Technology diffusion and uneven development

We propose a conceptualisation of the process of technology adoption that takes into account the uneven relative costs of technology implementation, especially country differences in wage levels. The novelties and contributions of our approach are the following. First, we introduce a dynamic macroeconomic model of technology diffusion, which is the first to directly account for the difference in factor cost proportions in an endogenous cross-country setting. Second, we utilise the Cross-country Historical Adoption of Technology (CHAT) and Penn World Table databases to calibrate the model using non-linear approximation across countries and technologies, which explains about 50% of the variability in technology density. Third, the results of the calibrated model offer a new insight into the dynamics and patterns of technology diffusion of differently developed countries, offering both an approximation of the average technology adoption across differently developed countries over time and an approximation of the relative technology density adoption curves, which are country specific generalisations of the logistic curves and depend highly on the general level of development and wage levels.


Introduction
In this article, we address the complex interdetermination of technological diffusion and uneven development, understood as persisting differences in economic development between countries.The main contribution is bridging the gap between theories of technology adoption that focus on the shape and pattern determining technology adoption curves (Comin and Hobijn, 2010;Griliches, 1957;Stokey, 2021) but abstract from uneven development, and more aggregate approaches to technology and path-dependent uneven development (Fagerberg and Godinho, 2018;Myrdal, 1957;Verspagen, 1991).Our conceptualizations aims to endogenously explain the emergence of technology adoption curves in the setting of uneven development.We aim to analyse the endogenous economic differences across countries that are relevant for technology adoption in the single dimension of relative technology implementation costs.The main hypothesis is that the relative wage level is one of the most important endogenous socioeconomic determinants of the relative cost of technology implementation and thus significantly determines and shapes the socially uneven process of technology diffusion and overall development.Generalising the concept of spatial distance in diffusion processes, we conceptualise the space of technology adoption costs and use it to present a novel dynamic mathematization of the technological diffusion process in the form of a macroeconomic technology diffusion model that incorporates the economic and social effects of uneven development as primary determinants of technology diffusion.We test our model and our main hypothesis that relative wage levels are a determinant of relative technology adoption costs with nonlinear and mixed effects regressions using the CHAT and PENN databases.
The first main contribution of the article lies in the novel reconceptualization of technology diffusion in the international setting and its mathematical representation.Although the diffusion analogy is widely used in conceptualising and modelling technology adoption, the mathematical form of technological diffusion in both mainstream and heterodox theories, as well as in micro and macro approaches, does not take the form of the diffusion equation.The reason is that technology diffusion is a complex process determined by technical, economic, social, and institutional factors, and simple spatial distance has very little significance in this process.We aim to link the mathematical property of the diffusion equation and the broader social and economic constraints in a dynamic model that offers an endogenous explanation of technology diffusion in the international setting of uneven development.This requires a comprehensive reconceptualization.
Rather than focusing on the notion of spatial distance that determines physical and thermal diffusion, we create a concept of the space of relative costs of technology implementation.Within this space, there exists a generalized notion of distance that separates countries by the relative cost of technology adoption.With this generalized distance, we aim to account for both social and economic elements, such as the ratio between wage and capital costs, institutions, and the overall level of development.The central idea is that the disorderly and chaotic process that leads to diffusion in physical space also exists in the form of microeconomic interactions that lead to technology diffusion.However, the main dimension that determines the likelihood of the microeconomic interaction leading to technology adoption is distance in the relative cost of technology implementation, rather than simple geographic distance.Technology spreads very unevenly -flowing rapidly to countries with low relative costs of technology implementation and spreading only with considerable delay to countries with higher relative costs, leaving the most distant countries almost entirely behind.
The main hypothesis is that the generalized distance in the relative costs of technology implementation is primarily determined by differences in gross nominal wages.Since nominal wages are country-specific and the cost of technology adoption includes capital investment, for which the law of one global price is a more appropriate approximation, the relative cost of technology adoption might be approximated by relative wage levels.We test this hypothesis with nonlinear regressions and mixed effects regressions using the CHAT and PENN databases and obtain robust results showing that technology diffusion can be well represented by the diffusion equation and that there are global economic conditions that endogenously perpetuate uneven technology diffusion and hence uneven development.

Literature review and conceptualization
The narrow focus of our analysis is on the adoption patterns of various concrete technologies in the international setting.Griliches (1957) was the first to introduce the concept of an individual cumulative technology adoption curve, which became the benchmark for theories of technology diffusion.It is assumed that logistic form corresponds to the nonlinear transition dynamics between the two discrete values of technology adoption -zero and saturation level.In the initial phase, only a minority of early adopters use the new technology and adoption rates remain low.Near the inflection point, adoption rates peak and the technology becomes dominant.In the late stages, the laggards are the last to adopt the new technology until it reaches the saturation point and is eventually replaced by the new technology, repeating the cycle.In this conceptual framework, cross-country differences in technology adoption rates may be due to the shift in the inflection point (delayed onset of adoption), differences in the slope of the logistic curves (different adoption rates), and differences in the long-run level of technology adoption (different steady state of technology intensity).
