数理解析研究所講究録 1264 巻 2002 年 83-90

83
Generic Inefficiency of Equilibria with Incomplete Markets
Ryo Nagata

School of Political Science and Economics
Waseda University
In the incomplete markets model with real assets and one good we show Pareto inefficiency ofequilibrium allocations generically in all agents’ utility functions and endowments. All we assume
on their utility ffinctions is monotonicity. No convexity is needed on them. We use Thorn
Transversality Theorem in 1-jet space to generically characterize the set of allocations fulfilling

first order necessary conditions for Pareto optimum. We also show that it is not utility
maximization behavior of each agent but his budget constraint thm yields inefficiency ofequilibrium allocations with incomplete markets.

1. Introduction
It is well known that with incomplete asset markets, the equilibrium allocations are generally
Pareto inefficient. “For along time the Pareto inefficiency of equilibrium allocations whenmarkets are incomplete was regarded as aFolk Theorem. The intuition was simply that barring
unusual initial circumstances, such as starting at aPareto optimum, with an insufficient number ofmarkets to trade on, agents’ marginal rates of substitution will not be equalized by trade. Formal
proofs are however recent ... and establishing the result requires techniques from differential
for studying qualitative properties of
topology. ...these techniques were introduced in thethey have turned out to be crucial for studying thethe standard general equilibrium model
model with incomplete markets since many of its properties me only generic.” (Magill M. and
M.Quinzii Theory of Incomplete Markets Volume 1’ p. 136. 1996.)
In the above mentioned book, Magill and Quinzii show that in one good -two period exchange
economy model the equilibrium allocation with incomplete markets is generically Pareto
inefficient with respect to the agents’ initial endowments. Their claim is, however, restrictive in
two points. One is that they assume strict quasi-concavity on an agent’s utility function. The otheris that they demonstrate inequality of the present-valued (normalized marginal rates ofsubstitution)vectors of all agents, which is too demanding for the proof of invalidity of Pareto
optimum.
We also consider one good-two period exchange economy model like Magill and Quinzii’s. But
differently from their approach we take not only agents’ endowments but also their utility$1970\mathrm{s}$–‘functions as economy parameters. Moreover, we require only monotonicity to the agents’ utility
functions. In the analysis first order necessary conditions for Pareto optimum play acrucial role intwofold sense. First of all, given the utility function space, the particular form of the conditions
enables us to use Thorn Transversality Theorem (product ffinctional form)in 1-jet spac

84
Secondly to show invalidity of the conditions implies impossibility theorem of
optimum.
By using transversality theorem recursively (this is the approach developed in Nagata (1998,$\mathrm{P}\mathrm{m}\mathrm{o}$2000), we obtain the result that for almost all agents’ monotone utility functions and endowments,

the equilibrium allocations with incomplete markets are Pareto inefficient In the course ofinvestigation, we also obtain as aby-product the &t about the cause of inefficiency of theequilibrium allocations. That is, it is not optimization behavior of each agent but his budget
constraint on assets that yields inefficiency of equilibrium allocations with incomplete markets.

2. The Model
The model to be considered here is the simplest two period exchange economy under
uncertainty. First and second period is each specified by
and 1and one of states of nature$\mathrm{E}$, S) occurs$\mathrm{S}$date 1. For simplicity, we call date, state $$o that in total there are$+1$ states. The economy consists of Iconsumers(, ,and asingle consumption good.
Thus the commodity space is.
The characteristics of each agent consist of three ingredients, that is, his consumption set ,his utility function and his initial endowment. We make assumptions on those ingredients asfollows. For each,
Assumption 1. X. is.
Assumption 2.
satisfies(1)
for sufficiently lrge
(2)
each
.
Assumption 3.
.
For simplicity , let ( , , u) and, , ) in the following.
There are real assets, , J) in the economy. Our interest is in the case of incomplete
asset markets so that. Each asset can be purchased for the pricedate 0and delivers areturnacross the states at date 1where each indicates ace1uin amount of good.
We see each
, , J) as acolumn vector and combine than to form
matrix of rehons. Ld ( , co ;v) denote the economy composed of 4 and .
We investigate inefficiency of equilibria with incomplete mmkets fiom a generic viewpoint withrespect to and. Thus the asset structure is fixed on which we may assume
rank
without loss of generality.
Since andare only pmmneters that specify the economy we consider the space thg consistsof permissible and. Let be the set of functions satisfying Assumption 2and let %be
I-product of , that is, U. Then the space of permissible andis %iscalled the economy space. To
standmd Euclidean topology is given whereas
is endowed with the Whitney $C$ topology so thatis considered tobe aproduct topological space.
Given the asset structure , each agent has achance to purchase some amounts of assets andadjust his income stream so that he can optimize his intertemporal consumptions. Let
denote the number of units of the assets purchased by agent . is called aportfolio of($\Leftrightarrow 1$,$\ldots$$\mathrm{E}$$\alpha$$\mathrm{i}=1$$\mathrm{H}$$\mathrm{S}$$\mathrm{D}$$\ldots$$\mathrm{R}^{\mathrm{s}+1}$$\mathrm{i}$$\mathrm{x}.$$\varpi \mathrm{i}$$\mathrm{u}^{i}$$(\mathrm{i}=1, \ldots, 1)$$\mathrm{i}$$\mathrm{R}^{\mathrm{s}+\iota}++$$\mathrm{u}^{\mathrm{i}}$$\mathrm{u}^{\mathrm{i}}\in C(\mathrm{R}^{*+1}++, \mathrm{R})$$\mathrm{r}>0$$\in \mathrm{R}^{\mathrm{s}+1}++$$\mathrm{D}\mathrm{u}^{\mathrm{i}}(\mathrm{x})\in \mathrm{R}^{\mathrm{s}+\iota}++\mathrm{f}\mathrm{o}\mathrm{r}$$\mathrm{x}$$\omega i$$\in \mathrm{R}^{\mathrm{s}+1}++$$\omega\leq\omega 1$$\mathrm{u}^{1}$$\mathrm{u}=$$\ldots$$0^{=1}.$$\mathrm{J}$$\Phi$$\ldots$$\ldots$$\mathrm{J}<\mathrm{S}$$\mathrm{j}$$\mathrm{v}^{1}=(\mathrm{v}^{\mathrm{i}}1,\ldots,\mathrm{v}^{1}\mathrm{s})$$\mathrm{v}^{1}$.

