Andrea Michiorri

The Climate Change Impact on Power Grid Transmission Capacity Montana

Keywords: Climate change, Dynamic Thermal Rating, Power Systems

Proposing practical solutions to mitigate climate change effects on the electrical power system requires a comprehensive understanding and quantification. By conducting an assessment at high grid resolution, this article explores the impact of climate on transmission network capacity, employing established thermal models and a regional expansion plan. The results indicate average reductions of 1.53%, 2.3%, and 0.2% for overhead lines, power transformers, and underground cables, respectively. We propose a quasi-Dynamic Thermal Rating method to counter these effects, estimating maximum capacity. This approach enhances component capacity by an average of up to 35% during winter at the power transformers and up to 14% during nighttime hours for overhead lines. This solution constitutes a viable alternative for electricity operators, to address the dilemma between the necessity of reducing the failure rate/decrease in capacity, and the imperative need for new investments in transmission assets.

Introduction

Climate change projections estimate an average atmospheric temperature increase of 2-4 o C until the end of the century [START_REF]Climate and energy indicators for Europe from 2005 to 2100 derived from climate projections[END_REF][START_REF] Bell | ERA5 hourly data on single levels from 1940 to present[END_REF]. This will directly and negatively impacts the electric power system, affecting transmission capacity, generation, demand, and congestion.

Regarding transmission, which is the main focus of this study, the current carrying capacity of Overhead Lines (OHL), Power Transformers (PT), and Underground Cables (UGC) is determined, among other factors, by their ability to dissipate joule losses into the external environment. In turn, this depends on ambient temperature: the lower the external temperature, the higher the transmission capacity, and vice versa. For instance, in the United States, the impact of global warming is anticipated to cause a reduction in OHL capacity within the range of 1.9% to 5.8% [START_REF] Matthew | Impacts of rising air temperatures on electric transmission ampacity and peak electricity load in the united states[END_REF].

In reference to power generation, higher temperatures leads to a reduction in production capacity: on the one hand, a higher ambient temperature increases the sink temperature in thermodynamic cycles, reducing overall conversion efficiency. On the other hand, it reduces air density, in turn reducing, the mass flow intake of fossil fuel generators. Furthermore, factors such as water discharge temperatures and diminishing water flows are anticipated to impact over 80% of the world's thermal power plants due to drought and shifting seasonal patterns, as detailed in [START_REF] Seleshi G Yalew | Impacts of climate change on energy systems in global and regional scenarios[END_REF].

Regarding power demand, this tends to grow with higher ambient temperature due to load thermosensitivity driven by air conditioning. This is in turn accentuated by the growing penetration of air cooling in power systems, including in developing countries. In [START_REF] Romitti | Heterogeneous climate change impacts on electricity demand in world cities circa mid-century[END_REF], the expected annual demand variation in different scenarios ranges from -2.7% to 5.7% on average and is further exacerbated during heat waves, culminating in an increase of up to 21% [START_REF] Añel | Impact of cold waves and heat waves on the energy production sector[END_REF].

The combination of lower transmission capacities, lower generation, and higher load can increase the likelihood of congestion in the transmission and distribution infrastructure. This leads to inefficiency and spikes in local power prices, exacerbating current trends which, for example, resulted in a cost of roughly $4.8 billion in 2016, as reported by the U.S. Department of Energy (DOE) in 2018 [7].

Among the solutions proposed to alleviate network congestion problems, Dynamic Thermal Rating (DTR) [START_REF] Karimi | Dynamic thermal rating of transmission lines: A review[END_REF][START_REF] Keyvani | Indirect weather-based approaches for increasing power transfer capabilities of electrical transmission networks[END_REF][START_REF] Douglass | A review of dynamic thermal line rating methods with forecasting[END_REF][START_REF] Karimi | Dynamic thermal rating of transmission lines: A review[END_REF] is actively being deployed on critical lines. This is a technology aiming at identifying the real-time current carrying capacity of network components, which is, in general, higher than its Static Thermal Rating (STR). On the one hand, this value is strongly weather-dependent; on the other, DTR allows the removal of network congestions and associated curtailments and delays, and can remove network reinforcements whilst improving reliability.

Various studies have incorporated DTR into power system expansion plans [START_REF] Glaum | Enhancing the german transmission grid through dynamic line rating[END_REF][START_REF] Trpovski | A comparative analysis of transmission system planning for overhead and underground power systems using ac and dc power flow[END_REF] and highlighted its importance in RES integration and penetration [START_REF] Hajeforosh | Enhancing the hosting capacity of distribution transformers for using dynamic component rating[END_REF][START_REF] David | Dynamic rating assists cost-effective expansion of wind farms by utilizing the hidden capacity of transformers[END_REF][START_REF] Teh | Reliability impacts of the dynamic thermal rating and battery energy storage systems on wind-integrated power networks[END_REF][START_REF] Lai | Network topology optimisation based on dynamic thermal rating and battery storage systems for improved wind penetration and reliability[END_REF][START_REF] Morozovska | Dynamic rating with applications to renewable energy[END_REF].These studies employ control and sensing devices [START_REF] Michiorri | Forecasting for dynamic line rating[END_REF][START_REF] Douglass | Real-time overhead transmission-line monitoring for dynamic rating[END_REF] or data-driven probabilistic methods to calculate the rating of OHL [START_REF] Mahmoudian | Real time congestion management in power systems considering quasi-dynamic thermal rating and congestion clearing time[END_REF]. All these studies yield generalized findings on the efficacy of DTR, among which the following can be emphasized: a) Decreased system congestion costs due to less generator re-dispatching. b) Reduction or postponement of investments required for reinforcing or expanding existing assets. c) Grid operational flexibility to enhance facilitated integration of RES.

1

In this study, we leverage historical and projected meteorological variables at different geographical points on the power system component's location. Through this data, we calculate the maximum current that the system can carry at any point in time without any section's temperature surpassing the predefined maximum threshold, employing DTR methodology. By doing so, we emphasize the significant potential of DTR in enhancing power system flexibility. To reinforce this assertion, we analyze power system planning within the Generation and Transmission Expansion Plan (G&TEP) framework using a tool for modeling hybrid power systems. This is made possible by the availability of widely accepted component thermal models [START_REF]2018, power transformers -part 7: Loading guide for mineral-oil-immersed power transformers[END_REF][START_REF]Electric cables-calculation of the current rating-part 1-1: Current rating equations (100% load factor) and calculation of losses-general[END_REF][START_REF] Iglesias | Guide for Thermal Rating Calculations of Overhead Lines[END_REF], open energy data models [START_REF] Brown | PyPSA-Eur: An open optimisation model of the european electricity system[END_REF] and quantitative climatic projections such as the Representative Concentration Pathways (RCP) [START_REF] Taylor | An overview of cmip5 and the experiment design[END_REF] and Shared Socio-economic Pathways (SSPs) [START_REF] Riahi | The shared socioeconomic pathways and their energy, land use, and greenhouse gas emissions implications: An overview[END_REF], which constitute a valuable toolset for assessing regional climate changes and their specific impacts on the energy sector.

As a main contribution, this research aims to quantify the impact of climate change on power system transmission capacity. It also proposes quasi-Dynamic Thermal Rating (qDTR) as the primary solution to recover lost transmission capacity, facilitating the connection of renewable resources and reducing network costs.

Result and Discussion

This study applies qDTR to estimate the maximum allowable currents of power transmission components using historical (qDTRH , from 1970 to 2022) and future weather projections for Europe (qDTRRCPx, where x represents the RCP scenario evaluated from 2023 to 2070).

Firstly, the use of qDTRH instead of STRH increases average transmission capacity in the region of 14.2%, 17.4% and 3.7% for OHL, PT and UGC respectively (see Table 1, first row). The results indicate an average reduction in the transmission capacity of the network, ranging from 1.3% to 8.6% as shown in Figure 1. This outcome is a consequence of an increase in ambient temperature of between 1.3% to 2.4%.

