Shall the Northern optimal R&D subsidy rate inversely respond to Southern Intellectual Property Protection?

1. Introduction

Innovative research and development (R&D) is key to technological advances and long-run growth. A central question is whether or not a decentralized economy only sustains a socially suboptimal level of investment in R&D, as was early recognized in the industrial organization literature (e.g., Dasgupta and Stiglitz 1980). To resolve this ambiguity, many empirical productivity studies based on neoclassical growth models find that knowledge spillovers are present and R&D's social rates of return considerably exceed its private rates. (1) So underinvestment in R&D does prevail in reality.

These earlier findings largely conform to recent studies using R&D-based endogenous growth models. Jones and Williams (1998, p. 1119), for instance, estimate that "optimal R&D investment is at least two to four times actual investment." (2) By simulating a calibrated R&D-based endogenous growth model without so-called "scale effects," (3) Jones and Williams (1999, p. 2) further find that "in the absence of taxes and subsidies, the decentralized economy underinvests in R&D, with the primary impetus coming from the surplus appropriability problem." Thus, after knowledge spillovers were found to be present and significant in earlier productivity studies, the main force promoting underinvestment now is found to be the monopoly pricing of innovative goods that prevents innovators from appropriating the entire consumer surplus associated with these goods.

Governments often subsidize R&D investment in advanced countries (North). In fact, the World Trade Organization (WTO) already completed a framework enabling governments to establish--free from counteractions by other countries--certain subsidy programs aimed at promoting R&D activities of firms, other than helping disadvantaged regions and protecting the environment. (4) It is common that Northern innovations are frequently pirated by low-cost developing countries (South), depending on their enforcement of intellectual property protection (IPP). It is therefore important to ask how the North should adjust its socially optimal R&D subsidy rate in response to Southern IPP standards. Indeed, the North has long striven to extend stricter IPP into the South under the GATT/WTO framework. Yet, as Helpman (1993) concludes, it is never in the South's interest to strengthen IPP. This conflict-of-interest problem had affected North-South relations from time to time. Other studies in the recent IPP/innovation literature include Chin and Grossman (1990), Diwan and Rodrick (1991), Deardorff (1992), Taylor (1993, 1994), and Grinols and Lin (2002). (5,6) But these studies are not aimed at addressing the plausibility of whether the North's socially optimal R&D subsidy rate tends to decline when Southern IPP standards are raised.

In the present paper we examine this plausibility problem using a familiar North-South trade model that was developed along the line of Krugman (1979), Dollar (1986), Jensen and Thursby (1987), Grossman and Helpman (1991c), and Helpman (1993). In the model, imitative activity is viable in the low-wage South but not in the high-wage North, whereas innovative activity is active only in the North. The rate of product innovation is endogenously determined by profit-driven research inputs but the rate of product imitation is parameterized by Southern IPP standards.

Using comparative static analysis, I find that Southern IPP and Northern R&D subsidies are not substitutable in keeping up the North's long-run innovation rate. This finding is not surprising. On the one hand, strengthening IPP serves to decrease the imitation rate, prolong the expected duration of monopoly power, free up less Northern resources from old goods production to support R&D, and thereby decrease the innovation rate. (7) On the other hand, reducing Northern R&D subsidy rates serves to increase innovation costs and thereby also decreases the innovation rate. Hence, substitutability does not hold true in the North-South framework.

More striking are the findings from our numerical simulations. First of all, our simulation results indicate that the North's socially optimal R&D subsidy rate may either positively or negatively correspond to Southern IPP standards, depending on the elasticity of demand for innovative products. If the demand is more elastic, a tightening of Southern IPP is found to induce the self-interested North to increase the optimal subsidy rate. Conversely, if the demand is less elastic, a tightening of Southern IPP is found to invite the North to decrease the optimal subsidy rate, as one's intuition predicts. Note that Jones and Williams's (1999) numerical simulations single out the surplus appropriability problem as the main force promoting underinvestment in R&D, as mentioned earlier. Here my simulation results further show that demand elasticity, which inversely measures the degree of monopoly power and the surplus appropriability problem, can determine whether or not the Northern optimal R&D subsidy rate inversel y corresponds to Southern IPP standards.

The market mechanism accounting for our above simulation results involves complicated opposing forces. First, a loosening of Southern IPP can increase the pace of imitation and can relocate more Northern production sites to the low-cost South while also spurring innovation, as my comparative analysis indicates. Faster innovation then brings more product varieties generating welfare gains to both the North and South. But faster innovation requires higher saving rates, thereby hurting Northern welfare at the same time. If the demand is more elastic, product differentiation is less significant and thus the beneficial product varieties effect and other effects tend to be dominated by the adverse saving effect. In this case, to respond to laxer Southern IPP, the self-interested North must decrease the subsidy rate to discourage R&D investment, slow down product innovation, and mitigate the adverse saving effect. This certainly implies that if the demand is more elastic, the North may need to increase the subsidy rate when the South strengthens IPP.

Another important finding from my numerical simulations concerns welfare implications. I find that Southern welfare declines but Northern welfare rises at the steady state, as long as Southern IPP is strengthened. This finding once again highlights the stark North-South conflict of interest, as was earlier formalized by Helpman (1993) in which the North does not subsidize R&D at all. Further, I find that worldwide welfare maximization requires a regime of Southern IPP that is neither as stringent as the North favors nor as lax as the South prefers. This finding appears robust to the elasticity of demand for innovative goods.

Section 2 outlines a North-South model of Krugman-Grossman-Helpman type. Section 3 analyzes substitutability between IPP and R&D subsidies, respectively, with respect to steady-state innovation rates and Northern utility maximization. This section heavily relies on numerical simulations when I attempt to compute the North's optimal R&D subsidy rates, given various standards of Southern IPP. Concluding remarks are given in section 4.

