Effects of Patents on the Transition from Stagnation to Growth

This study provides a growth-theoretic analysis on the effects of intellectual property rights on the endogenous takeoff of an economy. We incorporate patent protection into a Schumpeterian growth model in which takeoff occurs when the population size crosses an endogenous threshold. We find that strengthening patent protection has contrasting effects on economic growth at different stages of the economy. Specifically, it leads to an earlier takeoff but also reduces economic growth in the long run. JEL classification: O31, O34


Introduction
The di¤erential timing of countries experiencing a transition from stagnation to growth has governed patterns of comparative economic development across the world and contributed signi…cantly to the divergence in income across the world over the past two centuries. 1 Given the importance of intellectual property rights (IPR) to the pace of technological progress and therefore to the transition from stagnation to growth, this study explores the role that the patent system may have played in the pace of this transition and on economic growth in the long run.
The UK experienced this transition during the late 18th/early 19th century. Figure 1 plots real GDP per capita in the UK. 2 Figure 2 plots the log level of real GDP per capita, in which the slope shows the growth rate of income. In the 18th century, income in the UK grew very slowly. Speci…cally, the average annual growth rate of income in the UK from 1701 to 1800 was 0.4%. Then, the average growth rate from 1801 to 1900 increased to 1.0%. From the 20th century onwards, the average growth rate stabilized at about 1.7%. We incorporate patent protection into the Schumpeterian growth model of endogenous take-o¤ in Peretto (2015). In this model, the economy …rst experiences stagnation with zero growth in output per capita when the market size is small. Here population size plays the crucial role of determining the market size, which in turn implies that population growth gives rise to an expansion of the market. As the market size becomes su¢ciently large, innovation takes place and the economy gradually experiences growth. In the long run, the economy converges to a balanced growth path (BGP) with steady-state growth. Within this growth-theoretic framework that is consistent with the growth pattern in Figure 1 and 2, we obtain the following results.
Strengthening patent protection leads to an earlier take-o¤. Incentives for innovation to take place depend on the market value of inventions, which in turn depends on the level of patent protection and the market size. Therefore, when stronger patent protection increases the market value of patents by reducing price competition and making …rms more pro…table, it also reduces the market size required for innovation to take place. As a result, the economy starts to experience innovation and growth at an earlier time (i.e., an earlier industrial revolution). Our …nding that stronger IPR protection leads to an earlier (but not necessarily immediate) take-o¤ is consistent with historical evidence on the e¤ects of IPR on the industrial revolution. 3 However, stronger patent protection eventually reduces innovation and growth as recent studies tend to …nd. 4 Intuitively, although stronger patent protection encourages entry and increases the number of products in the economy, this larger number of products reduces the market size of each product and redirects resources away from the quality-improving innovation of each product, which determines long-run growth. 5 This study relates to the literature on innovation and economic growth. Romer (1990) develops the seminal variety-expanding growth model in which innovation is driven by new products, whereas Aghion and Howitt (1992) develop the Schumpeterian quality-ladder growth model in which innovation is driven by higher-quality products. Peretto (1998Peretto ( , 1999 and Smulders and van de Klundert (1995) combine the two dimensions of innovation and develop a Schumpeterian growth model with endogenous market structure. This study explores the e¤ects of IPR in this vintage of the Schumpeterian growth model.
In the literature on IPR and innovation, other studies also explore the e¤ects of IPR in the innovation-driven growth model. 6 These studies mostly focus on either variety expansion or quality improvement. Only a few studies, such as Chu, Cozzi and Galli (2012) and Chu, Furukawa and Ji (2016), explore the e¤ects of IPR in the Schumpeterian growth model with both dimensions of innovation. However, these studies do not consider the case in which the e¤ects of IPR can change at di¤erent stages of the economy. Iwaisako (2013), Chu, Cozzi and Galli (2014) and Chu, Cozzi, Fan, Pan and Zhang (2019) show that the growth or welfare e¤ects of IPR can depend on, respectively, the level of public services, the distance to the technology frontier and the level of …nancial development in the economy. However, none of these studies consider how IPR a¤ects the endogenous take-o¤ of an economy. The novel contributions of this study are to explore the e¤ects of IPR in a Schumpeterian growth model of endogenous take-o¤ and to highlight the contrasting e¤ects of IPR on economic growth at di¤erent stages of the economy with di¤erent dimensions of innovation.
This study also relates to the literature on endogenous take-o¤ and economic growth. 3 See e.g., North and Thomas (1973), North (1981), Dutton (1984) and Khan (2005). 4 Galor and Weil (2000) provide the seminal study and develop uni…ed growth theory, 7 which explores how the quality-quantity tradeo¤ in childrearing and human capital accumulation allow a country to escape from the Malthusian trap and lead to the endogenous take-o¤ of the economy. 8 Peretto (2015) develops a Schumpeterian growth model of endogenous take-o¤, which features exogenous population growth and does not capture the Malthusian trap; instead, it describes an economy in which take-o¤ is driven by innovation, which also relates to the industrial revolution and is suitable for our analysis of patent policy. The Peretto model features both quality improvement and variety expansion, under which endogenous growth in the number of products provides a dilution e¤ect that removes the scale e¤ect of population size on long-run growth. Therefore, although the population size a¤ects the timing of the take-o¤, it does not a¤ect the steady-state growth rate. We incorporate patent protection into the Peretto model to explore its e¤ects on endogenous take-o¤.
The rest of the paper is organized as follows. Section 2 presents the model. Section 3 explores the e¤ects of patent policy at di¤erent stages of the economy. Section 4 concludes.

