
ν i = 1/λi
p` = µ`
Outline

1 Choice Theory

2 Choice under Uncertainty

3 General Equilibrium
Introduction
Exchange economies: the Walrasian Model
The Welfare theorems
Characterizing optimality and equilibrium with first-order conditions
Existence of Walrasian equilibria
Properties of Walrasian equilibria
A useful restriction: the “gross substitutes” property
General equilibrium with production
Limitations of the Welfare Theorems: Externalities
An example of inefficiency
Public goods
Walrasian equilibrium

Definition (Walrasian equilibrium)

Prices p and quantities (xi )i∈I are a Walrasian equilibrium iff
1 All agents maximizing their utilities; i.e., for all i ∈ I,

xi ∈ argmax ui (x);
x∈B i (p)

2 Markets clear; i.e., for all ` ∈ L,

X X
xi` = ei` .
i∈I i∈I
Do Walrasian equilibria exist for every economy?

Theorem
Suppose for all i ∈ I,
1 ui (·) is continuous;
2 ui (·) is increasing; i.e., ui (x0 ) > ui (x) for any x0  x;
3 ui (·) is concave; and
4 ei  0; i.e., every agent has at least a little bit of every good.

There exist prices p ∈ Rl+ and allocations (xi )i∈I such that p and
x are a Walrasian equilibrium.
Excess demand

Definition (excess demand)

The excess demand of agent i is

z i (p) ≡ xi (p, p · ei ) − ei ,

where xi (p, w) is i’s Walrasian demand correspondence.

Aggregate excess demand is
X
z(p) ≡ z i (p).
i∈I

If p ∈ RL i i


+ satisfies z(p) = 0, then p and x (p, p · e ) i∈I
are a
Walrasian equilibrium
A few notes on excess demand I

X X
z(p) ≡ xi (p, p · ei ) − ei
i∈I i∈I

Under the assumptions of our existence theorem (ui (·) is

continuous, increasing, and concave, and ei  0 for all i):

z(·) is continuous
Continuity of ui implies continuity of xi
A few notes on excess demand II

X X
z(p) ≡ xi (p, p · ei ) − ei
i∈I i∈I

z(·) is homogeneous of degree zero

xi (p, wi ) is homogeneous of degree zero
i.e. xi (λp, λwi ) = λ0 xi (p, wi ) = xi (p, wi ).
xi (p, p · ei ) is homogeneous of degree zero in p
z i (p) ≡P xi (p, p · ei ) − ei is homogeneous of degree zero
z(p) ≡ i z i (p) is homogeneous of degree zero

This implies that only relative prices matter and we can normalize
one price.
A few notes on excess demand III

X X
z(p) ≡ xi (p, p · ei ) − ei
i∈I i∈I

p · z(p) = 0 for all p (Walras’ Law for excess demand)

By Walras’ Law, p · xi (p, wi ) = wi
p · xi (p, p · ei ) = p · ei

p · z i (p) ≡ p ·Pxi (p, p · ei ) − ei = 0
p · z(p) ≡ p · i z i (p) = 0
Suppose all but one market clear; i.e., z2 (p) = · · · = zL (p) = 0

p · z(p) = p1 z1 (p) + p2 z2 (p) + · · · + pL zL (p) = 0

| {z }
=0

by Walras’ Law; hence z1 (p) = 0 as long as p1 > 0

Thus if all but one market clear, the final market must also clear
W.E. requires a solution to z(p) = 0 I

Consider a two-good economy

Normalize p2 = 1 by homogeneity of degree zero of z(·)
As long as the good one market clears, the good two market
will as well (by Walras’ Law)

We can find a W.E. whenever z1 (p1 , 1) = 0

z1 (·, 1) is continuous
As p1 → 0, excess demand for good one must go to infinity
since preferences are increasing and ei2 > 0 for all i
As p1 → ∞, excess demand for good one must be negative
since preferences are increasing and ei1 > 0 for all i
W.E. requires a solution to z(p) = 0 II

By an intermediate value theorem, there is at least one W.E.

z1

z1 (·, 1)

0 p1
p∗1
Corner Endowments-Example I

Suppose there are two goods a, b and two agents 1, 2.

1/2
1/2
u1 (a1 , b1 ) = a1 + b1 u2 (a2 , b2 ) = b2
e1 = (20, 0) e2 = (0, 30)
Corner Endowments-Example II

1’s utility function

Corner Endowments-Example II

1’s marginal utility (in either good)

Corner Endowments-Example I

Suppose there are two goods a, b and two agents 1, 2.

1/2
1/2
u1 (a1 , b1 ) = a1 + b1 u2 (a2 , b2 ) = b2
e1 = (20, 0) e2 = (0, 30)
Corner Endowments-Example I

Suppose there are two goods a, b and two agents 1, 2.

1/2
1/2
u1 (a1 , b1 ) = a1 + b1 u2 (a2 , b2 ) = b2
e1 = (20, 0) e2 = (0, 30)

Agent 1 has infinite marginal utility from a good when

consuming none of it.
Agent 1 will demand a strictly positive amount of good b at
any prices such that pa /pb > 0.
Corner Endowments-Example I

Suppose there are two goods a, b and two agents 1, 2.

1/2
1/2
u1 (a1 , b1 ) = a1 + b1 u2 (a2 , b2 ) = b2
e1 = (20, 0) e2 = (0, 30)

Agent 1 has infinite marginal utility from a good when

consuming none of it.
Agent 1 will demand a strictly positive amount of good b at
any prices such that pa /pb > 0.
Agent 2 will demand at least 30 units of good b at any prices.
Corner Endowments-Example I

Suppose there are two goods a, b and two agents 1, 2.

1/2
1/2
u1 (a1 , b1 ) = a1 + b1 u2 (a2 , b2 ) = b2
e1 = (20, 0) e2 = (0, 30)

Agent 1 has infinite marginal utility from a good when

consuming none of it.
Agent 1 will demand a strictly positive amount of good b at
any prices such that pa /pb > 0.
Agent 2 will demand at least 30 units of good b at any prices.
So there is no equilibrium in which pa /pb > 0.
Corner Endowments-Example I

Suppose there are two goods a, b and two agents 1, 2.

1/2
1/2
u1 (a1 , b1 ) = a1 + b1 u2 (a2 , b2 ) = b2
e1 = (20, 0) e2 = (0, 30)

Agent 1 has infinite marginal utility from a good when

consuming none of it.
Agent 1 will demand a strictly positive amount of good b at
any prices such that pa /pb > 0.
Agent 2 will demand at least 30 units of good b at any prices.
So there is no equilibrium in which pa /pb > 0.
If pa /pb = 0, then agent 1 will demand an infinite amount of
good a.
Corner Endowments-Example I

Suppose there are two goods a, b and two agents 1, 2.

1/2
1/2
u1 (a1 , b1 ) = a1 + b1 u2 (a2 , b2 ) = b2
e1 = (20, 0) e2 = (0, 30)

Agent 1 has infinite marginal utility from a good when

consuming none of it.
Agent 1 will demand a strictly positive amount of good b at
any prices such that pa /pb > 0.
Agent 2 will demand at least 30 units of good b at any prices.
So there is no equilibrium in which pa /pb > 0.
If pa /pb = 0, then agent 1 will demand an infinite amount of
good a.
So there is no equilibrium.
Corner Endowments and Non-Existence

20

Agent 1 b 30
Corner Endowments and Non-Existence

20

Agent 1 b 30
Corner Endowments and Non-Existence

20

Agent 1 b 30
Corner Endowments and Non-Existence

20

Agent 1 b 30
Corner Endowments and Non-Existence

20

Agent 1 b 30
Corner Endowments and Non-Existence

20

Agent 1 b 30
Corner Endowments and Non-Existence

30 b Agent 2

20
Corner Endowments and Non-Existence

30 b Agent 2

20
Corner Endowments and Non-Existence

30 b Agent 2

20
Corner Endowments and Non-Existence

30 b Agent 2

20
Corner Endowments and Non-Existence

30 b Agent 2

20
Corner Endowments and Non-Existence

b Agent 2

Agent
1
Corner Endowments-Example II

Suppose there are two goods a, b and two agents 1, 2.

u1 (a1 , b1 ) = min(a1 , b1 ) u2 (a2 , b2 ) = min((a2 )1/2 , b2 )

e1 = (20, 0) e2 = (0, 30)
Corner Endowments-Example II

Suppose there are two goods a, b and two agents 1, 2.