2. The Model

We formulate the world to consist of an innovating region (North) and a noninnovating region (South). In both regions labor is the only factor of production and is inelastically supplied. We let regional labor endowments and other structural parameters combine to mimic a North-South environment in which labor wages are lower in the South than in the North. All goods are invented in the North, for only Northern producers engage in R&D. Since enforcement of intellectual property rights (IPR) is imperfect in the world, Northern innovations are to be pirated or imitated over time by low-cost Southern producers. Interregional trade thereby occurs with the North exporting pricy innovative goods in exchange for cheap imitated goods from the South. The North-South trade model laid out below is along the line of Krugman's (1979) seminal paper and the ensuing works, such as Dollar (1986), Jensen and Thursby (1987), Grossman and Helpman (1991a), and Helpman (1993). (8)

Innovation and Imitation

In the North many identical firms devote labor services to R&D. The flow of innovation, n(t), from research and development activities at time t is determined by

n(t) = [L.sup.N.sub.I](t)/a/n(t) (1)

where an overdot indicates a time derivative of a variable throughout the paper, [L.sup.N.sub.I] represents labor services devoted to R&D, n is the number of products or the stock of knowledge formerly invented in the North, and a denotes a parameter inversely measuring research productivity. Thus a/n(t) measures the instantaneous labor requirement per innovative product and implies the effect of "learning by doing" or "intertemporal knowledge spillovers." As knowledge accumulates (increase in n), a/n(t) declines and labor becomes more productive for next-generation innovations. The innovation rate g in terms of Equation 1 is

g(t) = n(t)/n(t) = [L.sup.N.sub.I](t)/a (2)

The way we endogenize the innovation rate is the same as in Romer (1990), Grossman and Helpman (1991b, c), and Helpman (1993), where g is linearly related to research inputs. (9) Thus, intertemporal knowledge spillovers do not peter out as knowledge accumulates over time and the equilibrium allocation of Northern labor between manufacturing and research determines the equilibrium innovation rate.

Knowledge of manufacturing innovative goods is under global IPR protection. Such protection is imperfect, however. As time passes, Southern labor will eventually learn to produce imported goods by dismantling and studying them (i.e., so-called "reverse engineering"). Such learning or imitating costs less than does innovating and enables low-cost Southern producers to underprice their Northern rivals. (10) This gives rise to North-South trade, with the South exporting imitated products in exchange for unimitated products from the North. Following Krugman (1979), the flow of technology transferred to the imitating South is given by

[n.sup.S](t) = [mn.sup.N](t) (3)

where [n.sup.N] is the number of products that have not yet been imitated at time t and m is the imitation rate or the probability that the South will learn to imitate a patented good over the next instant. Hence the inverse of the imitation rate, 1/m, reflects the expected duration of an innovator's monopoly position. In this paper Southern IPP determines the patent life, denoted by [delta] = (=1/m), which then dictates a Northern innovator's expected monopoly duration. If IPP is perfect, the patent life is infinity ([delta] [right arrow] [infinity]) or the imitation rate is zero (m = 0); then Northern innovators can confer permanent monopoly power. If IPP is not in place at all, the patent life is zero ([delta] = 0) or the imitation rate is infinity (m [right arrow] [infinity]); then any innovation is copied as soon as it is available and the North cannot keep a viable research sector in this polar case. We assume imperfect IPP prevails in the North-South model, where innovations and monopoly power are prot ected during a finite patent life (0 < [delta] = 1/m < [infinity]). For ease of notations, we omit time t from most variables in what follows.

Production, Pricing, and Resource Allocation 

Any product, imitated or unimitated, is produced with one unit of labor in each region. Once a product is imitated, its manufacturing technology is known worldwide. The market for imitated products is perfectly competitive and production of these products is viable only in the low-wage South. The Southern wage rate [w.sup.S] therefore determines the competitive price [p.sup.S] for all goods produced in the South,

[p.sup.S] = [p.sup.S](i) = [w.sup.S], i [member of] [0, [n.sup.S]]. (4)

Imitated products are indexed by 0 through [n.sup.S] on the real line. The remaining products that have not yet been imitated are indexed by [n.sup.S] through n ([n.sup.S] + [n.sup.N]). Northern producers engage in monopolistic competition and earn monopoly profits to cover one-time R&D costs. Each of them faces a downward sloping demand curve having a constant elasticity of demand, [epsilon], for an individual product. The marginal cost of an unimitated product is the Northern wage rate [w.sup.N]. Hence a representative monopoly price [p.sup.N] is given by

[p.sup.N] = [p.sup.N](i) = ([epsilon]/[epsilon] - 1)[w.sup.N], i [member of] ([n.sup.S], n]. (5)

Equations 4 and 5 imply that every producer produces the same output level [x.sup.j] = [x.sup.j](i), j = N or S, in each region. Further, Equation 5 implies that the flow of a Northern producer's operating profit is given by [[pi].sup.N] = ([p.sup.N] - [w.sup.N])[x.sup.N] = [p.sup.N][x.sup.N]/[epsilon]. Innovation requires resources (see Eqns. 1 or 2) but imitation does not. So Northern and Southern full employment condition are given by

ag + [n.sup.N][x.sup.N] = [L.sup.N] (6)

[n.sup.S][x.sup.s] = [L.sup.S]. (7)

