Two-sector growth models with productive public goods: equilibrium (in)determinacy.

• Author: Chen, Been-Lon

1. Introduction

   Models of economic growth with multiple equilibrium paths offer an explanation as to why we often observe diverse growth performances among countries with similar economic environments. A central question is "Under what conditions is it possible to establish local indeterminacy?" In this paper, we study a Lucas (1988) human capital-based growth model and investigate whether there are multiple equilibrium paths and, thus, local equilibrium indeterminacy in the neighborhood of a balanced growth path.

   Equilibrium indeterminacy in existing two-sector, human capital-based growth models emerges based upon two mechanisms. One of the mechanisms is differential factor tax rates, initiated in the work of Bond, Wang, and Yip (1996) and followed by that of Raurich (2001). The other mechanism is sector-specific externalities, pioneered in the work of Benhabib and Farmer (1996) in an exogenous growth model and followed by the work of Benhabib, Meng, and Nishimura (2000) and Mino (2001) in endogenous growth models. Ben-Gad (2003) combined the two mechanisms. These two mechanisms establish local indeterminacy when the mechanisms cause the factor intensities at the social perspective and those at the private perspective to reverse.

   The purpose of this paper is twofold. First, we argue that once productive public goods are considered, the two existing mechanisms are not robust in creating local indeterminacy. Then we take into account congestion in public services, so a two-sector model with the two existing mechanisms can generate local indeterminacy. Our purpose is motivated by the following.

   In the existing models with sector-specific externalities or distorted factor taxes, there is either no government spending or, if the government spending is taken into account, only consumptive public spending and government transfers are considered. Recently, a number of authors have documented the positive impact of public expenditure upon the productivity of private capital. (1) When there is productive public spending, the production technology is externally enhanced by the public goods. Productive public goods as an engine of economic growth have received much attention in endogenous growth models. In particular, a number of studies in this line of modeling strategy have obtained local indeterminacy. (2) Thus, it is important to determine whether or not differential factor tax rates and sector-specific externalities create local indeterminacy in models with productive public goods.

   However, as we will find, the two existing mechanisms no longer create local indeterminacy in a two-sector growth model with productive public goods unless the difference in the factor tax rates and the gap in the sector-specific externalities are implausibly large. Intuitively, productive public goods introduce the effects of intersectoral externalities. They offset the effects of the two existing mechanisms that would have otherwise reversed the factor intensity at the social viewpoint from the factor intensity at the private viewpoint. As a consequence, the two existing mechanisms are not robust in the establishment of local indeterminacy in models with productive public goods.

   This paper, then, introduces the congestion effect in the use of public goods, an effect offsetting the effect from public spending, to salvage the two existing mechanisms. (3) Congestion is a prevalent experience in the use of public services, such as irrigation systems, highways, harbors, airports, and other facilities. Such an effect has received much attention in public and spatial economics (e.g., Starrett 1988; Edwards 1990). The congestion effect has recently been taken into account extensively in endogenous growth models (e.g., Barro and Sala-i-Martin 1992; Glomm and Ravikumar 1997; Fisher and Turnovsky 1998).

   With a congestion effect in the use of public goods, we find that local indeterminacy arises. Intuition suggests that under constant returns, the congestion effect reduces an otherwise strong intersectoral external effect as a result of the marginal contribution of public goods services and increases the marginal contribution of aggregate physical, as well as human, capital. This makes marginal social products deviate from the marginal private products. The otherwise negative sloping relative marginal products of labor between the good and the human capital sectors are now positive sloping. As a result, a higher expected price in the sector that is more capital intensive reduces, rather than increases, the fractions of capital and labor allocated to that sector, thereby making the expectations about the higher price self-fulfilling.

   Elsewhere, it was shown that congestible public goods can generate indeterminacy in a two-sector, constant-return human capital--enhanced growth model (Chen and Lee 2007). Unlike the work of Chen and Lee (2007), the current paper is an attempt to show that the arguments based on the mechanisms of sector-specific externalities and factor taxes that would have otherwise established indeterminacy are not robust and become fragile once there are productive public goods. In an effort to salvage these two popular, existing mechanisms, in the end of the paper we will follow Chen and Lee (2007) and introduce the congestion mechanism. As expected, indeterminacy is regained in the model with sector-specific externalities and factor taxes.

