To "B" or Not to "B"
A Welfare Analysis of Breaking Up Monopolies in an Endogenous Growth Model
Author: Danyang Xie1
  • 1 0000000404811396https://isni.org/isni/0000000404811396International Monetary Fund

Contributor Notes

Author’s E-Mail Address: dxie@imf.org

This paper studies the welfare consequences of a government regulation that forces a patented equipment to be supplied by a number of independent producers. On the one hand, such a regulation hurts the value of a patent and therefore reduces activities in the R&D sector. On the other hand, the enhanced competition for the equipment improves efficiency in the manufacturing sector. Should monopolies protected by intellectual property rights be broken up? The answer is “no” in a Romer-type growth model, but there is sufficient reason to believe that the answer could be “yes” in a model advocated by Jones (1995).

Abstract

This paper studies the welfare consequences of a government regulation that forces a patented equipment to be supplied by a number of independent producers. On the one hand, such a regulation hurts the value of a patent and therefore reduces activities in the R&D sector. On the other hand, the enhanced competition for the equipment improves efficiency in the manufacturing sector. Should monopolies protected by intellectual property rights be broken up? The answer is “no” in a Romer-type growth model, but there is sufficient reason to believe that the answer could be “yes” in a model advocated by Jones (1995).

I. Introduction

The “B” in the title stands for “Break up.” “Breaking up” monopolies protected by a patent involves a clear trade-off. On the one hand, a “break up” enhances competition and reduces deadweight loss. On the other hand, a “break up” reduces the value of a patent and hence adversely affects the incentive of conducting R&D. This trade-off is essentially the conflict between the patent law and the antitrust law. A formal model would help the policy makers to strike a proper balance between the two in order to maximize the social welfare. “To Break up or Not To Break up,” that indeed is the question.

The currently heated debate on the U.S. Government’s case against Microsoft focuses on the anticompetitive act of Microsoft to extend its legitimate monopoly of operating system to other product markets. This paper deals with a more general question: should there be a government regulation that forces any patented product to be supplied by n independent producers? We conduct a welfare analysis of such a regulation in an endogenous growth model.

The main results are obtained with a R&D-based Romer model (1990). Clearly, if we are only concerned with the long-term rate of growth, a “break up” is never an attractive option: a “break up” hurts the value of a patent and therefore reduces the activities in the R&D sector, leading to a lower long-term growth rate. What we should be concerned with, however, is welfare. A “break up” enhances competition among the suppliers of specialized equipment so that the manufacturing sector will benefit from a lower input price.

What are the relevant factors for a trade-off? Will a “break up” ever have a chance to raise the welfare of a representative agent?

Endogenous growth models have been widely used to evaluate government fiscal policies (see, for example, Barro 1990, Rebelo 1991, Barro and Sala-i-Martin 1992, Glomm and Ravikumar 1994, Stokey and Rebelo 1994 and Mino 1996). Most of these analyses focus on the long-term rate of growth. Welfare analysis is usually limited to models of AK type. Devarajan, Xie and Zou (1998) investigate the welfare implications of government intervention when a lump-sum tax is not available. To obtain analytical results, they use logarithmic utility functions, the Cobb-Douglas production function and the 100 percent rate of capital depreciation.

The endogenous growth models are the natural vehicles to study government policies besides those dealing with fiscal issues. A number of papers examine the issue of optimal patent length (see, for example, Davidson and Segerstrom 1992, Horowitz and Lai 1996, and Futagami, Mino and Ohkusa 1999). This paper represents a first attempt at examining a government regulation that breaks up monopolies of patented equipment. It turns out that in Romer (1990), a “break up” can be very costly if the long-lived representative agent’s intertemporal rate of substitution is high. In a numerical example, a regulation that breaks up a monopoly into two independent firms lowers welfare by 63 percent.

The literature on the equity premium, however, indicates that a low intertemporal rate of substitution receives more empirical support. When the representative agent’s intertemporal rate of substitution is low, a “break up” of a monopoly into two independent suppliers reduces the welfare by only 0.3 percent. Hence, “To Break up or Not To Break up,” is essentially a draw.