Various microeconomic and game-theoretic approaches focus on modelling firms' strategic decisions to adopt a new technology under different conditions and attempt to explain the pattern of technology adoption characterised by the logistic curve.These approaches examine the expectation of a reduction in the technology supplier's costs to explain the delay in technology adoption (Stoneman and Ireland, 1983;Ireland and Stoneman, 1986) model rivalry in a duopoly (Fudenberg and Tirole, 1985;Reinganum, 1981;Riordan, 1992), or focus on the uncertainty of the benefits of the new technology (Jensen, 1982;McCardle, 1985), while some approaches combine these explanations (Huisman and Kort, 2000;Jensen, 1992;Stenbacka and Tombak, 1994).
A common feature of all microeconomic and game theory approaches is that they abstract a firm's strategic behaviour from the macroeconomic environment.Consequently, they can provide some insight into the operating mechanisms for lagged technology adoption in a relatively homogeneous economic environment.These approaches allow us to understand the pattern of technology adoption and the dynamic mechanisms that produce early adopters and technologically lagging firms within a homogeneous country among similar firms, but not between unevenly developed regions, since in this case much more fundamental macroeconomic differences determine the relative patterns of technology adoption.
Despite the widespread use of the logistic curve to explain the innovation process and the subsequent process of technology adoption (Stokey, 2021), its utility as a proxy for technology adoption is empirically controversial.In general, concrete technologies can have either extensive or intensive measures.The extensive measure of technology adoption captures the proportion of adoption among potential adopters and is individually a categorical variable of 0 or 1. Intensive measures also measure the extent to which the technology is used.Comin and Hobijn (2009), using the Cross-country Historical Adoption of Technology database (CHAT), showed that the logistic curve is a good approximation of the extensive margin of technology adoption but does not account for the intensive margin.To provide a comprehensive theoretical explanation for the empirically observed patterns at the level of specific technologies, as well as for total growth and the stylized facts associated with growth theories, Comin and Hobijn (2004;2010) introduced a theoretical model that brings together a neoclassical one-sector growth model and the process of technological diffusion to endogenously explain total factor productivity growth.The endogenously determined production of intermediate goods exhibits slow diffusion of new technologies (diffusion of new intermediate goods), which reflects the main theoretical and empirical results of the microeconomic and game-theoretic models, i.e., the technology adoption curves are logistic and provide an endogenous explanation of total factor productivity at the sectoral and aggregate levels.The aggregate results of the model are consistent with the neoclassical Solow-Swan model.When the model is empirically calibrated with the CHAT database, the main result is that the average lag of technology adoption is 45 years since invention, measured globally (Comin and Hobijn 2004;2010).
The main theoretical explanation for the delay in technology adoption in this model lies in the introduction of monopoly capital goods producers with fixed entry costs.Although the authors claim that this is a macroeconomic model, it is actually based on microeconomic dynamics.The central framework of the model is a closed economy in general equilibrium.Thus, the same considerations apply as in microeconomic and game-theoretic approaches.The model abstracts from all cross-country differences in institutions, education, labour power skills, and (most importantly) factor prices.The technology lag explanation is based on a fixed-cost monopolistic barrier to entry, which (similar to microeconomic approaches) can explain the dynamics of technology adoption within a country but not between countries.
Technology analyses with cumulative technology adoption curves contrast sharply with the approaches of less mainstream macroeconomic theories of development, trade, and technology that examine cross-country differences in institutions, factor prices, trade elasticities, functional specialisation, and their effects on technology diffusion and uneven development that cannot be reduced or explained by mere fixedcost barriers of monopoly producers, as suggested by Comin and Hobijn (2010).
The basic framework of the vast majority of neostructuralist and balance of payments constrained approaches to international development is the mathematization of Kaldor's stylized North-South approach with export-led growth defined by Dixon and Thirlwall (1975) and the benchmark growth model with balance of payments constraint presented by Thirlwall (1979).The conceptual extensions introduce a North-South model with demand-driven uneven development and path-dependent trajectories (Dutt, 2002;Spinola, 2020;Vera, 2006).The main conceptual result of these approaches is that the country-specific ratios between export growth and the income elasticity of import demand can lead to multiple steady states of long-run growth rates across countries.
Evolutionary technology gap theory aims to explain economic convergence and divergence processes and focuses on technology diffusion and adoption as primary factors determining uneven development (Fagerberg and Godinho, 2018;Fagerberg and Verspagen, 2002;Fagerberg et al., 2010;Nelson and Pack, 1999;Verspagen, 1991).This approach assumes that the potential for high productivity growth and catch-up grows linearly with the size of the technological gap.Much of the explanation for why economic convergence is the exception rather than the rule in the global economy comes from earlier theories of development and institutionalist economics (Rosenstein-Rodan and Bhagwati, 1973;Rostow, 1959).Abramovitz introduced the concept of technological congruence, which aims to capture the social capabilities that determine the ability to implement and adopt technology (Abramovitz, 1986;Dosi et al., 1990).This social capacity for technology adoption is primarily explained by extra-economic characteristics such as educational attainment, institutional environment, political stability, labour market structure, financial market development, and effective demand.In technology gap theories, it is referred to as learning capacity, which decreases exponentially with the technology gap, which, combined with linear catch-up potential, creates both a convergence potential and a low steady-state development trap (Verspagen, 1991).The core aspect of evolutionary approaches is to understand technology and its development and diffusion not as a purely technical phenomenon, but as dependent on and co-evolving with the institutional framework and broader social and structural changes (Fagerberg and Verspagen, 2002;Freeman, 2019).
