The Mesolevel Economy in Nineteenth-Century England and Wales: Applying Input-Output Accounting and Spatial Interaction Modelling to the Historical Study

Tis paper is ofered as a contribution to understanding historical trade. Te method of input-output modelling is utilised to evaluate the evolution of cities (and their associated regions) in England and Wales between 1851 and 1911. Te construction of input-output accounts for each city region is based on “heroic” data assumptions, which enable the construction of a demonstration model illustrating a new iterative approach to historical analysis. In its current application, the model enables estimates to be made of mesolevel trade between cities, which enhances our analysis of urban evolution in this period.


Introduction
One of the major contributions of history is to examine change and evolution in systems of interest over time.In our case, we take this focus and apply it to cities in the nineteenth century-the process of their change and evolution, the key drivers of these processes, and the implications of these fndings for wider understandings of the industrial revolution.In urban history, there have been extensive qualitative studies and a more limited number of quantitative approaches-in the latter case, often under the aegis of economic history (for example, [1][2][3]).Tere is also a signifcant literature which connects urban modelling developed in contemporary contexts to applications in history (for example, [4][5][6][7]).Our contribution is rooted in combining the output of the former with the analytical framework of the latter.
We take the-oft neglected-mesoscale as our focus.Tis means we are seeking to understand the economy of England and Wales in a way which allows us to concentrate on fner spatial scales (below the national level), while also simultaneously considering the relationships between these smaller units in the economy as a whole.We do this by constructing a set of input-output accounts for each town and city in England and Wales, by combining smaller "parishes" into urban units and their associated rural areas (which together we term a "city region").Tis enables us to compute sector-specifc trade fows between each pair of city regions in our dataset, which in turn facilitates analysis of their internal supply-demand relationships (including the balance-of-trade between a city region and the broader national economy), as well as geographic concentrations in the production and trade of diferent economic sectors.In doing so, we are able to examine the interdependence of the economies of towns and cities over the period 1851 to 1911-as well as how this evolved over a pivotal period of economic growth, which has important legacy efects for the British economy in the modern day.
In formal modelling terms, our ultimate goal is to build dynamic models-in which cities (and their associated regions) are nonlinear dynamical systems and as such are path dependent with change occurring through a series of phase transitions at critical values of model parameters (for a short overview, see Wilson [8]).Here, we present an important step in laying the foundations for this process: a demonstration model for a series of (mostly) decadal time points over the period 1851 to 1911.Representing this change through a time series of equilibrium models can fail to account for the phase changes present in a dynamic system, but it can nonetheless yield useful insights regarding the evolution of cities over the nineteenth century.By establishing equilibria at a series of points in time-which result from change in exogenous variables-we provide the basis for a future dynamic version (the algebraic basis of which we outline in Appendix 1, which draws on ideas in Wilson and Dearden [9] and applies them to this context).
Te combination of historical analysis and computational modelling has signifcant potential for a mutually benefcial research framework.Te former provides a starting point by identifying the drivers of change which need to be incorporated into the model.Te model can then be used to assess the extent to which these drivers are interdependent, as well as highlight phase transitions.Further historical analysis at a microscale can be employed to examine how these changes came about.Te outcomes of this research can then be used to further refne the model, and the process can be repeated.
A model-based analysis can therefore be employed to demonstrate points in time at which phase changes become possible, and historical study can then investigate their precise conditions.In the general area of our research, there have been several major contributions which might explain the growth trajectories of cities-entrepreneurs who recognised opportunities for innovation, pre-1850 industrial specialisation which facilitated later changes (either through creating necessary infrastructure or human capital formation), and the relationship between scientifc developments and industrial applications-on both a macroscale and a microscale, but examining these factors systematically poses a signifcant research challenge [1,2,10,11].
While others have focus on transport networks or endowments (for example, [12][13][14]), we focus more widely on input-output accounts for a system of cities-and then on trade as a feature of economic development.Both national accounting and international trade have been a major focus of historical research, and there are many examples of localor frm-level studies of specifc industries (for example, [15][16][17][18][19]), but estimating intercity trade (either by building a system of input-output accounts or some other means) remains a signifcant challenge for historians, primarily as a consequence of limited direct data.We therefore anchor our work in building a time series of input-output accounts and associated trade fows to fll this gap in the modelling base.Here, we adopt a broad defnition of "cities" to mean "towns and cities" with more than 10,000 population and use the terms "urban" and "city" more or less interchangeably.
To understand this focus, it is necessary to note that the main subsystems representing cities in a comprehensive urban model are interdependent-as outlined by Figure 1.Te left-hand (red) side of the diagram represents the demography of the city, generating individuals and households who live, work, and use a variety of services.Te right-hand (green) side represents the economy of the city: generating housing, employment, products, and services.Tese subsystems are all linked through the (blue) transport and communications subsystems.
Tese models can be built at various spatial scales-from a system of cities in a regional or national setting to single city regions with a fne-grain spatial structure (for example, [22][23][24]).In our case, the economic model on the right-hand side of the diagram is based on input-output accounts that represent both the internal economy of a city and trade between cities in a broader-scale system of cities, but it is informed by the demographic information on the left-hand side.Adopting the spatial scale where cities (and their associated regions) are the focal units is the most efective way of adding a spatial dimension to a "whole country" analysis.
In the sections that follow, we outline the core concepts-with particular reference to input-output accounts and modelling at diferent scales-and explain the process of creating demonstration models that both ofer insights in their own right and lay the foundations for future research.While we focus on a scale with towns and cities (including their "city regions") as the spatial units, the model specifcation allows for the potential of extending this research to fner spatial scales in the future.

