Advanced Microeconomics - Lecture Slides

講義 1：最適化理論と消費者問題
上級ミクロ経済学 — 財務省理論研修
安田洋祐
大阪大学大学院経済学研究科
yasuda@econ.osaka-u.ac.jp
2015 年 4 月 6 日
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Announcement | お知らせ
Course website You can ﬁnd my corse websites from the link below:
https://sites.google.com/site/yosukeyasuda2/home/lecture/mof15micro
Lecture slides Will be uploaded on the website before the lecture.
Textbooks There are two main textbook (JR, K) and two related books:
JR *Jehle and Reny, Advanced Microeconomic Theory, 3rd, 2011.
K 神取道宏, 『ミクロ経済学の力』日本評論社, 2014.
G ギボンズ, 『経済学のためのゲーム理論入門』創文社, 1995.
OY 尾山大輔・安田洋祐『改訂版経済学で出る数学』日本評論社, 2013.
Symbols that we use in lectures
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¢

¡Ex : Example,
§
¦
¤
¥
Fg : Figure,
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¢

¡Rm : Remark,
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¢

¡Q : Question.
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Readings | 参考文献
テキスト
*JR: AP1; AP2; Ch1
尾山・安田: 5-7 章
神取: 1 章; 補論 A; B
配布資料
*Vohra, Advanced Mathematical Economics, 2004: Ch1
関連図書
*武隈慎一『数理経済学』新世社, 2001.
*Sundaram, A First Course in Optimization Theory, 1996.
ディキシット『経済理論における最適化（第 2 版）』勁草書房, 1997.
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What and How to Optimize? | 何をどうやって最適化するのか？
Optimization (最適化) is a set of mathematical procedures to ﬁnd the optimal
value of some function.
We frequently adopt the assumption that an economic agent seeks to maximize
(最大化) or minimize (最小化) some function, for example:
効用最大化：Consumer maximizes her utility function.
費用最小化：Firm minimizes its cost function.
収入最大化：Seller at the auction maximizes (expected) revenue function.
社会厚生最大化：Government tries to maximize the social welfare function.
We study and apply static optimization (静学的最適化).
→ Dynamic optimization (動学的最適化), a common tool in modern
macroeconomics, will not be covered in our lectures. . .
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Equality Constraints | 等号制約
Consider choosing x1 and x2 to maximize f(x1, x2) , when x1 and x2 must
satisfy some particular relation to each other that we write implicit form as
g(x1, x2) = 0 .
Formally, we write this problem as follows:
max
x1,x2
f(x1, x2) subject to g(x1, x2) = 0.
f(x1, x2): objective function (目的関数).
x1 and x2: choice variables (選択変数).
g(x1, x2): constraint (制約).
Set of all (x1, x2) that satisfy the constraint: feasible set (実行可能集合).
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¡Q Does solution always exist?
→ 補論参照．
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Lagrange’s Method | ラグランジュ（の未定乗数）法 (1)
There are two approaches to solve this type of optimization problems with
equality constraints: substitution (代入) and Lagrange’s method.
Lagrange’s method is a powerful way to solve constrained optimization
problems, which essentially translates them into unconstrained problems.
等号制約付き (constrained) 問題 ⇒ 制約無し (unconstrained) 問題
Again, consider the following problem:
max
x1,x2
f(x1, x2) subject to g(x1, x2) = 0.
Let us construct a new function L(x1, x2, λ), called the Lagrangian function
(ラグランジュ関数) as follows:
L(x1, x2, λ) = f(x1, x2) + λg(x1, x2).
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Lagrange’s Method | ラグランジュ（の未定乗数）法 (2)
Then maximize this Lagrangian function, that is, derive the ﬁrst order
conditions (一階条件):
∂L
∂x1
=
∂f(x∗
1, x∗
2)
∂x1
+ λ∗ ∂g(x∗
1, x∗
2)
∂x1
= 0
∂L
∂x2
=
∂f(x∗
1, x∗
2)
∂x2
+ λ∗ ∂g(x∗
1, x∗
2)
∂x2
= 0
∂L
∂λ
= g(x∗
1, x∗
2) = 0.
Lagrange’s method asserts that if we ﬁnd values x∗
1, x∗
2, and λ∗
that solves
these three equations simultaneously, then we will have a critical point of
along the constraint g(x1, x2) = 0.
制約条件を“ あたかも ”忘れて解くことができる！
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Practice of Lagrange’s Method | ラグランジュ法の練習
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¡Ex Example A2.11 (see JR, pp.606)
max
x1,x2
x1x2 subject to a − 2x1 − 4x2 = 0
Forming the Lagrangian, we get
L = x1x2 + λ(a − 2x1 − 4x2),
with ﬁrst order conditions:
∂L
∂x1
= x2 − 2λ = 0,
∂L
∂x2
= x1 − 4λ = 0.
∂L
∂λ
= a − 2x1 − 4x2 = 0.
These can be solved to ﬁnd
x1 =
a
4
, x2 =
a
8
, λ =
a
16
.
Note that the solution of the problem is a function of parameter a.
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Envelope Theorem | 包絡線定理 (1)
Consider the following constrained optimization problem P1:
P1 : max
x
f(x, a) s.t. g(x, a) = 0.
where x is a vector of choice variables, and a := (a1, ..., am) is a vector of
parameters (パラメータ) that may enter the objective function and constraint.
Suppose that for each vector a, the solution is unique and denoted by x(a).
A maximum-value function (最大値関数), denoted by M(a), is deﬁned
as follows:
M(a) := max
x
f(x, a) s.t. g(x, a) = 0,
or equivalently, M(a) := f(x(a), a).
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Envelope Theorem | 包絡線定理 (2)
If the objective function, constraint, and the solutions are differentiable in the
parameters, there is a very powerful theorem that shows how the solutions vary
with the parameters.
Theorem 1 (Envelope Theorem (包絡線定理))
Consider P1 and suppose the objective function and constraint are
continuously differentiable in a. For each a, let x(a) 0 uniquely solve P1
and assume that it is also continuously differentiable (連続微分可能) in the
parameters a. Then, the Envelope theorem states that
∂M(a)
∂aj
=
∂L
∂aj
|x(a),λ(a) j = 1, ..., m,
where the right hand side denotes the partial derivative of the Lagrangian
function with respect to the parameter aj evaluated at the point (x(a),λ(a)).
Proof.
See JR, pp.604-606.
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Practice of Envelope Theorem | 包絡線定理の練習
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¡Ex Example A2.11 (again)
max
x1,x2
x1x2 s.t. a − 2x1 − 4x2 = 0.
We form the maximum-value function by substituting the solutions for x1 and
x2 into the objective function. Thus,
M(a) = x1(a)x2(a) =
a
4
·
a
8
=
a2
32
.
Diﬀerentiating M(a) with respect to a, we get
dM(a)
da
=
a
16
,
which tells us how the maximized value varies with a.
Applying the Envelope theorem, we directly obtain
dM(a)
da
=
∂L
∂a
|x(a),λ(a) = λ(a).
Since λ(a) =
a
16
, we veriﬁed that the Envelope theorem works.
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What is Consumer Problem? | 消費者問題とはなにか？
Assume is a consumer’s preference relation (選好関係) on the consumption
set (消費集合) X = Rn
+ where
Rn
+ := {(x1, ..., xn)|xi ≥ 0, i = 1, ..., n} ⊂ Rn
.
For any x, y ∈ X, x y means x is at least as preferred as y.
Consumption set contains all conceivable alternatives.
A budget set (予算集合) is a set of feasible consumption bundles,
represented as B(p, ω) = {x ∈ X|px ≤ ω}, where p is an n-dimensional
positive vector interpreted as prices, and ω is a positive number
interpreted as the consumer’s wealth.
We assume that the consumer is motivated to choose the most preferred
feasible alternative according to her preference relation. That is, she seeks
x∗
∈ B(p, ω) such that x∗
x for all x ∈ B(p, ω).
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Utility Maximization | 効用最大化
Deﬁnition 1
We refer to the problem of ﬁnding the best bundle in B(p, ω) as the
consumer problem (消費者問題).
Function U : X → R represents (表現する) the preference if for all x
and y ∈ X, x y if and only if U(x) ≥ U(y).
If U represents a preference relation , we call it a utility function (効用
関数), and we say that has a utility representation (効用表現).
Utility functions are useful since it is often more convenient to talk about
the maximization of a numerical function than of a preference relation.
Given utility representation, consumer problem becomes:
max
x∈B(p,ω)
U(x) or max
x∈X
U(x) s.t. x ∈ B(p, ω).
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Remarks on Utility Function | 効用関数に関する注意
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¡Rm Utility function has NO meaning other than that of representing a
preference relation .
Theorem 2
If U represents , then for any strictly increasing function f : R → R, the
function V (x) = f(U(x)) represents as well.
Proof.
Note that increasing function implies the second step.
a b ⇔ U(a) ≥ U(b) ⇔ f(U(a)) ≥ f(U(b)) ⇔ V (a) ≥ V (b).
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¡Q Under what conditions do utility representations exist?
→ 講義 3 で確認．
Theorem 3
If is represented by continuous utility function U, then any consumer
problem has a solution.
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Monotonicity | 単調性
Monotonicity says that “more is better (than less).”
Deﬁnition 2
A preference relation satisﬁes monotonicity (単調性) at the bundle y if for
all x ∈ X,
If xk ≥ yk for all k, then x y, and
If xk > yk for all k, then x y.
Monotonicity can be expressed by a increasing utility function U, which is
assumed in most of consumer problems.
Theorem 4
U(x) is strictly increasing if and only if is monotonic.
Monotonicity says that if one bundle contains at least as much of every
commodity as another bundle, then the one is at least as good as the other.
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Inequality Constraints | 不等号制約
Consider a two-variable optimization problem in which the only constraint is
given by the inequality g(x1, x2) ≥ 0. Formally, our problem is
max
x1,x2
f(x1, x2) s.t. g(x1, x2) ≥ 0.
Let us deﬁne Lagrangian function as if the constraint holds with equality.
L = f(x1, x2) + λg(x1, x2).
The optimal solutions must satisfy the following Kuhn-Tucker conditions
(クーン-タッカー条件):
∂L
∂x1
=
∂f
∂x1
+ λ
∂g
∂x1
= 0,
∂L
∂x2
=
∂f
∂x2
+ λ
∂g
∂x2
= 0.
λg(x1, x2) = 0, λ ≥ 0, g(x1, x2) ≥ 0.
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Solving Consumer Problem | 消費者問題を解く (1)
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¡Ex Practice: A consumer problem with n goods.
max
x∈Rn
+
u(x) s.t. px ≤ ω.
The corresponding Lagrangian function is:
L = u(x) + λ(ω − px) + λ1x1 + · · · + λnxn.
The Kuhn-Tucker conditions are as follows:
∂L
∂x1
=
∂u
∂x1
− λp1 + λ1 = 0.
...
∂L
∂xn
=
∂u
∂xn
− λpn + λn = 0.
λ(ω − px) = 0, λ ≥ 0, ω − px ≥ 0,
λ1x1 = 0, ..., λnxn = 0, λi ≥ 0, xi ≥ 0 for all i.
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Solving Consumer Problem | 消費者問題を解く (2)
Suppose that solution x∗
i and ∂u(x∗
)
∂xi
is strictly positive for all i. Then, the
corresponding Lagrangian multipliers (ラグランジュ乗数) λi must be 0 for all i,
which implies λ, pi > 0 for all i. Therefore, for any two goods j and k, we can
combine the conditions to conclude that
∂u(x∗
)
∂xj
∂u(x∗)
∂xk
=
λpj
λpk
=
pj
pk
.
This says that at the optimum, the marginal rate of substitution (MRS, 限界
代替率) between any two goods must be equal to the ratio of the goods’ prices.
Note that, for two goods case along the indiﬀerence curve (無差別曲線):
0 = du =
∂u(x∗
)
∂x1
dx1 +
∂u(x∗
)
∂x2
dx2 ⇐⇒
∂u(x∗
)
∂x1
∂u(x∗)
∂x2
= −
dx2
dx1
.
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Indirect Utility Function | 間接効用関数
To construct the indirect utility function (間接効用関数), we ﬁx market prices
and “initial wealth,” and seek the maximum level of utility that the consumer
could achieve.
Deﬁnition 3
The indirect utility function is the maximum-value function corresponding to
the consumer’s utility maximization problem (UMP, 効用最大化問題), and it
is denoted by v(p, ω). That is,
v(p, ω) = max
x∈Rn
+
u(x) s.t. px ≤ ω,
or, equivalently
v(p, ω) = u(x(p, ω))
where x(p, ω) is the solution of the UMP, known as Marshallian demand
functions (マーシャルの需要関数).
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Example of UMP | 効用最大化問題の例
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¡Ex Cobb-douglas (コブ-ダグラス) utility function with two goods
max
x∈R2
+
xα
1 x1−α
2 s.t. px (= p1x1 + p2x2) ≤ ω.
where α ∈ (0, 1).
Expenditure of each good is proportional to α and 1 − α:
p1x1 = αω, p2x2 = (1 − α)ω.
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¡Rm u(x) = xβ
1 xγ
2 (β, γ > 0) can express the identical preference.
Note that xα
1 x1−α
2 is a monotone transformation (単調変換) of u(x) = xβ
1 xγ
2 .
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Expenditure Function | (最小) 支出関数
To construct the expenditure function (支出関数), we ﬁx prices and a “level
of utility,” and seek the minimum level of money expenditure the consumer
must make to achieve this particular level of utility.
Deﬁnition 4
The expenditure function is the minimum-value function corresponding to the
consumer’s expenditure minimization problem (EMP, 支出最小化問題), and
it is denoted by e(p, u). That is,
e(p, u) = min
x∈Rn
+
px s.t. u(x) ≥ u,
or, equivalently
e(p, u) = pxh
(p, u)
where xh
(p, u) is the solution of the EMP, known as Hicksian
(Compensated) demand functions (ヒックスの (補償) 需要関数).
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【補論】 Example of EMP | EMP の例
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¡Ex Cobb-douglas utility function with two goods
min
x∈R2
+
p1x1 + p2x2 s.t. xα
1 x1−α
2 ≥ u
where α ∈ (0, 1).
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¡Rm EMP is the mirror image of UMP. This property is formally established as
duality (双対性)
→ 講義 2 を参照.
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【補論】 Indirect Utility Function | 間接効用関数
There are several properties that the indirect utility function possesses.
Theorem 5
If u(x) is continuous and strictly increasing on Rn
+, then v(p, ω) is
.
1 Continuous in p and ω.
.
2 Homogeneous of degree zero (0 次同次) in (p, ω).
.
3 Strictly increasing in ω.
.
4 Decreasing in p.
.
5 Quasiconvex in (p, ω).
6 Roy’s identity (ロアの恒等式): If v(p, ω) is diﬀerentiable at (p0
, ω0
), then
xi(p0
, ω0
) = −
∂v(p0
,ω0
)
∂pi
∂v(p0,ω0)
∂ω
, i = 1, ..., n.
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【補論】 More on Roy’s Identity | もっとロアの恒等式
Roy’s identity says that the consumer’s Marshallian demand for good i is
simply the ratio of the partial derivatives of indirect utility with respect to pi
and ω after a sign change.
By the envelope theorem and Lagrangian method,
∂v(p0
, ω0
)
∂pi
= −λxi(p0
, ω0
)
and
∂v(p0
, ω0
)
∂ω
= λ,
which implies
−
∂v(p0
,ω0
)
∂pi
∂v(p0,ω0)
∂ω
= xi(p0
, ω0
).
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【補論】 Expenditure Function | 支出関数
There are several properties that the expenditure function possesses.
Theorem 6
If u(x) is continuous and strictly increasing on Rn
+, then e(p, u) is
.
1 Continuous in p and ω.
.
2 Homogeneous of degree 1 (1 次同次) in p.
.
3 Strictly increasing in u, for all p 0.
.
4 Increasing in p.
.
5 Concave in p.
If, u(·) is strictly quasi-concave, we have
6 Shephard’s lemma (シェファードの補題): e(p, u) is diﬀerentiable in p at
(p0
, u0
) with p0
0, and
∂e(p0
, u0
)
∂pi
= xh
i (p0
, u0
), i = 1, ..., n.
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【補論】 More on Shephard’s Lemma | もっとシェファードの補題
To prove Shephard’s Lemma, we appeal to the Envelope theorem. Note that
EMP and expenditure function can be rewritten as:
−e(p, u) = max
x∈Rn
+
−px s.t. u(x) ≥ u,
Applying the Envelope theorem to this maximization problem,
∂ − e(p0
, u0
)
∂pi
=
∂L
∂pi
= −xh
i (p0
, u0
).
where L is the corresponding Lagrangian function:
L = −px + λ(u(x) − u).
Canceling out negative signs, we obtain the Shephard’s lemma.
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【補論】 (Marshallian) Demand Function | (マーシャルの) 需要関数
Theorem 7
The demand function x(p, ω) satisﬁes the following:
. 1 Homogeneous of degree zero (0 次同次性):
x(p, ω) = x(λp, λω) for any λ > 0.
.
2 Walras’s Law (ワルラス法則): If the preferences are monotonic, then any
solution x to the consumer problem B(p, ω) is located on its budget line,
i.e., px(p, ω) = ω.
.
3 Continuity (連続性): If is a continuous preference, then the demand
function is continuous in p and in ω.
Proof.
We give the sketch of the proof for 1 and 2.
1 The budget sets are identical, i.e, B(p, ω) = B(λp, λω).
2 If px(p, ω) < ω, there must exist some consumption bundle x with x
x and px(p, ω) ≤ ω. By monotonicity, x must be strictly preferred to x,
which contradicts x being a solution of the consumer problem.
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補足資料 — 講義 1：最適化理論と消費者問題
安田洋祐
2015 年 4 月 6 日
1 / 9

