Innovation and Patentability Requirement in a Globalized World∗
Davin Chor†

Edwin L.-C. Lai ‡First draft: February 2009This draft: 18 November 2009
Preliminary Draft

Abstract
One important element of a patent regime is the non-obviousness requirement, which captures
the minimum improvement to the best patented technology a new invention is required to make inorder for it to be patentable. We call it the required inventive step. We explore the implicationsof a patenting regime based on required inventive step, by incorporating such intellectual property
protection considerations in the quality-improvement model of technology, trade and growth developedby Eaton and Kortum (European Economic Review 2001). In considering whether to increase theinventive step, policy-makers trade oﬀ the benefits of a higher rate of innovation against the lossof consumer surplus as patent-holders enjoy their monopoly pricing power for a longer duration on
average. We first establish that there exists an optimal binding inventive step that maximizes country
welfare under certain reasonable conditions. We then proceed to formulate an open-economy model,
in which countries interact through cross-border trade and firms patent internationally. We focus
on the central case that domestic goods and foreign goods are highly substitutable with each otherso as to contrast with earlier work of Grossman and Lai (American Economic Review 2004), whoanalyzed international IPR issues under an alternative regime based on patent length protection andproduct innovation. We find that there exists a stable Nash equilibrium of the inventive-step-setting
game between national governments that features over-protection of intellectual property from a global
perspective. Moreover, country governments set higher inventive steps in the open-economy relative
to the closed-economy benchmark. We discuss how these results relate to Grossman and Lai (2004).
Of note, globalization in the form of reduced trade frictions leads countries to raise their equilibrium
inventive steps. Consequently, globalization leads to higher equilibrium research intensities in all
countries.

Keywords: Intellectual property rights protection, Non-obviousness, Patentability, Required inventive step
JEL Classification: O31, O34, F13∗

We thank Angus Chu as well as participants of the Federal Reserve Bank of Dallas - Southern Methodist University
Joint Economics Conference, the Midwest International Economics Conference in University of Iowa, the Asia Pacific Trade
Seminars, and seminar participants in University of Leipzig. The work in this paper has been partially supported by theHong Kong Research Grants Council (Project No. CityU 1476/05H). All errors are our own.
†
School of Economics, Singapore Management University. Address: 90 Stamford Road, Singapore 178903. E-mail:davinchor@smu.edu.sg‡(Corresponding author) Department of Economics, School of Business and Management, Hong Kong University of
Science and Technology, Clear Water Bay, Kowloon, Hong Kong. Email: elai@ust.hk

1

Introduction and Motivation

As globalization deepens, the protection of intellectual property has become increasingly important todeveloped countries. This is because the law of comparative advantage has steered developed countries
towards specialization in R&D- and skill-intensive products, the profits from which are usually sensitive
to intellectual property rights (IPR) protection. Not surprisingly, this has created political pressures ondeveloped country governments to push for stronger IPR protection from less developed countries. A
major step in this direction was the signing of the TRIPS agreement in 1994, which sought to provide
a set of universal minimum standards for IPR protection and facilitate coordination among countries inthis area. It is widely recognized that TRIPS required developing countries (the South) to substantiallystrengthen their IPR standards to catch up with those of the developed world (the North).
Many studies suggest that the North would gain, while the South lose, from a strengthening of
IPR standards as a result of a TRIPS-like agreement. Nevertheless, would the world as a whole gainfrom better IPR protection in the South? Is there any theoretical basis for the view that international
coordination in the protection of IPR can improve global welfare? Grossman and Lai (2004) shed lighton this issue by modeling the international protection of IPR in a setting where innovation expands theavailable variety of diﬀerentiated products, and where newly-patented products are legally protected from
imitation for a specified duration termed the patent length. A key result in their paper is that a country
with both a larger domestic market for IPR-sensitive goods and a higher innovative capability would
implement stronger IPR protection in equilibrium. Moreover, they identify the existence of a positive
cross-border externality from raising protection, and hence conclude that there is global under-protectionof IPR in Nash equilibrium when country governments set their (enforcement-adjusted) patent lengths
non-cooperatively to maximize their own national welfare.
In practice however, patent protection is accorded in more ways than through the patent length.
Whereas “patent length” captures the duration of monopoly power granted by law, we focus instead in
this paper on an important alternative mode of IPR protection, which we term the “required inventivestep”: Many attempts at innovation often seek to build upon and improve existing products, rather
than to engineer a radically new item. To serve its purpose of encouraging innovation, it is therefore
also imperative for the IPR regime to protect patent-holders from incremental innovation that would
compete away their profits too easily. To this end, patent laws often include clauses – called “nonobviousness” requirements in the US patent code — that disqualify an innovator from obtaining a newpatent on the basis of minor or trivial changes to existing patents. The concept of “required inventivestep” therefore refers to how much improvement the patent regime requires an innovator to make overthe best patented technology before the new invention is eligible for a patent. In order to focus on theeﬀects of patentability, we assume that any invention satisfying the inventive step requirement will not1

infringe on any existing patents. In other words, the “leading breadth” is suﬃciently small that it isnever greater than the required inventive step.1 To the best of our knowledge, no work has yet been
done to study the incentives for governments to increase required inventive steps in an open-economy
setting, and the question remains to what extent the Grossman-Lai results would continue to hold when
IPR protection takes this form. This paper presents a first attempt to fill this gap in the literature.
We develop a framework to analyze international IPR issues when protection is based on required inventive step, by incorporating such patenting considerations into Eaton and Kortum’s (2001) benchmark
model of technology, growth and trade. The underlying model of technology in this framework is thesearch-based model of innovation of Kortum (1997), in which research eﬀort raises the cumulative stockof ideas, and innovation takes the form of new ideas that improve the quality of existing goods (rather
than an expansion in the measure of varieties).2 Here, the term quality is synonymous to the laborproductivity of the production technique for the good, as is the common usage in the endogenous growth
literature. In this setting, the natural mode of IPR protection by which to protect existing patent-holders
would be to stipulate a minimum quality improvement (or inventive step) that any innovation would needto satisfy in order to be eligible for a new patent. We introduce these considerations into the model in thefollowing way: If z is the quality (labor productivity) of the current best patented idea for a given variety,any innovation needs to be of quality at least Bz for it to be granted a new patent, where B ≥ 1 is acountry policy parameter. B therefore captures the required inventive step, along the quality dimension
specifically, that is required under the country’s IPR regime, since it is the minimum quality-improving
step that an innovation must attain before it can be considered suﬃciently non-obvious to be patentable.3
In particular, if B is strictly greater than one, then the inventive step requirement has some real bite.
We show how to build this formulation of IPR protection based on inventive step into a fully-specified
quality-improvement growth model, and use this to investigate how inventive step policy aﬀects innovation
and welfare in both closed and open economy settings. Of note, the stochastic treatment of the innovation
process gives rise to a distribution of the best patented idea that is a function of the current cumulative
stock of ideas and the inventive step policy parameter. The derivation of this patent frontier distribution
is a key step that makes the entire model and the subsequent welfare analysis tractable. This patent1

In the formal patenting literature, a “non-obviousness” requirement captures how much improvement an invention has
made relative to the prior art in order for it to be eligible for a new patent. On the other hand, “leading breadth” refers
to how much a new innovation needs to better an existing patent before it can be considered not infringing on the latter’spatent rights. (The innovation need not necessarily be seeking a patent, but the innovator would have to pay royalties tothe existing patent-holder in order to market her innovation.) These two concepts are closely related but distinct. Since
patenting is assumed to be costless in our model, any innovator whose invention satisfies the required inventive step willautomatically patent her new idea.
2
The model of Eaton and Kortum (2001) can be viewed as an extension of Kortum (1997) to a multi-country, openeconomy setting.
3
We assume that royalty payments are prohibitive, so that no innovator would consider infringing an existing patent. On
the other hand, the cost of filing a patent is assumed to be negligible, while it is easy for competitors to imitate a production
technique. This implies that any innovator would obtain a patent for her technology whenever and wherever it is patentable.

2

frontier is closely related to the concept of the technology frontier — the distribution of the most productive
idea currently available — in the standard Eaton and Kortum framework without patenting. (We discuss
this relationship at more length in Section 2.)
We first study the closed-economy version of our model, to obtain insights on the incentives forextending inventive step protection in this benchmark case. In the closed-economy setting, there is alreadya well-developed argument for the existence of an optimal patent length when innovation expands theavailable measure of varieties and IPR protection takes this alternative form (see for example, Nordhaus
1969, Tirole 1988, Grossman and Lai 2004): An increase in the patent length raises the incentives forand hence the rate of innovation, and this is traded oﬀ at the margin against the consumer surplus
conceded to innovating firms.4 We show that a similar existence result holds in our quality-based growthmodel, even though a more careful argument is now needed to establish that an increased inventive step
will indeed raise the rate of innovation when innovation improves upon existing varieties rather than
expanding the range of available products. This is because three distinct eﬀects are now at play. First,
for a given patent frontier, a higher inventive step B means that more resources need to be expended
to obtain a new idea that surpasses the inventive step requirement, an eﬀect which tends to lower the
incentives for innovation. On the other hand, a higher B itself leads to a patent frontier distribution thatreflects a lower average quality, since a larger margin of technologically-feasible ideas cannot be marketed;this actually increases the probability that a new idea can better the patent frontier by a given inventive
step, and thus raises the incentives for innovation. In our model, it turns out that these first two eﬀects
oﬀset each other exact, so that the probability that a new idea improves upon the patent frontier by astep of at least B is in fact independent of B. This leaves the third eﬀect as the salient one, this being
the direct role of a higher B in raising expected profits once a patent has been obtained, by allowing thepatent-holder to charge a higher markup, while making it more diﬃcult for competitors to leapfrog thepatent.
In sum, the benefit of raising the inventive step is that it increases the overall incentives for innovation,
so that consumers enjoy quality improvements at a faster rate. However, the cost of a higher B is that itconfers patent-holders the ability to charge prices above marginal cost for a longer duration on average,
until a better idea that surpasses the inventive step arrives. Our first key result shows that there exists
a unique inventive step that resolves this tradeoﬀ, and that this optimal B is binding (B ∗ > 1) undercertain reasonable conditions. In other words, there is a scope for government intervention through theuse of the inventive step as an instrument in IPR protection to improve country welfare.
Turning to the open-economy case, we now consider how cross-border interactions will aﬀect each
4

There have been several studies examining the optimal combination of patent breadth and patent length when the
IPR regime features both forms of protection. For example, O’Donoghue, Scotchmer and Thisse (1998) study this in aquality-improvement model with some dynamic considerations, but theirs is not a fully specified dynamic growth model.

3

country’s inventive step policies and equilibrium research intensities. We allow for international tradein goods, subject to the standard iceberg trade costs. Similarly, we allow for international patenting
by firms, whereby firms always have incentives to obtain a patent in each potential destination country
before they market there. We focus on the central case that domestic goods and foreign goods are
diﬀerentiated but highly substitutable with each other so as to contrast with earlier work of Grossman
and Lai (American Economic Review 2004), who analyzed international IPR issues under an alternative
regime based on patent length protection and product innovation. We examine a world with national
treatment in the granting of patents, so that countries do not discriminate across foreigners and award
patents by the same rules to both their own nationals and foreigners alike. We focus our attention on thestrategic interaction in the two-country case (for ease of tractability), when their governments set theirrespective inventive step policies non-cooperatively. We show that there exists a stable Nash equilibriumof this inventive step-setting game between national governments. Moreover, a formal analysis of thewelfare properties of this equilibrium confirms that it features over-protection of IPR, in the sense that
there is a scope for all countries to lower their inventive steps while achieving a Pareto improvement. This
result is in sharp contrast with that of Grossman and Lai (2004), who find that there is under-protectionof patent length in an expanding-varieties innovation model.
To understand the underlying intuition, it is important to grasp the nature of the cross-border externalities from a country’s choice of inventive step. In this paper, the measure of varieties is fixed andinnovation takes place along the quality dimension, and the natural concept of IPR protection is theinventive step. Firms from diﬀerent countries compete directly with each other by selling goods that arehighly substitutable with each other, and this direct competition gives rise to the negative cross-border
externality. When Country 1 raises its inventive step, B1 , this raises incentive for Country 1’s firms toinvest in R&D and hence also the relative productivity position that Country 1 has in the diﬀerentiatedgoods sector. On the other hand, it discourages Country 2’s firms from investing in R&D, as the profitability of research in country 2 now diminishes as the relative productivity position of Country 2’s firms
falls – as a result of the trade barriers and high substitutability between goods developed by Country-1
and Country-2. Overall, the technological progress enjoyed by consumers in Country 2 actually slows
down, leading to lower welfare for their consumers. This is the source of the negative externality ofCountry 1’s policy action on Country 2 welfare (Country 1 does not take this spillover into account when
deciding, B1 ). This underpins the over-protection result that we have derived regarding inventive step
policy.
Grossman and Lai (2004) on the other hand, consider product innovation that expands the existingmeasure of varieties in the diﬀerentiated goods sector, in which the IPR protection instrument consideredis the duration of the patent protection of each new variety. As a result, an increase in Country 1’s

4patent length benefits both domestic and foreign firms by increasing the value of any global patent.
This induces more product innovation from firms in all countries, even in the presence of trade barriers.
The increased utility from the larger set of varieties that Country 2 can consume as well as increased
profits for Country-2 firms selling in Country 1 generate a positive cross-border externality from raising
Country 1’s patent length, and hence implies a tendency towards the under-provision of patent length
protection. Consequently, the conclusions with regards to whether patent protection is under- or overprovided by country governments can be sensitive to the underlying structure of innovation process, thepatent instrument being considered, as well as the substitutability between domestic and foreign goods.
Our present model also facilitates an analysis of the eﬀects of globalization (the reduction of distancebarriers) on patent policies.5 It turns out that, when domestic goods and foreign goods are diﬀerentiatedbut highly substitutable with each other, the equilibrium inventive step of a country is larger when it
is open to trade and patenting by foreigners than when it is under autarky. In this sense, globalization
increases the incentive that each government has to increase its inventive step. Evidently, the onset ofglobalization leads to a higher marginal benefit from raising inventive step protection relative to theclosed-economy benchmark. Intuitively, globalization leads to increased competition for Country 1 firms,
as they engage in direct competition with Country 2 firms. In this setting, a higher inventive stepincreases the profitability of home firms for two reasons. First, domestic patent-holding firms enjoy alonger average duration of monopoly profits, just like under autarky. Second, as the home inventive stepincreases, home firms’ incentive to invest in R&D increases, while foreign firms’ incentive to invest in
R&D deceases. This in turn improves domestic firms’ productivity relative to foreign firms, thus allowing
domestic firms to compete more eﬀectively in the product market, increasing the profitability of researchin the home country. This latter eﬀect is naturally absent in the closed-economy setting, hence explaining
why inventive step protection confers a higher marginal benefit when countries can trade. Of note, thisresult also stands in contrast to Grossman and Lai (2004).
The concept of required inventive step that we apply in this paper draws upon an extensive literature
on patenting in industrial organization. O’Donoghue (1998) models patent design in the context of
cumulative innovation comprising a long sequence of product improvements. He finds that a patentability
requirement can lead to an improvement in social welfare, namely that the benefit from a higher rateof innovation can exceed the loss from conferring more monopoly power to firms. This is consistent
with our closed-economy result, although it is derived in a partial equilibrium setting.6 Green and
5

Grossman and Lai (2004) did not explicitly consider the role of distance barriers, so that all newly-innovated varieties
are equally accessible by consumers in all markets.
6
O’Donoghue (1998) also clarifies the distinction between a minimum patentable inventive step and a leading breadth
requirement. The former governs the need for a new invention to be suﬃciently novel before it can qualify for a patent.
The latter governs whether a new invention can be considered as infringing upon an existing patent, and hence whether theinventor will need to pay royalty payments to the existing patent-holder.