Most neo-Schumpeterian models of creative destruction make similar assumptions.They model the interaction of innovation and lagged technology adoption as a function of the size of the technology gap.While the core model examines technological growth within a closed economy (Aghion, 2004;Aghion and Howitt, 1992), this approach focuses on explaining cross-country differences in capital accumulation and also productivity differences.However, similar to the technology gap theory, productivity differences arising from innovation and spillovers in a cross-country setting are not explored endogenously, but are assumed to be determined exogenously by parameters affecting the ability to adopt technology or the capacity to innovate, which are determined by the political and institutional environment (Acemoglu et al., 2006;Parente and Prescott, 1994).
Thus, on the one hand, we have the theories of technology diffusion that aim to capture the properties of concrete technology adoption through technology adoption curves that are detached from the broader macroeconomic determinants of uneven development and from the core-periphery heterogeneities that perpetuate uneven technology diffusion.On the other hand, macroeconomic and growth theories that focus on explaining uneven development, development traps, and conditions for convergence or divergence do not conceptualise and reproduce technology adoption curves that would correspond to the uneven development that is the subject of their study.Their conceptualization of technology is broad and often reduced to a onedimensional parameter, conceptualised either as a set of possible production techniques (Fagerberg et al., 2010;Gomulka, 1990) or even as information, human knowledge, and know-how, which often take complex and tacit forms (Arrow, 1962;Pavitt, 1999).
The main contribution of the article is an attempt to bridge this gap by formulating a model of technology diffusion that both reproduces the pattern of technology adoption curves for individual technologies in individual countries and simultaneously endogenizes the persistence of uneven technology diffusion across differently developed countries.

Conceptualisation of the diffusion process and model derivation
In our conceptualisation of technology diffusion, we rely on the analogous derivation of the physical diffusion process as conceived in physics, namely Fick's law and the diffusion equation, which describe the processes of mass diffusion in liquid or gaseous matter and heat transfer (Fick, 1855).
The question arises why all discussions, conceptualisations, mathematisations, and models dealing with technology adoption only use the concept of diffusion as a broad descriptive analogy, while the technological diffusion process in the mathematical form never takes the form of the diffusion equation?The reason for this is the following.The physical process of heat or particle transfer is fundamentally characterised by the spatial dimension.In other words, an exogenous imbalance (heat or particle source) leads to an evolution of distribution (either heat or density), which is the parameter of time and space.In economic theory location and spatial dimension can play an important role.Transport costs can influence the pattern of urban and rural development as first proposed by von Thünen (2009) and later by new economic geography and urban economics (Krugman, 1991;Krugman and Venables, 1995).In these models spatial dimension plays a fundamental role and transport costs combined with scale effects in production create a diverging rural and urban patterns of industrialization.However, when it comes to the diffusion of technology on a global scale, such urbanrural agglomeration dynamics cannot be the core explanations of uneven diffusion of technology, as country-specific factor costs (Zeira, 1998), institutions (Freeman, 2019;North, 2005), labour power skills (Abramovitz, 1986;Stiglitz and Greenwald, 2014;Young, 1991) and overall development (Myrdal, 1957;Verspagen, 1991) are more fundamental than mere spatial distance between technology users.
Instead of modifying spatial diffusion model to account for all the social and economic determinants of the technology diffusion, we generalise the spatial parameter to account for the trajectories across which technology diffusion unfolds.This is not the first attempt to generalize spatial parameter in economics, the most notable being the conceptualization of a generalized product space by Hidalgo et al. (2007), which serves as the basis for their empirical evaluation of the economic complexity. 1 Our spatial generalization aims to capture the conditions for the technology flow -we argue that the main dimension along which the process of technological diffusion takes place is the space of relative technology implementation costs.This generalized spatial dimension aims to endogenously grasp the differences in the condition for technological adoption, including not only direct economic costs, but also broader social and institutional preconditions for technology transfer.The central conceptual idea is that the analogy of the disorderly and chaotic process that leads to diffusion in physical space also exists in terms of microeconomic interactions that lead to technology diffusion.The main difference is that the microeconomic interaction that is relevant to technology diffusion does not occur on the basis of simple geographic distance, but is rather defined by distance in the relative cost of technology implementation.This derivation follows the logic of analogy between thermodynamics and economics proposed by various econophysicists (Chatterjee et al., 2005;Dimitrijević and Lovre, 2015), while being the first to treat diffusion of technology as an actual diffusion dynamics mathematically.
We begin our model derivation by defining the object of technology diffusion.Similar to physical diffusion, which works with density, we work with technology density, which is indirectly used in most contemporary empirical examinations of technology (Comin et al., 2006;Dosi and Nelson, 2016;Gomulka, 1990).Regardless if concrete technology is measured in intensive (for example number of combined harvester) or extensive form (share of population with access to internet) it can be defined as a technology density (Comin et al., 2006).While extensive measures of technology can already be interpreted as a measure of technology density, intensive measures of technology must be expressed in per capita form.We divide technology density of each technology by its world average to define a dimensionless quantityrelative technology density (x, t).
