A Theorem in the Pure CompetitionMorii,Shoken

Hiroshima University of Economics Academic Repository

A Theorem in the Pure Competition 
Shoken Morii 
The term Competition is included with the follows; atomistic competi-
tion， free competition， perfect competition， and pure competition. A-
tomistic competition is representative term that brings about， asthe mat-
ter constitutes of atomics， inthe case of competition constructing the 
business with comming together a number of small business. 
Free competition is showed the competition doing behavioral freely bet-
ween many producers and consumers with exclusive of the feudalistic 
restriction and mercantile protective system. However， the methodolo-
gical hypothesis in the modern economics takes up an idea of perfect com-
petition. 
Perfect competition satisfies the following conditions; (1) both firms and 
consumers are numerous; (2) firms produce a homogeneous commodity， 
and consumers are identical from the seller's point of view in that there are 
no advantages or disadvantages associated with selling to a particular con-
sumer; (3) both firms and consumers possess perfect information about 
acknowledgment of al commodities; (4) each individual unit is small in rela-
tion to the effectiveness into the whole markets， therefore they take in the 
market pric巴asa parameter; (5) entry into and exit from the market is free 
for firms and consumers. 
(1) cif. Henderson， ]. M. & R. E. Quandt; Microeconomic Theory， 2ed. 
Mcgraw-hill Kogakusha， 1971， p.104. 
2 第11巻第3号(経済学・経営学編)
Pure competition is similar to perfect competition， but its difference is 
that the prices of products are determinative with supply and demand in 
the whole market. However， each producer is not possible to bring about 
any effect to the market price， although they make alone increment or 
decrement in the quantity of production. They regard， therefore， as 
given to the market price and must behave it adaptabily. 
In this artic1e we mainly deal with firm's behavor in the perfect competi-
tion and pure competition. Finally we derive a theorem of adding-up pro-
blem. 
I. Optimizing Behavor of the Firm 
We assume that the firm is purchase two factors under a constant factor 
prices and that he produce a product. Designating a product by X and 
two factors; i. e. labor by L and capital by K. Production function is show-
ed as follow. 
X=F(L， K) (1) 
Production cost constitutes of variable cost and fixed cost. Manifesting 
cost required to produce any production X by C， and factor price by wand 
r respectively， cost equation is the following. 
C=wL十rK+B (2) 
where B is fixed cost. 
Now， we assume the firm take the behavor of optimum under a given 
cost. Transforming equation (2)， we have 
? ?? ?
?
?
(3) 
Substituting equation (3) to (1)， the equation can rewrite as follow. 
????
?
?? ??
?? ????
(4) 
A Theorem in the Pure Competition 3 
Differentiating equation (4) with respect to L， and setting it equal to 
zero， 
? ?? ???
?????? ?
? ? ?
(5) 
where FL and FK are a partial derivatives. From equation (5) we make 
up the fol1owing results. 
FL W 
FK r 
(6) 
This means that the relative percentage of marginal product to labor and 
capital equals relatively ratio of wage and rental. We cal1 a condition of 
first degree for the firm's optimization. 
Next， inorder to establish a condition of second degree for the optimiza-
tion of the firm， differentiating equation (5) with respect to L， we have the 
fol1owing. 
。2X ~W~ ， i W\~ ~;;=FLL←2~-FLK十 1:: I~FKK dL2 ~ LL ~r~ L/¥. ' ¥r} (7) 
Because we assume the Law of Decreasing Return to Scale， equation 
(7) becomes to have negatively a sigh. Namely， this means that， ifwe 
assume the Law of Decreasing Return to Scale， then an isoquant has to the 
curve sloping downwards and having downward convexity. 
General1y， firm may behave toward a target as the fol1owing; first， an en-
trepreneur produces the product maximization under constant cost; se-
cond， an entrepreneur produces with minimizing cost under constant pro-
ducts; third， aswe know better， a firm takes her profit maximization. 羽Te
are final1y possibl巴tosay that， ifan entrepreneur produces the product 
maximization with minimum cost， then entrepreneur's profit results in the 
maxlmum. 
To prove on describing the above， we can solve with Lagrangian equa-
4 第11巻第3号(経済学・経営学編)
tions as follow. 
First method is; 
L=F(L， K)+λ[C-(wL+rK+B)] (8) 
At equation (8)， first term on the right side is a production function and 
second term a production cost designating λby Lagrangian U nknown. 
Secood method is; 
L=(wL十rK十B)十λ[X-F(L，K)J (9) 
We shall know that both equations (8) and (9) become similar condi-
tions with first degree and second degree for the firm's optimum as 
follow. That is to say， first order condition is， 
FL W 
FK r 
?? ????
