I

ArticlesProducer Decision and
Input Demand Theory
A Modern Approach

Yasuhiro Sakai

Shiga University / Professor Emeritus

This paper is a drastically abridged version of Sakai(2023), with many equations and figures shortened andeven omitted. I am grateful to Dr. Masashi Tajima fortechnical assistance.

048

The Firm’s Expansion
Effect versus
the Consumer ’s
Income Effect

The purpose of this paper is to rigorously
analyze producer decision and production possibility sets, thus exploring the axiomatic
foundations of production theory. In older
days, Hicks (1946, 1953), Samuelson (1947),
Morishima (1953a, 1953b), and others, elaborated upon the well-established analysis of profitmaximizing firm's demand for inputs. Theirmain mathematical tools were classical differential calculus.
In more recent times, however, there haveemerged a series of papers which bravely study
the axiomatic foundations of production theory
on the basis of more modern topological tools.
For this point, see Scott (1962), Bear (1965),
Ferguson (1966, 1968, 1969), Friedman(1972),
Rader (1968), Basett and Borcherding (1969),
Hirota & Sakai (1969), Syrquin (1970), Shephard (1970), Arrow & Hahn (1971), Sakai (1973,
1974, 1975), and Malinvaud (1985). After the
1990s until the present day, unifying the axiomatic foundations of both consumption andproduction theories, brand new approaches
named duality approaches to microeconomic theory have emerged and widely flourished , with
Diewert (1982, 2018), Varian (1999, 2009), andMcKenzie (2002) being eminent accomplishments.
A number of those economists above mentioned have bravely attempted to clear up a
matter of long-standing confusion concerning
the analogy between consumer's income effectand producer's expansion effect. While consumer demand theory and input demand theory
1) In the 1960s and the 1970s, I myself was a graduate

student at the University of Rochester. And completing
my Ph.D. thesis, I became an assistant professor at the
University of Pittsburgh. In hindsight, I was so fortunate to be surrounded by many outstanding professors
such as Lionel McKenzie, Edward Zabel, Ronald Jones,
Hugh Rose, James Friedman, Walter Oi, and A kiraTHE HIKONE RONSO

Winter / Jan. 2024 / No.438

appear to be analogous, they are not the sameat all. Even a small difference at start may produce a big distinction at goal. The question
how and to what extent they are really different
is our main concern here.1)
As was independently shown by Hirota &Sakai (1969) and also by Syrquin (1970) , theeffect of a change in the price of a certain input
on the demand for another input can be divided into the following two separate effects.
They are a substitution effect along the old isoquant and an expansion effect along the new
expansion path. Unfortunately, the mathematical tool they employed was simple calculus,
thus lacking mathematical rigor and fineness.
The main purpose of this paper is to make our
input demand theory mathematically more sophisticated than the previous attempts, yetdeveloping it in many possible applications toeconomic reality.
we obtain a sort of decomposition equation in input demand theory which appears tocorrespond well to the famous Slutsky equation in consumer demand theor y. The
correspondence between the two decomposition equations, however, is not quite exact:
indeed, they look similar but are not identical.
We must pay special attention to the criticaldifference between the word "similar" and theword "identical." 2)
In this paper, we widely apply the powerful
method of McKenzie (1957), which was originally used for the development of consumption
theory, to the new area of input demand theory. Then, we can successfully derive various
kinds of decomposition equations. And in sodoing, we develop the new idea of a compensated change in output price when a certain input
price varies. This idea is analogous to, but notexactly the same as, the familiar concept of acompensated change in income when a certain
commodity price varies in the traditional consumption theory à la Hicks (1946).
So far, input demand theory has been developed in connection with the problem ofinferior inputs. An input is called normal (orinferior) if a rise in output price causes an increase (or a decrease) in the demand in that
input. We can obtain the following results. (i)While it is possible that all inputs are normal,
it is not possible that they are all inferior. (ii)
In case a certain input is inferior, it is not possible that all other inputs are gross complementswith it although it is possible that they are all
gross substitute for it.
The contents of this paper are as follows.
Section II is addressed to a system or a production technology. In Section III, the definitionof cost and profit functions will be given, witha careful discussion of their properties. The topological approach of Shephard (1970) to theduality principle between cost and production
is developed in these two sections. The total
effect of a change in the price of a certain input
on the demand for another input is decomposed into substitution and scale effects. In
Section IV, we are first concerned with theproperties of substitution and total effects, andexamine the question how and to what extent
they are similar to, or different from, those of
substitution and total effects in consumption
theory. The problems of inferior inputs and of
net and gross substitutability are also discussed.
Section V is devoted to various types of decomposition equations in input demand theory— one finite increment and two differential
versions. It is rigorously shown that the demand cur ve of an input is not positively

Takayama, and by many brilliant students including Jerr y Green, Jose Sheink man, Masayoshi Hirota, andMichihiro Ohyama. Though perhaps too late, I would
like to say my sincere thanks to all of them.

2) Historically speaking, the fundamental difference

Producer Decision and Input Demand Theory

between consumer demand theory and input demand
theory was first noticed by Hicks (1946), Chapter 7, andmathematically sophisticated by Samuelson (1947),
Chapter 4. It seems, however, that the similarity between them was not fully developed by these authors, resulting in later development of input demand theory.

Yasuhiro Sakai

049

sloping, and the substitution and expansion effects always go in the same direction. In
Section VI, we will apply our analysis to show
the validity of the Le Chatelier-Samuelson
principle in input demand theory.

II

Production Possibility Sets

We are concerned with a firm that is facedwith the problem of producing a single output
from a combination of a finite number of inputs subject to a production technology. Let
us suppose that there are n inputs and that input-output prices are competitively determinedin the market, being independent of the firm's
individual behavior.
In what follows, we will make full use of apowerful topological method. For this method, see McFadden (1966), Malinvaud (1985),
Mas-Colell & Whinston & Green (1995).
McKenzie (2002), and Mitra & Nishimura(2009).
Let us denote an input bundle by x = (x1,x2, ... , xn) . The set of all conceivable input
bundles is denoted by X. X is the set of all nonnegative n - vectors :X = {x = (x1, x2, ... , xn) : x ≧ 0 } .(1)

Let us assume that for any x ∈ X , the largest output is conceivable and conveniently
denoted by a production function f (x) . Let Y
be the range of the production function :Y = { y : y ≧ 0 and y = f (x) for some x ∈ X} .

