Dr Dominique Vollet
email: dominique.vollet@irstea.fr.pr.

Francis Aubert
email: francis.aubert@dijon.inra.fr.

Dr Quentin Frère
email: quentin.frere@uha.fr.

Dr Stéphanie Truchet
email: stephanie.truchet@irstea.fr.

The Importance of Integrating Supply Side Factors in Economic Base Models DOMINIQUE VOLLET, FRANCIS AUBERT, QUENTIN FRERE, DENIS LEPICIER AND STEPHANIE TRUCHET

ABSTRACT :

Economic base (or export-base) theory (EBT) is often presented as a Keynesian-inspired theory that only accounts for demand-side factors. However, the recent integration of supply-side factors into economic base models (EBMs) can contribute to renewing applications of this type of model in regional economics. This paper proposes a literature review focused on new EBMs that explicitly take into account supply effects and includes both North American and European studies. It first explains why integrating supply-side factors is relevant for EBMs. Second, the paper shows how to integrate these supply-side factors into the various types of models, focusing on two production factors: land and its amenities and labor. Third, the paper discusses the conditions for integrating such supplyside factors into EBMs. KEYWORDS: regional economics, economic base models, regional impact evaluation

Introduction

Since the 1950s, regional economics have provided specific models to represent regions. In particular, these models allow economists to account for the limited size of regional economies on the one hand and the significance of the interregional mobility of production factors on the other hand. Multiregional input-output (MRIO) models have been developed at the regional scale, while economic base models (EBM), based on the key role played by external demand in regional growth, have been developed at the interregional scale.

The former has led to regional models of computable general equilibrium (CGE). These are built as a disaggregation of national models and have been further improved to create multiregional CGE models. First created in the 1980s [START_REF] Liew | A Johansen model for regional analysis[END_REF], multiregional CGE models benefit from the analytical power of CGE models and treat a variety of regional development issues [START_REF] Giesecke | Regional Computable General Equilibrium Modeling[END_REF].

The latter evolved separately to take into account various dimensions of regional economic systems beyond demand effects, including supply effects and production factor endowments [START_REF] Mac Ferlane | From Agriculture to Mining: The Changing Base or Rural Economy and Implications for Development[END_REF]). Today, EBMs may be considered, to some extent, to be monoregional CGE models. In comparison to multiregional CGE models, they present two specificities: theoretical straightforwardness and ease of implementation. Indeed, the simplification of the regional economy into only two sectors, exporting and residential, facilitates the interpretation of results by focusing on the degree of openness and integration of regional economies. Moreover, monoregional CGE models, such as EBMs, do not necessitate building a complicated social accounting matrix (SAM), identifying interregional flows and estimating interregional exchange elasticities. These advantages are obtained at the cost of the model's ability to take into account the neighboring regions' effects and adjustments, particularly on the factors market.

These models are still frequently used in regional economics, despite the various critiques leveled over the past 30 years [START_REF] Richardson | Input-Output and economic base multipliers: Looking backward and forward[END_REF][START_REF] Krikelas | Why regions grow: A review of research on the economic base model[END_REF][START_REF] Kilkenny | Export sectors and rural development[END_REF]. The main issue concerns their inability to improve the estimations' robustness and their lack of adaptability to various regional and sector contexts, e.g., closure of industries, such as defense [START_REF] Droff | Assessing the regional economic impacts of defense activities: a survey of methods[END_REF], sustainable management of natural resources in Europe (Guimaraes et al. 2014), or tourism and culture in the USA [START_REF] Mulligan | Economic base multipliers: A comparison of ACDS and IMPLAN[END_REF][START_REF] Markusen | A consumption base theory of development: An application to the rural cultural economy[END_REF].

This article proposes a literature review focused on new EBMs that explicitly takes into account supply effects. These EBMs benefit from the great adaptability inherited from regional impact models on the one hand and from the analytical ability of CGE models on the other hand. Successive improvements of EBMs, however, have been made at the expense of various hypotheses for the regional economy. First, the size of regions conditions the validity of revised EBMs. Indeed, small geographical dimensions make regions particularly sensitive to spillover effects, commuting flows, and spatial interactions between neighboring regions after a demand shock. Moreover, few mechanisms may be considered to be endogenous within a small region. Second, the time horizon of EBMs may be questioned: is economic base theory a theory of long-run growth and development or a theory of short-run fluctuations? By estimating economic base multipliers, EBMs are relevant to identifying the specificities of regional economies in the short run. Difficulties may arise in the long run, however, when adjustments occur on the factor market, especially through the labor factor.

EBMs do not own the necessary dynamic analysis ability, while the static comparative analysis may limit their explanatory power.

The remainder of the paper is organized as follows. Section 1 explains the relevancy of taking into account supply-side factors in EBMs from the theoretical basis of the economic base theory initiated in the 1950s. Section 2 shows how to integrate these supply-side factors into the various types of EBMs, focusing on two production factors: land and its amenities and labor. Section 3 discusses the conditions for integrating such supply-side factors into

EBMs. Finally, we argue that time horizon and region size play a strategic role when implementing EBMs.

Teachings from the forgotten debates between founders: Why integrating supply-side factors in economic base theory is important

Seminal distinction between basic and non-basic activities

EBMs are founded on the distinction between two different spheres in the regional economy: basic versus non-basic activities.

Regarding the former activities, the founders of economic base theory 1 identified two broad categories of "primary" or "basic" activities in the 1920s: those that sold a physical product outside the city or region and those that lived on "imports" of revenues originating from the outside (students, external visitors, or renters in the previous examples). In fact, they had already distinguished what would later be called productive basic activities and residential basic activities.

In an initial article on economic base theory published in 1955 in the Journal of Political Economy, North suggested "that a region could be defined as a territory within which there was a common export base" [START_REF] Parr | Neglected aspects of regional policy: a retrospective view[END_REF]. Then, he clearly distinguished basic activities from non-basic activities or "residentiary activities." For him, the term refers to an activity linked to the local market and whose resulting products or services are intended for a population that lives and consumes locally. In his view, it is therefore the location of consumption that determines the non-basic character that is assimilated with "residentiary" character. 2 In this article, North also addressed the particular role of non-earning incomes and highlighted their ambiguous status. The author then recognized that these "residentiary" activities are more basic than non-basic in nature.

In an article published in response to [START_REF] North | Location theory and regional economic growth[END_REF], Tiebout (1956b) contributed to clarify the concept of an "export-base." 3 Like North, he defined the export base as activities coming from the outside, but he took care to explicitly include revenues from commuting as well as private and public capital flows (central government transfers).

Debates on the respective role of basic and non-basic activities in regional growth

In the 1950s, the theoretical debates between North and Tiebout went far beyond the discussion of the notions of basic and non-basic activities and addressed their respective influence on regional growth. One of the main sticking points between the authors is the place allocated to the export base in regional growth.

For [START_REF] North | Location theory and regional economic growth[END_REF], the sole source of regional growth is external demand: "Clearly, the export base plays a vital role in determining the level of obsolete and per capita income in a region… residentiary industry depends entirely on demand within the region." 4 For the author, the possible outside economies within regional exports reinforce their competitiveness. The development of organizations specialized in marketing, ease of transportation, credit, a well-trained workforce, and subcontracted activities then support the development of the export base [START_REF] North | Exports and regional economic growth: A reply[END_REF]).

Conversely, Tiebout (1956a) challenged the idea that exports are the only driver of regional growth: "There is no reason to assume that exports are the sole or even the most important autonomous variable determining regional income… However, in terms of causation, the nature of the residentiary industries will be a key factor in any possible development." He even referenced Adam Smith, for whom an increasing division of work can lead to regional growth without the development of exports. Further in the article, Tiebout discussed the export-base to non-basic activities ratio. He explained that, if we consider a region's limits, transportation network, market, and production factor endowments as given, the region could then channel its efforts into the breakdown between basic and non-basic activities. In the cases where export activities are too great (on this point, Tiebout provides no criteria for judgment), it could be pertinent to move a part of the resources toward non-basic activities. Possibilities of regional growth could therefore exist with a decrease in exports, representing a complete reversal of perspective in relation to the usual recommendations put forward by base models, notably in the perspectives raised by North.

In his reply, [START_REF] North | Exports and regional economic growth: A reply[END_REF] acknowledged that the authors agreed on the fact that the export base does not constitute the only source of regional growth. Contrary to his seminal article of 1955, he even recognized that non-basic activities do not play a purely passive role. North continued to think, however, that the basic export base represents the most important regional growth factor and denied the simultaneous possibility of regional growth and a decline in exports while proposing arguments involving international commerce.

In a final article, Tiebout (1962) maintained that regional economies can also grow thanks to "non-basic" activities in cases where these activities retain revenue for the local economy.

Exogeneity versus endogeneity of the costs of production factors

Tiebout's argument defending the existence of regional growth factors other than exports is mainly based on supply-side factors. Indeed, Tiebout (1956) noted that, given the transport network, the size and location of markets, and factor endowments, the export base could be a necessary and sufficient condition for regional growth only if the region could compete with other regions in the export market. This would imply an ability to produce outputs at a lower cost. According to him, a region's ability to find an export base thus depends on the value of units of output as well as on the cost of inputs. Furthermore, he considered that the determination of the factor costs depends in part on the nature of regional non-basic activities.

