Imitation as a Stepping Stone to Innovation
Amy Jocelyn Glassa∗a Department of Economics, Texas A&M University, College Station, TX 77845

Abstract
By creating a knowledge base that reduces innovation costs, imitation can serve
as a stepping stone enabling Þrms from lagging countries to undertake innovation.
This paper constructs a product cycle model with this feature to determine what
forces could cause a lagging country to shift its R&D from imitation toward innovation. Increasing resources or subsidizing R&D in the lagging South increases
Southern innovation, while increasing resources or subsidizing R&D in the Northdecreases Southern innovation. Eﬀects on Northern innovation, Southern imitation, aggregate expenditure, and the relative wage are compared with the casewhere innovation does not occur in the South.
Keywords: Innovation, Imitation, Appropriate technology
JEL classiÞcation: F1; F4; O3

1.

Introduction

Learning through imitation may enable Þrms to improve existing technologies. South Korea, Taiwan, and increasingly China, Malaysia, Indonesia and Thailand, have begun to follow Japan’s path by imitating technologies from abroad (see Carolan et al 1998). Firms from these countries are
now beginning to innovate.
The World Competitiveness Yearbook (2002) indicates that, as of 2001,
29.34% of Korea’s manufactured exports are high-tech, a value similar to
the United States (29.70%) and Japan (26.60%). Korea’s expenditure on
R&D in 2001 was $12,489 million, which ranks eighth and is similar inmagnitude to Canada ($13,517 million) but well below the United States
and Japan at $282,293 million and $142,014 million, respectively. Mainland China’s R&D in 2001 was $12,595 million (rank 7), and Taiwan’s was$6,063 (rank 10). Korea’s R&D is 2.921% of GDP, which ranks second,
close behind Japan at 2.981% and just above the United States at 2.800%.
Taiwan’s R&D is 2.156% of GDP (rank 6, above Canada and the United
Kingdom), and China’s R&D is 1.087% of GDP (rank 11, above Italy andSpain). An average of 34,052 patents were granted to Korean residents per
year over the period 1998-2000, which ranks third, well behind Japan and∗Tel.: 1-979-845-8507; fax: 1-979-847-8757.
E-mail address: aglass@econmail.tamu.edu

Imitation as a Stepping Stone to Innovation

2

the United States at 123,978 and 83,090, respectively. Taiwan ranks fourthwith 20,094 patents a year, and China ranks tenth with 3,742. In 2000,
Korean residents secured 7,032 patents abroad, ranking eighth and similar
to Canada at 7,473 (rank 7). Taiwan secured 2,486 patents abroad (rank10), and China 308 (rank 15).
What determines the degree that Þrms in a lagging country innovate?
For lagging economies, innovation may initially be prohibitively diﬃcult.
By bringing practical knowledge of the current state-of-the-art into thecountry, imitation can provide the vital knowledge base needed to makeinnovation attractive. Knowledge assimilation stemming from imitationis endogenous due to imitation being endogenous. Therefore, changes in
the incentives for (or capacity) for imitation are linked to changes in the
incentives for Þrms in lagging countries to innovate.
What are the eﬀects on the United States and other advanced countriesof the increasing integration of developing countries into the world trading system? How do these eﬀects depend on the level of development of
the lagging countries and, in particular, on whether they engage in imitation or innovation? As countries enter into freer trade with the rest of
the world, they seek to absorb technology from abroad through imitation
and, ultimately, to be able to achieve state-of-the-art technology throughinnovation. One can think of the entry and further integration of developing countries into the world trading system as an expansion in the size(or resources) of the developing world that is open to trade. One can also
think of subsidies to R&D as capturing policies countries pursue in order
to try to promote R&D.
This paper builds a product cycle model to determine the eﬀects offorces such as resource accumulation or government R&D incentives on
the mix of a lagging country’s R&D between innovation and imitation.
The model has two countries, the North (an advanced country) and theSouth (a lagging country). The results indicate that increasing the supplyof Southern resources can provide the added resources needed to increase
Southern innovation. Government incentives to R&D could also encourage
Southern Þrms to increase innovation. Such incentives need not be able
to distinguish innovation from imitation. Increasing Northern resources ora Northern R&D subsidy shifts Southern R&D out of innovation and into
imitation.

Imitation as a Stepping Stone to Innovation

3

Most of the product cycle literature, such as Grossman and Helpman(1991a) and Segerstrom et al (1990), assumes Þrms from the lagging country
only imitate and thus do not address what determines the degree that theSouth innovates. One exception is Currie et al (1999), where spillovers thatlower the cost of Southern innovation can occur directly from the knowledge
base built by Northern innovation. That model is based on innovations thatare new varieties (rather than new qualities) and does not consider a generalsubsidy to Southern R&D (both innovation and imitation). Considering ageneral Southern R&D subsidy is important because distinguishing between
innovation and imitation when subsidizing R&D is nearly impossible inpractice.
Van Elkan (1996) also provides a model where the South shifts from
imitation to innovation. There, productivity in imitation depends on thediﬀerence between the body of world knowledge and the stock of humancapital (the knowledge gap), while productivity in innovation depends onpast behavior through learning-by-doing. Imitation is easier the larger theknowledge gap due to the larger pool of potential imitations (larger world
knowledge stock), so as the knowledge gap closes with successive imitation, imitation becomes relatively more diﬃcult and innovation relatively
more attractive. The van Elkan model has the unrealistic trait that a technological advantage in imitation assures higher per capital output than atechnological advantage in innovation.
The paper is organized as follows. After establishing the behavior of
consumers and Þrms (Section 2), we Þnd the steady-state equilibrium whenSouthern innovation awaits the knowledge base generated by Southern imitation (Section 3). Then we examine what forces promote Southern innovation relative to imitation and compare those eﬀects to the case where theSouth only imitates (Section 4). Proofs of results appear in the Appendix.
2.

The Economy

The economy is composed of two countries, each containing a representative consumer and many Þrms. Firms diﬀer in their R&D abilities so that
Northern Þrms innovate while Southern Þrms imitate and only innovate if
imitation has lowered the costs of innovation. The Southern wage falls below the Northern wage, so imitation is proÞtable for Southern Þrms but notImitation as a Stepping Stone to Innovation

4for Northern Þrms. A Þxed supply of one factor, labor, is used for R&D
and production in each country. In equilibrium, expected proÞts from theproduct market compensate Þrms for their R&D costs, and resources are
fully employed in each country.
2.1.