Intercity Input-Output Accounts
2.1.Overview.Te adaptation of the input-output accounting framework [25] developed here originates from a sketch in Oléron-Evans [26], which translates a model from Levy, Oléron-Evans, and Wilson [27] focused on the whole world (with countries as the spatial units) to a single country with cities (and their associated rural areas) as the spatial units.Te specifcation-which we further develop here-roots the model in input-output accounts for each city region, including both a representation of trade fows between each pair of city regions in the network, and between these city regions and the rest of the world.Te model itself is novel in terms of its methodology: we estimate city region inputoutput accounts, building on the information provided by national accounts in earlier work by linking these with other data sources [28,29].As might be expected, such an ambitious exercise involves making "heroic" assumptions, but we believe this to be worthwhile enabling us to build a distinctive demonstration model which provides a platform for further work.Our fndings must therefore be interpreted with this in mind-and there are many refnements which might be made-but they nonetheless generate insights that add to conventional historical analysis due to their grounding in highly detailed demographic information.

Te Accounting Framework.
Te key variables which will be the basis of our model (all for an unstated time period) are listed in Figure 2. Note that for any variable, if an index i is replaced with an asterisk, this implies summation over i.
Teoretically, E m i and M m i can be dealt with by adding a rest-of-the-world zone to the spatial interaction model discussed below.However, in this implementation, these fows are specifed exogenously.

Complexity
While most of the variables listed in Figure 2 are selfexplanatory, the X m i , F m i , and x mn i terms require some additional explanation.X m i can be understood as total production less imports for i (where i is a geographic unit).It is therefore a measure of what is actually produced in a location, but it is not the total value of the sector, which is made up of X m i + m m i + M m i .F m i is more complex."Final demand" in a national accounting framework has a defned meaning, and while the major elements are present in our input-output data, the list is not quite so extensive.F m i is taken to mean the sum of consumption and investment; as outlined below, exports are specifed separately.
We can then defne x mn i (the total demand for m to produce a unit of n in i) in relation to a mn i (the amount of m needed to produce a unit of n in i), such that Te core input-output accounting equation can then be given as  [20].It can also be found in Rees et al. [21].
Complexity 3 Cities (or any other geographic unit) balance their accounts through external imports and exports.We therefore introduce a balancing term V m i which can be used to identify the extent of the imbalance when (2) is in its initial state (prior to establishing domestic trading relationships, with m m i � 0, e m i � 0): In other words, the "balance" is the total production plus imports, minus all demand.If the balance is positive, the geographic unit (i) can generate exports to balance its accounts (in the present application, the city region exports the surplus production to other city regions within England and Wales); if the balance is negative, instead imports (from other city regions) are necessary to meet the demand.For the sake of simplicity, we ignore the possibility of stockpiling.
We then predict the reconciliation of these imbalances through a doubly constrained spatial interaction model.Tis could be specifed in a number of ways, but here, we assume that city regions either export when V m i > 0 or import when V m i < 0. In efect, this zeros out the attractiveness term where a city region does not require imports to balance the demands of a particular sector (and vice versa).We acknowledge that this is a drastic oversimplifcation, but it also creates a signifcantly less complex mathematical problem, as many potential "city pairs" are eliminated.Te specifc sectors included are given in Figure 3, and the number of towns and cities for each year is given in Figure 4-a full list of named towns and cities is provided in Smith and Bennett [30]-but in the most computationally intensive year (1911), there are approximately 2.1 million potential city pair-sector combinations; employing this approach reduces the actual city pair-sector combinations to be balanced through the spatial interaction model to much more manageable 425,000.
For the city level, we have both estimates of population (P i ) and a proxy (Q m i ) for employment (weighted by demographic characteristics to account for the relative importance of women and children in the workforce of the mid-nineteenth century).As has been stated, "cities" are actually "city regions," as this allows for complete geographical coverage of England and Wales-and the process of estimating each "city region" is described below.We then use a national input-output table and scale each of the national variables down to a city level using the following equations-utilising population where the relationship is governed by demand and employment where this is determined by supply: As noted, we assume that a city can only "export" (within England and Wales) if In other words, the imbalance is made up by exports or imports (as appropriate) from other cities. e m i and m m i are therefore determined by the initial conditions-although their destinations/origins (and therefore, the fows of trade) are determined by the spatial interaction element: where which ensures that and Te spatial interaction model is doubly constrained, to ensure that the total value of (domestic) import fows is equal to the total value of (domestic) export fows-in other words, because we control for international imports and exports in the national input-output table (and account for these in ( 6) and ( 7)), imbalances in the supply of, or demand for, particular sectors in an individual city region must be reconciled through the domestic economy.4 Complexity Equations ( 10)-( 14) ensure that this is carried out in the most efcient manner-city regions will be predicted to import or export locally if they can do so but can also have a trading relationship with a distant city region if necessary.Tis is particularly important for sectors where production is geographically concentrated-such as mining or textiles-and examples of this are discussed below.Due to the limited information regarding transport infrastructure for the period of study, Euclidean distance between the centroid of each city (rather than the centroid of the city region, as the actual urban area is considered the organisational hub) is used as a proxy measure.As a further consequence of this limited information, we do not diferentiate between different methods of transport in the current application of the model-although the model specifcation retains the fexibility to adopt this approach in future work.
Tis method also allows us to estimate "value added" at a city region level.Although we refer to this aspect of the generated input-output tables as "value added" as a shorthand, in this instance, it is in essence the diference between the total value of the sector and the sum of intermediate production and imports: all of which are ascertained from (2).While this is a fairly coarse measure, it nonetheless provides a good indicator of the general value of a sector to the economy of the city region.

