Catch-up and fall-back through innovation and imitation

Will fast growing emerging economies sustain rapid growth rates until they “catch-up” to the technology frontier? Are there incentives for some developed countries to free-ride off of innovators and optimally “fall-back” relative to the frontier? This paper models agents growing as a result of investments in innovation and imitation. Imitation facilitates technology diffusion, with the productivity of imitation modeled by a catch-up function that increases with distance to the frontier. The resulting equilibrium is an endogenous segmentation between innovators and imitators, where imitating agents optimally choose to “catch-up” or “fall-back” to a productivity ratio below the frontier.

1. Chapter 5 of the World Bank Golden Growth publication (Gill and Raiser 2012) focuses on “Europe’s innovation deficit,” discussing the American “innovation machine” and the European “convergence machine.” Some of the main questions in Golden Growth are “whether Europe has fundamental flaws in its economic environment that make its innovation deficit a fact of life” and “what should European governments do to increase innovation?” Indeed, market failures and political pressures may well prevent optimal investments in technology adoption and innovation. By contrast, from the perspective of this theory, imitation and fall-back may be an optimal outcome from the perspective of the social planner or representative consumer. We show that countries close to the technology frontier may in fact optimally choose to limit their investments in innovation and technology adoption in favor of higher immediate consumption, and rely on technology diffusion externalities to sustain their growth rate, even if their distance to the technology frontier initially slips a bit. This force is discussed in the context of the “benefits to backwardness” in Klenow and Rodriguez-Clare (2005).

2. Along similar lines, see Howitt and Mayer-Foulkes (2005). For an exposition see (Aghion and Howitt (2009), pp. 152–158).

3. Acemoglu et al. (2006) introduce a model where high skill entrepreneurs can enhance productivity growth through innovation but not through technology adoption. When the distance to the frontier is large, firms may retain a low-skill entrepreneur but replace the entrepreneur at some cost when the distance closes and innovation becomes important. This introduces productivity growth that depends on the distance to the frontier. In our model, we allow the agents to optimally choose any level of investment in innovation as well as in technology adoption, rather than having firms make discrete decisions on the retention of low skill entrepreneurs. In both models, productivity growth and optimal actions depend on the distance to the frontier.

4. The literature on technology and knowledge spillovers also explores a similar trade-off between an agent’s own investment in R&D and exploiting externalities by investing in imitating the innovations of more productive firms. In Eeckhout and Jovanovic (2002) the total factor productivity of an agent depends on both its own capital and that of all better firms in the economy. Kortum (1997) has agents investing in a stochastic research technology, where the efficiency of new technologies depends on a spillover from the entire stock of existing research.

5. Technologies to produce goods could consist of various processes, not all of which are instantly adopted. For example, production may start with an assembly plant and the manufacturing of parts. Associated processes may be gradually adopted as know-how develops and accumulates. In 1977, on visiting a Samsung research lab in Korea, Ira Magaziner wrote: “[It] reminded me of a dilapidated high-school science room\(\ldots \)They’d gathered color televisions from every major company in the world—RCA, GE, Hitachi—and were using them to design a model of their own” (see (Magaziner and Patinkin 1990, p. 24)).

6. Jones (2005) explores idea based models of growth and the role of non-rivalrous inputs in generating endogenous growth. In our model, innovating agents have non-rivalrous productivity but are given no legal method to profit from imitation and face no negative effects from being copied. Moreover, technology diffusion is slow, copying modest improvements of the frontier sequentially as described in Footnote 5. This is in contrast to models of directed imitation at lower levels of aggregation, where agents make a rapid jump to the frontier by copying the most productive individual firms. In general, patent systems typically require active monitoring and expensive litigation that are only optimal for cutting-edge products with high rents. Interpretations of our model with, and without, scale effects are discussed in Sect. 6.

7. Segerstrom (1991) allows monopolistically competitive ex-ante identical firms to choose investment in innovation and imitation. Given the structure of costs and competition in Segerstrom (1991), in contrast to our model, the firms on the technology frontier choose not to invest in innovation and are eventually overtaken by firms who are behind the frontier and invest in innovation.

8. Other significant works that address the size distribution of innovations and/or productivity across sectors/firms are in Aghion and Howitt (1998), Thompson (2001) and Laincz (2009). Aghion and Howitt (1998) show that under some conditions there exists a stationary distribution of productivity levels in an otherwise standard quality-ladder endogenous growth model. Thompson (2001) derives in closed-form the steady state distribution of firms size in a model that builds on Thompson and Waldo (1994). Laincz (2009) studies an ambitious computational model that integrates into growth economics the approach of Ericson and Pakes (1995) to industry dynamics.

