Physical and human capital accumulation, R&D and economic growth.

I. Introduction

The fast-growing literature on the new growth theory can be broadly divided into two categories (capital-based and idea-based models) according to the underlying sources of growth, as discussed in Romer [9]. Capital-based models base growth on endogenous accumulation of physical or human capital and correspondingly emphasize investment in physical or human capital (e.g., Lucas [5], Rebelo [6] and Romer [7]). Idea-based models take endogenous technological changes resulting from R&D as the source of growth by treating the product of the R&D as a commodity (e.g., Aghion and Howitt [1], Grossman and Helpman [2; 3] and Segerstrom, Anant, and Dinopoulos [10]). The former focuses on the externalities of capital accumulation leaving aside the intentional R&D activity, which is the focus of the latter, while the latter assumes fixed factor endowments. Both categories capture one important aspect of economic growth and are able to generate sustained growth without relying on any exogenous factor growth.

However, physical and human capital accumulation and technological changes driven by innovative R&D are two integrated elements in driving economic growth in a real world economy. On the one hand, physical and human capital are two essential factors in R&D activities and in applying the new technologies resulting from successful R&D to production. On the other hand, the new technologies open up new economic opportunities for investment in physical and human capital to take place. If these two can be integrated into one single framework,(1) then we will be able to see the interaction between these two types of forces in pushing economic growth and therefore bring the theory a step closer to the reality.

The objective of this paper is to develop a synthesized endogenous growth model, in which both factor accumulation and technology change are endogenously determined and growth is driven by the interaction between these two types of economic forces, by integrating the two distinct categories of growth models mentioned above. Our model is a vertical product differentiation model. In the model economy, there are four types of activities - final good production, intermediate good production, physical and human capital accumulation and innovative R&D. Quality improvement of intermediate goods through innovative R&D is the source of growth. Successful innovations have two types of "creative destruction" effects. On the one hand, they discover new intermediate goods but destroy the old counterparts. On the other hand, they create new knowledge but make the existing human capital less effective. We assume that innovative R&D is the most human capital intensive activity. In the model specification, we make an extreme assumption that innovative R&D uses only human capital while intermediate good production requires only unskilled labor. Human capital accumulation is necessary because each successful innovation reduces the effectiveness of the existing human capital. So is physical capital investment because we assume that final good production uses capital and intermediate goods as inputs, the quality improvement of intermediate goods raises the productivity of final good production, which provides new opportunities for physical capital investment.

We analyze both the free market equilibrium and the social planner's problem. We find that the monopolist's market power (measured inversely by [Alpha]) plays a critically important role. It is the market power that determines whether the laissez faire equilibrium growth is too fast or too slow compared with the socially optimal growth.(2) It is also the market power that determines whether a tax or subsidy scheme is needed to support the optimal growth. Propositions 1-4 give the findings of this paper.

The rest of this paper is organized as follows: The next section describes the environment and sets up the model. Section III describes the laissez faire equilibrium conditions and the condition of equilibrium existence. We also perform comparative-static analysis in this section. The welfare property of the laissez faire equilibrium and policy implications are discussed in section IV. Finally, some concluding remarks are given in the last section.

II. The Model

The basic framework is due to Aghion and Howitt [1]. We consider a closed economy populated with a continuum of identical infinitely lived households with measure 1. Each household is endowed with N unit flow of time which is inelastically supplied to the production sectors and devoted to human capital accumulation activities.

Preferences

We assume that the household's preferences are given by

[integral of] [e.sup.-[Rho][Tau]][([C.sup.1 - [Sigma]] - 1)/(1 - [Sigma])]d[Tau] between limits [infinity] and 0, (1)

where C is consumption; [Rho] the constant rate of time preference; [Sigma] the relative risk aversion coefficient and [Tau] represents time. For simplicity, the time subscripts are omitted whenever no confusion can arise and the final good will be used as the numeraire. Furthermore, since we will deal only with the stationary equilibria, we will implicitly use the stationary conditions in the derivations of relevant equations. Given the household's total discounted lifetime income M and the interest rate r (which will be endogenously determined and will be constant in a stationary equilibrium), the household's lifetime budget constraint is

[integral of] [e.sup.-r[Tau]]Cd[Tau] [less than or equal to] M between limits [infinity] and 0. (2)

Maximizing the household's utility (1) subject to its budget constraint (2) gives the optimal time path of consumption, i.e.,

[Mathematical Expression Omitted], (3)

where [Mathematical Expression Omitted] is the time change rate of consumption C.

Technologies

There are four types of production activities in this economy: final good production, intermediate good production (a continuum of sectors located on [0, 1]), physical and human capital accumulation and R&D. It is assumed that perfect competition prevails in all sectors except the intermediate good sectors where there exists temporary monopoly power. The following describes each type of activities.

Final Good Production. The final good production uses the intermediate goods and. physical capital as its inputs subject to a constant-returns-to-scale (CRS) technology with the Cobb-Douglas form

Y = [K.sup.1 - [Alpha]] [integral of] [[A(i)x(i)].sup.[Alpha]] di between limits 1 and 0, (4)

where Y is the output of final good production; K and x(i) are respectively the physical capital and the flow of intermediate good i used in the final good production; [Alpha] is a parameter which measures the contribution of an intermediate good to the final good production and inversely measures the intermediate monopolist's market power; A(i) is the productivity coefficient of intermediate good i. Assume A(i) = [[Gamma].sup.i - 1]A, i [element of] [0, 1], where [Gamma] [greater than] 1 is the size of each innovation and A is the productivity of the most advanced intermediate good sector. This assumption implies that (intermediate good) sector 1 is the most advanced sector and sector 0 is the least advanced sector. Profit maximization of the final good sector gives the demand for the capital and the intermediate good i, i.e.,

(1 - [Alpha])[K.sup.-[Alpha]] [integral of] [[A(i)x(i)].sup.[Alpha]]di between limits 1 and 0 = r, (5)

[Alpha][K.sup.1 - [Alpha]]A[(i).sup.[Alpha]]x[(i).sup.[Alpha] - 1] = p(i), [for every]i [element of] [0, 1], (6)

where p(i) is the price of intermediate good i in terms of the final good.

