An economic allocation ( x
1
; ::; x
I
; y
1
:::; y
J
) is feasible if
X x li
I
!
l
+
X y lj
J
An allocation ( x; y ) is Pareto e¢ cient if there is no alternative allocation ( x 0 ; y 0 ) such that u i
( x
0 i
) u i
( x i
) for all i with u i
( x
0 i
) > u i
( x i
) for some i
A competitive equilibrium is an allocation ( x ; y ) and a vector of prices p such that
(i) For all …rms j y j solves max y j
2 Y j p y j
(ii) For all consumers i s.t.
p x i x i solves max u i
X p !
i
+ ij
(
( p y j x i
)
)
J
1

(iii) Market clearing: For each good l we have
X x li i
= !
l
+
X y lj
J
Note here that ij is consumer i 0 s ownership of …rm j . Clearly, any …rm j is totally owned by consumers
X ij
= 1 i
To solve this model for all commodities is left for a later chapter. Now we consider the case where one good is small in the economy, so we can ignore income e¤cts.
Consumers preferences are given by the utility functions ( m the numeraire) u i
( m i
; x i
) = m i
+ i
( x i
)
Where we normalize the price of commodity 1 to 1 and set the price of x as p:
Firms produce q j using the numerair as input, and the amount used is c ( q j
) they thus maximize
Note that the budget condition for the consumer is m i
+ p x i
!
mi
+
X j ij
( p q j c j
( q j
))
Where it is assumed that consumer only have endowment in the m commodity.
Equilibrium conditions p p
X x i
I
= c
0
( q j
) equality if q j
> 0
0
( x i
X q j
) with = if x i
J
> 0
Problem 1 What about market clearing in the m commodity?
Some important things to note:
2

The equilibrium allocation and price is independent of the inital allocation of endowment and ownership!
Theorem 2 (The First Fundamental Theorem of Welfare Economics) If the price p and the allocation ( x ; y ) constitutes a competitive equilibrium, then this allocation is
Pareto optimal.
Theorem 3 For any pareto e¢ cient levels of utility u = ( u
1
T i with
P
T i
= 0
; :::; u
I
) , there are transfers
, such that the competitive equilibrium with endowments !
i
+ T i yield utility u .
Can show:
X u i
!
+
X i
( x i
)
X c j
( q j
)
De…ne the Marshallian surpus
S ( x
1
; :::x
I
; q
1
; :::q
J
) =
X i
( x i
)
X c j
( q j
)
Follows that
S =
Z
0 x
( P ( s ) C
0
( s )) ds which yields the standard measurement of consumer and producer surplus. Note that as there is no income e¤ect, the walrasian demand yield the correct consumer surplus.
We focus on the supply side, and take aggregate demand as given, x ( p ) , but this can be as above. There is a potentially in…nite number of …rms that can ente, all with the same technology and hence same cost function. The free entry equilibrium is then given as
(i) Pro…t maximization q solves max p q c ( q )
(ii) Market clearing x ( p ) = J q
3

(iii) Free entry condition (no incentives to enter/exit) p q c ( q ) = 0
With constant returns to scale, q is indeterminate, and hence the equilibrium can arise for any number J of …rms.
Proposition 4 If c 0 ( q ) > 0 for all q 0 and c is strictly convex, then there is no free entry equilibrium provided x ( c 0 (0)) > 0 .
Proof: If p > c 0 (0) pro…t is positive and supply in…nite.
( p ) =
Z
0 q
( p c
0
( s )) ds >
Z
0 q
( p c
0
( q )) ds = 0
If p c
0
(0) ; then q = 0 while x ( p ) > 0 :
Remark 5 The lecture will focus mostly on subjective probabilities
A simple lottery is
L = ( p
1
; ::::p
N
) With p n
0 and
X p n
= 1 with N possible outcomes.
Compound lotteries
( L
1
; :::L
K
;
1
; :::
K
) With k
0 and
X k
= 1 p n
= equivalently
X k p k n k
1
L
1
+
2
L
2
::: +
K
L
K xperiments in Kahneman and Tversy show that the majority has preferences L
1
=
(3000) L
2
= (4000 ; 80%) , and ( L
1
; 0; 25% ; 75%) ( L
2
; 0; 25% ; 75%) while (4000 ; 20%)
(3000 ; 25%) .
4

