MD Catarina Almqvist
email: catarina.almqvist@ki.se

MD Hans-Olov Adami

Paul W Franks Phd

MD PhD Leif Groop

MD Erik Ingelsson

PhD Juha Kere

PhD Lauren Lissner

PhD Jan-Eric Litton

MD PhD Markus Maeurer

MD Karl Michaëlsson

PhD Juni Palmgren

MD Göran Pershagen

PhD Alexander Ploner

MD PhD Patrick F Sullivan

PhD Gunnel Tybring

PhD Nancy L Pedersen
email: nancy.pedersen@ki.se

LifeGene -population-based cohort

Keywords: Biobank, cohort study, epidemiology, prospective study, questionnaires, population genetics

Studying gene-environment interactions requires that the amount and quality of the lifestyle data is comparable to what is available for the corresponding genomic data. Sweden has several crucial prerequisites for comprehensive longitudinal biomedical research, such as the personal identity number, the universally available national health care system, continuously updated population and health registries and a scientifically motivated population. LifeGene builds on these strengths to bridge the gap between basic research and clinical applications with particular attention to populations, through a unique design in a research-friendly setting.

LifeGene is designed both as a prospective cohort study and an infrastructure with repeated contacts of study participants approximately every five years. Index persons aged 18-45 years old will be recruited and invited to include their household members (partner and any children). A comprehensive questionnaire addressing cutting-edge research questions will be administered through the web with short follow-ups annually. Biosamples and physical measurements will also be collected at baseline, and re-administered every five years thereafter. Event-based sampling will be a key feature of LifeGene. The household-based design will give the opportunity to involve young couples prior to and during pregnancy, allowing for the first study of children born into cohort with complete pre-and perinatal data from both the mother and father. Questions and sampling schemes will be tailored to the participants' age and life events. The target of LifeGene is to enrol 500,000 Swedes and follow them longitudinally for at least 20 years.

Introduction Background

In only a few years the methods used to study genetics of complex traits have evolved almost beyond recognition. The result has been unprecedented advances in our understanding of the genetic architecture of almost all common complex diseases. The main driving force behind these discoveries has been the rapid development and application of genome-wide association studies (GWAS).

Although there has been remarkable recent progress in defining the genetic basis of common complex diseases, a new generation of epidemiological studies that undertake standardised, finescale phenotyping and exposure assessments of many thousands of individuals measured repeatedly during many years of follow-up will be required to maintain this trajectory. New approaches for storing, processing, and charactering the tissue samples collected in these individuals will be necessary, as will the application of novel, comprehensive, and multi-factorial statistical methods to analyse data harvested from these biosamples.

Worldwide, there are several large-scale genetic epidemiological studies such as the UK Biobank [START_REF] Ollier | UK Biobank: from concept to reality[END_REF], deCODE in Iceland [START_REF] Gulcher | Population genomics: laying the groundwork for genetic disease modeling and targeting[END_REF], Millennium in Japan [START_REF] Nakamura | The BioBank Japan Project[END_REF], the Kadoorie study of Chronic Diseases in China [START_REF] Chen | Cohort profile: the Kadoorie Study of Chronic Disease in China (KSCDC)[END_REF] and CONOR/HUNT in Norway [START_REF] Naess | Cohort profile: cohort of Norway (CONOR)[END_REF]. These studies focus on chronic diseases diagnosed later in life, and some also collect detailed data on lifestyle factors. There are, however, very few large prospective studies focusing on disorders that emerge early in life, in some instances during infancy or early adolescence. LifeLines in Holland [START_REF] Stolk | Universal risk factors for multifactorial diseases: LifeLines: a three-generation population-based study[END_REF], a three-generation population-based study with a household recruitment approach differs from most other ongoing cohorts because data on diseases with onset early in life is collected. The Norwegian Mother and Child Cohort study (MoBa) [START_REF] Nilsen | Selfselection and bias in a large prospective pregnancy cohort in Norway[END_REF], ALSPAC (the Avon Longitudinal Study of Parents and Children) [START_REF] Golding | ALSPAC--the Avon Longitudinal Study of Parents and Children. I. Study methodology[END_REF] and the US National Child Study (9) have followed pregnant mothers from early pregnancy and their offspring throughout childhood, whilst other prospective birth cohort studies were brought together under the Global Asthma and Allergen European Network (Ga2len) [START_REF] Keil | European birth cohort studies on asthma and atopic diseases: II. Comparison of outcomes and exposures--a GA2LEN initiative[END_REF]. The Southampton Women's Survey [START_REF] Inskip | Cohort profile: The Southampton Women's Survey[END_REF] is of particular interest because it collected parental exposure data before the pregnancy and thus could assess associations to perinatal and infant outcomes. Pre-pregnancy exposure data are also likely to be valuable in longterm studies of chronic disease later in life, although to date no such studies have been undertaken. Sweden and other Nordic countries have several crucial prerequisites for comprehensive longitudinal biomedical research, due to the personal identity number, the universally available national health care system, continuously updated population and health registries and a scientifically motivated population. As a complement to the register-based epidemiologic tradition, Sweden is also at the forefront of developing and implementing information technology, cutting edge biotechnology and biobanking. In order to adequately address gene-environment interactions, acquisition of phenotype and exposure information repeatedly, biosamples and physiological assessment are important. A prospective study combining information on exposures as early as pre-conception and in early life throughout young and mid-adulthood, with the possibility to study a variety of diseases with onset relatively early in life and co-morbidities later in life will open up new research opportunities.

LifeGene is designed as a prospective cohort study with an infrastructure that allows repeated contacts of study participants approximately every five years, and short follow-ups annually.

Recruitment of index people aged 18-45 years old who are invited to include their household members (other adults and any children) increases the opportunity to involve young couples prior to and during pregnancy, allowing for the first study of children born into cohort with complete preand perinatal data from both the mother and father. Other types of event-based sampling, i.e. data collection initiated as a result of a relevant event, such as an accident or influenza, is a key feature of LifeGene. A comprehensive web-based questionnaire comprised of multifaceted questions concerning phenotypes and exposures is administered at baseline, briefly followed-up annually, and re-administered every five years thereafter. Biosamples are collected and stored in a manner that facilitates the full range of 'omics´, and physical measurements are also sampled repeatedly at five year intervals. Questions and sampling schemes will be tailored to the participants' age and life events. The target of LifeGene is to enrol 500,000 Swedes and follow them for at least 20 years.

Methods and planning process

In the early phases of planning Life Gene, multiple workshops and plenary sessions with invited international and national experts in a variety of diseases, genetic epidemiology, environmental epidemiology, and molecular genetics were organised during which the optimal LifeGene sample design, research questions that could be addressed with a prospective cohort, biological measures and phenotype and exposures were discussed.

Separate working groups focusing on six broad phenotypic domains: infections, inflammation and allergy, malignant diseases, metabolic and cardiovascular disorders, musculoskeletal disorders and neuro-psychiatric disorders were established. In addition, a working group was established to discuss lifestyle factors such as physical activity and diet. Each working group was invited to present their area of research and the rationale for participation in LifeGene, along with preferred study design and optimal measures to include in questionnaires, biosamples and physical assessments. Three working groups focusing on infrastructure issues were also created: Biostatistics, biosampling and biobanking, and IT-infrastructure. Based on the combined recommendations of all groups, study design, recruitment and assessment schemes and protocols were established.

Infectious diseases

Outcomes of exposures to infectious agents are shaped by genetic background, exposure history to pathogens (or antigens in general, including vaccines), along with innate and adaptive immune response. Exposure to infectious agents has a broader impact on health and disease as compared to infections strictu sensu. Chronic inflammation, in part induced by infection, may impact on a number of diseases; prime examples of infections associated with malignant transformation are infections with Helicobacter pylori or human papilloma virus type 16. Perhaps the most recent report which links a retroviral infection with chronic fatigue syndrome shows that gauging infections entails far more than just the examination of acute clinical infections. It also showed that retrospective analysis of old diseases may lead to the identification of novel pathogens [START_REF] Lombardi | Detection of an infectious retrovirus, XMRV, in blood cells of patients with chronic fatigue syndrome[END_REF]. Novel and more sensitive methods to detect infectious pathogens as well as gauging relevant immune responses will help elucidate associations between infections and clinical syndromes.

Pathogenicity of infectious agents is in part determined by the intricate host-pathogen relationship, which can be captured in a LifeGene grid of household interviews and biological sampling. An eventbased sampling study design allows linkage of clinically and biologically relevant events with the detection of infectious agents. An exercise in this context was proposed for the LifeGene pilot, which addressed Influenza-like-illnesses (ILI), described further on page 24.

Other information relevant to infectious and viral diseases, such as travel, sexual behaviour, and vaccination will also be collected.

Inflammation and allergy

Inflammation is a key factor in allergy and asthma as well as in rheumatic diseases, multiple sclerosis and other autoimmune diseases. It also plays an important role in cardiovascular, musculoskeletal and periodontal diseases. Important research questions include the role of factors pre-conception for disease development in the offspring, which may be mediated by epigenetic mechanisms. A LifeGene birth cohort design would also be uniquely suited for addressing questions related to fetal growth. Furthermore, research questions on gene-environment interactions are paramount and a wide variety of early life exposure and conditions are of interest, such as nutrition, infections, maternal stress, smoking and obesity, as well as exposure to allergens and ambient air pollution. Special focus for protocol development was given to recruitment of pregnant mothers and their partners, including questionnaires and biosamples before and during pregnancy, as well as on extensive long-term follow-up of children born into the cohort. This group is of prime interest also for many other research questions, such as testing and providing mechanistic explanations for the Barker hypothesis [START_REF] Barker | Fetal origins of coronary heart disease[END_REF]. Women in the LifeGene cohort who become pregnant were proposed to answer extensive questionnaires at gestational week 10-12 and at week 26-28, and biological samples to be taken using the same methodology as for the full cohort. Cord blood and maternal blood at delivery were also proposed and the child to be followed up with questionnaires at 4 months and then at 1 year of age, followed by yearly questionnaires. In subgroups additional biosamples may be obtained on several occasions during the first two years of life.

