1

Price Theory Encompassing

MIT 14.126, Fall 2001

2

Road Map

Traditional Doctrine: Convexity-Based Disentangling Three Big Ideas � Convexity: Prices & Duality � Order: Comparative Statics, Positive

Feedbacks, Strategic Complements � Value Functions: Differentiability and

Characterizations, Incentive Equivalence Theorems

3

Convexity and Traditional Price Theory

4

Convexity at Every Step

Global or local convexity conditions imply � Existence of prices � Comparative statics, using second-order conditions � Dual representations, which lead to… � Hotelling’s lemma � Shephard’s lemma � Samuelson-LeChatelier principle

“Convexity” is at the core idea on which the whole analysis rests.

5

Samuelson-LeChatelier Principle

Idea: Long-run demand is “more elastic” than short-run demand. Formally, the statement applies to smooth demand functions for sufficiently small price changes. Let p=(px,w,r) be the current vector of output and input prices and let p’ be the long-run price vector that determined the current choice of a fixed input, say capital. Theorem: If the demand for labor is differentiable at this point, then:

′=

∂ ≤

∂ ,

0 L

p p p

l w

6

Varian’s Proof

Long- and short-run profit functions defined:

Long-run profits are higher:

So, long-run demand “must be” more elastic:

π π π′ ≥ ( ) ( , ) for all , and ( ) ( , )L L Sp p p p p p p p

( π

π

= −

′ ′= −

,

*

( ) max ( , )

( , ) max ( ), ( )

L xk l

S xl

p f k l wl rk

p p f k p l wl rk p

π

′ =

∂ ∂ ∂≥ ≤ ≤

∂ ∂ ∂

2

2 ,

0 and 0 L L S

p p p p p p p p

l w w w

7

A Robust “Counterexample”

The production set consists of the convex hull of these three points, with free disposal allowed:

Fix the price of output at 9 and the price of capital at 3, and suppose the wage rises from w=2 to w=5. Demands are:

Long run labor demand falls less than short-run labor demand. Robustness: Tweaking the numbers or “smoothing” the production set does not alter this conclusion.

120

111

000

OutputLaborCapital

= =(2) 2, (5,2) 0, (5) 1L Ll l

8

An Alternative Doctrine

Disentangling Three Ideas

∂≤

∂

S

p

l w

π ′ =S

) −

′ − *

p

p

π

′=

∂ ≥

∂

2

2 ,

S l w

= S l

9

Separating the Elements

Convexity � Proving existence of prices � Dual representations of convex sets � Dual representations of optima

Order � Comparative statics � Positive feedbacks (LeChatelier principle) � Strategic complements

Envelopes � Useful with dual functions � Multi-stage optimizations � Characterizing information rents

10

Invariance Chart

One-to-oneValue function derivative formula:

Order-preservingLong-run optimum change is larger, same direction

Order-preservingOptimal choices increase in parameter

Linear (“convexity preserving”)

Supporting (“Lagrangian”) prices exist

Transformations of Choice Variable

Conclusions about ∈max ( , )x S f x t

* 2( ) ( ( ), )V t f x t t′ =

11

Convexity Alone

12

Pure Applications of Convexity

Separating Hyperplane Theorem � Existence of prices � Existence of probabilities � Existence of Dual Representations � Example: Bondavera-Shapley Theorem � Example: Linear programming duality

“Alleged” Applications of Duality � Hotelling’s lemma � Shephard’s lemma

13

Separating Hyperplane Theorem

Theorem. Let S be a non-empty, closed convex set in RN and x∉S. Then there exists p∈RN such that

Proof. Let y∈S be the nearest point in S to x. Let

� Argue that such a point y exists. � Argue that p.x>p.y. � Argue that if z∈S and p.z>p.y, then for some small

positive t, tz+(1-t)y is closer to x than y is.

{ ⋅ > ⋅ max |p x y y S

= − ( /p x x

14

Dual Characterizations

Corollary. If S is a closed convex set, then S is the intersection of the closed “half spaces” containing it. � Defining

� it must be true that { π∈ = ⋅ ≤∩ | )Np RS p x p

{ π = ⋅ ( ) max |p p x x S

15

Convexity and Quantification The following conditions on a closed set S in RN is are equivalent � S is convex � For every x on the boundary of S, there is

a supporting hyperplane for S through x. � For every concave objective function f

there is some λ such that the maximizers of f(x) subject to x∈S are maximizers of f(x)+λ.x subject to x∈RN.

