Fig 3 - uploaded by Sergey V Buldyrev
Content may be subject to copyright.
Size-variance relationship (S) for various V with P(K) K 2 (A) and real P(K) (B). A sharp crossover from 0 to 1/2 is seen for the power-law distribution even for large values of V. In case of real P(K) one can see wide crossover regions in which (S) can be approximated by a power-law relationship with 0 1/2. Note that the slope of the graphs () decreases with the increase of V. The graphs of (KS) and their asymptotes are also shown with squares and circles, respectively.
Source publication
The relationship between the size and the variance of firm growth rates is known to follow an approximate power-law behavior σ(S) ≈ S−β(S) where S is the firm size and β(S) ≈ 0.2 is an exponent that weakly depends on S. Here, we show how a model of proportional growth, which treats firms as classes composed of various numbers of units of variable s...
Contexts in source publication
Context 1
... further explore the effect of the P(K) on the size-variance relationship we select P(K) to be a pure power law P(K) K 2 (Fig. 3A). Moreover, we consider a realistic P(K) where K is the number of products by firms in the pharmaceutical industry (Fig. 3B). This distribution can be well approximated by a Yule distribution with 2 and an exponential cutoff for large K. Fig. 3 shows that, for a scale-free power-law distribution P(K), the size-variance relationship ...
Context 2
... further explore the effect of the P(K) on the size-variance relationship we select P(K) to be a pure power law P(K) K 2 (Fig. 3A). Moreover, we consider a realistic P(K) where K is the number of products by firms in the pharmaceutical industry (Fig. 3B). This distribution can be well approximated by a Yule distribution with 2 and an exponential cutoff for large K. Fig. 3 shows that, for a scale-free power-law distribution P(K), the size-variance relationship depicts a steep crossover from V given by Eq. 2 for small S to V/K S given by Eq. 3 for large S, for any value of V ...
Context 3
... effect of the P(K) on the size-variance relationship we select P(K) to be a pure power law P(K) K 2 (Fig. 3A). Moreover, we consider a realistic P(K) where K is the number of products by firms in the pharmaceutical industry (Fig. 3B). This distribution can be well approximated by a Yule distribution with 2 and an exponential cutoff for large K. Fig. 3 shows that, for a scale-free power-law distribution P(K), the size-variance relationship depicts a steep crossover from V given by Eq. 2 for small S to V/K S given by Eq. 3 for large S, for any value of V ...
Similar publications
The article investigates three ways of forming a final estimate using two different estimators with different variances and possibly also biases: (1) taking the estimate produced by the better estimator, (2) taking their simple average, (3) taking their weighted average. It is shown that if there is no serious positive correlation, using both estim...
Citations
... However, this approximation works only if units are about the same size x. Empirically, high heterogeneity has been observed to be persistent in time and across different scales of economic systems [21,17]. Furthermore, the variance is empirically found to decrease much slower, as S −2β with 2β < 1. ...
... In a recent contribution [7], the model was extended in several directions, for example to take into account a stable economy, the contribution of the change in the number of units K i and multiple levels of aggregation. For the scaling law, the model predicts a crossover with a variance that does not depend on size for small firms to S −1/2 in the limit S → ∞ [17]. This model can be extended to exponential mixtures of Gaussians [6]. ...
We review models of compositional growth, which were introduced to explain the growth statistics of various quantities ranging from firm sizes to GDP. In these models, entities are decomposed into units that grow independently. Thus, the growth rate of the entity is the addition of the growth rates of the composing units, with possibly heterogeneous weights. We review such models and show that they can be understood through a unifying theoretical framework, explaining the resulting growth rate distributions using mixtures of Gaussians.
... There has been a relatively recent interest in the parametric description of the log-growth rates of the sizes of firms and of countries' outputs in its relation to Gibrat's law [2,3,6,7,8,10,11,14,15,16,17,36,37,40,44]. In this strand of the literature, the asymmetric double Laplace distribution is typically studied, instead of the normal distribution, to account for the observed exponential tails exhibited by the log-growth rates. ...
... The body is approximately normal, although it is not possible to exactly delineate the switch between the normal and the exponential behaviours, since the adLn distribution is the convolution of an asymmetric double Laplace with a normal distribution. We will describe here the asymmetric version [3] of a distribution initially introduced by [45] and later considered by [2,6,11,14,15,16,30,36,37,44]. We will call it the asymmetric Subbotin distribution (aSub). ...