Asset Market Equilibrium

Northern firms invest in R&D and receive subsidies from the Northern government. The subsidy rate, denoted by [theta], determines the fraction of R&D costs borne by the government. Innovation of a new product requires a/n units of labor as noted earlier, so a firm's net R&D costs equal (1 - [theta]) [w.sup.N]a/n, which measures the fixed entry cost. A Northern firm's stock market value [v.sup.N] reflects the present value of future operating profits. It cannot be less than the entry cost, (1 - [theta])[w.sup.N]a/n, or rational agents would never invest in R&D; neither can it exceed the entry cost, or the R&D demand for labor would be unbounded. Thus, in equilibrium a Northern firm's stock market value is tied to its after-subsidy R&D costs,

[v.sup.N] = (1 - [theta])[w.sup.N] a/n. (8)

Northern firms face the risks that their products are to be imitated by Southern producers over time. (11) When an innovative product is imitated, the Southern imitator displaces a Northern firm and makes this firm's shareholders lose [v.sup.N]. During an instant dt, the probability of a product being imitated by the South is mdt. Therefore the expected capital gain is equal to (1 - mdt)[v.sup.N] dt. With the flow of operating profits equal to [[pi].sup.N] dt at an instant, the expected rewards for a Northern firm's shareholders equals [[pi].sup.N] dt + (1 - mdt)[v.sup.N] dt - (mdt)[v.sup.N]. Assuming well-functioning financial markets in the North, these rewards must equal [r.sup.N][v.sup.N] dt, which would be incurred if shareholders liquidated equity shares and invested in bonds paying the nominal rate of interest [r.sup.N]. Dividing the equality equation by [v.sup.N] and letting dt approach zero, we get

[[pi].sup.N]/[v.sup.N] + [v.sup.N]/[v.sup.N] = [r.sup.N] + m. (9)

This is a no-arbitrage condition in asset markets. (12) It states that the dividend rate plus the rate of capital gain on equity holdings equals the risk-adjusted interest rate ([r.sup.N] + m) at any time.

Demands and North-South Trade

Individuals have identical preferences in the world. In each region the representative consumer derives utility from an infinite stream of consumption beginning at time t. We assume a well-known logarithmic utility function,

[U.sup.j](t) = [[integral].sup.[infinity].sub.t] [e.sup.-[rho]([tau]-t)]log [u.sup.j]([tau])d[tau], j = N, S (10)

where [rho] is the constant rate of time preference and u([tau]) is the flow of utility at time [tau] determined by consumption of available products via a constant elasticity of substitution (CES) preference structure (Spence 1976; Dixit and Stiglitz 1977),

[FORMULA NOT REPRODUCIBLE IN ASCII] (11)

[FORMULA NOT REPRODUCIBLE IN ASCII] (12)

where [M.sup.j](*) and [C.sup.j](*) represent consumption of imports and domestically produced goods in region j, respectively. The consumer holds financial assets, the value of which is measured by A(t) at time t, and is subject to the dynamic budget constraint,

[A.sup.j] = ([r.sup.j][A.sup.j] + [w.sup.j][L.sup.j]) - [E.sup.j] - [theta][I.sup.j], j = N, S (13)

with the initial condition [A.sup.j](0) = [A.sup.j.sub.0] * [r.sup.j] is the nominal rate of interest prevailing in the North or South. [r.sup.j] [A.sup.j] measures expected capital gains. [w.sup.j][L.sup.j] is the flow of labor income. [E.sup.j] is the flow of aggregate consumption spending. [I.sup.j] is the flow of aggregate R&D investment with [I.sup.s] = 0 and [I.sup.N] = [w.sup.N] [L.sup.N.sub.I] = [w.sup.N] ag (see Eqn. 2), so [theta][I.sup.j] represents the total subsidy cost and is financed by a nondistortionary lump-sum tax paid by consumers. The term of expected capital gains in Equation 13 deserves further comment. Note that in the North [A.sup.N] is the total of the values of all Northern incumbent firms (i.e., [A.sup.N] = [n.sup.N][v.sup.N]). According to Equation 9, the total of expected rewards on asset holdings [r.sup.N][A.sup.N] [=[n.sup.N][v.sup.N])] is equal to [n.sup.N]([[pi].sup.N] + v - mv), where [n.sup.N](v - mv) is the book value of expected capital gains and [n.sup.N][[pi].sup.N] is the total flow of instantaneous operating profits. so the North's national income accounting is [GDP.sup.N] = [n.sup.N][[pi].sup.N] + [w.sup.N][L.sup.N]. By contrast, the South's counterpart is [GDP.sup.S] = [w.sup.S][L.sup.S], for Southern imitative firms can never generate monopoly profits in the present context.

We now solve a representative household's static optimization problem by maximizing Equations 11 and 12, subject to a given level of aggregate consumption spending [E.sup.j]. In so doing, Northern demands for imitated and unimitated products are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

and Southern demands for these products are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

where upper-cases P measures the familiar CES price index,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

where [eta] (=[n.sup.N]/n) denotes the fraction of unimitated goods, which are produced in the North. Markets clear at any instant. Thus, using Equations 6 and 7, symmetry among regional firms implies the following market-clearing conditions:

[n.sup.N]([C.sup.N] + [M.sup.S]) = [n.sup.N][x.sup.N] = [L.sup.N] - ag (19)

[n.sup.S]([C.sup.S] + [M.sup.N]) = [n.sup.S][X.sup.S] = [L.sup.S]. (20)

As implied by demand Equations 14-17, the North exports unimitated products and the South exports imitated products. Such North-South trade must be balanced at all times, since the model assumes away international borrowing. Accordingly, aggregate expenditure must be tied to aggregate income in each region. So, using Equations 19 and together with [GDP.sup.S] = [E.sup.S] and [GDP.sup.N] = [E.sup.N] + [I.sup.N], we can obtain the following macroeconomic equilibrium conditions, (13)