$\mathrm{q}\alpha$$j\mathfrak{h}.=1$$\mathrm{S}\cross \mathrm{J}$$\ldots$$\mathrm{v}=[\mathrm{v}^{1},\ldots,\mathrm{v}^{\mathrm{J}}]$$\mathscr{C}$$\omega$$\mathrm{u}$$\omega$$\mathrm{u}$$\mathrm{u}$$\mathrm{v}$$\mathrm{t}\mathrm{h}\mathrm{g}$$\mathrm{v}$$\mathrm{v}=\mathrm{J}$$\omega$$\mathrm{u}$$\mathrm{U}$$\omega$$\mathrm{U}$$\mathrm{u}$$\mathrm{x}$$\omega$$\mathrm{R}^{(l+11}++\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{h}$$\mathrm{R}^{(*+}1’++\mathrm{a}$$C(\mathrm{R}^{\mathrm{s}+1}++, \mathrm{R})$$C(\mathrm{R}^{\mathrm{s}+1}++, \mathrm{R})^{1}$$\mathrm{J}$$\mathrm{v}$$\mathrm{z}^{1}=(\mathrm{z}^{l}1,\ldots,\mathrm{z}_{\mathrm{J}}^{i})$$\in \mathrm{R}’$$\mathrm{J}$$\mathrm{z}^{\mathrm{i}}$$\mathrm{i}$85agent . Then the problem he has to solve is as follows.
$\mathrm{i}$${\rm Max}_{\mathrm{n}.\mathrm{a}}$$\mathrm{s}.\mathrm{t}$.

$\mathrm{u}^{\mathrm{i}}(\mathrm{x}^{1})$,i$\mathrm{I}^{\dot{1}}1=\omega$........... $(*)$$\mathrm{z}^{\mathrm{i}}\in \mathrm{R}^{\mathrm{J}}$$\mathrm{x}_{0}=\omega \mathrm{o}\mathrm{i}\mathrm{i}-\mathrm{q}\mathrm{z}\mathrm{i}$$+\mathrm{v}\mathrm{z}^{\mathrm{i}}$lSrespectively. Note th# there is only one
l denote
where andand ( il, , i)good in the economy so that the price of the good is normalized to be one (the good at date 0is
interpreted as anumeraire).
.
Now we define the equilibrium for the economyi$\mathrm{I}^{\mathrm{i}}1$$(\chi_{1}^{\mathrm{i}}, \ldots, \chi_{S}^{\mathrm{i}})$$\emptyset$$\omega$$\omega$$\ldots$$\mathscr{C}(\mathrm{u},\omega ; \mathrm{v})$Definition 1. An asset market equilibrium for ( ,co ;v) is atuple (, , I.
is asolution of the problem ($).
(i)
i-l on$\mathscr{S}$$(\mathrm{I}^{\mathrm{i}}, \mathrm{z}^{\mathrm{i}}\rangle., \mathrm{q})$$\mathrm{u}$$(\mathrm{i})$such thm$\mathrm{i}=1$$(\mathrm{x}^{\mathrm{i}}, \mathrm{z}^{\mathrm{i}})$$\ldots$$\mathrm{i}$$\Sigma_{\mathrm{i}-1}\mathrm{x}^{\mathrm{i}}=\mathrm{I}$$\Sigma_{i-1}\mathrm{z}^{\mathrm{i}}=0$$(\ddot{\mathrm{m}})$Next we consider efficiency of allocations. Givenand$\mathrm{u}$$\omega$, aPareto optimal allocation isdefined as follows.
Definition 2. An allocation

is aPareto optimum if$\mathrm{x}=(\mathrm{z}^{1}-, \ldots, -\mathrm{I}^{1})\in \mathrm{R}^{(\mathrm{s}+\prime \mathrm{p}}++$$\mathrm{i}$$\Sigma_{\mathrm{i}\cdot\prime}\mathrm{x}^{\mathrm{i}}=-\Sigma_{\mathrm{i}-\prime}\omega$$(\mathrm{i})$$(\dot{\mathrm{n}})$,$\mathrm{u}^{\mathrm{i}}(\mathrm{x}^{\mathrm{i}})-$

there does not exist$\mathrm{i}=1$,$\mathrm{x}=(\mathrm{z}^{1}, \ldots, \mathrm{x}^{1})\in \mathrm{R}^{\mathrm{t}\+1\rangle}++\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}$

that

t-l

$\Phi 1$and$\mathrm{u}^{\mathrm{i}}(\mathrm{x}^{\mathrm{i}})\geqq$, Iwith astrict inequality for at least one .
$\mathrm{i}$$\ldots$