A Schumpeterian model of endogenous take-o¤
The theoretical framework is based on the Schumpeterian growth model with both varietyexpanding innovation and quality-improving innovation in Peretto (2015). In this model, labor is used as a factor input for the production of …nal good. Final good is used for consumption and as a factor input for entry, in-house R&D, the production and operation of intermediate goods. We incorporate a patent policy parameter into the model and analyze its e¤ects on the take-o¤, transitional dynamics and the BGP of the economy.

Household
The representative household has a utility function given by where c t C t =L t denotes per capita consumption of …nal good (numeraire) at time t, and > 0 is the subjective discount rate. Population grows at an exogenous rate 2 (0; ) with initial population normalized to unity (i.e., L t = e t ). The household maximizes (1) subject to where a t A t =L t is the real value of assets owned by each member of the household, and r t is the real interest rate. Each member supplies one unit of labor to earn w t . Standard dynamic optimization yields

Final good
Final output Y t is produced by competitive …rms using the following production function: The productivity of X t (i) depends on its quality Z t (i) and the average quality of all intermediate goods Z t R Nt 0 Z t (j) dj=N t capturing technology spillovers. The private return to quality is determined by , and the degree of technology spillovers is determined by 1 . The parameter 1 captures a congestion e¤ect of variety, and hence, the social return to variety is measured by .
Pro…t maximization yields the following conditional demand functions for L t and X t (i):