[p.sup.S][L.sup.S] = [E.sup.S] (21)

[p.sup.N]([L.sup.N] - ag) = [E.sup.N] (22)

where ag measures the units of labor engaged in R&D so that [L.sup.N] -- ag measures the units of labor used in production (see Eqn. 6). Note that the flow of utility [u.sup.j] in Equation 11 (or 12) is in effect equal to the flow of real spending [E.sup.j]/[P.sup.j] in terms of demand Equations 14-17. Hence, Equation 10 can be rewritten as an indirect utility function,

[U.sup.j] = [[integral].sup.[infinity].sub.t] [e.sup.-[rho]([tau]-t)][log[E.sup.j]([tau])--log P([tau])]d[tau]. (23)

Maximization of Equation 23 subject to Equation 13 yields the standard result,

[E.sup.j]/[E.sup.j] = [r.sup.j] - [rho], j = N, S (24)

where as is well known, the growth of consumption spending requires the nominal interest rate exceeding the rate of time preference.

The Autonomous System 

Now, noting that [w.sup.N] = [p.sup.N]([epsilon] - 1)/[epsilon], [pi] = [p.sup.N] [x.sup.N/[epsilon] and [zeta] = [n.sup.N]/n, and using Equations 8, 22, and 24, we can transform the no-arbitrage Equation 9 into

g(t) = ([L.sup.N]/a - g(t)) ([rho] + m + g(t) - (1/1 - [theta]) (1/[epsilon] - 1) [[L.sup.N]/a - g(t)] (1/[zeta](t))) (25)

and the fraction of unimitated products [zeta](t) evolves over time according to

[zeta](t) = g(t) - [m + g(t)][zeta](t). (26)

The two equations constitute a general equilibrium autonomous system, which reduces to that of Helpman (1993) if the R&D subsidy rate [theta] is set equal to zero.

3. Substitutability between IPP and R&D Subsidies

In what follows we examine whether IPP enforced by the South is substitutable for Northern R&D subsidies, respectively, with respect to long-run innovation rates and Northern welfare maximization.

Imitation, Innovation, and R&D Subsidies

We now replace m with [delta] = 1/m and set g = 0 in Equation 25 and [zeta] = 0 in Equation 26. Then with some rearrangement, we can easily obtain two steady-state equilibrium conditions as follows,

[zeta] = 1/(1 - [theta])([epsilon] - 1) ([L.sup.N]/a - g) [delta]/1 + [delta]([rho] + g) (27)

[zeta] = [delta]g/1 + [delta]g. (28)

Equation 27 is represented by curve AA and Equation 28 is represented by curve OB in Figure 1. The intersection of curves AA and OB at point 1 represents the original steady-state equilibrium

([g.sub.1], [[zeta].sub.1]), subject to [delta], [theta], and the other parameters. It is clear from Equations 27 and 28 that a tightening of Southern IPP (increase in patent life, [delta]) shifts curve AA upward to the right and curve OB downward to the right and the downward shift must dominate, as indicated in Figure 1. Thus increased patent life must shift the steady-state equilibrium from ([g.sub.1], [[zeta].sub.1]) to ([g.sub.2], [[zeta].sub.2]). Namely, tighter IPP (i.e., longer patent life) in the South must lead to a smaller innovation rate and a larger proportion of unimitated products in the long run. In contrast, the R&D subsidy rate [theta] can shift curve AA only. As also indicated in Figure 1, if the North increases the R&D subsidy rate, curve AA shifts upward to the right while curve OB stays intact, shifting the equilibrium from ([g.sub.1], [[zeta].sub.1]) to ([g.sub.3], [[zeta].sub.3]). As a result, both g and [zeta] are increased by an increase in [theta]. The above comparative analysis show s that either a loosening of IPP by the South or a larger Northern R&D subsidy rate can spur innovation (increase in g), although the two policy instruments have opposing effects on the proportion of goods produced in each region. We can further characterize such relations by substituting Equation 28 into Equation 27,

[theta]([delta];g) = 1 - (1/[epsilon]-1) ([L.sup.N]/ag - 1) (1 + [delta]g/1 + [delta]([rho] + g)) (29)

where it is clear that [partial][theta]/[partial][delta] > 0 and [[partial].sup.2][theta]/[partial][[delta].sup.2] < 0 for any given g. Thus, [theta](*) is monotonically increasing and concave in [delta], given a positive steady-state innovation rate, g. These functional relations imply that to achieve the same steady-state innovation rate, the North must increase (decrease) the R&D subsidy rate in response to tighter (taxer) Southern IPP. We therefore obtain the following:

PROPOSITION 1. In the context of North-South trade, to attain a given long-run innovation rate, a tightening of Southern IPP (increase in patent life) is not substitutable for Northern R&D subsidies.

This seemingly counterintuitive result stems from labor migration between production and research in the North. As imitation slows down (due to fighter IPP), incumbent firms face a smaller risk of being displaced by Southern firms and can enjoy a longer duration of monopoly power, thereby freeing up less Northern resources from goods production to sustain R&D activity. A tightening of Southern IPP therefore ends up translating into a force against Northern innovation, as was found in Grossman and Helpman (1991c). In this case, a larger R&D subsidy rate is certainly needed to maintain the same long-run innovation rate.

Optimal R&D Subsidy Rate

However, welfare implications are policy makers' primary concern. In this subsection, we shall examine the North's optimal innovation rate, which then dictates the necessary R&D subsidy rate for a specific standard of Southern JPP. This examination is restricted to cases where IPP is such that patent life [delta] always stays above a critical level (as we will derive below) to ensure a North-South regime having [w.sup.N] > [w.sup.S].