Existence of Solutions | 解の存在
Compact Set (コンパクト集合)
A set S in Rn
is called bounded (有界) if there exists some ε > 0 such
that S ⊂ Bε(x) for some x ∈ Rn
.
A set S in Rn
is called compact if it is closed (閉) and bounded.
Theorem 1 (Thm A1.10 Weierstrass Existence of Extreme Values)
Let f : S → R be a continuous real-valued function where S is a non-empty
compact subset of Rn
. Then f has its maximum and minimum values. That
is, there exists vectors x and x such that
f(x) ≤ f(x) ≤ f(x) for all x ∈ S.
§
¦
¤
¥
Fg Figure A1.18 (see JR, pp.522)
Most problems in economics have compact domains (定義域) and
continuous objective functions. ⇒ Solutions guaranteed!
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Intuitions of Lagrange’s method | ラグランジュ法の直観 (1)
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¢

¡Q Why does Lagrange’s method work?
Take the total diﬀerential of the Lagrangian:
dL =
∂L
∂x1
dx1 +
∂L
∂x2
dx2 +
∂L
∂λ
dλ.
When (x1, x2, λ) = (x∗
1, x∗
2, λ∗
), it can be re-written as follows:
dL =
„
∂f(x∗
1, x∗
2)
∂x1
+ λ∗ ∂g(x∗
1, x∗
2)
∂x1
«
dx1
+
„
∂f(x∗
1, x∗
2)
∂x2
+ λ∗ ∂g(x∗
1, x∗
2)
∂x2
«
dx2 + g(x∗
1, x∗
2)dλ = 0.
Since g(x∗
1, x∗
2) = 0,
0 = dL =
∂f(x∗
1, x∗
2)
∂x1
dx1 +
∂f(x∗
1, x∗
2)
∂x2
dx2
+ λ∗
„
∂g(x∗
1, x∗
2)
∂x1
dx1 +
∂g(x∗
1, x∗
2)
∂x2
dx2
«
for all dx1 and dx2 that satisfy the constraint g.
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Intuitions of Lagrange’s method (2) | ラグランジュ法の直観 (2)
Note that,
dg =
∂g(x∗
1, x∗
2)
∂x1
dx1 +
∂g(x∗
1, x∗
2)
∂x2
dx2 = 0.
So, we can show that
dL =
∂f(x∗
1, x∗
2)
∂x1
dx1 +
∂f(x∗
1, x∗
2)
∂x2
dx2 = 0
for all dx1 and dx2 that satisfy the constraint g. Thus, (x∗
1, x∗
2) is indeed a
critical point of f given that the variables must satisfy the constraint.
Lagrange’s method is very clever and useful. In eﬀect, it oﬀers us an algorithm
for identifying the constrained optima in a wide class of practical problems.
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Convex Sets and Concave Functions | 凸集合と凹関数
Deﬁnition 1
S ⊂ Rn
is a convex set if for all x1
, x2
∈ S,
tx1
+ (1 − t)x2
∈ S ∀t ∈ [0, 1].
Let D be a convex set and xt
= tx1
+ (1 − t)x2
. f : D → R is a concave
function if for all x1
, x2
∈ D,
f(xt
) ≥ tf(x1
) + (1 − t)f(x2
) ∀t ∈ [0, 1].
§
¦
¤
¥
Fg Figures A1.5 and A1.27 (see JR, pp.502 and pp.534)
The points below the graph of all concave regions appear to be convex.
Formally, the next theorem holds.
Theorem 2
Let A := {(x, y)|x ∈ D, f(x) ≥ y} be the set of points “on and below” the
graph of f : D → R, where D ⊂ Rn
is a convex set. Then,
f is a concave function (凹関数) ⇐⇒ A is a convex set (凸集合).
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Convex Preference | 凸選好 (1)
There are three equivalent definitions of convex preferences (凸選好).
Theorem 3
The preference relation satisfies convexity (凸性) if
.
1 x y and α ∈ (0, 1) implies that αx + (1 − α)y y.
.
2 For all x, y and z such that z = αx + (1 − α)y for some α ∈ (0, 1), either
z x or z y.
.
3 For all y, the set AsGoodAs(y) := {z ∈ X|z y} is convex.
The notion of convex preferences captures the following intuitions:
1 If x is preferred to y, then going part of the way from y to x is also an
improvement upon y.
2 If z is between x and y then it is impossible that both x and y are better
than z.
3 If both x1 and x2 are better than y, then the average of x1 and x2 is
definitely better than y.
6 / 9

Convex Preference | 凸選好 (2)
Convexity has a stronger version.
Definition 2
The preference relation satisfies strict convexity (強凸性) if a y, b y,
a = b and λ ∈ (0, 1) imply that λa + (1 − λ)b y.
£
¢

¡Q How is the convexity of preferences translated into properties of the utility
function?
Definition 3
A function u(·) : X → R is quasi-concave if for all x, y ∈ X,
u(αx + (1 − α)y) ≥ min{u(x), u(y)}
holds for all α ∈ (0, 1). u(·) is strictly quasi-concave if for all x = y in X,
u(αx + (1 − α)y) > min{u(x), u(y)}
holds for all α ∈ (0, 1).
7 / 9

Properties of Convex Preference | 凸選好の性質
Convex preferences expressed by utility functions.
Theorem 4
Suppose is represented by utility function u(·). Then,
.
1 u(x) is quasiconcave if and only if is convex.
.
2 u(x) is strictly quasiconcave if and only if is strictly convex.
Convexity induces a simple solution structure.
Theorem 5
Convex preference satisﬁes the following properties.
1 If is convex, then the set of solutions for a choice from B(p, ω) is
convex.
2 If is strictly convex, then every consumer problem has at most one
solution.
8 / 9

Proof | 証明
We give the proof for each property in the second theorem.
.
1 Assume that both x and y maximize given B(p, ω). By the convexity
of the budget set B(p, ω) we have αx + (1 − α)y ∈ B(p, ω) and, by the
convexity of , αx + (1 − α)y x z for all α ∈ [0, 1] and z ∈ B(p, ω).
Thus, αx + (1 − α)y is also a solution to the consumer problem.
.
2 Assume that both x and y (where x = y) are solutions to the consumer
problem B(p, ω). By the convexity of the budget set B(p, ω) we have
αx + (1 − α)y ∈ B(p, ω) and, by the strict convexity of ,
αx + (1 − α)y z for all α ∈ (0, 1) and z ∈ B(p, ω), which is a
contradiction of x being optimal in B(p, ω).
9 / 9

講義 2：双対性と厚生評価
安田洋祐
2015 年 4 月 6 日
1 / 30

テキスト
*JR: Ch1; Ch4
神取: 1 章; 3 章; 補論 C
配布資料
Deaton and Muellbauer, Economics and Consumer Behavior, 1980: Ch2
関連図書
*Varian, Microeconomic Analysis, 3rd edition, 1992.
*奥野正寛・鈴村興太郎『ミクロ経済学（1）』岩波書店, 1985.
フェルドマン・セラーノ『厚生経済学と社会選択論』CAP 出版, 2009.
2 / 30

Connection between UMP and EMP | UMP と EMP の関係
There is a strong link between the utility maximization problem (UMP, 効用最
大化問題) and the expenditure minimization problem (EMP, 支出最小化問題).
Let us ﬁrst consider the following practice question.
£
¢

¡Q A consumer has the following indirect utility function:
v(p1, p2, ω) =
ω2
2p1p2
.
1 What is the consumer’s Marshallian demand for good 1?
2 What is the expenditure function?
3 What is the consumer’s Hicksian demand for good 1?
3 / 30