More importantly, if the economy works as in a model advocated by Jones (1995a), a “break up” may actually be the better choice. In Jones (1995a), there is no scale effect in R&D production. Hence, a “break up” affects only the short-term growth rate of R&D but it has no long-term growth effect. This considerably lowers the welfare cost of a “break up.” We speculate that in this model, an optimal n > 1 exists.

The computing technique we used in this paper is called “reverse shooting” (Judd 1998, pp. 355-57). In a Romer-type model, reverse shooting works well because the equilibrium path can be described by a system of three differential equations with one initial condition and two terminal conditions. In a Jones-type model, reverse shooting is problematic because the equilibrium path is described by a system of four differential equations with two initial conditions and two terminal conditions. Although Judd (1998) addresses briefly the issue of multidimensional reverse shooting (pp. 360-61), the computation is in fact much more difficult than it appears.2

The rest of the paper is organized as follows. In Section 2, an extended version of the Romer model is presented, allowing for imperfect substitutability among different types of equipment. Intuitively, if different types of equipment are highly substitutable, the welfare cost of a “break up” would be insignificant. This is confirmed in our comparative dynamics analysis in Section 3. Section 4 presents the conclusions.

II. An Extended Romer Model

The model is an extended version of Romer (1990). It deviates from his original work in two aspects. First, in the manufacturing sector, the production function is modified so that different types of equipment are no longer independent; they are assumed to be imperfect substitutes. Second, the government makes a regulation that requires each type of equipment e supplied by n independent producers.

A. The Manufacturing Sector

The production function in the manufacturing sector is given by

Y=HYαLβ(0Axaγ/ξda)ξ,(1)

where Hy is the amount of human capital (managers and engineers) employed in the manufacturing sector, L is the unskilled labor, which is in a fixed supply, xa is the amount of type a equipment and A is the number of different types of equipment. We assume that a + β + γ = 1 so that the production function exhibits constant returns to scale in (HY,L,{xa}0A). When ξ = γ, the production function is neoclassical in the sense that all equipment are treated as perfect substitutes. When ξ = 1, it is reduced to Romer model. When ξ is in interval (γ, 1), different types of equipment are imperfect substitutes.

A similar production function is used in Benhabib, Perli and Xie (1994). There, the attention was on the case when ξ > 1. They show that when different types of equipment display high enough complementarity, the modified Romer model exhibits indeterminacy.

B. The R&D Sector

The production of new designs in the R&D sector is given by

A˙=δHAA,(2)

where HA is the amount of human capital employed in the R&D sector and δ is a productivity parameter.

C. The Preferences

Assume that there is a long-lived representative individual with preferences, as shown by

max0[C1σ11σ]eptdt,(3)

where C is the consumption of the single final good, σ is the inverse of the intertemporal rate of substitution and ρ is the rate of time preference.

With this CES utility function, we know that the optimality consumption path satisfies the following condition:

C˙C=rρσ,(4)

where r is the real interest rate and could be time varying.

D. The Market Structure

The government makes a regulation that requires each type of equipment to be supplied by n independent producers. We assume that these n producers play a Cournot-Nash oligopoly game. An increase in n certainly lowers the value of a patent. Kremer (1998) has advocated a patent buyout by government, which eliminates the monopoly price distortions while increasing incentives for original research. The implementation of a buyout however is problematic because the price of a patent is subject to collusion. Furthermore, large-scale patent buyouts would be unthinkable if the government has to use distortionary taxes to finance these purchases. In this paper, the value of a patent is market driven and the only decision that the government makes is to set n, the number of licenses. There are an infinite number of potential producers, each of which (including the inventor, if he/she wishes) will submit a bid for one of the n licenses to become the supplier. The highest n bidders win the licenses and the value of the patent is given by the sum of the n winning bids.

To see the exact relationship between the number of licenses and the value of the patent, note that the demand curve for type a equipment is given by

pa=γHYαLβ(0Axaγ/ξda)ξ1xaγ/ξ1,(5)

where pa is the rental price of type a equipment.

Given the demand curve (5), supplier i of type a equipment, taking the quantity supplied by its n – 1 competitors (-iXa) as given, chooses its optimal quantity (xai) by solving the following problem:

maxπai=pa(xai,ixa)xairxai

where we follow Romer (1990) in assuming that one unit of final goods can be converted into one unit of type a equipment as long as the producer holds the patent. To see that the marginal rental cost of the producer is the real interest rate r, note that a piece of equipment is assumed to be perfectly durable. In this case, if we use letter m to denote the marginal rental cost of a supplier, the one-to-one conversion between a unit of final consumption goods and the equipment requires

1=tm(τ)etτr(s)dsdτ

for any time t. Differentiating the above identity with respect to t, we obtain that m(t) = r(t).

