A common trajectory recapitulated by urban economies

We characterize the economies of 350 U.S. cities between 1998 and 2013 by industries’ relative size using the North American Industry Classification System (NAICS) (see Materials and Methods and SI. I for full description on data). Revealed Comparative Advantage (RCA), also known as location quotient, is a widely used measure for a region’s industrial specialization  Hidalgo et al. (2007); Isserman (1977); Glaeser et al. (1992); Markusen and Schrock (2006). It is the city’s relative employment share to the national average for the industry $i$, and we define characteristic industry if its share in city $c$ is greater than national level at time $t$, $rca(c,i,t)>1$ (solid color in Fig. 1A)  Balassa (1965); Hidalgo et al. (2007).

Scaling theory remains as a powerful framework for size-dependent nature of urban characteristics Bettencourt et al. (2007); Youn et al. (2016); Bettencourt and Lobo (2016); Gomez-Lievano et al. (2016) and dynamics Bettencourt et al. (2010); Alves et al. (2015) despite its statistical caveat Arcaute et al. (2015); Leitao et al. (2016). Urban quantities are expressed as

where pre-factor $Y_{0}(i,t)$ and scaling exponent $\beta(i,t)$ are fitted given empirical observations of employment $Y(c,i,t)$ and population $N(c,t)$. We find the Eq. 1 is mostly in a good agreement with our dataset ($R^{2}>0.65$, see SI. III). Then, $\beta(i)$ characterizes the size-dependency of industry $i$ (sublinear, linear and superlinear).

Figure 1 summarizes our analyses in a network representation where characteristic industries are colored by $\beta$ and sized by RCA. Two industry $i$ and $j$ are connected by co-occurrence similarity, $\psi(i,j)$ across US cities (see Materials and Methods)  Hidalgo et al. (2007); Neffke and Svensson Henning (2008); Muneepeerakul et al. (2013); Shutters et al. (2016). As population grows, where the city’s characteristic industries occupying in the network moves (from agriculture and mining to managerial and technology-based industries), and this transition seems to be predicted by scaling exponents  Youn et al. (2016).

City size determines what are the characteristic industries and their directions are manifested as scaling exponents. For example, small cities are characterized by sublinear industries ($\beta(i)<1$), such as agriculture and mining, while large cities are by superlinear industries ($\beta(i)>1$) such as management and professional services. This result is consistent with existing observations that small cities are relying more on manual labor Henderson (1986) while large cities on cognitive labor Michaels et al. (2013); Florida (2004); Youn et al. (2016), suggesting an industrial transition according to $\beta(i)$ from sublinear to superlinear sectors (see Fig. 1A). Fig. 1B confirms this overall trend at the level of individual U.S. cities. The average scaling exponent $\beta(i)$ for the characteristic industries in each city is strongly correlated with population size (Pearson correlation $\rho=0.59$).

So far, our analysis uses a static snapshot of urban economies, but how do these cross-sectional relationships relate to temporal change in cities? Here, we explore urban recapitulation through the industrial composition of different sized cities over time. Cities are grouped into 20 equally-populated city groups (denoted by $g$) according to city size, and we represent the industrial characteristic of group $g$ as industry vector $\vec{G}(g,t)$ composed of logarithmic RCA values averaged for cities in group $g$ (see Materials and Methods). Then, the Pearson correlations $\phi(g,g^{\prime},t,\tau)$ of $\vec{G}(g,t)$ and $\vec{G}(g^{\prime},t+\tau)$ describes how the industrial compositions of city groups $g$ and $g^{\prime}$ relate over different time periods $t$ and $t+\tau$. A lead-follow score aggregates these industrial changes over each starting year in our data set, according to

This score is basically the average similarity change between reference group $g$ and observed group $g^{\prime}$ over a 10 year period as default, and a lead-follow matrix for the scores of all pairs summarizes the overall trend (see SI. IV for other time periods).

The resulting lead-follow matrix provides temporal evidence for urban recapitulation (see Fig. 1C). Groups of small cities become more similar to groups of large cities over a ten year period while groups of large cities become increasingly dissimilar to groups of small cities. This observation suggests that there exists a general pathway explored by large cities and followed by small cities over time. We interpret the lead-follow matrix as strong evidence for urban recapitulation of industrial compositions, and we endeavor to explain the phenomenon further. City size determines both characteristic industries therein and the future growth.

