# On the Uniqueness and Stability of the Equilibrium Price in Quasi-Linear
Economies

Yuhki Hosoya
Faculty of Economics, Chuo University
742-1 Higashinakano, Hachioji-shi Tokyo 192-0393, Japan 1-50 1601 Miyamachi,
Fuchu-shi, Tokyo 183-0023, Japan.TEL: +81-90-5525-5142, E-mail:
ukki(at)gs.econ.keio.ac.jp

###### Abstract

In this paper, we show that if every consumer in an economy has a quasi-linear
utility function, then the normalized equilibrium price is unique, and it is
locally stable with respect to the tâtonnement process. If the dimension of
the consumption space is two, then this result can be expressed by the
corresponding partial equilibrium model. Our study can be seen as that extends
the results in partial equilibrium theory to economies with more than two
dimensional consumption space.

JEL codes: C62, C61, D41, D51.

Keywords: Walrasian tâtonnement process, quasi-linear utility, excess demand
function, uniqueness of equilibrium price, local stability.

##  1 Introduction

How many competitive equilibrium prices are there? Since Arrow and Debreu
(1954) showed that there is at least one equilibrium price, this issue had
become of great interest to economists. If the equilibrium price is unique,
then it would greatly increase the predictive accuracy of the model. However,
even in textbook-level examples, the economy can have multiple equilibria
(e.g., see exercise 15.B.6 of Mas-Colell, Whinston, and Green (1995)). In this
issue, economists had devided into two positions. The first position is one
that tries to guarantee the uniqueness of the equilibrium price by making
strong assumptions about the economy, and the second position is one that
gives up the uniqueness of equilibrium prices. In particular, the position of
Arrow and Debreu were clearly divided on this matter.

Gerard Debreu took the latter position. He said that all the known conditions
for guaranteeing the uniqueness of equilibrium prices are “exceedingly strong”
(Debreu, 1972). On the other hand, he proved the Sonnenschein-Mantel-Debreu
theorem (Debreu, 1974). This theorem implies that any compact set in positive
orthant could be included in the set of equilibrium prices when only the
usual, widely accepted assumptions of economy were made. Thus, without any
assumptions that he said to be “exceedingly strong”, we know nothing about the
number of equilibria. The number of equilibria may be one, a million, or
infinity. To solve this problem, Debreu (1970) treated purturbation of the
economy by initial endowment vector, and showed that in “almost all” economy,
the number of equilibria is at least not infinite. This result had been
refined and developed into the theory of regular economy.

On the other hand, Kenneth Arrow took the former position. He treated this
problem in combination with another traditional problem, called the theory of
tâtonnement process. In 19th century, Walras (1874) explained why equilibrium
price is realized in the competitive equilibrium model as follows. First, when
the price is higher than the equilibrium price, then supply will be high and
demand will be low. This situation is called a state of excess supply. In this
situation, a lot of goods remain unsold and inventory increases. Therefore,
the price will go down. On the other hand, if the price is lower than the
equilibrium price, the opposite will occur: that is, demand will be high and
supply will be low. This situation is called a state of excess demand. In this
instance, there will be many sellouts and the price will be high. As a result,
the equilibrium price would attract the actual price, and the economy tends to
trade by the equilibrium price. This idea was refined and expressed in
differential equations, called tâtonnement process.

Arrow et al. (1959) showed that if the excess demand function is gross
substitute, then any equilibrium price is a globally stable steady state with
respect to tâtonnement process. Because there can be only one globally stable
steady state, they actually showed simultaneously that there is only one
equilibrium price. Then, the next problem is to determine the conditions of
the economy under which the excess demand function satisfies the gross
substitution. However, this problem has not yet been resolved. As a result,
Debreu’s view that “there is a little that can be said about the number of
equilibria in the general environment” is now common among economists.

By the way, there is another theory of equilibrium than the general
equilibrium theory, namely, the partial equilibrium theory. In fact, the
partial equilibrium theory has a foundation in general equilibrium theory,
where there are only two types of goods, numeraire good and traded good, and
consumer’s utility function must be quasi-linear. Instead, in partial
equilibrium model, it is easy to show by drawing a diagram that there is only
one equilibrium price and this price is globally stable (see our section 2).
In other words, in a quasi-linear economy with two commodities, the
equilibrium price is unique, and it is globally stable with respect to the
tâtonnement process.

The purpose of this paper is to extend this result to a quasi-linear economy
with more than two commodities. That is, the purpose of this study is to see
whether the above result holds when considering a general equilibrium model in
which the utility remains quasi-linear and the dimension of the consumption
space may be more than two. The results are as follows: first, the equilibrium
price is unique in an economy where all consumers have a quasi-linear utility
functions. Second, this equilibrium price is locally stable with respect to
the tâtonnement process. Global stability could not be derived in this paper.
This is related to the inherent difficulty of quasi-linear economies: see our
discussion in subsection 3.5.

The structure of this paper is as follows. First, in order to help the readers
better understand the purpose of this study, we provide an explanation of
partial equilibrium theory and the general equilibrium model behind this
theory in section 2. In section 3, we first define an economy, and then define
the type of economy we call the “quasi-linear economy”. There are two types of
economies we call “quasi-linear economy”, one permits negative consumption
with respect to the numeraire good,111Note that, this is one of the
traditional treatment of quasi-linear utility function. For example, see the
definition of the quasi-linear preference in Mas-Colell et al. (1995). and the
other assumes that the initial endowment of numeraire good is sufficiently
large. On the basis of the above preparations, we prove the main result.
Section 4 argues several topics concerning our main result. In subsection 4.1,
we consider the ‘consumer’s surplus’ in this general equilibrium setup. In
subsection 4.2, we discuss the relationship between this paper and related
research. Section 5 includes the conclusion. The proofs are placed in section
6.

##  2 Motivation

Consider the classical partial equilibrium model. The market is described as
the following figure. (Figure 1)

$x$$p$$D$$S$$p^{*}$$x^{*}$ Figure 1: The Market

The decreasing line is called the demand curve, and the increasing line is
called the supply curve. Two curves crosses at $(p^{*},x^{*})$. $p^{*}$ is
called the equilibrium price, and $x^{*}$ is called the equilibrium output. In
this model, trade is done by equilibrium price, and the amount of traded
commodity is determined by $x^{*}$.

We explain the hidden structure that determines both curves. The hidden
structure is written by general equilibrium model with two commodities. The
first commodity is that is traded in this market, and the second commodity is
the money. We assume that the price of the money is always one. Let $p$ denote
the price of the first commodity.

First, suppose that there are $M$ suppliers of the first commodity, where the
cost function of $i$-th supplier is denoted by $c_{i}(x_{i})$. We assume that
$c_{i}$ is twice differentiable. Then, by first- and second-order necessary
condition of profit maximization, we must have,

$p=c_{i}^{\prime}(x_{i}),\ c_{i}^{\prime\prime}(x_{i})\geq 0.$ (1)

The supply curve $S(x)$ is determined as follows:

$S(x)=p\Leftrightarrow\exists x_{1},...,x_{M}\mbox{ s.t.
}x=\sum_{i=1}^{M}x_{i},\ c_{i}^{\prime}(x_{i})=p.$

If $S^{\prime}(x)<0$, then $c_{i}^{\prime\prime}(x_{i})<0$ for some $i$, which
contradicts (1). Therefore, we have that $S^{\prime}(x)\geq 0$, and thus the
supply curve is nondecreasing.

Second, suppose that there are $N$ consumers, and every consumer has a quasi-
linear utility function: that is, the utility function $U_{i}(x_{i},y_{i})$ is
written as $u_{i}(x_{i})+y_{i}$. We assume that $U_{i}$ is strictly quasi-
concave, which is equivalent to the strict concavity of $u_{i}$. Moreover, we
assume that $u_{i}$ is twice continuously differentiable. By Lagrange’s first-
order condition, if $(x_{i}^{*},y_{i}^{*})$ maximizes the utility, then there
exists $\lambda>0$ such that

$\displaystyle u_{i}^{\prime}(x_{i})-\lambda p=$ $\displaystyle~{}0,$
$\displaystyle 1-\lambda=$ $\displaystyle~{}0,$ $\displaystyle m-px+y=$
$\displaystyle~{}0,$

which implies that $\lambda=1$, and thus $p=u_{i}^{\prime}(x_{i})$. Now,
because $u_{i}$ is strictly concave, we have that $u_{i}^{\prime}$ is
decreasing. Moreover, the demand curve $D(x)$ is determined by the following
equation:

$D(x)=p\Leftrightarrow x=\sum_{i=1}^{N}(u_{i}^{\prime})^{-1}(p),$

which implies that $D(x)$ is decreasing.

The above arguments says that $S(x)$ is nondecreasing and $D(x)$ is
decreasing, and thus the equation $D(x)=S(x)$ has at most one solution
$x^{*}$. Define $p^{*}=D(x^{*})=S(x^{*})$. Then, $(p^{*},x^{*})$ is the unique
equilibrium in this market. Moreover, this is stable in Walras’ sense by the
following reason. Choose any $p>p^{*}$, and suppose that
$D(\hat{x})=S(\bar{x})=p$. Then, by the above arguments, we must have that
$\hat{x}<\bar{x}$. That is, this economy is in the state of excess supply.
Hence, we expect that price goes down. Meanwhile, choose any $p<p^{*}$, and
suppose that $D(\hat{x})=S(\bar{x})=p$. Then, we must have that
$\hat{x}>\bar{x}$, and thus, this economy is in the state of excess demand.
Hence, we expect that price goes up. This arguments implies that the
equilibrium price has a power that attracts actual price, and thus this price
is called stable, and we can consider that $p^{*}$ is the realized price in
long-run economy. In conclusion, in quasi-linear economy with two commodities,
the equilibrium is unique and stable in Walras’ sense.

Is this result general in equilibrium theory? The answer is NO, even in pure
exchange economies. Debreu (1974) showed that for every compact subset in
$\\{p\in\mathbb{R}^{n}_{++}|\|p\|=1\\}$, there exists a pure exchange economy
in which the set of equilibrium prices includes this set. Therefore, the
uniqueness of the equilibrium is broken. Moreover, even if there uniquely
exists an equilibrium price, this price may be unstable. Therefore, the above
result is broken in a general situation.

Meanwhile, Arrow et al. (1959) showed that if the excess demand function is
gross substitute, then there uniquely exists an equilibrium price vector, and
it is stable on the tâtonnement process. Therefore, if the excess demand
function is gross substitute, then the results in partial equilibrium analysis
revive. However, this result is criticized by Debreu (1972). Debreu said that
the requirement of gross substitution is “exceedingly strong”. Actually, there
is no known natural assumption on an economy under which the excess demand
function becomes gross substitute.

Our next question is as follows: is the above result is general in quasi-
linear economy with more than two commodities? Our purpose is to answer this
question affirmatively. Consider an economy in which the utility function is
quasi-linear for every consumer. The purpose of this work is to show that
there uniquely exists a normalized equilibrium price vector, and it is locally
stable on Walrasian tâtonnement process. In other words, we will show that
almost all results in partial equilibrium theory also holds in every quasi-
linear economy.

##  3 Result

###  3.1 Preliminary: General Setups of Economies

In this paper, for $x,y\in\mathbb{R}^{M}$, $x\geq y$ means $x_{i}\geq y_{i}$
for every $i\in\\{1,...,M\\}$, and $x\gg y$ means $x_{i}>y_{i}$ for every
$i\in\\{1,...,M\\}$, respectively. Define the sets
$\mathbb{R}^{M}_{+}=\\{x\in\mathbb{R}^{M}|x\geq 0\\}$ and
$\mathbb{R}^{M}_{++}=\\{x\in\mathbb{R}^{M}|x\gg 0\\}$ as usual. If $M=1$, then
we abbreviate this symbol, and simply write these set as $\mathbb{R}_{+}$ and
$\mathbb{R}_{++}$. The notation $e_{j}$ denotes the $j$-th unit vector.

Let $L\geq 2$. We call a septuplet

$E=(N,M,(\Omega_{i})_{i\in N},(U_{i})_{i\in N},(\omega^{i})_{i\in
N},(Y_{j})_{j\in M},(\theta_{ij})_{i\in N,j\in M})$

an economy if,

  1. (1)

$N=\\{1,...,n\\}$ is a finite set of consumers, and $M=\\{1,...,\mu\\}$ is a
finite set of producers. We admit $\mu=0$ and in this case, this economy is
called a pure exchange economy.

  2. (2)

$U_{i}:\Omega_{i}\to\mathbb{R}$ denotes the utility function of $i$-th
consumer, where the set $\Omega_{i}\subset\mathbb{R}^{L}$ denotes the set of
all possible consumption plans for $i$-th consumer,

  3. (3)

$\omega^{i}\in\Omega_{i}$ denotes the initial endowment of $i$-th consumer,

  4. (4)

$Y_{j}\subset\mathbb{R}^{n}$ denotes the production set of $j$-th producer,
and

  5. (5)

$\theta_{ij}\geq 0$ denotes the share of $i$-th consumer to $j$-th producer.

We always assume that $\sum_{i\in N}\theta_{ij}=1$ for all $j\in M$.

Consider the following utility maximization problem:

$\displaystyle\max~{}~{}~{}~{}~{}$ $\displaystyle~{}U_{i}(x),$ subject to.
$\displaystyle~{}x\in\Omega_{i},$ (2) $\displaystyle~{}p\cdot x\leq m,$

where $p\gg 0$ and $m>0$. Let $f^{i}(p,m)$ denote the set of all solutions to
(2). This set-valued function $f^{i}(p,m)$ is called the demand function of
consumer $i$.