Intermediate Good Production. Each intermediate good, i, is produced using only (unskilled) labor, l(i), with each unit of labor producing one unit of intermediate good i, i.e., x(i) = l(i), [for every]i [element of] [0, 1]. Given the wage rate W, each intermediate monopolist maximizes its profit, i.e., [Alpha][K.sup.1 - [Alpha]][[A(i)x(i)].sup.[Alpha]] - Wx(i). The first-order condition for this maximization problem is

W = [[Alpha].sup.2][K.sup.1 - [Alpha]]A[(i).sup.[Alpha]]x[(i).sup.[Alpha] - 1], [for every]i [element of] [0, 1]. (7)

Solving the above equation gives intermediate sector i's optimal output

x(i) = k[[[[Gamma].sup.(1 - i)[Alpha]][Omega]/[[Alpha].sup.2]].sup.1/([Alpha] - 1)], [for every]i [element of] [0, 1],(8)

where [Omega] = W/A and k = K/A are the productivity-adjusted wage of unskilled labor and the productivity-adjusted capital stock, respectively. Let [Pi](i) denote the corresponding maximum profit, then

[[Pi].sub.t](i) = [Alpha](1 - [Alpha])A(t)k[[Gamma].sup.(i - 1)[Alpha]][[[[Gamma].sup.(i - 1)[Alpha]][Omega]/[[Alpha].sup.2]].sup.[Alpha]/([Alpha] - 1)], [for every]i [element of] [0, 1]. (9)

Innovative R&D. Attracted by the market incentive, i.e., the temporary monopoly profit obtained by monopolizing an intermediate good sector once an innovation succeeds, firms invest in R&D. Success of innovation in any intermediate good sector leads to a new intermediate good in that sector which can be used to replace the old one in the final good production and increase the productivity of the final good sector by a factor [[Gamma].sup.[Alpha]].(3) As in Aghion and Howitt [1, Section 8], we assume different sectors experience innovations in a deterministic order and the innovations always occur in the least advanced sectors.(4) We suppose that innovation follows the Poisson process with the arrival rate A, where A depends positively on the human capital devoted to the R&D activities and negatively on the sophistication of the current technology, which can be measured by the technology coefficient of the final good production A:

[Lambda] = [Lambda](H/A), (10)

where H is the human capital used in the R&D activities and [Lambda] [greater than] 0 is a productivity parameter. As mentioned above, we assume that innovative R&D is the most human capital intensive activity. Here, we take the extreme case where innovative R&D requires only human capital as its input. It seems reasonable to make this specification by observing that the more the human capital devoted to the R&D, the stochastically faster the innovations come and the more sophisticated the current technology the more difficult to improve it. Here, the Schumpeterian terminology "creative destruction" has double meanings: On the one hand, successful innovations create new intermediate goods which make the final good production more productive but destroy the old ones; on the other hand, they create new knowledge which helps human capital accumulation (see equation (13) below) but also make the existing human capital less effective. An R&D firm maximizes [Lambda](H/A)V - [[Omega].sub.H]H. The first-order condition for this maximization problem is

([Lambda]/A)V = [[Omega].sub.H], (11)

where [[Omega].sub.H] is the wage rate for skilled labor (human capital) and V is the value of innovation. To derive the value of innovation, we need to calculate an intermediate monopolist's profit flow at different ages. Since we assume that different sectors experience innovations in a deterministic order; the innovations always occur in the least advanced sectors; and innovation follows the Poisson process with the arrival rate [Lambda], each intermediate monopolist enjoys the monopolist's profit for a period of length 1/[Lambda]. An intermediate monopolist of age 0 has the highest productivity. As the intermediate monopolist becomes older, its productivity rank falls. When the intermediate monopolist reaches the end of its life (age 1/[Lambda]), its productivity rank falls to 0. So we have the relationship between the productivity ranking i and the age [Tau]: i = 1 - [Lambda][Tau]. From equation (9), we know that the profit flow for an intermediate monopolist of age [Tau] at time (t + [Tau]) is(5)

[[Pi].sub.t+[Tau]]([Tau]) = [Alpha](1 - [Alpha])A(t + [Tau])k[[Gamma].sup.-[Alpha][Lambda][Tau]][[[[Gamma].sup.[Alpha][ Lambda][Tau]][Omega]/[[Alpha].sup.2]].sup.[Alpha]/([Alpha] - 1)]

= [Alpha](1 - [Alpha])A(t)k[[Gamma].sup.(1 - [Alpha])[Lambda][Tau]][[[[Gamma].sup.[Alpha][Lambda][Tau]][Omega] /[[Alpha].sup.2]].sup.[Alpha]/([Alpha] - 1)], [for every][Tau] [element of] [0, 1/[Lambda]].

Then the value of innovation is the total discounted expected profit during the period of length 1/[Lambda] [starting from time t until time (t + 1/[Lambda])], which is given by(6)

[Mathematical Expression Omitted], (12)

because the innovator's productivity-adjusted flow of profits at time (t + [Tau]) is

[Mathematical Expression Omitted].

