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Endogenous growth - A dynamic technology augmentation of the Solow model

In this paper, I endeavour to construct a new model, by extending the classic exogenous economic growth model by including a measurement which tries to explain and quantify the size of technological innovation ( A ) endogenously. I do not agree technology is a"constant"exogenous variable, because it is humans who create all technological innovations, and it depends on how much human and physical capital is allocated for its research. I inspect several possible approaches to do this, and then I test my model both against sample and real world evidence data. I call this method"dynamic"because it tries to model the details in resource allocations between research, labor and capital, by affecting each other interactively. In the end, I point out which is the new residual and the parts of the economic growth model which can be further improved.


The problems of exogenous growth
The Solow-Swan model (1956) is a simplified way to express economic growth: In this model, the following variables are included: L -Labor . Measured in counts of physical people which actively engage in work.
A -Technology . A variable explaining technology level, including innovation, ideas, research, know-how. Helps to create more output with the same L and K .  (1-⍺) × K ⍺ (2) However, even the augmented model does not explain large variations in GDP per capita in different countries, based only on s (savings from capital which are reinvested), n (population growth rates) and H (quality of the human capital), and leaves this, again, to the so-called "residual", which is technology ( A ). This problem of Solow model's simplification becomes even more obvious as the share technological innovations continues to grow at present. 10 Therefore, I propose a new model, which treats technology as an endogenous variable. My approach is different by not only by considering human capital accumulation ( H ) but also physical capital's contribution ( K ), together with the introduction a depreciation model for human capital ( H ). I also include in my considerations the patent discovery rates, incentives provided for research and the size of active labor force or researchers available (which turns out to affect the model greatly). Empirical part: I test the scaling of my proposed models with sample and real world data sets. I try to determine which aspects of technology ( A ) affect output per capita ( Y/L ) most significantly. I also present the use of machine learning methods on how to distinguish anomalies in subsets of data.

Endogenous growth: Technology as an endogenously influenced function
Closing thoughts on general growth models and their limitations: I try to explain why growth models are imperfect by design and which are the areas left for further improvement.
In order to achieve all these difficult tasks, during our research we will also benefit from the service of our virtual assistant, Lee Chong, who has just returned to Singapore, after getting his Engineering degree in MIT. He is now employed at "ACME CORP -Singapore" and is asked which division he would like to join: Production department, where the engineers write code for future businesses, or the smaller R&D department, in which code is not being written, but existing algorithms are improved and new ones are created. Lee decides to go with the second, one which initially does not make his boss happy. Working in R&D means Lee will not be able to really "earn" money for the company, or at least not until he achieves his first patent. But the question is, will the company profit from hiring Lee? 10 I acknowledge that I have drawn my inspiration from Romer's paper on the topic (1990), however my approach on interpreting and modeling the problem are very different. I try to quantify some of the simplifications Romer has made. The main difference is I try to not focus so much on the division of human capital ( H ) but its combination with physical capital in order to allocate their product to research or production. Also, I use the terms "design", "patent" and "idea" interchangeably, because to my knowledge ideas cannot be measured in any other reliable way.

Overview of existing theoretical definitions on technology (A)
During the years, there have been different definitions of technological advancement ( A ). Solow (1956) treats it as global exogenous good, available to all countries.  defines it as a public 11 input, provided by the government. However, I do not consider these realistic enough, as even if technology was provided by an external party "for free", it would still cost time for labor force to learn it, and then to re-adapt the existing processes to it. Arrow (1962) assumes that an increase in capital ( K ) leads to increase in knowledge as well, due to "learning by doing", however he still treats knowledge and technology ( A ) as public good. Romer (1987 presents two models, which includes private, maximizing behavior's role in generating technological change, by introducing patent prices, monopoly options and human capital division, by referencing to the consumption goods theory of Dixit and  and  . 12 13 He models technology ( A ) as different "recipes" with which consumption goods ( K ) can be combined with labor ( L ), to produce output ( Y ). He also acknowledges that research has a cost: "Once the cost of creating a new set of instructions has been incurred, the instructions can be used over and over again at no additional cost. Developing new and better instructions is equivalent to incurring a fixed cost", however, he does not include this in his calculation for simplicity purposes (this is one of the points which I address in my model).

