endogenous product cycles economic journal 1991
TRANSCRIPT
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The Economic Journal, ioi {September
1 9 9 1 ) , 1 2 1 4 - 1 2 2 9
Printed in Great Britain
ENDOGENOUS PRODUCT CYCLES
Gene
M.
Grossman and Elhanan tJelpman
The product cycle features prominently
in
trade between
the
Northern
developed economies and the Sou thern newly indu striahsing econom ies. In a
much-celebrated article, Vernon (1966) described
the
'life cy cle'
of a
typical
manufactured product. Invention and initial ma nufacturing of new produc ts
occur
in
the North,
he
argued, because R& D capabilities
are
well developed
there,
and
because proximity
to
large, high-incom e m arkets facilitates
innovation. After
a
while, production method s may become more standard ised.
Then, technology transfer
or
imitation
by
Sou thern firms takes place,
whereupon the bulk of production migrates to the low-wage South.
Interregional trade
in
ma nufac tured goods involves exch ange
of the
latest,
innovative goods, produced only
in the
Nor th,
for
older, mo re established
goods, produced predominantly or entirely in the South.
The first attempt
at
formal mo delling of this pheno me non was carried out by
Krugman (1979). He posited an exogenous rate, g our notation) , of
introduction
of
new products
in the
North,
and an
exogenous rate,
/*, of
technology transfer
to the
South.
By
hypothesis, then,
the
totai n um ber
of
products known
to the
world evolves ac cord ing
to
ri/n = g , while
the
number
of products that
the
South
is
able
to
prod uce evolves accord ing
to n^
= /m^
where n^, is the n um ber of products in which the No rth tem porarily ma intains
exclusive productive capacity. These exogenous processes ensure the existence
of a steady state in which the share of the North in the total num ber of
products, 0 ,^
=
«,v/«, is ec|ual to g/ g + /i)- Adding some economic structure
to
the model, Krugman finds
a
positive relationship between
the
relative wage
paid to Northern labour w^./Wf^) and g/fJ- and an inverse relationship betwee
the relative wage
and the
relative size
of
the No rthern labo ur force.
Krugman's work has been extended by Do llar (1986) and Jensen and
Thursby (1986, 1987). Dollar maintains Krugman's assumption
of an
exogenous rate of product innovation, but relates the rate of technology
transfer
to
the Nort h-S ou th terms of trade, albeit
in an
entirely ad hoc manner .
Jensen
and
Th ursby (1986) attem pt
to
capture
the
resource costs
of
product
development and technology transfer, and the decision processes that determine
these expenditures,
but
they assume tha t
all
innovation
is
carried
out by a
single entrepreneur in the North, and that the allocation of resources to reverse
engineering
in
the S outh
is
made
by a
social plan ner. T he ir later (1987) p ap er
does allow
for a
fixed number (perhaps greater than one)
of
innovators
in the
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[ S E P T . I99 1] ENDOGENOUS PRODUCT CYCLES I2 15
imitation. Moreover, their analysis in both papers is partial equilibrium in
nature, inasmuch as they take the interest rate as given.^
In this paper we build upon our earlier work on product development and
international trade (1989^, 1990) to construct a model ofthe product cycle
featuring endogenous innovation and endogenous technology transfer. In our
model, competitive entrepreneurs in the North expend resources to bring out
new products whenever the expected present discounted value of subsequent
oligopoly profits exceeds current product development costs. Each Northern
oligopolist continuously faces the risk that its product will be copied by a
Southern imitator, at which time its profit stream comes to an end. Thus the
length o fthe initial ph ase in the life cycle of each p rod uct (i.e. wh en pro du ction
occurs in the North) is rand om . In the South, entrep reneu rs m ay devote
resources to learning the production processes that have been developed in the
North. There too, costs (of reverse engineering) must be covered by a
subsequent .stream of operating profits. In all this, interest rates are determined
endogenously so as to equate saving.s and investment.
Our approach enables us to study the determinants of the long-run rate of
grow th ofth e w orld economy an d the long-run rate of technological diffusion.
We find steady-state values for
g
and
/i
and relate these to underlying
structural characteristics of the world economy (the sizes of the two trading
blocs,
the productivities of resources in their various uses, and the nature of
dem and for the differentiated ma nufactured goods), and to the comm ercial
and industrial policies enacted by the two governments. Also, we provide an
analysis o fth e effects of exogenous events and of public policy on re lative w age
rates in the two regions, and find that Krugman s (1979) results derived for the
case of ^ and /^ exogenous m ay in fact be m isleading. For e xam ple, when we
allow for the changes in the steady-state rates of innovation and imitation that
are induced by variations in the two labour forces, we find that the direction
of movem ent in relative w ages in th e .steady state is exactly the opposite of th at
predicted by Krugman.
