Human capital use, innovation, patent protection, and economic growth in multiple regions

Abstract

We provide the first theoretical analysis of the effects of human capital use, innovative activity, and patent protection, on economic growth in a model with many regions. In each region, consumers have constant relative risk-aversion preferences, there is no human capital growth, and there are three kinds of manufacturing activities involving the production of blueprints for inputs (machines), the inputs themselves, and a single final consumption good. Our analysis generates four results. For any given region, we first describe the balanced growth path (BGP) equilibrium and show that the BGP growth rate depends negatively on the rate at which patents expire. Second, we characterize the transitional dynamics in our model. Third, we determine the value of the patent expiry rate that maximizes the equilibrium growth rate of a region. Finally, we show that a policy of offering perpetual patent protection does not necessarily maximize social welfare in a region.

Keywords:

• dynamic
• human capital
• innovation
• multi-region economy
• patent
• stochastic

JEL Classifications:

• R11
• O31
• L52

Acknowledgements

For their helpful comments on a previous version of this paper, we thank the Editor Cristiano Antonelli, two anonymous referees, participants in an International Workshop in the Tinbergen Institute, Amsterdam, The Netherlands, and seminar participants at the Delhi School of Economics, New Delhi, India, the Indian Statistical Institute, New Delhi, India, and the Indian Institute of Technology, Mumbai, India. In addition, Batabyal acknowledges financial support from the Gosnell endowment at RIT. The usual disclaimer applies.

Notes

In the remainder of this paper, we shall use the terms ‘input’ and ‘machine’ interchangeably.

The term ‘region’ is in general an imprecise term and it can refer to alternate geographical entities. Therefore, to fix ideas, the reader may want to think of the aggregate economy as the European Union (EU), the regions as the various nations in the EU, and the spatial units as the provinces within these individual EU member nations. In an alternate interpretation, the aggregate economy would be the USA, the regions would correspond to the various US states, and the spatial units would denote the counties in the individual US states. Clearly, in the first interpretation, the region corresponds to a nation whereas in the second interpretation, it corresponds to a state. As such, our model of an aggregate economy is general enough to accommodate alternate interpretations of what constitutes a region. Finally, note that within each region, we focus on the analysis of a growth model in a single sector. Consistent with the discussion in this note, growth models of this sort can be used to describe regions when they are smaller than and when they are equal to nations.

See Acemoglu (2009, 308–10) or Mas-Colell, Whinston, and Green (1995, 194) for more on the properties of CRRA utility functions. Second, as noted in Acemoglu (2009, 308), this CRRA utility function has constant intertemporal elasticity of substitution. Third, because the intertemporal elasticity of substitution is constant, it is clear that we are not accounting for uncertainty in our model and, in this regard, it is important to understand that it is not our objective in this paper to analyze the behavior of consumers in region i under uncertainty. Finally, as explained in Section 2.2, the only way in which we account for uncertainty in this paper is that the rate at which patents expire in region i is given by the non-negative Poisson rate λ _i .

If we allow the stock of human capital in region i to grow then balanced economic growth in this region will not be possible. In particular, the growth rate of the ith region will gradually increase and the model will ‘explode’. Put differently, there will be no closed-form solution to the model under study. This is why there is no growth in the stock of human capital in our model. See Acemoglu (2009, 446) for a more detailed corroboration of this point.

Since we are working with a model of endogenous technology, firms and individuals in region i must ultimately have a choice between different kinds of technologies and, in this regard, greater effort, investment, or R&D spending ought to lead to the invention of better technologies. These features tell us that there must exist a meta production function or a ‘production function over production functions’ which tells us how new technologies are generated as a function of various inputs. Following Acemoglu (2009, 413), we refer to this meta production function as the ‘innovation possibilities frontier’.

See Theorem 7.10 and the related discussion in Acemoglu (2009, 244, 435–36) for more on the technical details of this procedure.

In what follows, the model solution techniques we employ are similar to those employed by Peters and Simsek (2009, 144–48).

Acemoglu (2009, 309) points out that ‘balanced growth is only possible with preferences featuring a constant elasticity of intertemporal substitution \ldots’. This means that our study of the BGP equilibrium in this paper makes sense only if the coefficient of relative risk aversion is constant. Put differently, our model and our analysis of this model require that the representative household's utility function takes the CRRA form.

There is some similarity between our analysis of the dynamics of the two categories of machine varieties and the analysis undertaken in Krugman (1979) to determine the number of ‘new’ and ‘old’ goods in his model.

The unstable solutions can be safely discarded because they either violate the transversality condition or the resource constraint in our model.

The notion of stability associated with this equilibrium is that of saddle path stability. As noted by Acemoglu (2009, 271), the ‘notion of saddle-path stability is central in most growth models’. This accounts for our use of this notion in this paper. Note that our use of this saddle path stability concept precludes any need to demonstrate the global stability of the equilibrium described by Equations (21)–(26).

Many of these models are discussed thoroughly in Acemoglu (2009) and in Aghion and Howitt (2009).

The subsequent discussion in Section 4 will be brief because our purpose in this paper is not to undertake a detailed analysis of the dynamics along a transition path.

The claims in this and the preceding sentence in this paragraph can be verified formally by splitting the social planner's problem into two sub-problems and then solving these sub-problems sequentially. The first sub-problem or the static problem involves choosing the machine varieties x _i (v _i , t) to maximize net output in region i subject to the resource constraint given by Equation (4) and given the level of machines N _i (t). This maximization exercise gives us a closed-form expression for the social planner's optimal net output Ŷ _i (t). Tedious but straightforward computations show that this value of net output exceeds the value of net output in the decentralized BGP equilibrium obtained by subtracting X _i (t) in Equation (12) from Y _i (t) in Equation (10). The second sub-problem or the dynamic problem involves choosing consumption C _i (t) to maximize the region i representative household's utility function subject to a suitably rewritten version of the differential equation constraint given by Equation (2). This gives us the standard Euler equation from which we derive the rate at which the economy of region i grows. Comparing this social planner's growth rate with the BGP growth rate in Equation (16) tells us that the growth rate in the socially planned region i exceeds that in the BGP equilibrium.