Intermediate goods and in-house R&D
Monopolistic …rms produce di¤erentiated intermediate goods with a linear technology that requires X t (i) units of …nal good to produce Therefore, the marginal cost for the …rm in industry i to produce X t (i) with quality Z t (i) is one. The …rm also incurs Z t (i) Z 1 t units of …nal good as a …xed operating cost. To improve the quality of its products, the …rm devotes I t (i) units of …nal good to in-house R&D. The innovation process is and the …rm's (before-R&D) pro…t ‡ow at time t is The value of the monopolistic …rm in industry i is The monopolistic …rm maximizes (9) subject to (7) and (8). We solve this dynamic optimization problem in the proof of Lemma 1 and …nd that the unconstrained pro…t-maximizing markup ratio is 1= . To analyze the e¤ects of patent breadth, we introduce a policy parameter > 1, which determines the unit cost for imitative …rms to produce X t (i) with the same quality Z t (i) 9 as the monopolistic …rm in industry i. 10 Intuitively, a larger patent breadth increases the cost of imitation and allows the monopolistic producer of X t (i), who owns the patents, to charge a higher markup without losing her market share to potential imitators; 11 see also Li (2001), Goh and Olivier (2002) and Iwaisako and Futagami (2013). The equilibrium price becomes We assume that < 1= . In this case, increasing patent breadth raises the markup. We follow previous studies to consider a symmetric equilibrium in which Z t (i) = Z t for i 2 [0; N t ] and the size of each intermediate-good …rm is identical across all industries X t (i) = X t . 12 From (6) and p t (i) = , the quality-adjusted …rm size is We de…ne the following transformed variable: which is a state variable determined by the quality-adjusted …rm size and not directly a¤ected by (but indirectly via N t ). In Lemma 1, we derive the rate of return on quality-improving R&D, which is increasing in x t and .

Lemma 1
The rate of return on quality-improving in-house R&D is 13 Proof. See the Appendix.

Entrants
Following previous studies, we assume that entrants have access to aggregate technology Z t to ensure symmetric equilibrium at any time t. A new …rm pays X t units of …nal good to enter the market with a new variety of intermediate goods and set up its operation. > 0 is an entry-cost parameter. The asset-pricing equation implies that the return on assets is When entry is positive, free entry implies Substituting (7), (8), (12), (15) and p t = into (14) yields the return on entry as where z t _ Z t =Z t is the growth rate of aggregate quality.

Equilibrium
The equilibrium is a time path of allocations fA t ; Y t ; C t ; X t ; I t g and prices fr t ; w t ; p t ; V t g such that the household maximizes utility taking fr t ; w t g as given; competitive …rms produce Y t and maximize pro…ts taking fw t ; p t g as given; incumbents for intermediate goods choose fp t ; I t g to maximize V t taking r t as given; entrants make entry decisions taking V t as given; the value of all existing monopolistic …rms adds up to the value of the household's assets such that A t = N t V t ; and the following market-clearing condition of …nal good holds:

Aggregation
Substituting (6) and p t = into (4) and imposing symmetry yield aggregate output as The growth rate of output per capita is where y t Y t =L t denotes output per capita. Its growth rate g t is determined by both the variety growth rate n t _ N t =N t and the quality growth rate z t .

Dynamics of the economy
The dynamics of the economy is determined by the dynamics of x t = 1=(1 ) L t =N 1 t . Its initial value is x 0 = 1=(1 ) =N 1 0 . In the …rst stage of the economy, there is neither variety expansion nor quality improvement. At this stage, x t increases solely due to population growth. When x t becomes su¢ciently large, innovation begins to happen. The following inequality ensures the realistic case in which the creation of products (i.e., variety-expanding innovation) happens before the improvement of products (i.e., quality-improving innovation).
Variety-expanding innovation happens when x t crosses the …rst threshold x N de…ned as which is the value of x t that yields n t = 0 when z t = 0. Then, quality-improving innovation also happens when x t crosses the second threshold x Z de…ned as which is the value of x t that yields z t = 0 when n t > 0. The inequality in (20) implies x N < x Z . In the long run, x t converges to its steady-state value x . The following inequalities ensure that when the economy is on the BGP, the variables fx ; z ; g g are positive: The following proposition adapted from Peretto (2015) summarizes the dynamics of x t .
Proposition 1 When the initial condition of the economy satis…es 14 the dynamics of x t is given by 15 where v 1 1 + 1 , 14 The inequality x 0 > 1=(1 ) = ( 1) implies that 0 > 0. 15 It can be shown that (20) and (23) Proof. See the Appendix.