16

Order Alone

}∈p

−) y y

}( x

}∈

17

“Order” Concepts & Results Order-related definitions Optimization problems � Comparative statics for separable objectives � An improved LeChatelier principle � Comparative statics with non-separable “trade-offs”

Equilibrium w/ Strategic Complements � Dominance and equilibrium � Comparative statics � Adaptive Learning � LeChatelier principle for equilibrium

18

Two Aspects of Complements

Constraints � Activities are complementary if doing one enables doing the

other… � …or at least doesn’t prevent doing the other.

� This condition is described by sets that are sublattices.

Payoffs � Activities are complementary if doing one makes it weakly

more profitable to do the other… � This is described by supermodular payoffs.

� …or at least doesn’t change the other from being profitable to being unprofitable � This is described by payoffs satisfying a single crossing condition.

19

Definitions: “Lattice”

Given a partially ordered set (X,≥), define �

�

(X,≥) is a “lattice” if

Example: X=RN,

{ }∨ = ∈ ≥ ≥The " " : inf | , .join x y z X z x z y

{ }∧ = ∈ ≤ ≤The " " : sup | , .meet x y z X z x z y

( ∀ ∈ ∨ ∈, x y X x y x y X

≥ =

∧ =

∨ =

if , 1,..., ( ) min( , ); 1,..., ( ) max( , ); 1,...,

i

i i

i i

x y y i N

x y y i N

x y y i N

20

Definitions, 2

(X,≥) is a “complete lattice” if for every non-empty subset S, a greatest lower bound inf(S) and a least upper bound sup(S) exist in X.

A function f : XÆR is “supermodular ” if

A function f is “submodular” if –f is supermodular.

( ∀ + ≤ ∧ + ∨, ( ) ) ( ) )x y X f x f y f x y f x y ) ∧ ,

≥

=

=

i

i

i

x x x

)∈ ( (

21

Definitions, 3

Given two subsets S,T⊂X, “S is as high as T,” written S≥T, means

A function x* is “isotone” (or “weakly increasing”) if

� “Nondecreasing” is not used because…

A set S is a “sublattice” if S≥S.

[ ] [ ]

∈

⇒ ∈ ∧ ∈

and and

x y T

x y S x y T

* ( ) ( )t t x t x t′ ≥ ⇒ ≥

22

Sublattices of R2

x

y

x∨y

x∧y

z

23

Convexity, order and topology are mostly independent concepts. However, in R, these concepts coincide � Topology: S = compact set with boundary {a,b} � Convexity: � Order:

Not Sublattices

= ≤ ≤[ , ] { | }S b x a x b { }α α = − ∈(1 ) | [0,1]S b

24

“Pairwise” Supermodularity

Theorem (Topkis). Let f :RNÆR. The following are equivalent: � f is supermodular � For all n≠m and x-nm, the restriction f (.,.,x-nm):R2ÆR is

supermodular.

∈

∨

S

* ′

= a

α+ a

25

Proof of Pairwise Supermodularity

⇒ This direction follows from the definition. ⇐ Given x≠y, suppose for notational simplicity that

Then,

QED

= = = +

max( , ) for 1,..., min( , ) for 1,...,

i i

i

x y i n x

x y n N

( + =

+ =

−

∨ = −

≥

−

= ∧

∑ ∑

1 1 11

1 11

1 1

( ) ( ) ( ,..., , ,..., ) ( ,..., , ,..., )

[ ( ,..., , ,... , ,... )

( ,..., , ,... , ,... )] ( ) ( )

n

i N i i Ni n

i n n Ni

i n n N

f x y f y f x x y y f x x y y

f x x y y x x

f x x y y x x f x f x y

26

“Pairwise” Sublattices

Theorem (Topkis). Let S be a sublattice of RN. Define

Remark. Thus, a sublattice can be expressed as a collection of constraints on pairs of arguments. In particular, undecomposable constraints like

can never describe in a sublattice.