Consumers

The speciÞcation of the consumer’s problem follows Grossman and Helpman (1991a). Consumers choose from a continuum of products j ∈ [0, 1].
Quality level m of product j provides quality qm (j) ≡ λm . By the definition of quality improvement, new generations are better than the old:qm (j) > qm−1 (j) → λm > λm−1 → λ > 1. All products start at time t = 0
at quality level m = 0, so the base quality is q0 (j) = λ0 = 1.
A consumer from country i ∈ {N, S} has additively separable intertemporal preferences given by lifetime utility
Z ∞e−ρt log ui (t)dt,
(1)
Ui =
0

where ρ is the common subjective discount factor. Instantaneous utility is
Z 1
X mlog ui (t) =log(λ) xim (j, t)dj,
(2)
0

m

where xim (j, t) is consumption by consumers from country i of quality level
m of product j at time t.
Consumers maximize lifetime utility subject to an intertemporal budget
constraint. Since preferences are homothetic, aggregate demand is found by
maximizing lifetime utility subject to the aggregate intertemporal budget
constraint
Z ∞
Z ∞
−R(t)
e
Ei (t)dt ≤ Ai (0) +e−R(t) Yi (t)dt,
(3)
0

0

Rt

where R(t) = 0 r(s)ds is the cumulative interest rate up to time t and Ai (0)is the aggregate value of initial asset holdings by consumers from countryi. Individuals hold assets in the form of ownership in Þrms, but with adiversiÞed portfolio, any capital losses appear as capital gains elsewhere so
only initial asset holdings remain. Aggregate labor income of all consumers
from country i is Yi (t) = Li wi (t), where wi (t) is the wage in country i atImitation as a Stepping Stone to Innovation

5time t and Li is the labor supply there, so Li wi (t) is total labor income incountry i at time t. Aggregate expenditure of all consumers in country i is#
Z 1 "Xpm (j, t)xim (j, t) dj,(4)
Ei (t) =0m

where pm (j, t) is the price of quality level m of product j at time t, andEi (t) is aggregate expenditure of consumers in country i, where aggregate
expenditure is E(t) = EN (t) + ES (t). Due to assumed free trade, pricelevels do not vary across countries.
A consumer’s maximization problem can be broken into three stages:
the allocation of lifetime wealth across time, the allocation of expenditureat each instant across products, and the allocation of expenditure at each
instant for each product across available quality levels. In the Þnal stage,
consumers allocate expenditure for each product at each instant to thequality level m(j,e t) oﬀering the lowest quality-adjusted price, pm (j, t)/λm .
Consumers are indiﬀerent between quality level m and quality level m − 1
if the relative price equals the quality diﬀerence pm (j, t)/pm−1 (j, t) = λ.
Settle indiﬀerence in favor of the higher quality level so the quality levelselected is unique. Only the highest quality level available of each productwill sell in equilibrium.
In the second stage, consumers then evenly spread expenditure acrossthe unit measure of all products, Ei (j, t) = Ei (t), as the elasticity of
substitution between any two products is constant at unity. Consumers
e t) of productdemand xime (j, t) = Ei (t)/pm
e (j, t) units of quality level m(j,j and no units of other quality levels of that product. In the Þrst stage,
consumers evenly spread lifetime expenditure across time, Ei (t) = Ei , as
the utility function for each consumer is time separable and the aggregate
price level will not vary across time log pme (j, t) = log pm
e (j). Since aggregate
expenditure is constant across time, the interest rate at each point in time
reßects the discount rate r (t) = ρ, so R(t) = ρt in the intertemporal budget
constraint.
In summary, consumers evenly spread expenditure across time andproducts. For each product, they are willing to pay a premium λ for aquality increment and will purchase only the highest quality level available.

Imitation as a Stepping Stone to Innovation

2.2.

6

Production

The Þrm’s problem can be broken down into two stages: R&D and thenproduction. In the production stage (once successful in R&D), each Þrmthen chooses the price of its product to maximize its value, given prices andR&D intensities of other Þrms. Assume that Þrms have access to discarded
technology (technologies that no longer yield proÞts from production, as
described later). This assumption assures that Southern Þrms have theability to produce lower quality levels.
Normalize the labor supplies to make the labor requirement in production be one in each country. Normalize prices to make the Southern wage
be one wS = 1. DeÞne w ≡ wN /wS as the relative wage. Consequently, themarginal cost of production is cS = cM = 1 for Southern Þrms (innovators
or imitators) and cN = w for Northern Þrms. ProÞts are price minus costtimes sales π = (p − c) x.
Consider Þrst markets where a Northern Þrm has just succeeded ininnovation. The Northern Þrm engages in limit pricing against the SouthernÞrms with access to discarded technology one quality level below, charging apremium reßecting consumer’s valuation of the quality improvement pN =λ and making sales xN = E/λ. This price is just low enough to keep
Southern Þrms from being able to sell the lower quality level at a proÞt,since consumers would be willing to pay pN /λ = 1 for the lower quality
level, which is the production cost for a Southern Þrm. InstantaneousproÞts for the Northern Þrm are
³
w´
.
(5)
πN = E 1 −λCall these markets Northern-led since only Northern Þrms have the ability
to produce the state-of-the-art technology.
Now consider markets where a Southern Þrm has just imitated. The
Southern Þrm engages in limit pricing against the Northern Þrm at the same
quality level by charging a price equal to the Northern Þrm’s cost pM = w
and makes sales xM = E/w. Instantaneous proÞts for the Southern Þrm
are
¶
µ
1
.
(6)
πM = E 1 −w
Call these markets imitated since both Northern and Southern Þrms havethe ability to produce the state-of-the-art technology.

Imitation as a Stepping Stone to Innovation

7

Finally consider markets where a Southern Þrm has just succeeded ininnovation. The Southern Þrm engages in limit pricing against the other
Southern Þrms with access to discarded technology one quality level below,
charging a premium reßecting consumer’s valuation of the quality improvement pS = λ and making sales xS = E/λ. Instantaneous proÞts for theSouthern Þrm are
µ
¶
1
.
(7)
πS = E 1 −λCall these markets Southern-led since only Southern Þrms have the ability
to produce the state-of-the-art technology.
Northern Þrms target all markets for innovation. Southern Þrms target only Northern-led markets for imitation; they do not target imitatedor Southern-led markets since they have no cost advantage against other
Southern Þrms. Southern Þrms target imitated and Southern-led markets
for innovation; they do not target Northern-led markets since lack of a
suﬃcient knowledge base makes innovation targeting Northern-led markets
prohibitively diﬃcult (to be shown).
A successful Northern innovator gains the value from producing in a
Northern-led market vN , a successful Southern imitator gains the value
from producing in an imitated market vM , and a successful Southern innovator gains the value from producing in a Southern-led market vS . To
Þnd the value of producing Þrms, discount the ßow of instantaneous proÞts
to account for the chance that the proÞt stream will be terminated due toinnovation or imitation.
Northern-led markets are targeted by both imitation and Northern innovation, so the value from producing in a Northern-led market is
vN =

πN
.
ρ + ιN + µS

(8)The eﬀective discount rate (the denominator) is the subjective discount rateρ plus the sum of all the chances that the proÞt stream will be terminatedat an instant in time. Imitated markets are targeted by both Northern andSouthern innovation, so the value from producing in an imitated market is
vM =

πM
.
ρ + ιN + ιS

(9)Imitation as a Stepping Stone to Innovation

8

Southern-led markets are also targeted by both Northern and Southern
innovation, so the value from producing in a Southern-led market is
vS =

πS
.
ρ + ιN + ιS

(10)These values indicate the reward to successful Northern innovation, imitation and Southern innovation. These rewards must oﬀset the costs ofinnovation and imitation for innovation and imitation to occur in equilibrium.
2.3.