He illustrated this point by a hypothetical peninsula in New Jersey where a coalmine was found. If the zone is completely surrounded with sand without the possibility of cultivation and if transport costs are found to be very high, there is nothing to indicate that this mining activity can develop because of the high cost of providing non-basic services. In other words, it is the capacity to develop an export base that determines regional growth, but in terms of causality, it is nature and the costs affecting non-basic activities that constitute key factors in their development. He concluded by remarking that, in North American and European economic history, cities developed around sites surrounded by fertile plains and not in lands that were unfavorable for cultivation.

In his reply, [START_REF] North | Exports and regional economic growth: A reply[END_REF] recognized that Tiebout's argument on the costs of production factors was valid and that the region's ability to export depends both on the labor supply and the transfer costs. [START_REF] Merrifield | A note on the general mathematical equivalency of economic base and aggregate input-output multipliers: fact of fiction[END_REF][START_REF] Merrifield | A practical note on the neoclassical economic-base marginal multiplier[END_REF] works allowed for further discussion of the influence of production factors. According to the author, the fact that the multiplier is constant in equilibrium constitutes an important limitation of EBMs. Indeed, this leads to the assumption that productive factors are exogenous, that is to say, available in unlimited quantities and at constant prices. This assumption would be reasonable if the factor mobility was perfect (which is not realistic for labor supply) or if a close substitute could replace less mobile factors (which is not possible in the short term). Thus, the author proposed an examination of the economic base multiplier from a neoclassical perspective and built a model with endogenous factor prices. Among others, [START_REF] Merrifield | A note on the general mathematical equivalency of economic base and aggregate input-output multipliers: fact of fiction[END_REF] showed that the marginal multiplier decreases when the wage elasticity of labor supply, the elasticities of factor substitution, and the price elasticity of demand for non-basic outputs increase, whereas it increases with the income elasticity of demand for non-basic outputs.

Validity conditions of economic base models: Time and space

As shown in the debates between [START_REF] North | Location theory and regional economic growth[END_REF][START_REF] North | Exports and regional economic growth: A reply[END_REF] and Tiebout (1956b), considering the exogeneity or endogeneity of the supply-side factor costs raises questions about the validity conditions of the EBM in relation to time and space.

Regarding the spatial conditions, Tiebout (1956b) considered that the validity of an EBM depends on the size of the region and its neighborhood. Indeed, for the author, the smaller the region, the more open it will be and the more the regional growth will depend on exports.

Moreover, the larger the region, the more frequently the determinants of income (including factor costs) will be endogenous. Furthermore, Tiebout (1956b) assumed that exports often result from exchanges between the region and its neighborhood and that export receipts decrease with distance. Consequently, the exports of a region will depend on the characteristics of the neighboring areas, and their volume will increase with the incomes of the neighboring areas. Tiebout (1956b) concluded that the idea of the export base is more relevant for small regions and particular areas, such as satellite cities in the suburban fringe located near the market. This point was not really discussed by [START_REF] North | Exports and regional economic growth: A reply[END_REF], who also considered it essential to take into account the size of the region and its neighborhood.

The debate between [START_REF] North | Exports and regional economic growth: A reply[END_REF] and Tiebout (1956b) regarding the short or long-term validity of the EBM was more noteworthy. Before developing controversial considerations 5 on the differences between regional growth and regional development, Tiebout (1956b) mentioned the short-term validity of the economic base. He considered that, in the long term, other factors affecting non-basic activities, such as consumer behavior, must be integrated (for example, by means of equations describing structural balances) [START_REF] Tiebout | The community economic base study, Urban and Regional Economics[END_REF]. Although he considered Tiebout's distinction between regional growth and economic development spurious, [START_REF] North | Exports and regional economic growth: A reply[END_REF] also concluded that EBMs are not relevant to analyzing long-run economic growth. In his view, the study of long-run growth requires integration of the determinants of efficiency changes as well as those of labor and capital inflows.

How to integrate supply-side factors into economic base models

One of the most recent articles strongly attacking the use of EBMs is that of [START_REF] Kilkenny | Export sectors and rural development[END_REF]. Nevertheless, it remains unclear whether they are not themselves convinced of the interest of those EBMs that integrate the supply characteristics of services. We can interpret the objective of their article as showing the unsuitable character of economic base models that are limited to traditional farming and industrial bases, especially rural policies limited to supporting farming activities. The authors are also aware of the significant limits of the assignment method used in their article, as we have noted: "The data show that service sector employment can grow a rural economy. Therefore, policies… that focus on the factor supply-side, such as local amenities, immigration, factor accumulation are likely to be more effective than export sector support" (924).

In this context, we propose to successively examine two sets of supply-side factors that we consider crucial for increasing the relevance of EBMs: amenities and labor.

The influence of amenities on basic and non-basic activities

Amenities and natural endowments impact regional growth through both basic and nonbasic activities. Many studies have highlighted the major role played by place-specific natural attributes in the growth of rural or non-metropolitan areas [START_REF] Power | Lost Landscape and Failed Economies: The search for Value of Place[END_REF][START_REF] Deller | The role of amenities and quality of life in rural economic growth[END_REF][START_REF] Olson | Natural amenities and rural development in new urbanrural spaces[END_REF] and in basic activities, in particular. [START_REF] Power | Lost Landscape and Failed Economies: The search for Value of Place[END_REF] was one of the first to argue that the natural landscape is an essential part of a community's economic base.

Furthermore, amenities play a role in non-basic activities at two levels: on the one hand, they cause differences in local consumption characteristics and encourage matching between a supply of new types of goods and services and latent local demand; on the other hand, they stimulate the import-substitution phenomenon. [START_REF] Cortright | The economic importance of being different: Regional variations in tastes, increasing returns, and the dynamics of development[END_REF] most explicitly asserted the differences in the characteristics of local consumption as a source of growth. The characteristics of local consumption can indeed support the conception of new products that are first developed locally and then widely exported. [START_REF] Cortright | The economic importance of being different: Regional variations in tastes, increasing returns, and the dynamics of development[END_REF] pointed out the example of beer consumption and the development of micro-breweries in Portland, Oregon, in North America. He was the first to reintroduce regional differences in consumer spending preferences and behaviors, making explicit reference to the seminal reflections of [START_REF] Tiebout | The community economic base study, Urban and Regional Economics[END_REF]. He based his reasoning on a comparison of two sets of elements concerning the region of Portland, Oregon (USA): the specific characteristics of consumer spending behavior from survey data and the location quotients of employment of certain employment categories (see Appendix A). He drew a number of conclusions about regional politics. For him, it was essential to cultivate regional specificities in terms of consumer preferences and spending behaviors, which create new knowledge and sources of innovation.

The possibility of substituting external purchases of services by local purchases thanks to amenities (green or urban) was first studied by [START_REF] Rutland | The growing localness of the Canadian city, or, on the continued (ir)relevance of economic base theory[END_REF]. Starting with a review of the debate between North and Tiebout, these authors proposed an examination of the role of potential exchanges of services that have become increasingly significant for the validity of EBMs. [START_REF] Rutland | The growing localness of the Canadian city, or, on the continued (ir)relevance of economic base theory[END_REF] provided a method for estimating the "localness" of a regional economy from a method based on minimum requirements (174). To measure whether the economies of Canadian cities were becoming more "local," they proposed a measure of the total degree of "localness" for each category of city size (see Appendix B). This was obtained by multiplying the minimum share of employment for each category of activity by the proportion of jobs of this activity for the category of size as a whole. It is, in fact, a weighted average of the different minimum shares of each size category, such that:

𝐿 = (𝑚 /𝑢 )𝑢 (1)
where 𝐿 denotes the degree of "localness" or local character, 𝑚 is the minimum share of employment in activity 𝑖, and 𝑢 is the proportion of jobs in activity 𝑖 for the category of size Therefore, amenities made limits between basic and non-basic activities more permeable. [START_REF] Markusen | A consumption base theory of development: An application to the rural cultural economy[END_REF] expressed this idea more explicitly through the concept of the "consumption base," which is defined as the part of local consumption sold to local residents in place of the external provision of imported goods or services because of amenities and which can be a source of regional growth for this reason. She replaces the formalization of the multiplier of the base adopted hitherto to express a link between total employment and basic employment using an equation linking total employment with basic employment plus employment linked to consumption base:

∆𝐸 = 1 1 -𝛼 (∆𝐸 + ∆𝐸 ) (2)
where ∆𝐸 denotes the variation in total employment, α is the marginal propensity to spend locally, and ∆𝐸 and ∆𝐸 denote the variation in basic employment and in consumption base employment, respectively. The author applied this notion to the cultural sector in several rural zones of Minnesota and showed the extent to which a more diversified cultural supply could Using data from three different census years, [START_REF] Mulligan | Place-specific economic base multipliers[END_REF] estimated the effect of natural amenities on non-basic employment and introduced interaction effects between basic employment and natural amenities:

𝑁 = 𝛽 + 𝛽 𝑇 + ⋯ + 𝛽 𝐶𝑜𝑛𝑡𝑥 + 𝛽 𝐶𝑜𝑛𝑡𝑒 + 𝜀 (3)
where the multiplier effect is:

𝑀 = 1 (1 -𝛽 -𝛽 𝐶𝑜𝑛𝑡𝑒) ⁄ (4)
𝐶𝑜𝑛𝑡𝑥 represents a vector of contextual variables that reflect the local attributes (i.e., natural amenities with two separate variables: standardized county-level index generated from six environmental subindices and a dummy variable distinguishing counties having coastal locations).

