Endogenous Growth and North-South Technology Transfer with. Two Intermediate Inputs

Transcription

1 Endogenous Growth and North-South Technology Transfer with Two Intermediate Inputs Yuchen Shao Keith Maskus January 21, 2017 Abstract We present a North-South general-equilibrium model in which final goods are made from two intermediate inputs of differing technological levels, with each subject to dynamic quality improvements. The model generates endogenous input specialization in global production and R&D activities. Specifically, in steady-state equilibrium a leading Southern intermediate firm produces the low-technology input, which it had successfully imitated, and other Southern firms strive to imitate the high-technology version. A leading Northern firm specializes in producing the high-technology input that it had successfully developed, while others innovate against the low-technology variant. Thus, product-cycle dynamics appear only in the low-technology input, while production of the high-technology version remains in the North. Simulation of the calibrated model finds that an increase in Southern intellectual property protection raises the Northern relative wage while reducing the rate of technology transfer. In contrast, a higher relative quality improvement in the high-technology good expands technology diffusion. Keywords: global production; input specialization; quality ladders JEL classification: F10, O3, O4 We are grateful to Xiaoping Chen, Yongmin Chen, Thibault Fally, Terra McKinnish, James Markusen and all participants in the Research Method Workshop of CU-Boulder and the 2011 Western Economic Association Annual Conference for their very helpful comments. All remaining mistakes are our own. Department of International Economics and Trade, School of Economics, Nanjing University, Nanjing, Jiangsu Province , China; shaoych04@gmail.com Corresponding author. Department of Economics, University of Colorado, Boulder CO 80309; keith.maskus@colorado.edu 1

2 1 Introduction One of the primary responses to sustained trade liberalization and reduction in transport costs in recent decades has been the extensive fragmentation of production both within global networks and via outsourcing (Baldwin and Lopez-Gonzalez, 2013; Antras, et al, 2012). Empirical evidence shows that international outsourcing of vertical production stages accounts for a major share of recent growth of world trade in manufactures (Bems, et al, 2013). Our point of departure in this analysis is the observation that within this vertical structure there is a specialized pattern of some manufacturing segments. Specifically, some intermediate inputs, which are typically more technologically standardized, tend to follow the classical product-cycle structure of innovation in advanced countries, followed by the transfer of production to emerging economies, with the inputs often exported back to the innovator for incorporation into a final good. Others, which have more sophisticated technologies, are less likely to enter this cycle in favor of production remaining within advanced economies. One example is the semiconductors and integrated circuits industry (HS 8542). The products of this group, including electronic integrated circuits, processors, controllers, memories and the like, are components of most electronic devices. On the one hand, fabrication activities that remain in the advanced regions tend to yield products with higher unit values, which is taken as an indication of higher quality in empirical studies (Schott, 2004; Hummels and Klenow, 2005; Krishna and Maloney, 2011). On the other hand, assembly operations in Asia dominate production of lower-value electronic inputs. The dynamic read-write random access memory (DRAM, HS ) is one illustration. Invented in the United States decades ago, China now is among the largest producers in the world. However, DRAM size (the amount of memory) is a good indicator of product quality and technical sophistication. As illustrated in Figure 1, the customs value of Chinese exports to the United States in 2007 compared with that of Germany clearly declined as the memory capacity of such chips increased. <Insert Figure 1 here.> 1

3 The data in Figure 1 suggest that trade patterns among similar inputs of varying quality depend on national origin. However, this is a cross-sectional story and our product cycle analysis is dynamic, involving trade over time in two inputs, one involving a higher level of technical sophistication than the other. In that context, Figure 2 depicts the evolution of U.S. imports in aircraft engines (HS ), among the most sophisticated components of airplanes, from China and Brazil, on the one hand, and Germany, the United Kingdom, and Canada, on the other, from 1989 to It is evident that the former pair, despite their significant relative growth in overall manufactures over this period, failed to achieve inroads in this product in the U.S. market. The import share of the three industrialized nations, in contrast, remained relatively stable. In Figure 3 we depict similar computations for aircraft parts (HS ), many of which have become standardized over time. As indicated, the two developing countries saw steady growth in the U.S. import share, while that of the developed economies fell by over 40 percent. <Insert Figure 2 here.> <Insert Figure 3 here.> This relative specialization, or differential proclivity of inputs toward product cycles, has yet to be studied in a dynamic, general-equilibrium model that can unearth insights about implications for growth, relative wages, and welfare. Thus, we examine the possibility of input specialization in an extended quality-ladders framework. Based on Grossman and Helpman (1991b), we build a dynamic North-South general-equilibrium model in which every final good is made from two intermediate inputs, each subject to quality improvements. At any point in time, the high-technology input is further up its quality ladder than the low-technology one. Firms in both intermediates try to climb the ladder simultaneously and independently. Thus, in contrast to previous models, R&D activities take place in the intermediate sectors. To make the model tractable, we permit innovation to occur only in the North and imitation only in the South. The final-good assembler procures inputs from the global upstream market, searching for the lowest quality-adjusted prices. With these assumptions, the quantities of all goods depend on a combination of instantaneous input prices and qualities. 2

4 With that structure in place, we manipulate the ratio of quality increases, the result of innovation, between the two inputs and analyze the effects on steady-state equilibrium choices. What emerges is a specialized pattern of global input production in the steady state. Due to their technological disadvantage, Southern firms are only able to imitate and produce the low-technology input, though they also invest in copying the more advanced input. Firms in the North are capable of producing both intermediates, but choose to specialize in the high-technology version, in which they have a comparative advantage, while also innovating new higher-quality versions of the low-technology input. Hence, the product cycle in our model endogenously arises only in the low-technology intermediate good. This outcome offers a model-based justification for the common assumption that production locations of high-technology versus low-technology goods are given exogenously (Glass and Saggi, 2001; Antras, 2005). Reaching this conclusion requires calibration of the model, which permits computation of regional welfare impacts and yields the following additional implications. First, the relative wage between the North and South increases with the quality difference between the high-technology and low-technology inputs. A rise in this difference also increases technology diffusion and expands welfare in both regions. Second, the relative Northern wage rises in the degree of Southern patent protection and the Northern labor endowment, but decreases with the Southern labor endowment. These relationships between labor supplies and the relative wage are counterintuitive and emerge in the calibrated model from the interplay between production specialization and Bertrand pricing. Each of these changes affects the equilibrium rate of technology transfer and economic well-being as well. Interregional information flows are enhanced by an increase in either labor endowment due to an expenditure effect. Endowments are not neutral in terms of welfare, however. A greater Northern labor supply increases local welfare but reduces it in the South, while a larger Southern labor force expands well-being in both regions. Regarding policy, interest focuses on the role of IPRs. In this model, stronger Southern patent protection limits technology transfer by raising imitation costs and reducing imitation intensity, in turn lowering local welfare. However, it generates greater Northern profits and a higher innovation intensity, expanding welfare in that region. Finally, in this model the aggregate (firm- 3

5 measure-weighted) steady-state rate of innovation (which supplants the current generation of the low-technology input with a higher quality version) is equal to the analogous aggregate rate of imitation. Consequently, an increase in Southern patent rights ends up slowing down the equilibrium total rates of innovation and technology diffusion, a result reminiscent of that in Helpman (1993) for a single final good. The rest of the paper is organized as follows. After a brief literature review in Section 2, we discuss the basic setup of our modified quality-ladders model with two inputs in Section 3. The main results in the steady-state equilibrium are derived in Section 4. Then we perform extensive simulation analysis on the calibrated model in Section 5 to illustrate key results. We conclude in Section 6. 2 Literature Review The concept of an international product life cycle, involving innovation of a new product in an advanced region, followed over time by standardization of the technology and the purposeful transfer of production to developing countries, was initiated by Vernon (1966). Formal modeling of the process began with Krugman (1979), in which Northern innovation and Southern imitation proceeded at exogenous rates in labor-only economies. These rates determined impacts on welfare and wage differentials between regions. 1 Subsequent analyses attempt to explain the product cycle in general equilibrium, recognizing the explicit dynamics. An important framework that formalizes the process is the quality-ladders model, first set out in trade by Grossman and Helpman (1991b, henceforth GH). In these models, after each successful R&D investment the quality of a product increases, typically by a fixed increment, within a given number of product varieties. The GH paper endogenizes the rate of technology transfer as a function of Southern imitation activity, with successful imitation modeled as a Poisson arrival rate. Imitation is the only form of technology diffusion in this model and increases in its rate ultimately accelerate Northern innovation in order to profit from faster arrival of market 1 Dollar (1986) extended these concepts to a two-factor context, while Jensen and Thursby (1986, 1987) were early attempts to endogenize the speed of the product cycle, but without full general-equilibrium effects. 4

