Economic growth and economic development 244

Introduction to Modern Economic Growth
Theorem 5.4. (Representative Firm Theorem) Consider a competitive
production economy with N ∈ N∪ {+∞} commodities and a countable set F of

firms, each with a convex production possibilities set Y f ⊂ RN . Let p ∈ RN
+ be the

price vector in this economy and denote the set of profit maximizing net supplies of
firm f ∈ F by Yˆ f (p) ⊂ Y f (so that for any yˆf ∈ Yˆ f (p), we have p · yˆf ≥ p · y f for
all y f ∈ Y f ). Then there exists a representative firm with production possibilities
set Y ⊂ RN and set of profit maximizing net supplies Yˆ (p) such that
ˆ ∈ Yˆ (p) if and only if yˆ (p) =
for any p ∈ RN
+, y

X

f ∈F

yˆf for some yˆf ∈ Yˆ f (p) for each f ∈ F.

Proof. Let Y be defined as follows:
(
)
X
Y =
y f : y f ∈ Y f for each f ∈ F .
f ∈F

P
ˆ = f ∈F yˆf for

To prove the “if” part of the theorem, fix p ∈ RN
+ and construct y
/ Yˆ (p),
some yˆf ∈ Yˆ f (p) for each f ∈ F. Suppose, to obtain a contradiction, that yˆ ∈

so that there exists y 0 such that p · y 0 > p · yˆ. By definition of the set Y , this implies
â ê
that there exists y f f ∈F with y f ∈ Y f such that
p·

Ã
X
f ∈F

X

f ∈F

y

f

!

> p·

p · yf >

Ã
X

X

f ∈F

f

yˆ

f ∈F

!

p · yˆf ,

so that there exists at least one f 0 ∈ F such that
0

0

p · y f > p · yˆf ,
which contradicts the hypothesis that yˆf ∈ Yˆ f (p) for each f ∈ F and completes

this part of the proof.

To prove the “only if” part of the theorem, let yˆ ∈ Yˆ (p) be a profit maximizing
choice for the representative firm. Then, since Yˆ (p) ⊂ Y , we have that
yˆ =

X

f ∈F

230

yf