The present paper studies resource allocation in an expanding product variety framework à la Romer (1990). We consider the limiting case where net returns to specialization are zero and, therefore, R&D is completely useless, socially. Nevertheless, the market equilibrium involves allocation of ressources to R&D, and this wasteful allocation may take place even in a steady state.Research and development (R&D); Expanding product variety; Creative destruction.

Christian Groth

María J. Alvarez-Peláez

English

Research Papers in Economics

E2003/42
Profitable Unproductive Innovations
María J. Alvarez-Peláez
Christian GrothbcentrA:  
Fundación Centro de Estudios Andaluces 
 
Documento de Trabajo 
Serie Economía E2003/42 
 
Profitable Unproductive Innovations
* 
 
María J. Alvarez-Peláez
a  
Universidad de Málaga y CentrA 
Christian Groth
b
University of Copenhagen
 
 
RESUMEN 
Este trabajo estudia la distribución de recursos en un modelo de crecimiento 
endógeno basado en la expansión de la variedad de producto a la Romer 
(1990). Consideramos el caso extremo en el que los rendimientos netos de 
la especialización son nulos y por tanto I+D es completamente inútil desde 
un punto de vista social. Sin embargo, la economía de mercado en equilibrio 
destina recursos a I+D y ese desperdicio de recursos puede darse incluso 
en un estado estacionario. 
 
Palabras clave: Investigación y desarrollo (I+D); Expansión de la variedad 
de producto; Destrucción creativa. 
 
 
 
ABSTRACT 
The present paper studies resource allocation in an expanding product 
variety framework à la Romer (1990). We consider the limiting case where 
net returns to specialization are zero and, therefore, R&D is completely 
useless, socially. Nevertheless, the market equilibrium involves allocation of 
ressources to R&D, and this wasteful allocation may take place even in a 
steady state. 
 
Keywords: Research and development (R&D); Expanding product variety; Creative 
destruction. 
 
JEL classification: O33, O41 
 
 
 
 
                                                 