9. Using the frontier (the maximum statistic) to summarize the distribution provides a great deal of analytical tractability. Perla and Tonetti (2013) and Lucas and Moll (2013) use random search models to focus on the feedback relationship between the entire endogenously determined productivity distribution and optimal technology adoption decisions. Alvarez et al. (2012) combine a similar random search structure with a model of trade to explicitly micro-found technology diffusion across countries. These papers do not feature an innovation technology or an endogenously evolving technology frontier. Dependence only on the frontier can be thought of as directed search with no differential search costs. The effect of directed investment in imitation, with imitation costs increasing as the distance to the imitated grows, is left for future research.

10. Note that in the logistic case \((m=1)\) the increase in productivity \(\dot{z}\) from diffusion, \(\tilde{D}(t,z) z=cz \left( 1-\left( \tfrac{z}{F(t)}\right) \right) \), is zero when \(z =0\) or when \(z =F\left( t\right) \), and it is maximized at \(z =F(t) /2\). By contrast in the Nelson–Phelps case \((m=-1),\,\tilde{D}(t,z) z =cz \left( \left( \tfrac{F(t)}{z}\right) -1\right) =c\left( F(t)-z\right) \), the larger the distance to the frontier, the larger is the diffusion flow.

11. If \(\frac{m\sigma }{c}>1\) then \(x^{*}=0\) and there is no imitation region. In this case, all agents choose to innovate at the same rate, as is shown in Sect. 4.2.2.

12. Since \(x\) and \(\lambda \) are continuous on an optimal path, at the threshold \( x^{*} = \left( 1 - \frac{\sigma m}{c} \right) ^{1/m}\), from the left,

    $$\begin{aligned} \frac{\dot{x}}{x}=s \frac{c}{m} \left( 1- x^{m}\right) -g =s \frac{c}{m}\left( \frac{m\sigma }{c} \right) -g =s\sigma -g, \end{aligned}$$

    and from the FOC, coming from the left,

    $$\begin{aligned} B-s&= \left( \lambda x \frac{c}{m} \left( 1-x^{m}\right) \right) ^{-1}=\left( \frac{1}{r}\frac{c}{m}\left( \frac{m\sigma }{c}\right) \right) ^{-1},\\ s&= B-\frac{r}{\sigma }>0, \\ \frac{\dot{x}}{x}&= s\sigma -g = B\sigma -r -g = 0. \end{aligned}$$

    So the growth rate is continuous and equal to the leader’s growth rate from the left as \(x \rightarrow (1-\frac{m\sigma }{c})^{1/m}\). Note though that \(s\) will not be continuous at the threshold.

13. For a general strictly monotone \(D(x)\) function, from Eq. (27), \(x^{*} = D^{-1}(\sigma )\). Using Eqs. (26) and (21) and solving for the optimal control in the stationary equilibrium, \(\bar{s} = \frac{B D(\bar{x}) - r}{D(\bar{x}) - \bar{x}D'(\bar{x})}\). Assuming the existence of a stationary equilibrium with imitation and using the law of motion in Eq. (15a) gives the implicit equation defining \(\bar{x}\),

    $$\begin{aligned} B \sigma - r = \frac{B D(\bar{x})- r}{1 - \bar{x}D'(\bar{x})/D(\bar{x})}. \end{aligned}$$

14. The solution for the \(m=0\) case needs to be solved with the Hamiltonian directly in terms of \(D(x) = -c \ln (x)\) as discussed in Sect. 2. Using similar methods to those in Sect. 4.2, it can be shown that the stable solution is:

    $$\begin{aligned} x^{*} \!=\! \exp \left( \!-\!\frac{\sigma }{c}\right) ,\quad \bar{x} \!=\! \exp \left( \!-\!\frac{\sigma }{2 c} \!+\!\frac{\sqrt{\!-\!4 c r\!+\!4 B c \sigma \!+\!B \sigma ^2}}{2 c\sqrt{B}}\right) ,\quad \bar{s} \!=\! \frac{-B \sigma +\sqrt{B} \sqrt{-4 c r+4 B c \sigma +B \sigma ^2}}{2 c}. \end{aligned}$$

    Finally, it can be shown that this solution is the same as the limit of the \(m \ne 0\) case, ensuring that there is no discontinuity at \(m=0\).

15. The productivity improvement technology does not generate mixing, in that the relative productivity rankings of agents is constant for all time. Additionally, innovators never become imitators and imitators never become innovators. This is because the model is deterministic and investments gradually improve growth rates as opposed to facilitating discrete advances in productivity levels. Shocks to productivity would introduce leapfrogging and switches between innovation and imitation.