6 leadership profits 2. This insight was extended by Helpman (1993), who analyzed the welfare implications of stronger Southern protection of intellectual property protection (IPP), taken to mean a decline in the imitation rate. He found that reduced imitation would diminish Northern innovation in general equilibrium by sustaining leadership for longer periods, lowering incentives to invest in quality improvements. Yang and Maskus (2001) added licensing as a form of technology transfer to the quality-ladders model, noting that if strengthened IPP were to cut licensing costs then Helpman s (1993) conclusions would be overturned. The extended model in Glass and Saggi (2002) added foreign direct investment (FDI) as another channel, with innovation, imitation, and FDI made endogenous processes. Here, Southern firms can copy from either multinational corporations located in the South or innovators in the North, with different likelihoods of success. They showed that a more rigorous IPP regime in the South results in a decline in FDI and further diminishes the innovation incentive. An alternative to investigating product-cycle dynamics within vertical quality improvements is to consider innovation in horizontal product varieties, as in Grossman and Helpman (1991a). In this framework Lai (1998) introduced FDI into the general-equilibrium product-cycle literature. Rather than directly targeting innovative firms in the North, imitation can aim also at copying the technologies multinational enterprises transfer to their Southern affiliates. He found that stronger IPP reduces the rates of innovation and technology diffusion in the case of direct imitation but has opposite effects when copying targets FDI. The wage differential between the North and South is reduced under FDI, even when both channels co-exist. Glass and Wu (2007) combined the horizontal and vertical frameworks into a hybrid model, taking innovation to be endogenous and imitation to be exogenous. They found that raising the strength of IPP can shift innovation from quality improvement to variety expansion. The vertical-differentiation models above consider a quality ladder within a single product, or stable set of products, subject to a given quality increment. However, it is evident that these dynamics could vary across sectors. One form of heterogeneity is in Glass (1997), who investigated 2 Segerstrom, et al (1990) significantly contributed to this literature by incorporating Schumpeterian invention lotteries in order to endogenize innovation in the context of probabilistic R&D races, but sustained a deterministic time length between innovation successes. 5

7 the product cycle from the demand side. There are two quality levels for each product due to differences in heterogeneous consumers willingness to pay. The South manufactures the low-quality good at the beginning of the process but through production learns the know-how of the high-quality product and is ultimately able to penetrate the world market for the latter. Working from the insights of Taylor (1993), Lu (2007) established a model of supply-side heterogeneity in final goods, in which the quality increment of each innovation is positively related to the industry s relative technological ranking. Once an innovator loses its Bertrand market dominance, it chooses between further innovation and FDI, depending on its industry technology. Here, productcycle processes emerge only in sectors with medium technology levels. Innovation and production of high-technology goods remain in the North, while low-technology products migrate permanently to the South with no risk of obsolescence because it is not profitable to target innovation at them once they have diffused. Glass and Saggi (2001) first examined the role of outsourcing in the general-equilibrium framework, permitting a quality ladder in a single final good. In this context, outsourcing is the endogenous choice between home and foreign inputs of given quality. Production consists of two stages. A basic stage takes place in the South and the input it produces is combined with that of an advanced stage in the North to assemble the final good. A lower adaption cost and a larger share of production content at the basic stage give rise to a greater extent of international outsourcing in that input. Increased outsourcing encourages a faster rate of innovation because it reduces the share of Northern labor devoted to manufacturing, making more resources available for R&D. Moreover, greater input sourcing from the South reduces the wage differential between regions in equilibrium 3. Missing from the literature so far is the possibility that there are multiple intermediate inputs that may be subject to product-cycle dynamics. Thus, in this paper we extend the quality-ladders idea by combining two heterogeneous inputs, each subject to cycles of innovation and imitation in a vertical framework, in the production of a final good. Specifically, we incorporate vertical quality increments in two intermediate inputs, which are themselves characterized by differences in technological sophistication, to examine the resulting product-cycle dynamics. Our emphasis on 3 See Glass (2004) and Sayek and Sener (2006) for extensions of this work. 6

8 intermediates stems from the obvious and growing importance of such goods in international trade and their complex combinations into final goods (Baldwin and Lopez-Gonzalez, 2013; Antras, et al, 2012). Moreover, rapid innovation characterizes some intermediate inputs more than others. The model permits production of these inputs in different locations, offering additional perspective on the nature of fragmented production stages across borders. In this context, we borrow the production structure set out in Antras (2005), in which one final good is made with two intermediate inputs. Production of the high-technology input is restricted to the North, while production of the low-technology input may be contracted to either region. The tradeoff between an expensive incomplete contract and cheap manufacturing cost in the South creates the product cycle. Over time, outsourcing to the South raises final-producer profits in the share of low-technology inputs. Antras permitted the share of this low-technology input to increase exogenously over time to rationalize Vernon s (1966) standardization process. Ultimately, all production of this basic input is transferred to the South. This framework delivers many insights about product standardization and international firm structure. However, it sets aside the informative product-cycle dynamics available in the qualityladders approach. To explore this idea we embed the basic production elements of the Antras model within a dynamic general-equilibrium model featuring two intermediate inputs. Specifically, we extend the production function to a situation where both inputs independently climb a quality ladder and examine the resulting product-cycle dynamics. The cost of this complexity is that we cannot embed and analyze the contracting problem underlying Antras product cycle. Rather, we characterize the steady-state equilibrium emerging from simultaneous dynamic competition in two inputs. Unlike Antras, we find that the low-technology input remains subject to recurring product cycles. 3 The Model We build a model in which there are two regions, a developed North and a less-developed South. These countries differ in technology levels and population sizes. Labor is the only factor and is 7

9 assumed to be immobile across borders. Trade in goods is assumed to be costless. 3.1 Consumers Let there be a continuum of final goods in the world, indexed by y(i) with i [0, 1]. Consumers in both regions share identical and homothetic constant elasticity of substitution (CES) preferences. The intertemporal utility function for an infinitely-lived representative consumer is denoted as U = 0 e ρt logu(t)dt (1) where ρ is the common subjective discount factor. Instantaneous utility at time t is ( 1 1/α u(t) = y(i, t) di) α, 0 < α < 1 (2) 0 Here, ɛ = 1/(1 α) > 1 is the elasticity of substitution among final goods and y(i, t) is the quantity of final good i consumed at time t 4. Consumers maximize lifetime utility subject to the intertemporal budget constraint given by 0 e R(t) E(t)dt = 0 e R(t) Y (t)dt (3) where R(t) = t 0 r(s)ds is the cumulative interest rate from time 0 to time t and Y (t) is aggregate factor income. Total world expenditure on all final goods is given by E(t) = 1 0 p(i, t)y(i, t)di (4) where p(i, t) is the price of final good i. Consumers maximize utility in two stages. First, they allocate lifetime spending across time periods and, second, they decide how to spread instantaneous expenditure across final goods at each point in time. With CES utility the expenditure at every instant is the same and there is a common expenditure on each final good. 4 The quality parameters here do not show up directly in this utility function because we model quality-improving R&D in the intermediate inputs rather than the final good. The quality increment parameters will appear in the final-good production function instead. 8

10 With this maximization the final good faces the following iso-elastic demand function at time t: y(i, t) = φ(t)p(i, t) 1/(1 α) (5) where φ(t) = E(t) 1, defined across all final goods m, is taken as given by the assembler. 0 p(m,t) α/(1 α) dm Put differently, the share of final good i in total world expenditure is negatively related to its own price and positively related to the global price index. The number of final goods is fixed and all varieties of such goods are identical in the quality-ladder framework. Thus, for simplicity we suppress variety index i from this point forward. 3.2 Firms Following Antras (2005), let there be one final-good assembler in each product and many manufacturing plants making intermediate inputs. The assembly of final good y requires two inputs, a low-technology input x l and a high-techology input x h, which are specific to the particular final product. They are distinct in their quality increments and R&D requirements. Specifically, the high-technology input requires more resources to improve upon through innovation or to copy through imitation than does the other input. We assume that research activities take place within manufacturing plants of the intermediate goods instead of the final good. We also posit that plants located in the North have the capacity to innovate and push forward the quality frontier of either type of input, while the Southern plants are only able to imitate existing quality levels from the North. Production of one unit of either input demands one unit of labor. Let the South have the larger labor endowment and normalize its wage to 1, with the relative wage between North and South denoted as ω > 1. Thus, South has a comparative advantage in manufacturing, which offers a tradeoff with its inefficiency in innovation. 9

