Price Theory Encompassing
Transcript of Price Theory Encompassing
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Price Theory Encompassing
MIT 14.126, Fall 2001
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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
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Convexity and Traditional Price Theory
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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.
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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
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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
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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
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An Alternative Doctrine
Disentangling Three Ideas
∂≤
∂
S
p
l w
π ′ =S
) −
′ − *
p
p
π
′=
∂ ≥
∂
2
2 ,
S l w
= S l
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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
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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′ =
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Convexity Alone
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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
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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
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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
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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.
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Order Alone
}∈p
−) y y
}( x
}∈
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“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
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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.
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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
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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
)∈ ( (
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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′ ≥ ⇒ ≥
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Sublattices of R2
x
y
x∨y
x∧y
z
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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
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“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
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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
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“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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
) + ≥(
(
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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 +
-
+
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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
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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
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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
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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
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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
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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
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Introduction to Supermodular Games
))
)
t
)
) )
−
− U
)−(
) )
x
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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 ).
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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