Endogenous Product Cycles Economic Journal 1991

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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



[ 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



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)



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

wâ^,

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



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

Wfâ^/aL

(the

monopoly price) falls short

of the

marginal cost

of

Northern

production, wâj^,

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



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



1220

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<ô/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



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



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.



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 > î^ 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



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



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 Â ' 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



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



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.



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



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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Documents

Endogenous Product Cycles Economic Journal 1991