The profit maximization problem yields a first-order condition

r=(γ/ξ1)paxaxai+pa.(6)

Note that all these suppliers are identical, namely at equilibrium,

xai=1nxa for all i.

Hence, the equilibrium rental price of the Cournot-Nash oligopoly game can be obtained from (6):

pa=r1(1γ/ξ)/n(7)

which reduces to pa r/γ if ξ = 1 and n = 1, the same as in Romer (1990).

The total profit that can be made by these suppliers of type a equipment is

πa=i=1nπai=[(1γ/ξ)/n1(1γ/ξ)/n]rxa.

Thus the value of a patent for a new design, a, invented at time t is equal to the present value of the entire profit stream in the future:

pa(t)=tπa(τ)etτr(s)dsdτ.

Because of the symmetry in all the new equipment invented at time t, we can replace the lower-case letter a by the upper-case letter A in the equations above; and the subscript is sometimes omitted all together when it will not cause any confusion. In particular, we have

PA(t)=tπ(τ)etτr(s)dsdτ=t[(1γ/ξ)/n1(1γ/ξ)/n]rxetτr(s)dsdτ,(8)

where x can be read from equations (5) and (7):

r1(1γ/ξ)/n=γHYαLβAξ1xγ1.(9)

E. Human Capital Allocation

We assume that total stock of human capital is fixed, and is denoted by H. In equilibrium, the allocation of human capital is such that the marginal revenue product of human capital is equalized in the two sectors. That is,

αHYα1LβAξxγ=PAδA,(10)

where Hy+ HA = H.

F. Transitional Dynamics

Understanding transitional dynamics is essential in our evaluation of “To Break up or Not To Break up.” If we are only concerned with long-term growth, “Not To Break up” (n = 1) is clearly the best choice. From the welfare point of view, however, it is questionable whether n = 1 continues to be optimal because an increase in n benefits the manufacturing sector.

The question therefore is the following: Can the benefit to the manufacturing sector be large enough to compensate for the reduced rate of growth in R&D?

To calculate the level of welfare, we need to study the transitional dynamics. The traditional method of linearization around the steady state does not work well here because, when n changes, the corresponding steady state also changes. Our starting point can only be close to one but not all the steady states. The errors from linear approximation may contaminate the comparisons among the welfare levels attainable under different n.

The reverse shooting computing technique, described in Judd (1998), works well when the system of differential equations has only one initial condition. To proceed, we follow Mulligan and Sala-i-Martin (1991) and define state-like and control-like variables as follows:

z=AξγKγ1a state-like variable,(11)
q=C/Ka control-like variable,(12)

and HY is a control variable, where K=0Axada=Ax is the total stock of physical capital.

Once we obtain the paths of these three variables, we can easily solve for other variables, such as, A(t), K(t), x(t), C(t) and r(t). To obtain the differential equations governing the three variables, note that from equation (8),

P˙APA=rπPA=r(1γ/ξ)/γnαδHYuse equation (9)=(1(1γ/ξ)/n)γHγαLβz(1γ/ξ)γnαδHY.

Rewrite equation (10) as

αHYα1LβAξγ1Kγ=PAδ.(13)

Computing growth rates from both sides of the above equation, we obtain

(α1)H˙YHY+(ξγ1)A˙A+γK˙K=P˙APA.

Hence,

H˙YHY=11α[(ξγ1)A˙A+γK˙KP˙APA]=11α[(ξγ1)δ(HHY)γq+(1γ/ξ)γnHYαLβz+(1γ/ξ)γnαδHY].(14)

Obtaining the dynamic equations for z and q is straightforward:

z˙z=(ξγ)A˙A+(γ1)K˙K=(ξγ)δ(HHY)+(γ1)(HYαLβzq)(15)

and

q˙q=C˙CK˙K=(1(1γ/ξ)/n)γHYαLβzρσ(HYαLβzq).(16)

To use the reverse shooting method, we need to know the steady-state values of q, z, and HY, which are denoted q*, z* and HY*. If z(0) = z*, the economy is on a balanced growth path.