. K and L: Physical capital and Labor force
Romer suggests that any employee ( L ) can choose to participate by either investing his human capital (H) to work in production of final output (Y) (which generates output for the current period t ) or in human capital accumulation for technology (A) (which generates output in future period t + x , where x is the 14 amount of periods needed to reach the break-even point from the investment in research): However, he does not acknowledge the role of physical capital consumption, by accepting the following   simplification of nonrivalry: "For simplicity, the arguments here will treat designs as idealized goods that are not tied to any physical good and can be costlessly replicated, but nothing hinges on whether this is literally true or merely close to being true." While I do agree with a simplification, which would ignore the cost of the paper, on which the design is printed or multiplied, the capital which I would like to focus on is the one "invisibly consumed" until the paper with the design is created.
But let's go back to our assistant Lee. His company "ACME CORP -Singapore" is famous around the world, thanks to it's unique human capital of intelligent and young engineers. However, would those engineers even consider to do research for the company, if it wasn't for its large and quiet testing labs, free hardware and infinitely "free" testing samples? While Lee has been investing mostly his human capital ( H ) only in the research, he is happily unaware of the research equipment costs, or even the cost of him not producing direct output for the present. Therefore, it costs capital ( K ) to invest in the creation of a new design ( A ), since it implies an opportunity cost. And then again it costs time and resources for the firm to apply the new design in their process. By conducting research in order to improve technology ( A ), certain portion of the capital ( K ) and labor ( L ) need to be allocated to research and development. Thus, investing in research to improve technology ( A ), actually decreases the capital ( K ) and labor ( L ) for immediate output. And this is why Lee's boss is unhappy he chose to work in the R&D department.
While Romer includes interest rate in his model, as a way of measuring how 'costly' it would be to invest in research at present, in order to benefit from it in future, I have chosen to follow a different approach, by dividing physical capital ( K ) and labor ( L ) to production and research parts:

H: Human capital:
Romer suggests human capital ( H ) is to be divided to two parts: research and production. In my model, I cannot consider how "one can be trained or study for innovation or production", so I express human capital ( H ) in its classic form. One might ask why human capital ( H ) cannot be infinitely effective, just like technology ( A ) in cutting-edge growth. I observe that as people become older, at first their human capital ( H ) increases steeply, then it stagnates and starts to depreciate slowly with age. No worker ( L ) is able to preserve his human capital ( H ) beyond his mortal life. Even worse, people start to underperform as they get older and after the person dies, the human capital ( H ) is lost (with some part of it, hopefully, transferred to the permanent technology ( A ) during his lifetime -this link is modeled with formulas (10) and (11). Therefore, I think it is more realistic to model human capital by adding a slight depreciation factor to it. I introduce a log-normal distribution function in order to model more realistically human capital's ( H ) following attributes: until a certain critical inflex point (which can be a certain age or IQ), workers benefit exponentially from increase in their human capital ( H ). Beyond this inflex point, they start to experience diminishing returns, and even slow decrease in their current productivity. An university student, who excessively studies (not to be confused with doing research!) until his early thirties without actually putting to practice his learned skills at active labor, thus producing no output. He is well beyond his human capital's ( H ) inflex point and will need to recoup his extra time put in study (while hoping for a higher salary), unlike his peers, who engaged in the labor market earlier.
The second example is with one of the professors of Lee in MIT, named Mr. Stein, who is 79 years old. Lee remembers that while he enjoyed his professor, he often noted that he tends to forget some facts due to his age, together with some of his knowledge already being outdated.
Thus, we need to add to human capital ( H ) a depreciation rate : d H d (6) With the depreciation rate being defined recursively as: d d = √ e lnH lnH (7) applies is log-normal distribution function which models the aforementioned anomalies with human d capital ( H ) as it grows: to diminishing returns, then is slowly counterproductive after reaching an inflex point G . is the optimal human AG ︿ capital accumulation (with subject to diminishing returns).
is the ineffective human capital accumulation. GB