Innovation and Imitation

In the R&D stage, each Þrm chooses its intensity of R&D to maximize its expected value, given the R&D intensities of other Þrms. R&D
races occur simultaneously for all products. R&D races are like lotteries
in that certain costs lead to uncertain outcomes. A Northern Þrm undertaking innovation intensity ιN for a time interval dt requires aN ιN dt unitsof Northern labor at cost waN ιN dt and leads to success with probabilityιN dt. Southern innovation and imitation operate similarly but with diﬀerent labor requirements (and Southern workers are paid the Southern wageof one). A Southern Þrm undertaking imitation intensity µS for a time interval dt requires aM µS dt units of Southern labor at cost aM µS dt and leads
to success with probability µS dt. A Southern Þrm undertaking innovation
intensity ιS for a time interval dt requires aS ιS dt units of Southern laborat cost aS ιS dt and leads to success with probability ιS dt. Imitation is easier than Southern innovation, and thus requires relatively fewer resources:γ ≡ aM /aS < 1. The quality increment λ is Þxed.
Assume each country is at worst one step below the world technology
frontier for each product. Such a situation occurs if knowledge of the designof quality levels no longer produced (discarded technology) spreads globally.
A technology becomes discarded when the Þrm that invented that quality
level no longer earns any proÞts from producing it; therefore, no Þrm would
have any reason to protect its design. Thus, there are the three possible
situations. Either a Southern Þrm has a one quality level lead, a NorthernÞrm has a one quality level lead, or a Northern Þrm and a Southern Þrm
are at the same quality level.

Imitation as a Stepping Stone to Innovation

9

Southern Þrms innovate only appropriate technologies, quality levels
that have already been imitated by a Southern Þrm. The underlying assumption is that the resource requirement aS applies to innovation targetingan imitated or Southern-led market but a substantially higher resource requirement A > aS applies to innovation targeting a Northern-led market,
so that Southern innovation costs exceed the expected reward to innovation targeting Northern-led markets in equilibrium: A > vS . Successful
prior imitation of a quality level of a product generates a knowledge basethat makes subsequent Southern innovation for that product easier. The
North might well enjoy lower innovation costs following successful imitation of products in Southern-led markets, but Northern imitation does notoccur in equilibrium because the North does not have a production costadvantage relative to the South.
The assumption that Southern innovation proceeds only after successful
imitation resembles Glass (1997), which assumes (in a setting where twoquality levels sell) that imitation of the low quality level provides a knowledge base for imitation of the high quality level of each product. Aghion
et al (2001) and Aghion et al (1997) also make a similar assumption in a
one-country model of innovation: innovation in their models must proceed
in a step-by-step fashion.
Firms engage in positive rates of innovation or imitation whenever the
expected gains are no less than their costs. As usual, only Þrms thatare not currently producing engage in R&D. To generate Þnite rates ofinnovation and imitation, the expected gains must not exceed their cost.
Northern Þrms conduct innovation at intensity ιN , earning the reward vN
if successful.
vN ≤ waN , ιN > 0 ⇔ vN = waN(11)vM ≤ aM , µS > 0 ⇔ vM = aM

(12)vS ≤ aS , ιS > 0 ⇔ vS = aS

(13)Southern Þrms conduct imitation at intensity µS , earning the reward vM if
successful.
Southern Þrms also conduct innovation at intensity ιS , earning the rewardvS if successful.
Resource constraints and conditions for the measure of markets to remain
constant complete the model.

Imitation as a Stepping Stone to Innovation

3.

10

Steady-State Equilibrium

The purchasing decisions of consumers and the R&D and pricing decisions of Þrms have been established. Now R&D and production must
be constrained according to resource availability in each country. Further,
the measures of Northern-led, imitated and Southern-led markets must remain constant. We also establish important relationships between variablesof interest in a steady-state equilibrium, both with and without Southern
innovation.
Focus mainly on the case where all three forms of R&D occur in equilibrium, so that these three R&D conditions all hold with equality. Section
4.5 will conÞrm that parameters leading to such an equilibrium do indeed
exist. Section 4.6 will consider the case where innovation is prohibitively
costly in the South even following imitation.
3.1.

Resource Constraints

Let nN be the measure of Northern-led markets, nM be the measure ofimitated markets and nS be the measure of Southern-led markets, which
together sum to one. The measure of Northern-led markets is the fractionof products that have a Northern Þrm serving the market. Similarly, themeasure of Southern-led markets is the fraction of products that have aSouthern innovator serving the market (and likewise for imitated markets).
Labor is used for R&D and production. DeÞne δ ≡ 1/λ. In the North,labor demand for innovation is aN ιN , while labor demand for production
is nN Eδ (sales of xN = Eδ for the fraction nN of markets). In the South,labor demand for imitation is aM µS nN , labor demand for innovation isaS ιS (nM + nS ), while labor demand for production is nM E/w + nS Eδ.
For equilibrium in the labor markets, the demand for labor must equal theÞxed supply of labor in each country.
aN ιN + nN Eδ = LN

(14)

µ ¶
E
aM µS nN + aS ιS (nM + nS ) + nM
+ nS Eδ = LSw

(15)

The resource constraints limit the amount of R&D and production performed in each country.

Imitation as a Stepping Stone to Innovation

3.2.

11

Constant Measures

Product markets change between Northern-led, imitated, and Southernled over time as innovation or imitation occurs; however, each market typemust have ßows in match ßows out for the measure of each market typeto remain constant at the aggregate level in a steady-state equilibrium.
For Northern-led markets, the measure of products newly innovated by
Northern Þrms must match the measure of products newly imitatedιN (nM + nS ) = µS nN .

(16)

For Southern-led markets, the measure of products newly innovated bySouthern Þrms must match the measure of Southern products innovatedover by Northern ÞrmsιS nM = ιN nS .

(17)

Finally, the measures of market types must sum to one
nN + nM + nS = 1.