Stage 1: Stagnation
When the market size is not large enough (i.e., x t x N ), there are insu¢cient incentives for …rms to develop new products or improve the quality of existing products. In this case, output per capita is and the growth rate of y t is g t = 0. In this regime, strengthening patent protection decreases y t due to monopolistic distortion that reduces intermediate production X t . However, stronger patent protection also leads to an earlier (but not necessarily immediate) take-o¤ by decreasing x N in (21). Intuitively, stronger patent protection increases the pro…tability of …rms and provides more incentives for …rms to develop new products. As a result, the economy starts to experience innovation at an earlier time.
Proposition 2 When x t x N , stronger patent protection reduces the level of output per capita but leads to an earlier take-o¤.
Proof. Use (21) and (26) to show that x N and y t are decreasing in . Given that x t increases at the exogenous rate when x t x N , a smaller x N implies an earlier take-o¤.

Stage 2: Variety expansion
When the market size is su¢ciently large (i.e., x t > x N ), …rms have incentives to develop new products. In this case, output per capita is and the growth rate of y t is g t = n t . In the Appendix, we show that whenever n t > 0, c t =y t always jumps to a steady state. Therefore, we can substitute r e t in (16) into the Euler equation r t = + g t = + n t in (3) and also use (12) to derive the variety growth rate as 16 16 Note from (21) and (28)  For a given level of x t , a larger patent breadth raises the rate of return on variety-expanding innovation and increases the equilibrium growth rate g t = n t as in previous studies, such as Li (2001)

and O'Donoghue and Zweimuller (2004).
Proposition 3 For a given x t 2 (x N ; x Z ), stronger patent protection increases the equilibrium growth rate.
Proof. Use (28) to show that g t = n t is increasing in for a given x t .

Stage 3: Quality improvement and variety expansion
When the market size becomes even larger (i.e., x t > x Z ), …rms have incentives to improve the quality of products in addition to inventing new products. Then, output per capita is and the growth rate of y t is g t = n t + z t . We can then substitute r q t in (13) into the Euler equation r t = + g t = + n t + z t in (3) to derive the quality growth rate as 17 For a given level of x t , a larger patent breadth raises the rate of return on quality-improving innovation and continues to increase the equilibrium growth rate g t = n t + z t = r q t , where r q t = Proposition 4 For a given x t 2 (x Z ; x ), stronger patent protection increases the equilibrium growth rate.
Proof. Use (30) to show that g t = n t + z t is increasing in for a given x t .

Stage 4: Balanced growth path
In the long run, x t converges to x . Then, the steady-state quality growth rate is where n = =(1 ) > 0 and which is decreasing in . Intuitively, stronger patent protection increases the number of products, which leads to a smaller market size for each product. This smaller …rm size x in turn reduces the incentives for quality-improving innovation and the steady-state equilibrium growth rate g = n + z . This result generalizes the one in Chu et al. (2016), who assume zero social return to variety (i.e., = 0).

Proposition 5
On the BGP (i.e., x t = x ), stronger patent protection decreases the steadystate equilibrium growth rate.
Proof. Use (31) and (32) to show that g = n + z is decreasing in .

Summary
We summarize the dynamics of the economy in the following …gures. Figure 3 plots the relationship between the quality-adjusted …rm size x t and the equilibrium growth rate g t . It shows that when x t is below the …rst threshold x N , the economy does not grow due to the absence of variety-expanding innovation (and also quality-improving innovation). When x t crosses the …rst threshold x N , variety-expanding innovation begins to happen. When x t crosses the second threshold x Z , quality-improving innovation also happens. A larger patent breadth shifts the curve to the left giving rise to a higher growth rate for any given x t . Figure 3: Relationship between …rm size and growth Figure 4 plots the transition path of the quality-adjusted …rm size x t . 18 It shows how x t evolves from an initial state x 0 to the steady state x , which is decreasing in the level of patent breadth . Finally, Figure 5 summarizes the transition path of the equilibrium growth rate g t and shows that strengthening patent protection leads to an earlier take-o¤ (by decreasing x N ) but also lower long-run growth (by decreasing x ).