We then quantify the climate impact of transmission network capacity for three main components: PT, OHL, and UGC. The results are shown in Table 1. Demonstrate how average component ratings can drop from 0.4% to 1.53% (OHL), creating additional costs for the network in the region of 0.006% to 0.064%. Figure 4 reports the average OHL rating variation expected in Europe in the RCP 2.6 Scenario.

Figure 1 visually illustrates the trend, presenting the rating evolution over the next five decades for the European network's OHL [START_REF] Matke | SciGRID -An Open Source Reference Model for the European Transmission Network[END_REF] under low, medium, and high emission scenarios. These ratings are then incorporated into a G&TEP, enabling the calculation of optimal investments and operating costs. Across the three scenarios, not considering climate evolution would result in costs underestimations in the region of 0.1% to 0.2% for CAPEX, -6.2% to 3.1% for OPEX and 9.3% to 70.2% for renewable curtailment.

The next three sections show firstly the increased transmission capacity allowed by qDTR, then the impact of climate change on network transmission capacity calculated by qDTR and finally the effect of climate change on network planning.

Benefits of qDTR vs STR

We start by analyzing the benefit of using qDTR instead of STR in network operations. Figure 2 reports the percentage variation in average transmission capacity for OHL, PT, and UGC, respectively, in Europe. In this case, qDTRH and STRH are calculated using historical values from 1970 to 2020. Note that qDTRH are always above STRH in all studied regions. Figure 2: Geographical distribution of the mean Historical ∆ % (qDTRH, STRH) difference at the country level. This is performed for the three main power components (OHL, PT and UGC). Fig. 3 shows the yearly and hourly variations of qDTRH for the three components in southeast France. For OHL and PT, we can see that in daily summer hours, qDTRH are lower than STRH. On the contrary, in winter and night hours, qDTRH are far higher, resulting in a larger overall transmission capacity. Concerning UGC, as expected, their qDTR do not change during the day, due to the high thermal inertia of the soil. Another aspect worth mentioning is the daily and yearly variations of qDTRH. For the three components considered, the yearly qDTR variation range in the region is ±15% from the highest to the lowest value, driven by temperature variations.

Impact of climate change on network transmission capacity

When qDTR are calculated using projections for the next decades instead of weather data relative to past years, the effect of a temperature rise predicted by climatic projections becomes apparent.

Table 1 reports the average and extreme values for the difference in transmission capacity calculated using data on historical weather and expected future weather. In all three scenarios, transmission capacity is expected to drop. Transformers are the component with the highest variation (from -1.0% to -2.3%) followed by OHL (from -0.4% to -1.53%) and UGC (from -0.1% to -0.2%). This is explained by the fact that the OHL rating is mainly influenced by air temperature and wind speed, whilst the PT rating is only influenced by air temperature. Often hottest hours are also characterized by not null wind speeds, reducing the derating effect of temperature. For UGC, the much narrower temperature variation of the soil prevents large rating drops. The worst cases simulated show a maximum reduction of -3.9%, -5.1% and -1% for the ratings of OHL, PT and UGC, respectively.

Figure 4 shows the spatial distribution of the rating variations summarized in 1. The spatial variation of average transmission capacity for the three components is reported above. Figure 4: Geographical distribution of the mean qDTR difference at the country level over fifty years for the RCPs and the historical reanalysis. This is performed for the three main power components (OHL -PT and UGC). The first row reflects the difference in the variation of the historical average for the region, and the subsequent rows illustrate the variation in the average for each of the RCPs. The first row shows the spatial variability of qDTRH in Europe, which is in the region of 20%, 30% and 90% for OHL, PT and UGC. The lower variability of OHL is given, as explained above, by the double dependency from wind and air temperature of its rating. The very high variability of UGC ratings on the contrary is given by the variety of soils in the different regions, which considerably impacts thermal diffusivity and moisture retention. The following three rows show the percentage variation in qDTR in Europe according to the different climate scenarios considered. The variations are almost unanimously negative, with peaks in central Spain, the Arctic and mountainous regions. As mentioned above, UGC presents lower variations because of the high soil inertia.

Finally, the heat map in Fig. 5 represents the temporal variation by month and hour of qDTRRCP on a specific node of the network. The effect of increased ambient temperature in the three RCP scenarios is shown in Fig. 5. The observed variations, which are moderately significant, for instance translate into an average rating reduction of -2.3% in July for qDTRRCP 8.5 and -0.7% for qDTRRCP 4.5 in the PT. This could be translated in terms of variation in the risk level regarding the equipment's lifetime for the network operator. On the other hand, the opposite effect can also be observed in specific months and hours, such as -0.9% on February mornings for qDTRRCP 8.5, +0.7% for qDTRRCP 4.5 and +0.02 qDTRRCP 2.6.

Impact of climate change on network costs

This section examines investment decisions within the Generation and Transmission Expansion Plan, incorporating climate-variant supply and qDTR for power system components.

To achieve this, we use a network model from the PyPSA [START_REF] Brown | PyPSA-Eur: An open optimisation model of the european electricity system[END_REF] package, applying STRH and qDTRRCP calculated with historical or climate projections using weather data. The network model is then used to carry out a G&TEP in order to estimate changes in CAPEX, OPEX, and renewable curtailment

The results are summarized in Table 2, which shows: 1 the results of a baseline simulation with STR, 2 the improvements obtained using qDTRH instead of STRH, and 3 the error incurred when using historical weather instead of climate projections in qDTRRCP calculations. In each case, the CAPEX and OPEX are reported for different assets, such as fossil fuel generators, renewable generators, nuclear generators, transmission lines, and storage. For renewable generators, the amount of expected curtailment is also reported.

When comparing the minimum and maximum emission scenarios in Europe under the same projection, a difference of 154 MWh is observed in the dispatch energy for onshore wind. Conversely, in France, electricity generation CO2 targets remain relatively stable for nuclear, run of river, solar, coal lignite,

The average additional transmission capacity provided by qDTR in place of STR, calculated from historical data. The spatial variation on the continent of qDTR, calculated from historical data. The variation of transmission capacity calculated with the specific climatic scenario versus historical weather nuclear, and oil, indicating their consistent performance in the context of the analyzed DTR scenarios (from -0.25% to 0.32%). The impact of climate change is linked to qDTR in Transmission Expansion Plans (TEPs). In scenarios marked by increased greenhouse gas emissions, the transmission capacity requirements are subject to a substantial increase of 49% in Europe and 98% in France for the 2050 scenario, as evidenced in the highemission scenario RCP 8.5. However, It must be noted that using qDTR instead of static rating reduces curtailment by up to 60% in renewables in the French case for the 2.6 scenario due to the massive expansion of renewables, and 17% in the European case.

Concomitantly, as emissions continue to escalate, resulting in elevated temperatures across Europe, the vulnerability to energy curtailments becomes more pronounced. In specific scenarios like RCP 8.5, characterized by high emissions, the likelihood of curtailments increases from 9.3% to 70% in Europe and from 2.6% to 36.6% in France. In the field of renewable technology, onshore wind curtailment displays notable variability across CO2 budgets ranging from up to 1.25 GW for Europe in the low-emission scenario employing STRH, to 1.02 GW in the high-emission scenario using qDTRH. Other technologies, such as solar PV, also manifest changes, albeit with less magnitude, across CO2 budget scenarios qDTRRCP 2.6 and qDTRRCP 8.5, indicating a shift from 233.64 to 129.52 MW.

