General Equilibrium under Uncertainty
The Arrow-Debreu Model
General Idea:
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this model is formally identical to the GE model
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commodities are interpreted as contingent commodities
(commodities are contingent to a state of nature, date,
location)
Definition of the list of commodities
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S states of nature s = 1, ..., S
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L physical commodities l = 1, ..., L
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→ LS contingent commodities (index ls)
The Arrow Debreu model = the GE model with LS goods (the
goods are numbered ls)
Interpreting the model: what does ”trading contingent goods”
mean?
2 periods:
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at t = 0, agents ignore s, they trade contingent goods
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to exchange xls units of good ls against xl 0 s 0 units of good l 0 s 0
= to sign a contract committing to deliver xls units of physical
good l if state s occurs at t = 1 in exchange of receiving xl 0 s 0
units of physical goods l 0 if state s 0 occurs at t = 1
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At t = 1, no contract is signed (markets are closed): the state
of nature s becomes public, contracts contingent to state s
are executed (= physical goods are delivered and consumed),
contracts contingent to other states are destroyed (they have
no value)
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endowment (ωils )ls of agent i: at the beginning of date 1, in
state s, agent i receives a quantity ωils of physical good l
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optimal demand (xils )ls of agent i is a decision taken at t = 0:
at t = 0, agent decides that, at t = 1, if state s occurs, he
will consume xils units of physical good l (at t = 0, he signs a
contract committing him to buy zils = xils − ωils units of
physical good l if state s occurs at t = 1)
Trading contingent goods allows agents to ”transfer wealth across
states”
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The demand of i maximizes his utility function subject to the
budget constraint p.x = p.ω, that is:
X
X
pls xils =
pls ωils
l,s
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l,s
This rewrites
X
ts = 0
s
where ts is the wealth that needs to be transferred to state s:
X
X
ts =
pls xils
−
pls ωils
| l {z
}
value of the consumption in s
(ts ≤ 0 or ts ≥ 0)
| l {z
}
value of the endowment in s
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one can have a model with more than 2 periods and 1 state of
nature or more as well (all the contingent goods are traded at
t = 0 and not later, these are commitments to deliver physical
goods at a certain date t in a certain state s in exchange of
receiving something at a date t 00 ≥ 1 in a state s 0 )
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Interest of the model: to discuss risk sharing and intertemporal
trades (saving, borrowing) with all the tools (concepts and
theorems) of the GE Theory (for example, equality between
the MRS at equilibrium across all dates and states)
Remark on the utility of i:
This is a function with LS variables xils
One can assume an expected utility form:
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at t = 0, agent i assigns probability πis to state s
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vi is a function with L variables ( = a bundle of L physical
goods)
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the agent chooses his demand by maximizing
X
πis vi (xi1s , ..., xiLs )
ui (xi11 , ..., xiLS ) =
s
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vi can depend on s as well:
X
πis vi (xi1s , ..., xiLs , s)
max
s
About production of contingent goods
Example with L = S = 2 (2 physical goods, 2 states of nature),
y = (y1a , y1b , y2a , y2b )
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the firm transforms l = 1 into l = 2, whatever s = a, b is
(production function f ). The production set is
y ∈ IR 4 /y1a = y1b , y2a = y2b , y2a ≤ f (−y1a )
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the firm’s technology is not the same in the 2 states (2
production functions fa and fb ) and the firm decides the
amount of input before s is revealed. The production set is
y ∈ IR 4 /y1a = y1b , y2a ≤ fa (−y1a ) , y2b ≤ fb (−y1a )
Arrow Debreu Equilibrium
This is exactly the usual equilibrium:
I
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ILS , (y ∗ , ..., y ∗ ) ∈ IR JLS ,
an allocation (x1∗ , ..., xI∗ ) ∈ IR+
1
J
LS
prices p ∈ IR+
such that
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Individual optimality
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∀i, xi∗ solves max ui (xi ) subject to the budget constraint
X
θij πj∗
p.xi ≤ p.ωi +
j
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∀j, yj∗ solves max p.yj subject to yj ∈ Yj (denote πj∗ = p.yj∗ )
Market clearing:
∀l, ∀s,
X
i
∗
xils
=
X
i
ωils +
X
j
∗
yjls
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The Welfare Theorems apply (1st Theorem means that risk
sharing across states is efficient)