𝐶𝑜𝑛𝑡𝑒, when used, is composed of the pooled dataset. The results show that natural amenities have become increasingly important in determining the size of the multiplier. For Mulligan and Vias (2011, 999), "at least one observer has even called for a new, consumption-driven base theory to replace traditional export base theory."

Differences in consumption behaviors can be integrated into other kinds of models, such as input-output models. [START_REF] Billings | The mathematical identity of the multipliers derived from the economic base and the input-output model[END_REF] and [START_REF] Merrifield | A note on the general mathematical equivalency of economic base and aggregate input-output multipliers: fact of fiction[END_REF] demonstrated theoretically that both economic base and input-output approaches produce identical multipliers. [START_REF] Mulligan | Economic base multipliers: A comparison of ACDS and IMPLAN[END_REF] showed that they generate comparable economic base multipliers (the comparison was made across 577 non-metropolitan US counties in the year 2000). Nevertheless, Hughes (1997) noted that "the IO approach is most activity inclusive incorporating, for example, household consumption effects beyond the first round."

Strategic integration of the labor supply in the latest economic base models

The reasoning behind economic base theory starts with the regional economy being in a static state in the absence of exogenous incomes; sales of export goods provide the injection of additional revenues needed to trigger regional growth. In the steady state, revenue and population are considered constant; in a dynamic state, monetary flow growth is introduced through export revenues, while the population remains constant. Thus, in a given area and in the short run, any increase in employment in the basic activities sector, following growth in exports, lowers employment in the non-basic activities sector once labor is mobile between sectors [START_REF] Polzin | A two-sector approach[END_REF]). In the long run, a rise in employment in the basic activities sector is likely to create employment growth in the residential sector (non-basic activities), depending on the elasticity of the labor supply. Polzin's regression estimates for selected areas (SMSAs) from 1950 to 1970 suggest that increases in basic activities induce more than sufficient inmigration to offset the effects on the local labor market. This is one of Tiebout's criticisms with regard to North's work -the neglect of factorial adjustments by keeping to them in the short run -which corresponds to a consensus giving economic base theory an effective role in explaining short-run fluctuations but an inability to provide a complete explanation of longrun regional growth.

Labor supply considerations can affect regional growth in two ways to be examined through economic base theory. The first is factorial adjustment on the job market; the second is the possibility of a relatively independent population dynamic in relation to local employment.

Employment growth in the basic sector (a new firm or new external demand) will create non-basic sector jobs to a magnitude that will depend on the qualification level of the jobs (employee skill level and the firm's technological level). This effect comes from an increase in demand for non-basic sector goods and services due to the increase in the number of employees and rising wage rates. On the other hand, the effects are weak for other basic sectors because they are subjected to rising wages, while their prices are determined outside the area, although this is offset by a demand effect in intermediate consumption (according to the location of the supply chain) and agglomeration externalities. Using metropolitan-level data from the [START_REF] Haining | Intraregional estimation of central place population parameters[END_REF], 1990, and 2000US censuses, Moretti (2010) estimated the sensitivity of spillover effects according to employee skill level by using an instrumental variable built from national sectorial data to control for the effects of exogenous shifts on demand for work.

He estimated the following models:

∆N = α + β∆N + 𝛾𝑑 + 𝜀 (5) ∆N = α′ + β′∆N + 𝛾′𝑑 + 𝜀′ (6)
where ∆N and ∆N are, respectively, the log first difference of the number of jobs in the tradable and non-tradable sectors in city c, ∆N is the log first difference of jobs in a randomly selected part of the tradable sector, and ∆N is the log first difference of jobs in the rest of the tradable sector.

By considering skilled workers with varying college education levels versus unskilled workers with a high school education or less, he demonstrated that one additional skilled job in the basic sector generates 2.5 jobs in the non-basic sector, while adding one additional unskilled job in the basic sector generates only 1.04 jobs in the non-basic sector.

The market adjustment of production factors presupposes an elastic supply. When applied to labor, this elasticity depends on migration and the population's rate of participation in the activity. Concerning labor mobility, if the function of workers is solely dependent on the rate of net wages relative to the cost of living in each region, then mobility will be perfect and the labor supply infinitely elastic. More realistically, however, if we take account of workers' preferences for certain locations and criteria, such as amenities, independent of the job market, households' residential destinations are also influenced by regional living environment characteristics. Research by migration specialists has shown that since the 1970s, climatic and quality-of-life factors have assumed relative importance in relation to economic determinants in migratory plans [START_REF] Greenwood | Lagged Response in the Decision to Migrate[END_REF][START_REF] Cebula | A Note on Migration, Economic Opportunity, and the Quality of Life[END_REF][START_REF] Graves | A Reexamination of Migration, Economic Opportunity, and the Quality of Life[END_REF]).

This decision-making process was summed up by [START_REF] Mulligan | Place-specific economic base multipliers[END_REF]: "Households are often willing to substitute low-wage, high amenity locations for high-wage, low amenity locations" (1000).

At a smaller spatial level, the labor supply depends on the population's rate of participation in the activity. Here, too, the economic determinants influencing whether an individual will decide to go on the job market are compounded by other determinants. [START_REF] Moretti | Local multipliers and human capital in the United States and Sweden[END_REF], assuming perfect labor mobility across sectors within a region and identical amenities for all cities, compared the effects of rising demand for basic-sector work in the United States and Sweden. They built on Moretti's work on the United States (2010) with a similar study of Sweden (1994Sweden ( -2008) ) at the level of "labor market regions." The empirical results converged to show that the value of the multiplier depends on the technological level of activities and jobs as well as on the elasticity of the labor supply -"higher labor mobility means smaller wage increases and therefore a larger multiplier" (344). Differences between the countries also emerged, albeit mainly in local labor supply elasticity. Employment multipliers are less differentiated in Sweden due to lower wage differentials (which reduces the effect of skill levels) and/or its social insurance system, which, being more advantageous for employees, reduces variations in the labor supply. At a more sharply spatial level and for small areas, the area of the labor market may be questioned because of the substantial interregional population movements of commuters.

In the long run, labor supply thus plays an important role in explaining employment dynamics in the basic and non-basic sectors as well as how they interact. Some significant changes are underway, however, that challenge the idea that demand for labor can be derived from final demand. A large body of empirical research has shown that population changes precede job changes in both the United States and Europe. Such relative autonomy of population growth in relation to local jobs gives further fodder to the thesis of the "local consumption base," since these new households' incomes frequently rely on social transfers of external origin. As Storper's position attests, however, the debate over causality is far from settled: although "a great deal of urban economics and regional science claims that migration is the chicken […], the contrary view that I will defend […] is that individuals and households

[…] are the egg, with the chicken being the location of jobs and opportunity to earn income" (Storper 2013, 15).

Empirical analysis of validity conditions of economic base theory

Determining basic and non-basic activities: the relevance of bifurcation methods

As shown in Section 1, the delimitation between basic and non-basic activities is a controversial point, and the foundations of economic base theory hold the possibility of integrating the physical export of goods and the local spending of outside revenue in the base.

It is important to include non-earning income, which varies across different types of economies [START_REF] Bain | Transfer payment impacts on rural retail markets: A regression analysis[END_REF][START_REF] Mulligan | Regression estimates of economic base multipliers for small communities[END_REF][START_REF] Kendall | Non employment income and the economic base of Michigan counties: 1959-1986[END_REF][START_REF] Nesse | Expanding the Economic Base Model to Include Nonwage Income[END_REF][START_REF] Hodge | The economic impact of retirees on smaller communities, concepts and findings from three Canadian studies[END_REF]). To determine the basic nature of revenue, two sets of conceptions are used. For some, geographic origin is the only thing that counts, while others select another criterion: the dependence or independence of this revenue on the level of regional production and on a set of its characteristics [START_REF] Stabler | Embodied inputs and the Classification of basic and non basic activity: implications for economic base and regional growth analysis[END_REF]. The level of regional production directly influences the number of households in a region, and consequently the number of tourists or travelers visiting local households, as well as the number of households eligible for different forms of transfer (unemployment or retirement benefits). This link will be stronger when local households are immobile, thereby explaining the phenomenon of "depreciation" observed locally after the closing of establishments that are big local employers -cf. the applications of EBMs by [START_REF] Rioux | Economic impact of a military base on its surrounding economy: The case of CFB Esquimalt, Victoria, British Columbia[END_REF]. Inversely, if retirees and the unemployed are very mobile, leaving and arriving from certain regions, the dependence on or links between the regional capacity of production and level of transfers are much weaker. For example, retirees have a greater proclivity for moving to regions that are deemed pleasant places to live. Farness (1989, 321) noted the following: "If the economic base model is to be analytically useful, it must be recognized that a share of the visitor population and components of the government transfer income are endogenous. They are functionally related to regional levels of economic activity and therefore functionally related to the truly exogenous regional production."