Discussion

By conducting an assessment at high grid resolution, this analysis takes a close look at the climate impact on transmission network capacity by utilizing established thermal models and a regional expansion plan for generation and transmission by 2050. Using the qDTR method to estimate maximum capacity with a low-risk exceedance probability, our findings reveal substantial reductions in the mid-term high-emission horizon for power transformers and overhead lines. In contrast, for underground cables, the reductions are more modest, highlighting their lower sensitivity to potential climate changes; this demonstrates that the heat balance has a memory, i.e., soil moisture anomalies may persist for weeks or even months (monthly rate change less than 5%.).

Mitigating the risk of exceeding the Static Season Rating, attributed to the likelihood of overload in unfavorable meteorological conditions that impede heat dissipation (e.g., low wind speed, high temperature, low precipitation), is achieved with a reduction of up to 99.998%. This improvement positively contributes to the reliability of the transmission system. However, as part of a trade-off, qDTR allow us to optimize energy utilization during the colder nighttime hours in both summer and winter. This is particularly crucial for managing nighttime winter load peaks. The observed contrast is more prominent in summer than in winter, attributable to the elevated temperature variations.

Regional advantages in electricity sector investments, linked to atmospheric carbon concentration, are predominantly centered around renewables. This trend is evident in the analysis of twelve distinct scenarios tailored for all EU countries with a time horizon of 2050, ensuring that investment strategies align with the dynamic nature of climate-affected power supply and the thermal characteristics of critical system components. Furthermore, the use of qDTRH data reveals a reduction in transmission planning costs, harnessing the flexibility of the generation system, with a particularly notable effect in the scenarios tailored for France.

The introduction of qDTR opens a new avenue for curtailing reductions. Our results demonstrate, across all CO2 targets and emission scenarios, the pivotal role of qDTR in enabling greater utilization of renewables without the need for storage technology. As an illustration, in France, with the CO2 and qDTRRCP 2.6 scenarios, curtailment sees a reduction of up to 59% compared to STRH. This reduction translates into enhanced network transmission capacity and a higher capacity expansion for renewable energy.

Conclusions

Overall, this work confirms that 1) predicted climate, with higher ambient temperatures, causes a reduction in power system transmission capacity, in the region of 1.53%, 2.3%, and 0.2% for OHL, PT, and UGL, respectively. 2) the qDTR approach proposed, allows for higher transmission capacities, overcoming the rating reduction caused by climate change, as it improves component capacity by up to 35% on average during winter for PT and up to 14% during nighttime hours for OHL. 3) Overall network costs change modestly and we achieve a consistent reduction in renewable production curtailment.

Method

Overview

We develop a procedure to quantify the impact of climate change on power grid transmission capacity. This method is described in Fig. 7 and can be divided into two steps:

1) Firstly, DTR and qDTR are estimated for OHL, PT, and UGC [START_REF] Montana | Long-term dynamic thermal ratings of underground cables integrating soil dynamics and climate projections[END_REF] using thermal models of the components and weather data from historical reanalysis and climatic projections. This allows us to quantify the variation in transmission capacity due to climate change. 2) Secondly, a G&TEP is calculated with a horizon of 2050, using transmission capacities calculated with historical weather reanalysis, RCP, STR, and qDTR. This allows us to quantify the impact of climate change on transmission capacity calculated in the previous step.

Data

The data used in this study can be divided into two broad categories: (c) component data (static) and (a) environment data (static and dynamic). Historical meteorological conditions in Europe for the period 1970-2020 are obtained from ERA reanalysis [START_REF] Bell | ERA5 hourly data on single levels from 1940 to present[END_REF], whilst climatic projections for the period 2020-2070 are obtained from the Copernicus Climate Change Service (C3S) [START_REF]Climate and energy indicators for Europe from 2005 to 2100 derived from climate projections[END_REF]. Additional (b) soil properties for underground cable rating calculations are obtained from [START_REF] Dorigo | The international soil moisture network: serving earth system science for over a decade[END_REF][START_REF] Hiederer | Mapping Soil Properties for Europe -Spatial Representation of Soil Database Attributes[END_REF]. A list of the parameters used and their source is reported in Table . 3. Component parameters, relative to the most popular elements, are obtained from the existing literature or data sheets provided by the main manufacturers such as [START_REF] Kiessling | Overhead Power Lines Planning, Design, Construction -Hard-drawn AL1 ACSR. PublisherName[END_REF] for OHL, [START_REF] Kable | High and extra high voltage cables. www.tfkable.com[END_REF] for UGC, and [34] for PT.

Preprocessing

The raw data described are preprocessed as follows: (e) time series are uniformed by linear interpolation to a common time step of 1h. When a specific coordinate is required, parameters are interpolated linearly from the four nearest available grid points. (d, f) soil temperature and soil moisture are calculated using a dedicated machine-learning-based model for each coordinate. This is necessary since the available values from [START_REF]Climate and energy indicators for Europe from 2005 to 2100 derived from climate projections[END_REF] are either not obtainable at the typical UGC burial depth (1-5m, in the case of soil temperature) or absent (in the case of soil moisture).

Rating estimation

The estimation of ratings starts with the use of (h) component thermal models based on the thermal balance between the heat generated by the Joule effect I 2 R and the heat dissipated in the environment by convection or conductivity Qc, radiation Qr and the solar heat gain Qs. This is shown in Eq. 1, where the parameters are influenced by: surface absorptivity for α, surface emissivity, maximum allowable temperature Tc and air temperature Ta for β. The parameter γ depends on atmospheric values such as wind speed Ws and Ta for OHL, and soil parameters such as soil temperature Ts and soil moisture ψs for UGC.

I 2 R + αQs = βQr + γQc (1)
The state of the art thermal models for OHL [START_REF] Iglesias | Guide for thermal rating calculations of overhead lines[END_REF], UGC [36] and PTR [START_REF]2018 -power transformers -part 7: Loading guide for mineral-oil-immersed power transformers[END_REF] are used to compute the DTR (i) for each power component at each coordinate and each hourly time steps available in the datasets [START_REF]Climate and energy indicators for Europe from 2005 to 2100 derived from climate projections[END_REF][START_REF] Bell | ERA5 hourly data on single levels from 1940 to present[END_REF].

At this point, (j) the simulated historical or future DTRs are grouped by time interval (yearly, monthly, monthly/hourly, etc.). For each group, a power law function is fitted to the lowest tail of the distribution (k). Finally, an accepted risk for thermal overload is chosen, x = 0.1% in this work, and the qDTR in terms of current intensity I are calculated (l) as in Eq. 2. For DTR, a thermal model is utilized to compute the maximum capacity, taking into account real-time variables. Contrastingly, for STRs, a fixed value is applied throughout the year as a constrained capacity limit, without considering changing conditions. qDTR, calculates a 0.1% overload risk over the lower tail values of DTR across a 50-year time horizon. Notably, cooler forced temperatures and lower irradiation levels increase the capacity during the winter season and nighttime hours.

I(x) = Ax α (2)

Costs estimation

The financial impacts resulting from the reduction in DTR due to climate change are assessed through a series of G&TEP using [START_REF] Brown | PyPSA: Python for Power System Analysis[END_REF] and CO2 budgets for the RCP from [START_REF] Van Vuuren | Stabilizing greenhouse gas concentrations at low levels: an assessment of reduction strategies and costs[END_REF][START_REF] Clarke | Scenarios of greenhouse gas emissions and atmospheric concentrations[END_REF][START_REF] Smith | Multi-gas forcing stabilization with the minicam[END_REF][START_REF] Wise | Implications of limiting co2 concentrations for land use and energy[END_REF][START_REF] Fujino | Multi-gas mitigation analysis on stabilization scenarios using aim global model. Multigas Mitigation and Climate Policy[END_REF][START_REF] Hijioka | Global ghg emissions scenarios under ghg concentration stabilization targets[END_REF][START_REF] Riahi | Scenarios of long-term socio-economic and environmental development under climate stabilization[END_REF] . These studies rely on the use of yearly STRs, and monthly/hourly DTRs (mhDTR), equivalent to 288 DTRs calculated for each month/hour combination. Several comparisons are carried out: 1) G&TEP are carried out using STRH calculated on a historical reanalysis and qDTRRCP on the three climatic projections. The comparison shows the error incurred when not considering the impact of climate change on transmission capacity.