The remainder ofthis paper is organised as follows. We develop our model
ofthe product cycle in the next section. In Section II we solve for the steady-
state equilibrium and discuss its dependence on structural features ofthe world
economy. Section III contains policy analysis. We analyse the long-run effects
of subsidies to innovation in the North, of subsidies to reverse engineering (or
learning) in the South, and of trade policies in both regions. The concluding
section provides a summary of the findings.
I. A M O D E L O F T H E P R O D U C T CY C L E
We study a world economy com prising two regions, denoted by N o rt h and
South . The regions differ only in their abilities to innovate. The North enjoys
absolute (and comparative) advantage in developing new products and
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I 2 l 6 THE ECONOMIC JOURN AL [SEPTEMB ER
bringing them to ma rket. Indeed , we shall assume that So uthern productivity
in pro du ct innovation is sufficiently low tha t the Sou th performs none of this
activity in a trade equilibrium.
W e consider a world of symm etrically d ifferentiated prod ucts. Th ere exists
a continuum of potential goods that are desirable to consumers, but only a
subset of th es e (of finite mea sure) a re produ ced at any one time. Before an y
pro du ct can be ma nufactu red and sold to consum ers, resources must be devoted
to 'dev elo pin g' the prod uct ; that is, the good must be designed, the p roduction
techniques perfected, and so on.
Households worldwide share identical preferences for the differentiated
products. Each household seeks to maximise the time-separable intertemporal
utility function
r 7 ) ] r f r , (I)
w h e r e p is t h e s u b j e c t i v e d i s c o u n t r a t e a n d u{-) is t h e i n s t a n t a n e o u s s u b - u t i l i t y
f u n c t i o n g i v e n b y
r
fn l l / a
U {
=
x{(o)
da)\ ,
o<oc<i .
( 2 )
Uo J
In (2), .v(w) denotes cons um ption of differentiated pro du ct w, and n (a function
of
T)
is the number (measure) of varieties available on the market.
The representative consumer maximises (i) subject to an intertemporal
budget constraint
r
3
where R{t) is the cumulative interest factor from time o to time / that the
consumer faces on the local capital market; £(7) and Y{T) are his spending and
factor income at time T, respectively; and A{1) represents the value of his asset
holdings at t. Our results concerning the steady state do not hinge on whether
capital is traded internationally or not; but for ease of exposition we shall
assume that all agents face the same interest rate.
As is well known (see, for example, Grossman and Helpman (19896)), the
solution to the intertemporal maximisation problem requires
R-p, (4)
while (2) generates an instantaneous demand for variety o) given by
^^ ^ ^ £ , (5)
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EN DO GE NO US PR OD UC T CYCLES 2 7
random. But all risks are idiosyncratic, so tbat households can earn a safe
retu rn by holding well-diversified portfolios of equities. Th en arb itrag e ensures
that, in equilibrium, the return to such a portfolio of Northern firms equals the
riskless interest rate, which in turn equals tbe (certain) return on a share of any
Southern firm.
The production sector comprises two distinct activities. Before a firm can
begin to manufacture any variety, it must learn the production technique
specific to that variety . If the pro duc t is a new one (i.e. not previously ava ilable
in the marketplace), then this learning represents
innovation.
If, in.stead, the
product already exists on the market, then the learning represents
imitation.
In
either case, tbe learning activity requires an expenditure of resources by the
entrepreneur (presumably more for innovation than for imitation), with
productivity parameters that vary by region. After a production technique has
been mastered, the firm can manufacture the chosen product according to a
constant-returns-to-sca e production function.
We suppose that the regions are endowed with a single primary input, which
we call labour. Consider first the manufacturing activity. Production of any
variety of consumer good in either country requires
a^
units oflabo ur per unit
of output. Hence the marginal cost of any good produced in country
i
is
w^a^,
where f is the wage there, for i = S (South) and A (N orth) . At any pa rticu lar
moment, the set of available products and the number of firms in either
location able to produce every product is given by history. The producers
behave as Bertrand competitors, taking the prices of other firms' products and
the level of aggregate spending as fixed. They maximise profits by setting
marginal revenue equal to marginal cost.
