Complex Economies Have a Lateral Escape from the Poverty Trap

We analyze the decisive role played by the complexity of economic systems at the onset of the industrialization process of countries over the past 50 years. Our analysis of the input growth dynamics, considering a further dimension through a recently introduced measure of economic complexity, reveals that more differentiated and more complex economies face a lower barrier (in terms of GDP per capita) when starting the transition towards industrialization. As a consequence, we can extend the classical concept of a one-dimensional poverty trap, by introducing a two-dimensional poverty trap: a country will start the industrialization process if it is rich enough (as in neo-classical economic theories), complex enough (using this new dimension and laterally escaping from the poverty trap), or a linear combination of the two. This naturally leads to the proposal of a Complex Index of Relative Development (CIRD) which shows, when analyzed as a function of the growth due to input, a shape of an upside down parabola similar to that expected from the standard economic theories when considering only the GDP per capita dimension.


Basic Solow Model of Growth and Growth Accounting
In this appendix we give a simple version of the models of economic growth based on [2] and [5]. Even if the aim of the paper is an empirical analysis, we thought it might be useful to the potential reader new to the macroeconomic analysis to have a simple version of economic model aimed at explaining growth. Moreover this derivation is required in our analysis to decompose the growth in input growth and exogenous growth.
We start writing a production function as generic as possible, where Y c,t is the production of country c at time t, A c,t is an efficiency measure and, for different js, I j c,t are the different inputs of the production (Physical Capital, Labor, Human Capital, ...). The production function gives the output of the economy for different levels of inputs and efficiency. The growth of output, that we will identify with GDP in the following, can therefore be the consequence of an efficiency and technological growth, i.e. a growth of A, or an input growth.
In growth models some inputs are accumulated in an endogenous way: a part of the output is invested to build new physical capital, a part of the working time of the laborers is spent to train new workers and accumulate human capital. Their level in equilibrium is the result of their accumulation and their depreciation. Growth due to input accumulation is therefore called endogenous growth. On the opposite side, growth due to technology and efficiency, a growth of A, is called exogenous growth.
In the following we will use a minimal case. We will take the standard Cobb-Douglas production function with the combination of two inputs, physical capital K c,t and labor L c,t : If a fraction s of the output Y is invested in the production of new physical capital K and a fraction δ of K decades at each time step due to depreciation, the time evolution of physical capital is Since in this simple model A c,t and L c,t are assumed to be exogeneous processes (for a case in which the technological progress is seen as an endogenous process see [3,7]), we will have as the only endogeneous variable K c,t . Notice therefore that any equilibrium we find in these coupled equations for K c,t is actually an equilibrium path, a value K * (A c,t , L c,t ) such that K c,t = K * do not change if the exogeneous variables do not change. Therefore, since the equilibrium point K * depends on A c,t and L c,t , even at the equilibrium K c,t will still grow if those exogeneous variables grow.
In Fig 1(a) we show that this equation has only one stable equilibrium path. The equilibrium is possible due to the decreasing returns on capital: the more capital a country has, the less output the country gains with an additional unit of capital. However already in [2] there were the idea that multiple equilibria in the capital levels are possible. If for example the investing rate s is not independent from the per capita income of the country, but it depends on the achievement on a minimal level of subsistence, the capital accumulation function becomes non linear. We can for example assume a functional form of the investing rate like, where s c is a country-dependent parameter and K F a minimum threshold to achieve subsistence and start investing. As a consequence of this non linearity the system can present a behavior similar to Fig 1(b).
After overcoming a barrier, in Fig 1(b) represented by point K * 3 , the country capital would move endogenously to K * 2 . The out-of-equilibrium dynamics from one equilibrium to another has to be characterized by fast input accumulation and, therefore, fast endogenous economic growth. A way of looking the results of our empirical analysis (see the Results section) is that K F depends on the Fitness of the country.
In this setting we can also understand the decomposition From 2, the GDP per capita is equal to where L is now written as E, the number of employees, times H, a factor related to the average human capital of the work force, and both sides of the equation are divided by the total population P of the country. Therefore, defining with the lowercase letters the growth rates of the respective uppercase variables and with the hat the division by population, In equation 5 we dropped the hats notation to simplify the reading, and y, k, and e are presented directly as the GDP per capita, physical capita per capita, and employment rate.

Robustness checks with the 1995-2010 database
A natural question is whether our results are robust when considering other time ranges and, as a consequence, different countries that experience the industrialization process and the catching up. As a consequence, in this appendix we reproduce the main results for a different database, described in [8], which covers a time span which ranges from 1995 to 2010 (the results in the main text being obtained with the database described in [1], which covers the period 1963-2000). Given the shorter time period (which has been approximately reduced to one third with respect the previous analysis), we would expect a noisier structure but, as we will see in the following, this is not the case, establishing that our results are not only sound but also replicable for smaller datasets.
In Fig 2 we replicate the same analysis of Fig 2 of the main text. One can easily see the different performance, in terms of input growth, of high fitness countries with respect to the low fitness countries at a low level of GDP per capita. After the catching up, on the contrary, the input growth becomes very similar.
The next step is to study the dependence of the input growth from all the fitness ranges, that is, to go from a discrete classification in high and low fitness countries to the study of all the fitness spectrum. This analysis leads to     We plot in blue the isolevels of GDP growth due to input growth, ans in shades of grays, the estimation error of GDP growth due to input, where black means a standard error of 0.4% or more, and white a standard error of 0.2% or less.