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The existence theorem applies as well (convexity of preferences
and production sets implies existence of an equilibrium)
Risk sharing: an example
an exchange economy with 2 consumers i = A, B, L = 1, S = 2 (2
contingent goods: wealth in state s = 1, 2)
utility has an EU form: for i = A, B, denote
π1i ui (x1i ) + π2i ui (x2i )
with ui C 2 (u 00 < 0 < u 0 )
3 cases
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no aggregate uncertainty ($1 = $2 ) and objective
probabilities (π1A = π1B )
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no aggregate uncertainty and subjective probabilities
(π1A < π1B )
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aggregate uncertainty ($1 < $2 ) and objective probabilities
Consider an interior equilibrium: MRSA = MRSB . This implies:
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Case $1 = $2 and π1A = π1B
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Case $1 = $2 and π1A < π1B
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complete insurance (xi1 = xi2 )
p2
π2
p1 = π 1
partial insurance xA1 < xA2 and xB1 > xB2
p2
π2B
π2A
π1B < p1 < π1A
Case $1 < $2 and π1A = π1B
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xA1 < xA2 and xB1 < xB2
p2
p1
π2 < π1
The example generalizes to an arbitrary number of states and
consumers
Sequential Trade
Preliminary remark: Mas-Colell et alii present an example of
sequential trade (19D) before the introduction of asset markets
(19E). The case carried in 19D (Definition 19D1 and Proposition
19D1) is an example of a model with asset markets (where the
Arrow assets only are available)
This is why I skip most of this section. I give 2 remarks only
Remark 1
Opening of the markets at t = 1 is useless
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the above interpretation of the AD model: contingent goods
markets at t = 0 and no market at t = 1
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one could open the markets at t = 1: spot markets for the L
physical goods
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1st Welfare Theorem implies that, at equilibrium, no
additional trade occurs at t = 1
An alternative interpretation is: there would be ”too many”
markets
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that is: agents could trade good ls against good l 0 s at t = 1
(if state s occurs) instead of trading at t = 0
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when the spot markets exist, not all the markets at t = 0 are
necessary for the equ alloc to be PO. Markets at t = 0 are
required to transfer wealth across states only (not to exchange
goods ”within the same state”).
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This is what the example in 19D shows: trading the S
contingent goods 1s at t = 0 is enough (”one good/state”).
In the context of asset markets, these goods are named
”Arrow assets”
Remark 2: rational expectations
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when trade is sequential, there are opened markets at
differents dates
Apparently, the decision that is made at t = 0 concerns goods
traded at t = 0 only
But, to make an optimal choice at t = 0, the agent needs to
determine at the same moment what he will do at t = 1 (the
decisions made at t = 0 and t = 1 both concern max ui )
Hence, at t = 0, he needs to know the prices of the goods
traded at t = 1
The AD model assumes that the agents perfectly anticipates
the prices. Perfect expectations are the so-called ”rational
expectations” in this model (in other models, rational
expectations are not perfect: some uncertainty remains)
Rational Expectations are assumed in most definitions of
equilibrium. A general definition is beyond the scope of this course
Asset Markets
Consider an exchange economy with I consumers, L physical
goods, S states of nature, K assets
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at t = 0, s is unknown and assets are exchanged
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at t = 1, s becomes public, asset payoffs are paid, goods are
exchanged
What is an asset?
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a title (price qk ) to receive an amount rsk ≥ 0 units of good 1
at t = 1 in state s
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this is a real asset (payoff in good 1 only is for convenience) 6=
nominal assets
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rsk < 0 could be considered as well
Examples of assets
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safe asset r = (c, ..., c)
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the S Arrow assets rs = (0, ..., 0, 1, 0, ..., 0) (rss = 1, rsk = 0
otherwise)
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option (example of a derivative asset) with strike price c,
primary asset rk
call :
roption = (max (0, rks − c))1≤s≤S
put :
roption = (max (0, c − rks ))1≤s≤S
The agent chooses at t = 0
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a portfolio of assets zi = (zi1 , ..., ziK ) ∈ IR K
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L in each state s
a commodity bundle xis = (xis1 , ..., xisL ) ∈ IR+
max
X
πis ui (xis1 , ..., xisL )
s
s.t.