In Appendix A, we show that the Poisson process with a deterministic innovation order for a continuum of intermediate sectors gives rise to a deterministic result: the length of each interval is 1/[Lambda] and thus the relative rank of each intermediate good's productivity decreases exponentially. Therefore, the direct formulation of the value of innovation is equivalent to the limiting case (m [approaches] [infinity] of the m intermediate good model in Aghion and Howitt [1].

Physical and Human Capital Accumulation. Finally, we describe the physical and human capital accumulation processes. For physical capital accumulation, we assume that each unit of consumption good foregone can produce one unit of capital and there is no capital depreciation.

The formulation of human capital accumulation in the human capital literature has reached a high degree of sophistication. For the problem at hand, we assume that the growth of human capital depends on the time devoted to human capital accumulation activities and the current stock of knowledge which is measured by A:

[Mathematical Expression Omitted], (13)

where [Mathematical Expression Omitted] is the time change rate of human capital stock H; B [greater than] 0 is a technology coefficient; and S is the time spent on the human capital accumulation. As mentioned above, successful innovations increase the stock of knowledge and therefore speed up the human capital accumulation.

At each point in time, individuals have two choices:(7) supply unskilled labor to intermediate good production or accumulate human capital which will be used in future R&D. To choose one of the two activities, each individual compares the earnings of these two activities. If the individual supplies his unskilled labor to the intermediate good production, then he earns a wage W per unit of time. If he chooses to accumulate human capital, then each unit of time devoted to this activity will increase his human capital stock by an amount BA, which will bring him an earning [[Omega].sub.H] per unit of human capital forever. Since earnings in the future have to be discounted at the rate of interest r, and in equilibrium, the earnings of these two activities have to be equal, then we have

Ba [integral of] [e.sup.-r[Tau]] [[Omega].sub.H]d[Tau] between limits [infinity] and 0 = W. (14)

The left-hand side of equation (14) is the discounted earnings per unit of time devoted to human capital accumulation activities and the right-hand side of equation (14) is the earnings per unit of time supplied to the intermediate good production.

Labor Market. Assume full employment, then we have the labor market clearing condition (time constraint)

X + S = N, (15)

where X is the total employment in the intermediate good production, i.e.,

X = [integral of] x(i)di between limits 1 and 0, (16)

because we assume that one unit of labor produces one unit of intermediate good in any intermediate good sector.

Final Good Market. The assumption that each unit of final good foregone can produce one unit of capital and there is no capital depreciation implies the following market clearing condition

[Mathematical Expression Omitted], (17)

where [Mathematical Expression Omitted] is the time change rate of physical capital stock K.

Capital Market. We assume that there exists a perfect capital market on which capital for production and R&D is raised.

We have completed the description of the model economy's environment and basic economic activities. Now we turn to the equilibria of this model economy.

III. Equilibrium

We consider only balanced growth stationary equilibria (BGSE). A BGSE is a collection of constant values of {c, k, [Omega], [[Omega].sub.H], r}, constant values of {p(i), x(i)} for each i [element of] [0, 1] and a constant growth rate for {C, K, H, W, A} such that

(i). Each household maximizes its dynasty utility by allocating its time between human capital accumulation activities and intermediate good production and income between consumption and savings (equations (3) and (14));

(ii). Each firm (intermediate good, final good and R&D) maximizes its profit (equations (5), (6), (7) and (11));

(iii). Capital, labor and final good markets clear (equations (15) and (17)).

To simplify the equilibrium analysis, we reduce the relevant equilibrium conditions to the following three conditions that involve three variables {h, [Omega] and k} (See Appendix B for derivation):

(1). Zero R&D profit condition:

[Alpha](1 - [Alpha])[Lambda]k[([Omega]/[[Alpha].sup.2]).sup.[Alpha]/([Alpha] - 1)][1 - [[Gamma].sup.(2[Alpha] - 1)/([Alpha] - 1)][e.sup.-([Sigma] ln [Gamma] + [Rho]/([Lambda]h))]] / [([Sigma] - (2[Alpha] - 1)/([Alpha] - 1))[Lambda]h ln [Gamma] + [Rho]] = ([Sigma] [Lambda]h ln [Gamma] + [Rho])[Omega]/B, (18)

where h = H/A is the productivity-adjusted human capital stock. Appendix C shows that [Delta]h/[Delta][Omega] [less than] 0. Therefore, in equilibrium, R&D firms will employ less human capital (skilled labor) h as the wage rate [Omega] for unskilled labor rises. This is because the wage rate [[Omega].sub.H] for human capital (skilled labor) also increases as the wage rate [Omega] for unskilled labor increases.

(2). Labor market clearing condition:

N - [Lambda][h.sup.2] ln [Gamma]/B = [(1 - [Alpha])(1 - [[Gamma].sup.[Alpha]/([Alpha] - 1)])k/([Alpha] ln [Gamma])][([Omega]/[[Alpha].sup.2]).sup.1/([Alpha] - 1)]. (19)

We know from Appendix C that [Delta]h/[Delta][Phi] [greater than] 0. That is, in equilibrium, more human capital (skilled labor) h will require a larger fraction of the population to be involved in human capital accumulation activities, which will make unskilled labor more scarce in manufacturing, which will raise the equilibrium wage rate [Omega].