2) G&TEP are carried out using STRH and monthly/hourly qDTRH calculated from historical reanalysis. The comparison shows the benefits of using frequently changing qDTR to recover the lost transmission capacity.

Figure 1 :

 1 Figure 1: Climate impacts on transmission capacity, difference in capacity for qDTR in the month of June under the average RCP 2,6, 4.5 and 8.5 scenario for OHL.

Figure 3 :

 3 Figure 3: qDTRH calculated for each month/hour combination for OHL, PT and UGC calculated in the node of Tavel (southeast France). Colors represent months from the coldest (blue) to the warmest (red). Yearly static ratings are represented by a dashed black line.

Figure 5 :

 5 Figure 5: Difference in the calculated qDTR month/hour between historical values and RCP projections 2.6, 4.5, 8.5. This is performed for the three main power components (OHL -PT and UGC), in a node located in Tavel, southeast France.

  (a) European electricity system infrastructure (b) French electricity system infrastructure

Figure 6 :

 6 Figure 6: Regional G&TEP evaluated with Static Annual Rating at 0.1% probability of overload

Figure 7 :

 7 Figure 7: Visual representation of the procedure used in this study

Figure 8 :

 8 Figure8: Conceptual illustrations of power system component rating capacities can be categorized into different methods. For DTR, a thermal model is utilized to compute the maximum capacity, taking into account real-time variables. Contrastingly, for STRs, a fixed value is applied throughout the year as a constrained capacity limit, without considering changing conditions. qDTR, calculates a 0.1% overload risk over the lower tail values of DTR across a 50-year time horizon. Notably, cooler forced temperatures and lower irradiation levels increase the capacity during the winter season and nighttime hours.

Figure 9 :

 9 Figure 9: Meteorological change projection , evaluated rolling mean for 2.6, 4.5, 8.5 RCP and the 2nd percentile values for each region

Table 1 :

 1 Variation for the three scenarios with respect to historical values for qDTR for the European region.

	Description		Component
		OHL	PT	UGC
	∆ % (qDTR H , STR H ) 1
	Mean	14.2	17.4	3.74
	∆ % (qDTR H )
	Max	10.7	17.4	29.6
	Min	-9.4	-16.4	-57.8
	∆ % (qDTR RCP2.6 , qDTR H ) 3
	Mean	-0.4	-1.0	-0.1
	Max	-1.4	-2.4	-0.5
	Min	0.75	0.3	0.8
	∆ % (qDTR RCP4.5 , qDTR H ) 3
	Mean	-0.7	-1.7	-0.2
	Max	-1.7	-3.2	-0.6
	Min	0.9	-0.6	0.5
	∆ % (qDTR RCP8.5 , qDTR H ) 3
	Mean	-1.53	-2.3	-0.2
	Max	-3.9	-5.1	-1
	Min	0.32	-0.8	1.0

Table 2 :

 2 Variation (in %) for the three scenarios with respect to yearly fix rating and costs, the scenarios covering historical static (STR

	, historical (qDTR H ) 2
	H ) 1

Table 3 :

 3 Summary of data and sources. a:ERA[START_REF]Climate and energy indicators for Europe from 2005 to 2100 derived from climate projections[END_REF][START_REF] Bell | ERA5 hourly data on single levels from 1940 to present[END_REF], b:ISMN[START_REF] Dorigo | The international soil moisture network: serving earth system science for over a decade[END_REF], c:ESDAC[START_REF] Hiederer | Mapping Soil Properties for Europe -Spatial Representation of Soil Database Attributes[END_REF]. Soil composition: Silt, sand, clay, organic, bulk, texture

	Parameter, Units	Unit	Source
	Air temperature at 2 m	• C -θa	a
	Total precipitation	mm -σ	a
	Net surface solar radiation	Jm -2 -H	a
	u -v -wind at 10 m	ms -1 -W s	a
	Soil composition	%	b, c
	Soil Temperature	Jm -2 -θs	b
	Soil Moisture	Jm -2 -ψs	b


Oleg Korablev

Franck Montmessin

Alexander Trokhimovskiy

Anna Fedorova

Nikolay Ignatiev

Alexey Shakun

Alexey Grigoriev

The Atmospheric Chemistry Suite (ACS) on board the ExoMars Trace Gas Orbiter

teaching and research institutions in France or abroad, or from public or private research centers.

The Atmospheric Chemistry Suite (ACS) on board the ExoMars Trace Gas Orbiter

Oleg Korablev [START_REF] Korablev | The Atmospheric Chemistry Suite (ACS) of Three Spectrometers for the ExoMars 2016 Trace Gas Orbiter[END_REF], Franck Montmessin (2), Alexander Trokhimovskiy (1), Anna Fedorova (1), Nikolay Ignatiev (1), Alexey Shakun (1), Alexey Grigoriev (1), and the ACS Team.

(1) Space Research Institute (IKI), Moscow, Russia, (2) LATMOS-CNRS, Guyancourt, France (korab@iki.rssi.ru)

Abstract

The Atmospheric Chemistry Suite (ACS) package is an element of the Russian contribution to the ESA-Roscosmos ExoMars 2016 Trace Gas Orbiter (TGO) mission. ACS consists of three separate infrared spectrometers [START_REF] Korablev | The Atmospheric Chemistry Suite (ACS) of Three Spectrometers for the ExoMars 2016 Trace Gas Orbiter[END_REF]. This ensemble has been designed and developed in response to the Trace Gas Orbiter mission objectives that specifically address the requirement of high sensitivity instruments to enable the unambiguous detection of trace gases of potential geophysical or biological interest. ACS embarks a set of instruments achieving simultaneously very high accuracy (ppt level), very high resolving power (>10,000) and broad spectral coverage (0.7 to 17 µm -the visible to thermal infrared range).

The near-infrared (NIR) channel is a versatile spectrometer covering the 0.7-1.6 µm spectral range with a resolving power of ≥20,000. The overall status of the ACS experiment and key findings available by the time of the conference will be reported.

  nadir aperture with the spectral resolution of 0.13 cm -1 giving access to trace gases [Grigoriev et al. EPSC 2018].

	surface up to 50-60 km and to monitor dust and water
	ice clouds in nadir. Also the surface temperature is
	measured [Ignatiev et al. EPSC 2018]. In solar
	occultation TIRVIM is mostly operated in
	"climatology" mode, with spectral resolution of 0.8
	cm -1 , delivering profiles of CO 2 , CO, H 2 O and
	aerosols. In a more "sensitive" mode, which requires
	dedicated spacecraft pointing TIRVIM observes
	trough the full
	This channel is
	operated in solar occultation and nadir. In nadir NIR
	is mostly measuring water vapor and dayside oxygen
	emission. In solar occultation NIR provides profiling
	of CO 2 , H 2 O, and the molecular oxygen O 2
	[Fedorova et al. EPSC 2018]. NIR can observe
	occultations together with the two other ACS
	channels MIR and TIRVIM, or together with another
	spectrometer aboard TGO, NOMAD, and TIRVIM.
	The mid-infrared (MIR) channel is a high spectral
	resolution (resolving power of ≥50,000) instrument
	dedicated to solar occultation measurements in the
	2.2-4.4 µm range. MIR targets to accomplish the
	most sensitive measurements of the trace gases in the
	Martian atmosphere. Also the abundant components,
	such as CO 2 , CO, H 2 O are profiled in a broad altitude
	range [Trokhimovskiy et al. EPSC 2018].
	The thermal-infrared channel (TIRVIM) is a Fourier-
	transform spectrometer encompassing the spectral
	range of 1.7-17 µm. TIRVIM is being continuously
	operated in nadir to profile the temperature from the

Acknowledgements

ExoMars is the space mission of ESA and Roscosmos. The ACS experiment is led by IKI Space Research Institute in Moscow. The project acknowledges funding by Roscosmos and CNES. Science operations of ACS are funded by Roscosmos and ESA. OK, AT, AF, NI acknowledge support from the Ministry of Education and Science of the Russian Federation, Grant 14.W03.31.0017.