(2)
We assume that Y is nonempty, convex,
and open above. Y need not be the nonnegative real line. It may be any subset of Y.

We now let define the following sets.
W = {w = (w1, w2, ... , wn) : w ＞0 }.

(3)

P = { p : p ＞ 0 }.

(4)Evidently, W is the input price space or theset of all input vectors, and P is the output
price space or the set of all conceivable output
prices.
There is an alternative useful description of
the production function f (x). To show this, let
us define the following production possibility
sets .
Ā(y) = {x : x ∈ X and f (x) ≧ y }.

(5)A(y) = {x : x ∈ X and f (x) ≦ y }.

(6)I(y) = Ā (y) ∩ A(y)= {x : x ∈ X and f (x) = y }.

(7)

It is noted that the upper possibility set Ā(y)is the set of all input vectors which are capableof producing at least the output y, whereas thelower possibility set A(y) is the set of all input
vectors which are capable of producing at mostthe output y. The set I(y), which is the intersection of Ā(y) and A(y), is clearly the familiar
isoquant frontier corresponding to y.3)
We are ready to define a production technolog y as a family of those production
possibility sets satisfying the following four assumptions.
Assumption (A1) If 0 ∈ I(y), then y = 0.
Assumption (A2) For each y ∈ Y, Ā(y)and A(y) are nonempty and closed in X.
3) For a topological and convex approach to production

theor y, see Uzawa (1964), Nikaido (1968), Shephard(1970), and others.

050

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Winter / Jan. 2024 / No.438

Assumption (A3) Let us take x ∈ Ā(y )and x1 ∈ Ā(y1) where x0 ≠ x1. For any t ∈ (0,1), if we put xt = (1–t) x0 + tx1 and yt = (1–t)
y0 + ty1, then we must find xt ∈ Int Ā(yt) or theinterior of the set Ā(yt).
00Assumption (A4) If x1 ≧ x0 ∈ and x0 ∈Ā(y), then x1 ∈ Ā(y).
These assumptions require detailed explanations. (A1) states that a positive output
cannot be obtained from a null input bundle.
In short, nothing comes from nothing. (A2)asserts that any output level y is attainable forsome input bundle x, and that the production
function f (x) is continuous. Therefore, there
should exist neither gaps nor jumps on the production curve .
Remarkably, (A3) together with (A2)clearly implies that the set Ā(y) is a "strictly
convex set," meaning that for any x 0 and x1,
namely two points of Ā(y) , if xt = (1–t)x0 +tx1, then xt must belong to the interior of theset Ā(y), not to the boundary I(y).
And finally, (A4) insures that free disposalof inputs is possible. In other words, the producer can dispose of any extra inputs with nocosts. Although this may not necessarily reflect
the reality, it is theoretically a very convenientassumption.

III

The Cost and Proﬁt
Functions:Deﬁnitions and Properties

cost and profit functions will be given and theirproperties will carefully be investigated.
First of all, for any (w, y) ∈ W×Y , we define a cost function as follows:c(w, y) = Min {wx : x ∈ Ā(y)} .

To see that the cost function c(w, y) is welldefined, we take any arbitrary x* ∈ Ā(y) andlet H(w) = {x : x ∈ X and wx ≦ wx*}. Since Ā(y) is closed by (A2) and H(w) is clearly compact, it follows that Ā(y) ∩ H(w) is compact aswell. Therefore, the continuous function wx
takes on a minimum on Ā(y) ∩ H(w). Notethat for any x ∈ Ā(y)–H(w), we have wx ＞
wx*. Hence, the function wx attains a minimum on Ā(y) .
In connection with the cost function c(w,y), we define a compensated input function as
follows:u(w, y) = {x : x ∈ Ā(y) and wx = c(w, y)}.

(9)The newly-defined function u(w, y) indicates the input bundle demanded by the firm
when input prices are w and output is to be y.
Under Assumptions (A1)–(A4), u(w, y) is theunique element of X minimizing wx subject tox ∈ Ā(y). Consequently, we should have thefollowing equation:c(w, y) = w・u (w, y).(10)We are now in a position to establish thefollowing useful lemma:

In what follows, we assume that Assumptions (A1)–(A4) are met for the production
technology. In this section, the definition of

Producer Decision and Input Demand Theory(8)