Answers to the Question | 問題への答え
.
1 Using Roy’s identity (ロアの恒等式), we obtain
x1(p, ω) = −
∂v(p,ω)
∂p1
∂v(p,ω)
∂ω
= −
− ω2
2p2
1p2
ω
p1p2
=
ω
2p1
.
.
2 By duality (双対性, explained formally later), the indirect utility function
can be translated into the following expenditure function:
u =
e(p, u)2
2p1p2
⇔ e(p, u) =
p
2p1p2u.
3 Using Shephard’s lemma (シェファードの補題), we obtain
xh
1 (p, u) =
∂e(p, u)
∂p1
=
r
p2u
2p1
.
4 / 30

Dual Problem - Example | 双対問題 - 例 (1)
£
¢

¡Ex A dual turtle problem.
.
1 The maximal distance a turtle can travel in 1 day is 1 km.
.
2 The minimal time it takes a turtle to travel 1 km is 1 day.
Let M(t) be the maximal distance the turtle can travel in time t.
£
¢

¡Q What kinds of conditions (on M(t)) are needed for the above statements
to be equivalent? (By deﬁnition, M(t) is non-decreasing, but this is not
enough to establish “duality” (双対性).)
5 / 30

Dual Problem - Example | 双対問題 - 例 (2)
Assume M is “strictly increasing” and “continuous.” Then,
The maximal distance a turtle can travel in t∗
is x∗
is equivalent to
The minimal time it takes a turtle to travel x∗
is t∗
Proof.
(⇒): If the maximal distance that the turtle can pass within t∗
is x∗
, and if the
minimal time to cover the distance x∗
is strictly less than t∗
, then by strict
monotonicity the turtle would cover a distance strictly larger than x∗
.
(⇐): If it takes t∗
for the turtle to cover the distance x∗
and if the turtle
passes a larger distance than x∗
in t∗
, then by continuity the turtle will be
beyond the distance with strictly less time than t∗
.
6 / 30

Dual Problem - Theory | 双対問題 - 理論 (1)
Applying the duality idea to the consumer problem, we can establish the close
relationship between the indirect utility and expenditure functions, and between
the Marshallian and Hicksian demand functions.
Let v(p, ω) and e(p, u) be the indirect utility function and expenditure
function. Then, by deﬁnition, the following property must hold:
e(p, v(p, ω)) ≤ ω for all (p, ω) 0.
v(p, e(p, u)) ≥ u for all (p, u) ∈ Rn
++ × R.
The next theorem demonstrates that under certain conditions on preferences,
both of these weak inequalities must be equalities.
7 / 30

Theorem 1
Suppose the consumer’s preference satisfy continuity and monotonicity. Then
for all p 0, ω ≥ 0 and u ∈ R:
e(p, v(p, ω)) = ω (1)
and
v(p, e(p, u)) = u. (2)
Note that, holding prices in both functions constant, we can invert the indirect
utility function (note this is strictly increasing in ω) in its income variable.
Applying the inverse function (逆関数), denoted by v−1
(p : ·), to both sides of
(2), we obtain
e(p, u) = v−1
(p : u).
Similarly, applying the inverse function (逆関数) of the expenditure function,
denoted by e−1
(p : ·), to both sides of (1), we obtain
v(p, ω) = e−1
(p : ω).
8 / 30

Theorem 2
Suppose the consumer’s preference is continuous, monotone and strictly
convex. Then, we have the following relations between the Hicksian and
Marshallian demand functions for p 0, ω ≥ 0 and u ∈ R, and i = 1, 2, ..., n:
xi(p, ω) = xh
i (p, v(p, ω))
and
xh
i (p, u) = xi(p, e(p, u)).
.
1 Marshallian demand at prices p and income ω is equal to the Hicksian
demand at those prices and the maximum utility level that can be
achieved at those prices and income ω.
2 Hicksian demand at any prices p and utility level u is the same as the
Marshallian demand at those prices and an income level equal to the
minimum expenditure necessary at those prices to achieve utility level u.
9 / 30

Slutsky Equation | スルツキー方程式 (1)
When the price of a good declines, there are two conceptually separate
reactions: The consumer is expected to substitute the relatively cheaper good
for the now relatively more expensive good (= substitution effect (代替効果)),
and to arrange her purchases of all goods due to the expansion of her effective
income, i.e., the budget set (= income effect (所得効果)).
Theorem 3
Suppose the consumer’s preference is continuous, monotone and strictly
convex, and all the relevant functions are differentiable. Let u∗
be the level of
utility the consumer achieves at prices p and income ω∗
. Then, for
i, j = 1, ..., n,
∂xi(p, ω∗
)
∂pj
TE
=
∂xh
i (p, u∗
)
∂pj
SE
−xj(p, ω∗
)
∂xi(p, ω∗
)
∂ω
IE
.
10 / 30

By duality, xh
i (p, u∗
) = xi(p, e(p, u∗
)).
Since this equality holds for all p 0, diﬀerentiating both sides with respect
to pj preserves the equality.
∂xh
i (p, u∗
)
∂pj
=
∂xi(p, e(p, u∗
))
∂pj
+
∂xi(p, e(p, u∗
))
∂ω
∂e(p, u∗
)
∂pj
.
By duality and Shephard’s lemma,
e(p, u∗
) = e(p, v(p, ω∗
)) = ω∗
∂e(p, u∗
)
∂pj
= xh
j (p, u∗
) = xh
j (p, v(p, ω∗
)) = xj(p, ω∗
).
Substituting these relations into the second equation,
∂xh
i (p, u∗
)
∂pj
=
∂xi(p, ω∗
)
∂pj
+
∂xi(p, ω∗
)
∂ω
xj(p, ω∗
).
11 / 30

The above equation is called the Slutsky equation (スルツキー方程式),
sometimes called the “Fundamental Equation of Demand Theory”, which
provides neat analytical expressions for substitution and income effects.
When j = i, the Slutsky equation shows the response of the Marshallian
demand to a change in own price.
∂xi(p, ω)
∂pi
=
∂xh
i (p, u∗
)
∂pi
− xi(p, ω)
∂xi(p, ω)
∂ω
.
Although substitution effects are not observable, demand theory can provide
some strong properties on own-price effects and cross-substitution effects. The
first claim says that own-price effects can never be positive.
12 / 30

Substitution Effects | 代替効果 (1)
Theorem 4
Suppose e(p, u) is twice continuously differentiable in p. Then, for i = 1, ..., n,
∂xh
i (p, u)
∂pi
≤ 0.
Proof.
By Shephard’s lemma,
xh
i (p, u) =
∂e(p, u)
∂pi
.
Differentiating again with respect to pi, we obtain
∂xh
i (p, u)
∂pi
=
∂2
e(p, u)
∂p2
i
.
The right hand side must be non-positive since the expenditure function is a
concave function of p.
13 / 30

The non-positive own-price effects give us some implication to the response of
the Marshallian demand as well.
A good is called normal (正常財) (resp. inferior (下級財)) if consumption
of it increases (resp. declines) as income increases, holding prices constant.
A decrease in the own price of a normal good will cause quantity
demanded to increase. If an own price decrease causes a decline in
quantity demanded, known as Giffen’s paradox (ギッフェン・パラドク
ス), the good must be inferior.
The next theorem says that “cross-substitution effects” are symmetric.
14 / 30

Theorem 5
Suppose that e(p, u) is twice continuously diﬀerentiable in p. Then, for
i, j = 1, ..., n,
∂xh
i (p, u)
∂pj
=
∂xh
j (p, u)
∂pi
.
Proof.
By Shephard’s lemma,
∂xh
i (p, u)
∂pj
=
∂
∂pj
(
∂e(p, u)
∂pi
) =
∂2
e(p, u)
∂pj∂pi
, and
∂xh
j (p, u)
∂pi
=
∂
∂pi
(
∂e(p, u)
∂pj
) =
∂2
e(p, u)
∂pi∂pj
.
By Yong’s theorem (ヤングの定理), we complete the proof:
∂2
e(p, u)
∂pj∂pi
=
∂2
e(p, u)
∂pi∂pj
.
15 / 30

How to Measure Welfare Change | 厚生の変化をどうはかるか？
When the economic environment or market outcome changes, a consumer may
be made better off (改善) or worse off (悪化). Economists often want to
measure how consumers are affected by these changes, and have developed
several tools for the assessment of welfare (厚生).
The obvious measure of the welfare change involved in moving from (p0
, ω0
) to
(p1
, ω1
) is just the difference in indirect utility:
v(p1
, ω1
) − v(p0
, ω0
).
If the utility difference is positive, then the policy change is worth doing,
at least as far as this consumer is concerned.
If it is negative, the policy change is not worth doing.
£
¢

¡Q Is there any monetary measure that quantifies welfare changes?
16 / 30

Consumers’ Surplus | 消費者余剰
Suppose that the price of some good moves from p0
to p1
while the prices of
other goods and initial wealth remain unchanged.
Deﬁnition 1
The classical measure of welfare change is consumers’ surplus (CS, 消費者余
剰), which is the area below the Marshallian demand curve and above market
price. The change of CS is deﬁned as
∆CS := CS(p1
, ω) − CS(p0
, ω) =
Z p0
p1
x(p, ω)dp.
This is simply the area to the left of the Marshallian demand curve
between p0
and p1
.
Although CS is intuitive and simple, it is an exact measure of welfare
change only in special circumstances.
17 / 30

Beyond Consumers’ Surplus | 消費者余剰を超えて
Depending on how to quantify utility changes, we have two different measures.
Definition 2
The compensating variation (CV, 補償変分) and equivalent variation (EV,
等価変分) are defined as follows:
v(p1
, ω + CV ) = v(p0
, ω),
v(p0
, ω − EV ) = v(p1
, ω).
EV uses the current prices as the base and asks what income change at
the current prices would be equivalent to the proposed change in terms of
its impact on (indirect) utility.
CV uses the new prices as the base and asks what income change would
be necessary to compensate the consumer.
That is, EV (resp. CV ) requires to keep a consumer’s utility constant
before (resp. as a result of) a price change.
§
¦
¤
¥
Fg Figure 4.5 (see JR, pp.181)
18 / 30

Compensating Variation | 補償変分
Using the deﬁnitions of CV and expenditure function, CV can be written by
e(p1
, v(p0
, ω)) = e(p1
, v(p1
, ω + CV ))
= ω + CV
⇒ CV = e(p1
, v(p0
, ω)) − e(p0
, v(p0
, ω))
By Shepard’s lemma, we obtain
CV = e(p1
, v(p0
, ω)) − e(p0
, v(p0
, ω))
=
Z p1
p0
∂e(p, v(p0
, ω))
∂p
dp =
Z p1
p0
qh
(p, v(p0
, ω))dp.
This is simply the area to the left of the Hicksian demand curve between p0
and p1
, when the target utility level is v(p0
, ω).
19 / 30

Equivalent Variation | 等価変分
Similarly, EV can be expressed by
e(p0
, v(p1
, ω)) = e(p0
, v(p0
, ω − EV )) = ω − EV
⇒ EV = e(p1
, v(p1
, ω)) − e(p0
, v(p1
, ω))
=
Z p1
p0
qh
(p, v(p1
, ω))dp.
This is simply the area to the left of the Hicksian demand curve between p0
and p1
, when the target utility level is v(p1
, ω).
The absolute value of ∆CS is always between that of CV and EV . These
three measures coincide if and only if there is no income eﬀect, for instance,
when the utility function is quasi-linear (準線形): u(x1, x2) = f(x1) + x2.
20 / 30

Exchange Economy | 交換経済
Next, we introduce the welfare measures that do not rely on any quantitative
assessment.
Consider an exchange economy (交換経済) with I people and n goods where
all of the economic agents are consumers and production is absent.
Let e = (e1
, ..., eI
) denote the economy’s initial endowment (初期保有)
vector, where ei
= (ei
1, ..., ei
n) denotes i’s initial endowment.
Deﬁne an allocation (配分) as a vector, x = (x1
, ..., xI
), where
xi
= (xi
1, ..., xi
n) denotes i’s consumption bundle according to the
allocation.
The set of feasible allocations (達成可能配分) is this economy is given by
F(e) = {x |
X
i∈I
xi
=
X
i∈I
ei
}.
21 / 30

Edgeworth Box | エッジワース・ボックス
The most useful example of an exchange economy is one in which there are two
people and two goods. This economy’s set of allocations can be illustrated in
an Edgeworth box (エッジワース・ボックス) diagram.
The length of the horizontal axis measures the total amount of good 1.
The height of the vertical axis measures the total amount of good 2.
Each point in this box is a feasible allocation.
§
¦
¤
¥
Fg Figures 5.1 and 5.2 (see JR, pp.196-197)
£
¢

