Law Of Proportionate Effect Research Articles

The search for a universal law of crystal growth: The law of proportionate effect?

The search for a universal law of crystal growth: The law of proportionate effect?

Crystal growth according to the law of proportionate effect

Abstract This paper summarizes an approach to crystal growth that was published in parts in various articles over the course of 25 years by the present author and his colleagues. Evidence for this approach, which is confirmed in detail by data in the cited publications and in the figures and equations in the supplementary material that accompanies this paper (see the Online Materials1), comes mainly from the shapes of crystal size distributions (CSDs). Such distributions reveal the growth histories of natural minerals and synthetic compounds, histories that can be used to make geological interpretations and to guide industrial syntheses. CSDs have three fundamental shapes: log-normal, asymptotic, and Ostwald. These shapes result from different degrees of supersaturation near the time of nucleation. The first two distribution shapes form according to the Law of Proportionate Effect (LPE) at moderate supersaturation, and the latter rare distribution forms by Ostwald ripening at large supersaturation. Initially, the first two distributions have mean diameters of up to tens of nanometers and grow by surface-limited growth kinetics. The slow step in this reaction is the incorporation of nanoparticles (bits of crystal or adparticles) onto the crystal surface. As the crystals become larger, their demand for nutrients, as calculated by the LPE, increases exponentially. Then the slow step in the reaction changes to the rate of transfer of nutrients to the crystal (supply- or transport-limited growth). Crystal diameters often grow the most during this latter stage, and the initial CSD shapes that originally formed during surface-limited growth are retained and scaled up proportionately. Proportionate growth during the supply-limited stage can be simulated approximately by multiplying the diameter of each crystal in a distribution by a constant. Crystals can also grow by a constant rate law in which a constant length is added to each crystal diameter in the distribution. This rare process causes the original CSD to narrow so that its initial shape is not preserved. The growth law that prevails, either proportionate or constant, is determined by the manner in which nutrients are supplied to the crystal. Supply is by advective flow during proportionate growth, with the nutrient solution moving with respect to the crystals. Constant growth relies on the random diffusion of nutrients through a quiescent solution. Proportionate growth is by far the most common growth law, and therefore, nutrient supply by diffusion alone during crystal growth is uncommon. Distributions formed by Oswald ripening and those formed by other rare processes are also discussed. During Ostwald ripening, nucleation caused by mixing reactants at large supersaturation forms crystals that are extremely fine and numerous. The larger crystals grow at the expense of the finer, less stable crystals, thereby forming, on completion, the universal steady-state CSD shape predicted by the Lifshitz-Slyozov-Wagner (LSW) theory. This unique CSD shape, as well as other rare shapes, then are scaled up to larger sizes by supply limited proportionate growth.

Do Firms’ Growth Rates Follow a Random Walk? Evidence from Incubated Small and Medium Enterprises in South Africa

Debate on the validity of the Law of Proportionate Effect (LPE) on firm growth is ongoing decades after it was postulated by Gibrat in 1931. The theoretical model which asserts that firm growth follows a random walk has been largely tested in developed economies using data from non-incubated firms, with scanty research in developing regions like Africa. This paper, therefore, aims to address this gap by being the first to assess the validity of Gibrat's law on incubated small, medium, and micro enterprises (SMMEs) in South Africa. The study utilised four-year panel data from 300 incubated SMMEs across the country, for the period between 2018 to 2021. Utilising the Law's generalised growth rate model, the generalised least square regression modelling was harnessed, using R Software. The findings, using sales as firm size proxy, confirmed Gibrat’s Law. The results showed that firm size had no effect on the sales growth rate of incubated firms, on the other hand when employment proxied performance the LPE was rejected. The findings provide important implications for both practitioners and pertinent stakeholders in the SMME sector in South Africa.

How Innovation Impacts Firm Growth Dynamics in Natural Resource Industries

Growth has generally been considered a crucial objective for a firm, and, consequently, it has become one of the most researched subjects in economic sciences. This paper aims to provide an in-depth study of how incremental innovation, a ubiquitous factor, affects the growth of small and large-sized firms differently. Specifically, this work examines firm growth dynamics in natural resource industries. In these industries, innovation is mainly based on processes in the form of incremental changes, and adoption of innovations has significant sunk costs. We argue that, before incremental process innovation, firm growth is directly proportional to firm size. However, in the presence of incremental innovation events, firm growth is indirectly proportional to firm size, since smaller firms pose higher strategic flexibility and can adopt innovations faster. Our empirical findings not only confirm the dependency of growth rate on firm size, rejecting Gibrat’s Law of Proportionate Effect, but highlight the relevance of incremental innovation as an inflection point of firm growth, creating a competitive opportunity window for small firms.