16. For some numerical examples that fulfill Assumptions 1, 2, and 3:

    • Nelson–Phelps \((m=-1)\): \(x^{*} =0.5263\) and \(\bar{x}= 0.3148\), when \(\sigma =1;c=.9;B=2;r=0.05.\)

    • Gompertz \((m=0)\): \(x^{*} = 0.4066\) and \(\bar{x} = 0.2258\), when \(\sigma =.9;c=1;B=2;r=0.05.\)

    • Logistic \((m=1)\): \(x^{*}=0.1\) and \(\bar{x} =0.0520\), when \(\sigma =.9;c=1;B=2;r=0.05.\)

17. We may alternatively consider \(B\) as an intrinsic parameter of per capita productivity. In that case, \(s\) and \(\gamma \) would be interpreted as per capita investment in intensity in imitation and innovation. If every agent within a country is identical, then the number of agents within a country would not affect productivity dynamics and there would be no size or scale effects. There may, however, be other factors that affect per capita productivity and induce heterogeneous \(B\) across countries. Another alternative interpretation is that scale effects may be avoided by subdividing countries with different factor endowments into economic regions that equalize relative factor endowments across regions.

18. If \(B > \bar{B}/\hat{B}\), then imitating is not optimal and the agent would want to innovate. However, if the agent innovates, it does so at a faster rate since \(B > \hat{B}\), and hence they become the frontier, invalidating equilibria with \(B > \bar{B}(\hat{B})\).

19. This result is similar to the force in Benhabib and Spiegel (2005).

20. The parameters used in this simulation are \(\bar{\sigma } = 1.25, \bar{c} = 1, B = 2\) and \(r = 0.05\). Using these parameters, Fig. 4 plots \(\bar{x}(A)\) for \(m=-.5\) and \(m=.5\). When \(m=0.5,\,\frac{\bar{x}(\infty )}{\bar{x}(1)} = 0.989\). When \(m = -.5,\,\frac{\bar{x}(\infty )}{\bar{x}(1)} = 0.991\). For visual clarity, the plotted axes do not intersect at the origin.

21. To see this, first note, using a first order Taylor expansion of the convex function \(x^{m}\) around \(m=0\), that \(1-x^{m} > -mx^{m} \ln x\). Then using this inequality, \(\frac{d\left( \frac{c}{m} \left( 1-x^{m}\right) \right) }{dm}= \frac{c}{m^{2}}\left( x^{m} -1 -m x^{m} \ln x \right) < \frac{c}{m^{2}}\left( x^{m}-1+\left( 1-x^{m}\right) \right) = 0\).

22. The first derivative of the term is \(\ln (1- \frac{\sigma m}{c})\) and the second derivative is positive: \(\frac{1}{1- \frac{\sigma m}{c}}>0\) for \(1- \frac{\sigma m}{c}>0\).

23. Note: \(\lim _{m \rightarrow 0^{+}} \frac{\mathrm{d}x^{*}}{\mathrm{d}m} = \lim _{m \rightarrow 0^{-}} \frac{\mathrm{d}x^{*}}{\mathrm{d}m} = \frac{-\sigma ^{2}}{2c^{2}}e^{\frac{-\sigma }{c}} < 0\).

24. The parameters used in this simulation are \(\sigma = 0.9, \hat{c} = 2, B = 2\) and \(r = 0.05\). The normalization parameter \(\hat{c}\) is chosen such that \(x^{*}\) and \(\bar{x}\) are equal across the two experiments at \(m=1\). Figures are not drawn to scale.

25. These arguments are loose in that the sufficient conditions for concavity do not necessarily hold for large \(c\). Also, note that the flow costs \(s(t) z(t)\) of an instantaneous jump to the frontier are negligible in our setup as \(c \rightarrow \infty \). If there is a discrete as opposed to flow consumption cost proportional to \(z(t)\) for enabling technology diffusion, whether a jump occurs or not will depend on the discrete benefit of jumping to the frontier technology relative to the discrete cost.

26. By Assumption 1, \(\sigma B > r >0\), so it is sufficient to show \(B \frac{1}{m}(1 - \bar{q}) - \frac{\sigma B}{c} > 0\). By Lemma 1, in this region \(\frac{1}{m}\left( 1 - \bar{q}\right) > \frac{\sigma }{c}\).

We would like to thank Boyan Jovanovic and Tom Sargent for useful feedback.

Authors and Affiliations

1. Department of Economics, New York University, 19 W 4th St, 6th Floor, New York, NY, 10012, USA

   Jess Benhabib

2. Vancouver School of Economics, University of British Columbia, 997-1873 East Mall, Vancouver, BC, V6T 1Z1, Canada

   Jesse Perla

3. Graduate School of Business, Stanford University, 655 Knight Way, Stanford, CA, 94305, USA

   Christopher Tonetti

Corresponding author

Correspondence to Jess Benhabib.