Drivers and Interdependence
Te model specifcation here stops short of a fully dynamic implementation, but it nonetheless contains many of the drivers associated with economic change.In Figure 5, we outline how these relate to the model variables.Te frst column notes the potential driver of change ("driver"), the second details how this driver can be observed ("observable aspects"), and the third maps this onto model variables ("representation").Many of these aspects are currently proxied or implicitly modelled by variables (especially the a mn coefcients), but it is possible that these could be specifed directly (using submodels) in the future.
As is evident from Figure 5, the a mn coefcients play an important role in defning the relationships between sectors [25].Te data sources employed for these are discussed in   6 Complexity Section 4, but-as outlined in equation ( 1)-the inputoutput method hinges upon a mn terms accounting for the intermediate demands of each sector.By incorporating these intermediary sector fows, we are able to demonstrate the interdependence of diferent sectors-and this has important implications which are not immediately apparent when only fnal demand is considered.First, some sectors used a notable proportion of their own output to meet their demands-this is particularly important for "textile, clothing, and leather goods" and "metal manufacturing."However, these proportions were not static and represent changes in manufacturing production processes.Notably, increased technological innovation did not inherently mean a reduction in these requirements-both previously mentioned sectors increased their internal consumption per unit production over the period as they became more technologically sophisticated, seemingly as an outcome of increasingly complex products and broader consumer choice.
Second, some sectors were closely reliant on others.Tis is informative at a macrolevel, but at a mesolevel, it also incorporates a distinctly spatial element.For example, "metal manufacturing" understandably had a notable relationship with "mining and quarrying," which often led the two to be closely located to one another-indeed, in south Wales, coal pits and iron foundries often functioned under the same enterprise.However, if two related sectors were not closely geographically located, then this generates a trade fow-which would not be highlighted by measures of fnal demand/output.