Yasuhiro Sakai

051

LEMMA 1 (Properties of c(w, y) and u(w, y))
For all (w, y) ∈ W×Y, we have the following properties:(1) c(w, y) is continuous in (w, y).
(2) c(w, y) is differentiable in w, and ∂c(w,y) / ∂wi = ui(w, y) for i = 1, 2, ..., n.
(3) c(w, y) is homogeneous of degree one
in w.
(4) c(w, y) is concave in w.
(5) c(w, y) is strictly convex in y.
(6) If y0 ＞ y1, y0, y1 ∈ Y, then c(w, y0) ＞
c(w, y1).
(7) u(w, y) is continuous in (w, y).
(8) u(w, y) is homogeneous of degree zero
in w.
The proof of this lemma is rather simple
yet considerably lengthy. Because of space constraint, it is omitted here. For details, see Sakai(2023).
At first glance, the contents of Lemma 1
appear to be rather technical and even too
mathematical. It should be noticed, however,that it discusses many important properties of
the cost and compensated input functions
which play critical roles in the theory of costand production. According to Property (1),the cost function c(w, y) is continuous in (w, y),so that the cost curve as a graphical expressionof the cost function is overall smooth and hasneither gaps nor jumps throughout. Property(2) tells us that focusing on w only, c(w, y) is extremely smooth, having no kinks at all.
Moreover, it also indicates a very nice bridge
between the two functions, namely the costfunction c(w, y) and the compensated input
function u(w, y). Exactly speaking, for any i,
we should have ∂c(w, y) /∂wi = ui(w, y), demonstrating that the rate of change of total costwhen the price of an input changes is equal tothe amount of compensated demand for that
input.
According to Properties (3), (4), (5) and(6), while the cost function c(w, y) is homogeneous one and concave in w, it is strictly convex
in y and also strictly increasing. Such concaveand convex relations should be worthy of attention. Finally, in the light of Properties (7)and (8), whereas the compensated input demand function u(w, y) is continuous in (w, y),it is homogeneous of degree zero in w only.
This teaches us that u(w, y) reflects the niceproperties of u(w , y) .
In addition to Assumptions (A1)–(A4),
we are now ready to postulate the following assumption.
Assumption (A5) Y is bounded fromabove.4)
As will be seen later, the purpose of Assumption (A5) is to force the profit function
to be defined later for any (w, p) ∈ W×P. The
newly added Assumption (A5) together withthe previous assumptions on Y implies that Y= [0, y*) for some y* ＜ + ∞. Let E(w, p) = {x: x ∈ X and pf (x) – wx ≧ 0} . Then, under Assumptions (A1)–(A5) , the newly defined set
E(w, p) is clearly bounded and closed; therefore, the continuous function pf (x) – wx takes
on maximum on E(w, p).
Note that by Assumption (A1), pf (x) –wx= 0 for x = 0 ; pf (x) – wx ＜ 0 for any x ∈ X –E(w, p). Thus, the maximum attained on E(w,p) is actually the maximum on the whole X.
Now, for any (w, p) ∈ W×P, let us define a
profit function π(w, p) as follows.
π(w, p) = Max { pf (x) – wx : x ∈ X}. (11)4) The purpose of Assumption (A5) is to force the profit
function π(w, p) = {p・f (x) – wx : x ∈ X} to be defined forany (w, p) ∈ W×P.

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Winter / Jan. 2024 / No.438

Clearly, for any given price vector (w, p),π(w, p) stands for the maximum profit which
can be obtained from the production technolo g y. Since f (x) is strictly concave by
Assumption (A3), it is clear that π(w, p) is positive for some (w, p) ∈ W×P. Thus, an
optimum input function and x(w, p) and an optimum output function y(w, p) ∈ W×P aredefined for all (w, p). Evidently, x(w, p) andy(w, p) respectively indicate the input bundle
as follows.
x(w, p) = { x : x ∈ X and π(w, p)= pf (x) – wx} .

(12)y(w, p) = f (x(w, p)) .

(13)

Evidently, x(w, p) and y(w, p) respectively
indicate the input bundle demanded by thefirm, and the output bundle supplied by thefirm. It is noted that the input-output vector(x(w, p), y(w, p)) is uniquely determined foreach price vector (w, p) ∈ W×P since the profit π(w, p) is strictly concave in x .
The properties of the profit function alongwith those of the optimum input-output function will be seen in the following lemma.
LEMMA 2 (Properties of π(w , p))
For all (w, p) ∈ W×P, we have following
properties .
(1) π(w, p)is differentiable in (w, p) , and∂π(w, p)/∂wi = –xi (w, y) for i = 1, 2, ..., n ,∂π(w, p)/∂p = y(w, y ) .
(2) π(w, p) is homogeneous of degree one
in (w, p).
(3) π(w, p)is convex in (w, p).

Producer Decision and Input Demand Theory(4) If w 0 ≧ w 1 ＞ 0 , then π(w 0 , p) ≦π(w1, p), with strict inequality if x(w0,p) ＞ 0.
(5) If p0 ＞ p1 ＞ 0, then π(w, p0) ＞ π(w,p1), with strict inequality if y(w0, p1) ＞
0.
(6) x(w, p) and y(w, p) are continuous in(w, p).
(7) x(w, p) and y(w, p) are homogeneousof degree zero in (w, p).
The proof of Lemma 2 is omitted here. For
details, see Sakai (2023). It is interesting inmaking a bridge between the profit and optimum input-output functions. The essence ofProperty (1) is expressed by the two equations;∂π/∂w i = – x i (w, p) and ∂π/∂p = y(w, y).
Graphically, it can easily be seen in Fig. 1.
In plain English, on the one hand, the rateof change of total profit when the price of an
input price changes is equal to the amount of
the input demanded by the firm, multiplied by(–1) . On the other hand, the rate of change of
total profit when the price of an output changes is equal to the amount of the output
supplied by the firm.
According to Property (1), the profit function π(w, p) is not only continuous but also
differentiable in (w, p), so that the profit curve
as a graphical expression of the profit function
is overall very smooth, with no kinks, having
neither gaps nor jumps throughout. Property(2) implies that when (w, p) doubles, π also
doubles. It follows from Property (3) that forany (w, p) ∈ W×P, and for any fraction t ∈ (0,1), the following inequality holds.
(1–t)π(w0, p0) + tπ(w1, p1)＞ π ((1–t)w0 + tw1 , (1–t)p0+tp1).

Yasuhiro Sakai

053

(14)

Fig. 1 Nice relationship between profit and input-output : ∂π/∂wi = –xi and ∂π/∂p = y .

Fig. 1 Nice relationship between profit and input-output : ∂π/∂w i = —ｘi and ∂π/∂p = y . .