5

Scotchmer (1995) are among the first to model patent breadth in a quality-improvement innovation
setting. Similarly, Gallini (1992) proposed viewing the patent breadth as governing the cost of inventing
around the patent.7
Several existing papers have built on Kortum’s (1997) model of innovation to extend it to an openeconomy setting. Eaton and Kortum (1999) provides an alternative way to incorporate patenting in aquality-based innovation model, where patenting reduces the hazard rate of imitation by potential competitors. On a similar note, Eaton and Kortum (2006) explores issues related to the cross-border diﬀusionof ideas, but in the absence of patenting constraints. Our contribution is instead to model the patenting
regime explicitly through a inventive step requirement that specifies the minimum quality improvement
needed for a new idea to be patentable and therefore to be marketed without being imitated.8 It isalso useful to clarify that while our model features quality-improving innovation, it is strictly speaking
not a full model of cumulative innovation along the lines of the quality-ladder models of Grossman andHelpman (1991) and Aghion and Howitt (1992). In those papers, successful innovation confers a positive
externality on future research eﬀort since this eﬀort is concentrated on ideas that would strictly improveon the technology frontier. In contrast, our framework follows Kortum (1997) in that the ongoing search
process for future innovations can yield unsuccessful ideas that fall short of the patent frontier.
The paper proceeds as follows. Section 2 describes the innovation process based on Kortum (1997),
and derives the patent frontier distribution. Section 3 sets up the full model of patenting, growth andtrade. Section 4 analyzes the benchmark case of the closed-economy model. Section 5 tackles the
Nash equilibrium properties in the open-economy setting, focusing on the two-country symmetric case tohighlight the strategic interactions at play. Section 6 concludes. All detailed proofs are in the Appendix.

2

Innovation and the patent frontier

We first describe the innovation process, and show how this delivers a stochastic formulation of thepatent frontier. This will be a key building block when we spell out the full model in Section 3. We
derive the patent frontier in the closed-economy case, and then expand on this to develop useful results
for the multi-country case. Not surprisingly, the patent frontier distribution will depend on (among otherthings) the inventive step B enacted by the country government.
7

Note that our notion of required inventive step diﬀers from that in Gilbert and Shapiro (1990) and Klemperer (1990),
who study patent breadth under horizontal diﬀerentiation of products, rather than innovation along a quality dimension.
8
In this light, the model of Kortum (1997), in which any research idea that advances the technological frontier is
automatically patented, is essentially the special closed-economy case in which B ≡ 1.

6

2.1

The patent frontier in the closed economy

Consider a diﬀerentiated goods sector in which quality-based R&D takes place. There is a continuum ofvarieties in this sector, indexed by j ∈ [0, 1]. Since our focus is on innovation along the quality dimension,
we hold the measure of varieties constant at 1. Labor is the sole factor of production, and all production
technologies feature constant returns-to-scale. As a starting point, consider the closed-economy case. For
now, suppose that the prevailing wage in the country, w, is pinned down by conditions in the economy
that are exogenous from the perspective of this sector.
A defining feature of this diﬀerentiated goods sector is that productivity levels diﬀer across varieties.
Let zt (j) denote the productivity of the best patented idea for producing variety j at time t. Using
this best patented idea, 1/zt (j) is the labor input required to produce each unit of this variety, so thatthe corresponding unit production cost is: ct (j) = w/zt (j). On the other hand, denote by z̃t (j) the
productivity of the best technologically-feasible idea for producing this variety at time t. Note thatzt (j) ≤ z̃t (j), as not all feasible technologies will be eligible for a patent. Given the stochastic features of
the innovation process which we shall specify below, for each j ∈ [0, 1], zt (j) and z̃t (j) will be realizationsof random variables, whose cumulative distribution functions (cdfs) we denote by F (z; t) and F̃ (z; t)respectively.9 We refer to F (z; t) as the patent frontier distribution, and to F̃ (z; t) as the technology
frontier distribution.
Innovation: The innovation process that generates these technology and patent frontiers is basedon Kortum (1997) and Eaton and Kortum (2001). Innovation takes place along the quality dimension.
The actual innovation event itself is the outcome of a stochastic search process for ideas that potentially
improve upon the productivity of the current technologies. These ideas are generated by the researchsector in this economy. Specifically, each worker in the research sector obtains ideas at a Poisson arrival
rate α. These ideas represent productivity levels, q, drawn from a stationary search distribution, H(q).
The specific variety j to which each idea applies is determined by a random draw from the uniform
distribution on [0, 1], so that the varieties are symmetric from the perspective of the innovation process.10
Let Rt ≤ Lt be the number of workers employed in the research sector at time t, with Lt being the total
workforce of the country. (The remaining Lt − Rt workers are employed in production.) The cumulative
Rt
stock of ideas in the knowledge pool of the economy at time t is thus: T (t) = α 0 Rs ds, where we

initialize time at t = 0, with T (0) = 0.

To complete the description of the innovation process, we follow Kortum (1997) in specifying thestationary search distribution to be the standard Pareto distribution: H(q) = 1 − q −θ , q ≥ 1.11 As is9

F (z; t) and F̃ (z; t) do not depend on j, since the varieties are ex ante symmetric.
We rule out the possibility of directed search or innovation targeted towards specific goods, which would significantly
complicate the considerations involved.
11
That this search distribution is fixed across time periods means that researchers may sample ideas that are below
10

7

well-known, the Pareto search distribution gives rise to a technology frontier distribution across varieties©
ªj ∈ [0, 1] which is Fréchet with cdf: F̃ (z; t) = exp −T (t)z −θ .12 This distribution is characterized by
two key parameters: The scale parameter, T (t), governs the mean of the distribution, so that an economy

with a larger stock of ideas will on average feature a more productive technology frontier. On the other
hand, a larger shape parameter, θ, corresponds to a H(q) distribution with a lower variance, meaningthat researchers draw from a search distribution that places less weight on high productivity draws.13
For this reason, θ is also referred to as the inverse spread parameter, as it leads to a technology frontier
distribution with a thinner right-tail.
Patenting: We extend this treatment of the innovation process by incorporating IPR considerationsin the form of a inventive step requirement. As imitation costs of domestically sold goods are zero,
innovators with new ideas need to apply for a patent before the idea can be marketed without being
imitated. To improve the incentives for innovation, the patent authority or government protects existing
patent-holders by stipulating that a new idea can only be patented if it improves upon the productivityof the current best patented idea by a minimum inventive step of B ≥ 1. Formally, if the current best
patented idea for variety j is of productivity z, a new patent will only be awarded if an idea of productivity
at least Bz is drawn.
We refer to B as the required inventive step of the country’s IPR regime. This inventive step requirement oﬀers protection to patent-holders in that one cannot obtain a new patent with a quality improvement smaller than the required inventive step, as only ideas that exceed Bz in quality are awarded a newpatent. Thus, at any one time, for each variety j, there is (generically) one patent-holder who commands
the best available patent and the market for that variety. (We assume that there are no costs in applying
for and enforcing a patent, so that all innovators will patent their technology whenever and wherever it
qualifies for a patent.)
The stochastic arrival of ideas and the inventive step requirement jointly give rise to a patent frontier
distribution, F (z; t), across varieties j ∈ [0, 1]. Our first key result derives a parametric form for thisthe current technological frontier, so that innovation is not truly cumulative in the sense of the quality-ladder models ofGrossman and Helpman (1991) and Aghion and Howitt (1992). Nevertheless, Kortum (1997) shows that parameterizing
the search distribution as Pareto gives rise to a steady state that features a constant rate of patenting in spite of a constantpositive growth rate in the research workforce, matching the US growth experience in the past half-century.
12
See Eaton and Kortum (2001). This is derived by considering the probability that exactly k ideas with quality ≤ z are
drawn when the stock of ideas is T (t), and summing over all k ≥ 0:F̃ (z; t) =∞[exp {−T (t)} (T (t))kk=0k!

H(z)k = exp {−T (t)}∞[(T (t)(1 − z −θ ))k
k=0

k!

r
q
= exp −T (t)z −θ

The above holds for z ≥ 1, which is the support of the Pareto search distribution. Kortum (1997) shows that this cdf
remains asymptotically valid for all z > 0, as the probability mass for 0 < z < 1 becomes infinitesimally small as the stockof ideas grows large.
13
The mean of log Z is θ1 (γ + log T (t)), which is increasing in T (t); γ is the Euler-Mascheroni constant. Also, the standardsdeviation of log Z is π/(θ (6)), which is decreasing in θ.

8patent frontier distribution:
Lemma 1: Suppose that T 0 (t) is bounded. Then, as the stock of ideas grows arbitrarily large (T (t) → ∞),the patent frontier at time t across varieties j ∈ [0, 1] has a Fréchet distribution with cdf F (z; t) =onexp −T (t) (Bz)−θ , where B is the inventive step established by the country.
Proof. See the Appendix. Heuristically, the proof sets up a diﬀerential equation with respect to t for
F (z; t) that captures the evolution of the patent frontier over small time periods. Solving this diﬀerential
equation yields the above functional form for F (z; t). (This is similar to the solution strategy in Evenson
and Kislev (1976), Kortum (1997), and Alvarez et al. (2008).)
The patent frontier therefore belongs to the family of Fréchet distributions with the same shapeparameter as the technology frontier, but with a (weakly) smaller scale parameter, T (t)B −θ , since B ≥ 1.
Intuitively, the patent frontier features ideas that are on average less productive than those on thetechnology frontier, as some ideas which are technologically feasible are not patented because they failto surpass the inventive step requirement. Of note, a larger inventive step (a higher B) tends to restrict
more ideas from being patented, and hence results in a patent frontier distribution with a lower mean
productivity.

2.2

The patent frontier in the open-economy

The above discussion demonstrates how one can introduce inventive step considerations into the qualityimprovement growth model of Kortum (1997), while retaining much of the convenience of the stochastic
formulation of innovation in that model. We now extend this discussion to a setting with N ≥ 1 countries.
When there is international trade, the diﬀerentiated goods sector in each country now has available alarger set of varieties of goods than that available in a closed economy. In particular, firms in eachof the N countries are producers of their own unit measure of varieties ji ∈ [0, 1], i = 1, 2, ...N , andsell them to all countries in the world. In doing so, they compete with not only varieties produced byfirms in their own countries but also those produced by foreign firms. Instead of having varieties ofmeasure one available to them, consumers in each country now are faced with a wider choice of varieties
with a measure of N . The aggregate utility derived from the sets of varieties originated from diﬀerent
countries are diﬀerentiated technologically and utility-wise. In the central case we consider, they arehighly substitutable with each other.
Innovation: Production ideas emerge from the research sectors in all the N countries. The ideagenerating process within each country is similar: Each worker in the research sector obtains production
ideas with a Poisson arrival rate of αi , where i denotes the country. When an idea arrives, the specific9variety j to which it applies is a random draw from a standard uniform distribution, while the qualityof the idea is drawn from the stationary Pareto search distribution, H(q) = 1 − q −θ , q ≥ 1. Letting
Rit denote the number of workers in country i employed in its research sector at time t, the current
Rt
cumulative stock of ideas generated by country i is then: Ti (t) = αi s=0 Rs ds. This innovation process

yields a technological frontier within each country, defined as the most productive idea obtained by thatcountry’s research sector. As we have seen, this technological frontier across varieties j is described by the
Fréchet cdf, exp{Ti (t)z −θ }, where the relevant scale parameter is the cumulative idea stock attributable
to country i. Our task moving forward is to understand how patenting aﬀects this innovation process.
Patenting: The open-economy model features a global market for ideas. Varieties originated from
diﬀerent countries are technologically and utility-wise diﬀerentiated from each other. They are therefore
treated as distinct varieties by all patent authorities. A producer seeking to market a diﬀerentiatedvariety in a country can obtain a patent from that country’s patent authority (regardless of the countryof origin of the producer). IPR protection takes the form of an inventive step in each country: If theexisting best patented idea in country n for a particular variety is of productivity z, a new patent will
only be awarded if the quality of the new idea is at least Bn z, where Bn ≥ 1 is the required inventive
step established by country n’s patent board.
We adopt a setting in which international patenting is governed by the principles of non-discriminationand national treatment. Each country therefore accords all ideas the same treatment under its patenting
regime, regardless of their country of origin. In particular, the same procedure for determining whether
an idea qualifies for a patent — namely, whether that idea exceeds the minimum inventive step of Bn — isapplied to innovations from all countries. Apart from the patenting requirement, we assume that there
are no other impediments to the global flow of ideas. In particular, there are no costs to obtaining apatent in any country. This implies that producers will seek out and obtain a patent for their productsin every market where it is patentable. The imitation cost of a good that is sold locally is zero, but theimitation cost of a good that is sold only in foreign countries is prohibitive, even if the technology has
been patented there.14 Goods from diﬀerent countries are diﬀerentiated and treated as technologicallydistinct by patent authorities.
The following lemma states the distribution of the patent frontier in a country n fro varieties originatedfrom each source country i:
Lemma 2: Suppose that Ti0 (t) is bounded for all countries i. Then, as the stock of ideas from eachcountry grows arbitrarily large (Ti (t) → ∞), under national treatment, the patent frontier in countryn at time t across varieties originated from country i ji ∈ [0, 1] has a Fréchet distribution with cdf:
14

Specifically, the cost of imitating a technology of a good that is sold only in a foreign country (even if it is patented
there) and marketing it at home is higher than the expected profit from obtaining a new patent in the home country.

10

o
n
exp −Ti (t) (Bn z)−θ , where Bn is the inventive step established by country n.
Proof. This follows directly from Lemma 1. Since there are no barriers to international patenting of ideas
in the open-economy, the result follows by replacing T (t) by Ti (t) and B by Bn in the closed-economy
patent frontier distribution from Lemma 1.
This is a close parallel of the expression for the patent frontier in the closed-economy case, beingfrom the same family of Fréchet distributions with shape parameter θ. Note that the scale parameter is
now Ti (t)Bn−θ , since the relevant stock of ideas is that which originates from country i only. Naturally,
the mean of this best patented idea distribution is increasing in the source country stock of ideas, anddecreasing in the inventive step in the recipient country.