This is same as equation (6). Now， second order condition follows， 
時FLL-2FLFKFLK+ FiFKK< 0 (1l) 
This is same as equation (例7わ)悶dju刷s
rK 
Third method puts into the following， 
日=p.F(L， K) -(wL+rK + B) (12) 
where n isthe firm's profit， P a price per product. Differentiating equa-
tion (12)， and setting zero in the derivative， 
。n
一一=PF，-w=OaL ~ ~ L (13) 
an -一=PF，-r=OaK ~~ 11. (14) 
Then， we have a following equation called on the first conditions for 
firm's optimization. 
A Theorem in the Pure Competition 5 
?
???
? ?
(15) 
Now， differentiating equations (13) and (14)， we get the following equa-
tion designed the conditions of the second degree. 
iPFLL PhKlidL l_ìdw-hd~ 
LPFKL PFKKJLdKJ Ldr-FKdPJ 
(16) 
Solving Hessian matrix， 
P2(FLLFKK-FA) >0 (17) 
This manifests to have in sign positively in order to FLL and FKK<O or 
both P and FLK>O. 
To summary; An entrepreneur devices to maximize her profit according 
to product maximum with cost minimum. Therefore， the conditions of 
profit maximum result as fol1ow. 
dL FL w 
dK FK r 
(18) 
That is to say， this describes to equate the rate of technical substitution， 
the ratio of the marginal product of labor to the marginal product of 
capital， and the ratio of the wage to rent respectively. This is the first 
order conditions for the profit maximum and represent the important. 
1. Adding-up Theorem 
We have a proposition that marginal revenue， so that the price of pro-
duct must be equal marginal cost under perfect competition. Then， we 
have some equations; production function， cost equation and profitable 
equation as follow. 
X=F(L，K) 
C=wL十rK十B
H二 PX-C
(19) 
(20) 
(21) 
6 第11巻 第3号(経済学ー経営学編)
The proposition for the profit maximum under perfect competition is， 
differentiating equation (21) and setting up it zero， the following equation. 
??? ?
? ? ?
??? ?
? ? ?
(22) 
where cost depends on productive goods (X) become the cost increases or 
decreases in proportion to X. 
Or， this equation can be able to rewrite as follows. 
??
????? ??? ? ? (23) 
Therefore， itbecomes， 
?
???
? ? ?
(24) 
Equation (24) means that the price of product， namely average revenue 
must be equal to average cost 80 as a pure revenue to zero. This is possi-
ble to be exposition with the homogeneity of production function and ad-
ding~up problem. Adding~up problem postulates the pure competition; it 
means that demand curve for producer is infinitely elastic and the sale 
from producer exerts not influence upon the price. Under those provi-
sions the condition of initial equilibrium is to equate relatively factor prices 
with the marginal productivity times the selling price. 
Well， differentiating equation (21) in respect to L and K respectively， 
and setting to equal it zero， we fol1ow， 
dII d (PXl dC ~ 
dL dL dL ~ (25) 
dII d (PX) dC A 
dK dK dK ~ (26) 
(2) cif. Samuelson， P.A.; Foundation of Economic Analysis， Atheneum， 1970， pp. 
81~89 
A Theorem in the Pure Competition 7 
Then， differentiating totally to first term aside right hand in equation 
(25)， 
d (PX) ~ðX ， ，;Jp ~ðXん， X dP¥ 。L ~ dL '.~ðL ~ dV ~ ， P dXJ (27) 
Next， differentiating partially to revenue， R=PX inrespect to X， we are 
given the following. 
dR d(PX) ~(.， X dP¥ 
一一三一一一一二Pl1十一一一ldX dX • ¥. ， P dXJ (28) 
Moreover， equation (27) manifests as follow， 
d(PX) dX dR 
dL dL dX 
(29) 
Because the demand elasticity of product definits the following， 
?
? ?
? ?
? ?? ? (30) 
Therefore， equation (28) is replace as follow. 
d(PX) dX dR ~ðXI. 1¥ 一一一一=一一 二P:，，~'11-'::'1。X dL dX ~ dV. "J (31) 
Then， equation (31) suggests marginal revenue must be equal the 
marginal revenue product， so that the marginal product times marginal 
revenue， and equate to the marginal value products. 
Well， we differentiate second term aside right hand of equation (25) in 
respect to L. 
?????????? ??????↑? ???? (32) 
。L
where el圭一一一 isthe supply elasticity to productive factor L. L dw 
To final， from equations (31) and (32)， 
8 第11巻第3号(経済学経営学編)
PZ(1ーか司(1十台 (33) 
Then， equation (33) means the marginal value products equals the 
marginal cost. 