Applications
We now turn to the task of applying the model to historical data.Te model outlined in Section 2 requires us to specify time points, spatial units, and sectors.Much of the challenge is the assembly of these inputs from data sources which must be aligned with one another.Where data are not available with sufcient resolution or granularity, it is necessary to infer or estimate the required information from the closest available.
Te simplest of these is perhaps time points: we take the decadal census years, starting with 1851 (the frst surviving complete national census for England and Wales) and ending in 1911 (the most recent historical census available to academic researchers).We have to exclude 1871, as this was not digitised as part of the Integrated Census Micro-Data (I-CeM) project-which created digitised and codifed records for each person in the census, including their basic demographic (name, age, sex, and family structure), geographic (the parish in which they were found at the time of the census), occupational, and other information [31].Te I-CeM database therefore provides us with population and occupation (used as the basis for our employment/labour measure) by parish for England and Wales.We presently exclude Scotland and Ireland-although there are complete data available for the former-due to the availability of other data sources.
I-CeM classifes individuals on the basis of their occupation and parish, but it does not inherently allow for analysis of city units.For this, coding from the BBCE (the British Business Census of Entrepreneurs)-which directly builds upon the I-CeM data-is employed.Te BBCE identifes "urban" parishes and groups them based on a list of towns and cities identifed by the Robson-Law database [32].Tis is an imperfect method for our purposes as city boundaries did not usually conform to parish ones, but it does permit identifcation of predominantly urban areas.
By aggregating the individuals associated with each urban unit, we can therefore produce a labour estimate for each city identifed in the data.As demographic factors (including child labour) played a more pronounced role in determining productivity in this period, each sector in each city is weighted into an "equivalent labour value"-which in essence accounts for the proportion of men, women, and children which constituted the labour force for each city.For the present model, these weightings are constant (see Appendix 2) although it is possible to vary these weightings by sector and city.
Notably, a signifcant proportion of national economic activity in this period took place outside of urban centres, particularly for certain industries (such as agriculture and mining and quarrying).One approach to solving this problem would be to create rural zones in addition to urban ones.However, this would drastically increase the size of the zonalpair matrix; if we assume that each city had at least one rural zone, then the number of ij pairs would increase approximately fourfold-and often a single rural zone would fail to represent the complexity of the urban-hinterland exchange structure.
Instead, we opt to expand the geographical city unit to include its surrounding rural areas.Tere are several established methods for this type of operation.Within the geography literature, both market areas and central place theory might be applied [33,34].Percolation theory might also be applicable [35].Here, given the paucity of sufcient data, we opt for a simpler approach.Terefore, each nonurban geographic unit (in this case, a historic census parish) is assigned to the city which maximises the function: where K and L are weighting terms.Te weighting of K � 8 and L � 0.003 in this instance is determined experimentally to ensure that rural parishes are assigned to a city region primarily on the basis of their proximity to the urban boundaries of each city, but that larger cities (as determined by their productive capacity) have a longer reach.Tis is necessary because of the diversity in "urban" units in the dataset-the infuence of a small town (the minimum for inclusion being 10,000 population) on its surrounding area is not necessarily equal to that of a large urban centre with population in millions.Nonetheless, weighting favours the former, and therefore, the calculated city regions are typically contiguous (although there are a very small number of exceptions).In areas with a high density of towns (which particularly occurs towards the end of the period), city regions may consist of a single (urban) parish.Conversely, in isolated areas, towns may provide the central point for Complexity multiple rural parishes in addition to the urban area of the town itself.Tis method has the notable shortcoming that each city region is distinct-in reality, these would have overlapped-but such a simplifcation is necessary with the available information.As there are limited accessible road data for the nineteenth century, major roads from the OpenStreetMap project are used as a proxy for the most likely transport routes between each parish and the centre of each urban unit, with the data obtained and routes determined using the OSMnx Python package [36].
An overview of the results of this exercise is visualised in Figure 6.By establishing which parishes can be classifed as belonging to each city, we can then link these to the geospatial units provided by a set of census boundary shapefles [37].Tere are no parish boundary datasets for the entirety of our period, so in this instance, we approximate the position of parish units to their "registration subdistrict"-this is less precise than the parish but still provides a sufciently accurate approximation for our purposes and allows us to avoid the caveats and complications of working with approximated "constant" parish units [38].
Te distinction between a "city" and a "city region" is important to consider and is therefore discussed in relation to the results below.However, it is necessary to highlight that the cities likely played an important organisational role in this period-particularly with the limitations of the transport network-and were therefore very important to their local economies.As a consequence, although there are some city regions with a notable proportion of their population outside of the city boundaries-with Norwich being perhaps the most signifcant example-they still likely had an important economic organising efect.
Tis approach allows us to segment the entirety of the national economy (once Scotland and Ireland have been subtracted) -and because the parish units are tied to the Census data, this also allows us to infer population and occupational numbers for each city region.By classifying occupations into sectors which map onto our national inputoutput data (which we do by classifying the 797 "occupational codes" given to each employed individual in I-CeM to the sectors listed below), we can infer the economic contribution of each city region using the method outlined in Section 2.
Such an approach still requires national-level economic data.Tere is a signifcant body of literature concerning British industrial output during the nineteenth century.Tis addresses both national (typically, GDP) and sectoral levels.Tis debate has typically been framed around the rate of growth, in an attempt to understand the timing and speed of the industrial revolution [16,39].But comparatively little research has concentrated on intersectoral linkages within the British economy.Nonetheless, the literature provides two major contributions which (mostly) bookend our period of interest-Horrell, Humphries and Weale's work (HHW) for 1841 [29], and the Tomas thesis for 1907 [28].Both are signifcant historical contributions, combining a vast array of published sources to create national input-output tables for their year of interest.We are therefore able to extract both the input-output tables and technical coefcients from their work and use them to inform our own estimates.
Te frst issue with this process is that-despite the HHW (1841) table being signifcantly infuenced by Tomas' (1907) table-the sectors do not directly map onto each other.Te latter originally contains 41 sectors and the former 17, likely as a refection of the comparative complexity of the economy in both periods.It is therefore necessary to construct the sector map given in Figure 3).Tis is an imperfect process and comes with the following caveats: Te second issue is that these input-output tables are constructed for the United Kingdom.Due to data limitations (outlined above), our work focuses on England and Wales.It is therefore necessary to defate these values accordingly.We do this in accordance with the Scottish and Irish shares of GDP for this period, taken from Geary and Stark's regional GDP work [40].Geary and Stark do not give a value for 1851; it is likely that the Irish fgure was infuenced by the famine for this year, but we adopt an estimate in-line with Geary and Stark's other years.In any case, the results are not heavily infuenced by this fgure, and it is only relevant when directly comparing the value of trade between 1851 and another year.
HHW and Tomas therefore provide much of the data necessary for dynamically estimating an input-output table for each year between 1841 and 1911.However, some further information is required: In the case of a), we assume a linear relationship between our two data points.It is likely that individual industries difered from this growth curve, and that this would also be refected at the sector level (albeit to a lesser extent).However, this should provide a reasonable approximate over growth over the 70 year span.Tis can be expressed as follows: 8 Complexity where a mn t(x) is the amount of sector m required to produce one unit of sector n at time x, and t (x) = the time point, where integers denote known datapoints (in years) and letters denote target years.a mn t(x) < 0 can potentially occur in cases where t (x) < t (1) and a mn t(1) < a mn t( 2) , (17) or t (x) > t (2) and a mn t(1) > a mn t( 2) . ( In practice, this means that a few a mn terms could be predicted as negative at the very end of the period; these are substituted to 0. For b), we assume an exponential curve between each of the two known data points.With regards to c), HHW provides both imports and exports for 1841.Tomas provides imports for 1907 but not exports.Te latter are therefore estimated using data from Mitchell's work on collating macroeconomic statistics, the MeasuringWorth datasets for price/cost comparisons, and other specialist historical sources; the specifc sources are outlined in Figure 7. Decadal estimates are then calculating using the same method described for value added above.
Tese data can all be combined to iteratively generate an input-output table for any year in the period of interest (1851-1911).Te technical coefcients are treated as fxed (once calculated, as above), but the ultimate production of a sector is not.Te total for each sector is initially set to equal the sum of value added (as calculated above) and international imports, and this is multiplied by the technical coefcients to generate an initial value for each a mn fow.Te consumption and investment column can then be calculated as the sector total less the sum of exports and intermediate demand.We then enter an iterative cycle, where the  [37].Note.Te total number of city regions for each year is summarised in Figure 4.
Complexity technical coefcients are multiplied by the previous sector total until this total reaches convergence (defned as all sector totals changing by less than one percent).