(18)

These conditions can be solved to Þnd an expression for each market measure in terms of the intensities of Northern innovation, imitation and Southern innovation.
Since Northern innovation targets all markets, the aggregate rate ofNorthern innovation is the intensity of innovation ιN . Since imitation targets only Northern-led markets, the aggregate rate of imitation is the intensity of imitation times the measure of Northern-led markets µ ≡ µS nN .
Since Southern innovation targets imitated and Southern-led markets, theaggregate rate of Southern innovation is the intensity of Southern innovation times the measure of these markets: IS ≡ ιS (nM + nS ). The rate ofSouthern innovation relative to imitation isψ≡ιS (nM + nS )ιS (nM + nS )
ιS
IS
=
.
=
=
µ
µS nN
ιN (nM + nS )
ιN

(19)

By (16), the rate of Southern innovation relative to imitation reßects theintensity of Southern innovation relative to the intensity of Northern innovation. The rate of Southern innovation can be recovered from the rate ofimitation µ and the ratio ψ using IS ≡ ψµ. Overall, the aggregate rate

Imitation as a Stepping Stone to Innovation

12of innovation is the sum of the Northern and Southern rates of innovationι ≡ ιN + ιS (nM + nS ) ≡ ιN + IS . The aggregate rate of innovation is thefraction of all products that experience quality improvement at each point
in time.
3.3.

With Southern Innovation

The equations reduce to a system of Þve equations in Þve unknowns.
Assuming Northern innovation ιN > 0, Southern innovation ιS > 0, andimitation µS > 0 all occur, three conditions are required to ensure that costs
equal beneÞts for each form of R&D. Plugging proÞts (5) and producing
Þrm value (8) into the return to R&D condition (11) yields the Northern
innovation condition.
E (1 − wδ) = waN (ρ + ιN + µS )Similarly (6), (9), and (12) yield the imitation condition¶µ
1= aM (ρ + ιN + ιS )
E 1−
w

(20)

(21)

and (7), (10), and (13) yield the Southern innovation condition.
E (1 − δ) = aS (ρ + ιN + ιS )(22)

The resource constraints (14, 15), with the market measures substituted
out using the constant measure conditions (16, 17, 18), form the remaining
two equations.
As shown in the Appendix, the deÞnitions of the rate of imitation µ
and rate of Southern innovation relative to imitation ψ can be applied
to translate the system into rates rather than intensities, which proves
convenient for directly determining the eﬀects of the parameters on theaggregate rates of R&D. Also, the system can be reduced to four equations,with the relative wage determined separately.
The relative wage is exclusively determined to ensure that the returns
to Southern innovation and imitation are equal in equilibrium. If both
imitation and Southern innovation occur in equilibrium (µS > 0, ιS > 0),
the R&D conditions (21, 22) must hold with equality. The relative wage
must equal
1
(23)
w=
1 − γ (1 − δ)Imitation as a Stepping Stone to Innovation

13

so that the proÞts from imitation are less than proÞts from Southern innovation in proportion to the relative ease of imitation.
πM
aM
=
≡γ<1
πS
aS

(24)

Southern Þrms all have the same unit costs of production, but imitators sell
at price w due to competition from Northern Þrms capable of producing thesame quality level whereas innovators sell at price λ reßecting their quality
lead. The relative wage is assured of being above one w > 1 (wages higher
in North than South) provided innovations are quality improvements λ > 1
and imitation is easier than innovation in the South γ ≡ aM /aS < 1. The
relative wage is assured of being smaller than the quality increment w < λ(imitators charge a lower price than innovators) provided imitation is easierthan innovation.
3.4.

Without Southern Innovation

Before considering parameter changes, the key equations when Southern
innovation does not occur should be described so that the situations withand without Southern innovation can later be compared. If the resource
requirement in Southern innovation (following imitation) is large enoughaS > aS , no Southern Þrms will innovate ιS = 0 and thus there would
be no markets led by Southern Þrms nS = 0. The Southern innovation
condition would be an inequality.
E (1 − δ) < aS (ρ + ιN )(25)

The Northern resource constraint (14) is unaﬀected, and the Southern resource constraint (15) reduces toµ ¶E= LS .
(26)
aM µS nN + nM
w
The Northern innovation condition (20) is unaﬀected, and the imitation
condition reduces to
¶
µ
1
(27)
= aM (ρ + ιN ) .
E 1−wImitation as a Stepping Stone to Innovation

14

The steady-state conditions for the market measures to be constant areµS nN = ιN nM and nN + nM = 1, which lead to the substitutions nN =1 − nM and µS = ιN nM /(1 − nM ).
The two R&D conditions and the two resource constraints then determine Northern innovation ιN , the measure of imitated markets nM , aggregate expenditure E, and the relative wage w. The rate of imitation can be
determined from Northern innovation and the measure of imitated marketsµ ≡ µS nN = ιN nM . This scenario closely resembles the case of ineﬃcient
Northern followers in Grossman and Helpman (1991a). The key diﬀerenceis that with access to discarded technologies, Northern innovators target
all markets here, not just markets in which imitation has occurred.
Now the eﬀects of parameters such as resource availability on the endogenous variables can be determined to Þnd what forces can cause Southern innovation to increase.
4.

Innovation Through Imitation

In this section we explore how imitation, Northern innovation, Southern
innovation, Southern innovation relative to imitation, aggregate spending
and the relative wage depend on the parameters of the model. Southerncountries such as the East Asian Tigers are striving to push forward thetechnology frontier. They want to increase the mix of innovation in their
R&D. What parameters increase Southern innovation relative to imitation?
Can any of these forces increase Southern innovation without reducing aggregate innovation? The propositions below address these questions for thecase where Southern innovation does occur, but only following imitation.
The Southern government might have means of inßuencing the Southernlabor supply or the cost of conducting R&D in the South. The Southern labor supply represents all factors of production and is measured in eﬃciency
units (recall one unit of labor produces one unit of output). So increasing
the supply of Southern labor represents increasing the supply of any resource useful for R&D and production. It could stem from an increase in
human capital due to educational programs, for example. An expansion inSouthern resources could stem from entry of additional developing countries
into the world trading system. The Southern government could also oﬀer
subsidies to lower the cost of R&D. Since imitation is hard to distinguishImitation as a Stepping Stone to Innovation

15

from innovation, we examine a general R&D subsidy (to both innovationand imitation) in the South. Changes in resources and R&D subsidies in
the North could also aﬀect the amount of innovation done in the South.
To determine their eﬀects, introduce the parameter φN to represent anincrease in Northern resources, φS to represent an increase in Southern
resources, σN to represent a subsidy to Northern R&D, and σS to represent a subsidy to Southern R&D. As a result, (1 + φN ) LN is the Northern
resource supply, and (1 + φS ) LS is the Southern resource supply. Additionally, the term (1 − σN ) is added to the cost side of the Northern innovation
valuation condition (20), as the Northern government bears some of Northern Þrms’ innovation costs; the term (1 − σ S ) is added to the cost sideof the imitation and Southern innovation conditions (21) and (22), as theSouthern government bears some of the costs of Southern R&D.
4.1.