Since c(w, y) is concave in y by Lemma 1(4), it must be almost everywhere differentiaAccording to Properties (3) and (4), while
π tends to decrease as w rises, but it tends to inble in y.5)
0 , p 0 ) + tπ(w 1 , p 1 )(1—t)π(w
Further, Eq. (15) asserts that the function
crease as p rises, thus agreeing with common
Theresense. ＞ π ((1—t)w 0 + t w 1 , (1—t)p 0 + t p 1 ) .py – c(w, y) is maximized at y = y(w, p).(14)Finally, by help of Property (6), inputs andfore, the first-order characterization for the
outputs continuously change in response to a
maximum yields the following .
According to Properties (3) and (4), whileπtends to decrease as w rises, but it
small change in (w, p). And Property (7)p rises,thus
tends
increaseshows tous thatwhen asall input
pricesandagreeing
all out- with common∂(py – c(w,sense.
y))/∂y = 0 , almost everywhere.
put prices
changethe same
proportion,(6),all inputs and outputs continuously change in
Finally,
by inhelpof Properties
This implies
that p –us
∂c(w,
y)/∂y
= 0 all
at yinput=
inputsandalloutputare
expectedtoremainresponse to a small change in (w ,p ) . And Property(7) showsthat
wheny(w, p) almost everywhere, from which immeunaffected, agreeing with our common sense.
prices and all output prices change in the same proportion, all inputs and all output are
Now, we are ready to establish the followdiately ∂c(w, y(w, p))/∂y = p almost everywhere.
expected
to remaincommon sense.
ing interestinglemma. unaffected, agreeing our Q.E.D.
Now, we are ready to establish the following interesting lemma.
Lemma 3 is an important lemma, saying
LEMMA 3 (Marginal cost and output price)that the marginal cost of an output is equal to
For all (w, p) ∈ W×P, we obtain ∂c(w,output price. Surely, it demonstrates profity(w, p)) /∂y = p, almost everywhere .
maximizing behavior of the firm from adifferent angle. The essence of Lemma 3 is 9
Proof. By the definition of c(w, y(w, p)), wegraphically illustrated in Fig. 2 .
note that the following inequality must hold:We are now in a position to discuss how
the optimum and compensated input functions
p・y(w, p)– c(w, y(w, p)) ≧ py – c(w, y)
are related with each other.
for all (w, y) ∈ W×Y.(15)5) For the properties of convex and concave functions,

see Fenchel (1953) , Rockafellar (1970), Shephard (1970)
and Varian (1999, 2009).

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Winter / Jan. 2024 / No.438

Fig. 2 At equilibrium, the following relation holds almost everywhere:

Fig. 2∂c(w, y)/∂y = p at y = y(w, p).

At equilibrium, the following relation holds almost everywhere:∂c ( w , y ) /∂y = p at y = y (w , p ) .
From this lemma, the economic signifiLEMMA 4 (Properties of x(w, p) and y(w , p))
For all (w, p) ∈ W×P, we obtain the folcance of u(w, y(w, p)) is quite clear. For an
arbitrary price vector (w, p), the optimum inlowing equations .
(1) x(w, p) = u(w, y(w, p)).
put-output combination (x(w, p)) is uniquely(2) c(w, y(w, p)) = w・x (w, p).
determined from the production technology.
(3) π(w, p) = p・y(w, p) – c(w, y(w, p)).
Then, c(w, y(w, p)) denotes the cost level whichgoverns the output level y(w, p), and u(w, y(w,p)) is the corresponding input bundle. It is also
Proof. To see Property (1), we first noteseen that the profit gained by the firm is equalx(w, p) ∈ Ā(y(w, p)). Let us next take xs ∈s
to the difference between p・y(w, p) and c(w,
Ā(y(w, p)), and x ≠x (w, p). This is possible byy(w, p)).
Assumptions (A2) and (A4). Now, let a "midtsdle point" x = (1–t)x(w, p) + tx , t ∈ (0, 1).
Then, by means of Assumptions (A2) and (A3),
Properties oftt
we obtain x ∈ Int Ā(y(w, p)), so that wx ＞ w
the Substitution andx(w,p).
Total Effects
・Letting t → 0 yields xt → x(w, p), and w x ≧w・x (w, p). Since x(w, p) ∈ Ā(y(w, p)), and u
In the last section, we sought the informa(w, y(w, p)) is uniquely determined for each (w,tion about conditions governing inputsp), it must follows that u(w, y(w, p)) = x(w, p).
demanded by the firm at given input-output
This proves Property (1).
prices. In this section, we are now ready to use
From (1), we find c(w, y(w, p)) = w・x(w, p)
it to discover how the inputs will change when
= u(w, y(w, p)), and π(w, p) = p・y(w, p) – c(w,these prices change.
y(w, p)), assuring Properties (2) and (3).
Now, let us consider a change in wj. Then,
Q.E.D.
we will see that its impact on xi can be divided
into the following two effects: a substitution ef-

IV

Producer Decision and Input Demand Theory

Yasuhiro Sakai

055

(3)

∂u i /∂w j =∂u j /∂w i for any i, j .

(4)