De…nition 6 Closedness f 2 [0 ; 1] : L + (1 f 2 [0 ; 1] : L
00
) L
0
L
00 g
L + (1 ) L
0 g are closed
De…nition 7 Independence axiom
L L
0 () L + (1 ) L
00
L
0
+ (1 ) L
00
Now preferences over lotteries may be represented by a utility function, that is
L L
0 i¤ U ( L ) > U ( L
0
)
De…nition 8 Expected utility
U ( L ) =
X p n u n n
Proposition 9 Expected utility i¤
U
X k
L k
=
X k
U ( L k
)
Proposition 10 Linear transformation only. If
~ and U represent the same ranking of lotteries:
~
( L ) = U ( L ) +
Proposition 11 If rational preferences over lotteries satis…es independence and continuity, then they allow an expected utility representation
In general a money lottery is represented by a cumulative distribution function F ( x ) and the expectation is written
Z xdF ( x ) while expected utility is
Z u ( x ) dF ( x )
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To interpret this
X
Z xdF ( x ) = x i p i when F ( x ) =
Z xf ( x ) dx if the distribution in continuous
X p i
(Note that F ( x i
) = p i
) x i x
But we can also have intermediate cases.
The following properties are equivalent
(i) The decision maker is risk avers: Always prefer Ex =
R xdF ( x ) to F ( x )
(ii) u () is concave
(iii) The certainty equivalent c ( F; u ) Ex ( u ( c ( F; u )) =
R u ( x ) dF ( x ) )
(iv) ( x + "; p ; x "; 1 p ) x for " > 0 only if p > 0 : 5
The Arrow-Pratt measure of Absolute Risk aversion.
r
A
( x ) = u 00 ( x ) u 0 ( x )
A justi…cation for this measure is the following theorem
Proposition 12 Let u
1 and u
2 be two Bernoulli utility functions (von Neumann & Morgenstern) The following are equivalent r
A
( x ; u
2
) r
A
( x ; u
1
)
There is an increasing concave function / such that u
2
( x ) = ( u
1
( x ))
For any lottery F () c ( F ; u
2
) c ( F ; u
1
)
( x; "; u
2
) ( x; "; u
1
)
( x; "; u ) : u ( x + " ) + (1 ) u ( x " ) = u ( x )
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For any lottery F ()
Z u
2
( x ) dF ( x ) u
2
( x ) = )
Z u
1
( x ) dF ( x ) u
1
( x )
First order stochastic dominance is when we always prefer F to G for any increasing utility function
Z u ( x ) dF ( x )
Z u ( x ) dG ( x ) For any increasing utility function u
: This is the case if and only if
F ( x ) G ( x ) for all x
More interesting is second order stochastic dominance.
We say F second order stochastically dominate G if:
Z Z u ( x ) dF ( x ) u ( x ) dG ( x ) for any increasing and concave utility function
We say that G is a mean preserving spread of F if
G is the cumulative distribution of y where y = x + z ( x ) where F is the cumulative distribution of x z ( x ) is a random variable with Ez ( x ) = 0 for any realization of x .
the outcome in
The following are equivalent
F second order stochastically dominates G
G is a mean preserving spread of F .
Z x
F ( t ) dt
0
Z x
G ( t ) dt
0
7