Malignant diseases

Progress in cancer epidemiology has been remarkable during the last decades. Several malignancies -including those predominantly caused by tobacco, alcohol and certain oncogenic infections (e.g. hepatitis virus B and C, human papilloma virus and helicobacter pylori) -are now largely preventable.

At the same time, progress has been limited or non-existent in other areas. Indeed, the causes of major cancer sites and types remain enigmatic. Prostate cancer, lymphomas, leukaemia and testicular cancer are salient examples.

The cancer working group agreed that LifeGene would provide unparallel novel opportunities to reveal genetic and environmental causes of cancer. No other existing study combines the size of LifeGene with opportunities for repeated exposure measurement starting early in life, an extensive biobank allowing studies of biomarkers and genes with the convenience in Sweden of retrieving tumor tissue from pathology departments. As a corollary, LifeGene provides the opportunities to take a life course approach to cancer etiology; to accommodate emerging hypotheses through repeated exposure assessment by creative application of e-epidemiology; and to apply novel molecular techniques for refined -and perhaps etiologically more relevant -phenotypic subgrouping of cancer sites and types. The intense methodologic development in studies of gene-environment interactions will only make LifeGene more informative over time.

Metabolic and cardiovascular disorders

The current global increase of obesity and type 2 diabetes incidence has taken epidemic proportions, and even if this is almost certainly due to a global shift towards more obesogenic behaviours, such as physical inactivity and caloric dense diet, susceptibility to these conditions varies greatly from one person to the next and tends to segregate within families, irrespective of lifestyle. Furthermore, the response to lifestyle interventions differs markedly from one person to the next, which also reinforces the hypothesis that metabolic and cardiovascular disorders emerge through a complex interaction of genetic and environmental factors. LifeGene provides a unique opportunity to study such interactions in detail with the possibility to address primordial, early-life factors and the role of longitudinal changes for the development of these diseases.

The collection and storage of DNA, serum, plasma and urine allows for a core set of selected biomarkers, of which most are highly relevant for metabolic and cardiovascular disorders, to be analysed in front-end chemistry, and for subsequent analyses of genetic markers, proteins and metabolites in frozen specimens. As HbA1c is introduced for diagnosis of diabetes in many countries, measurement of HbA1c could provide a valuable estimate of the proportion of people with clearly abnormal glucose tolerance in the age group.

Musculoskeletal disorders

Musculoskeletal problems are commonly related to injuries, degenerative changes or from still unknown causes. With impaired functioning and pain as the dominant consequences for the individual, they bring high costs for the society [START_REF] Brooks | The burden of musculoskeletal disease--a global perspective[END_REF].

The high incidence of osteoporotic fractures in Sweden, and other Scandinavian countries, is an observation that cannot readily be explained by known lifestyle or genetic determinants, climate or longevity. Osteoarthritis, back and shoulder pain are common reasons for disabling pain and sick leave. There is also emerging evidence for an interaction between bone and the endocrine system.

LifeGene will provide opportunities to combine metabolic information with information on bone density and fracture risk. LifeGene will examine changes in bone mineral density, bone structure and body composition in a subgroup of individuals. The diagnoses of clinically manifest arthritis can be obtained from Swedish registries, but LifeGene will provide an opportunity to detect pre-clinical biomarkers for the diseases, and to furthermore evaluate the relative importance of genetic susceptibility of arthritis compared to life-style behaviours.

Worldwide, injuries are one of the major causes of death, disability and health care consumption at all ages below 60. Sports, play, traffic accidents and intentional injuries (violence and self-harm) are the most common causes. A special interest will be to follow children from pre-puberty until they reach 25 years of age to evaluate genetic and environmental predictors for peak bone mass. Detailed questionnaire information on causes (event-based) and consequences of the injuries, along with genetic and lifestyle characterisation of individuals at high risk of severe or repeated injuries will be enabled with detailed prospective characterisation of participants at an especially high risk for the development of these diseases.

Neuro-psychiatric disorders and hearing

The etiology of most neuro-psychiatric disorders remains cryptic although extant data suggest that most have roots in childhood. The national health registers capture inpatient admissions relatively well although these constitute a small fraction of population burden. Patient charts from general medical settings seldom include psychiatric diagnoses. In the LifeGene neuro-psychiatric working group, initial focus was thus on which neuro-psychiatric disorders to consider and, second, which might be reasonable to assess with a web-based instrument. Using Swedish data on disabilityadjusted life year estimates obtained from the World Health Organisation (WHO) Global Burden of Disease (15) as a guide, the working group defined a list of disorders that were both important and practical to assess, along with details about the age ranges of assessment and assessment method.

Focus was mainly on lifetime prevalence (not current or interval), and need to capture age of onset, age of offset, and number of episodes. A set of impulse control disorders of clear public health relevance are also assessed (e.g., pathological gambling). Important disorders such as schizophrenia, bipolar disorder, and autism are poorly assessed by the web-based format around which LifeGene will be based, so these will be captured via national registers (i.e., the National Patient Register and the Prescribed Drug Register).

The most common cause of hearing loss is genetic (50-70 %). Studies have shown that carriers of gene deficits also influence the sensitivity to other risk factors, such as occupational exposures.

Hearing loss, and its progression, can be caused and/or modified by many different external (e.g. exposure at work, day care centres and free time) as well as internal risk factors like genetic and life style factors. The prevalence of hearing loss has not been investigated, using audiometric measurements, in a large sample of the general population. Hearing tests in LifeGene would provide robust knowledge whether the increasing prevalence of hearing loss is true and also validate the selfestimated hearing. Inclusion of children and young people in LifeGene would also help answering the question whether hearing loss in the "MP3/I-pod" generation is increasing and, if so, how this is influenced by age and other factors. A prospective study design would also give knowledge of the progression of hearing loss over time, both in the population and in individuals [START_REF] Shargorodsky | Change in prevalence of hearing loss in US adolescents[END_REF].

Physical Activity, Nutrition and Dietary Assessment

The broad objectives of the Physical Activity, Nutrition and Dietary Assessment (PANDA) working group were to investigate the feasibility, validity and reliability of a range of methods for the assessment of diet and physical activity suitable for LifeGene. The results from new (LifeGene sponsored) and existing studies were summarised, and recommendations on how the measurement instruments might be applied in LifeGene were provided. More detail is provided in the pre-pilot section, page 22.

E-epidemiology

E-epidemiology is the science underlying the acquisition, maintenance and application of epidemiological knowledge and information using digital media such as the Internet, mobile phones, digital paper, and digital TV. E-epidemiology also refers to the large-scale science that will increasingly be conducted through distributed global collaborations enabled by the Internet [START_REF] Ekman | New times, new needs; e-epidemiology[END_REF]. A critical condition in introducing the Internet and other communication technologies in populationbased studies is the degree of access to the techniques among the population. In 2008, up to 84% of the Swedish population used the Internet regularly according to Statistics Sweden. For all age groups, using these tools is more common among men than women, but these differences are decreasing year to year. A web-based application for collection of data is easy and inexpensive to construct and maintain [START_REF] Ekman | Feasibility of using web-based questionnaires in large population-based epidemiological studies[END_REF][START_REF] Ekman | Optimizing the design of webbased questionnaires--experience from a population-based study among 50,000 women[END_REF][START_REF] Bexelius | Evaluation of an internet-based hearing test--comparison with established methods for detection of hearing loss[END_REF]. Web technologies also bring possibilities for enhanced data collection not possible through the traditional approaches, including real-time data collection, interactivity, tailored and personalised questionnaires and repeated measures. Hence, a key feature of LifeGene will be exploitation of modern digital tools.

Sample design and participants

Baseline study

Based on the working groups' recommendations, suggested sample design and recruitment of participants was set up. In the sample design finally selected, index persons (aged 18-45 years) will be randomly sampled from the general population, with oversampling of twins from the Swedish twin registry who have recently been screened with an instrument similar to the LifeGene questionnaire. Participants will be invited to include their household members (partner and any children). By collecting information on households, shared environmental risks and exposures can be quantified. Recruiting children living in the shared household introduces information on persons who share environment plus varying degrees of relatedness, as well as contribute to the life course information 0-45 years. There will also be provision for spontaneous sign-up, to improve power and include participants in a cost-efficient way.

The planned cohort size of 500,000 participants is the same as recommended by an expert panel assembled by the US National Human Genome Research Institute in 2004 [START_REF] Collins | The case for a US prospective cohort study of genes and environment[END_REF]. For a moderately rare disease (yearly incidence of 50-100 cases per 100,000) and a follow-up of five years, this is expected to generate a sufficient number of cases to detect a gene-environment interaction with an odds-ratio of 2.5-3 with 80% probability at a conservative significance level of =0.0001 (to protect against false positives). This holds under fairly conventional assumptions about allele frequencies, mode of inheritance and prevalence of environmental exposures [START_REF] Burton | Size matters: just how big is BIG?: Quantifying realistic sample size requirements for human genome epidemiology[END_REF].

Annual follow-ups

The study participations will be prompted annually to respond to a short, web-based questionnaire for updates on changes in household composition, changes in symptoms, injuries and pregnancy.