Appendix 1: Solving for policy functions

1.1 1.1 Optimal investment in innovation by the leader

The leader must choose optimal investment in innovation by solving

with

and

Defining \(\tilde{\mu } =\tilde{\lambda } z\) gives

The solution is

and the transversality condition,

immediately requires

Maximizing the Hamiltonian with respect to \(\gamma \) yields

Using Eq. (57), optimal investment in innovation satisfies

which holds with equality since \(\gamma > 0\) under Assumption 1. Finally, it is straightforward to show that the maximized Hamiltonian is concave, so that this is indeed an optimal solution. See Appendix 4 (Supplementary Material) for details.

1.2 1.2 Optimal investment in imitation

In the region \(x < x^{*}\), only imitation will occur, as stated in Lemma 1. To solve for the policy functions of agents in this region it is useful to perform another change of variables, where the relative distance of the agent to the frontier is distorted by the diffusion parameter \(m\): \(q(t) \equiv x(t)^m\). Using the chain rule:

The law of motion for \(q\) is obtained using this change of variables together with the law of motion for \(x\) (Eq. 15a) when \(\gamma = 0\):

The first order necessary condition in Eq. (22), with \(\gamma = 0\), gives

The law of motion for \(\mu \) in terms of \(q\) is obtained from Eq. (25)

Equations (60), (61), and (62) form a system in the variables \(\mu ,q,s\). Eliminating \(\mu \) via substitution will create a 2x2 system. First, differentiating Eq. (61) provides

Substituting Eqs. (61) and (63) into (62), and rearranging for \(\dot{s}\) gives

Substituting for \(\dot{q}\) from the law of motion in Eq. (60), generates a first order ODE in \(s\) and \(q\):

Equations (60) and (65) are a system of 2 nonlinear first-order ODEs in \(q\) and \(s\). Stacking the equations and defining the vector valued function \(\Psi \) yields

To find the stationary solution, evaluate the first order condition in Eq. (61) at the stationary value \(\bar{s}\),

Substituting for \(\bar{\mu }\) from Eq. (26) and solving for \(\bar{s}\) gives

From Assumption 1 and Lemma 1 it follows that \(\bar{s}> 0\).^{Footnote 26} To determine the stationary solution for \(q(t),\,\bar{q}\), substitute this expression into the stationary law of motion in Eq. (60) to find

Using the known value of \(g\), this becomes a quadratic equation in \(\bar{q}\):

The roots of this quadratic are

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As is proven in Appendix 2.1, the unique interior stationary solution, \(\bar{x} \in (0,x^{*})\), is

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Appendix 2: Stability and uniqueness

1.1 2.1 Uniqueness of interior stationary \(\bar{x}\)

Lemma 2

For any quadratic of form \(y^{2} -a y +b\) with roots \(y_{1}\) and \(y_{2}\), if \(a, b > 0\) and \(\text {Discriminant}>0\), then both roots are real and positive. Furthermore, \(y_{1},y_{2} \notin [\min \{b,1\}, \max \{b,1\} ]\) if \(a > 1 + b\).

Proof

The result that roots are real and positive is a standard property of quadratic equations, as \(b\) is the product of the roots and \(a\) is the sum of the roots. Assume that there exists a root \(\hat{y} \in [\min \{b,1\}, \max \{b,1\} ]\). Note that the product of the roots is given by \(P(\hat{y}) = \hat{y}(a-\hat{y})\). Since \(P(\hat{y})\) is concave, the argmin of \(P\) is either \(b\) or 1. If we show that \(P(b)>b\) and \(P(1)>b\), then we have a contradiction. \(P(1) = a-1\) and \(P(b) = ba -b^{2}\), so \(P(1) > b\) if \(a > 1 + b\) and \(P(b) > b\) if \(a>1+b\). Thus if \(a > 1 + b,\,\hat{y}\) can not be in \([\min \{b,1\}, \max \{b,1\} ]\).

To show that a unique root \(\bar{x} \in (0,x^{*})\) exists, we take Eq. (72) in the form of \(\bar{q}^2 - a \bar{q} + b\), where

\(b>0\) from Assumption 2. \(a > 0\) using Assumptions 1 and 2. Additionally, the discriminant is positive:

Thus the \(\bar{q}\) roots are real and positive, and hence \(\bar{x}\) roots are real and positive. It remains to show that there is a unique \(\bar{x} \in (0,x^{*})\). That is, there exists a unique \(\bar{q}\) s.t. \(\bar{q}^{\frac{1}{m}} < b^{1/m}\). Using Assumption 1, the conditions in Lemma 2 are satisfied, i.e., \(a > 1+b\):

Thus, by Lemma 2, \(\bar{q}_{1}, \bar{q}_{2} \notin [\min \{1-\frac{m\sigma }{c},1\}, \max \{1-\frac{m\sigma }{c},1\} ]\). Note for \(m>0\), if both roots were less than \(1-\frac{m\sigma }{c}<1\) or both roots were greater than one, the product could not be \(1-\frac{m\sigma }{c}\). Note for \(m<0\), if both roots were greater than \(1-\frac{m\sigma }{c}>1\) or both roots were less than one, the product could not be \(1-\frac{m\sigma }{c}\). Let convention be that \(\bar{q}_{1} < \bar{q}_{2}\). Thus \(\bar{q}_{1} < 1-\frac{m\sigma }{c}\) and \(\bar{q}_{2} > 1-\frac{m\sigma }{c}\).