Lastly we consider aparticular feasibility for the economy

Definition 3. An allocation$(\mathrm{i})$$\Sigma_{\mathrm{i}-1}\mathrm{x}^{\mathrm{i}}=\Sigma$$\mathrm{x}=(\mathrm{I}^{1},\ldots,\mathrm{I}^{1})\in \mathrm{R}^{\mathrm{t}+1\mathrm{X}}++\mathrm{i}\mathrm{s}\mathrm{v}$$\mathscr{S}$( , co ; ).
$\mathrm{v}$$\mathrm{u}$-feasible if$\Sigma_{i-1\mathrm{I}^{\mathrm{i}}}=\Sigma_{\mathrm{i}-1}\omega \mathrm{i}$(ii)$\mathrm{I}^{\mathrm{i}}1\in$(v)$+\omega$il,$\mathrm{i}$

$=1$

,

$\ldots$,I

indicates a vector subspace spanned by the columns of .
denote the set of -feasible allocations with respect to . If atuple
an asset market equilibrium for ( , ; v) , then obviously ( . is an element ofwill play avery critical role in the analysis.
where
Let ,
$\mathrm{F}$$\langle \mathrm{v}\rangle$$\mathrm{v}$$(\omega)$$((\mathrm{x}^{\mathrm{i}}, \mathrm{z})$$\omega$$\mathrm{v}$$\mathscr{C}$$\mathrm{u}$$\omega$$\mathrm{x}^{\mathrm{i}}\rangle$$\mathrm{F}\not\in\omega$,).

$\mathrm{q})$is$\mathrm{F},(\mathrm{c}\mathrm{o} )$3. Main Result
Our analysis proceeds according to the following scenario. First we consider the set ofallocations which satisfy the first order necessary condition for aPareto optimum. To characterize
the set we can use Thorn Transversality Theorem in 1-jet space and obtain the result that the set.
in . Then we turn to ,constitutes adefinite dimensional manifold inevery
It can be easily seen that. To examine
is amanifold intransversality between
and the set previously mentioned we can also use atransversality
the intersection of those setstheorem (another form)again and conclude that for almost allis empty, which implies the consequence to be shown.
Now let us consider the first order necessary conditions of Pareto optimal allocations.
$\mathrm{F}$$\mathrm{R}^{(\mathrm{s}+\iota \mathrm{X}}++\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}$$\mathrm{u}$$\mathrm{R}^{(\mathrm{s}+1\mathrm{X}}++\mathrm{f}\mathrm{o}\mathrm{r}$$\omega$$\mathrm{p}_{\mathrm{v}}(\omega)$$\mathrm{p}_{\mathrm{v}}(\omega)$$\omega$$(\omega)$

86
Proposition 1. Given andfulfilling assumption 2and 3and let an allocation
Pareto optimal. Then the following equation holds.
$\omega$$\mathrm{u}$$\overline{\mathrm{x}}=(\mathrm{x}^{1}-, \ldots, -\mathrm{z}^{1})\in$$\mathrm{R}^{\mathrm{t}\+\iota\mu}++\mathrm{k}$$\mathrm{D}\mathrm{u}^{1}(\mathrm{x}^{1}\mathrm{y}\Sigma---\triangleright 1\mathrm{D}\mathrm{u}^{1}\langle \mathrm{I}^{1})=\ldots=\mathrm{D}\mathrm{u}^{1}(\mathrm{I}^{1})/\Sigma-1\mathrm{D}\mathrm{u}^{1}\langle_{\mathrm{I}^{1}}^{-})$

Since a Pareto optimal allocation can be characterized as asolution of the next
maximization problem,.

$\mathrm{P}\mathrm{r}\infty \mathrm{f}$${\rm Max} \mathrm{u}^{1}(\mathrm{z}^{1})$$\mathrm{s}.\mathrm{t}$.

$\mathrm{u}^{\mathrm{i}}(\mathrm{I}^{1})\geqq\overline{\mathrm{u}}^{i}$$\mathrm{i}=2$,$\ldots$,I$i$$\Sigma_{i-1}\mathrm{x}^{\mathrm{i}}=\Sigma_{\mathrm{i}- 1}\varpi$

the consequence is easily deduced by Lagrange multiplier method.
Q.E.D.
Let the set of allocations which satisfy the equation in the proposition be. Note that anelement of
is not required to fulfill feaslbility (i.e. condition (i)of dffinition 2).
We are going to apply Thorn Transversality Theorem to ch
generically in . To
this end, some preparations are needed.
Since the structure of
requires us to consider all agents’ utility functions simultaneously, weneed aproduct functional version of original Thorn Transversality TheoremLet
be product ofandbe product of -jet space
where
and
differential manifolds, -product function of fi
and its r-jet) are defined as follows.
extension
X...
fi
by fi X... fi( X...
))by ( X...
$\mathrm{A}(\mathrm{u})$$\mathrm{A}(\mathrm{u})$$\ovalbox{\tt\small REJECT}$.