Event-based sampling

The LifeGene study design allows event-based sampling of specific populations. In the early phases of LifeGene, three types of event-based sampling will be included.

1) The LifeGene-Influensa Like Illness (ILI) study addressing influenza in event-based sampling from volunteer participants was launched during the LifeGene pilot in September 2009. The LifeGene infrastructure and possibilities to sample in response to events was planned before the H1N1 viral epidemic emerged. Event-based reports in the LifeGene-ILI allowed sampling prior to disease onset (and vaccination), a household sampling triggered by a positive H1N1 or Corona virus diagnosis confirmed by PCR and followed by a post-seasonal sampling. DNA, serum and viable peripheral mononuclear cells were stored for further analysis. Other tests gauge the immune response directed against the current H1N1 strains as well as previous influenza pathogens.

2) "Born into Cohort":

The LifeGene study design includes recruitment of individuals who may become pregnant during the first 6 years of baseline data collection. Thus, LifeGene plans to contact pregnant women (and their partners) with additional web-based questionnaire and biosamples during pregnancy and delivery. With a target of 500,000 individuals in the cohort, a simulation indicates that approximately 21,000 children will be born into the cohort and followed prospectively.

3) Injuries: During the annual follow-ups of LifeGene, participants will be asked about injuries during the past year. Positive response to these items will generate a more in-depth assessment of causes and consequences of these injuries.

Measures and study procedures

Based on the recommendations of the working groups, LifeGene was designed to include collection of data through web-based questionnaires and in person testing to collect biosamples and a range of physical tests, Figure 1. In preparation for the LifeGene pilot, a protocol for web-based questionnaires and in-person testing (biosamples and physical examinations) was developed. Names and addresses of potential index participants will be randomly sampled from the general population, and an invitation letter with personal log-in information sent out. After agreement and consent on the LifeGene webpage, they will be able to respond to the LifeGene survey, book an appointment at the test centre for in-person testing and invite household members to participate. After the first invitation letter, up to three reminders will be sent out. Questions that potential participants may have can be directed to the toll-free number of the LifeGene Support Centre, or via the internet.

Questionnaires

The LifeGene core survey is designed to collect information about the physical, mental and social well-being in the Swedish population, and LifeGene therefore collects data on the diseases and disorders most common in Sweden. Extensive sections of the survey are devoted to assessment of exposures that may be relevant for these outcomes. All survey data are collected through a web portal.

The LifeGene survey is based on a library that distributes relevant questions to the LifeGene participants through a web portal. There are three main categories of study participants entering the web portal: adults aged 18 to 45 years (index persons) or older, children invited by index persons and the parents to these children. The adult library holds approximately 1,350 questions and the child / parent library approximately 1,150 questions.

The questions are available to the study participants through a web portal, showing a circular clocklike menu with questionnaire themes on the dial. Nine themes are shown to adults: Lifestyle, Selfcare, Woman's health, Living habits, Health history, Asthma and allergy, Injuries, Mental health and Sociodemography, and between four to nine themes to the partners and children (Figure 2). Parents answer for their children aged 0 to 14 and children answer for themselves from 11 and up, which means that there are parallel questions to children and parents between the ages of 11 and 14.

In-person testing

At the test centres, the study participants are examined for weight, height, waist, hip and chest circumference, bioimpedance, heart rate and blood pressure, along with audiometry and spirometry, with adaptations in individuals above or below 7 years of age. Blood and urine samples are also taken at the test centre for analysis and bio-banking. Table 1 gives details on in-person testing in the LifeGene pilot. In adults, EDTA whole blood for DNA and front-end chemistry will be obtained, along with tubes of EDTA with gel, citrate, lithium heparin and trace metals. In individuals 7 -18 years old, only tubes with EDTA for DNA, EDTA with gel, lithium heparin, trace metals and urinary samples will be collected, and for those under 7 years only EDTA whole blood for DNA and a urinary sample. All samples will be aliquoted for subsequent processing and storage. DNA will be extracted concurrent with other sample processing initially, to be available for subsequent genotyping. The extent of genotyping and choice of platform will be dictated by researcher interest and methodological developments.

Infrastructure

Biobanking

The biological samples collected during in-person testing will be transported in a refrigerated chain with temperature monitors to a biobank facility where primary sample tubes will be processed and stored. The ISO accredited Biobank at Karolinska Institutet is currently establishing new and costeffective automated processes to allow for novel high throughput techniques. The input processes include registration, volume measurement, sorting with respect to sample type, centrifugation, de-capping and aliquoting of multiple 225 µl plasma and urine fractions into 2D bar-coded tubes and DNA extraction from 400 l EDTA blood. Planned processes also include RNA extraction, cell isolation, and front-end chemistry based on blood and urine. The storage solution consist of three elements:

(1) a farm of liquid nitrogen tanks housing tens of millions of samples (2) one or several robotic repicking stations to retrieve desired samples and (3) in the future a large-scale automatic freezer.

The analysis of samples for research takes place after retrieval from storage. Through stringently standardized sampling protocols and modern biobanking, LifeGene will provide an opportunity to evaluate the full ´omics´ -set including proteomics, metabolomics, genomics and epigenomics.

Researchers interested in the LifeGene materials will be offered to use external quality-approved analytical centres. Some analysis platforms in high demand will be set up in-house.

IT structure

For LifeGene IT structure, it will be important to provide a long term platform for data collection and withdrawal. We have developed an IT infrastructure and functionality based on process analysis applying modular, rather than single, integration solution approach, built on proven industrial solutions. Integration has been done using web services based on Microsoft.Net and all research data for collection are stored in a central repository with no sensitive data stored at LifeGene test centres.

Biostatistics

The household based longitudinal design of LifeGene allows great flexibility in the type of research questions that can be addressed. It was emphasised that classical concerns of observational epidemiology, such as bias due to selection, confounding and reverse causation are a reality, as well as issues of incomplete data due to non-compliance, drop-out and measurement errors.

For many studies based on LifeGene it may prove useful also to draw on modern statistical methods for observational data, as discussed by Hernán and Robins [START_REF] Hernán | Observational studies analyzed like randomized experiments: Best of both worlds[END_REF]. They set up a framework for causal modelling, which mimics the rationale for controlled randomised trials, and which can disentangle complex dependencies between time varying exposures and confounders. Referring to the rationale in Prentice et al [START_REF] Prentice | Statistical issues arising in the Women's Health Initiative[END_REF] for the Women's Health Initiative, it is also worth paying attention to the systematic and random errors in the assessment of exposures, in particular assessment of dietary intake and physical activity, and the implications that these measurement errors may have on the estimates of associations between exposure and outcome.

Special design, quality control and analysis issues arise when dealing with high dimensional multiomics data. These may provide biomarker profiles for early detection or a basis for disease classification, prognosis and treatment prediction. In particular, metabolomics could meet nutritional epidemiology in the search for novel exposure biomarkers that could calibrate traditional measurements for diet and physical activity. Mendelian randomisation could be a useful tool to infer causal relationship between a modifiable exposure and disease [START_REF] Smith | Mendelian randomization': can genetic epidemiology contribute to understanding environmental determinants of disease?[END_REF].

In large randomised controlled trials, it is common practice to augment the original study design with observational components. Conversely, randomised components of lifestyle interventions could be superimposed on the LifeGene observational cohort [START_REF] Ioannidis | Nested randomized trials in large cohorts and biobanks: studying the health effects of lifestyle factors[END_REF]. This would provide analytic advantages for the study of gene-lifestyle interactions, and such "multi-life" interventions would allow volunteer participants to take more active part in the LifeGene assembly of data.

Results

In preparation for the pilot, a number of pre-pilots were performed.

Pre-pilots

1) Physical activity and dietary assessment methods were validated in pregnant and non-pregnant women (aged 20-35 yrs). A physical activity assessment feasibility study was also conducted in young mothers and their four month old babies. The studies compared estimates of physical activity energy expenditure derived from three activity monitors with validation obtained from the standard criterion method of doubly labelled water (DLW).

2) A meal-based food frequency questionnaire (Meal-Q) was developed for the web-based questionnaire and validated against DLW, nutrient-related biomarkers and/or seven days of food records in different age groups. Due to limitations on numbers of food items that can be assessed in a short time, it is important to adapt the method to diets currently consumed by adults, teens, adolescents, children and infants. The selection of food items to be included in Meal-Q was therefore based on age-specific studies using 24 hour recall or food records. Physical activity questionnaires (Active-Q) were developed that were similarly adapted to the usual domains of leisure and occupational activity as well as regular transportations in the various age groups, and validated using DLW, accelerometers and/or seven days records of activities in different age groups.

3) In another LifeGene pre-pilot, dust samples were collected to assess the stability of allergens and endotoxins over time and the variability in allergen content related to fresh, frozen and thawed extracts.

LifeGene Pilot

Based on the recommendations of the working groups, we performed a full "dress-rehearsal" pilot of 1% of the target population, i.e. 5,000 tested individuals. Based on the Swedish population, the proposed pilot size allows us to estimate the response rate in each age and sex stratum within 5% with 95% probability. The testing sites were chosen to reflect various combinations of geographical location, population density, and association with a university. Thus, we chose a large city (Stockholm, capital of Sweden with almost 2 million inhabitants), a university city (Umeå, in the north with 111,000 inhabitants) and a smaller town (Alingsås, in the west with 36,000 inhabitants). The LifeGene pilot study was launched in Stockholm in October, 2009, followed by the Umeå pilot in late November 2009 and Alingsås in January 2010, and all testing was stopped on March 31, 2010. The testing schedule was designed to generate the number of participants at the proposed testing centres that would stretch the limits of the biobanking facility's ability to process samples within specified parameters and fully test logistics regarding chilled transport. Figure 3 shows the expected age distribution of participants in the pilot.