Findings
Tis all leads to a modelled set of cities (and their associated regions), sectors, and trade fows.Te generality of the model means that this could be analysed in any number of ways.Here, we focus on the "major" centres identifed by the model and their relationship to the national economy through trade.Section 5.1 considers the properties of the city regions contained within the system, Section 5.2 then employs these units to examine the spatial characteristics of the economy (with an emphasis of sectoral specifcity), and Section 5.3 focuses on value added as a key indicator of economic specialisation and development.Together, these fndings demonstrate how national accounting measures fail to adequately account for the distributed nature of the British economy in this period and highlight the long-term legacies of regional specialisation in explaining the uneven capabilities of the present-day economy.

Te System of City Regions.
As the only condition for including a town or city for any given year was a minimum population of 10,000 people, the number of town and city regions grows over the period, such that it more than doubled between 1851 and 1911 (Figure 4).Tis is a consequence of the combined efects of urbanisation and intrinsic population growth over the nineteenth century.Te population (P i ) of each city region is used to estimate an "equivalent labour value" (a proxy for employment)-i.e.(Q m i )-by weighting the population employed in each economic sector for the productive capacity of each demographic group, as outlined in Appendix 2. Tis accounts for the proportion of men, women, and children in Q m i is primarily used in ( 4) and ( 6), but independent analysis of  m Q m i by city region facilitates a sector-agnostic overview of the labour force of England and Wales over this period.It shows that London is by far the biggest labour centre throughout the period, being (approximately) three times as large as the second biggest city region-Liverpool (and Birkenhead)-in 1851, and growing to almost six times as large by 1911.Liverpool is consistently predicted to have a large city region in the early years of the model.Tis is due to its geographical positioning and the lack of large towns in north Wales until 1881-this is consistent with the cultural importance of the former to the latter (being often known as "the Capital of North Wales"), even well into the midtwentieth century [46].Although relatively sparsely populated by comparison, the size of the Liverpool city region diminishes over the period of study, so that the labour value term is much closer to those of third placed Manchester (and Salford) and fourth placed Birmingham (and Smethwick) by 1911.
Figure 8 show the top 15 city regions for each year by estimated labour value (i.e., it ranks  m Q m i by city region).While the gap between fourth and seventh is small in 1851, Birmingham (seventh in that year) had established itself as far bigger than ffth placed Leeds by 1911 and was very comparable to Liverpool and Manchester.Plymouth and

Complexity
Norwich-ranking fourth and ffth in 1851-were quickly surpassed by Bristol, Leeds, and (later) Shefeld as the century progressed-the latter two on the back of their growing metal industries.Both Plymouth and Norwich were historically important regional centres for the relatively isolated agricultural production of Devon and Norfolk, and while the general city regions remained broadly consistent over the period, the signifcance of this trade appears to have diminished.
Tis emphasises the importance of understanding the diferent roles that cities played within the economy.While urban centres are often associated with service sectors, the idea of cities as a gateway to hinterland economies complicates this urban-rural dichotomy.Te method employed here suggests that there were "agricultural" cities (even if the urban areas themselves did not create agricultural products), just as there were cities with little surrounding farmland that specialised in other areas.Similarly, while Newcastle contained urban collieries, its historic association with coal comes as much from its surrounding area as the city itself.Of course, it has long been the case that certain industries were associated with particular places-Nottingham was synonymous with lace, just as Shefeld was with steel-but the extent to which such relationships should be quantifed and analysed at scale is more limited.