( { }= ℜ ∃ ∈ = =| N ij i i j jS z S x z x z

= ∩ , Then, .iji j S

+ ≤1 3 1x x

27

Proof of Pairwise Sublattices

QED

( ⊂

∈ ∈ = =

= ∨ ∈ ≥ =

= ∧ ∈ =

∩ ∩

,

,

It is immediate that . For the reverse,

suppose . Then, and .

Define For all , , .

So, satisfies for all .

iji j

ij ij ijij i ji i j

i j j i j i ii

i i i

S

x S S z x z x

z S j z x z x

z S z x i

28

Complementarity

Complementarity/supermodularity has equivalent characterizations: � Higher marginal returns

� Nonnegative mixed second differences

� For smooth objectives, non-negative mixed second derivatives:

y

x

x∨y

x∧y

∨ ≥ − ∧( ( ) ( ) ( f x y f x f y f x y

[ ] [ ]∨ − − ∧ ≥( ( ) ( ) ( 0f x y f x f y f x y

∂ ≥

∂ ∂

2

0 for i

f i x x

i

i i

)−

+

+

−

−

1

1

1

i

i

i

)∈ , x

S

+ 2 x

) ∃

.

j

ij

i

S

z

z

z

− ) )

− ) )

≠ j

j

29

Monotonicity Theorem

Theorem (Topkis). Let f :X×R ÆR be a supermodular function and define

If t ≥t’ and S (t ) ≥ S (t’ ), then x*(t ) ≥ x*(t’ ). Corollary. Let f :X×R ÆR be a supermodular function and suppose S (t) is isotone. Then, for each t, S (t) and x* (t ) are sublattices. Proof of Corollary. Trivially, t ≥t, S (t ) ≥ S (t ) and x*(t ) ≥ x*(t ). QED

∈ ≡*

( )( ) argmax ( , ).

x S t x t f x t

30

Proof of Monotonicity Theorem

If either any of these inequalities are strict then their sum contradicts supermodularity:

QED

′ ′∈ > *

Suppose that f is supermodular and that ( ), ( ), .x x t x x t t t

′ ′∧ ∨ ∈

′ ′ ′ ′≥ ≥ ∧

Then, ( ) ( ),( ) ( ) So, ( , ) ( , ) and ( , ) ( , ).

x x S t x x S t f x t f x x t f x t f x x t

′ ′ ′ ′ ′+ ∧ + ∨( , ) , ) ( , ) , ).f x t f x t f x x t f x x t

31

Necessity for Separable Objectives

Theorem (Milgrom). Let f :RN×R ÆR be a supermodular function and suppose S is a sublattice.

Then, the following are equivalent: � f is supermodular �

Remarks: � This is a “robust monotonicity” theorem. �

∈ =

≡ ∑* ,

1 Let ) argmax ( , ) ( ).

N

g S n nx S n

x f x t g x

ℜ→ ℜ * 1 For all ,..., : , ( ) is isotone.N Sg x t

≡

+ ∧ + ∨

∑The function ( ) ( ) is "modular": ( ) ( ) ( ( .

n g x g x g x g y g x y g x y

32

Proof

⇒ Follows from Topkis’s theorem. ⇐ It suffices to show “pairwise supermodularity.” Hence, it is sufficient to show that supermodularity is necessary when N=2. We treat the case of two choice variables; the treatment of a choice variable and parameter is similar.

Fix −∞ ∉ ∧ = = =

∧ = =

if { , } ( ( ) if , 1

( ) ( ( ) if y , 2

0 otherwise

i i

i i

i

z y f x y f x z x i

g z f x y f y z i

( + ∧ + ∨ = ∧

¬

*

*

If ( ) ( ) ( ) ( ), then { , , }

is not a sublattice, so ( ) ( ) . QED gf x f y f x y f x y x x y x y

x t x t

∈ℜ > <2 1 2 2Let , be unordered: ,x y y x y

so

′ ∈ *

′ ∈

′ ∨

> ( (

+ ( t

, g g

= ) )n

−

−

) )

i

i i

i

x

) >

≥ *

1 x

33

Application: Production Theory

Problem:

Suppose that L is supermodular in the natural order, for example, L(l,w)=wl. � Then, -L is supermodular when the order on l is reversed. � l*(w) is nonincreasing in the natural order.

If f is supermodular, then k*(w) is also nonincreasing. � That is, capital and labor are “price theory complements.”

If f is supermodular with the reverse order, then capital and labor are “price theory substitutes.”