He has proposed one of the most accomplished classifications of regional production from the point of view of economic base theory. Regional production is classified according to the place of residence of the buyer, the place of delivery of the production, the source of revenue spent, and the (in)dependence of regional production. We end up with a classification of regional production in nine categories:

I. Goods and services sold and delivered to external buyers II. Goods sold to non-resident visitors (travelers)

A. Dependent on regional levels of production (sold to business travelers with regional destinations and visitors of friends and relatives) B. Independent of regional levels of production (sold to travelers with no particular link to the region) III. Goods sold to residents who finance their purchases with legal revenues derived from outside the region A. Private interregional transfers (income, revenue from investments, etc.): 1-Dependent on regional levels of production; 2-Independent of regional levels of production;

B. Public transfers (old age, unemployment and job security allowances, agricultural, etc.): 1-Dependent on regional levels of production; 2-Independent of regional levels of production;

IV. Goods sold to residents who finance their purchases with revenues derived from sales of illegal goods to nonresidents V. Goods sold to residents (households, businesses, local governments) who finance their purchases with income earned from regional production These nine categories are then classified according to their basic or non-basic status and according to the place of residence of the buyer, product delivery, the origin of revenue expended, and its dependence on regional production. Comparing these criteria with each of the nine categories defined previously, we obtain the classification presented in Appendix C1.

Only categories I and V -for which the buyer's residence, the place of delivery, and the source of the flow of money are either regional or extra-regional -can be classified without any ambiguity as basic or non-basic. For others that have mixed characteristics, identification and classification problems arise. The share of revenues from extra-regional sources that depend on regional characteristics is thus classified as non-basic because of its endogenous character in relation to the local economy. This is typically the case with spending by travelers who are visiting friends who live in the region, revenues from investments placed by local households, or national benefits for old age/solidarity for which the volume depends directly on the number of local households and therefore the level of regional production (categories II-A, III-A1, III-B1).

In the same vein as [START_REF] Farness | Detecting the economic base: New challenges[END_REF], articles aiming to locally estimate the diversity of regional bases note "local" retirees ("aging in place retirees") and "outsiders" (i.e., attracted by the quality of life) [START_REF] Hodge | The economic impact of retirees on smaller communities, concepts and findings from three Canadian studies[END_REF][START_REF] Serow | Measuring the economic impact of retirement migration: The case of Western North Carolina[END_REF][START_REF] Vollet | Impact of retirees on rural development: Some observations on the situation in the South of France[END_REF][START_REF] Nesse | Expanding the Economic Base Model to Include Nonwage Income[END_REF][START_REF] Calafati | On the Economic Base of European City[END_REF].

The applications conducted along the same lines as the work by [START_REF] Davezies | La crise qui vient : la nouvelle fracture territoriale[END_REF], more on a national level, consider the revenues of all retirees as basic, even if the definitions proposed for the residential base are more open from this point of view: "the residential base regroups the revenue captured by territories thanks to their residential 'assets'" and "we speak more of residential base to indicate all of the revenue captured by territories independent of their productive capacity" (Pecqueur and Talandier 2011, 35). For support in making public decisions, some authors [START_REF] Vollet | Impact of retirees on rural development: Some observations on the situation in the South of France[END_REF] provide estimations of basic revenues in a fairly broad range according to whether they include "local" retirees or are limited to "external" retirees. In addition to the difficulty of data collection, each estimation presents a different interest: a wide range highlights a reliance upon national redistribution mechanisms, and a narrow range underlines the importance and potential of attracting new retired populations to local activities on the basis of the attractiveness of the zone over the recent period.

It is more difficult to address the case of public or non-profit jobs, whether they are linked to local authorities or to the state. If we refer to the spirit of the founders, Sombart identified among the "city formers" or basic revenues "a king who collects taxes" [START_REF] Krumme | Werner Sombart and the economic base concept[END_REF]).

North American applications [START_REF] Hirschl | Cash transfers and the export base of small communities[END_REF][START_REF] Mulligan | Place-specific economic base multipliers[END_REF]) consider a "Public Administration (PADMN)" as able to correspond to a basic and non-basic demand according to the type of need it is answering. For Talandier (2008, 38), in France, the public base comprises "salaries of non-commuting civil servants"; however, he pointed out that "We estimate that a third of the territorial civil service is directly paid through local taxes and not by the state, subtracting from the public base." Segessmann and Crevoisier (2013, 716) distinguished between three scenarios for Swiss cantons: local and canton administrations come under the residential economy (respectively "local" and "urban"), while the federal administration is exporting.

Finally, to integrate supply-side factors into EBM, the choice of bifurcation method plays a fundamental role (see the results of [START_REF] Gerking | Bifurcation and the time pattern of impacts in the economic base model[END_REF] or [START_REF] Nishiyama | Exports' contribution to economic growth: empirical evidence for California, Massachusetts, and Texas, using employment data[END_REF]). It is necessary to exclude excessively simplistic bifurcation methods (e.g., the assignment method) that are based on wrong assumptions, especially because of the increase in mobility and the role of supply-side factors in this mobility (as a reminder, in the assignment method, it is assumed that misallocations of industry as a basic activity are offset by those assigning all tertiary activities as non-basic). The mobility of retirees and commuters depends on the characteristics of various regional supplies (work, services, and businesses), the roles of which have become increasingly important in contemporary regional economics.

Interactions between the local supply of amenities and the sensitivity of workers or of the population to these amenities complicate the previous nine category classifications. A low supply of amenities makes the area less attractive to external workers, while the marginal multiplier effect of job categories rather than basics will tend to be lower than the average multiplier effect and vice versa (see Appendix C2). Moreover, a high supply of amenities makes non-resident visitors less sensitive to their initial visiting reasons (e.g., visits to friends or family, as in category II-A). As a result, a significant share of their spending is basic (because it depends on non-resident demand, that is, external demand), whereas it is considered to be basic because it was mainly devoted to the resident population. Therefore, increasing mobility tends to make straightforward bifurcation methods unreliable as an assignment method.

The time horizon of economic base models

The time horizon of the base multiplier may be questioned. Because of the seminal theoretical debates initiated by [START_REF] North | Location theory and regional economic growth[END_REF] and Tiebout (1956) (cf. Section 1.2), the question of whether economic base theory is a theory of long-run growth and development or a theory of short-run fluctuations remains controversial.

In an initial study, [START_REF] Mcnulty | A test of the time dimension in economic base analysis[END_REF] used income data for 41 SMSAs in the southeastern United States to estimate a set of differential regional multipliers, such as:

∆N = β + β ∆B + ⋯ + β ∆B + ∆ε (7)
where ∆N is the change in non-basic activity, ∆B is the change in the ith basic activity sector, the βs are the parameters to be estimated, and ε is the random term. Estimating the model over various time periods, [START_REF] Mcnulty | A test of the time dimension in economic base analysis[END_REF] showed that the model explained the data better (higher R2) when long time differences were used rather than short time differences. This result was interpreted as meaning that economic base theory is more a long-term than a short-term regional analysis theory.

Gerking and Isserman (1981), however, highlighted various methodological flaws in [START_REF] Mcnulty | A test of the time dimension in economic base analysis[END_REF] study, the main one being that the model design did not test for the existence of a lagged impact of basic activities on non-basic activities. As there were no timelagged variables, the estimates merely tested immediate effects over a shorter or longer time period.

Instead, the model to be estimated should be dynamic, as in [START_REF] Sasaki | Military expenditures and the employment multiplier in Hawaii[END_REF] or [START_REF] Moody | The empirical verification of the urban base multiplier: traditional and adjustment process models[END_REF], where non-basic activities are explained by the present and past values of nonbasic and basic activities. More precisely, in [START_REF] Sasaki | Military expenditures and the employment multiplier in Hawaii[END_REF] model, non-basic employment in period t (N ) is explained by previous total employment (T = N + B ):

N = β + ρ T + ε (8)
Similarly, [START_REF] Moody | The empirical verification of the urban base multiplier: traditional and adjustment process models[END_REF] specified and estimated an economic base model with an adjustment mechanism. Although this second approach is more convincing, the two studies drew opposite conclusions: [START_REF] Sasaki | Military expenditures and the employment multiplier in Hawaii[END_REF] concluded that the multiplier effect was felt in the short run (within a year), while Moody and Puffer found it in the long run (several decades) (1970). Beyond the differences in the data samples and lag model designs, the bifurcation method itself may also explain these opposite results.

Gerking and Isserman (1981) estimated a model similar to [START_REF] Sasaki | Military expenditures and the employment multiplier in Hawaii[END_REF] model (equation 2) for three SMSAs (Los Angeles, Miami, and Pittsburgh) using three different bifurcation methods: the minimum requirement, the location quotient, and the assignment method.

Comparing the estimation results obtained for each of the nine cases, they observed that the bifurcation method dramatically drives the conclusion. The minimum requirement and the location methods both indicate short lags in all three SMSAs, while the assignment method always indicates a long-run effect. Los Angeles is the most illustrative case: the entire impact multiplier is felt within the first year when using the minimum requirement or the location method, while 95% of the total impact multiplier is experienced in the second and subsequent years when using the assignment method. Therefore, the apparently contradictory results of [START_REF] Sasaki | Military expenditures and the employment multiplier in Hawaii[END_REF] and [START_REF] Moody | The empirical verification of the urban base multiplier: traditional and adjustment process models[END_REF] studies could merely be due to the different bifurcation methods chosen.

Using a dynamic version of the traditionally static location quotient [START_REF] Lesage | The dynamic relationship between export, local, and total area employment[END_REF], [START_REF] Lesage | Forecasting metropolitan employment using an export-base errorcorrection model[END_REF] used the error-correction mechanism methodology to identify both long-run equilibrium relations and short-run dynamic interactions between basic and nonbasic activities. The results showed evidence of cointegration, indicating a long-run relationship between basic and non-basic activities. It also revealed that the inclusion of the error-correction variable improved forecasting accuracy, which emphasizes the importance of short-run dynamics. A few years later, however, [START_REF] Nishiyama | Exports' contribution to economic growth: empirical evidence for California, Massachusetts, and Texas, using employment data[END_REF] contradicted this initial result, as he found no cointegration for the states of California, Massachusetts, or Texas; however, he confirmed the observation of [START_REF] Gerking | Bifurcation and the time pattern of impacts in the economic base model[END_REF] regarding the sensitivity of the results to the bifurcation method used.