$I(\mathrm{N}\mathrm{N})^{\mathrm{r}}$$\mathrm{k}$$/(\mathrm{N},\mathrm{M})$$\mathrm{r}$$\cross\ldots$$\mathrm{k}$$\mathrm{x}\mathrm{g}$$\in C(\mathrm{N}\mathcal{M})^{\mathrm{k}}$$.\cdot \mathrm{N}^{*}arrow \mathrm{M}$$\mathrm{x}\mathrm{g}$$\cross \mathrm{f}\mathrm{i}$$.\cdot \mathrm{N}$$\cross$$arrow I(\mathrm{N},\mathrm{M})^{\mathrm{k}}$$(\mathrm{x}1, \ldots, \mathrm{r}\mathrm{H} \mathrm{f}\mathrm{i}(\mathrm{x}\mathrm{i}), \ldots, \propto \mathrm{n})$$\mathrm{f}\mathrm{i}$$\cross \mathrm{f}\triangleright$$\mathrm{j}\mathrm{r}\mathrm{k}$$)$$(\mathrm{x}\mathrm{i}, \ldots, \mathrm{n}\mathrm{H} \mathrm{j}^{\mathrm{r}}\mathrm{f}_{1}(\mathrm{x}\mathrm{i}), \ldots, \mathrm{j}^{\mathrm{r}}\propto \mathrm{x}\iota))$

Theorem 1. Thorn Transversality Theorem (product fimcriond form)
Let be asubmanifold of. Then{ fi) is transversal with respect to}$C$
is aresidual subset ofin the Whitney product topology. So
is dense in
Proof. See Theorem 7.3. of Nagata (2001).
Since our product function space is notitself but asubset (i.e. %\subset $C$(
we need to check if the above theorem is applicable to the subset.
$\Gamma(\mathrm{N},\mathrm{M})^{\mathrm{k}}$$\mathrm{Q}$$\mathscr{T}=$$\in C(\mathrm{N}\mathcal{M})^{\mathrm{k}}|$$\cross\ldots\cross \mathrm{f}\mathrm{i}$$\mathrm{j}^{\mathrm{r}}4\mathrm{f}_{1}\cross\ldots\cross \mathrm{f}\mathrm{i}$$\mathrm{Q}$$C(\mathrm{N}\mathcal{M})^{1}$$\mathscr{T}$

Proposition 2. %is an open subset of$C(\mathrm{R}^{\mathrm{s}+\iota}++, \mathrm{R})^{1}$$C(\mathrm{R}^{\mathrm{s}+\prime}++, \mathrm{R})$$\mathrm{U}$))$\mathrm{R}$in the Whitney$\mathrm{C}^{1}$$\{ (l\mathrm{b}, \mathrm{c}1, \ldots, \mathrm{C}\mathrm{s} +1)\in \mathrm{J}^{1}(\mathrm{R}^{\mathrm{s}+1}++, \mathrm{R})|\alpha >0, \mathrm{i}=1, \ldots, \mathrm{S}+1\}$

Then obviously$\mathrm{U}=$$\{\mathrm{f}\in C(\mathrm{R}^{\mathrm{s}+1}++, \mathrm{R})|\mathrm{J}^{1}\mathrm{f}(\mathrm{R}^{\mathrm{s}+1}++)\subset \mathrm{V}\}$

substantially equal to$\mathrm{t}\mathrm{h}\mathrm{A}$,.

Proof. It is sufficient to show th#is an open subset of
topology. Consider the following set V.
$\mathrm{V}=$$C(\mathrm{N},\mathrm{M})^{l}$$\mathrm{R}^{\mathrm{s}+1}++$$C(\mathrm{N}\mathrm{N})^{\mathrm{r}}$implies$\mathrm{N}$$\cross \mathrm{f}\mathrm{i}$$\mathrm{j}^{\mathrm{r}}4\mathrm{f}1$$\mathrm{f}_{1}$$\mathrm{k}$$\mathrm{C}(\mathrm{N}’ \mathrm{M})$$\mathrm{M}\pi \mathrm{e}$$\mathrm{j}^{\mathrm{r}}*$$\mathrm{u}$$\mathrm{A}(\mathrm{u})$$C(\mathrm{N}\mathrm{J}\emptyset$$\cross\ldots$$\mathrm{A}(\mathrm{u})$$\mathrm{U}$$\mathrm{R}^{\mathrm{s}+1}++\cross \mathrm{R}\cross \mathrm{R}^{\mathrm{s}+1}$

is an open subset of, it is obvious that

$C(\mathrm{R}^{\mathrm{s}+1}++, \mathrm{R})$$\mathrm{V}$. Noting thatis open in$\mathrm{J}^{1}(\mathrm{R}^{\mathrm{S}+1}++, \mathrm{R})$is$\mathrm{J}^{1}(\mathrm{R}^{\mathrm{S}+1}++, \mathrm{R}),\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{h}$.

Q.E.D.

In the sequel we are solely concerned with 1-jet space

identified with. Let
which is obviously an open submanifold of$\mathrm{R}^{\mathrm{s}+1}++\cross \mathrm{R}\cross \mathrm{R}^{\mathrm{s}+\iota}$$\mathrm{J}^{1}(\mathrm{R}^{\mathrm{S}+1}++, \mathrm{R})$$\mathrm{J}^{1}+(\mathrm{R}^{\mathrm{s}+1}++, \mathrm{R})$$\mathrm{J}^{1}(\mathrm{R}^{l+1}++, \mathrm{R})$denotewhich is naturally$\mathrm{R}^{\mathrm{s}+1}++\cross \mathrm{R}\cross \mathrm{R}^{\mathrm{s}+\mathrm{I}}++$. Then we define the map