(3). Final good market clearing condition:

[Lambda]h ln [Gamma] + [Omega]N/k = [(1 - [Alpha])(1 - [[Gamma].sup.[Alpha]/([Alpha] - 1)])/([Alpha] ln [Gamma])][([Omega]/[[Alpha].sup.2]).sup.[Alpha]/([Alpha] - 1)]. (20)

From Appendix C, we have [Delta]h/[Delta][Omega] [less than] 0 and [Delta]k/[Delta][Omega] [greater than] 0. That is, as the wage rate [Omega] rises, the intermediate goods become more expensive. So final good producers will substitute physical capital for intermediate goods. R&D firms hire less human capital (skilled labor) because of lower profit resulting from lower demand for intermediate goods. As a result, in equilibrium, physical capital k rises and human capital (skilled labor) falls.

Since we will focus on the stationary growth rate [g.sub.c],(8) we further simplify the above three equilibrium conditions to the following condition:

[Mathematical Expression Omitted], (21)

which determines a constant growth rate [g.sub.c]. Now let's look at the condition under which an equilibrium exists. Appendix D proves the following proposition.

PROPOSITION 1. If BN[Lambda] ln [Gamma]/[[[Rho].sup.2](1 - [[Gamma].sup.[Alpha]/([Alpha] - 1)])] [less than or equal to] 1, there is no growth ([g.sub.c] = 0); if BN[Lambda] ln [Gamma]/[[[Rho].sup.2](1 - [[Gamma].sup.[Alpha]/([Alpha] - 1)])] [greater than] 1, there is a unique equilibrium growth rate [g.sub.c] [element of] (0, [(BN [Lambda] ln [Gamma]).sup.1/2]).

The intuition behind this proposition is very straightforward. It simply states that if one or more of the following situations occur: (a) the size of the economy (measured by the total time endowment N) is too small; (b) the accumulation of human capital is too inefficient (B is too small); (c) innovation is too difficult ([Lambda] is too small); (d) the size of innovation [Gamma] is too small; (e) the degree of the monopolist's market power is too low ([Alpha] is too large); (f) the economy is too impatient ([Rho] is too large), then there will be no investment in innovative R&D and thus there is no growth. The reason for this is that these situations will either reduce the expected benefit of innovative R&D or increase the cost of innovative R&D or both to such an extent that no firms invest in this activity. As a result, there is no growth. Otherwise, there always are firms investing in innovative R&D, hence the economy experiences a positive growth.

The comparative-static analysis of the equilibrium (see Appendix E) shows the following results.

PROPOSITION 2. The laissez faire equilibrium growth rate depends positively on the efficiency of human capital accumulation B, the size of the economy N, the arrival rate parameter [Lambda], the size of innovation [Gamma] and the monopolist's market power (1/[Alpha]) and negatively on the risk aversion coefficient [Sigma] and the rate of time preference [Rho].

These results are intuitive. Each of these parameters (N, B, [Lambda], [Gamma], [Alpha], [Rho], [Sigma]) directly and/or indirectly affects the investment in innovative R&D and growth by changing either the marginal benefit or the marginal cost of innovative R&D or both. For example, an increase in the market power (a decrease in [Alpha]) increases the marginal benefit of R&D; an increase in the size of the economy N both increases the marginal benefit of R&D and reduces the marginal cost of R&D.

IV. The Social Planner's Problem

In order to examine the welfare property of the laissez faire equilibrium, in this section, we solve the social planner's problem. The social planner maximizes

[integral of] [e.sup.-[Rho][Tau]] [([C.sup.1 - [Sigma]] - 1)/(1 - [Sigma])]d[Tau] between limits [infinity] and 0,

subject to

[Mathematical Expression Omitted],

[Mathematical Expression Omitted],

[Mathematical Expression Omitted],

given [A.sub.0], [H.sub.0] and [K.sub.0].

The Hamiltonian for this maximization problem is

H = [e.sup.-[Rho][Tau]] ([C.sup.1 - [Sigma]] - 1)/(1 - [Sigma]) + [Xi][Lambda]H ln [Gamma] + [Mu]BA(N - X) + [Nu][[K.sup.1 - [Alpha]][A.sup.[Alpha]] [integral of] [([[Gamma].sup.i - 1]x(i)).sup.[Alpha]] - C],

where [Xi], [Mu], and [Nu] are the co-state variables. Then the necessary conditions for a maximum are

[Delta]H/[Delta]C = [e.sup.-[Rho][Tau]][C.sup.-[Sigma]] - [Nu] = 0,

[Delta]H/[Delta]x(i) = -[Mu]BA + [Nu][Alpha][K.sup.1 - [Alpha]][A.sup.[Alpha]][[Gamma].sup.(i - 1)[Alpha]] [(x(i)).sup.[Alpha] - 1] = 0,

[Mathematical Expression Omitted],

[Mathematical Expression Omitted],

[Mathematical Expression Omitted],

and the transversality conditions (TVCs): [lim.sub.[Tau][approaches][infinity]] [Xi]A = [lim.sub.[Tau][approaches][infinity] [Mu]H = [lim.sub.[Tau][approaches][infinity] [Nu]K = 0. We consider only balanced growth, i.e., [Mathematical Expression Omitted], then [Mathematical Expression Omitted] implies

[g.sub.p] = [[(BN [Lambda] ln [Gamma]).sup.1/2] - [Rho]]/[Sigma], (22)

where [g.sub.p] is the socially optimal balanced growth rate. A positive growth rate requires [(BN [Lambda] ln [Gamma]).sup.1/2] [greater than] [Rho] an the TVCs imply [(BN [Lambda] ln [Gamma]).sup.1/2] [less than] [Rho]/(1 - [Sigma]). So the optimal growth rate must satisfy 0 [less than] [g.sub.p] [less than] [Rho]/(1 - [Sigma]). Obviously, we have

PROPOSITION 3. The optional growth rate increases with an increase in the efficiency of human capital accumulation B, the size of the economy N, the arrival rate parameter [Lambda] and the size [Gamma] of innovation and decreases with an increase in the risk aversion coefficient [Sigma] and the rate of time preference [Rho].