G. Balanced Growth Path

To find the balanced growth path, set H˙Y, z˙ and q˙ to zero, We find:

HY*=ρ+1ξ+σ(ξγ)1γδH[(1γ/ξ)γnα+1ξ+σ(ξγ)1γ]δ(17)
z*=ρ+σ(ξγ)1γδ(HHY*)(1(1γ/ξ)/n)γHY*αLβ(18)
q*=HY*αLβz*(ξγ)1γδ(HHY*).(19)

The above formulae are only valid when HY* is an interior solution. Since the technology in the R&D sector is linear in HY, a corner solution such as HY*=H may arise. For example, when n is too large, the profit generated by a patent is so tiny that no R&D is done. In the following discussion on the comparative dynamics, we implicitly assume

ρ<(1γ/ξ)γnαδH

to ensure an interior solution.

To avoid repeating what is discussed in Romer (1990), we will focus on the effect of n and ξ on the long-term rate of growth of output, which can be expressed as

gY=gC=gK=[ξγ1γ]δ(HHY*).(20)

An increase in n reduces the value of a patent and hence reduces HA*. Thus HY* increases with n, which is confirmed by our formula (17). The rate of growth of output is thus lower. To see the effect of ξ on gY, let us rewrite the expression (20) as,

gY=[(1γ/ξ)γnαδHρ][(1γ)γnαξ+1ξξγ+σ]

which states that as ξ increases, the denominator gets smaller whereas the numerator gets bigger, hence gY increases. This conforms with our intuition that as different types of goods become less substitutable, the value of R&D activity becomes more valuable and the long-term economic growth is greater.

The real interest rate on the balanced growth path can be calculated as follows:

r*=(1(1γ/ξ)/n)γHY*αLβz*.(21)

Suppose, for simplicity, that A(0) = 1. If the economy starts with z(0) = z*, we can compute the level of welfare easily, because, in this case,

C(0)=q*K(0)=q*[z*/A(0)ξγ]1/(γ1)=q*[z*]1/(γ1).

Thus, welfare can be directly computed as

V*(n)=11σ[(q*[z*]1/(γ1))1σρ(1σ)(r*ρ)/σ1ρ],(22)

which is used in the comparative dynamics analysis below.

III. Comparative Dynamics Analysis

We will show later that the result is sensitive to the value of σ. To begin with, we let σ = 0.8. The values for other parameters in the model are either standard or set by normalization: α = 1/3, β = 1/3, γ = 1/3, ξ = 1, δ = 0.001, H = 100, L = 1, ρ= 0.01, A(0) = 1 and K(0) is chosen such that z(0) = z*|n=5. We would like to find out how welfare depends on n. When n is large, we will encounter comer solutions that require delicate treatment in welfare calculation. Thus we restrict ourself to n = 1, 2, …, 5.

Given that we let z(0) = z* |n=5, the welfare for n = 5 can be calculated directly from (22). V(n) for n = 1, 2, 3, 4 are calculated using the reverse shooting method. This choice of z(0) is to make sure that a corner solution does not arise in calculating V(n) for n = 1, 2, …5.

Table 1 shows that as we break up the monopoly to 2, 3, 4, or 5 independent companies through regulations, welfare drops by 63, 73, 77, and 78 percent. The drop in welfare is very substantial.

Table 1:

σ = 0.8, ξ = 1, H = 100

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If we change the value of ξ to ξ = 0.8, in other words, when the different types of equipment are substitutes, how much will the drop in welfare as resulting from the break up be moderated?

Table 2 gives the answer. Welfare drops by 33, 45, 48, and 51 percent, respectively. The drop is still substantial, but quite dramatically moderated compared with the case of ξ = 1. Of course, as ξ approaches γ, the R&D sector becomes less important because the incentive for R&D is low in any case, and the drop in welfare will become insignificant.

Table 2:

σ = 0.8, ξ = 0.8, H = 100

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The discussion above seems to hint that as long as different types of equipment are not close substitutes, the benefit from increased efficiency in the manufacturing sector can hardly compensate for the harm done to long-term growth.