When we assume now it is that the commodity market and the factor 
market exist in the pure competition， because of η=∞and Clニ∞， equa-
tions (31) and (32) consist of the following equality. 
d (PX) dX dR ~ðX dC 
dX 瓦夜 I瓦瓦 W (34) 
where this suggests that the marginal revenue must be equal the marginal 
revenue products， the marginal value product， the marginal cost and fac-
tor pnce. 
Now， we derive out of equation (34) as follow. 
(1)ax aR w[l十一=一一一一
Cl) dL dX 
(35) 
Hence， equation (35) is rewritten as follow 
δX 
dL dR wニ
1 dX 
1+ 
el 
(36) 
In this case， estimating it as yet with similar method， we belive to be 
able to get in the following equation. 
? ?
?
?? ? ? ?
? ?? (37) 
Here， ifwe imagine the profit is zero， then the equation becomes the 
A Theorem in the Pure Competition 9 
following equality. 
px=ωL+rK (38) 
Substituting equation (38) to equations (36) and (37)， 
ax ax 
px=竺I~ιL十1竺KI
vι¥1十二 1十二 j 
(39) 
Therefore， replacing it， 
aR， ax ax ¥ 
x=立I~ιL+ aK"KI 
Ph十1uzl+1J
(40) 
This is the general form of the adding-up theorem. 
When we assume the pure competition in the commodity market， 
aR 
because of ~~=P， equation (40) is as follow. ax 
ax ax 
aL T ， aK X二一一~L+ v'~"K 1 ~ ， " ， 1 
1十 1十-
el e2 
(41) 
Moreover， we suppose the pure competition in the factor market， inthis 
case Cl二∞ ande2二∞， equation (41) is rewritten as the following. 
? ????
(42) 
This equation is similar equation with Mr.羽Ticksteedin his Essay on the 
Coordination of the Law of Distribution published in 1894， asfollow. 
(:3) cif. Yamane， T.; Mathematics for Economists， 2ed. Prentice-hall， 1968， pp 
226~228. 
(4) cif. Walras， L.; Elements of Pure Economics， translated by William， ].， George 
Allen & Unwin， 1965， pp. 489~495. 
10 第11巻第3号(経済学ー経営学編)
P=F(A，B， C) 
subject to the condition that 
mP=F(mA， mB， mC) 
whence he derives 
dP..dP~.dP P=ー~A+"'，~rB+"'，~~C
dA.' dF" dC 
where P is the product and the factors of production by A， B， C. Mr. 
Wicksteed regards the product as a function of the factors of production. 
Walras said 1 wish to cal1 attention to a s1ight difference between Mr 
Wicksteed's and my own modifications of the English theory. Mr. 
Wicksteed's equation may be supposed non-linear and non-homogeneous 
and P isa physical quantity， not a value， ofthe product. 羽Talrasspecifies 
the following; (1) free competition leads to mimimum cost of production; 
(2) under the rule of free competition， the rate of remuneration for each ser-
vice is equal to the partial derivative of the production function， i. e. to its 
marginal productivity; and (3) the total quantity of the output is distributed 
among the productive services. This is precisely what 1 said. 
Arround the equation (42) which we derive， Samuelson described as 
follow; This looks superficially like Euler's theorem， but it is not. For 
Euler's theorem is an identity， whereas this is merely a condition of 
equilibrium holding at a single output. Moreover， inrelation to our equa-
tion (40)， i. e. adding-up theorem; This formulation differs from that of 
Walras， Wicksell， and Hicks in that the condition of minimum unit cost is 
derived as a theorem from the condition that total revenue equal total ex-
penditure. 
June 30， 1988. 
(5) cif. SumueIson， P.A.; Fundations of Economic Analysis， Atheneum， 1970， p. 
81~89. 
A Theorem in the Pure Competition 11 
References 
[ 1 J Henderson， ].M. & R. E. Quandt; Microeconomic Theory， 2ed. Mcgraw-hill 
Kogakusha， 1971 
[ll] Hicks，]. R.; Value and Capital， 2ed. Maruzen， 1957 
l:rn= Kogiku， K. C.; Microeconomic Models， Harper & Row， 1971. 
[N] Leon Walras; Elements of Pure Economics， translated by William Taffe， 
George Allen & Unwin， 1965. 
[V] Samuelson， P. A.; Foundation of Economic Analysis， Atheneum， 1970. 
[V[ Yamane， T.; Mathematics for Economists， 2ed. Prentic-hall， 1968. 


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