   As developed below, section 2 sets up a model that features factor taxation, sector-specific externalities, and productive public goods, while section 3 studies the optimization and the Balanced Growth Path (BGP) equilibrium. Section 4 intuitively explains the reasons underlying determinacy and indeterminacy of a BGP in different set-ups, and the transitional dynamics are analyzed in section 5, in which the first subsection notes economies without congestion effects and the subsequent subsection discusses economies with congestion effects. Finally, some concluding remarks are made in section 6.

2. The Model

   The model is based upon the work of Lucas (1988), Rebelo (1991), Bond, Wang, and Yip (1996), and Ben-Gad (2003) and is general enough to include both the features of the sector-specific externalities and the distortionary factor taxation in existing literature. The economy consists of a continuum of representative consumers/producers with two sectors, referred to as the goods sector, Y, and the education sector, X. Human and physical capital are both used in the two sectors, with their productivity enhanced by public good services h la Barro (1990).

   More specifically, we assume both sectors are of constant returns to scale and, for simplicity, of the following parametric forms:

   Y = [A(vk).sup.[alpha]][(uh).sup.[beta]] [[bar.(uH)].sup.[epsilon]] [[??].sup.[gamma]], K(0) > 0, H(0) > 0, (1)

   given

   X = B[[(1 - v)k].sup.[eta]][[(1 - u)h].sup.[theta]][[[bar.(1-u)H]].sup.[xi]][[??].sup.[delta]], (2)

   in which variables with an upper bar indicate externalities and where 0 < v < 1 (resp. 0 < u < 1) is the fraction of physical capital, k (resp. human capital, h), allocated to the goods sector. Parameters A > 0 and B > 0 are productivity coefficients; 0 < [alpha] < 1 (0 < [eta] < 1) and 0 < [beta] < 1 (0 < [theta] < 1) are the shares of physical and human capital in the goods (education) sector. Moreover, both sectors are subject to sector-specific externalities, with the productivity in each sector enhanced by the average amount of aggregate human capital (denoted as H) employed in each sector, and the contribution of externality in the goods (education) sector being [epsilon] > 0 ([xi] > 0). Finally, the public services, [??], affect both sectors, with parameter [gamma] ([delta]) representing the degree to which the public services enhance the productivity in the goods (education) sector.

   Let us comment on the sector-specific externalities generated only by human capital. In general, physical and/or human capital may create sector-specific externalities (e.g., Benhabib, Meng, and Nishimura 2000; Mino 2001). The appearance of only one source of sector-specific externalities in the basic model is to maintain the insights into the existing models with sector-specific externalities while making the model as simple as possible in order to be easily tractable. In Appendix D, we have considered sector-specific externalities of both physical capital and human capital in the technology. As the Appendix shows, the dynamic equilibrium system is exactly the same as the model using the technology in Equations 1 and 2. As a result, the model with two sources of sector-specific externalities is more complicated, but the main results are expected to be the same as those in the model with one source of sector-specific externalities. We thus present in the text the model with one source of sector-specific externalities in order to keep the model simple, while leaving the model with two sources of sector-specific externalities in the Appendix for references.

   There is a congestion effect in that the amount of public goods services received by the representative agent in Equations 1 and 2 may be less than the actual amount of public spending, G, with the perceived amount decreasing in the aggregate activity. Specifically, we assume that public services received by the representative agent are as follows:

   [??] = G x [(1/K).sup.[psi]] [(1/H).sup.[zeta]], (3)

   in which K is aggregate physical capital, [psi] indicates the degree of congestion due to the aggregate use of physical capital, and [zeta] is the degree of congestion due to the aggregate use of human capital. If [psi] = [zeta] = 0, the public service is non-rival and non-excludable and is therefore a pure public good.

   The following parameter restrictions are imposed for the technologies:

   ASSUMPTION 1

   (i) [alpha] - [gamma][psi] > 0, [beta] + [epsilon] - [gamma][zeta] > 0, [eta] - [delta][psi] > 0, and [theta] + [delta][xi] > 0;

   (ii) [alpha] + [beta] +...