A Northern firm with the unique ability to produce some variety faces,
according to (5) and our assumptions about market structure, a demand curve
with constant elasticity equal to
e
Such a firm maximises profits by setting
a price
p^
that is a fixed mark-up over marginal cost, or
P N =
"^^flx/a- (6)
The producer earns instantaneous profits given by
r •
= {i-cc PsXs^
(7)
where
x^^
is the equilibrium output level, calculated from (5). If two Northern
firms happ ene d to be capab le of produ cing a variety in comm on (e.g. if one h ad
imitated the innovation ofthe other), then as Bertrand competitors they would
each set a price equal to marginal cost and earn zero profits. It follows that no
Northern entrepreneur can recoup the fixed costs of imitation, so none of this
activity takes place in the North.
The equilibrium wage rate in the South is assumed to be less than that ofthe
North.^ We rule out the possibility that a Southern firm will have exclusive
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I2l8
THE
ECONOMIC JOURNAL [SEPTEMBER
preneurs. This implies that all innovation occurs in the North, and that
knowledge acqui.sition
in the
South
is
confined
to
imitation.
If two
Southern
firms have copied
the
same variety
of
consumer good, these
two
will
set
prices
equal
to
their marginal costs
and
earn zero profits.
So the
second
of the
imitators could never justify bearing
the
cost
of
reverse engineering. Therefore,
the only market structure that
we
need
to
consider
for
Southern producers
is
one where
a
single firm
(an
imitator) competes with
a
single Northern firm
(the
innovator)
in the
market
for
some variety.
Two outcomes
may
result from this competition, depending
on the
size ofthe
gap between Northern
and
Southern wages.
If the gap is
large,
the
Southern
firm can charge
its
monopoly price without regard
for the
competition from
the
Northern innovator. This price for Southern products prevails whenever
Wf^a^/aL
(the
monopoly price) falls short
of the
marginal cost
of
Northern
production, w^aj^,
or
when w_^
^ aw^. We
shall refer
to
this
as the
wide-gap cas
If relative wages
in the
South
are
somewhat higher,
a
Southern firm charging
its monopoly price would
be
undercut
by its
Northern rival.
In
this narrow-gap
case^
the
Southern firm prices just below
the
marginal cost
of the
Northern
producer, thereby capturing
the
entire market. Thus,
we
have
.s = Wgd^/a if Wg ^ aw; , {8a
Ps
~ ^N^x if
Ws
^
ccw^.
(8i)
The instantaneous profits ofa Southern firm are
where
x^
represents
the
firm s sales (calculated from
(5)).
Notice that,
in
either
case,, the
Southern firm uses
its
cost advantage
to
capture
the
entire
sub-
market. Northern firms make
no
further sales once their varieties have been
copied abroad. This feature ofthe model captures
the
migration
of
production
from North to South, as first described by Vernon (1966}.
We turn next
to the
learning activities.
As in
Romer (1990)
and
Grossman
and Helpman (1990, 1991),
we
assume that
the
resources devoted
to
industrial
research generate
two
sorts of outputs. First, when
an
entrepreneur hires labour
for purposes of innovation or imitation, he derives an appropriable output in
the form ofa blueprint for producing a particular variety. This blueprint is
the entry ticket into
the
product market,
and
carries
the
reward
ofa
stream
of oligopoly profits.
At the
same time,
the
development activity
(in the
North)
and
the
imitation activity
(in the
South) create non-appropriable benefits
in
the form
of
additions
to
general knowledge. Such knowledge includes scientific
and technical information that
has
widespread appHcability,
and
that
contributes
to the
productivity
of
subsequent learning efforts.
We
assume that
the stocks of industrial knowledge are specific to the regions in which they have
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1990 ENDOGENOUS PRODUCT CYCLES I219
must devote ajjK units of labour to the task, w here aj is a fixed produc tivity
parameter ( / for imitation) and
K^
is the stock of disembodied knowledge
capital in the South. We take the stock of knowledge to be proportional to
cumulative experience in the learning sector in the South, and choose units so
that
K^
=
R5,
where ^5 is the num ber of varieties th at have been imitated
in the
past. Under this specification,^
ri^ = rif-LjIa,,
(10)
where L, represents labour employed in reverse engineering in the South.
Entry into imitation can be financed by bond issue or by an equ ity offering.
If this activity ever were to offer a pure profit, incipient entry by entrepreneurs
would generate excess demand for Southern labour. It follows that, in an
equilibrium with some labour devoted to im itation in the South, the present
value
of
Sou thern profits from m anu factu ring just eq uals
the
cost
of
reverse
engineering, or
Differentiating this break-even condition with respect to /, we find
Equation (ii) expresses a no-arbitrage condition, equating the sum of the
instantaneous profit rate (first term
on the
left-han d side)
and the
rate
of
capital gain (second term on the left-hand side) to the instantaneo us interest
rate. The capital-gain term reflects the fact that the value of a Southern firm
equals the cost of imitation, and so varies positively with the wage rate and
negatively with productivity in the learning activity.