11 3.2.1 Manufacturing plants With this setup there are four categories of plants: Northern low-technology and high-technology plants and their Southern counterparts. The firm measures are n N l, n N h, ns l and n S h, respectively. Plants can either engage in manufacturing or conduct R&D activities aimed at improving their own input type but, for tractability, we do not permit plants to target inputs of the other type 5. As mentioned above, innovation only takes place in the North. There are profit incentives for Northern plants to invest in better quality. We assume that both low-technology and hightechnology inputs stochastically and independently climb their own quality ladder and that quality levels may be promoted indefinitely. The probability of successful innovation in the next instant of time, the hazard rate, depends on firms current R&D investment. Label these hazard rates ι l and ι h for the two input types. A Northern low-technology (high-technology) input supplier undertaking innovation intensity ι l (ι h ) during time interval dt requires a l ι l dt (a h ι h dt) units of labor, at a cost of ωa l ι l dt (ωa h ι h dt). Parameters a l and a h are the fixed labor requirements for every unit of innovation intensity of the two inputs, with a l < a h. These processes generate stochastic innovation outcomes given the hazard rates. Each successful innovation generates one step up the quality ladder for the associated input. The quality increment for each step is a parameter λ l > 1 for the low-technology input and λ h > λ l > 1 for the high-technology input. Note that differences in technology levels are thereby characterized by both costs of innovation and the resulting quality improvement. Without loss of generality, assume that the initial qualities for both inputs are 1. Then the quality levels are q l = λ η l after η improvements in the low-technology input and q h = λ θ h after θ improvements in the other input. The difference in quality increments has a substantial impact on our steady-state predictions. Though Southern input firms are similar to their Northern counterparts in manufacturing technologies, we assume they cannot invent the next generation of inputs. Thus, they may reverse engineer the Northern variants and learn first-level technologies by imitation, which we assume to be the sole source of international technology diffusion. The hazard rates for imitation are given 5 As we will see, in equilibrium South (North) does not produce the high-technology (low-technology) inputs. This means that n N l in fact refers to Northern firms investing R&D in innovation and n S h to Southern firms investing R&D in imitation. 10

12 by µ l and µ h for the low-technology and high-technology inputs, respectively. Thus, the cost of imitation during any time period is (1 + κ)c l µ l dt and (1 + κ)c h µ h dt for the respective goods, where c l and c h capture the fixed per-unit labor costs of imitation. We use parameter κ > 0 to measure the degree the South s patent enforcement, which is imperfect. An increase in IPP protection raises κ, thereby making imitation more costly for any hazard rate. Any firm that successfully discovers the best technology, denoted as the leader for that input type, earns positive temporary profits. We label all other plants in both regions as followers. It is convenient to follow the framework of an inefficient follower (Grossman and Helpman, 1991b; Maskus and Yang, 2001; Glass, 2002). We suppose that Northern innovation aims at improving technologies currently produced in the South, while Southern imitation only targets the Northern innovators. That is, Northerners choose not to improve further upon an input until the most recent generation has been copied. Similarly, once a Southern firm wins the imitation competition, no other Southern plant undertakes R&D activities until the North reclaims leadership in the input. To summarize, North and South alternately control production, reminiscent of the product cycle. We now describe the behavior of manufacturers under these circumstances. Manufacturers maximize instantaneous profits by choosing output price. With the quality-ladders structure, the firm with the lowest quality-adjusted price dominates the market. These prices are p l /q l and p h /q h, which make the final-good assembler indifferent between the leading and lagging quality of each input. This feature results in Bertrand competition in each input, as each plant tries to set a price that just prevents its rivals from making a profit and drives them from the market. In equilibrium this price is equivalent to the second highest marginal cost in quality-adjusted terms (Glass and Saggi, 2002, p. 394). This situation supports two possible scenarios, which we call S and N, for each input. Under scenario S, if the Southern plant successfully copies the state-of-the-art technology of either input via imitative investment, it sets a price equal to the Northern wage rate ω (or an infinitesimal δ below it) and takes over the entire market. Its marginal cost is unity, the Southern wage. Other Southern input suppliers, with lower quality levels, and Northern competitors with equal production technology, earn zero profits. The Southern imitator s instantaneous profits, 11

13 accounting for variable costs, if it imitates the low-technology input are π S l = (ω 1)x S l (6) Its profits if it imitates the high-technology input are π S h = (ω 1)xS h (7) Note that p S l = p S h = ω and xs l and x S h are quantities of inputs sold, which are determined by the final-good assembler and discussed in the following section. Under scenario N, a Northern firm successfully upgrades the quality level of an input that was being produced in the South. This firm charges a price equal to the Southern competitor s marginal cost after quality adjustment (or δ below it). Thus, the instantaneous profits of this case for the low-technology and high-technology inputs are, respectively: π N l = (λ l ω)x N l (8) π N h = (λ h ω)x N h (9) Similarly, ω is the marginal production cost and the quantity variables are determined by the finalgood assembler. Here, the input prices are p N l = λ l and p N h = λ h in light of their respective quality increments. We emphasize that this situation can pertain only when the wage gap is not too large. That is, we only consider the case where λ h > λ l > ω, or the narrow-wage-gap situation described in Grossman and Helpman (1991a) 6. These two scenarios cannot exist at the same time, as either input is provided by exactly one plant in equilibrium, either in North or South. The firm measures of each type must sum to 1. 6 Grossman and Helpman (1991a) consider both the wide-wage-gap case (ω > λ) and the narrow-gap case (ω < λ). But in Grossman and Helpman (1991b) they analyze just the latter case, which is common in the quality-ladders literature. 12

14 Thus, for the input types we must have n S l + n N l = 1 (10) n S h + nn h = 1 (11) We next incorporate equilibrium zero-profit conditions by noting that the expected value of a manufacturing firm cannot exceed its R&D costs, for that would drive infinite investment. Neither can its expected value be less than R&D cost in order to sustain the possibility of profitable investment under successful innovation or imitation. These conditions generate the following zeroprofit conditions in scenario S: vl S = (1 + κ)c l if µ l > 0 (12) v S h = (1 + κ)c h if µ h > 0 (13) Equation (12) holds for Southern low-technology plants if imitation takes place and equation (13) for Southern high-technology plants analogously. The corresponding conditions for Northern firms in scenario N if innovation happens are: v N l = ωa l if ι l > 0 (14) v N h = ωa h if ι h > 0 (15) Final-good assembler Next we introduce the final-good assembly firm and characterize its equilibrium choices. Adding this kind of agent is new to the quality-ladders literature. This assembler purchases the two inputs from the global market, searching in each case for the lowest quality-adjusted price. It produces the final good from these inputs with no additional costs. We posit a Cobb-Douglas production function at this stage: y jk = ζ z (q l x j l )z (q h x k h )1 z, 0 z 1 (16) 13

15 where ζ z = z z (1 z) (1 z) is a parameter that normalizes final-good quantity. Parameter z in equation (16) is the exogenous share of the low-technology input used in production. Note that by inserting this equation into the utility function (equation (2)) the input qualities enter indirectly into consumer welfare. Because each input can follow two pricing scenarios, S and N, the Northern final-good assembler could encounter four combinations of input prices. Define the price of the low-technology input as p j l (j {S, N}) and of the high-technology input as p k h (k {S, N}). Then xj l and x k h are the corresponding input quantities. Recall from earlier that in free trade we have p S l = p S h = ω, pn l = λ l and p N h = λ h. Production function (16) is borrowed from Antras (2005), who then imagined a product cycle emerging from an exogenous rise in the low-technology input share in combination with contracting problems. In his model all decisions are made by the Northern final producer and Southern firms are passive. At a point in time the Northern final producer outsources the low-technology input to the South and it is the consequent potential hold-up problem that generates a product cycle. In contrast, we embed this function into a quality-ladders framework in which both Northern innovators and Southern imitators play an active role by endogenously choosing their R&D investments. These investments determine the lowest quality-adjusted prices, which the final producer chooses to combine in this production function. The combination of these choices establishes production locations. The final-good firm chooses inputs to maximize expected instantaneous profits, which are its sales revenues minus input costs: max x l, x h π jk = φ 1 α ζ α z (q l ) αz (x j l )αz (q h ) α(1 z) (x k h )α(1 z) p j l xj l pk h xk h (17) From this problem we can write the optimal final-good price in terms of input prices and qualities 7 7 Derivations are in Appendix A. p jk = 1 α (pj l q l ) z ( pk h q h ) 1 z (18) 14