3

Setting up the full model of innovation, patenting and trade

Having laid out the underlying innovation and patenting process, we proceed now to set up the full model.
Here, we build on the generality that the Eaton-Kortum framework aﬀords us to develop the model for
the N -country case (where N ≥ 1). The case of N = 1 corresponds to the closed-economy setting, which
will be analyzed in Section 4. We also focus on the two-country case in more detail in Section 5, in orderto derive more explicit results regarding inventive steps in the Nash equilibrium, although much of theintuition here is applicable to the general N -country case.

3.1

The market for goods

Utility: There are two sectors in the world economy: (i) a homogeneous goods sector, and (ii) adiﬀerentiated goods sector featuring N sets of continuum of varieties ji ∈ [0, 1] for i = 1, 2, ...N where icorresponds to country i. The preferences of the representative consumer in each country n ∈ {1, . . . , N }at time t are given by following Cobb-Douglas utility function:·´β ¸ β1 1−ηR1
PN ³
µ ¶η 

i=1 exp{ 0 ln xnit (j)dj}
ynt


Unt =


η
1−η



(1)

Here, i indexes the country of origin of the diﬀerentiated variety, while n denotes the country of consumption. The elasticity of substitution that prevails between the consumption of varieties that originate from
diﬀerent countries is: φ = 1/(1 − β), where 0 < β < 1. For each country, utility of varieties is aggregated,with elasticity of substitution equal to 1. The aggregates of diﬀerentiated goods from diﬀerent countries
are imperfect substitutes of each other. Here, ynt denotes consumption of the homogenous good, andxnit (j) denotes consumption of variety j of diﬀerentiated goods from country i. A constant share of11

income η ∈ (0, 1) is devoted to purchases of the homogenous good. In equilibrium, the expenditure on
varieties originated from country i is distributed equally among these varieties. The diﬀerentiated goods
sector consists of varieties from all N countries. The diﬀerentiated goods originated from each country
is as described in Section 2, and features innovation and patenting over time. On the other hand, there
is no technological progress in the homogenous goods sector; for this reason, we shall also refer to this
sector as the “traditional” sector, in the sense that no R&D eﬀort is channeled towards it.
Production and market structure: The traditional sector has a simple constant returns-to-scale
structure. In country n, production of this good uses an units of labor. Trade in this good encounters nocross-border frictions, so its price is equalized across all countries and can be normalized to one. In what
follows, we shall focus on equilibria in which a positive quantity of this numeraire good is produced ineach country, so that the wage in country n is given by wn = 1/an , being pinned down by technologicalconditions in this traditional sector.15 In practice, this means that the labor force Ln in each country,
as well as the consumer demand for homogenous goods (captured by η), must all be suﬃciently large, tosupport such an equilibrium. (We shall make these parameter conditions more explicit when we explore
the two-country model in Section 5.)
Production in the diﬀerentiated goods sector likewise takes place under constant returns-to-scale.
In the multi-country world, however, countries will in general diﬀer in their productivity levels. These
underlying productivity diﬀerences generate a Ricardian basis for trade between countries, as varieties
can potentially be sourced from each of the N countries. However, the flow of goods across borders isimpeded by transport costs of the iceberg form: dni ≥ 1 units of a goods must be shipped out fromcountry i in order for one unit to arrive in country n.16 We make the standard assumptions that: (i)dii = 1 for all i, so that there are no shipping costs within each country; and (ii) dni ≤ dnk dki for all
k 6= i, n, so that transhipment through a third-country port is not pursued.
(1)

Let znit (j) denote the productivity of the best idea originating from country i which has successfully
obtained a patent in country n at time t for variety ji . The marginal cost of production for this patentholder is: cnit = wi dni /znit (j), after taking the labor and shipping costs into account. To determine themarket outcome in country n, note that only producers who hold a country-n patent will participate
in that market in equilibrium. For each variety ji from country i, all producers from i who hold acountry-n patent for that variety engage in head-to-head Bertrand competition. The country i producer
who possesses the patented idea with the lowest marginal cost will thus command the entire country-nmarket for variety ji , and limit-prices at the level of the patent-holder with the second-lowest marginal
15

It will turn out that when all countries have the same labor force growth rate, the steady state exhibits a constant wagein each country that does not vary over time, so that we need not attach a time subscript to wn . The assumption of aconstant marginal product of labor is thus consistent with the long-run equilibrium of the model.
16
We follow the convention in EK, where the second subscript denotes the exporting country, while the first subscript
refers to the importing country.

12cost. This rule ensures that all other country i patent-holders with less cost-eﬃcient ideas cannot earn
positive profits, and therefore leave the market to the lowest-cost patent-holder as the sole supplier.
We seek to evaluate the distribution of market prices for diﬀerentiated varieties, in order to later
derive an expression for welfare. Given the limit-pricing rule, it follows that we need to specify thestochastic distribution that jointly describes both the best and second-best patented ideas that originate
from country i in country n. We have already seen in Lemma 2 that the distribution of this best patented
(1)

(1)

idea from i in n, denoted by the random variable Znit (j), is given by the Fréchet cdf: P r(Znit (j) ≤ z) =onBPni (z; t) = exp −Ti (t) (Bn z)−θ , with Bn being the inventive step required by country n. Moving
forward, following Bernard et al. (2003), we specify the joint distribution of the best and second-bestpatented ideas from country i in country n to be the multivariate analogue of the Fréchet distribution in
Lemma 2:
´i
n
o
h
³
(1)
(2)
P r(Znit ≤ z1 , Znit ≤ z2 ) = 1 + Ti (t) (Bn2 z2 )−θ − (Bn z1 )−θ exp −T (Bn2 z2 )−θ , 0 ≤ Bn z2 ≤ z1 (2)Note that we require Bn z2 ≤ z1 , since the best patented idea needs to satisfy the inventive step require´³(1)
ment vis-à-vis the second-best patented idea. Also, as Bn z2 → z1 , (2) reduces to: Pr Znit ≤ z1 =©
ªexp −Ti (t)(Bn z1 )−θ , which is precisely the best patented idea distribution from Lemma 2.
Welfare: The joint productivity distribution for the best and second-best patented ideas from eachcountry given by (2) implies that the price of each variety ji originated from country i in country n, pnit (j),R1can be viewed as a realization from a price distribution. Define Pnit = exp{ 0 ln pnit (j)dj}. This is theideal price index of country i diﬀerentiated goods in country n. Then, solving the utility-maximization
problem based on (1), one can show that the demand xnit (j) for this variety by a country-n individual

who earns wage wn is:
xnit (j) =1−φ(1 − η)wn Pnit
1PN
1−φ p (j)nit
k=1 Pnkt(3)

In particular, if φ > 1, then when Pnit increase, the demand level for each variety from country i incountry n will fall (holding pnit (j) constant).
Let’s now specialize to the two-country case. Define d as the iceberg trade cost between the twocountries. Based on the expression for the ideal price index in the closed-economy case derived in the−1

Appendix (setting N = 1), we have: Pnit = Ti θ wi dni Bn2 γ, where γ is a constant that does not dependon Bn . In particular:−1

P11t = T1t θ w1 B12 γ−1

P12t = T2t θ (w2 d)B12 γ−1

P21t = T1t θ (w1 d)B22 γ−1

P22t = T2t θ w2 B22 γ
13(4)

Based on (1), welfare (the real wage) is now given by:
Un = ³PN

wn

1−φ
i=1 Pni

´ 1−η

(5)

1−φ

The task of maximizing welfare thus boils down to the task of maximizing

³P
N

1−φ
i=1 Pni´− 11−φ, ie mini-

mizing the price index. Using the expressions for the price indices, this reduces in the two-country caseto:max B1−2
B1

µ
¶− 1
1−φ
− 1θ
− θ1
1−φ
1−φ
(T1 w1 )+ (T2 w2 d)(6)

max B2−2
B2

µ
¶− 1
1−φ
− 1θ
− θ1
1−φ
1−φ
(T1 w1 d)+ (T2 w2 )

(7)

and

The above welfare expressions highlight the key tradeoﬀ involved in inventive step policy. On the one
hand, as we shall soon see below, a larger inventive step (a higher Bn ) will raise research incentives inall countries, and will hence increase the stock of ideas (each country’s Ti ) in the aggregate. However,this decreases consumer surplus, as a larger inventive step protects incumbent patent-holders for a longer
duration on average, by preventing a larger margin of lower-priced ideas from securing a patent. This
loss is captured by the Bn−2 term (n = 1, 2) in (6) and (7), which reflects the eﬀect of a higher Bn on
welfare holding the stock of ideas constant. The optimal patent policy will thus be the choice of Bn that
equates the marginal benefit of a larger idea stock against the marginal cost of the consumer surplus lost.

3.2

Research incentives and the country idea stocks

We turn now to the task of understanding this steady-state relationship between the inventive step policiesof the N countries and the country idea stocks. This is a key step towards characterizing the benefitof raising IPR protection through a higher inventive step, and hence towards evaluating the optimal
inventive step. This requires us to consider the incentives faced by the research sector, specifically how
innovation activities are aﬀected when the inventive step is raised.
Markups and profits: Since innovation eﬀort is motivated by the expectation of a future flowof profit returns, we compute the expected profits of diﬀerentiated varieties for firms in each country.
Recall that varieties originated from diﬀerent countries are technologically diﬀerentiated and are treatedas diﬀerent goods by the patent authorities. Therefore, only innovators from i who comes up with an
idea for variety ji ∈ [0, 1] will be considered for a patent in country n for variety ji . Let bnit (m) be theprobability that a given idea originating from country i undercuts the current best patented idea in nfrom i by a factor of at least m, where m > 1. Then:(µ ¶θ )
Z ∞m1bnit (m) =exp −Ti (t)Bn−θθq −θ−1 dq ' −θq
Bn mθ Ti (t)
1
14

(8)

where we have used the fact that the probability of the best patented idea in country n from country i©
ªat time t being ≤ c is equal to: 1 − exp −Ti (t)Bn−θ cθ .17 The approximation taken in (8) is valid when

Ti (t) → ∞.18

Observe in particular that the probability of a new idea from country i being eligible for a patentin country n is: bnit (Bn ) = Ti1(t) . Not surprisingly, the larger the eﬀective stock of ideas available incountry i (a higher Ti (t)), the more diﬃcult it is for a new idea to secure a patent in country n. Moreinterestingly, this probability turns out to be independent of the inventive step, Bn . There are in fact two
countervailing eﬀects of an increase in Bn . First, given the current best patented idea from i in n, a larger
inventive step tends to reduce the probability that a new idea from i would qualify for a patent in n. On
the other hand, however, this would also lead to a patent frontier distribution of technologies from i inn with a lower mean productivity level, since a larger margin of ideas which are technologically feasible
are not patented when Bn is high; this tends to raise the probability of an idea from i of successfully
qualifying for a patent in n. In our model, it turns out these two eﬀects oﬀset each other exactly, sothat Bn does not aﬀect the probability of a new idea from i obtaining a country-n patent, as asserted in
the introduction. As discussed in the Introduction, the inventive step set by a country thus aﬀects the
expected value of an idea only through its impact on the profits accruing to the innovator conditionalon successfully obtaining a patent in that country. On this last count, we know that a higher Bn tends
to raise flow profits, both because the successfully-patented idea would command a higher markup on
average, and because it makes it harder for competitor ideas to displace the patent-holder. A larger
inventive step therefore raises the overall incentive to innovate in our model.
The markup that a new patented idea from country i can command in country n is stochasticallydetermined, and is given by:bnit (Bn ) − bnit (m)Pr [M ≤ m | M ≥ Bn ] ==1−bnit (Bn )

µ

m
Bn

¶−θ

,

m ≥ Bn(9)This markup thus inherits a Pareto distribution; naturally, its support is defined only for m ≥ Bn , since
any newly-patented idea must by definition secure a markup of at least Bn vis-à-vis the second-bestpatented idea. Of note, conditional on the idea surpassing the inventive step, the distribution of themarkup that the new idea commands does not depend on its country of origin.
We are now in a position to compute the profits that accrue to successful innovators.
Let Xnit be the total expenditure by country-n residents on diﬀerentiated goods originated fromcountry i. Let Πnit denote the profits earned by country i producers from sales in the country n market.
17

Let Gni (c) denote the cost distribution for the best patented idea in country n from country i. Then Gni (c) =1 − exp −Tit Bn−θ cθ .
18
There is a discrepancy in that the stationary search distribution is defined for ideas q ≥ 1, whereas the patent frontier
distribution is defined for all z ≥ 0. Kortum (1997) shows that the probability mass for ideas with productivity between 0
and 1 can be ignored as Ti (t) → ∞.

15

Using the markup distribution (9), we have:
·
µ
¶¸
Z ∞
¡
¢
Bn − 1Xnit
1+θΠnit = Xnit
(10)
1 − m−1 θm−θ−1 Bnθ dm =
1+θ
Bn
Bn
h
³
´i
1
Profits are therefore equal to the fraction 1+θ
1 + θ BBn −1∈ (0, 1) of consumer expenditures on thendiﬀerentiated goods. Note the natural properties that a more protective patent regime (a higher Bn ) anda larger variance for the distribution of ideas (a smaller θ) both lead to a larger profit share.19
Using (3), (4) and (10), we can deduce that:
X11t
X12t
X21t
X22t

=

Π11t
=
Π12t

=

Π21t
=
Π22t

µ
µ

P11t
P12t
P21t
P22t

¶1−φ
¶1−φ

=
=

µ
µ

T1
T2
T1
T2

¶ φ−1 µθ

¶ φ−1 µ
θ

w2 d
w1
w2
w1 d

¶φ−1

(11)

¶φ−1

(12)

These expenditures and profits in the diﬀerentiated goods sector are further linked as a matter ofnational income accounting. Specifically, consumption expenditure on all goods (including the numerairegood) from all countries must be equal to the sum total of labor income in the production sector andincome earned from profits. In the two-country case, this means that:
X11t + X12t
1−η
X21t + X22t1−η

= (1 − r1 ) w1 L1t + Π11t + Π21t
= (1 − r2 ) w2 L2t + Π12t + Π22t

Note that rit = Rit /Lit is the share of the labor force engaged in the research sector in country i at timet, so that 1 − rit is the relevant share of workers who earn the labor wage wi .