X
qk zik
= 0
pls xils
=
k
∀s,
X
l
X
l
pls ωils + p1s
X
rsk zik
k
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Typically, zk ≥ 0 for some k and zk ≤ 0 for some other k
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no endowment of asset (this assumption for convenience only)
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usual endowment ωils of physical good l in state s (received at
the beginning of t = 1)
Radner Equilibrium
LSI and prices (q, p) ∈ IR K +LS such
An allocation (z, x) ∈ IR KI × IR+
+
that:
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Individual optimality: ∀i, (zi , xi ) solves the max problem on
the previous slide
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Market clearing:
∀k,
X
zik
= 0
xils
=
i
∀l, ∀s,
X
i
X
i
ωils
Comments:
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ps : spot prices in state s
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w.l.o.g. ∀s, ps1 = 1
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equilibrium of plans, prices and price expectations (portfolio
and consumption plans, the definition includes an assumption
about the ”price expectations”: see the previous slide on
”rational expectations”)
Fundamental property 1 of the equilibrium asset prices
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At equilibrium, q is arbitrage-free
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The converse implication is not true: an arbitrage-free price
vector q is not always an equilibrium price vector
Proof
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An arbitrage portfolio is zi∗ such that q.zi∗ ≤ 0,
X
∀s,
rsk zki∗ ≥ 0 with strict inequality for (at least) one s
k
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If there is an arbitrage portfolio, then no agent i has an
optimal decision:
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consider a decision (xi , zi )
zi + zi∗ gives a larger wealth in every state s
0
0
there is xi0 such that ui (xis1
, ..., xisL
) > ui (xis1 , ..., xisL ) in every
s (given that ui is increasing)
loosely speaking, i wants to buy the portfolio λzi∗ with
λ = +∞ in order to get an infinite wealth in every state s (so
that no market clearing is possible - assets or goods -)
Fundamental property 2 of the equilibrium asset prices
At equilibrium, there are (µ1 , ..., µS ) ≥ 0 such that, for every asset
with returns rk
X
qk =
µs rsk
s
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µs is the price of the Arrow asset s
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Proof: Farkas’ lemma (derived from a separating hyperplane
theorem, see Mas-Colell et alii)
Proof
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consider the space IR K
z ∈ IR K /q.z < 0 is a convex set
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the
intersection
P of the Sconvex sets (one for each asset)
K
z ∈ IR / s rsk zk ≥ 0 is a convex set
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the 2 sets do not intersect (the asset prices are arbitrage-free)
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there is (µ1 , ..., µS ) ≥ 0 such that



q1
X
 .. 

µs 
 . =
s
qK

rs1
.. 
. 
rsK
The fundamental characteristic of the market structure
It is either complete or incomplete
Definition: an asset structure (K assets, associated with a S × K
return matrix R = (rsk )s,k ) is complete iff the rank of R is S.
Comments:
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S is the maximal possible rank for a S × K matrix
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this means that there are S ”linearly independent” assets
(example: the S Arrow assets)
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S assets are enough to get a complete market structure (but
the markets may be incomplete even if K > S)
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With S linearly independent assets, further assets are
redundant: their rk is a linear combination of the rk of the S
first assets
Interpretation: all the transfers of wealth across states are feasible
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Budget constraint ”over the S states” (this is the BC of the
associated Arrow Debreu economy)
X
X
pls xils =
pls ωils
l,s
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l,s
define a ”transfer of wealth to state s”
X
X
∀s, ts =
pls xils −
pls ωils
l
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l
question is: are these transfers feasible through an appropriate
portfolio zi , that is: is there zi such that
X
∀s, p1s
rsk zik = ts
k
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if the return matrix R has rank S, then the answer is yes
the remaining question is: does zi satisfy q.zi ≤ 0? (see next
slides)
A formal statement of the above interpretation
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This statement = 2 converse implications (stating that
Radner equilibrium = Arrow Debreu equilibrium)
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an Arrow Debreu economy with no asset and LS contingent
goods can be ”associated with” every economy with complete
asset markets and sequential trades,
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conversely, with an Arrow Debreu economy, we can associate
a market structure (typically the S Arrow assets) and consider
the economy with sequential trade where assets are traded at
t = 0 and the L physical goods are traded at t = 1 (once the
state s is revealed)
Implication 1: Consider an economy with complete asset markets.