Using the relative technology density as our main function of investigation is a novel way to address technology measurements in a cross-country setting and we argue for it due to its three main benefits: 1.) It can be the object of the diffusion equation in unmodified form.2.) It reduces all different technologies, regardless if they are measured with intensive or extensive measure and regardless of the unit in which they are measured, to a dimensionless and comparable scale that represents each country's technology density in the units of the world average density.3.) It avoids conceptual and modelling complications in introducing production as a source of technology.The production sources that lead to the global absolute increases in the same technology only contribute to the relative changes in technology density through their uneven distribution and adoption.Because of the relative definition of (x, t), the function behaves like a probability density function and remains permanently standardised over time ( ∞ 0 (x, t)dx = 1), regardless of the absolute changes in technology use and potential production sources at the global level for each specific technology.Thus we can focus entirely on the process of diffusion.
The relative technology density (x, t) describes the distribution of the relative technology density of each concrete technology along the time dimension (t) and the relative technology implementation cost dimension (x), which theoretically ranges from 0 to infinity on the real number scale.Equality x = 1 represents the world average of relative technology implementation costs, while higher or lower values represent deviations from the average.
As emphasised earlier, the dimension represented by x is not a standard spatial parameter but is a dimension representing relative country-specific technology implementation costs x.Countries are indirectly related to relative technology density through their country-specific relative technology implementation costs.We treat the space of relative technology implementation costs as continuous, conceptually representing the wide variety of countries, regions, and subregions with different technology implementation costs.
We begin our dynamic conceptualisation with an analogy to Fick's Law, which concerns mass diffusion (Fick, 1855).We define the flow of relative technology density J (x, t) as determined by the gradient of relative technology density (x, t) along dimension x and weighted by the diffusion constant D.
The logic behind this is the following: the greater the difference in relative technology density and the smaller the difference in technology implementation costs (shorter the distance in x), the greater and faster will be the flow of technology density from a technologically denser to a technologically less dense country.We can see how the introduction of relative technology implementation costs affects the flow of technology in the context of the narrative of the technology gap theory, which defines two opposing forces that can either close or widen the technology gap.On the one hand, we have the potential that is determined with the size of the gap (the larger the gap, higher the catch-up potential).On the other hand, we have the learning capacity, conversely defined as having negative impact on technology adoption and increasing with the size of the gap (Verspagen, 1991).In the context of our equation, both forces are endogenised in the expression that flow of technology is proportional to It is both the technology gap (x, t) and the proximity of technology implementation costs x that constraint and determine technology flows.The main difference in the narrative is that we aim to treat technology implementation costs as economically endogenous, as opposed to exogenous and extra-economic learning capacity defined by Verspagen (1991).
From the perspective of a single infinitesimal point in the space of relative technology implementation costs, the difference between the relative inflow (from more developed countries) and outflow (to less developed countries) of technology density must lead to equal changes in relative technology density: Using the continuity Eq. (3.3), which simply establishes the predefined fact that relative technology density can only change relatively and cannot increase or decrease absolutely, we derive the classical diffusion Eq. (3.4) for our main variable (x, t): Equation 3.4 represents the final general solution of the proposed model, and any particular solution can be derived from the initial state of the relative technology density function by solving the partial differential equation.
Despite these general possibilities, we propose a comprehensive particular solution to the model.We claim that the initial implementation of each technology starts in the countries and regions with the lowest possible implementation costs (x = 0).Where the relative cost of technology implementation is lowest, there is not only the highest economic rationality for implementing the given technology (the benefit-cost ratio would be highest) but there are also the most economically fertile conditions for putting resources and human effort into solving the problem that leads to the invention and implementation of the given new technology.
Therefore, to solve the partial differential equation describing technology diffusion, we assume that the initial state of technology density at the time of invention is concentrated at Here δ(x) is a Dirac delta impulse function -a generalised function that has zero value everywhere except at x = 0 and whose integral over the entire set of real numbers is equal to one, corresponding to our predefined standardisation.The initial state of relative technology density is normalised and completely centered at the origin of our space of relative technology implementation costs.In this sense, it technically does not matter where the technology is invented because its spread is always determined from the origin of our space of relative technology implementation costs in the absence of additional economic barriers, such as patents.2Technology density then spreads according to the diffusion equation, initially leading to implementation of the technology by early adopters (the most developed countries with the lowest relative costs of technology adoption), while slowly spreading over time to a broader and broader group of countries (Fig. 1).
The dynamic solution of our technology diffusion problem (Eq.3.4) with initial condition (Eq.3.5) is the fundamental solution of the diffusion equation, which is derived in detail in Appendix A: Equation 3.8 represents the dynamic solution of the technology diffusion process over time, characterised by the diffusion constant D and the dimension of relative technology implementation costs x specific to different countries or regions.
Up to this point we have treated our generalized spatial dimension of technology implementation costs as a broad indicator of the conditions for technology adoption.Our main hypothesis is that relative wage level is one of the most important determinants of this dimension, which shapes the uneven technology diffusion.We argue that there exists a direct first order economic relationship between relative wages and technology adoption costs.