How does city size characterize urban economic structures and their dynamics? Characterized industries are determined by city size and scaling exponents. First, revealed comparative advantage of city is expressed as an industry’s relative size to a national level. The industry’s size is explained by scaling relation Eq. 1. Therefore, we can express RCA as a function of scaling exponents as following:

The equation can be further approximated by Zipf’s law $P(N(c)=N)\propto N^{-2}$ Rosen and Resnick (1980) (see SI. V):

where $N_{max}(t)$ and $N_{min}(t)$ are size of the largest and smallest city at time $t$, respectively (see SI. V for a complete derivation).

This framework determines the characteristic industries and predicts which industry will be characteristic given scaling relations. When cities grow, superlinear industries, $\beta(i)>1$, will be characteristic industries as a result of increasing RCA value, as shown in Fig. 1A&B. We exemplify this observation empirically by examining the growth of individual U.S. cities from 1998 to 2013 compared to their RCA values for the Education industry (see Fig. 2A), which has $\beta(i)=1.21$, and the Manufacturing industry (see Fig. 2B), which has $\beta(i)=0.94$.

Both our analytic framework and empirical observations imply the existence of urban recapitulation by city size. Now, how can we measure the degree of recapitulation for individual industries and cities? The size-dependency $\beta(i)$ estimates the employment growth when the population grows in time. We compare the cross-sectional ratio $\beta(i)$ and the observed longitudinal growth ratio $\hat{\beta}(i)$ as a measure of recapitulation. From Eq. 1, we have

where $\Delta\log(Y(c,i))=\log(Y(c,i,2013)-\log(Y(c,i,1998))$, and the same for $\log{Y_{0}}$ and $\log{N}$. This calculation decomposes employment growth into each city’s size-dependent growth (i.e. $\beta(i)\cdot\Delta\log(N(c))$) and global time-dependent growth (i.e. $\Delta\log(Y_{0}(i))$. See the schematic in Fig. 2B). By fitting Eq. 5 to empirical employment data, we derive the longitudinal size-dependency $\hat{\beta}(i)$ and the global trend $\Delta\log(\hat{Y}_{0}(t))$ across all cities between 1998 and 2013, and we compare $\hat{\beta}(i)$ with the cross-sectional size-dependency $\beta(i)$. If an individual industry is perfectly recapitulated by growing cities, then cross-sectional and longitudinal size-dependencies should be equal ($\hat{\beta}(i)=\beta(i)$). Therefore, we employ an industry recapitulation score according to

The industry recapitulation score becomes one when every city perfectly recapitulates, and equals to zero when employment growth has no longitudinal size-dependency.

We observe high recapitulation scores (i.e. $S(i)>0.50$) for industries that are well-described by urban scaling (i.e. $R^{2}>0.65$). This strong longitudinal size-dependency is a sufficient condition of the constant deviation from a common trajectory Bettencourt et al. (2010). In contrast to the recent research on traffic congestion Depersin and Barthelemy (2018), the common trajectory is well explained by city size. The most significant difference is teasing out the global trend, $\Delta\log{(Y_{0})}$, which makes common upward or downward drifts as shown in manufacturing in Fig. 2B. Its recapitulation score around $0.7$ shows that decomposition is essential to properly measure the size-dependent growth.

Analogous to industries, do cities recapitulate with the same strength? We generalize the decomposition in Eq. 5 for individual cities, and measure the recapitulation scores of city groups, accordingly. We extract the size-dependent growth of each city by subtracting the global growth $\Delta\log(\hat{Y}_{0}(t))$ estimated from Eq. 5. Then, we measure the longitudinal size-dependency $\hat{B}(g,i)$ averaged for the cities in city group $g$ and industry $i$ similar to $\hat{\beta}(i)$ (see SI. VI for the details).

where $c\in g$ is the set of cities in city group $g$, and the groups are determined in the order of populations. Finally, we measure a city group recapitulation score according to

where $i\in I$ is the set of all industries. Similar to before, larger values of $S(g)$ mean large tendency of recapitulation.

In addition to industries, a strong tendency of recapitulation is also observed for city groups. Fig. 2D demonstrates that recapitulation occurs across all city groups as $\langle S(g)\rangle_{g}\approx 0.70$. Most city groups exhibit a high strength of recapitulation, except for a few groups of very small cities. From the measured recapitulation scores of industries and cities, we conclude that cities recapitulate a general pathway ruled by city size, in agreement with the observed lead-follow matrix and our framework connecting RCA and scaling relation.