− ,

max ( , ) ( , ) ( , )k l pf k l L l w K k r

34

Application: Pricing Decisions

A monopolist facing demand D (p,t) produces at unit cost c.

p*(c,t ) is always isotone in c. It is also isotone int if log(D (p,t)) is supermodular in (p,t), which is the same as being supermodular in (log(p),t), which means that increases in t make demand less elastic:

(

( ( )>

=

= +

* ( ) argmax ( , )

argmax log log ( , ) p

p c

p t p c D p t

p c p t

∂

∂

log ( , ) nondecreasing inlog( ) D p t t p

35

Application: Auction Theory

A firm’s value of winning an item at price p is U(p,t), where t is the firm’s type. (Losing is normalized to zero.) A bid of p wins with probability F(p). Question: Can we conclude that p(t) is nondecreasing, without knowing F?

Answer: Yes, if and only if log(U (p,t)) is supermodular.

( ( )

=

=

* ( ) argmax ( , ) ( )

argmax log ( , ) log ( )

F p

p

p t U p t F p

U p t F p

36

Long v Short-Run Demand

Notation. Let l S(w,w’ ) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is w’. Setting w =w’ in l S gives the long run demands. Samuelson-LeChatelier principle:

� which can be restated revealingly as:

≥ 10 , ) ( , ). dl w w l w wdw

≥ 20 , ).l w w

− )

)

−

− D

) + ≥(

(

37

Milgrom-Roberts Analysis

Remarks: � This analysis involves no assumptions about convexity,

divisibility, etc. � For smooth demands, symmetry of the substitution matrix

implies that, locally, one of the two cases above applies.

Complements Substitutes

Labor

Capital

Wage -

-

-Labor

Capital

Wage +

-

+

38

Improved LeChatelier Principle

Theorem (Milgrom & Roberts). x* is isotone in both arguments. In particular, if t >t’ , then

Let ( , , ) be supermodular and a sublattice.H x y t S

( ∈

= *

( , ) Let ), ( ) max arg max ( , , )

x y S x t y t H x y t

{ ′ ′ ′∈ ′ =

*

*

|( , ( )) Let , ) max arg max ( , ( ), )

x x x y t S x t t H x y t t

′ ′ ′ = ≥ = * * * *( ) ( , ) ( , ) ( , ) ( )x t x t t x t t x t t x t

39

Proof

By the Topkis Monotonicity Theorem, Applying the same theorem again, for all t,t” >t’

Setting t=t” completes the proof.

{

{

{

′ ′ ′∈

′ ′ ′∈

′ ′ ′′∈

′ ′ ′ ′ =

′ ≤

′′ ′′≤

*

*

*

*

|( , ( ))

*

|( , ( ))

*

|( , ( ))

( , ) max arg max ( , ( ), )

( , ) max arg max ( , ( ), )

( , ) max arg max ( , ( ), )

x x x y t S

x x x y t S

x x x y t S

x t t H x y t t

x t t H x y t t

x t t H x y t t

′≥* ( ) ( )y t y t

40

Long v Short-Run Demand

Theorem. Let w>w’. Suppose capital and labor are complements, i.e., f (k,l ) is supermodular in the natural order. If demand is single-valued at w and w’ , then

Theorem. Let w>w’. Suppose capital and labor are substitutes, i.e., f (k,l ) is supermodular when capital is given its reverse order. If demand is single-valued at w and w’ , then

′ ′≤ ( , ) ( , ) ( , )S Sl w w l w w l w w

′ ′≤ ( , ) ( , ) ( , )S Sl w w l w w l w w

) * (

}∈ ′ * (

′ ≥ *

}

}

}

∈

∈

∈

′=

=

*

*

*

*

′ ≤S

′ ≤S

41

Non-separable Objectives

Consider an optimization problem featuring “trade-offs” among effects. � x is the real-valued choice variable � B (x) is the “benefits production function” � Optimal choice is

( π ∈

= * ( ) argmax , ( ),B x X x t x B x t

42

Robust Monotonicity Theorem

Define:

Theorem. Suppose π is continuously differentiable and π2 is nowhere 0. Then:

(

(

π π

π π

∀

⇒

⇒ ∀

1

2

*

1

2

( , , ), is increasing in( , , )

For all , ( ) is isotone

( , , ), is nondecreasing in( , , )

B

x y tx y t x y t

B x t

x y tx y x y t

( π ∈

= * ( ) argmax , ( ),B x X x t x B x t

43

Application: Savings Decisions

By saving x, one can consume F (x ) in period 2.