In statistics classes you may have learned not to talk about the probability distribution of parameters. We talk about
Pr(^ > T j = 0) but not about
Pr( > 0)
Now, in may economic situations the payo¤ depend on some parameter. The success of a product depend on the share of the consumers who would like to buy it over the competitors product. In games the optimal strategy may depend on the type of your opponent (high or low cost). So what do we do in these situations.
We may stick to traditional inference theory. The product is a success if > and then we test the null hypothesis that . We construct an estimator
^ and compute
Pr(^ > T j )
But what level should we require – why is it always 5% or 1%. The product may be a small success, or a huge success. What are the consequences of failing. Is failing badly worse than failing barely? A more satisfactory approach is to start with preferences. But we have no probabilities and hence what are preferences de…ned over?
The approach in Savage is to de…ne acts x : !
R
That is there is a list of possible outcomes !
2 : For each outcome there is a consequence, e.g. in terms of payo¤. And then we de…ne preferences over acts.
Savage introduced the Sure Thing Principle , that is, if we compare two acts x and x 0 who are equal except in the event E , that is x ( !
) = x
0
( !
) for ! = E then the ranking of x and x
0 is independent of the consequences outside E .
Consider a horse lottery, where you win (1000 kroner) or loose (0). The horses are
Rex, Rodney, other...
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x : you win if Rex win x
0
: you win if Rodney win y : you win unless Rodney win y
0
: you win unless Rex win
Then x x
0 () y y
0
Now, we can interpret the preferences as
Pr ( Rex ) > Pr ( Rodney )
If there are at least three states and preferences over acts are continuous and satisfy the sure thing preferences, there are state probabilities and utilities such that x x
0
()
X u ( x ( !
)) p
!
>
X u ( x
0
( !
)) p
!
!
2 !
2
I will not prove this theorem, but rather consider the idea of Anscombe and Auman
(1963) but in the wrapping of Blume et al: The setup is:
We have a space of pure consequences C and the set of lotteries over this space, called
P : Let us denote these F 2 P ; presenting the probability distributions over consequences c 2 C: So this is the standard setup for expected utility. Now, the act can have such lotteries as outcome x : 7 ! P thus for any !
2 we have x ( !
) 2 P
The set of such functions are denoted
P
We add acts the natural way
( x + y ) ( !
) = x ( !
) + y ( !
)
Now we add some axioms
9

Axiom 13 Preferences over acts are rational
Axiom 14 The independence axiom applied to acts x y then x + (1 ) z y + (1 ) z
Axiom 15 Nontriviality: There are x and y such that x y
Axiom 16 Archimedian x y z then there exist 0 < < < 1 such that x + (1 ) z y x + (1 ) z
Next, we de…ne conditional prefernces.
x
S y if for some z : ( x
S
:z
S
) ( y
S
:z
S
)
Some outcome may be ignored by the individual, these are called Savage null, this is when x
S y for all acts x and y
Note that if
( x
S
:z
S
) ( y
S
:z
S
) and ( y
S
:w
S
) ( x
S
:w
S
)
Then
1
2
1
(( x
S
:z
(( y
S
:w
2
S
S
) + ( x
S
) + ( x
S
:w
:z
S
S
))
))
2
1
1
(( y
S
:z
(( x
S
:w
2
S
) + ( x
S
:w
S
) + ( x
S
:z
S
)) and
S
)) which is a contradition, so the choice of z does not matter.
Axiom 17 (Non-null State independence) For constant acts: x
!
y i¤ x
!
0 y for all non-nul !
and !
0
I’ll comment more on the axiom as we need them. First let me give the result and indicate the idea of the proof:
10

Proposition 18 If Axiom 1-5 is satsi…ed, then there are probabilities p
!
and a utility function u such that x y i¤
X
Z
!
2 u ( c ) dF ( x ( !
)) p
!
>
X
Z
!
2 u ( c ) dF ( y ( !
)) p
!
That is expected utility using the joint probabilities from P and p
!
.
How do we do it. Let x; y and z be constant acts. E.g.
x ( !
) = x ( !
0
) for all !
and
!
0 . Then these acts simply collapses to standard lotteries, and the independence and archimedian axiom yields expected utility for these lotteries. Axiom 5 now ensures that these ranking of classical lotteries are independent of !
. Now, we get the probabilities p
!
as follows.(Normalize u (0) = 0 ; or with non-monetary prices choose some other consequence)
Consider two lottery: x is a constant act vinning c 0 with probability y with y ( !
) = c and y ( !
0
) = 0 for !
0
= !
Chose such that x y then p
!
= .
11