The time horizon of the base multiplier remains controversial [START_REF] Bloomquist | A comparison of alternative methods for generating economic base multipliers[END_REF][START_REF] Harris | A historical application of the export base model: Economic impacts of an Indian reservation[END_REF]Kaarlsson et al. 2015). Empirical results have been contradictory, depending on the data samples, designs, and especially the bifurcation method used, but these seminal works called for additional studies on the temporal path of regional growth (e.g., [START_REF] Kraybill | A dynamic intersectoral model of regional economic growth[END_REF].

Integration of city size and sub-regional movements in economic base models

For many years, central place theory and economic base theory developed separately and in parallel. Since the late 1970s, however, the literature has sought to establish connections between these theories in order to better take into account the influence of regional size and geographical characteristics. Following [START_REF] Dacey | Population of places in a central place hierarchy[END_REF] and [START_REF] Beckmann | City size distribution in a central place hierarchy: an alternative approach[END_REF], [START_REF] Parr | City size models and the economic base: a recent controversy[END_REF], [START_REF] Nourse | Equivalence of central place and economic base theories of urban growth[END_REF][START_REF] Berry | Market Centers and retail Location Englewood Cliffs[END_REF] showed that both theories are equivalent at the city level and that the resulting multipliers are the same from a formal point of view. Thus, according to [START_REF] Nourse | Equivalence of central place and economic base theories of urban growth[END_REF], the central place population is a multiple of the proportion needed in the city to serve the hinterland, which may be considered to be the base population. The empirical implications of these theoretical works are three-fold.

A first strand of the empirical literature has analyzed the influence of location by integrating city size and distance to the city among other community characteristics. [START_REF] Shahidsaless | Community characteristics and employment multipliers in nonmetropolitan counties, 1950-1970[END_REF] thus estimated local multipliers, such as:

(∆𝐸 ) = 𝛽 + (𝛽 -1)∆𝐸 + 𝑓(𝑧)∆𝐸 + 𝜀 (9)
where 𝐸 is non-basic employment in community 𝑖; 𝐸 is basic employment in community 𝑖; 𝜀 is the random error term; and 𝑓(𝑧) = 𝑓(𝐿 , 𝑃 , 𝑀 ), with 𝐿 being the location of the community 𝑖, 𝑃 its population size, and 𝑀 the type of manufacturing in community 𝑖. The results show that distance from a central place affects local multipliers, but this influence depends on the basic sector (agriculture, manufacturing, or third sector).

A second strand of empirical studies has sought to refine analysis of intra-regional interdependencies by distinguishing between export incomes according to their origin. In line with their previous work, [START_REF] Olfert | Multipliers in a central place hierarchy[END_REF] calculated cross-community multipliers in addition to the own community multiplier for the same 598 communities classified in six functional levels. The cross-community multiplier is defined as the impact on community j resulting from an autonomous expenditure increase initiated in community 𝑖. Thus, the combination of own and cross-community multipliers produces system-wide multipliers, which show the system-wide impact of spending initiated at any level in the hierarchy. The results show that cross-community multiplier effects are small for intermediate-sized rural communities and suggest that the central place role of these communities is weak and probably declining. Conversely, larger multiplier effects are observed at higher levels of the trade center system, which may generate conditions conducive to the development of agglomeration economies. Regarding the existence of agglomeration economies, [START_REF] Coulson | Urban economic base as catalyst for movements in real estate prices[END_REF] examined the link between economic bases and the price of real estate. The authors used indices of employment growth and integrated the possibility of agglomeration economies derived directly from the first intuitions of Tiebout (1955): "Particular firms are seen as drivers for local economies, not only because they create exogenous spending in and of themselves but also because they often act as triggers for the agglomeration of their industries in particular areas" (1024). They thus showed that the diversity of regional bases decreases the volatility of housing prices. We should note that the question of the importance of the diversity of local bases is the subject of growing investigation in the literature. For example, one of the very first models aims at measuring the effects of the diversification of the local economic base and the stability of local tax revenue by associating EBMs and portfolio theory from corporate finance [START_REF] Yan | The interactive effect of revenue diversification and economic base on US local government revenue stability[END_REF]. According to this theory, diversification helps to reduce the risk or variability of different crises: high levels of revenues and a diversified economic base are said to protect the region from external crises and economic changes and to increase its capacity to repay debt. A study conducted by [START_REF] Yan | The interactive effect of revenue diversification and economic base on US local government revenue stability[END_REF] in counties in the North American state of Georgia using tax data from 1985 to 2004 showed that the diversification of tax revenue (which is often presented as an objective for the management of local finances)

has highly variable effects according to the more or less stable nature of the local economic base.

In an analysis of residential economics in Swiss cantons, Segessmann andCrevoisier (2013, 2016) explicitly admitted the role of factors relating to the supply of services and implicitly recognized the existence of economies of agglomeration in this sector. They distinguished employment that comes from the "urban residential economy" and that is divided spatially according to the hierarchy of urban centers in a dynamic of interregional competition (hospitals, supermarkets, etc.) from that coming from the "local residential economy" that is located close to customers, without the possibility of playing on the agglomeration forces (i.e., retail stores, local banks, and local administration). Another approach was adopted by [START_REF] Haining | Intraregional estimation of central place population parameters[END_REF][START_REF] Haining | Small area aggregate income models: Theory and methods with an application to urban and rural income data for Pennsylvania[END_REF], where exogenous incomes are subdivided into long-distance income movements (extra-regional) and short-distance movements (intercommunity). According to the author, investment income, government outlays, and export income earned by the sale of primary and secondary products tend to be associated with extraregional transfers, whereas the wages of commuting workers represent leakages out of one community but internal to the region. Using employment data for 276 non-metropolitan counties and autoregressive response models, [START_REF] Haining | Small area aggregate income models: Theory and methods with an application to urban and rural income data for Pennsylvania[END_REF] estimated different models related to rural \ or urban income. Moreover, each model consists of both a long-distance exogenous component and an inter-county exogenous component. Finally, in the urban model, the inter-county component is broken down into several functions of income levels in order to distinguish between the flows according to the different order levels. The first-order urban model is thus:

Y , = β + β X , + k Ψ ; , + ε ( 10
)
where Y is the total income of a place of first order, k Y is the income-creating propensity to consume in the first-order center, X is the export sector income in the 1 st order center, and 1 is the total rural income in the hinterland of the i first-order place. The results ascertain the influence of geographical organization within the region and suggest a possible loss of income in the first-order centers located between or close to two areas of highest urbanization, which may be due to better opportunities for spending in those higher order urbanized areas.

A third strand of empirical studies has analyzed the relationship between the multiplier value and the size and functional level of cities. These theoretical works conclude that as city size increases and higher orders are attained, the multiplier becomes larger. This relationship may arise from consumer spatial behavior. Indeed, the size of the multiplier increases as the propensity to outshop in other communities decreases, and the latter is expected to occur as the functional level of the city increases [START_REF] Olfert | Community level multipliers for rural development initiatives[END_REF]. The link between the multiplier size and the city functional level may also depend on the labor market. A larger labor pool will reduce in-commuting.

Several empirical studies have sought to test the relationship between multiplier values and city functional levels. Among them, [START_REF] Olfert | Community level multipliers for rural development initiatives[END_REF] The time horizon of EBMs is frequently questioned and is still a debated issue for both EBMs and models that include supply-side factors. The empirical literature has shown contradictory results depending on the data samples, designs, and especially the bifurcation method used. The use of rudimentary bifurcation methods, such as the assignment method, does not appear to be appropriate. By contrast, classifications based on various local mobility features and amenity endowments are relevant to improving the accuracy and validity of EBMs. Moreover, the use of certain econometric techniques (such as error-correction models in time-series data) could provide an even more dynamic character to the models (see notably [START_REF] Lesage | Forecasting metropolitan employment using an export-base errorcorrection model[END_REF]Droff and Paloyo 2015 390). Thus, contrary to [START_REF] Margarian | A Constructive Critique of the Endogenous Development Approach in the European Support of Rural Areas[END_REF], the effect of coordination is not necessarily convergent but could be path-dependent unequal development.

The size of the region determines the agglomeration forces in both basic and non-basic activities, as revealed by Segessmann andCrevoisier (2013, 2016) with their distinction between the "urban residential economy" and "local residential economy." Moreover, the choice of the size of the region reveals a dilemma in the theoretical scheme for EBMs that includes supply-side factors: either it is a small region with a small degree of economic integration, where the interdependency between production factors, population, and the rest of the national economy is so intense that monoregional modeling becomes difficult; alternatively, it is a bigger region with a higher but limited degree of economic integration to ensure that the region does significantly impact the national economy; or, finally, multiregional modeling would be more relevant than monoregional modeling. Future works implementing EBMs should address this dilemma.