87$arrow \mathrm{R}^{\mathrm{s}+1}++\cross \mathrm{R}\cross\Delta \mathrm{s}$$\emptyset:\mathrm{J}^{1}+(\mathrm{R}^{\mathrm{S}+1}++, \mathrm{R})$by$(\mathrm{a}, \mathrm{b}, \mathrm{c}1, \ldots, \mathrm{c}_{\mathrm{S}+1})=$$\emptyset$( ,$\mathrm{a}$$\mathrm{b}$,$\mathrm{c}_{1}/\Sigma$,

i-l$\mathrm{G}$$\ldots$,$\mathrm{C}\mathrm{s}$$+$$+\iota$$/\Sigma$

i-l

positive -simplex. I-product function of $(denoted by$\+)^{1})$ is defined in the same way as fi, i.e.
Considering Theorem 1, we obtain the following proposition.

where

)

$\mathrm{G}$$\Delta^{\mathrm{s}}+\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{s}$strictly$\Phi:\mathrm{J}^{1}+(\mathrm{R}^{S+1}++, \mathrm{R})^{1}arrow(\mathrm{R}^{\mathrm{s}+1}++\mathrm{X}$$\mathrm{S}$$\Phi(\mathrm{y}^{1}, \ldots, \mathrm{y}^{1})=(\emptyset (\mathrm{y}^{1}), \ldots, \phi (\mathrm{y}^{\mathrm{I}}))$$\cross\ldots\cross \mathrm{f}_{\mathrm{k}}$$\cross\Delta$$\mathrm{R}$.

. Then
Proposition 3. Let be asubmanifold of}is transversal with respect to$!=$ {u\in %is dense in %.
Proof. See appendix.
.
Then we have aconsequence regarding$(\mathrm{R}^{\mathrm{s}+1}++\cross \mathrm{R}\cross\Delta +)^{\mathrm{I}}$$\mathrm{W}$$|\Phi$$\cdot$$\mathrm{W}$$\mathrm{j}_{1}^{1}\mathrm{u}$$\mathrm{A}(\mathrm{u})$Proposition 4There exists adense subset %*0f %such that for any.
dimensional submanifold in$\mathrm{u}$in %*,$\mathrm{A}(\mathrm{u})$constitutes a$\mathrm{R}^{(3+11}++$$\mathrm{S}+\mathrm{I}$. It is obvious that
Proof. Let be the set. From proposition
dimensional submanifold of
is a
with
is
2there exists adense subset %*0f %such that for any in %*,dimensional submanifold ofconstitutes arespect to W. Thus for those , (!), the proof is completed.
. Since (!)$\{(\mathrm{a}^{\mathrm{i}}, \mathrm{b}^{\mathrm{i}}, \mathrm{c})\mathrm{i} \in(\mathrm{R}^{\mathrm{s}+1}++\cross \mathrm{R}\cross\Delta \mathrm{s} +)^{\mathrm{I}}|\mathrm{c}^{1}=\mathrm{c}^{2}=\ldots-arrow^{1}\}$$\mathrm{W}$$(\mathrm{R}^{\mathrm{s}+\iota}++\cross \mathrm{R}\cross\Delta \mathrm{s} +)^{1}$$(\mathrm{S}+1)\mathrm{I}+\mathrm{I}+\mathrm{S}$$\mathrm{W}$$\Phi\cdot$.

$\mathrm{R}^{(S+1\mu}++$.

$\mathrm{u}$$\mathrm{t}\mathrm{m}\mathrm{s}\mathrm{v}\mathrm{e}\mathrm{r}\ovalbox{\tt\small REJECT}$$\mathrm{j}_{1}^{1}\mathrm{u}$$\mathrm{u}$$\mathrm{S}+\mathrm{I}$$\mathrm{j}^{\iota_{1\mathrm{u}}-1}(\mathrm{W})$$\mathrm{j}^{\iota_{1\mathrm{u}}-1}(\mathrm{W})=\mathrm{A}(\mathrm{u})$Q.E.D.

. Aparticular property ofassumption on the number of agents, assets and states.
Now we turn toAssumption 4.

$(\mathrm{J}$$\mathrm{F}_{\mathrm{v}}(\omega)$\dagger 1 $)\mathrm{I}>\mathrm{S}+1$ .

Proposition 5. Under assumption 4,
eachsubmanifold of$\mathrm{R}^{(\mathrm{s}+1\mu}++\mathrm{f}\mathrm{o}\mathrm{r}$Proof. Let$\mathrm{X}$is obtainable with an additional$\mathrm{p}_{\mathrm{v}}(\omega)$be the set { ( ,$\mathrm{x}\mathrm{o}$$\mathrm{x}\mathrm{i}$,$\mathrm{p}_{\mathrm{v}}(\omega)$

