Endogenous growth and technological progress with innovation driven by social interactions

Abstract

We analyze the implications of innovation and social interactions on economic growth in a stylized endogenous growth model with heterogeneous research firms. A large number of research firms decide whether to innovate or not, by taking into account what competitors (i.e., other firms) do. This is due to the fact that their profits partly depend on an externality related to the share of firms which actively engage in research activities. Such a share of innovative firms also determines the evolution of technology in the macroeconomy, which ultimately drives economic growth. We show that when the externality effect is strong enough multiple BGP equilibria may exist. In such a framework, the economy may face a low growth trap suggesting that it may end up in a situation of slow long-run growth; however, such an outcome may be fully solved by government intervention. We also show that whenever multiple BGP exist, they are metastable meaning that the economy may cyclically fluctuate between the low and high BGP as a result of shocks affecting the individual behavior of research firms.

Appendices

Appendix 1: The rationale behind random utility models

In this appendix, we briefly summarize the main ideas recovered by Brock and Durlauf (2001) and leading to the profit structure defined in (1). Suppose that a research firm faces the binary decision to innovate or not to innovate. We define the binary random variable $\omega \in \{0,1\}$ accordingly. The main assumption behind random utility models is that the profit $\pi$ related to the innovation has the following general structure:

$$\begin{aligned} \pi (\omega _i)=R(\omega _i, \mu _i^e(\omega _{-i}), h)-\zeta (\omega _i), \end{aligned}$$

where revenues R depend on the choice made by the firm, on the price h received by the buyer of the innovation and by an externality term. Indeed, each firm i estimates the conditional probability measure $\mu _i^e$ on the choices of others, where $\omega _{-i}$ denotes the vector of actions deprived of the i-th component. As seen in Sect. 2, costs are random and denoted by $\zeta$. For the moment, we set $z=0$ for simplicity.

We now make some further (minimal) assumptions to came up with a tractable profit structure.

• (i) $\pi (0)=0$. This is an obvious normalization. Both R and $\zeta$ are zero if no research activity is in place. Therefore, we concentrate on $\pi (1)$ (we call it simply $\pi$). Rearranging variables and notations, we have:

  $$\begin{aligned} \pi =R( \mu _i^e(\omega _{-i}), h)-\zeta _i. \end{aligned}$$

• (ii) Externalities due to the behavior of competitors only depend on the average action of others’ choice. This implies that $\mu _i^e(\omega _{-i})$ is substituted by the (simpler) statistics $x_i^e=\frac{1}{N-1}\sum _{j\ne i} x_{ij}^e$, where $x_{ij}^e={\mathbb {E}}^{(i)}[\omega _j]$ denotes the expectation of firm i about the choice of competitor j. Therefore,

  $$\begin{aligned} \pi =R(x_i^e, h)-\zeta _i. \end{aligned}$$

  Concerning the information structure of the model, we also assume that ${\mathbb {E}}^{(i)}[\cdot ]={\mathbb {E}}^{(j)}[\cdot ]$ for all $i,j=1,\dots , N$. This amounts in saying that all firms share the same expectations about others’ choices.

• (iii) We assume that $\frac{\partial \pi }{\partial x^e_i}=J$. This simplifying assumption introduces a unique parameter J measuring the degree of dependence (or the force of externality) due to the others’ actions. Note that $J>0$ resembles a staying-on-the-shoulder situation, whereas $J<0$ a fishing-out case. Secondly, as obvious, $\frac{\partial \pi }{\partial h}>0$.

• (iv) We assume that the pecuniary effects due to the sale of the technology and the externalities are additive. Moreover, for sake of simplicity, we assume a linear dependence. This fact, together with assumption iii), produces the following payoff:

  $$\begin{aligned} \pi =h+J x_i^e-\zeta _i. \end{aligned}$$

• (v) Finally, we slightly correct $x^e_i$ by substituting it with $x^e_i-\frac{1}{2}$. The reason is that we want the decision to be driven by what the majority of the population of firms is doing. The quantity $x_i^e-\frac{1}{2}$ reflects exactly this goal: it is positive if and only if the majority of the research firms produces an innovation. Therefore, in case of a positive J, the single firm is more prone to align with the majority. On the contrary, if $J<0$, the firm will tend to behave in the opposite direction. We obtain:

  $$\begin{aligned} \pi =h-\zeta _i+J \, \left( x_i^e-\frac{1}{2}\right) . \end{aligned}$$

