研究ノート

A Note on Aggregate Production Eﬃciency
under Increasing Returns
Tadashi HAMANO
Abstract
In this note we consider an economy with one output and many inputs whereproduction technologies are expressed by production functions and show that, if
each production function is superadditive, it is necessary and suﬃcient formonopoly to achieve aggregate production eﬃciency that the function of themaximum value of individual production functions at each input is superadditive.

1 Introduction
It is well known that in an economy with convex technologies, proﬁt-maximizing behavior ofﬁrms leads to both individual and aggregate production eﬃciency1）. In other words, given
prices, if each ﬁrm operates at a proﬁt-maximizing production plan, it is on the individual
production frontier ; and the aggregate production plan is on the frontier of the aggregate
production set.
By contrast, under non-convex or increasing returns to scale technologies, aggregate
production eﬃciency cannot be achieved through individual maximizing behavior2）. Under
these technologies monopoly production may be required for aggregate production
eﬃciency. This is explained by the fact that increasing returns yield scale merits ofproduction and favor monopoly over decentralized production where two or more ﬁrms
（technologies）are simultaneously active. That is, under these technologies monopoly may
produce more output than two or more ﬁrms combined can do.
Hamano（1996）considers an economy where there are many ﬁrms with increasing
returns to scale technologies and attempts to derive suﬃcient conditions ensuring thatmonopoly achieves aggregate production eﬃciency. More specially, Hamano （1996）
examines an economy with one output and many inputs where production technologies are169

A Note on Aggregate Production Eﬃciency under Increasing Returns

expressed by production functions, and obtains such suﬃcient conditions which can beinterpreted as non-decreasing “generalized” average productivity of inputs for each ﬁrm.
In this note, we examine an economy with increasing returns technologies and derive anecessary and suﬃcient condition for monopoly to achieve aggregate production eﬃciency.
More precisely we consider an economy with one output and many inputs where productiontechnologies are expressed by production functions which is the same setting as in Hamano（1996）. We show that, if each production function is superadditive, it is necessary andsuﬃcient for monopoly to achieve aggregate production eﬃciency that the function ofmaximum value of individual production functions at each input is superadditive3）.
The organization of the paper is as follows : Section 2 presents a basic framework and aconcept of an aggregate production function. In Section 3 we derive a necessary andsuﬃcient condition that monopoly achieves production eﬃciency in an economy whereproduction technologies are expressed by production sets.

2 The ModelLet us consider an economy with one output, m inputs and H ﬁrms where H ≥2. Thetechnology of the h -th ﬁrm（h=1, 2, ⋯, H ) is represented by a production function f deﬁned from R to R . That is, given a vector of inputs z= ( z, z, ⋯, z ) ∈R, f  ( z ) is a
maximum amount of output. For each h, a production function f  is assumed to be nondecreasing and to satisfy the condition f  (0) =0.
We recall the deﬁnition of a superadditive function.
Deﬁnition 1 A function f : RR  is called superadditive if, for all z and z′,f ( z+z′) ≥ f ( z ) +f ( z′) .
We next present the deﬁnition of the aggregate production function at each inputz∈R, given individual production functions.
Deﬁnition 2

Given individual production function f  (h=1, 2, ⋯, H ) , the aggregate

production function F : RR  is deﬁned asF (z) ≡max∑ f ( z).

       （1）We also deﬁne a function of the maximum value of the individual output at each inputz∈R.
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第 295 号Deﬁnition 3 Given individual production function f  (h=1, 2, ⋯, H ) , the function of themaximum value of the individual output, G : RR , is deﬁned as
G (z) ≡max f  ( z ) .

   （2）Note that, if G ( z )=F ( z ) for all z∈R, then the monopoly achieves aggregate
production eﬃciency.

3 The Result
Our result is that the monopoly achieves aggregate production eﬃciency if and only if thefunction G is superadditive.
Theorem 1 Suppose that f  is superadditive for all h. Then, the function F ( z ) is equal toG ( z ) for all z∈R if and only if G is superadditive.
Proof.
Let us ﬁrst show that, if F ( z )=G ( z ) for all z∈R, then the function G ( z ) issuperadditive. Fix z′, z″∈R. It suﬃces to show thatG ( z′+z″) ≥ G ( z′) +G ( z″) .

（3）

We now pick up indices of ﬁrms h′, h″ associated with z′, z″, respectively, as follows :G ( z′) = f  ( z′) ,G ( z″) = f



（4）

( z″) .

（5）
When we choose { z  }satisfying z′+z″=∑  z , it follows from the equality of F ( z ) withG ( z ) thatG ( z′+z″) = F ( z′+z″) ≥ ∑ f  ( z  ) .

（6）F ( z′+z″) ≥ f  ( z′) +f  ( z″) .

（7）

If h′≠h″, then we have

Moreover, if h′=h″, then it follows from superadditivity of f  thatF ( z′+z″) ≥ f  ( z′+z″) ≥ f  ( z′) +f  ( z″) = f  ( z′) +f  ( z″) .

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A Note on Aggregate Production Eﬃciency under Increasing Returns

In either case, we haveG ( z′+z″) ≥ f  ( z′) +f  ( z″) .

（9）

Therefore, it follows from（4）,（5）and（6）that
G ( z′+z″) ≥ G ( z′) +G ( z″) ,（10）

which is to be shown.
Conversely, suppose that G ( z ) is superadditive. We need to show that F ( z ) =G ( z ) forall z∈R. Note from the deﬁnition of F that F ( z ) ≥G ( z ) . Suppose on the contrary thatz )>G ( z ). If we set F ( z ) =∑  f  ( z  ) where z =∑  z ,there exists z ∈R such that F ( then it follows from superadditivity of G that

∑ f (z ) = F ( z ) > G (z ) ≥ ∑ G (z ).

（11）Now, by the deﬁnition of G, we have the relationship
G (z ) ≥ f ( z )

for all h,（12）and, therefore,∑ G (z ) ≥ ∑ f ( z ).

（13）Thus, we combine inequalities（11）and（13）to yield the relationship∑ f (z ) > ∑ f ( z ),

which is a contradiction.

（14）
Q.E.D.

Notes
1 ）See Mas-Colell et al.（1995, pp. 147-149）for example.
2 ）See Hamano（2011）for details.
3 ）Without superadditivity of individual production functions, Ginsberg（1974）considered a nicely
convex-concave production function ; and derived a condition on aggregate production
eﬃciency in a case of one input and one output.

Acknowledgement
I am indebted to Yoshiaki Ushio for helpful comments. As usual all remaining errors are mine.

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ReferencesGinsberg, W.（1974）“The multiplant ﬁrm with increasing returns to scale.” Journal of Economic
Theory 9 : 283-292.
Hamano, T.（1996）“Increasing returns and aggregate production eﬃciency by a monopoly.”
Journal of Economics（Zeitschrift für Nationalökonomie）64 : 155-161.
Hamano, T.（2011）“Generalized Cobb-Douglas Production Functions and Aggregate Production
Eﬃciency by a Monopoly.” Tokyo Keizai Gakkaishi（東京経済学会誌―経済学―）269 : 155-164.
Mas-Colell, A., Whinston, M. D., Green, J. R.（1995）Microeconomic Theory. Oxford University Press,New York.

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