16 In turn, the final-good quantity is y jk = φ[ 1 α (pj l q l ) z ( pk h q h ) 1 z ] 1 1 α (19) Substituting these expressions into the profit function (17) we derive the expected profits of the final-good assembler in equilibrium, which are π jk = φ(1 α)[α( q l p j l ) z ( q h p k ) 1 z ] α 1 α h = φ(1 α)[α( λ η l p j l ) z ( λθ h p k ) 1 z ] α 1 α (20) h The final-good price increases with input prices and decreases with input quality levels, but the final-good quantity falls with input prices and rises with qualities. When these effects are combined in the profit function the quantity effect dominates. Thus, the final-assembler?s profits rise with higher input quality levels but are diminished by higher input prices. We can now write the input quantities chosen for the two quality levels as x j l = φzα 1 1 α (p j l ) 1 ( q l p j ) z α l 1 α ( q h p k h ) (1 z) α 1 α (21) x k h 1 = φ(1 z)α 1 α (p k h ) 1 ( q l p j ) z α l 1 α ( q h p k h ) (1 z) α 1 α (22) Recall again that in scenario S the input price is the endogenous relative wage ω. Detailed comparative-static analysis will be performed in Section Market-clearing conditions The labor endowments are L N and L S, with L S > L N, and we assume these are constant over time. Workers cannot migrate between regions. Manufacturing plants allocate laborers either to production or R&D activities. Constant returns to scale ensure that the number of workers engaged in production equals the quantity of inputs sourced, while research costs were detailed above. We also assume that final-good assembly does not require labor. Thus, the resource constraint in the 15

17 South is n N l µ l (1 + κ)c l + n N h µ h(1 + κ)c h + n S l xs l + n S h xs h = LS (23) The first two terms capture labor employed in imitation (aimed at the Northern top-tier producers of both inputs) and the second pair of terms embody labor used in manufacturing inputs in the South. To clarify, the first term indicates that Southern imitation targets each of n N l Northern producers of the low-technology input at intensity µ l, at unit imitation cost (1 + κ)c l. Similarly, the labor constraint in the North is n S l ι la l + n S h ι ha h + n N l x N l + n N h xn h = LN (24) The first two terms are R&D aimed at innovating beyond the best inputs that have diffused to the South and the second two terms are manufacturing production. In equilibrium, aggregate expenditures E should equal total labor incomes plus instantaneous profits in final assembly, or Y = L S + ωl N + π jk. 8 The input plants just break even because their operating profits must cover R&D costs. We therefore have E = L S + ωl N + φ(1 α)[α( λη l p j l ) z ( λθ h p k ) 1 z ] α 1 α (25) h 4 Steady State Equilibrium We first explore the general conditions under which all types of R&D investments take place and then proceed to analyze the four particular cases arising from different combinations of input prices. In models like ours, all nominal values must move at the same constant rate in steady state, which rate is the difference between the interest rate and the consumers subjective discount factor: v j l /vj l = v h k /vk h = Ė/E = r ρ (26) Equations (27) to (30) below demonstrate the no-arbitrage conditions for each category of plants 8 This is a consequence of the CES utility function and equation (3). 16

18 in the steady state. In each equation, the first term is a manufacturing plant s dividend ratio, calculated as its instantaneous profits over its expected firm value. The next term is its instantaneous return to capital on equity holdings. In all, the left-hand side represents the total expected return to a plant. This return equals the instantaneous interest rate plus the risk of loss from suffering successful imitation or innovation. π S l /vs l + v S l /vs l = r + ι l (27) π S h /vs h + vs h /vs h = r + ι h (28) π N l /v N l + v N l /v N l = r + µ l (29) π N h /vn h + vn h /vn h = r + µ h (30) Combining equations (26) and (27) determines the steady-state relationship between the Southern low-technology plant s profits and and its expected value in scenario S: v S l = πs l ρ + ι l (31) A similar combination generates the condition for the high-technology plant: v S h = πs h ρ + ι h (32) Next, inserting equations (6) and (12) into (31) we generate a free-entry condition for the Southern low-technology plant (ω 1)x S l = (ρ + ι l )(1 + κ)c l (33) Similarly, combining (7), (13) and (32) we get the condition for the Southern high-technology plant (ω 1)x S h = (ρ + ι h)(1 + κ)c h (34) In these two conditions the left-hand sides involve two endogenous variables, ω and x S l (or x S h ) and 17

19 the right-hand side has just one, ι l (or ι h ). Other things equal, an increase in either innovation intensity would raise the demand for Northern labor, raising the relative wage. In turn, this would expand Southern input manufacturing because the relative wage is the price at which inputs are sold. Similar analysis under scenario N generates these valuation and free-entry conditions: v N l = πn l ρ + µ l (35) v N h = πn h ρ + µ h (36) (λ l ω)x N l = (ρ + µ l )ωa l (37) (λ h ω)x N h = (ρ + µ h)ωa h (38) Here, when Southern imitation intensity in either input goes up there is more labor devoted to imitative R&D, raising the Southern relative wage and diminishing ω. In turn, Northern firms engage in more manufacturing due to this decline in wage costs, raising instantaneous profits. From equation pairs (33) and (34) and (37) and (38) we can determine reduced-form expressions for imitation and innovation intensities. Further, as may be seen in equations (21) and (22), the quantities of inputs are functions of input prices, quality levels and other parameters. Recall that input prices equal the relative wage in scenario S or the quality increments in scenario N. The quality increments are predetermined parameters. Thus, the following set of equations, (39) to (42), list the R&D intensities in each category in terms of parameters (λ l, λ h, ρ, a l, a h, c l, c h, κ, z, α, L N, L S ) and one endogenous variable, ω. 9 µ l = (λ l ω)x N l a l ω µ h = (λ h ω)x N h a h ω ρ (39) ρ (40) ι l = (ω 1)xS l (1 + κ)c l ρ (41) 9 The input quantities contain a specific term φ, which is instantaneous expenditure E divided by the price index for all final goods. Thus, the relative wage is the single endogenous variable. 18

20 ι h = (ω 1)xS h (1 + κ)c h ρ (42) Because it is impossible to observe negative R&D expenditures in equilibrium, each of these four endogenous reduced-form investment intensities should be non-negative. As we demonstrate in Appendix B, there is a negative effect of the relative wage on imitation intensities, µ l / ω < 0 and µ h / ω < 0. As noted above, a higher relative wage pushes more labor into Southern production and out of imitation. However, we are unable to sign the innovation partial derivatives ι l / ω and ι h / ω. A rise in the relative wage ordinarily would be expected to increase the Northern innovation incentive by making production activities less attractive. However, it also increases the labor cost of innovation, making the overall effect ambiguous. Finally, we define the aggregate rate of innovation as ι = ι l n S l + ι h n S h. This is measured as the sum of the innovation intensities for each input times the relevant Southern firm measure. Similarly, we can define the aggregate rate of imitation as µ = µ l n N l + µ h n N h. In steady state all flows into a market must equal all flows out of that market, requiring equality in these two aggregate rates, or ι = µ. Thus, the pace at which production diffuses to the South through imitation is the same as the speed at which production returns to the North due to innovation per unit time period. Consequently, the flow measure of inputs in the integrated market must satisfy µ l n N l + µ h n N h = ι ln S l + ι h n S h (43) An important observation is that both µ and ι characterize the frequency of the product cycle. As in GH, the average length of the product cycle is captured by the inverse of either of these rates. These inverses describe the time period during which a typical input is manufactured in the North before being copied in the South. To summarize, we can write a steady-state system of equations in (44). These consist of the market-structure conditions (10) and (11), the international knowledge-flow equation (43), and the 19

21 labor-market clearing conditions (23) and (24). n S l + n N l = 1 n S h + nn h = 1 µ l n N l + µ h n N h = ι ln S l + ι h n S h n N l µ l (1 + κ)c l + n N h µ h(1 + κ)c h + n S l xs l + n S h xs h = LS n S l ι la l + n S h ι ha h + n N l x N l + n N h xn h = LN (44) In addition, the R&D intensity rates can be solved from (39) to (42) and the input quantities are given by (21) and (22). These conditions characterize the general-equilibrium of the model. Again, the final-good assembler in principle could face four configurations of input prices in the upstream market, since each input could reside in either scenario S or N. 10 The steady-state outcomes could vary depending on the situation at hand. We next study these cases in turn. 4.1 Case 1: all production in South We assume from now on that the subjective discount factor ρ is sufficiently close to zero to prevent an economically meaningless negative R&D intensity (e.g., see equation (39)). 11 Now consider the case in which both inputs are in scenario S and produced in the South (j = k = S). In this situation both input prices equal the relative wage and no input manufacturing takes place in the North. Because there is no Northern production the imitation rates µ l and µ h are zero. In economic terms, all Northern plants are engaged in trying to innovate beyond the Southern inputs and the Southern producing plants, which supply the entire market, have no incentive to engage in imitation. The final-good assembler would expect to receive an instantaneous profit of π SS = φ(1 α)[α( q l ω )z ( q h ω )1 z ] α 1 α (45) 10 It is not necessary to consider the assembler in steady state since there is no entry in the downstream market and all other firms in the economy are in equilibrium. 11 To be precise we assume it is such a small number that it approaches zero in the limit and may be ignored in steady state. While we recognize that a zero discount rate leads to unbounded inter-temporal utility it is not sensible for the imitation rates to be negative. In robustness checks on the simulations below we find that comparative statics are unchanged with ρ > 0. 20