u1 (a1 , b1 ) = min(a1 , b1 ) u2 (a2 , b2 ) = min((a2 )1/2 , b2 )

e1 = (20, 0) e2 = (0, 30)

For prices pa , pb > 0, 1 optimally consumes an amount

a1 = b1 .
For prices pa , pb > 0, 2 optimally consumes an amount
a2 = (b2 )2 .
Finally, in such an equilibrium a1 + a2 = 20 and b1 + b2 = 30.
Corner Endowments-Example II

Suppose there are two goods a, b and two agents 1, 2.

u1 (a1 , b1 ) = min(a1 , b1 ) u2 (a2 , b2 ) = min((a2 )1/2 , b2 )

e1 = (20, 0) e2 = (0, 30)

For prices pa , pb > 0, 1 optimally consumes an amount

a1 = b1 .
For prices pa , pb > 0, 2 optimally consumes an amount
a2 = (b2 )2 .
Finally, in such an equilibrium a1 + a2 = 20 and b1 + b2 = 30.
There is no feasible allocation that solves this system of
equations.
Corner Endowments-Example II

20

Agent 1 b 20 30
Corner Endowments-Example II

20

Agent 1 b 20 30
Corner Endowments-Example II

20

Agent 1 b 20 30
Corner Endowments-Example II

20

Agent 1 b 20 30
Corner Endowments-Example II

20

Agent 1 b 20 30
Corner Endowments-Example II

1’s Offer Curve

20

Agent 1 b 20 30
Corner Endowments-Example II

30 b Agent 2

20
Corner Endowments-Example II

30 b Agent 2

20
Corner Endowments-Example II

30 b Agent 2

20
Corner Endowments-Example II

30 b Agent 2

20
Corner Endowments-Example II

30 b Agent 2

20
Corner Endowments-Example II

30 b Agent 2

20
Corner Endowments-Example II

b Agent 2

Agent 1
Corner Endowments-Example II

Suppose there are two goods a, b and two agents 1, 2.

u1 (a1 , b1 ) = min(a1 , b1 ) u2 (a2 , b2 ) = min((a2 )1/2 , b2 )

e1 = (20, 0) e2 = (0, 30)

What if pa /pb = 0?
Corner Endowments-Example II

Suppose there are two goods a, b and two agents 1, 2.

u1 (a1 , b1 ) = min(a1 , b1 ) u2 (a2 , b2 ) = min((a2 )1/2 , b2 )

e1 = (20, 0) e2 = (0, 30)

What if pa /pb = 0?
1 has no wealth and must consume b1 = 0.
But can then optimally choose any a1 ≥ 0.
2 optimally consumes b2 = 30 and a2 ≥ 900.
Infeasible.
Corner Endowments-Example II

Suppose there are two goods a, b and two agents 1, 2.

u1 (a1 , b1 ) = min(a1 , b1 ) u2 (a2 , b2 ) = min((a2 )1/2 , b2 )

e1 = (20, 0) e2 = (0, 30)

What if pb /pa = 0?
Corner Endowments-Example II

Suppose there are two goods a, b and two agents 1, 2.

u1 (a1 , b1 ) = min(a1 , b1 ) u2 (a2 , b2 ) = min((a2 )1/2 , b2 )

e1 = (20, 0) e2 = (0, 30)

What if pb /pa = 0?
2 has no wealth and must consume a2 = 0.
But can then optimally choose any b2 ≥ 0.
1 optimally consumes a1 = 20 and b1 ≥ 20.
Some such allocations are feasible!
We have a continuum of Walrasian equilibria.
Corner Endowments-Example II

20

Agent 1 b 20 30
Corner Endowments-Example II

20

Agent 1 b 20 30
Corner Endowments-Example II

20

Agent 1 b 20 30
Corner Endowments-Example II

20

Agent 1 b 20 30
Corner Endowments-Example II

b Agent 2

a
Corner Endowments-Example II

b Agent 2

Agent 1
Corner Endowments-Example II

b Agent 2

Agent 1
Corner Endowments-Example II

b Agent 2

Walrasian
Equilibria a

Agent 1
Corner Endowments-Example III

Suppose there are two goods a, b and two agents 1, 2.

u1 (a1 , b1 ) = min(a1 , b1 ) u2 (a2 , b2 ) = min((a2 )1/2 , b2 )

e1 = (20, 0) e2 = (0, 5)
Corner Endowments-Example III

Suppose there are two goods a, b and two agents 1, 2.

u1 (a1 , b1 ) = min(a1 , b1 ) u2 (a2 , b2 ) = min((a2 )1/2 , b2 )

e1 = (20, 0) e2 = (0, 5)

For prices pa , pb > 0, 1 optimally consumes an amount

a1 = b1 .
For prices pa , pb > 0, 2 optimally consumes an amount
a2 = (b2 )2 .
Finally, in such an equilibrium

a1 + a2 = 20 and b1 + b2 = 5.

There is a feasible allocation that solves this system of

equations.
Corner Endowments-Example III

20

Agent 1 b 5
Corner Endowments-Example III

5 b Agent 2

20
Corner Endowments-Example III

b 2

a
a

1
b
Outline

1 Choice Theory

2 Choice under Uncertainty

3 General Equilibrium
Introduction
Exchange economies: the Walrasian Model
The Welfare theorems
Characterizing optimality and equilibrium with first-order conditions
Existence of Walrasian equilibria
Properties of Walrasian equilibria
A useful restriction: the “gross substitutes” property
General equilibrium with production
Limitations of the Welfare Theorems: Externalities
An example of inefficiency
Public goods
Other properties of Walrasian equilibria

We have established that an economy satisfying certain properties,

at least one Walrasian equilibrium exists

Other questions include:

1 How many Walrasian equilibria are there?

2 How does an economy (as distinct from an economist) “find”

equilibrium?

3 Can we test the Walrasian model in the data?

Uniqueness of Walrasian equilibria: Edgeworth box

Question 1: Uniqueness
Is there a unique Walrasian equilibrium (up to price
normalization)? If not, how many Walrasian equilibria are there?

x12
x21 Agent 2

Agent 1 x11
2
Uniqueness of Walrasian equilibria: Edgeworth box

Question 1: Uniqueness
Is there a unique Walrasian equilibrium (up to price
normalization)? If not, how many Walrasian equilibria are there?

x12
x21 Agent 2

Agent 1 x11
2
Uniqueness of Walrasian equilibria: Edgeworth box

Question 1: Uniqueness
Is there a unique Walrasian equilibrium (up to price
normalization)? If not, how many Walrasian equilibria are there?

x12
x21 Agent 2

OC1

Agent 1 x11
2
Uniqueness of Walrasian equilibria: Edgeworth box

Question 1: Uniqueness
Is there a unique Walrasian equilibrium (up to price
normalization)? If not, how many Walrasian equilibria are there?

x12
x21 Agent 2

OC2

Agent 1 x11
2
Uniqueness of Walrasian equilibria: Edgeworth box

Question 1: Uniqueness
Is there a unique Walrasian equilibrium (up to price
normalization)? If not, how many Walrasian equilibria are there?

x12
x21 Agent 2

OC2
OC1

Agent 1 x11
2
Uniqueness of Walrasian equilibria I

There could be one Walrasian equilibrium

z1

z1 (·, 1)

0 p1
p∗1
Uniqueness of Walrasian equilibria II

There could be two W.E. (although this is “non-generic”)

z1

z1 (·, 1)

0 p1
p∗∗
1 p∗1
Uniqueness of Walrasian equilibria III

There could be three W.E.

z1

z1 (·, 1)

0 p1
p∗∗∗
1 p∗∗
1 p∗1
Uniqueness of Walrasian equilibria IV

There could be infinite W.E. (although again, not generically)

z1

z1 (·, 1)

0 p1
Observations on multiplicity of Walrasian equilibria

It seems (and can be rigorously shown) that:

W.E. are generally not globally unique

W.E. are locally unique (generically)

There are a finite number of W.E. (generically)

There are an odd number of W.E. (generically)

Stability of Walrasian equilibria

Question 2: Stability
Is a Walrasian equilibrium “stable,” in the sense that a reasonable
dynamic adjustment process converges to equilibrium prices and
quantities?
Stability of Walrasian equilibria

Question 2: Stability
Is a Walrasian equilibrium “stable,” in the sense that a reasonable
dynamic adjustment process converges to equilibrium prices and
quantities?

