The age distribution of business firms

We investigate upon the shape and the determinants of the age distribution of business rms. By employing a novel dataset covering the population of French businesses, we highlight that a geometric law provides a reasonable approximation for the age distribution. However, relevant systematic deviations and sectoral heterogeneity appear. We develop a stochastic model of rm dynamics to explain the mechanisms behind this evidence and relate them to business dynamism. Results reveal a long-term decline in entry rates and lower survival probabilities of young rms. Our ndings bear important implications for aggregate outcomes, notably employment growth. JEL Codes: L11, L22, M13, M21.

1 Introduction is obtained. Moving to more complete scenarios, more apt at coping with violations of the geometric law, our model suggests that di erent slopes of the age log-distribution are related to variations in entry rates, to heterogeneity in the survival probabilities, or to sectoral di erences. Third, our analysis highlights how the slope of the cross-sectional age log-distribution -pictured at a single given point in time -is an important structural parameter that has signi cant links with the long-term evolution of business dynamism and the life-cycles of industries, with relevant implications for economic outcomes, notably employment growth.
The remainder of the paper is organised as follows. Section 2 provides a brief overview of the literature related to this study. Section 3 describes the data used for the empirical analysis and presents three key stylised facts related to the age distribution of business rms. Section 4 formalizes a stochastic model of rm dynamics, solved under di erent assumptions on survival probabilities and entry rates, and presents several comparative statics exercises that guide the interpretation of stylised facts. Section 5 presents a calibration exercise of the stochastic model and con rms its validity. Section 6 discusses the stylised facts, in the light of both model prescriptions and its calibration and suggests that sectoral di erences in age distributions have important implications for employment growth. Section 7 concludes and discusses avenues for further research.

A brief overview of the literature
Only few empirical analyses provide comprehensive insights on the rm age distribution. These existing studies include the work by Coad (2010b), Barba Navaretti et al. (2014), Kinsella (2009), and Grazzi and Moschella (2018). In particular, Coad (2010b) investigates and compares the age distributions of small scale Indian businesses, of young establishments in the United States, of samples of Italian and Spanish rms, and of rms in the international airline industry, providing a rst broad investigation of cross-sectional age distributions using di erent data sources. Furthermore, Barba Navaretti et al. (2014) estimates the age distribution for a sample of rms in Italy, France and Spain, Kinsella (2009) instead focuses on a sample of Irish rms, while Grazzi and Moschella (2018) as a side exercise to their core analysis, provide some insights on the age distribution of Italian exporters vs. non-exporters.
These studies suggest that the rm age distributions of the samples or groups of rms analysed approximate an exponential law, the continuous counterpart of the geometric distribution. 1 Despite being highly informative, these existing studies do not provide a complete characterisation of the aggregate age distribution of business rms. As a mater of fact, they are mainly based on samples rather than whole populations of businesses. Furthermore, at the best of our knowledge, discussions about the mechanisms behind the shape of the age distribution, systematically linking it to entry rates and survival, to the debate on the evolution of business dynamism and industry life-cycles, or more generally to industrial dynamics, have been very limited.
A relevant exception is the work by Steindl (1965) that employs the rm age distribution to provide an explanation of the emergence of a Pareto rm size distribution in a stochastic model of rm dynamics. A similar approach is taken by Coad (2010a), who takes instead an exponential age distribution as an assumption, to derive the Pareto rm size distribution. In particular, Coad (2010a) suggests that there are two possible frameworks under which a perfect exponential age distribution would emerge. The rst case is a model in which a constant number of rms enter the market, but a constant proportion of these entrants exit the market in each year. The second case, instead, is a model in which rms do not exit and an exponentially increasing number of rms enter the market in each period (see Simon, 1955). Our approach generalizes the previous ones considering entry rates and introducing stochastic exit. Moreover, allowing for dynamics in entry rates, heterogeneity in survival probabilities, and sectoral di erences, we are able to account for deviations from the exponential (or geometric) benchmark.
Indeed, this paper directly links the age distribution of businesses to the dynamics of entry and survival. In that, it is closely related to the recent literature focusing on the evolution of business dynamism. In particular, there is a lively debate on the extent to which business dynamism, and notably entry rates, have been declining in the last decades. Signi cant declines have been observed in the United States (see Decker et al., 2014 among the others) and across countries (Criscuolo et al., 2014;Calvino and Criscuolo, 2019). However, the lack of availability of very long and consistent time series of entry and exit rates, especially at detailed sectoral levels of aggregations, has been a signi cant limit for this stream of research. Focusing on the cross-sectional age distribution represents an indirect way to analyse the longer-term evolutions of business dynamism, also detailing sectoral patterns, as will be further discussed along this paper.
Our analysis also links sectoral di erences in age distributions with industry evolution, revealing that sectors may be at di erent phases of their life-cycles. We therefore relate to the literature on the evolution of industries, and importantly to the legacy of Klepper (1996). In particular, the seminal contribution by Klepper (1996) suggests that at the beginning of the life-cycle of many industries entry is high, there is a growing number of producers, with market shares changing rapidly and signi cant product innovations. As industries evolves, entry starts declining, there is a shake-out reducing the number of producers, and the industry leadership stabilises. In this stage product innovation and the diversity of product varieties decline, with rms' innovative activity shifting to process innovation. In this framework, analysing di erences in sectoral age distributions is particularly promising to identify the stages at which di erent sectors stand, as will be further discussed in the following.

Data and stylised facts
The analysis is based on a novel data source produced and published by the French National Institute of Statistics (INSEE), the Sirene database. The data covers the population of all enterprises and of all their establishments in France and encompasses about 10 million establishments in all sectors of the economy. 2 This is the most appropriate database to study rm age distributions, given the comprehensiveness of its coverage and the ability to provide information on establishment and enterprise age also for very old businesses. Using this data source represents a signi cant novelty with respect to previous empirical investigations on the topic.
The dataset includes information on enterprise and establishment identi ers, detailed sectoral and geographical codes, and -most importantly for the aim of this study -information on the date of creation of each enterprise and establishment. 3 Our analysis, thus, focuses on all businesses in France métropolitaine and we exclude regions oversea, called in French the départements d'outre-mer, i.e., Guadeloupe, Guyane, Martinique, Mayotte et Réunion and Saint-Pierre et Miquelon. Also, given the focus on business rms, the main body of the paper excludes the non-business sectors and focuses mainly on manufacturing and non-nancial market services. 4 2 Di erently from balance-sheet data (e.g., FARE) or matched employer-employee data (e.g., DADS), the coverage of Sirene also comprehensively includes self-employed individuals and rms that are not subject to the presentation of detailed balance sheets. The database is timely available and updated monthly (https://www.data.gouv.fr/fr/datasets/base-sirene-des-entreprises-et-de-leursetablissements-siren-siret/). This may limit the identi cation of inactive rms.