22 In this case, however, equation (43) becomes 0 = ι l n S l + ι h n S h, which would imply zero innovation rates and is inconsistent with this market structure. Case 1 would be unstable in the steady state and so we rule out this possibility. 4.2 Case 2: all production in the North In this case all production lies in the North and Southern plants focus on imitative R&D. The prices are p N l = λ l and p N h = λ h for the two inputs, leading to instantaneous profits of the final assembler of π NN = φ(1 α)[α( q l λ l ) z ( q h λ h ) 1 z ] α 1 α (46) The current producing firms have no incentive to undertake research and so innovation rates are zero. By an analogous argument to the prior one there is a contradiction in equation (43) in this case as well. Thus, we rule out Case 2 also. 4.3 Case 3: First mixed regime Let the low-technology input be produced in the South and the high-technology input in the North (j = S, k = N). In terms of economic intuition, this case corresponds to a situation in which it is cheap to reverse engineer the low-technology input so the South continues to produce it after successful imitation. However, since the high-technology version is difficult to copy the North can keep its technology hidden and production remains in the advanced region. Innovation targets the low-technology Southern plants and imitation targets the high-technology Northern plants. The final-good assembler faces these prices: p S l = ω and p N h = λ h. The assembler s instantaneous profits are then π SN = φ(1 α)[α( q l ω )z ( q h λ h ) 1 z ] α 1 α (47) The currently producing (leading) plants in each region do not invest in R&D, meaning that µ l = 0 and ι h = However, the non-producing (trailing) plants do invest, hoping to acquire the market, 12 In the limiting case of ρ = 0 we can derive these zero rates by substituting x S h = 0 and x N l = 0 into (39) and 21

23 in which case µ h > 0 and ι l > 0. The currently producing plants in both countries no longer take part in research, which leads to µ l = 0 and ι h = The non-producing plants participate in R&D, hoping to acquire the market. That is, µ h > 0 and ι l > 0. The assembler acquires zero inputs of the following types since they are not produced x S h = xn l = 0 (48) It purchases the following quantity from the Southern low-technology plant x S l = φzα 1 1 α (ω) 1 ( q l ω )z α q 1 α h ( ) (1 z) α 1 α (49) λ h and this quantity of the high-technology input from the Northern producer x N h = φ(1 z)α 1 1 α (λh ) 1 ( q l ω )z α q 1 α h ( ) (1 z) α 1 α (50) λ h With these input prices the price of the final good is p SN = 1 α ( ω q l ) z ( λ h q h ) 1 z (51) with corresponding final sales quantity to consumers of y SN = φ[ 1 α ( ω q l ) z ( λ h q h ) 1 z ] 1 1 α (52) Recall that with Bertrand pricing the relative wage is the price of the low-technology input. Thus, an increase in ω raises the cost of this input for the final-good producer, inducing it to source a lower volume and cut sales. Despite the fact that the final-good price increases, the negative quantity impact dominates and the assembler s profits decline. 14 (42), respectively. 13 In the limiting case of ρ = 0, µ l = 0 and ι h = 0 are derived by substituting (48) into (39) and (42) respectively. 14 See Appendix B.1. 22

24 Now the steady-state system in (44) is simplified to the following set of linear equations n S l + n N l = 1 n S h + nn h = 1 µ h n N h = ι ln S l (53) n N h µ h(1 + κ)c h + n S l xs l n S l ι la l + n N h xn h = LN = L S From system (53) we can solve for two equations illustrating the measure of the Southern lowtechnology plants. n S l = L S c l (ω 1)x S l c h + x S l c l (54) n S l = L N (1 + κ)c l (ω 1)x S l (a l + a hω λ h ω ) ( 54 ) We also obtain the measure of Northern high-technology plants using (54) n N h = z(ω 1)λ h a h L S (1 z)x S l [(ω 1)c h + c l ](1 + κ)(λ h ω) (55) the measure of Southern firms attempting to imitate the Northern high-technology input n S h = 1 L S c l (ω 1)x S l c h + x S l c l (56) and the measure of Northern firms trying to innovate a new version of the low-technology input n N l = 1 z(ω 1)λ h a h L S (1 z)x S l [(ω 1)c h + c l ](1 + κ)(λ h ω) (57) Finally, by equating (54) and (54 ) we should, in principle, be able to solve for the equilibrium 23

25 relative wage from the reduced-form quadratic equation (58) below. [a l L S L N (1 + κ)c h a h L S ]ω 2 + [a h L S (1 + κ)c l L N (1 + λ h )a l L S + (1 + λ h )(1 + κ)c h L N ]ω + [a l L S (1 + κ)c h L N + (1 + κ)c l L N ]λ h = 0 (58) The solution for ω would then involve both labor endowments, the vector of unit innovation and imitation costs, Southern IPR protection, and the quality increment of the high-technology input. 15 Input quantities do not appear because they cancel in the corresponding calculations. This reflects the idea that changes in input quantities do not affect the reduced-form relative wage. Calibration in the next section will show that only one root for this quadratic equation is economically meaningful and we denote this root by ω. It does not depend on the low-technology quality increment (λ l ), the quality jump levels (η and θ), or the share of the low-technology input (z). The presence of the high-technology increment in the solution stems from the pricing equation p N h = λ h. The analytical solutions to the endogenous variables of interest are complex and we leave details to the calibration exercise in the next section. Upon determining the relative wage we can compute aggregate expenditure as E = L S + ω L N + φ(1 α)[α( λη l ω )z ( λθ h λ h ) 1 z ] α 1 α (59) Finally, by incorporating (43) and equation (54) or (54 ) we achieve an expression for the aggregate technological flows, which can be represented in two ways ι = L N a l + a hω λ h ω ; µ = (ω 1)L S (1 + κ)[(ω 1)c h + c l ] ; ι = µ (60) We refer to the left-hand equation as the NN curve and the right-hand equation is the SS curve, which capture the full-employment constraints. Setting these equal to each other determines the aggregate innovation and imitation rates, required to be the same in steady state. These equations lay bare the notion that both stronger patent protection in the South and higher imitation costs 15 See Appendix B.3 for the general analytical solution. 24

26 reduce technology transfer. 4.4 Case 4: Second mixed regime Next we consider the situation where the Southern plants produce the high-technology input and Northern plants manufacture the low-technology version. The input prices in this situation are p N l = λ l and p S h = ω, respectively. With this configuration the price in Bertrand competition of the low-technology input is higher than that of the high-technology input. As in case 3 we can specify the final-good assembler s profit function π NS = φ(1 α)[α( q l λ l ) z ( q h ω )1 z ] α 1 α (61) with the price of the final good equal to p NS = 1 α (λ l q l ) z ( ω q h ) 1 z (62) The total quantity of the final good sold is y NS = φ[ 1 α (λ l q l ) z ( ω q h ) 1 z ] 1 1 α (63) The quantity of the low-technology input purchased from the producing Northern plant is x N l = φzα 1 1 α (λl ) 1 ( q l λ l ) z α q 1 α h ( α ω )(1 z) 1 α (64) and the quantity of the high-technology input procured from the Southern plant is x S h = φ(1 z)α 1 1 α (ω) 1 ( q l λ l ) z α q 1 α h ( α ω )(1 z) 1 α (65) The remaining plants in the North invest in innovative R&D to target the high-technology input, while those in the South invest in imitative R&D to reverse engineer the low-technology version. Thus, ι h = (ω 1)xS h (1+κ)c h ρ > 0 and µ l = (λ l ω)x N l a l ω ρ > 0. And, because x S l = x N h = 0 it must be that 25