Underlying question is: How does the economy “find” prices?

Hard to say in real world where prices come from
Proposed idea: a dynamic adjustment mechanism that
converges to W.E. prices
Walrasian tatonnement

One possibility
1 “Walrasian auctioneer” suggests prices
2 Agents report demand at these prices
3 If excess demand is non-zero, return to step 1
Possible price adjustment rule:


p(t + 1) = p(t) + α(t) z p(t)
Walrasian tatonnement

One possibility
1 “Walrasian auctioneer” suggests prices
2 Agents report demand at these prices
3 If excess demand is non-zero, return to step 1
Possible price adjustment rule:


p(t + 1) = p(t) + α(t) z p(t)
Big problems:
Unrealistic description of how the economy really works
No incentives to honestly report demand
Not necessarily stable
Some equilibria are unstable.
Adjustment rule can cycle and never converge.
(Scarf, H., “Some Examples of Global Instability of the
Competitive Equilibrium,” International Economic Review,
1960.)
Possible stability of Walrasian tatonnement

z1

z1 (·, 1)

0 p1
p∗1
Possible stability of Walrasian tatonnement

z1

z1 (·, 1)

0 p1
p∗1
Possible instability of Walrasian tatonnement

z1

z1 (·, 1)

0 p1
p∗∗∗
1 p∗∗
1 p∗1
Possible instability of Walrasian tatonnement

z1

z1 (·, 1)

0 p1
p∗∗∗
1 p∗∗
1 p∗1
Revived interest

Tatonnement is a very old topic.

Auction design literature has generated new interest.

How to run multi-unit auctions like the radio spectrum

auctions.

Instead of the mythical Walrasian auctioneer, there is a real

auctioneer.

Indivisible goods, but that can be worked into the theory.

Under what conditions do we get convergence to a Walrasian

equilibrium

When are agents incentivized to report their demand

truthfully.
Quasilinear Economies

Let good 1 be a numeraire good:

ui (x) = xi1 + v i (xi−1 ),
where xi−1 = (xi2 , . . . , xiL ).
Agents have enough of good 1 that the non-negativity
constraint on it never binds. Equivalently, we can allow
negative consumption of good 1.
Typically normalize p1 = 1.
Normally interpreted as partial equilibrium models.
Want to study a subset of markets and model the rest
through the numeraire good.
Only makes sense when:
wealth effects are small.
complementarities/substitutabilties with unmodeled goods are
small
Quasilinear Economies: Demand Functions

max xi1 + v i (xi−1 ) xi1 + pi−1 · xi−1 − ei−1 ≤ ei1

subject to
xi ≥0
Quasilinear Economies: Demand Functions

max xi1 + v i (xi−1 ) xi1 + pi−1 · xi−1 − ei−1 ≤ ei1

subject to
xi ≥0

X
Li = xi1 + v i (xi−1 ) + µi (ei1 − xi1 − pi−1 · xi−1 − ei−1 ) + ζ`i xi`

`6=1
Quasilinear Economies: Demand Functions

max xi1 + v i (xi−1 ) xi1 + pi−1 · xi−1 − ei−1 ≤ ei1

subject to
xi ≥0

X
Li = xi1 + v i (xi−1 ) + µi (ei1 − xi1 − pi−1 · xi−1 − ei−1 ) + ζ`i xi`

`6=1

FOC for good 1:

1 − µi = 0
Quasilinear Economies: Demand Functions

max xi1 + v i (xi−1 ) xi1 + pi−1 · xi−1 − ei−1 ≤ ei1

subject to
xi ≥0

X
Li = xi1 + v i (xi−1 ) + µi (ei1 − xi1 − pi−1 · xi−1 − ei−1 ) + ζ`i xi`

`6=1

FOC for good 1:

1 − µi = 0
So,

xi−1 ∈ argmax v i (xi−1 ) − p−1 · xi−1 − ei−1 + ei1

xi−1 ≥0

= argmax v i (xi−1 ) − p−1 · xi−1

xi−1 ≥0
Quasilinear Economies

Get demand functions

xi−1 ∈ argmax v i (xi−1 ) − p−1 · xi−1

xi−1 ≥0

Note that this demand does not depend on the consumer’s

endowment.

Pareto optimal allocations also have a much simpler form:

Consider a feasible profile (u1 , u2 , . . . , un ) ∈ U.

Then the profile (u1 + c1 , uP

2 + c2 , . . . , un + cn ) ∈ U

for any vector c such that i ci = 0.

Graphical Illustration of Pareto Problem

𝑢1 (𝑥 1 )

𝑢2 (𝑥 2 )
Graphical Illustration of Pareto Problem

𝑢1 (𝑥 1 )

𝑢2 (𝑥 2 )
Graphical Illustration of Pareto Problem

𝑢1 (𝑥 1 )

𝑢2 (𝑥 2 )
Graphical Illustration of Pareto Problem

𝑢1 (𝑥 1 )

𝑢2 (𝑥 2 )
Graphical Illustration of Pareto Problem

𝑢1 (𝑥 1 )

𝑢2 (𝑥 2 )
Graphical Illustration of Pareto Problem

𝑢1 (𝑥 1 )

𝑢2 (𝑥 2 )
Quasilinear Economies: Pareto Optimality and Equilibria

So the planner must maximize the sum of utilities and use transfers
of the numeraire good to move around the Pareto frontier.

In all Pareto efficient allocations consumption of non-numeraire

goods is identical!

Thus in all Walrasian equilibria consumption of the non-numeraire

goods is identical.

Moreover, it can be shown that there is a unique Walrasian

equilibrium.
Testable restrictions implied by the Walrasian model

Question 3: Testability
Does Walrasian equilibrium impose meaningful restrictions on
observable data?

We noted several properties of excess demand z(p):

Continuity
Homogeneity of degree zero
Walras’ Law (p · z(p) = 0 for all p)
Limit properties
Testable restrictions implied by the Walrasian model

Question 3: Testability
Does Walrasian equilibrium impose meaningful restrictions on
observable data?

We noted several properties of excess demand z(p):

Continuity
Homogeneity of degree zero
Walras’ Law (p · z(p) = 0 for all p)
Limit properties
Actually, this is all we get
Anything goes

Theorem (Sonnenschein-Mantel-Debreu)
Consider a continuous function f : B → RL on an open and
bounded set B ⊆ RL ++ such that
f (·) is homogeneous of degree zero, and
p · f (p) = 0 for all p ∈ B.
Then there exists an economy (goods, agents, preferences, and
endowments) with aggregate excess demand function z(·)
satisfying z(p) = f (p) for all p ∈ B.

Often interpreted as “anything goes” in terms of comparative

statics. . . actually not quite right
Restrictions on possible prices

What if we can observe how equilibrium prices changes as we

change endowments. Is everything permissable?
Theorem (Brown-Matzkin)
There exist prices and endowments (p, (ei )ni=1 ) and (p̂, (êi )ni=1 )
such that it is impossible to find strictly monotone preferences
(ui )ni=1 with the property that p is a Walrasian equilibrium price
vector for the economy (ui , ei )ni=1 and p̂ is a Walrasian equilibrium
price vector for the economy (ui , êi )ni=1 .
Graphical proof, 2 goods 2 consumers

Agent 2

Agent 1
Graphical proof, 2 goods 2 consumers

Agent 2

Some point a’ on a
𝑒 is preferred by 1 to
𝑎′
any point on b
𝑎

𝑏
Agent 1
Graphical proof, 2 goods 2 consumers

Agent 2
𝑒′

Agent 1
Graphical proof, 2 goods 2 consumers

Some point c’ on c is
preferred by 1 to
any point on d
𝑑

Agent 2
𝑒′
𝑐 𝑐′

Agent 1
Graphical proof, 2 goods 2 consumers

Agent 2
By monotonicity,
point d’ is
𝑑 𝑑′
preferred by 1 to a’
𝑎′
𝑎 Agent 2
𝑏
𝑐 𝑐′

Agent 1
Graphical proof, 2 goods 2 consumers

Agent 2
By monotonicity,
point b’ is
𝑑 𝑑′
preferred by 1 to c’
𝑎′
𝑎 Agent 2
𝑏
𝑐 𝑐′
𝑏′

Agent 1
Graphical proof, 2 goods 2 consumers

Agent 2
So:
𝑎′ ≻1 𝑏′ ≻1 𝑐′ ≻1 𝑑′ ≻1 𝑎′
𝑑 𝑑′
𝑎′
𝑎 Agent 2
𝑏
𝑐 𝑐′
𝑏′

Agent 1
Outline

1 Choice Theory

2 Choice under Uncertainty

3 General Equilibrium
Introduction
Exchange economies: the Walrasian Model
The Welfare theorems
Characterizing optimality and equilibrium with first-order conditions
Existence of Walrasian equilibria
Properties of Walrasian equilibria
A useful restriction: the “gross substitutes” property
General equilibrium with production
Limitations of the Welfare Theorems: Externalities
An example of inefficiency
Public goods
Gross substitutes and the gross substitutes property I

Recall that
Definition (Gross substitute—partial equilibrium)
Good ` is a (strict) gross substitute for good m iff x` (p, w) is
(strictly) increasing in pm .