3 Dates of creation have been systematically collected from 1980. Before that date, rms and establishment creation dates are based on the di erent business and commercial registries (registres du commerce, des socétés, de l'industrie, etc.). We will take this into account in our empirical exercise. 4 Sectoral codes (AP ET700) retained include manufacturing, non-nancial market services, as well as computer repairers and personal services, so that the following condition AP ET700 ∈ [1001, 3320]; [4110,6399]; [6810,8299]; [9511,9609] is met.
We focus on the cross-sectional stock of enterprises and establishments at the 1 st of January 2018. Additionally, the data have been cleaned to take into account common issues, such as avoiding situations in which a multi-establishment business owned an establishment which was registered before the enterprise itself. The ltering procedure of sectors, in addition, has been performed at the establishment level. This allows us to keep in our clean dataset also the single establishments operating in business sectors, of the multi-establishment enterprises whose core activity does not fall into business industries. The nal population used for the analysis, includes 6 519 413 enterprises and 7 155 860 establishments belonging to 608 di erent 4-digit sectors. Age is de ned as the number of years between the creation date and 2018. Thus, in what follows age shall be intended as a discrete variable. 5 This choice is pretty natural and allows us to avoid possible seasonality issues. A rst set of descriptive statistics concerning the age of these rms is provided in Table 1 Table 1 highlights that aggregate age distributions are right-skewed, as the average establishment and enterprise ages lie at the right than the two medians. Reasonably, enterprises are in general older than establishments due to the possibility that multi-establishment businesses might have built new plants after the creation of the enterprise itself. Based on these data, the reminder of this section presents more in detail three key stylised facts on the age distribution of businesses in France from a purely empirical viewpoint. The discussion of their implications through the lenses of an economic model is left to Section 6.
3.1 Fact 1: geometric law as a rst-order approximation for the age distribution The rst part of the empirical analysis focuses on the aggregate cross-sectional age distribution of businesses in France in 2018. As a rst exercise, we show the plots of the age log-distributions at the establishments and enterprises levels in Figure 1, where the ordinary least squares (OLS) line of best t is also depicted. 6 In black we depict the empirical distribution and in blue the OLS t. 5 Businesses with age equal to zero are disregarded as the 1 st of January is not representative of the whole year. 6 Unreported analysis suggests that a qualitatively similar shape is also observed when excluding self-employed.
From Figure 1 -where we depict log-frequencies on the y-axis -one can observe that a geometric law can provide a reasonable rst-order approximation of the two empirical age distributions. As a matter of fact, in a perfect geometric case, the log-distribution would be represented by a perfectly straight line. Plots similar to the ones observed in Figure 1 are also visible when taking a slightly lower aggregation level. Indeed, even by separating rms and establishments between manufacturing or services, the aggregate results remain unchanged both qualitatively and quantitatively. We show this in a simple regression framework in which we estimate by means of ordinary least squares the linear model: where the subscript i ∈ {Aggregate, Manufacturing, Services} denotes the level of aggregation. We collect the results in Table 2 for establishments and enterprises. It is possible to observe that not only at aggregate level, but also when focusing separately on manufacturing and services, the estimated coe cients for the slope of the log-distribution is between −0.10 and −0.12. This suggests that for each additional year of life, on average, there are 10.4% less rms. The OLS estimation shall only be considered as an approximation. As a matter of fact, recent contributions to the literature highlight that there may be some challenges when carrying out this type of OLS estimation (see Bottazzi et al., 2015 for a critical discussion). 7 An alternative way to estimate the slope parameter is to assume ex-ante that the overall age distribution follows a geometric law. Under this strong assumption, one can consistently and e ciently estimate the parameter λ characterizing the geometric distribution λ(1−λ) x−1 by means of maximum likelihood (ML). A relevant di erence between the OLS and the geometric ML t lies in the way observations are weighted. The former assigns the same weight to every age pro le disregarding their size in terms of observations. The latter, instead, takes into account the fact that many more observations are available for low age pro les. Thus, the ML t of a geometric law is designed to better capture the slope of the age distribution for younger rms, at the cost of a poor t in the right tail of the distribution, for the few very old rms. Results from this exercise support the intuition provided by the simpler but biased OLS (see Figure A.1).
In general, also considering the very high levels of adjusted R 2 from the regression exercises equally weighting all age bins, we conclude that at broad levels of aggregations -similarly to what pointed out by Coad (2018) among the others -the geometric distribution provides a reasonable rst-order approximation of the age distribution of businesses. However, as already evident in Figure 1, the geometric t is not perfect. A formal quantitative statistical evaluation carried out exploiting the ML framework (e.g. by a χ 2 test, which is presented in Appendix A) leads to a strong rejection of the null hypothesis of a perfect geometric shape. 8 This is further explored in the next stylised fact.

Fact 2: signi cant deviations from the geometric benchmark in the aggregate
Examining more closely Figure 1, one can however notice that the geometric t, despite being reasonable at rst glance, has limitations. The deviations from the geometric benchmark display, as a matter of fact, a clustered pattern over the age support (or over time reading Figure 1 from the right to the left). As we move from the youngest rms to the oldest ones in Figure 1 we can observe i) a atter distribution for rms with age between 1 and about 30 years old and ii) a steeper distribution for rms in the 30-50 years old region.
After that age, we observe more signi cant variations from the geometric benchmark. The distribution appears to some extent even steeper -but more volatile -for rms older than 50 but younger than 75 years old, and much atter after that. These patterns need to be taken with more caution due to the absence of 7 Bottazzi et al. (2015) focus on the Zipf law, however their arguments can be also adapted to our case considering an exponential transformation of data. 8 This rejection and the equal weighting of all age bins further convinced us to keep referring to the OLS framework in the main text. This will also better t with the intuitions behind the theoretical model presented below.  a su ciently high number of observations and the fact that the dates of creation of rms have been more systematically collected only since 1980.