¡Q How will agents trade their goods in voluntary exchange?
⇒ If they trade both eﬃciently and in mutually beneﬁcial way, then the
allocation must be on the contract curve (契約曲線).
22 / 30

Pareto Efficiency | パレート効率性 (1)
A situation is called Pareto efficient (パレート効率的) if there is no way to
make someone better off without making someone else worse off.
That is, there is no way to make all agents better off.
To put it differently, each agent is as well off as possible, given the utilities
of the other agents.
This central welfare notion in Economics is formally defined as follows.
Definition 3
A feasible allocation, x ∈ F(e), is (strongly) Pareto efficient if there is no
other feasible allocation, y ∈ F(e) such that
ui(yi
) ≥ ui(xi
) for all i ∈ I, and
ui(yi
) > ui(xi
) for at least one i ∈ I.
23 / 30

Pareto Efficiency | パレート効率性 (2)
Definition 4
A feasible allocation, x ∈ F(e), is weakly Pareto efficient if there is no other
feasible allocation, y ∈ F(e) such that
ui(yi
) > ui(xi
) for all i ∈ I
It is straightforward that an allocation that is (strongly) Pareto efficient is also
weakly Pareto efficient.
In general, the reverse is not true. However, under some additional weak
assumptions, the reverse implication is true.
Theorem 6
Suppose that preference relations are continuous and monotonic. Then an
allocation is weakly Pareto efficient if and only if it is strongly Pareto efficient.
Let S ⊂ I denote a coalition (subset) of consumers.
24 / 30

Block and Core | ブロックとコア
Definition 5
A coalition S blocks (ブロックする) x ∈ F(e), if there is an allocation (among
S) y such that
X
i∈S
yi
=
X
i∈S
ei
,
ui(yi
) ≥ ui(xi
) for all i ∈ S, and
ui(yi
) > ui(xi
) for at least one i ∈ S.
That is, an allocation x is blocked whenever some group can do better than
they do under x by simply going it alone.
Definition 6
The core (コア) of an exchange economy C(e) is the set of all feasible
allocations which cannot be blocked by any coalition.
£
¢

¡Rm The core must be Pareto efficient, since the core cannot be blocked by
any allocation, in particular by the grand coalition, i.e., S = I.
25 / 30

Not in Core but Pareto Eﬃcient Allocation | コアに含まれない効率配分
Suppose there are two goods and three individuals whose initial endowments are
e1
= (5, 0), e2
= (0, 5), e3
= (1, 1).
The preferences of these individuals are symmetric and convex, represented by
u1
= u2
= u3
=
√
x1x2.
Then, the following allocation x is Pareto eﬃcient but not in the Core.
x1
= x2
= x3
= (2, 2).
Since individuals 1 and 2 can block x by y1
= y2
= (2.5, 2.5).
In this example, there is a unique allocation in the Core,
x1
= x2
= (2.5, 2.5), x3
= (1, 1).
26 / 30

【補論】 General Environmental Changes | 一般的な環境変化
Consider the move from (p0
, ω0
) to (p1
, ω1
). Then,
Deﬁnition 7
The compensating variation (CV) and equivalent variation (EV) are deﬁned
as follows:
v(p1
, ω1
+ CV ) = v(p0
, ω0
),
v(p0
, ω0
− EV ) = v(p1
, ω1
).
£
¢

¡Rm CV and EV can be expressed as follows:
e(p1
, v(p0
, ω0
)) = e(p1
, v(p1
, ω1
+ CV )) = ω1
+ CV
⇒ CV = e(p1
, v(p0
, ω0
)) − ω1
,
e(p0
, v(p1
, ω1
)) = e(p0
, v(p0
, ω0
− EV )) = ω0
− EV
⇒ EV = ω0
− e(p0
, v(p1
, ω1
)).
27 / 30

【補論】 Pareto Efficiency and Calculus | パレート効率性と計算
Theorem 7
A feasible allocation x∗
is Pareto efficient if and only if x∗
solves the following
maximization problems for i = 1, ..., I:
max
x
ui(xi) s.t.
IX
h=1
xk
h ≤ ek
k = 1, ..., n
uj(x∗
j ) ≤ uj(xj) for all j = i.
Proof.
(⇐) Suppose x∗
solves all maximization problems but x∗
is not Pareto
efficient. This means that there is some allocation x where someone i is
strictly better off. But then x∗
cannot solve the problem for i, a contradiction.
(⇒) Suppose x∗
is Pareto efficient, but it does not solve one of the problems.
Instead, let x solve that particular problem. Then x makes one of the agents
strictly better off without hurting any other agents, a contradiction.
28 / 30

【補論】 Social Welfare Function | 社会的厚生関数 (1)
Deﬁnition 8
A social welfare function (社会的厚生関数) W : RI
→ R is a hypothetical
scheme for ranking potential allocations of resources based on the private
utilities they provide to individuals:
Social Welfare = W(u1(x1), · · · , uI (xI ))
Assume that W is increasing in each of its arguments. Then we immediately
obtain the following theorem.
Theorem 8
If x∗
solves the following (social welfare) maximization problem, then x∗
is
Pareto eﬃcient.
max
x
W(u1(x1), · · · , uI (xI ))
s.t.
IX
h=1
xk
h ≤ ek
for k = 1, ..., n.
29 / 30

【補論】 Social Welfare Function | 社会的厚生関数 (2)
Imposing additional assumptions, we can completely characterize Pareto
eﬃcient allocations by the maximization problem of weighted average (荷重平
均) of individual utilities.
Theorem 9
If ui is an increasing and concave function for all i ∈ I. Then, x∗
is Pareto
eﬃcient if and only if x∗
solves the following maximization problem for some
(λ1, · · · , λI ) ∈ RI
+ {0}.
max
x
IX
i=1
λiui(xi)
s.t.
IX
h=1
xk
h ≤ ek
for k = 1, ..., n.
£
¢

¡Q What happens if a consumer’s utility function is not concave?
30 / 30

講義 3：意思決定と不確実性
安田洋祐
2015 年 4 月 7 日
1 / 34

テキスト
*JR: Ch1; Ch2
神取：1 章; 6 章
尾山・安田: 9 章
配布資料
*Laﬀont, The Economics of Uncertainty and Information, 1989: Ch2
関連図書
*ギルボア『不確実性下の意思決定理論』勁草書房, 2014.
*Kreps, Notes on The Theory of Choice, 1988.
*Rubinstein, Lecture Notes in Microeconomic Theory: The Economic
Agent, 2nd edition, 2012.
Binmore, Rational Decision, 2009.
ギルボア『合理的選択』みすず書房, 2013.
2 / 34

Preferences | 選好
To construct a model of individual choice, the notion of preferences (選好)
plays a central role in economic theory, which speciﬁes the form of consistency
or inconsistency in the person’s choices.
We view preferences as the mental attitude of an individual toward alternatives
independent of any actual choice.
We require only that the individual make binary (二項の) comparisons,
that is, that she only examine two choice alternatives at a time and make
a decision regarding those two.
For each pair of alternatives in the choice set X, the description of
preferences should provide an answer to the question of how the agent
compares the two alternatives.
Considering questionnaire (アンケート) R, we formulate the consistency
requirements necessary to make the responses “preferences”.
3 / 34

Questionnaire R | アンケート R
§
¦
¤
¥
R(x, y) for all x, y ∈ X, not necessarily distinct:
£
¢

¡Q Is x at least as preferred as y? Tick one and only one of the following two
options:
.
1 Yes (or, x is at least as good as y): x y.
.
2 No (or, x is strictly worse than y): x y.
Deﬁnition 1
Preferences (R) on a set X is a binary relation on X satisfying the following
two axioms.
Axiom 1: Completeness (完備性)
For any x, y ∈ X, x y or y x.
Axiom 2: Transitivity (推移性):
For any x, y, z ∈ X, if x y and y z, then x z.
4 / 34

Remarks on the Axioms | 公理に関する注意
Completeness formalizes the notion that the individual can make
comparisons, that is, that she has the ability to discriminate and the
necessary knowledge to evaluate alternatives. It says the individual can
examine any two distinct alternatives.
Transitivity gives a very particular form to the requirement that the
individual choices be consistent. Although we require only that she be
capable of comparing two alternatives at a time, the axiom of transitivity
requires that those pairwise comparisons be linked together in a consistent
way.
£
¢

¡Rm The money pump argument when transitivity is violated.
→ 推移性を満たさない個人からはお金をいくらでも吸い取ることができる！
5 / 34

Questionnaire P | アンケート P
§
¦
¤
¥
P(x, y) For all distinct x and y in the set X. How do you compare x and y?
Tick one and only one of the following three options:
.
1 I prefer x to y, or x is strictly preferred (強く選好される) to y: x y
.
2 I prefer y to x, or y is strictly preferred to x: y x
.
3 I am indiﬀerent, or x is indiﬀerent (無差別である) to y: x ∼ y
Note that we implicitly assume that the elements in X are all comparable, and
ignore the intensity of preferences.
A legal answer to the questionnaire P can be formulated as a function f which
assigns to any pair (x, y) of distinct elements in X exactly one of the three
values: x y, y x or x ∼ y. That is,
f(x, y) =
8
<
:
x y
y x
x ∼ y
.
6 / 34

Preference P | 選好 P (1)
Preferences are characterized by axioms (公理) that are intended to give
formal mathematical expression to fundamental aspects of choice behavior and
attitudes toward the objects of choice.
The following basic axioms are (almost) always imposed.
Definition 2
Preferences (P) on a set X are a function f that assigns to any pair (x, y) of
distinct elements in X exactly one of the three values: x y, y x or x ∼ y
so that for any three different elements x, y and z in X, the following two
properties hold:
1 No order effect: f(x, y) = f(y, x).
2 Transitivity:
1 if f(x, y) = x y and f(y, z) = y z, then f(x, z) = x z, and
2 if f(x, y) = x ∼ y and f(y, z) = y ∼ z, then f(x, z) = x ∼ z.
7 / 34

Preference P | 選好 P (2)
The ﬁrst property requires the answer to P(x, y) being identical to the answer
to P(y, x), and the second requires that the answer to P(x, y) and P(y, z) are
consistent with the answer to P(x, z) in a particular way.
£
¢

¡Ex Non-preference relation
For any x, y ∈ R, f(x, y)(= f(y, x)) = x y if x ≥ y + 1 and f(x, y) = x ∼ y
if |x − y| < 1. This is not a preference relation since transitivity is violated. For
instance, suppose x = 1, y = 1.8, z = 2.6. Then,
f(x, y) = x ∼ y and f(y, z) = y ∼ z, but f(x, z) = z x,
which violates transitivity (2-2).
8 / 34

Equivalence of the Two Preferences | 2 つの選好の同値性
We can translate one formulation of preferences to another by the following
mapping (bijection). Note that completeness guarantees “x y and y x”
never happen.
f(x, y) = x y ⇔ x y and y x.
f(x, y) = y x ⇔ y x and x y.
f(x, y) = x ∼ y ⇔ x y and y x.
In our lectures, we take the second deﬁnition, i.e., preference (R), and denote
x y when both x y and y x, and x ∼ y, when x y and y x.
Deﬁnition 3
A preference (R) is called a preference relation (選好関係).
9 / 34

Utility Representation | 効用表現
Definition 4 (Review)
Function U : X → R represents (表現する) the preference if for all x and
y ∈ X, x y if and only if U(x) ≥ U(y). If the function U represents the
preference relation , we refer to it as a utility function and we say that has
a utility representation (効用表現).
£
¢

¡Q Under what conditions do utility representations exist?
Theorem 1
If is a preference relation on a finite set X, then has a utility
representation with values being natural numbers.
Proof.
There is a minimal (resp. maximal) element (an element a ∈ X is minimal
(resp. maximal) if a x (resp. a x) for any x ∈ X) in any finite set A ⊂ X.
We can construct a sequence of sets from the minimal to the maximal and can
assign natural numbers according to their ordering.
10 / 34

Revealed Preferences | 顕示選好
Important diﬀerence between choice (demand) and preferences or utility is that
the former is observable while the latter cannot be.
We may want to develop the theory which is based on the observable choice
behaviors, not on preferences or utility.
We say that the preferences (fully) rationalize the demand function x if
for any (p, ω) the bundle x(p, ω) is the unique best bundle within
B(p, ω).
We say that a is revealed to be better than b, if there is (p, ω) so that
both a and b are in B(p, ω) and a = x(p, ω).
£
¢

¡Q What are general conditions guaranteeing that a demand function x(p, ω)
can be rationalized?
→ Present two axioms of revealed preferences (顕示選好).
11 / 34

Weak Axiom of Revealed Preferences | 顕示選好の弱公理
Deﬁnition 5 (Weak Axiom)
The weak axiom of revealed preferences (WA) is a property of choice
function which says that it is impossible that a be revealed to be better than b
and b be revealed to be better than a. That is,
if px(p , ω ) ≤ ω and x(p, ω) = x(p , ω ),
then p x(p, ω) > ω .
§
¦
¤
¥
£
¢