The observed recapitulation predicts the structural change of urban economy, represented by characteristic industries, as city size grows. Can we specify the characteristic size at which a city transitions from a small city economy into the innovative and productive economy of large cities? Specifically, given the analytical relationship between city size and corresponding characteristic industries in Eq. 4, at what size do cities begin to leverage innovative industries Youn et al. (2016) with $\beta(i)>1$? Fig. 3A compares cities according to size and the Pearson correlation of logarithmic RCA values (i.e. $\vec{I}(c,t)=(\dots,\log(rca(c,i,t)+1),\dots)$) for city pairs averaged for entire time span. We observe a transition at around the 50th largest city indicating that the 50 largest cities are characterized by qualitatively different industries compared to the remaining smaller cities.

Our framework can explain this observed transition through a fixed point in the $\beta-N$ plane. From Eq. 4, we observe a saddle point at $\beta^{*}(i)=1$ and city size $N^{*}\approx 1.2$ million, which is around the population of Louisville, KY, the 43rd largest city (see SI. V for a complete derivation). The existence of this fixed point suggests that characteristic industries in growing cities will change based on $\beta(i)$ as cities achieve this critical size. We expect that small cities with $N(c)<N^{*}$ will be characterized by industries with $\beta(i)<1$ while large cities with $N(c)>N^{*}$ will be characterized by industries with $\beta(i)>1$. Fig. 3B empirically confirms that scaling exponents distinguish the trends of characteristic industries by city size. The characteristic industries of growing cities transition from industries with $\beta(i)<1$ to industries with $\beta(i)>1$ at precisely the predicted city size $N^{*}=$ 1.2 millions.

This study uses employment data by North American Industry Classification System (NAICS) industry code and 350 U.S. metropolitan statistical areas (referred to as “city” throughout the manuscript) according to the U.S. Bureau of Labor Statistics. The sizes of industry set are 19, 86, 289, 642 and 978 according to the depth of classification denoted by 2-6 digits. Data includes annual employment measurements for the years 1998 through 2013.

We employ revealed comparative advantage (RCA) Balassa (1965); Hidalgo et al. (2007) to identify characteristic industries in each U.S. city. Let $I$ denote the set of 2-digits NAICS industry codes and $C$ denote the set of U.S. cities, then $Y(c,i,t)$ denotes the employment in industry $i\in I$ in city $c\in C$ at year $t\in[1998,2013]$. The revealed comparative advantage is given by

where $c$, $i$ and $t$ denote city, industry and time, respectively.

We can characterize industry $i$ as a vector of RCA values of cities as $\vec{I}(i,t)=(\dots,\log(rca(c,i,t)+1),\dots)$. The time-averaged inter-industry relatedness is defined by the Pearson correlation of two industry vectors $\vec{I}(i,t)$ and $\vec{I}(j,t)$ as

where $\rho$ is the Pearson correlation function. Figure 1 A shows links that satisfy $\psi(i,j)>0.15$.

In the analysis on lead-follow relationship, we bin cities into 20 equally-populated city groups (denoted by $g$) according to city size, and, for each city group, we calculate the average RCA values of each city according to

where $I$ denotes the set of 2-digit NAICS industry codes.

The North American Industry Classification System (NAICS) is used by Federal statistical agencies to classify business establishments for the purpose of collecting, analyzing, and publishing statistical data related to the U.S. business economy. In addition to the temporal evolution of business sectors, this taxonomy itself is periodically revised to account for new industries. These updates are informative of the features that reshape the U.S. economy, including technological change for example. Nevertheless, it is possible that some sectors may remain under the same name in the nomenclature with a completely new content, and we cannot capture innovation in these cases. Instead, we leave them as background technological progress encompassing entire sectors. This ambiguity adds noise to our findings.

In our analysis, we use annual employment data from 1998 to 2013. We identify industries based on 6-digit NAICS codes, and the temporal changes of classification in 1997, 2002, 2007 and 2012 are harmonized into the classification in 2012. The intervals sharing the same NAICS classifications are 1998-2002, 2003-2007, 2008-2011, and 2012-2013 given by U.S. Census Bureau. This scheme allows us to employ a single industry taxonomy for the entire time period of our study. There are some cases of inexact matching between two consecutive NAICS revisions. For example, “Business to Business Electronic Markets” is a newly created sector by combining some functional parts of 68 wholesaler sectors. There are only 4 cases of creation throughout the entire period, which are “New Housing For-Sale Builders”, “Residential Remodelers”, “Business to Business Electronic Markets” and “Wholesale Trade Agents and Brokers” in 2003.