Analysis. If MRSxy increases with x, then optimal savings are isotone in wealth:

This is the same condition as found in price theory, when F is restricted to be linear. Here, F is unrestricted. � Also applies to Koopmans consumption-savings model.

( (

≤ ≤

≤ ≤

=

=

0 *

0

( ) max , ( )

( ) max argmax , ( ) x w

F x w

V w U w x F x

x w w x F x

( π = Define: , , ) ,x y t U t x y

( (

⇒

1 *

2

, increasing in ( ) isotone

, F

U x y x w

U x y

44

Introduction to Supermodular Games

))

)

t

)

) )

−

− U

)−(

) )

x

45

Formulation

N players (infinite is okay) Strategy sets Xn are complete sublattices �

Payoff functions Un(x) are � Continuous

� “Supermodular with isotone differences”

= min , maxn n nx x X

( ( ) ( )− − ′ ∀ ∈ ∀ ∈

′ ′+ ∧

≥

+

, ( ) ( ) ( ) ( )

n n n n n

n n n

n x X x x X

U x U x U x x U x x

46

Bertrand Oligopoly Models

Linear/supermodular Oligopoly:

Log-supermodular Oligopoly:

( ≠

= − +

=

∂ =

∂

∑Demand: )

Profit: ( ) ( )

( ) which is increasing in

n j jj n

n n n

n m n n

m

Q x A ax b x

U x x c Q x

U b x c x x

( = +

∂ ≥ ⇔ ≥

∂ ∂ ∂ ∂

2

log ( ) log log ( )

log ( )0 log log

n n n

n

m n m

U x x c Q x

U x

x x x x

47

Linear Cournot Duopoly

Linear Cournot duopoly (but not more general oligopoly) is supermodular if one player’s strategy set is given the reverse of its usual order.

= − −

=

∂ = −

∂

1 Inverse Demand: ( ) ( ) ( ) ( )n n n

n n

m

P x A x x U x x P x C x U x x

48

Analysis of Supermodular Games

Extremal Best Reply Functions

� By Topkis’s Theorem, these are isotone functions.

Lemma:

Proof.

( (

−′ ∈

−′ ∈

′=

′=

( ) max argmax ( , )

( ) min argmax ( , )

n

n

n n n x

n n n x

B x U x x

b x U x x

[ [ ¬ ⇒ ∨( ) is strictly dominated by ( )n n n nx b x x b x x

− − −∨ ≥ − ∧ >( ( ), ) , ) ( ( ), ) ( ), ) 0n n n n n n n n n n n nU x b x x U x x U b x x U x b x x

[ ]¬ If ) , thenn x x

=n X

) − ′∀

′ ≤ ∨ n

n

x

) −

−

( n

n

n

)−

∂2

0

n

n

n

Q

− 2

n ) )

n

n

n X

n X

] ]≥ n

− − ( ( n n

≥ ( nb

49

Rationalizability & Equilibrium

Theorem (Milgrom & Roberts): The smallest rationalizable strategies for the players are given by

Similarly the largest rationalizable strategies for the players are given by

Both are Nash equilibrium profiles.

→∞ = lim ( )k

k z x

→∞ = lim ( )k

k z x

50

Proof

Notice that bk(x) is an isotone, bounded sequence, so its limit z exists. By continuity of payoffs, its limit is a fixed point of b, and hence a Nash equilibrium. Any strategy less than zn is less than some bk

n(x) and hence is deleted during iterated deletion of dominated strategies. QED

51

Comparative Statics

Theorem. (Milgrom & Roberts) Consider a family of supermodular games with payoffs parameterized by t. Suppose that for all n, x-n, Un(xn,x-n;t) is supermodular in (xn,t). Then

Proof. By Topkis’s theorem, bt(x) is isotone in t. Hence, if t >t’,

and similarly for QED

( ), ( ) are isotone.z t z t

′

′ →∞ →∞

≥

′=

( ) ( ) ( ) lim ( ) lim ( ) ( )

k t

k t k

b x b x

z t b x b x z t

z

52

Adaptive Learning

Player n’s behavior is called “consistent with adaptive learning” if for every date t there is some date t’ after which n does not play a strategy that is strictly dominated in the game in which others are restricted to play only strategies they have played since date t. Theorem (Milgrom & Roberts). In a finite strategy game, if every player’s behavior is consistent with adaptive learning, then all eventually play only rationalizable strategies.

b

B

.