Meanwhile, consider the following profit maximization problem:

$\displaystyle\max~{}~{}~{}~{}~{}$ $\displaystyle~{}p\cdot y,$ subject to.
$\displaystyle~{}y\in Y_{j},$ (3)

where $p\gg 0$. Let $y^{j}(p)$ denote the set of all solutions to (3). This
set-valued function $y^{j}(p)$ is called the supply function of producer $j$.
Moreover, the value of (3) is denoted by $\pi^{j}(p)$, which is called the
profit function.

We write

$m^{i}(p)=p\cdot\omega^{i}+\sum_{j\in M}\theta_{ij}\pi^{j}(p).$

For given demand function $f^{i}$ and supply functions $y^{1},...,y^{m}$,
define

$X^{i}(p)=f^{i}(p,m^{i}(p))-\omega^{i}.$

This function is called the excess demand function of consumer $i$. Define

$X(p)=\sum_{i\in N}X^{i}(p),$

and

$\zeta(p)=X(p)-\sum_{j\in M}y^{j}(p).$

This set-valued function is called the excess demand function in this economy.
We call $p^{*}\in\mathbb{R}^{L}_{++}$ an equilibrium price if
$0\in\zeta(p^{*})$.

If the demand function $f^{i}$ is single-valued and differentiable at $(p,m)$,
then we can define

$s^{i}_{jk}(p,m)=\frac{\partial f^{i}_{j}}{\partial p_{k}}(p,m)+\frac{\partial
f^{i}_{j}}{\partial m}(p,m)f^{i}_{k}(p,m).$

The $L\times L$ matrix-valued function
$S_{f^{i}}(p,m)=(s^{i}_{jk}(p,m))_{j,k=1}^{L}$ is called the Slutsky matrix.

Consider the following differential inclusion:

$\dot{p}_{j}(t)\in a_{j}\zeta_{j}(p(t)),\ p_{j}(0)=p_{0j},\
j\in\\{1,...,L\\},$ (4)

where $a_{1},...,a_{L}>0$ and $p_{0}\in\mathbb{R}^{L}_{++}$. This inclusion is
called the tâtonnement process in this economy $E$. We call a set
$I\subset\mathbb{R}$ an interval if it is convex and contains at least two
different points. A function $p:I\to\mathbb{R}^{L}_{++}$ is called a solution
to (4) if $I$ is an interval including $0$, $p(t)$ is absolutely continuous on
any compact set $C\subset I$, $p(0)=p_{0}$, and

$\dot{p}_{j}(t)\in a_{j}\zeta_{j}(p(t)),$

for all $j\in\\{1,...,L\\}$ and almost all $t\in I$. It is well known that if
$\zeta$ is single-valued and continuous, then every solution to (4) is
continuously differentiable.

An equilibrium price $p^{*}$ is called locally stable if there exists an open
neighborhood $U$ of $p^{*}$ such that 1) for every $p_{0}\in U$, there exists
a solution $p:I\to\mathbb{R}^{L}_{++}$ to (4) such that $\mathbb{R}_{+}\subset
I$, and 2) for every solution $p:I\to\mathbb{R}^{L}_{++}$ to (4) such that
$\mathbb{R}_{+}\subset I$, $\lim_{t\to\infty}p(t)=ap^{*}$ for some $a>0$. If
we can choose $U=\mathbb{R}^{L}_{++}$, then $p^{*}$ is called globally stable.

Define the set $Y=\sum_{j\in M}Y_{j}$. This set is called the aggregate
production set.

Let $(x,y)=(x^{1},...,x^{n},y^{1},...,y^{m})\in\mathbb{R}^{(n+\mu)L}$. We call
this vecter a feasible allocation in $E$ if 1) $x^{i}\in\Omega_{i}$ for all
$i\in N$, 2) $y^{j}\in Y_{j}$ for all $j\in M$, and 3) $\sum_{i\in
N}x^{i}=\sum_{i\in N}\omega^{i}+\sum_{j\in M}y^{j}$. Let $\bar{A}_{\omega}$
denote the set of all feasible allocations in the economy $E$, where
$\omega=(\omega^{1},...,\omega^{n})$ is given. We call this set the attainable
set in this economy.

Finally, throughout this paper, the symbol $Dh(x)$ refer to either the Fréchet
derivative of $h$ at $x$ or its transpose, whichever is more convenient. We
believe that there is no danger of causing confusion by this omission of the
transposition sign.

###  3.2 Economies with Quasi-Linear Environments

We make assumptions for economies with quasi-linear environment. However,
there are two styles on quasi-linear environment. The first style is in Mas-
Colell et al. (1995), in which the consumption space $\Omega_{i}$ is assumed
to be $\mathbb{R}^{L-1}_{+}\times\mathbb{R}$. That is, in this style,
consuming the negative amount of the numeraire good is admitted.222If we
consider that $x_{L}$ denotes the amount of money, the negative $x_{L}$
indicates a debt. In the second style, we assume that as usual
$\Omega_{i}=\mathbb{R}^{n}_{+}$, and the initial endowment of the numeraire
good is sufficiently large for every consumer. We treat both styles, and thus
we separate assumptions for these environments.

Throughout this paper, for every $x\in\mathbb{R}^{L}$, $\tilde{x}$ denotes
$(x_{1},...,x_{L-1})\in\mathbb{R}^{L-1}$. Here, we make assumptions on
economies with quasi-linear environment.

Assumption F. For every $i\in N$,
$\Omega_{i}=\mathbb{R}^{L-1}_{+}\times\mathbb{R}$, and the function $U_{i}$
can be written as follows:

$U_{i}(x)=u_{i}(\tilde{x})+x_{L},$ (5)

where $u_{i}$ is continuous, nondecreasing, and concave on
$\mathbb{R}^{L-1}_{+}$. Moreover, $u_{i}$ is twice continuously differentiable
on $\mathbb{R}^{L-1}_{++}$, $Du_{i}(\tilde{x})\gg 0$, and the Hessian matrix
$D^{2}u_{i}(\tilde{x})$ is negative definite for every
$\tilde{x}\in\mathbb{R}^{L-1}_{++}$.333This assumption is equivalent to the
non-zero Gaussian curvature requirement of Debreu (1972) for every
indifference hypersurface of $U_{i}$ passing through the interior of
$\Omega_{i}$. Debreu (1972) showed that this requirement is equivalent to the
differentiability of the demand function at any price and money such that
every coordinate of the demand is positive.

Assumption S1. For every $i\in N$, $\Omega_{i}=\mathbb{R}^{L}_{+}$ and the
function $U_{i}$ can be written as (5), where $u_{i}$ is continuous,
nondecreasing, and concave on $\mathbb{R}^{L-1}_{+}$. Moreover, $u_{i}$ is
twice continuously differentiable on $\mathbb{R}^{L-1}_{++}$,
$Du_{i}(\tilde{x})\gg 0$ and the Hessian matrix $D^{2}u_{i}(\tilde{x})$ is
negative definite for every $\tilde{x}\in\mathbb{R}^{L-1}_{++}$.

Assumption S2. Let $\hat{\omega}^{i}=(\omega^{i}_{1},...,\omega^{i}_{L-1},0)$,
and define

$\alpha_{i}=\sup\\{u_{i}(\tilde{x}^{i})-u_{i}(\tilde{\omega}^{i})|(x,y)\in\bar{A}_{\hat{\omega}},\
U_{j}(x^{j})\geq U_{j}(\hat{\omega}^{j})\mbox{ for all }j\in
N\setminus\\{i\\}\\}.$ (6)

Then, $\omega^{i}_{L}>\alpha_{i}$ for every $i\in N$.444The finiteness of
$\alpha_{i}$ is confirmed later.

Assumption Q. For every $i\in N$, $\tilde{p}\in\mathbb{R}^{L-1}_{++}$, and
$m>0$, the following problem

$\displaystyle\max~{}~{}~{}~{}~{}$ $\displaystyle~{}u_{i}(\tilde{x})$ subject
to. $\displaystyle~{}\tilde{x}\in\mathbb{R}^{L-1}_{+},$ (7)
$\displaystyle~{}\tilde{p}\cdot\tilde{x}\leq m$

has an inner solution $\tilde{x}^{*}\gg 0$. Moreover, the equation

$Du_{i}(\tilde{x})=\tilde{p}$ (8)

also has an inner solution $\tilde{x}^{+}\gg 0$. In addition, if
$u_{i}(\tilde{x})>u_{i}(0)$, then $u_{i}$ is strictly increasing on
$\tilde{x}+\mathbb{R}^{L-1}_{+}$.555This requirement admits both
$u_{i}(x)=(x_{1}x_{2})^{1/3}$ and $u_{i}(x)=\sqrt{x_{1}}+\sqrt{x_{2}}$.

Assumption U. $\omega^{i}\in\mathbb{R}^{L}_{+}\setminus\\{0\\}$ and
$\sum_{i\in N}\omega^{i}\gg 0$.

Assumption P. For every $j\in M$, $Y_{j}\cap\mathbb{R}^{L}_{+}=\\{0\\}$,
$Y_{j}$ is closed and convex including $-\mathbb{R}^{L}_{+}$, and
$Y\cap(-Y)=\\{0\\}$. Moreover, if $y^{1},y^{2}\in Y_{j}$, $y^{1}\neq y^{2}$,
$0<t<1$, and $y^{1},y^{2}\notin-\mathbb{R}^{n}_{+}$, then $(1-t)y^{1}+ty^{2}$
is in the interior of $Y_{j}$. Furthermore, there exists
$p^{+}\in\mathbb{R}^{L}_{++}$ such that for every $j\in M$, $y^{j}(p^{+})$ is
a nonempty set.666The first assumption says that the possibility of inaction
and no free-lunch condition. The second assumption implies the free disposal
property, that is, $Y_{j}=Y_{j}-\mathbb{R}^{L}_{+}$. See Problem 5.B.5 of Mas-
Colell et al. (1995). The third assumption says that the aggregate production
set is irreversible. The fourth assumption indicates that this production set
is decreasing return to scale.

In this section, we call an economy $E$ that satisfies Assumptions F, Q, U,
and P a first-type quasi-linear economy, and that satisfies Assumptions S1,
S2, Q, U, and P a second-type quasi-linear economy, respectively. We call $E$
a quasi-linear economy if it is either a first-type quasi-linear economy or a
second-type quasi-linear economy.

###  3.3 Propositions

We first present several basic results on quasi-linear economies.

Proposition 1. Suppose that $E$ is a quasi-linear economy. Then, for every
$i\in N$, $f^{i}$ is a single-valued continuous function. Moreover, if
$x=f^{i}(p,m)$, then $\tilde{x}\in\mathbb{R}^{L-1}_{++}$. Furthermore, Walras’
law

$p\cdot f^{i}(p,m)=m$ (9)

and homogeneity of degree zero

$f^{i}(ap,am)=f^{i}(p,m)\mbox{ for all }a>0$ (10)

hold.

Proposition 2. Suppose that $E$ is a quasi-linear economy. If either $E$ is
first-type or $f^{i}_{L}(p,m)>0$, then $f^{i}$ is continuously differentiable
at $(p,m)$, and

$\frac{\partial f^{i}_{j}}{\partial m}(p,m)=\begin{cases}0&\mbox{if }1\leq
j\leq L-1,\\\ \frac{1}{p_{L}}&\mbox{if }j=L.\end{cases}$ (11)

Proposition 3. Suppose that $E$ is a quasi-linear economy, and fix
$(p,m)\in\mathbb{R}^{L}_{++}\times\mathbb{R}_{++}$. Suppose also that either
$E$ is first-type or $f^{i}_{L}(p,m)>0$. Then, the Slutsky matrix
$S_{f^{i}}(p,m)$ satisfies the following three properties.

(R) The rank of $S_{f^{i}}(p,m)$ is $L-1$. Moreover,
$p^{T}S_{f^{i}}(p,m)=0^{T}$ and $S_{f^{i}}(p,m)p=0$.

(ND) For every $v\in\mathbb{R}^{L}$ such that $v\neq 0$ and $p\cdot v=0$,
$v^{T}S_{f^{i}}(p,m)v<0$.

(S) The matrix $S_{f^{i}}(p,m)$ is symmetric.

Proposition 4. If Assumptions S1, Q, U, and P hold, then $\alpha_{i}$
appearing (6) is finite.

Proposition 5. Suppose that Assumption P holds, and $y^{j}(p)$ is nonempty.
Then, $y^{j}(p)$ is a singleton.

Recall that the definition of the function $X$:

$X(p)=\sum_{i\in N}X^{i}(p)=\sum_{i\in N}[f^{i}(p,m^{i}(p))-\omega^{i}].$

Because of Proposition 1, we have that $X$ is a single-valued continuous
function defined on $\mathbb{R}^{L}_{++}$. Moreover, because

$\zeta(p)=X(p)-\sum_{j\in M}y^{j}(p),$

the excess demand function $\zeta$ is also single-valued continuous function
on the set in which $y^{j}(p)$ is nonempty for every $j\in\\{1,...,\mu\\}$.

Proposition 6. Suppose that $E$ is a quasi-linear economy, and $\zeta$ is the
excess demand function of this economy. Then, this function $\zeta$ satisfies
the following Walras’ law

$p\cdot\zeta(p)=0,$ (12)

and the homogeneity of degree zero

$\zeta(ap)=\zeta(p)\mbox{ for all }a>0.$ (13)

Moreover, there is at least one equilibrium price $p^{*}$ in this economy.

Proposition 7. Suppose that $E$ is either a second-type quasi-linear economy
or a first-type quasi-linear economy that satisfies assumption S2, and let
$p^{*}$ be an equilibrium price of this economy. Then,
$f^{i}_{L}(p^{*},p^{*}\cdot\omega^{i})>0$ for every $i\in N$.