... And if the tails are exponential, the plot at the upper/lower tails should be linear. What we see at Figures 1 and 2 is that the discrepancies are not very remarkable but that the graphs are mainly curved, as a difference with reported properties of firm growth rates [2,3,6,10,11,14,15,16,17,36,37,44]. Table 5. ML estimators and standard errors (SE) for the dmeGB2 and the European samples. ...
We have studied the parametric description of the distribution of the log-growth rates of the sizes of cities of France, Germany, Italy, Spain and the USA. We have considered several parametric distributions well known in the literature as well as some others recently introduced. There are some models that provide similar excellent performance, for all studied samples. The normal distribution is not the one observed empirically.
... In fact, we show that a parsimonious theoretical framework, as the model proposed by Refs. 19,[41][42][43] , can fit the mutual fund sample. For this reason, we refer to such theoretical framework and we propose to use the corresponding slope of the sizevariance relationship as a synthetic indicator for evaluating the aggregate conditions of the system arising from the micro-level behaviors of its participants. ...
... We refer to the stochastic model of generalized proportional growth proposed in 19,[41][42][43] to derive the stylized facts in the mutual fund industry. This model is based on two sets of empirically verifiable assumptions regarding the structure of economic entities. ...
... The size-variance relationship. The size-variance relationship describes the behavior of the standard deviation of economic entities' growth rates versus their sizes, which can be roughly characterized as a power law σ r = S −β with an exponent β ≈ 0.20 (see, e.g., refs 15,16,18,19,[41][42][43] ). In the model proposed by 19,[41][42][43] , the sizevariance relationship theoretically follows from the assumption of uncorrelated fluctuations of units and a sufficiently broad distribution of unit sizes. ...
We show that some key features of the behavior of mutual funds is accounted for by a stochastic model of proportional growth. We find that the negative dependence of the variance of funds’ growth rates on size is well described by an approximate power law. We discover that during periods of crisis the volatility of the largest funds’ growth rates increases with respect to mid-sized funds. Our result reveals that a lower and flatter slope provides relevant information on the structure of the system. We find that growth rates volatility poorly depends on the size of the funds, thus questioning the benefits of diversification achieved by larger funds. Our findings show that the slope of the size-variance relationship can be used as a synthetic indicator to monitor the intensity of instabilities and systemic risk in financial markets.
... Verifying previously hypothesized stylized facts and checking the mathematical consistency of independently known power-law exponents help to validate or refute possible theories regarding business firms. Numerous models [35,[48][49][50][51][52][53][54][55][56][57][58][59][60][61] have been suggested to explain only a few stylized facts relating to firms, and it appears that new criteria are needed to select the models empirically. Thus, our results for multivariate scaling should be incorporated into subsequent theoretical considerations. ...
Although the sizes of business firms have been a subject of intensive research, the definition of a “size” of a firm remains unclear. In this study, we empirically characterize in detail the scaling relations between size measures of business firms, analyzing them based on allometric scaling. Using a large dataset of Japanese firms that tracked approximately one million firms annually for two decades (1994–2015), we examined up to the trivariate relations between corporate size measures: annual sales, capital stock, total assets, and numbers of employees and trading partners. The data were examined using a multivariate generalization of a previously proposed method for analyzing bivariate scalings. We found that relations between measures other than the capital stock are marked by allometric scaling relations. Power–law exponents for scalings and distributions of multiple firm size measures were mostly robust throughout the years but had fluctuations that appeared to correlate with national economic conditions. We established theoretical relations between the exponents. We expect these results to allow direct estimation of the effects of using alternative size measures of business firms in regression analyses, to facilitate the modeling of firms, and to enhance the current theoretical understanding of complex systems.
... In fact, not all new firms pursue growth Financing, selection, and value-adding effects of venture capital 253 as a strategic objective (Cabral and Mata 2003;Hurst and Pugsley 2011) and may remain small in order to avoid taxes, to maintain workforce flexibility, or because of control aversion (see Chittenden et al. 1996 on high-tech spinoffs andCressy andOlofsson 1997). Consistently, in Bottazzi and Secchi (2006), Riccaboni et al. (2008), and Bottazzi et al. (2011), one finds that the standard deviation of firm growth rates is negatively correlated with lagged firm log-size (variance-size scaling). 4 Although it has recently been argued that variance-size scaling is a statistical artefact of mixing Gaussian variates (Manas 2012), heteroskedasticity needs to be accounted for in statistical estimation and testing, for instance, when running Hausman tests to decide between fixed and random effects. ...