These results are also easy to understand. Each of these parameters (N, B, [Lambda], [Gamma], [Alpha], [Rho], [Sigma]) affects the optimal growth rate by changing either the marginal social benefit or the marginal social cost of R&D or both. For example, an increase in the size of innovation increases the marginal social benefit of R&D, therefore, the society should allocate more resource to R&D and increase the growth rate.

Notice that, unlike the laissez faire growth rate, the optimal growth rate does not depend on the monopolist's market power (1/[Alpha]).(9)

Comparing equations (21) and (22), we know that the laissez faire equilibrium growth rate can be less than, equal to or greater than the optimal growth rate depending on the degree of monopoly power. However, the optimal growth can be supported by a tax/subsidy policy. Let [t.sup.*] be optimal the tax/subsidy rate on the wage of unskilled labor, then we have

PROPOSITION 4. The optimal growth can be supposed by a tax/subsidy policy with a tax/subsidy rate on the wage of unskilled labor [t.sup.*] = 1 - [Psi]([g.sub.p])[Phi]([g.sub.p]).

We show in Appendix F that [Delta][g.sub.c]/[Delta][Alpha] [less than] 0, [Psi]([g.sub.p])[Phi]([g.sub.p]) [where] [Alpha] = 0 = +[infinity] and [Psi]([g.sub.p])[Phi]([g.sub.p]) [where] [Alpha] = 1 = 0, there exists [Alpha] = [[Alpha].sup.*] (the critical point) such that [g.sub.c] = [g.sub.p]. So if the degree of monopoly power is lower than the critical point ([Alpha] [greater than] [[Alpha].sup.*]), then a tax is required ([t.sup.*] [greater than] 0); if the degree of monopoly power is higher than the critical point ([Alpha] [less than] [[Alpha].sup.*]), then the optimal growth can be achieved through a subsidy ([t.sup.*] [less than] 0).

As explained above, the degree of the monopolist's market power has a positive effect on the marginal benefit of R&D. If the degree of the monopolist's market power is too low, then R&D firms do not have enough incentive to invest in R&D. A tax on unskilled labor will increase the supply of human capital (skilled labor), which will reduce the wage rate for skilled labor (the marginal cost of R&D) and thus induce R&D firms to invest more in R&D. If the degree of the monopolist's market power is too high, then a subsidy is required. The subsidy will do exactly the opposite.

V. Conclusions

This paper integrates models of capital-based and idea-based economic growth, developing a dynamic general equilibrium growth model. In the model, both physical and human capital accumulation and investment in R&D are endogenously determined, and successful innovations not only discover new goods and destroy the old counterparts, but also create new knowledge and make the existing human capital less effective.

The model shows that both the laissez faire equilibrium and the optimal growth rates depend positively upon the efficiency of human capital accumulation, the size of the economy, the productivity of R&D and the size of innovation and negatively upon the risk aversion coefficient and the rate of time preference; but the monopoly power does not affect the optimal growth rate while it tends to increase the laissez faire growth rate. It also shows that under laissez faire the growth rate may be more or less than optimal, and there always exists a tax/subsidy system which can be used to achieve the optimal growth.

Appendix A: Proof of Equation (12)

This appendix is to show, through a Cobb-Douglas example, that the direct formulation is equivalent to the limiting case of the m intermediate good model in Aghion and Howitt [1]. The structure of the Cobb-Douglas case is as follows:

Y = [K.sup.1 - [Alpha]] [summation of] [[[A.sub.i][x.sub.i]].sup.[Alpha]] where i = 1 to m, 0 [less than] [Alpha] [less than] 1, (final good production technology)

[x.sub.i] = [l.sub.i], i = 1, 2, ..., m, (intermediate good production technology)

m[Lambda] = m[Lambda](H/A), (R&D technology)

[Mathematical Expression Omitted], (human capital accumulation technology)

[A.sub.i] = [[Gamma].sup.(1 - i)/m]A. (ith most advanced sector, i = 1, 2, ..., m)

The final good sector and intermediate good sector i's optimization conditions give intermediate sector i's optimal output

[x.sub.i] = k[[[[Gamma].sup.(i - 1)/m][Omega]/[[Alpha].sup.2]].sup.1/([Alpha] - 1)].

Then the average optimal output is

[Mathematical Expression Omitted].

Solving the above equation for [Omega], we have

[Mathematical Expression Omitted].

Since intermediate sector i's profit is

[[Pi].sub.i] = [Alpha](1 - [Alpha])Ak[[Gamma].sup.(i - 1)(1 - [Alpha])/m][[[[Gamma].sup.(i - 1)[Alpha]/m][Omega]/[[Alpha].sup.2]].sup.[Alpha]/([Alpha] - 1)],

we have intermediate sector i's productivity-adjusted profit

[Mathematical Expression Omitted].

Then the value of innovation is

[Mathematical Expression Omitted].

Thus we have

[Mathematical Expression Omitted],

and

[Mathematical Expression Omitted],

because

lim [[m[Lambda]/(r + m[Lambda])].sup.m] where m [approaches] [infinity] = lim [[1 + (r/[Lambda])/m].sup.-m] where m [approaches] [infinity] = [e.sup.-r/[Lambda]],

and

[Mathematical Expression Omitted]

(by multiplying the numerator and denominator by [m.sup.2])

= 1/[r + (2[Alpha] - 1)[Lambda] ln [Gamma]/(1 - [Alpha])].