Influenza pilot

The LifeGene study design allows event-based sampling. An exercise in this context is the LifeGene pilot addressing 'Influenza-like-illnesses', ILI. This study was planned before the H1N1 viral epidemic emerged. The LifeGene-ILI pilot study included sampling of 2,000 adults prior to disease onset (and vaccination), mechanisms for reporting influenza-like symptoms and providing nasal swabs for viral analysis, household sampling triggered by a positive H1N1 diagnosis confirmed by PCR and followed by a post-seasonal sampling. DNA, serum, plasma and viable peripheral mononuclear cells are stored for further analysis; other tests gauge the immune response directed against the current H1N1 strains, as well previous influenza pathogens.

The pre-pilots were concluded during the spring of 2009, the actual pilot was stopped by design at the end of March, 2010 and is currently being evaluated. Recruitment for the main study will start November, 2010 with a gradual expansion of testing centres in the remaining four cities with major universities followed by selected locations in smaller towns, and is planned to take six years.

Discussion

LifeGene is a prospective cohort study with the aim to combine advances in modern biotechnology and information on individuals' health and lifestyle. A comprehensive baseline questionnaire designed to accommodate cutting-edge research questions, state of the art biosampling, repeated follow-ups including event-based sampling are key features of this ambitious initiative. The wide array of high-tech tools encompassed by e-epidemiology will facilitate the longitudinal aspects of the study. Recruitment of index people 18-45 years old, who are invited to include their adult household members and children, will increase the opportunity to involve young couples prior to and during pregnancy, allowing for the first study of children born into a cohort with complete pre-and perinatal data from both mother and father. Furthermore, by focusing on index individuals in young and mid-adulthood in a household structure, we will be able to assess exposures and disease progression for a myriad of disorders that have considerable consequences not only for health care demand, but also for future development of chronic diseases.

Large population-based studies with repeated collection of questionnaire data and sampling over time are extremely costly and the funding has to be well invested. Given the numerous targets that have already been identified through other studies throughout the world, care has to be taken when designing and planning yet another large-scale study. Future studies have to consider carefully whether it is most cost-effective to monitor large populations in prospective cohort studies or undertake well-designed case-control studies on a particular exposure or disease [START_REF] Collins | The case for a US prospective cohort study of genes and environment[END_REF]. It has been argued that population-based cohorts are necessary to account for methodological challenges such as standardisation and generalisation, to allow for new methodologies and to include younger age groups [START_REF] Collins | Merging and emerging cohorts: necessary but not sufficient[END_REF]. Others suggest that initiating new large population-based cohorts are not worth the wait and should be replaced by merging ongoing smaller cohort studies [START_REF] Willett | Merging and emerging cohorts: not worth the wait[END_REF]. In this context, LifeGene has been designed to address a wide variety of research questions, including the possibility of detecting pre-clinical markers for disease, event-based sampling and a cohort of children born into the study.

Tables and Figures

Figure 1 .

 1 Figure 1. Flow chart on the structure of the study. Questionnaires (Q), parental questionnaires (pQ) and In Person Testing (IPT) in adults, adolescents and children.

  Figure 1. Flow chart on the structure of the study. Questionnaires (Q), parental questionnaires (pQ) and In Person Testing (IPT) in adults, adolescents and children.

Figure 2 .

 2 Figure 2. Questionnaire modules in adults (a) and children (b) a)

Figure 3 .

 3 Figure 3. Population distribution of the LifeGene pilot at the end of the recruitment period. Red line: median over 100 simulations, dark grey are: 95% for 100 simulations.

Table 1 .

 1 Biosamples and physical measurements collected during the pilot

	Biosamples	Purpose	Adults	7-18 yrs	<7 yrs
	EDTA whole blood	DNA	x	x	x
	EDTA whole blood	WBC+diff/HbA1c	x
	LiHep with gel	Frontend chemistry	x
	EDTA with gel	Aliquots	x	x
	EDTA with gel	Aliquots	x
	Citrate 3.8%	Aliquots	x
	LiHep with gel	Aliquots	x	x
	Trace metal	Aliquots	x	x	x
	Urine	Aliquots	x	x	x
	Physical measures	>7 yrs	6 yrs	5 yrs	<5 yrs
	Height	x	x	x	x
	Weight	x	x	x	x
	Bioimpedance	x
	Waist circumference	x	x	x	x
	Hip circumference	x	x	x	x
	Thorax circumference	x	x
	Heart rate	x	x	x
	Blood pressure	x	x	x
	Audiometry	x	x
	Spirometry	x	x

Acknowledgment

The authors would like to gratefully acknowledge all the scientists who have contributed with their time and expertise, without which this study would not have been possible to prepare. Their names and affiliations are listed on the LifeGene website: www.LifeGene.se. Start-up funding has been received from Karolinska Institutet, the Stockholm County Council and the Swedish Research Council.

Funding has also been obtained from the Torsten and Ragnar Söderbergs Foundation and AFA Försäkringar.

Ethics synopsis

A document on LifeGene ethics policy including details on recruitment, consent, data and sample access and governance has been developed. Further details on the ethics document can be found at www.lifegene.se/For-Scientists1/Ethics/ . Permission for the pilot study was obtained from the Regional Ethical Review Board of Stockholm, Sweden. Researchers requesting to use the data and/or samples will be required to apply for Ethics approval. An impartial data access committee will vet requests for access to de-identified data and samples.

We would also like to take this opportunity to invite the scientific community to participate in LifeGene, by proposing sub-studies and possible add-ons. The LifeGene project is a national population-based study of global importance.

Website

www.LifeGene.se


Greqam

Karine Gente

Carine Nourry

Integration, real exchange rate and growth

Keywords: Two-sector model, OLG model, real exchange rate, economic growth. Journal of Economic Literature Classification Numbers: E21, F15, F43, O41

This paper deals with the relationship between real exchange rate and growth in the process of economic integration. Using a 2x2x2 model of overlapping generations, we show that growth depends on the real exchange rate (RER) through human capital accumulation. Integration leads to convergence in growth rates only in presence of cross-border externalities in human capital. Otherwise, divergence is likely to occur and integration may be good (bad) for growth if the integrated RER is higher (lower) than the autarky's RER. In reality, since capital mobility prevents the real exchange rate from adjusting, economic integration may lead to income divergence if countries are too different in terms of preference, altruism or productivity.

Introduction

How does economic integration affect growth? This article addresses this question theoretically developing a 2-country 2-sector endogenous growth model with education. This framework allows to consider the relationship between real exchange rate (RER) and growth which is specially relevant in the process of european economic integration since RER still move even between eurozone members [START_REF] Berka | What determines european real exchange rates[END_REF]). In a recent work, [START_REF] Galor | Trading population for productivity: theory and evidence[END_REF] highlight the influence of international trade on human capital. They show that trade exerts opposite effects in developed and developing countries. Education increases (decreases) in OECD (non-OECD) countries. Economic integration goes beyond trade integration allowing capital mobility between European countries. The model we develop highlights consequences of this capital mobility in a two-sector setting. We assume there are two sectors in the economy: a traded one and a nontraded one. The way capital mobility affect human capital accumulation depends on the traded Total Factor Productivity (TFP) gap between countries. This TFP gap reflects differences in relative prices of nontraded to traded goods which can be considered as an internal real exchange rate. We find that countries whose RER appreciates (depreciates) with integration accumulate more (less) human capital and experience higher (lower) economic growth than in autarky. In a general way, capital mobility prevents the RER from freely adjusting and may be growth damaging.

The literature on capital mobility, exchange rate and growth is not yet well documented. [START_REF] Aghion | Exchange Rate Volatility and Productivity Growth: The Role of Financial Development[END_REF] show that exchange rate volatility matters for growth but especially for countries with relatively low levels of financial development. The literature on export led growth models often shows the benefits for growth in maintaining a low RER. First, Rodrik (2008) concludes that the depreciation is growth enhancing when there exist some economic distorsions in the traded sector. This explanation holds especially for Asian countries in which stages of growth accelerations have followed RER depreciation. Elsewhere, evidence is mixed and growth acceleration does not systematically follow episodes of depreciation. [START_REF] Eichengreen | The Real Exchange Rate and Economic Growth, Commission on Growth and Development[END_REF] concludes that for developing countries -before the beginning of the growth process -depreciation is growth enhancing, driving resources out of the nontraded to the traded sector. The common starting point of these papers is that there exist some distorsions that prevent optimal resources allocation between the traded and nontraded sectors detrimental to the traded sector [START_REF] Krueger | Foreign capital and economic growth[END_REF]]. However, these models do insist on the fact that what is important for growth is to maintain the "correct" RER instead of targeting an artificially low level of the RER1 .

Second, [START_REF] Harris | Is there a case for exchange-rate-induced productivity changes?[END_REF] shows empirically that the US Dollar appreciation against the Canadian Dollar in the 1990's is responsible for the technological spread between US and Canada. His argument is based on the fact that when RER appreciates, firms of the traded sector are forced to invest in R&D to maintain competitiveness despite the fact that their traded prices are higher. There were more incentives to invest for US firms than for Canadian ones. This may explain part of the technological difference. Finally, the relationship between relative prices and relative productivities is positive for developed countries [START_REF] Canzoneri | Relative labor productivity and the RER in the long run: evidence for a panel of OECD countries[END_REF]]. This means that growth episodes in developed countries are frequently associated with real appreciations. This is not the case for developing countries [START_REF] Ito | Economic growth and real exchange rate: an overview of the Balassa-Samuelson hypothesis in Asia[END_REF]].