Northern Resources

Larger Northern resources support a faster rate of Northern innovation and more Northern production. Faster Northern innovation erodes
the South’s knowledge base, so the rate of Southern innovation relative toimitation falls. Aggregate expenditure rises ,which increases the resources
used in Northern production by increasing the sales of each Northern producer. The labor used in Northern production also rises due to the increasein Northern-led markets.
Proposition 1 The rate of imitation, the rate of Northern innovation, andthe aggregate rate of innovation increase while the rate of Southern innovation relative to imitation and the rate of Southern innovation decrease
with larger Northern resources φN . Aggregate expenditure rises, while therelative wage is unaﬀected. More markets become led by Northern Þrms,and fewer by Southern Þrms, with fewer imitated products.
4.2.

Southern Resources

Similarly, larger Southern resources support more Southern R&D andmore Southern production. An expansion in Southern resources causes
both Southern innovation and imitation to rise. Since imitation is needed
to provide the knowledge base for innovation, Southern innovation relative

Imitation as a Stepping Stone to Innovation

16

to imitation grows. Aggregate expenditure rises, which increases resource
demand in Southern production by increasing the sales of each Southern
producer. The labor used in Southern production also rises due to thedecrease in Northern-led markets (so more production occurs in the South).
Proposition 2 The rate of imitation, the rate of Southern innovation relative to imitation, the rate of Southern innovation, and the aggregate rateof innovation all increase with larger Southern resources φS . Aggregate expenditure rises, while the relative wage and the rate of Northern innovation
are unaﬀected. More markets become led by Southern Þrms, and fewer by
Northern Þrms, with more imitated products.
4.3.

Northern R&D Subsidy

A subsidy to Northern innovation reduces the cost of Northern innovation, which encourages Northern innovation. Aggregate expenditure falls to
reduce the reward to innovation to match the lower cost of innovation bornby Northern Þrms. A subsidy to Northern innovation encourages imitation
but reduces the mix of Southern innovation in Southern R&D.
Proposition 3 The rate of imitation, the rate of Northern innovation, andthe aggregate rate of innovation increase while the rate of Southern innovation relative to imitation and the rate of Southern innovation decrease witha larger Northern innovation subsidy σN . Aggregate expenditure falls, while
the relative wage is unaﬀected. Fewer markets become led by Northern andSouthern Þrms, so more products have been imitated.
4.4.

Southern R&D Subsidy

Imitation may be diﬃcult to distinguish from innovation when allocating R&D subsidies. The results below suggest that R&D incentives canencourage innovation, even if innovation cannot be distinguished from imitation when making subsidy payments.
A subsidy to Southern R&D causes Southern innovation to rise but
imitation to fall. Although the subsidy reduces the costs of imitation, theincrease in Southern innovation reduces the reward to imitation by making
the proÞts of imitators shorter in duration. The reward to imitation isImitation as a Stepping Stone to Innovation

17

further reduced by a reduction in aggregate expenditure, so proÞts are
smaller as well as shorter in duration.
Proposition 4 The rate of Southern innovation relative to imitation, therate of Southern innovation and the aggregate rate of innovation increase
while the rate of imitation decreases with a larger Southern R&D subsidy
σS . Aggregate expenditure falls, while the relative wage and the rate ofNorthern innovation are unaﬀected. More markets become led by Northernand Southern Þrms, so fewer products have been imitated.
Table 1 summarizes the sign of the eﬀects on the endogenous variables.
Table 1. Analytical Results

LN
LS
σN
σS

ιN
+
0
+
0

ψ
+
+

µ
+
+
+
-

IS
+
+

ι
+
+
+
+

E
+
+
-

w
0
0
0
0

nN
+
+

nM
+
+
-

nS
−
+
−
+

To sign the derivatives, the analytical results are done in the limit as thediscount rate approaches zero ρ → 0. The discount rate is presumably close
to zero and thus this practice should not much alter the results. However,
the eﬀects on Northern innovation that are zero above could be nonzero
for positive discounting. The eﬀects on the relative wage are necessarily
zero since they were determined prior to taking the limit. To explore therobustness of the results for ρ > 0, we turn to numerical solutions.
4.5.

Numerical Examples

When the discount rate approached zero, larger proÞts with a shorter
duration yielded the same reward to innovation when Southern resources
expand. For a positive discount rate, the larger size of the proÞts would
then dominate their shorter length (as the termination of proÞts occurs
at a lag), causing the reward to innovation to rise. So intuitively, for an
increase in Southern resources, Northern innovation should rise slightly.
On the other hand, for a Southern R&D subsidy, the smaller but longerproÞts should yield a somewhat smaller reward to innovation with discounting, so Northern innovation should fall slightly. Since the decrease inImitation as a Stepping Stone to Innovation

18

Northern innovation should be slight, the increase in Southern innovation
should still ensure that aggregate innovation increases.
Indeed, numerical solutions conÞrm this intuition regarding the eﬀects
on Northern innovation for a positive discount rate. The following results
are calculated for discount rate ρ = 1/25 = 4%, a quality increment λ = 3,
Northern resources LN = 4, Southern resources LS = 2, resource requirement in Northern innovation aN = 2, resource requirement in Southern
innovation aS = 3, and resource requirement in imitation aM = 1.
Starting from the base case, the parameter shifts considered are a 50
percent increase in Northern resources φN = 1/2, a 50 percent increase
in Southern resources φS = 1/2, a Northern R&D subsidy that pays onequarter of the R&D costs of Northern Þrms σN = 1/4, or a Southern R&D
subsidy that pays half of the R&D costs of Southern Þrms σS = 1/2. Table
2 reports the equilibrium values and the implied percentage changes in theendogenous variables relative to the base values (the relative wage remainsÞxed at w = 1.3).
Table 2A. Numerical Results

base
LN
LS
σN
σS

ιN
1.130
1.701
1.132
1.270
1.129

ψ
0.39
0.26
0.58
0.09
1.55

µ
0.32
0.35
0.42
0.40
0.23

IS
0.12
0.09
0.24
0.03
0.36

ι
1.25
1.79
1.37
1.31
1.49

E
7.2
9.8
8.2
6.4
6.6

nN
0.72
0.79
0.63
0.68
0.79

nM
0.20
0.16
0.23
0.29
0.08

nS
0.08
0.04
0.14
0.03
0.13

Table 2B. Percentage Changes

LN
LS
σN
σS

ιN
50%
0.13%
12%
-0.11%

ψ
-33%
50%
-77%
302%

µ
11%
32%
27%
-26%

IS
-26%
98%
-71%
196%

ι
43%
9%
4%
19%

E
36%
14%
-12%
-9%

nN
10%
-12%
-5%
10%

nM
-19%
16%
44%
-60%

nS
-46%
74%
-68%
61%

The eﬀects on Northern innovation, which are zero when ρ → 0, are allquite small when ρ > 0. Northern innovation rises slightly with Southern
resources and falls slightly with Southern R&D subsidies. Even thoughsubsidizing Southern R&D does reduce Northern innovation, Southern innovation expands by more so that aggregate innovation rises.