In order to explore more in detail the deviations from the geometric benchmark we perform further analysis of the residuals and their correlation. This is reported in Figure 2, which shows that, the deviations from the perfect geometric benchmark are non-random over time and they rather display very high levels of auto-correlation at several lags (i.e., age di erences). This indicates that the patterns of the age distribution identi ed (e.g., the increasing residuals between 1 and about 30 and the decreasing ones between 30 and 50) are likely due to some long-run trends in the economic system that characterize the degree of market turbulence. This will be further discussed in the following sections.  We then enrich the analysis based on simple OLS estimates complementing the residuals analysis with the maximum likelihood estimation of a piecewise geometric distribution. We de ne it as the distribution that assigns to every x ∈ N a probability where k = (k 1 , k 2 , . . . , k K ) is a vector of K breaks (that is, points in which the distribution changes its shape), λ = (λ 1 , λ 2 , . . . , λ K+1 ) is a vector containing the parameters which determine the shape of the di erent pieces, H(x) represents the Heaviside function (such that H(x) = 1 if x > 0 and H(x) = 0 otherwise), and g 0 (λ, k) is a normalization term reading The results are reported in Figure 3, where we set k = (10, 20, 30, 40) and zoom on rms younger than 50 years old. 9 The slope of the geometric t is steeper in the interval (0, 10) than in [10, 30), i.e., when contrasting younger with older businesses. Similarly, the distribution is steeper for rms with age a ≥ 30 with respect to rms with age a ∈ [10, 30), i.e. when contrasting rms born during the 1970s and 1980s with rms born during the great moderation. 10 So far the analysis has focused on aggregate or very broad macro-sectoral features of the age distribution of businesses. But these aggregate behaviours can hide important compositional e ects. For this reason, we focus on more disaggregated sectoral groups and analyse their age distributions in the next sub-section.

Fact 3: signi cant sectoral heterogeneity, with stronger deviations at ner grained levels
In this section we further explore the properties of the age distribution of French businesses in 2018 focusing on lower levels of aggregation. As previously mentioned, this is relevant to better understand the extent to which sectoral heterogeneity a ects the aggregate distribution.
In particular, we plot the empirical age distributions and perform OLS geometric ts separately for different 2-and 3-digit sectors. 11 Although we have plotted age distributions of every single 2-and 3-digit sectors, we report the plots of the empirical distributions with geometric ts for nine selected 2-digit sectors, in Figure 4 below, and nine selected 3-digit sectors in Figure A.2 in Appendix A.
Focusing on those gures, it is already possible to observe that the geometric t still appears as a reasonable rst-order approximation of the empirical distribution in many sectors, but that the coe cient estimates are heterogeneous across sectors. For example, Figure 4 suggests that the slope of the log-distribution of the food manufacturing industry (10) is signi cantly atter than the slope of the computer programming, consultancy and related activities industry (62). Furthermore, Figure A.2 highlights that this result holds true at more disaggregated levels, when comparing the manufacturing of dairy products (105) with computer programming, consultancy and related activities (620). Signi cant heterogeneity of the slopes of the log-distributions with relatively good geometric t are also observed at the 2-digit level. Results of the linear regression estimates are reported by Table A.2. In order to summarise the sectoral heterogeneity in the slopes of the log-distributions and the goodness of their geometric t, we present in Figure 5 the distributions of the slope coe cients and the adjusted R 2 (retrieved from the estimation of equation (1) at the 2-and 3-digit levels). The results suggest that the more one looks at the microeconomic level, by focusing on narrower sectors, the higher the heterogeneity in the estimated slope and the lower the quality of the geometric t. The goodness of t seems to con rm that the geometric distribution appears as a reasonable rst-order approximation also when focusing on 2-digit sectors, but at the 3-digit sectoral disaggregation, more important deviations are revealed, as highlighted by the fatter left tail of the adjusted R 2 distribution (bottom right panel).
Sectoral heterogeneity and deviations from the geometric t are likely related to sectoral characteristics and industrial dynamics. An important dimension to consider appears to be related to the phase of their life-cycle, as will be further discussed in the following. Furthermore, the signi cant sectoral heterogeneity suggests that the generalized geometric pattern observed at the aggregate level (see Table 2) is therefore conceivable as an emergent property and that compositional e ects play an important role.

Model
In order to better understand the mechanisms behind the shape of the age distribution, its deviations from the geometric benchmark, and the sectoral heterogeneity discussed in the previous section, we propose a stylised stochastic model of rm dynamics. The model suggests that the age distribution is nothing but an emergent property -an unconditional object using the language of Brock (1999) -whose cross-sectional distributional shape is determined by the evolution of the rates of entry and survival of businesses as well as by their di erence.
This model abstracts from several aspects of the reality -e.g. it does not account for growth dynamics nor considers rm size -but has the ability of supporting us in a better understanding of the stylised facts discussed in the previous section. This approach is consistent with some of the intuitions put forward by Coad (2010a).
We consider an economy in discrete time indexed by t ∈ N. De ne N t ∈ N as the number of rms present in the economy at the end of t. We assume that at the beginning of time t = 1 a number N 0 ∈ N of rms are created, while, in every period t > 1, φ t N t−1 new rms enter the economy. 12 Thus, φ t is the entry rate. Each rm is uniquely identi ed by a number i ∈ N which is progressively assigned. De ne F t as the set of rms present in the economy at the end of time t (such that N t = |F t |) and f t as the set of new entrants at time t. Each rm i ∈ {F t−1 ∪ f t } has a probability ρ i,t of surviving time t and, thus, of belonging to F t . It follows that the probability of rm i to exit the economy at time Finally, the age of rm i ∈ F t , which entered the economy at time t i , is de ned as a i,t = t − t i + 1 and the number of rms of age a at time

Solving the model
We solve the model for the expected number of rms of a given age at a generic time period t under di erent assumptions on survival probabilities and entry rates -i.e. di erent scenarios. Looking at the expected number for di erent age levels, one recovers a sort of theoretical histogram that can be easily interpreted as a theoretical age distribution. In the simplest scenario we are able to go further and derive asymptotic relative frequencies. This exercise helps in understanding how the geometric behaviour originates. The fundamental idea underlying our analysis is that the event " rm i survives time t" can be expressed in terms of a Bernoulli random variable s i,t ∈ {0, 1} with Prob{s i,t = 1} = ρ i,t . As we shall explain, even if such process let the age 12 We assume that φt ∀t > 1 and N0 are such that φtNt−1 ∈ N ∀t > 1.
of a rm at the moment of failure be geometrically distributed, obtaining that the distribution of rms' age at a given time is geometric is not trivial.
