Hakuoh University Repository / 白鷗大学機関リポジトリ

Devon,Arthurson

English

Using Reflection Journals to Explore the Interdisciplinary Practice of Community Development and English as a Foreign Language

呉庭・李丹

Japanese

The Literature of Kahu Nagai in the History of Anti-Modern Japanese Literature

浅木,尚実

佐々木,美和

乳幼児の探索意欲と絵本との関係： ―愛着形成との関わりと発語期―

大木,俊英

言語生成AIを用いた英語学習日記の分析の信頼性： テキストマイニングとの比較より

齋藤,正憲

南島素描 06：奄美のカミサマ

佐藤,ちひろ

松倉,佳子

保育実習（施設）における実習プログラムに関する研究

神吉, 尚男

ルネ・レモンのアカデミー・フランセーズ入会演説　1999年11月4日

白
　
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39
巻
第
１
号
２
０
２
４
年
９
月
白
鷗
大
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経
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学
部
CONTENTS
HAKUOH DAIGAKU RONSHU
(THE HAKUOH UNIVERSITY JOURNAL)
HAKUOH UNIVERSITY
Faculty of Business Administration
OYAMA, TOCHIGI, JAPAN
Vol.39, No.1 September, 2024
Articles
Note
Shuichiro KOIKE’s Embellishment and Aesthetics 
　－ A Study of Three Musicals ……………………………………SEKIDO Fuyuhiko（１）
Using Reflection Journals to Explore the Interdisciplinary Practice of Community 
Development and English as a Foreign Language ……………… Devon Arthurson（19）
The Literature of Kahu Nagai in the History of Anti-Modern Japanese Literature
　…………………………………………………………………………Wu Ting ,  Li Dan（35）
Examining the Part-of-Speech Classification in School Grammar
　……………………………………………………………………………… UENO Yoshio （49）
A note on envelope theorems and subderivatives
　…………………………………………………………………………… YAO Masayuki（85）
白鷗大学論集
白鷗大学経営学部
第39巻第１号
2024年９月
ISSN  0913－7661
研究ノート
論 文
演出家・小池修一郎の潤色と美学
　―３つのミュージカル作品を分析して………………… 関　戸　冬　彦（ １ ）
Using Reflection Journals to Explore the Interdisciplinary Practice of 
Community Development and English as a Foreign Language 　
　…………………………………………………………… Devon Arthurson（ 19 ）
反近代文学史上の永井荷風文学……………………………呉　庭  ・  李　丹（ 35 ）
学校英文法の品詞分類を検討する………………………… 上　野　義　雄（ 49 ）
包絡線定理と劣導関数についてのノート………………… 八　尾　政　行（ 85 ）