Imitation as a Stepping Stone to Innovation

4.6.

19

No Southern Innovation

We now brießy consider how the model behaves if Southern innovation
is too diﬃcult to occur even after imitation. Aggregate innovation is simply
Northern innovation when Southern innovation is absent. The results here
are identical to the case of ineﬃcient Northern followers in Grossman andHelpman (1991a): an increase in either Southern resources or a subsidy toimitation increases both innovation and imitation.
• The rate of innovation and the rate of imitation increase with larger
Northern resources φN . Aggregate expenditure and the relative wage
both increase. More markets become led by Northern Þrms, so fewer
products have been imitated.
• The rate of innovation and the rate of imitation increase with larger
Southern resources φS . Aggregate expenditure increases, while therelative wage decreases. Fewer markets become led by Northern Þrms,so more products have been imitated.
• The rate of innovation and the rate of imitation increase with alarger Northern imitation subsidy σN . Aggregate expenditure decreases, while the relative wage increases. Fewer markets become led
by Northern Þrms, so more products have been imitated.
• The rate of innovation and the rate of imitation increase with a larger
Southern imitation subsidy σS . Aggregate expenditure and the relative wage both decrease. More markets become led by NorthernÞrms, so fewer products have been imitated.
Table 3 summarizes the signs of these eﬀects for the case when Southern
innovation does not occur.
Table 3. Results When No Southern Innovation

LN
LS
σN
σS

ι
+
+
+
+

µ
+
+
+
+*

E
+
+
-

w
+*
-*
+*
-*

nN
+
+

nM
+
+
-

Imitation as a Stepping Stone to Innovation

20

The sign of cells marked by an asterisk * in Table 3 depends on whether
there is Southern innovation. The eﬀects on the relative wage, which were
zero with Southern innovation, are not in its absence. Increasing Northern
resources or a Northern R&D subsidy increase the relative wage, while
increasing Southern resources or a Southern R&D subsidy decreases therelative wage. When both Southern innovation and imitation occur, therelative wage adjusts to equalize their returns. Any policy that aﬀects
Southern R&D symmetrically (the same consequences for innovation as forimitation) will therefore not aﬀect the relative wage.
In the absence of Southern innovation, a subsidy to imitation lowers
the cost of imitation, and so the beneÞt of imitation must fall to restore
equality between the costs and beneÞts. A lower relative wage is one meansof lowering the reward to imitation: imitators charge a price equal to thecosts of the Northern Þrm whose product they imitated, which is the relative
wage, so the proÞts of an imitator are lower when the relative wage falls.
When Southern resources increase, a lower relative wage leads to a lower
price charged by imitators and thus larger sales, and thus more resources
absorbed into production in the South. A lower relative wage, by increasing
the demand for Southern resources, helps restore equality in the Southern
resource constraint when the supply of Southern resources expands.
The eﬀects on aggregate innovation are the same regardless of whether
innovation occurs in the South. Increased Southern resources or a subsidy
to Southern R&D increase aggregate innovation. However, the eﬀects on
imitation do depend on whether some innovation occurs in the South. A
subsidy to Southern R&D clearly expands imitation if there is no Southern innovation, but decreases imitation if Southern innovation then targets
products after an imitation has occurred. A subsidy, even though it lowers
the cost of imitation, may increase the onslaught of innovation to such a
degree that less imitation occurs in equilibrium.
The equilibrium without Southern innovation can be solved for the same
parameter values as for Table 2 but for aS > aS = 3.79 as needed for thecost of innovation to exceed its expected beneÞt in the South. Table 4reports the equilibrium values and their corresponding percentage changes
relative to the base values.

Imitation as a Stepping Stone to Innovation

21

Table 4A. Numerical Results When No Southern Innovation

base
LN
LS
σN
σS

µ
0.34
0.37
0.46
0.41
0.36

ι
1.18
1.75
1.20
1.28
1.25

E
6.9
9.5
7.8
6.3
6.3

w
1.21
1.23
1.19
1.27
1.11

nN
0.71
0.83
0.61
0.68
0.71

nM
0.29
0.17
0.39
0.32
0.29

Table 4B. Percentage Changes When No Southern Innovation

LN
LS
σN
σS

µ
8.5%
35%
20%
-5.0%

ι
49%
1.6%
8.8%
5.8%

E
37%
13%
-9.0%
-8.5%

w
1.6%
-2.1%
4.3%
-8.3%

nN
16%
-13%
-4.0%
0.3%

nM
-40%
33%
10%
-0.7%

These numerical values conÞrm the analytical results reported in Table
3 for the case without Southern innovation. Table 4 can also be used tocontrast the magnitude of eﬀects with and without Southern innovation
when the eﬀects share the same sign.
5.

Conclusion

This paper develops a model of Southern Þrms switching R&D from
imitation towards innovation. Imitation may be necessary for the South tobuild a suﬃcient knowledge base to make innovation attractive to Southern
Þrms. Absent the knowledge base from imitation, the expected reward to
Southern innovation would not cover the innovation cost.
In this setting, how does a Southern country, such South Korea or
Taiwan, shift its R&D toward innovation? One option is to encourage resource accumulation, which releases the Southern resource constraint andpermits all Southern R&D activities to grow. Subsidizing Southern R&D
also promotes both innovation and imitation in the South. Thus, Southern innovation can be promoted even if it is not possible to distinguishinnovation from imitation when allocating subsidies. However, when innovation and imitation both occur in the South, a general subsidy to SouthernImitation as a Stepping Stone to Innovation

22

R&D decreases Southern imitation due to the surge in Southern innovation(whereas subsidizing Southern imitation in the absence of Southern innovation increases imitation). Expanding Northern resources or Northern R&D
subsidies discourage Southern innovation.
Further research should explore other speciÞcations of how imitation canhelp to promote innovation in developing countries. Here, it was assumed
that every Northern innovation had to be imitated by a Southern Þrm
before innovation would occur for that product in the South. An alternative
speciÞcation could assume that imitation of a product by a Southern Þrm
permanently lowers the cost of innovation in the South, not just for thecurrent generation, but also for all further improvement, even if a Northern
innovation intercedes. Southern imitation would thus fall over time as fewer
products remain that have not yet been imitated. The interaction between
imitation and innovation in the South could diﬀer in such a setup.
A

Appendix

Substitute for the quality increment λ = 1/δ, the intensity of imitationµS = µ/nN , the intensity of Southern innovation ιS = ψµ/ (nM + nS ), themeasure of Northern-led markets nN = 1 − µ/ιN , the measure of imitatedmarkets nM = µ/ [ιN (1 + ψ)], the measure of Southern-led markets nS =
ψµ/ [ιN (1 + ψ)], and the relative wage (23). The remaining system of four
equations isaN ι2N + Eδ (ιN − µ) − (1 + φN ) ιN LN = 0(28)µιN aS (1 + ψ) (aM + ψaS ) − (1 + ψ) (1 + φS ) aS ιN LS+µE [aS (1 + ψδ) − aM (1 − δ)] = 0(29)