There is a substantial amount of economic theory that links the technology diffusion with factor prices -primarily wage levels -which together endogenously determine productivity growth (Amin, 1974;Sylos-Labini, 1984;Zeira, 1998).These theories attempt to explain the relationship between wage levels and productivity growth by examining and modelling the capital-labour relationship as a choice by firms between employing labour and purchasing labour-saving machinery.They derive the direct effect of the relationship between the relative costs of labour power and machinery on technical progress and productivity driven by the introduction of machinery.Acemoglu (2010) has analytically demonstrated that although the majority of canonical neoclassical and endogenous growth models conceptualise technology as labourcomplementary, there also exists a plausible theoretical socioeconomic environment in which technology is strictly labour-saving.In such an environment, factor costs are one of the most important determinants of technology development and adoption.Our conceptualisation and subsequent empirical investigation aims to prove that in reality new technologies predominantly take the form of labour-saving improvements and that factor costs are important determinant of the technological diffusion process.Although these approaches work within the framework of a closed economy, they have explicit implications for all theories that attempt to study technological growth and diffusion in the setting of differently developed countries.If the relationship between the relative prices of factors of production -primarily the relative price of labour power and the relative price of fixed capital -are determinants of technological progress, as studied by Sylos-Labini (1984) and Zeira (1998), then the cross-country differences in relative factor prices that are determined by initial differences in productivity and technological development could be a source of long-run structural differences in the relative costs of technology implementation.Therefore, they may explain, at least in part, the long-run patterns of technology diffusion.
Given that a technology is adopted only if it is profitable, the adoption of new labour-saving technologies depends directly on the relationship between the nominal wage level and the fixed costs of adopting the technology (Kaldor and Mirrlees, 1962;Sylos-Labini, 1984;Zeira, 1998).Because nominal wage differentials are greater than differences in fixed costs of technology adoption, especially fixed capital costs, the countries with lower wages face higher relative costs of technology adoption and investment (Hsieh and Klenow, 2007;Jovanovic and Rob, 1997).Eaton and Kortum (2001) even find that prices of equipment and machinery are higher in countries with lower wages.Thus, the country-specific wage level and the global price of the technology that is embodied in the machines cause different countries to be more or less far apart in terms of the relative costs and benefits of technology adoption.
The relative costs of technology implementation are country and time specific.According to empirical analyses (Hsieh and Klenow, 2007;Jovanovic and Rob, 1997) we assume that the technology adoption depends on the fixed capital element of investment that has a single global price k and country-specific mean wage w ct .This leads to the following relative factor cost ratio: Our hypothesis translates to the proposition that the relative technology implementation costs are directly proportional to the ratio of global average and country-specific mean wage: Due to limited data for some of the available technologies that coincides with the data on labour costs (the cross section being zero or having small number of observations) and the relative country-specific asymmetry of some technologies, 4 we used 3 A detailed description of the panel, countries and technologies included is provided by Comin and Hobijn (2009) and can be accessed at: https://www.nber.org/papers/w15319. 4For example, steel production as measured in tonnes, even when converted to per capita figures, still shows a large variability that has more to do with international specialisation than with the actual process of technology diffusion.Similarly, irrigation data are highly dependent on geographic and other countryspecific conditions rather than pure technology intensity.36 technologies from the dataset for our empirical analysis.We conduct our empirical evaluation using a set of 36 technologies: four agricultural (number of combined harvesters, tractors, milking machines, and fertilisers), seven infrastructural (electricity generation, railroads, ship transportation), two steel (electric arc furnaces, oxygen blast furnaces), eight information technologies (cable TV, Internet, mail, newspaper, radio, telegram, telephone, TV), 12 health technologies (hospital beds, mammography, radiation, transplantation, dialysis), automobile use (2), and tourism capacity (2).Each technology has a corresponding specific time variable that measures the time since invention t T .The time of invention is set as the first non-missing observation in the dataset, which is standard in similar studies of technology diffusion (Comin and Hobijn, 2010).

Empirical calibration
Inserting relationship between relative technology adoption costs and wage levels (Eq.3.10) into our model (Eq.3.8) we obtain the following nonlinear model for empirical evaluation: Using nonlinear least squares estimation, we obtain the results in Table 1.Highly significant results indicate a general relevance of the proposed conceptualisation of technology diffusion.To further analyse stability of the results, especially potential sensitivity to sample selection, we make additional random sampling robustness tests, which are presented in detail in Appendix B. The bimodal normal distribution of the repeated sampling estimations of the diffusion constant show concealed heterogeneity in the technology diffusion process for different technologies and indicates that we should control for technology specific random effects in our estimations (Fig. 3b).
For that reason we perform an additional estimation of nonlinear mixed effects based on maximum likelihood estimation.We modify our model to account for random effects at the level of the diffusion process of each technology.We define T as a technology-specific random effect of the parameter α and δ T as a technology-specific random effect of the diffusion constant D. The technology implementation costs remain country and time specific, as defined in Eq. 3.10.