27 ι l = µ h = 0. This configuration simplifies system (44) to n S l + n N l = 1 n S h + nn h = 1 µ l n N l = ι h n S h n N l µ l (1 + κ)c l + n S h xs h = LS n S h ι ha h + n N l x N l = L N (66) Then the firm-measure relationships are given by n N l = L S µ l (1 + κ)c l + µ l ι h x S h = a l ω(ω 1)L S (λ l ω)(1 + κ)[(ω 1)c l + c h ]x N l (67) n N l = L N µ l a h + x N l = a l ωl N (λ l ω)a h x N l + a l ωx N l (68) n S l = 1 L N µ l a h + x N l = a l ωl N (λ l ω)a h x N l + a l ωx N l (69) n S h = (1 + κ)(λ l ω)c h L N [a h (λ l ω) + a l ω](ω 1)x S h (70) n N h = 1 (1 + κ)(λ l ω)c h L N [a h (λ l ω) + a l ω](ω 1)x S h (71) Equations (68) and (70) are firm measures in production, while equations (69) and (71) capture imitation and innovation efforts, respectively. From equations (67) and (68) we can, in principle, determine the relative wage ω from the following quadratic equation (a h L S c l L N a l L S )ω 2 + [(1 + λ l )c l L N (1 + κ)c h L N (1 + λ l )a h L S + a l L S ]ω +[a h L S c l L N + (1 + κ)c h L N ]λ l = 0 (72) Aggregate world expenditure is E = L S + ω L N + φ(1 α)[α( λη l λ l ) z ( λθ h ω )1 z ] α 1 α (73) 26

28 Finally, the total rate of international technology flows is ι = µ = (ω 1)L S (1 + κ)c l (ω 1) + (1 + κ)c h (74) 4.5 Economic welfare In this subsection we state expressions for economic welfare in North and South, leaving the full mathematical details to Appendix C. As shown there we can analyze welfare using the instantaneous utility functions and can therefore state these expressions for South and North, respectively, as follows: log u S = log L S + log α z log ω (t)di + z log λ l η(i)di + (1 z) log λ h (θ(i) 1)di (75) log u N = loge N log p(i)di (76) 0 5 Model Calibration and Comparative Statics The complexity of the model precludes developing comparative-static exercises in which analytical solutions may be signed. Thus, we now carry out numerical simulations based on Case 3, in which there is a continuous product cycle in the low-technology input but Northern specialization in the high-technology version Calibration The benchmark parameter values we select for the analysis are shown in Table 1. To avoid the possibility of negative R&D intensities we must set the discount rate ρ = 0. Consistent with the model, we select a larger Southern labor force than Northern (L S = 10, L N = 3). The innovation cost of the high-technology input (0.8) is higher than that of the low-technology input (0.5). In line with received thinking from the literature, both of these are assumed to be higher than imitation 16 We also perform simulation analysis for Case 4. However, because they correspond to the unlikely case in which the South (North) produces the high-tech (low-tech) input we do not report them here, and they do not generate additional insights in any case. Results are available on request. 27

29 cost parameters, where imitating the high-tech version is far costlier than the low-tech input (0.5 versus 0.05). 17 Next, the quality increment of the high-tech good is assumed to be twice that of the low-tech good. In the benchmark we assume that both versions have been improved twice. The Cobb-Douglas share of the low-technology input in final goods production is held fixed at Next, we assume an elasticity of substitution of 6.6, which is consistent with the central estimate in Broda-Weinstein (2006) for varieties defined at the 5-digit SITC level. Finally, we assume that the initial parameter capturing Southern patent protection is κ = 0.6, meaning that the patent rules raise imitation costs by 60 percent over their basic levels (Mansfield, et al, 1981; He and Maskus, 2012). We note that the results of the steady-state comparative statics we perform below are robust even if we adjust the key parameters within reasonable ranges. < Insert Table 1 here.> < Insert Table 2 here.> With these parameter values, the simulated benchmark results are listed in Table 2. Recall that in any steady state the aggregate rate of innovation targeting the South must equal the corresponding rate of imitation targeting the North (ι = µ ). Thus, we first calibrate the pair of equations in (60) to solve for the benchmark relative wage between the North and South. To illustrate, we depict the NN and SS curves in Figure 4. Regarding the former, a higher relative wage in the North makes innovation costlier, producing a downward-sloping N N relationship between the rate of technology transfer and ω. As for the latter, a higher relative wage, which is equivalent to a higher price for the inputs they produce, gives Southern imitators more incentive to conduct imitative R&D. Thus, the SS curve is an upward-sloping relationship between the relative wage and the aggregate technology diffusion rate. As shown in Figure 4, these monotonic curves intersect only once at a positive ω, implying a unique solution for the relative wage. < Insert Figure 4 here.> 17 Mansfield, al (1981) review estimates of imitation costs, which are a substantial proportion of original innovation costs in patented (i.e., high-technology) goods but lower in other goods. 28

30 In the simulated benchmark ω = 1.035, as noted in Figure 4. At that wage, total expenditure (15.446) is made up of the Southern wage bill, the Northern wage bill, and the final-good assembler s profits. Input producers make positive operating profits but these are just offset in equilibrium by R&D costs and do not enter overall income. Among firms engaged with the low-technology input, 89.9 percent are Southern producers and 10.1 percent are Northern innovators. Regarding the hightechnology input, 89.7 percent are Northern producers and 10.3 percent are Southern imitators. Finally, the benchmark aggregate rate of technology transfer is In Figure 4 we also depict welfare curves for both regions. Ceteris paribus, Northern (Southern) welfare rises (falls) with increases in the relative wage. For the explanations to come, it is instructive to consider the relationship between the relative wage and input manufacturers operating profits. First, an increase in the relative wage directly raises the relative cost of Northern production, lowering instantaneous profits of those firms ( πn h ω < 0). Second, because the relative wage equals the price of the low-tech input, an increase in ω directly raises the low-tech input manufacturers profits ( πs l ω > 0). Indirectly, however, a rise in ω induces the final good assembler to substitute away from low-tech inputs toward more high-tech inputs ( xs l ω < 0, xn h ω > 0), tending to offset the direct impacts. These adjustments also imply that a higher relative wage in the North makes the final good assembler more profitable ( πsn ω > 0). 5.2 Comparative statics Starting now from the calibrated value of the relative wage, we investigate the determinants of key endogenous variables in the model. The remaining columns in Table 1 detail in bold the parameter changes analyzed, with percentage impacts on endogenous variables listed in the corresponding columns in Table 2. We perform comparative-statics experiments with respect to changes in the quality-ladder parameters, the level of patent protection in the South, and the relative sizes of the two regions Impacts of changes in unit R&D costs for innovation and imitation are straightforward. 29

31 5.3 Quality parameters The novel element of this model is the existence of two inputs, differentiated by quality, each of which independently climbs a quality ladder. In principle, therefore, all market outcomes are affected by these quality differences. Note that equation (58) expresses the relative wage as a quadratic equation in several exogenous parameters. However, this equation does not include the quality increment of the low-technology good (λ l ), nor the two quality-jump parameters η and θ. The first parameter fails to appear because it does not affect either of the Bertrand input prices, which are given by p S l = ω and p N h = λ h in Case 3, nor does it enter the input-demand functions in equations (49) and (50). The intuition is that the final-good producer selects inputs with the lowest quality-adjusted price, implying that it is concerned with the relative quality of inputs, rather than actual quality levels. Similarly, the magnitudes of the quality jumps do not affect input pricing, and therefore the relative wage, in steady state. However, all three parameters appear in Equation (59), the expenditure expression, through their impacts on the final assembler s instantaneous profits. In this context, technological progress through innovations in quality accrue to profits of the final-good producer, which we assume are passed on to consumers. Thus, our initial experiment is to increase λ h from three to four. This change has two offsetting impacts on the final-goods producer. First, it raises the price of the high-tech input, tending to reduce demand for that input and diminishing profits. Second, for the given amount of inputs purchased, the higher quality reduces cost, tending to raise final output and profits. Overall, the second effect dominates, implying that the final good assembler s output and profits increase, by 10.2 percent and 0.2 percent, respectively. However, the quality increment is sufficiently large that the use of both input quantities declines in steady state, with a particularly large drop in the high-technology variant. Next, note that the rise in λ h results in a higher relative wage in equilibrium. To see this, note that if Southern R&D is successful at imitating the high-technology input, the price the Southern leader can charge for its input gets higher, implying a greater relative wage. This incentivizes an increase in the imitation intensity in the South. Nevertheless, the measure of Southern firms engaged in imitation actually falls because of higher costs, as does local input production. Thus, 30