* The authors are grateful for valuable comments from Christian Schultz, José-Víctor Ríos-Rull and 
participants at the II Jornadas Béticas de Macroeconomía Dinámica, Sevilla 2003. María J. Alvarez-
Peláez thanks the Institute of Economics, University of Copenhagen, for the kind hospitality while 
working on this project. 
a Dpto. Teoría e Historia Económica, Universidad de Málaga, El Ejido s/n, 29702 Málaga, Spain.                     
Email: alvarezp@uma.es. 
b Institute of Economics, University of Copenhagen, Studiestraede 6, DK-1455, Copenhagen K, 
Denmark. Email: Chr.Groth@econ.ku.dk.  Proﬁtable Unproductive Innovations∗
Maria J. Alvarez-Pelaez† Christian Groth‡
Universidad de Malaga University of Copenhagen
CentrA
May 2003
Abstract
The present paper studies resource allocation in an expanding product
variety framework à la Romer (1990). We consider the limiting case where net
returns to specialization are zero and, therefore, R&D is completely useless,
socially. Nevertheless, the market equilibrium involves allocation of resources
to R&D, and this wasteful allocation may take place even in a steady state.
JEL Classiﬁcation: O33, O41
Key words: R&D; Expanding product variety; Creative destruction.
∗The authors are grateful for valuable comments from Christian Schultz, José-Víctor
Ríos-Rull and participants at the II Jornadas Béticas de Macroeconomía Dinámica, Sevilla
2003. María J. Alvarez-Peláez thanks the Institute of Economics, University of Copen-
hagen, for the kind hospitality while working on this project.
†Dpto. Teoría e Historia Económica, Universidad de Málaga, El Ejido s/n, 29702 Málaga,
Spain. Email: alvarezp@uma.es
‡Institute of Economics, University of Copenhagen, Studiestraede 6, DK-1455, Copenhagen K,
Denmark. Email: Chr.Groth@econ.ku.dk. Tel: (+45) 35323928.
11 Introduction
The paper studies resource allocation in an expanding product variety framework à la
Romer (1990), but with the parameters for returns to specialization, market power,
and share of capital, respectively, disentangled from each other. We examine the
limiting case where, at the aggregate level, indirect negative eﬀects of specialization
completely oﬀset the immediate private beneﬁts of innovation so that returns to
specialization are zero.
Let manufacturing output be
Y = A
ηX
αNY
1−α,
where A indicates the level of technical knowledge in society, NY is labor input, and
X is a CES aggregate of quantities, xi, of specialized capital goods with elasticity
of substitution equal to 1/(1 − ε). The parameter η captures ”net returns to spe-
cialization”, i.e., the degree to which society beneﬁts from specializing production
in an increasing number of branches. The original Romer (1990) article had implic-
itly the three parameters linked by η =1− α and α = ε.1 As a result, the model
had the particular feature that the amount of R&D is always insuﬃcient compared
to the social optimum. In later articles Benassy (1998), Groot and Nahuis (1998),
and Alvarez and Groth (2002) showed that disentangling one or both of Romer’s
parameter links could lead to too much R&D in the market economy. But none
of these papers considered the possibility that η =0 , that is, the case where, at
the aggregate level, indirect, negative eﬀects of specialization completely oﬀset the
immediate positive eﬀects of an enlarged spectrum of input varieties. We are going
to show that in spite of R&D being, in this case, completely useless from a social
point of view, there may exist a market equilibrium involving allocation of resources
to R&D, indeed this is so even in a steady state.
2 The model
The economy is populated by a constant number, L, of households each of which
supplies one unit of labor inelastically and has the utility function
U0 =
Z ∞
0
e
−ρtc1−θ − 1
1 − θ
dt, ρ > 0,θ>0, (2.1)
1That is, Y =(
R A
0 xi
αdi)N
1−α
Y , ignoring that Romer considered two kinds of labor.
2where c = c(t) is consumption at time t.2
There are two production sectors, here called the basic-goods sector and the
specialized capital-goods sector. In the basic-goods sector the production function
is as above, though with η =0:
Y = X
αNY
1−α, 0 <α<1, (2.2)
There is a continuum of specialized capital goods, measured on the interval [0,A],
where A indicates the stock of engineering principles that grows through research.
The composite factor X is given by
X = A(
1
A
Z A
0
xi
εdi)
1
ε, 0 <ε<1. (2.3)
The output of basic goods is used for consumption, C ≡ cL, and investment in
”raw” capital. The stock of raw capital K changes according to
˙ K = Y − C − δK, δ ≥ 0,K (0) = K0 > 0.( 2 . 4 )
In the specialized capital-goods sector, which is also the innovative sector, a unit of
raw capital can immediately be transformed to a specialized capital good on the
basis of a given technical design. The number of new designs created per time unit
is assumed proportional to the existing stock of knowledge, as in Romer (1990),
˙ A = γNAA, γ > 0,A (0) = A0 > 0, (2.5)
where γ is a productivity parameter, and NA is aggregate research work.
Because of the strict concavity of X in xi and the symmetric cost structure,
static eﬃciency requires xi = x for all i ∈ [0,A].3 Hence, assuming static eﬃciency,
X = Ax from (2.3), and when demand for raw capital equals supply we have