B(1+θ)
1
Define h(B) = 1−ηB(1+θ)−θ . Then, using (10), we can re-write the above equations as:h(B1 )(Π11t + Π12t ) = (1 − r1 ) w1 L1t + Π11t + Π21t
h(B2 )(Π21t + Π22t ) = (1 − r2 ) w2 L2t + Π12t + Π22t

(13)

Research incentives: To close the model fully, we need to specify how profits motivate innovation
activities. Let Vi denote the expected value of an idea that originates in country i. In equilibrium, theexpected value of such an idea multiplied by the arrival rate of ideas per worker needs to be equal to thedomestic wage: αi Vit = wi , so that workers are indiﬀerent between engaging in production or research
activities. In turn, the expected value of an idea originated from country i, Vit , is the expected discounted
flow of future profits in each of the N potential consumer markets:
Vit = PitZ ∞t−ρ(s−t)eN
X

Πnis bnis (1)ds =bnit (Bn ) ×
Pis bnit (Bn )
n=1

19

Z ∞
t

−ρ(s−t) PiteNX

Pis n=1

Πnis

Bnθ
ds
Ti (s)(14)

In the limiting case where Bn = 1, (10) implies a profit share of 1/(1 + θ), consistent with Eaton and Kortum (2001)’smodel without patenting.

16

The term ΠPns
captures the real profits earned in country n at time s; these are pre-multiplied by Pit
isnis (1)
to express these profits in terms of time-t prices. Note also that bbnit(Bn ) is the probability that theinnovation continues to have the market to itself at time s > t, conditional on its having secured themarket at time t. (The numerator is bnis (1) since the incumbent patent-holder does not need to satisfy
the required inventive step constraint vis-à-vis new entering ideas.) For simplicity, all countries share thesame discount rate parameter ρ, where ρ satisfies: ρ > gθL (1 − η), to guarantee the existence of a steadystate where the country idea stocks are finite.
We evaluate Vit for a steady state in which: (i) the labor force grows in all countries at a constantrate, gL (so that L̇i /Li = gL for all i), and (ii) the share of labor in the research sector is constant in
Rteach country (so that rit = ri ). Since Ti (t) = αi 0 Ris ds, the time path of the stock of ideas is therefore

governed by: Ṫi (t) = αi ri Lit . This implies that T /L converges to a constant value, αgiLri , in the steadystate. The stock of ideas therefore grows at the same rate as the labor force: Ṫ /T = L̇/L = gL . Now,µ¶ 1−η© 1−γ ª´1−φ 1−φPN ³ − θ12
wi dni Bn exp θ.
observe that the country price index derived in the Appendix is
i=1 Ti1−ηThis implies that: Pit /Pis = (Ti (t)/Ti (s))−( θ ) since idea stocks in all countries are growing at the same
n³
o
´
1−η
(s
−
t)
. Also, a quick inspection of (10)rate gL in steady state. Therefore, Pit /Pis = expgLθand (13) implies that profits earned in each country market, Πnit , will grow in tandem with Li and Ti atrate gL . In particular, this means that Πnit /Ti (t) is constant over time.
Using these properties, the expected value of an idea from country i from (14) can be expressed as:
Vit =

Z ∞

g

−[ρ− θL (1−η)](s−t)

e

t

N
X

n=1

N

X
Bθ
Bθ
1
¤
Πnt n ds = £Πnt ngLTi (t)Ti (t)
ρ − θ (1 − η)n=1

Substituting this into the equilibrium condition αi Vit = wi , we have, in the two country case:B1θ Π11 + B2θ Π21 =B1θ Π12 + B2θ Π22 =´gL
w1 T1 ³
(1 − η)
ρ−
α1
θ
´
gL
w2 T2 ³ρ−
(1 − η)
α2
θ

(15)
(16)

This delivers a set of equations in (16) that capture how profits, Πnit , and the stock of ideas, Tit , must be
related in the steady state in which each country features a constant share of the workforce, ri ∈ (0, 1),
engaged in the research sector.
Moreover, we can now substitute the steady state expression for ri = gαLi LTitit back into the accounting
equations (13) that equate national expenditure to income. This yields, in the two-country case:w1 T1t gL+ Π11t + Π21t
α1
w2 T2t gLh(B2 )(Π21t + Π22t ) = w2 L2t −+ Π12t + Π22t
α2

h(B1 )(Π11t + Π12t ) = w1 L1t −

17

(17)
(18)

3.3

Equilibrium

The equilibrium in the two-country model is now fully-determined by the equations (11)-(12) and (15)(18), which represent a system of 6 equations in the 6 unknowns, Π11 , Π12 , Π21 , Π22 , T1 , and T2 . The
profit levels, expenditure levels, and stock of ideas in the diﬀerentiated goods industry in each country
can be solved for as a function of the underlying conditions in the two countries, including the labor
force situation (Lit , wi , gL ), the deep parameters of the model related to the innovation process (αi , θ),preferences (η, ρ), distance barriers (d), as well as the country patent policies chosen by each country
government (the Bn ’s).
Several remarks are worth highlighting. First, the share of labor eﬀort in the economy devoted towards
innovation activities is an endogenous outcome of this equilibrium, being equal to ri = gαLi LTitit . We shallrefer to ri as the research intensity of this country. Second, using a labor market-clearing condition, one
can show that the wages in each country, wi , are in fact constant in this steady state; the argument forthis is analogous to that in Eaton and Kortum (2001). Finally, the equilibrium of the model does featurebalanced trade: Countries that are net exporters of diﬀerentiated goods are in turn net importers of the
traditional good (and vice versa), so that the overall trade balance is zero for each country.

4

The closed-economy benchmark

We first study the closed-economy case of the model, which provides an important benchmark. This
allows us in particular to establish a baseline result on the existence of a non-trivial optimal inventive
step that improves country welfare. We will build on this result when we turn to the two-country modelwith strategic interaction in Section 5.

4.1

Equilibrium

It is fairly straightforward to solve explicitly for the equilibrium in the closed-economy case. When N = 1,(10) and (13) are a system of two equations in precisely two unknowns, Xt and Πt , which are respectively
the consumption expenditures and profits for the diﬀerentiated goods sector in the sole country. (Wedrop the country subscript for convenience.) Solving these two equations simultaneously yields:"#
B (1 + θ)Xt =(1 − rt )wt Ltη1−η B (1 + θ) + θ"#
B (1 + θ) − θΠt =(1 − rt )wt Ltη1−η B (1 + θ) + θ

(19)
(20)

Based on (19) and (20), one can show that for a given country income level, wt Lt , and for a given
research intensity, rt , both consumption and profits in this diﬀerentiated goods sector will increase if: (i)18the country’s inventive step, B, is raised; or (ii) the stationary search distribution of ideas has a larger
spread (1/θ is higher). Naturally, both Xt and Πt fall as the share of income spent on the numeraire
good, η, rises.
From the closed-economy analogue of (16), the steady-state stock of ideas for this country is therefore:
T (t) =B θ [B (1 + θ) − θ]
αLt
h
i
ih
·
η
gL B θ [B (1 + θ) − θ] + ρ − 1−η
B (1 + θ) + θ
gL

θ

(21)

1−η

where we have used the expression for Πt from (20). Observe that the larger the labor force, Lt , thegreater the stock of ideas the economy is able to generate in equilibrium, by virtue of being able to devote
more workers to the research sector. It is moreover straightforward to diﬀerentiate (21) and verify that
T (t) is in fact increasing in B. Thus, a more stringent inventive step requirement indeed expands thestock of research ideas accessible to the economy.
Since T /L = gαrL in steady state, the share of the labor force employed in the research sector is:r=

B θ [B (1 + θ) − θ]´h³iηB θ [B (1 + θ) − θ] + gρL − 1−η
B
(1
+
θ)
+
θ
θ
1−η

(22)

¡ ¢
Note that r ∈ (0, 1), so long as ρ − gθL (1 − η) > 0. Importantly, the equilibrium value of r also increases
with the inventive step. Intuitively, as B increases, if we hold r (and hence also T ) constant, this will
increase the value of a patent Vt . This is evident by re-writing (15) in the closed-economy case:Bθ1¤Π
Vt = £tTt
ρ − gθL (1 − η)With a higher B, the per-period profits Πt increase as we saw from (20); this happens because themore stringent inventive step requirement raises the expected markup which existing patent-holders cancharge. Moreover, the B θ term also increases, which reflects the fact that the higher B enables theincumbent patent-holder to retain the market for a longer expected duration. (Recall from (8) thatθbs (1) = BTs is equal to the probability that an idea secures a patent at date t and continues to hold themarket at date s > t.) An increase in B therefore tends to raise the value of a patent through these twochannels. However, we know that Vt must stay constant in steady-state, as dictated by the labor market
equilibrium condition, αVt = w. Thus, workers move out of production into the R&D sector in responseto the higher inventive step, so that r and T both rise endogenously; in turn, the expanded stock of ideas
lowers the expected duration over which the patent-holder can command the market, which provides thecountervailing force that keeps Vt constant in equilibrium.

4.2

Optimal patent policy

We now analyze the standpoint of a government that seeks to maximize consumer welfare through its
choice of patent policy instrument. The closed-economy analogue for welfare can be obtained from the19expression for the real wage in (5) by setting N = 1 and wn = w. Simplifying, we have:
³ 1
´1−η
w
= γ 0 wη T θ B −2
P

(23)

where we have suppressed the time argument for notational simplicity. (In steady state, maximizing thepresent discounted flow of real wages over time is equivalent to maximizing the per period real wage.)
Since γ 0 and w are constants, the optimal inventive step needs to equate the marginal benefit of anincrease in B to the economy (as captured by an increase in T ) with the marginal cost from paying
higher prices for goods that are under patent protection (as captured by a decrease in the term B −2 ).
Substituting in the expression for the steady-state stock of ideas from (21), and ignoring various
constants that do not aﬀect the maximization outcome, one can show that the optimal inventive step
policy is given by:"

B ∗ = arg max B −2θ 1 +
B

µ

1−η
ρ
−
gL
θ

¶

subject to B ≥ 1

#−1
η
B
(1
+
θ)
+
θ
1−η
B θ [B(1 + θ) − θ](24)Solving this maximization problem delivers the first key result of this paper, which addresses thescope for inventive step protection to improve outcomes in the closed-economy setting:
Proposition 1 [Optimal Patent Policy in the Closed Economy] If gρL − 1−ηθ > 2, the optimal

policy calls for a binding inventive step, B ∗ > 1. This optimal inventive step, B ∗ , is moreover unique.
∗
Otherwise, for 0 < gρL − 1−ηθ ≤ 2, the optimal policy is B = 1.

The above proposition is in line with a set of results from the endogenous growth literature on how
there can be under-investment in innovation activities in the laissez-faire equilibrium in such models(Aghion and Howitt 1992, Grossman and Helpman 1991). In our model, it is useful to inspect thecondition gρL − 1−ηθ > 2, to obtain some intuition for understanding when such under-investment is more
likely. Research activities are more likely to be ineﬃciently low from a social planner’s perspective when:(i) ρ is large, so that consumers weigh the short-term consumer surplus loss heavily, leading to a low
incentive to undertake R&D; (ii) gL is small, so that the stock of ideas does not advance suﬃciently
fast in steady state ( ṪT is small); (iii) 1 − η is small, namely when expenditure on diﬀerentiated varieties
and hence the profit motive for innovation in this sector are both small; and (iv) 1/θ is small, so thatthe right-tail of the stationary search distribution for ideas is thinner, hence lowering the likelihood of
obtaining a very high productivity innovation.
It should moreover be stressed that this condition is readily satisfied by standard parameter values.
For example, consider ρ = 0.05 and gL = 0.02, which imply an annual discount factor of about 0.95
and a 2% labor force growth rate respectively. Eaton and Kortum (2002) oﬀer a range of estimates for20

θ from fitting their model of trade to the OECD data, of which θ = 8.28 is a commonly-used centralρ1value. Taken together, we have: gρL − 1−ηθ > gL − θ = 2.38 > 2. If we further identify the diﬀerentiatedgoods sector with manufacturing industries, then 1 − η is the share of income spent by consumers on
manufactured goods, which in Eaton and Kortum (2002) is calibrated to equal 0.13. The relatively smallvalue of this parameter makes it even easier to satisfy the desired condition for an interior solution to
the welfare maximization problem. In short, there is good reason to believe that a binding inventive step
will indeed be the optimal policy for a wide range of relevant parameterizations.

5

Nash equilibrium analysis for the two-country caseWe turn next to a detailed analysis of the open-economy model. This will allow us to expand thediscussion into issues relating to the cross-border externalities from national inventive step policies, andthe eﬀect of globalization on the domestic research intensity and the overall incentive to extend inventive
step protection.

5.1

Solving for the equilibrium

While specifying the equilibrium is conceptually straightforward, this being described in Section 3.3,
solving explicitly for closed-form expressions is algebraically complicated for the general N -country case.
For the sake of tractability, we will therefore focus most of our attention on the two-country case. This
will allow us to derive some sharp results concerning the policy game between the two countries in termsof how strongly each chooses to protect inventive step. It will moreover facilitate a more direct comparison
with the benchmark results in Grossman and Lai (2004) regarding optimal patent length policy in their
two-country setting.
Suppose that there are only two countries, i = 1, 2, in the global economy. From (11) and (12), define:
K1 =
K2 =

µ
µ

T1
T2
T1
T2

¶ 1−φ µθ

¶ φ−1 µ
θ

w2 d
w1
w2
w1 d

¶1−φ

¶φ−1

Then, we have that Π12 = K1 Π11 , and Π21 = K2 Π22 . (Moreover, observe that K1 K2 = d2(1−φ) .)Substitute Π12 = K1 Π11 , and Π21 = K2 Π22 into (15) and (16). This gives a system of two equations
in two unknowns, Π11 and Π22 . The solution to this system is:µ¶ρ − gθL (1 − η) 1 T1 w1
T2 w2
Π11 =
− K2
α1
α2
1 − d2(1−φ) B1θ
µ¶gLρ − θ (1 − η) 1 T2 w2
T1 w1
− K1
Π22 =
α2
α1
1 − d2(1−φ) B2θ
21

(25)
(26)

First, substitute Π12 = K1 Π11 , and Π21 = K2 Π22 into (17) and (18). Then, substitute the expressionsfor Π11 and Π22 into (17) and (18). Finally, using the fact that ri = gαLi LTii , we get the following two
equations in the two unknowns T1 and T2 .