These temporal changes in industry classification define a bipartite network connecting old NAICS codes to new ones. For convenience, we refer to old codes as parents and we refer to new codes as children. All codes are classified into 7 groups: Inheritance (I), Creation (C), Merging (MG), Divided (D), Inherited Division (ID), Non-inherited Division (ND), and Merged (MD). This scheme exclusively and entirely classifies every code change in our dataset. The classification is defined by out-degrees $k_{out}(i)$ of old codes and in-degrees $k_{in}(j)$ of new codes in the connection structure.

Figure S2 shows the connection scheme for classification changes. Inheritance is the basic connection defined for simple one-to-one conversions of NAICS code. It is also defined for a code pair when the new code has $k_{in}(j)=1$, and the other new codes connected to the old code have $k_{in}(j)\geq 2$. Briefly, only one new code exclusively inherits its parent as the old code is the only parent of the child. Inherited Division and Non-inherited Division are defined for old codes with $k_{out}(i)\geq 2$ (division), and depends on whether the old code has any inherited child or not. Creation, Merging and Divided are defined for new codes. A new code is determined as Creation when each of its parents has another inherited child. Merging is defined for the non-creation case that its parents has no other inherited child. A new node is considered as Divided when it has $k_{in}(j)=1$ and is the child of a Non-inherited division node. Finally, Merged is defined for an old code that has $k_{out}(i)=1$ and is the parent of a Merging code.

We translate all industry classifications to the most recent classification (2012) to make a harmonized time series of number of employees. In the case of inheritance (black arrows in Fig. S2), all employees in the old classification are transferred to the Inheritance node in the new classification. The division of employees colored by light grey is not transferred to the latter nodes, which means these connections are ignored. As a special case, Non-inherited division equally distribute its employment to the divided nodes (dark grey).

Similar to the NAICS revisions, the Metropolitan Statistical Areas (MSAs) are revised about every 5 years. The exact time intervals are 1993-1999, 2000-2002, 2003-2006, 2007-2011 and 2012-2013. By using the conversion table obtained from American Association of State Highway and Transportation Officials (AASHTO), we harmonize all MSAs to a single reference. In 1998 and 1999, the MSAs are classified into two types: Consolidated Metropolitan Statistical Area (CMSA) and Primary Metropolitan Statistical Area (PMSA) where a CMSA contains several PMSAs. From 2000, MSAs have been revised to Combined Statistical Area (CBSA). For example, New York City is classified as ‘New York-Northern New Jersey-Long Island, NY-NJ-CT-PA’ CBSA, and ‘New York-Northern New Jersey-Long Island, NY-NJ-PA’ CMSA with 15 PMSAs including ‘Bergen-Passaic, NJ’, ‘Bridgeport, CT’, ‘Danbury, CT’, ‘Dutchess County, NY’, ‘Jersey City, NJ’, ‘Middlesex-Somerset-Hunterdon, NJ’, ‘Monmouth-Ocean, NJ’, ‘Nassau-Suffolk, NY’, ‘New Haven-Meriden, CT’, ‘New York, NY’, ‘Newark, NJ’, ‘Newburgh, NY-PA’, ‘Stamford-Norwalk, CT’, ‘Trenton, NJ’ and ‘Waterbury, CT’. For 1998 and 1999, we use CMSAs as the delineation of cities since CMSAs has a similar scale to CBSAs. When an MSA is divided into several MSAs or merged together with another MSA, we aggregated these MSAs to the largest one. Some exceptions are modified by using MSA definition provided by U.S. Census Bureau, and some megalopolises, such as New York, Los Angeles and Chicago, are cross-checked with the definitions in Wikipedia.

We normalize the size of urban employment using RCA in Eq. S2. By using the RCA values, we characterize the industries in a city as industry vector $\vec{I}(c,t)$ of city $c$ in year $t$. The spatial vector of industry $i$ can be defined in a similar way.

where $lrca=\log{(rca+1)}$, $I$ and $C$ denote the logarithmic RCA, the set of industries, and the set of cities, respectively. The length of the industry vector is equal to the number of industries (e.g. 19 for 2-digit NAICS industries). The industrial similarity between two cities $c$ and $c^{\prime}$ with a time lag $\tau$ is now measured by a Pearson correlation between two vectors $\vec{I}(c,t)$ and $\vec{I}(c^{\prime},t+\tau)$.

where $\rho$ measures the Pearson correlation between two vectors. The inter-industry locational similarity $\psi(i,t;i^{\prime},t+\tau)$ can also be obtained for the location vector $\vec{L}(i,t)$.