The following matrix is negative semi-definite :fect along the old isoquant and an expansion
 u1 / w1  u1 / wn effect(or a scale effect) along the new expansion
path (or scale path). Although the naming of .
 u / w  u / w 
the first effect is fairly traditional, that of the
1
n
n
 n
second effect might be quite new. The composition of the total effect into the substitution
Proof. Recalling that c(w, y) is concave inand the expansion (or scale) effect in producw by Lemma 1 (4) , it should be almost everyProof. Recalling that c (w , y ) is concave in w by Lemma 1 (4) , it should be altion theory was first noticed by Hirota andwhere twice differentiable in w. Note that by
twice
in 1w(2). Notethatthat
by Lemma(2)i ,=we find that ∂clater developedby differentiable
Sakai
Lemma
, we find∂c(w, y) /1 ∂w
Sakai (1969), and everywhere
)/ ∂
wi =
u i compo(w , y ) foruall
Therefore,
it follows
that(1973), Sakai (1974),yandothers.
Such
i. Therefore,
it follows
thatforforany i , u i (w , yi(w, y)i .for all
any i, ui insition in productionalmost
seems toeverywhere
be similar to,differentiableyet(w,wy). is almost
everywhere differentia6)
not as the same as the composition into theble in w.6)
With those preparations in mind, we also recall that u (w , y ) is homogeneo
substitution and income effects in consumpWith those preparations in mind, we also
w by(8). u(w,Here,
we apply the
famous Euler theorem
recall1 thaty) isifhomogeneousof degreetion theory. Howdegree
and tozero
whatinextent
theLemma
in w by Lemma(8).following
Here, if we
apply
expansion effect differsfrom the income
effect wezero
homogeneous
equations.
immediately
find 1theequation.
the famous Euler theorem on homogeneous
is a very important question, thus composing
equations. we immediately find the following
the main theme of this paper. A detailed dis(∂u i /∂w 1 ) w 1 + ... + (∂u i /∂w n ) w n = 0 for any i , almost everywhere ,equation.
cussion on this point will be made in the next
section. Although "the expansion effect" has(∂ui / ∂w1)w1 + ... + (∂ui / ∂wn)wn = 0 forbeen frequently usedin the previous
literatureassuring
Property(1).
and also in present paper, we would like to say
any i , almost everywhere ,that "the scale effect" may be an equally appealing name because it deals with "expansion" (orassuring Property (1).
larger scale) as well as "contraction" (or smaller
To prove Property (2), it is noted that byscale).
definition of the compensated function u, theThe properties of the substitution effectfollowing inequality holds.
can be summarized in the following theorem.
w・u(ws, y) ≧ w・u(w, y) for all ws ∈ W.
THEOREM 5 ( Differential properties of u(16)(w, y))For all (w, y) ∈ W×Y, we have the followEq. (16) tells us that the function w・u(w,y) attains minimum among all the functions ofing properties almost everywhere.
the form w・u (ws, y) at ws = w. Hence, the(1) (∂ui / ∂w1) w1 + ... + (∂ui / ∂wn) wn =0 for any i .
first order characterization for the minimum(2) w1(∂u1 / ∂wj) + ... + wn(∂un / ∂wj) = 0
yields the following .
for any j.
(3) ∂ui / ∂wj = ∂uj / ∂wi for any i, j.
∂ (w・u(ws, y)) / ∂wsj = 0 at ws = w, j(4) The following matrix is negative semi= 1, ... , n , almost everywhere,
definite :6) Note that any concave or convex function is almost

everywhere twice differentiable. See Alexandroff (1939).

056

THE HIKONE RONSO

Winter / Jan. 2024 / No.438

tween two inputs. Naturally, this agrees withfrom which Property (2) follows.
common sense.
To see Property (3), we recall the wellknown Young theorem on differential calculus,
The properties of the total effect will bewhich says that whenever a function is twice
summarized in the following theorem.
differentiable, the order of differentiation is
not important.7)
THEOREM 6 (Differential properties of x
(w, p) and y (w, p))
Therefore, by Lemma 1 (2), for any i, j, we
For all (w, p) ∈ W×P, we have the followobtain ∂ui / ∂wj = ∂2c / ∂wj ∂wi = ∂2c / ∂wi ∂wj
Theorem6(Differentialof x (w , p ) and y (w , p= ∂uj / ∂w, almost everywhere, which provesing properties, properties
almost everywhere.
Property (3).
+ (∂xi / ∂w
×P(∂x
, we
have
followingproperties,
almo
i / ∂w1)wthe1 + ...
n) w n +For all ( w , p )∈W (1)Finally, since the Hessian matrixof xthe(∂xi / ∂p) p = 0 for any i.
(1) (∂i /∂w 1 )w 1 + ... + (∂x i /∂w n ) w n + (∂x i /∂p ) p =concave function c(w, y) with respect to w is al(2) (∂y / ∂w1) w1 + ... + (∂y /∂wn) wn + (∂y(2) (∂y /∂w 1 ) w 1 + ... + (∂y /∂w n ) w n + (∂y /∂p ) p = 0 .
/ ∂p) p = 0 .
most everywhere negative semi-definite,(3)Lemma∂x i 1/∂w j =∂w ji = ∂xforj /allj . all i, j.
(3) x∂xj i/∂
Property (4) is easily derived from/ ∂w
∂wii ,for(2). Q.E.D.
(4)y
 ∂x
+0
∂y /for∂wiall= 0i .for all i.
i /w∂pi =(4)∂x i /∂p +∂/∂We are in a position to discuss a new con(5) The following matrix is positive semi(5) The following matrix is positive semi-definite :definite :cept of net substitution and net complementarity
in contrast to a more traditional concept of  x1 / w1   x1 / wn x1 / p gross substitution and gross complementarity.


 

We say that the input i is a net substitute for the  x / w   x / w x / p  .
input j if ∂ui / ∂wj ＞ 0, and a net complement
1
n
n
n
  n



for it if ∂ui / ∂wj ＜ 0. Possibly as a rare case, we


y

w


y

w

y

p
/

/
/
1
n


could have the special situation in which ∂ui /Proof. We first note that the profit func∂wj ≦ 0 for all i. Generally speaking, when theprice of a certain input price rises,
thedemandtion
π(w,p) theis almostywhereProof. We first notethat
profit everfunctionπ(twicew , p ) is afor that input tends to decrease if output is todifferentiable since it is convex by Lemma 2 (3).
differentiable since it is convex by Lemma 2 (3). Since∂π/∂w
be constant. In other words, the compensatedSince ∂π / ∂wi = –xi for any i, and ∂π / ∂p = y,
∂π
/
∂
p
=
y
,
this
obviouslythat
both
input func
demand curve tends to be negatively sloping.
thisobviously
impliesimplies
that both
input
function
Coupled with this result, Property(1) of Theooutput
functiony(w, p) are aly ( p)w and, pthe) arealmosteverywhere
differenti
outputfunction x(w,rem 5 implies that whereas it is possible that all
most e ver y where differentiable. Suchproperties are quite important for further derivations.
other inputs are net substitutes for an input, it
differential properties are quite important for
From Lemma
2 (7),
we see that x (w , p ) and y ( w ,is not possible that they are net complements
furtherderivations.
degree one. Therefore,
by applyingtheEuler thfor it. In short, although net substitutability
From Lemma
2 (7), we see
thatfamous
x(w, p) andcan be seen everywhere, net complementarity
y(w,
p)
are
homogeneousofdegreeone.
Therefunctions here , we can immediately obtain Properties (1) and (2is a rare phenomenon. Property (2) also showsfore, by applying the famous Euler theorem on
Next, Let us recall the famous Young theorem, which sa
further limits on the possibility of net complehomogenous functions here , we can immeditwice
differentiable,
the orderof differentiation
does not mattmentarity. Property (3) asserts that theately obtain
Properties(1) and (2).
be- if we apply
recall the
famous Young
substitution effect should be symmetrical
Next, Let
result. So,
theusYoungtheoremto thetheoprofit func
rem,
which
says
a functionis twice equati
convex by Lemma
2 (7), thenwethatcanifobtainthe following7) See Takagi (1961), pp. 57-58.