Baseline scenario: constant entry rate and survival probability. Let us begin from the simplest case by assuming that the survival probability is constant for all rms and all periods, ρ i,t = ρ ∀t, i, and the entry rate is constant over time, φ t = φ ∀t > 1. The number of rms in the economy at the end of time t can be expressed using the Bernoulli random variables just introduced. More speci cally one has (3) Iterating the map one gets Then, exploiting the de nition of the Bernoulli random variables, the number of rms with age a at time t can be written as Recalling that |f t | = N t−1 φ for t > 1 and N 0 for t = 1, its expectation for the case a < t is and, substituting with eq. (4) computed at time t − a, the expected number of rms with age a reads The case a = t is derived from eq. (6) noticing that the expected number of rms created at time t = 1 is indeed known and equal to N 0 . Notice that log E[N t (a)] is linear in a and the coe cient of a is −(1+φ). This means that the slope of the age log-distribution, depends only on the entry rate, while the survival probability in uences only the log-distribution intercept.
To better understand how the results originate, we compute the asymptotic relative frequencies of the age variable, under the assumption that the number of rms in the economy is su ciently large in every period. In particular, it is enough to assume N 0 → +∞ and φ ≥ (1 − ρ)/ρ, such that the expected number of rms cannot converge to zero for large t. 13 Under those assumptions, the Strong Law of Large Numbers implies that, almost surely with respect to the probability measure of the Bernoulli trials (s i,t ) i∈N,t∈N , it is lim N 0 →∞ N 1 /N 0 = ρ in the rst period and lim N 0 →∞ N t /N 0 = (1 + φ) t−1 ρ t for any t > 1. 14 The relative frequency of rms with age a at time t can be computed as the ratio of the number of rms born at t − a + 1 and survived until t over the sum of all the rms survived until t. Calling F t (a) such quantity, we have and, by the Strong Law of Large Numbers, we almost surely obtain Factorizing and taking the limit for t → ∞, we get the asymptotic relative frequency of rms with age a, F(a), which almost surely reads Solving for the geometric series at the denominator and de ning λ = φ/(1 + φ), we almost surely have which clearly shows the geometric behaviour of the rm age distribution. Given the way in which the random variable s i,t is de ned, obtaining that age is geometrically distributed may appear trivial. But this is not the case. First, because the age of a rm at a given time t cannot be interpreted as the number of Bernoulli trials (i.e. periods) needed to have a rm to fail. In other words, age is not computed as how old a rm is at the time of exiting the economy; a rm can in fact continue to survive even in the following periods. Second, because the number of new rms entering the economy changes over time and it is strictly connected to the amount of incumbent rms. Thus, we cannot see cohorts of rms with di erent ages as random samples from the same underlying process observed at di erent stages. This is evident by noticing that F(a) only depends upon the entry rate φ, and not on the survival rate ρ.
The emergence of the geometric behaviour in our framework can be understood considering the following intuition. Notice that λ is the fraction of new rms in the economy at the beginning of a given period -i.e. λ = |f t |/|{F t−1 ∪ f t }| ∀t ∈ N -while 1 − λ is the fraction of incumbents. Hence, in order to show an age equal to a, a rm has to belong to the new entrants group for 1 period, and to the incumbents group for a − 1 periods. Multiplying such fractions accordingly, the asymptotic relative frequency F(a) is obtained and the geometric behaviour emerges. This also explains why the slope of the log-distribution is only in uenced by the entry rate. Ultimately, φ is the unique determinant of relative sizes of di erent age pro les.
Unconditional scenario: time-dependent entry rate and homogeneous survival probabilities.
We consider now the case in which survival probabilities and entry rates change over time. We keep, however, the assumption that survival probabilities are homogeneous and do not depend on rms' characteristics. That is, ρ i,t = ρ t ∀i, t. Concerning the entry rate, we distinguish two cases. In the rst one, φ t is an independent random variable such that E[φ t ] = φ ∀t. In the second one, φ t varies over time according to a certain deterministic process, such that (φ 2 , φ 3 , . . . , φ t , . . .) is a known sequence of parameters. For what concerns the expected number of rms with a given age at a generic time t, the second case is equivalent to assume that every entry rate is an independent random variable with time-dependent expectation. Under this interpretation, we abuse notation letting φ t be interpreted as the expected entry rate at time t. In the rst case, taking the expectation of eq. (3) we obtain and by iterating the map backward, the expected number of rms at time t becomes Then, since the number of rms with age a at time t can be written as in eq. (5), we obtain and, substituting with eq. (8) computed at time t − a while recalling the di erent speci cation for rms of age t, one has Taking the log of E[N t (a)] one immediately notices that, as in the previous case, the expected numerosity of rms with age age a is a linear map of a with coe cient − log(1 + φ). Thus, once again, we infer that the survival probability ρ t does not impact on the slope of log E[N t (a)] but it in uences the intercept.
For the second case, we solve using the same steps, but taking into account that φ t varies deterministically over time. Thus, the expected number of rms is E[N t ] = E[N t−1 ](1 + φ t )ρ t and, by iteration, one obtains The expected number of rms with age a at time t now reads E Taking Thus, if entry rates change deterministically over time -or, equivalently, they are independent random variables with time-changing expectations -for any age pro le we have that the coe cient of a is an arithmetic average of − log(1 + φ τ ) for the periods ranging from 2 to the one before entry. Also this case implies that the variations in the survival probability ρ t have no age speci c e ects on the log-distribution.