Modeling of HDO in the Martian atmosphere

HDO and the D/H ratios are important species to study to understand Mars past and present climate, in particular with regards to the evolution through ages of the Matian water cycle. We present here new modeling efforts aimed at rejuvinating the representation of HDO in the LMD Mars GCM motivated by the future comparison with the new observations provided by the Atmospheric Chemistry Suite (ACS) on board the ESA/Roscosmos Trace Gas Orbiter.

Introduction

Mars is known to have had a significant liquid water reservoir on the surface and the D/H ratio is an important tool to estimate the abundance of the early water reservoir on Mars and its evolution with time. The D/H ratio is a measure of the ratio of the current exchangeable water reservoir to the initial exchangeable water reservoir. Many observations from the ground have shown that the current ratio is five times that of the reference in Earth's oceans [START_REF] Encrenaz | New measurements of D/H on Mars using EXES aboard SOFIA[END_REF][START_REF] Krasnopolsky | Variations of the HDO/H2O ratio in the martian atmosphere and loss of water from Mars[END_REF][START_REF] Villanueva | Strong water isotopic anomalies in the martian atmosphere: Probing current and ancient reservoirs[END_REF].

H and D atoms in the upper atmosphere are coming from H 2 O and HDO, their sole precursor in the lower atmosphere. The lower mass of H over D atoms and the fact that H 2 O is preferentially photolysed over HDO [START_REF] Cheng | Photo-induced fractionation of water isotopomers in the Martian atmosphere[END_REF] explain the differential escape of these two elements. Finally, due to differences in vapor pressure for HDO and H 2 O, the solid phase of water is enriched in deuterium compared to the vapor phase. This effect is known as the Vapor Pressure Isotope Effect (VPIE) and can reduce the D/H ratio above the condensation level to values as low as 10% of the D/H ratio near the surface [START_REF] Bertaux | Isotopic fractionation through water vapor condensation: The Deuteropause, a cold trap for deuterium in the atmosphere of Mars[END_REF][START_REF] Fouchet | Vapor Pressure Isotope Fractionation Effects in Planetary Atmospheres: Application to Deuterium[END_REF].

Modeling HDO

These fractionation processes can affect the amount of HDO with latitude, longitude, altitude and with the season. In particular, previous models [START_REF] Montmessin | Modeling the annual cycle of HDO in the Martian atmosphere[END_REF] have shown that an isotopic gradient should appear between the cold regions where condensation occurs and the warmer regions. This leads to a latitudinal gradient of D/H (with variations greater than a factor of 5) between the warm and moist summer hemisphere and the cold and dry winter hemisphere. Yet some observations [START_REF] Villanueva | Strong water isotopic anomalies in the martian atmosphere: Probing current and ancient reservoirs[END_REF] also show longitudinal variations of H/D ratios which are not explained so far. It is therefore essential to model the HDO cycle and the associated processes of fractionation in a 3D general climate model (GCM).

Previous work has been done on 3D GCMs of HDO, in particular around the LMD Mars GCM [START_REF] Montmessin | Modeling the annual cycle of HDO in the Martian atmosphere[END_REF]. Since the GCM has considerably evolved since this first HDO introduction in the modeled water cycle, a reappraisal of HDO predictions is needed to account for the detailed microphsyics that control cloud formation and thus HDO fractionation. With the TGO mission now entering its mission phase, very strong and precise constraints will be soon available to evaluate the GCM prediction capability.

Contribution of ACS

The Trace Gas Orbiter, part of the ExoMars mission, has now reached its operational orbit. Onboard is the Atmospheric Chemistry Suite, a set of spectrometers designed to study the atmosphere of Mars with a specific focus on trace gases such as methane.

ACS combines high resolving power (> 10,000), high accuracy (ppb level) and large spectral coverage (0.7 -17 µm) to a large variety of observation modes [START_REF] Korablev | The atmospheric chemistry suite (ACS) of three spectrometers for the ExoMars 2016 Trace Gas Orbiter[END_REF].

Thanks to its solar occulation capabilities, ACS will be able to provide vertical profiles of HDO and H 2 O from a few kilometers above the surface up to 60 km, reaching precisions up to 1 ppb. It will thus be possible to detect when the abundance of the two species decreases due to condensation, to observe the cloud formation and to measure the subsequent fractionation.

These measurements will constrain the spatial and We will describe here our preliminary work on the update of the HDO model in the LMD Mars GCM and will attempt first comparison with the early TGO/ACS observations.

  European Planetary Science Congress temporal variabilities and help improve our understanding of the HDO and H 2 O cycles.

	EPSC Abstracts Vol. 12, EPSC2018-786, 2018 European Planetary Science Congress 2018 c Author(s) 2018	EPSC

Acknowledgements

ExoMars is the space mission of ESA and Roscosmos. The ACS experiment is led by IKI Space Research Institute in Moscow. The project acknowledges funding by Roscosmos and CNES. Science operations of ACS are funded by Roscosmos and ESA. Science support in IKI is funded by Federal agency of science organizations (FANO). L.R. acknowledges support from the Excellence Laboratory "Exploration Spatiale des Environnements Planétaires (ESEP)".


Hai-Ha Pham
email: npham@em-normandie.fr

Ngoc-Sang Pham
email: pns.pham@gmail.com

Linh Trung

Effects of Wariness on Economic Growth in Overlapping Generations Models

Keywords: wariness, overlapping generations, dynamic efficiency, economic growth, endogenous growth. JEL Classifications: D14, D5, E71, O41

We introduce the notion of wariness in overlapping generations models and explore its effects on economic growth. In an exogenous growth model, under standard assumptions, we prove that the capital stock converges to a steady state. We then explore conditions under which this steady state is increasing (or decreasing) in the wariness level. We also provide a necessary and sufficient condition for the dynamic efficiency of the intertemporal equilibrium. In an endogenous growth model à la Romer (1986), we show that the growth rate of capital stock per capita in the economy with wariness is lower (higher, respectively) than that in the economy without wariness if and only if the capital return is high (low, respectively).

Introduction

Uncertainty plays an important role in our society, including economic activities. An example of uncertainty is the emergence of the COVID-19 pandemic. 1 In a world with uncertainty, some people may worry about the future, prefer to live better now, and enjoy the present. However, others may save more in order to face uncertainties in the future. This behavior concerns the notion of wariness. Motivated by these observations, we aim to investigate the role of wariness in economic growth.

To model the wariness, we consider that individuals take care not only of the discounted sum of utilities but also the minimum utility across time. Formally, when a consumer lives for two periods, we assume that her intertemporal utility is given by Intertemporal utility = u(c) + βu(d) + γ min u(c), u(d) ,

(1)

where c, d represent the consumer's consumption in the present and the future respectively, β is the rate of time preference. The parameter γ ∈ [0, ∞) represents the wariness of the household. When λ = 0, we recover the standard case.

In order to understand whether wariness harms or promotes economic development, we embed the modeling (1) of wariness in a standard overlapping generations model (OLG model, henceforth). We then study the role of wariness on the dynamics of capital stocks and the interplay between wariness and economic development.

First, we study how the saving function of the household (with the capital return given) depends on the wariness. Interestingly, we find that the household's saving under the presence of wariness can be higher or lower than that in the case without wariness. It depends on the relationship between the capital return and the wariness level. The intuition is the following. When the capital return is high, the household's income when old would be high while her income when young would be low. So, in the presence of wariness, the household cares more about her consumption when young, which implies that the household consumes more and saves less when young with respect to the case without wariness. By contrast, the household saves more when the capital return is low.