[h(B1 )(1 + K1 ) − 1]·¸
T2 w2 /α2
1 T2 w2 /α211+ d2(1−φ) θ1−
K− K2 θ
2
T1 w1 /α1
B1θB2 T1 w1 /α1
B2

=

µ

α1 L1
− gL
T1

¶

[h(B2 )(1 + K2 ) − 1]·¸1
T1 w1 /α1
1 T1 w1 /α1
1+ d2(1−φ) θ1−
K− K1 θ
1
T2 w2 /α2
B2θB1 T2 w2 /α2
B1

=

µ

α2 L2
− gL
T2

¶

1 − d2(1−φ)ρ − gθL (1 − η)(27)
1 − d2(1−φ)ρ − gθL (1 − η)(28)Note the symmetry in the two equations. Now, observe from the definition of K1 that K1 is decreasingin T1 /T2 when φ > 1. On the other hand, K2 TT21 is increasing in T1 /T2 when φ > 1 + θ. Also, K2 isincreasing in T1 /T2 when φ > 1, while K1 TT12 is decreasing in T1 /T2 when φ > 1 + θ.
In order to determine the sign of T̂1 and T̂2 , we need to totally diﬀerentiate the system of equationsin (27) and (28) with respect to B1 . To simplify the notation, let us define the following:

A11 ≡
A12 ≡
A13 ≡
A21 ≡
A22 ≡
A23 ≡

·
¸
T2 w2 /α21[h(B1 )(1 + K1 ) − 1] θ 1 − K2
T1 w1 /α1
B1
1 T2 w2 /α2−K2 θ
B2 T1 w1 /α1
1d2(1−φ) θB2·¸
T1 w1 /α11[h(B2 )(1 + K2 ) − 1] θ 1 − K1
T2 w2 /α2
B2
1 T1 w1 /α1−K1 θ
B1 T2 w2 /α2
1d2(1−φ) θ
B1Note that A11 , A12 , and A13 are respectively the first, second and third summands on the left-handside of (27), while A21 , A22 , and A23 are the corresponding summands in (28). We focus our attention onequilibria in which diﬀerentiated products firms earn positive profits from their sales in both countries.
T1 w1 /α12 /α2From (25) and (26), we require that 1 − K2 TT21 w
w1 /α1 > 0 and 1 − K1 T2 w2 /α2 > 0 in such equilibria.

Moreover, note that K1 , K2 ≥ 0, and that h(B1 ), h(B2 ) > 1 for all B > 1. It follows that A11 , A21 ≥ 0,
A12 , A22 ≤ 0, and A13 , A23 ≥ 0.

dTi
. To facilitate the diﬀerentiation with respect to B1 , observe that:
Define T̂i ≡ T1i dB
1

22

dK1
dB1
dK2
dB1
´

³
d K1 TT12dB³ 1 ´
d K2 TT21dB1

1−φ(T̂1 − T̂2 )θ
φ−1(T̂1 − T̂2 )
= K2
θ
= K1

= K1

T1 1 − φ + θ(T̂1 − T̂2 )
T2
θ

= K2

T2 φ − θ − 1(T̂1 − T̂2 )T1
θUsing these expressions, we can now totally diﬀerentiate (27) and (28) with respect to B1 . This yields






2 /α2
K2 TT21 w
h(B
1
−
φ
)K
1
−
φ
+
θ
1
−
φ
+
θ
w
/α
1
1
1
1
A11 
 − A12 (T̂1 − T̂2 )+h(B1 )(1 + K1 ) − 1 θθθ
1 − K2 T2 w2 /α2T1 w1 /α12(1−φ)

α1 L1 1 − dT̂1T1 ρ − gθL (1 − η)
µ 0¶
h (B1 )(1 + K1 )θ
= −A11−h(B1 )(1 + K1 ) − 1 B1
+

(29)

and




A21 

1 /α1
K1 TT12 w
w2 /α2h(B2 )K2
φ−1
φ − θ − 1
φ − θ − 1+
− A22
(T̂1 − T̂2 )T1 w1 /α1h(B2 )(1 + K2 ) − 1 θθθ
1 − K1T2 w2 /α22(1−φ)

α2 L2 1 − dT̂2T2 ρ − gθL (1 − η)θ= (A22 + A23 )
B1
+

(30)

For notational ease, let us define:

D1

D2



2 /α2
K2 TT21 w
w1 /α1h(B1 )K1
φ−1
φ − θ − 1
φ−θ−1
+
− A12
,
T2 w2 /α2h(B1 )(1 + K1 ) − 1 θθθ
1 − K2 T1 w1 /α1
T1 w1 /α1K
1 T2 w2 /α2h(B2 )K2
φ−1
φ − θ − 1
φ−θ−1
≡ A21 +− A22.
T1 w1 /α1h(B2 )(1 + K2 ) − 1 θθθ
1 − K1≡ A11 T2 w2 /α2Thus, D1 is the negative of the coeﬃcient of T̂1 − T̂2 in (29), and D2 is the coeﬃcient of T̂1 − T̂2
in (30). Let us consider a case where the diﬀerentiated varieties from the two countries are suﬃciently
strong substitutes. In particular, assume that: φ > 1 + θ. This would imply that D1 , D2 ≥ 0 (bearing in
mind that A11 , A21 ≥ 0 and A12 , A22 ≤ 0).
A slight rearrangement of (29) and (30) yields:

23

·

¸
α1 L1 1 − d2(1−φ)− D1 T̂1 + D1 T̂2
T1 ρ − gθL (1 − η)·¸
α2 L2 1 − d2(1−φ)
D2 T̂1 +− D2 T̂2
T2 ρ − gθL (1 − η)

= −A11µh0 (B1 )(1 + K1 )θ−h(B1 )(1 + K1 ) − 1 B1

= (A22 + A23 )

¶

(31)

θ
B1

(32)

which is a system of linear simultaneous equations in T̂1 and T̂2 . Using Cramer’s Rule, it follows that:af −ceT̂1 = de−bf
ad−bc and T̂2 = ad−bc , where:¶·¸θh0 (B1 )(1 + K1 )α2 L2 1 − d2(1−φ)θ− D1 (A22 + A23 )
−
−
D
2
gL
h(B1 )(1 + K1 ) − 1 B1
T2 ρ − θ (1 − η)
B1
·
¸
µ 0
¶
2(1−φ)h (B1 )(1 + K1 )θ α1 L1 1 − dθaf − ce = (A22 + A23 )D2− D1 + A11−
B1
T1 ρ − gθL (1 − η)h(B1 )(1 + K1 ) − 1 B1·¸·¸
α1 L1 1 − d2(1−φ)α2 L2 1 − d2(1−φ)
ad − bc =−D− D2 − D1 D2
1
T1 ρ − gθL (1 − η)
T2 ρ − gθL (1 − η)de − bf= −A11µFor tractability, from now on, we shall assume symmetry, i.e. w1 = w2 = w, α1 = α2 = α, L1 = L2 =L, B1 = B2 = B, d > 1. Therefore, T1 = T2 = T . It will be seen that the results we derive in the rest of
the paper do seem to be robust to generalization to the asymmetric cases.
In the appendix, it is shown that a suﬃcient condition for T̂2 < 0 is·¸³´iθB θ h(B) − 1
2d1−φ h
gL
1−φ
2
(φ
−
1)
−
θ
1
+
d
>
+
h(B)
Bθ
ρ − gθL (1 − η)
1 − d2(1−φ)(33)

For any given B, θ, η, ρ, and gL , it can be easily shown that this holds when φ is suﬃciently large, ord is suﬃciently large. The fact that Tb1 + Tb2 > 0 and Tb1 > Tb2 means that Tb2 < 0 implies that Tb1 > 0.
Therefore, the following lemma establishes how the country stock of ideas responds to inventive step
policy:Lemma 3: Consider equilibria of the two-country model in which both countries produce a positive∂T∂Tiamount of output in both the homogeneous and diﬀerentiated goods sectors. Then, ∂B> 0 and ∂Bji < 0,iwhere i ∈ {1, 2}, i 6= j as long as φ is suﬃciently large or d is suﬃciently large.
Proof: See the Appendix for the derivation of (33) and an analysis of its property.
Thus, an increase in the inventive step raises the domestic stock of ideas, so that ceteris paribus thispatent instrument does indeed have its presumed intended eﬀect of promoting innovations. On the other
hand, the model features a negative cross-border eﬀect, in that a higher inventive step lowers the stock ofideas in the foreign country. Intuitively, an increase in B1 promotes domestic innovation and raises the24average productivity of domestic firms in the diﬀerentiated goods sector. This erodes the profitability of
R&D for foreign firms in this sector, leading to a shift of labor out of the foreign diﬀerentiated goods
sector into the traditional good sector there. Therefore, the research intensity and hence the stock ofideas falls in the foreign country.
To address the issue of welfare, refer to equation (6).
To facilitate exposition, define
µ
¶− 1
1
1
1−φ
−
−
Ψi ≡ (Ti θ wi )1−φ + (Tj θ wj d)1−φ
where i 6= j. Diﬀerentiating (6) with respect to B1 , we

get:

−1

−1

d ln B1−2 Ψ1−2θ (T1 θ w1 )1−φ T̂1 + (T2 θ w2 d)1−φ T̂2
θ
=
+
−1
−1
dB1
B1
(T θ w )1−φ + (T θ w d)1−φ
1

1

2

2

Treating the inventive step as a policy instrument, what then is the nature of the tradeoﬀ facing eachcountry in deciding on its optimal patent policy? Without loss of generality, let us illustrate this forcountry 1 based on (6). A more active patenting regime corresponding to a higher B1 should in principle
raise the aggregate stock of ideas available, hence providing a boost to consumption in country 1. This
benefit is reflected in the Ψ1 term in (6) for country 1 welfare, although it is not a priori obvious fromthe algebraic expression how this will vary with B1 . Since an increase in B1 would raise T1 , but lower T2 ,so the net eﬀect on Ψ1 needs to be formally worked out from the above equation. Assuming symmetry
as before, we get

d ln B1−2 Ψ1−2θ d1−φ T̂2 + T̂1
(34)
=
+
dB1
B1
d1−φ + 1´³´³
1−φ T̂ + T̂ . Since T1
b1 + Tb2 > 0 and Tb1 > Tb2 , it is clear that=sgnd
It is clear that sgn dΨ
2
1
dB1
θ

d1−φ T̂2 + T̂1 > 0. Therefore, we have the following:

Lemma 4: Consider equilibria of the two-country model in which both countries produce a positiveiamount of output in both the homogeneous and diﬀerentiated goods sectors. We have: ∂Ψ∂Bi > 0, wherei ∈ {1, 2}.
Thus, it is indeed the case that an increase in B1 would raise Ψ1 and thus confer a net benefit to country
1’s consumers. In particular, the increase in the domestic stock of ideas T1 outweighs the negative eﬀect
on T2 , intuitively because the eﬀect of T2 on Ψ1 is more muted due to the distance barrier, d.
On the cost side however, ideas that improve on the current best patented idea by a factor smaller
than B1 are not eligible for a patent and hence cannot enter the market, so that consumers have to
pay a higher price for these particular varieties even though a more productive technology has become
feasible. The higher is B1 , the larger the consumer surplus loss due to the protection that the inventive
step aﬀords to existing patent holders. This cost of patent protection is captured by the B1−2 term in (6).
From the perspective of country 1, the optimal patent policy choice thus entails trading oﬀ the benefit

25

of more innovation against this short-term consumer surplus loss.20
Incidentally, it is straightforward to verify as a consistency check that as d → ∞, (5) simplify to theexpression for welfare in the closed-economy case, namely (23). In other words, the cross-border spilloverof ideas vanishes if goods cannot be traded due to a prohibitive transport cost, and so welfare would
simply be given by the closed-economy expression with the domestic country stock of ideas.

5.2

Nash equilibrium analysis

Having derived the country welfare expressions, we are now ready to tackle the question of how inventive
step policies are determined. Following Grossman and Lai (2004), we model this policy question asa one-shot non-cooperative game, with B1 and B2 chosen simultaneously by the respective country
governments.
We assume that each country government seeks to maximize their respective country’s welfare (theyare national welfare-maximizers), but that they do not internalize the spillover eﬀects which their policy
choice has on the foreign country. More formally, given the current choice of inventive step protection
being exercised by Country 2, the Country 1 government would do best by choosing: B1∗ = arg max
B1

B1−2 Ψ1 . This delivers Country 1’s best-response function (BRF), which expresses B1∗ as a function of B2 .
Likewise, given B1 , Country 2’s BRF is given by solving B2∗ = arg max B2−2 Ψ2 . The Nash Equilibrium
B2

outcome is then determined as a point of intersection of the two best-response functions.
At first glance, this view of how inventive steps are set may seem at odds with the observation that dejure patent laws are rarely changed in practice. Nevertheless, we take B1 and B2 in our model to reflect
the actual de facto implementation of these formal laws, namely how stringently patent laws are applied.
The degree of this enforcement clearly varies more substantially across countries and across time, and isplausibly more responsive to the policy choices made by a country’s trading partners.
∂ ln(B1−2 Ψ1 )∂B1

To obtain the BRF for Country 1 (BRF1 ), we calculate

given B2 , and set this partial

derivative to zero. This computation lead to the following equation for BRF1 :−1−1−2θ (T1 θ w1 )1−φ T̂1 + (T2 θ w2 d)1−φ T̂2
+
=0
− θ1
− 1θ
B1
1−φ
1−φ
(T w )+ (T w d)
1

1

2

(35)

2

In particular, the marginal benefit term is a weighted average of T̂1 and T̂2 , namely a weighted averageof the percentage change in the idea stocks in each country. Likewise, solving for∂ ln(B2−2 Ψ2 )= 0 yields∂B2

the BRF for Country 2 (BRF2 ):−1−1−2θ (T2 θ w2 )1−φ T̂20 + (T1 θ w1 d)1−φ T̂10
+
=0
−1
−1
B2
(T θ w )1−φ + (T θ w d)1−φ
2

2

1

20

(36)

1

It should also be clear that the problem of maximizing the present discounted flow of real wages will boil down tochoosing B1 to maximize the per-period flow welfare given by (6).