It turns out the this conclusion is pre-mature. The above conclusion largely depends on the low value of σ, i.e. the high intertemporal rate of substitution. Empirical studies on equity premium indicate that a large σ (low intertemporal rate of substitution) is more likely the case. Once I raise the value of σ to 2, welfare drop is almost zero as Table 3 and 4 illustrate.

Table 3:

σ = 2, ξ = 1, H = 100

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Table 4:

σ = 2, ξ = 0.8, H = 100

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Table 3 shows the case of σ = 2 and ξ = 1. Welfare drops by 0.3 percent only when a monopoly is broken into 2 independent firms. Even when it is broken into 5 firms, the loss is only 1.2 percent. Table 4 shows the case of σ= 2 and ξ = 0.8. Here, the loss is only 1.0 percent when a monopoly is broken up into 5 firms.

Why a large σ tends to raise the static efficiency gains in the manufacturing sector from “breaking up,” and to lower the damage on the long-run growth? A large σ means a low elasticity of intertemporal rate of substitution. Other things being equal, an agent would prefer to have a more smoothed consumption pattern instead of a steeply rising one. As a result, the amount of human capital devoted in R&D is lower and the growth rate is lower. “Breaking up”, indeed, acts to raise the current production and to reduce the long-term rate of growth. In the extreme case when σ is close to infinity, growth rate will be close to zero with or without break up. Hence, the static efficiency gains from breaking up should be more than sufficient to overcome the loss of growth. For reasonable parameter values of σ, however, the static efficiency gains are still not enough to compensate for the loss of growth. Table 4, however, shows that the welfare cost is insignificant.

Table 5 puts the comparison in a clearer perspective. In Table 5, the total stock of human capital is increased to H = 500. This magnifies the growth differences. Indeed, in the case of n = 1, the long-term growth rate of output is 8.64 percent, much higher than the corresponding rate (1.86 percent) in the case of n = 5. This huge loss in long-term growth is however almost fully compensated by the static efficiency gains in the manufacturing sector; welfare when n = 5 is only 0.5 percent lower, compared with the welfare in the case of n = 1.

Table 5:

σ = 2, ξ = 0.8, H = 500

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This allows us to speculate that the tiny disadvantage in “breaking up” in the case of a realistically low intertemporal rate of substitution may be overcome in Jones’ R&D-based growth model (1995a). Jones (1995b) uses time series evidence to discredit the linearity assumption in the R&D technology in Romer (1990). He advocates that the technology should be modified as in Jones (1995a), namely,

A˙=δHAAϕ,

where he imposes the condition that 0 < φ < 1. An immediate implication of this type of technology is that the long-term rate of growth is independent of HA and in fact it only depends on the rate of population growth and the parameter value φ. Hence, “breaking up” will have no impact on the long-term rate of growth but it will still keep the static efficiency gains intact. Intuitively, since the long-term rate of growth will be the same, the static efficiency gains that occur every period should overcome the loss in the short-term rate of growth. There should exist an optimal degree of “breaking up” in such a model.

IV. Conclusion

In Romer’s endogenous growth model (1990), whether there is a serious welfare loss from a regulation that breaks up a monopoly into independent companies depends critically on the intertemporal rate of substitution. With a low intertemporal rate of substitution, the representative agent dislikes consumption variability across time, and the harm done to long-term growth is almost fully compensated by the benefit received in the manufacturing sector. Table 5 is particularly telling.

This exercise hints that in a growth model such as Jones (1995a), a regulation that breaks up every monopoly might even enhance welfare if the intertemporal rate of substitution is low and the substitutability of different types of machines is sufficiently high. The reason is that in Jones’ model, there is no scale effect in R&D production. The long-run growth is determined by the population growth rate and a fixed parameter in the R&D sector. Thus, a break-up will have no impact on the long-run rate of growth.

Indeed, “To Break up or Not To Break up” depends on which model we believe in, the endogenous growth model by Romer (1990), or a semi-endogenous one by Jones (1995a). It is important to identify a model that better describes the real world because policy recommendations drawn in an endogenous growth model and those drawn in a semi-endogenous one are likely to be just the opposite.

If we interpret our analysis as one for a particular industry, the conclusion is as follows. The monopoly for specialized equipment in this industry should be broken up into a larger number of independent suppliers if the intertemporal rate of substitution for the final consumption good in the industry is lower; the substitutability of specialized equipment is higher; and the parameter φ is lower.