Basic or non-basic status

Place of the buyer's residence Place of delivery Source of revenue Basic Non-basic Regional Extraregional Regional Extraregional Regional Extra-regional dependent on regional production Extra-regional independent of regional production

I x x x x II -A x x x x II -B x x x x III -A1 x x x x III -A2 x x x x III -B1 x x x x III -B2 x x x x IV x x x x V x x x x
1 Werner Sombart is generally recognized as the founder of the concept [START_REF] Krumme | Werner Sombart and the economic base concept[END_REF]). He made the distinction between "actual city founders," whose activities "enabled them to bring in money from outside the city," and the "passive or secondary city founders," whose activity depended on the former. [START_REF] Krumme | Werner Sombart and the economic base concept[END_REF] highlighted that Sombart had already identified several categories of "city founders": "a manufacturer who sells industrial products to the outside;" "a student who is supported by his parent in another place;" "a merchant who profits from trade with the outside;" "an author whose writings are being bought outside the gates." In the same way, [START_REF] Alexander | The basic-non basic concept of urban economic functions[END_REF] came to the conclusion that authors such as Aurousseau or Olmsted (1921[START_REF] Alexander | The basic-non basic concept of urban economic functions[END_REF] distinguished between two types of activities: "primary" economic activities (currently understood as basic) or "ancillary" activities (currently understood as non-basic or domestic activities).

2 [START_REF] North | Location theory and regional economic growth[END_REF] borrowed the term "residentiary" from [START_REF] Vining | The region as a concept in business-cycle analysis[END_REF], who spoke of "residentiary industry" and had already borrowed the term from the non-published work of P.S. Florence (cf. North 1955, 249).

3. Tiebout (1956) distinguished the concept of export-base, developed in regional analysis to identify the main autonomous variable determining the level of regional revenue, from that of the economic base, rather in vogue with planners and researchers interested in urban issues.

4. North's work in the 1980s and 1990s insisted on the role of institutional factors as key factors in regional growth: "The most fundamental long-run source of change is learning by individuals and entrepreneurs of organizations…" (North 1994, p. 361).

5. This point was to be the subject of sharp criticism from his detractors, with North leading the way in an article in the form of a counterargument.

  as a whole. The authors mentioned the inherent limitations of the minimum requirements method, notably the assumption of homogeneity of spending behavior and of the level of productivity. They showed that, in the period 1986-2001, Canadian cities were increasingly oriented toward the satisfaction of local needs. The authors drew both theoretical and empirical conclusions. From a theoretical point of view, they explicitly rehabilitated Tiebout's contributions concerning the possibility of an endogenous source of growth from non-basic activities (181). From an empirical point of view, they recommended continuing to stimulate growth by external revenues but also limiting leaks by substituting external purchases by local manufacturing or provisions of goods and services (182).Christofakis and Gkouzos (2014) used the same index L as[START_REF] Rutland | The growing localness of the Canadian city, or, on the continued (ir)relevance of economic base theory[END_REF] to measure the local character of employment catchment areas for all Greek prefectures between 2000 and 2009. For Canadian cities, the index increased for all size categories.

  constitute a consumption base by avoiding external spending on travel for local inhabitants.[START_REF] Dissart | Landscapes and territory-specific economic bases[END_REF] also applied this concept in France for a set of activities linked to landscape amenities. Based on the "forming of a consumption base theory,"[START_REF] Ruault | Déplacements de consommation et transferts de richesses en Ile-de-France[END_REF] highlighted the role of the fragmentation of places of consumption between the "departments" of the Paris region (food and leisure industries in Paris and specialization of the periphery in everyday purchases). Guimaraes et al. (2014) proposed an original model of the economic base by combining their model with an ecological analysis (Systems Approach Framework or SAF) for the integrated management of a coastal zone (the Guadiana Estuary in Portugal). The authors underlined that "Markusen (2007) pointed out the role of tourism activities that also serve local consumption as a strategy to enhance the contribution of tourism sector to the local development. The importance of golf activities in Castro Marim is an example that demonstrates how local consumption of tourism activities has not been an implemented strategy…" (68).

  classified 598 communities into six functional levels according to the number and complexity of their functions. Using estimates of the marginal propensity to import and the marginal propensity to consume locally, they calculated community level multipliers for each of the six functional levels and compared the resulting values. The results confirm the theoretical predictions and show that the smallest communities have the smallest multipliers. The authors conclude that development expenditures have a better chance of success at stimulating the local economy in intermediate-sized rural centers than in the smallest ones, as in this latter case, the initial expenditures are likely to be almost entirely leaked from the community. Conclusion Presented as a theoretical framework attentive to demand and inspired by Keynes, the inclusion of supply-side factors in EBMs has been the subject of debate since the 1950s, initiated by Douglas North and Charles Tiebout, making contemporary EBMs closer to monoregional CGE. Two features are strategic for implementing EBMs with supply-side factors: the time horizon and the size of the region.

Appendix A Relationships between behaviors and location quotients of linked activities (>1) for the region of Portland in 1996

Distinctive behaviors

Location quotient (value in parentheses), possible presence of industry cluster Recreation and fitness More outdoor activities (e.g., goes camping twice as often, 60% more hiking, 40% more golf, 50% more hunting, 20% more fishing than average Americans).

Sporting goods manufacturing (1.28), sporting goods and bikeshop (1.18); location of many specialized firms (Nike, Columbia Sportswear, Jantzen).

Literacy

More reading (+37%) and less TV (-10%).

Bookstores (1.58), commercial printing (1.52); cluster of publishers and printing firms.

Technical competence

More regular use of personal computer (15% more), ranked 11 th in metropolitan areas for internet access.

Semi-conductors (9.43), computers (2.36); location of many firms specialized in software and hardware (In-Focus, Sequent) and independent service providers.

Environmental orientation

Buying of organic food (+14%), committed to wildlife preservation or to the environment (+15%).

Lawn and garden services (1.28), horticulture (1.94).

Drinking to your health

Portland = ranked 3 rd (after Seattle and San Francisco) for the number of bars, high consumption of alcohol (+17%).

Brewing industry (2.93), wineries (2.73).

Source: from [START_REF] Cortright | The economic importance of being different: Regional variations in tastes, increasing returns, and the dynamics of development[END_REF]


Han Han

Vincent Lostanlen

Mathieu Lagrange

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

PERCEPTUAL-NEURAL-PHYSICAL SOUND MATCHING

Han Han, Vincent Lostanlen, and Mathieu Lagrange Nantes Université, École Centrale Nantes, CNRS, LS2N, UMR 6004, F-44000 Nantes, France ABSTRACT Sound matching algorithms seek to approximate a target waveform by parametric audio synthesis. Deep neural networks have achieved promising results in matching sustained harmonic tones. However, the task is more challenging when targets are nonstationary and inharmonic, e.g., percussion. We attribute this problem to the inadequacy of loss function. On one hand, mean square error in the parametric domain, known as "P-loss", is simple and fast but fails to accommodate the differing perceptual significance of each parameter. On the other hand, mean square error in the spectrotemporal domain, known as "spectral loss", is perceptually motivated and serves in differentiable digital signal processing (DDSP). Yet, spectral loss is a poor predictor of pitch intervals and its gradient may be computationally expensive; hence a slow convergence. Against this conundrum, we present Perceptual-Neural-Physical loss (PNP). PNP is the optimal quadratic approximation of spectral loss while being as fast as P-loss during training. We instantiate PNP with physical modeling synthesis as decoder and joint time-frequency scattering transform (JTFS) as spectral representation. We demonstrate its potential on matching synthetic drum sounds in comparison with other loss functions.

Index Termssound matching, auditory similarity, scattering transform, deep convolutional networks, physical modeling synthesis.

INTRODUCTION

Given an audio synthesizer g, the task of sound matching [START_REF] Horner | Wavetable matching synthesis of dynamic instruments with genetic algorithms[END_REF] consists in retrieving the parameter setting θ that "matches" a target sound x; i.e., such that a human ear judges the generated sound g(θ) to resemble x. Sound matching has applications in automatic music transcription, virtual reality, and audio engineering [START_REF] Shier | Manifold learning methods for visualization and browsing of drum machine samples[END_REF][START_REF] Esling | Universal audio synthesizer control with normalizing flows[END_REF]. Of particular interest is the case where g(θ) solves a known partial differential equation (PDE) whose coefficients are contained in the vector θ. In this case, θ reveals some key design choices in acoustical manufacturing, such as the shape and material properties of the resonator.

Over the past decade, the renewed interest for deep neural networks (DNN's) in audio content analysis has led researchers to formulate sound matching as a supervised learning problem [START_REF] Gabrielli | End-to-end learning for physics-based acoustic modeling[END_REF]. Intuitively, the goal is to optimize the synaptic weights W of a DNN f W so that f W (xn) = θn approximates θn over a training set of pairs (xn, θn). Because g automates the mapping from parameter θn to sound xn, this training procedure incurs no real-world audio acquisition nor human annotation. However, prior publications have pointed out that the approximation formula θn ≈ θn lacks a perceptual meaning: depending on the choice of target xn, some deviations ( θn -θn) may be judged to have a greater effect than others [START_REF] Masuda | Synthesizer sound matching with differentiable DSP[END_REF][START_REF] Roth | A comparison of parametric optimization techniques for musical instrument tone matching[END_REF][START_REF] Yee-King | Automatic programming of vst sound synthesizers using deep networks and other techniques[END_REF].