Welfare Decomposition 

We now turn to welfare analyses for the North, South, and world economics, respectively. First of all, in utility Equation 23, let us replace [E.sup.S] with Equation 21, [E.sup.N] with Equation 22, and P with Equation 18, respectively. Then on a per worker basis, the steady-state utility functions of the North and South can be rewritten as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

where the initial number of available products n(0) is assumed to be one and so its logarithm is suppressed. Using Equations 14-17 together with symmetric properties (Eqns. 4, 5, and 18) and noting [zeta] = [n.sup.N]/n and 1 - [zeta] = [n.sup.S]/n, we can divide Equation 19 by Equation 20 to derive the steady-state relative price equation,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

which dictates that for [w.sup.N] > [w.sup.S], patent life must be such that

[delta] > [([epsilon]/[epsilon] - 1).sup.[epsilon]] ([L.sup.N] - ag/[gL.sup.S]).

Equation 30 decomposes Northern welfare into three terms. The first term captures the contribution of the consumption rate (measured by 1 - ag/[L.sup.N]) to welfare. The second term measures the contribution of product availability to welfare, which increases with the innovation rate, g. The third term reflects the combined contributions of terms of trade, [p.sup.N]/[p.sup.S], and manufacturing location, [zeta]. If [p.sup.N]/[p.sup.S] rises, the terms of trade move in favor of the North. (15) If [zeta] rises, a larger proportion of goods is produced in the high-cost North and is sold at monopoly prices. This production/consumption inefficiency is welfare reducing worldwide. If [zeta] falls, a larger proportion of goods is instead produced in the low-cost South and is sold at competitive prices. Such relocation of manufacturing sites from the North to South is welfare improving worldwide. Equation 31 decomposes Southern welfare into two terms only. The South does not devote resources to product innovation, so the consumption rate is always fixed at one and its logarithm (equal to zero) is washed out in Equation 31. Like the North, the South gains from increased product availability and relocation of manufacturing sites from the North to South. Yet, the terms-of-trade effect always beggars thy neighbor, causing interest conflicts between the North and South.

First Order Conditions

Now, substitution of Equations 28 and 32 into Equation 30 allows us to express the steadystate Northern lifetime utility function in terms of g, subject to all parameters (See the Appendix for derivations):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

Though the R&D subsidy rate, [theta], does not directly enter Equation 30', it can affect welfare via the innovation rate in terms of Equation 29. The derivation of the optimal R&D subsidy rate therefore requires two steps. The first step involves the derivation of the socially optimal innovation rate, [g.sup.*] to maximize Equation 30'. The second involves substitution of the optimal innovation rate into Equation 29 to solve for the optimal R&D subsidy rate, [[theta].sup.*]. The market mechanism that justifies existence of the optimal innovation rate merit attention, though. Note that g is positively associated with [theta] (see Figure 1). As g is increased, Northern welfare increases with improved product availability and terms-of-trade but decreases with adverse consumption and production location effects. Hence, the optimal R&D subsidy rate, [g.sup.*] must be such that it delivers the optimal innovation rate that balances the beneficial product availability and terms-of-trade effects against the adverse c onsumption and production-location effects on the margin. For this purpose, we derive the first-order condition of maximizing Equation 30' with respect to g:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This condition does not have a closed-form solution for [g.sup.*]. We must resort to numerical approximation. Note that the optimal innovation rate [g.sup.*] must fall into the feasible open interval, (0, [L.sup.N]/a) [subset] [R.sup.+], since Northern research employment, [ag.sup.*], cannot exceed [L.sup.N]. Further, f(*) is continuous in [g.sup.*] [member of] (0, [L.sup.N]/a) and approaches positive infinity as [g.sup.*] [right arrow] [0.sup.+] or negative infinity as [g.sup.*] [right arrow] [L.sup.N]/a. Thus there must exist a real, positive root [g.sup.*] such that f(*) = 0.

Numerical Simulations

We first find a feasible root [g.sup.*] such that f([g.sup.*]) approximates zero. (16) Once this root is found, we begin to compute the optimal R&D subsidy rate ([[theta].sup.*]) with Equation 29, the proportion of unimitated products ([zeta]) with Equation 28, the terms of trade ([p.sup.N]/[p.sup.S]) with Equation 32, Northern welfare ([U.sup.N]) with Equation 30', Southern welfare ([U.sup.S]) with Equation 31, and the combined worldwide welfare ([U.sup.N] + [U.sup.S]). Computation of these variables uses a benchmark set of parameter values ([L.sup.N] = 1, [L.sup.s] = 12, a = 20) (17) while allowing [delta] (patent life), [epsilon] (elasticity of demand for individual innovative goods), and [rho] (constant rate of time preference) to change. Simulation results on the basis of [rho] = 0.025 are reported in Table 1. To analyze these results, we take [delta] = 20 years as the benchmark patent life and summarize the most important findings as follows:

In the panel with [epsilon] = 3, we see that when the South strengthens IPP by increasing patent life from 20 to 25 years, the North's socially optimal R&D subsidy rate decreases from 74.28% to 73.88%; and when the South loosens IPP by decreasing patent life from 20 to 15 years, the North's optimal subsidy rate increases from 74.28% to 74.66%. Hence, in this panel, the Northern optimal R&D subsidy rate is found to inversely correspond to Southern IPP standards. In the next panel with [epsilon] = 5, however, this inverse relation is overturned: When patent life is increased from 20 to 25 years, the Northern optimal R&D subsidy rate increases from 58.49% to 58.90%; when patent life is decreased from 20 to 15 years, the subsidy rate decreases from 58.49% to 57.94%. We therefore obtain the following:

PROPOSITION 2. A tightening of Southern IPP (increase in patent life) does not necessarily allow the self-interested North a smaller socially optimal R&D subsidy rate, depending on whether demand for innovative goods is less elastic. If the demand is elastic enough, a tightening of Southern IPP instead gives rise to a larger optimal subsidy rate.