A potential Northern innovator faces a similar, though som ewhat more
complex decision problem. The development of a new variety requires a^/K^
units of labour, where a , is another productivity param eter ( Z) for
development) and A ;^. represents the level of scientific knowhow in the N orth.
We assume that each development project contributes
a
similar amount
to the
stock of knowledge in the North, so that
K^^
is proportional to cumulative
experience in innov ation. Units are chosen so that
K^.
=
n.
Then the m easure
of the set of available products grows according to
fi
=
nLJao,
(12)
where
L^
is the aggreg ate am oun t of labou r hired for product development. A
Northern innovator who brings out a new product at time
t
faces thereafter a
positive probability that the pro duc t will be targeted by a Southern
* Some alternative specifications may be equally plausible. First, productivity in imitation m ight depen d
on both imitation experience
and
knowledge areumu lated
in the
No rth. This specification would ap ply
if
iiirormation dis.seminated internationally, and if the knowledge generated in the course of innovation were
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THE
ECONOMIC JOURNAL [SEPTEMBER
entrepreneur
for
imitation.
If
the product
is
copied
at
time
T., the
innovator's
stream
of
monopoly profits ends at T. In that event,
the
innovator earns, in
total, a
sum
whose present discounted value at /
is
At
the
time that
a
particular product
is
developed,
the
date
T at
which
imitation will occur is unknown. However, as we noted before, shareholders
can diversity away this product-specific risk
by
holding a portfolio of Northern
shares. Thus the individual firm maximises its stock market value by
maximising
the
expected present discounted value
of
the stream
of
monopoly
profits less innovation costs. We will assume that Northern agents have rational
expectations. Letting F(/,
T
denote the cumulative distribution function
for
T
for a product developed at t (i.e.
the
probability that monopoly power wili
be
lost to a Southern imitator before time T), we can write the expected present
value
of
profits
for
a time-/ innovator
as
Since we allow free entry by Northern entrepreneurs into product development,
a positive rate
of
innovation implies
V{t =wM<^o/n[t .
13)
The evolution of imitation activity in the South after time / determines the
distribution of the terminal date Tfor
a
product developed
at
/. Since Southern
entrepreneurs choose their targets
at
random, each existing Northern monopoly
faces
the
same risk of being imitated.
The
instantaneous rate of imitation,
li{T =n_^{T ln {T ,
gives
the
hazard rate
for F(/, 7 ),
or F>p/{i
—F .
This
Fit, T = i-t-r/i[T]dT. 14)
Using (14),
and the
definitions of
V{t
and n(/,
7 ),
we
calculate
(15)
Now, differentiating
(13)
with respect
to t, and
using (15),
we
find
Equation
(16)
expresses
a
no-arbitrage relationship similar
to (n) . It
equates
the sum of
the instantaneous profit rate (first term
on the
left-hand
side)
and the
capital gain (second term on
the
left-hand side) to
the
risk-
adjusted interest rate.
The
capital gain here
is the
increase in
the
value
of a
typical Northern firm, which equals
the
rate
of
increase in Northern wages
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I O
ENDOGENOUS PRODUCT CYCLES
I I
clearing conditions.
In
each region, lab ou r is employed
in
both ma nufacturing
and learning activities. Letting X^ = ^^.v,, denote aggregate output in region i,
I = , S and /,; denote the exogenous labour supply there, we equ ate labo ur
supply and demand in each region in the following equations:
iaj/ns ns
+
a^Xs = Lsl
17)
{aj,/n n-\-a^X^
= L .
18)
The equations that we have derived in this section fully de term ine the
evolution
of
the world econom y from any initial con ditions (i.e. nu m bers
of
goods produced in the No rth an d Sou th), provided that we choose an initial
level of spending, £ (0), consistent with long-run convergence to
a
steady state.
We proceed now
to
exam ine the steady-state properties ofth e model.
II.
DETERMINANTS OF IMITATION AND INNOVATION IN THE LONG
RUN
In the steady state, the number of varieties grows
at
Me constant rate ^, and
Southern firms imitate at the constant rate //. We are interested in the
determ inants ofthese long-run rates of innovation and imitation. We are also
concerned with growth of log U { T , since this measures instantaneous utility in
our model. But it is easy to show that t/fiog u{T ]/dt = (i a) g/a, so the factors
that affect the long-run rate of innovation similarly influence the steady-state
growth in utility.
We normali.se nominal prices so that
w^ = n.