X = Ax = K. (2.6)
Inserting into (2.2) gives output of basic goods as
Y = K
αNY
1−α. (2.7)
Af e a s i b l ep a t h(K,A,C,Y,NY,N A)∞
t=0 is called a steady state if K, C, and Y
are strictly positive and grow at constant (though not necessarily equal or positive)
2In case θ =1 , the expression c1−θ−1
1−θ should be interpreted as lnc.
3Therefore, obsolescence of old capital goods never occurs.
3rates. Let the rate of growth of a strictly positive variable x be denoted gx, i.e.,
gx ≡ ˙ x/x. Let u be the fraction of total labor supply employed in the basic-goods
sector, i.e., with full employment, NY ≡ uL, 0 ≤ u ≤ 1.B y( 2 . 5 ) ,gA ≥ 0 always.
Lemma 1 (i) In a steady state gc = gY = gK =0 . (ii) Further, gA is constant and
satisﬁes 0 ≤ gA = γ(1 − u)L<γ L .
Proof. (i) Consider a steady state. By deﬁnition of a steady state, Y> 0;
hence, from (2.7), NY ≡ uL > 0 and
gY = αgK +( 1− α)gu. (2.8)
Therefore, since in a steady state, gY and gK are constant, gu is also constant,
implying gu =0in view of 0 <u≤ 1 for all t ≥ 0. Now (2.8) gives gY = gK =0 .
Then, by (2.4), cL/K is constant, hence gc = gK =0 . (ii) From (2.5), gA = γ(1−u)L
is constant, since u is constant. Further, since 0 <u≤ 1, we have 0 ≤ gA <γ L .
Now, we will embed this economic system in a market economy. Apart from
allowing ε 6= α and η =0 , the set-up is similar to Romer (1990).
The representative ﬁrm in the basic goods sector rents labor at the wage w and
specialized capital goods at the rental rate Ri,i∈ [0,A]. Using basic goods as our
numeraire, proﬁt maximization under perfect competition yields
∂Y
∂NY
=( 1− α)
Y
NY
= w, (2.9)
∂Y
∂xi
= α
Y
X
∂X
∂xi
= Ri. (2.10)
From (2.10) we can express the demand for the specialized capital good i conditional
on a given X as:
xi =
X
A
(
Ri
R
)
− 1
1−ε,i =1 ,2,...,A, (2.11)
where R =( A−1 R A
0 R
ε
ε−1
i )
ε−1
ε is the minimum cost per unit of X.
The supply of capital good i is decided by the ﬁrm that invented its design. The
ﬁrm gets compensated for the sunk research cost through retention of monopoly
power which is supported by patents of inﬁnite duration4.G i v e nd e s i g ni, to deliver
x(i) units of capital good i, it takes x(i) units of raw capital. At each instant of
time, ﬁrm i, facing the demand curve (2.11) and taking X and R as given, sets the
4At initial time, t =0 , the number of ﬁrms, A(0), is large enough so that each ﬁrm’s action is
negligible in the aggregate economy.
4rental rate Ri so that current proﬁt πi ≡ Rixi − (r + δ)xi is maximized, i.e.,
Ri =
1
ε
(r + δ), (2.12)
where r i st h er e a lr a t eo fi n t e r e s t .
Since, by (2.12), all ﬁrms in the specialized capital-goods sector set the same
rental price, they supply the same quantity, x, and they earn the same proﬁt
π =(
1
ε
− 1)(r + δ)x. (2.13)
The equilibrium value, p, of a patent satisﬁes the no-arbitrage condition
π + ˙ p
p
= r. (2.14)
There is free entry to research activity. Research is done by new ﬁrms wanting
to enter the specialized capital-goods sector. Proﬁt maximization subject to (2.5)
entails, in equilibrium,
w = pγA, with = if NA > 0. (2.15)
By increasing A, research has a positive external eﬀect on the productivity of future
research5. But since aggregate returns to specialization are zero, research does
not contribute to total factor productivity in manufacturing (in contrast to the
endogenous growth literature).
Households consume and save, and savings can be either in capital or in shares
of the monopoly ﬁrms. Financial wealth of the representative household is v ≡
(K + pA)/L. The household makes a plan (c)∞
t=0 to maximize U0 subject to ˙ v =
w + rv − c, v(0) = v0, and the standard no-Ponzi-game condition. Necessary and
suﬃcient conditions for a solution are that the Keynes-Ramsey rule,
gc =
1
θ
(r − ρ), (2.16)
and the transversality condition, limτ→∞ ve−
R τ
t rds =0 , hold for all t ≥ 0.
Given the clearing conditions, K = X = xA, L = NA + NY, and the deﬁnitions
k ≡ K/NY and u ≡ NY/L we have, from (2.9), (2.10), and (2.12),
w =( 1− α)
Y
uL
=( 1− α)k
α, (2.17)
1
ε
(r + δ)=
∂Y
∂K
= α
Y
K
= αk
α−1. (2.18)
5Each research ﬁrm is small and therefore perceives, correctly, its contribution to aggregate ˙ A
to be practically negligible.
5If u<1, i.e., NA > 0,t h e nw also equals the value of the marginal product of
labor in research so that (2.15) reduces to w = pγA. This together with (2.17) gives
the market value of a patent as
p =
1 − α
γA
k
α. (2.19)
An interior equilibrium is an equilibrium such that for all t ≥ 0, 0 <u<1
(there is positive employment in both sectors). In an interior equilibrium, by (2.19),
gp = αgk − gA. Inserting this together with (2.13), (2.18), and (2.19) into (2.14),
using the fact that γuL = γL− gA, from (2.5), gives the market interest rate as
r =
1 − ε
1 − α
α(γL− gA)+αgk − gA. (2.20)
3T h e m a i n r e s u l t
By Lemma 1, in a steady state, gc =0 , implying r = ρ, from (2.16); and since also
gk =0in a steady state, inserting into (2.20) gives
gA =
(1 − ε)αγL − (1 − α)ρ
1 − εα
> 0, (3.1)
presupposing the parameter restriction
ρ<
1
1 − α
(1 − ε)αγL. (A1)
This restriction comes from the interiority condition u<1. If A1 is violated, then
impatience is so large that R&D activity cannot be supported in a steady state