Therefore, by reintroducing a private cost z and recalling that $\pi (0)=0$, we obtain the general expression for $\pi$ as it appears in (1):

$$\begin{aligned} \pi (\omega _i)=\omega _i \left[ h-(z+\zeta _i)+ J\left( x_i^e-\frac{1}{2}\right) \right] . \end{aligned}$$

By applying the payoff structure defined above, we can verify that

$$\begin{aligned} \omega _i=1 \, \iff \, \pi (1)\ge \pi (0) \, \iff \, h-(z+\zeta _i)+ J\left( x_i^e-\frac{1}{2}\right) \ge 0. \end{aligned}$$

The probabilistic structure of the model implies that for all $i=1,\dots , N$,

$$\begin{aligned} {\mathbb {P}}(\omega _i=1)= & {} {\mathbb {P}} \left( h-(z+\zeta _i)+ J\left( x_i^e-\frac{1}{2}\right) \ge 0\right) \\= & {} {\mathbb {P}} \left( \zeta _i \le h-z+ J\left( x_i^e-\frac{1}{2}\right) \right) . \end{aligned}$$

Since agents receive different private signals, agents may have a different feeling about the best choice. Heterogeneity gives rise to the non-trivial equilibria and the (possible) multiplicity discussed in Proposition 2. In the case of a completely deterministic model (i.e., $\zeta _i=0$ for all i), agents would be homogeneous and we would obtain:

$$\begin{aligned} {\mathbb {P}} \left( 0 \le h-z+ J\left( x_i^e-\frac{1}{2}\right) \right) \in \{0;1\}, \end{aligned}$$

meaning that either $\omega _i=0$ or $\omega _i=1$ for all $i=1,\dots ,N$. The same reasoning extends to the continuous-time counterpart described by (4): assuming no randomness, there would be no space for any dynamics, and the outcome would be to a large extent trivial with all firms deciding either to innovate or not to innovate. In the body of the paper, we thus focus on the most interesting situation in which agent heterogeneity gives rises to non-trivial dynamics. Most of our qualitative results in Proposition 2 would still hold true in the absence of heterogeneity case, but in this case the BGP equilibrium would be necessarily unique and characterized by either one of the two extreme long-run growth rates $\gamma =\frac{\phi }{1-\alpha }$ if $\overline{x}=1$ or $\gamma =0$ if $\overline{x}=0$.

Appendix 2: Proof of Proposition 3

The proof of Proposition 3 basically follows Theorem 4.6 in Olivieri and Vares (2005). In order to make this reading as much self-consistent as possible, we sketch the proof rearranged to match our model and our notations. We firstly specify the functional form for $\varDelta$. To this aim, we introduce the so-called Gibbs free energy:²¹

$$\begin{aligned} f_{\beta ,J,h}(m)=-\left( \frac{J}{4}m^2+{hm} \right) + \frac{1}{\beta }\cdot \varepsilon (m), \end{aligned}$$

(35)

where

$$\begin{aligned} \varepsilon (m)=\frac{1+m}{2}\ln \left( \frac{1+m}{2}\right) +\frac{1-m}{2}\ln \left( \frac{1-m}{2}\right) . \end{aligned}$$

Finally, define

$$\begin{aligned} \varDelta =\beta \,\left( f(m_M)-f(m_L) \right) \end{aligned}$$

where f is as defined in (35), $m_M=2\bar{x}_M-1$ and $m_L=2\bar{x}_L-1$ and where $\bar{x}_L$ and $\bar{x}_M$ are, respectively, the smallest and the middle solutions (recall that, under our assumptions on the values of the parameters, this equation admits three real solutions $\bar{x}_L<\bar{x}_M<\bar{x}_H$) to

$$\begin{aligned} \frac{1}{2}\,\tanh \left\{ \beta \left[ h-z+J\left( x_t-\frac{1}{2}\right) \right] \right\} -x_t+\frac{1}{2}=0. \end{aligned}$$

In what follows, we organize the proof Proposition 3 into four steps. In the first step, we provide a lower bound for $T_N$, in the second an upper bound. Finally, we prove part (a) and part (b) of the proposition. As said, we only sketch the main results and refer the reader to Olivieri and Vares (2005) for further details. Our aim is mainly to let the reader appreciate the probabilistic properties, which this proposition relies on.