The model we develop in this paper is a two-sector generalization of the Michel and Vidal (1999) framework. In the Michel and Vidal one-sector model, determinants of growth were patience, altruism and the size of the external effect in human capital accumulation. Introducing a second sector of production, the RER matters for growth as well. The way the RER is determined depends on the assumption concerning capital mobility; so do growth determinants. In autarky, this RER clears the nontraded goods market and as a result depends on the propensity to consume services, the degree of altruism, the saving rate. In autarky, the more services the country consumes, the higher the relative price, the higher the return on education -if nontraded sector is labor intensive-, the higher the growth rate. Conversely, when capital is perfectly mobile between countries, RER no longer depends on demand and supply factors but only depends on TFP gap between traded and nontraded sectors. This derives from the fact that there are as many mobile factors as sectors in this economy. Then, the return on human capital -the wage-is no longer a function of global capital intensity. Instead it only depends on the RER and capital mobility implies that marginal products of capital are the same between countries. The higher traded productivity, the more appreciated the RER [Balassa-Samuelson effect].

Integration leads to convergence in growth rates only in presence of cross-border externalities.

Nevertheless empirical evidence does not suggest that integration in Europe has promoted convergence of growth rates. Indeed, the divergence of growth rates looks higher over the recent period (1991-2005) than it was before economic integration [See Table 1].

When we reject the cross-border externality assumption, integration leads effectively to divergence: integration may be good (bad) for growth if the integrated RER is higher (lower) than the autarky's RER. The intuition of this result is that a higher RER stimulates human capital accumulation. Capital mobility prevents the RER from adjusting and may lead to an inapropriate resource allocation between the two sectors. In this case, it will not be growth enhancing. This result is in line with [START_REF] Eichengreen | The Real Exchange Rate and Economic Growth, Commission on Growth and Development[END_REF] in the sense that the policy recommendation would be to allow the relative price to keep the autarky level which corresponds to the "correct" Per Capita GDP growth rates 1970-1990 1991- The rest of the paper is as follows. Section 2 presents the model. Section 3 deals with autarky whereas Section 4 deals with integration introducing capital mobility between the two countries. Section 5 concludes.

The traded good is a mixed good which can either be consumed or invested. This two-sector production structure is a generalization of the standard two-sector setting in which one good is a pure consumption while the other is a pure investment good [START_REF] Galor | A Two-Sector Overlapping-Generations Model: a Global Characterization of the Dynamical System[END_REF], [START_REF] Venditti | The Two Sector Overlapping Generations Model: a Simple Formulation[END_REF]]. The world consists of two countries which accumulate human capital and experiment endogenous growth.

Production

The representative firm produces in two sectors: the tradable, and the non-tradable one. Production in the tradable (Y T ) and in the non-tradable (Y N ) sector resulting from two Cobb-Douglas production technologies, using two inputs, human capital H, and physical capital K. Let K i and L i , i = T, N , be respectively the quantities of capital and labor used by sector i, production is given by

Y T = A T K α T T H 1-α T T (1)
Y N = A N K α N N H 1-α N N (2)
with α T , α N ∈ (0, 1), A T > 0 and A N > 0.

Investment instantaneously transforms a unit of tradable good into a unit of installed capital: K t+1 = I t and capital fully depreciates after one period. Both inputs are perfectly mobile between the two sectors provided that:

H T + H N ≤ H, K T + K N ≤ K (3)
K being the total stock of physical capital and H the total amount of human capital.

Let k i = Ki/Hi be the capital intensity of sector i, h i = H i /H be the share of human capital allocated to sector i, i = T, N , and k = K/H the physical to human capital ratio. Equations ( 2), ( 3) and ( 5) can be rewritten:

h T + h N ≤ 1, k T h T + k N h N ≤ k (4)
y T = A T k α T T ( 5
)
y N = A N k α N N ( 6
)
where y T and y N are the production per unit of human capital in each sector. Denoting w the wage rate, R the gross rental rate of capital and P N the price of the non tradable good, profit maximization over the two sectors implies that production factors are paid their marginal product:

R t = α T A T k T α T -1 t = P N t α N A N k N α N -1 t (7)
w t = (1 -α T )A T k T α T t = P N t (1 -α N )A N k N α N t (8)
From which we derive the physical to human capital ratios as functions of the price of the non tradable good:

k T t = B(P N t ) 1 α T -α N k N t = α N (1-α T ) α T (1-α N ) B(P N t ) 1 α T -α N with B = α N α T α N α T -α N A N A T 1 α T -α N 1-α T 1-α N α N -1 α T -α N (9)
And thus the input prices are:

w t = (1 -α T )A T B α T P N α T α T -α N t ≡ w(P N t ) R t = α T A T B α T -1 P N α T -1 α T -α N t ≡ R(P N t ) (10)

Consumption, savings and children's education

The economy consists, in each country, of a sequence of three individual life periods.

In the second period of his life, each individual gives birth to 1 + n children so that population grows at rate n. We assume the population growth rate is the same in the two countries. Each generation born in period t consists of N t identical individuals who make decisions concerning consumption, children's education, and savings. During childhood, individuals make no decision: their consumption is included in their parent's consumption. They are reared by their parents who decide on their level of educational attainment. When adult, they work and receive the market wage, consume, save, and rear their own children. When old they retire, and consume the proceeds of their savings.

Individuals care about their children's education. They exhibit a kind of paternalistic altruism whereby they value their child's human capital. Our modeling of intergenerational altruism follows Glomm and Ravikumar (1992) who assume that the parental bequest is the quality of education received by their children. The preferences of an individual belonging to generation t are represented by:

U (c t , d t+1 , h t+1 ) = (1 -β)lnc t + βlnd t+1 + γlnh t+1 (11)
where c t , d t+1 and h t+1 are respectively consumption when adult, consumption when old, and the child's human capital; β ∈]0, 1[ denotes individuals' thrift and γ is the altruism factor. When adult, each agent born at t supplies inelastically h t+1 units of efficient labor. The level of human capital of each adult depends on his parent's decision on education during his childhood:

h t+1 = b t e a t ( 12
)
where b t is an externality, e t the amount of resources a parent devotes to his child's education, and a ∈]0, 1[ the elasticity of the technology of human capital formation. Let x = c, d denote individual consumption at each period of life, x N and x T be respectively the spending allocated to nontraded and traded goods. Instantaneous preferences over the two goods are defined according to:

x = x µ T x 1-µ N ( 13
)
with µ ∈ (0, 1). We denote π the consumer price index in terms of traded good. Adults distribute their earnings that consist of labor income, w t h t , among own consumption spending, investment in child's education, and savings, s t ,

w t h t = π t c t + e t + s t (14)
When old, individuals retire and consume the proceeds of their savings:

R t+1 s t = π t+1 d t+1 (15)
An individual born in period t -1 is endowed with h t units of human capital at the beginning of adulthood, and chooses e t and s t so as to maximize his life-cycle utility [START_REF] Krueger | Foreign capital and economic growth[END_REF] under his budget constraints ( 12), ( 14) and ( 15). An individual's optimal choice is characterized by the first order conditions:

- 1 -β π t c t + βR t+1 π t+1 d t+1 = 0 (16) - 1 -β π t c t + γa e t = 0 (17)
c T t = µπ t c t P N t c N t = (1 -µ)π t c t π = φ (µ) ≡ µ -µ (1 -µ) -(1-µ) (18)
Equation ( 16) characterizes the optimal allocation of consumption for an individual over his lifetime. Equation (17) gives the optimal investment in the offspring's human capital. An adult reduces his consumption spending until his loss equates the increment in the utility he derives from his child's level of human capital out of altruism. Equations (18) give the static allocation of consumption spending between the two goods. Plugging ( 14) and (15) into ( 16) and (17) yields:

s t = β 1 + γa w t h t ( 19
)
e t = γa 1 + γa w t h t (20)
As usual in overlapping generation models with paternalistic altruism, savings increase with individual's thrift and decrease with altruism. The more altruistic parents are, the more they invest in their offspring's education.

Cross-border external effects in human capital

Throughout the analysis, foreign variables are denoted by an asterisk. We assume cross-border externalities in human capital formation. An individual's investment in his child's human capital generates a positive externality for his country's fellows.

We assume an externality of the form:

b t = b(pe t + p * e * t ) λ e t 1-a-λ and b * t = b(pe t + p * e * t ) λ e * t 1-a-λ (21)
where b > 0, λ ∈ [0, 1 -a], p = N/(N + N * ) and p * = 1 -p. Since population grows at the same rate in the two countries, p and p * , the shares of each country in the world population, are constant. We denote respectively e t and e t * the average levels of investment in children's human capital in the home and the foreign country. Since individuals are identical within each country, in equilibrium: e t = e t and e * t = e t * . The magnitude of these crossborder external effects is given by λ. The term (pe t + p * e t * ) λ is intended to capture the strength of international spillover of knowledge. The higher λ, the more the home country benefits from the foreign country's private expenditures in education.