As the dynamics of each city fluctuates, we group the cities by their populations, and use the average population as the reference.

where $\langle\cdot\rangle_{c\in g}$ denotes the average for the cities $c$ in group $g$. We usually group the cities into 20 groups of equal sizes. The industrial similarity between groups can also be defined using their correlations as in Eq. S7.

Then, we fix one group $g$ as a reference group, let another group $g^{\prime}$ evolve in time ($\tau$), and measure their industrial similarity $\phi(g,g^{\prime},t,\tau)$. By following the change of industry similarity $\phi$ as time lag $\tau$ changes, we can observe if group $g^{\prime}$ is getting similar or dissimilar to group $g$ in time. To focus on the effect by time lag $\tau$, we average the change over the reference times $t$.

where $\Delta\phi(g,g^{\prime},\tau)$ is the average change of similarity between reference group $g$ and observed group $g^{\prime}$ for time difference $\tau$. Figure S6A and Figure S6B show the trends of $\Delta\phi(g,g^{\prime},\tau)$ versus on lag $\tau$ for a fixed reference group $g$ and various observed group $g^{\prime}$ denoted by colored lines. In Fig. S6A, the similarity generally increases for the largest reference city, which means that small cities get similar to the largest cities in time. On the other hand, large cities get dissimilar to the smallest group in Fig. S6B. These trends show the directional evolution of urban economy led by large cities and followed by small cities. We summarize these trend in the form of matrix.

Finally, we measure the average similarity change between groups for $T$ years, which is called “Lead-follow matrix” denoted as $LF$. We set $T$ as 10 years considering the time span of our data. The average rate is obtained by the least squares method.

Figure S6C summarizes the time-lag similarities across cities as a “Lead-Follow Matrix”. The general pattern observed in the figures is that small cities get similar to larger cities whereas large cities get dissimilar from smaller cities. As the city groups are ordered by their population sizes, the upper triangle represents the temporal behavior of small cities on the larger cities on x-axis, whereas the lower triangle describes the behavior of larger cities. The positive values in the upper triangle mean that small cities become more and more similar to the past of large cities, recapitulating, in terms of their industrial characteristics, while the negative values in the lower triangle represent that large cities grow away from the past of smaller cites as time goes. The asymmetric shape indicates that the evolution of urban industry has a clear direction that large cities lead the change and small cities follow the path from behind. In other words, the largest 3 city groups (top 60 cities) are considered as main leaders from their significant patterns in the lead-follow matrix.

We check the robustness of the lead-follow matrix for various conditions: length of time lags, NAICS level, and number of city groups. We test the time lags for 3 and 5 years, NAICS for 3- and 4-digits, and city groups for 50 groups and 70 groups based on the the reference parameters for 10-years time lags, 2-digit NAICS, and 20 city groups. As a result, the lead-follow matrix is robust for the variants of time lags, NAICS digits, and city group sizes.

By the scaling relation, the size $Y(c,i)$ of industry $i$ in a city $c$ with population size $N(c)$ is given as follows:

The pre-factor $Y_{0}$ and the scaling exponent $\beta$ are obtained by taking a regression on $\log{Y(c,i)}$ and $\log{N(c)}$ for each industry $i$. By definition, the revealed comparative advantage (RCA) can be expressed as a function of population and scaling exponent.

where $\sum_{c,i}Y(c,i)$ is a constant. Since $\sum_{i}Y(c,i)$ is the total size of industries in a city, it can be approximated to be proportional with $N(m)$. The total industry size for all cities, $\sum_{c}N(c)^{\beta(i)}$, is estimated by continuous population approximation, whose distribution follows $P(N)\sim N^{-\gamma}$. For $\beta\neq\gamma-1$, the summation becomes,

For $\beta=\gamma-1$, the summation can be determined by either taking limitation $\beta\to\gamma-1$ or changing the integration to $\int_{N_{min}}^{N_{max}}dNN^{-1}$.

where $\delta=\beta-\gamma+1$ and $\log x=\lim_{\delta\to 0}(\frac{x^{\delta}-1}{\delta})$ by definition. The same result can be achieved by changing the integration as,

As a result, the RCA is approximated as,

The scaling exponent of population distribution $\gamma$ generally equals to 2 for cities following the Zipf’s law. For $\gamma=2$, the RCA is approximated as,

The derived RCA has different trends by population sizes. Approximately, RCA is proportional to $\beta$ for large $N$ while it is inversely proportional to $\beta$ for small $N$. In that case, if we correlate the RCA vectors of small and large cities, the correlation becomes negative while the correlations inside small cities and large cities remain positive. This property generates two clusters of small and large cities.