Te Mesolevel Economy.
Te relative size of diferent city regions-and their trajectory over the period-allows us to make a number of inferences regarding the nature of the system (i.e., the economy of England and Wales) as a whole.While these fndings are both useful and valuable, they do not rely on the main analytical power provided by the model.A particularly insightful application is assessing the necessary mobility of goods within each sector.A national-level input-output table can highlight the fow of goods from one sector to another within the economy, but by scaling this to the mesolevel, we are also able to consider the geographic fow of goods in addition.
Te fows discussed here are the output of the doubly constrained spatial interaction model-in other words, y m ij from (10).For simplicity, these are defated (using [47]), with a base year of 2015.As has been outlined above, the fows Complexity between diferent cities should be considered the minimum necessary to balance the demands of the economy, and trade may well have taken place beyond this.Nonetheless, understanding this "minimum" allows us to ascertain the extent to which production was evenly distributed (in relation to demand) throughout the country, or-conversely-the extent to which demand in certain areas outstripped supply in that region.
Tere are notable continuities between 1851 (Figure 9) and 1911, (Figure 10) albeit with some signifcant diferences.In both cases, the "mining and quarrying" sector shows the highest trade fow per unit of production, although the actual value fell signifcantly over the period--likely due to the rise of the mining industry outside the major regions in the second half of the nineteenth century."Agriculture"-one of the largest sectors by production-also exhibits a high tendency to move its produce in both years, which intuitively makes sense given the concentration of agricultural land in certain regions (although this only increased marginally over the period as a whole)."Public administration and defence" show a similar pattern, which is a consequence of its concentration in administrative centres.
At the other end of the scale, "construction" remains a lowmovement sector in both years; this makes sense, given the nature of construction in this period (and note that some quasiconstruction projects-such as shipbuilding-are allocated to other sectors)."Distribution and transport"-although increasing its fows/production rank in relation to other sectors between 1851 and 1911-remains relatively constant in terms of fows/production value.While this might initially seem odd given the focus of the sector on the movement of people and goods, this can be explained by two main factors: (a) transport workers were relatively few compared to those concerned with the distribution of goods, most of which took place within city regions, and (b) although the nature of the sector involved intercity movement, this does not mean that fow of value needed to move between cities; they were relatively good at exporting their own goods services.
Tere are two major surprises in these fndings: frst, the relatively low movement of goods produced by the "textile, clothing, and leather goods" sector and second, the static Complexity fows/production value for the "domestic and other service" sector-despite its increase in ranking between 1851 and 1911.In the case of the former, this may be explained by the relatively high value of international exports; if textiles were in large part an export good, then lower internal movement might be expected.
For the latter, although there is signifcant growth in the size of the sector between 1851 and 1911, fows/production remains remarkably constant throughout the period, hovering between 0.10 and 0.12.Tis consistency may suggest that the service sector grew largely in proportion to geographic demand-in other words, it was largely created and consumed at the local level-or it may indicate that the signifcant diversity within the service sector meant that the movement of its value is masked by false equivalency.Te latter might be corrected by an entropy-maximising "internal-likeness" coefcient, but such an exercise lies beyond the immediate scope of this work.Tis analysis has signifcant implications for understanding market integration in this period.By estimating the extent to which goods might be expected to move internally within the economy, the model demonstrates the importance of internal trade networks to economic growth and regional specialisation.To further this analysis, we can consider the geographic concentrations in trade fows.
Te frst thing to note-which has already been suggested above-is the dominant position of London in the economy of England and Wales during the whole of this period.As outlined in the model specifcation, cities (and their associated hinterlands) were either exporters or importers in each sector, determined by their shortfalls and surpluses.can only be described as a service powerhouse.Te sectors in which it imported were generally those producing raw materials: "agriculture," "mining and quarrying," and "metal manufacture"-the exceptions being 14 Complexity the manufacturing sectors of "metal goods" (which was a fairly small sector throughout the period) and "textiles, clothing, and leather goods" (which was heavily concentrated in the north-west of England, and "internal" London production-although signifcant-was likely outstripped by the sheer size of the population).
On the other hand, London's largest export sectors were generally service-based.Te "domestic and other service" sector, although large in the earlier years of the study (third largest export in 1851 and second in 1861), really took of towards the end of the century; in 1881, it (narrowly) became London's largest export but totalled roughly the same value as the exports of all other sectors combined by 1911."Distribution and transport"-a sector with a signifcant service component-was the second largest export sector for every year other than 1861 (when it was third)."Food, drink, and tobacco"-again with a notable, albeit comparatively smaller, service element (e.g., through hospitality)-were the other notable export, the largest in 1851 and 1861, but largely stagnant (with a slight decline) from 1881 onwards.Te other sectors ("other manufacturing," "gas and water," "public administration and defence," "construction," and Complexity "soaps, candles, and dyes") were much smaller exporters, but this was more a consequence of the size of the sector than London's role in them.
It is signifcant that of the four major cities (London, Manchester and Salford, Liverpool and Birkenhead, and Birmingham and Smethwick), none share the same importexport profle.Liverpool's exports were heavily weighted towards "distribution and transport," exceeding the sum total of all other sectors for each year.Manchester, despite its geographic closeness to Liverpool, also primarily exported "distribution and transport" (at least, after 1861) but also had a notable "textile, clothing, and leather goods" component, although the latter declined over the period of study.Birmingham was perhaps the most distinct of the four, with a clear emphasis on "metal goods" and "metal manufacturing," particularly after 1861.Tis indicates that although these cities (in the case of Manchester and Liverpool, somewhat jointly) played a signifcant role in the economies of their local regions, they did not fulfl precisely the same function, and thus, the importance of intercity trade to balance accounts becomes of particular importance.
Tis is not to say that the largest exports did not come from these cities (Figure 11).London accounted for the largest export fow for six to eight (out of the thirteen) sectors (depending on the year) and Birmingham a further two; Liverpool and Manchester rank as the largest exporters for any of the given sectors more rarely, but this is primarily due to their largest export industry ("distribution and transport") being overshadowed by London, and they are therefore relegated to smaller industries (in the case of    16 Complexity Liverpool) or fghting with Bradford over textile exports (in the case of Manchester).Te remainder were more specialist producers; Newcastle representing the Great Northern Coalfeld in the "mining and quarrying" sector-which was then subsequently displaced by Rhondda in south Wales-should be no surprise (although the relatively early importance of Merthyr Tydfl might be considered unexpected) nor should the role of Aldershot and Portsmouth in "public administration and defence."Norwich as a large agricultural exporter is also perhaps unsurprising.
Te relative importance of Bradford in the textile sector is more unexpected; while its production was smaller than that of Manchester in each year (sometimes by as much as 25 percent), the relative populations meant that Bradford had a greater surplus to export.As for services, it is important to note that there are some imperfect substitution efects to which the model (in its present specifcation) is not very sensitive-Bradford was known for its woolens, whereas Manchester was primarily known for producing cotton products.Te model presents these to be directly comparable and interchangeable, but these products were to some extent consumed by diferent markets.With any level of aggregation, this type of issue is likely to occur; a more specifc sectoral classifcation would help to avoid this issue, but the data required to do this are not presently available until 1907.