Conditional scenario: time-dependent entry rate and heterogeneous survival probabilities. Here we analyse the case in which survival probabilities depend on rms' characteristics, focusing on age. Indeed, empirical evidence suggests that young rms have, on average, lower survival rates (see for instance . We account for the presence of such a stylised fact assuming that whereā ∈ N represents the age after which a rm is considered as old. Setting the condition ρ Y < ρ O provides coherence between our modelling framework and the above mentioned empirical evidence. We assume that every φ t is an independent random variable with E[φ t ] = φ ∀t > 1. 15 When survival probability is age-dependent, deriving the expected number of rms with a given age becomes more complicated. Indeed, one has that the expected number of rms at time t is 15 Equivalently, one can assume that the entry rate is constant over time, φt = φ ∀t > 1.
where the rst component represents the expected number of young rms at time t and the second accounts for the old ones. Considering the map for E[N t−1 ] and rearranging terms, one gets and, substituting in eq. (11), the expected number of rms at time t ≥ā can be obtained iterating the recursive map . . ,ā−1} and, with a slight abuse of notation, 16 Once the sequence of expected number of rms in each period is available, the expected number of rms for a given age can be derived as In this case interpreting the analytical distribution is more complicated; it is also more di cult to understand how variations in entry or survival rates a ect the emerging distribution. Thus, we rely on comparative statics exercises presented in Section 4.2. As we shall see, age-dependent heterogeneity in survival probabilities signi cantly a ects the shape of the log-distribution, with a steeper slope of the log-distribution on the left of its support, corresponding to the e ect of lower survival probabilities of young rms. That is, contrary to the previous scenarios, a di erent survival probability for young rms with respect to old ones has age-speci c consequences on the age distribution.
Sectoral scenario: sector-speci c time dependent entry rates and sector-speci c survival probabilities. To cope with the signi cant sectoral heterogeneity highlighted in the previous section, we consider a case in which each rm belongs to one (and only one) among K sectors. One can thus consider a rm as a single establishment, rather than a multi-establishment enterprise. In this case, each sector k ∈ {1, 2, . . . , K} is characterized by a speci c entry rate φ k and all the rms of such sector have survival probability ρ k . Also in this case, we can alternatively (and equivalently) assume that entry rates are sector speci c independent random variables with E[φ k,t ] = φ k ∀k and ∀t > 1. We further assume that once a rm is born in a sector it remains in that sector for all of its life and that the number of entrants is in uenced only by the number of existing rms in the sector. In this way, the expected number of rms with age a in sector k, E[N k,t (a)], can be derived following the same mathematical steps of the baseline scenario (if a constant entry rate is assumed) or adapting those of the unconditional scenario ( rst case, if a random entry rate is assumed). Indeed, denoting by N k,0 the initial number of rms in sector k, one obtains The expected number of rms with age a at time t in the whole economy is obtained by simple aggregation over the K sectors This setting allows for composition e ects. To shed light on the e ects of aggregation of heterogeneous sectors, we consider the function g : R → R, x → g(x), de ned as Applying the Mean Value Theorem, for any x > 0 we obtain with ξ ∈ (0, x). The function g(x) can be used to compute the logarithm of E[N t (a)] for any age level a. That is, for any a ∈ {1, 2, . . . , t − 1}, there exists a ξ a ∈ (0, a) such that 17 According to our model at the sector level, the logarithm of E[N k,t (a)] is linear in a and can be written as log E[N k,t (a)] = β 0,k + β k,1 a, with β k,0 = log(N k,0 φ k (1 + φ k ) t−1 ρ t k ) and β k,1 = − log(1 + φ k ). Then, at the economy level we obtain Notice that the coe cient of a is an age dependent convex combination of the sectoral coe cients. This implies that at the aggregate level the slope of the log-distribution is bounded from above by the maximum sectoral slope while it is bounded from below by the minimum one. Thus, an approximated geometric behaviour can emerge at the economy level from perfect sector-wise geometric distributions. At the same time, aggregate deviations can also be generated by the sectoral heterogeneity.

Comparative Statics
In this section we rstly analyse the comparative statics of the age distribution for di erent values of entry rate and survival probabilities. We then study how structural shocks might a ect the shape of the theoretical age distribution. We present results for both unconditional and conditional shocks to survival rates. Coherently with the previous section, we consider the former as shocks that a ect all the rms alike (independently of their age); the latter as shocks that a ect di erent groups of rms (e.g. young and old rms) in an heterogeneous manner. 18 In all the following exercises we compute the age distribution at time t = T = 100 and we set N 0 = 10 6 .
Comparative statics with no shocks. We start considering the baseline scenario and compare the distributions that emerge under di erent assumptions on the parameter values. In Figure 6 we plot the expected number of rms with age {1, 2, . . . , 100} both for the particular case φ = 1 − ρ (which we denote as baseline) and for cases in which either the entry rate or the survival probability is higher.
The exercise con rms our analytical ndings. In particular, keeping φ constant, if the survival probability is higher, the unique observable variation is that the equilibrium distribution becomes more populated and moves outward, with the slope of the distribution remaining unchanged (see the short-dashed line of Figure 6). Alternatively, if the entry probability is higher, one can observe a variation in the slope of the distribution 17 The expected number of rms with age t is omitted because its functional form cannot be expressed by means of g(x). The logarithm of its expected total number can be trivially computed and it does not signi cantly a ect the log-distribution. 18 We focus on one-time persistent shocks, but similar insights can also be derived in cases in which transitions from one scenario to the other occur in a smoother way. which becomes steeper (long-dashed line in Figure 6). Conversely, if the entry probability were lower, the distribution would be atter.
Unconditional shocks. A more interesting situation is a dynamic one, in which either the entry rate or the survival probability are hit by a shock at a speci c point in time. How does the shock a ect the age distribution in the medium run? To answer this question we use the unconditional scenario (second case) presented in advance assuming that φ t and ρ t maintain the baseline values (solid line) until time t c and shift to di erent values for t ≥ t c . 19 In this way we capture the e ects of a persistent shock -e.g. a structural change, possibly generated by a new policy regime, that hits either the survival probability (short-dashed line) or the entry rate (long-dashed line). We x the critical period at which the shock hits our economy at t c = 80.
When the shock is unconditional there are two main conclusions that we can grasp (see Figure 7). First, a fall (an increase) in the survival rate shifts downward (upward) the distribution without however changing the slope of the distribution (cfr. the short dashed line translated below with respect to the solid line representing the baseline without shocks). This is due to the fact that all the rms are hit homogeneously. Both the rms that were already in the market before the shock has been hitting (i.e. at the right of the dotted blue vertical line) and the ones that have entered the market after the shock has hit (i.e. at the left of the shock line) are equally a ected. This is also consistent with the intuition provided in Section 4 and with the comparative statics observed in Figure 6: survival rates only impact upon the intercept of the distribution. Second, a fall (an increase) in the entry rate shifts downward (upward) the distribution only after the shock hits (i.e. at the left of the shock vertical line) and also a ects the slope of the distribution which becomes relatively atter (steeper).