... Understanding the distribution of orders of sellers is conducive to taking in the growth trend, which can ultimately enhance the prediction results. There are sufficient literatures in enterprise increasing theory recognizing the expansion mechanism of firm sizes as the power law (Ashton, 1926;Little, 1962;Takayasu and Okuyama, 1998;Buldyrev et al., 2007;Riccaboni et al., 2008). On that basis, an assumption is proposed that Online retailing platform the sellers' order size subjects to a power-law distribution to catch the fact that there are fewer big sellers while most are small scale. ...
Purpose
Nowadays, some online retailing platforms emerge to integrate transport capacity to provide standard distribution service for sellers. Such an integrated form of service is defined as delivery alliance. To have a better understanding of how to price the service, this study aims to fixate on the seller’s problems and builds a series of profit maximization models in accordance with the two-sided market pricing theory within a platform business model.
Design/methodology/approach
In the present study, some optimization models are built in the two-sided market type and the optimal solutions are found in a three-dimensional decision space. By using the basic model as the benchmark, some optimization problems of DA in realistic situations are discussed. Particularly, a power-law-distribution model is established to deal with the uncertainty in forecasting. Also, a price-sensitive model and a loss-aversion model are presented to describe the various reactions of sellers to charging modes. Finally, some combined situations are discussed and the strategies are compared under the mentioned models.
Findings
By selecting the basic model as the benchmark, the specific pricing strategies are found for each context to yield the optimal profits. The flexibility of pricing strategy in the basic model and rigid pricing strategies in extended models, are discussed. As a result, the guidelines for the online retailing platforms are developed on designing and pricing the DA service.
Research limitations/implications
First, it would be interesting to expand the pricing plan of the platform. For instance, menu pricing and quantity discount have not been considered, which are common in practice. The time discounting has also been ignored. If the time value were calculated, the contract fees would be more critical due to the earliest of collecting money. Finally, those joiners who have huge order sizes are crucial for the ecosystem indeed, but arouse no attention. While in reality, they may have more power to bargain with the platform. Thus, how the platform competition affects the pricing strategies needs future research.
Originality/value
The optimal pricing strategies under these models are analytically found out, and it is shown that the presented models result in the same scale of joiners and profits in optimization. This suggests that DA works well in various behavioral contexts. This also suggests that DA is a significant controller in service quality improvement. Then, the optimal pricing strategies are compared among all the models. During this, it is discovered that the realistic contexts might reduce the profit, whereas an appropriate pricing strategy can pull this back without loss of service quality.
... In terms of researcher population growth, measured by S(t) for each country, non-EU (EU) countries are growing on average at a 5.3% (3.5%) annual rate. Extensive research shows that growth rates are typically inversely proportional to the enterprise size (Riccaboni et al., 2008). As such, the higher average value for non-EU countries largely reflects the emerging economies that have entered the scientific enterprise at the global scale only as of relatively recently. ...
Quantitative research evaluation requires measures that are transparent, relatively simple, and free of disciplinary and temporal bias. We document and provide a solution to a hitherto unaddressed temporal bias – citation inflation – which arises from the basic fact that scientific publication is steadily growing at roughly 4% per year. Moreover, because the total production of citations grows by a factor of 2 every 12 years, this means that the real value of a citation depends on when it was produced. Consequently, failing to convert nominal citation values into real citation values produces significant mis-measurement of scientific impact. To address this problem, we develop a citation deflator method, outline the steps to generalize and implement it using the Web of Science portal, and analyze a large set of researchers from biology and physics to demonstrate how two common evaluation metrics – total citations and h-index – can differ by a remarkable amount depending on whether the underlying citation counts are deflated or not. In particular, our results show that the scientific impact of prior generations is likely to be significantly underestimated when citations are not deflated, often by 100% or more of the nominal value. Thus, our study points to the need for a systemic overhaul of the counting methods used evaluating citation impact – especially in the case of researchers, journals, and institutions – which can span several decades and thus several doubling periods.
... On business companies, for example, models have highlighted several mechanisms or factors as diverse as hierarchical organization [30], stochasticity in competition [31,32], financial [33] or hiring/ firing [34] behaviors, preferential attachment of company units [35], social networks [36] and multiple independent components in companies [35,[37][38][39][40]. Although they often agree with the data at the level of distributions calculated from a snapshot of the systems, these models lack validation by direct observation at microscopic levels, or fail when their microscopic dynamics is explored [41]. The situation is similar in biodiversity research: two classes of models with radically different assumptions on dynamics at microscopic levels both can explain the rank-size distribution of abundance [42,43] and spatial distributions [44,45] of natural species. ...