City-Level Economic Indicators.
All of this brings us to an estimate of value added by city.Tis has enormous potential for understanding the nature of the economy in England and Wales during this period.We estimate value added by drawing data from the input-output table for each city region and calculating the diference between the total value of the sector and the sum of intermediate fows used in its production and imports.Tis is a fairly coarse measure of value added, and it is also infuenced by sectoral composition-in part, as a consequence of the model assumptions-but identifes several signifcant trends.
First, the scale of London's economy continues to show through.As Figure 12 demonstrates, it was three and a half times as large as the next biggest city region-Liverpool and Birkenhead.Tis is broadly in line with the size of the workforce.Indeed, the highest value added city units for 1851 largely correspond to the biggest producers, with London, Liverpool and Birkenhead, Manchester and Salford, Bristol, Norwich, and Birmingham and Smethwick shown as the leaders.
City-level value added having the potential to become particularly insightful is to understand the evolution of city-level economies over the period.To continue with London as an example, in 1851 (Figure 13), its most valuable sector (in terms of value added) was "distribution and transport," followed by "domestic and other service" and "food, drink, and tobacco."In 1911 (Figure 14), these remain the biggest contributors to value added, but the order is shufed, with the relative importance of other smaller sectors also altered.
If we compare this to a major established northern city-such as Leeds (Figures 15 and 16)-we are still able to observe growth in the importance of "distribution and transport" and "domestic and other service" as well as a decline in the contributions of "agriculture" (common to many of the larger economic centres), but the relative importance of manufacturing sectors (in this case, "textiles, clothing, and leather goods" and "metal goods") is much greater.