###  3.4 Main Results

Suppose that $E$ is a quasi-linear economy, and recall the tâtonnement process

$\dot{p}_{j}(t)=a_{j}\zeta_{j}(p(t)),\ p_{j}(0)=p_{0j},$ (14)

where $a_{1},...,a_{L}>0$ and $p_{0}\in\mathbb{R}^{L}_{++}$. Note that,
because the excess demand function $\zeta$ is single-valued in quasi-linear
economies, the tâtonnement process is not a differential inclusion but an
ordinary differential equation in such economies, and thus we can use usual
techniques on ordinary differential equations. In particular, any solution
$p:I\to\mathbb{R}^{L}_{++}$ to (14) is continuously differentiable.

We now complete the preparation of our main result.

Theorem 1. Suppose that $E$ is a quasi-linear economy, and $\zeta$ is the
excess demand function of this economy. Choose any equilibrium price $p^{*}$
in this economy. Then, every equilibrium price is proportional to $p^{*}$.
Moreover, for any parameters $a_{1},...,a_{L}>0$, this equilibrium price is
locally stable.

###  3.5 Remarks on Global Stability of Equilibria

Actually, we want to show the global stability of an equilibrium price
$p^{*}$. That is, we want to show that for every
$p_{0}\in\mathbb{R}^{L}_{++}$, there exists a solution $p(t)$ to (14) defined
on $\mathbb{R}_{+}$, and for such a solution, $\lim_{t\to\infty}p(t)$ exists
and is proportional to $p^{*}$. This cannot be proved because we do not assume
that $y^{j}(p)$ is always defined, and thus if $p_{0}$ is sufficiently far
from the equilibrium price, then $\zeta(p_{0})$ may be undefined. This problem
can be avoided by strengthening assumption P or, more simply, considering a
pure exchange economy. However, even in a pure exchange economy, there are two
hard difficulties that could not easily be solved.

First, suppose that $p(t)$ is a solution to (14) for some
$p_{0}\in\mathbb{R}^{L}_{++}$ defined on $\mathbb{R}_{+}$. We want to show
that $\lim_{t\to\infty}p(t)$ exists. However, this problem is difficult if
$p_{0}$ is too far from the half-line $\\{ap^{*}|a>0\\}$, and we could not
prove this result. For example, we could not exclude the possibility that the
trajectory of $p(t)$ consists of a cycle. If $L=2$, then we can show that such
a case vanishes, because, by the intermediate value theorem, the sign of
$\zeta_{1}(p_{1},p_{2})$ coincides with that of
$\frac{p_{2}}{p_{1}}-\frac{p_{2}^{*}}{p_{1}^{*}}$. However, this is a too
simple case, and if $L>2$, such a rough argument cannot be done.

The second problem is more serious. That is, we could not show the existence
of a solution $p(t)$ to (14) defined on $\mathbb{R}_{+}$. The essential
problem is the following: because $\mathbb{R}^{L}_{++}$ is open,
$C\subset\mathbb{R}^{L}_{++}$ may be not compact even if it is bounded and
closed in the relative topology of $\mathbb{R}^{L}_{++}$. The existence of a
solution to (14) whose trajectory is included in $\mathbb{R}^{L}_{++}$ is
called the ‘viability problem’. To solve this viability problem, we can use
the strong boundary condition of the excess demand function. Consider a pure
exchange economy $E$ such that the excess demand function $\zeta$ is a single-
valued function. We say that $\zeta$ satisfies the strong boundary condition
if and only if for every sequence $(p^{k})$ in $\mathbb{R}^{L}_{++}$ such that
$p^{k}\to p\in\mathbb{R}^{L}_{+}\setminus(\mathbb{R}^{L}_{++}\cup\\{0\\})$ as
$k\to\infty$,

$\sum_{j:p_{j}=0}\zeta_{j}(p^{k})\to+\infty.$

If the set $\Omega_{i}$ is bounded from below, then $\zeta$ usually satisfies
the strong boundary condition. Theorem 7 of Hosoya and Yu (2013) states that
if $\zeta$ is single-valued, continuous, and homogeneous of degree zero, and
it satisfies Walras’ law and the strong boundary condition, then there exists
a solution to (14) defined on $\mathbb{R}_{+}$. Therefore, if the economy $E$
is second-type quasi-linear, then we can solve the viability problem
positively.

However, in first-type quasi-linear economies, there is no known method for
solving the viability problem. Therefore, it is hard to prove the existence of
a solution $p(t)$ to (14) defined on $\mathbb{R}_{+}$, and thus the global
stability is also difficult to verify.

##  4 Discussion

###  4.1 On the Consumer’s Surplus

One of the virtue of the partial equilibrium theory is to be able to calculate
the change of consumer’s utility from the demand curve. That is, we can
consider the consumer’s surplus instead of the utility function directly. If
$L=2$ and the background model is quasi-linear, then it is known that change
of the consumer’s surplus coincides with change of the sum of the utility of
consumers in this economy. We can extend this result for the case in which
$L>2$.

To simplify the arguments, we assume that $E$ is a first-type quasi-linear
economy. We show in the proof section that the domain of $f^{i}$ can be
extended to $\mathbb{R}^{L}_{++}\times\mathbb{R}$, and thus in this section,
we assume that the domain of $f^{i}$ is $\mathbb{R}^{L}_{++}\times\mathbb{R}$.
Suppose that $\tilde{p},\tilde{q}\in\mathbb{R}^{L-1}_{++}$. Define the
following function:

$V_{i}(\tilde{p},\tilde{q})=\int_{0}^{1}\tilde{f}^{i}(c(t),1,m)\cdot
c^{\prime}(t)dt,$

where $c:[0,1]\to\mathbb{R}^{L}_{++}$ is a continuously differentiable
function such that $c(0)=\tilde{p}$ and $c(1)=\tilde{q}$. The following
theorem holds.

Theorem 2. $V_{i}(\tilde{p},\tilde{q})$ is invariant from the choice of $c(t)$
and $m$. Moreover, if we set $p_{L}=q_{L}=1$, then

$V_{i}(\tilde{p},\tilde{q})=u_{i}(\tilde{f}^{i}(q,m))-u_{i}(\tilde{f}^{i}(p,m)).$

Let $D(\tilde{p})$ be the aggregate demand function: that is, for
$\ell\in\\{1,...,L-1\\}$,

$D_{\ell}(\tilde{p})=\sum_{i\in N}\tilde{f}^{i}_{\ell}(\tilde{p},1,0).$

Then, we can obtain the following corollary.

Corollary 1. Choose any $\tilde{p},\tilde{q}\in\mathbb{R}^{L-1}_{++}$, and
suppose that $c:[0,1]\to\mathbb{R}^{L-1}_{++}$ is a continuously
differentiable function such that $c(0)=\tilde{p}$ and $c(1)=\tilde{q}$, and
$p_{L}=q_{L}=1$. Then, for all $m_{1},...,m_{n}$,

$\int_{0}^{1}D(c(t))\cdot c^{\prime}(t)dt-[\tilde{q}\cdot
D(\tilde{q})-\tilde{p}\cdot D(\tilde{p})]=\sum_{i\in
N}[U_{i}(f^{i}(q,m_{i}))-U_{i}(f^{i}(p,m_{i}))].$ (15)

That is, we can calculate the change of the sum of the utility in this trade
using only the aggregated demand $D(\tilde{p})$, which arises in the left-hand
side of (15).

###  4.2 Comparizon with Related Works

Most studies on the uniqueness of equilibrium do not use primitive assumptions
on the economy itself, but rather make assumptions on what is derived from the
economy. Mas-Colell (1991) summarised several classical results in this area.
He argued that the gross substitution of the excess demand function is no
longer a sufficient condition for the uniqueness of the equilibrium price when
production is introduced. Instead, the paper focused on the weak axiom of
revealed preference for the excess demand function, and pointed out that the
set of equilibrium prices is a convex set when this condition holds. If the
economy is regular, then the set of equilibrium prices becomes discrete, but
any discrete convex set is a singleton. Thus, the uniqueness of the
equilibrium price is obtained in this case. On the other hand, if the weak
axiom of revealed preference is not satisfied, an economy with multiple
equilibrium prices can easily be created by introducing production technology
that is constant return to scale. In this sense, Mas-Colell’s conclusion is
that the weak axiom of revealed preference is approximately ‘necessary and
sufficient’ for equilibrium price to be unique. This argument was discussed
again in Chapter 17 of Mas-Colell et al. (1995).

However, both the gross substitution and the weak axiom of revealed preference
on the excess demand function have the weakness that many economists have
never found good and acceptable conditions in the economy to guarantee them.
Recently, Giménes (2022) proved the uniqueness of equilibrium prices in a pure
exchange economy with two commodities and two consumers, given an assumption
on the offer curves. However, as already mentioned in the introduction, in a
two-commodity, two-consumer case, Exercise 15.B.6 of Mas-Colell et al. (1995)
present a numerical example of a pure exchange economy such that there are
multiple equilibrium prices. In this economy, the utility functions of the
consumers are both CES type, and thus this economy is not so strange.
Therefore, even in a pure exchange economy with two commodities and two
consumers, it is not easy to find conditions that guarantee the uniqueness of
the equilibrium prices.

On the other hand, this paper provides conditions for uniqueness of
equilibrium price in an economy where there are $L$ commodities, $n$
consumers, and $\mu$ producers. Moreover, unlike the various conditions
discussed above, our conditions are requirements on the utility function and
production set, not on the excess demand function or the offer curve. This is
a feature of this paper in this context. Although it is unclear how strong
Debreu considered this condition to be, we believe that the quasi-linear
assumptions given in this paper are not so strong. Therefore, as far as we can
assess, the theorem in this paper is a good enough result in this context. One
may think that in Theorem 1, only the uniqueness of the equilibrium price is
important, and its local stability is not so important. However, as the proof
shows, local stability is in fact necessary for the proof of the uniqueness.
It is incorporated into one important step of the proof for a completely
different reason than in Arrow et al. (1959).

We would like to discuss this point in some detail. Suppose that $p^{*}$ is
the equilibrium price of a quasi-linear economy, and the production function
$y^{j}(p)$ is differentiable at this point. By an extension of Hotelling’s
lemma, we can show that $y^{j}(p)$ is in fact a gradient vector field of the
profit function $\pi^{j}(p)$. Since the profit function is convex, we have
that $Dy^{j}(p)$ is positive semi-definite. Moreover, for a quasi-linear
economy, we can show by a direct culculation that

$DX(p^{*})=\sum_{i\in N}S_{f^{i}}(p^{*},m^{i}(p^{*})),$ (16)

where $X$ is the sum of the excess demand functions for each consumers. From
these two facts, we can show that $D\zeta(p^{*})$ is negative definite on the
space of all vectors normal to $p^{*}$, and from here, we can obtain the
uniqueness result by using an extension of the Poincaré-Hopf theorem presented
in Mas-Colell (1985).

However, the above is the argument when $y^{j}$ is differentiable at $p^{*}$.
Actually, our assumption P is too weak and can only prove the continuity of
$y^{j}$. Hence, we use approximation by using mollifier, which is frequently
used in Fourier analysis. As is well known, when a convex function is
approximated by a convolution of this with a mollifier, then the approximate
function becomes a smooth convex function. Using this approximation for the
profit function $\pi^{j}(p)$, we can obtain the ‘approximated’ profit
function, and using this function, we construct a smooth approximation of the
excess demand function that satisfies (16), and applying the above theorem, we
obtain the desired result.

However, in order to perform this approximation properly, it must be shown
first that the set of equilibrium prices is discrete. And local stability is
crucial to show this fact. As can be easily seen, if every equilibrium price
is locally stable, then the set of equilibrium prices is discrete under
appropriate normalization. Therefore, we first show that every equilibrium
price is locally stable, and then use the above logic to derive the result.
This is why local stability became necessary for the derivation of our result.

So, why are all equilibrium prices locally stable in a quasi-linear economy?
The answer is obtained by the theory of no-trade equilibria. Balasko (1978,
Theorem 1) showed that in a pure exchange economy, any no-trade equilibrium
price is locally stable. This result was in fact substantially shown in
Kihlstrom et al. (1977, Lemma 1). Namely, they showed that if the initial
endowments become the equilibrium allocation, then (16) holds for the
corresponding equilibrium price. If the economy is not quasi-linear, (16) may
not hold at some equilibrium price because the income effect that arises from
the gap between the initial endowments and the equilibrium allocation has a
non-negligible power. In a quasi-linear economy, however, the income effect
affects only the numeraire good, and when we sum up the excess demand function
of each consumer, the error with (16) is equal to the value of the excess
demand function devided by $p_{L}$. (See Propositions 2-3 and (20) in Step 2
of the proof of Theorem 1.) Thus, this effect is canceled out when the price
is an equilibrium price. As a result, the result that holds at the no-trade
equilibrium price is restored at any equilibrium price.

Hayashi (2017) provided details on studies discussing the relationship between
partial and general equilibrium. In this connection, let us evaluate our
assumption within the partial equilibrium framework. We find that Assumption P
is a sufficiently weak. This assumption prohibits the case where the supply
curve is horizontal, but prohibits almost nothing else. For example, the
supply curve may diverge at finite output level. Or, conversely, it may rise
only to a finite price-level even if output diverges. In contrast, the
assumptions for consumers are a little heavy. Actually, Assumption Q implies
that the range of the demand curve $D(x)$ must be $\mathbb{R}_{++}$. Although
this assumption is strong, we could not remove this in producing general
results. For some special cases, weaker assumptions could be sufficient.