The paradigm of differentiable digital signal processing (DDSP) has brought a principled methodology to address this issue [START_REF] Engel | DDSP: Differentiable Digital Signal Processing[END_REF]. The key idea behind DDSP is to chain the learnable encoder f W with the known decoder g and a non-learnable but differentiable feature map Φ. In DDSP, f W is trained to minimize the perceptual distance In this article, we propose a new learning objective for sound matching, named perceptual-neural-physical (PNP). Our main contribution is to compute the Riemannian metric M associated to the Jacobian ∇ (Φ•g) over each sample θn (see Section 2.1). With M(θn), we train f W to minimize a locally linear approximation of spectral loss, making PNP comparable to DDSP. Yet, unlike in DDSP, the computation of ∇ (Φ•g) is independent from the encoder f W : thus, it may be parallelized and cached during DNN training. A second novelty of our paper resides in its choice of application: namely, differentiable sound matching for percussion instruments. This requires not only a fine characterization of the spectral envelope, as in the DDSP of sustained tones; but also of attack and release transients. For this purpose, we need g and Φ to accommodate sharp spectrotemporal modulations. Specifically, we rely on original differentiable implementations of the functional transformation method (FTM) for g and the joint time-frequency scattering transform (JTFS) for Φ. 1 2 2. METHODS

Riemannian metric g Φ f W g Φ g f W ∇ (Φ•g) Fig.

Accelerating spectral loss with Riemannian geometry

We assume the synthesizer g and the feature map Φ to be continuously differentiable. Let us denote by L DDSP the "spectral loss" associated to the triplet (Φ, f W , g). Its value at a parameter set θ is:

L DDSP θ (W) = 1 2 ∥Φ(x) -Φ(x)∥ 2 2 = 1 2 (Φ • g • f W • g)(θ) -(Φ • g)(θ) 2 2
(1) by definition of x and x. Using θ as shorthand for (f W • g)(θ), we conduct a first-order Taylor expansion of (Φ • g) near θ. We obtain:

Φ(x) = Φ(x) + ∇ (Φ•g) (θ) • ( θ -θ) + O(∥ θ -θ∥ 2 2 ), (2)
where the Jacobian matrix ∇ (Φ•g) (θ) contains P = dim Φ(x) rows and J = dim θ columns. The manifold formed by differentiable map (Φ • g) and the open set θ ⊂ R J induces a Riemannian metric M, i.e., an inner product on the tangent space at each point θ:

M(θ) j,j ′ = P p=1 ∇ (Φ•g) (θ)p,j ∇ (Φ•g) (θ) p,j ′ . (3)
The real-valued square matrix M(θ) defines a positive semidefinite kernel which, once plugged into Equation 2, serves to approximate L DDSP θ (W) in terms of a quadratic form over ( θθ):

∥Φ(x) -Φ(x)∥ 2 2 = θ -θ M(θ) θ -θ + O ∥ θ -θ∥ 3 2 . ( 4
)
The advantage of the approximation above is that the metric M may be computed over the training set once and for all. This is because Equation 3 is independent of the encoder f W . Furthermore, since θ is low-dimensional, we may store M(θ) on RAM. From this perspective, we define the perceptual-neural-physical loss (PNP) associated to (Φ, f W , g) as the linearization of spectral loss at θ:

L PNP θ (W) = 1 2 (f W • g)(θ) -θ M(θ) (f W • g)(θ) -θ = L DDSP θ (W) + O ∥(f W • g)(θ) -θ∥ 3 2 . ( 5
)
According to the chain rule, the gradient of PNP loss at a given training pair (xn, θn) with respect to some scalar weight Wi is:

∂L PNP θ ∂Wi (θn) = f W (xn) -θn M(θn) ∂f W ∂Wi (xn) . (6)
Observe that replacing M(θn) by the identity matrix in the equation above would give the gradient of parameter loss (P-loss); that is, the mean squared error between the predicted parameter θ and the true parameter θ. Hence, we may regard PNP as a perceptually motivated extension of P-loss, in which parameter deviations are locally recombined and rescaled so as to linearly approximate a DDSP objective. The matrix M(θ) is constant in W. Hence, its value may be cached across training epochs, and even across hyperparameter settings of the encoder. In comparison with P-loss, the only computational overhead of PNP is the bilinear form in Equation 6. However, this computation is performed in the parametric domain, i.e., in low dimension (J = dim θ). Hence, its cost is negligible in front of the forward (f W ) and backward pass (∂f W /∂Wi) of DNN training.

Damped least squares

The principal components of the Jacobian ∇ (Φ•g) (θ) are the eigenvectors of M(θ). We denote them by vj and the corresponding eigenvalues by σ 2 j : for each of them, we have M(θ)vj = σ 2 j vj. The vj's form an orthonormal basis of R J , in which we can decompose the parameter deviation ( θθ). Recalling Equation 5, we obtain an alternative formula for PNP loss:

L PNP θ (W) = 1 2 J j=1 σ 2 j ⟨(f W • g)(θ) -θ vj⟩ 2 (7)
The eigenvalues σ 2 j stretch and compress the error vector along their associated direction vj, analogous to the magnification and suppression of perceptually relevant and irrelevant parameter deviations. In practice however, when σ 2 j cover drastic ranges or contain zeros, as presented below in Section 4.3, the error vector is subject to extreme distortion and potential instability due to numerical precision errors. These scenarios, commonly referred to as M being ill-conditioned, can lead to intractable learning objective L PNP θ . Reminiscent of the damping mechanism introduced in Levenberg-Marquardt algorithm when solving nonlinear optimization problems, we update Equation 5as

L PNP θ (W) = 1 2 θ -θ M(θ) + λI θ -θ (8)
The damping term λI up-shifts all eigenvalues of M by a constant positive amount λ, thereby changing its condition number. At the limit of λ → 0, L PNP θ reduces to a quadratic form which is asymptotically equivalent to spectral loss as P-loss approaches zero. At the limit of λ → ∞, M is negligible in front of λI thus L PNP θ boils down to P-loss. Alternatively, Equation 8 may also be viewed as a L2 regularization with coefficient λ.

To further address potential convergence issues, λ may be scheduled or adaptively changed according to epoch validation loss. We adopt delayed gratification mechanism to decrease λ by a factor of 5 when epoch validation loss is going down, and fix λ otherwise.

APPLICATION TO DRUM SOUND MATCHING

Perceptual: Joint time-frequency scattering (JTFS)

The joint time-frequency scattering transform (JTFS) is a nonlinear convolutional operator which extracts spectrotemporal modulations in the constant-Q scalogram [START_REF] Andén | Joint time-frequency scattering[END_REF][START_REF] Mathieu Andreux | Kymatio: Scattering transforms in Python[END_REF]. Its kernels proceed from a separable product between two complex-valued wavelet filterbanks, defined over the time axis and over the log-frequency axis respectively. After convolution, we apply pointwise complex modulus and temporal averaging to each JTFS coefficient. These coefficients are known as scattering "paths" p. We apply a logarithmic transformation to the feature vector JTFS(xn) corresponding to each sound xn, yielding

Sn,p = Φ(xn)p = (Φ • g)(θn)p = log 1 + JTFS(xn)p ε , (9)
We set ε = 10 -3 , which is the order of magnitude of the median value of JTFS across all examples xn and paths p.

The multiresolution structure of JTFS is reminiscent of spectrotemporal receptive fields (STRF), and thus may serve as a biologically plausible predictor of neurophysiological responses in the primary auditory cortex [START_REF] Chi | Multiresolution spectrotemporal analysis of complex sounds[END_REF]. At a higher level of music cognition, a recent study has shown that Euclidean distances in Φ space predict auditory judgments of timbre similarity within a large vocabulary of instrumental playing techniques, as collected from a group of professional composers and non-expert music listeners [START_REF] Vincent Lostanlen | Timefrequency scattering accurately models auditory similarities between instrumental playing techniques[END_REF].

We use the GPU implementation of [START_REF] Muradeli | Differentiable time-frequency scattering in kymatio[END_REF] to compute JTFS with the same parameters as [START_REF] Vincent Lostanlen | Timefrequency scattering accurately models auditory similarities between instrumental playing techniques[END_REF]: Q1 = 12, Q2 = 1, and Q fr = 1 filters per octave respectively. We set the temporal averaging to T = 3 seconds and the frequential averaging to F = 2 octaves; hence a total of P = 20762 paths. We refer to [START_REF] Vahidi | Mesostructures: Beyond spectrogram loss in differentiable time-frequency analysis[END_REF] for further details on the ability of ∇ (Φ•g) to extract "mesostructures" in nonstationary audio signals.

Neural: Deep convolutional network (convnet)

EfficientNet is a convolutional neural network architecture that balances the scaling of the depth, width and input resolution of consecutive convolutional blocks [START_REF] Tan | EfficientNet: Rethinking model scaling for convolutional neural networks[END_REF]. Achieving state-of-the-art performance on image classification with significantly less trainable parameters, its most light-weight version EfficientNet-B0 also succeeded in benchmarking audio classification tasks [START_REF] Zeghidour | LEAF: A learnable frontend for audio classification[END_REF]. We adopt EfficientNet-B0 as our encoder f W , resulting in 4M learnable parameters. We append a linear dense layer of J = dim θ neurons and a 1D batch normalization before tanh activation. The goal of batch normalization is to gaussianize the input, such that the activated output is capable of uniformly cover the normalized prediction range. The input to f W is the log-scaled CQT coefficients of each example, spanning 10 octaves with 12 filters per octave.

Physical: Functional transformation method (FTM)

We are interested in the perpendicular displacement X(t, u) on a rectangular drum face, which can be solved from the following partial differential equation defined in the Cartesian coordinate system u = (u1, u2).