¡Rm Note that any choice function rationalized by some preference relation
must satisfy WA.
12 / 34

Weak Axiom ⇒ Law of Demand | 弱公理 ⇒ 需要法則
Theorem 2
Let x(p, ω) be a choice function satisfying Walras’s Law and WA. Then,
. 1 x(·) is homogeneous of degree zero, and
.
2 if ω = p x(p, ω), then either x(p , ω ) = x(p, ω) or
(p − p)(x(p , ω ) − x(p, ω)) < 0.
Proof.
The proof for 1 is left for the assignment. Assume that x(p , ω ) = x(p, ω). By
Walras’s Law and the assumption that ω = p x(p, ω):
(p − p)(x(p , ω ) − x(p, ω))
= p x(p , ω ) − p x(p, ω) − px(p , ω ) + px(p, ω)
= ω − ω − px(p , ω ) + ω
= ω − px(p , ω ).
By WA the right hand side is less than 0.
13 / 34

Strong Axiom of Revealed Preferences | 顕示選好の強公理
The previous theorem implies that the compensated (Hicksian) demand
function y(p ) = x(p , p x(p, ω)) satisfies the law of demand (需要法則),
that is, yk is decreasing in pk.
WA is not a sufficient condition for extending the binary relation (defined
from the choice function) into a complete and transitive relation. The following
stronger condition than WA is known to be necessary and sufficient.
Definition 6 (Strong Axiom)
Choice function satisfies the strong axiom of revealed preferences (SA) if for
every sequence of distinct bundles x0
, x1
, ..., xk
, where x0
is revealed preferred
to x1
, and x1
is revealed preferred to x2
, ..., and xk−1
is revealed preferred to
xk
, it is not the case that xk
is revealed preferred to x0
.
14 / 34

Decision under Uncertainty | 不確実性下の意思決定
We have so far not distinguished between individual’s actions (行動) and
consequences (帰結), but many choices made by agents take place under
conditions of uncertainty (不確実性).
This lecture discusses such a decision under uncertainty, i.e., an environment in
which the correspondence between actions and consequences is not
deterministic (確定的) but stochastic (確率的).
To discuss a decision under uncertainty, we extend the domain of choice
functions. The choice of an action is viewed as choosing a “lottery” (く
じ) where the prizes are the consequences.
An implicit assumption is that the decision maker does not care about the
nature of the random factors but only about the distribution (分布) of
consequences.
15 / 34

Lotteries | くじ (1)
We consider preferences and choices over the set of “lotteries.”
Let S be a set of consequences or prizes (賞). We assume that S is a
ﬁnite set and the number of its elements (= |S|) is S.
A lottery p is a function that assigns a nonnegative number to each prize
s, where
P
s∈S p(s) = 1 (here p(s) is the objective probability (客観確率)
of obtaining the prize s given the lottery p).
Let α ◦ x ⊕ (1 − α) ◦ y denote the lottery in which the prize x is realized
with probability α and the prize y with 1 − α.
Denote by L(S) the (inﬁnite) space containing all lotteries with prizes in
S. That is, {x ∈ RS
+|
P
xs = 1}.
We will discuss preferences over L(S).
16 / 34

Lotteries | くじ (2)
We impose the following three assumptions on the lotteries.
.
1 1 ◦ x ⊕ (1 − 1) ◦ y ∼ x: Getting a prize with probability one is the same as
getting the prize for certain.
.
2 α ◦ x ⊕ (1 − α) ◦ y ∼ (1 − α) ◦ y ⊕ α ◦ x: The consumer does not care
about the order in which the lottery is described.
.
3 β ◦ (α ◦ x ⊕ (1 − α) ◦ y) ⊕ (1 − β) ◦ y ∼ (βα) ◦ x ⊕ (1 − βα) ◦ y: A
consumer’s perception of a lottery depends only on the net probabilities of
receiving the various prizes.
The ﬁrst two assumptions appear to be innocuous.
The third assumption sometimes called “reduction of compound lotteries (複
合くじ)” is somewhat suspect.
There is some evidence to suggest that consumers treat compound
lotteries diﬀerent than one-shot lotteries.
17 / 34

St Petersburg Paradox | セントペテルスブルグのパラドックス (1)
The most primitive way to evaluate a lottery is to calculate its mathematical
expectation (数学的期待値), i.e., E[p] =
P
s∈S p(s)s.
Daniel Bernoulli first doubt this approach in the 18th century when he
examined the St Petersburg paradox (セントペテルスブルグのパラドックス).
£
¢

¡Ex St Petersburg Paradox
A fair coin is tossed until it shows heads for the first time. If the first head
appears on the k-th trial, a player wins $2k
. How much are you willing to pay
to participate in this lottery?
£
¢

¡Rm The expected value of the lottery is infinite:
2
2
+
22
22
+
23
23
+ · · · = 1 + 1 + 1 + · · · = ∞.
18 / 34

St Petersburg Paradox | セントペテルスブルグのパラドックス (2)
The St Petersburg paradox shows that maximizing your dollar expectation may
not always be a good idea. It suggests that an agent in risky situation might
want to maximize the expectation of some “utility function” with decreasing
marginal utility:
E[u(x)] = u(2)
1
2
+ u(4)
1
4
+ u(8)
1
8
+ · · ·,
which can be a finite number.
£
¢

¡Q Under what kinds of conditions does a decision maker maximizes the
expectation of some “utility function”?
£
¢

¡Rm By utility theory, we know that for any preference relation defined on the
space of lotteries that satisfies continuity, there is a utility representation U:
L(S) → R, continuous in the probabilities, such that p q if and only if
U(p) ≥ U(q).
19 / 34

Expected Utility Theory | 期待効用理論 (1)
We will use the following two axioms to isolate a family of preference relations
which have a representation by a more structured utility function.
Independence Axiom (I, 独立性公理): For any p, q, r ∈ L(S) and any
α ∈ (0, 1), p q ⇔ α ◦ p ⊕ (1 − α) ◦ r α ◦ q ⊕ (1 − α) ◦ r.
Continuity Axiom (C, 連続性公理): If p q r, then there exists
α ∈ (0, 1) such that
q ∼ [α ◦ p ⊕ (1 − α) ◦ r].
Theorem 3
Let be a preference relation over L(S) satisfying the I and C. There are
numbers (v(s))s∈S such that
p q ⇔ U(p) =
X
s∈S
p(s)v(s) ≥ U(q) =
X
s∈S
q(s)v(s).
20 / 34

Expected Utility Theory | 期待効用理論 (2)
Sketch of the proof.
Let M and m be a best and a worst certain lotteries in L(S). When M ∼ m,
choosing v(s) = 0 for all s we have
P
s∈S p(s)v(s) = 0 for all p ∈ L(S).
Consider the case that M m. By I and C, there must be a single number
v(s) ∈ [0, 1] such that
v(s) ◦ M ⊕ (1 − v(s)) ◦ m ∼ [s]
where [s] is a certain lottery with prize s, i.e., [s] = 1 ◦ s.
In particular, v(M) = 1 and v(m) = 0. I implies that
p ∼ (
X
s∈S
p(s)v(s)) ◦ M ⊕ (1 −
X
s∈S
p(s)v(s)) ◦ m.
Since M m, we can show that
p q ⇔
X
s∈S
p(s)v(s) ≥
X
s∈S
q(s)v(s).
21 / 34

vNM Utility Function | ノイマン-モルゲンシュテルン効用関数 (1)
Note the function U is a utility function representing the preferences on L(S)
while v is a utility function deﬁned over S, which is the building block for the
construction of U(p). We refer to v as a vNM (Von Neumann-Morgenstern)
utility function (ノイマン-モルゲンシュテルン効用関数).
£
¢

¡Q How can we construct the vNM utility function?
Let si(∈ S), i = 1, ..., K be a set of consequences and s1, sK be the best and
the worst consequences. That is, for any i,
[s1] [si] [sK ].
Then, construct a function v : S → [0, 1] in the following way:
v(s1) = 1 and v(sK ) = 0, and
[sj] ∼ v(sj) ◦ [s1] ⊕ (1 − v(sj) ◦ [sK ] for all j.
By continuity axiom, we can ﬁnd a unique value of v(sj) ∈ [0, 1].
22 / 34

£
¢

¡Q To what extent, vNM utility function is unique?
The vNM utilities are unique up to positive affine transformation (アフィン変
換), i.e., multiplication by a positive number and adding any scalar, and are not
invariant to arbitrary monotonic transformation (単調変換).
Theorem 4
Suppose is a preference relation defined over L(S) and let v(s) be the vNM
utilities representing the preference relation. Then, defining w(s) = αv(s) + β
for all s (for some α > 0 and some β), the utility function
W(p) =
P
s∈S p(s)w(s) also represents .
23 / 34

Proof.
For any lotteries p, q ∈ L(S), p q if and only if
X
s∈S
p(s)v(s) ≥
X
s∈S
q(s)v(s).
Now, the followings hold.
X
s∈S
p(s)w(s) =
X
s∈S
p(s)(αv(s) + β) = α
X
s∈S
p(s)v(s) + β.
X
s∈S
q(s)w(s) =
X
s∈S
q(s)(αv(s) + β) = α
X
s∈S
q(s)v(s) + β.
Thus,
X
s∈S
p(s)v(s) ≥
X
s∈S
q(s)v(s)
holds if and only if
X
s∈S
p(s)w(s) ≥
X
s∈S
q(s)w(s) (for α > 0).
24 / 34

Allais Paradox | アレのパラドックス (1)
Many experiments reveal systematic deviations from vNM assumptions. The
most famous one is the Allais paradox (アレのパラドックス).
£
¢

¡Ex Allais paradox
Choose ﬁrst the between
L1 = [3000] and L2 = 0.8 ◦ [4000] ⊕ 0.2 ◦ [0]
and then choose between
L3 = 0.5 ◦ [3000] ⊕ 0.5 ◦ [0] and L4 = 0.4 ◦ [4000] ⊕ 0.6 ◦ [0].
Note that L3 = 0.5 ◦ L1 ⊕ 0.5 ◦ [0] and L4 = 0.5 ◦ L3 ⊕ 0.5 ◦ [0]. Axiom I
requires that the preference between L1 and L2 be the same as that between
L3 and L4. However, a majority of people express the preferences L1 L2 and
L3 ≺ L4, violating the axiom.
25 / 34

Allais paradox | アレのパラドックス (2)
Assume L1 L2 but α ◦ L ⊕ (1 − α) ◦ L1 ≺ α ◦ L ⊕ (1 − α) ◦ L2. (In our
example of Allais paradox, α = 0.5 and L = [0].)
Then, we can perform the following trick on the decision maker:
.
1 Take α ◦ L ⊕ (1 − α) ◦ L1.
.
2 Take instead α ◦ L ⊕ (1 − α) ◦ L2, which you prefer (and you pay me
something...).
.
3 Let us agree to replace L2 with L1 in case L2 realizes (and you pay me
something now...).
4 Note that you hold α ◦ L ⊕ (1 − α) ◦ L1.
5 Let us start from the beginning...
This argument may make the independence axiom looks somewhat reasonable
(and Allais paradox unreasonable).
26 / 34

Zeckhouser’s Paradox | ゼックハウザーのパラドックス (1)
Allais paradox can be viewed as a violation of independence axiom. The
following paradox also shows that many people do not necessarily follow the
expected utility maximization behavior.
£
¢

¡Ex Zeckhauser’s paradox
Some bullets are loaded into a revolver with six chambers. The cylinder is then
spun and the gun pointed at your head.
Would you be prepared to pay more to get one bullet removed when only one
bullet was loaded, or when four bullets were loaded?
£
¢

¡Q People usually say they would pay more in the ﬁrst case, because they
would then be buying their lives for certain.
Is this decision reasonable?
£
¢

¡Rm Note that you cannot use your money once you die...
27 / 34

Zeckhouser’s Paradox | ゼックハウザーのパラドックス (2)
Suppose $X (resp. $Y ) is the most that you are willing to pay to get one
bullet removed from a gun containing one (resp. four) bullet. Let L mean
death, and W mean being alive after paying nothing. Let C mean being alive
after paying $X, and D mean being alive after paying $Y . Note that
u(D) < u(C) ⇔ D ≺ C ⇔ X < Y .
Let u(L) = 0 and u(W) = 1. Then,
u(C) =
1
6
u(L) +
5
6
u(W) =
5
6
, and
1
2
u(L) +
1
2
u(D) =
2
3
u(L) +
1
3
u(W) ⇒ u(D) =
2
3
.
Since u(D) < u(C), you must be ready to pay less to get one bullet removed
when only one bullet was loaded than when four bullets were loaded.
28 / 34