Finally, the study discussed in section 4.1 is related to section 4 of Osana
(1992). This paper, written in Japanese, discusses not only consumer surplus
but also the relationship between this and equivalent and compensating
variations. The relationship between Stokes’ theorem and these results is also
discussed.

##  5 Concluding Remarks

We showed that in a quasi-linear economy, the equilibrium price is uniquely
determined and it is locally stable. Compared with similar results, a feature
of this result is that there is no assumption imposed on the excess demand
function. Moreover, we exhibited that in this economy, consumers’ surplus can
be defined, and coincides with the change of the sum of utilities.

There are a few future tasks. First, we assumed that $u$ is twice continuously
differentiable and the Hessian matrix is negative definite. This is needed for
assuring the differentiability of the demand function. However, it can be
shown by using some approximation techniques that our results still hold for
non-smooth demands.

Second, we prohibit the case in which the boundary of the production set
includes a line without $-\mathbb{R}^{L}_{+}$. This assumption is needed for
preventing the multi-valuedness of $y^{i}(p)$. However, for some accurate
approximation technique, we can overcome this technical difficulty. In
particular, we may be able to show the same result for economies with
technologies of constant return to scale.

##  6 Proofs in Results of Section 3

###  6.1 Lemmas

In this subsection, we show several lemmas.

First, suppose that $E$ is a quasi-linear economy and choose any $s\geq 0$.
Let

$\Omega_{i}^{s}=\Omega_{i}\cap(\mathbb{R}^{L-1}_{+}\times[-s,+\infty[),$
$Y_{j}^{s}=Y_{j}\cap\\{y\in\mathbb{R}^{L}|\|y\|\leq s\\}.$

Consider the modified economy

$E_{s}=(N,M,(\Omega_{i}^{s})_{i\in N},(U_{i})_{i\in N},(\omega^{i})_{i\in
N},(Y_{j}^{s})_{j\in M},(\theta_{ij})_{i\in N,j\in M}).$

Let $f^{i}_{s}$ be the demand function of consumer $i$ in the economy $E_{s}$,
$y^{j}_{s},\pi^{j}_{s}$ be the supply function and the profit function of
producer $j$ in the economy $E_{s}$, $X^{i}_{s}$ be the excess demand function
of consumer $i$ in the economy $E_{s}$, $X_{s}=\sum_{i\in N}X^{i}_{s}$, and
$\zeta_{s}$ be the excess demand function of the economy $E_{s}$.

Lemma 1. Suppose that $E$ is a quasi-linear economy. Then, there exists a
continuously differentiable function
$\tilde{x}^{i}:\mathbb{R}^{L-1}_{++}\to\mathbb{R}^{L-1}_{++}$ such that

$\tilde{y}=\tilde{x}^{i}(\tilde{p})\Leftrightarrow
Du_{i}(\tilde{y})=\tilde{p}.$

Proof. Let $\tilde{p}\in\mathbb{R}^{L-1}_{++}$. By Assumption Q, there exists
a solution $\tilde{x}^{*}\in\mathbb{R}^{L-1}_{++}$ of the following equation:

$Du_{i}(\tilde{x})=\tilde{p}.$ (17)

Consider the following optimization problem:

$\displaystyle\max~{}~{}~{}~{}~{}$
$\displaystyle~{}u_{i}(\tilde{x})-\tilde{p}\cdot\tilde{x}$ subject to.
$\displaystyle~{}\tilde{x}\in\mathbb{R}^{L-1}_{++}.$

Because of either Assumption F or Assumption S1, we have that $u_{i}$ is
strictly concave on $\mathbb{R}^{L-1}_{++}$, and thus 1) any solution to (17)
is also a solution to this problem, and 2) the solution to the above problem
is unique. Therefore, $\tilde{x}^{*}$ is the unique solution to the equation
(17), and thus we can define $\tilde{x}^{i}(\tilde{p})=\tilde{x}^{*}$. Because
$D^{2}u_{i}(\tilde{x}^{*})$ is negative definite, it is regular, and thus by
the inverse function theorem, we have that $\tilde{x}^{i}(\tilde{p})$ is
continuously differentiable. This completes the proof. $\blacksquare$

For any $\tilde{p}\in\mathbb{R}^{L-1}_{++}$ and $m\in\mathbb{R}_{++}$, let

$x_{L}^{i}(\tilde{p},m)=m-\tilde{p}\cdot\tilde{x}^{i}(\tilde{p}).$ (18)

Lemma 2. Suppose that $E$ is a first-type quasi-linear economy and $s\geq 0$,
and choose any $(p,m)\in\mathbb{R}^{L}_{++}\times\mathbb{R}_{++}$. Define
$(q,w)=p_{L}^{-1}(p,m)$. Then,

$f^{i}(p,m)=(\tilde{x}^{i}(\tilde{q}),x_{L}^{i}(\tilde{q},w)),$

and if $x_{L}^{i}(\tilde{q},w)\geq-s$, then

$f^{i}_{s}(p,m)=(\tilde{x}^{i}(\tilde{q}),x_{L}^{i}(\tilde{q},w)).$

Proof. By Assumption F, we have that $u_{i}(\tilde{x})$ is strictly concave on
$\mathbb{R}^{L-1}_{++}$. Note that, because $u_{i}$ is concave on
$\mathbb{R}^{L-1}_{+}$, $U_{i}$ is also concave on $\Omega_{i}$.

Define

$x^{*}=(\tilde{x}^{i}(\tilde{q}),x_{L}^{i}(\tilde{q},w)).$

First, we show the latter claim of this lemma. Suppose that $x_{L}^{*}\geq-s$.
Then, we have that $x^{*}\in\Omega_{i}^{s}$. Moreover,

$p\cdot
x^{*}=p_{L}(\tilde{q}\cdot\tilde{x}^{i}(\tilde{q})+x_{L}^{i}(\tilde{q},w))=p_{L}w=m.$

By Lagrange’s multiplier rule, we have that $x^{*}\in f^{i}_{s}(p,m)$. Suppose
that $y^{*}\in f^{i}_{s}(p,m)$ and $x^{*}\neq y^{*}$. Then,
$U_{i}(x^{*})=U_{i}(y^{*})$. If $\tilde{x}^{*}=\tilde{y}^{*}$, then
$x_{L}^{*}=y_{L}^{*}$ by equation $U_{i}(x^{*})=U_{i}(y^{*})$, which
contradicts that $x^{*}\neq y^{*}$. Thus, we have that
$\tilde{x}^{*}\neq\tilde{y}^{*}$. Define $z(t)=(1-t)x^{*}+ty^{*}$. Then, for
every $t\in[0,1]$, we have that $p\cdot z(t)\leq m$, and thus $U_{i}(z(t))\leq
U_{i}(x^{*})$. Because $U_{i}$ is concave, we have that
$U_{i}(z(t))=U_{i}(x^{*})$ for all $t\in[0,1]$. Set $t_{1}=\frac{1}{2}$ and
$t_{2}=\frac{1}{4}$, and let $z^{*}=z(t_{1}),z^{+}=z(t_{2})$. Then,
$\tilde{z}^{*}\in\mathbb{R}^{L-1}_{++}$, $p\cdot z^{*}\leq m$, and
$U_{i}(z^{*})=U_{i}(x^{*})$. However, because $u_{i}$ is strictly concave on
$\mathbb{R}^{L-1}_{++}$, we have that

$u_{i}(\tilde{z}^{+})>\frac{1}{2}u_{i}(\tilde{x}^{*})+\frac{1}{2}u_{i}(\tilde{z}^{*}),$

which implies that $U_{i}(z^{+})>U_{i}(x^{*})$. This contradicts that
$U_{i}(z^{+})=U_{i}(x^{*})$. Therefore, $f^{i}_{s}(p,m)=\\{x^{*}\\}$, as
desired.

Next, we show the former claim of this lemma. Clearly $x^{*}\in\Omega_{i}$.
Again by Lagrange’s multiplier rule, we have that $x^{*}\in f^{i}(p,m)$. If
$y^{*}\in f^{i}(p,m)$ and $x^{*}\neq y^{*}$, then choose $s>0$ so large that
$x^{*},y^{*}\in\Omega_{i}^{s}$. Then, $y^{*}\in f^{i}_{s}(p,m)$, which is a
contradiction. This completes the proof. $\blacksquare$

Lemma 3. Suppose that $E$ is a first-type quasi-linear economy, and $s\geq 0$.
Choose any $(p,m)\in\mathbb{R}^{L}_{++}\times\mathbb{R}_{++}$, and define

$x^{*}=(\tilde{x}^{i}(p_{L}^{-1}\tilde{p}),x_{L}^{i}(p_{L}^{-1}\tilde{p},p_{L}^{-1}m)).$

Suppose that $x^{*}_{L}<-s$. Let $\bar{m}=m+p_{L}s$, and
$\tilde{x}^{+}\in\mathbb{R}^{L-1}_{++}$ be a solution to the problem

$\displaystyle\max~{}~{}~{}~{}~{}$ $\displaystyle~{}u_{i}(\tilde{x})$ subject
to. $\displaystyle~{}\tilde{x}\in\mathbb{R}^{L-1}_{+},$
$\displaystyle~{}\tilde{p}\cdot\tilde{x}\leq\bar{m},$

and define $x_{L}^{+}=-s$. Then,
$f^{i}_{s}(p,m)=(\tilde{x}^{+},x_{L}^{+})$.777Note that, the existence of such
$\tilde{x}^{+}$ is assumed in Assumption Q.

Proof. First, because of Assumption F, we have that $u_{i}$ is continuous and
strictly concave on $\mathbb{R}^{L-1}_{++}$. Therefore, we have that if
$\tilde{y}\in\mathbb{R}^{L-1}_{+}$ and $\tilde{p}\cdot\tilde{y}\leq\bar{m}$,
then either $\tilde{y}=\tilde{x}^{+}$ or
$u_{i}(\tilde{y})<u_{i}(\tilde{x}^{+})$. Moreover, because $u_{i}$ is
increasing on $\mathbb{R}^{L-1}_{++}$, we have that
$\tilde{p}\cdot\tilde{x}^{+}=\bar{m}$. Because $f^{i}(p,m)=x^{*}$, we have
that $U_{i}(x^{*})>U_{i}(x^{+})$.

Suppose that there exists $y^{+}\in\Omega_{i}^{s}$ such that $x^{+}\neq
y^{+}$, $p\cdot y^{+}\leq m$ and $U_{i}(y^{+})\geq U_{i}(x^{+})$. Then,
$\tilde{p}\cdot\tilde{y}^{+}\leq\bar{m}$. If $y_{L}^{+}=-s$, then we have that
$\tilde{y}^{+}\neq\tilde{x}^{+}$, and thus
$u_{i}(\tilde{y}^{+})<u_{i}(\tilde{x}^{+})$, which is a contradiction.
Therefore, we have that $y_{L}^{+}>-s$, and there exists $t\in]0,1[$ such that
for $z^{+}=(1-t)x^{*}+ty^{+}$, $z_{L}^{+}=-s$. Because $u_{i}$ is concave on
$\mathbb{R}^{L-1}_{+}$, we have that
$u_{i}(\tilde{z}^{+})\geq(1-t)u_{i}(\tilde{x}^{*})+tu_{i}(\tilde{y}^{+})$,
which implies that $U_{i}(z^{+})>U_{i}(x^{+})$, and thus
$u_{i}(\tilde{z}^{+})>u_{i}(\tilde{x}^{+})$. However,
$\tilde{p}\cdot\tilde{z}^{+}\leq m+p_{L}s=\bar{m}$, which contradicts the
definition of $\tilde{x}^{+}$. Thus, we have that
$f^{i}_{s}(p,m)=\\{x^{+}\\}$, as desired. This completes the proof.
$\blacksquare$

Lemma 4. Suppose that $E$ is a second-type quasi-linear economy, and $\hat{E}$
is the first-type quasi-linear economy such that all components except for
$\Omega_{i}$ are the same as $E$. Then, the demand function $f^{i}$ in $E$
coincides with $f^{i}_{0}$, where $f^{i}_{0}$ is the demand function in
$\hat{E}_{0}$.

Proof. Obvious by definition. $\blacksquare$

Lemma 5. Suppose that $E$ is a quasi-linear economy. If $y^{j}(p)$ is
nonempty, then it is a singleton. The same holds for $y^{j}_{s}(p)$ for $s>0$.
Moreover, $y^{j}_{s}$ is a single-valued continuous function defined on
$\mathbb{R}^{L}_{++}$, and the profit function $\pi^{j}_{s}$ is convex and
satisfies $D\pi^{j}_{s}(p)=y^{j}_{s}(p)$ for all
$p\in\mathbb{R}^{L}_{++}$.888This is a variety of Hotelling’s lemma.

Proof. First, suppose that $y^{1},y^{2}\in y^{j}(p)$ and $y^{1}\neq y^{2}$.
Define $y^{3}=\frac{1}{2}(y^{1}+y^{2})$. Because $p\cdot y^{3}=p\cdot
y^{1}+p\cdot y^{2}$, we have that $y^{3}\in y^{j}(p)$. Because $0\in Y_{j}$,
$p\cdot y^{1}=p\cdot y^{2}\geq 0$, and thus either $y^{i}=0$ or
$y^{i}\notin-\mathbb{R}^{L}_{+}$ for $i\in\\{1,2\\}$. If $y^{1}\neq 0$ and
$y^{2}\neq 0$, then $y^{3}$ is in the interior of $Y_{j}$. This implies that
$y^{3}\notin y^{j}(p)$, which is a contradiction. Therefore, either $y^{1}=0$
or $y^{2}=0$. We assume that $y^{1}=0$, and define
$y^{4}=\frac{1}{2}(y^{3}+y^{2})$. Then, $y^{4}\in y^{j}(p)$. Because
$y^{2}\notin-\mathbb{R}^{L}_{+}$, we have that
$y^{3}\notin-\mathbb{R}^{L}_{+}$, and thus $y^{4}$ is in the interior of
$Y_{j}$. This implies that $y^{4}\notin y^{j}(p)$, which is a contradiction.
Therefore, $y^{j}(p)$ is either a singleton or the empty set.