∂ 2 X ∂t 2 (t, u) -c 2 ∇ 2 X(t, u) + S 4 ∇ 4 X(t, u) + ∂ ∂t d1X(t, u) + d3∇ 2 X(t, u) = 0 (10)
In addition to the standard traveling wave equation in the first above parenthesis, the fourth-order spatial and first-order time derivatives incorporate damping factors induced by stiffness, internal friction in the drum material and air friction in the external environment, rendering the solution a closer simulation to reality. Specifically, α, S, c, d1, d3 designate respectively the side length ratio, stiffness, traveling wave speed, frequency-independent damping and frequencydependent damping of the drum. Even though real world drums are mostly circular, a rectangular drum model is equally capable of eliciting representative percussive sounds in real world scenarios. The circular drum model simply requires a conversion of Equation 10into the Polar coordinate system. We bound the four sides of this l by lα rectangular drum at zero at all time. For simplicity, we simulate the excitation by setting the initial condition at t0 = 0 to be X(t0, u = (0.4l, 0.4lα)) = 0.03 meters, and 0 otherwise.

We implement generator g as a PDE solver to this high-order damped wave equation, namely the functional transformation method (FTM) [START_REF] Trautmann | Digital Sound Synthesis by Physical Modeling Using the Functional Transformation Method[END_REF][START_REF] Schäfer | Physical modeling in sound synthesis: Vibrating plates[END_REF]. FTM solves the PDE by transforming the equation into its Laplace and functional space domain, where an algebraic solution can be obtained. It then finds the time-space domain solution via inverse functional transforms, expressed in an infinite modal summation form

x(t) = X(t, u) = m∈N 2 Km(u, t) exp(σmt) sin(ωmt) (11)
The coefficients Km(u, t), σm, ωm are derived from the original PDE parameters in the following ways.

ω 2 m = (S 4 - d 2 3 4 )Γ 2 m 1 ,m 2 + (c 2 + d1d3 2 )Γm 1 ,m 2 - d 2 1 4 (12)
Fig. 2. Distributions of the five sorted eigenvalues of M(θn). For the sake of comparison between PNP and P-loss, the bold line indicates the eigenvalues of the identity matrix (see Equation 6).

σm = d3 2 Γm 1 ,m 2 - d1 2 (13) Km(u, t) = y m u δ(t) sin( πm1u1 l ) sin πm2u2 lα ( 14
)
where

Γm 1 ,m 2 = π 2 m 2 1 /l 2 + π 2 m 2 2 /(lα) 2
, and y m u is the m th coefficient associated to the eigenfunction sin(πmu/l) that decomposes yu(u).

Without losing connections to the acoustical manufacturing of the drum yet better relating g's input with perceptual dimensions, we reparametrize the PDE parameters {S, c, d1, d3, α} into θ = {log ω1, τ1, log p, log D, α}, detailed in Section 3.4 of [START_REF] Han | wav2shape: Hearing the Shape of a Drum Machine[END_REF]. We prescribe sonically-plausible ranges for each parameter in θ, normalize them between -1 and 1, uniformly sample in the hyper-dimensional cube, and obtain a dataset of 100k percussive sounds sampled at 22050 HZ. The train/test/validation split is 8 : 1 : 1.

In particular, fundamental frequency ω1, duration τ1 falls into ranges [40, 1000] Hz and [0.4, 3] seconds respectively. Inhomogeneous damping rate p, frequential dispersion D and aspect ratio α ranges are [10 -5 , 0.2], [10 -5 , 0.3], and [10 -5 , 1].

RESULTS

Baselines

We train fW with 3 different losses -multi-scale spectral loss [START_REF] Steinmetz | auraloss: Audio focused loss functions in PyTorch[END_REF], parameter loss, and PNP loss. We use a batch size of 64 samples for spectral loss, and 256 samples for P-loss and PNP loss. The training proceeds for 70 epochs, where around 20% of the training set is seen at each epoch. We use Adam optimizer with learning rate 10 -3 . Table 1 reports the training time per epoch on a single Tesla V100 16GB GPU.

Evaluation with JTFS-based spectral loss

We propose to use the L2 norm of JTFS coefficients error averaged over test set for evaluation. As a point of reference, we also include the average multi-scale spectral error, implemented as in Section 4.1. One of the key distinctions between Euclidean JTFS distance and multi-scale spectral error is the former's inclusion of spectrotemporal modulations information. Meanwhile unlike mean squared parameter error, both metrics reflect the perceptual closeness instead of parametric retrieval accuracy for each proposed model.

Discussion

Despite being the optimal quadratic approximation of spectral loss, it is nontrivial to apply the bare PNP loss form as Equation 5 experimental settings. On one hand, Φ • g potentially has undesirable property that exposes the Riemannian metric calculations to numerical precision errors. On the other hand, extreme deformation of the optimization landscape may lead to the same numerical instability facing stochastic gradient descent with spectral loss. We report on a few remedies that helped stabilize learning with PNP loss, and offer insights on future directions to take. First and foremost, our preliminary experiments show that training PNP loss without damping λ = 0 subjects to convergence issues due to the high condition numbers in empirical Ms as illustrated in Section 2.2. Fig. 2 shows the sorted eigenvalue distribution of all Ms in test set, where Ms are rank-2,3 or 4 matrices with eigenvalues ranging from 0 to 10 20 . This could be an implication that entries of θ contain implicit linear dependencies in generator g, or that local variations of certain θ fail to linearize differences in the output of g or Φ • g. As an example, the aspect ratio α influences the modal frequencies and decay rates via [START_REF] Han | wav2shape: Hearing the Shape of a Drum Machine[END_REF], where in fact (1/α + 1/α 2 ) could be a better choice of variable that linearizes g.

To address Ms' ill conditions we attempted at numerous damping mechanisms to update λ: namely, constant λ, scheduled λ decay, and adaptive λ decay. The intuition is to have L PNP θ start in the parameter loss regime and move towards the spectral loss regime while training. The best performing model is achieved with adaptive λ decay (see Section 2.2). We propose to divide λ by a factor of 5 if the model breaks the best epoch validation loss record and keep it the same otherwise. In practice, we initialize λ to match the largest empirical σ 2 j ≈ 10 20 , and then adaptively decay it to 3 × 10 14 in 20 epochs. This indicates that f W is able to learn with damped PNP loss if λ is large enough to compensate for rank deficiency in M.

The diagonal elements of M(θ) can be regarded as both the applied weights' magnitudes and proxies for the perceptual importance of θ's accuracy. Inspecting the results of τ = θ[iτ ] regression, we observe in Fig. 3 that in comparison with P-loss model, PNP model improves retrieval accuracy for sounds inducing larger perceptual difference to changes in τ (in blue), at the expense of lowered accuracy for the opposite (in yellow). This suggests a trade-off behavior aligned with PNP loss' weighting scheme.

We believe that more of PNP loss' mathematical potential can be exploited in the future, notably in cases where parameterization without domain-specific knowledge renders the failure of P-loss, and its use in hybrid optimization schemes. We plan to investigate the scalability of each loss function under reparameterizations, as well as other damping schemes and optimizers. The current update mechanism, originated from the Leverberg-Marquardt algorithm, aims to improve the conditioning of a matrix inversion problem in the Gauss-Newton algorithm. However when used jointly with stochastic gradient descent, each λ update may change the optimization landscape drastically. The resulting optimization behavior is thus not fully understood. We consider interfacing nonlinear least squares solver with SGD and forming a hybrid learning scheme in future work.

CONCLUSION

Knowledge on human auditory perception aiding data-driven approaches to machine listening tasks have been exemplified in a multitude of applications [START_REF] Heller | Synergy between human and machine approaches to sound/scene recognition and processing: An overview of ICASSP special session[END_REF]. In this article we presented another case of this synergy named Perceptual-Neural-Physical (PNP) autoencoding, a bilinear form learning objective for sound matching task. In our application, PNP optimizes the retrieval of physical parameters from sounds in a perceptually-motivated metric space, enabled by differentiable implementations of physical model and computational proxy of neurophysiological construct of human auditory system.

We demonstrated PNP's mathematical relationship to spectral loss and parameter loss. Using this formulation, we motivated and established one way of interpolating between optimizing in parameter and spectral loss regimes. We presented damping mechanisms to facilitate its learning under ill-conditioned empirical settings and provided future plans for further exploiting its mathematical potential.

Fig. 3 .

 3 Fig. 3. Histogram of log squared τ estimation error for perceptually significant (largest 20% M(θ)[iτ , iτ ]) and less significant (smallest 20% M(θ)[iτ , iτ ]) sounds. The yellow and blue regions indicate the weighting-induced difference in retrieval accuracy.

Table 1 .

 1 Report of average JTFS distance and MSS metrics evaluated on test set. Six models are trained with two modalities: 1. the inclusion of pitch retrieval i.e. regressing θ = {τ, log p, log D, α} vs. θ = {log ω1, τ, log p, log D, α}, and 2. the choice of loss function: P-loss, MSS loss, or PNP loss with adaptive damping mechanism. The best performing models with known and unknown pitch are P-loss and PNP loss respectively. Training with MSS loss is more time consuming than training with P-loss or PNP loss. Training with differentiable JTFS loss is unrealistic in the interest of time.

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Companion website: https://github.com/lylyhan/perceptual neural physical

Audio examples: https://pnp.cargo.site/