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

• Author: Zeng, Jinli

1. 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.

2. 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...