The same proof can be applied for $y^{j}_{s}(p)$. Because $Y_{j}^{s}$ is
compact, we have that $y^{j}_{s}(p)$ is always single-valued. By Berge’s
maximum theorem, we have that $y^{j}_{s}(p)$ is continuous.

Choose any $p,q\in\mathbb{R}^{L}_{++}$ and let $0\leq t\leq 1$ and
$r=(1-t)p+tq$. Then,

$r\cdot y^{j}_{s}(r)=(1-t)p\cdot y^{j}_{s}(r)+tq\cdot
y^{j}_{s}(r)\leq(1-t)p\cdot y^{j}_{s}(p)+tq\cdot y^{j}_{s}(q),$

which implies that

$\pi^{j}_{s}(r)\leq(1-t)\pi^{j}_{s}(p)+t\pi^{j}_{s}(q).$

Therefore, $\pi^{j}_{s}$ is a convex function. Recall that $e_{\ell}$ denotes
the $\ell$-th unit vector, and let $q=p+he_{\ell}$. Then, $p\cdot
y^{j}_{s}(q)\leq p\cdot y^{j}_{s}(p)$, and thus

$\displaystyle\pi^{j}_{s}(q)-\pi^{j}_{s}(p)=$ $\displaystyle~{}q\cdot
y^{j}_{s}(q)-p\cdot y^{j}_{s}(p)$ $\displaystyle=$
$\displaystyle~{}p\cdot(y^{j}_{s}(q)-y^{j}_{s}(p))+hy^{j}_{s,\ell}(q)$
$\displaystyle\leq$ $\displaystyle~{}hy^{j}_{s,\ell}(q).$

This implies that

$\lim_{h\uparrow 0}\frac{\pi^{j}_{s}(q)-\pi^{j}_{s}(p)}{h}\geq
y^{j}_{s,\ell}(p)\geq\lim_{h\downarrow
0}\frac{\pi^{j}_{s}(q)-\pi^{j}_{s}(p)}{h}.$

Because $\pi^{j}_{s}$ is convex, the above inequalities turn into the
equalities, and thus

$D\pi^{j}_{s}(p)=y^{j}_{s}(p),$

as desired. This completes the proof. $\blacksquare$

Lemma 6. Suppose that $E$ is a quasi-linear economy. Define

$B_{\omega}=\\{(x,y)\in\bar{A}_{\omega}|U_{i}(x^{i})\geq
U_{i}(\omega^{i})\mbox{ for each }i\in N\\}.$

Then, $B_{\omega}$ is compact.

Proof. If $E$ is a second-type quasi-linear economy, then this property can be
shown by the usual arguments.999See, for example, subsection 3.3.1 of Arrow
and Debreu (1954). Hence, we assume that $E$ is a first-type quasi-linear
economy.

Suppose that $B_{\omega}$ is not compact. Because $B_{\omega}$ is closed, this
implies that $B_{\omega}$ is unbounded. Therefore, there exists a sequence
$(x^{k},y^{k})$ in $B_{\omega}$ such that $\|(x^{k},y^{k})\|\to\infty$ as
$k\to\infty$.

First, suppose that there exists $s\geq 0$ such that $x^{ik}_{L}\geq-s$ for
all $i$ and all sufficiently large $k$. Then, we can assume that for all $k$,
$(x^{k},y^{k})$ is admissible in the economy $E_{s}$. Thus, by the same
argument as in subsection 3.3.1 of Arrow and Debreu (1954), we can show that
$(x^{k},y^{k})$ is bounded, which is a contradiction. Therefore, we can assume
that there exists $i\in N$ such that $x^{ik}_{L}$ is unbounded from below. Let
$I$ be the set of such $i$, and by taking a subsequence, we can assume that
for all $i\in I$, $x^{ik}_{L}\to-\infty$ as $k\to\infty$. Define
$y^{k*}=\sum_{j\in M}y^{jk}$, and

$\tilde{z}^{k}=\sum_{i\in N}\tilde{\omega}^{i}+\tilde{y}^{k*}.$

Then, $u_{i}(\tilde{x}^{k})\leq u_{i}(\tilde{z}^{k})$. Choose $p^{+}$ in the
assumption P. Then,

$p^{+}\cdot y^{k*}\leq\sum_{j\in M}\pi^{j}(p^{+})\equiv m^{*},$

which implies that

$\tilde{p}^{+}\cdot\tilde{y}^{k*}\leq m^{*}-p^{+}_{L}y^{k*}_{L}.$

For $i\in N\setminus I$, define $w^{i}=\inf_{k}x^{ik}_{L}$. Moreover, define

$x^{k*}_{L}=\sum_{i\in I}x^{ik}_{L},\ w^{*}=\sum_{i\in
N}\omega^{i}_{L}-\sum_{i\in N\setminus I}w^{i}.$

Because

$\sum_{i\in N}x^{ik}_{L}=\sum_{i\in N}\omega^{i}_{L}+y^{k*}_{L},$

we have that

$x^{k*}_{L}-w^{*}\leq y^{k*}_{L}.$

Now, choose any $i\in I$, and let $g^{i}(\tilde{p},m)$ be the solution to (7).
Then, by the above arguments,

$u_{i}(\tilde{z}^{k})\leq
u_{i}(g^{i}(\tilde{p}^{+},m^{+}(p^{+}_{L}x^{k*}_{L}))),$

where

$m^{+}(a)=m^{*}+\sum_{i\in
N}\tilde{p}^{+}\cdot\tilde{\omega}^{i}+p_{L}^{+}w^{*}-a.$

Define $c_{i}(m)=u_{i}(g^{i}(\tilde{p}^{+},m))$. Because (7) has an inner
solution, we have that $c_{i}$ is continuously differentiable. Moreover, by
envelope theorem, we have that $c_{i}^{\prime}(m)=\frac{\frac{\partial
u_{i}}{\partial x_{1}}(g^{i}(\tilde{p}^{+},m))}{p_{1}^{+}}$. Choose any
$t\in[0,1]$ and $m_{1},m_{2}>0$ with $m_{1}<m_{2}$, and let
$m_{3}=(1-t)m_{1}+tm_{2}$. Because $u_{i}$ is concave, we have that

$c_{i}(m_{3})\geq
u_{i}((1-t)g^{i}(\tilde{p}^{+},m_{1})+tg^{i}(\tilde{p}^{+},m_{2}))\geq(1-t)c_{i}(m_{1})+tc_{i}(m_{2}),$

which implies that $c_{i}$ is concave. By assumption Q, there exists
$\tilde{x}^{i*}\in\mathbb{R}^{L-1}_{++}$ such that
$Du_{i}(\tilde{x}^{i*})=\frac{1}{2|I|p_{L}^{+}}\tilde{p}^{+}$, where $|I|$
denotes the number of the elements included in $I$. Note that, because of
Lagrange’s multiplier rule,
$\tilde{x}^{i*}=g^{i}(\tilde{p}^{+},\tilde{p}^{+}\cdot\tilde{x}^{i*})$. If
$m_{0}=\max_{i\in I}\tilde{p}^{+}\cdot\tilde{x}^{i*}$, then we have that
$c_{i}^{\prime}(m)\leq\frac{1}{2|I|p_{L}^{+}}$ for all $i\in I$ and $m\geq
m_{0}$. Because $x^{k*}_{L}\to-\infty$, we have that
$m^{+}(p_{L}^{+}x^{k*}_{L})\geq m_{0}$ for any sufficiently large $k$, and
thus

$\displaystyle\sum_{i\in I}U_{i}(x^{ik})=$ $\displaystyle~{}\sum_{i\in
I}u_{i}(\tilde{x}^{ik})+x^{k*}_{L}$ $\displaystyle\leq$
$\displaystyle~{}\sum_{i\in I}u_{i}(\tilde{z}^{k})+x^{k*}_{L}$
$\displaystyle\leq$ $\displaystyle~{}\sum_{i\in
I}c_{i}(m^{+}(p^{+}_{L}x^{k*}_{L}))+x^{k*}_{L}\to-\infty$

as $k\to\infty$, which contradicts the definition of $B_{\omega}$. This
completes the proof. $\blacksquare$

Lemma 7. Suppose that $E$ is a quasi-linear economy and choose any $s>0$.
Then, $\zeta_{s}$ is a single-valued function that satisfies the following.

  1. 1)

$\zeta_{s}$ is continuous and satisfies (12) and (13).

  2. 2)

There exists $S>0$ such that $\zeta_{s,j}(p)>-S$ for every
$p\in\mathbb{R}^{L}_{++}$ and $j\in\\{1,...,L\\}$.

  3. 3)

If $(p^{k})$ is a sequence in $\mathbb{R}^{L}_{++}$ such that $p^{k}\to p\neq
0$ as $k\to 0$ and the set $J=\\{\ell|p_{\ell}=0\\}$ is nonempty, then
$\|\zeta_{s}(p^{k})\|\to\infty$ as $k\to\infty$.

Proof. Recall again that $e_{\ell}$ denotes the $\ell$-th unit vector. First,
we treat a first-type quasi-linear economy. By Lemmas 2-3, we have that
$f^{i}_{s}$ is a single-valued function. Because of Berge’s maximum theorem,
we have that $f^{i}_{s}$ is continuous. Moreover, by Lemma 4, $y^{j}_{s}$ is
also single-valued and continuous. Therefore, $\zeta_{s}$ is single-valued and
continuous.

It is easy to prove that $\zeta_{s}$ satisfies (12) and (13), and thus we omit
the proof of this fact. Thus, $\zeta_{s}$ satisfies 1).

Because $\zeta_{s}(p)\gg-\sum_{i\in
N}\omega^{i}-((\mu+1)s+1,(\mu+1)s+1,...,(\mu+1)s+1)$ for all
$p\in\mathbb{R}^{L}_{++}$, we have that $\zeta_{s}$ satisfies 2).

Therefore, it suffices to show that 3) holds for $\zeta_{s}$. Suppose that
$(p^{k})$ is a sequence in $\mathbb{R}^{L}_{++}$ such that $p^{k}\to p\neq 0$
as $k\to 0$ and the set $J=\\{\ell|p_{\ell}=0\\}$ is nonempty, but
$\|\zeta_{s}(p^{k})\|\not\to\infty$. By taking a subsequence, we can assume
that $\zeta_{s}(p^{k})\to x$ as $k\to\infty$ for some $x\in\mathbb{R}^{L}$.
Define

$z^{k}=\zeta_{s}(p^{k}),\ y^{jk}=y^{j}_{s}(p^{k}),$
$x^{ik}=f^{i}_{s}(p^{k},p^{k}\cdot\omega^{i}+\sum_{j\in
M}\theta_{ij}p^{k}\cdot y^{jk}).$

Because $(y^{jk})$ is bounded for each $j\in\\{1,...,\mu\\}$, we must have
that $(x^{ik})$ is also bounded for each $i\in\\{1,...,n\\}$, and thus we can
assume that $x^{ik}\to x^{i}$ and $y^{jk}\to y^{j}$ as $k\to\infty$. Note that
$p^{k}\cdot y^{jk}\geq 0$, and thus $p\cdot y^{j}\geq 0$.

Suppose that $p_{L}=0$. Because $p\cdot x^{i}=p\cdot\omega^{i}+\sum_{j\in
M}\theta_{ij}p\cdot y^{j}$ and $\sum_{i\in N}\omega^{i}\gg 0$, we have that
there exists $i$ such that $x^{i}_{\ell}>0$ for some $\ell$ with $p_{\ell}>0$.
Define $v^{i}=x^{i}-\varepsilon e_{\ell}+e_{L}$, where $\varepsilon>0$ is
sufficiently small that $U_{i}(v^{i})>U_{i}(x^{i})$. Then,
$U_{i}(v^{i})>U_{i}(x^{ik})$ and $p^{k}\cdot v^{i}<p^{k}\cdot x^{ik}$ for some
$k$, which is a contradiction.