Appendix I

Transitional Dynamics in Jones’ R&D-based Model (1995)

Jones (1995b) criticizes the scale effect presented in R&D-based Romer-type models. He reports times series evidence, showing that the long-term implications of these models are strongly rejected by data. He proposes to replace the technology of the R&D sector by

A˙=δHAAϕ,

where he imposes the condition that 0 < φ < 1.

If human capital stock were fixed as in Romer (1990), then the above change of R&D technology would make long-term growth impossible due to the restriction that φ< 1. To allow for long-run growth, Jones assumes that human capital stock grows at a constant rate

H˙=gH.

It is then easy to see that in the long run,

A˙A=g1ϕ,

which will also determine the rate of growth of capital and output.

The rest of the model is the same as in our extended Romer model. Before going into the details of the analysis, it is worthwhile to work on the intuition. Since a forced break up, albeit reducing the value of a patent, has no impact on the long-term rate of growth, such a policy is more likely to raise welfare as a result of improved efficiency in the manufacturing sector and an optimal degree of breaking up should exist.

Again, following Mulligan and Sala-i-Martin (1991), we redefine our variables as follows:

hY=HY/H
z=Aξγ+α(1ϕ)Kγ1
q=CK

and

a=A1ϕ/H.

We know that these variables will all converge to their steady-state values in the long run.

The equilibrium path is characterized by following differential equations in a, z, hY and q. The first two variables arc state-like and the latter two are control-like.

a˙a=(1ϕ)δ(1hY)ag
z˙z=(ξγ+α(1ϕ))δ(1hY)a+(γ1)((hYa)αzq)
hYhY=11α[γ((hYa)αzq)+(ξϕγ)δ(1hY)a(1α)g+δ[(1γξ)n]γα(hYa)[1(1γξ)n]γ(hYa)αz]=11α[(ξϕγ)δ(1hY)a+δ[(1γξ)]γnα(hYa)+(1γξ)nγ(hYa)αzγq]q˙q=γ(hYa)αz[1(1γξ)n]ρσ((hYa)zq)=[[1(1γξ)n]γσ](hYa)αzσ+qρσ.

Let us now calculate the steady-state values of these variables. From a˙/a=0, we find

δ(1hY*)a*=g1ϕ.(23)

From z˙/z=0and q˙/q=0, we obtain

(ξγ+α(1ϕ))g1ϕ=(1γ)((hY*a*)αz*q*)r*=ρ+σξγ+α(1ϕ)1γg1ϕ(hY*a*)αz*=q*+(ξγ+α(1ϕ))1γg1ϕ.(24)

From h˙Y/h˙Y=0, we know that

δγ[(1γ/ξ)]nαhY*a*+(1γ/ξ)nγ[q*+(ξγ+α(1ϕ))1γg1ϕ]=γq*[ξγ+α(1ϕ)1]g1ϕ.(25)

Since

r*=γ[1(1γ/ξ)/n](hY*a*)αz*

can be used to solve for

(hY*a*)α=1[1(1γ/ξ)/n]γr*,

substitute this back into equation (24), and we can compute

q*=1[1(1γ/ξ)/n]γr*(ξγ+α(1ϕ))1γg1ϕ.

Then from equation (25), we solve for

hY*a*=γq*(1γ/ξ)n1[1(1γ/ξ)/n]γr*[ξγ+α(1ϕ)1]g1ϕδγ(1γ/ξ)nα.(26)

which can then be used to solve for

z*=1[1(1γ/ξ)/n]γr*(hy*a*)α.(27)

Finally, a* and hY* can be solved from (23) and (26).

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1

I would like to thank Professor Kazuo Nishimura for organizing a Kyoto workshop on growth and indeterminacy during which the idea of this paper was conceived. I am grateful to Saleh M. Nsouli, Sergio Rebelo, Guofii Tan, Reza Vaez-Zadeh, Eden Yu, and seminar participants at CUHK for helpful comments. Financial support from HKUST DAG and a grant from Research Grants Council of the HKSAR (Project No HKUST6004/99H) are gratefully acknowledged.

2

Detailed differential equations for the Jones model are given in the appendix.