• (i) There exists a positive constant $c_1$ such that, for N large enough $\bar{,}$ and each positive integer T,

  $$\begin{aligned} {\mathbb {P}}(T_N\le T)\le c_1 T e^{-N\varDelta }. \end{aligned}$$

  (36)

  This fact follows from the properties of the stationary distribution of a Markov chain. Indeed, let us define the stationary measure of $(x^N_t)_{t\ge 0}$ as $\nu _N$. It can be proved that

  $$\begin{aligned} {\mathbb {P}}(T_N\le T)\le T \cdot \frac{\nu _N(x_M^N)}{\nu _N(x_L^N)}=T\cdot e^{-N\beta (f(m_M)-f(m_L))}. \end{aligned}$$

  On the other hand, for N large enough, $\beta \, f(m_M)-f(m_L)\ge \varDelta -\frac{c_2}{N}$ for a suitable constant $c_2$. Therefore, (36) easily follows by putting $c_1=e^{c_2}$.

• (ii) For any positive sequence $(\varphi _N)_{N\ge 1}$ such that $\varphi _N \rightarrow \infty$,

  $$\begin{aligned} {\mathbb {P}}(T_N\ge e^{N\varDelta } N \varphi _N )=0. \end{aligned}$$

  (37)

  This follows from the fact that, for suitable constants $c_3$ and $c_4$,

  $$\begin{aligned} c_3 e^{N\varDelta } \le {\mathbb {E}}(T_N)\le c_4 N^2 e^{N\varDelta }. \end{aligned}$$

  (38)

  For details on the proof of (38), we refer to Corollary 4.9 in Olivieri and Vares (2005). From (38) and applying the Markov inequality, we obtain (37).

• (iii) Point (a) of Proposition 3 follows from the fact that

  $$\begin{aligned}&1-{\mathbb {P}}\left( \frac{1}{\varphi _N} e^{N\varDelta }< T_N<e^{N\varDelta } N \varphi _N \right) \nonumber \\&\quad ={\mathbb {P}}\left( T_N\le \frac{1}{\varphi _N}e^{N\varDelta }\right) +{\mathbb {P}}\left( T_N\ge e^{N\varDelta } N \varphi _N \right) . \end{aligned}$$

  Both terms of the RHS go to zero for any positive sequence $(\varphi _N)_{N\ge 1}$ such that $\varphi _N\rightarrow \infty$ due to (i) and (ii), respectively. This proves part (a) in Proposition 3.

• (iv) Define the sequence of random variables $(\tilde{T}_N)_{N\ge 2}$, where $\tilde{T}_N{:=}\,T_N/\gamma _N$ and where $\gamma _N$ is such that

  $$\begin{aligned} \lim _{N \rightarrow \infty } N^{-1}\ln ( \gamma _N)=\varDelta . \end{aligned}$$

  It can be shown that this sequence is tight and its limits $\tau$ along subsequences have the property that

  $$\begin{aligned} {\mathbb {P}}(\tau>t+s)={\mathbb {P}}(\tau>t)\,{\mathbb {P}}(\tau >s). \end{aligned}$$

  This, in turns, shows that $\tilde{T}_N$ is asymptotically exponential, thus, memoryless. Finally,

  $$\begin{aligned} \lim _{N\rightarrow \infty } {\mathbb {E}} [\tilde{T}_N] =\int _0^{+\infty } \lim _{N\rightarrow \infty } {\mathbb {P}}(T_N>s\, \gamma _N) \, ds=\int _0^{+\infty }e^{-s}ds=1\,, \end{aligned}$$

  and this concludes the proof of Proposition 3. To be precise, what we have shown in point iv) is true for a stopped version of $x^N$: consider $\tilde{x}^N$ where $\tilde{x}^N_t$ has the same transition probabilities of $x^N_t$ for $t\le T_N$ and $\tilde{x}^N_t\equiv \tilde{x}^N_{T_N}$ for $t\ge T_N$ (it is a stopped version of the original process at time $T_N$). It can be proved that the two processes are coupled up to $T_N$, so that the their probabilistic features are the same. Since we are interested in the trajectories up to $T_N$, working with $x^N$ or $\tilde{x}^N$ is exactly the same to our purposes. $\square$

Note, finally, that the transition from $\bar{x}_L$ to $\bar{x}_H$ can be analyzed exactly in the same way, by simply considering $\varDelta =\beta \,\left( f(m_M)-f(m_H) \right)$. We would like to stress the fact that, differently from the common notion of cycles in macroeconomics in which their periodicity is highly irregular and stochastic, in probability theory the notion of cycles requires the periods of the transitions to be deterministic and constant. According to this latter view, the tunneling time $T_N$ should converge to 1, rather that to an exponential random time with average 1 as stated in Proposition 3. Therefore, even if this is not totally correct from a probabilistic point of view, in our discussion we adopt the macroeconomic view and terminology by referring to the metastability property as a cycling behavior.