Therefore, the R&D firm's optimization condition (in the limiting case) implies

m[Lambda]V/A = [[Omega].sub.H],

which, combining with the other equilibrium conditions and the balanced growth restriction, gives the same equilibrium growth rate as in section III.

Appendix B: Derivation of Equilibrium Conditions

We get [Lambda] = [Lambda]h and [[Omega].sub.H] = r[Omega]/B respectively from equations (10) and (14), where h = H/A is the productivity-adjusted human capital stock. By the definition of balanced growth stationary equilibrium, we have [Mathematical Expression Omitted] [Mathematical Expression Omitted],(10) which along with equation (3) implies r = [Sigma][Lambda]h ln [Gamma] + [Rho]. Then substituting [Lambda], [[Omega].sub.H], r and V (equation (12)) into equation (11), we get the zero R&D profit condition [equation (18)].

From equation (13), we have [Mathematical Expression Omitted] ln [Gamma]/B.(11) From equation (8), we have

X = [integral of x(i)di] between limits 1 and 0 = [(1 - [Alpha])(1 - [[Gamma].sup.[Alpha]/([Alpha] - 1)])k/([Alpha] ln [Gamma])][([Omega]/[[Alpha].sup.2]).sup.1/([Alpha] - 1),

where X is the total unskilled labor employment in intermediate good sectors. Then equation (15) implies the labor market clearing condition [equation (19)].

Finally, since A(i) = [A.sub.[[Gamma].sup.i - 1] and x(i) = k[[[Gamma].sup.(1 - i)[Alpha]][Omega]/[[Alpha].sup.2]].sup.1/([Alpha] - 1), the right-hand side of equation (17) is

[Mathematical Expression Omitted]. (A1)

The left-hand side of equation (17) is

[Mathematical Expression Omitted], (A2)

where c = C/A is the productivity-adjusted consumption. Since the household's discounted life-time income is

[Mathematical Expression Omitted],

we can solve equation (2) for the productivity-adjusted consumption c. The left-hand side of equation (2) is

[Mathematical Expression Omitted].

Then equation (2) implies c = [Omega]N. Finally, from equations (A1) and (A2) with c = [Omega]N, we get the final good market clearing condition.

Appendix C: The Effects on h and k of [Omega]

From equation (18), we have

[Delta]h/[Delta][Omega] = -{B[Lambda]k[[Alpha].sup.2]/(1 - [Alpha])] [1 - [[Gamma].sup.(2[Alpha] - 1)/[([Alpha] - 1)][e.sup.([Sigma] ln [Gamma] + [Rho]/([Lambda]h))]]

/[([Sigma] - (2[Alpha] - 1)/([Alpha] - 1))[Lambda]h ln [Gamma] + [Rho]]}/[[Alpha][Sigma] [Lambda] ln [Gamma] + + (1 - [Alpha])B[Lambda]k[([Omega]/[[Alpha].sup.2]).sup.1/([Alpha] - 1)]Q] [less than] 0,

where

[Mathematical Expression Omitted],

and

[N.sub.p] = 1 - [[Gamma].sup.(2[Alpha] - 1)/([Alpha] - 1))[Lambda]h ln [Gamma] + [Rho].

and

[D.sub.p] = ([Sigma] - (2[Alpha] - 1)/([Alpha] - 1))[Lambda]h ln [Gamma] + [Rho].

Similarly, from equation (19), we have

[Delta]h/[Delta]q = B(1 - [[Gamma].sup.[Alpha]/([Alpha] - 1)k[[Omega].sup.(2 - [Alpha])/([Alpha] - 1)]/[2[Lambda]h[(ln [Gamma]).sup.2][[Alpha].sup.([Alpha] + 1)/([Alpha] - 1)]] [greater than] 0.

Finally, from equation (20), we have

[Delta]h/[Delta][Omega] = -{(1/(1 - [Alpha]))[([Omega]/[[Alpha].sup.2]).sup.[Alpha]/([Alpha] - 1)][(1 - [Alpha])(1 - [[Gamma].sup.[Alpha]/([Alpha] - 1)])/([Alpha] ln [Gamma])] - [Lambda]h ln [Gamma]}/([Lambda][Omega] ln [Gamma]) [less than] 0,

and

[Delta]k/[Delta][Omega] = {(1/(1 - [Alpha]))[([Omega]/[[Alpha].sup.2]).sup.[Alpha]/([Alpha] - 1)][(1 - [Alpha])(1 - [[Gamma].sup.[Alpha]/([Alpha] - 1)])/([Alpha] ln [Gamma])] - [Lambda]h ln [Gamma]}

/[[[([Omega]/[[Alpha].sup.2]).sup.[Alpha]/([Alpha] - 1)](1 - [Alpha])(1 - [[Gamma].sup.[Alpha]/([Alpha] - 1)])/([Alpha] ln [Gamma]) - [Lambda]h ln [Gamma]].sup.2] [greater than] 0.