In equilibrium, human capital depends both on domestic and foreign investment in education and on cross-border externality in human capital formation:

h t+1 = b t e t a = b(pe t + p * e * t ) λ e 1-λ t ( 22
)
Let ρ t = e * t /e t be the ratio of foreign over home average investment in children's human capital and g t = e t /e t-1 -1 the economy growth rate. Using equations (20), (22) and finally [START_REF] Ito | Economic growth and real exchange rate: an overview of the Balassa-Samuelson hypothesis in Asia[END_REF], we obtain:

1 + g t = γab 1 + γa (1 -α T )A T k T t α T (p + p * ρ t-1 ) λ ≡ G t (23)

The nontradable market clearing condition

Since there exists a nontraded good, we should consider a market clearing condition for that good:

P N t Y N t = N t P N t c N t + N t-1 P N t d N t (24)
This equation simply states that production equals total consumption in nontraded goods. We can rewrite this condition with only wage, interest factor and physical to human capital ratios:

Lemma 1 The home country non tradable market clearing condition can be written

1 -µ 1 + γa ((1 -β)w t + βR t 1 + n 1 + γa γab(p + p * ρ * t-1 ) λ ) = P N t A N Dk T α N -1 (k t -k T t ) (25) With D = (α N (1-α T )) α N (α T (1-α N )) 1-α N α N -α T . Proof: see Appendix 7.1.
It can be noted that expression D is the same for both countries as we assume home and foreign technologies have identical elasticities of substitution between production factors.

Autarky

As we first consider autarky, we rule out any interactions between countries. Investments in human capital in one country do not result in an external effect that enhances the formation of human capital in the other (λ = 0). The human capital externality depends only on the average level of education. From equation ( 21), we have with λ = 0: b t = be t 1-a . From equation ( 22), since individuals are identical, social returns on human capital investment are constant in equilibrium h t+1 = be t . Young people's savings finance the following period's physical capital:

K t+1 = H t+1 k t+1 = N t s t (26)
The labor market clears:

H t = N t h t (27)
Combining ( 19),(20), ( 22), ( 26) and ( 27), we obtain the constant equilibrium physical to human capital ratio:

k ≡ k A = β γab(1 + n) (28)
We then compute the equilibrium autarkic growth rate g A .

Lemma 2

The autarkic growth factor is:

1 + g A = γab 1 + γa (1 -α T )A T k T α T (29) with 2 k T = k A α T 1 -α T 1 -µα T -(1 -µ)α N 1-µ 1+γa (1 -β)(α N -α T ) + α T (30)
Proof: see Appendix 7.2.

The following proposition contains some comparative statics results relating the longrun growth rate to preference parameters.

Proposition 1 The more patient individuals are, the higher the growth rate. The growth rate is first increasing and then decreasing with γ reaching a maximum in

γ = 1-2α T + √ 1+4((1-µ)(1-β)(α N -α T )) 2aα T . Moreover, if α T > α N (respectively α N > α T )
, the growth rate is decreasing (respectively increasing) in µ.

Proof: see Appendix 7.3. The more altruistic individuals are, the higher their investment in children's education and the lower their consumption. We obtain, as in Michel and Vidal (1999), that excessive as well as weak altruism can lead to poor growth records. When the tradable sector is capital intensive (α T > α N ), the growth rate decreases with the preference for tradable goods (µ). This is a consequence of the Stolper-Samuelson Theorem. An increase in the propensity to consume the traded good (µ) leads to an RER depreciation. Since the nontraded good is labor intensive, the real depreciation entails a fall in the wage. Then, the return to human capital decreases and so does the growth rate. Hereafter, to fit empirical evidence [START_REF] Ito | Economic growth and real exchange rate: an overview of the Balassa-Samuelson hypothesis in Asia[END_REF]], we consider:

Assumption 1 α N < α T .

Economic integration and growth

We consider a two-country overlapping generations world in which countries differ in both levels of patience and altruism. We establish the growth implications of world economic integration. In the integrated economy, as we assume no labor mobility between the two countries, the labor market clearing condition of the domestic country is given as in autarky by equation (27). Equation (25) gives the non tradable market clearing conditions for the home country. The foreign country equations are obtained if we denote by * foreign variables. In a two-country integrated world, there are capital flows between countries and the equality between domestic savings and domestic investment -equation (26)-no longer holds. The amount saved by adults in the home and the foreign country in period t is equal to the physical capital available as productive input in period t + 1 :

K t+1 + K * t+1 = N t s t + N * t s * t ( 31
)
Dividing by the world population, the world capital market clearing condition is:

(1 + n) pk t+1 h t+1 + (1 -p)k * t+1 h * t+1 = ps t + (1 -p)s * t (32)

Convergence of countries' growth rates

With perfect capital mobility, the interest rate is the same for both countries:

R t = R * t (33)
Using [START_REF] Ito | Economic growth and real exchange rate: an overview of the Balassa-Samuelson hypothesis in Asia[END_REF], we can determine the ratio between domestic and foreign relative prices:

P N t P * Nt = A * N A N A T A * T 1-α N 1-α T (34)
This ratio reflects the bilateral real exchange rate between these two countries. Using equation ( 20) we can compute the ratio of foreign over home average investment in children's human capital:

ρ t+1 = e * t+1 e t+1 = γ * γ 1 + γa 1 + γ * a w * t+1 h * t+1 w t+1 h t+1 (35)
Integrating human capital accumulation from equation ( 22), we have:

ρ t+1 = γ * γ 1 + γa 1 + γ * a w * t+1 w t+1 ρ t 1-λ (36)
Using expression of wages as functions of P N and P * N , from equation [START_REF] Ito | Economic growth and real exchange rate: an overview of the Balassa-Samuelson hypothesis in Asia[END_REF], we obtain:

ρ t+1 = γ * γ 1 + γa 1 + γ * a P * N t+1 P N t+1 α T α T -α N A * N A N α T α T -α N A T A * T α N α T -α N ρ t 1-λ (37)
Integrating equation (34), we can then determine the foreign relative to home growth on the steady path:

ρ t+1 = γ * γ 1 + γa 1 + γ * a A * T A T 1 1-α T ρ t 1-λ (38)
To understand the importance of cross-border externalities, we consider first the case without cross-border external effects (λ = 0), which corresponds to the case where countries can exchange capital and goods but the education level in one country does not affect the human capital formation in the other country. Second, we focus on the case with cross-border external effects.

Divergence without cross-border external effects

Economic integration without cross-border external effects corresponds to the case where λ = 0 and thus b t = be 1-a t as in autarky. Equation (38) becomes:

ρ t = 1 + g * t 1 + g t ρ t-1 (39)
With

1 + g * t 1 + g t = γ * γ 1 + γa 1 + γ * a A * T A T 1 1-α T (40) Convergence of growth rates means that (1 + g * t )/(1 + g t ) = 1.
If the altruism factor is the same in both countries γ = γ * , convergence is obtained if and only if A * T = A T . Thus if countries differ with respect to their total factor productivity (TFP), convergence is no longer possible. If altruism factors differ, convergence means:

γ * γ 1 + γa 1 + γ * a A * T A T 1 1-α T = 1 (41)
We have thus proved a first result concerning convergence of growth rates :

Proposition 2 Without cross-border external effects, for identical altruism factors (γ = γ * ), if there exists a gap between traded TFP A * T = A T , perfect capital mobility does not result in convergence in growth rates. Convergence in growth rates may occur for very special altruism factors only, namely 3 γ = Γ(γ * ) with

Γ(γ * ) = 1 γ * + a A T A * T 1 1-α T -a -1 .
We may notice that if (1 + g * t )/(1 + g t ) < 1, then lim t→∞ ρ t = 0 meaning that the foreign education level e * becomes in the long run very small compared to the home education level e. Symmetrically, if (1 + g * t )/(1 + g t ) > 1, then lim t→∞ ρ t = +∞, meaning that e * becomes much higher than e. In the literature, the one-sector model concludes that integration systematically promotes convergence of growth rates when both countries have the same degree of altruism. This is no longer the case in a two-sector model. Introducing a 2-sector setting, we show here that what matters for economic growth is not only the altruism factors but also the traded productivity spread between the home and the foreign country which determines the RER through equation (34). The higher productivity in the traded sector, the higher the growth rate. This is an illustration of the Stolper-Samuelson theorem when the traded sector is capital intensive. Indeed, the RER affects wages and then education and growth. A higher RER increases return on human capital and stimulates education. The proposition states that if both countries have the same traded total factor productivity (TFP), the more altruistic the country, the higher the growth rate in an integrated world. Assume that there is a technological gap between countries and let the foreign country be the technologically "advanced" country. The foreign country will have the highest growth rate except if the domestic country has a sufficient degree of altruism (γ > Γ (γ * )). Altruism can

3 if A T ≥ A * T or γ * < a A * T A T 1 1-α T -1 -1
then γ is positive.

be interpreted in terms of education spending. An "advanced" country continues to have the highest growth except if the other country invests enough in education to catch up on this technological gap.

Convergence with cross-border external effects

Introducing cross-border external effects, each country can benefit from a higher level of education in the other country. Using equation (38) with 0 < λ < 1 -a, there exist two steady state values for ρ. The trivial long run value ρ = 0, and a positive one ρ :

ρ = γ * γ 1 + γa 1 + γ * a A * T A T 1 1-α T 1 λ (42)
As λ ∈ (0, 1), the sequence of {ρ t } tends towards ρ and growth rates converge. The result is true for any value of altruism degrees, and especially with same altruism factors (γ = γ * ). We have thus proved: Proposition 3 With cross-border external effects, perfect capital mobility results in convergence in growth rates for any possible altruism factors.

Thus the existence of cross-country external effects in human capital guarantees convergence of growth rates. Nevertheless, empirical evidence suggests that such external effects are not observed. As a result, cross border external effects are generally absent in models where labor is not mobile between countries. In line with this literature, we focus on the realistic case without cross-border effects.

Growth rates and integration without cross-border effects

Without cross-border effect, i.e. when λ = 0, Proposition 2 states that convergence in growth rates never occurs except for a very special ratio of altruism degrees. As a result, if there is a spread between technologies, the divergence in growth rates may last forever.

