## Abstract

We incorporate the external effects of capital in production and endogenous technology choice into the standard overlapping generations model. We demonstrate that our model can exhibit a poverty trap, a middle-income trap, and perpetual growth paths. We also show that, under some economic conditions, an economy exhibits all three of these phenomena, depending on its initial capital level, and that the economy caught in the middle-income trap can exhibit chaotic fluctuations in the long run. This is a stark contrast to the result obtained by Umezuki and Yokoo (J Econ Dyn Control 100:164-175, 2019a) that we cannot observe any chaotic fluctuations. Because the model of Umezuki and Yokoo (2019) is a special case of our model in the sense that there is no externality in production, our result means that we need the combination of technology choice and externalities in production to obtain chaotic fluctuations in the standard overlapping generations model with Cobb-Douglas technologies.

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## Notes

In endogenous business cycle theory, economic fluctuations occur spontaneously as a result of nonlinear factors within the economy, without any shocks from outside the economy. Early studies in this line include Benhabib and Day (1982), Grandmont (1985), Benhabib and Nishimura (1985), and Nishimura and Yano (1995).

Iwaisako (2002) is an exception. He considered two possible technologies: a constant returns to scale technology and an increasing returns to scale technology. However, he relied exclusively on graphical analysis.

In general, external effects or externalities mean that the action of an agent affects other agents’ costs or benefits without going through the market transactions. For example, knowledge accumulation has positive externalities for society (e.g., newly obtained mathematical theorems are freely available to everyone). On the other hand, crime is an example of a negative externality with social costs.

For example, see Azariadis and John (2005) for a survey.

As a recent theoretical work, Hu et al. (2022) show that the degree of externalities plays a significant part in technology choice and makes it possible to explain the empirically observed development patterns (a poverty trap, a middle-income trap, and a flying geese pattern of economic development) in a unified way. See also Asano et al. (2022a) and references therein for details.

This means that even if the degree of external effects observed on average in the economy is relatively small, we can have chaotic fluctuations, because, in our model, the selected technology continues to switch between technology with externalities and that without externalities.

Regarding another possible interpretation, we may assume that the firm chooses its production technology in a discrete manner in the first stage and then chooses optimal inputs in the second stage.

Several studies measure external effects by estimating the percentage increase in a firm’s output caused by a 1% increase in aggregate inputs (or aggregate output), keeping an individual firm’s inputs unchanged. Caballero and Richard (1989), Caballero and Richard (1992)) estimated the external effect in the US manufacturing industry and obtained values ranging from 0.49 to 0.89 and from 0.32 to 0.49 in their 1989 and 1992 studies, respectively. Caballero and Lyons (1990) also provided estimates for European countries ranging from 0.29 to 1.40. Moreover, the values estimated by Lindström (2000) for Swedish manufacturing range from 0.16 to 0.53. By contrast, using the industry-level manufacturing data for the United Kingdom, Oulton (1996) found no evidence of either external effects or increasing returns to scale. These results show that the degree of external effects varies across countries and industries.

Let

*I*be a compact interval and \(f:I\rightarrow I\) be a piecewise-smooth map \(x_{t+1}=f(x_{t})\). The Lyapunov exponent \(\lambda\) of*f*on*I*is defined by$$\begin{aligned} \lambda =\lim _{n\rightarrow \infty }\frac{1}{n}\sum _{t=0}^{n}\log |f^{\prime }(x_{t})|,\quad \text { whenever the derivative exists}. \end{aligned}$$If the Lyapunov exponent is positive, then the map is chaotic in the long run.

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## Acknowledgements

The authors acknowledge an anonymous reviewer for comments on this paper. The authors would also like to thank Real Arai, Keita Kamei, and seminar participants at Seinan Gakuin University for their helpful comments and suggestions. This research was financially supported by International Joint Research Center of Advanced Economic Research of Kyoto Institute of Economic Research and the Grants-in-Aid for Scientific Research, Japan Society for the Promotion of Science (20H05631, 21K01388, and 23K01468).