With this choice of numeraire,
all prices and wages grow
at
the common rate g
in
the steady state,
as
does
nominal spending E. Then
4)
implies
(19)
In the steady state, the share ofthe North in the total number of varieties,
^.\ = «A-/ > is constant and equal to g/{g fi . Using this fact, and substituting
(6),
7) and
(ig ) into (16), we find
^
20)
Now we combine (18)
and
(20)
to
derive
where h = L ja^
is the
'effectiv e' lab ou r force
of
the N orth, measured
in
terms of productivity
in
innovation.
Equation (21) expresses
a
steady-state relationship between g and ji.
We
depict this relationship by the curve NN in Fig. i.* The curve shows
combinations
of
steady-state rates
of
innovation
and
imitation that
are
consistent with labour-market clearing
in
the North
and a
profit rate there
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1222 THE ECONOMIC JOURNAL [SEPTEMBER
into
the
learning sector, thereby decreasing aggregate output
in the
North,
and
hence the sales base over which mark-up profits
are
earned. Second,
an
increase
in
g
raises
the
share
of
Northern products
in the
total number
of
varieties,
and
thus lowers
the
output
per
Northern firm
for a
given
n and X . At the
same
time, an
increase
in the
rate of innovation raises
the
interest rate,
ceteris paribus
So
an
increase
in g
opens
a
positive
gap
between
the
risk-adjusted interest rate
and
the
profit rate.
An
increase
in the
rate
of
imitation
is
needed, then,
to
restore equality between
the two. The
increase
in fi
raises
the
risk premium,
thereby exacerbating
the
disequilibrium,
but it
also raises
the
profit rate.
As
can
be
seen from
21), the
effect
on the
profit rate
the
left-hand side)
dominates.
The
effect
of/< on the
Northern profit rate stems from
the
implied
reduction
in a and
thus
the
increase
in
sales
for
each Northern firm
at
given
n
and
X .
To derive a second relationship between g and //, we must brirtg in the
equilibrium conditions for the South. The nature of this second relationship
varies according to the size ofthe gap between Northern and Southern wages.
We take up the wide-gap and narrow-gap cases in turn, discussing in each
instance the determinants of steady-state) g and {i.
The Wide-gap Case
R eca l l t ha t , when
w_^/w^
< a, a S ou t he r n i m i t a t o r cha r ges
the
mo nopo l y p r i ce
wi thout f ear
of
com pet i t ion f rom
the
o r i g ina l No r t he r n deve l ope r
of
t h a t
va r i e t y . In this case, we sub s t i tute 8f l) , 9a) and 19) in to the n o - a r b i t ra g e
cond i t i on
for
So uthe rn f irms ,
11),
wh ich gives
{i a axXs =
aa,{g
p . 22)
N ow
we use the
l abo u r - m ar ke t c l ea r i ng cond i t ion
for the
S o u t h ,
17), to
subs t i tu te
for X in 22).
R ecogn i s i ng t ha t
the
n u m b e r
of
S o u t h e r n p r o d u c t s
mus t g row
at the
r a te
^ in a
s t eady s t a te ,
we
have
g= [\-a. hs-CLp, 23)
where h^
=
Z-s/fl/ represents
the
effective labour force ofthe South measured
in
terms of its productivity in imitation.
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EN DO GE NO US PRO DUC T CYCLES 223
We represent equation (23) by the horizontal line labelled SS in Fig. i. The
steady-state rates of innovation and imitation for the wide-gap case are given
by the intersection ofthe curves .S and V.V in the figure.^ Of course, the wide -
gap case applies only if
li;
^ aw^- is implied by the point of intersection. (More
on this point below.)
We note first the effects of international trade on long-run growth in the two
regions. The North's autarky rate of innovation is found at the intersection of
th e NN curve and the horizontal axis, where fi = o. Since the A iV curve slopes
upward, the North innovates faster in the steady state of a wide-gap
equilibrium than it does in the absence of trade. Imitation causes resources to
be released from the Northern manufacturing sector, which find their way into
the research laboratory. This reallocation of resources is mediated by an
increase in the profit rate, which results when, at given n a smaller number of
firms competes in the Northern labour market. In autarky, the South would be
forced to develop its own varieties from scratch. The resource requirements for
this surely would exceed those needed for assimilating and adapting Northern
technologies. To the extent that this is true, the opportunities for imitation
contribute to a more rapid pace of technological progress in the South. In
particular, the region's autarky rate of learning is given by an equation like
(23), except that aj is replaced by a pa ram ete r reflecting S outhern productivity
in
innovation
say aj^g As long as a^s > ^i^ the South also grows faster with trade
than without.