≥ ≥

k t

k tk

53

Equilibrium LeChatelier Principle

Formulation � Consider a parameterized family of supermodular games

with payoffs parameterized by t. Suppose that for all n, x-n, Un(xn,x-n;t) is supermodular in (xn,t).

� Fixing player 1’s strategy at z1(t’) induces a supermodular game among the remaining players. Let y(t,t’) be the smallest Nash equilibrium in the induced game, with y1(t,t’)=z1(t’).

Theorem. �

�

′ ′> ≥If then ( ) ( , ) ( ).t t z t y t t z t ′ ′< ≤If then ( ) ( , ) ( ).t t z t y t t z t

…and a similar conclusion applies to the maximum equilibrium. 54

Proof

Observe (exercise) that

Suppose t>t’. � By the comparative statics theorem, z is isotone, so:

� Hence, by the comparative statics theorem applied again, y is isotone, so:

QED

′ ′ = ( ) ( , ), ( ) ( , ).z t y t t z t y t t

′≥( ) ( ).z t z t

′ ′≥ ( , ) ( , ) ( , ).y t t y t t y t t

55

Envelope Functions Alone

Based on “Envelope Theorems for Arbitrary Choice Sets” by Paul Milgrom & Ilya Segal

56

What are “Envelope Theorems”?

Envelope theorems deal with the properties of the value function: Answer questions about… � when V is differentiable, directionally differentiable,

Lipschitz, or absolutely continuous � when V satisfies the “envelope formula”

“Traditional” envelope theorems assume that set X is convex and the objective f (.,t ) is concave and differentiable.

∈ ≡( ) max ( , )

x X V t x t

′ = ∈ *( ) ( , ) for ( )tV t f x t x x t

′ ≥ , ′ ≤ ,

′ =

′ ≥

f

57

Intuitive Argument

When X={x1,x2,x3}… � V is left- and right-

differentiable everywhere

� if ft (x,t) is constant on x∈x*(t), then V is differentiable at t

� envelope formulas apply for � V’ (t )=ft (x* (t ),t ) � V’ (t+) and V’ (t-)

f (x1,t)

f (x3,t)

f (x2,t)

V (t)

t

58

Envelope Derivative Formula

Theorem 1. Take t ∈[0,1] and x∈x*(t), and suppose that ft(x,t) exists. � If t<1 and V’ (t+) exists, then V’ (t+) ≥ ft (x,t). � If t>0 and V’ (t -) exists, then V’ (t-) ≤ ft (x,t). � If t∈(0,1) and V’ (t ) exists, then V’ (t ) = ft (x,t ).

Proof:

V

t

f (x,t) V

t

f (x,t)

59

Absolute Continuity

Theorem 2(A). Suppose that � f (x,.) is is differentiable (or just

absolutely continuous) for all x∈X with derivative (or density) ft .

� there exists an integrable function b(t ) such that |ft (x,.)| ≤ b(t ) for all x∈X and almost all t∈[0,1].

Then V is absolutely continuous with density satisfying |V’ (t )|≤ b(t ).

60

Proof of Theorem 2(A)

Define

Then for t” > t’ :

It suffices to prove the theorem for intervals, because open intervals are a basis for the open sets. QED

∈

′′ ′′

′ ∈

′′

′

′′ ′ ′′ ′− −

=

′′ ′≤ −

∫

∫

| ( ) ( ) | sup | ( , ) ( , ) |

sup ( , ) sup ( , )

( ) ( ) ( )

x X

t t

t x X x X

t

t

V t V t f x t f x t

f x t dt f x t dt

b t dt B t B t

= ∫0( ) ( ) t

B t b s ds

′ ∈

≤

≤

=

∫ t

t t

61

Why do we need b (.)? Let X=(0,1] and f (x,t )=g (t /x), where g is smooth and single-peaked with unique maximum at 1. � V (0)=g (0), V (t )=g (1): V is discontinuous at 0. � This example has no integrable bound b (t ):