A: Technology, ideas, know-how
Another important effect is the cost of introducing or learning a new technology. It takes time and resources to produce each new innovation or patent. If Lee needs to learn a new library, in order to work with the new software, he is going to lose some days to learn it. However, it must be noted, that thanks to his higher human capital ( H ), he is able to learn it quicker (since he has more experience). Thus, we can point that there should be a relation between human capital ( H ) and costs of introducing a new technology ( A ). In order to model technology ( A ) reliably, I have tested few alternatives. The first one is suggested by Tabarrok and Cowen (2013) : 15 A 1 = Population × Incentives × Ideas per hour (8) The problem with this version is that it makes no sense to represent ideas per hour, as it does not change the model much. Another problem is that it takes as a measure the total population count, while Romer (1990) points out his error in his previous paper (1987) that it is more accurate to measure the human capital devoted to research instead. But perhaps the worst part is, if you try this formula with real data, it tends to scale too much, primarily due to Population. If Population is replaced with percent of population engaged in research (refer to Appendix 2 ), then it has better results, but it still overscales slightly, while introducing the problem of not taking the total population in account (which again leads to some distortions in very big or small countries). It also lacks human capital ( H ), since a more educated labor force ( L ) would be able to grasp technology ( A ) faster. Below, trying to avoid the aforementioned problems, I offer a better way of quantifying technology ( A ): A 2 = 1 + (Researchers × Incentives × Ideas per capita) (9) However, this model has a fundamental weakness: It does not scale with human capital ( H ).

How can a country with a low human capital ( H ) benefit from latest technological innovations ( A )? Who
is going to harness this knowledge, without education? The better trained the labor force ( L ) is, the faster they can develop new or embrace existing technologies ( A ). This is why, we are also adding a relation of So technology accumulation, augmented with the level of human capital becomes: (10) The last model includes the full benefit from incentives and government spending on research. It scales well when simulated with sample data (refer to Experiment 1 and 2 ). My simulations with sample and real world data show the third model ( A 3 ) to be the most stable, both with rich and poor countries, without any significant distortions (at least according to findings so far). Expressed in a stricter mathematical notion: Where P is patent applications (ideas) , L is size of labor force available, L research is labor force engaged 16 in research and K research is capital size spent on research and H is human capital level.
After comparing models A 2 and A 3 with sample data, the results is they differ only at their speeds of scaling (influenced via human capital ( H )). Both of them display the same pattern: Graph 3: Comparison of technology functions growths. 16 I choose to use patent applications, instead of approved patents, since an idea can be rejected by the patent office.
The actual number of ideas should be even larger, since not all researchers and inventors patent their ideas.
Both models A 2 ( ) and A 3 ( ) are scaling decently, with A 2 scaling at a slower rate than model A 3

Endogenous growth: Formulating the complete model
So, let's inspect again the classical production function again: After applying the above models of each variable, the dynamic endogenous growth function becomes:

Part III: Description of methods, data and tools used
In this section I describe which methods and tools I use in testing the model against data evidence in the empirical section.

Model assessment methods:
The following variables and methods will be used to determine the fit of collected data and if the compared models are better or worse than their alternatives: 1. t -statistic : Departure ratio of a parameter, which is estimated from its notion value and its standard error . We will assume a critical value of 1.97 , meaning that it has 95% chance of being statistically 5. Adjusted R 2 -The explanatory power of the regression (explained variation to total variation ratio).
Interpretation : worst: 0, best: 1. It is to be noted, that since in the current paper we will not filter our 18 country data by groups (Oil countries, developing countries and etc.) the data set will contain more "contrasting" data, which will result in imperfect fits, leading to overall lower level of R-squared, but with the tradeoff of testing more universally applicable theories.
6. Log-likelihood : Cannot be used standalone as measure of good fit, but as subjective measure, the higher value, the better the model.

Software tools used:
• Data is stored in tables via Google Sheets . 19 • Regression analysis is done via Gretl .

21
• The online service "Blockspring " has been used for the machine learning algorithms calculation 22 (K-Means clustering ).

Variable building:
Error  α -Capital's share income (assumed as 0.3333, based on recommendation of the RMW paper) Pa -Patent applications (not to be confused with approved patents).
Pc -Ideas per capita. Calculated by dividing patent applications ( Pa ) to L (Labor).
Ph -Ideas per hour. Calculated by dividing patent applications ( Pa ) to 8760 (total hours in one year).
Hd -Human capital, with its depreciation rate applied. Calculated by the formulas (6) and (7).  Data has been filtered at 5 year intervals (1965, 1970, 1975, 1980, 1985, 1990, 1995, 2000 and 2005) in order to be able to run regressions involving capital savings rate and human capital (education attainment) The cleaned up files and original sources have been attached in the Appendix 4 , with data collected and summarized for the period between 1960-2009.