E (1 − δ) − aS (1 − σS ) [ρ + ιN (1 + ψ)] = 0

(31)

£
¤
E (1 − δ) [aS + aM ] (ιN − µ)−(1 − σN ) aN aS ρ (ιN − µ) + ι2N = 0(30)The four remaining variables are: aggregate spending E, the rate of Northern innovation ιN , the rate of imitation µ, and the rate of Southern inno-Imitation as a Stepping Stone to Innovation

23

vation relative to imitation ψ. Derivatives are generated by the system


 
 
dιN
0
b11 0 −δE δ (ιN − µ)
 b21 b22 b23
  dψ   c2 
b24


 
 
(32)
 b31 0  dµ  =  0 
b33
b34
b41 b42
0
1−δ
c4
dE
where

b11 = 2aN ιN + δE − (1 + φN ) LN
b21 = −aS (1 + ψ) [(1 + φS ) LS − µ (ψaS + aM )]b22 = −aS [(1 + φS ) ιN LS − µιN (aM + (1 + 2ψ) aS ) − µδE]
b23 = ιN aS (1 + ψ) (aM + ψaS ) + E [aS (1 + ψ) + (aS − aM ) (1 − δ)]b24 = µ [aS (1 + ψ) + (aS − aM ) (1 − δ)]b31 = E (1 − δ) [aS + aM ] − (1 − σN ) aN aS (ρ + 2ιN )b33 = −E (1 − δ) [aS + aM ] − (1 − σN ) aN aS ρ
b34 = (1 − δ) [aS + aM ] (ιN − µ)b41 = −aS (1 + ψ) (1 − σS )b42 = −aS ιN (1 − σS )c1 = ιN LN ∂φN
c2 = ιN (1 + ψ) LS ∂φS£¤
c3 = −aN aS ρ (ιN − µ) + ι2N ∂σN
c4 = − [ρ + ιN (1 + ψ)] aS ∂σS

The following derivatives are evaluated at σ N = φN = σS = φS = 0 and in
the limit as ρ → 0 for expositional ease since the discount rate should benear zero.
|B| =E 5 (1 − γ) (1 − δ)4 (ιN − µ) [1 − γ (1 − δ)]>0ι2N (1 + ψ)2(33)

Imitation as a Stepping Stone to Innovation

24

A1. Proof of Proposition 1
An increase in Northern resources increases the rate of Northern innovation∂ιN= ι2N > 0(34)
∂φN
has no eﬀect the relative wage ∂w/∂φN = 0, decreases the rate of Southern
innovation relative to imitation∂ψµ (1 + ψ)
=−
<0
(35)
∂φN
ιN
increases the rate of imitation
∂µ
µ2
=
>0
∂φN
ιN

(36)

increases aggregate spending∂E
E (ιN − µ)
=
>0
∂φN
ιN

(37)

as ιN − µ > 0 for nN > 0, decreases the rate of Southern innovation
∂IS
µ2
=−
<0
∂φN
ιN

increases the rate of innovation∂ι(ιN + µ) (ιN − µ)
=
>0
∂φN
ιN
increases the measure of Northern-led marketsµ∂nN=
nN > 0∂φNιNdecreases the measure of imitated markets
∂nM
µ
<0
=−
∂φN
ιN (1 + ψ)and decreases the measure of Southern-led markets.
∂nS
=
∂φN

(38)

(39)

(40)

(41)

(42)

Imitation as a Stepping Stone to Innovation

25

A2. Proof of Proposition 2
An increase in Southern resources has no eﬀect on the rate of Northern
innovation or the relative wage ∂ιN /∂φS = ∂w/∂φS = 0, increases the rateof Southern innovation relative to imitation
1+ψ
∂ψ
=
>0
∂φS
E

(43)

increases the rate of imitation
ιN − µ
∂µ
=
>0
∂φS
E

(44)

increases the rate of Southern innovation and the aggregate rate of innovation∂ιµ + ψιN
∂IS
>0
=
=
∂φS
∂φS
E

(45)

increases aggregate spending
∂E
=1>0
∂φS
decreases the measure of Northern-led markets
µ ¶
∂nN
∂µ
1
=−
<0
∂φS
ιN ∂φS
increases the measure of imitated markets
´
³
∂µ
µ
∂ψ
−
∂φ
1+ψ
∂φS
∂nM
1
1 − 2nNS==−> 0 ←→ nN >∂φSιN (1 + ψ)
E (1 + ψ)2
and increases the measure of Southern-led markets.
´
³
∂µ
µ
∂ψ
−
ψ
∂φS
1+ψ ∂φS
∂nS
>0
=
∂φS
ιN (1 + ψ)

(46)

(47)

(48)

(49)

Imitation as a Stepping Stone to Innovation

26

A3. Proof of Proposition 3
An increase in the subsidy to Northern innovation increases the rate ofNorthern innovation
ιN δ
∂ιN
=
>0
(50)
∂σN
1 − γ (1 − δ)
has no eﬀect on the relative wage ∂w/∂σN = 0, decreases the rate ofNorthern innovation relative to imitation∂ψ(1 + ψ) [(ιN − µ) [1 − γ (1 − δ)] + µδ]
<0
(51)
=−
∂σN
ιN [1 − γ (1 − δ)]increases the rate of imitationµ [(ιN − µ) [1 − γ (1 − δ)] + µδ]
∂µ
=
>0
∂σN
ιN [1 − γ (1 − δ)](52)

decreases aggregate spending
E (1 − δ) (ιN − µ) (1 − γ)
∂E
=−
<0
∂σN
ιN [1 − γ (1 − δ)](53)

decreases the rate of Southern innovationµ [(ιN − µ) [1 − γ (1 − δ)] + µδ]
∂IS
<0
=−
∂σN
ιN [1 − γ (1 − δ)](54)

and increases the rate of innovation (if λ is not too large)∂ι(ιN − µ) [διN − µ (1 − γ) (1 − δ)]=>0∂σNιN [1 − γ (1 − δ)]decreases the measure of Northern-led marketsµ (1 − δ) (1 − γ) nN
∂nN
=−
<0
∂σN
ιN [1 − γ (1 − δ)]increases the measure of imitated markets
∂nM
=
∂σN
and decreases the measure of Southern-led markets.
∂nS
=
∂σN

(55)

(56)

(57)

(58)