( ∈ ∈ ∈

′ = 1

(0, ) (0, ) (0, ) sup ( , ) sup ( ) sup ( )t

t t x t x x

f x t g xg x

t

V

f (x,t)

g(1)

g(0) 62

Envelope Integral Formula

Theorem 2(B). Suppose that, in addition to the assumptions of 2(A), the set of optimizers x*(t) is non-empty for all t. Then for any selection x (t)∈x*(t),

= ∫0( ) (0) ( ( ), ) . s

tV s f x t t dt

63

Equi-differentiability

Definition. A family of functions {f (x,.)}x∈X is “equi -differentiable” at t∈(0,1) if

If X is finite, this is the same as simple differentiability.

→′

−′ − −′ ( , ) ( , )lim sup ( , ) 0tt t x

f x t f x t f x tt t

64

Directional Differentiability

Theorem 3. If � (i) {f (x,.)}x∈X is equi-differentiable at t0, � (ii) x* (t ) is non-empty for all t, and � (iii) supx|ft (x,t0)|<∞,

then for any selection x (t )∈ x* (t ), V is left- and right-differentiable at t0∈(0,1) and the derivatives satisfy

→ +

→ −

′ + =

′ − = 0

0

0

0

( ) lim ( ( ), )

( ) lim ( ( ), )

tt t

tt t

V t x t t

V t x t t

)∞ ∞ ∞

′= 1 t x

x

+ V

=

0

0

f

f

65

Role of “Equi-differentiability”

-1

-0.5

0

0.5

1

Simple differentiability (rather than equi-differentiability) is not enough for V to have left-and right-derivatives: �

�

π π= > − −

= −

=

Let ( ) sinlog( ), ( , ) ( ) if exp( / 2 2 ), ( , ) otherwise.

Then, ) ( )

g t t f x t g t t x f x t t

V t g t

66

Continuous Problems Theorem 4. Suppose X is a non-empty compact space, f is upper semi-continuous on X and ft is continuous in (x,t). Then, � V is directionally differentiable

� In particular, � V is differentiable at t if any of the following hold: � V is concave (because V’(t+)≤V’(t-)) � t is a maximum of V(.) (because V’ (t+) ≤ V’ (t-) � x*(t ) is a singleton (because V’ (t+)=V’ (t-))

∈

∈

′ + = ∈

′ − = ∈

*( )

*( )

( ) max ( , ) for [0,1)

( ) min ( , ) for (0,1]

t x x t

t x x t

V t x t t

V t x t t

′ + ≥ −( ) ( ).V t t

67

Contrast to a “Traditional” Approach

In some approaches, the differentiability of x* is used in the argument. However,V can be differentiable even when x* is not. This often happens, for example, in strictly convex problems:

X

f

68

Applications

=

(

t f

f

′V

69

Hotelling’s Lemma

Define:

Theorem. Suppose X is compact. Then, π ′(p) exists if and only if x*(p) is a singleton, and in that case π ′(p) = x*(p).

π ∈

∈

=

= *

( ) max

( ) argmax x X

x X

p p x

x p p x

70

Shephard’s Lemma

Define:

Remark: The variable x1 represents “output” and the other variables represent inputs, measured as negative numbers. Theorem. Suppose X is compact. Then, ∂C/∂p exists if and only if x*(p) is a singleton, and in that case ∂C/∂p = x*(p).

− ∈

− ∈

= ⋅

= ⋅ 1

1

1 ,

* 1 ,

( , ) min

( ) arg min

x X x y

x X x y

C y p p x

x p p x

71

Multi-Stage Maximization

Stage 1: choose investment t ≥ 0. Stage 2: choose action vector x∈X ≠∅ Assume: � f(x,t) is equidifferentiable in t and t*>0 � f(x,t) is u.s.c. in x and X is compact Conclusion: the value function V (t ) is differentiable at t* and V’ (t* )=0. � Proof: Apply theorem 4.

72

Mechanism Design

Y=set of outcomes Agent’s type is t, utility is f (x,t). M=message space. h:M→Y is outcome function. X=h(M) is set of “accessible outcomes.”