In the steady state ofa wide-gap equilibrium, the growth rate is proximately
determined by economic forces in the South. An expansion in the North's
labo ur force or a rise in the produ ctivity of N orthe rn lab our in the resea rch
laboratory shifts the NN curve upward. This reduces the rate of Southern
imitation a nd hence the steady-state share of produc ts manufac tured by the
South, but it has no effect on the steady-state growth rate.* The explanation for
this lies in the determination of a Southern firm's profit rate,, which in the
steady state must equal
g-\-p.
Consider a shock in the North that alters the
derived demand for Southern labour, hence the equilibrium relative wage
Wff/w^ . S ince the prices of S outh ern goods in the wid e-gap case are a fixed
mark-up over production costs there, the change in w^/wy alters profits per
variety and the cost of imitation equ iprop ortion ately . T he shock in the N orth
also may ch ange the fraction of produc ts m anufactured in the S outh. But given
aggregate So uthern o utput of manufactures X^ a change in n,y affects similarly
the profits of a given variety and productivity in itnitation. So, by either
channel, the net effect on the profit r te in the South is nil. It follows that the
* If the
SS
curve lies everywhere above the A^^ curve, then th e narrow -gap case, rathe r than the wide-
gap case, applies. If the .55 curve lies everywhere below the A'A'curve, then there can be no steady slaie with
a positive rate of imitation in the South, Instead, the South imitates for a while, then produces a fixed set
of goods. An equilibrium with ongoing imitation requires that the
A'JV
curve em anate s from a point on the
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1224 ^^^ ECONOMIC JOURN AL [SEPTEMB ER
initial values of ^ and .^5 contin ue to satisfy the con dition s for a stea dy -state
equil ibr ium.
An Improvement in productivity at manufacturing, a^, has no effect on SS,
hence no effect on the steady -state values of// o r^ . But an expansion of effective
labour in the South, precipitated either by an increase in L,, or a decline in a,,
causes the SS curve to shift upward, and generates an increase in the steady-
state rates of imitation and innovation. The impact effect entails a rise in the
rate of imitation. In the North this raises the risk premium for product
deve lopm ent, but also boosts profits for each surviving mo nopo ly, as ou tpu t per
Northern brand expands. As we mentioned before, the latter effect dominates,
so innovation responds positively.
W e close ou r discussion ofth e w ide-gap case by considering the de term ina nts
ofthe relative wage. We evaluate {5) at the equilibrium prices given in (6) and
{8a), an d then take the ratio of ou tpu ts per variety in the N orth and Sou th, to
derive
K^2
Substituting for ^^7X5 using (20) and (22), and noting
IT^
= g/{g + /f-), we find
pj
The right-hand side of (24) is increasing in and dechning in g and in g/{i.
An expansion ofthe labour force in the North alters the right-hand side of
(25(7) only via its effect on /*. We see, therefore, that this shock causes Wg/w^
to fall. A larger labour force in the South., on the other hand, implies
acceleration of both innovation and imitation, but a fall in the ratio ol g to /i
(the slope of a ray from the origin to E falls as we move up along a given NN
curve), hence a larger value of
W^/K'^J-
We conclude that the relative wage of
either region varies directly with the size of that region.
The response of relative wages to the productivity parameters is similar. An
improvement in the productivity of Northern researchers reduces the steady-
state rate of imitation, and also has a direct negative effect on the right-hand
side of (250), so the relative wage of the South falls in response. An
improvem ent in Southern prod uctivity iu imitation raises//, depresses^//*, and
directly increases the right-hand side of (25a), and so cau.scs Wg/w^ to rise.
Our results on the relationship between labour supplies and relative wages
stand in sharp contrast to those reported by Krugman (1979). The sources of
the difference can be seen in the equation
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EN DO GE NO US PR OD UC T CYCLES 225
nu m ber of differentiated goods, an expansion of a region's labou r force
increases the relative supply of a typical good m anu facture d there relative to
the supply of a good man ufactured abro ad. T his exerts dow nw ard pressure on
the relative price of goods em an atin g from the expa nd ing region, and so the
relative wage ofthe region tends to fall. But (256) points to two additional
channels through which a change in labour supply might affect relative wages
in the long run. First, the endowment change might induce a reallocation of
resources between manufacturing and re.search in each country. If the steady-
state size ofthe research sector exp and s in both regions (i.e. if^ rises), resources
must be drawn from manufacturing in each one. This tends to increase the
relative wage of the North, because the per-unit labour requirements for
research are greater there in relation to the size of the region's labour force.'