The Dynamics of the Profitability and Growth of Restaurants; The Case of Norway

The restaurant industry is quite similar across borders. It is a labour-intensive industry that is important for tourism and employment. It consists mainly of many small businesses that are regionally dispersed. There are many studies that have analysed this sector. However, rather few articles have focused on the dynamics of growth and profit. The purpose of this paper is to apply the theory of profit persistence and the law of proportionate effect (LPE) to Norwegian restaurants by using publicly available public panel data from 2010 to 2019. The sample includes 866 restaurants. One important finding is that Gibrat’s law (LPE) does not seem to hold, meaning the growth is not independent of the size of the firms. Small businesses grow faster than the others, and they are also more profitable. There is some degree of profit persistence in the restaurant industry. Profitability is negatively linked to debt ratios but positively related to working capital. The study shows there is a trade-off between size and profit. These findings are useful for the industry and for others (public planning, lenders, and more).

Gibrat’s Law, Incremental Innovation and Firm Growth Dynamics in Natural Resource Industries

This paper examines firm growth dynamics in natural resource industries. In these industries, innovation is mainly based on processes in the form of incremental changes, and adoption of innovations has significant sunk costs. We argue that, in a steady state before an incremental process innovation, firm growth is directly proportional to firm size. However, in the presence of incremental innovation events firm growth is indirectly proportional to firm size, since smaller firms pose higher strategic flexibility and are able to adopt innovations faster. Our findings confirm the dependency of growth rate on firm size, highlighting the relevance of incremental innovation as a determinant factor of firm growth, rejecting Gibrat’s Law of Proportionate Effect.

The size and growth of firms: new evidence on law of proportionate effect from Asia

PurposeThis paper aims to investigate the relationship between firm size and growth under the framework of Law of Proportionate Effect (LPE) for Asian firms.Design/methodology/approachAn unbalanced panel data for about 12,001 unique non-financial listed and active firms from 1995 to 2016 for 12 industrial and emerging Asian economies was examined. Total assets and net sales were used as size variables. Firm-specific variables such as return on equity, leverage and liquidity ratio were used along with macroeconomic variables such as GDP growth and two financial development indicators. The fixed effects and random effects approach were used to estimate the dynamic growth model after taking into account econometric issues such as correlation between the cross-country-specific error component and the regressors and heteroscedasticity.FindingsThe estimated coefficient of firm size was found to be always significant and negative rejecting the Gibrat’s law for Asian firms confirming that the small-sized firms are growing faster than larger-sized firms. Also, the persistence of growth coefficient suggested that a positive persistence of firm growth does not exist for the selected Asian firms. Gibrat’s LPE was also rejected across small, medium- and large-sized companies. For the aggregate sample, the coefficient of leverage was found to be negative and significant, whereas liquidity ratio, GDP growth, banking sector and stock market variables are found to have positive and significant relationship with growth of firms. For individual economies, a mix of positive and negative (significant and insignificant) estimated coefficient was observed.Practical implicationsAt macro-level, the examination of firm growth is likely to have significant policy implications for the regulators and various government agencies as firm growth may increase economic activity in general and employment opportunities in particular. The policymakers can control economic and employment activity by designing specific firm growth policies. At micro-level, the study will have significant implications for managerial decision-making.Originality/valueTo the best of the authors’ knowledge, this is one of the first studies to test the validity of Gibrat’s LPE for large Asian economies and firms using recent data under a dynamic growth framework using firm-specific and macroeconomic variables. Also, persistence of growth of firms under LPE (that growth does not persist from one period to the next) is uniquely examined for Asian firms.

Firm growth and Laplace distribution: The importance of large jumps

Abstract Recent empirical studies have shown that firm growth rate distribution is not Gaussian but closely follows a Laplace distribution. This robust feature of the growth rate distribution challenges existing models based on Gibrat’s model because it predicts a Gaussian distribution. First, we analyze more than 100,000 Japanese firms and empirically show that the Laplace shape can be observed for the Japanese firms. Then, by using the theory of stochastic processes, we theoretically show that the absence of jumps causes the discrepancy between Gibrat’s model and the Laplace shape. In particular, based on the Laplace shape and the law of proportionate effect, we show that the firm growth process is a jump process. In other words, firm growth cannot be explained by the consequence of many small shocks but is determined by a few large jumps. The widely observed Laplace distribution reflects this jump property of firm growth dynamics.