Therefore, we have that $p_{L}>0$. Next, suppose that for some $i$,
$u_{i}(\tilde{x}^{i})=u_{i}(0)$. Because $U_{i}(x^{i})\geq U_{i}(\omega^{i})$,
we have that $x^{i}_{L}\geq 0>-s$. Fix any
$M^{+}>2\frac{\|\tilde{p}\|}{p_{L}}$. By Assumption Q, there exists
$\tilde{x}^{+}\in\mathbb{R}^{L-1}_{++}$ such that $\frac{\partial
u_{i}}{\partial x_{j}}(\tilde{x}^{+})>M^{+}$ for all $j$. Define
$\tilde{v}=\frac{1}{\|\tilde{x}^{+}\|}\tilde{x}^{+}$ and
$v_{L}=-\frac{2}{p_{L}}(\tilde{p}\cdot\tilde{v})$. Because $u_{i}$ is strictly
concave and increasing on $\mathbb{R}^{L-1}_{++}$, we have that
$g(t)=u_{i}(t\tilde{v})$ is increasing and $\lim_{t\downarrow
0}g^{\prime}(t)>M^{+}$. Thus, there exists $t>0$ such that for
$w^{i}=(0,x^{i}_{L})+tv$, $U_{i}(w^{i})>U_{i}(x^{i})$, $w^{i}_{L}>-s$, and
$p\cdot w^{i}<p\cdot x^{i}$. This implies that $U_{i}(w^{i})>U_{i}(x^{ik})$
and $p^{k}\cdot w^{i}<p^{k}\cdot x^{ik}$ for some $k$, which is a
contradiction. Therefore, for every $i$, $u_{i}(\tilde{x}^{i})>u_{i}(0)$.
Choose any $\ell$ such that $p_{\ell}=0$. Because $\zeta_{s}$ satisfies (12),
there exists $i$ and $\ell^{\prime}$ such that $p_{\ell^{\prime}}>0$ and
$x^{i}_{\ell^{\prime}}>0$. Then, for $v^{i}=x^{i}+e_{\ell}-\varepsilon
e_{\ell^{\prime}}$, we have that $v^{i}_{\ell^{\prime}}>0$ and
$U_{i}(v^{i})>U_{i}(x^{i})$ if $\varepsilon>0$ is sufficiently small. Because
$p\cdot v^{i}<p\cdot x^{i}$, we have that there exists $k$ such that
$U_{i}(v^{i})>U_{i}(x^{ik})$ and $p^{k}\cdot v^{i}<p^{k}\cdot x^{ik}$, which
is a contradiction. This completes the proof in the case of first-type quasi-
linear economies. Because almost the same arguments can be executed for
second-type quasi-linear economies, we omit the proof of such cases.
$\blacksquare$

Lemma 8. Suppose that $\xi:\mathbb{R}^{L}_{++}\to\mathbb{R}^{L}$ is a
continuous function, and define

$S^{*}=\\{p\in\mathbb{R}^{L}_{++}|\|p\|=1\\}.$

Suppose also that $\xi$ satisfies the following five properties:

  1. 1)

The function $\xi$ satisfies (12) and (13).

  2. 2)

There exists $s>0$ such that $\xi_{j}(p)>-s$ for every
$p\in\mathbb{R}^{L}_{++}$ and $j\in\\{1,...,L\\}$.

  3. 3)

If $(p^{k})$ is a sequence in $\mathbb{R}^{L}_{++}$ such that $p^{k}\to p\neq
0$ as $k\to 0$ and the set $J=\\{j|p_{j}=0\\}$ is nonempty, then
$\|\xi(p^{k})\|\to\infty$ as $k\to\infty$.

  4. 4)

If $\xi(p)=0$, then $\xi$ is continuously differentiable around $p$.

  5. 5)

If $\xi(p)=0$, then

$\chi(p)=\begin{vmatrix}D\xi(p)&p\\\ p^{T}&0\end{vmatrix}\neq 0.$

Define $E=\xi^{-1}(0)\cap S^{*}$, and for $p\in E$,

$\mbox{index}(p)=\begin{cases}+1&\mbox{if }(-1)^{L}\chi(p)>0,\\\ -1&\mbox{if
}(-1)^{L}\chi(p)<0.\end{cases}$

Then, the set $E$ is finite, and

$\sum_{p\in E}\mbox{index}(p)=+1.$

Proof. Omitted. See Propositions 5.3.3, 5.3.4, and 5.6.1 of Mas-Colell (1985).

###  6.2 Proof of Proposition 1

If $E$ is first-type, then by Lemma 2, we have that $f^{i}$ is single-valued
and continuously differentiable, and
$\tilde{f}^{i}(p,m)\in\mathbb{R}^{L-1}_{++}$. If $E$ is second-type, then by
Lemmas 2-4, $f^{i}$ is single-valued and
$\tilde{f}^{i}(p,m)\in\mathbb{R}^{L-1}_{++}$, and by Berge’s maximum theorem,
$f^{i}$ is continuous.

It is clear that $f^{i}$ is homogeneous of degree zero. Because $U_{i}$ is
locally non-satiated, we have that $f^{i}$ satisfies Walras’ law. This
completes the proof. $\blacksquare$

###  6.3 Proof of Proposition 2

Let $\tilde{\Omega}_{i}$ denote the interior of $\Omega_{i}$. Suppose that
$x=f^{i}(p,m)$ and either the economy is first-type or $x_{L}>0$. Then,
$x\in\tilde{\Omega}_{i}$, and because of Lemmas 2 and 4, we have that there
exists an open neighborhood $V$ of $(p,m)$ such that if $(q,w)\in V$, then

$f^{i}(q,w)=(\tilde{x}^{i}(q_{L}^{-1}\tilde{q}),x_{L}^{i}(q_{L}^{-1}\tilde{q},q_{L}^{-1}w)),$

where $x_{L}^{i}$ is defined in (18). Therefore, $f^{i}$ is continuously
differentiable on $V$ and (11) holds. This completes the proof. $\blacksquare$

###  6.4 Proof of Proposition 3

First, choose an open neighborhood $U$ of $(p,m)$ such that $f^{i}$ is single-
valued and continuously differentiable on $U$. Because of Proposition 1, we
have that

$q\cdot f^{i}(q,w)=w$

for every $(q,w)\in U$ and

$f^{i}(ap,am)=f^{i}(p,m)$

for every $a>0$. Hence, by differentiation,

$f^{i}_{j}(p,m)+\sum_{k=1}^{L}p_{k}\frac{\partial f^{i}_{k}}{\partial
p_{j}}(p,m)=0,$ $p\cdot\frac{\partial f^{i}}{\partial m}(p,m)=1,$
$\sum_{j=1}^{L}p_{j}\frac{\partial f^{i}_{k}}{\partial
p_{j}}(p,m)+m\frac{\partial f^{i}_{k}}{\partial m}(p,m)=0,$

and thus we have

$p^{T}S_{f^{i}}(p,m)=0^{T},\ S_{f^{i}}(p,m)p=0.$

Second, for $x=f^{i}(p,m)$, define

$E^{x}_{i}(q)=\inf\\{q\cdot y|U_{i}(y)\geq U_{i}(x)\\}.$

If the economy $E$ is second-type, we can apply Lemma 1 of Hosoya (2020)
directly, and obtain that $E^{x}_{i}$ is concave and positive, and the
following equation (Shephard’s lemma) holds:

$DE^{x}_{i}(q)=f^{i}(q,E^{x}_{i}(q)),\ E^{x}_{i}(p)=m.$ (19)

Suppose that the economy is first-type. First, we show that for every
$q\in\mathbb{R}^{L}_{++}$, $E^{x}(q)$ is finite. Define $r=\frac{1}{q_{L}}q$,
and

$x^{*}=(\tilde{x}^{i}(r),U_{i}(x)-u_{i}(\tilde{x}^{i}(r))).$

Then, $U_{i}(x^{*})=U_{i}(x)$. Let $q\cdot x^{*}=w$. By lemma 2,
$x^{*}=f^{i}(q,w)$. Therefore, if $q\cdot y\leq w$ and $y\neq x^{*}$, then
$U_{i}(y)<U_{i}(x)$. This implies that $E^{x}_{i}(q)=w>-\infty$.

If $q^{1},q^{2}\in\mathbb{R}^{L}_{++}$ and $0\leq t\leq 1$, then $U_{i}(y)\geq
U_{i}(x)$ implies that

$[(1-t)q^{1}+tq^{2}]\cdot y=(1-t)q^{1}\cdot y+tq^{2}\cdot
y\geq(1-t)E^{x}(q^{1})+tE^{x}(q^{2}),$

and thus, we have that

$E^{x}((1-t)q^{1}+tq^{2})\geq(1-t)E^{x}(q^{1})+tE^{x}(q^{2}).$

This implies that $E^{x}$ is concave. Because every concave function defined
on $\mathbb{R}^{L}_{++}$ is continuous, we have that $E^{x}$ is continuous. It
is easy to show that $E^{x}_{i}(p)=m$, and thus there exists a neighborhood
$U$ of $p$ such that $E^{x}_{i}(q)>0$ for all $q\in U$. By the same proof as
in that of Lemma 1 of Hosoya (2020), again we can show that (19) holds on $U$.
Therefore, under our assumtions, we have that (19) holds on some neighborhood
of $p$. Because $f^{i}$ is continuously differentiable around $(p,m)$, to
differentiate both sides of (19), we have that

$D^{2}E^{x}_{i}(p)=S_{f^{i}}(p,m).$

Therefore, (S) holds because of Young’s theorem, and $S_{f^{i}}(p,m)$ is
negative semi-definite.

Third, let $\tilde{\Omega}_{i}$ denote the interior of $\Omega_{i}$, and for
$y\in\tilde{\Omega}_{i}$, define

$g^{i}(y)=DU_{i}(y),$

and for $j,k\in\\{1,...,L-1\\}$, define

$a^{i}_{jk}(y)=\frac{\partial g^{i}_{j}}{\partial x_{k}}(y)-\frac{\partial
g^{i}_{j}}{\partial x_{n}}(y)g^{i}_{k}(y).$

The $(L-1)\times(L-1)$ matrix-valued function
$A_{g^{i}}(y)=(a^{i}_{jk}(y))_{j,k=1}^{L-1}$ is called the Antonelli matrix.
Samuelson (1950) showed that if $y=f^{i}(q,w)\in\tilde{\Omega}_{i}$, then
$A_{g^{i}}(y)$ is regular, and the inverse matrix of $A_{g^{i}}(y)$ coincides
with $(s_{jk}^{i}(q,w))_{j,k=1}^{L-1}$. By assumption of this proposition, we
have that if $x=f^{i}(p,m)$, then $x\in\tilde{\Omega}_{i}$, and thus, (R)
holds. Moreover, because $S_{f^{i}}(p,m)$ is symmetric, there exists an
orthogonal matrix $P$ such that

$S_{f^{i}}(p,m)=P^{T}\begin{pmatrix}\lambda_{1}&0&...&0\\\
0&\lambda_{2}&...&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&...&\lambda_{L}\end{pmatrix}P,$

where each $\lambda_{j}\leq 0$ is an eigenvalue of $S_{f^{i}}(p,m)$. Because
of (R), we have that there exists exactly one $j$ such that $\lambda_{j}=0$,
and $\lambda_{k}<0$ whenever $k\neq j$. This implies that (ND) holds, which
completes the proof. $\blacksquare$

###  6.5 Proof of Proposition 4

This is just an immediate consequence of Lemma 6. $\blacksquare$

###  6.6 Proof of Proposition 5

This has already been shown in Lemma 5. $\blacksquare$

###  6.7 Proof of Proposition 6

To show (12) and (13) is easy, and thus we omit its proof. By Lemma 6, we have
that $B_{\omega}$ is compact. Choose any $s>\sup\\{\|(x,y)\||(x,y)\in
B_{\omega}\\}$, and consider the economy $E_{s}$. By Lemma 7 and Proposition
17.C.1 of Mas-Colell et al. (1995), we have that there exists an equilibrium
price $p^{*}$ of $E_{s}$. By Lemma 5, $y^{j}_{s}$ is defined on
$\mathbb{R}^{L}_{++}$ and single-valued. Define

$y^{j}=y^{j}_{s}(p^{*}),\
x^{i}=f^{i}_{s}(p^{*},p^{*}\cdot\omega^{i}+\sum_{j\in
M}\theta_{ij}\pi^{j}_{s}(p^{*})).$

Because $0\in Y^{j}$, we have that $\pi^{j}_{s}(p^{*})\geq 0$, and thus,

$U_{i}(x^{i})\geq U_{i}(\omega^{i}).$

Moreover,

$\sum_{i\in N}x^{i}=\sum_{i\in N}\omega^{i}+\sum_{j\in M}y^{j}.$

This implies that $(x,y)\in B_{\omega}$, and thus $x^{i}_{L}>-s$ for all $i\in
N$ and $\|y^{j}\|<s$. We can easily check that the latter implies that
$y^{j}=y^{j}(p^{*})$, and thus $\pi^{j}_{s}(p^{*})=\pi^{j}(p^{*})$.
Therefore,101010Recall that $m^{i}(p)=p\cdot\omega^{i}+\sum_{j\in
M}\theta_{ij}\pi^{j}(p)$.

$x^{i}=f^{i}(p^{*},m^{i}(p^{*})),$

which implies that $\zeta(p^{*})=0$ and $p^{*}$ is an equilibrium price of the
economy $E$. This completes the proof. $\blacksquare$

###  6.8 Proof of Proposition 7

First, note that if $x^{i}=f^{i}(p^{*},m^{i}(p^{*}))$ and
$y^{j}=y^{j}(p^{*})$, then

$\sum_{i\in N}x^{i}=\sum_{i\in N}\omega^{i}+\sum_{j\in M}y^{j}.$

Therefore, we have that $(x,y)\in B_{\omega}$, and thus

$u_{i}(\tilde{x}^{i})\leq
u_{i}(\tilde{\omega}^{i})+\alpha_{i}<U_{i}(\omega^{i}).$

Because $U_{i}(x^{i})\geq U_{i}(\omega^{i})$, we have that $x^{i}_{L}>0$. This
completes the proof. $\blacksquare$

###  6.9 Proof of Theorem 1

First, suppose that this theorem holds for any first-type quasi-linear
economy, and $E$ is a second-type quasi-linear economy. Let $\hat{E}$ be a
first-type quasi-linear economy in which all components except for
$\Omega_{i}$ are the same as that of $E$. By definition, the supply function
of the $j$-th producer in the economy $\hat{E}$ is the same as $y^{j}(p)$.
Moreover, if $p^{*}$ is an equilibrium of $\hat{E}$, then for the excess
demand function $\hat{X}^{i}$ of the $i$-th consumer in the economy $\hat{E}$,
by Lemmas 2-4 and Proposition 7,

$\hat{X}^{i}(p^{*})\in\mathbb{R}^{L}_{++}$

which implies that $X^{i}(p)=\hat{X}^{i}(p)$ on some neighborhood of $p^{*}$.
By the same arguments, if $p^{*}$ is an equilibrium of $E$, then
$X^{i}(p)=\hat{X}^{i}(p)$ on some neighborhood of $p^{*}$. Therefore, by our
initial assumption, we obtain this theorem holds for this economy $E$. Hence,
it suffices to show that this theorem holds for any first-type quasi-linear
economy, and hereafter, we assume that $E$ is first-type.