Second, we investigate the dynamics of intertemporal equilibrium to understand how it depends on wariness. Under mild conditions, we prove that the equilibrium capital path converges to the unique steady state and provide comparative statics. We call the stationary capital return R * the capital return of the economy with stationary consumption (i.e., the consumption when young equals the consumption when old and it does not depend on time). We show that when the wariness is high enough, the steady-state capital stock equals the capital of the economy with stationary consumption. The intuition behind this is that when households really care about the worst consumption over two periods of life, they should choose the same level of consumption when young and old.

The more interesting case is when the wariness level is low. In this case, the steady-state capital stock is increasing (respectively, decreasing) in the wariness level if the stationary capital return is low (respectively, high). Interestingly, the steady-state capital in the economy with wariness may be higher or lower than that in the economy without wariness. It is higher if either (1) the wariness level and the stationary capital return are low or (2) the wariness level is high and the capital return of the economy without wariness is low. We illustrate our theoretical findings with several simulations.

Third, we analyze the effects of wariness in an endogenous growth model à la [START_REF] Romer | Increasing returns and long-run growth[END_REF]. We point out that when the wariness level is quite low, the growth rate of capital is increasing (respectively, decreasing) in the wariness level if the productivity is low (respectively, high). However, when the wariness level is high, the growth rate of capital does not depend on the wariness. We also prove that the capital growth rate in the economy with wariness is higher than that in the economy without wariness if and only if the capital return is low.

Our results in both exogenous and endogenous growth models lead to an important implication: Taking into account the worst situation does not necessarily harm economic growth because it may enhance savings and economic growth, particularly in cases of low capital returns.

Related literature

Our paper is related to the literature on endogenous discounting. [START_REF] Araujo | General equilibrium, wariness and efficient bubbles[END_REF], [START_REF] Araujo | On the efficiency of monetary equilibrium when agents are wary[END_REF] consider the utility function of the form t≥0 β t u(c t )+a inf t≥0 u(c t ) in an infinite-horizon general equilibrium model. However, they focus on the effects of parameter a on asset bubbles while we investigate the growth issue.

Ha-Huy and Nguyen (2022) investigate the optimal capital path of an infinite-horizon model with Ramsey-Rawls criterion where the objective function is t≥0 β t u(c t ) + a inf t≥0 u(c t ). They interpret parameter a as the the importance of equity in the inter-temporal generational evaluation in the choice of the economic agent. They provide conditions under which the optimal capital stock is constant over time or it coincides with the solution of the Ramsey problem (i.e., when the parameter a = 0). Our paper differs from Ha-Huy and Nguyen (2022) in several points. First, we employ a OLG model instead of a model with an infinitely-lived agent as in Ha-Huy and Nguyen (2022); by consequence, the economic meaning of the wariness in our paper is quite different from that of parameter a in Ha-Huy and Nguyen (2022). Second, and more importantly, the capital path of equilibrium in our paper is never constant over time (excepted the case where the initial capital is at the steady state). Third, we also investigate the issue of endogenous growth while Ha-Huy and Nguyen (2022) do not.

Our paper is also related to the literature on the impact of uncertainty. Several papers document a negative relationship between high uncertainty and firms' investment. [START_REF] Kumar | The Effect of Macroeconomic Uncertainty on Firm Decisions[END_REF] use a survey of firms in New Zealand and show that firms' perceived macroeconomic uncertainty leads them to reduce their investment. [START_REF] Bloom | Investment and Subjective Uncertainty[END_REF] use a survey measure of subjective uncertainty collected by the U.S. Census Bureau for approximately 25,000 manufacturing plants and find that there is a strong and robust negative relationship between investment and high subjective uncertainty of plant managers. [START_REF] Fukuda | Knightian uncertainty and poverty trap in a model of economic growth[END_REF] uses a OLG model but focuses on the profit maximization of the firm under Knightian uncertainty. He considers that the productivity of firm is random and the producer faces "uncertainty" in its productivity. By analyzing the effects of this uncertainty on the dynamic properties of equilibrium, he finds that there may be multiple steady states and a poverty trap may arise. [START_REF] Bianchi | The Origins and Effects of Macroeconomic Uncertainty[END_REF] find sizable negative effects of stochastic changes in the volatility of demand-side (rate of time preference) and supply-side (TFP) shocks on consumption, investment, and output.

Different from these papers focusing on the behavior of firms, we study how the wariness affects the savings of households and then the economic growth. Our novel point is that if uncertainties generate a wariness in the households' preferences, then, in some cases, it may be beneficial to the economy because households may save more and by the way improve the investment and economic growth.

The remainder of the paper is organized as follows. In Section 2, we introduce wariness in an exogenous growth model and provide basic properties of households' saving. Section 3 presents the effects of wariness in the exogenous growth model while Section 4 discusses the role of wariness in an endogenous growth model à la [START_REF] Romer | Increasing returns and long-run growth[END_REF]. Section 5 concludes. Formal proofs are gathered in the appendix section.

An OLG Model with wariness

We introduce wariness in a standard two-period OLG model.2

Household and wariness. At period t, N t individuals are born. We assume that the population growth rate is constant over time and denote n ≡ N t+1 /N t . Each consumer-worker lives two periods. When young, she supplies one unit of labor, earns a labor income, consumes c t and saves s t . When old, she receives the income from her saving and consumes d t+1 .

Assume that the utility of this agent represents by the following function

(1 -λ) u(c t ) + βu(d t+1 ) + λ min u(c t ), u(d t+1 )
where λ ∈ [0, 1] represents the wariness of this individual. When λ = 0, we recover the standard case. When λ = 1, the agent only cares about the minimum of her(his) consumption min(u(c t ), u(d t+1 )). Denote γ = λ 1-λ . Then, γ varies between 0 and +∞. It also represents the wariness of the household. The maximization problem of household born at date t is given by (P c,t ) : max

(ct,d t+1 ,st) U (c t , d t ) ≡ u(c t ) + βu(d t+1 ) + γ min(u(c t ), u(d t+1 )) c t + s t ≤ w t (2) d t+1 ≤ R t+1 s t (3) c t , d t+1 , s t ≥ 0, (4)
where R t represents the capital return between time t and t + 1. We assume that:

Assumption 1. The function u is twice continuously differentiable, strictly increasing, strictly concave. u (0) = ∞ and u (∞) = 0.

Production. Technology is represented by a constant returns to scale production function F (K, L) where K t and L t are the aggregate capital and the labor forces.

The firm's profit maximization problem is

(P f,t ) : max Kt,Lt≥0 F (K t , L t ) -R t K t -w t L t (5)
where R t and w t represent the return on capital and the wage rate. Denote k t ≡ K t /L t denotes the capital intensity, f (k t ) ≡ F (k t , 1). We require standard assumptions.

Assumption 2. The function f is twice continuously differentiable, strictly increasing, strictly concave. f (0) = 0, f (0) = ∞ and f (∞) = 0.

The profit maximization implies that

R t = f (k t ) and w t = ω (k t ) , where we denote ω (k t ) ≡ f (k t ) -k t f (k t ) (6)
Under a Cobb-Douglas technology represented by F (K, L) ≡ AK α L 1-α where A > 0 represents the total factor productivity. In this case, we have R t = αAk α-1 t and w t = (1 -α) Ak α t . (2) market clearing conditions:

Intertemporal equilibrium

physical capital : K t+1 = N t s t labor : L t = N t consumption good : s t + c t + d t /n = f (k t ) ,

Properties of the individual saving function

Definition 2. Given β > 0, w > 0, R > 0, we define s β (w, R) the unique solution of the following equation u (w -s) = βRu (Rs) . Lemma 1. Let w > 0, R > 0 be given. If β > β > 0, then s β (w, R) > s β (w, R).