Note if \(m>0\), then \(\bar{x}< (1-\frac{m\sigma }{c})^{1/m}\) iff \(\bar{q} < (1-\frac{m\sigma }{c})\) and if \(m<0,\,\bar{x}< (1-\frac{m\sigma }{c})^{1/m}\) iff \(\bar{q} > (1-\frac{m\sigma }{c})\). Therefore, the smaller root, \(\bar{q}_{1}\), is the unique stationary solution if \(m>0\) and the larger root, \(\bar{q}_{2}\), is the unique stationary solution if \(m<0\). Thus there exists a unique root \(\bar{x} \in (0,x^{*})\).

1.2 2.2 Stability of BGP

The proof of stability for the entire parameter space is completed as follows: First, prove stability for a particular \(r\) by ensuring the determinant of the Jacobian of the dynamic system \(\Psi (\cdot ,\cdot ),\,J_{\Psi }(\cdot ,\cdot )\), is negative. Then, use a homotopy argument to show that the system must remain stable as \(r\) is varied throughout the rest of the valid parameter space.

Repeating Eq. (36),

Taking the Jacobian of this and simplifying yields

For the starting point of the homotopy argument, take the limit as \(r \rightarrow \sigma B\), where \(g \rightarrow 0\). As proven in Appendix 2.1, \(\bar{q}_1\) is the unique stationary root if \(m > 0\), and \(\bar{q}_2\) is the unique stationary root if \(m < 0\). In either case, the stationary solution, \(\Psi (\bar{s},\bar{q})=0\), as \(r \rightarrow \sigma B\) is

Substituting these into the the Jacobian in Eq. (77) generates

Taking \(r \rightarrow \sigma B\) and simplifying further gives

The determinant of this Jacobian evaluated at the steady state in the limit as \(r \rightarrow \sigma B\) is

The parameter restriction in Assumption 2 ensures that \((m\sigma - c) < 0\), and thus \(\det \left( J_{\bar{\Psi }}\right) < 0\). Thus for a small neighborhood around \(r = \sigma B\) the system has local saddle-point stability. For clarity of exposition, let \(\Gamma (s,x)\) be the equivalent system to \(\Psi (s,q)\) in \(x\) space. Since \(\det \left( J_{\bar{\Psi }}\right) <0\) near \(r=B\sigma \), then \(\det \left( J_{\bar{\Gamma }}\right) <0\) near \(r=B\sigma \).

To complete the proof, use a homotopy argument for varying \(r\), by parameterizing the system of equations by \(r\): \(\Gamma _{r}(s(r),x(r))\). Let \(y(r) = (s(r),x(r))\) and the steady state, \(\bar{y}(r) =\left( \bar{s}(r), \bar{x}(r) \right) \), be defined by \(\bar{\Gamma }_{r} \equiv \Gamma _{r}\left( \bar{y}\right) = 0\) for \(r\in S\) where \(S=\left\{ \varepsilon ,B\sigma -\varepsilon \right\} \) for any small \(\varepsilon >0\). Since \(\det \left( J_{\bar{\Gamma }_{r}}\right) <0\) near \(r=B\sigma \), it is negative at \(r = B \sigma -\varepsilon \) for small \(\varepsilon >0\).

Let \(\left\{ \bar{y}(r) \right\} =\left\{ \Gamma _{r}^{-1}(0) \right\} \) for isolated zeros of \(\Gamma _{r}(\bar{y})\). For \(r \in S\) let \(\bar{y}(r) \in \mathcal {D}\,\subset (0,B) \times (0,1)\), where \(\mathcal {D}\) is chosen so that there are no \(\bar{y}(r)\in \partial \mathcal {D}\). Then the homotopy invariant Poincare–Hopf degree on the boundary \(\partial \mathcal {D}\) of \(\mathcal {D}\) is \(\mathop {\displaystyle \sum }_{y \in \left\{ \bar{y}(r) \right\} } \mathrm{sign} \det \left( J_{\Gamma _{r}}(y)\right) = 1\) or \(-1\). See Milnor (1965). Since, from Eq. (38), the stationary point \(\bar{x}(r)\), and therefore \(\bar{y}(r)\), is always unique for \(r \in S\), then \(\mathrm{sign}\det \left( J_{\Gamma _{r}}(\bar{y}(r) ) \right) =\mathrm{sign} \det J_{\Gamma _{r}}(\bar{y}\left( B\sigma -\varepsilon \right) )=-1\).