Economic Reasoning

To justify this proposition, we weigh multiple conflicting forces stemming from changes in IPP and R&D subsidies. First of all, we shall note that as was analyzed earlier, an increase in patent life, [delta], leads to a decrease in [g.sup.*] but an increase in [[zeta].sup.*] (see Figure 1). As a result, the North gains from both improved terms of trade (due to increase in [[zeta].sup.*]) and decreased consumption rate (due to decrease in [g.sup.*]), while also losing to both adverse product availability (due to decrease in [g.sup.*]) and manufacturing sites relocating (due to increase in [[zeta].sup.*]). In general, the net welfare effect is ambiguous. If demand for innovative goods is less elastic (smaller [epsilon] implies a higher degree of product differentiation), then the adverse product availability effect tends to dominate and reduce Northern welfare. (18) In this case (smaller [epsilon]), the North shall counteract tighter IPP by raising the R&D subsidy rate to spur innovation. To the contrary, if de mand for innovative goods is more elastic (larger [epsilon] implies a smaller degree of product differentiation), then new and old goods do not differ as much and the adverse product availability effect caused by tightening Southern IPP is not of great concern. Thus, in the opposing case (larger [epsilon]), the North shall lower the socially optimal R&D subsidy rate to reinforce the South's IPP impact by further slowing down innovation so as to enhance the consumption rate. These justifications accord with our intuition. Innovation that brings new goods too similar to old goods is not as valuable, after all.

The above analysis remains valid when the constant rate of time preference [rho] is kept as low as 0.015 (See Table 2). The only difference is that a smaller rate of time preference requires a larger elasticity of demand for innovative goods, [epsilon], to overturm the substitutability relation between Southern IPP and the Northern optimal R&D subsidy rate. For instance, as [rho] 0.025, this relation gets overturned when [epsilon] rises to 5 (Table 1). But as [rho] = 0.015, such a reversal occurs when [epsilon] rises to 7 (Table 2).

Wage Gaps and Worldwide Welfare

Our numerical results also have interesting implications on North-South wage gaps and the world's combined welfare. As indicated in Tables 1 and 2, a tightening of Southern IPP unambiguously widens the North-South wage gap ([w.sup.N] > [w.sup.S]), whether the Northern optimal R&D subsidy rate positively or negatively corresponds to the duration of patent life. The same effects are found in Grossman and Helpman (1991c) and Grinols and Lin (2002), where R&D activity is not subsidized, however.

Moreover, our simulations (see Table 3) indicate that maximization of the combined North--South welfare requires patent life equal to about 15 years. Namely, the optimal IPP for the global economy is neither as stringent as the North favors nor as lax as the South prefers. This finding is found to be robust to the elasticity of demand for innovative goods and the coefficient of time preference. It builds on the fact that Northern firms cannot fully appropriate R&D's social benefits and Southern firms bear none of R&D's resource costs. Jensen and Thursby (1987) also obtain similar results. The present study, however, shows that these results still hold even when the North optimizes R&D subsidies to various standards of Southern IPR protection. Once again, our results highlight the North--South conflict of interest as in Helpman (1993).

4. Concluding Remarks

The paper has falsified one's intuition that the innovating North's socially optimal R&D subsidy rate should decline if the imitating South tightens IPR enforcement. I have found that the relation between Southern IPR protection and Northern socially optimal R&D subsidies is sensitive to the elasticity of demand for innovative goods, which determines the degree of monopoly power and the surplus appropriability problem that Jones and William (1999) find to be the main force promoting underinvestment in R&D.

It should be noted that like earlier R&D-based endogenous growth models, the dynamic North--South trade model I use has the same "scale effects" property that larger economies grow faster than smaller ones. This is because in the model of the paper intertemporal knowledge spillovers do not diminish as knowledge accumulates over time. As Jones (1995) criticizes, this feature is inconsistent with postwar U.S. empirical experience. However, removal of the scale effects property from the model should not invalidate the role of demand elasticity in governing the relation between Southern IPR protection and Northern socially optimal R&D subsidies. The reason is that the elasticity of demand mainly determines the size of the static monopoly pricing distortion, regardless of whether scale effects exist or not.

However, to address the robustness problem, it would be helpful to extend the present study by allowing regional consumers to purchase the goods that most closely match their preferences. In such a context, Northern innovation activity is aimed at inventing goods differentiated for Northern and Southern consumers. Diwan and Rodrik (1991) deal with regionally differentiated goods in a static setting, whereas Grinols and Lin (2002) address this issue in a dynamic context. Lastly, I am also not sure whether the results of the present study are robust to other features, such as process innovation, product replacement, and oligopoly.

Appendix

Derivation of the Optimal R&D Subsidy Rate

Note that in the steady state

[zeta] = g/(m + g) and [zeta]/1 - [zeta] = g/(m + g)/1 - g/(m + g) = g/m.

So. Equation 30 can be rewritten,

[p.sup.N]/[p.sup.S] = [(g/m).sup.1/[epsilon]] [([L.sup.S]/[L.sup.N] - ag).sup.1/[epsilon]].