We separate the proof into several steps.

Step 1. There exists $s>0$ such that for every equilibrium price $p^{*}$,
there exists a neighborhood $U$ of $p^{*}$ such that the following results
hold:

  1. 1)

The set of equilibrium prices in $E$ is the same as that of $E_{s}$,

  2. 2)

For every equilibrium price $p^{*}$, there exists a neighborhood $U$ of
$p^{*}$ such that if $p\in U$, then $\pi^{j}(p)=\pi^{j}_{s}(p)$ and
$X^{i}(p)=X^{i}_{s}(p)$.

Proof of Step 1. By Lemma 6, $B_{\omega}$ is compact, and thus choose
$s>\sup\\{\|(x,y)\||(x,y)\in B_{\omega}\\}$. Suppose that $p^{*}$ is an
equilibrium price in $E$. Define $y^{j}=y^{j}(p^{*})$ and
$x^{i}=f^{i}(p^{*},m^{i}(p^{*}))$. Then,

$\sum_{i\in N}x^{i}=\sum_{i\in N}\omega^{i}+\sum_{j\in N}y^{j}.$

Moreover, because $0\in Y^{j}$ for each $j$, we have that
$\pi_{j}(p^{*})=p^{*}\cdot y^{j}(p^{*})\geq 0$. Therefore, $p^{*}\cdot
x^{i}\geq p^{*}\cdot\omega^{i}$, and thus

$U_{i}(x^{i})\geq U_{i}(\omega^{i}),$

which implies that $(x,y)\in B_{\omega}$, and thus $x^{i}_{L}>-s$ and
$\|y^{j}\|<s$. Therefore, we have that $y^{j}=y^{j}_{s}(p^{*})$ and
$x^{i}-\omega^{i}=X^{i}_{s}(p^{*})$, and thus $p^{*}$ is an equilibrium price
in $E_{s}$. By the symmetrical proof, we can show that any equilibrium price
in $E_{s}$ is that in $E$, and 1) holds.

Next, choose any equilibrium price $p^{*}$. By Lemma 5, $y^{j}_{s}$ is defined
on $\mathbb{R}^{L}_{++}$ and continuous, and thus there exists an open
neighborhood $U^{\prime}$ of $p^{*}$ such that if $p\in U^{\prime}$, then
$\|y^{j}_{s}(p)\|<s$ for all $j\in M$. Suppose that $y^{j}_{s}(p)\notin
y^{j}(p)$ for some $j\in M$ and $p\in U^{\prime}$. Then, there exists $y\in
Y^{j}$ such that $\|y\|>s$ and $p\cdot y>p\cdot y^{j}_{s}(p)$. Define
$y(t)=(1-t)y^{j}_{s}(p)+ty$. Then, for sufficiently small $t>0$, $\|y(t)\|<s$
and $p\cdot y(t)>p\cdot y^{j}_{s}(p)$, which is a contradiction. Therefore,
$y^{j}_{s}(p)=y^{j}(p)$ for all $p\in U$ and $j\in M$. Then, there exists an
open neighborhood $U\subset U^{\prime}$ of $p^{*}$ such that if $p\in U$, then
$X^{i}_{s,L}(p)+\omega^{i}_{L}>-s$. By Lemma 2, we have that
$X^{i}_{s}(p)=X^{i}(p)$ for all $p\in U$. This completes the proof of Step 1.
$\blacksquare$

Step 2. For every $p\in\mathbb{R}^{L}_{++}$, if $y^{j}(q)$ is defined on some
neighborhood of $p$ for all $j\in M$, then

$\frac{\partial X^{i}_{\ell}}{\partial p_{k}}(p)=\begin{cases}s^{i}_{\ell
k}(p,m^{i}(p))&\mbox{if }\ell\neq L,\\\ s^{i}_{\ell
k}(p,m^{i}(p))-\frac{X^{i}_{k}(p)-\sum_{j\in
M}\theta_{ij}y^{j}_{k}(p)}{p_{L}}&\mbox{if }\ell=L.\end{cases}$ (20)

Proof of Step 2. First, by almost the same arguments in the proof of Step 1,
we can show that $y^{j}(q)=y^{j}_{s}(q)$ on some neighborhood of $p$. By Lemma
5, we have that $D\pi^{j}(q)=y^{j}(q)$ on this neighborhood.

If $\ell\neq L$, then by Proposition 2,

$\frac{\partial X^{i}_{\ell}}{\partial p_{k}}(p)=\frac{\partial
f^{i}_{\ell}}{\partial p_{k}}(p,m^{i}(p))=s^{i}_{\ell k}(p,m^{i}(p)),$

as desired. If $\ell=L$, then by Walras’ law (9),

$X^{i}_{L}(p)=f^{i}_{L}(p,m^{i}(p))-\omega^{i}_{L}=\frac{m^{i}(p)}{p_{L}}-\frac{1}{p_{L}}(\tilde{p}\cdot\tilde{f}^{i}(p,m))-\omega^{i}_{L}.$

Since $\frac{\partial\tilde{f}^{i}}{\partial m}(p,m^{i}(p))=0$, we have that
for $k\neq L$,

$\frac{\partial X^{i}_{L}}{\partial p_{k}}(p)=\frac{\omega^{i}_{k}+\sum_{j\in
M}\theta_{ij}y^{j}_{k}(p)}{p_{L}}-\frac{f^{i}_{k}(p,m^{i}(p))}{p_{L}}-\frac{1}{p_{L}}\left(\tilde{p}\cdot\frac{\partial\tilde{f}^{i}}{\partial
p_{k}}(p,m^{i}(p))\right).$

Because of (R), we have that $\sum_{\ell=1}^{L}p_{\ell}s_{\ell
k}(p,m^{i}(p))=0$, and thus,

$s^{i}_{Lk}(p,m^{i}(p))=-\frac{1}{p_{L}}\left(\tilde{p}\cdot\frac{\partial\tilde{f}^{i}}{\partial
p_{k}}(p,m^{i}(p))\right).$

This implies that

$\frac{\partial X^{i}_{L}}{\partial
p_{k}}(p)=s^{i}_{Lk}(p,m^{i}(p))-\frac{X^{i}_{k}(p)-\sum_{j\in
M}\theta_{ij}y^{j}_{k}(p)}{p_{L}},$

as desired. By similar arguments, we can show that

$\frac{\partial X^{i}_{L}}{\partial
L}(p)=s^{i}_{LL}(p,m^{i}(p))-\frac{X^{i}_{L}(p)-\sum_{j\in
M}\theta_{ij}y^{j}_{L}(p)}{p_{L}},$

as desired. This completes the proof of Step 2. $\blacksquare$

Let $a_{1},...,a_{L}>0$ be fixed. Choose any equilibrium price $p^{*}$ in this
economy. Note that, there exists such an equilibrium price because of
Proposition 6. By Step 1, we have that $y^{j}$ is defined and continuous on
some neighborhood of $p$. Recall the tâtonnement process (14):

$\dot{p}_{j}(t)=a_{j}\zeta_{j}(p(t)),\ p_{j}(0)=p_{0j}.$

Because $p^{*}$ is an equilibrium price in this economy, we have that $p^{*}$
is a steady state of (14). Define

$h(p)=\sqrt{a_{1}^{-1}p_{1}^{2}+...+a_{L}^{-1}p_{L}^{2}}.$

We note that $h(p)$ satisfies all requirements of the norm. In particular, we
have that $h(ap)=ah(p)$ for every $p$ and $a>0$, and thus $Dh(p)p=h(p)$ if
$p\neq 0$. Define

$S(b)=\\{p\in\mathbb{R}^{L}_{++}|h(p)=h(bp^{*})\\}.$

Choose a sufficiently small $\varepsilon>0$ such that if
$\|v\|\leq\varepsilon$ and $t\in[-1,1]$, then
$p^{*}+tv\in\mathbb{R}^{L}_{++}$. Define

$S=\\{v\in\mathbb{R}^{L}|\|v\|=\varepsilon,\ Dh(p^{*})v=0\\},$

and

$p(t,v)=\frac{h(p^{*})}{h(p^{*}+tv)}(p^{*}+tv).$

Step 3. Define $m_{i}^{*}=m^{i}(p^{*})$ and the following function

$g^{i}(t,v)=\begin{cases}\frac{1}{t^{2}}(p(t,v)-p^{*})\cdot(f^{i}(p(t,v),m^{i}(p(t,v)))-f^{i}(p^{*},m_{i}^{*}))&\mbox{if
}t\neq 0,\\\ v^{T}DX^{i}(p^{*})v&\mbox{if }t=0.\end{cases}$

Then, $g^{i}$ is continuous on $[-1,1]\times S$.

Proof of Step 3. Clearly, $g^{i}$ is continuous at $(t,v)$ if $t\neq 0$.
Therefore, it suffices to show that $g^{i}$ is continuous at $(0,v)$ for all
$v\in S$.

Choose any $\varepsilon^{\prime}>0$. Note that, $p(0,v)=p^{*}$ and $h$ is
continuously differentiable without $0$. We can easily check that

$\displaystyle\frac{\partial p}{\partial t}(0,v)=$ $\displaystyle~{}v,$ (21)
$\displaystyle\frac{\partial p}{\partial v_{j}}(0,v)=$
$\displaystyle~{}0\mbox{ for all }j\in\\{1,...,L\\}.$ (22)

Define

$q(t,v)=\|p(t,v)-(p^{*}+tv)\|.$

By (21) and (22), we have that $q(0,v)=0$ and $D(q(0,v))^{2}=0^{T}$ for all
$v\in S$. Hence, by the formula of finite increments, for all $v\in S$, there
exists an open and convex neighborhood $U_{v}$ of $(0,v)$ such that if
$(t^{\prime},v^{\prime}),(t^{\prime\prime},v^{\prime\prime})\in U_{v}$, then

$|q(t^{\prime},v^{\prime})-q(t^{\prime\prime},v^{\prime\prime})|\leq\varepsilon^{\prime}\|(t^{\prime},v^{\prime})-(t^{\prime\prime},v^{\prime\prime})\|/2.$
(23)

We can assume without loss of generality that

$U_{v}=\\{(t^{\prime},v^{\prime})\in[-1,1]\times S||t^{\prime}|<2\delta_{v},\
\|v^{\prime}-v\|<\delta_{v}\\}.$

Define

$W_{v}=\\{v^{\prime}\in S|\|v^{\prime}-v\|<\delta_{v}\\}.$

Then, $(W_{v})$ is an open covering of $S$, and thus, there exists a finite
subcovering $(W_{v_{1}},...,W_{v_{K}})$. Let
$\delta^{*}=\min\\{\delta_{v_{1}},...,\delta_{v_{K}}\\}$. Then, we have that

$\sup_{t\in]0,\delta^{*}]}\frac{q(t,v)}{t}<\varepsilon^{\prime}$

for all $v\in S$.

Fix any $v\in S$. Since $f^{i}(ap,am)=f^{i}(p,m)$ and $m^{i}(ap)=am^{i}(p)$,
we have that

$f^{i}(p(t,v),m^{i}(p(t,v)))=f^{i}(p^{*}+tv,m^{i}(p^{*}+tv)).$

Define

$\hat{g}^{i}(t,v)=\frac{1}{t}v\cdot(f^{i}(p^{*}+tv,m^{i}(p^{*}+tv))-f^{i}(p^{*},m_{i}^{*})).$

Then, by chain rule, we can easily show that there exists $\delta>0$ such that
if $0<|t|<\delta$ and $v\in S$, then

$|\hat{g}^{i}(t,v)-v^{T}DX^{i}(p^{*})v|<\varepsilon^{\prime}.$

Therefore, if $0<|t|<\min\\{\delta,\delta^{*}\\}$, we have that

$\displaystyle|g^{i}(t,v)-v^{T}DX^{i}(p^{*})v|\leq$
$\displaystyle~{}|g^{i}(t,v)-\hat{g}^{i}(t,v)|+|\hat{g}^{i}(t,v)-v^{T}DX^{i}(p^{*})v|$
$\displaystyle<$
$\displaystyle~{}\frac{1}{t}\varepsilon^{\prime}\|f^{i}(p(t,v),m^{i}(p(t,v)))-f^{i}(p^{*},m_{i}^{*})\|+\varepsilon^{\prime}$
$\displaystyle\leq$ $\displaystyle~{}(K+1)\varepsilon^{\prime},$

where $K>0$ is some constant independent of $(t,v)$.111111Use the formula of
finite increments. Let $K^{*}>0$ be the operator norm of $DX^{i}(p^{*})$. If
$\|v-v^{\prime}\|<\sqrt{\varepsilon^{\prime}}$ and
$0\leq|t|<\min\\{\delta,\delta^{*}\\}$, then

$\displaystyle|g^{i}(t,v^{\prime})-g^{i}(0,v)|\leq$
$\displaystyle~{}|g^{i}(t,v^{\prime})-g^{i}(0,v^{\prime})|+|g^{i}(0,v^{\prime})-g^{i}(0,v)|$
$\displaystyle<$ $\displaystyle~{}(K+K^{*}+1)\varepsilon^{\prime}.$

Therefore, $g^{i}$ is continuous at $(0,v)$. This completes the proof of Step
3. $\blacksquare$

Note that, because $S_{f^{i}}(p^{*},m_{i}^{*})$ satisfies (ND), by Step 1, we
have that $\max_{v\in S}\sum_{i\in N}g^{i}(0,v)<0$, and thus there exists
$\delta_{1}>0$ and $L^{+}>0$ such that if $|t|\leq\delta_{1}$, then
$\max_{v\in S}\sum_{i\in N}g^{i}(t,v)\leq-L^{+}$.