The estimated mixed effects model yields quite similar results when compared to the nonlinear least squares regression, but are much more stable and non-sensitive to random sample perturbation (Table 2; Appendix B).The additional information gained by introducing a technology-specific random distribution of diffusion constants is that the differences in diffusion constants are relatively small.The variance of the technology-specific random deviation from the general technology diffusion constant is relatively small when compared to the diffusion constant estimate.This means that our approach to technology diffusion is relatively universal for all tested technologies and that their diffusion process is very well described by the single diffusion constant and the distance based on relative nominal wage costs.
The calibrated model results that are shown in Fig. 1 describe the dynamic evolution of technology diffusion and are dynamic in two ways.First, the relative density of technology use spreads from developed countries to less developed countries depending on how close they are in terms of technology implementation costs.Second, countries can exogenously change their position in terms of their relative costs of technology implementation and with this improve the conditions for endogenous technology diffusion.
Figure 2 shows the evolution of technology density for countries with different wage levels.While the theory of technology adoption, which abstracts from uneven development, almost universally models technology adoption in the form of a logistic curves, our dynamic examination presents a generalisation.The main difference from most examinations of technology adoption is in the object of study.Our object of study is the relative technology density as opposed to either extensive or intensive technology measures, which are the subject of the majority of existing empirical work on innovation and technology adoption.In this sense, our results do not contradict existing research on technology adoption.In absolute terms, the technology adoption of each country can be approximated quite well by different logistic curves if measured in the extensive margin, as shown by Comin et al. (2006).The relative technology density adoption curves (Fig. 2) provide new information about the dynamics of technology adoption in the relative sense, which is the most important in the context of uneven development.2) The most developed countries, with average productivity and wages above the world average, are the early adopters of new technologies and increase their relative technology density in the early period after invention.The relative technology density of the most developed early adopters only gradually begins to decline when the technology spreads to the majority of countries, even though it is still potentially increasing in absolute terms.The decline in the relative technology density adoption curve of the most developed countries represents a period of slower adoption rate when compared to new adopters from less developed countries.The countries with average productivity and wage levels close to the world average exhibit a technology adoption function that is quite similar to the logistic pattern described by the theory that abstracts from uneven development.The main differences between the technology adoption curves in the less developed countries is the much more gradual slope of technology adoption, and the longer period between the time of invention and the beginning of the economically relevant adoption rate.For example, the relative technology density for a country with twice the world average wage increases almost immediately after the invention; for a country with 50% of the world average wage level, it remains close to zero for more than 10 years since the invention; while for an underdeveloped country with wages equal to only 25% of the world average, it remains close to zero for many decades.Underdeveloped countries with wage levels close to subsistence level do not  2).Linear relationship between wages and GNI per capita in 2021 is assumed to infer how country clusters defined by UN and WB methodology would correspond to relative technology adoption curves.The GNI per capita of high income economies is higher than 13.206$, for upper middle-income it is between 4.256$ and 13.205$, for lower middle-income it is between 4.256$ and 1.086$, with low income economies below that adopt new technologies at all and, according to our model, are practically technologically blocked.A general result can thus be summarised in the following statement: while each country in isolation experiences absolute technology adoption in the form of the logistic curve when measured intensively, both the horizontal translation and the slope of the logistic curve are highly dependent on the relative costs of technology implementation, which are primarily determined by the wage level.
The relative technology density adoption curves of differently developed countries are thus highly uneven over time.Relative differences in technology use are largest initially (due to differences in the time between invention and the start of adoption for differently developed countries) and gradually decrease but still remain stable over long periods of time, indicating multiple steady states of the relative intensity in the technology use for differently developed countries.Even after long periods in which technology becomes obsolete and is replaced by a new innovation cycle, the relative density of the use of old technologies remains very unevenly distributed.

Discussion
The explanatory power of the proposed model bridges the gap between contemporary studies of technology adoption through the prism of technology adoption curves and broader macroeconomic theories of uneven development by linking technology adoption conditions to relative nominal wage levels.The empirical results can be seen as evidence that a broad conceptualization of technology needs to incorporate its predominantly labour-saving effect (Acemoglu, 2010) in order to adequately account for the dynamic effects of relative wage and investment costs on technology adoption and diffusion.
While we have based our derivation of endogenous technology diffusion on the direct relationship between the relative costs of technology adoption and wage levels, there are still many indirect relationships behind the link between wages, labour skills, productivity, overall development and institutional framework, all of which are endogenously intertwined.
There is ample literature on endogenous growth that emphasises the scale effects of learning and attempts to explain modern economic growth through the prism of human capital accumulation (Lucas, 1988;Romer, 1990), learning (Arrow, 1962;Stiglitz and Greenwald, 2014), and quality improvements (Grossman and Helpman, 1991).According to these approaches, wage levels reflect aggregate productivity and the scale effects of learning and skills.This would imply an indirect link between wage levels and the accumulation of human capital and skills, which in some cases are a prerequisite for effective technology adoption.The relationship between skills, wages, and endogenous technology growth has also been examined in a task-based framework.This has shown how endogenous technology growth can be influenced by skills and affect the distribution of income and, conversely, how the composition of skills endogenously determines technological upgrading and productivity growth (Acemoglu and Autor, 2010;Acemoglu and Restrepo, 2019).Institutional development, structural change, economic growth, and wages are inextricably linked (Acemoglu and Robinson, 2013;Freeman, 2019;North, 2005).Relative wages may thus indirectly signal and reflect also the institutional environment and other broader socioeconomic conditions for growth, providing another endogenous feedback loop through which relative wages indirectly influence technology adoption.Our finding that relative wages are a statistically significant and robust empirical equivalent of the relative costs of technology adoption can therefore be interpreted as reflecting both direct effects through the relative economic costs of investment and indirect effects of the broader social conditions for technology adoption that are indirectly signalled by relative wages.