constitutes a $(\mathrm{J}+1)\mathrm{I}-(\mathrm{S}+1)$ dimensional$\omega\in \mathrm{R}^{\mathrm{t}\S+1)1}++$$\ldots$,$\mathrm{X}\mathrm{s})\in \mathrm{R}^{(S+1}’++|(\mathrm{x}1$,$\ldots$,$\mathrm{X}\mathrm{s}$$)\in$$\langle$$\mathrm{v}\rangle+\omega$rankdimensional submanifold of’. Sincebymap
is surjective at eachby assumption 4, Dg,
isthat is asubmersion. Since 0is aregular value ofeasily seen that$\mathrm{g}:\Pi X$$\mathrm{X}^{\mathrm{i}}$is a$\mathrm{R}^{(\mathrm{s}+1)}++\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{e}$$\mathrm{J}+1$$arrow \mathrm{R}^{(\+1)}$$(\mathrm{x}^{1}, \ldots, \mathrm{x}^{1})=\Sigma$il} . It can be. Define a$\mathrm{v}=\mathrm{J}$$\dim\Pi \mathrm{X}$ $=(\mathrm{J}+1)\mathrm{I}>$$\mathrm{i}-^{\mathrm{i}}-\Sigma_{\mathrm{I}}\omega$$\mathrm{g}$$\dim \mathrm{R}^{(S+1)}=\mathrm{S}+1$$(\mathrm{I}^{1}, \ldots, \mathrm{I}’)\in\Pi \mathrm{i}\mathrm{X}$$(\mathrm{I}^{1}, \ldots, \mathrm{x}^{1})$$\mathrm{g}$$\mathrm{g}-1(0)=\mathrm{F}$$(\omega)$so$\mathrm{a}(\mathrm{J}+1)\mathrm{I}-(\mathrm{S}+$$\mathrm{g}$1)dimensional submanifold of$\mathrm{R}^{(\mathrm{s}+\iota \mathrm{I}}++$.

Q.E.D.

for. To this end, we fix anandFinally we investigate transversality between
generically in . We take someis transversal toany given of % * and check if
fix it. Then define the map $:out ofdevice to do so. First pick an arbitrarywhere$$\mathrm{A}(\mathrm{u})$$\mathrm{u}$$\mathrm{A}(\mathrm{u})$$\mathrm{F}_{\vee}(\omega)$$\mathrm{A}(\mathrm{u})$$\mathrm{p}_{\mathrm{v}}(\omega)$$\omega$$\mathrm{R}^{(\mathrm{s}+1\mathrm{X}}++\mathrm{a}\mathrm{n}\mathrm{d}$$\omega$$\mathrm{p}_{\mathrm{v}}(\omega)$$\cross \mathrm{N}_{1}+(\omega)arrow \mathrm{R}^{(\mathrm{s}+1)\mathrm{I}}++\mathrm{b}\mathrm{y}$$-\omega||<1\}$$(\mathrm{x}, \mathrm{y})=\mathrm{x}+\mathrm{y}$$\mathrm{N}_{1}+(\omega\succ^{-}\{\mathrm{y}\in \mathrm{R}^{(S+\iota\mu}++|||\mathrm{y}$88Proposition 6. For almost all, $($\mathrm{y}\in \mathrm{N}_{1}+(\omega)$$\cdot$, y): F.

is transversal with$(\omega)arrow \mathrm{R}^{(\mathrm{S}+\iota 1}++$.
Proof. It is obvious from the structure of $thm $is asubmersion. Thus $is transversal withrespect to any submanifold of, especially. Here we apply the transversality
theorem (Guillemin and Polladc (1974)p.68)to 0 md obtain the result.
respect to$\mathrm{A}(\mathrm{u})$$\mathrm{R}^{\mathrm{O}\star 1}’++$$\mathrm{A}(\mathrm{u})$Q.E.D.
Proposition 7. For each
Proof. Since F.
is the set of$\mathrm{y}\in \mathrm{N}_{1}+(\omega)$$(\omega)$$\langle \mathrm{v}\rangle+\Phi$$(\dot{\mathrm{n}})\mathrm{I}^{\mathrm{i}}|\in$F.

$(\omega+\mathrm{y})$il,$\mathrm{i}$$=1$, $(F.

$(\omega),\mathrm{y}\mathrm{F}$F.

.

$(\omega+\mathrm{y})$smisfics, 1, it is easily seen thm the image of$\mathrm{R}(\mathrm{I}^{1},\ldots’)\in \mathrm{R}^{\alpha+\iota t}++\mathrm{U}]\#$,$\ldots$$(\mathrm{i})$$i$$\Sigma..\iota \mathrm{I}^{}=\Sigma_{\mathrm{i}-1}\Phi$,$(\omega)$$\mathrm{F}$,by $is.
Q.E.D.

Now we are in aposition to state our main claim.

Theorem. Under assumptions 1\sim 4, for almost all and , each equilibrium allocation of theeconomy $P$ ( , ; v) is Pareto indficimt.
Proof Let be an element of 9 . For any given, define. Let beelement ofsuch that $ ,is transversal with respect to. Note that fiom
proposition 6such an is an element of adense set. Now supposeF.
rl. Since $ ,is transversal with respect to, it follows fiom proposition 7that
any,’where indicates atangent space,. But considering proposition 4, proposition 5and assumption 4,$\omega$$\mathrm{u}$$\mathrm{u}$$\omega$$\omega\in \mathrm{R}^{l+1}$$\iota$$\mathrm{u}$$\mathrm{N}_{1}+(\omega)$$(\cdot$$\mathrm{g}$$\mathrm{m}\mathrm{d}$$\mathrm{x}$$\Leftarrow\phi$.

$\in \mathrm{F}$$(\cdot$$(\cdot$$\mathrm{N}_{1}+(\omega)$$++$$\mathrm{y}$$\mathrm{y})$$\mathrm{m}$$\mathrm{A}(\mathrm{u})$$\mathrm{o}\mathrm{f}\mathrm{N}_{1}+(\omega)$$\mathrm{y}$$\mathrm{A}(\mathrm{u})\neq\emptyset$.