【補論】 Choice Function | 選択関数
We consider an agent’s “behavior” as a hypothetical response to the following
questionnaire, one for each A ⊆ X:
§
¦
¤
¥
Q(A) Assume you must choose from a set of alternatives A. Which
alternative do you choose?
A choice function C assigns to each set A ⊆ X a unique element of A
with the interpretation that C(A) is the chosen element from the set A.
£
¢

¡Rm Here are a couple of remarks on choice functions:
1 We assume that the agent selects a unique element in A for every
question Q(A).
2 The choice function C does not need to be observable.
3 The agent behaving in accordance with C will choose C(A) if she has to
make a choice from a set A.
29 / 34

【補論】 Continuous Preferences | 連続な選好
To guarantee the existence of a utility representation over consumption set,
i.e., an infinite subset of Rn
, we need some additional axiom.
Definition 7
A preference relation on X is continuous (連続, Axiom 3) if {xn
} (a
sequence of consumption bundles) with limit x satisfies the following two
conditions for all y ∈ X.
.
1 if x y, then for all n sufficiently large, xn
y, and
.
2 if y x, then for all n sufficiently large, y xn
.
The equivalent definition of continuity is that the “at least as good as” and
“no better than” sets for each point x ∈ X are closed. This axiom rules out
certain discontinuous (不連続な) behavior and guarantees that sudden
preference reversals do not occur.
§
¦
¤
¥
Fg Figures 1.2 and 1.3 (see JR, pp.9)
30 / 34

【補論】 Continuous Utility | 連続な効用
Given axioms 1-3, we can establish the existence of the (continuous) utility
function.
Theorem 5
Assume that X is a convex subset of Rn
. If is a continuous preference
relation on X, then is represented by a continuous utility function.
Here are two remarks on continuity.
.
1 If on X is represented by a continuous function U, then must be
continuous.
2 The lexicographic preferences (辞書的選好) are not continuous.
Theorem 6
The lexicographic preference relation L on [0, 1] × [0, 1], i.e.,
(a1, a2) L (b1, b2) if a1 > b1 or both a1 = b1 and a2 ≥ b2, does not have a
utility representation.
31 / 34

【補論】 Solutions to Consumer Problems | 消費者問題の解
Theorem 7
If is a continuous preference relation, then all consumer problems have a
solution.
Proof.
Since the budget set is convex, we can apply the ﬁrst theorem in the previous
slide to establish that the preferences are represented by a continuous utility
function.
Then, by the Weielstrass theorem (ワイエルシュトラスの定理), there exists a
maximum (and minimum) value of continuous functions if the domain is a
compact (that is, closed and bounded) set and a range is R. Since every
budget set is compact and a utility function is continuous, there must exist a
consumption bundle which gives a maximum utility value, a solution of the
consumer problem.
32 / 34

【補論】 Rational Choice | 合理的な選択
We (Economics) assume that when the agent has in mind a preference relation
on X, then given any choice problem Q(A) for A ⊆ X, she chooses an
element in A which is “ optimal.”
Definition 8
An induced choice function C is the function that assigns every nonempty
set A ⊆ X the -best element of A.
A choice function C can be rationalized (合理化される) if there is a preference
relation on X so that C = C , i.e., C(A) = C (A) for any A in the domain
of C.
£
¢

¡Q Under what conditions any choice functions can be presented “as if (あた
かも)” derived from some preference relation?
Definition 9
Choice function C satisfies condition α if for any A ⊂ B, C(B) ∈ A implies
C(A) = C(B).
33 / 34

【補論】 Choice ⇐⇒ Preference | 選択 ⇐⇒ 選好
Theorem 8
Assume C is a choice function with a domain containing at least all subsets of
X of size 2 or 3. If C satisfies condition α, then there is a preference relation
on X so that C = C .
Proof.
Define by x y if x = C({x, y}). Let us first show that satisfies
completeness and transitivity.
Completeness: Follows from that C = ({x, y}) is well-defined.
Transitivity: If x y and y z, then by definition of we have
C({x, y}) = x and C({y, z}) = y. If C({x, z}) = z, then, by condition α,
C({x, y, z}) = x. Similarly, by C({x, y}) = x and condition α,
C({x, y, z}) = y, and by C({y, z}) = y and condition α, C({x, y, z}) = z. A
contradiction to C({x, y, z}) ∈ {x, y, z}.
Next we show that C(A) = C (A) for all A ⊆ X. Suppose on
contrary C(A) = C (A). That is, C(A) = x and C (A) = y(= x). By
definition of and y x, this means C({x, y}) = y, contradicting condition
α.
34 / 34

補足資料 — 講義 3：意思決定と不確実性
安田洋祐
2015 年 4 月 7 日
1 / 13

Risk Aversion | リスク回避 (1)
We continue to assume that a decision maker satisﬁes vNM assumptions and
that the space of prizes S is a set of real numbers.
s ∈ S is interpreted as “receiving s dollars.”
assume is monotone (単調), i.e., a > b implies [a] [b].
Deﬁnition 1
The individual is said to be
risk averse (リスク回避的) if [E(p)] p or u(E(p)) > u(p)
risk neutral (リスク中立的) if [E(p)] ∼ p or u(E(p)) = u(p)
risk loving (リスク愛好的) if [E(p)] ≺ p or u(E(p)) < u(p)
for any non-degenerated lottery p where u(p) =
P
s∈S psu(s).
Theorem 1
Let be a preference on L(Z) represented by the vNM utility function u. The
preference relation is risk averse if and only if u is strictly concave.
2 / 13

Risk Aversion | リスク回避 (2)
The certainty equivalent (確実性等価) of a lottery p, denoted by CE(p),
is a prize satisfying [CE(p)] ∼ p.
The risk premium (リスクプレミアム) of p is the diﬀerence
P(p) := E(p) − CE(p).
The preference relation 1 is “more risk averse” than 2 if
CE1(p) ≤ CE2(p) for all p.
£
¢

¡Rm The individual is risk averse if and only if P(p) > 0 for all p.
§
¦
¤
¥
Fg Figures 9.2 and 9.3 (see Rubinstein, pp.112-113).
It is often convenient to have a measure of risk aversion. Intuitively, the more
concave the expected utility function, the more risk averse the consumer.
3 / 13

Absolute Risk Aversion | 絶対的リスク回避度 (1)
Thus, one might think risk aversion could be measured by the second derivative
of the expected utility function.
However, the second derivative is not invariant to the positive linear
transformation of the expected utility function.
Therefore, some normalization (正規化) is needed.
Depending on the way of normalization, there are at least two reasonable
measures of risk aversion.
Definition 2
The first measure is called the (Arrow-Pratt measure of) absolute risk
aversion (絶対的リスク回避度), defined by
r(x) = −
u (x)
u (x)
.
4 / 13

Absolute Risk Aversion | 絶対的リスク回避度 (2)
The next proposition gives a rationale for this measure.
Theorem 2 (Pratt’s Theorem)
Let u1 and u2 be twice differentiable, increasing, and strictly concave vNM
utility functions. Then, the following properties are equivalent.
(i) CE1(p) ≤ CE2(p) for all p.
(ii) the function ϕ, defined by u1(t) = ϕ(u2(t)), is concave.
(iii) r1(x) ≥ r2(x) for all x, where ri(·) is the absolute risk aversion of ui.
Note that (i) is the definition of “more risk averse”. The equivalence between
(i) and (iii) means that a decision maker has higher absolute risk aversion if
and only if she is more risk-averse.
5 / 13

Constant Absolute Risk Aversion | 絶対的リスク回避度一定
Deﬁnition 3
We say that preference relation exhibits invariance to wealth if
(x + p1) (x + p2) is true or false independent of x.
Theorem 3
If u is a vNM continuous utility function representing preferences that are
monotonic and exhibit both risk aversion and invariance to wealth, then u must
be exponential,
u(x) = −ce−θx
+ d for some c, θ > 0 and d.
r(x) becomes θ and is therefore constant, i.e., independent of x:
invariance to wealth ⇔ constant absolute risk aversion.
However, it is commonly observed that each person becomes less risk
averse when she has more wealth.
That is, absolute risk aversion is decreasing function of x.
The second measure ﬁxes this problem to some extent.
6 / 13

Relative Risk Aversion | 相対的リスク回避度
Deﬁnition 4
The second measure is called the (Arrow-Pratt measure of) relative risk
aversion (相対的リスク回避度), deﬁned by
rr(x) = −
u (x)x
u (x)
.
This measure turns out to be appropriate measure of evaluating the risk
attitude towards the following type of proportional (相対的な) risk:
with probability p, a consumer with wealth x will receive a times of her
current wealth x
with probability 1 − p she will receive b times of x.
Theorem 4
Assume that the assumptions of Pratt’s Theorem holds. Then, for any
proportional risk, the decision maker 1 is more risk averse than 2 if and only if
rr1(x) ≥ rr2(x) for all x, where rri(·) is the relative risk aversion of ui.
7 / 13

Further Properties on vNM Function | vNM 関数の更なる性質 (1)
Deﬁnition 5
We say that p ﬁrst-order stochastically dominates (FOSD) q, denoted by
pD1q, if p q for any on L(S) satisfying vNM assumptions as well as
monotonicity in money. That is, pD1q if Eu(p) ≥ Eu(q) for all increasing u.
For any lottery p and a number x,
Let G(p, x) =
P
s≥x p(s), the probability that the lottery p yields a prize
at least as high as x.
Let F(p, x) denote the cumulative distribution function of p, that is
F(p, x) =
P
s≤x p(s).
The next theorem characterizes FOSD (1 次確率支配).
Theorem 5
pD1q if and only if for all x, G(p, x) ≥ G(q, x), or alternatively,
F(p, x) ≤ F(q, x).
8 / 13

Further Properties on vNM Function | vNM 関数の更なる性質 (2)
Proof.
We only show (⇐): Let x0 < x1 < · · · < xK be the prizes in the union of the
supports of p and q. Then, the expected utility attached to p is written by
Eu(p) =
KX
k=0
p(xk)u(xk) = u(x0) +
KX
k=1
G(p, xk)(u(xk) − u(xk−1)).
Now, if G(p, xk) ≥ G(q, xk) for all k, then for all increasing u,
Eu(p) = u(x0) +
KX
k=1
G(p, xk)(u(xk) − u(xk−1))
≥ u(x0) +
KX
k=1
G(q, xk)(u(xk) − u(xk−1)) = Eu(q).
§
¦
¤
¥
Fg Figure 9.1 (see Rubinstein, pp.110)
9 / 13

Properties on Concave vNM Function | 凹 vNM 関数の性質
Risk aversion is closely related to the concavity of the vNM utility function. Let
us recall some basic properties of concave functions (凹関数):
.
1 An increasing and concave function must be continuous (but not
necessarily differentiable).
.
2 Jensen Inequality (ジェンセンの不等式): If u is concave, then for any
finite sequence of α1, α2, ..., αK of positive numbers that sum up to 1, the
following inequality must hold:
u
KX
k=1
αkxk
!
≥
KX
k=1
αku(xk).
3 If u is twice differentiable, then for any a < c, u (a) ≥ u (c), and thus
u (x) ≤ 0 for all x.
10 / 13

Proof of Pratt’s Theorem | プラットの定理の証明 (1)
Sketch of the Proof.
To establish (i) ⇔ (iii), it is enough to show that P is positively related to r.
Let ε be a “small” random variable with expectation of zero, i.e., E(ε) = 0.
The risk premium P (at initial wealth x) is deﬁned by
E(u(x + ε)) = u(CE(x + ε)) = u(x − P(ε)).
For any value ˆε of ε by the second order Taylor series expansion (テイラー展開),
u(x + ˆε) ≈ u(x) + ˆεu (x) +
ˆε2
2
u (x),
from which it follows that
E(u(x + ε)) ≈ u(x) +
σ2
ε
2
u (x) (1)
where σ2
ε is the variance (分散) of random variable ε (note E(ε) = 0).
11 / 13

Proof of Pratt’s Theorem | プラットの定理の証明 (2)
Sketch of the Proof (cont.)
On the other hand, the following approximation (近似) holds
u(x − P(ε)) ≈ u(x) − P(ε)u (x), (2)
since P(ε) is small due to ε being “small”. By (1) and (2),
P(ε) = −
1
2
σ2
ε
u (x)
u (x)
=
σ2
ε
2
r(x).
That is, the (coeﬃcient of) absolute risk aversion at the level of wealth x,
r(x) = −
u (x)
u (x)
, is twice of the risk premium per unit of variance for small risk
ε.
£
¢