In our G.E. framework, wealth depends on prices (w = e · p) so

Definition (Gross substitute—general equilibrium)

Good ` is a (strict) gross substitute for good m iff x` (p, e · p) is
(strictly) increasing in pm .
Gross substitutes and the gross substitutes property II

Definition (Gross substitutes property)

Marshallian demand function x(p) ≡ x(p, e · p) has the (strict)
gross substitutes property if every good is a (strict) gross
substitute for every other good.

More generally. . .
Definition (Gross substitutes property)
A function f (·) has the (strict) gross substitutes property if f` (p)
is (strictly) increasing in pm for all ` 6= m.
Gross substitutes and the gross substitutes property III

Suppose each individual’s Marshallian demand satisfies the gross

substitutes property; i.e., xi` (p) is increasing in pm for all ` 6= m

Then
Individual excess demands also satisfy it: z`i (p) ≡ xi` (p) − ei` is
increasing in pm
Aggregate excess demand also satisfies it: z` (p) ≡ i z`i (p) is
P
increasing in pm
Uniqueness of Walrasian equilibrium: 2 goods

Theorem
If aggregate excess demand z(·) satisfies the strict gross
substitutes property, then the economy has at most one Walrasian
equilibrium (up to price normalization).
Uniqueness of Walrasian equilibrium: 2 goods

Towards contradiction, two non-collinear Walrasian equilibrium

prices p = (p1 , p2 ) and p0 = (p01 , p02 ).
Uniqueness of Walrasian equilibrium: 2 goods

Towards contradiction, two non-collinear Walrasian equilibrium

prices p = (p1 , p2 ) and p0 = (p01 , p02 ).
Let λ1 = p01 /p1 and let λ2 = p02 /p2 . By non-collinearity λ1 6= λ2 .
Uniqueness of Walrasian equilibrium: 2 goods

Towards contradiction, two non-collinear Walrasian equilibrium

prices p = (p1 , p2 ) and p0 = (p01 , p02 ).
Let λ1 = p01 /p1 and let λ2 = p02 /p2 . By non-collinearity λ1 6= λ2 .
Suppose λ1 > λ2 .
Uniqueness of Walrasian equilibrium: 2 goods

Towards contradiction, two non-collinear Walrasian equilibrium

prices p = (p1 , p2 ) and p0 = (p01 , p02 ).
Let λ1 = p01 /p1 and let λ2 = p02 /p2 . By non-collinearity λ1 6= λ2 .
Suppose λ1 > λ2 .
Let p̃1 = λ1 p1 = p01 and p̃2 = λ1 p2 > λ2 p2 = p02 .
Uniqueness of Walrasian equilibrium: 2 goods

Towards contradiction, two non-collinear Walrasian equilibrium

prices p = (p1 , p2 ) and p0 = (p01 , p02 ).
Let λ1 = p01 /p1 and let λ2 = p02 /p2 . By non-collinearity λ1 6= λ2 .
Suppose λ1 > λ2 .
Let p̃1 = λ1 p1 = p01 and p̃2 = λ1 p2 > λ2 p2 = p02 .
We have z1 (p0 ) = 0. Consider moving to p̃. As p01 = p̃1 this only
requires increasing p02 to p̃2 .
Uniqueness of Walrasian equilibrium: 2 goods

Towards contradiction, two non-collinear Walrasian equilibrium

prices p = (p1 , p2 ) and p0 = (p01 , p02 ).
Let λ1 = p01 /p1 and let λ2 = p02 /p2 . By non-collinearity λ1 6= λ2 .
Suppose λ1 > λ2 .
Let p̃1 = λ1 p1 = p01 and p̃2 = λ1 p2 > λ2 p2 = p02 .
We have z1 (p0 ) = 0. Consider moving to p̃. As p01 = p̃1 this only
requires increasing p02 to p̃2 .
By the strict gross substitutes property, increasing p2 strictly
increases demand for p1 so z1 (p̃) > 0.
Uniqueness of Walrasian equilibrium: 2 goods

Towards contradiction, two non-collinear Walrasian equilibrium

prices p = (p1 , p2 ) and p0 = (p01 , p02 ).
Let λ1 = p01 /p1 and let λ2 = p02 /p2 . By non-collinearity λ1 6= λ2 .
Suppose λ1 > λ2 .
Let p̃1 = λ1 p1 = p01 and p̃2 = λ1 p2 > λ2 p2 = p02 .
We have z1 (p0 ) = 0. Consider moving to p̃. As p01 = p̃1 this only
requires increasing p02 to p̃2 .
By the strict gross substitutes property, increasing p2 strictly
increases demand for p1 so z1 (p̃) > 0.
But z1 (p̃1 , p̃2 ) = z1 (λ1 p1 , λ1 p2 ) = z1 (p1 , p2 ) = 0.
Uniqueness of Walrasian equilibrium: 2 goods

Towards contradiction, two non-collinear Walrasian equilibrium

prices p = (p1 , p2 ) and p0 = (p01 , p02 ).
Let λ1 = p01 /p1 and let λ2 = p02 /p2 . By non-collinearity λ1 6= λ2 .
Suppose λ1 > λ2 .
Let p̃1 = λ1 p1 = p01 and p̃2 = λ1 p2 > λ2 p2 = p02 .
We have z1 (p0 ) = 0. Consider moving to p̃. As p01 = p̃1 this only
requires increasing p02 to p̃2 .
By the strict gross substitutes property, increasing p2 strictly
increases demand for p1 so z1 (p̃) > 0.
But z1 (p̃1 , p̃2 ) = z1 (λ1 p1 , λ1 p2 ) = z1 (p1 , p2 ) = 0. So both p and
p0 cannot be equilibria.
Uniqueness of Walrasian equilibrium: 2 goods Basic Idea

Take one of the price vectors p.

Scale it up until all prices but one are greater than the other price
vector p0 .
Now consider the good that has the same price.
There must be excess demand for this good after all the other
prices have been increased.
By the gross substitutes property and because we were in
equilibrium before.
But then we scaled up prices, without changing relative prices, and
found that we were out of equilibrium!
Works with m goods too.
Other implications of gross substitutes

The gross substitutes property can be used to show a number of

other properties of Walrasian equilibrium; e.g.,
Walrasian tatonnement will converge to the unique equilibrium

Any change that raises the excess demand of a good will

increase its equilibrium price
Outline

1 Choice Theory

2 Choice under Uncertainty

3 General Equilibrium
Introduction
Exchange economies: the Walrasian Model
The Welfare theorems
Characterizing optimality and equilibrium with first-order conditions
Existence of Walrasian equilibria
Properties of Walrasian equilibria
A useful restriction: the “gross substitutes” property
General equilibrium with production
Limitations of the Welfare Theorems: Externalities
An example of inefficiency
Public goods
Producer theory: simplifying assumptions

Standard model: firms choose production plans (technologically

feasible lists of inputs and outputs) to maximize profits

Simplifying assumptions include:

1 Firms are price takers (both input and output markets)
2 Technology is exogenously given
3 Firms maximize profits; should be true as long as
The firm is competitive
There is no uncertainty about profits
Managers are perfectly controlled by owners
Production sets

Exogenously given technology applies over n commodities (both

inputs and outputs)
Definition (production plan)
A vector y = (y1 , . . . , yn ) ∈ Rn where an output has yk > 0 and
an input has yk < 0.