Proof. Denote H(s, β) = u (w -s)-βRu (Rs) . Then, we have ∂H ∂s (s, β) = -u (w -s)-βR 2 u (Rs) > 0. We also observe that ∂H ∂β (s, β) < 0. So, s β (w, R) is strictly increasing in β.

Denote β 1 ≡ β + γ, β 2 ≡ β 1+γ .
The following result shows the optimal solution of households.

Proposition 1. The optimal saving of the household problem (P c,t ) is given by

s t =          s β 1 (w t , R t+1 ) if R t+1 < 1 γ+β wt 1+R t+1 if 1 γ+β ≤ R t+1 ≤ 1+γ β s β 2 (w t , R t+1 ) if R t+1 > 1+γ β . ( 7
)
Proof. See Appendix A.

When there is no wariness, the optimal saving is s

t = s β (w t , R t+1 ). Since β 1 ≥ β ≥ β 2 , Lemma 1 implies that s β 1 (w t , R t+1 ) ≥ s β (w t , R t+1 ) ≥ s β 2 (w t , R t+1 ). ( 8
)
It means that the saving of the household under the presence of wariness can be higher or lower than that in the case without wariness. It depends on the relationship between the interest rate R t and the wariness level γ, as shown in (7).

Let us look at the condition R t+1 < 1 γ+β in (7). This condition can be rewritten as R t+1 < 1 β (the capital return is low) and γ < 1 R t+1 -β (the wariness is low). So, when the capital return and wariness are low, the saving is s β 1 (w t , R t+1 ) which is higher than the saving in the case without wariness s β (w t , R t+1 ). The intuition is that when the capital return is low, the household's expected income when old would be low while her income when young would be high. So, in the presence of low wariness, the household cares more about her consumption when old. This implies that the household consumes less and saves more when young.

A similar interpretation applies for the case R t+1 > 1+γ β (the capital return is high βR t+1 -1 > 0 and the wariness is low γ < βR t+1 -1.

Condition

1 γ+β ≤ R t+1 ≤ 1+γ β is equivalent to γ ≥ max(βR t+1 -1, 1 -βR t+1
) which can be interpreted as the high wariness. In this case, the saving equals wt 1+R t+1 at which the consumptions when young and old are the same.

The following result explores the effects of the wariness on the optimal saving.

Proposition 2. Given ω t > 0 and R t+1 > 0. We denote s t (γ) the optimal saving of the household with the wariness level γ. Let γ 1 < γ 2 . We have different situations.

1. If R t+1 < 1 γ 1 +β then s t (γ 1 ) > s t (γ 2 ). 2. If 1 γ 1 +β ≤ R t+1 ≤ 1+γ 1 β then s t (γ 1 ) = s t (γ 2 ) = wt 1+R t+1 3. If 1+γ 1 β < R t+1 , then s t (γ 1 ) > s t (γ 2 ).
Proof. See Appendix A.

By comparing βR t+1 with 1, we obtain the following result showing the monotonicity of the saving function with respect to the wariness level γ.

Corollary 1.

(1) If βR t+1 < 1 then the optimal saving is increasing in γ for γ > 0.

(2) If βR t+1 > 1 then the optimal saving is decreasing in γ for γ > 0.

Proof. See Appendix A.

In some particular cases, we can explicitly compute the household's saving. Let us firstly consider the logarithmic utility function.

Corollary 2. If u(c) = ln(c), then s β (w t , R t+1 ) = βwt 1+β , ∀β > 0.
In this case, we obtain the optimal saving

s t =          β 1 wt 1+β 1 if R t+1 < 1 γ+β wt 1+R t+1 if 1 γ+β ≤ R t+1 ≤ 1+γ β β 2 wt 1+β 2 if R t+1 > 1+γ β .
When there is no wariness (γ = 0), we recover the standard function of savings:

s t = βwt 1+β .
Let us denote the optimal allocation when there is no wariness.

c o t ≡ w t 1 + β , d o t+1 ≡ βw t 1 + β R t+1 , s o t ≡ βw t 1 + β . ( 9
)
Corollary 2 provides an interesting insight. If the return is high in the sense that R t+1 > 1+γ β , then

c t = 1 + γ 1 + β + γ w t > c o t ≡ w t 1 + β (10) d t+1 = β 1 + β + γ R t+1 w t < d o t+1 ≡ β 1 + β R t+1 w t ( 11
)
s t = β 1 + β + γ w t < s o t ≡ β 1 + β R t+1 w t . ( 12
)
By contrast, when the return rate is low, in the sense that R t+1 < 1 β+γ , then

c t = 1 1+β+γ w t > d t+1 = β+γ 1+β+γ R t+1 w t , s t = β+γ 1+β+γ w t .
When the return rate R t+1 takes a middle value, i.e., 1 β+γ ≤ R t+1 ≤ 1+γ β , then households consume the same amount in both date:

c t = d t+1 = w t R t+1 1+R t+1 , s t = wt 1+R t+1 .
Under the CRRA utility function, we have a similar result.

Corollary 3. If u(c) = c 1-σ 1-σ for 0 < σ, then, by solving the equations (w t -s) -σ = βR t+1 (R t+1 s) -σ , we obtain that s β (w t , R t+1 ) = wt 1+R -1 σ +1 t+1 β -1 σ
, ∀β. By consequence, the optimal saving is determined as follows:

s t =              wt 1+R -1 σ +1 t+1 β -1 σ 1 if R t+1 < 1 γ+β wt 1+R t+1 if 1 γ+β ≤ R t+1 ≤ 1+γ β wt 1+R -1 σ +1 t+1 β -1 σ 2 if R t+1 > 1+γ β .
3 Wariness and exogenous growth

Intertemporal equilibrium: convergence and steady state

We now investigate the intertemporal equilibrium. The market clearing condition requires that nk t+1 = s t which becomes

nk t+1 = s(w t , R t+1 ) =          s β 1 (w t , R t+1 ) if R t+1 < 1 γ+β wt 1+R t+1 if 1 γ+β ≤ R t+1 ≤ 1+γ β s β 2 (w t , R t+1 ) if R t+1 > 1+γ β . ( 13
)
We are interested in the convergence of (k t ). To deal with this question, we need intermediate steps.

Lemma 2. Assume that kf (k) is increasing on the interval (0, ∞). Given k t , let k t+1 determined by consider the Euler equation u (w t -nk t+1 ) = β 1 R t+1 u (R t+1 nk t+1 ) and R t+1 = f (k t+1 ) < 1 γ+β . Then k t+1 is a strictly increasing function of w t = f (k t ) -k t f (k t )
, and so k t .

Proof. It suffices to prove that w t is increasing in k t+1 . Taking the derivative of both sides of the equation u (w t -nk t+1 ) = β 1 R t+1 u (R t+1 nk t+1 ) with respect to k t+1 and noting that R t+1 = f (k t+1 ), we have

( ∂w t ∂k t+1 -n)u (c t ) = β 1 ∂ R t+1 u (R t+1 nk t+1 ) ∂k t+1
or, equivalently,

∂w t ∂k t+1 u (c t ) = nu (c t ) + β 1 ∂ R t+1 u (R t+1 nk t+1 ) ∂k t+1 Since u < 0 and f (k)u nkf (k) is decreasing in k for any k > 0, we have ∂wt ∂k t+1 > 0. Note that w t = f (k t ) -k t f (k t ) is increasing in k t . By consequence, we get that k t+1 is an increasing function of k t . Lemma 3. Assume that f (k)u nkf (k) is decreasing in k for any k > 0. Then k t+1 is a strictly increasing function, denoted by G(k t ), of k t . By consequence, the capital path (k t ) converges.
Proof. By using the same argument in Lemma 2, we can prove that k t+1 is an increasing function of k t in each case in the formula (13).