Conclusions
Te paucity of data for intercity trade in the nineteenth century hinders understandings of regional economic development.Tis is important because the legacies of this pivotal period survive into the present day-the major cities in 1851 by and large remain the major cities over 150 years Complexity later.But more than this, in many cases, the growth trajectory of a city or city region was established by its specialisation and development in the second half of the nineteenth century.Understanding the origins of this growth is therefore key to explaining geographic socioeconomic divisions in modern Britain.
Unsurprisingly, the mesolevel economy demonstrates a complex picture over the period of study.Te demonstration model presented here is a frst step in unravelling this complexity, but it is only the beginning of the work necessary to adequately understand this phenomenon.Calibration is particularly challenging, and it is reliant on very general data and heroic assumptions-both of which can be superseded and refned by specialists in the future.However, despite the potential future refnements and modifcations, several major themes emerge.
First, the spatial intricacy of the nineteenth century mesolevel economy is a severely underexplored theme, but it is central to explaining regional economic development.While some related sectors were located to take advantage of proximity, in each sector, there was still signifcant transportation of goods around the city system in order to fulfl the local supply and demand requirements in each instance.Historians have often focused on transport networks or frm-level movement of goods, but sector-level fows are difcult to directly observe.Tis is where the method outlined here can make a valuable contribution: we demonstrate how certain sectors (for example, textiles) were largely centralised, and others (for example, mining and quarrying) were heavily concentrated in multiple regions-both of which necessitated notable movement of goods around the domestic economy.Tis is not in itself a new idea, but the interactions between diferent sectors in diferent places have hitherto received very little attention and rarely at scale.Tese interchanges of goods are not something which can be ignored-there were associated frictional costs, mesolevel competitive advantages leading to particular paths of economic development, and sociopolitical ramifcations.
Second, the general signifcance of London is often acknowledged in the literature, but the fndings here demonstrate just how important it was to the economy of England and Wales in this period.Its relative economic (and associated political and cultural) power dwarfed competing regional centres, and it played a signifcant role (either as an importer or an exporter) in every sector of the economy in England and Wales.Tere are other major centres which played key roles in particularly sectors, but none with the economic breadth displayed by London.Te general dominance of London over the south-east (and south England in general) can also be contrasted with the relationship of other major centres (such as Birmingham and Manchester) with their respective regions: although the latter were undoubtedly important, the economic diversity and relative population size meant that their ability to pull in trade was more limited by comparison.
Finally, we move towards establishing value added statistics at a city region level.While these may beneft from future refnement, this is nonetheless an important step in establishing the regional wealth disparity characteristic of the modern day "north-south divide."We demonstrate that some sectors were greater contributors to city-level value added than others and that these diferences go some way to explaining the relative wealth of cities.
Tere are many areas in which this model might be developed-this is the nature of attempting to specify something so general.However, we would conclude by highlighting the areas in which we perceive the most signifcant gains might be made.First, relaxing the relationship between employment and output to allow for regional variations in productivity would facilitate a better representation of geographic disparities in production, as well as demonstrate greater sensitivity in city-level metrics-these are important for understanding the roots of geographic inequality in modern Britain.Second, highlighting the role of investment (both government and private) would improve the capacity of the model to demonstrate the infuence of capital formation.Tird, an integrated transport network-and costs varied by transport method-would better account for the economic penalties to the movement of goods and integrate the rapid infrastructural development taking place in this period.Finally, removing the calculation of international imports and exports and allowing these fows to be modelled dynamically (including the specifcation of port cities as points of ingress and egress) would allow for an improvement in the realism of both international and domestic fows.In the longer run, a dynamic model (such as that specifed in Appendix 1) should be the ultimate goal--but incremental refnements to the work presented here can yield new insights into the development of the economy in England and Wales during this pivotal period.

Data Availability
Te census data used in this study were supplied by UKDS [SN:7481] under license and so cannot be made freely available.Requests for access to these data should be made to UKDS.Te RSD boundary data were supplied directly by Joe Day at the University of Bristol and Alice Reid at the University of Cambridge.Tey are in the process of being deposited with UKDS, and the citation may change to refect this in due course.

Disclosure
A preprint version of this paper was made available as Solomon and Wilson [48].

Figure 4 :
Figure 4: Te number of towns and cities include in the model, 1851-1911.
(a) a mn coefcients (b) Value added, by sector (c) Exports, by sector, for 1911

Figure 6 :
Figure 6: An illustration of calculated city regions, 1851-1911.Te boundary data are provided by[37].Note.Te total number of city regions for each year is summarised in Figure4.

Figure 8 :
Figure 8: City region ranking by labour force size and year.

Figure 11 :
Figure 11: Largest single exporter by sector.

Figure 13 :Figure 12 :
Figure 13: London value added by economic sector in 1851.

Figure 15 :
Figure 15: Leeds value added by economic sector in 1851.

Figure 14 :
Figure 14: London value added by economic sector in 1911.

Figure 16 :
Figure 16: Leeds value added by economic sector in 1911.
Tomas' table indicates small linkages without monetary values.Tese are replaced with 0.1 (essentially rounding up).Tis approach may overweight transactions including textiles due to small sector aggregation, but it does follow the HHW method.(ii) "Domestic service" does not feature as a separate sector in Tomas' table, so needs to be combined into "other service" in 1841.(iii) To match Tomas' work, "distribution" and "transport" in the HHW table are combined.Tis is nonoptimal, given their distinct roles in an economy.(iv) With the lack of a suitable comparator in Tomas' table, "housing services" are added to general service sector in HHW.It is not entirely clear where the equivalent falls in Tomas' table.(v) Public utility in Tomas' table has relatively little description and may not include defence spending (in contrast to HHW).