$\mathrm{y})$$\mathrm{h}\alpha$$(\omega+\mathrm{y})$$\mathrm{A}(\mathrm{u})$$(\omega+\mathrm{y})\cap \mathrm{A}(\mathrm{u})$$\mathrm{T}*$$\mathrm{F}.(\omega)+\mathrm{T}A(\mathrm{u})=\mathrm{R}^{\sigma+1}$$\mathrm{T}$$\mathrm{y})^{-1}(\mathrm{x})$$\dim \mathrm{F}\mathrm{T}(0)$$\dim \mathrm{A}(\mathrm{u})=(\mathrm{J}+1)\mathrm{I}-(\mathrm{S}+1)+\mathrm{S}+\mathrm{I}$$\leqq \mathrm{S}\mathrm{I}+1-1$$<(\mathrm{S}+1)\mathrm{I}$.

$=\dim \mathrm{R}^{\mathrm{t}\+\mathrm{l}}’$which is acontradiction. Thus$\mathrm{F}$,, that is to say, there exist no Pareto$(\omega+\mathrm{y})\cap \mathrm{A}(\mathrm{u})=\emptyset$

optimal allocations fulfilling -feasibility. Since each equilibrium allocation of ( ,obviously -feasible, it is Pareto inefficient. Noting thatis arbitrarily taken from$\mathrm{e}$$\mathrm{v}$$\mathrm{u}$$\mathrm{v}$and$\mathrm{u}$and$\mathrm{y}$$\omega$$\omega$; v) is$\mathrm{R}^{(*+1\mathrm{x}}++$

are respectively arbitrary elements of the dense sets, our claim follows.
Q.E.D.

4. Concluding$\mathrm{R}\mathrm{e}\mathrm{m}[] \mathrm{r}\mathrm{k}$We have investigated inefficiency of equilibrium allocations with incomplete markets from thegeneric viewpoint with respect to all agents’ utility
and endowments in one good-two
period exchange economy model. One of our findings is that as long as all agents’ utility$\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{c}\dot{\mathrm{h}}\mathrm{o}\mathrm{n}\mathrm{s}$ffinctions are strictly monotone and endowments
strictly positive, for almost all utility
fimctions and mdowmmts each equilibrium allocation with incomplete markets is, if any, Pgeto$\alpha \mathrm{e}$89Magill and Quinzii have also shown inefficiency of equilibrium allocations with incomplete
markets. In their argument, however, the agents ’endowments are only considered as economy
parameters given their utility functions satisfying monotonicity and some convexity (Magill andQuinzii (1996) . Our result substantially extends theirs in that (i)all agents’ utility functions are$)$no convexity is needed on the functions.
included in the genericity analysis andAnother finding is that it is not utility maximization behavior of each agent but his budget
constraint ( -feasibility)that yields Pareto inefficiency of equilibrium allocations withgenerically in andincomplete markets. This is easily seen from the fact that$(\dot{\mathrm{n}})$$\mathrm{v}$$\mathrm{u}$$\mathrm{F}_{\mathrm{v}}(\omega)\cap \mathrm{A}(\mathrm{u})=\emptyset$$\omega$, which is shown in the proof of the theorem . To put it in another way, as soon as the agents

participate in the incomplete asset markets, they are kept away from Pareto optimal allocations
regardless of their optimization behavior In this connection, we may add another remark that is, thenmore suggestive. It is about likelihood of inefficiency of equilibrium allocations. Whenthe incomplete markets model turns into complete markets model so that every equilibrium$\mathrm{J}=\mathrm{S}$allocation is Pareto optimal by the first fundamental theorem of welfare economics. Therefore,
even if the asset markets are incomplete, likelihood of inefficiency of equilibrium allocations is
intuitively expected to be dependent on how much differs from S. It is intuitively inferred thatequilibrium allocations with incomplete markets are more likely to be Pareto optimal when isthan when is way below S. But it is revealed from our argument that such an
close togenerically in andintuition is wrong. The reason is this. It is always true that, that is, the more the number grows, the bigger. The number has an effect on$\mathrm{J}$$\mathrm{J}$$\mathrm{J}$$\mathrm{S}$$\mathrm{p}_{\mathrm{v}}(\omega)*$$\mathrm{A}(\mathrm{u})$$\mathrm{u}$$\dim \mathrm{p}_{\mathrm{v}}(\omega)$$\mathrm{J}$$\omega$is. However, as long as.
necessarily yields that$\dim \mathrm{F}_{\mathrm{v}}$$\mathrm{J}<\mathrm{S}$$(\mathrm{c}\mathrm{o})$,,$\dim \mathrm{F}_{\mathrm{v}}(\omega)+\dim \mathrm{A}(\mathrm{u})<(\mathrm{S}+1)\mathrm{I}$which$\mathrm{p}_{\mathrm{v}}(\omega)\cap \mathrm{A}(\mathrm{u})=\emptyset$

Appendix.
To prove Proposition 2, we give two lemmas.

manifolds withand beLemma 1. Let ,be asubmersion. Then
mapping. Let
Proof. See Nagata (2001) , lemma 7.3, p. 124.
$\mathrm{X}$$\mathrm{Y}$$\mathrm{Z}$$\mathrm{C}^{\mathrm{r}}$asubmanifold of$\mathrm{W}$$\mathrm{f}\overline{r\mathrm{h}}\mathrm{g}(-1\mathrm{W})$$\mathrm{g}:\mathrm{Y}arrow \mathrm{Z}$Lemma 2. Asmooth map”$\mathrm{f}:\mathrm{R}^{\ln}++arrow\Delta^{\prime 1}$$1\mathrm{r}$$+\mathrm{d}\mathrm{e}\mathrm{f}_{1}\mathrm{n}\mathrm{e}\mathrm{d}$$\mathrm{Z}$if and only if

and
ffiW.