∂x i /∂w j = —∂2π/∂w j∂w i = —∂2π/∂w i∂w j =∂x j /∂x i /∂p +∂y /∂w i = —∂2π/∂p∂w i +∂2π/∂w i∂p= —∂2π/∂w i∂p +∂2π/∂w i∂p = 0

for all i .

Therefore, Properties (3) and (4) are surely assured.
Producer Decision and Input Demand Theory

Yasuhiro) is convex, its H
Finally, since
theSakairisk functionπ(w , p 057

differentiable, the order of differentiation does
not matter, leading to the same result. So, if we
apply the Young theorem to the profit function
π(w, p) which is convex by Lemma 2 (7) , thenwe can obtain the following equations almost
everywhere:.
∂xi / ∂wj = –∂2π / ∂wj∂wi = –∂2π / ∂wi∂wj= ∂xj / ∂wi for all i, j,∂xi / ∂p +∂y / ∂wi = –∂2π /∂p ∂w i + ∂2π /∂wi ∂p= –∂2π / ∂wi ∂p +∂2π / ∂wi∂p = 0 for all i.
Therefore, Properties (3) and (4) are surely
assured.
Finally, since the risk function π(w, p) isconvex, its Hessian matrix must be positive
semi-definite, almost everywhere. We notethat the following set of equations hold.
∂2π /∂wj ∂wi = ∂xi / ∂wj, ∂2π / ∂p ∂wi = ∂xi/ ∂p.
∂2π / ∂wj ∂p = ∂y / ∂wj, ∂2π / ∂p2 = ∂y / ∂p.
Hence, Property (5) is definitely assured.
Q.E.D.
To see the economic significance of Theorem 6, let us introduce several useful concepts
here. We say that the input i is a gross substitutefor the input j if ∂xi / ∂wj ＞ 0 ; a gross complement for it if ∂xi / ∂wj ＜ 0. We also say that the
input i is a normal input if ∂xi / ∂p ＞ 0 ; an inferior input if ∂xi / ∂p ＜ 0. If we employ theseconcepts, we can give new light on the relationship between inputs and outputs on the one
hand and input prices and output prices on theother hand.

First of all, Property (4) of Theorem 6 tells
us that the "supposedly normal situation" underwhich ∂xi / ∂wi ≦ 0 for any i and ∂y / ∂p ≧ 0
is certainly plausible but not inevitable. As caneasily be expected, while the input demand
curve tends to be negatively sloping, the output supply curve tends to be positively sloping.
Property (4) means that ∂xi / ∂p ＞ 0 (or ＜ 0)
if and only if ∂y / ∂wi ＜ 0 (or ＞ 0). In plain
English, this means that the input i is a normal
input (or am inferior input) if and only if a fallin the price of the input leads to an increase (ora decrease) in output.
In the light of Property (5), we find ∂y / ∂p
≧ 0. So, Property (2) implies that although it
is likely that all of inputs are normal, it is a mission impossible that they are all inferior. Now
suppose that a certain input, say xi, is an inferior input, so that ∂xi / ∂p ＞ 0. It follows from
Property (1) that, in such a case, it is not possible at all that all other inputs are grosscomplements for the input i although it is truly
possible that they are all gross substitutes for it.
Because of Property (3), the total effect is symmetrical between two inputs. However, wealready know from Theorem 5 (3) above thatthe substitution effect is symmetrical between
them. Therefore, the expansion effect, as thedifference of the total and substitution effects,
must be symmetrical as well.8)
Finally, in the light of Properties (3) and(5), we obtain the following inequalities for alli ≠ j.
(∂xi / ∂wi) (∂xj / ∂wj) ≧ (∂xi/ ∂wj)2.
(–∂xi / ∂wi) (∂y / ∂p) ≧ (∂xi / ∂p) (–∂y /∂w1).

8) This markedly contrasts with the income effect inconsumption theory since the latter may be non-symmetrical.

058

THE HIKONE RONSO

Winter / Jan. 2024 / No.438

Clearly, these inequalities indicate dominance of the own effects over the cross effects.
Summing up, the differential properties of x(w,p) and y(w, p) are so important that they will
be utilized for further discussions on input demand theory.

V

Decomposition
Equations in
Input Demand Theory

As stated above, the total effect of a change
in the price of an input on the demand for another input can be split up into the two
separate effects, namely the substitution andscale effects. The purpose of this section is to
use the previous results to derive various typesof decomposition equations in input demand
theory. Our attention will be mainly devoted
to seeing how and to what extent they are analogous to , or distinct from, the famous Slutsky
equations in consumer demand theory (seeSlutsky (1915) and McKenzie (1957) ).
Let us attempt to decompose the change ofthe demand for the input i responding to achange in the price of the input j. For that purpose, let us consider the following increment .
Δ
 xi(w, p) = xi(w +Δjw, p) – xi(w, p), with
Δjw = (0, ... , 0, Δwj, 0, ... , 0).
Then, by 4 (1) above, we obtain the following.
Δ xi(w, p) = ui(w+Δjw, y(w+Δjw, p)) –ui(w,y(w, p)).
= ui(w+Δjw, y(w, p)) – ui(w, y(w, p))
+ ui(w+Δjw, y(w+Δjw, p)) – ui(w+Δjw,y(w, p)).(17)Producer Decision and Input Demand Theory