EMPRESAS TECNOLÓGICAS Y POLÍTICAS PÚBLICAS DE DESARROLLO REGIONAL EN BRASIL

This study aims to analyze the growth of technology companies in Brazil in 2002-2013. To do so, we use Gibrat's law, otherwise known as the Law of Proportionate Effect (lpe) and the panel data model, which allows us to control unobservable heterogeneity. However, unlike the previous empirical evidence, this study takes into account the effect of the recent financial crisis and profitability on business growth. The main results show that in times of crisis, public policies could allot some resources available to help some of the smaller, more profitable, and least indebted technology companies to boost economic growth.

Systematic Differences and Random Rates: Reconciling Gibrat’s Law with Firm Differences

A fundamental premise of the strategy field is the existence of persistent firm-level differences in resources and capabilities. This property of heterogeneity should express itself in a variety of empirical “signatures,” such as firm performance and arguably systematic and persistent differences in firm-level growth rates, with low cost firms outpacing high cost firms. While this property of performance differences is a robust regularity, the empirical evidence on firm growth and Gibrat’s law does not support the later conjecture. Gibrat’s law, or the “law of proportionate effect,” states that, across a population of firms and over time, firm growth at any point is, on average, proportionate to size of the firm. We develop a theoretical argument that provides a reconciliation of this apparent paradox. The model implies that in early stages of an industry history. firm growth may have a systematic component, but for much of an industry’s and firm’s history should have a random pattern consistent with the Gibrat property. The intuition is as follows. In a Cournot equilibrium, firms of better “type” (i.e., lower cost) realize a larger market share, but act with some restraint on their choice of quantity in the face of a downward sloping demand curve and recognition of their impact on the market price. If firms are subject to random firm-specific shocks, then in this equilibrium setting a population of such firms would generate a pattern of growth consistent with Gibrat’s law. However, if broader evolutionary dynamics of firm entry, and the subsequent consolidation of market share and industry shake-out is considered, then during early epochs of industry evolution, one would tend to observe systematic differences in growth rates associated with firm’s competitive fitness. Thus, it is only in these settings far from industry equilibrium that we should see systematic deviations from Gibrat’s law.

Quantitative analysis of the phase-separated structure and mechanical properties of acrylic copolymer/epoxy thermosetting resin composites

The effects of thermal curing reactivity on the phase-separated structures of acrylic copolymer/epoxy thermosetting resin composites containing various accelerator amounts were investigated to elucidate their phase separation behavior. These composites exhibited a sea-island structure, in which the island size decreased as the amount of accelerator increased. The island diameter distribution was represented as a lognormal plot. Island formation is explained by the law of proportionate effect. As observed during the last stage of phase separation via spinodal decomposition, small domains formed prior to coalescing. The effects of material composition on the phase structures and mechanical properties of acrylic copolymer/epoxy thermosetting resin composites were also examined. The morphologies varied depending on the components, and in particular, the total island area fractions and fracture surface shapes after the tensile tests differed. A small fraction of island area resulted in a lower fracture surface roughness and higher mechanical strength, indicating that in stronger composites, many epoxy components remained in the acrylic copolymer-rich sea phases. In addition, the destruction of resin composites was propagated through phase boundaries because the network formation states of the sea and island phases do not significantly differ. The effects of thermal curing reactivity on phase-separated structures of acrylic copolymer/epoxy thermosetting resin composites were investigated to clarify their phase separation behavior. These composites displayed a sea-island structure, for which the island size altered as the amount of accelerator. The island diameter distribution was represented as a lognormal plot. This phase separation is explained by the law of proportionate effect as observed during the last stage of phase separation via spinodal decomposition.

The Distribution of Firm Size in Africa's Manufacturing Sector and Its Implication for Industrial Policy

This paper analyzes the distribution of Kenyan firms into four size categories: micro, small, medium and large size. Four questions are investigated. First, do small firms grow faster than large ones or is firm growth independent of size as stipulated by Gibrat's “Law of Proportionate Effect” Second, what is the steady state size of firms? Third, how long does it take to reach the steady state size? Fourth, does the use of bank credit affect a firm's growth and the process of firm size convergence? On the basis of ergodic probability distributions, we derive information on firms' steady state growth and convergence. Using data on Kenya's manufacturing sector, empirical results suggest that firm growth is associated with overall economic performance. Before the early 1990s, firms had a high growth potential. In equilibrium, 70 percent of surviving firms converged to large size. In contrast, growth and convergence in the 1990s reflected the economic crisis that hit the Kenyan economy: the steady state distribution of firm size was concentrated in the first two quartiles, with suggestive evidence that smaller firms recorded the highest rates of failure. Also, the use of credit increased the growth of surviving firms but appears to have contributed to precipitating firm failure. These results suggest that support policies such as special credit schemes are not a panacea for a firm's survival and growth.