LSI × IR KI × IR LS × IR K is a Radner equilibrium
If (x ∗ , z ∗ , p, q) ∈ IR+
+
+
(normalization condition: ∀s, p1s = 1), then there is
S such that (x ∗ , µ p , ..., µ p ) ∈ IR LSI × IR LS is
(µ1 , ..., µS ) ∈ IR+
1 1
S S
+
+
S is the vector of prices of
an Arrow Debreu equilibrium (ps ∈ IR+
contingent goods ls, µs is ”the value of state s”, that is the price
of the Arrow asset s)
Fundamental consequence of Implication 1: the 2 welfare theorems
apply
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when asset markets are complete, the allocation x ∗ of a
Radner equilibrium is PO
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a PO allocation x ∗ is the allocation of a Radner equilibrium
(with appropriate wealth transfers)
LSI × IR LS is an Arrow Debreu
Implication 2: If (x ∗ , p) ∈ IR+
+
equilibrium, then there is a market structure with K assets (of
returns rk ) and there are portfolios z ∗ ∈ IR KI and asset prices
K such that (x ∗ , z ∗ , p, q) is a Radner equilibrium
q ∈ IR+
Fundamental consequence of Implication 2: the existence theorem
applies
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when the asset markets are complete, there is a Radner
equilibrium (given concavity of the ui )
Proofs
R = 1)
Consider the 2 budget sets (p1s
LS /∃z ∈ IR K /q.z ≤ 0
xiP
∈ IR+
i P
i P
BR =
and ∀s, l plsR xils ≤ l plsR ωils + k rsk zik




X
X
LS
BAD =
xi ∈ IR+
/
plsAD xils ≤
plsAD ωils


l,s
l,s
We show that, for properly chosen p AD and p R , these 2 sets
coincide
Inclusion BR ⊂ BAD
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consider the µs such that qk =
I
for xi ∈ BR
P
s
µs rsk
!
X
µs
s
X
plsR xils
!
≤
µs plsR
xils
≤
l,s
I
X
plsR ωils +
l
µs plsR
ωils +
l,s
that is: xi ∈ BAD
X X
k
X
l,s
rsk zik
!
hence, with plsAD = µs plsR ,
plsAD xils ≤
X
k
l,s
X
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µs
X
s
l
X
X
plsAD ωils
s
µs rsk
zik
Inclusion BAD ⊂ BR
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consider the S Arrow assets (complete asset markets)
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AD and p R = p AD /p AD (notice p R = 1)
define qs = p1s
1s
1s
ls
ls
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for xi ∈ BAD , define
∀s, zis =
X
AD
p1s
l
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check that q.zi ≤ 0
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and check that
∀s,
X
l
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that is: xi ∈ BR
!
1
plsR xils =
plsAD xils
−
X
plsAD ωils
l
X
l
plsR ωils +
X
k
rsk zik
About redundant assets
If x ∗ is the allocation of a Radner equilibrium with an asset
structure associated with a return matrix R, then x ∗ is the
allocation of a Radner equilibrium with any other asset structure
associated with a return matrix R 0 such that rangeR 0 = rangeR
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range of a (return) S × K matrix: rangeR = v ∈ IR S /v = Rz, z ∈ IR K
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redundant assets can be deleted without changing the
allocation of Radner equilibrium
About incomplete markets
When markets are incomplete
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there can be no equilibrium
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equilibrium can be suboptimal
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equilibrium is sometimes not even ”constrained optimal”
(constrained optimality: Pareto optimality among the
allocations x that are feasible given the asset structure)
The end of the chapter