Conditional shocks. Relaxing the assumption that the survival rate is independent from age provides additional relevant insights. Hence, we consider the case in which at a given time t c the economy moves from the baseline to the conditional scenario. 20 Thus, in Figure 8 we investigate the case in which young rms (without loss of generality, less than or equal to 10 years old) have the same survival probability of older ones. But when a shock hits, only the survival rate of these young rms declines. 19 Concerning the entry rate we could alternatively and equivalently assume that it is an independent random variable whose expectation changes around tc. 20 See section B in the appendix for all the mathematical details. Here the shock hits at t c = 60. Firms that have a i,T > 50 in Figure 8 were already old when the shock hits; hence for age level larger than 50, the distribution resembles the perfect geometric shape of the baseline case without shock. Since when the shock hits it decreases the survival rate of the young rms, the cohorts of rms that were still young at t c are partially hit. For example, the set of rms with a i,tc−1 = 9 is a ected for only one period while the set of rms with a i,tc−1 = 3 is a ected for 7 periods. This leads to the declining slope observed in the interval (40, 50] of Figure 8. Further on the left in the same panel -i.e. for all rms born after the shock -we can clearly observe a piecewise geometric behaviour: one for old rms with a i,T ∈ (10, 40]whose rate of exit is lower -and one for young rms with a i,T ≤ 10 -whose rate of exit is higher. Consistently with the analytical results outlined before, in the presence of conditional e ects, also the survival rate a ects the slope of the age distribution.

Model Calibration
An important exercise to validate our modelling framework and to relate it with the empirical evidence on business dynamism is to analyse the entry and exit rates implied by our model. We calibrate the unconditional scenario ( rst case) speci cation of Section 4 assuming that the survival probability is constant over time: ρ t = ρ ∀t. For this exercise we also employ the empirical estimates presented in Section 3. We have observed that the linear t in Section 3 provides correlated residuals (cfr. Figure 2). This exercise has to be thought as a rough indication of what values are reasonable for the average entry rate and survival probability in order to obtain a log-distribution comparable to the real one, but one shall keep into account that requires the additional assumption of time-independence of entry rates. Still, this is very informative as it allows to infer rough averages of entry and exit rates over a very long time period. With this caveat in mind, our tted model readŝ from which, knowing that the slope is determined only by the entry rate, we can directly derive the calibrated value of the average entry rate. This writes φ * = e −β 1 − 1. Concerning the other two parameters to calibrate, we have to rely on a more sophisticated methodology because of an under-identi cation problem. We thus employ the indirect inference technique. The literature on calibration and estimation of heterogeneous agents models has prospered during the last decades with the development of new techniques for accomplishing these computationally heavy tasks (see Fagiolo et al., 2019). Many contributions, in particular aimed at enriching the indirect inference technique originally developed by Gourieroux et al. (1993), have allowed the estimation of models for nancial markets (Franke and Westerho , 2012), for industry dynamics (Alfarano et al., 2012) and for aggregate macroeconomics dynamics (Grazzini et al., 2017). In this case, given the stylised nature of our model and its small dimensionality in both the parameter space and the output, we use a simple calibration framework. We employ the indirect inference method exploiting the analytical closed form equation for the target variable to be matched by the model. Hence, we numerically calibrate the model by plugging a nite combinations of parameters, selected from a regular two-dimensional lattice, into this equation.
We design a strategy aiming at minimizing the absolute deviations between the model prediction and the empirical estimates. Thus, we de ne as target variableŷ the empirically estimated interceptβ 0 . We instead de ne as elements of the vector of key structural parameters to be calibrated θ * the survival rate ρ and the initial number of rms N 0 . 21 We then explore the parameter space by selecting a discrete set Θ of values and we compute the model generated target variableȳ(θ) for each parameter combination θ ∈ Θ. We select the optimal parameter vector such that θ * = arg inf θ∈Θ |ȳ(θ) −ŷ| (16) whereȳ(θ) represents the intercept predicted by the model under the speci c parameter choice -i.e.β 0 (θ * ) = β 0 (N * 0 , ρ * )). The model prediction for this intercept is formally described by the following equation: Our design selects the optimal parameters by minimizing the euclidean distance of the model predicted interceptβ 0 and the empirically estimated oneβ 0 . Before discussing the calibration results, it is important to remark that the number of statistics to be matched is lower than the number of parameters to be calibrated. This gives rise to an under-identi cation problem and to the possibility of a multiplicity of optimal points. The result of our calibration exercise for the establishments of all aggregated sectors is presented in Figure A.3 in Appendix A. A single optimal value (N * 0 , ρ * ) can be found, and is represented as a red dot over the N 0 , ρ plane. However, its relative performance with respect to a countable number of other parameter con gurations, represented along the blue corridor, is relatively small. This is the result of the under-identi cation 21 We assume that t matches the maximum age recorded in the data, that is 100.
issue. However, the set of points close to the local optimum are all located along an almost horizontal line, except for very small values of N 0 . They are therefore characterized by a survival rate ρ between 90% and 95% and by a large variability in terms of N 0 . But the value of N 0 does not have a direct economic interpretation, while ρ does. We conclude therefore that it can be almost arbitrary xed in order to accommodate a a reasonable value of the survival rate. This is a good property for the possible empirical application of our model to datasets from other national economies, with the aim of performing a comparative exercise.
All the results of the entry and exit rates stemming from our calibration exercise at broad aggregation levels (i.e. for the aggregate, manufacturing, services) and at the enterprise and establishment levels, are presented in Table 3. These calibrated values provide a rough indication of the level of entry and exit rates over the period of time spanning from the date of birth of the oldest rm in the sample (i.e. around 1918) to the year of the observation (i.e. 2018).
Over this long time-horizon our results point to higher entry rates and lower exit rates for services rather than manufacturing. This is consistent with the fact that over the last century a structural transformation a ected the French economy, as the economies of many advanced countries. In particular, a process of deindustrialization took place with the heavy industrial activity slowing down and leaving the space to the creation of new business in the service sectors, as the entry costs for the generation of new businesses declined. Overall, we conclude that our model is su ciently good for the combination of an empirically estimated value of the entry rate and a reasonable value of exit rates. Comparing our results with the values of entry and exit rates for establishments in France over the period 1993-2014 reported by Aghion et al. (2018) (cfr. Figure 9a in particular), we notice that our calibrated entry rate is in line with their estimates, while our exit rate results are slightly below the estimates therein provided. Even if there are di erences in the datasets employed and in the considered periods, this result con rms the good performance of our model for identifying entry rates. sector entry rate (φ * ) exit rate (1 − ρ * ) Manufacturing (establishments) 0.11 0.11 Manufacturing (enterprises) 0.11 0.10 Services (establishments) 0.13 0.07 Services (enterprises) 0.12 0.08 Aggregate (establishments) 0.12 0.08 Aggregate (enterprises) 0.12 0.07 Table 3: Calibrated entry (φ * ) and exit (1 − ρ * ) rates at the largest levels of aggregation.