Circularity and endogeneity within a conceptual model are not flaws, however, if circularity reflects the complex and non-linear dynamics of reality.In this context, our model aims to explain both technology adoption curves in the context of uneven development, as well as the broader aggregate results of cumulative causation models (Myrdal, 1957), industrialization approaches that use the demand-driven Kaldor-Veerdorn law and examine path-dependent uneven growth (McCombie and Spreafico, 2016), and evolutionary technology gap approaches (Fagerberg et al., 2010;Fagerberg and Godinho, 2018;Verspagen, 1991).Since wage levels are at least very roughly related to productivity and development that are largely determined by the available technology, this implies an indirect feedback loop through which wages can influence development outcomes both through their direct impact on the cost of technology adoption and as a mediator of aggregate demand, while being constrained by the given technology, which is a typical path-dependant evolutionary pattern.
Compared to Verspagen (1991) study of the technology trap and the conditions for catching up and lagging behind, our model also exhibits nonlinear dynamic feedbacks that cause a path-dependent evolution of the technology gap between differently developed countries.The main differences are two.First, our model features an entire continuum in the scale of technology adoption states, rather than discrete, multiple steady states.This better reflects the complexity of technology adoption, in particular the fact that in addition to the near-zero technology trap, there are also states of medium and high industrial technology development that do not fully converge, indicating the phenomenon of the middle income trap.While various conceptual, modelling, and empirical analyses have proposed different factors that explain the emergence of a middle-income trap (Eichengreen et al., 2013;Hartmann et al., 2021;Krūminas et al., 2019;Myant, 2018), we differentiate from them by deriving it fully endogenously in the context of technology adoption.Second, the narratives describing the social and economic forces preventing catch-up and closing the technology gap are almost entirely extra-economic and exogenous (e.g., learning capacity, barriers to technology adoption, institutions) (Comin and Hobijn, 2010;Freeman, 2019;Parente and Prescott, 1994;Verspagen, 1991).Conversely, our model and its result show that uneven technology adoption can be understood endogenously, as an economic functioning of the global market-based economy, as there is no such endogenous force that would close a technology gap between highly unevenly developed countries.Our approach shows (both theoretically and empirically) that technology diffusion cannot be abstracted from relative production factor costs, especially relative wage levels, which put differently developed countries in a substantially different structural position when it comes to the process of adopting concrete technologies.Since local profitability is the endogenous driver of investment and technological change, differences in relative wage costs result in quite different amounts of feasible technology being available in different countries (Hsieh and Klenow, 2007;Jovanovic and Rob, 1997).Thus, the endogenous process that determines the closing of the technology gap is conditional.The technology gap closes quickly when differences in technology density are large and the relative costs of technology implementation are similar, while the technology gap may persist for decades and even centuries in cases where relative cost structures and wage levels differ widely.
Our approach to technology diffusion also sheds new light on the issue of development and industrial policy.While improving education, institutional stability, and research spending are fundamental preconditions for growth and catch-up, the pattern of technology diffusion under study requires more than passive creation of preconditions and letting the "invisible hand of the market" close the technology gap.Our results could indicate that, in addition to passive, broad-based societal prerequisites, closing the technology gap also requires active, deliberate intervention in the form of active industrial policy with a focus on technology transfer.
The relevance of technology transfer-oriented industrial policy in the context of our results can be briefly discussed in the example of China, which used its labour cost advantage and market size to attract administratively conditioned sector-and technology-specific FDI and used political and extra-economic means (such as conditioning domestic market entry with joint ventures and technology transfers) to promote and expand its own domestic capacity within state-owned or state-subsidised enterprises (Kenderdine, 2017;Mao et al., 2021).In contrast, the new Central and Eastern European EU member states also attracted a lot of FDI because of their labour cost advantage, but did not have a targeted industrial policy or influence on technology transfer to domestic producers because of the EU regulatory framework and the small size of their domestic market (Myant, 2018).Thus, on the one hand, China used targeted industrial policies, administratively mediated technology transfers, and subsidised domestic industrial champions to facilitate technology adoption, despite relatively unfavourable endogenous conditions related to their lower wages, positively affecting both wages and technology adoption in the long run.Conversely, the new CEE EU member states were and are caught in a typical middle-income trap, as technology diffusion is left to endogenous forces (Krūminas et al., 2019).This could indicate that a targeted exogenous and extra-economic push is needed to break out of path-dependent development explored by our model, and that relying solely on the labour cost advantage does not lead out of the technology trap and unfavourable functional specialisation.5

Conclusion
The proposed dynamic model of technology diffusion aims to address the two main drawbacks of the existing literature dealing with technology transfer and adoption.The first drawback is the abstraction from the uneven and highly asymmetrically developed world economy practised by the micro-and game-theoretic approaches that focus on technology adoption curves.The second major drawback is the almost exclusive reliance on extra-economic, exogenous and country-specific factors to explain the uneven distribution of technology in the world.In our approach, the process of technology diffusion is conceptualised as fundamentally dependent on the cost of technology implementation, which depends directly on relative wage levels.We derive the dynamic diffusion equation that describes the process of change in relative technology density that depends on the gradient of technology density with respect to distance in terms of the relative wage level.
