A competitive idea-based growth model

ABSTRACT

In this paper, we present a model in which endogenous growth arises in competitive markets. Knowledge is described as a factor used directly in the final goods' production. Firms demand both basic nonrival knowledge contents, which are supplied jointly and inelastically with raw labor, and further contents supplied by patent holders. This fact, together with Lindahl prices for knowledge, allows competition to work, while it also implies that workers' income share declines overtime. In a first version of the model with constant cost of knowledge production, the first best is attained. In a further version of the model, in which the cost of knowledge production is allowed to change over time and thus intertemporal externalities arise, in a decentralized economy a second best equilibrium occurs in the transitional period, while in the long run there is convergence to efficiency.

KEYWORDS:

• Endogenous growth
• competitive markets
• intangible inputs
• Lindahl prices

JEL Classification:

• C61
• E10
• O31
• O41

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Carla Marchese http://orcid.org/0000-0001-9935-2038

Fabio Privileggi http://orcid.org/0000-0002-8837-2006

Notes

1 This trend for the U.S. can be dated back to 1998, when in the so called State Street Bank case a business method was declared patentable. Many other similar rulings followed with respect to software. For patentability in general see Eckert and Langinier (2013).

2 For the case in which labor and individual licenses for accessing disembodied knowledge are jointly demanded, see Marchese and Privileggi (2018).

3 We envisage a potential turning point and thus depart from Acemoglu and Restrepo (2016) who maintain that new and more complex tasks for labor can always be created.

4 Arrieta Ibarra et al. (2018) stress also the role of data produced by human interactions in the generation of information useful for Machine Learning. They envisage a world of ‘Data as Labor’, where workers are identified with the suppliers of such information, i.e. of basic intangible intellectual contents, and paid accordingly.

5 The idea that market power is necessary for inducing costly research dates back to Schumpeter (1911). It was brought up again in the '90s by many papers (besides Romer 1990, see, e.g. also Grossman and Helpman 1991; Peretto 1996; Aghion and Howitt 1998) and subsequently it has become a widely shared tenet in the literature on endogenous economic growth.

6 For an extension in which population growth is considered see the working paper version of this paper (Marchese and Privileggi 2016).

7 By a continuity argument, the knowledge endowment of workers at the initial instant of time provides a marginal benefit and commands a price equal to that of the further contents produced under patents' protection.

8 For the sake of simplicity it is assumed that labor L is a continuous variable.

9 See, e.g. pp. 428–429 in Acemoglu (2009).

10 F-firms cannot re-rent knowledge or share it with other firms; that is, no arbitrage is allowed.

11 All firms employ the same intermediate good/labor ratio, $x\equiv x_i=X_i/L_i$, as will become clear in the sequel.

12 Alternatively, one can assume that there is a continuum of output levels $Y(i,t)\ge 0$, $i\in [0,N(t\left)\right]$, each produced by a density $m(i,t)\ge 0$ of identical F-firms, so that the economy is populated by an absolutely continuous distribution of firms over the compact support $[0,N(t\left)\right]$, and total output is given by $Y\left(t\right)={\int }_0^{N\left(t\right)}m(i,t)Y(i,t)\ast di$. Note that when $m(i,t)>0$ there is a continuum of firms each producing $Y(i,t)$, thus assuring that such sub-market is competitive, while if $m(i,t)=0$ there are no firms producing the $Y(i,t)$ level.

13 A similar approach has been pursued in Chantrel, Grimaud, and Tournemaine (2012) with reference to knowledge demanded by producers of differentiated goods, each one with its specific sub-market and Lindahl price. They also show that under perfect information and full excludability this is the only market equilibrium.

14 See Example 1 in the working paper version of this paper (Marchese and Privileggi 2016) for an illustration on how Lindahl prices are determined when production is Cobb-Douglas.

15 As $m_1=m_2=1$, in Figure 1 the representative firm's demand coincides with the total sub-market demand.

16 Note that agents share the ownership of the cultural heritage, which is collective and not excludable for them. Property rights on subsets of $A_0$ cannot be assigned.

17 In Cozzi (1998) the dynamic of such common cultural endowment is endogenized by a market for non-technical education.

18 If we are interested in an asymptotic balanced growth path (ABGP) it is sufficient to assume that $\underset{t\to +\mathrm{\infty }}{lim}\left[1/{\epsilon }_u\right(C\left)\right]=\underset{t\to +\mathrm{\infty }}{lim}\{-u^{\mathrm{\prime }}(C\left)/\right[u^{\mathrm{\prime }\mathrm{\prime }}\left(C\right)C\left]\right\}=1/\sigma$. This approach will be pursued in Section 5.

19 As a matter of fact, all mainstream extensions of Romer's model implicitly assume time-dependent costs of producing new ideas. In such a framework, increasing research costs can even offset scale effects (Cozzi and Spinesi 2004). However, like Romer's one, all these models are based on knowledge production functions that use labor as a main input factor, so that the cost of new knowledge depends on the equilibrium wage. Because, according to (3(3) $\dot{A}=\frac{J}{\eta },$(3) ) and (14(14) $\dot{A}\left(t\right)=\frac{J\left(t\right)}{\eta \left(t\right)},$(14) ), in our setting we assume that knowledge is produced through financial investment rather than labor, these contributions are not directly comparable with our model.

20 Note that (14(14) $\dot{A}\left(t\right)=\frac{J\left(t\right)}{\eta \left(t\right)},$(14) ) encompasses also the case in which new knowledge is being produced by decentralized $R\mathrm{\&}D$-firms for a price $\eta \left(t\right)=\psi \left[A\right(t\left)\right]$ that includes a mark-up over the Tsur and Zemel (2007) first-best cost $\varphi \left[A\right(t\left)\right]$, as in Marchese et al. (2017).

21 This might correspond to the introduction of a limited duration of patents, an extension of the model that we leave for future research. We thank an anonymous referee for suggesting this direction of further investigation.

22 See Section 6 in the working paper version of this paper (Marchese and Privileggi 2016) for a worked out example under different assumptions on the cost of knowledge production.