26

dTi
where T̂i0 ≡ T1i dB.
2Focusing first on BRF1 , we show in the Appendix that, under symmetry, for a given level of B2 ,the expression in the curly brackets on the left-hand side of (35) is in fact a monotonically decreasing
function of B1 , so that the value of B1 that maximizes Country 1 welfare is in fact unique. Given
the underlying symmetry between (35) and (36), it follows that for each given level of B1 , there is aunique B2∗ that maximizes welfare in Country 2. In short, the BRFs for both countries are well-defined
functions, and the Nash Equilibrium inventive steps are determined by the intersection point of the two
BRFs in B1 -B2 space. We have found through extensive numerical simulations that B1 and B2 are
typically strategic complements, in the sense that both BRFs tend to be upward-sloping functions when
B1 , B2 ≥ 1. Moreover, for a wide range of standard parameterizations, we have found that there is aunique, stable Nash equilibrium to the inventive step-setting game. (We have found no examples to thecontrary, and it is our conjecture that this can be proven formally.)
A central question in this line of research concerns the eﬃciency of the patent policies that countries
adopt in the strategic equilibrium. A key result in Grossman and Lai (2004) is that the Nash equilibriumin their patent policy game features under-protection of patent lengths, in the sense that a Pareto
improvement can be achieved if both countries could jointly coordinate to raise their patent length
protection. This results rests on the existence of a net positive cross-border externality as a country
extends its patent length: This raises the incentives to innovate in the foreign country, but this benefit
is not fully internalized in the setting of the domestic patent length. What then is the nature of thecross-border externality in our context, where the nature of innovation and the instrument of patent
protection is markedly diﬀerent? To tackle this question, we need to understand the nature of the crossborder externality present in our model. Observe from (7) that Country 1’s choice of inventive step B1
aﬀects the welfare of Country 2 through the term Ψ2 , namely through its eﬀect on the stock of ideas that
Country 2 eﬀectively consumes. In fact, it can be shown that−1

−1

(T θ w1 d)1−φ T̂1 + (T2 θ w2 )1−φ T̂2
d1−φ T̂1 + T̂2
d ln B2−2 Ψ2
= 1 1
=
θ
−
−1
dB1
d1−φ + 1(T θ w d)1−φ + (T θ w )1−φ1

1

2

under symmetry

2

Therefore, there is negative (positive) externality when d1−φ T̂1 + T̂2 is less (greater) than zero. In the
appendix, it is shown that a suﬃcient condition for d1−φ T̂1 + T̂2 < 0 is
·
¸
£
¡
¢¤
¡
¢¤
gL
θB θ h(B) − 1
2d1−φ £
1
1−φ
+
2
(φ
−
1)
−
θ
1
+
d
>
2 (φ − 1) − θ 1 + d1−φ
>
gL
1−φ
θ
2(1−φ)1−dh(B)Bρ − θ (1 − η)
1−d
(37)

In fact, it can be shown that the expression on the left hand side increases with φ, while the one on
the right hand side has a hump-shaped relationship with φ as well as decreases with d. The expression on
27

the right hand side tends to zero as φ gets large, whereas the expression on the left hand side increases
without bound with φ. This is shown in Figure 1. It can be seen that for any given B, θ, η, ρ, gL , theabove set of inequalities (37) holds when d or φ is suﬃciently large.21[FIGURE 1 ABOUT HERE]
Therefore, we have:
Proposition 2[Negative externality of inventive step protection] Consider equilibria of thetwo-country model in which both countries produce a positive amount of output in both the homogeneousand diﬀerentiated goods sectors. We have:

∂Uj
∂Bi

< 0, where i ∈ {1, 2} and j 6= i as long as φ or d is

suﬃciently large .
Thus, an increase in Country 1’s inventive step tends to lower Ψ2 when φ is suﬃciently large andhence lower welfare in Country 2. In other words, we in fact have a negative cross-border spillover ofCountry 1’s inventive step policy on Country 2 , and vice versa when foreign and domestic goods arehighly substitutable with each other. In the Appendix, we show that it is possible to find conditions
under which positive externality occurs, as Figure 1 indicates.22
Now we turn to the highly policy-relevant issue of whether there is global over- or under-protectionof IPR protection in Nash equilibrium. To answer this question, we show below that under symmetry
the Nash equilibrium inventive step is greater than the globally optimal one for any given d as long as
there is negative cross-border externality of raising the inventive step requirement.
First, define
1 dTi
Ti dB2T̂i0 ≡

Due to symmetry, to calculate the globally optimal B, we only need to focus on maximizing B1−2 Ψ1by choosing B, bearing in mind that (1) both B1 and B2 change as B changes, (2) dB1 = dB2 , and (3)each of B1 and B2 aﬀects T1 as well as T2 . Therefore, to obtain the globally optimal B under symmetry,
we need to solve

dB1−2 Ψ1 dB1−2 Ψ1
+
=0
dB1
dB2

and then set B1 = B2 = B. Note thatdB1−2 Ψ1dB2

and

dB2−2 Ψ2dB1

are mirror images of each other. Therefore,

the above equation is equivalent to




−1
−1
−1
−1
B1  −2θ (T1 θ w1 )1−φ T̂1 + (T2 θ w2 d)1−φ T̂2  B2  (T2 θ w2 d)1−φ T̂20 + (T1 θ w1 )1−φ T̂10 
+
+
=0
−1
−1
−1
−1
θ
B1
θ
(T θ w )1−φ + (T θ w d)1−φ(T θ w d)1−φ + (T θ w )1−φ
1

1

2

2

2

2

1

1

21
To be more exact, there is negative externality when d is suﬃciently large so that case (a) shown in Figure 1 occurs,and φ is suﬃciently large so that the externality falls into the negative range indicated in case (a).
22
It is shown in the Appendix that a positive cross-border externality occurs when d and φ are intermediate in values.

28Symmetry of the two countries implies that T̂1 = T̂20 and T̂2 = T̂10 . So, we have
"
¡
¢
¡
¢ #
1
1
B −2θ (T − θ w)1−φ 1 + d1−φ T̂1 + (T − θ wd)1−φ 1 + d1−φ T̂2
+
=0
1
θ
B
(T − θ w)1−φ (1 + d1−φ )⇔−2θ
+ T̂1 + T̂2 = 0
B

On the other hand, the condition that determines the Nash equilibrium B under symmetry is obtainedby setting the expression in (34) to zero, which becomes−2θ T̂1 + d1−φ T̂2+=0
B
1 + d1−φNow, it can be easily shown thatT̂1 + d1−φ T̂2> T̂1 + T̂2
1 + d1−φ⇐⇒T̂2 + d1−φ T̂1 < 0

for any given d > 1

for any given dwhich is true iﬀ there is negative cross-border externality, as shown in (??).
Therefore, Nash B > globally optimal B for any given d iﬀ there is negative cross-border externality
from raising B. ¥Therefore, we have
Proposition 3 [Global over-provision of inventive step requirement in equilibrium] Undersymmetry, when φ or d is suﬃciently large, and suppose that both countries set strictly binding inventive
steps in Nash Equilibrium (B1∗ , B2∗ > 1 where B1∗ = B2∗ = B ∗ ). There is over-provision of inventive
step requirement in Nash Equilibrium, in that there exist B1∗∗ and B2∗∗ (where B1∗∗ = B2∗∗ = B ∗∗ ) with
1 < B ∗∗ < B ∗ , such that welfare in both countries will increase if their inventive step policies are lowered
to B ∗∗ .
We illustrate this finding in Figure 2, which highlights the contrast between our result and Grossman
and Lai (2004). In Figure 2a, we sketch the BRFs in our present model for the typical case in which
the BRFs are upward-sloping (inventive steps are strategic complements). Given the negative natureof the eﬀect of a higher B1 on Country 2’s welfare, the iso-welfare curves along BRF2 correspond tohigher levels of welfare as we move in a leftward direction (the direction of decreasing B1 ). Similarly, theiso-welfare curves along BRF1 reflect increased levels of welfare as we move downward (in the direction of
decreasing B2 ). The shaded area to the southwest of the Nash equilibrium thus corresponds to a regionof stronger inventive step protection in both countries that nevertheless leads to a Pareto improvement
in welfare. In short, there is over-provision of inventive step protection in Nash equilibrium. This stands
29

in contrast with the situation in Grossman and Lai (2004), which is shown in Figure 2b. There, patent
length policies are in fact strategic substitutes in the central case they consider, hence the downwardsloping best-response functions.23 Here, as the patent length policy in Country 1, Ω1 , increases, welfarein Country 2 also increases, reflecting the positive cross-border externality. The direction of increasing
welfare for the Country 2 iso-welfare curves is thus to the right (the direction of higher Ω1 ). An analogous
argument applies for Country 1’s iso-welfare curves. Taken together, there is a region to the northeastof the Nash equilibrium that illustrates combinations of patent length policies that would yield Pareto
superior outcomes, so that the Nash equilibrium features under-protection of patent lengths instead.
[FIGURE 2 ABOUT HERE]
The contrast between Proposition 3 and the conclusions in Grossman and Lai (2004) have an intuitive
explanation that hinges on the diﬀerences in the nature of innovation in the two papers. In this paper, themeasure of varieties from each country is fixed and innovation takes place along the quality dimension,and the natural concept of IPR protection is the inventive step. Firms from diﬀerent countries compete
directly with each other by selling goods that are highly substitutable with each other, and this directcompetition gives rise to the negative cross-border externality. When Country 1 raises its inventive step,
B1 , this raises T1 and hence also the relative productivity position that Country 1 has in the diﬀerentiatedgoods sector. As domestic and foreign goods are highly substitutable, and Country-2 firms competes at an
disadvantage in Country-1 market (due to trade barriers), the profitability of R&D for firms in Country 2
falls, leading to a decrease in T2 . Overall, consumers in Country 1 gain because Ψ1 increases overall; theincrease in T1 outweighs the eﬀect of the fall in T2 , as consumers can only access Country 2 technologies
after paying an iceberg transport cost d. Conversely, for Country 2, the eﬀect of T2 falling dominates theincrease in T1 for much the same reason, and Ψ2 falls, generating a negative externality of Country 1’s
policy action on Country 2 welfare (Country 1 does not take this spillover into account when deciding
B1 ). This underpins the over-protection result that we have derive inventive step policy.
Grossman and Lai (2004) on the other hand, consider product innovation that expands the existingmeasure of varieties in the diﬀerentiated goods sector, in which the IPR protection instrument consideredis the duration of the patent protection of each new variety. As a result, an increase in Ω1 (Country
1’s patent length) benefits both domestic and foreign firms by increasing the value of any global patent.
This induces more product innovation from firms in all countries, even in the presence of trade barriers.
The increased utility from the larger set of varieties that Country 2 can consume as well as increased
profits for Country-2 firms selling in Country 1 generate a positive cross-border externality from raising
23

It is stated in Grossman and Lai (2004) that the patent lengths can be strategic complements too for some values of
β in their model (where the elasticity of substitution between skilled and unskilled labor in R&D is 1/(1 − β). But thosecases are considered to be rarer.

30

Ω1 , and hence implies a tendency towards the under-provision of patent length protection.
In sum, the verdict with regards to whether there is over- or under-protection of IPR in Nash equilibrium of the patent-setting game hinges on the sign of the cross-border externality that the patent
instrument generates. In the model which we have presented above, the sign of this spillover is negative,
in contrast with Grossman and Lai (2004), because goods from diﬀerent countries are highly substitutablewith each other and an increase in home inventive step disproportionately favors the competitiveness ofhome firms versus foreign firms because of the existence of trade barriers. Consequently, the conclusions
with regards to whether patent protection is under- or over-provided by country governments can besensitive to the underlying structure of innovation process and the patent instrument being considered.
We turn next to the key question of how globalization aﬀects research intensity in each country.
The can be two approaches to answering this question. First, we can assume that countries respond to
globalization by changing their patent protection policies in response to it. Within the context of ourmodel, this boils down to a question of how d aﬀects the equilibrium inventive steps, which would in turnaﬀect research intensities. The second approach will be based on the assumption that countries do not
adjust their required inventive steps as globalization deepens, perhaps because history and institutional
inertia create hurdles for governments to change their patent laws. Under such circumstances, what
would be the eﬀect of globalization on research intensities of countries? To analyze this case, we haveto assume that B1 and B2 are exogenous and determine the eﬀects of a decrease in d on T1 and T2 . We
shall start with the second approach, as it is very simple to analyze.
(27) and (16) imply that under symmetryαLh(B) − 1
T − gL=θB
ρ − gθL (1 − η)for any given d(38)

This implies that globalization (decrease in d) does not aﬀect research intensities in both countries if
B1 = B2 = B are exogenous given. Therefore, we have
Proposition 4 [Eﬀect of globalization when inventive steps are exogenous] Given that countries set the same exogenous inventive steps, the research intensities of all countries remain unchanged
as distance barriers are lowered.
What is the intuition for this result? As Eaton and Kortum (2001) point out, there are two eﬀects
on the incentive to allocate resource to R&D as d decreases: 1. an increase in profitability from R&D
as the increased access to the foreign market increases profit opportunities; 2. a decrease in profitability
from R&D as there is increased competition from foreign firms in the home market. Each of these eﬀects
increases with the required inventive step of the market in question. When B1 = B2 , the two eﬀects areequal.
31

Note that equation (38) holds for d = ∞, which is the closed economy. Therefore, the globally optimal
B for any d is the same as the optimal B in a closed economy. Since we have proved previously thatwhen φ or d is suﬃciently large the Nash B > globally optimal B for any given d, it must be true thatNash B (∞ > d > 1) > closed economy optimal B = Nash B (d = ∞). Consequently, the equilibrium
inventive steps chosen in the open-economy setting will be strictly higher than that chosen in autarky.
We record this finding in:
Proposition 5 [Eﬀect of globalization when inventive steps are endogenous] The equilibrium
inventive step of a country is larger when it is open to trade and patenting by foreigners than when it isunder autarky when φ or d is suﬃciently large.
This result merits some discussion. Observe first that the −2θ
B1 term on the left-hand side of (35)comes from the log-derivative of the B1−2 term in the real wage, and hence represents the marginal cost
to Country 1 of raising its inventive step. On the other hand, the remaining summand on the left-handside of (35) correspond to the marginal benefit of raising B1 , namely from the increase in the stock ofideas, Ψ1 , accessible to Country 1’s consumers. These marginal cost and marginal benefit functions areillustrated in Figure 1 for a given value of B2 ; both curves are downward-sloping, but the marginal benefit
function descends more steeply and cuts the marginal cost curve from above.24[FIGURE 3 ABOUT HERE]
Evidently, the onset of globalization leads to a higher marginal benefit from raising inventive step
protection relative to the closed-economy benchmark. Intuitively, when domestic goods and foreign
goods are highly substitutable with each other, globalization leads to increased competition for Country
1 firms, as they engage in direct competition with Country 2 firms . In this setting, a higher inventive
step encourages domestic innovation at the expense of that in the foreign country (T1 increases, while T2
decreases), improving domestic firms’ productivity relative to foreign firms, and thus allowing domestic
firms to compete more eﬀectively in the product market. This eﬀect is naturally absent in the closedeconomy setting, hence explaining why inventive step protection confers a higher marginal benefit whencountries can trade. (Refer to Figure 1 again)
Of note, this result also stands in contrast to Grossman and Lai (2004). In their Proposition 1, theyfind that globalization actually leads to a decreased propensity to extend patent length protection, theintuition being that patent policy in an open-economy setting is less eﬀective at stimulating innovation
as firms earn part of their profits in overseas markets where the patent protection extended in their
24

To see this, note that after factoring out Bθ1 , the marginal cost component is a constant (−2). On the other hand, themarginal benefit component remains a decreasing function of B1 ; this follows from the proof that the expression in the curly
brackets on the left-hand side of (35) is monotonically decreasing in B1 .

32

home countries does not apply. Therefore, the marginal benefit of extending patent length is lower in an
open economy than in autarky. Moreover, the marginal cost of extending patent length is higher in an
open economy than in autarky, as a fraction of goods consumed is imported, and an increase in patent
length benefits foreign firms instead of home firms. It appears therefore that inventive step protectionin quality-improvement innovation in the presence of direct competition between domestic and foreign
goods that are highly substitutable in our open-economy model in fact raises the marginal benefit ofpatent protection without changing the marginal cost, leading to the converse result that the incentives
to protect inventive steps are higher in the open-economy.