Imitation as a Stepping Stone to Innovation

27

A4. Proof of Proposition 4
An increase in the subsidy to Southern R&D has no eﬀect on the rateof Northern innovation or the relative wage ∂ιN /∂σS = ∂w/∂σ S = 0,
increases the rate of Southern innovation relative to imitation∂ψ= 1 + ψ − (1 − δ) (1 − nN ) > 0
∂σS

(59)

decreases the rate of imitation∂µµnN (1 − δ) (γ + ψ)
=−
<0
∂σS
1+ψ

(60)

increases the rate of Southern innovation and the aggregate rate of innovation
·
µ
¶µ
¶¸
∂IS
∂ι
µ/ιN + ψaM /aS + ψ== µ (1 + ψ) 1 − (1 − δ)
> 0(61)
∂σS
∂σS
1+ψ
1+ψ
decreases aggregate spending∂E= −µ (aM + aS ψ) < 0∂σSincreases the measure of Northern-led markets
µ ¶
∂nN
∂µ
1
=
>0
∂σS
ιN ∂σS
decreases the measure of imitated markets
´
³
∂µ
µ
∂ψ
−
∂σ S
1+ψ ∂σS
∂nM
=
<0
∂σS
ιN (1 + ψ)

(62)

(63)

(64)

and increases the measure of Southern-led markets, with C ≡ γ + ψ.
´
³
∂µ
µ
∂ψ
µ
¶
ψ
−
∂σ S
1+ψ ∂σS
1 − nN∂nS==[1 − (1 − δ) (1 − nN ) C] > 0(65)∂σSιN (1 + ψ)
1+ψImitation as a Stepping Stone to Innovation

28

A5. Northern Resources When No Southern InnovationDeÞne D ≡ 1 − wδ + (w − 1) [1 − nM (1 − wδ)] > 0. An increase in
Northern resources increases the relative wage
µw (w − 1) (1 − wδ)
∂w
=
>0
∂φN
D

(66)

increases the rate of imitation∂µµ2 (1 − wδ)
=
>0
∂φN
D

(67)

increases the aggregate rate of innovation
ι [(ι − µ) (w − 1) + ι (1 − wδ)]
∂ι
>0
=
∂φN
D

(68)

increases aggregate spending∂EEw (1 − δ) (ι − µ)
=
>0
∂φS
D
increases the measure of Northern-led markets∂nN
nN [(1 − wδ) + µ (w − 1)]
=
>0
∂φN
D

(69)

(70)

and therefore decreases the measure of imitated markets ∂nM /∂φN =−∂nN /∂φN < 0.
A6. Southern Resources When No Southern Innovation
An increase in Southern resources decreases the relative wage
wnM (w − 1) (1 − wδ)
∂w
=−
<0
∂φS
D
increases the rate of imitation
ιwnM [(1 − nM ) (1 − δ) + nM δ (w − 1)]
∂µ
=
>0
∂φS
D

(71)

(72)

increases the aggregate rate of innovation∂ιιwnM δ (w − 1)
>0
=
∂φS
D

(73)

Imitation as a Stepping Stone to Innovation

29

increases aggregate spending
EnM [1 − wδ + wδ (w − 1)]
∂E
=
>0
∂φS
D

(74)

increases the measure of imitated markets∂nM
wnM (1 − δ) (1 − nM )
=
>0
∂φS
D

(75)

and therefore decreases the measure of Northern-led markets ∂nN /∂φS =−∂nM /∂φS < 0.
A7. Subsidy to Northern R&D When No Southern Innovation
An increase in the subsidy to Northern R&D increases the relative wagew (w − 1) (1 − wδ) [ι − µ (1 − wδ)]
∂w
>0
=
∂σN
D

(76)

increases the rate of imitation∂µµ (1 − wδ) [ι − µ (1 − wδ)]
=
>0
∂σN
D

(77)

increases the aggregate rate of innovation∂ιwι2 δ (1 − wδ)
=
>0
∂σN
D

(78)

decreases aggregate spending
E (ι − µ) (1 − wδ)2
∂E
=−
<0
∂σN
D

(79)

decreases the measure of Northern-led markets∂nNµnN (1 − wδ)2
<0
=−
∂σN
D

(80)

and therefore increases the measure of imitated markets ∂nM /∂σN = −∂nN /∂σN >0.

Imitation as a Stepping Stone to Innovation

30

A8. Subsidy to Southern R&D When No Southern Innovation
An increase in the subsidy to Southern R&D decreases the relative wage
(w − 1) (1 − wδ) [w (1 − nM ) + nM ]
∂w
<0
=−
∂σS
D

(81)

increases the rate of imitation∂µιnM δ (w − 1)
=
>0
∂σS
D

(82)

increases the aggregate rate of innovation∂ιιδ (w − 1) [w (1 − nM ) + nM ]
=
>0
∂σS
D
decreases aggregate spending, with N ≡ w (1 − nM ) + nM
µ
¶µ
¶
∂E
1
wnM δ (w − 1) + (1 − wδ) N= −E 1 −
<0
∂σS
w
D

(83)

(84)

decreases the measure of imitated markets∂nM
nM δ (1 − nM ) (w − 1)2
<0
=−
∂σS
D

(85)

and therefore increases the measure of Northern-led markets ∂nN /∂σS =−∂nM /∂σS > 0.
A9. Innovation Independent of Southern Resources and R&D Subsidy
Propositions 2 and 4 showed that an increase in Southern resources or in
the Southern R&D subsidy had no eﬀect on the rate of Northern innovationin the limit as the discount rate approaches zero ρ → 0. To see why, solve
the Northern innovation valuation condition (20) for aggregate expenditureE=

waN (ρ + ιN + µS ).
1 − wδ

Insert this expression into the Northern labor constraint (14)¸¶·µ
waN (ρ + ιN + µS )ιN
δ = LN ,aN ιN +
ιN + ιS
1 − wδ

(86)(87)

Imitation as a Stepping Stone to Innovation

31

where the conditions for constant market measures require nN = ιN / (ιN + ιS ),and simplify
·
µ
¶µ
¶¸
ρ + ιN + µS
wδ
(88)
= LN
aN ιN 1 +
ιN + ιS
1 − wδ
·
µ
¶µ
¶¸
ρ
wδ
(89)
= LN .
aN ιN 1 + 1 +ιN + ιS
1 − wδ
As ρ → 0, the Northern labor constraint approaches·¸
wδ
aN ιN 1 += LN
1 − wδaN ιN= LN ,
1 − wδ

(90)(91)

which solves for a Northern rate of innovation ofιN =LN(1 − wδ) .
aN

(92)

This solution is independent of any endogenous variables yet to be determined other than the relative wage (23), which depends only on the size ofthe quality increment and the diﬃculty of imitation relative to Southern
innovation. This solution is also independent of Southern resources and theSouthern R&D subsidy. The solution for the rate of Northern innovation
does depend on Northern resources and the Northern R&D subsidy (thelatter through the solution for aggregate expenditure).
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