Definition (production set)

Set Y ⊆ Rn of feasible production plans; generally assumed to be
non-empty and closed.
Example: 1 input 1 output

y = (y1 , y2 ), where Y ≡ {y : y1 = (−y2 )0.5 , y2 ∈ [−1, 0]}

1.0

0.8

0.6

0.4

0.2

-1.0 -0.8 -0.6 -0.4 -0.2

Example: 2 inputs 1 output

y = (y1 , y2 , y3 ), where
Y ≡ {y : y1 = (−y2 )0.5 (−y3 )0.5 , y2 ∈ [−1, 0], y3 ∈ [−1, 0]}
Properties of production sets I

Definition (shutdown)
0∈Y.

Definition (free disposal)

y ∈ Y and y 0 ≤ y imply y 0 ∈ Y .

=⇒
Properties of production sets II

y = (y1 , y2 , y3 ), where
Y ≡ {y : y1 ≤ (−y2 )0.5 (−y3 )0.5 , y2 ≤ 0, y3 ≤ 0}
Properties of production sets III

Definition (nonincreasing returns to scale)

y ∈ Y implies αy ∈ Y for all α ∈ [0, 1].

Implies shutdown

Definition (nondecreasing returns to scale)

y ∈ Y implies αy ∈ Y for all α ≥ 1.

Along with shutdown, implies π(p) = 0 or π(p) = +∞ for all p

Definition (constant returns to scale)

y ∈ Y implies αy ∈ Y for all α ≥ 0; i.e., nonincreasing and
nondecreasing returns to scale.
Properties of production sets IV

Nonincreasing, Nondecreasing or Constant Returns to Scale?

Properties of production sets V

Definition (convex production set)

y, y 0 ∈ Y imply ty + (1 − t)y 0 ∈ Y for all t ∈ [0, 1].

Vaguely “nonincreasing returns to specialization”

If 0 ∈ Y , then convexity implies nonincreasing returns to scale

Strictly convex iff for t ∈ (0, 1), the convex combination is in the
interior of Y
Properties of production sets VI

(Strictly) convex?
Adding production to the Walrasian model

So far our exchange economy has treated the stock of goods

available as fixed through endowments

Now add K firms k ∈ K ≡ {1, . . . , K}

Firm k has production set Y k ⊆ RL
Will make a number of “standard” producer theory
assumptions
Also need some additional assumptions to make sure economy
is well behaved

Final primitive: what happens to firms’ profits? We typically

assume firms are owned by consumers
The Walrasian Model of the production economy I

Primitives of the model

L goods ` ∈ L ≡ {1, . . . , L}
I consumers i ∈ I ≡ {1, . . . , I}
Endowments ei ∈ RL + ; consumers do not have monetary
wealth, but rather an endowment of goods which they can
trade or consume
Preferences represented by utility function ui : RL
+ →R
K firms k ∈ K ≡ {1, . . . , K}
Production sets Y k ⊆ RL
Ownership structure (αki )k∈K,i∈I where αki is consumer i’s
share of firm k
Endogenous prices p ∈ RL
+ , taken as given by each consumer
and firm
The Walrasian Model of the production economy II

Each consumer i solves

max ui (x)
x∈B i (p)

where
n X o
B i (p) ≡ x ∈ RL i
+: p · x ≤ p · e + αki (p · y k )
k∈K

Each firm k solves

max p · y k
y k ∈Y k
The Walrasian Model of the production economy III

Definition (Aggregate production set)

The set of feasible aggregate production plans:
X X
Y ≡ Y k = {y ∈ Rn : y = y k , y k ∈ Y k for all k}.
k∈K k
Walrasian equilibrium

Definition (Walrasian equilibrium)

Prices p and quantities (xi )i∈I and (y k )k∈K are a Walrasian
equilibrium iff
1 All consumers maximize their utilities; i.e., for all i ∈ I,

xi ∈ argmax ui (x);
x∈B i (p)

2 All firms maximize their profits; i.e., for all k ∈ K,

y k ∈ argmax p · y;
y∈Y k

3 Markets clear; i.e., for all ` ∈ L,

X X X
xi` = ei` + y`k .
i∈I i∈I k∈K
Pareto optimality

Definition (feasible alocation)

Allocations (xi )i∈I ∈ RI·L k
+ and production plan (y )k∈K ∈ R
K·L
k k
are feasible iff y ∈ Y for all k ∈ K, and for all ` ∈ L,
X X X
xi` ≤ ei` + y`k .
i∈I i∈I k∈K

Definition (Pareto optimality)

Allocations (xi )i∈I and production plan (y k )k∈K are Pareto
optimal iff
1 x and y are feasible, and
2 There are no other feasible allocations x̂ and ŷ such that
ui (x̂i ) ≥ ui (xi ) for all i ∈ I with strict inequality for some i.
The First Welfare Theorem

Theorem (First Welfare Theorem)

Suppose ui (·) is increasing (i.e., ui (x0 ) > ui (x) for any x0  x) for
all i ∈ I.

If p and (xi )i∈I and (y k )k∈K are a Walrasian equilibrium, then the
allocations (xi )i∈I and (y k )k∈K are Pareto optimal.
The Second Welfare Theorem

Theorem (Second Welfare Theorem)

Suppose for all i ∈ I,
1 ui (·) is increasing; i.e., ui (x0 ) > ui (x) for any x0  x;
2 ui (·) is concave; and
3 ei  0; i.e., every agent has at least a little bit of every good.
Further suppose that production sets Y k are closed and convex for
all k ∈ K, which rules out increasing returns to scale.
Suppose (xi )i∈I and (y k )k∈K are Pareto optimal, and that xi  0
for all i ∈ I.
Then there exist prices p ∈ Rl+ , ownership shares (αki )k∈K,i∈I , and
transferred endowments (ẽi )i∈I where i ei = i ẽi such that p
P P
and (xi )i∈I and (y k )k∈K are a Walrasian equilibrium in the
economy where endowments are (ẽi )i∈I .
Do Walrasian equilibria exist for every economy?

Theorem
Suppose
ui (·) is continuous, increasing, and concave for all i ∈ I;
ei  0 for all i ∈ I;
Production sets Y k are closed and convex, and have shutdown
and free disposal for all k ∈ K;
X   X 
k
Y ∩ − Y k = {0},
k∈K k∈K

which rules out the possibility that firms can cooperate to

produce unlimited output.
Then there exists a Walrasian equilibrium.
No production pump assumption

𝑦1

𝑌𝑖

𝑦2

𝑌𝑗
No production pump assumption

𝑦1

𝑦2
No production pump assumption

𝑦1

−𝑌 𝑖

𝑦2
𝑗
−𝑌
No production pump assumption

𝑦1

−𝑌 𝑖
𝑌𝑖

𝑦2
𝑗
−𝑌
𝑌𝑗
Firms with constant returns to scale technology

Suppose a firm has CRS production technology; i.e.,

y ∈ Y =⇒ βy ∈ Y for all y and all β > 0

What can we say about its profit?

Can it be strictly positive?
Firms with constant returns to scale technology

Suppose a firm has CRS production technology; i.e.,

y ∈ Y =⇒ βy ∈ Y for all y and all β > 0

What can we say about its profit?

Can it be strictly positive? No. . . otherwise it could scale up
production arbitrarily and achieve infinite profit
Could be zero due to prices
Could be zero due to shutdown
Firms with constant returns to scale technology

Suppose a firm has CRS production technology; i.e.,

y ∈ Y =⇒ βy ∈ Y for all y and all β > 0

What can we say about its profit?