Now, we find the characteristic city size that distinguishes the small and large cities. The characteristic size is given as a saddle point of Eq. S19. The saddle point should satisfy both $\partial rca/\partial\beta=0$ and $\partial rca/\partial N=0$. Since $rca$ is continuous, we only consider the case for $\beta\neq\gamma-1$ in the calculation of the saddle point.

Therefore, $\beta^{*}=1$. The above conditions mean that the saddle point exists only if the probability of scaling exponent around $\beta=1$ is not zero. For example, if every scaling exponent is larger than 1 or smaller than 1, the saddle point needed for clustering of cities does not exist.

where $n=N_{max}^{\beta-\gamma+1}$ and $m=N_{min}^{\beta-\gamma+1}$. Therefore, the saddle point population is given at,

where $N_{norm}=\exp(\frac{n\log{N_{max}}-m\log{N_{min}}}{n-m})$. The saddle-point population $N^{*}$ is calculated as 1.2 millions for $\gamma=1.9$, $N_{min}=10^{5}$ and $N_{max}=10^{7}$, which is similar with the observation in data.

As a result of connection between RCA and scaling relations, we can expect the clustering of cities by their industrial compositions. Fig. S8A is the result obtained by the Pearson correlations between industry vectors of cities characterized by RCA with 2-digit industry classifications. The clustering of cities is clearly observed with a fixed point near 50th largest city.

We can also figure out the fixed point using the trends in Fig. S8B. RCA as a function of $N$ and $\beta$ in Eq. S19 can be simplified by taking logarithm on both sides.

.

Logarithm of RCA becomes a linear function of $N$ with a slope $\beta-1$. It makes the trend inverted when $\beta$ changes from $\beta<1$ to $\beta>1$, and generates two clusters by their populations.

To understand the time-dependent behavior of RCA following the scaling relation, we derive the time-dependent equation of RCA under several assumptions. In addition to two assumptions (i.e., (1) power-law distributed population, (2) total employee size in a city) used in the time-independent derivation, we include two additional assumptions: (3) total national employee size proportional to the summation of the total populations, and (4) time-independent $Y_{0}$, $\beta$, $\gamma$, $N_{max}$ and $N_{min}$. The third assumption is given from the second assumption by definition.

where the second assumption is given as $\sum_{i}Y(c,i,t)\simeq C(t)N(c)$ for time-dependent constant $C(t)$. For the fourth assumption, the scaling exponent $\beta$ and the pre-factor $Y_{0}$ do not change much in time as in Fig. S5, and the scaling exponent of the population distribution $\gamma$ is also not expected to change much in time. For the maximum and minimum populations, the average urban population change between the first year (1998) and the last year (2013) is about 23%, so we can marginally assume its time-independency when we observe the dynamics in a short period. Four assumptions are listed as follows:

1. 1.

   $\begin{aligned} P(N)\simeq\frac{1-\gamma}{N_{max}^{1-\gamma}-N_{min}^{1-\gamma}}N^{-\gamma}\\ \end{aligned}$

2. 2.

   $\begin{aligned} \sum_{i}Y(c,i,t)\simeq C(t)N(c,t)\\ \end{aligned}$

3. 3.

   $\begin{aligned} \sum_{c,i}Y(c,i,t)\simeq C(t)\sum_{c}N(c,t)\\ \end{aligned}$

4. 4.

   Time-independent $Y_{0}$, $\beta$, $\gamma$, $N_{max}$ and $N_{min}$.

We can derive the time-dependent RCA by substituting the time-dependent industry size $Y(c,i,t)$ in the definition.

The summations can be obtained using the approximation of power-law distributed continuous populations (assumption 1).

Finally, we can obtain the closed-form time-dependent RCA as a function of $\beta$, $N$, and $t$.

where $C$ is a time-invariant constant for simplifying the equation. Since the only time-variant term is $N(t)^{\beta-1}$, the time-derivative of RCA depends only on the time-derivative of population. For easy calculation, we take logarithm on both sides and take the derivatives.