Given all other parameters, [zeta] and [p.sup.N]/p are subject to g. Thus, the North's lifetime utility can be expressed as a function of g:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A1)

where g is subject to [theta], m, and the other parameters [rho], [epsilon], a, [L.sup.N], [L.sup.S]. So, [U.sup.N] = [U.sup.N](g) = [U.sup.N][g([theta]), m, [rho], [epsilon], a, [L.sup.N], [L.sup.S]]. To find the optimal R&D subsidy rate, one can first maximize Equation A1 by finding and substituting the utility-maximizing innovation rate into Equation 27 to solve for [theta].

For the utility-maximizing optimal innovation rate, one can solve the first-order condition,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A2)

Factoring out [([g.sup.*]/m).sup.-1/[epsilon]] [([L.sup.S]/[L.sup.N]-[ag.sup.*]).sup.[epsilon]-1/[epsilon]] we can rewrite the last bracketed term of Equation A2 as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

With this result, the first-order condition reduces to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A3)

Multiplying Equation A3 by [rho]([epsilon] - 1) and replacing m with 1/[delta] (i.e., inverse of patent fife), one can get the first-order condition, f([g.sup.*], m, [rho], [epsilon] a, [L.sup.N], [L.sup.S]) = 0, which appears in the text.

[FIGURE 1 OMITTED]

Table 1

Southern IPP and Northern Optimal R&D Subsidy Rate ([rho] = 0.025)

                           Southern IPP       Northern Optimal
Elasticity of Demand  (Years of Patent Life)  R&D Subsidy Rate
[epsilon]                    [delta]              [theta]         g

                                25                 0.7388       0.0239
3                               20                 0.7428       0.0246
                                15                 0.7466       0.0254
                                25                 0.5890       0.0124
5                               20                 0.5849       0.0128
                                15                 0.5794       0.0133


Elasticity of Demand
[epsilon]             [zeta]  [p.sup.N]/[p.sup.S]  [w.sup.N]/[w.sup.S]

                      0.3742        2.3963               1.5975
3                     0.3300        2.2669               1.5113
                      0.2761        2.1037               1.4025
                      0.2365        1.3764               1.1012
5                     0.2040        1.3283               1.0627
                      0.1663        1.2666               1.0133


Elasticity of Demand
[epsilon]             [U.sup.N]  [U.sup.S]  [U.sup.N] + [U.sup.S]

                        20.66      11.74            32.40
3                       19.12      13.52            32.64
                        16.87      15.53            32.40
                         4.48       3.09             7.57
5                        3.15       3.64             6.79
                         1.34       4.25             5.59

Common parameter values used for each panel are: [L.sup.S] = 12,
[L.sup.N] = 1, and a = 20.
Table 2

Southern IPP and Northern Optimal R&D Subsidy Rate ([rho] = 0.015)

                    Southern IPP    Northern Optimal
Elasticity of     (Years of Patent    R&D Subsidy
Demand [epsilon]   Life) [delta]      Rate [theta]      g     [zeta]

                         25              0.8371       0.0314  0.4400
3                        20              0.8392       0.0318  0.3890
                         15              0.8413       0.0323  0.3263
                         25              0.6842       0.0167  0.2943
5                        20              0.6872       0.0172  0.2556
                         15              0.6902       0.0177  0.2098
                         25              0.5753       0.0103  0.2046
7                        20              0.5739       0.0106  0.1748
                         18 (a)          0.5732       0.0107  0.1618


Elasticity of
Demand [epsilon]  [p.sup.N]/[p.sup.S]  [w.sup.N]/[w.sup.S]  [U.sup.N]

                        2.9385               1.9590           59.26
3                       2.7597               1.8398           57.18
                        2.5414               1.6942           53.98
                        1.4968               1.1974           13.88
5                       1.4438               1.1550           11.87
                        1.3760               1.1008            9.08
                        1.2140               1.0406            3.51
7                       1.1821               1.0132            1.82
                        1.1670               1.0003            1.01


Elasticity of
Demand [epsilon]  [U.sup.S]  [U.sup.N] + [U.sup.S]

                    53.41           112.67
3                   56.98           114.16
                    61.01           114.99
                    14.06            27.94
5                   15.42            27.29
                    16.94            26.02
                     5.94             9.45
7                    6.54             8.36
                     6.80             7.81

Common parameter values used for each panel are the same as in Table 1.

(a) Reminds that as patent life is approximately less than 18 years,
North-South wages are equalized ([w.sup.N]/[w.sup.s] = 1) by laxer
Southern IPP.
Table 3

Maximization of Worldwide Welfare ([rho] = 0.015)

                    Southern IPP    Northern Optimal
Elasticity of     (Years of Patent    R&D Subsidy
Demand [epsilon]   Life) [delta]      Rate [theta]      g     [zeta]

                         30              0.8352       0.0311  0.4824
                         25              0.8371       0.0314  0.4400
3                        20              0.8392       0.0318  0.3890
                         15 (a)          0.8413       0.0323  0.3263
                         10              0.8435       0.0328  0.2472
                          5              0.8458       0.0335  0.1435


Elasticity of
Demand [epsilon]  [p.sup.N]/[p.sup.S]  [w.sup.N]/[w.sup.S]  [U.sup.N]

                        3.0911               2.0607           60.28
                        2.9385               1.9590           59.26
3                       2.7597               1.8398           57.18
                        2.5414               1.6942           53.98
                        2.2560               1.5040           48.54
                        1.8265               1.2176           37.16


Elasticity of
Demand [epsilon]  [U.sup.S]  [U.sup.N] + [U.sup.S]

                    50.19           110.87
                    53.41           112.67
3                   56.98           114.17
                    61.01           114.99 (a)
                    65.60           114.14
                    70.92           108.08

Common parameter values used for each panel are the same as in Table 1.