Can it be strictly positive? No. . . otherwise it could scale up
production arbitrarily and achieve infinite profit
Could be zero due to prices
Could be zero due to shutdown

So ownership structure (αki )ki doesn’t matter

General Equilibrium: Key conceptual takeaways I

Challenge: Simultaneously clearing many interrelated markets

with optimizing agents.
Concept: Walrasian equilibrium (exchange and production
economies).
Think of it just as a fixed point.
An equilibrium of the system in which everyone is happy with
their decisions.
Doesn’t prescribe how to get there.
Concept: Pareto optimality (exchange and production
economies)
First Welfare Theorem (exchange and production economies).
Second Welfare Theorem (exchange and production
economies).
General Equilibrium: Key conceptual takeaways II

Saw how the social planners problem, consumers’ problem and

Pareto problems are interlinked.
Found conditions for existence (exchange and production
economies).
Multiplicity and uniqueness under gross substitutes and in
quasilinear economies.
Tatonnenment–generally problematic, but does work under
gross substitutes.
Testability.
General Equilibrium: Key tools I

Edgeworth box.
Offer curves as a way of finding equilibria.
Pareto sets
The contract curve — Pareto optimal and individually rational.
Simplifications under Quasilinearity.
Simplifications due to Walras’ law.
Excess demand function as a way to characterize Walrasian
equilibria.
Pareto problem.
Optimization approaches to finding Walrasian equilibria under
functional form assumptions.
Simplification under gross substitutes.
Constant returns to scale and zero profits.
Implicit assumptions

Welfare theorems formalize the efficiency of markets.

Straw man.
What economically important things are assumed away?
Implicit assumptions

Welfare theorems formalize the efficiency of markets.

Straw man.
What economically important things are assumed away?
No externalities.
No missing markets.
Buyers and sellers have symmetric information.
No market power.
Externalities

Definition
An externality is present whenever the well-being of a consumer or
the production possibility of a firm are directly affected by the
actions of another agent in the economy.

The word directly is important.

Affects that occur indirectly through prices are excluded.
These are sometimes called pecuniary externalities to differentiate
them from real externalities.
Example 1: Consumption Externalities

Without any consumption externalities consumer problem is:

max ui (xi ) subject to p · xi ≤ w i .

xi ≥0
Example 1: Consumption Externalities

Without any consumption externalities consumer problem is:

max ui (xi ) subject to p · xi ≤ w i .

xi ≥0

If there are consumption externalities, i’s problem becomes

max ui (xi , x−i ) subject to p · xi ≤ w i .

xi ≥0

where x−i is the consumptions of agents other than i.

Example 2: Production Externalities

Without any production externalities firm problem is

max p · y k .
y k ∈Y k
Example 2: Production Externalities

Without any production externalities firm problem is

max p · y k .
y k ∈Y k

If there are production externalities then k’s problem becomes

max p · yk .
y k ∈Y k (y −k )

where y −k is the productions of agents other than k.

Example 3: Production Externalities

Without any production externalities consumer problem is:

max ui (xi ) subject to p · xi ≤ w i .

xi ≥0
Example 3: Production Externalities

Without any production externalities consumer problem is:

max ui (xi ) subject to p · xi ≤ w i .

xi ≥0

With production externalities, i’s problem becomes

max ui (xi , y) subject to p · xi ≤ w i .

xi ≥0

where y is the production of all firms.

Example 4: Consumption Externalities

Without any consumption externalities firm problem is

max p · y k .
y k ∈Y k
Example 4: Consumption Externalities

Without any consumption externalities firm problem is

max p · y k .
y k ∈Y k

With production externalities, i’s problem becomes

max p · y k .
y k ∈Y k (x)

where x is the consumptions of all consumers of all goods.

Consumer problem with consumption externalities

Suppose utility functions are increasing in own consumption,

differentiable in own consumption and concave in own
consumption.

max ui (xi , x−i ) subject to p · xi ≤ p · e i .

xi ≥0

This gives Lagrangians (for the individual problems)

L
X
i i i −i i i i
L = u (x , x ) + ν p · (e − x ) + ζ`i xi`
`=1
Consumer problem with consumption externalities

Suppose utility functions are increasing in own consumption,

differentiable in own consumption and concave in own
consumption.

max ui (xi , x−i ) subject to p · xi ≤ p · e i .

xi ≥0

This gives Lagrangians (for the individual problems)

L
X
i i i −i i i i
L = u (x , x ) + ν p · (e − x ) + ζ`i xi`
`=1

and (summarized) FOCs

∂ui
≤ ν i p` with equality if xi` > 0
∂xi`
Ignores the affect of xi on uj for j 6= i.
Public Goods

A special case of externalities.

Non-rivalrous, and for our purposes non-excludable.

Think of consumption of this good as a contribution to the

public good.

There are then positive externalities that take a particular

form.
Private Provision of Public goods

Move back to partial equilibrium framework for simplicity.

i
P
n consumers, one public good θ = iθ (and a numeraire good).

Utilities ui (θ) = φi (θ) − θi , where φ(·) is strictly increasing,

differentiable and concave.

max φi (θ) − θi
θi ≥0

FOC:

∂φi
−1≤0 with equality if θi 6= 0
∂θi
Private Provision of Public goods

Suppose we can order the agents in terms of their marginal

benefits so that for all aggregate provision levels θ

∂φ1 ∂φ2 ∂φn

(θ) > (θ) > · · · > (θ).
∂θ ∂θ ∂θ
Claim: Then only agent 1 contributes to the public good.
Private Provision of Public goods

Suppose we can order the agents in terms of their marginal

benefits so that for all aggregate provision levels θ

∂φ1 ∂φ2 ∂φn

(θ) > (θ) > · · · > (θ).
∂θ ∂θ ∂θ
Claim: Then only agent 1 contributes to the public good.

If agent i 6= 1 contributed to the public good:

∂φ1 ∂φi
> = 1,
∂θ ∂θ
contradicting the first order condition for 1.

Free riding!
Lindahl Equilibria

Idea: Consider each externality imposed by one agent on another.

Take each externalities and pretend it is a tradeable good.
Add a production sector where the externalities are produced as
biproducts.
For example, suppose consumption by i of good 1 generates
positive externalities for j and k.
Then add a production sector.
A unit of goods 1ii, 1ij and 1ik are produced by a unit of
good 1.
These goods replace xi1 in i, j and k’s utility functions.
Lindahl Equilibria

Find the Walrasian equilibrium in this world with no externalities.

As there are no externalities, the Walrasian equilibria are efficient.

So we have found an efficient outcome.

Conceptual importance: All externalities are just missing markets.

If we could just replace these missing markets we’d be good!

Lindahl Equilibria: Limitations

Externalities are priced bilaterally. There are no market forces to

clear these markets.
Lindahl Equilibria: Limitations

Externalities are priced bilaterally. There are no market forces to

clear these markets.
“no decentralized pricing system can serve to determine
optimally these levels of collective consumption. . . It is in
the selfish interest of each person to give false signals, to
pretend to have less interest in a given collective
consumption activity than he really has”
Samuelson, The Pure Theory of Public Expenditure (1954)
Lindahl Equilibria: Limitations

Externalities are priced bilaterally. There are no market forces to

clear these markets.
“no decentralized pricing system can serve to determine
optimally these levels of collective consumption. . . It is in
the selfish interest of each person to give false signals, to
pretend to have less interest in a given collective
consumption activity than he really has”
Samuelson, The Pure Theory of Public Expenditure (1954)

Nevertheless, there are literatures on implementing Lindahl

outcomes through bargaining and through mechanism design.
Outline

1 Choice Theory

2 Choice under Uncertainty

3 General Equilibrium

4 Limitations of the Welfare Theorems: Asymmetric information

Adverse Selection
Moral Hazard
Implicit Assumptions Underlying the Welfare Theorems

What if there is not symmetric information?

Say seller knows the quality of a good, but buyer doesn’t?

This is pre-contractual opportunism—Adverse Selection.

Say a seller of a service can chooses to provide a bad service?

This is post-contractual opportunism—Moral Hazard.

Question: Suppose someone wants a business loan. Which

type of asymmetric information should a bank worry about?
Outline

1 Choice Theory

2 Choice under Uncertainty

3 General Equilibrium

4 Limitations of the Welfare Theorems: Asymmetric information

Adverse Selection
Moral Hazard
Akerlof, 70

Akerlof, George A. “The market for “lemons”: Quality uncertainty and

the market mechanism.”
The quarterly journal of economics (1970): 488-500.
Some Examples

Buying a used car (seller has the informational advantage).

Buying insurance (buyer has the informational advantage).

TARP: Governmental program to buy “toxic assets.”

Buying a house.

Many more.
Simple Model of Health Insurance I

Can markets efficiently provide health insurance?

Welfare theorems tell us they can if certain assumptions hold.

Let’s relax one of these assumptions and see what happens.

Suppose the current health of a buyer is private information.

A buyer’s health is given by θ ∈ [0, 1].

Healthier buyers have a higher θ.