Secondly, the expanding region captures in the long run an increased share of
the total num ber of differentiated prod ucts. Th at is,
cr ^
varies directly with L^,
bu t inversely with Z, ., reflecting the fact th at p rod uc ts accum ula te relatively
faster along the transition path to a new steady state in the region that has
experienced the endowment growth. With relatively more goods to produce,
the expanding region sees growth in the relative demand for its labour.
Evidently, these latter effects dominate. In a sense, labour supply creates its
own (long-run) demand in the research laboratory.
Tke Narrow-gap Case
In Fig. 2 we have reproduced the
A''JV
curve (equation (21)), which c ontinues
to apply. We find a second relationship between the long-run values of/i and
g for the narrow-gap case by substituting the equilibrium prices in (6) and {8b)
into (5), taking the ratio of aggregate outputs in the North and South, and
using the result together with (17) and (18). This gives
^ ~—71 r~ - (26)
^
[fi-g]
g ^
We plot the com binations of and fi that satisfy (26) as XX in Fig. 2. This
curve gives the rates of innovation and imitation that are consistent with
simultaneous clearing of the labour markets and product markets in each
region. It is easy to show that the XX curve slopes upward, once we recall that
.s > A.v is requ ired for the ex istence of a steady -state eq uilib rium with a
positive rate of imitation. We prove in our working paper (Grossman and
H clpm an , 1989(2) that the XA curve must be steeper tha n the A ^A ' curve at any
point of intersection, and that the curves intersect exactly once. This
intersection (at Q) represents the unique narrow-gap equilibrium, provided
that the relative wage associated with that point satisfies Wf^/w^ > a.^
Fig. 2 can be used to study the relationship between the effective sizes ofthe
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1226
THE ECONOMIC JOURNAL
[SEPTEMBER
Fig. J The narrow-gap case.
two regions and the long-run rates
of
imitation and innovation.
To
conserve
space, we refer the interested reader to our (1989 a working paper, and present
here only the results of the analysis. We find that the long-run rate of
innovation rises
in
response
to an
expansion
in
the size of either labour force,
or
a
reduction
in the
labour requirements
for
learning new technologies
in
either region. The rate of imitation, on the other hand, varies directly with tbe
effective size of tbe S ou th, b ut inversely w ith tbe size of the N ortb . As in tbe
narrow-gap case, tbe relative wage of a region increases with the relative size
of that region's labour force.
I I I . TRADE AND INDUSTRIAL POLICIES
Nations often contemplate tbe use of policies to speed their growtb
or to
slow
the rate of
loss
of markets to foreign competitors. We can use our model of tbe
product cycle
to
study the effects
of
policy
on
long-run rates
of
growtb and
imitation, and on relative wages in tbe steady state. In tbis section we sbal
consider subsidies to the learning activities and protective trad e policies. We
limit
our
analysis here
to
positive issues; G rossm an
and
He lpm an (1991)
contains
a
complete welfare analysis
for a
small country witb
an
economic
structure similar
to
tbe one described here.
We let I — Aj, i
= S, N^
represe nt tbe fraction of learn ing costs borne by tb
government in country
i
We assume that subsidies are financed by lump-sum
taxes. Since Ricardian neutrality applies in our model, we need not specify the
intertemporal pattern of the tax collections, so long as tbe present value of tbe
government's casb flow
is
zero.
The subsidies to learning alter tbe private incentives for researcb, hence the
no-arbitrage conditions tbat apply in eacb region
in
tbe long-run equilibrium
Witb these policies in place, we need to multiply the rigbt-band side of (21) by
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199^] ENDOGENOUS PRODUCT CYCLES I22 7
to tbe left, wbile leaving tbe curve in place. Tbis policy reduces the long-run
rate of imitation and boosts the steady-state share of varieties produced in the
North, but has no effect on tbe long-run growtb rate. Using a modified version
of
(25(1),
it is easy to see that tbe Northern government's intervention raises ihe
relative wage of its workers in the steady state.
A subsidy to imitation or to technology adaptation in the South shifts the
curve ofthe wide-gap case upward. The growth rate and the imitation rate rise,
as does tbe sbare of varieties produced in tbe South. Like an improvement in
productivity in imitation, this serves to raise tbe Soutb's relative wage in the
long run. The positive effect on the growth rate should be well understood by
now. Although tbe speeding of the product cycle directly reduces the
profitability of product development, tbis effect is more tban offset by the
expansion of sales for Northern products tbat survive. So the incentive to
innovate is strengthened by faster imitation in tbe South.