Appendix D: Existence of Equilibrium

First of all, we show that the LHS of equation (21) is a decreasing function of [g.sub.c]. Let

[Mathematical Expression Omitted],

and

[Phi] = [1 - [[Gamma].sup.-([Alpha] - (2[Alpha] - 1)/([Alpha] - 1) + [Rho]/[g.sub.c])]/[([Sigma][g.sub.c] + [Rho]) - (2[Alpha] - 1)[g.sub.c]/([Alpha] - 1)],

then equation (21) becomes [Psi]([g.sub.c])[Phi]([g.sub.c]) = 1. For simplicity, the following notations will be used:

[Mathematical Expression Omitted],

[D.sub.[Psi]] =([Sigma][g.sub.c] + [Rho])(1 - [[Gamma].sup.[Alpha]/([Alpha] _ !)]),

[N.sub.[Phi]] = 1 - [[Gamma].sup.-([Sigma] - (2[Alpha] - 1)/([Alpha] - 1) + [Rho]/[g.sub.c]),

[D.sub.[Phi]] = [Sigma][g.sub.c] + [Rho] - (2[Alpha] - 1)[g.sub.c]/([Alpha] - 1),

F = [[Gamma].sup.-([Sigma] - (2[Alpha] - 1)/([Alpha] - 1) + [Rho]/[g.sub.c]),

Since

[Mathematical Expression Omitted],

and

[Mathematical Expression Omitted],

where we assume [Sigma] [greater than] (2[Sigma] - 1)/([Alpha] - 1), the LHS of equation (21) is decreasing in [g.sub.c]. Moreover, we have LHS ([g.sub.c] = 0) = BN[Lambda] ln [Gamma]/[[[Rho].sup.2] (1 - [[Gamma].sup.[Alpha]]/([Alpha] - 1))] and LHS ([g.sub.c] = (BN[Lambda]ln [Gamma]).sup.1/2]) = 0. Then if BN[Lambda] ln [Gamma]/[[[Rho].sup.@}(1 -[[Gamma].sup.[Alpha]/([Alpha] - 1)])] [less than or equal to] 1, there is no growth; if BN[Lambda] ln [Gamma]/[[[Rho].sup.2](1 - [[Gamma].sup.[Alpha]/([Alpha] - 1)])] [greater than] 1, there is a unique equilibrium where [g.sub.c] [element of] (0, (BN [Lambda] ln [Gamma]).sup.1/2]).

Appendix E: Comparative Statics 

From Appendix D, we have the laissez faire equilibrium condition

[Psi][Phi] = 1.

Differentiate the above equation with respect to [Xi], where [Xi] = N, B, [Lambda], [Gamma], [Alpha], [Rho], [Sigma], we have

[Phi][([Delta][Psi]/[Delta][g.sub.c])([Delta][g.sub.c]/[Delta][Xi]) + ([Delta][Psi]/[Delta][Xi])] + [Psi][([Delta][Phi]/[Delta][g.sub.c])([Delta][g.sub.c]/[Delta][Xi]) + [Delta][Phi]/[Delta][Xi]] = 0,

which implies

[Delta][g.sub.c]/[Delta][Xi] = [([Delta][Psi]/[Delta][Xi])[Phi] + ([Delta][Phi]/[Delta][Xi])[Psi]]/[-(([Delta][Psi]/[Delta][g.sub.c ])[Phi] + ([Delta][Phi]/[Delta]/[Delta][g.sub.c])[Psi])].zz

Since [Psi] [greater than] 0, [Phi] [greater than] 0 and we have shown in Appendix D that [Delta][Psi]/[Delta][g.sub.c] [less than] 0 and [Delta][Phi]/[Delta][g.sub.c] [less than] 0, then the denominator of [Delta][g.sub.c]/[Delta][Xi], i.e.,

- [([Delta][Psi]/[Delta][g.sub.c])[Phi] + ([Delta][Phi]/[Delta][g.sub.c])[Psi]] [greater than] 0.

Therefore, the sign of [Delta][g.sub.c]/[Delta][Xi] is determined by the sign of the numerator of [Delta][g.sub.c]/[Delta][Xi], i.e., [([Delta][Psi]/[Delta][Xi])[Phi] + ([Delta][Phi]/[Delta][Xi])[Psi]]. To determine the sign of [([Delta][Psi]/[Delta][Xi])[Phi] + ([Delta][Phi]/[Delta][Xi])[Psi]], we derive the relevant derivatives:

[Delta][Psi]/[Delta]N = B[Lambda]ln [Gamma]/[D.sub.[Psi] [greater than] 0,

[Delta][Phi]/[Delta]N = 0,

[Delta][Psi]/[Delta]B = B[Lambda] ln [Gamma]/[D.sub.[Psi]] [greater than] 0, [Delta][Phi]/[Delta]B = 0, [Delta][Psi]/[Delta][Lambda] = BN ln [Gamma][D.sub.[Psi] [greater than] 0,

[Delta][Phi]/[Delta][Lambda] = 0,

[Mathematical Expression Omitted] (See Note 1),

[Delta][Phi]/[Delta][Gamma] = [[Sigma] - (2[Alpha] - 1)/([Alpha] - 1) + [Rho]/[g.sub.c]][[Gamma].sup.[[Sigma] - (2[Alpha] - 1)/([summation of] - 1) + [Rho]/[g.sub.c]]-1] [greater than] 0,

[Delta][Psi]/[Delta][Alpha] = -[[Psi]([Sigma][g.sub.c] + [Rho]) ln ([Gamma])[[Gamma].sup.[Alpha]/([Alpha] - 1)]]/[[(1 - [Alpha]).sup.2] [D.sub.[Psi]]] [less than] 0,

[Mathematical Expression Omitted] (See Note 3),

[Delta][Psi]/[Delta][Rho] = -[Psi]/([Sigma][g.sub.c] + [Rho]) [less than] 0,

[Mathematical Expression Omitted] (See Note 3),

[Delta][Psi]/[Delta][Sigma] = -[g.sub.c][Psi]/([Sigma][g.sub.c] = [Rho]) [less than] 0,

[Mathematical Expression Omitted] (See Note 4).

Then we can determine the sign of [([Delta][Psi]/[Delta][Xi])[Phi] + ([Delta][Phi]/[Delta][Xi])[Psi]]. The results are shown in Table I.