There are 4 contributions of our approach: 1.) We provide a conceptual and modelling framework that describes the diffusion of technology in terms of a diffusion equation equivalent to physical and heat diffusion.
2.) We generalize the spatial parameter to account for endogeneities in the conditions of technology adoption and show that relative wages are the main determinant of this generalized spatial dimension.3.) We derive relative technology adoption curves that reflect technology adoption in the context of uneven international development.4.) The main aggregate results are consistent with the macroeconomic evolutionary and cumulative causation approaches to uneven development.
We thus close a gap between approaches that focus more on concrete technologies and their adoption curves and generally abstract from uneven development, and more macroeconomic approaches that reflect similar path-dependent dynamics at the aggregate level but lack more specific technology-related dynamics.
The endogenous process of technology diffusion and adoption can thus be understood as one of the endogenous structural mechanisms that contribute to the perpetuation of uneven development, dynamically acting in both directions.On the one hand, technology adoption is determined by the relative differences in wage levels and the different cost structures of technology implementation.On the other hand, the uneven distribution of technology determines uneven development and relative differences in wage levels.Although this may seem like a simplistic circular tautology, it is an alternative explanation for why the majority of middle-income countries remain middle-income, why the majority of high-income countries remain high-income, and why the majority of low-income countries remain low-income than the prevailing explanations that rely on country-specific exogenous capacity, institutional frameworks, culture, political stability, and similar extra-economic factors.While it is clear that institutions and other extra-economic factors are fundamental to the catch-up process, it is also clear that the long-term stability of global uneven development requires endogenous explanations.
The model also has some limitations and potential for further research and extension.The proposed model and its empirical calibration abstract from the specifics of each technology and cannot be used to explain technology adoption curves specific to each technology or country-specific patterns of adoption.Further research would be needed to combine the proposed framework with country-and technology-specific analyses and approaches that could potentially produce more detailed results and account for additional heterogeneities in both the characteristics of different technologies and country-specific institutions.random sampling experiment.We draw without replacement 10 random technologies out of the CHAT database and make both the nonlinear least squares estimation, as well as mixed effects estimation on the sample.Repeating such sampling for 2000 times, we obtain the distribution of the estimated model parameters (Figs. 3 and 4).
In the case of nonlinear least squares estimation (Fig. 3) the treatment of all technologies as having a homogeneous diffusion process is demonstrated to be insufficient.This is particularly visible in the case of estimating the diffusion constant D as there is a hidden heterogeneity among different technologies, characterized by the bimodal normal distribution observed in Fig. 3b.This is result is expected, as its was shown already by Comin and Hobijn (2010) that earlier technologies required approximately twice as much time to diffuse than more modern technologies.As CHAT database includes both old (e.g.railways, tractor) and modern (e.g.internet) technologies, bimodal normal distribution roughly corresponds to differences in the diffusion process for different technologies in different periods To obtain a stable global result a technology specific random effects must be introduced.This is corroborated by much higher stability of the sampling estimations in the case of mixed effects estimation that include technology specific random effects for both α and the diffusion constant D (Fig. 4 and Table 3).

Fig. 1
Fig. 1 Diffusion of technology in relation to technology implementation costs.The diffusion of technology is represented by the relative technology density function at different stages of the diffusion process.Low values of x correspond to developed countries with high wage levels, while higher values of x correspond to underdevelopment, low wages, and (in the case of extremely high values) subsistence economies.The graph is plotted as the result of the diffusion Eq. 2 with parameters D = 0.00318 and α = 0.477 taken from the results of the nonlinear regression, which takes into account the random effect deviations of different technologies (Table2) Fig. 2 Country-specific relative technology density adoption curves.Diffusion of technology through time from the perspective of differently developed countries.The graph is plotted as a result of the diffusion Eq. 3.8 with parameters D = 0.00318 and α = 0.477 taken from the results of the nonlinear regression, which takes into account the random effect deviations of different technologies (Table2).Linear relationship between wages and GNI per capita in 2021 is assumed to infer how country clusters defined by UN and WB methodology would correspond to relative technology adoption curves.The GNI per capita of high income economies is higher than 13.206$, for upper middle-income it is between 4.256$ and 13.205$, for lower middle-income it is between 4.256$ and 1.086$, with low income economies below that

Fig. 4
Fig. 4 Mixed effects with technology specific random effects random sampling distribution

Table 1
Non-linear least squares regression results

Table 2
Non-linear mixed effects regression results

Table 3
Robustness test means and standard deviations