Now, let us newly define the quantities SEijand EEij as follows.
S Eij =【ui(w+Δjw, y(w, p)) –ui(w, y(w, p))】/ Δwj.(18)E E ij = 【 u i ( w + Δ j w , y ( w + Δ j w , p ) ) –ui(w+Δjw, y(w, p))】/ Δwj.(19)Then, it is easily seen from Eq. (17) that thefollowing equation holds.
Δxi / Δwj = SEij + EEij.(20)In our opinion, Eq. (20) has very important implications. It is noted that SEij and EEij
respectively show the substitution effect and the
expansion effect. At first appearance, it seems tobe the finite increment version of the famous
Slutsky equation in consumer demand theory.
In fact, a change Δwj in the price of the input jaffects the behavior of the firm in two different
ways. While it causes a change in input priceratios which induces technical substitution
among inputs along the old isoquant, it entails achange in the profit maximizing output along
the new scale path.
On the one hand, the first substitution effect stands for the variation in the optimum
combination of inputs within the isoquant class
to which the original x(w, p) belongs. Then, c(w+ Δjw, y(w, p)) represents the corresponding
level of cost which keeps the firm within thesame isoquant class as before the input pricechange, even in the new input price situation
w+Δjw. On the other hand, the scale effectrepresents the shift of the optimum input combination in the new input price situation w +Δjw,
responding to the change of output level from theold level w + Δjw to the new level y(w +Δjw, p).

Yasuhiro Sakai

059

Note that Eqs. (18)-(19) stand for the difference versions of the decomposition equations.
We are now ready to derive the differential versions, which are perhaps more interesting thanthe difference versions. To this end, it is necessary to make an additional assumption as
follows.
Assumption (A6) For all (w, p) ∈ W×P ,x(w, p) and y(w, p) are differentiable in (w, p).
This newly added assumption (A6) is not
so far from the previous assumptions (A1)–(A5). In fact, as was seen above, the latter ones
already assure almost everywhere differentiability of x(w, p) and y(w, p). The only difference
between the previous assumptions and the
newly added assumption comes down to thedifference between "almost everywhere" and"everywhere." Moreover, when (A6) is assumed, it is not hard to see from Lemma 4 thatu(w, y) and c(w, y) are also differentiable.
Now, we are in a position to establish andprove one of the most important theorems inthis paper.
THEOREM 7 (The first kind of decomposition equations)
For all (w, p) ∈ W×P, we have the following equations.
(1) ∂xi / ∂wj = ∂ui / ∂wj + (∂ui / ∂y) (∂y /∂wj) for all i, j.
(2) ∂xi / ∂p = (∂ui / ∂y) (∂y / ∂p) for all i.
Proof. In the light of Lemma 4 (1), theproof of (1) and (2) is easy and straightforward.
Indeed, we know that the following important
equation holds.

x(w, p) = u(w, y(w, p)).
If we apply the well-known rule on the differentiation of a composite function, then we
can immediately obtain the desired Properties(1) and (2). Q.E.D.
Property (1) indicates the first version of
decomposition equation in input demand theory in differential terms. Exactly speaking, wefind that ∂xi / ∂wj = ∂ui / ∂wj + (∂ui / ∂y) (∂y /∂wj). The first term (∂ui / ∂wj) on the righthand side tells us to what extent a change in wj
influences ui . The second term (∂ui / ∂y) (∂y /∂wj) on the right-hand , representing the remarkable scale effect term, shows the composite
effect containing the following two partial derivative terms :(i) The partial derivative term (∂y / ∂wj),
representing the change in the optimum output corresponding to the change in the price of
the input j,(ii) The partial derivative term (∂ui / ∂y),showing the change in the optimum input responding to the variation in the output above
mentioned.9)
Property (2) demonstrates that the total effect of a change in input price on the input ican be decomposed into the following two partial derivative terms :(i) The partial derivative term (∂y / ∂p),
representing the change in the optimum output as a result of the change in output price,(ii) The partial derivative term (∂ui / ∂y),showing that the change in the optimum input
corresponding to the variation in output above
mentioned.
At first appearance, the combination of the
firm's substitution and expansion effects seems
9) The f irst version of decomposition equation was

merely referred to by Basett & Borcherding (1969) without any proof, and later was given a formal proof by Hirota & Sakai (1969) and Syrquin (1970) by using simple
calculus. It is noted that the most exact form of decomposition equations in input demand theory is given here
in terms of the compensated input functions.

060

THE HIKONE RONSO

Winter / Jan. 2024 / No.438

to be analogous to that of the consumer's wellknown substitution and income effects. We
must say, however, that they are not exactly thesame.
Whereas in consumption theory we have
simply to consider the relations between thosecommodities which can be regarded as being
essentially similar, in production theory wehave two different kinds of commodities to consider — inputs and outputs. Therefore, theirmutual relations and their cross-relations take alittle more disentangling. We remind readersthat, to obtain a decomposition equation in
consumption theory, it is necessary to imagine
a compensated change in consumer's income soas to keep the same preference level as before,in spite of a change in the price of a certain
commodity. In contrast to such a compensatedchange in income in consumption theory, in
production theory we have to introduce theidea of a compensated change in output price,which was first introduced by Hirota & Sakai(1969) and later developed by Sakai (1973),
Otani (1982), Diewert (1982, 2018), and others.
More exactly speaking, when the price of a certain input price varies, we imagine such acompensated change in output price as would
induce the firm to maintain the output level as
before the input price change.
By making use of the above-mentioned
idea peculiar to production theory, we can now
derive the second version of decomposition equation. Compared with the first version, thissecond one will turn out to be more analogous
to the decomposition equation in consumption theory. For that purpose, Assumption(A6) needs to be a bit strengthened to the following assumption.