We now consider the world allocation of savings in the case without cross-border externalities. Denoting x ≡ β * γ/βγ * then, from equation (28), k * A = xk A . In this two-country two-sector model, only one good can be traded. Nontraded good is exclusively produced domestically. In some extreme cases, when the propensity to consume and/or the preference for non traded good are high, the representative firm tends to produce mainly nontraded goods, the traded good being imported. As we consider a human capital intensive non-traded production, to guarantee physical capital accumulation, we have to assume:

Assumption 2 -ζ 1-ζ < xη η * < 1-ζ ζ .
We obtain the following lemma which provides a simple expression of the world capital accumulation equation, and expressions of the physical to human capital ratios:

Lemma 3 In an integrated world, the international capital market clearing condition is:

pk t+1 + (1 -p)k * t+1 ρ t = pk A + (1 -p)k * A ρ t (43)
and, under assumption 2 the physical to human capital ratios are:

k t = ηP N 1 α T -α N t + ζk A ( 44
)
k * t = η * P N t 1 α T -α N + ζxk A (45)
The domestic price of the nontraded good is:

P N 1 α T -α N t+1 = k A (ρ t (1 -p)x + p) (1 -ζ) ρ t (1 -p)η * + pη (46)
with η > 0, η * > 0 and ζ of the sign of α N -α T . 4Proof: see Appendix 7.4.

From equation ( 46), the price of the nontraded good P N t+1 depends on time only through ρ t . But equations ( 39) and (40) show that education levels between countries diverge with ρ t tends to zero or infinity. We focus on the case of divergence, ρ t → 0, where education in the domestic country becomes very large with respect to education in the foreign country. The domestic growth rate 1 + g increases with k T through equation ( 23). Since k T is increasing with the global physical to human capital ratio k through the price of the nontraded good5 P N , then, 1 + g is increasing with k. By symmetry, 1 + g * is increasing with k * . We finally state:

Proposition 4 Under Assumption 1 and 2, the home country converges to its autarkic physical to human capital ratio, and growth rates diverge iff γ > Γ(γ * ) (Condition 1). Perfect capital mobility then results in a decrease (increase) in the foreign country growth rate iff xη η * < 1 (>) (Condition 2).

Proof: see Appendix 7.5.

Let us define the "leading" country as the country with the higher education spending and/or the higher traded productivity; this leading country is the home country if γ > Γ (γ * ). The proposition states on one hand that the growth rate with integration corresponds to the autarky growth rate of the "leading" country. On the other hand, integration may be growth reducing for the foreign country.

To sum up, with a sufficient degree of altruism (education spending), integration does not change the home country's growth rate whereas consequences of integration are mixed in the foreign country. If A * T /A T and/or β * /β are high (low) enough 6 , then the foreign country's growth rate is higher (lower) with integration.

This proposition means that integration may have dramatic consequences if the two countries are not similar enough in terms of productivity, altruism and saving rates. Indeed, it may lead to a growth reduction for the lowest traded TFP countryand/or the less altruistic country. This is a typically two-sector (2x2) setting result. In such a model, growth depends on the level of the RER. In autarky, this RER is directly linked to altruism, savings and nontraded consumption. Integration involves perfect capital mobility between countries. The 2x2 model becomes a 2x2x2 model. Capital mobility implies that relative prices no longer depend either on domestic savings, altruism or propensity to consume. Instead, relative prices only depend on relative traded and nontraded productivity through equation ( 34). Let us consider that β = β * and γ = γ * . Conditions (1) and ( 2) become A * T < A T . In this case, Proposition 4 simply states that the growth rate with integration is the autarky growth rate of the home country which is the leading one in terms of productivity -since we have assumed identical altruism degrees-and that economic growth in the foreign country is worst in the case of integration.

In autarky, factor returns and growth rate are lower in the foreign country :

w A /w * A = R A /R * A = P A N /P * A N = 1 + g A / 1 + g * A = A T /A * T > 1.
Capital mobility changes the way relative prices are determined: relative prices then result from equation (34). Relative prices depend only on relative traded TFP. Since A T > A * T , we still have P N > P * N from equation (34) but the ratio between these two prices is higher 7 :

P N /P * N = (A T /A * T ) (1-α N )/(1-α T )
. This means that P * N < P A * N , w * < w A * and the foreign country's growth rate is even lower in an integrated world. The intuition is as follows. With capital mobility, the domestic and foreign returns on capital converge to a common world return on capital. This common world return is the one of the "leading" country -the Home country in our simple case. Then, integration consists of an increase in the foreign return on capital compared to the foreign autarky case. This increase in the foreign return comes from a decrease in the foreign relative price. With capital mobility, the foreign relative price is stuck at a too low level compared to what it would be in autarky. As a result, the foreign country's growth rate will be lower than in autarky for the foreign country (g * < g A * ). This mechanism works when β = β * and γ = γ * , as soon as there is a traded TFP spread between the two countries. We focused on the simplest case (β = β * , γ = γ * ) but 6 Notice that Condition 2 is equivalent to 7 This ratio is higher when α T > α N .

A * T A T 1 1-α T < βγ * (1+γ * a)(ζ(1-α T )(1-β)+(1+γa)) β * γ(1+γa)(ζ(1-α T )(1-β * )+(1+γ * a))
conditions (1) and ( 2) allow all cases to be considered. The leading country can be the higher TFP country or the more altruistic country or the lower time-preference country, other things being equal.

Calibrations

Table 2 collects parameter values. Since we focus on the case without cross-border externality we set a = 1 and then λ = 0. Education technology b is assumed to be unity. As usual in two-sector models, we assume half of consumption is dedicated to traded goods: µ = 0.5. In the model, β/ (1 + γa) denotes the saving rate whereas γa/ (1 + γa) denotes education spending. Our calibration is consistent with figures collected in Table 3 where we assume each period of life lasts for 25 years. Using Beine et al. (2002), the rate of time preference is supposed to be around 3.7%, then we assume β = 0.4 and then γ * = 0.36. In Figures 2, the discount rates β and β * lie between 0.3 and 0.4 to consider differences in saving behavior between countries. Finally, population growth reflects French annual population growth: 0.4%. Notice that in Table 3, education spending represents the sum of public spending (4.64%) and private spending (0.67%). Assuming capital shares in production of traded and nontraded sectors respectively of 0.65 and 0.2, we obtain in autarky that production of traded goods corresponds to 40.5% of GDP. Factor allocation between the two sectors is such that k T = 1.58k N and the autarky interest rate is R = 2.77 which represents an annual real interest rate of 2.3%. These figures are consistent with empirical evidence since traded output represents 40% of total output [Mahbub Morshed and Turnovsky (2004)].

Figure 1 illustrates Proposition 2 which states that convergence in growth rates is obtained for very special rates of altruism only. The convergence line is the dotted line. If the two countries have the same traded TFP, identical rates of altruism guarantee convergence (A T /A * T = 1, γ = γ * = 0.36). If the domestic country has a higher traded TFP, then convergence occurs only for a domestic rate of altruism γ = 0.305 -as a function of the domestic rate of altruism γ. The dotted line represents the threshold Γ (γ * ) . Notice that on these figures, the domestic country is the leading country only when γ > Γ (γ * ). Otherwise, the foreign country becomes the leading country.

Figure 2a (2b) corresponds to the case where the domestic country has a lower traded (higher) TFP but a higher (lower) discount rate (β). The consequences of capital mobility on the relative price depend both on the technological gap between the two countries and on the rates of time preference. If the domestic country integrates with a high-TFP-low-saving foreign country and γ < Γ (γ * ) , its relative price depreciates (black line) compared to what it would be in autarky relative price (grey line). Conversely, if the domestic country integrates with a low-TFP-high-saving foreign country and γ < Γ (γ * ), its relative price appreciates (black line) compared In this two-sector model, economic growth depends only on the RER. Figure 2a' (2b') depicts the ratio k/k A8 to compare the benefits of integration on growth when the domestic country integrates with a high-traded-TFP-low-saving country (low-traded-TFP-high-saving country). We focus on the case where γ < Γ (γ * ) . The advantage of integration depends on the gap between the grey and the black lines on Figures 2a and2b. On Figure 2a (2b), integration leads to an RER depreciation (appreciation) compared to the autarky relative price; thus, integration is growth damaging (enhancing): G/G a < 1 (G/G a > 1). In a standard one-sector model, capital mobility affects economic growth in each country because growth no longer depends on the domestic saving rate. In this two-sector model, economic growth also depends on the relative price, and then on the traded TFP gap between the two countries. When a low-TFP-high-saving domestic country integrates with a high-TFP-low-saving foreign country, capital mobility entails a depreciation, and growth rate with integration (G) is lower than the autarky growth rate (G A ). Capital mobility is then growth reducing when the rate of altruism is too low (γ < Γ (γ * )) in a low-TFP-high saving country (see Figure 2a'). This means that a low-educationspending-low-traded-TFP-high-saving country has a higher growth rate in autarky. This calibration exercise shows that capital mobility prevents the relative price from adjusting. Sometimes this real exchange rate sluggishness may be damaging for growth. The only case where growth may be better with capital mobility is in the case of a low-altruism-high-traded-TFP-low-saving domestic country as depicted by Figure 2b'.

Conclusion

This paper deals with capital mobility, RER and growth. Introducing a 2-sector structure in Michel and Vidal (1999) we show that opening borders to trade and factor mobility does lead to divergence between countries growth rates except if there exist cross-border externalities in human capital accumulation. In this 2x2x2 model, consequences of integration are mixed and highly dependent on both altruism and traded TFP gap between countries. Convergence is far from being the rule even with capital mobility. When a low-traded-TFP-low-altruism domestic country integrates with a high-TFP foreign country, capital mobility entails a depreciation and a growth deterioration in the domestic country (compared to autarky). This means that a low-education-spending-low-traded-TFP country has a higher growth rate in autarky. When a high-traded-TFP-high-altruism domestic country integrates with a low-TFP foreign country, capital mobility does not affect the domestic growth rate but improves the foreign one (compared to autarky).