$\mathrm{f}:\mathrm{X}arrow \mathrm{Y}$be$\mathrm{C}^{\mathrm{r}}$$\mathrm{f}$$\mathrm{g}\cdot$by$\mathrm{f}(1^{1},\ldots,\mathrm{I}^{\mathfrak{n}})=(\mathrm{X}^{1}/\Sigma 1-1\mathrm{X}^{1}\ddagger, \ldots, \mathrm{x}^{\mathfrak{n}}/\Sigma_{\mathfrak{l}\cdot 1}\mathrm{x}^{\mathfrak{n}_{\mathrm{i}}})$

is asubmersion.
Proof. Considering the structure of , it is sufficient to show that a map-1Xi is asubmersion. To this end, we show that at any
defined byis surjective. It is easily seen that -th column vector of Dh (x) is$\mathrm{h}:\mathrm{R}^{\mathrm{I}}++arrow\Delta^{1-1}+$$\mathrm{f}$$(\in \mathrm{R}^{\mathfrak{l}}++)\mathrm{D}\mathrm{h}(\mathrm{x})$$\mathrm{I}$$\mathrm{h}(\mathrm{x})=\mathrm{x}/\Sigma \mathrm{i}$$.\mathrm{R}^{1}arrow \mathrm{R}^{1-1}$$\mathrm{i}$$(\mathrm{x}_{1}/(\Sigma \mathrm{i}\cdot \mathrm{l}\mathrm{x}\iota )^{2}, \ldots, \mathrm{x} -’/(\Sigma \mathrm{i}-\mathrm{l}\mathrm{x}\iota)^{2}, \Sigma_{1t\uparrow \mathrm{X}_{1/(\Sigma}}\mathrm{i}-\prime \mathrm{X}\iota)^{2}$,$+1/(\Sigma \mathrm{i}-\prime \mathrm{x}_{1})^{2}$$\mathrm{x}$,$\ldots$,$\mathrm{x}|/(\Sigma i-\prime \mathrm{x}_{\mathrm{f}})^{2})^{\uparrow}$designates the transpose. Thus we see that first 1–1 column vectors of Dh(x) are$-1/(\Sigma$ i-lx
,
i-tL
,
,
i-lx ,
linearly independent. Indeed, set

where$\mathrm{t}$$\Sigma \mathrm{i}-1’-1\lambda \mathrm{i}(\mathrm{x}_{1}/(\Sigma$$)^{2}$$)^{2}$$\mathrm{x}$$\ldots$$\Sigma_{\mathrm{J}\mathrm{s}1\mathrm{r}}/(\Sigma$$)^{2}$

90$\mathrm{x}$

$+1/$

i-lx

$(\Sigma$$)^{2}$

equation yields$(\mathrm{k}=1, \ldots, 1・1)$,$\ldots$$\Sigma$,$\chi\sqrt(\Sigma$

i-ll-li-lx $)^{2})’=0$ where each$\lambda \mathrm{i}=0$$\lambda$: is ascalar $(\mathrm{i}=1, \ldots, 1 -1)$ . Then its 1-th

, which in turn makes each

. It follows thm /1$1^{=}\lambda 2^{=\ldots=\lambda}1-’\triangleleft$. So$\mathrm{k}$-th equation turn into /1$\mathrm{D}\mathrm{h}$$(\mathrm{x})$$\mathrm{I}/(\Sigma$

i-iXi$\succ^{-}0$

is surjective.
Q.E.D.

Proof of Proposition 2.
Note that when is an element of %, then the range$\mathrm{u}$$\mathrm{J}^{1}+(\mathrm{R}^{S+\prime}++, \mathrm{R})^{1}$. Take$\mathrm{R}^{6+1\prime}++$,in Lemma 1. Then if we show that$\Phi$$\zeta$$\mathrm{J}^{1}(\mathrm{R}^{\mathrm{S}+}1 ++, \mathrm{R})^{1}$$(\mathrm{R}^{\mathrm{s}+1}++\cross \mathrm{R}\cross\Delta^{\mathrm{s}}+)’$$\mathrm{X}$$\mathrm{Y}$$\mathrm{Z}$$\Phi$

Theorem1. But by the sMcture ofmapby$\zeta:\mathrm{R}^{(S+\mathrm{t}}’++arrow\Delta \mathrm{s}\iota$

is not

but
and
for
and
is a submersion, the proof is completed by Lemma 1and, to show its submersiveness we have only to show that the$\mathrm{J}^{1}+(\mathrm{R}^{\mathrm{s}+1}++, \mathrm{R})’$$\mathrm{o}\mathrm{f}\mathrm{j}^{1}\prime \mathrm{u}$$+\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}$$(\mathrm{c}^{1}, \mathrm{c}^{2}, \ldots, \mathrm{c}’)=(\mathrm{c}’/\Sigma\ddagger-1\mathrm{c}’\mathrm{i}, \mathrm{c}^{2}/\Sigma \mathrm{I}-1\mathrm{c}_{\dot{b}}^{2}\ldots, \mathrm{c}^{1}/\Sigma 1-1\mathrm{C}’\iota)$

is asubmersion. By applying Lemma 2we see that this map is indeed asubmersion.
Q.E.D.

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