表紙

上野,義雄

Examining the Part-of-Speech Classification in School Grammar

八尾,政行

白鷗大学論集　第39巻　第１号
研究ノート
包絡線定理と劣導関数についてのノート
八　尾　政　行
A note on envelope theorems and subderivatives
YAO Masayuki1
１　Faculty of Business Administration, Hakuoh University
Abstract
In this note, we consider parameterized optimization problems with 
the abstract choice set. We characterize the value function using 
subderivatives. From this result, we derive one of Milgrom and Segal's ［1］ 
results.
1. Introduction
The envelope theorem states the sufficient condition for the value of a 
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八　尾　政　行
parameterized optimization problem to be differentiable with respect to 
the parameter and provides a formula for its derivative. The envelope 
theorem is a standard tool in economic analysis （see, for example, Mas-
Colell, Whinston, and Green ［2］）. Various approaches have been studied 
on this issue in recent years （e.g., Milgrom and Segal ［1］, Morand, 
Reffett, and Tarafdar ［3］, Oyama and Takenawa ［4］, Marimon and 
Werner ［5］）. 
　　　Among these, Milgrom and Segal ［1］ are the major milestone. The 
traditional envelope theorem requires the choice set to have convexity 
and topological properties （see, for instance Benveniste and Scheinkman 
［6］）. In contrast, Milgrom and Segal ［1］ make no such assumptions on 
the choice set. The reason for using such an abstract choice set is to 
answer the demands of the models used in recent economics, such as 
mechanism design.
　　　Milgrom and Segal ［1］ derive various results, one of which is the 
characterization of the left and right derivatives of the value function 
（Milgrom and Segal ［1］, Theorem 1）. It is known that various types of 
derivatives can be obtained by taking various limits of a function called 
the difference quotient. （see, for example, Rockafellar and Wets ［7］）. 
The right and left derivatives used by Milgrom and Segal are only one of 
these. They are special cases of the subderivatives.
　　　In this paper, we derive the characterization of the value function 
using subderivatives. We then derive the Milgrom and Segal’s ［1］ result 
as a corollary of our result.
　　　The reminder of the paper is organized as follows: In Section 2 
we prepare to derive our result. The content of this section relies on 
Rockafellar and Wets ［7］. In Section 3, we present our result and use it 
to show the Milgrom and Segal’s ［1］ result.
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A note on envelope theorems and subderivatives
2. Preliminaries
Let =（∞,∞） be a real line and let n be an n-dimensional Euclidean 
space. In this paper, we consider expanded real valued functions. That is, 
the functions take values in the expanded-real-line ¯ =［∞,∞］ instead of . 
 For a function f : n → ¯ at a point x‾ ∈ ¯ where f（ x‾ ） is finite, 
that is,  f（ x‾ ） ∈ , the difference quotient function is defined by 
By taking limits of different types of the difference quotient, we can get 
different types of derivatives.
 We use the notion τ↘０⇔τ→０ with τ>０. Using this notion, 
we define subderivatives as follows:
Definition 2.1. For a function f : n → ¯ at a point x‾ ∈ n where f（ x‾ ） is 
finite, the lower subderivative function d ‾ f（ x‾ ） : n → ¯  is defined by 
 d ‾ f（ x‾ ） （w‾ ）=liminfτ↘0, w→w‾ Δτ f（ x‾ ） （w）,
and the upper subderivative function d + f（ x‾ ） : n → ¯  is defined by 
 d+ f（ x‾ ） （w‾ ）=limsupτ↘0, w→w‾ Δτ f（ x‾ ） （w）.
The value of d ‾ f（x‾ ） （w‾ ） （resp. d+ f（x‾ ） （w‾ ） ） is called the lower （resp. 
upper） subderivative of f at x‾ for w‾ .
 We write d ‾ f（ x‾ ） simply as d f（ x‾ ） and call it the subderivative 
function when no confusion will arise. From the definition and the notion, 
the following relationship holds between the lower subderivative and the 
upper subderivative function :
 −d（−f ）（x‾ ） （w‾ ）  =−liminfτ↘0, w→w‾ Δτ f（ x‾ ） （w）
 　　　　　　　　= limsupτ↘0, w→w‾ Δτ f（ x‾ ） （w）
 　　　　　　　　=d+f（ x‾ ） （w）.
τ Δτ f（ x‾ ）（w）= ———————————  for τ≠0.
f（ x‾ +τw）− f（ x‾ ）
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 We also use the notion τ↗０⇔τ→０ withτ<０. Using this 
notion, we can introduce the following :
 −df（x‾ ） （w‾ ）  = liminfτ↗0, w→w‾ Δτ （−f  ）（ x‾ ） （−w）  ，
 d（−f ）（x‾ ） （w‾ ）  = limsupτ↗0, w→w‾ Δτ （−f  ）（ x‾ ） （−w）  .
 In the reminder of this section, we consider functions whose 
domain is . For a f unction f :  → ¯ at a point x‾ ∈  where f （ x‾ ）  is finite, 
the subderivative functions f （ x‾ ）  are determined from their values at w‾ =
±１. This holds because df （ x‾ ）  is positive homogeneous （see Rockafellar 
and Wets ［7］, Theorem 8.18）. There is no need to take the limits w→w‾ 
in the subderivative functions, so the following holds : 
  　　　df （ x‾ ） （1）= liminfτ↘0Δτ f（ x‾ ） （1）,
  −d（−f ）（ x‾ ） （1）= limsupτ↘0Δτ f（ x‾ ） （1）,