Table I. The Sign of [Delta][g.sub.c]/[Delta][Xi]

[Xi]   N   B   [Lambda]   [Gamma]   [Alpha]   [Rho]   [Sigma]

Sign   +   +       +         +         -         -       -

Note 1: First, notice that f (v) [approximately equal to] [[Gamma].sup.-v](1 +v ln(v)) [less than] 1, [for every]0 [less than] v [less than] [infinity], because f (v = 0) = 1 and f[prime](v) =

-v[(ln [Gamma]).sup.2] [[Gamma].sup.-v] [less than] 0. Let v = [Alpha]/(1 - [Alpha]), then we have

[Mathematical Expression Omitted].

Note 2: Let u = [Sigma] - (2[Alpha] - 1)/([Alpha] - 1) + [Rho]/[g.sub.c], then we get

[Mathematical Expression Omitted].

Note 3: Using the same definition of u as in Note 2, we obtain

[Mathematical Expression Omitted].

Note 4: Again, we use the same definition of u as in Note 2 to get

[Mathematical Expression Omitted].

Appendix F: Optimal Tax/Subsidy Scheme

From Appendix E, we have [Delta][g.sub.c]/[[Delta][Alpha] [less than] 0. Furthermore,

[Mathematical Expression Omitted]

and

[Mathematical Expression Omitted].

Therefore, there must exist [Alpha] = [[Alpha].sup.*] such that [g.sub.c] = [g.sub.p].

Consider a tax t levied on unskilled labor, then equation (14) becomes

BA [integral of [e.sup.-r[Tau]]] between limits [infinity] and 0 [[Omega].sub.H][d.sub.[Tau]] = W(1 - t).

As a result, equation (21) becomes

[Psi]([g.sub.c])[Phi]([g.sub.c]) = 1 - t.

Choose [t.sup.*] = 1 - [Psi]([g.sub.p])[Phi]([g.sub.p]), then the solution to the above equation gives the desired result: [g.sub.c] = [g.sub.p].

1. Romer [8] incorporates physical capital into a horizontal product differentiation model, in which, continuous capital investment is required for the production of new intermediate goods discovered through R&D. Grossman and Helpman [4, 112-43] also introduce physical and human capital separately into their idea-based model. Our model differs from these models in two aspects: first, we have both physical and human capital accumulation in a single model; second, unlike Grossman and Helpman [4, 112-43] (where human capital stock is fixed even though it is endogenously determined), human capital accumulates over time.

2. Aghion and Howitt [1] have a similar result. But as will be seen below, the optimal growth does not depend on the market power. This is different from Aghion and Howitt [1].

3. As will be seen below, the increase in the productivity of intermediate goods will be accompanied with an increase in physical capital input in the final good production. In a balanced growth equilibrium, physical capital input will increase by a factor [Gamma] and this increase raises the output of the final good sector by [[Gamma].sup.1 - [Alpha]]. As a result, the final good production will increase by [Gamma].

4. This assumption is initially taken from Shleifer [11].

5. Since the productivity A(t) of the most advanced sector grows at the rate [Lambda] ln [Gamma], we have A(t + [Tau]) = A(t)[[Gamma].sup.[Lambda][Tau]].

6. See Appendix A for proof.

7. Individuals also choose how much to save in the form of physical capital. Given the representative household's income stream, the optimal time path of consumption (3) implies the optimal time path of saving.

8. The growth rate [g.sub.c] is assumed to satisfy the condition [g.sub.c] [less than] [Rho]/(1 - [Sigma]) to guarantee that the household's utility is finite.

9. This result is different from that in Aghion and Howitt [1] in which there is no physical or human capital accumulation.

10. Since we assume that innovation follow the Poisson process with the arrival rate A and always occurs in the least advanced sector, the productivity of the most advanced sector will grow at the rate [Gamma] ln [Gamma] = [Lambda]h ln [Gamma].

11. In a balanced growth equilibrium, human capital stock H also grows at the same rate as the productivity A of the most advanced sector.

References

1. Aghion, Phillippe and Peter Howitt, "A Model of Growth Through Creative Destruction." Econometrica, March 1992, 323-51.

2. Grossman, Gene and Elhanan Helpman, "Quality Ladders in the Theory of Growth." Review of Economic Studies, January 1991, 43-61.

3. -----. "Quality Ladder and Product Cycles." Quarterly Journal of Economics, May 1991, 557-86.

4. -----. Innovation and Growth in the Global Economy. Cambridge, Mass.: MIT Press, 1991.

5. Lucas, Robert Jr., "On the Mechanics of Economic Development." Journal of Monetary Economics, July 1988, 3-42.

6. Rebelo, Sergio, "Long-Run Policy Analysis and Long-Run Growth." Journal of Political Economy, June 1991, 500-21.

7. Romer, Paul, "Increasing Returns and Long-Run Growth." Journal of Political Economy, October 1986, 1002-37.

8. -----. "Endogenous Technological Change." Journal of Political Economy, October 1990, S71-S102.

9. -----. "Two Strategies for Economic Development: Using Ideas and Producing Ideas." Working Paper, March 1992.

10. Segerstrom, Paul, T. C. A. Anant, and Elias Dinopoulos, "A Schumpeterian Model of the Product Life Cycle." American Economic Review, December 1990, 1077-91.

11. Shleifer, Andrei, "Implementation Cycles." Journal of Political Economy, December 1986, 1163-90.

12. Zeng, Jinli. "Essays on R&D and Economic Growth." Ph.D. Dissertation, the University of Western Ontario, 1995.