equilibrium. In that case the steady state solution of the model is like that of a
one-sector model without technical change, and gA =0 .
Deﬁning uniqueness of a steady state to be present if, given the parameters
(α,ε,θ,ρ,δ,γ,L), there exists only one pair (gc,Y/K) which is consistent with the
steady state conditions, we have:
Proposition 1 (i) There exists a unique steady state in the market economy; it has
gc =0 , and if A1 holds, then
NA =
(1 − ε)αγL − (1 − α)ρ
(1 − εα)γ
> 0, (3.2)
and gA = γNA > 0, while NA = gA =0otherwise. In any case Y/K =( ρ+δ)/(εα).
(ii) In contrast, the unique steady state of the social optimum has N∗
A = g∗
A = g∗
c =0 ,
and Y/K∗ =( ρ + δ)/α.
6Proof. (i) Given A1, (3.2) follows from (2.5) and (3.1). With r = ρ, (2.16) and
(2.18) give Y/K =( ρ+δ)/(εα), whether or not A1 holds. (ii) Since increasing A is
a waste of resources, the social planner’s problem reduces to a standard one-sector
Ramsey problem with no technical progress. Hence, in steady state, N∗
A = g∗
A =
g∗
c =0 , and, from the Keynes-Ramsey rule, g∗
c =( αY/K∗ − δ − ρ)/θ =0 , we get
Y/K∗ =( ρ + δ)/α.
Though from a social point of view resources applied to R&D are wasted, the
ﬁrms bearing the research costs are able to recover them by taking advantage of
their market power (granted by the patent legislation). As soon as a new specialized
capital good is invented and supplied to the market, there is a demand for it due to
the symmetry and the strict concavity of Y in xi,i∈ [0,A]. This demand reﬂects the
productive eﬀect of redistribution of a given amount of raw capital to an enlarged
spectrum of varieties − the ”direct eﬀect” of increased specialization. But at the
aggregate level this eﬀect is completely outweighed by a concomitant decrease in
the productivity of the old capital goods, since we have η =0 , i.e., net returns to
specialization are zero. This ”indirect eﬀect” of increased specialization (a negative
externality) may be interpreted as a result of higher complexity in a more specialized
world6.
As to the comparative statics of (3.2) we have
∂NA
∂ε
= −
α(1 − α)(γL+ ρ)
γ(1 − εα)2 < 0,
∂NA
∂α
=
(1 − ε)(γL+ ρ)
γ(1 − εα)2 > 0.
From this together with Proposition 1 it follows that both Y/Kand NA in the market
economy depend on the substitution parameter ε while the allocation in the social
optimum does not. When specialized capital-goods are close substitutes (ε high),
the markup over marginal cost in the specialized capital-goods sector becomes low,
making inventions of new designs less proﬁtable, thereby reducing R&D. Similarly,
the capital share parameter α aﬀects resource allocation in the market economy,
but not in the social optimum. Indeed, the higher is α, the lower is the wage share,
1 − α, which, given the factor prices, implies less room for proﬁtable employment
6The decomposition of the eﬀects of specialization into direct and indirect eﬀects is studied,
in a more general context, in Alvarez and Groth (2003). The indirect eﬀect can be seen as a
reminiscence of the ”creative destruction” eﬀect in the quality ladder models, cf. Aghion and
Howitt (1998).
7in the basic-goods sector7.T h e r e b y m o r e o f t h e ﬁx e dl a b o rf o r c ei sa v a i l a b l ef o r
employment in research.
W h e t h e ro rn o tNA > 0, the steady state of the market economy has a ”too high”
output/capital ratio. This is because monopoly pricing implies a wedge between the
price of the services of specialized capital goods and the marginal cost of providing
them.8 Correction of the market failures may be accomplished by a subsidy σ =
1 − ε to buyers of capital services and a tax τ =1− ε
1−ε
1−α
α
ρ
γL on monopoly prof-
its, supplemented by a lump-sum tax (or transfer) to cover the possible shortage
(surplus) of public revenue.
4 Final remarks
Since our result emerges without ambiguity in a straightforward way it may well hold
in a more general setting as to technologies in the two sectors, population growth,
slightly negative net returns to specialization, etc.
References
Aghion, P., and P. Howitt, 1998. Endogenous Growth Theory. MIT Press, Cambridge
(Mass.).
Alvarez-Pelaez, M. J., and C. Groth, 2003. Too little or too much R&D? European
Economic Review, Forthcoming.
Benassy, J. P., 1998. Is there always too little research in endogenous growth with
expanding product variety? European Economic Review 42 (1), 61-69.
Groot, H. L. F. de, and R. Nahuis, 1998. Taste for diversity and the optimality of
economic growth. Economics Letters 58, 291-295.
Romer, P.M., 1990. Endogenous technical change. Journal of Political Economy 98
(Suppl.), 71-102.
7(2.12), (2.17), and (2.18) give uL = 1−α
α K R
w.
8Notice, that, given A1, along the steady state path of the market economy, variety expands
at the rate gA > 0, while market size, x, and proﬁts, π, of each single monopolist contracts at the
same rate (in view of (2.13) where x = K/A).
8

Endogenous Growth Theory.

Endogenous technical change.

Is there always too little research in endogenous growth with expanding product variety?

Taste for diversity and the optimality of economic growth.

Profitable Unproductive Innovations

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Profitable Unproductive Innovations

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