θ is uniformly distributed on [0, 1].

The buyer’s cost of treatment is 1 − θ.

Simple Model of Health Insurance II

If the buyer does not have insurance, he pays 1 − θ, otherwise

the insurance company pays.

The consumer has utility function

u(θ, y) = v(θ) + y,
where y is consumption of a numeraire good (and negative
consumption is permitted).

When i has no insurance y = −(1 − θ).

If i has insurance then y = −p, where p is the price of the

insurance.
Simple Model of Health Insurance III

We assume that v(θ) is an increasing, concave, and

differentiable function (thus that the consumer is risk averse
with respect to their health, but not the numeraire).

Insurance companies cannot differentiate between buyers, so

offer them all the same price of insurance.

Suppose the insurance market is competitive, so a price p is

charged that satisfies a zero-profit condition.

So, if all buyers purchase insurance, p = 12 .

Would all buyers purchase insurance at this price?

1
A buyer with θ > p = 2 would rather self-insure.
Simple Model of Health Insurance IV

Suppose the remaining buyers purchase insurance.

3
Then insurance companies would need to charge a price p = 4
to break even.
1
But then buyers with θ > 4 will not buy the insurance.

And so on.

The market collapses!

How can we formalize this idea?

Simple Model of Health Insurance V

When there is some consumption, we can define an

equilibrium by a p such that

Θ∗ (p) ≡ {θ : 1 − θ > p}
p = E[1 − θ|θ ∈ Θ∗ (p)]

We are looking for a price that generates zero profit

conditional on firms correctly anticipating who will buy from
them.

In the previous example, there is no such equilibrium with

Θ∗ (p) 6= ∅.
A little more realism I

But we assumed buyers know their true health.

They might be uncertain too.

What would happen if buyers observe a signal of their health

θ̂ where θ̂ = θ with probability q > 0 and with probability
1 − q it is drawn uniformly at random from [0, 1]?

Does anything change?

Consumer’s vN-M utility conditional on buying insurance:

Z 1
qv(θ̂) + (1 − q) v(s)ds − p
0
A little more realism II

Consumer’s vN-M utility conditional on not buying insurance:

Z 1
q(v(θ̂) − (1 − θ̂)) + (1 − q) v(s) − (1 − s)ds
0

Thus no insurance is preferred if and only if

1
p > q(1 − θ̂) + (1 − q) .
2
So if all consumers buy insurance, p = 1/2 but then any
consumer with signal θ̂ > 1/2 would not want to buy it.

Nothing changes! A tiny amount of private information (any

q > 0) is enough for the market to collapse.
More General Utility Functions I

Let the consumer observe her health again, but consider a

more general utility function.
Health is not a good candidate for including a numeraire.
Suppose instead we had ui (θ, y) that is increasing,
differentiable and concave in both arguments
vN-M utility conditional on buying insurance:

u(θ, −p)

Consumer’s vN-M utility conditional on not buying insurance:

u(θ, −1 + θ)
More General Utility Functions II

Insurance is then preferred if

u(θ, −p) > u(θ, −1 + θ)

As u(θ, −p) is increasing in its second argument the above

condition holds if and only if −p > −1 + θ, or equivalently

p < 1 − θ.

So nothing changes!
Combining Changes I

Consumer’s vN-M utility conditional on buying insurance:

Z 1
qu(θ̂, −p) + (1 − q) u(s, −p)ds
0

Consumer’s vN-M utility conditional on not buying insurance:

Z 1
qu(θ̂, −1 + θ̂) + (1 − q) u(s, −1 + s)ds
0
Insurance is then preferred if

Z 1
(1 − q) (u(s, −p) − u(s, −1 + s)) ds
0
 
> q u(θ̂, −1 + θ̂) − u(θ̂, −p)
Combining Changes II

If the most healthy agent takes insurance, so will all others.

Thus there is now an equilibrium in which all agents take
insurance if:

Eθ [u(θ, −1/2) − u(θ, −1 + θ)]

q
> (u(1, 0) − u(1, −1/2)) ,
1−q

Requires q fairly small (limited asymmetric information) and u

sufficiently concave (high risk aversion).
Insurmountable Problems?

The prevailing view in the 1950s was that problems of informational

asymmetry were insurmountable. Recall the Samuelson quote:
“no decentralized pricing system can serve to determine
optimally these levels of collective consumption. . . It is in
the selfish interest of each person to give false signals, to
pretend to have less interest in a given collective
consumption activity than he really has”
Samuelson, The Pure Theory of Public Expenditure (1954)

Idea: What if a planner could commit to use the information they

receive in a certain way?
Overcoming Information Asymmetries

What if we can credibly commit to how private information

will be treated?
Perhaps in a dynamic context maintaining credibility is
important enough.
Might have long lived institutions.

Can we leverage this commitment power to get efficient

outcomes?

Have people reveal all we need to know to do this?

Possible ways of creating commitment power

Contracts that will credibly be enforced by courts of law.

Politicians who will not get re-elected if they renege on

promises.
Perhaps an institution has a well defined and credible
mandate. For example:
Central banks mandated to set interest rates to target inflation.
Auction houses that commit to sell at the agree upon price.

Online reviews or word of mouth incentivise vendors to

advertise truthfully.

Can this power be used to extract enough information to

implement efficient outcomes? When? How?
General Set up

Agent: i = 1, . . . , n

Types: θi ∈ Θi

Social outcome: x ∈ X

Bernoulli utility functions: ui (θi , x).

Types are drawn from some (commonly known) distribution

according to pdf φ(θ)

And then privately observed

Planner

Asks agents to report their types.

Uses this information to choose a desirable outcome.

If the planner assumes reports will be truthful, does anything

go wrong?
Example 1: An exchange economy

L goods, n consumers and interior endowments e. So,

n X X o
L
X = (x1 , . . . , xn ) : xi ∈ R+ and x`i ≤ e`i for ` = 1, . . . , L .
i

θi is i’s preference relation over the different goods.

Θi is the set of monotone, convex preference relations over xi .

Ex-post efficient social choice rule: select a Walrasian equilibria.

Example 1: Simplified problem

Let L = n = 2, and suppose 1’s preferences are known:

Θ1 = θ1

Θ2 is the set of monotone, convex preference relations over x2

Will 2 always report report truthfully?

Example 1: Mis-reporting of Preferences

Agent 2

Agent 1
Example 1: Mis-reporting of Preferences

Agent 2

Agent 1
Example 1: Mis-reporting of Preferences

Agent 2

Agent 1
Example 1: Mis-reporting of Preferences

Agent 2

Agent 1
Example 1: Mis-reporting of Preferences

Agent 2

Agent 1
So what will work?
So what will work?

If we want the truth we are going to have to incentivize

people to report truthfully.
So what will work?

If we want the truth we are going to have to incentivize

people to report truthfully.

Commit not to make them any worse off if they tell the truth.
So what will work?

If we want the truth we are going to have to incentivize

people to report truthfully.

Commit not to make them any worse off if they tell the truth.

But maybe it is too costly to make everyone always tell us the

truth?
So what will work?

If we want the truth we are going to have to incentivize

people to report truthfully.

Commit not to make them any worse off if they tell the truth.

But maybe it is too costly to make everyone always tell us the

truth?

Can we incentivise truth telling in general?

So what will work?

If we want the truth we are going to have to incentivize

people to report truthfully.

Commit not to make them any worse off if they tell the truth.

But maybe it is too costly to make everyone always tell us the

truth?

Can we incentivise truth telling in general?

If not, then under what conditions?

So what will work?

If we want the truth we are going to have to incentivize

people to report truthfully.

Commit not to make them any worse off if they tell the truth.

But maybe it is too costly to make everyone always tell us the

truth?

Can we incentivise truth telling in general?

If not, then under what conditions?

And how?
So what will work?

If we want the truth we are going to have to incentivize

people to report truthfully.

Commit not to make them any worse off if they tell the truth.

But maybe it is too costly to make everyone always tell us the

truth?

Can we incentivise truth telling in general?

If not, then under what conditions?

And how?

And what does being correctly incentivized mean?

The right incentives

Suppose we ask people to reveal their types to us.

The right incentives

Suppose we ask people to reveal their types to us.

And commit to implement outcomes than make it in

everyone’s own self-interest to truthfully reveal their types.
The right incentives

Suppose we ask people to reveal their types to us.