Assumption (A6') For all (w, p) ∈ W×P,
x(w, p) and y(w, p) are twice differentiable in(w,p).
When Assumption (A6') is made, it is obvious by Lemma 4 that u(w, p) and c(w, y) arealso twice differentiable. We are now ready toderive the following theorem.
THEOREM 8 (The second kind of decomposition equations)
For all (w, p) ∈ W×P, we have the following equations.
∂xi / ∂wj = ∂ui / ∂wj – (∂xi / ∂p) (∂p / ∂wj)
for all i, j ,where ∂p / ∂wj≡[dp / dwj]dwi = 0(i ≠ j), dy = 0.
The proof of this theorem is similar to thatof THEOREM 7, thus being omitted. For details, see Sakai (2023)
Theorem 8 gives us the second version of decomposition equations in input demand theory.
The scale effect here is meant to represent thecomposite effect of the following two terms:(i) The change in output price so as tomaintain the same output level as before, inspite of the change in the price of the input j.
(ii) The change in the input i corresponding to the variation in output price above
mentioned.10)
This second version of decomposition
equations are as important as the first version,
presumably being even more comparable to thefamous Slutsky equations in consumption theory (see Slutsky (1915)). As an immediate
result of this theorem, we can finally derive thefollowing.

10) By relying on the traditional calculus method, Hi-

may look a small step, but surely a giant jump indeed.

Producer Decision and Input Demand Theory

Yasuhiro Sakai

rota & Sakai (1969) succeeded in deriving the second version of decomposition equations. To our regret, however, their proof was rather sketchy and based on the Hessian matrix of the production function per se. We believe that a topological approach taken here is more general, and more elegant, than the previous method. This

061

THEOREM 9 ( Property of the expansion effect )For all (w, p) ∈ W× P, we have the following property.
EEii ≡– (∂xi / ∂p) (∂p / ∂wi) ≦ 0, i = 1, ... , n.
Proof. By making use of Theorem 8, Theorem 7 (2), and Lemma 1 (2), we can obtain thefollowing sequence of equations.
– (∂xi /∂p) (∂p / ∂wi) = – (∂xi / ∂p) (∂2c /∂y ∂wi )= –(∂ui / ∂y) (∂y / ∂p) (∂ui / ∂y) = – (∂y /∂p) (∂ui / ∂y)2 .
Clearly, it should be non-positive by Theorem
6 (5). Q.E.D.
The economic significance of this theorem
is quite clear. It is recalled by Theorem 5 (4)that SEii ≡∂ui / ∂wi ≦ 0.
In the light of the previous discussions, it isclear that the substitution effect SEij and thescale effect EEii should always go in the same
direction. Surely, this is also a remarkable contrast to consumption theory à la Hicks (1946),
in which the substitution and income effects
may go in opposite directions.
In short, input demand theory is input demand theor y, thus being distinct fromconsumer demand theory. Although those twotheories look somewhat similar, they are definitely different. We have to understand exactly
how and to what extent they are analogous or
distinct.

062

VI Final Remarkson the Le Chatelier Samuelson Principle

In the above, we have been manly concerned with the axiomatic foundations of input
demand theory. While the approach taken
here looks mathematical and rigorous, it has
useful economic implications. In particular, it
is noted that the total effect of a change in an
input can be decomposed into the substitution
and expansion effect. How and to what extent
such decomposition in input demand is comparable to that in consumer demand iscertainly a very important question to ask. To
our surprise, such comparison has been rather
neglected for long time in the economics literature. We do believe, however, that as the saying
goes, it is better late than never.
We are ready here to do some economic
applications and make final remarks. First, wenote that our decomposition in input demandis closely related to the famous Le Chatelier-Samuelson principle. Although this connection was pointed out by our friend Yoshihiko
Otani (1982), we are going our own way toconfirm it below.
Henri Louis Le Chatelier (1850-1936) wasa noted French scientist. He was best known
for his work on his chemical equilibrium,
which was to be called Le Chatelier principle
in the academia. In his classical work contained in Samuelson (1947, enlarged edition
1983), Samuelson boldly applied the principle
to economic equilibrium, so that naming of the
Le Chatelier - Samuelson principle has been so
popular in the academic world, especially inthe economic profession (see Stiglitz (1966)).11)

THE HIKONE RONSO

Winter / Jan. 2024 / No.438

In our setting of input demand theory discussed so far, we have only to compare
the case with no isoquant constraint andthe one with a single constraint, namely theconstraint that output y remains constant regardless of a change in wi. So, in terms of the
input demand setting, we must have (dxi / dwi)
0 ≦ (dxi / dwi ) 1 ≦ 0.
If we rather want to follow the decomposition equation formula above mentioned, then,we must find ∂xi / ∂wi = ∂ui / ∂wi – (∂xi / ∂p)(∂p / ∂wi), or in short, TEii = SEii + EEii
Since the terms TEii, SEii, and EEii are all
non-positive, we must have the following: TEii
≦ SEii (≦ 0 ). Needless to say, this inequality
clearly demonstrates the validity of the famous
Le Chatelier - Samuelson principle in input demand equilibrium. For this point, also refer toRader (1968).
In conclusion, the name of Chemist
Le Chatelier is brilliantly shining not only inthe chemical world but also in the economic
world as well. Life may be short, but science islong and spreading indeed !
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THE HIKONE RONSO

Winter / Jan. 2024 / No.438

Producer Decision and Input Demand Theory
A Modern Approach

Yasuhiro Sakai

This paper is concerned with an axiomatic
approach to input demand theory. By help ofgeneral production possibility sets, we intend
to derive decomposition equations in input demand theory, which have been rather neglected
so far in the economics literature. Special attention is paid to important comparison
between the firm’s expansion effect and theconsumer’s income effect. We discuss the question how and to what extent the expansion
effect is distinct from the income effect. In thisconnection, the Le Chatelier -Samuelson principle is also discussed,
Keywords: producer decision, comparative
static analysis, expansion effect, inferior input, Le Chatelier-Samuelson
principle

Producer Decision and Input Demand Theory

Yasuhiro Sakai

065