If the roots of \(J_{\Gamma _{r}}\left( \bar{y}(r) \right) \) are real, then they must be of opposite sign. Therefore \(\det \left( J_{\Gamma _{r}}\left( \bar{y}(r) \right) \right) <0\) for \(r\in S\) and the system is locally saddle-point stable: given \(x(0)\) we can choose \(s(0)\) on the stable manifold converging to \(\bar{y}\left( r\right) \). Note that the roots of \(\det \left( J_{\Gamma _{r}}\left( \bar{y}(r) \right) \right) \) cannot be complex since the determinant, the product of the roots, would have to be positive in that case. Therefore \(x\) is a continuous function of time described by the solution of \(\dot{x}=\sigma \gamma x+s\left( x\right) x \frac{c}{m}\left( 1- x^{m}\right) \) with \(s\left( x\right) \) optimally chosen for \(r\in S.\) Therefore the BGP has to be globally stable since \(\frac{dx}{dt} \ne 0\) at points other than at the unique \(\bar{x}(r)\), or the BGP would not be unique. Thus, it must be that \(\frac{dx}{dt}>0\) to the left of \(\bar{x}\) and \(\frac{dx}{dt}<0\) to the right of \(\bar{x}\), so the unique BGP is stable: all agents with productivities \(x(t)<x^{*}\) eventually end up at \(\bar{x}\left( r\right) \).

Appendix 3: Agents with heterogeneous \(B\)

Assume that the agents have idiosyncratic \(B\), s.t. \(B_{i}>0 \quad \forall i\). Define the \(B\) of the frontier agent at time \(t\) as \(\hat{B}(t) \equiv \left\{ {B_i | z_i = \max _{i'}\left\{ {z_{i'}}\right\} }\right\} \). In this Appendix section, we prove Proposition 3.

1.1 3.1 Eventually, the frontier agent’s type is constant

To show that there are no oscillations at the frontier in the stationary equilibrium, assume there was an oscillating set of frontier agents with different values of \(B\). Compare pairwise any two values for \(B\) within this set. In the instance where the higher \(B\) becomes the leader, the first order conditions show that it will continue to innovate at a rate faster than that of a lower B innovator, and thus never fall behind. By process of pairwise iteration through the set of potential frontier agents, the highest \(B\) within the set of potential frontier agents must eventually become the leader for all time. Thus, there exists a \(\hat{t}\) such that for all \(t > \hat{t}, \hat{B}(t) = \hat{B}\). Then, following the solution for the frontier agent in Sect. 3, \(g \equiv \hat{B}\sigma - r\) for all \(t > \hat{t}\).

To solve for full dynamics, the problem is no longer autonomous, as \(g\) depends on the growth rate of the current frontier agent. However, for \(t > \hat{t}\) the Hamiltonian for an agent with state \(x\) and parameter \(B\) is identical to Eq. (16).

1.2 3.2 Regions of innovation and imitation

Following identical algebra to Sect. 4.2.1, the indifference point between innovating and imitating is \(x^{*} \equiv \left( 1 - \frac{\sigma m}{c} \right) ^{1/m}\). Hence, the innovation threshold is independent of the particular \(B\) of an agent and the \(\hat{B}\) of the frontier agent.

1.3 3.3 Innovation only: \(x > x^{*}\)

Consider the stationary solution where an agent with parameter \(B\) innovates forever. The first order condition is identical to Eq. (32). As they innovate forever, the Lagrange multiplier is identical to Eq. (57). Combining these equations yields the optimal policy, \(\gamma (B) = B - r/\sigma \). Substituting into the law of motion generates,

If \(B \ne \hat{B},\,x\) cannot be stationary as it grows at a constant rate towards either the frontier or the imitation region. Hence, the only agents in the innovation region of the stationary solution are those with \(B = \hat{B}\).