Conclusion
In this study, we analyze the e¤ects of IPR in a Schumpeterian growth model with endogenous take-o¤ and …nd that strengthening patent protection causes an earlier take-o¤ by increasing the pro…tability of …rms and providing more incentives for …rms to innovate. However, stronger patent protection eventually slows down economic growth by increasing the number of products that reduces the market size of each product and the incentives for qualityimproving innovation. These contrasting e¤ects of IPR at di¤erent stages of the economy are consistent with historical evidence on the industrial revolution and recent evidence on the e¤ects of the patent system.
These results are also consistent with the fact that the UK implemented a patent system before the US and experienced an earlier industrial revolution but eventually lower economic growth than the US. Our analysis also addresses some critiques on the hypothesis that IPR contributed to the occurrence of the industrial revolution; see for example, Mokyr (2009).
These critiques can be summarized as follows. First, the emergence of the patent system occurred much earlier than the industrial revolution. Second, many inventions at that time were not patented. Our analysis shows that strengthening IPR does not necessarily lead to an immediate take-o¤ but only an earlier take-o¤. Furthermore, although our analysis does not feature unpatented inventions, the no-arbitrage condition in a model with both patented and unpatented inventions should imply that when the rate of return on patented inventions increases, the rate of return on unpatented inventions also increases.
Finally, this study considers a closed economy for simplicity. In an open economy, the strengthening of patent protection and the endogenous take-o¤ of one country may have the following e¤ects on other countries. On the one hand, it may lead to technology spillovers to other countries. On the other hand, it may cause the industrializing country to specialize in industrial production and other countries to specialize in agricultural production, resulting into a delay of their take-o¤. 19 We leave this interesting extension to future research.

Compliance with ethical standards
The authors declare that they have no con ‡ict of interest. 19

See Galor and Mountford (2008).
Proof of Lemma 1. The current-value Hamiltonian for monopolistic …rm i is where ! t (i) is the multiplier on p t (i) . Substituting (6)-(8) into (A1), we can derive If p t (i) < , then ! t (i) = 0. In this case, @ t (i) =@p t (i) = 0 yields p t (i) = 1= . If the constraint on p t (i) is binding, then ! t (i) > 0. In this case, we have p t (i) = . Therefore, Given that we assume < 1= , the monopolistic …rm sets its price at p t (i) = . Substituting (A3), (12) and p t (i) = into (A4) and imposing symmetry yield which is the rate of return on quality-improving in-house R&D.
Before we prove Proposition 1, we …rst derive the dynamics of the consumption-output ratio C t =Y t when n t > 0.
Lemma 2 When n t > 0, the consumption-output ratio always jumps to Proof. The total value of assets owned by the household is When n t > 0, the no-arbitrage condition for entry in (15) holds. Then, substituting (15) and X t N t = Y t into (A8) yields which implies that the asset-output ratio A t =Y t is constant. Substituting (A9), (3) and (5) into which can be rearranged as Therefore, the dynamics of C t =Y t is characterized by saddle-point stability, such that C t =Y t jumps to its steady-state value in (A7).
Proof of Proposition 1. Using (12), we can derive the growth rate of x t as When x 0 x t x N , we have n t = 0 and z t = 0. In this case, the dynamics of x t is given by When x N < x t x Z , we have n t > 0 and z t = 0. In this case, Lemma 2 implies that C t =Y t is constant and _ c t =c t = _ y t =y t . Therefore, we can substitute r e t in (16) and (A12) into r t = + n t in (3) to obtain (28). Substituting (28) into (A12) yields the dynamics of x t as De…ning v 1 1 + 1 and x 1=(1 ) 1 ( + 1 ) , we can express (A14) as When x t > x Z , we have n t > 0 and z t > 0. In this case, Lemma 2 implies that C t =Y t is also constant, and _ c t =c t = _ y t =y t . Then, we use (3), (19) and _ c t =c t = _ y t =y to obtain r t = + n t + z t .