This paper concludes that capital mobility may lead to real exchange rate misalignments which decrease growth in the less altruistic country with resulting income divergence. Education policy may be an appropriate instrument to make convergence more likely to occur. We leave this issue for future research.

Proof of Lemma 3

From equations ( 19) and (20), we obtain s t = βet γa and s * t =

β * e * t γ * a and thus the world capital market clearing condition (32) becomes:

(1 + n) pk t+1 h t+1 + (1 -p)k * t+1 h * t+1 = p βe t γa + (1 -p) β * e * t γ * a (60)
Substituting the individual level of human capital for equation ( 22) :

(1 + n)b pk t+1 e t + (1 -p)k * t+1 e * t = p βe t γa + (1 -p) β * e * t γ * a (61)
dividing by e t to write the equation as a function of ρ t = e * t et , we obtain:

(1 + n)b(p + p * ρ t ) pk t+1 + (1 -p)k * t+1 ρ t = p β γa + (1 -p) β * γ * a ρ t (62)
From equation ( 28 9) and [START_REF] Ito | Economic growth and real exchange rate: an overview of the Balassa-Samuelson hypothesis in Asia[END_REF] in equation (25), we obtain the nontraded goods market clearing condition for the home country:

Meaning that

k t = (1 -µ) 1-β 1+γa w (P N t ) + k A R (P N t ) P N t A N D [k T (P N t )] α N -1 + k T (P N t ) (64)
Substituting the expression of k T from equation ( 9), the factor prices w t and R t from equations [START_REF] Ito | Economic growth and real exchange rate: an overview of the Balassa-Samuelson hypothesis in Asia[END_REF], and simplifying by

P α T -1 α T -α N N t
, we obtain: Then we obtain the expressions for k t and k * t given by equations ( 44) and (45), where:

k t = (1 -µ) A T A N B α T -α N D (1 -β) (1 + γa) (1 -α T )BP N t 1 α T -α N + k A α T + BP N t
η ≡ ∆ A N A T 1 α T -α N (1 -µ) α N -α T α T 1 -β 1 + γa + 1 η * ≡ ∆ A N A * T 1 α T -α N A * T A T 1-α N (α T -α N )(1-α T ) (1 -µ) α N -α T α T (1 -β * ) (1 + γ * a) + 1 ζ ≡ (1 -µ) α N -α T 1 -α T Finally,

Proof of Proposition 4

If γ > Γ(γ * ) then 1+g * 1+g < 1 and ρ t converges to zero as t goes to infinity. But as from equation (43):

ρ t = p 1 -p k A -k t+1 k * A -k * t+1
then the capital stock of the home country k t converges to k A . From equations (45) and (46) with ρ = 0, we have:

k * t = k * A ζ + (1 -ζ) η * ηx (67)
In order to understand the effect of economic integration in the foreign country, it is important to compare the long term value of k * to the autarkic value k * A .

As we have assumed that α N < α T , we know 1 -ζ > 0, and Condition 2 follows from k * > k * A iff η * ηx > 1. Since ζ(1 -α T ) 1-β 1+γa + α T > 0 and ζ(1 -α T ) 1-β * 1+γ * a + α T > 0 from the proof of Proposition 1 and using the expressions of η, η * and x, we can rewrite Condition 2 as :

A * T A T

  0.2 1.64 0.5 0.105 0.36 0.3 or 0.4 0.3 or 0.4

Figure 1 :

 1 Figure 1: Domestic and foreign growth in an integrated world

  Figure 2a

Figure 2 :

 2 Figure 2a'Figure2b'

  -β) w (P N t ) + (1 + γa)k A R (P N t ) = P N t A N D [k T (P N t )] α N -1 (k t -k T (P N t ))

1 αTB 1 α

 11 -α N (65) and similarly for the foreign country:k * t = (1 -µ) A * T A * N B * αT -α N D (1 -β * ) (1 + γ * a) (1 -α T )B * P * N t 1 α T -α N + k A * α T + B * P * * αT -α N D = α N -α T α T (1 -α T )and B can be rewritten B = ∆ A N A T T -α N , B * = ∆ α N is the same for both country.

  substituting those expressions in the world market clearing condition (43), we obtain the expression of P N t given by equation (46). It is clear that ζ is of the sign of α N -α T . Thus if α N < α T , have ζ < 0 and we need conditions to get k > 0 and k * > 0. If E(ρ t ) ≡ ρt(1-p)x+pρt(1-p)η * +pη , then from equations (44), (45) and (46), and the monotonicity of E(.):k t > 0 ⇔ E(ρ t ) lim ρt→+∞ E(ρ t ) > -ζ η(1-ζ) (b) and k t * > 0 ⇔ E(ρ t ) > -ζx η * (1 -ζ) ⇔ E(0) > -ζx η * (1-ζ) (c) lim ρt→+∞ E(ρ t ) > -ζx η * (1-ζ) (d)Conditions (a) and (d) always hold when ζ is negative, and under assumption 2, conditions (b) and (c) hold.

1 -

 1 γ(1 + γ * a) (ζ(1 -α T )(1 -β) + α T (1 + γa)) βγ * (1 + γa) (ζ(1 -α T )(1 -β * ) + α T (1 + γ * a)) But as 1 + g * < 1 + g means -α T )(1 -β) + α T (1 + γa) ζ(1 -α T )(1 -β * ) + α T (1 + γ * a)This case is obtained under Assumption 2 when αT )(1 -β) + α T (1 + γa) ζ(1 -α T )(1 -β * ) + α T (1 + γ * a) < xη η * < 1with γ such that:β * β ζ(1 -α T )(1 -β) + α T (1 + γa) ζ(1 -α T )(1 -β * ) + α T (1 + γ * a) < 1

Table 1 :

 1

	2005

European GDP growth rates. Source: Eurostat and Author's calculations. level given the fundamentals (altruism, productivity, savings). In this model, capital mobility may lead to RER misalignment and growth reduction in the less altruistic country. Finally, this paper complements the existing literature. It points out that an additional cost of maintaining an RER too low -inflation and reserves accumulation are often blamed -decreases the incentive to accumulate human capital and is then bad for growth. Whereas the literature [Prasad et al. (

2007

) Rodrik (2008),

[START_REF] Eichengreen | The Real Exchange Rate and Economic Growth, Commission on Growth and Development[END_REF]

] suggests that a significantly overvalued RER is growth reducing as well.

Table 2 :

 2 Parameter values

	Education spending (% of GDP)	5.31%
	Households saving rate (% of GDP) 5.8%

Table 3 :

 3 Average on 27 European countries (2006). Source: Eurostat.

  ), we have k A = ab , we finally get the expression of the lemma, giving implicitly ρ t . Denoting x ≡ β * γ βγ * then k * A = xk A , and we can compute k t , k * t as functions of P N t for which we also give an expression. Using equations (

	β (1+n)γab and k * A =	β

*

(1+n)γ *

Maintaining an artificially low level of the RER would generate costs in terms of inflation and reserve accumulation and maybe benefits in terms of temporary growth.[START_REF] Eichengreen | The Real Exchange Rate and Economic Growth, Commission on Growth and Development[END_REF] states that the ratio between costs and benefits increases with the country's level of development.

Notice that k T > 0 if and only if α N > α T or α T > α N and µ < 1-α N α T -α N . If technology is the same in both sectors α T = α N and k T = k A .

The expressions of η, η * , and ζ are given in Appendix 7.4.

From equation (9), k T is increasing with P N iff α T > α N and from equation (44), P N is increasing with k iff α T > α N .

Notice that from equations (23), (29) and (30), G/G a = (k/k A ) α T .

Appendix

Proof of Lemma 1

Substituting equations ( 2) and (18) in equation ( 24), and dividing by N t , we obtain:

Integrating the budget constraints (14), ( 15) and the optimal level for s t and e t from equations (19) and (20) gives:

Moreover, from the optimal choice of investment in children's education(20) we know:

And thus using equation ( 22) and dividing (48) by h t we get:

As from equation (4), h N = k-k T k N -k T , the non tradable market clearing condition is:

From equations ( 9), we finally get the condition of the lemma.

Proof of Lemma 2

Considering autarky, and thus λ = 0 and k t = k A , the non tradable market clearing condition (25) is:

Substituting P N t from equations (9):

With factor prices from equations ( 7) and ( 8), we get:

Dividing by

(54) From straightforward computations, we finally obtain equation (30). The last task is to compute the equilibrium growth rate g A . Using equation (23), we readily obtain equation (29). 1+γa) , we can define the growth factor as a function of β and γ:

Proof of Proposition 1

The logarithmic derivative of G A with respect to β is:

Which is positive if and only if

The numerator is always positive. The denominator is positive if and only if

which is true iff α N ≤ α T or α N ≥ α T and β ≤ β with β = α N (1-µ)+α T (γa+µ) (1-µ)(α N -α T ) . Since β > 1, the growth factor is always increasing with β.

Concerning the variation of the growth rate with γ. The logarithmic derivative of G A with respect to γ is:

Which is zero for a unique positive value of γ, γ =

, positive for 0 ≤ γ ≤ γ and negative for γ > γ . Concerning the variations of the growth rate with µ. As The denominator is positive from the previous analyses and the positivity of k T . Thus this derivative is of the sign of α N -α T .