1.4 3.4 Imitation only: \(x < x^{*}\)

For the stationary solution, the algebra follows identically to Eq. (70). The growth rate of the frontier is \(g(\hat{B}) = \sigma \hat{B} - r\). In Eq. (70), define \(\hat{\sigma } \equiv \sigma \frac{\hat{B}}{B}\), then Eq. (71) becomes

The derivation is then identical to Appendix 1.2, with the new \(\hat{\sigma }(\hat{B})\). This yields the quadratic

The solution for the interior root is then \(\bar{x}(B)\), where

(84)

(85)

1.5 3.5 Existence of interior \(\bar{x}(B) \in \left( 0, 1 - \frac{m \sigma \frac{\hat{B}}{B}}{c} \right) \)

Using \(\hat{\sigma } = \sigma \frac{\hat{B}}{B}\), by Lemma 2, \(\bar{q}_{1},\bar{q}_{2}\notin [\min \{1-\frac{m\sigma \frac{\hat{B}}{B}}{c},1\},\max \{1-\frac{m\sigma \frac{\hat{B}}{B}}{c},1\}]\). Note for \(m>0\), if both roots were less than \(1-\frac{m\sigma \frac{\hat{B}}{B}}{c}<1\) or both roots were greater than one, the product could not be \(1-\frac{m\sigma \frac{\hat{B}}{B}}{c}\). Note for \(m<0\), if both roots were greater than \(1-\frac{m\sigma \frac{\hat{B}}{B}}{c}>1\) or both roots were less than one, the product could not be \(1-\frac{m\sigma \frac{\hat{B}}{B}}{c}\). Let convention be that \(\bar{q}_{1}<\bar{q}_{2}\). Thus \(\bar{q}_{1}<1-\frac{m\sigma \frac{\hat{B}}{B}}{c}\) and \(\bar{q}_{2}>1-\frac{m\sigma \frac{\hat{B}}{B}}{c}\).

Now note that if \(m>0\), then \(\bar{x}<(1-\frac{m\sigma \frac{\hat{B}}{B}}{c})^{1/m}\) iff \(\bar{q}<(1-\frac{m\sigma \frac{\hat{B}}{B}}{c})\) and if \(m<0, \bar{x}<(1-\frac{m\sigma \frac{\hat{B}}{B}}{c})^{1/m}\) iff \(\bar{q}>(1-\frac{m\sigma \frac{\hat{B}}{B}}{c})\). Therefore, the smaller root, \(\bar{q}_{1}\), is the unique stationary solution if \(m>0\) and the larger root, \(\bar{q}_{2}\), is the unique stationary solution if \(m<0\). Thus, there exists a unique root \(\bar{x}(B) \in \left( 0,1-\frac{m\sigma \frac{\hat{B}}{B}}{c}\right) \).

We can now derive \(\underline{B}\) and \(\bar{B}\).

1.6 3.6 Derivation of \(\underline{B}\)

From Eq. (85), solve for the \(B\) such that \(\bar{x} = 0\), which determines the lower crossing point. Solving for this root gives,

From Assumption 2, \(\underline{B} < \hat{B}\), and if the agent has the \(B\) of the frontier (i.e., \(\hat{B} = B\)), then this reduces directly to Assumption 2. Note, if \(m<0\), then \(\not \exists \; \; B>0 \, \text{ s.t. } \bar{x}=0\).

1.7 3.7 Derivation of \(\bar{B}\)

From Eq. (85), solve for the \(B\) such that \(\bar{x} = x^{*}\), which determines the upper crossing point. Solving for this root gives,

Rearranging yields

From Assumption 1, which guarantees positive growth at the frontier, \(\frac{r}{\sigma \hat{B}}<1\), and from Assumption 2, \(c-m\sigma >0\). Thus, \(1 < \frac{\bar{B}}{\hat{B}} < \infty \).

Lemma 3

For all \(B<\hat{B},\,\bar{x}(B)<x^{*}\).

Proof

Remember that \(x^{*}= (1-\frac{m\sigma }{c})^{\frac{1}{m}}\) and \(\bar{x}<(1-\frac{m\sigma \frac{\hat{B}}{B}}{c})^{1/m}\). To determine the relationship between \(x^{*}\) and \(\bar{x}\) there are four cases, \(m\) positive or negative and \(\frac{\hat{B}}{B}\) above or below one.

If \(B<\hat{B}\), for both \(m>0\) or \(m<0\), then

However, if \(B>\hat{B}\), for both \(m>0\) or \(m<0\), either

In brief, if \(B<\hat{B}\), then \(\bar{x}<x^{*}\), but if \(B>\hat{B}\) then it is possible but not necessary for \(\bar{x}>x^{*}\).

Lemma 4

For all \(B<\bar{B},\,\bar{x}(B)<x^{*}.\)

Proof

We have shown that given \(\hat{B}\) there is a unique \(\bar{B}\) such that \(\bar{x}=x^{*}\). Now we can show that for \(B<\bar{B},\,\bar{x}(B)<x^{*}\). From Lemma 3, for \(B<\hat{B},\,\bar{x}(B)<x^{*}\). But there is a unique \(\bar{B}>\hat{B}\) at which \(\bar{x}(B)=x^{*}\), so by continuity of \(\bar{x}\) in \(B,\,\bar{x}(B)<x^{*}\) for \(B \in (\hat{B},\bar{B})\).