¡Rm r(x) can serve as a local measure of risk aversion.
12 / 13

Relative Risk Aversion: Proof | 相対的リスク回避度：証明
Sketch of the Proof.
Let ˆP(ε) be a relative risk premium for any proportional risk ε (at the wealth
level x), defined by
E(u(x(1 + ε))) = u(x(1 − ˆP(ε))) = u(x − x ˆP(ε)).
Note that, by definition of the (absolute) risk premium P(ε),
E(u(x(1 + ε))) = E(u(x + xε)) = u(x − P(xε)).
where we set E(xε) = 0. Therefore,
x ˆP(ε) = P(xε) = −
1
2
x2
σ2
ε
u (x)
u (x)
⇒ ˆP(ε) = −
σ2
ε
2
xu (x)
u (x)
= −
σ2
ε
2
rr(x).
£
¢

¡Rm If the individual has a constant relative risk aversion, then preferences
over proportional gambles will not be affected by x.
13 / 13

講義 4：一般均衡理論（基礎編）
安田洋祐
2015 年 4 月 7 日
1 / 27

テキスト
*JR: Ch5
神取：3 章; 補論 D
配布資料
*Laﬀont, The Economics of Uncertainty and Information, 1989: Intro
関連図書
*二階堂副包『現代経済学の数学的方法―位相数学による分析入門』岩波書
店, 1960. （オンデマンド, 2012）
*長名寛明『ミクロ経済分析の基礎』知泉書館, 2011.
*Debreu, Theory of Value: An Axiomatic Analysis of Economic
Equilibrium, 1959.
Koopmans, Three Essays on the State of Economic Science, 1957.
(Reprint, 2013)
2 / 27

Market Economy | 市場経済
In previous lectures, we studied the behavior of individual consumers, describing
optimal behavior when market prices were fixed and beyond the agent’s control.
We begin to explore the consequences of that behavior when consumers (and
firms) come together in markets. First, we consider price and quantity
determination in a single market.
In a partial equilibrium (部分均衡) model,
individual consumers and firms determine their demands and supplies for
the good in question
all prices other than the good in question are fixed
the market price is adjusted to clear the market.
⇒ In the general equilibrium (一般均衡) model, prices of all goods vary and
all markets clear at the same time.
3 / 27

Perfectly Competitive Market | 完全競争市場
In (perfectly) competitive markets (競争市場), buyers and sellers are
sufficiently large in number to ensure that no single one of them, alone, has the
power to influence market price.
⇒ Market price is outside of their control: they are price takers (価格受容者).
A Consumer’s Problem
max
x,z
u(x, z) s.t. px + qz = ω
A Firm’s Problem
max
y
py − c(y)
where all prices other than p is assumed to remain fixed.
£
¢

¡Q How can we aggregate (集計する) individual demand or supply?
4 / 27

Market Supply Function | 市場供給関数
The (individual firm) supply function (個別供給関数) is straight-forwardly
derived from the profit maximization problem. Its first order condition
p = c (y)
(and second order condition c (y) ≥ 0) implies that supply function of a
competitive firm coincides with its marginal cost curve (限界費用曲線).
The market (or industry) supply function (市場供給関数) is simply the sum
of the individual firm supply function. If yj(p) is firm j’s supply function in an
industry with J firms, the market supply function is
Y (p) =
JX
j=1
yj(p).
5 / 27

From Partial to General Equilibrium | 部分から一般均衡へ
Partial Equilibrium Model (部分均衡モデル)
Each agent determines his or her demands and supplies for the good in
question, given that
All prices of other goods are assumed to remain ﬁxed.
Equilibrium requires that the market in question clears.
General Equilibrium Model (一般均衡モデル)
Each agent determines his or her demands and supplies for all the goods
simultaneously, where
All prices are subject to change.
Equilibrium requires that all markets clear at the same time.
For simplicity, let us consider a pure exchange economy (純粋交換経済): the
special case of the general equilibrium model where all of the economic agents
are consumers, i.e., there is no production.
6 / 27

Aggregation across Consumers | 消費者間での集計
Suppose there are I consumers, each of whom has a demand function for n
commodities; Consumer i’s demand function is
xi(p, ωi) = (x1
i (p, ωi), · · · , xn
i (p, ωi)).
Then, the aggregate demand function (集計需要関数) is deﬁned by
X(p, ω1, · · · , ωI ) =
IX
i=1
xi(p, ωi).
The market (or industry) demand function (市場需要関数) of good x is
simply the sum of all consumers’ individual demand for the good:
X(p) =
IX
i=1
xi(p)
assuming all the prices other than p as well as incomes ﬁxed.
7 / 27

Market Equilibrium | 市場均衡
The market supply function measures the total output supplied at any price;
the market demand function measures the total output demanded at any price.
Deﬁnition 1
An equilibrium price (均衡価格) in a partial equilibrium is a price where the
amount demanded equals the amount supplied:
IX
i=1
xi(p) =
JX
j=1
yj(p).
£
¢

¡Q Why does such a price deserve to be called an equilibrium?
£
¢

¡Rm At the equilibrium, no one would desire to change actions, while at any
other price some agent would ﬁnd it in her interest to unilaterally change its
behavior.
8 / 27

Aggregation across Goods | 財に関する集計
When we analyze the demand for a single good (partial equilibrium study), it
would be convenient to aggregate “all other goods”.
A Consumer’s Problem (again)
max
x,z
u(x, z) s.t. px + qz = ω
£
¢

¡Q Under what conditions can we study the demand problem for the z-goods
as a group, without worrying about how demand is divided among diﬀerent
components of the z-goods?
Suppose that the relative prices of the z-goods remain constant, so that
q = Pq0
where P can be interpreted as some price index.
V (P, p, ω) = max
x,z
u(x, z) s.t. px + Pq0
z = ω
9 / 27

Composite Commodity | 複合材
Given the indirect utility function V , we can recover the demand function for
the z-goods by Roy’s identity (and envelope theorem):
z(P, p, ω) = −
∂V (P,p,ω)
∂P
∂V (P,p,ω)
∂ω
= q0
z(p, q, ω).
This shows that z(P, p, ω) is an appropriate quantity index for the z-goods
consumption: such z is called composite commodity (複合財).
Since all prices q move together (by assumption) and the demand function is
homogeneous of degree 0, we can write
x = x(P, p, ω) = x(p/P, ω/P),
which says that the demand for good x depends on the relative price of x to
composite good and income divided by price index.
10 / 27

Properties of Aggregate Demand Function | 集計需要関数の性質
The aggregate demand function clearly satisﬁes homogeneous of degree
0 in all prices and the vector of buyers’ incomes.
It also becomes continuous whenever individual demand functions are
continuous.
Unfortunately, the aggregate demand function in general possesses no
interesting properties other than homogeneity and continuity.
For example, aggregate version of Slutsky equation or strong axiom of
revealed preference cannot hold.
The properties of aggregate demand function are completely diﬀerent from
those of individual demand function.
£
¢

¡Q Under what condition the aggregate demand may look as though it were
generated by a single “representative (代表的な)” consumer?
11 / 27

Representative Consumer | 代表的消費者 (1)
The next theorem shows that the aggregate demand function possesses nice
properties when the consumers’ indirect utility functions take a speciﬁc
functional form.
Theorem 1
The aggregated demand function can be generated by a representative
consumer if and only if indirect utility functions of all individual consumers take
the following Gorman form (ゴーマン型):
vi(p, ωi) = ai(p) + b(p)ωi.
Note that the ai(p) term can diﬀer across consumers, but the b(p) term is
assumed to be identical for all consumers.
Gorman form is imposed on an indirect utility function, not on a (direct)
utility function.
12 / 27

Proof of if part (⇐).
By Roy’s identity, the demand function for good j of consumer i will also take
the Gorman form
xj
i (p, ωi) = αj
i (p) + βj
(p)ωi
where
αj
i (p) = −
∂ai(p)
∂pj
b(p)
, βj
(p) = −
∂b(p)
∂pj
b(p)
.
Note that marginal propensity to consume good j
„
=
∂xj
i (p, ωi)
∂ωi
«
is
independent of the income level and constant across consumers.
In other words, income eﬀect is proportional to consumer’s income level, which
makes possible to aggregate individual incomes.
13 / 27

Proof of if part (⇐) cont.
The aggregate demand for good j then take the form
Xj
(p, ω1, · · · , ωI ) = −
0
@
IX
i=1
∂ai(p)
∂pj
b(p)
+
∂b(p)
∂pj
b(p)
IX
i=1
ωi
1
A .
This (market) demand function can be generated by the following
representative indirect utility function
v(p, ω) =
IX
i=1
ai(p) + b(p)ω
where ω =
PI
i=1 ωi shows the aggregate income of consumers.
£
¢

¡Q When does indirect utility function takes Gorman form?
⇒ A utility function is homothetic (ホモセティック) or quasilinear (準線形).
14 / 27

Competitive Equilibrium | 競争均衡 (1)
The allocation with the price vector constitute a competitive/Walrasian
equilibrium (競争/ワルラス均衡) if every consumer maximizes her utility and
all markets clear, i.e., supply equal to demand for every good. To state this
formally, let us deﬁne the excess demand function.
Deﬁnition 2
The (aggregate) excess demand function (超過需要関数) for good j is the
real-valued function,
zj(p) =
X
i∈I
xi
j(p, p · ei
) −
X
i∈I
ei
j.
The excess demand function is the vector-valued function,
z(p) = (z1(p), ..., zN (p)).
When zj(p) > 0, the aggregate demand for good j exceeds its aggregate
endowment; there is excess demand for good j.
When zj(p) < 0, there is excess supply of good j.
15 / 27

Competitive Equilibrium | 競争均衡 (2)
Lemma 1
If ui satisfies the assumptions in the theorem below, then for all p 0, the
consumer’s problem has a unique solution xi
(p, p · ei
). Moreover, xi
(p, p · ei
)
is continuous in p on Rn
++.
Theorem 2
Suppose that utility function ui is continuous, strongly increasing, and strictly
quasiconcave for all i ∈ I. Then, for all p 0, the excess demand function
satisfies,
.
1 Continuity (連続性): z(·) is continuous at p.
.
2 Homogeneity (同次性): z(λp) = z(p) for all λ > 0.
3 Walras’ law (ワルラス法則): p · z(p) = 0.
Definition 3
An allocation-price pair (x, p) where p 0 is called a competitive/Walrasian
equilibrium (競争/ワルラス均衡) if z(p) = 0.
16 / 27

Walras’ Law | ワルラス法則
Walras’ law states that the value of aggregate excess demand is identically (恒
等的に) zero at any set of positive prices.
Proof.
When ui is strongly increasing, each consumer’s budget constraint holds with
equality, i.e., pxi
(p, p · ei
) = pei
. Then,
pz(p) = p
X
i∈I
xi
(p, p · ei
) −
X
i∈I
ei
!
=
X
i∈I
“
pxi
(p, p · ei
) − pei
”
= 0.
Moreover, if at some set of prices n − 1 markets are in equilibrium, Walras’ law
ensures the nth market is also in equilibrium.
Corollary 1
If demand equals supply in n − 1 markets, and pn > 0, then demand must
equal supply in the nth market.
17 / 27

Competitive Equilibrium: Example | 競争均衡：例
£
¢

¡Ex Consider two agents (1 and 2) and two goods (x and y) exchange
economy. Suppose that agents’ utility functions and initial endowments are
given as follows:
u1(x1, y1) = xa
1y1−a
1 , u2(x2, y2) = xb
2y1−b
2
ω1 = (1, 0), ω2 = (0, 1)
Solve a competitive equilibrium price ratio py/px.
Answer Let us ﬁrst derive the demand function xi for i = 1, 2. Since utility
functions are both Cobb-Douglas, we obtain
pxx1 = aω1 = apx ⇒ x1 = a
pxx2 = bω2 = bpy ⇒ x2 = b
py
px
.
Since the excess demand for good x must be 0,
x1 + x2 = a + b
py
px
= 1 ⇒
py
px
=
1 − a
b
.
Note that by Walras’ law, the excess demand for good y is also 0.
18 / 27

First Theorem on Welfare Economics | 厚生経済学の第 1 定理 (1)
Lemma 2
Let (x, p) be a competitive equilibrium. If ui(yi) > ui(xi) for some bundles
yi, then
p · yi > p · xi.
If i has an increasing utility function and ui(yi) ≥ ui(xi) for some bundle yi,
then
p · yi ≥ p · xi.
The following theorem which shows the efficiency of competitive market is
called the first fundamental theorem of welfare economics (厚生経済学の第
1 基本定理), or first welfare theorem.
Theorem 3
If utility function ui is increasing for all i ∈ I and (x, p) is a competitive
equilibrium, then x is in the core (コア).
19 / 27