Assume that each type has an optimal choice

∈ ∈( ) argmax ( , )

x X x t f x t

⋅

⋅

−=

−=

−

−

1

1

73

Analysis

Corollary 1. Suppose that the agent’s utility function f (x,t) is differentiable and absolutely continuous in t for all x∈Y, and that supx∈Yft(x,t) is integrable on [0,1]. V in any mechanism implementing a given choice rule x must satisfy the following integral condition.

� This had previously been shown only with (sometimes “weak”) additional conditions.

= ∫0( ) (0) ( ( ), ) . t

tV t f x s s ds

74

Mechanism Design Applications

Models in which payoffs are v .p-π, so

Theorems � Green-Laffont Theorem � Uniqueness of Dominant Strategy Mechanisms

� Holmstrom-Williams Theorem � Bayesian Revenue Equivalence

� Myerson-Satterthwaite Theorem � Necessity of Bargaining Inefficiency

� Jehiel-Moldovanu Theorem � Impossibility of Efficiency with Value Interdependencies

= ⋅∫1 *

0( ) (0) ( ) .U v U v p sv ds

75

Green-Laffont Theorem

“Uniqueness of dominant strategy implementation.” Theorem (Holmstrom’s variation). Suppose that � M is a direct mechanism to implement the efficient outcome

in dominant strategies � the type space is smoothly path-connected.

Then, � the payment function for player j in mechanism M is equal to

the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(v-j) (which depends only on the other player’s types).

76

Green-Laffont Theorem

Given any value vector v, let {vj(t)|t∈[0,1]} be a smooth path connecting some fixed value vj to vj= vj(1). By the Envelope Theorem applied to the path parameter t ,

So, Xj is fully determined by the functions p and fj .

( ( ( ( ( )

( ( ( ) (

− −

−

− − −

−

= −

′= ⋅

′∴ = ⋅ − ⋅

= −

∫ ∫

0 1

0

( ), ( ), ( ),

, ), ( )

(1), ( ) (1), ( ), ( )

where ( ) ,

j j j j j j j j j

t

j j j j j j

j j j j j j j j j j j j

j j j

U v t v p v t v v X v t v

U v v p v s v v s ds

X v v f v p v v v p v s v v s ds

f v U v v

Then the agent’s equilibrium utility

+ V

+

) ) ) )

) ) )

−

−

−

−

⋅

+

+

(

j

j

j

j

77

Holmstrom-Williams’ Theorem

Theorem: Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism, plus some bidder-specific constant. Proof. Let {vj(s),s∈[0,1]} be a path from some fixed value vector to any other value vector. By the Envelope Theorem,

Hence, Xj(v) is uniquely determined by Uj(0). It is equal to Uj(0) plus the expected payment in the Vickrey mechanism.

( ( ( ( )

= −

′= ⋅∫0

( ) ( ) ( ) ( ( ))

(0) ( ) ( )

j j j j j

t

j j j j

U v t p v t v t X v t

U v p v s v s ds

78

Two-Person Bargaining

Assume � there is a buyer with value v distributed on [0,1] � there is a seller with cost c distributed on [0,1]

The Vickrey-Clarke-Groves mechanism � has each party report its value � entails p*(v,c)=1 if v>c and p*(v,c)=0 otherwise � payments are � if p*(v,c)=0, no payments � if p*(v,c)=1, buyer pays c and the seller receives v

79

Myerson-Sattherthwaite Theorem

Expected profits are: � UB(v)=E[(v-c)1{v>c}|v], so E[UB(v)]=E[(v-c)1{v>c}] � US(c)=E[(v-c)1{v>c}|s], so E[US(c)]=E[(v-c)1{v>c}] � each bidder expects to receive the entire social surplus.

Apply Holmstrom-Williams theorem: Theorem (Myerson-Satterthwaite). There is no mechanism and Bayesian Nash equilibrium such that the mechanism implements for all v,c with v>c and � UB(0)=US(1)=0 (“voluntary participation by worst type”) � E[UB(v)]+E[US(c)]≤E[(v-c)1{v>c}] (“balanced expected

budget”) 80

Subtleties

Consider a model in which:

Q: Why doesn’t simply trading at price p=1 violate the theorem in this model? A: Because it prescribes trade even when c>v!

{ } { }> < Pr 1 Pr 1 1v

) ) ) ⋅

+

j j

j

= =c