To prove that the solution is stable for \(t > t^{*}\), parameterize \(r(\tau )\) and \(B(\tau )\) to be continuous functions for \(\tau \in [0,1]\), where \(\tilde{r}(0) = \sigma \hat{B},\,\tilde{r}(1) = r,\,\tilde{B}(0) = \hat{B}\), and \(\tilde{B}(1) = B\). Then use the same homotopy argument as in Appendix 2.2, but over \(\tau \) instead of \(r\). As the set of \((B,r) \in [0, \bar{B}]\times [\sigma B, \infty )\) is connected, any point in the parameter space can be reached by appropriately chosen functions \(r(\tau )\) and \(B(\tau )\).

Finally, note that for any \(B_{1}\) and \(B_{2}\) such that \(B_{1} < \bar{B}(B_{2})\) and \(B_{2} < \bar{B}(B_{1})\), there exist initial conditions leading to stationary solutions, \((\bar{x}(B_{2}), 1)\) and \((1, \bar{x}(B_{1}))\). Hence, given a set of \(B\), there are several possible stationary solutions. Now, for each of these initial conditions, perturb an \(\epsilon \) such that \((\bar{x}(B_{2}) + \epsilon , 1)\) and \((1, \bar{x}(B_{1}) + \epsilon )\). Since \(\bar{x}(B) < x^{*}\) and the solutions are stable, these perturbed initial conditions will approach the respective stationary solutions. Hence, the system is not ergodic, and the stationary solution depends on the initial conditions.

1.8 3.8 Non-existence of innovators with \(B \ne \hat{B}\)

By contradiction, separately consider the cases where \(B > \hat{B}\) and \(B < \hat{B}\).

Assume that there exists a \(B\) such that \(B > \hat{B}\) with a stationary \(x^{*} \le x^s(B) < 1\). From Eq. (81), if the agent is an innovator and \(B > \hat{B}\), then \(x(t)\) has a constant, positive growth rate and hence is not stationary (i.e., will eventually overtake the existing \(\hat{B}\) frontier).

Alternatively, assume that there exists a \(B < \hat{B}\) such that \(x^{*} \le x^s(B) < 1\). From Eq. (81), if the agent is an innovator, then \(x(t)\) has a constant, negative growth rate and is not stationary (i.e., will eventually drop to \(x(t) \le x^{*}\), and hence enter the imitation region).

1.9 3.9 Non-existence of equilibria with imitators \(B>\bar{B}\)

We want to show that there does not exist an equilibrium in which agents with \(B>\bar{B}\) imitates. There are two possible cases: the agent invests only in imitation or the agent invests in some combination of imitation and innovation.

For case one, assume by contradiction that an equilibrium exists in which an agent with arbitrarily large \(B/\hat{B}\) invests only in imitation. As proved in the derivation of \(\bar{B} > \hat{B}\), since \(\bar{x}(B)\) crosses \(x^{*}\) at \(\bar{B}\), any agent with \(B > \bar{B}> \hat{B}\) would optimally choose to invest in innovation, contradicting that it optimally invests only in imitation.

For case two, assume there exists a stationary equilibrium such that \(B > \hat{B}\) and the agent invests in both innovation and imitation simultaneously. By using the two first-order necessary conditions in Eq. (27), this is only possible if \(x^s(B) = x^{*}\), where the agent optimally remains at the \(x^{*}\) threshold. Next, note that from the first order condition, \(\sigma = D(x^{*})\), so the law of motion is

Since the returns to innovation and imitation are identical at \(x^{*}\), the optimal investment composition between innovation and imitation is indeterminate. Nonetheless, total investment is \(\bar{\gamma } + \bar{s} = \hat{B} - \frac{r}{\sigma }\).

Rephrasing the contradiction, assume that for some \(B > \hat{B}\), the choice of \(x^s(B) = x^{*}\) and \(\bar{\gamma } + \bar{s} = \hat{B} - \frac{r}{\sigma }\) is the optimal solution. Then the optimal path of \(z\) is \(z(t) = z_0 e^{g(\hat{B}) t}\), where \(z_0\) is an initial condition and \(g(\hat{B}) \equiv \sigma \hat{B} - r\). Substituting into the agent’s optimization problem in Eq. (1) yields,

An alternative feasible policy would be for this agent to choose to become the frontier agent and grow at a rate \(g = \sigma B - r\). In that case \(z(t) = z_0 e^{g(B) t}\) and the value is

Comparing the value of the two policies by subtracting Eq. (89) from Eq. (91),

Since \(B - \hat{B}>0\),

This expression holds with equality iff \(B = \hat{B}\). Therefore, we have a contradiction as the choice to become the frontier agent dominates the policy of investing in innovation and imitation simultaneously to remain at \(x^{*}\) with leader \(\hat{B}\).

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