Proof.
Suppose, contrary to the theorem, that x is not in the core. Then some
coalition S ⊂ I can block x. That is, there exist reallocation y among S such
that
X
i∈S
yi
=
X
i∈S
ei
,
ui(yi
) ≥ ui(xi
) for all i ∈ S, and
ui(yi
) > ui(xi
) for at least one i ∈ S.
By Lemma, we obtain
p · yi ≥ p · xi for all i ∈ S, and
p · yi > p · xi for at least one i ∈ S.
Now adding these inequalities over all individuals in S,
X
i∈S
p · yi
>
X
i∈S
p · xi
=
X
i∈S
p · ei
.
20 / 27

cont.
This inequality can be rewritten as
X
i∈S
p · yi
= p ·
X
i∈S
yi
>
X
i∈S
p · ei
= p ·
X
i∈S
ei
⇒ p ·
X
i∈S
yi
−
X
i∈S
ei
!
> 0,
which contradicts to
P
i∈S yi
=
P
i∈S ei
.
Theorem 4
If utility function ui is increasing for all i ∈ I and (x, p) is a competitive
equilibrium, then x is Pareto efficient (パレート効率的).
§
¦
¤
¥
The first welfare theorem claims that a competitive equilibrium allocation is in
the core, and is Pareto efficient.
21 / 27

Second Theorem on Welfare Economics | 厚生経済学の第 2 定理 (1)
Theorem 5
Consider an exchange economy with
P
i∈I ei
0, and assume that utility
function ui is continuous, strongly increasing, and strictly quasiconcave for all
i ∈ I. Then, any Pareto efficient allocation x is a competitive equilibrium
allocation when endowments are redistributed to be equal to x.
Corollary 2
Under the assumptions of the preceding theorem, if ex is Pareto efficient, then ex
is a competitive equilibrium allocation for the price vector p after redistribution
of initial endowments to any feasible allocation ee, such that
p · eei
= p · exi
for all i ∈ I.
The existence of equilibrium price vector supporting x is guaranteed by
separating hyperplane theorem (分離超平面定理). In order to apply this
theorem, we need a convex environment, which is much more restrictive than
the one needed for the first welfare theorem.
22 / 27

Second Theorem on Welfare Economics | 厚生経済学の第 2 定理 (2)
The second theorem may look more difficult to understand intuitively than the
first theorem, but it is indeed fundamental to interpret decentralized planning
in a private property economy (私有経済).
Pareto efficiency of the competitive equilibrium (First theorem) is
satisfactory with respect to the efficiency criterion, but it may lead to
undesirable income distributions.
The second theorem states: whichever Pareto efficient allocation we wish
to decentralize, it is possible to decentralize (分権化する) this allocation
as a competitive equilibrium so long as the incomes of the agents are
chosen appropriately.
That is, a private property economy can achieve any Pareto efficient
allocation (パレート効率的配分) so long as the appropriate lump-sup
transfers (一括移転) are made.
23 / 27

【補論】 Homogeneity and Relative Prices | 同次性と相対価格
Aggregate excess demand function is homogeneous of degree 0.
⇒ Let us normalize (正規化) prices and express demands in terms of the
following relative prices:
pi =
ˆpi
Pn
j=1 ˆpj
where ˆpi is the original (absolute) price of good i.
Since the normalized prices pi must sum up to 1, i.e.,
Pn
i=1 pi = 1, we can
restrict our attention to price vectors belonging to the n − 1 dimensional unit
simplex:
Sn−1
=
(
p ∈ Rn
+ |
nX
i=1
pi = 1
)
24 / 27

【補論】 Existence of Competitive Equilibrium | 競争均衡の存在 (1)
Theorem 6
If z : Sn−1
→ Rn
is a continuous function that satisﬁes Walras’ law, pz = 0
for all p ∈ Sn−1
, then there exists some p∗
in Sn−1
such that z(p∗
) = 0.
To prove the theorem, let us deﬁne a function g : Sn−1
→ Sn−1
by
gi(p) =
pi + max(0, zi(p))
1 +
Pn
j=1 max(0, zj(p))
for i = 1, · · · , n.
Note that this function gi(p)
is continuous, since z and max function are continuous.
takes a value on Sn−1
, since
Pn
i=1 gi(p) = 1.
has an economic interpretation: if there is excess demand in some market,
then the relative price of that good is increased.
25 / 27

Applying Brouwer fixed-point theorem (ブラウワーの不動点定理), we obtain:
Lemma 3
g : Sn−1
→ Sn−1
has a fixed point, p∗
such that p∗
= g(p∗
). That is,
p∗
i =
p∗
i + max(0, zi(p∗
))
1 +
Pn
j=1 max(0, zj(p∗))
for i = 1, · · · , n.
We now have to show that p∗
is a competitive equilibrium.
Using Warlas’ law, we can show that zi(p∗
) = 0 for all i.
It is common to show the existence of equilibrium by applying a version of
fixed-point theorems in Economics.
£
¢

¡Q Can we establish the existence without assuming continuity of z (especially
at the boundary points where pj = 0 for some j)?
26 / 27

The excess demand function may not even be well deﬁned on the boundary of
the price simplex. However, this discontinuity can be handled by slightly more
complicated mathematical argument.
Theorem 7
Consider an exchange economy with
P
i∈I ei
0, and assume that utility
function ui is continuous, strongly increasing, and strictly quasiconcave for all
i ∈ I. Then, there exists at least one price vector p∗
0 such that z(p∗
) = 0.
£
¢

¡Rm The assumption of strongly increasing utilities is somewhat restrictive,
but it allows us to keep the analysis relatively simple.
Cobb-Douglas functional form of utility is not strongly increasing on Rn
+.
Competitice equilibrium with Cobb-Douglas preferences is nonetheless
guaranteed.
27 / 27

講義 5：一般均衡理論（発展編）
安田洋祐
2015 年 4 月 14 日
1 / 26

テキスト
*JR: Ch5
神取：3 章
配布資料
*Laﬀont, The Economics of Uncertainty and Information, 1989: Ch5
*Riley, Essential Microeconomics, 2012: Ch8
関連図書
*Duﬃe, Dynamic Asset Pricing Theory, 3rd edition, 2001.
*Ellickson, Competitive Equilibrium: Theory and Applications, 1994.
池田昌幸『金融経済学の基礎』朝倉書店, 2000.
野口悠紀雄・藤井眞理子『現代ファイナンス理論』東洋経済新報社, 2005.
2 / 26

From Pure Exchange to Production Economy | 交換経済から生産経済へ
Previous lecture considered the general equilibrium (一般均衡) in a pure
exchange economy (純粋交換経済) where all agents are consumers.
Now we expand our description of the economy to include production (生産) as
well as consumption.
£
¢

¡Rm We will find that most of the important properties of competitive market
systems uncovered earlier continue to hold.
In a general equilibrium model with production:
効用最大化 Consumers, 1, 2, ..., I, act to maximize utility subject to their
budget constraints.
利潤最大化 Firms, 1, 2, ..., J, seek to maximize profit.
競争市場 Both consumers and firms are price takers.
私有経済 Firms’ profits are shared among individual consumers.
3 / 26

Firm Behavior | 企業行動 (1)
Definition 1
Let Y denote the aggregate production possibilities set (集計された生産可能
集合), defined as the sum of the individual production possibility sets:
Y =
X
j∈J
Yj = {y | y =
X
j∈J
yj where yj ∈ Yj}.
Let y(p) be the aggregate net supply function (集計された純供給関数)
defined as the sum of the individual net supply functions:
y(p) =
X
j∈J
yj(p).
where yj(p) associates to each vector p the profit-maximizing net output
vector at those prices.
£
¢

¡Rm Y represents all production plans that can be achieved by some
distribution of production among J individual firms.
4 / 26

Firm Behavior | 企業行動 (2)
The next theorem says that if each firm maximizes profits, then aggregate
profits must be maximized. Conversely, if aggregate profits are maximized, then
each firm’s profits must be maximized.
Theorem 1
An aggregate production plan y maximizes aggregate profit, if and only if each
firm’s production plan yj maximizes its individual profit for all j ∈ J.
The theorem implies that there are two equivalent ways to construct the
aggregate net supply function:
1 Add up the individual firms’ net supply functions.
2 Add up the individual firms’ production sets and then determine the net
supply function that maximizes profits on this aggregate production set.
→ 証明は【補論】を参照．
5 / 26

Consumer Behavior | 消費者行動 (1)
Definition 2
Consumer i’s share of the profits of firm j is represented by
0 ≤ θij ≤ 1 for all i ∈ I and j ∈ J
where
X
i∈I
θij = 1 for all j ∈ J.
That is, each firm is completely owned by individual consumers.
Then, the budget constraint of each consumer i becomes:
pxi ≤ pei +
X
j∈J
θijpyj(p).
£
¢

¡Rm Given ei and θij, the budget set (予算集合) is characterized by p alone.
Hence, consumer i’s demand function can be written as a function of p,
denoted by xi(p).
6 / 26

Consumer Behavior | 消費者行動 (2)
Deﬁnition 3
Let x(p) =
P
i∈I xi(p) be the aggregate (consumer) demand function (集
計された需要関数), the sum of the individual demand functions.
The aggregate excess demand function (集計された超過需要関数) is
z(p) = x(p) − y(p) − e
where e is the aggregate supply from consumers, e =
P
i∈I ei.
Walras’ law (ワルラス法則) holds in the production economy for the
same reason that it holds in the pure exchange economy.
Each consumer satisﬁes her budget constraint, so the economy as a whole
has to satisfy an aggregate budget constraint (集計された予算制約).
Theorem 2 (Walras’ law)
If ui is strictly increasing for all i ∈ I, then pz(p) = 0 must hold for all p.
7 / 26

Competitive Equilibrium | 競争均衡
The production economy is represented by (ui, ei, θij, Yj)i∈I,j∈J .
Deﬁnition 4
An allocation-price pair (x, y, p) where p 0 is called a competitive
(Walrasian) equilibrium (競争均衡), if z(p) = 0.
The next theorem guarantees the existence of equilibrium (均衡の存在).
Theorem 3
Consider a production economy (ui, ei, θij, Yj)i∈I,j∈J . Suppose that the
following conditions are satisﬁed:
Utility function ui is continuous, strongly increasing, and strictly
quasiconcave for all i ∈ I.
0 ∈ Yj ⊆ Rn
, Yj is closed, bounded and strongly convex.
y +
P
i∈I ei 0 for some aggregate production vector y ∈ Y .
Then, there exists at least one price vector p∗
0 such that z(p∗
) = 0. That
is, the competitive equilibrium price exists.
8 / 26

Competitive Equilibrium: Example | 競争均衡：例 (1)
£
¢

¡Ex Consider a Robinson Crusoe economy (ロビンソン・クルーソー経済)
where a consumer has the following Cobb-Douglas utility function for
consumption x and leisure R and initial endowments e:
u(x, R) = xa
R1−a
The consumer is endowed with one unit of labor (労働)/leisure (余暇) and the
firm has a production function (生産関数) x = L1/2
. Let the price of x be
normalized by 1. Then, solve a competitive equilibrium price of labor, w∗
.
Answer Let us first solve the profit maximization problem:
max
L
L1/2
− wL
From the first order condition,
1
2
L−1/2
− w = 0, we obtain
L(w) =
1
(2w)2
, xs(w) =
1
2w
, π(w) =
1
4w
.
9 / 26

Competitive Equilibrium: Example | 競争均衡：例 (2)
Since the ﬁrm’s proﬁts are distributed to the consumer, she solves:
max
x,R
xa
R1−a
s.t. x + wR = w +
1
4w
By the property of the Cobb-Douglas utility function, we obtain
xd(w) = a
„
w +
1
4w
«
, R(w) =
1 − a
w
„
w +
1
4w
«
.
In a competitive equilibrium, the supply and demand for x coincide,
xs(w∗
) = xd(w∗
) ⇔
1
2w∗
= a
„
w∗
+
1
4w
«
⇒ w∗
=
„
2 − a
4a
«1/2
.
Note that, by Walras’ law, the labor market also clears.
財市場の均衡 ⇐⇒ 労働市場の均衡
10 / 26

First Welfare Theorem | 厚生経済学の第一定理
Theorem 4
If each ui is strictly increasing on Rn
+, then every competitive equilibrium is
Pareto efficient (パレート効率的).
Proof.
Suppose not, and let (x , y ) be a Pareto dominating allocation. Then, since
consumers are maximizing utility,
pxi > pei +
X
j∈J
θijpyj
must hold for all i ∈ I. Summing over consumers,
p
X
i∈I
xi >
X
i∈I
pei +
X
j∈J
pyj.
Feasibility of x implies
p
X
j∈J
yj +
X
i∈I
ei
!
>
X
i∈I
pei +
X
i∈J
pyj ⇔
X
j∈J
pyj >
X
j∈J
pyj,
which contradicts profit maximization by firms.
11 / 26

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