... Finding the novel stylized fact of agreement between the scaling and stability curves, we believe that our results could be also beneficial to future modeling and theoretical considerations. Although researchers have formulated numerous models [30][31][32][33][34][35][36][37][38][39][40][41], we are not aware of any work that could be used to explain the agreement between the scaling and stability curves demonstrated here. Thus, we suggest that future theoretical studies should incorporate this phenomenological multi-variate evolution of companies. ...
Although scaling relations of power law forms have been revealed in a wide variety of complex systems, the origin of scaling relations in real systems remains mostly unsolved. Based on a long tradition in physics, we here explore the phase space dynamics of business companies, whose different measures of size have been found to exhibit multiple scaling relations. Using a large scale dataset of Japanese companies covering over two million companies for over two decades (1994–2015), we compile the data of three measures of size, namely, annual sales, number of employee and number of trading partners. Tracking the historical time evolution of companies, we first show that there exists a stable region that attracts most of the data points in the long time scale, although the company dynamics in the three-dimensional phase space looks random in a shorter time scale. We then elaborate on ‘evolving flow diagrams’ of the averaged motion in the 3D space. Remarkably, the flow diagrams indicate that a 3D curve which represents the scaling relations between the three size measures can be regarded as an approximate attractor of the averaged flows. The results could serve for better modeling and understanding of companies dynamics and our method can be applied to other dynamics such as social and biological phenomena.
... Recently, the pursuit of statistical regularities in economics data and theoretical explanations have received increasing interest from both the physics and economics communities 1,2 . Using data on US manufacturing firms from 1974 to 1993, Stanley et al. 3,4 documented that the standard deviations of the growth rates obey a power law with a scaling exponent of approximately −1/5 [4][5][6][7] and that the distribution of growth rates conditional on initial size is exponential over seven orders of magnitude 3,4 . These results resemble the power laws that are robust statistical properties of many complex interacting physical systems 2,[8][9][10][11] . ...
The growth of business firms is an example of a system of complex interacting units that resembles complex interacting systems in nature such as earthquakes. Remarkably, work in econophysics has provided evidence that the statistical properties of the growth of business firms follow the same sorts of power laws that characterize physical systems near their critical points. Given how economies change over time, whether these statistical properties are persistent, robust, and universal like those of physical systems remains an open question. Here, we show that the scaling properties of firm growth previously demonstrated for publicly-traded U.S. manufacturing firms from 1974 to 1993 apply to the same sorts of firms from 1993 to 2015, to firms in other broad sectors (such as materials), and to firms in new sectors (such as Internet services). We measure virtually the same scaling exponent for manufacturing for the 1993 to 2015 period as for the 1974 to 1993 period and virtually the same scaling exponent for other sectors as for manufacturing. Furthermore, we show that fluctuations of the growth rate for new industries self-organize into a power law over relatively short time scales.
... Recently, the pursuit of statistical regularities in economics data and theoretical explanations have received increasing interest from both the physics and economics communities [1]. Stanley et al. [8,9] documented a strong empirical relationship in economics data; for US manufacturing firms between 1974 to 1992, the standard deviations of the growth rates were shown to obey a power law with a scaling exponent of approximately -1/5 [9][10][11]. Furthermore, it has been shown that for publicly traded US manufacturing firms in the 1990s, the distribution of growth rates conditional on initial size is exponential over seven orders of magnitude [8]. ...
Work in econophysics has shown the existence of robust structures in economics data, which had been largely ignored in past economic literature. Many complex interacting systems in nature, such as earthquakes, are described by power laws. In statistical physics, fluctuations for systems near the critical point follow a power law distribution. The exponents associated with the power law for fluctuations can be used to categorize the systems into specific universality classes. In economics, growth rates for firms depend on different factors for different industries. Here, we show that fluctuations in growth for companies follow a power law with very similar scaling exponents, irrespective of the economic sector. Our findings hold for many industries across all available data. Furthermore, we show that fluctuations of the growth rate for new industries self-organize into a power law distribution over relatively short time scales. Our results provide the first strong evidence of statistical mechanical universality in economic systems, and can be used as an empirical test for theories of microeconomic growth.