6

Conclusion

In the patent protection literature, it is widely recognized that patent length protection leads to a
tradeoﬀ between the dynamic gains from induced innovation and static losses from the increased durationof monopoly power of the patent-holders. Grossman and Lai (2004) conducted an analysis of a multicountry patent-length-setting game, based on a model of on-going product innovation. One of their major
findings is that there are positive cross-border externalities from extending patent length protection. Thisimplies that there is global under-protection of IPR, and that there is room for international coordinationin patent policy to improve global welfare by increasing the strength of global patent protection. Thisis because an increase in patent length protection for new products benefits both domestic and foreign
firms even with the presence of trade barriers. As firms do not invest in R&D so as to improve theirproductivity, there is no sense in which an increase in patent length hurt the competitiveness of foreign
firms. Thus, there are positive cross-border externalities from extending patent length.
In the present paper, we find that if one considers required inventive step (or non-obviousness) as the
instrument of patent protection in quality-improvement innovation (which we can call process innovation), there can be negative cross-border externalities from increaseing the strength of patent protection if
domestic and foreign goods are suﬃciently substitutable with each other. This means that there is global
over-protection of IPR and international coordination should aim at decreasing global IPR protection.
This is because an increase in inventive step requirement at home disproportionately increases the profitability of quality-improvement R&D at home vis-a-vis that of foreign firms, due the existence of tradebarriers. High substitutability between home and foreign goods exacerbate this disadvantage of foreign
firms, lowering the profitability their R&D activities and slowing their technological progress. Meanwhile,the existence of trade barriers limits the gains of foreign consumers from productivity improvement in
the home country. This is the cause of the negative cross-border externalities.
Consequently, the conclusions with regard to whether patent protection is under- or over-provided
by country governments can be sensitive to the underlying structure of innovation process, the patent
33

instrument being considered, as well as the substitutability between domestic and foreign goods.
Interestingly, the existence of negative externalities in our model also leads to the conclusion that
globalization leads to an increase in the Nash equilibrium inventive steps and higher equilibrium researchintensities in all countries.

34

7

References

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Econometrica 60: 323-351.
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Cambridge: MIT Press.
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36

8

Appendix: Details of Proofs

Proof of Lemma 1: [Patent Frontier in the Closed Economy]
Proof. The objective is to obtain an expression for F (z; t), the time-t distribution of the patent frontier. Consider
a short window of time between t and t + (with small) during which the patent oﬃce is open and ideas can bepatented. Let Z∆t be a random variable denoting the most productive idea that arrives between time t and t + .
During this short time period, the stock of ideas increases by ∆T (t) = T (t + ) − T (t). Thus:
P r(Z∆t < z) =∞X
exp {−∆T (t)} (∆T (t))k

k!

k=0

H(z)k = exp {−∆T (t)}∞X(∆T (t)(1 − z −θ ))k

k=0

k!

ª
©
= exp −∆T (t)z −θ(39)(To obtain this last expression, we sum over the probability that exactly k ideas ≤ z arrive between time t andt + , these being drawn from the incremental stock of ideas ∆T (t), where k ranges from 0 to ∞.)We write down an expression for F (z; t + ) in terms of F (z; t). The probability that the patent frontier is ≤ z
at time t + is the union of two events: (i) all ideas that arrive between t and t + do not surpass the required
inventive step, and the patent frontier at time t is itself already ≤ z, and (ii) the best idea that arrives between t
and t + surpasses the required inventive step, and this new best idea is itself ≤ z. Note that only the best ideathat arrives during this short time period of length will be considered for a patent.
Let P Ft be the random variable denoting the quality of the idea at the patent frontier at time t. We have:
F (z; t + ) =

Z z

ζ=0

P r(Z∆t < Bζ|P Ft = ζ) dP r(P Ft ≤ ζ) +

Z z/B
ζ=0

P r(Bζ ≤ Z∆t < z|P Ft = ζ) dP r(P Ft ≤ ζ)

The first summand captures the probability of event (i): Conditioning on the current value of the patent frontier(P Ft = ζ), we evaluate the probability that the best idea that arrives does not exceed Bζ. The integral sign
aggregates this probability over the possible values of ζ ∈ [0, z], these being consistent with P Ft+ being ≤ z. The
second summand captures event (ii): Conditioning once again on the value of the time-t patent frontier, we workout the probability that the best idea that arrives surpasses the required inventive step but nevertheless remains≤ z. (The upper limit of the integral is z/B, to ensure that there are ideas which surpasses the required inventive
step, but nevertheless remain ≤ z.)
Substituting in the cdf for Z∆t from (39), and simplifying, we get:Z z
F (z; t + ) =exp{−∆T (t)(Bζ)−θ } dP r(P Ft ≤ ζ)
ζ=0

+
=

Z z/B
ζ=0

Z z/B
ζ=0

¡
¢
exp{−∆T (t)z −θ } − exp{−∆T (t)(Bζ)−θ } dP r(P Ft ≤ ζ)

exp{−∆T (t)z −θ } dF (ζ; t) += exp{−∆T (t)z −θ }F (z/B; t) +Z z

= exp{−∆T (t)(Bz)−θ }F (z; t) −

Z z

ζ=z/B

ζ=z/B
Z z

exp{−∆T (t)(Bζ)−θ } dF (ζ; t)exp{−∆T (t)(Bζ)−θ } dF (ζ; t)

ζ=z/Bθ∆T (t)B −θ ζ −θ−1 exp{−∆T (t)(Bζ)−θ }F (ζ; t) dζ

where the last step follows from integrating by parts.
(z;t)= lim →0 1 [F (z; t + ) − F (z; t)], and apply
Now, substitute the above expression for F (z; t + ) into ∂F∂t
L’Hospital’s Rule. One obtains:
Z z∂ F (z; t)
T 0 (t)θB −θ ζ −θ−1 F (ζ; t) dζ= −T 0 (t)(Bz)−θ F (z; t) −
∂t
ζ=z/B

37

=⇒

Z t "
Z z
0
−θ
log F (z; t) =−T (s)(Bz) −s=0= −T (t)(Bz)−θ −
=⇒

Z t

0

T (s)θB−θ −θ−1 F (ζ; s)ζF (z; s)

ζ=z/B

T 0 (s)s=0

Z zθB −θ ζ −θ−1

ζ=z/B

ª
©
F (z; t) = exp −T (t)(Bz)−θ × exp

(Z
t

s=0

0

−T (s)#dζ ds

F (ζ; s)dz ds
F (z; s)Z zθB−θ −θ−1 F (ζ; s)

ζ=z/B

ζ

F (z; s)

)

dζ ds

(40)

Note that we adopt the normalization, T (0) = 0 (the stock of ideas is zero in ©the initial timeª period).
Since the integrand in (40) is non-positive, it follows that F (z; t) ≤ exp −T (t)(Bz)−θ . Together with theassumption that T 0 (t) is bounded with |T 0 (t)| ≤ k, we have:
Z z
Z z
©
ª
0
−θ −θ−1
T (t)θB ζ
F (ζ; t) dζ ≤T 0 (t)θB −θ ζ −θ−1 exp −T (t)(Bζ)−θ dζ
ζ=z/B

ζ=z/B

≤

ª #z
©
exp −T (t)(Bζ)−θ
k
T (t)
"

ζ=z/B

ª
©
ª¢
©
k ¡
=
exp −T (t)(Bz)−θ − exp −T (t)(z)−θ
T (t)
→ 0
as T (t) → +∞Hence, as the stock of ideas grows large, the second term in the product on the right-hand side of (40) tends to 1.
©
ª
The patent frontier distribution thus tends asymptotically to the Fréchet cdf: F (z; t) = exp −T (t)(Bz)−θ . This
distribution applies to each variety j ∈ [0, 1], since varieties are ex ante identical.

Proof of Lemma 2: [Best Patented Idea from each Country]
Proof. We present a short proof below. An alternative proof derived from first principles in a manner similar toLemma 1 above is available on request.
Recall that Bn is the inventive step in country n. Country n awards country-specific patents, namely where it
has a separate patenting system for ideas originating from each country. Let CSPni (t) denote the country-specificpatent frontier for ideas from country i in country n at time t. Thus, an idea from country i receives this countryspecific patent so long as it is more productive than the previous best idea holding a country-i specific patent by
an inventive step of Bn .
Moreover, applying Lemma 1, as Ti (t) → +∞ for all countries i, we have:
Pr (CSPni (t) ≤ z) = exp{−Ti (t)(Bn z)−θ }.

(41)

Derivation of the Price Index and Welfare Expressions:
Proof Recall that in the open-economy model, the price index for the diﬀerentiated goods sector in country

´ 1
³P
N
1−φ 1−φ, where Pni is the ideal price index for varieties originating from country i. We
n is given by:
i=1 Pnithus need to compute Pni , the derivation of which follows closely that in the Mathematical Appendix to Bernard
et al. (2003).
Consider a new idea originating from country i. (Without loss of generality, suppose this new idea is for somevariety j ∈ [0, 1].) This new idea needs to surpasses the minimum inventive step requirement of Bn in order toqualify for a patent in country n. Specifically, this means that the new idea needs to improve on the existing
best idea from country i for variety j which holds a patent in country n by an inventive step of at least Bn .
Note in particular that if this new idea satisfies this inventive step requirement, it would automatically secure the38market for that country-i variety in country n: If the current best patented idea from country i in country n forthat variety is z, then the current market price for that variety would be wi dni /z. A new idea that satisfies theminimum inventive step would have a market price of at most wi dni /Bn z, which is ≤ wi dni /z. Hence, a new ideathat is patentable would automatically be competitive and secure the country-n market for that variety.
(1)(2)
Let Cni and Cni be the unit production costs corresponding to the best and second-best patented ideas
respectively originating from country i in country n. These costs are random variables across varieties j fromcountry i’s diﬀerentiated goods sector. (We suppress the time subscripts for ease of notation.) Using the joint
distribution for the productivity of these two best ideas from (2), we have:
(1)

(2)

Gcni (c1 , c2 ) = Pr(Cni ≥ c1 , Cni ≥ c2 )¶µ
wi dni (2) wi dni(1), Zni ≤
= Pr Zni ≤
c1
c2
n
o
2
−θ θ
=
1 + Ti [(Bn wi dni ) c2 − (Bn wi dni )−θ cθ1 ] exp{−Ti (Bn2 wi dni )−θ cθ2 },0 ≤ c1 ≤c2
Bn

From this, the joint distribution of the lowest and second lowest costs corresponding to ideas from country ithat have been patented in country n, which we denote by Gni (c1 , c2 ), is:
(1)

(2)

Gni (c1 , c2 ) = Pr(Cni ≤ c1 , Cni ≤ c2 )
(1)

(2)

(1)

(2)

(1)

(2)

= 1 − Pr(Cni ≥ c1 , Cni ≥ c1 ) − Pr(Cni ≥ 0, Cni ≥ c2 ) + Pr(Cni ≥ c1 , Cni ≥ c2 )©
ª
= 1 − (1 + Ti (wi dni )−θ (Bn−2θ − Bn−θ )cθ1 ) exp −Ti (wi dni )−θ Bn−2θ cθ1 . . .
©
ª−Ti (wi dni )−θ Bn−θ cθ1 exp −Ti (wi dni )−θ Bn−2θ cθ2
Since the firm with the best patented idea sets a price equal to the marginal cost of the second-best patented idea
(limit-pricing), the prevailing market price for country i varieties in country n will be governed by the distribution(2)of Cni :
´
³
´
³
(2)
(1)
(2)
= Pr Cni ≤ c2 , Cni ≤ c2
Pr Cni ≤ c2©
©
ªª
= 1 − exp −Ti (wi dni )−θ Bn−2θ cθ2 − Ti (wi dni )−θ Bn−2θ cθ2 exp −Ti (wi dni )−θ Bn−2θ cθ2
The probability distribution function (pdf) of the market price pni (j) for variety j from country i in country
n is thus given by:
©
©
ª
ª¤
∂ £
1 − exp −Ti (wi dni )−θ Bn−2θ pθni − Ti (wi dni )−θ Bn−2θ pθni exp −Ti (wi dni )−θ Bn−2θ pθni
∂pni©
ª
= θp2θ−1
Ti2 (wi dni )−2θ Bn−4θ exp −Ti (wi dni )−θ Bn−2θ pθni
ni

g2 (pni ) =The ideal price index for diﬀerentiated varieties from country i in country n is therefore:
½Z 1
¾
½Z ∞
¾
exp
ln pni (j) dj= exp(ln p)g2 (p) dp
0
½Z0 ∞
¾
©
ª
2θ−1 2−2θ −4θ−θ −2θ θ= exp(ln p)θpni Ti (wi dni ) Bn exp −Ti (wi dni ) Bn pni dp½Z0 ∞ ·
¾
´¸
³ 1
1
−θ
2
= exp
ln x + ln Ti wi dni Bn · x exp {−x} dxθ0where we have performed the change of variables, x = Ti (wi dni )−θ Bn−2θ pθ , to get³ to this last step.
´
R∞−1
It is straightforward first to show through integration by parts that: 0 ln Ti θ wi dni Bn2 x exp {−x} dx =

39

³ 1
´
R∞
−
ln Ti θ wi dni Bn2 . As for 0 θ1 (ln x)x exp {−x} dx, perform the change of variables, y = − ln x:
Z ∞01(ln x)x exp {−x} dx =
θ

1
θ

Z ∞

−∞

− exp {−y} exp {− exp {−y}} y exp {−y} dy

1∞= − [exp {− exp {−y}} y exp {−y}]−∞
θ
Z
1 ∞
+
exp {− exp {−y}} [exp {−y} − y exp {−y}] dy
θ −∞
Z
1 ∞
=
exp {− exp {−y}} [exp {−y} − y exp {−y}] dy
θ −∞
Z
1
1 ∞
=
−
y exp {−y} exp {− exp {−y}} dy[exp {− exp {−y}}]∞
−∞
θ
θ −∞
Z
1 1 ∞
=
y exp {−y} exp {− exp {−y}} dy−θ θ −∞

R∞
But −∞ y exp {−y} exp {− exp {−y}} dy is the mean of the Gumbel distribution (with pdf: exp {− exp {−y}} dy).
R∞
This is precisely equal to γ, the Euler-Mascheroni constant. So, 0 θ1 (ln x)x exp {−x} dx = 1−γ
θ . The ideal pricenRo© 1−γ ª− θ1
12
index, Pni = exp 0 ln p(j)ni dj is thus equal to Ti wi dni Bn exp θ .
µ¶ 1© 1−γ ª´1−φ 1−φPN ³ − θ1
2
The diﬀerentiated goods sector price index for country n is thus:
w
d
B
exp
,
T
i ni nii=1θ1µ´1−φ ¶ 1−φ
PN ³ − θ1wdup to a multiplicative constant. In particular, for the twoT
which is equal to: Bn2
i ni
i
i=1

country case which we consider in Section 5 of the paper, this price index is equal to:
1
1
µ³
µ³
´1−φ ¶ 1−φ´1−φ ¶ 1−φ´1−φ ³ 1´1−φ ³ 1
− θ1
−θ
− θ1
−θ
2
2
B1
+ T2 w2 d
for country 1; and equal to: B2+ T1 w1 d
T1 w1
T2 w2

for country 2. The corresponding welfare expressions (the real wage) for country 1 and 2 respectively are:
1−η
1−η
µ³
µ³
´1−φ ¶− 1−φ´1−φ ¶− 1−φ´1−φ ³ 1´1−φ ³ 1
− θ1
−θ
− θ1
−θ
−2(1−η)−2(1−η)w1 B1+ T2 w2 d
and w2 B2+ T1 w1 d
.
T1 w1
T2 w2

Proof of Proposition 1: [Optimal inventive step in the Closed Economy]

Proof. Log-diﬀerentiating (24), one obtains the following first-order condition:ηη¢ (B(1+θ)2 −θ2 )( 1−η
¡B+1)+ 1−ηB
ρ − gθL (1 − η)B θ [B(1+θ)−θ]2
=0
−2 +
¡
¢ ( η )B(1+θ)+θ
gL + ρ − gθL (1 − η) B1−η
θ [B(1+θ)−θ]Note that the denominator of the fraction on the left-hand side is positive, since: (i) (1 + θ)B − θ > 0, as
B ≥ 1 > θ/(1 + θ); and (ii) gρL − 1−ηθ > 0. Upon further simplification, the first-order condition is equivalent to:2−2B θ [B(1 + θ) − θ] +

µ

ρ
1−η
−
gL
θ

¸
¶·
η
η
−
B 2 (1 + θ)2 +B(1 + θ)2 + B(1 − θ2 ) + θ2 = 0
1−η
1−ηLet Γ denote the expression on the left-hand side of the above equation. We have: Diﬀerentiating this withrespect to B yields:dΓdB= −2B θ−1 (B(1 + θ) − θ) [θ (B(1 + θ) − θ) + 2B(1 + θ)]
¸
¶·
µ
1−η
η
ρ
2
2
−
(1 − 2B)(1 + θ) + (1 − θ )
+
gL
θ
1−η
< 0

for all B ≥ 1 and θ > 1.