In Appendix A (cfr. Table A.3) we also present the detailed results for the 2-digit level of aggregation. But in this case a note of caution shall be spelled out. We shall indeed take into account the evidence presented in Figure 5 as the presence of deviations from a well behaved geometric distribution might limit the validity of the calibration exercise. Especially for young sectors, in which only few observations are available and the uncertainty in the calibration exercise can be relevant. Our results, however, point to some interesting features of business dynamism. We nd that sectors with higher business dynamism (the sum of entry and exit rates) are the ones concerning programming and broadcasting activities, telecommunications, computer programming, consultancy and the postal and courier activities, consistently with international evidence (Calvino and Criscuolo, 2019). Also in this case, we conclude that our simple model is able to capture some important features of the French business dynamism. In particular, we argue that the relationship between the age distribution and business dynamism is a solid one and is further discussed in the next section.

Discussion
The model with di erent scenarios, together with the comparative statics and calibration exercises carried out thereafter, are particularly helpful in guiding the interpretation of the stylised facts discussed in Section 3. In particular, building upon archetypal situations generated by our theoretical framework, we can provide some suggestive evidence that helps uncover the underlying mechanisms behind the shape of the age distribution.
The rst stylised fact presented, i.e., that the geometric law is a rst approximation for the aggregate and macro-sectoral age distributions can be interpreted in the light of the fact that, on average over a long time period, a roughly stable entry rate -around 12% -may not be too far from reality, at least as a rough rst-order approximation. However, the signi cant aggregate deviations from the geometric distribution discussed in the second stylised fact may be interpreted as indicative of di erences and variations in business dynamism, across rms with di erent characteristics and over time.
In particular, a rst relevant structural break highlighted in Section 3 indicates that the slope of the age distribution is steeper for a < 10 than for a ∈ [10, 30), as discussed in the second stylised fact and evident in Figure A.4. This can be explained by the presence of heterogeneous survival rates. In particular, according to our results and consistently with empirical evidence, a lower survival probability for young businesses seems to be the most plausible explanation. 22 Such e ect could actually be even stronger, considering that a decline in entry rates after the crisis (or a more secular one, as discussed next) would have acted in the opposite direction.
A second structural break highlighted in Section 3 divides the age distribution between rms with a < 30 (born during the great moderation) and rms with a ∈ [30, 50). The change of steepness, following the interpretations stemming from our model, would be consistent with a decline in entry rates. This is also evident when focusing on the estimated parameters of the piecewise geometric distribution (based on Equation 2), which depict a lower λ i towards the left of the support (see Figure A.4). 23 Furthermore, the downward jump at age equal to 10 would be consistent with a one period drop in entry rates which follows the beginning of the global nancial crisis.
The last three decades appear therefore characterized by a lower rate of entry with respect to the number of incumbents. This interpretation suggests that business dynamism in France has declined during the great moderation period and that it was instead larger between the beginning of the 1970s and the beginning of the 1990s. This result is also consistent with the evidence that Decker et al. (2016); Calvino and Criscuolo (2019) have put forward and might indicate a global pattern of declining entry rates in several advanced economies.
For a ≥ 50, deviations from the geometric benchmark are more relevant and may be more di cult to interpret, also due to the less systematic data collection procedures. However, the steeper and more volatile age log-distribution for rms older than 50 but younger than 75 years old may correspond to higher rates of entry over the post-war recovery period (Trente Glorieuses). Around the Second World War instead a significant downward jump in the log-distribution is evident, just before rms aged 75. This would be consistent with a notable drop in entry rates. A atter log-distribution in correspondence to rms older than 75 may instead re ect the e ects of the Great Depression.
Sectoral heterogeneity appears as a characteristic feature of age distributions, as highlighted discussing the third stylised fact in Section 3. Cross-sectoral variation in age distributions appears importantly related to di erences in the stage of the life-cycle in which di erent industries stand (see Klepper, 1996). Importantly, and as highlighted in the previous section, younger sectors -such as the Computer programming, consultancy and related activities industry (62) -tend to have a steeper age log-distribution, consistently with higher average entry rates over a given period of time. Di erently, more traditional sectors, such as the food manufacturing industry (10), tend to have atter age log-distributions, consistently with lower average entry rates that characterise more mature sectors.
Furthermore, focusing again on Figures 4 and A.2 and on Table A.2, one can observe that more mature industries tend to have larger average, median and maximal ages. They thus have a longer right-tail and tend to be more populated (see e.g. sector 105 representing the manufacture of dairy products). Young industries, have been born less than 100 years ago and display a short cut to the tail and a possible fast increase in the 22 This is consistent with the observation by Axtell (2018), who suggests for this very reason that a Weibull distribution may be a closer t than a geometric one. 23 For the youngest group of establishments the survival e ect described above seems to dominate.
log-count of rms, when reading the graph from the right to the left (see e.g. sector 620 representing computer programming, consultancy and related activities). 24 Aggregation of relatively broad sectors that are approximately geometrically distributed yields -as a reasonable approximation -an aggregate geometric distribution whose decay depends upon sector-speci c weight and decays. This suggests that the stage of life-cycle of di erent sectors also a ects the overall maturity of the economy and that cross-country comparisons of the slope of aggregate age log-distributions may be interesting.
More signi cant deviations from the geometric benchmark arise when analysing 3-digit sectors' dynamics. This may be indicative of the fact that -in more detailed sectors -entry rates are less constant in given time periods, and that evolutionary dynamics of entry and exit over the industry life-cycle become better observable at ner-grained levels of aggregation. In this context, a reasonable conjecture seems to be that what is observed at more aggregate levels can be reasonably approximated by a more constant quantity resulting from the composition of life-cycle evolutionary dynamics occurring at the more micro level. This may occur as the process of technological change and industry evolution is continuous, with new sectors being born as old ones get closer to maturity.