32 net labor demand falls in the South. Meanwhile, both innovation and production rise in the North in the new steady state. In addition, the higher relative wage implies that both input prices go up in these adjustments. In turn, we find that these price gains dominate the reduction in use of these inputs, permitting instantaneous profits in both the high-technology and low-technology intermediates to rise. Finally, because these profits of input manufacturers are exactly offset by imitation and innovation costs, both of the latter flows rise and there is a higher steady-state flow of technology transfer (ι = µ). We demonstrate this result graphically in Figure 5. The increase in λ h shifts up the NN curve but does not move the SS curve. In consequence, both the relative wage and the rate of technology flows are higher in the new equilibrium. 19 To summarize, a higher quality of the high-tech input sets in motion impacts that both raise the Northern relative wage and expand technology innovation and diffusion. This permits welfare in both regions to rise in the new equilibrium. These are novel results in the quality-ladders literature, stemming from the existence of two inputs and the associated competition. < Insert Figure 5 here.> 5.4 Stronger Southern patent protection Consider next the implications of an exogenous increase in the rigor of Southern intellectual property rights, from 0.6 to 1.0. This generates both a fall in the rate of Southern imitation and an increase in the rate of innovation in the North. Because the imitation risk is diminished, the relative wage rises as Northern firms devote relatively more labor to innovating against the low-technology input. There is also an increase in Northern production of the high-technology input. Note next that the higher relative wage mandates a higher price for the low-technology input, reducing demand for that good by the final producer as it substitutes into the other input. Nonetheless, instantaneous profits of the Southern low-tech input producer rise overall, as they must to offset the higher imitation costs. 19 Additional analysis finds that the marginal increase in the relative wage and expenditure decline as the quality increment gets larger. 31

33 These ideas are illustrated in Figure 6. Here, the stronger patent regime shifts the SS curve downward by raising the cost of imitation. The relative wage rises from to , or 1.21 percent. Aggregate technology flows fall by about 0.84 percent. Southern welfare falls and Northern well-being rises slightly due to the stronger patent regime. < Insert Figure 6 here.> 5.5 Changes in labor endowments Next, we consider changes in labor endowments in the two regions. Begin with an increase in the Southern labor force from 10 to 11, holding other parameters constant. As may be seen in equation (60), the impact effect is a higher imitation rate for any relative wage, implying a shift upward in the SS curve (Figure 7). Moreover, the higher endowment produces larger aggregate expenditure in the new equilibrium, generating higher output of the final good which, in this case, translates into higher quantity demanded for both inputs. Note, however, that the price of the low-tech input, which equals ω, must be lower in the new equilibrium, according to Figure 7. Indeed, this price falls sufficiently to more than offset the higher output of this good, generating lower instantaneous profits in Southern production. In contrast, both outputs and profits rise in Northern production. < Insert Figure 7 here.> This output shift is key to understanding the counterintuitive result that a rise in the Southern labor endowment generates a lower relative wage, implying that Northern labor becomes comparatively less expensive. Note that lower instantaneous profits in Southern low-technology production must be met by lower investment costs in Northern innovation against this good. Indeed, as noted in Table 2, innovators in the North decline by 44.9 percent. However, the measure of Southern imitators targeting the high-technology input increases by 64 percent, even as there is just a small decline in input production in that region. Overall these impacts raise labor demand relatively more in the South, lowering the relative wage. Cross-regional technology flows increase and, in this case, both locations enjoy higher welfare. 32

34 The last experiment is to increase the Northern labor force from 3 to 3.3. As noted in Table 2 this raises global expenditure but also increases the relative wage in the North. Again, this result is counterintuitive and comes from the interaction of key variables in general equilibrium. As in the prior case, outputs of both intermediates rise, however here the instantaneous profits of both input producers go up. This implies higher costs of both innovation and imitation at higher rates of those activities. Both firm measures in the South fall and those in the North rise, consistent with the increase in relative wage associated with this large hike in labor demand in the latter. Finally, there is a substantial increase in the rate of aggregate technology flows, as shown in Figure 8. Overall, the key message is that a larger market size generates faster innovation in the high-technology input and faster product cycles in the low-technology input. < Insert Figure 8 here.> 6 Concluding Remarks In this paper we set out a two-region, quality-ladders product cycle model in which final goods are produced with two inputs distinguished by their levels of technological sophistication. The high-technology input is characterized by greater quality jumps and is more costly to innovate and imitate than is the low-technology input. Both, however, independently climb quality ladders due to endogenous innovation, while technology flows between North and South through endogenous imitation. In its general form the model yields continuous rates of innovation and imitation targeted against both inputs. However, our solution focuses on the most likely case, in which there is Northern innovation against the low-technology input but its production shifts to the South after successful imitation, generating a product cycle in that variant. Production of the high-technology intermediate good remains in the North, implying endogenous specialization in production. As discussed at the outset, these dynamics are consistent with observed production and trade flows in various intermediate goods subject to quality improvement. The model also permits calculation of economic welfare in steady state in both regions. The model is sufficiently complex that analyzing shifts in steady-state equilibria associated with 33

35 various parametric changes requires calibration and simulation. This analysis delivers a number of novel results. For example, an increase in the quality level of the high-technology input raises the Northern wage relative to the Southern wage. However, it expands the aggregate rates of innovation and diffusion sufficiently that welfare rises in both regions. Next, if the South strengthens its patent protection, raising the costs of imitation, the net results involve a higher relative wage and lower technology transfer, consistent with the classic results in Helpman (1993), albeit in the context of intermediate goods. Finally, increases in either region s labor force generates more innovation and technology diffusion due largely to an expenditure effect. However, the welfare predictions vary between a rise in the Southern endowment, which is beneficial for both areas, and an increase in the Northern endowment, which disfavors the South. We emphasize that these results are specific to the model, which is highly stylized. However, they do offer food for thought about the effects of technical change and intellectual property reforms in a world of increasing production fragmentation. We hope that the model offers a framework for additional analysis of quality dynamics in the presence of endogenous input choices. References Antras, P., Incomplete contracts and the product cycle. American Economic Review 95(4), Antras, P., D. Chor, T. Fally and R. Hillberry, Measuring the upstreamness of production and trade flows. American Economic Review: Papers and Proceedings 102(3), Baldwin, R. and J. Lopez-Gonzalez, Supply-chain trade: a portrait of global patterns and several testable hypotheses. Cambridge: NBER working paper number Bems, R., R. C. Johnson and K-M Yi The great trade collapse. Annual Review of Economics 5, Broda, C. and D.E.Weinstein, Globalization and the gains from variety. Quarterly Journal of Economics 121(2),

36 Dollar, D., Technological innovation, capital mobility, and the product cycle in north-south trade. American Economic Review 76(1), Glass, A. J., Product cycles and market penetration. International Economic Review 38(4), Glass, A. J., Outsourcing under imperfect protection of intellectual property. Review of International Economics 12(5), Glass, A. J. and K. Saggi, Innovation and wage effects of international outsourcing. European Economic Review 45(1), Glass, A. J. and K. Saggi, Intellectual property rights and foreign direct investment. Journal of International Economics 56(2), Glass, A. J. and X. Wu, Intellectual property rights and quality improvement. Journal of Development Economics 82(2), Grossman, G. and E. Helpman, 1991a. Innovation and Growth in the Global Economy. Cambridge MA: MIT Press. Grossman, G. and E. Helpman, 1991b. Quality ladders and product cycles. Quarterly Journal of Economics 106(2), Helpman, E., Innovation, imitation, and intellectual property rights. Econometrica 61(6), Hummels, D. and P. J. Klenow, The variety and quality of a nation s exports. American Economic Review 95(3), Tanaka, H. and T. Iwaisako, Intellectual property rights and foreign direct investment: A welfare analysis. European Economic Review 67(2014), Jensen, R. and M. Thursby, A strategic approach to the product life cycle. Journal of International Economics 21(3), Jensen, R. and M. Thursby, A decision-theoretic model of innovation, technology transfer, and trade. Review of Economic Studies 54(4), Krishna, P. and W. F. Maloney, Export quality dynamics. World Bank Policy 35

37 Research Working Paper Krugman, P., A model of innovation, technology transfer, and the world distribution of income. Journal of Political Economy 87(2), Lai, E. L.-C., International intellectual property rights protection and the rate of product innovation. Journal of Development Economics 55(1), Lu, C.-H., Moving up or moving out? A unified theory of R&D, FDI, and trade. Journal of International Economics 71(2), Mansfield, E., M. Schwartz, and S. Wagner, Imitation costs and patents: an empirical study. Economic Journal 91(364), Sayek, S. and F. Sener, Outsourcing and wage inequality in a dynamic product cycle model. Review of Development Economics 10(1), Schott, P. K., Across-product versus within-product specialization in international trade.the Quarterly Journal of Economics119(2), Segerstrom, P. S., T. C. Anant and E. Dinopoulos, A Schumpeterian model of the product life cycle. American Economic Review 80(5), Taylor, M. S., Quality ladders and Ricardian trade. Journal of International Economics 34(3), Vernon, R., International investment and international trade in the product cycle. Quarterly Journal of Economics 80(2), Yang, G. and K. E. Maskus, Intellectual property rights, licensing, and innovation in an endogenous product-cycle model. Journal of International Economics 53(1), Appendix A. Final good assembler s problem The final good assembler faces this problem (equation (17) in the text) max π jk = φ 1 α ζz α (q l ) αz (x j x l, x l )αz (q h ) α(1 z) (x k h h) α(1 z) p j l xj l pk hx k h ( 17) 36