40

³
´
dΓ
Observe that: (i) Γ < 0 as B → ∞; and (ii) Γ = −2 + gρL − 1−η< 0 for all B > 1,
when B = 1. Since dBθ
the intermediate value theorem implies that a necessary and suﬃcient condition for a solution to the first-ordercondition to exist is: gρL − 1−ηθ > 2. When this holds, the graph of Γ as a function of B starts out at a positive
value at B = 1 and declines monotonically, eventually taking on a negative value for B suﬃciently big. Therethus exists a unique B ∗ > 1 that satisfies the first-order condition. Moreover, sign(Γ) > 0 for 1 ≥ B < B ∗ andsign(Γ) < 0 for B > B ∗ , which means that the slope of the original objective function is positive for 1 ≥ B < B ∗and negative for B > B ∗ . In other words, the turning point at B ∗ is indeed a maxima.
On the other hand, when 0 < gρL − 1−ηθ < 2, the Γ function always takes on a negative value, which means thatoriginal objective function is decreasing for all B ≥ 1. The maximum is therefore achieved at a corner solution,
B ∗ = 1.
Proof that there in negative cross-border externality when φ is suﬃciently large andpositive externality when φ and d are intermediate in value (or, externality would never be positive when d islarge)
Proof We can be easily show that under symmetry
K1
K2
A11
A12
A13
A21
A22
A23
D1
D2

= d1−φ= d1−φ¢¤¡¢£¡h(B) 1 + d1−φ − 1 1 − d1−φ
=
>0
Bθ
d1−φ
= − θ <0Bd2(1−φ)
=
>0
Bθ
= A11
= A12
= A13
¡
¢¤
d1−φ h(B) £=2 (φ − 1) − θ 1 + d1−φ > 0 given φ > θ + 1
θθB= D1 = D

Referring to (27) and (28), it can be easily shown that under symmetry⇒h(B) − 1
Bθ
αL
£
¤
gL
T ρ − θ (1 − η)==αL
T − gLρ − gθL (1 − η)for all dgLh(B) − 1+gLθBρ − θ (1 − η)for all dThis equation will be useful below.
Recall that there is negative (positive) externality when d1−φ T̂1 + T̂2 is less (greater) than zero.
1. Negative externality
A suﬃcient condition for negative externality (i.e. d1−φ T̂1 + T̂2 < 0) is ad − bc > 0 and (af − ce) +(de − bf ) d1−φ < 0
A suﬃcient condition for ad − bc > 0, af − ce < 0 and de − bf > 0 is·¸αL
1 − d2(1−φ)−D >DT ρ − gθL (1 − η)which is equivalent to

·
>

¸³
´
h(B) − 1gL+
1 − d2(1−φ)
gLθBρ − θ (1 − η)
1−φ
¡
¢¤
2d
h(B) £
2 (φ − 1) − θ 1 + d1−φ
θB θ

41

(42)

Given that αLTexternality) ish1−d2(1−φ)g
ρ− θL (1−η)

i−ce|> de − bf (so as to get negative− D > D , a suﬃcient condition for |af
d1−φ

D

d
which is equivalent to

>
1−φ

·
¸
αL
1 − d2(1−φ)−DT ρ − gθL (1 − η)

¢
¡
¡
¢¤
1 + d1−φ h(B) £2 (φ − 1) − θ 1 + d1−φ
θ
θB
·
¸³
´
h(B) − 1gL>+
1 − d2(1−φ)
gLθBρ − θ (1 − η)(43)

Combining (42) and (43), we have

which is equivalent to (37):

¢
¡
¡
¢¤
1 + d1−φ h(B) £2 (φ − 1) − θ 1 + d1−φ
θ
θB
·
¸³
´
h(B) − 1
gL
2(1−φ)
>
+
1
−
d
g
Bθ
ρ − θL (1 − η)¡¢¤
2d1−φ h(B) £>2 (φ − 1) − θ 1 + d1−φ
θ
θB

(44)

·
¸
£
¡
¢¤
¡
¢¤
gL
θB θ h(B) − 1
2d1−φ £
1
1−φ
+
2
(φ
−
1)
−
θ
1
+
d
>
2 (φ − 1) − θ 1 + d1−φ
>
g
L
1−φ
θ
2(1−φ)1−dh(B)Bρ − θ (1 − η)
1−dThe above condition (44) together with φ > θ + 1 would guarantee that
1. ad − bc > 0
2. de − bf > 0
3. af − ce < 0−ce|
4. |af> de − bfd1−φ
5. and therefore, an increase in B1 leads to a decrease in U2 – there is negative cross-border externality fromincreasing the inventive step requirement at home.
The eﬀect of φ on the LHS of (37) is always positive for all φ, since the eﬀects of φ − 1 and φ − θ − 1 are going
1+d1−φto dominate that of 1−d1−φ . Proof:"¢#¡∂ 2 (φ − 1) − θ 1 + d1−φ∂φ
1 − d1−φ¡¢2
1 − d1−φ + 1 − d2(1−φ) − 2d2(1−φ) (φ − θ − 1) ln d=(45)2
(1 − d1−φ )Note that the above expression is positive for all 1 < φ ≤ θ + 1. Now, taking derivative of the numerator,
i
¢2
∂ h¡
1 − d1−φ + 1 − d2(1−φ) − 2d1−φ (φ − θ − 1) ln d
∂φ
i
h
¡
¢
= ln d 2d2(1−φ) + 4d2(1−φ) (φ − θ − 1) (ln d) − 2d2(1−φ) + 2 1 − d1−φ d1−φhi¡¢= ln d 4d2(1−φ) (φ − θ − 1) (ln d) + 2 1 − d1−φ d1−φ > 0 for φ ≥ θ + 1
We conclude that"¢#¡
∂ 2 (φ − 1) − θ 1 + d1−φ
> 0 for all φ > 1∂φ1 − d1−φ42

2(φ−1)−θ(1+d1−φ )Moreover, the value ofranges from 0 to infinity as φ increases from small to large. Hence, for a1−d1−φ∗given d, we can always find a φ such that when φ > φ∗ condition (44) is satisfied. The graph of this expression
as a function of φ is shown in Figure 1.

2. Positive externality
A suﬃcient condition for positive externality (i.e. d1−φ T̂1 + T̂2 > 0) is ad − bc < 0 and (af − ce) +(de − bf ) d1−φ < 0.
The suﬃcient conditions for ad − bc < 0 and af − ce < 0 are·¸
1 − d2(1−φ)αL−D <DT ρ − gθL (1 − η)and·¸
1 − d2(1−φ)αL−D >0T ρ − gθL (1 − η)These two conditions together is equivalent to
h
0<

αL
T

1−d2(1−φ)g
ρ− θL (1−η)

D

i

−D

<1

(46)

As T̂1 > T̂2 , we know that T̂2 > 0 ⇒ T̂1 > 0 ⇒ dU2 /dB1 > 0. But (46) is already suﬃcient for ad − bc < 0 and
af − ce < 0, which implies that T̂2 > 0. Therefore, (46) is a suﬃcient condition for dU2 /dB1 > 0.
(46) implies that·¸αL
1 − d2(1−φ)
D<< 2DT ρ − gθL (1 − η)which is equivalent towhich is equivalent to (??):¡¢¤
d1−φ h(B) £2 (φ − 1) − θ 1 + d1−φ
θ
θB
·
¸³
´
h(B) − 1
gL
2(1−φ)<+1−dBθρ − gθL (1 − η)¡¢¤
2d1−φ h(B) £<2 (φ − 1) − θ 1 + d1−φ
θ
θB

·
¸
£
¡
¢¤
¡
¢¤
d1−φ
θB θ h(B) − 1
gL
2d1−φ £
1−φ
2
(φ
−
1)
−
θ
1
+
d
<
2 (φ − 1) − θ 1 + d1−φ
+
<
g
h(B)
Bθ
ρ − θL (1 − η)
1 − d2(1−φ)
1 − d2(1−φ)
The right hand side is exactly two times the left hand side. It can be shown that both left hand side andright hand side have a hump-shaped relationship with φ, and the expressions tend to zero as φ tends to infinity.
Moreover, for any given φ, both expressions decrease with d, and increase without bound as d decreases towards
one. The expressions are graphed as functions of φ in Figure 1. Note that the right hand side expression is the sameas the right hand side expression in the last set of inequalities. The left hand side expression is exactly one halftimes the right hand side expression. Therefore, it also has a hump-shaped relationship with φ and also decreases
with d. It tends to infinity as d tends to one. Therefore, from case (b) of Figure 1, we see that for any given φ,
it is possible for there to be positive externality when d is suﬃcient large but not too large. Therefore, we have:∂Uj∂Bi > 0 where i ∈ {1, 2} and j 6= i, i.e. there is positive cross-border externality of strengthening inventive steprequirement by a country, when φ and d are both intermediate in value.

Best Response Functions under Symmetry43

In this notes, we try to derive the best response function under symmetry, and then show that the second ordercondition is satisfied.
Tb1 (ad − bc)
£
¤
· 0
¸
− h(B1 )(1 + d1−φ ) − 1 (1 − d1−φ )h (B1 )(1 + d1−φ )θ=−h(B1 )(1 + d1−φ ) − 1 B1
B1θ
½·
¸¾
¸h
i · d1−φ h(B ) ¡¢h(B2 ) − 1
gL
d1−φ
2
2(1−φ)1−φ×+− 1 + (φ − θ − 1)−
φ − θd1−dρ − gθL (1 − η)
B2θ
θB2θ
θB1θ
#
"
¡
¢
¸
· 1−φ
¢
d1−φ 1 − d1−φθ
d1−φdh(B1 ) ¡
1−φ
φ
−
1
−
θd
+
(φ
−
θ
−
1)
+
B1
θB1θ
θB2θ
B1θ

Tb2 (ad − bc)"¡¢#
d1−φ 1 − d1−φ
θ
= −
θ
B1
B1
½·
¸¾
¸h
i · d1−φ h(B ) ¡¢h(B1 ) − 1
gL
d1−φ
1
2(1−φ)1−φ×+− 1 + (φ − θ − 1)−
φ − θd1−dρ − gθL (1 − η)
B1θ
θB1θ
θB2θ
¤
£
h(B1 )(1 + d1−φ ) − 1 (1 − d1−φ )
+
B1θ
· 0
¸ · 1−φ¸¢h (B1 )(1 + d1−φ )dd1−φθh(B2 ) ¡1−φ
×
φ − 1 − θd+ (φ − θ − 1)−h(B1 )(1 + d1−φ ) − 1 B1
θB2θ
θB1θ

The best response function of country 1 is given by (35), which can be re-written as: ·³ ´ 1

¸1−φ
−θ


T
w



× w21dB1 T̂1 + B1 T̂2 
 T12

=0
−2θ +
·³ ´ 1
¸1−φ


−θ


T1




× w1
+1
T2

w2 d

Since an increase in B1 leads to an increase in TT12 , the eﬀect of B1 on
Since T̂1 > T̂2 , the eﬀect of

·³

T1
T2

´− 1
θ

× ww21d

¸1−φ

·³

T1
T2

´− 1
θ

× ww21d¸1−φis negative.

on the second term on the LHS is positive. Therefore an·³ ´ 1
¸1−φ
−θ
T1
w1
× w2 dincrease in B1 leads to a decrease of the second term on the LHS so far as its eﬀect on T2·³ ´ 1
¸1−φ
−θ
T1
w1
is concerned. Under symmetry,× w2 d= dφ−1 > 1. From the first two equations above, we
T2
φ−1

1 T̂1 +B1 T̂2believe it is true that for any given B2 , an increse in B1 leads to a decrease in d dBφ−1. Therefore,+1
we believe that an increase in B1 leads to a decrease in the second term on the LHS, which means thatthe second order condition is satisfied.

44

Negative
externality(a) When B is large, d islarge, rho is small, g_L islarge and eta is small.

X
C

Y
Z

0

φ
Negative

Positive

Negative
externality(b) When B, d, rho, g_L andeta are intermediate invalues.

X
C
Y
Z
0

φ
Positive

Positive

Negative
externality(c) When B is small, d issmall, rho is large, g_L issmall and eta is large.

X

Y

C

Z
0

φ

Figure 1. Curve X = LHS of equation (37); curve Y = RHS ofequation (37); curve Z = LHS of equation (38); C is the value ofthe middle expression in both (37) and (38).

Increasing welfare:Country 1

B2

BRF1
45-degree line

BRF2

Increasing
welfare: Country 2

Nash Equilibrium (NE)

Pareto superior to NE

B1 (required inventive step)

Figure 2A
Nash Equilibrium under a Required Inventive Step IP Regime for Chor and Lai (2009)

Ω2

BRF1

Increasing welfare:Country 1
45-degree line

Pareto superior to NE

Increasing welfare:Country 2

BRF2
Nash Equilibrium (NE)

Ω1 (patent length)
Figure 2B
Nash Equilibrium under a Patent Length IP Regime for Grossman and Lai (2004)MB (closed-economy)

MB (open-economy)

MC

B1

Figure 3
The marginal benefit and marginal cost of increased required inventive step