     −df （ x‾ ） （−1）= liminfτ↗0Δτ （−f ）（ x‾ ） （−1）,　　　　　　　 （1）
 d（−f ）（ x‾ ） （−1）= limsupτ↗0Δτ （−f ）（ x‾ ） （−1）.
When df （ x‾ ） （1）=−d（−f ）（ x‾ ） （1）（resp. −df （ x‾ ） （−1）=d（−f ）（ x‾ ） （−1））
and the value is finite, we say that f is right （resp. left） differentiable at x‾. 
The value 
 　　f '+（ x‾ ） = d f （ x‾ ） （1）=−d（− f ）（ x‾ ） （1）
is called right derivative of f at x‾ and the value
 　　f '_（ x‾ ） = −d f （ x‾ ） （−1）=d（− f ）（ x‾ ） （−1）
is called left derivative of f at x‾ . If f '+（ x‾ ）= f '_（ x‾ ）, we say that f is 
differentiable at x‾ . We write the common value as f '（x‾ ） and call it the 
derivative.
3. Results
Let X denote the choice set and let Y be the parameter set. Letting f : X
×Y→ ¯  denote the parameterized objective function, the value function 
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A note on envelope theorems and subderivatives
v : Y→ ¯  and the optimal choice set-valued function M : Y  X are given by
 v（y）= sup x∈X f（x,y）,
 M（y）= ｛x∈X : f（x,y）=v（y）　｝.　　　　　　　　　　　 　　　  (2)
We denote by dyf （x‾ , y‾） the subderivative function associated with f（x‾,・） 
at y‾ .
Theorem 3.1. Suppose Y =  n. Take y‾∈Y and x*∈M（y）, and suppose 
that f（x*, y‾ ） is finite. Then,
 　　  dv（y‾ ）（w‾ ）　≥ dy f（x*, y‾ ）（w‾ ）,
 −d（−v）（y‾ ）（w‾ ）　≥ −dy （−f ）（x*, y‾ ）（w‾ ）,
   −dv（y‾ ）（−w‾ ）　≤ −dy f（x*, y‾ ）（−w‾ ）,                             (3)
  d（−v）（y‾ ）（−w‾ ）　≤ dy （−f ）（x*, y‾ ）（−w‾ ）.
Proof .  Using （2）, we have that for any w∈Y that converges to w‾ and 
τ>０,
 　　　v（y‾ +τw）≥ f（x* , y‾ +τw）.
Since f（x*, y）= v（y）,
 v（y‾ +τw）−v（y）≥ f（x*,y‾ +τw）−f（x*, y‾ ）
holds. Dividing both sides by τ,
Taking the lower limit as τ ↘ ０ and w → w‾ on both sides of this 
inequality, we get that 
 　　　dv（y）（w‾ ）≥ dy f（x*,y）（w‾ ）.
Taking the upper limit as τ↘０ and w→ w‾ on both sides of the inequality
（4）, we also get that 
 　−d（−v）（y）（w‾ ）≥ −dy（−f）（x*,y）（w‾ ）.
Using similar arguments, the remaining inequalities can be derived. □
 Using our result, we can derive the Milgrom and Segal’s result 
as a corollary.
（4）τ τ　　　　————————　 ≥　——————————————　.
 v（y+τw）−v（y）　　 f（x*,y+τw）−f（x*,y）
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Corollary 3.2. （Milgrom and Segal ［1］ , Theorem 1）. Suppose Y=［0,1］. 
Take t∈Y=［0,1］ and x*∈M（t）, and suppose  that  the partial derivative
　　　  of the objective function with respect to the parameter exists. If 
t < 1 and v is right differentiable at t, then
If t > ０ and v is left differentiable at t, then
If t∈（0,1） and v is differentialble at t, then 
Proof.  Since　　　　exists, f（x*,t） is finite. If t <１, take t '∈（t ,1） and τ=
t'−t > ０. Using the first two inequalities in （1） and （3）, we have that
 　　   dv（t）（1）≥ dy f（x*,t）（1）,
 −d（−v）（t）（1）≥ −dy （−f ）（x*,t）（1）.
Form the assumptions, the left-hand sides of these inequalities are equal to
v'+（t） and the right-hand sides are equal to　　　　 .  Then, we obtain that
If t > ０, take t'∈（0, t） and τ=t '−t < ０. Using the last inequalities in （1） 
and （3）, we have that
 　− dv（t）（−1）≤ −dy f（x*,t）（−1）,
 d（−v）（t）（−1）≤ dy（−f）（x*,t）（−1）.
Using the assumptions, we obtain the
∂f（x*,t）
∂t
∂f（x*,t）
∂t
∂f（x*,t）
∂t
v'+（t）≥ ———— .
∂ f（x*,t）
∂t
v'−（t）≤ ———— .
∂ f（x*,t）
∂t
v'（t）＝ ———— .
∂ f（x*,t）
∂t
v'＋（t）≥ ———— .
∂ f（x*,t）
∂t
v'−（t）≤ ———— .
∂ f（x*,t）
∂t
（5）
（6）
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A note on envelope theorems and subderivatives
References
［1］ Milgrom, Paul, and Ilya Segal. "Envelope theorems for arbitrary choice sets." 
Econometrica 70.2 （2002）: 583-601. 
［2］ Mas-Colell, Andreu, Michael D. Whinston, and Jerry R. Green. "Microeconomic 
theory." Oxford University Press. Caps 3.5 （1995）: 15. 
［3］ Morand, Olivier, Kevin Reffett, and Suchismita Tarafdar. "A nonsmooth 
approach to envelope theorems." Journal of Mathematical Economics 61 （2015）: 157-
165. 
［4］ Oyama, Daisuke, and Tomoyuki Takenawa. "On the （non-） differentiability 
of the optimal value function when the optimal solution is unique." Journal of 
Mathematical Economics 76 （2018）: 21-32. 
［5］ Marimon, Ramon, and Jan Werner. "The envelope theorem, Euler and Bellman 
equations, without differentiability." Journal of Economic Theory 196 （2021）: 105309.
［6］ Benveniste, Lawrence M., and Jose A. Scheinkman. "On the differentiability of 
the value function in dynamic models of economics." Econometrica: Journal of the 
Econometric Society （1979）: 727-732. 
［7］ Rockafellar, R. Tyrrell, and Roger J-B. Wets. Variational analysis. Vol. 317. 
Springer Science & Business Media, 2009.
（本学経営学部講師）
If t∈（0,1） and v is differentiable at t, v'（t）= v'+ （t）=v'_ （t）. Using （5） and
（6）, we have 
□
v'（t）= ———— .
∂ f（x*,t）
∂t
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A note on envelope theorems and subderivatives

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