In the narrow-gap case, a subsidy to imitation serves only to alter relative
wages {see Grossman and Helpman, 19890). As before, tbe relative wage ofthe
South rises when its gov ernm ent subsidises learnin g. An R D subsidy in tbe
Nortb causes tbe growtb rate and tbe rate of imitation both to increase. The
subsidy also serves to increase tbe relative wage of tbe North in tbe long
run.
We turn now to trade policy. For the wide-gap case, the analysis is quite
simple. An ad v lorem tariff or export subsidy imposed by either country does
not affect tbe elasticity of demand perceived by producers of any nationality.
The refore, it does not affect tbe prices charg ed by them . T b e profit rates do not
change with trade policy, nor do tbe no-arbitrage conditions. Of course, the
labour-market-clearing conditions, (17) and (18), continue to apply. It follows
that trade policies in either country do not affect the A JV or the of the wide-
ga p case, and therefore tbey do not alter the steady -state rates of inno vation or
imitation.
The conclusion for the narrow-gap case turns out the same. Tbe reader is
referred to our (19890) working paper for tbe details. In our model with only
one production sector, trade policies (and production incentives more
generally) do not affect tbe long-run allocation of resources to the learning
activity, because tbeir effect on investment incentives is fully offset by an
induced change in factor rewards. We should note, however, that tbe
conclusion would b e different In an ex tended version of ou r model tba t
included a second ma nufacturing sector, a second prim ary factor of prod uction ,
and different factor intensities in the various economic activities.
I V . C O N C L U S I O N S
Tbe product cycle describes an ever-evolving pattern of Inter-regional trade.
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1228 THE ECONOMIC JOUR NAL [SEPTEMB ER
average length of the cycle and the speed with which new products are
introduced to the market are both determined endogenousiy. We have used our
model to study tbe determinants of the long-run rates of imitation and
innovation, and the long-run distribution oflabour income.
As in previous studies of technology-driven growth (e.g. Romer (1990),
Grossman and Helpman (198913)), we found that tbe size ofthe resource base
and the productivity of resources in tbe learning activities are important
determinants of tbe steady-state growtb rate. Long-run growtb is faster tbe
larger is tbe resource base of the South, and the more productive are its
resources in learning the production processes for products originally developed
in the North. This is surprising, perbaps, because faster imitation by the South
means on average a shorter period over which a Northern firm can earn
monopoly profits. But profits during the monopoly phase are higher when a
smaller num ber of No rthe rn firms compete for resources in the m anufa cturing
sector. We found that the latter effect dominates, so faster imitation by the
South uhimately strengthens the incentive to innovate in tbe Nortb.
Steady-state growtb also increases witb the effective size of tbe North,
provided tbat the gap between wages in the North and South is not too large
(what we have called the 'narrow-gap case'). An increase in the North's
effective labour force always slows the rate of imitation (hence the average
length of tbe first stage of tbe product cycle) and raises tbe steady-state sbare
of varieties produced in the North. An increase in tbe effective labour force of
tbe South has jus t tbe opposite effect on the im itation rate an d on pro duc t
shares in the long run.
Perbaps surprisingly, we find tbat the relative wage in the Nortb rises when
tbe effective size of tbe North expands in relation to the effective size of the
South. Tbis result, which contradicts one derived by Krugman (1979) in his
product-cycle model with exogenous rates of innovation and imitation, stems
ultima tely from the increasing-returns na ture of tbe tecbnologies for pro duc tion
of goods and knowledge.
In comparing tbe product-cycle equilibrium to one witb autarky in eacb
region, we find that international trade always contributes to faster growth in
botb regions. Tbe migration of some products from North to South frees
Northern resources for use in product development. In the steady-state
equilibrium witb trade, Nortbern firms bave a greater incentive to undertake
R D th an in au tark y, because eacb earns a bigher profit ra te, albeit for a
shorter period of time. Th e South grows faster w itb trade tban witbo ut, because
tbe resources needed for learning and adapting Northern technologies are far
fewer tban those needed for developing new products from scratch.
We studied the long-run effects of two types of policies. Subsidisation of tbe
learning activities (innovation in tbe North or imitation in tbe South) tends to
boost long-run rates of growth. The only exception to this occurs for subsidies
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199^] ENDOGENOUS PRODUC T CYCLES 1229
policies serve only to alter relative wages and the steady-state levels of real
spending in tbe two regions. This finding, which is perhaps reminiscent of
similar results tha t app ly in neoclassical models of grow th, rehes strongly on the
two-sector structure of our regional economies. If, instead, we were to allow a
second manufacturing activity in each country in a manner that preserved the
existence of a steady state, trade policy would indeed play a role in determining
tbe long-run growth rate.
Princeton University
Tel Aviv University
Date of receipt of final typescript: Octob er iggo
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