(a) North-South welfare maximization occurs at approximately [delta] =
15 years.

Received November 2000; accepted March 2002.

(1.) See Cohen and Levin (1989), Griliches (1992), and Nadiri (1993) for review of this literature.

(2.) Note that in the productivity literature based on the neoclassical growth theory, R&D is merely treated as another form of capital investment and several R&D-related distortions are simply ignored. These distortions are well known as monopoly pricing, interteinporal knowledge spillovers, congestion externalities, and creative destruction and have been partially or fully fotmalized in R&D-based endogenous growth models, such as Romer (1990), Grossman and Helpman (1991a, b, c), Segerstrom (1991, 1998), Jones (1995), Young (1998), and Jones and Williams (1999).

(3.) First-generation R&D-based endogenous growth models, including Romer (1990), Segerstrom (1991), and Grossman and Helpman (1991a, b, c), feature the implausible "scale effects" property that larger economies grow faster than smaller ones. Second-generation models, including Jones (1995, 1997), Aghion and Howitt (1998), Segerstrom (1998), Young (1998), and Jones and Williams (1999), however, do not have this property.

(4.) See WTO Focus, World Trade Organization, July 1988.

(5.) Grinols and Lin (2002) show that Helpman's conclusion that it is never in the South's interest to strengthen IPP is not robust if innovative products are differentiated for regional markets.

(6.) Another strand of research focuses on R&D subsidy policy, innovation, and economic welfare, yet without taking into account the South's imperfect IPP. Romer (1990) and Young (1998) derive the socially optimal R&D subsidy rate in R&D-based endogenous growth models; the former has scale effects but the latter does not. Segerstrom (1991), Grossman and Helpman (1991a,b), Jones (1997), and Lin and Russo (1999) examine the relation between an increase in the R&D subsidy rate and the rates of innovation and economic growth in steady state.

(7.) In Jones (1997), where population growth is endogenous, decreased R&D subsidies can lead to faster population growth and thus a higher long-run innovation rate. This stands in stark contrast to most R&D-based growth models.

(8.) Krugman's pioneering North--South model, where both the rate of product innovation and the rate of product imitation are exogenously determined, has been extended for different research focuses in several ways. Dollar (1986) endogenizes the imitation rate by regional cost differences; Jensen and Thurshy (1987) and Helpman (1993) endogenize the innovation rate by research inputs; and yet, Grossman and Hclpman (1991c) endogenously determine both innovation and imitation rates. In this paper I shall endogenize the rate of innovation, while parameterizing the rate of imitation by standards of IPR protection, as in Jensen and Thurshy (1987) and Heiproan (1993).

(9.) This is in stark contrast to Jensen and Thurshy (1987) and Jones (1995, 1997), where the rate of innovation is concave in research inputs and thereby converges to zero in steady state.

(10.) According to Mansfield, Schwartz, and Wagner (1981), copying costs average only 65% of the original innovation.

(11.) These risks are idiosyncratic because they can be diversified away via portfolio holdings.

(12.) This condition is the same as Grossman and Helpman (1991c, Ch. 11) and Helpman (1993).

(13.) Equation 21 is straightforward for [w.sup.S][L.sup.S] = [GDP.sup.S] = [p.sup.S][L.sup.S] due to Equation 4. To let [n.sup.N][[pi].sup.N] + [w.sup.N][L.sup.N] = [GDP.sup.N] = [E.sup.N] + [I.sup.N] reduce to Equation 22, we need to note [n.sup.N][[pi].sup.N] = [n.sup.N]([p.sup.N][x.sup.N]/[epsilon]), [n.sup.N][x.sup.N] = [L.sup.N] - ag, [I.sup.N] = [w.sup.N]ag, and [p.sup.N] = ([epsilon]/([epsilon] - 1))[w.sup.N] due to Equation 5.

(14.) Note that measures the quantity of resources (i.e., labor) devoted to R&D and [L.sup.N] - ag measures the quantity of resources devoted to production of consumables. Thus, 1 - ag/[L.sup.N][=([p.sup.N] ([L.sup.N] - ag)/[E.sup.N] =[=([p.sup.N]([L.sup.N] - ag)/[p.sup.N][L.sup.N]) measures the Northern consumption rate.

(15.) As Equation 32 indicates, when a larger share of goods (increase in [zeta]) is produced in the North or when innovation speeds up (increase in g), the tenns of trade [p.sup.N]/[p.sup.S] moves in favor of the North because either situation can lead to stronger input demand for Northern labor, high Northern wages, and higher relative prices of Northern goods.

(16.) The allowance of error is set within [+ or -] [10.sup.-7] For computation, I wrote a Fortran program including an IMSL library subroutine, DZREAL, which uses Muller's method (Gerald and Wheatley 1999, pp. 52-5). The Fortran program is available from the author upon request.

(17.) The size of the South's labor force is set at 12 times as much as that of the North's. According to Labor Force Statistics, 1977-1999, OECD 2000 Edition, the population ratio of OECD to non-OECD is one to six. But considering the fact that most patents have been issued to a few countries centered in the United States, Japan, and Germany, we let the population ratio of North to South he one to twelve.

(18.) This is because both the second and third terms on the right-hand side of Equation 30 are divided by ([epsilon] - 1) but the first 15 not.

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Hwan C. Lin *

* Department of Economies, Belk College of Business Administration, University of North Carolina at Charlotte, 9201 University City Boulevard, Charlotte, NC 28223, USA; E-mail [e-mail].

I thank two anonymous referees for constructive comments and suggestions and acknowledge gratefully that the paper was partially funded by the BarclaysAmerican Summer Research Award of 1997.