Step 4. There exists $\delta>0$ such that $\delta\leq\delta_{1}$ and if
$0<|t|\leq\delta$ and $v\in S$, then for all $j\in M$,121212If $\mu=0$ and the
economy is pure exchange, then this step is not needed.

$\frac{1}{t^{2}}(p(t,v)-p^{*})\cdot(y^{j}(p(t,v))-y^{j}(p^{*}))\geq-\frac{L^{+}}{2\mu}.$

Proof of Step 4. By Lemma 5, there exists an open and convex neighborhood $U$
of $p^{*}$ such that $\pi^{j}(p)$ is defined, convex, continuously
differentiable, and $D\pi^{j}(p)=y^{j}(p)$ for all $p\in U$. Therefore, if
$v\in S$, then

$v\cdot\frac{D\pi^{j}(p^{*}+tv)-D\pi^{j}(p^{*})}{t}\geq 0$

for all $t\neq 0$ such that $|t|$ is sufficiently small. Choose any $v\in S$.
By Taylor’s theorem,

$p(t,v)-p^{*}=vt+\frac{\partial^{2}p}{\partial
t^{2}}(0,v)\frac{t^{2}}{2}+R(t,v)t^{2},$

where for $\ell\in\\{1,...,L\\}$,

$R_{\ell}(t,v)=\frac{\partial^{3}p_{\ell}}{\partial
t^{3}}(\theta_{\ell}t,v)\frac{t}{6}$

for some $\theta_{\ell}\in[0,1]$. Moreover, because $y$ is homogeneous of
degree zero, we have that $y^{j}(p^{*}+tv)=y^{j}(p(t,v))$. Therefore,

$\displaystyle~{}\frac{1}{t^{2}}(p(t,v)-p^{*})\cdot(y^{j}(p(t,v))-y^{j}(p^{*}))$
$\displaystyle=$
$\displaystyle~{}v\cdot\frac{D\pi^{j}(p^{*}+tv)-D\pi^{j}(p^{*})}{t}+\left(\frac{1}{2}\frac{\partial^{2}p}{\partial
t^{2}}(0,v)+R(t,v)\right)\cdot(y^{j}(p(t,v))-y^{j}(p^{*}))$
$\displaystyle\geq$
$\displaystyle~{}-\left\|\frac{1}{2}\frac{\partial^{2}p}{\partial
t^{2}}(0,v)+R(t,v)\right\|\|y^{j}(p(t,v)-y^{j}(p^{*})\|.$

This implies that there exists $\delta>0$ such that if $0<|t|\leq\delta$ and
$v\in S$, then

$\frac{1}{t^{2}}((p(t,v)-p^{*})\cdot(y^{j}(p(t,v))-y^{j}(p^{*}))\geq-\frac{L^{+}}{2\mu}.$

We can assume that $\delta\leq\delta_{1}$. This completes the proof.
$\blacksquare$

Step 5. There exists an open neighborhood $U$ of $p^{*}$ such that if
$p_{0}\in U$, then there exists a solution $p(t)$ to (14) defined on
$\mathbb{R}_{+}$, and for all such solutions, $\lim_{t\to\infty}p(t)=bp^{*}$
for $b=\frac{h(p_{0})}{h(p^{*})}$.

Proof of Step 5. First, suppose that $p(t)$ is a solution to (14). Then,

$\frac{d}{dt}(h(p(t)))^{2}=\sum_{\ell=1}^{L}a_{\ell}^{-1}p_{j}(t)a_{\ell}\zeta_{\ell}(p(t))=0,$

by Proposition 4. Therefore, we have that $h(p(t))=h(p_{0})$ for all $t$.

Next, let $W=\\{p\in\mathbb{R}^{L}_{++}|h(p)=h(p^{*}),\
h(p-p^{*})<\varepsilon^{\prime}\\}$, where $\varepsilon^{\prime}>0$ is so
small that for all $p\in W$, there exists $v\in S$ and $t\in[0,\delta]$ such
that $p$ is proportional to $p^{*}+tv$. Let
$U=\\{p\in\mathbb{R}^{L}_{++}|(h(p^{*})/h(p))p\in W\\}$. Define

$V(p)=(h(p-(h(p)/h(p^{*}))p^{*}))^{2}.$

Then, by Steps 3-4, we have that when $p(t)$ is not proportional to $p^{*}$,
then

$\frac{d}{dt}V(p(t))<0$

for every solution $p(t)$ to (14) such that $p_{0}\in U$. Because $V(p)>0$ if
$p$ is not proportional to $p^{*}$ and $V(bp^{*})=0$ for $b>0$, we have that
$V$ is a Lyapunov function of (14) on $U\cap S(b)$. Therefore, if $p_{0}\in
U$, then there exists a solution $p(t)$ to (14) defined on $\mathbb{R}_{+}$,
and for any such a solution $p(t)$, $\lim_{t\to\infty}p(t)=bp^{*}$ for
$b=\frac{h(p_{0})}{h(p^{*})}$. This completes the proof of Step 5.
$\blacksquare$.

Step 6. Let $p^{*}$ be an equilibrium price such that $\|p^{*}\|=1$. Then,
there exists a neighborhood $U$ of $p^{*}$ such that if $p\in U$ is an
equilibrium price such that $\|p\|=1$, then $p=p^{*}$.

Proof of Step 6. Choose any equilibrium price $p^{*}$ with $\|p^{*}\|=1$, and
let $U$ be defined in Step 5. If there exists an equilibrium price $p_{0}\in
U$ such that $\|p_{0}\|=1$ and $p_{0}\neq p^{*}$, then $p(t)\equiv p_{0}$ is a
solution to (14), which contradicts Step 5. This completes the proof of Step
6. $\blacksquare$

Step 7. Define $S^{*}=\\{p\in\mathbb{R}^{L}_{++}|\|p\|=1\\}$. Then, there
uniquely exists $p^{*}\in S^{*}$ that is an equilibrium price of this economy.

Proof of Step 7. Choose $s>0$ in Step 1. Let $E^{*}$ be the set of all
equilibrium prices in the economy $E_{s}$ whose norm is $1$. Because of Lemma
7, we have that $E^{*}$ is compact, and by Step 6, it is finite. Choose
$\varepsilon>0$ such that $U_{\varepsilon}(p^{*})\subset\mathbb{R}^{L}_{++}$
and $E^{*}\cap U_{\varepsilon}(p^{*})=\\{p^{*}\\}$ for all $p^{*}\in E^{*}$,
where

$U_{\varepsilon}(p)=\\{q\in\mathbb{R}^{L}|\|q-p\|\leq 4\varepsilon\\}.$

Now, choose a mollifier

$\varphi(p)=\begin{cases}Ce^{-\frac{1}{1-\|p\|^{2}}}&\mbox{if }\|p\|<1,\\\
0&\mbox{if }\|p\|\geq 1,\end{cases}$

where $C>0$ is chosen as

$\int_{\mathbb{R}^{L}}\varphi(p)dp=1.$

Define

$\varphi_{\delta}(p)=\delta^{L}\varphi(p/\delta).$

Construct a function $\pi^{j}_{\delta,p^{*}}$ such that

$\pi^{j}_{\delta,p^{*}}(p)=\int_{\mathbb{R}^{L}}\pi^{j}_{s}(p-q)\varphi_{\delta}(q)dq,$

and define

$t_{p^{*}}(p)=\min\\{1,\max\\{0,2-\varepsilon^{-1}\|p/\|p\|-p^{*}\|\\}\\},$
$y^{j}_{\delta}(p)=\left(1-\sum_{p^{*}\in
E^{*}}t_{p^{*}}(p)\right)y^{j}_{s}(p)+\sum_{p^{*}\in
E^{*}}t_{p^{*}}(p)[D\pi^{j}_{\delta,p^{*}}(p/\|p\|)-D\pi^{j}_{\delta,p^{*}}(p^{*})].$

Note that, $y^{j}_{\delta}(p)$ is continuous, homogeneous of degree zero, and
$y^{j}_{\delta}(p^{*})=0$ for all $p^{*}\in E^{*}$. Moreover, by general
theory of mollifiers,

$\lim_{\delta\to
0}\max\\{|\pi^{j}_{\delta,p^{*}}(p)-\pi^{j}_{s}(p)||\|p-p^{*}\|\leq
2\varepsilon\\}=0.$

Because $\pi^{j}$ and $\pi^{j}_{\delta,p^{*}}$ are convex, by Theorem 25.7 of
Rockafeller (1970), we have that

$\lim_{\delta\to
0}\max\\{\|D\pi^{j}_{\delta,p^{*}}(p)-y^{j}_{s}(p)\||\|p-p^{*}\|\leq
2\varepsilon\\}=0.$

Therefore, $y^{j}_{\delta}$ converges to $y^{j}_{s}$ uniformly as $\delta\to
0$. Define

$m^{i}_{\delta}(p)=p\cdot\omega^{i}+\sum_{j\in M}\theta_{ij}p\cdot
y^{j}_{\delta}(p),$
$X^{i}_{\delta}(p)=f^{i}_{s}(p,m^{i}_{\delta}(p))-\omega^{i},$
$\zeta_{\delta}(p)=\sum_{i\in N}X^{i}_{\delta}(p)-\sum_{j\in
M}y^{j}_{\delta}(p).$

Then, $\zeta_{\delta}(p^{*})=0$ for all $p^{*}\in E^{*}$, and if $\delta>0$ is
sufficiently small, then $\zeta_{\delta}(p)\neq 0$ for all
$p\in\mathbb{R}^{L}_{++}$ such that $\|p\|=1$ and $\|p-p^{*}\|\geq\varepsilon$
for all $p^{*}\in E^{*}$. Suppose that $p^{+}\in\mathbb{R}^{L}_{++},\
\|p^{+}\|=1$ and $\zeta_{\delta}(p^{+})=0$. Then, $\zeta_{\delta}$ is
continuously differentiable around $p^{+}$. Moreover, there uniquely exists
$p^{*}\in E^{*}$ such that $\|p^{+}-p^{*}\|<\varepsilon$, and by the same
arguments as in the proof of Step 2, we have that

$D\zeta_{\delta}(p^{+})=\sum_{i\in
N}S_{f^{i}}(p^{+},m^{i}_{\delta}(p^{+}))-\sum_{j\in
M}D^{2}\pi^{j}_{\delta,p^{*}}(p^{+}).$

Because $\pi^{j}_{\delta,p^{*}}$ is convex,
$D^{2}\pi^{j}_{\delta,p^{*}}(p^{+})$ is positive semi-definite. By using (ND)
in Proposition 3 and results in Debreu (1952), we have that
$\mbox{index}(p^{+})=+1$. Therefore, by Lemma 8,

$\sum_{p^{+}:\zeta_{\delta}(p^{+})=0}\mbox{index}(p^{+})=+1.$

This indicates that there uniquely exists $p^{+}$ such that
$\zeta_{\delta}(p^{+})=0$. On the other hand, by construction, we have that
$\zeta_{\delta}(p^{*})=0$ for all $p^{*}\in E^{*}$, which implies that $E^{*}$
is a singleton. This completes the proof of Step 7. $\blacksquare$.

Steps 5 and 7 states our claim of Theorem 1 is correct. This completes the
proof. $\blacksquare$

###  6.10 Proofs in Results of Subsection 4.1

Corollary 1 can be easily obtained by Theorem 2, and thus we only show Theorem
2. In Lemma 2, we showed that if $p_{L}=1$, then

$f^{i}(p,m)=(\tilde{x}^{i}(\tilde{p}),m-\tilde{p}\cdot\tilde{x}^{i}(\tilde{p})),$

where $\tilde{x}^{i}$ is the inverse function of $Du_{i}(\tilde{x})$. Suppose
that $c:[0,1]\to\mathbb{R}^{n}_{++}$ is a continuously differentiable function
such that $c(0)=\tilde{p}$ and $c(1)=\tilde{q}$. Define

$v_{i}(\tilde{p},m)=u_{i}(f^{i}(\tilde{p},1,m)).$

By Roy’s identity, we have that

$\frac{\partial v_{i}}{\partial
p_{\ell}}(\tilde{p},m)=\tilde{x}^{i}(\tilde{p}).$

Therefore,

$\displaystyle u_{i}(\tilde{f}^{i}(q,m))-u_{i}(\tilde{f}^{i}(p,m))=$
$\displaystyle~{}u_{i}(\tilde{x}^{i}(c(1)))-u_{i}(\tilde{x}^{i}(c(0)))$
$\displaystyle=$
$\displaystyle~{}\int_{0}^{1}\frac{d}{dt}u_{i}(\tilde{x}^{i}(c(t)))dt=\int_{0}^{1}D_{\tilde{p}}v_{i}(c(t),m)c^{\prime}(t)dt$
$\displaystyle=$
$\displaystyle~{}\int_{0}^{1}\tilde{x}^{i}(c(t))c^{\prime}(t)dt=V_{i}(\tilde{p},\tilde{q}).$

## Acknowledgement

I am grateful for Shinichi Suda and Toru Maruyama for their helpful lectures
on tâtonnement processes. I also thank to Shinsuke Nakamura and Chaowen Yu,
who advise me for constructing the proof of our main theorem.

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