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This is a revised version of our preprint, Kyoto Institute of Economic Research (KIER) Discussion Paper No.1075, which is referred to in References (Asano et al. (2022c)).

## Appendix

### Appendix

### Proof of Claim 2

Let \(\alpha _1A_1/\alpha _2A_2=1\) and \(1>\alpha _2>\alpha _1\). Then, all we need to show is that the following inequalities are possible:

Rewriting the above expression as:

we notice that \(\alpha _2/\alpha _1(1-\alpha _1)<1/(1-\alpha _2)\) always holds because \(\alpha _2>\alpha _1\). As \(sA_2\) can take any positive value, the claim is proven. \(\square\)

### Proof of Lemma 2

Let \(\alpha _1A_1/\alpha _2A_2=1\) and \(1>\alpha _2>\alpha _1\). Then, the first inequality in condition (9) can be rewritten as:

Similarly, the second inequality in condition (9) is expressed as

Letting \(V=1/(1-\alpha _2)\) and \(W=\alpha _1/\alpha _2(1-\alpha _1)\), we observe that \(V>W\) because \(\alpha _2>\alpha _1\). Because \(sA_2\) can take any positive value, it suffices to show that \(V^{1-1/\beta _1}W^{1/\beta _1}<V^{\beta _2}W^{1-1/\beta _2}\) or \(V^\gamma <W^\gamma\), where \(\gamma =1-1/\beta _1-1/\beta _2\). Thus, the last inequality holds if \(\gamma <0\) or \(1/\beta _1+1/\beta _2>1\). \(\square\)

### Proof of Lemma 3

Using the same notations as in the proof of Claim 2, it suffices to show that \(V^{1-1/\beta _1}W^(1/\beta _1)>V^{\beta _2}W^{1-1/\beta _2}\) or \((V/W)^\gamma >1\), where \(\gamma =1-1/\beta _1-1/\beta _2\). Because \(V/W>1\), the last inequality holds if we take \(\beta _1\) and \(\beta _2\) (\(\beta _2>\beta _1>1\)) such that \(\gamma >0\) or \(1/\beta _1+1/\beta _2<1\). \(\square\)

### Proof of Lemma 4

Let \(s\in (0,1)\) and \(\alpha _2\in (0,1)\) (hence, \(\eta _2=\beta _2-\alpha _2\)) be fixed. Let \(a_i\) (\(i=1, 2\)) be any numbers such that \(1<a_1<a_2\). Let \(\alpha _1A_1/\alpha _2A_2=1\). Then, inequalities (15) and (16) can be reduced to:

Solving the following simultaneous equations for \(A_2\) and \(\alpha _1\),

we obtain:

which verifies the assertion. \(\square\)

### Proof of Lemma 8

Let us begin with \(\tau _1(0)=1-c\beta _1>c\), which implies from Lemma 7 that any trajectory visits \(I_R\) at least once immediately after visiting \(I_L\). To ensure that the trajectory stays successively twice in \(I_R\), we require that:

To ensure that the trajectory stays in \(I_R\) successively at least three times, we have:

Repeating this up to *n* times, we obtain:

Solving the last inequality for *c* yields the result. \(\square\)

### Proof of Proposition 8

Note that for the value of the threshold *c* in the assumption of the proposition to be taken, it must hold that:

This inequality is equivalent to \(\beta _1\beta _2<1+\beta _1\), which is assured by assumption. Because Lemma 7 indicates that any trajectory (i.e., irrelevant to the initial conditions) of \(\tau\) visits \(I_L\) successively at most once, it follows for any initial condition \(x_0\in (0,1)\) that:

where the last inequality follows by assumption. Thus, \(\tau\) is eventually expanding and hence chaotic in the sense of Lasota and Yorke (1973). By conjugacy *h*, *T* is also chaotic. \(\square\)

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Asano, T., Shibata, A. & Yokoo, M. Technology choice, externalities in production, and a chaotic middle-income trap.
*J Econ* (2023). https://doi.org/10.1007/s00712-023-00844-9

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DOI: https://doi.org/10.1007/s00712-023-00844-9