38 From this we get x l : αzφ 1 α ζ α z (q l ) αz (x j l )αz 1 (q h ) α(1 z) (x k h) α(1 z) = p j l (A1) x h : α(1 z)φ 1 α ζ α z (q l ) αz (x j l )αz (q h ) α(1 z) (x k h) α(1 z) 1 = p k h (A2) Dividing (A1) by (A2) we derive the relationship between x k h and xj l x k h = 1 z z p j l x j p k l h (A3) Plugging (A3) into (A1), we get the reduced-form expression for x j l. (x l ) 1 α = αz 1 α φ 1 α (q l ) αz (q h ) α(1 z) (p j l )α(1 z) 1 (p k h) α(1 z) (A4) Incorporating text equation (5) we derive p jk = ( φ y jk ) 1 α = [ = [ φ ζ z(q l x j l )z (q h x k ]1 α h )1 z φ z z (1 z) (1 z) ql zq1 z h (x j l )z (1 z) 1 z z (1 z) (p j l )1 z (p k h ) (1 z) (x j ]1 α l )1 z = 1 α ( pj l q l ) z ( pk h qh ) 1 z = 1 α ( pj l λ η l ) z ( pk h ) 1 z λ θ h (A5) Now substituting (A5) into text equation (5) we obtain the function for the quantity of final good y jk = φ(p jk ) 1 1 α = φ[ 1 α (pj l q l ) z ( pk h q h ) 1 z ] 1 1 α (A6) Appendix B. Supplemental calculations B.1. Calculations related to the goods market Recall that φ = E 1 0 p(m) α/(1 α) dm where E = LS + ωl N + φ(1 α)[α( λη l ω )z ( λθ h λ h ) 1 z ] α 1 α considered exogenous by all firms. Taking derivatives with respect to the relative wage ω is 37

39 yields x S l 1 α ω = 1 + z α ω x N h 1 α ω = z α x S l < 0 (B1) ω xn h < 0 (B2) p SN ω = z ω psn > 0 (B3) y SN ω = z ω(1 α) ysn < 0 (B4) π SN ω = z α 1 α ω πsn < 0 (B5) π S l ω = xs l + (ω 1) xs l ω > or < 0 (B6) π N h ω = xn h + (λ h ω) xn h ω < 0 (B7) B.2. Calculations related to R&D intensities ι ω = a h λ h L N (a l λ h a l ω + a h ω) < 0 (B8) 2 µ ω = c l L S (1 + κ)[(ω + 1)c h + c l ] > 0 (B9) µ h ω = (λ h ω) xn h a ω hω λ h a h < 0 (B10) (a h ω) 2 ι l ω = (1 α zα zαω)xs l ω(1 α)(1 + κ)c l > or < 0 (B11) Note that these expressions refer to partial derivatives, not to the full steady-state equilibrium impacts. B.3. Analytical solution for the relative wage 38

40 Reproducing equation (58) from the text: [a l L S L N (1 + κ)c h a h L S ]ω 2 + [a h L S (1 + κ)c l L N (1 + λ h )a l L S + (1 + λ h )(1 + κ)c h L N ]ω + [a l L S (1 + κ)c h L N + (1 + κ)c l L N ]λ h = 0 ( 58) Set a = a l L S L N (1 + κ)c h a h L S b = a h L S (1 + κ)c l L N (1 + λ h )a l L S + (1 + λ h )(1 + κ)c h L N (B12) c = [a l L S (1 + κ)c h L N + (1 + κ)c l L N ]λ h The usual quadratic-equation solution applies and we ignore the negative root ω 1,2 = b ± b 2 4ac 2a (B13) Appendix C. Derivation of welfare expressions We follow the method in Tanaka and Iwaisako (2014) to calculate the welfare of the two regions. First, we add the country superscript j into the intertemporal utiltiy function as U j = 0 e ρt log u j (t)dt (C1) where j {S, N} denotes country. Similarly, rewrite the instantaneous utility for each country: u j (t) = [ Recall that y j (i, t) = φ j (t)p(i, t) (y j (i, t)) α di] 1 α, 0 < α < 1 (C2) 1 α = E j (t) 1 α 0 p(m,t) 1 α dm p(i, t) 1 1 α, where the final good price p(i, t) is the same across countries. Now from text equation (25) we have E S (t) = L S (C3) 39

41 E N (t) = ω(t)l N φ j (t)(1 α)[α( q l(i, t) p l (i, t) )z ( q h(i, t) p h (i, t) )(1 z) ] α 1 α di (C4) Note also that the top-quality versions of the two inputs are q l (i, t) = λ l (i, t) η, q h (i, t) = λ h (i, t) θ. As a result, we find that 1 log y j (i, t) = log E j (t) log p(i, t)di (C5) Applying this analysis specifically to Case 3 in the text (after substituting input prices) yields 1 0 log p(i, t)di = 1 0 log[ 1 α ( ω (t) q l (i,t) )z ( λ h q h (i,t) )1 z ]di = log 1 α + z 1 0 log ω (t)di z 1 0 log q l(i, t)di + (1 z) 1 0 log λ h q h (i,t) di = log α + z 1 0 log ω (t)di z log λ l 1 0 η(i, t)di + (1 z) log λ h 1 (1 θ(i, t))di 0 (C6) From this analysis we find that Southern instantaneous welfare is: log u S (t) = log L S +log α z log ω (t)di+z log λ l η(i, t)di+(1 z) log λ h (θ(i, t) 1)di where omega (t) is the positive root from the solution to equation (58). Using a similar method, we can calculate the instantaneous utility of the Northern consumers as: log u N (t) = log E N (t) 1 0 p(i, t)di As noted in (C4), E N is a complicated expression and we cannot determine an analytical solution to log u N (t). However, it is straightforward to calibrate, as we do in the text. (C7) (C8) Finally, note that it is difficult to directly calibrate the intertemporal utility (C1) since we must set the discount parameter ρ = 0. However, the effects of all parameter changes on U j are qualitatively the same as those on log u j (t). Thus, we analyze the comparative statics on log u j (t) in the simulation. 40

42 Table 1: Parameter Values for Simulation and Comparative-Static Analysis Economic Meaning Notation Benchmark Quality South IPR South Labor North Labor Change Increase Increase Increase Consumer subjective discount ρ Northern labor endowment L N Southern labor endowment L S Innovation cost of high-technology input ah Innovation cost of low-technology input al Imitation cost of high-technology input ch Imitation cost of low-technology input cl Quality increment of high-technology input λh Quality increment of low-technology input λl Quality jumps of the high-technology input θ Quality jumps of the low-technology input η Share of low-technology input in final good production z Elasticity of substitution among final goods ɛ Southern IPRs parameter κ

43 Table 2: Benchmark Numerical Values and Percentage Changes of Endogenous Variables Economic Meaning Notation Benchmark Quality South IPR South Labor North Labor Change Increase Increase Increase Relative wage between the North and the South ω % +1.21% -0.40% +0.46% Total world expenditures E % +0.28% +7.54% +2.49% Quantity of low-technology input x S l % -0.91% +7.96% +2.02% Quantity of high-technology input x N h % +0.29% +7.53% +2.49% Operating profits of Southern low-tech input manufacturers π l S % % -4.56% % Operating profits of Northern high-tech input manufacturers π h N % -0.35% +7.76% +2.24% Price of final good p SN % +0.78% -0.26% +0.30% Quantity of final good y SN % -0.49% +7.81% +2.19% Profits of the final good assembler π SN % +0.29% +7.54% +2.49% Imitation intensity of low-technology input µl 0 Imitation intensity of high-technology input µh % -1.54% +8.19% +1.78% Innovation intensity of low-technology input ιl % +7.31% -4.55% % Innovation intensity of high-technology input ιh 0 Measure of low-tech input manufacturers in the South n S l % -7.60% +5.06% -5.31% Measure of high-tech input imitators in the South n S h % -6.19% % % Measure of low-tech input innovators in the North n N l % % % % Measure of high-technology manufacturers in the North n N h % +0.71% -7.31% +7.74% Aggregate technological flows across borders ι = µ % -0.84% +0.28% +9.65% Southern instantaneous welfare log u S % -0.26% +3.23% -0.10% Northern instantaneous welfare log u N % +0.01% +1.33% +2.69% 42

44 Figure 1: Relative Exports of China and Germany in DRAMS of Varying Sizes to US,

45 Figure 2: Relative Exports of Selected Developing and Developed Countries in Aircraft Engines to the United States Figure 3: Relative Exports of Selected Developing and Developed Countries in Aircraft Parts to the United States 44