And commit to implement outcomes than make it in

everyone’s own self-interest to truthfully reveal their types.

We can formulate these constraints: incentive compatibility

constraints.
The right incentives

Suppose we ask people to reveal their types to us.

And commit to implement outcomes than make it in

everyone’s own self-interest to truthfully reveal their types.

We can formulate these constraints: incentive compatibility

constraints.

We can also make sure everyone wants to participate:

individual rationality constraints
The right incentives

Suppose we ask people to reveal their types to us.

And commit to implement outcomes than make it in

everyone’s own self-interest to truthfully reveal their types.

We can formulate these constraints: incentive compatibility

constraints.

We can also make sure everyone wants to participate:

individual rationality constraints

It is relatively straightforward when these constraints don’t

depend on the actions of other agents, but more complicated
when they do. In this case we need to use game theory.
Exchange Economy

Recall the planner knows θ2 but not θ1 .

Exchange Economy

Recall the planner knows θ2 but not θ1 .

The planner wants to implement the Walrasian equilibrium, so

asks 1 to report a type.
Exchange Economy

Recall the planner knows θ2 but not θ1 .

The planner wants to implement the Walrasian equilibrium, so

asks 1 to report a type.

Suppose 1 reports θb1 , the planner then implements a

Walrasian equilibrium assuming the report is true x∗ (θ2 , θb1 ).
Exchange Economy

Recall the planner knows θ2 but not θ1 .

The planner wants to implement the Walrasian equilibrium, so

asks 1 to report a type.

Suppose 1 reports θb1 , the planner then implements a

Walrasian equilibrium assuming the report is true x∗ (θ2 , θb1 ).

Absent transfers 1 will report θb1 ∈ argmax u1 (x∗ (θ2 , θb1 ), θ1 ).

Exchange Economy

Recall the planner knows θ2 but not θ1 .

The planner wants to implement the Walrasian equilibrium, so

asks 1 to report a type.

Suppose 1 reports θb1 , the planner then implements a

Walrasian equilibrium assuming the report is true x∗ (θ2 , θb1 ).

Absent transfers 1 will report θb1 ∈ argmax u1 (x∗ (θ2 , θb1 ), θ1 ).

Thus, to make truth telling optimal for 1, we need to set

transfers such that θ1 ∈ argmax u1 (x∗ (θ2 , θb1 ), θ1 ) + t1 (θb1 ).
Mechanism Design: A preview

This is one possible solution to the planner’s problem.

Incentivize everyone to report truthfully, and then use these reports

to implement desirable outcomes.

Is this the best solution? It is efficient to always elicit truthful

reports?
Mechanism Design: A preview

This is one possible solution to the planner’s problem.

Incentivize everyone to report truthfully, and then use these reports

to implement desirable outcomes.

Is this the best solution? It is efficient to always elicit truthful

reports?

Yes! Quite generally anything you can do with any other approach,
you can also do by eliciting truthful reports.
Outline

1 Choice Theory

2 Choice under Uncertainty

3 General Equilibrium

4 Limitations of the Welfare Theorems: Asymmetric information

Adverse Selection
Moral Hazard
Some of the key early contributors

Holmstrom Milgrom Mirrlees Stiglitz Zeckhauser

Some Examples

Labor: How much effort workers exert.

Traders who are paid large bonuses.

Mangers who want to “empire build” to further their careers.

Equity holders of failing firms.

Insurance.

Many more.
Contract Theory I

Maybe writing contracts before actions are taken can limit moral
hazard difficulties?
A manager employs a worker.
Worker exerts costly effort to produce output. But, the effort
exerted by the worker is private information.
Manager wants to incentivize the worker to worker hard, but
the worker prefers to slack off.
First the manager commits to a contract for the worker.
The worker decides whether to accept the contract.
If so the worker chooses effort e ∈ R+ at cost c(e) = e2 /2.
Contract Theory II

Output is then realized, y = e + ε

ε is a normally distributed, mean 0 variance σ 2 noise term.
ε is not observed, the agent observes e but principal doesn’t.
Thus the contract offered by the manager depends only on y.
Let w(y) be this wage offer.
Firm is risk neutral with payoff:

UM = E[y − w(y)].

Worker is risk averse with payoff:

UW = E[w(y) − c(e)] − (λ/2) Var[w − c(e)].

If effort was observed I

Special case, manager observes effort

Consider the effort that maximizes surplus:

e∗ ≡ argmax E[y − c(e)] = argmax e − e2 /2 = 1

e e

Can write a contract that only pays a worker who exerts this effort
level.
e∗2
So w = 0 if e 6= e∗ and otherwise, w = 2 .

Worker problem is then:

max E[w(e) − c(e)] − (λ/2) Var[w(e) − c(e)].

e

A worker’s expected payoff is 0 for e ∈ {e∗ , 0} and negative

otherwise.
If effort was observed II

We resolve the worker’s indifference in favor of e∗ , but this is not

important. Why?
Thus the manager incentivizes the worker to maximize the surplus,
and then extracts it all.
Linear Contracts I

Suppose the manager offers a linear contract:

w(y) = α + βy
Useful to solve from the back and work forwards.
In the second stage, the worker chooses effort to solve:

max E[α + β(e + ε) − c(e)] − (λ/2) Var[α + β(e + ε)]

e∈R+

simplifies to
max α + βe − e2 /2 − (λ/2)β 2 σ 2
e∈R+

This problem is concave, so the first order conditions will

characterize the optimum.
The first order condition requires that β − e = 0
Linear Contracts II

Worker’s expected utility from accepting the contract and

optimizing effort is then:

β2
α + β 2 − β 2 /2 − (λ/2)β 2 σ 2 = α + (1 − λσ 2 )
2
Accepting the contract is optimal for worker if and only if:

1 λσ 2
α + β2( − )≥0
2 2
We refer to this as the paticipation or individual rationality
constraint for the worker.
Linear Contracts III

Now to the first stage.

Manager anticipates worker response to contract w(y) = α + βy
Selects contract to maximize her payoff condition on this.
There are many ways to solve this problem.
We could use the Lagrangian method.
But, is time consuming—must find which constraints bind.
We’ll take a different approach, using “economic intuition.”

Eye ball the expected utility of the worker and the manager:
Linear Contracts IV

UM (y) = E [y] − E [α + βy]

= (1 − β)E [y] − α
λ
UW (y) = E [w(y)] − Var [w(y)] − c(e)
2
λ 2

= α + βE [y] − β Var [y] − c(e)
2
Where does α enters these expressions?
There’s a ‘−α’ term in the manager’expected s utility and a ‘+α’
term in the worker’s expected utility.
Moreover, UM (y) + UW (y) doesn’t depend on α.
Linear Contracts V

The manager can costlessly transfer utility between himself and the
worker!
So: set β to maximize joint surplus, subject to incentive
compatibility, then transfer it all to herself using α

 
2 λ 2 1
UM (y) + UW (y) = β − β σ +
2 2
Linear Contracts VI

So we solve the following problem:

β2
max β − (λσ 2 + 1),
β∈R 2
FOC implies that β = 1/(1 + λσ 2 ). So e = 1/(1 + λσ 2 ) < 1.
So the optimal (linear contract) places less emphasis on output
when the principals observation of effort is worse and when the
agent is more risk averse.
Now, to extract all the gains from trade make UW (y) = 0 by
setting

β2
α=− (1 − λσ 2 )
2
Linear Contracts VII

Although α is negative, we normalized the worker’s values to

rejecting the contract to 0.
So in practice we might expect to see an α > 0 but less than this
outside option.
Note that the manager can infer what effort the agent has taken.
Why not just directly compensate the worker for his effort?
This would avoid transferring risk to him.
However, the manager cannot do this.
If the manager inferred effort and rewarded this, the agent would
want to deviate.
The agent would put in low effort and claim a bad shock
happened.
Linear Contracts VIII

Remark
To create the right incentives, the principal has to punish agents
who receive bad shocks and reward agents who receive god shocks,
even though he knows these deviations are just because of the
shocks.
Linear Contracts IX

Why is the principal constrained to use linear contracts?

What if the agent performs multiple tasks with different levels

of observability and his allocation of effort to each tasks is
unobserved?

Empirically evidence is mixed. Incentives do seem to matter in

a variety of contexts, but the theory has not been especially
successful at predicting the optimal form of contracts.