The Validity of Gibrat's Law: Evidence from the Service Sector in Jordan

Abstract Gibrat's Law or law of proportionate effect (1931) suggested that the growth of the firm is independent of the firm size. This law has been widely studied over the last 40 years until the present time. A lot of studies on Gibrat's Law have been applied in developed countries; however, there is a lack of research on Gibrat's Law in the developing countries. The purpose of this study is to test Gibrat's Law for the service sector in Jordan by using the transition matrix over the period of 2009-2011. Furthermore, the data will be provided by the Department of Statistic, Jordan, using sales as a measurement of firm size.The study rejects Gibrat's Law for the service sector. This means that small firms grow faster than large firm. Consequently, small firms are an important consideration for the policy makers, firm's shareholders and international organizations to create employment in Jordan to solve the problems of unemployment.

MECHANISMS OF CRYSTAL GROWTH AND ZONING OF HYDROTHERMAL PYRITE FROM Sb-Au DEPOSITS IN NIZKE TATRY, SLOVAKIA

Pyrite is one of the most common ore minerals in hydrothermal ores and a hallmark of many types of hydrothermal deposits, e.g. , mesothermal quartz veins with pyrite, arsenopyrite, and visible or invisible gold. In addition to any other mineralogical investigation, it is possible to examine and interpret the crystal size distribution (CSD) of the mineral which is under investigation. The conditions of mineral-forming processes can be diverse in terms of temperature, pressure, and compositional variables, but the CSD shapes are usually convergent and by far the most common CSD type is the lognormal distribution. In this study, we determined the CSD for pyrite crystals from mesothermal quartz-pyrite-arsenopyrite-stibnite-gold ores from the Nizke Tatry Mountains (Western Carpathians, Slovakia). A large number of crystals were measured manually from ore sections or from stereographic images. All samples show a crystal size spectrum from ≈2 μm up to ≈550 μm. We discovered that all CSDs show a lognormal or pseudo-lognormal shape with low logarithmic size variances (β2). In order to understand the history of nucleation and growth of pyrite formed in hydrothermal milieus, we compared our measurements to simulations with different growth mechanisms using the GALOPER (Growth According to the Law of Proportionate Effect) program (after [Eberl et al. 1998][1]). From the shape of the distributions, the logarithmic mean (α) and β2, we inferred that the crystals grew in an open system with its associated growth mechanisms (surface-controlled growth and supply-controlled growth), since no evidence for ripening processes in the data or in the numerical simulations of our data could be found. Since ripening as a growth mechanism can be excluded, we have to assume that the growth of the pyrite crystals and so the formation of the whole ore body occurred on a relatively short geological time scale. This preliminary result opens more questions related to the emplacement, circulation, and supersaturation of hydrothermal fluids and to the formation of large ore bodies over short time scales. [1]: #ref-10

The Validity of Gibrat's Law in Developed and Developing Countries (2008–2013): Comparison based Assessment

The Gibrat's Law of Proportionate Effect, indicated that the growth rate of a given firm is independent of its size at the beginning of the examined period. While earlier studies tended to confirm the Law, more recent research generally rejects it. This paper aims to highlight important empirical studies that performed analysis of Gibrat's Law in which about 50 papers were taken into account for this study. The main objective of this paper is to investigate the validity of Gibrat's Law in developing countries based in developed countries. In this paper the literature on Gibrat's Law was reviewed based on three types of empirical results: the first type accepts Gibrat's Law, the second type rejects Gibrat's Law, and the third classification reconciles both acceptance and rejection of Gibrat's Law. It was found that in most of the manufacturing sector, Gibrat's Law fails to hold but for the service sector Gibrat's law was valid. Additionally, only a few empirical studies have investigated Gibrat's law in developing countries; most of the studies have been conducted in developed countries. Furthermore, most of the empirical studies that have been applied in the developed countries rejected the Gibrat's Law. Finally, based on the features of SMEs that it's a labour intensive, and has the ability to absorb a lot of labour due to the low cost of creating job opportunities for both developed and developing countries. From this overview, it can be deduced that it is possible to implement the law for developing countries, which in turn could probably have the same result that the Gibrat's Law is not valid as smaller firms tend to grow faster than larger firms.