Sample data set:
I have also generated a data set with sample, linearly increasing data. The purpose is to simulate a complete data set, without missing variables, in order to test the behavior of my models for which real life data was not available at present, due to incomplete reporting . I hope in future those models can be ran 25 with real life data as well. All variables in this data set are scaling linearly, with constant growth rates. 25 T he particular variables are amount of researchers per country and GDP percentage spending on research (prior year 1995).

Part IV: Empirical tests and results
In my tests, I start with the simplest model, and then I add regressors one by one, in order to confirm if the model becomes more accurate. In the end, I test with all variables of the model included.

Experiment 1 : Comparison of growth with constant and dynamic technology augmentation
In this experiment, we will compare Solow's classic growth ( ) function against our dynamically OB ︿ augmented model ( ). Both models are using same linearly increasing values for human capital (H), OA ︿ but the dynamic model is allocating a percentage of physical capital (4%) and labor (0.025%) to research (refer to Appendix 3 ). However, all of this comes at cost. As seen later in Experiment 5 , investing in technology ( A ) function requires a "critical mass" to be reached before starting to be beneficial.

Experiment 2 : Test of dynamic technology model ( A ) with real world data
In this experiment, I test the stability of my dynamic model for technology ( A ) (formula (11)) against real world data by using first classical and then human capital augmented Solow model.     At this point both population growth rates and capital savings rates are becoming overshadowed by research incentives, by being at the edge of statistical insignificance. Another find is that the importance of ideas per capita is also slightly diminishing . It is interesting to see that actually incentives for research 26 play a significant role for growth (refer to Appendix 10 ). Human capital ( H ) accumulation also seems to contribute to growth quite well, despite the depreciation. This is not conflicting with my notion for depreciation since we are using real life data (with highest data of H=13.10 ( ln(H)=2. 5 ) for year 2010 in USA) with depreciation applied to it it's Hd=1.125 ) which means that the data has experienced very little to no depreciation rate (since average 13 years of education means getting at most a 2-years college degree (or dropout), which is far from perfect, especially for the needs high-value added industries.
Therefore, we still have far to go by improving our human capital ( H ) globally.
research, however the data available is only for 38 countries since 1996 on this indicator, with a lot of missing reports. My explanation for this insignificance is that not all ideas are actually put into production, and in some countries it is mostly big corporations filing patents. Thus, we need an even better measurement of ideas. The cluster parameter is k=3 (in order to split data to 3 clusters of similarity via technology). The results confirm the theory of divergence of cutting-edge and catching-up growth, together with the existence of divergent group. However it also helps us spot anomalies and "hidden groups of data". When K-means adds a member to a cluster, which is significantly away from the other cluster members, this means that the parameter must be increased, since more subclusters exists in the data set, and this member is an anomaly for the cluster.

Conclusion
My stance is that technology is created by people, using time and resources, so using this links a model can be achieved. I wanted to prove that technology is affected endogenously, and it is too important and interconnected to be left out as exogenous constant.

Closing words on the Solow model's issues
The classic Solow  currently there is a trend of retiring from generalized models (the larger the model, the more data is being lost, due to the use of models) in favor of smaller specific ones.
The endogenous growth model I propose still has a residual which needs to be further examined. I speak about the local political factors and regulations which Sala-i-Martin has found to be statistically significant . During my research I have stumbled upon only one paper which tries to address institutions 29 in the Solow model, however, without empirical evidence . 30 Mathematics must not be abused: it allows us to model processes, but with loss of details. We need to find reliable ways of abstraction in order to model the role of institutions for collaboration and government policies, which are multi-vector compound variables. My sincere belief is this can -and should be -done with absolute data, not with questionnaires and ratings ("spending a month in a country and rating its institutions").
I consider these to be the important missing link and the real "residual" in our detailed growth model. Let us take a look at Japan's productivity levels . Japan has very productive manufacturing sector, which supplies 12% of the employment, combined with a huge low productivity retail sector (55% of employment). And this is a case, where even our endogenously augmented model would fail, as we are not measuring the microeconomics. The retail market in Japan is dominated by local "moms and pops shops", since it is restricted to local retailers, limiting competitiveness only to domestic level, while, for example, the US retailers are competing on the rough global markets .

31
While Japan has an exceptional human capital ( H ) and one of the highest rates of technology ( A ), they still combined it inefficiently, due to various regulations. This means that neither technology ( A ) or human capital ( H ) can be modeled to measure labor productivity. In order to measure productivity, or the lack of it, we need to consider also the local factors conditions. 31