Di erences in the age distribution of businesses have signi cant implications for economic outcomes. Indeed, the age composition of countries and sectors of activity is an important feature of industrial dynamics, with relevant implications in terms of job creation, innovation, and ultimately economic growth. Building upon the literature on the role of young rms and their importance for job creation (Haltiwanger et al., 2013;Criscuolo et al., 2014), and the literature on the evolution of industries, which highlights that younger sectors are more prone to labour-friendly product innovations (Klepper, 1996), we explore the relationship between employment growth in France and the calibrated sectoral entry and net entry rates derived in Section 5.
We source employment growth rate at sectoral level from the OECD STAN database, focusing on France for the period 2006-2016 (the last available years). 25 We then carry out an association exercise by estimating the simple linear model by means of ordinary least squares with robust standard errors: ∆ log(emp) s,t = α entry s + θ X s,t + γ t + s,t , where ∆ log(emp) s,t measures employment growth, entry s stands for the calibrated entry rate φ * s or, alternatively, for the net entry rate de ned as φ * s − (1 − ρ * s ), which are both reported in Table A.3; X s,t represent a set of control including sector size, measured in terms of employment (i.e. emp s,t ); γ t refers to year xed e ects and s,t is the usual error term. Subscripts s and t indicate sectors and time, respectively.  Table 4: Robust standard errors in parentheses. *** p<0.01, ** p<0.05, * p<0.1 24 Each age distribution plot can indeed be also read as time passing in the direction from the right to the left: rms of 100 years of age have been funded in 1918, while rms that are 20 years old have been funded in 1998, for example. 25 We replicated growth rates for few slightly more aggregated sectors in order to get to a full 2-digit sectors settings, compatible with the one of the calibration data. Employment data refer to the number of employees but similar results also hold when focusing on total persons engaged.
Results, reported in Table 4, highlight a positive association between sectoral employment growth rates and entry, suggesting that industries in the earlier phases of their life-cycle present more labour-friendly dynamics. This con rms the importance of young rms for employment growth, and is possibly related to a more signi cant share of product innovations in younger industries. Results also hold when controlling for the size of the sector, as measured by total sectoral employment.
Of course this last exercise is only a rst and suggestive application in order to highlight the potential and relevance of studying features of age distributions, and in particular its coe cients. As a matter of fact, in the framework of the model we have proposed, they have a relevant link with entry rates and the life-cycle of industries. Future analysis may further explore and relate the parameters of age distributions, and the calibrated entry and exit rates, to di erent economic outcomes.

Conclusions
In this paper we have studied in detail the shape and determinants of the age distribution of business rms. This has been a subject overlooked by most of the literature, although important given the key role of young rms in market economies. Using a new comprehensive cross-sectional database of French rms we presented three stylised facts. First, a geometric shape appears to be a reasonable rst-order approximation for the aggregate age distribution; second, there are however signi cant deviations from the geometric benchmark at aggregate level; third, signi cant cross-sectoral heterogeneity in the slope of the age log-distribution emerges, with larger deviations from the geometric benchmark at very ne levels of aggregation.
We have then developed and calibrated a stochastic model of rm dynamics to better understand the mechanisms behind the stylised facts and guide the interpretation of the empirical ndings through comparative statics exercises. Our modelling strategy provides a novel non-trivial mechanism to generate a geometric age distribution and to explain deviations from such a benchmark. We also introduce sectoral heterogeneity and we show that the aggregation of a nite number of geometric distributions with heterogeneous coe cients, each representing one sector, can lead to the emergence of an approximated geometric distribution at the aggregate level.
Combining the empirical evidence with the model prescriptions, we suggested the existence of a long-term decline in entry rates, which attens the age log-distribution for rms with less than 30 years, and of a lower survival probability for young rms, which induces a relatively steeper distribution for businesses with less than 10 years of age. The ndings also pointed to a short-term e ect of the Great Recession, which generates a blip in the age distribution for rms born around 2008. E ects of the Great Depression, the Second World War, and the subsequent post-war recovery were also visible, although should be taken more cautiously as older data may have been collected less systematically.
Further examination of the age distribution at lower level of aggregations revealed that the geometric benchmark broadly holds also within 2-digit sectors. But with sectors-speci c slopes that are signi cantly di erent from each other. This suggested that di erent sectors are at di erent stages of their life-cycle, providing an indirect way to measure them, and that the aggregate behaviour re ects important compositional e ects. More signi cant deviations from the geometric benchmark at very ne grained (3-digit) levels of aggregation may suggested that evolutionary life-cycle dynamics become even more observable in more detailed sectors. We have ultimately extensively discussed the main ndings and provided a rst application, in order to highlight the relevance of the age distribution -and its parameters -for economic outcomes, notably for employment growth.
This work can be extended along di erent dimensions. From an empirical perspective, slopes of the age log-distributions can be studied beyond France, across sectors of di erent countries (or across countries), in order to assess di erences in the maturity of sectors to their life-cycle. Furthermore, additional applications can relate the parameters of the age distributions to di erent economic outcomes, beyond employment growth. From a theoretical perspective, instead, a more ambitious extension would study the age distribution jointly with the size and growth distributions of rms. Building a stochastic model that has the ability of replicating and explaining all the three empirical patterns can disclose new information on the mechanism behind rms and industry dynamics which might be relevant also for policy makers.

Appendix B. Technical Annex
Conditional scenario -dynamical system setting Using matrix notation, eq. (12) becomes Initial conditions can be computed using eq. (4) with t ∈ {1, 2, . . . ,ā − 1} and ρ Y in the place of ρ. E[N 0 ] is not de ned, with an abuse of notation we impose it equal to N 0 /φ, such that the map in eq. (12) is well-de ned for each t ≥ā.
The dynamical system can be solved using the standard spectral decomposition technique and one obtains the expected number of rms in each period t. The formula for the expected number of rms with age a at time t in eq. (13) is derived adapting eq. (6) to the age-dependent survival probability.

From baseline to conditional
In what follows we assume that at time t c the economy shifts from the baseline scenario to the conditional scenario. Let us start from the expected number of rms in the economy and notice that for t < t c E[N t ] can be easily recovered from eq. (4). If t = t c one should carefully consider the fact that rms with a i,tc−1 <ā and newly created ones now have a di erent survival probability, hence it is with k ∈ {1, 2, . . . ,ā − 1}. Then, for t ≥ t c +ā one can use the dynamical system approach presented in advance, obtaining  with x 1 , x 2 , . . . , xā −1 de ned as before. Finally, eq. (6) can be adapted such that one gets