Dynamic Capabilities and Competitive Advantage into Mexican Firms: Testing Gibrat´s Law


 This paper inquires into the existence of dynamic capabilities and competitive advantage in Mexican firms. Dynamic capabilities refer to the particular nonimi­tability capacity firms possess to shape, reshape, configure, and reconfigure their assets so as to respond to changing technologies and market conditions for esca­ping the zero-profits condition. In turn, competitive advantage refers to the capa­city of firms for acquiring economic rents. In this sense, dynamic capabilities and competitive advantage are likely to be essential to the survival of firms in markets characterized to be innovative and in rapid technology change. It is argued that lo­cal firms ought to stimulate their own dynamic capabilities to successfully compete in markets. On the other hand, foreign firms would eventually transfer their own capabilities (technology and know-how) to local subsidiaries to successfully com­pete in these markets. Making use of econometric methods, it is corroborated the relation between firm performance (Law of Proportionate Effects) and dynamic capabilities into firms in Mexico.

Firm size and growth in Sweden's life insurance market between 1855 and 1947: A test of Gibrat's law

Using data for the period from 1855 to 1947 and the two sub-periods, 1855–1902 and 1903–47, the article examines whether the organic growth rates of 38 Swedish life insurance firms are independent of size, as predicted by Gibrat's (1931) Law of Proportionate Effects. Using panel unit root tests and panel Generalised Method of Moments (GMM) regression, the article finds a significant difference between the growth rates of small and large Swedish life insurance firms (with smaller firms tending to grow faster than larger firms), a result that clearly contradicts Gibrat's Law as a long-run tendency in the Swedish life insurance sector. significant influences were also found on firm growth from profitability, organisational form, reinsurance, the real rate of interest and the Swedish regulatory environment.

Innovation and Growth in Ugandan Manufacturing Firms

The aim of this study was to establish the effects of innovation on the growth of Ugandan manufacturing firms' using pseudo panel data. Little is known about the relationship between innovation and firm growth in Sub-Saharan African countries. This study adopts Gibrat's Law of Proportionate Effect and a learning model by Jovanovic (with some modifications) to analyse the relationship between innovation and firm growth in Uganda. Descriptive results show that innovative firms, as measured by computer usage and purchase of new machinery, compared to non-innovative firms on average grow faster. However, regression results showed that there is no significant difference between the growth rates of innovative and non-innovative firms.

Reconciling quantile autoregressions of firm size and variance–size scaling

The aim of this paper is to understand what economic mechanisms may cause the Law of Proportionate Effect to break down for fast-growing and shrinking firms. Recent evidence has highlighted that the first-order coefficients of quantile auto-regression of firm size decline across quantiles. Our theoretical results show that negative variance–size scaling is sufficient to yield a decline in quantile auto-regression coefficients if firm log-size is Laplace-distributed, conditional on size one period ahead. However, it is sufficient only for declining auto-regression coefficients for fast-growing firms under Asymmetric Laplace conditional log-size if skewness is decreasing with size. In other words, if the growth of large firms is less dispersed and more left-skewed, size is a disadvantage for the growth of fast-growers, but not necessarily an advantage for fast-decliners. Thus, size-related determinants of negative growth skewness, such as diseconomies of growth, market power, and managerial attention issues, impact on how the LPE is violated. Using data on Dutch manufacturing companies from the Business Register of Enterprises observed between 1994 and 2004, our empirical estimates of quantile regression models confirm the evidence of declining quantile regression coefficients for small–medium firms (20–199 employees) mainly in the right-most quantiles, and for the same subsample, we find that growth rates variance and skewness are decreasing with size. The theoretical propositions of the paper are thus corroborated.

Evolutionary model of the growth and size of firms

The key idea of this model is that firms are the result of an evolutionary process. Based on demand and supply considerations the evolutionary model presented here derives explicitly Gibrat’s law of proportionate effects as the result of the competition between products. Applying a preferential attachment mechanism for firms, the theory allows to establish the size distribution of products and firms. Also established are the growth rate and price distribution of consumer goods. Taking into account the characteristic property of human activities to occur in bursts, the model allows also an explanation of the size–variance relationship of the growth rate distribution of products and firms. Further the product life cycle, the learning (experience) curve and the market size in terms of the mean number of firms that can survive in a market are derived. The model also suggests the existence of an invariant of a market as the ratio of total profit to total revenue. The relationship between a neo-classic and an evolutionary view of a market is discussed. The comparison with empirical investigations suggests that the theory is able to describe the main stylized facts concerning the size and growth of firms.