TO Add: We must determine the image of the function G to ensure that a 1 > 0, a 2 > 0.

Lemma 4. Assume that f (k)u nkf (k) is decreasing in k for any k > 0. At equilibrium, the dynamics of capital is characterized by the following system

nk t+1 = s γ (w t , R t+1 ) =          s β 1 (w t , R t+1 ) if k t > a 1 wt 1+R t+1 if a 2 ≤ k t ≤ a 1 s β 2 (w t , R t+1 ) if k t < a 2 . ( 14
)
where

a 1 ≡ G -1 (f ) -1 1 γ+β and a 2 ≡ G -1 (f ) -1 1+γ β . Proof. By Lemma 3, we have k t+1 = G(k t ) is an increasing function in k t . The rate of return R t+1 = f (k t+1
) is an decreasing function in k t+1 and then decreasing in

k t . Hence the conditions R t+1 ≡ f (G(k t )) < 1 γ+β and R t+1 > 1+γ β are respectively equivalent to k t > a 1 ≡ G -1 (f ) -1 1 γ+β and k t < a 2 ≡ G -1 (f ) -1 1+γ β
. Thus, the system (13) becomes ( 14).

Lemma 5 (finding steady state). Assume that the function

w(k) k(1+f (k)) is strictly decreasing in k. Let k * i , i = 1, 2, 3 be satisfied the following equations nk * 1 = s β 1 ω(k * 1 ), f (k * 1 ) nk * 2 = s β 2 ω(k * 2 ), f (k * 2 ) nk * 3 = ω(k * 3 ) 1 + f (k * 3 )
.

We have that:

(1)f (k * 3 ) < 1 γ+β if and only if f (k * 1 ) < 1 γ+β , and (2) f (k * 3 ) > 1+γ β if and only if f (k * 2 ) > 1+γ β .
Proof. See Appendix A.

The level k * 1 and k * 2 represent the capital steady state of the economy with discount rate β 1 and β 2 respectively. k * 3 is the capital steady state of the economy where the consumptions of young and old generateions are the same.

We now state the main result of this section, which is a direct consequence of Lemmas 2-5.

Proposition 3. Suppose that f (k)u nkf (k) is decreasing in k for all k > 0. Then k t+1 is an increasing function, denoted by G(k t ), of k t . By consequence, the capital path (k t ) monotonically converges.
Assume, in addition, that the function w(k) k(1+f (k)) is strictly decreasing in k. Then, the steady state k * is determined as follows.

1. If f (k * 3 ) < 1 γ+β , then k * is a solution to the equation nk = s β 1 ω(k), f (k) . 2. If f (k * 3 ) > 1+γ β , then k * is a solution to the equation nk = s β 2 ω(k), f (k) . 3. If 1+γ β ≥ f (k * 3 ) ≥ 1 γ+β , then k * = k * 3 , where k * 3 is the unique solution to the equation n = w(k) k(1+f (k)) .
3 Proposition 3 provides a general condition for the global convergence of the equilibrium capital stock. It should be notice that assumptions in Proposition 3 are satisfied in standard specifications. Indeed, if the function cu

(c) is increasing in c or kf (k) is increasing in k, then f (k)u nkf (k) is decreasing in k. 4 Moreover, if f (k) = Ak α , then w(k) k(1+f (k)) = (1-α)Ak α-1 1+αAk α-1 , which is strictly increasing in k. Since k * 3 is the solution to the equation w(k) k(1+f (k)) = n, we have that n = (1-α)A(k * 3 ) α-1 1+f (k * 3 ) , i.e., f (k * 3 ) = αn 1-α-αn . Therefore, we see that 1 + γ β ≥ f (k * 3 ) ≥ 1 γ + β ⇐⇒ β 1 + γ + 1 ≤ 1 -α αn ≤ 1 + γ + β

Effects of wariness on the equilibrium capital path

In the following, we study the effect of wariness on the equilibrium capital path and the steady state.

Definition 3. Denote k s the steady state in the standard economy without wariness, i.e., k s is determined by

nk s = s(ω(k s ), f (k s )).
3 It should be noticed that k * 3 does not depend on γ. 4 Observe that

∂ f (k)u nkf (k) ∂k = f (k)u nkf (k) + f (k)u nkf (k) n f (k) + kf (k) . This is negative if f (k) + kf (k) > 0. Moreover, if u (c) + cu (c) > 0, ∀c, then we have ∂ f (k)u nkf (k) ∂k < u nkf (k) nf (k) -kf (k) + f (k) + kf (k) < 0.
Following de la Croix and Michel (2002) (Proposition 1.11), we introduce an additional assumption.

Assumption 3. For each β > 0, the function

s β (ω(k),f (k)) k is strictly decreasing in k and lim k→∞ s β (ω(k), f (k)) k < n < lim k→0 s β (ω(k), f (k)) k (15) Since s ≤ ω(k) ≤ f (k) and lim k→∞ f (k) k
< n, the first inequality in (15) always holds. It should also be noticed that Assumption 3 holds under standard specifications. Indeed, when f (k) = Ak α , α ∈ (0, 1) and u(c) = ln(c), we have

s β (ω(k),f (k)) k = β(1-α)Ak α-1 1+β
, which decreases from ∞ to 0 when k increases from 0 to ∞. When f (k) =

Ak α , α ∈ (0, 1) and u(c) = c 1-σ 1-σ with σ > 0, we have

s β (ω(k),f (k)) k = (1-α)Ak α-1 1+(αAk α-1 ) -1 σ +1 β -1 σ ,
which decreases from ∞ to 0 when k increases from 0 to ∞. Under Assumption 3, we have the following property.

Lemma 6. Let β > 0 be given. Under Assumption 3, then there exists a unique k satisfying n =

s β (ω(k),f (k)) k
. Moreover, such a k is increasing in β.

We are now ready to state the main result of this section, which provides a full comparative statics showing the effect of the wariness on the capital steady state.

* 2 < k s . 3. If 1+γ β ≥ f (k * 3 ) ≥ 1 γ+β , then the capital path converges to k * = k * 3 , where k * 3 is the unique solution to the equation n = w(k) k(1+f (k)) . Moreover, k * 3 > k s if

and only if the interest rate of the economy without wariness is low (in the sense

that βf (k s ) < 1) Proof. 1. If f (k * 3 ) < 1 γ+β , then k * is a solution to the equation nk = s β 1 ω(k), f (k)
. By Lemma 6, we get that k * = k * 1 is increasing in β 1 and hence in γ.

If f (k *

3 ) > 1+γ β , we use the same argument as in point 1.

Denote

H(k) ≡ w(k) k(1+f (k)) . Since H(k) is strictly decreasing and H(k * 3 ) = n, condition k * 3 > k s is equivalent to H(k s ) > H(k * 3 ) = n, i.e., w(k s ) -nk s > f (k s )nk s . This is equivalent to u (ω(k s ) -s) < u (f (k s )s), i.e., βf (k s ) < 1. First, condition f (k * 3 ) < 1 γ+β can be rewritten as βf (k * 3 ) < 1 and γ < 1 f (k *
3 ) -β, which can be interpreted as low capital return and low wariness respectively. Second, condition f (k * 3 ) > 1+γ β is equivalent to 0 < γ < βf (k * 3 ) -1, which can be interpreted as high capital return and low wariness. Note that k * 3 is the capital return of the economy with stationary consumption (i.e., the consumption when young equals the consumption when old and it does not depend on time).

Therefore, Proposition 4 indicates that under low wariness and low capital return (respectively, high capital return), the steady state capital stock is increasing (respectively, decreasing) in the wariness level.

However, if the wariness level is high enough (in the sense that 1+γ

β ≥ f (k * 3 ) ≥ 1 γ+β , or equivalently, γ ≥ max βf (k * 3 ) -1, 1-βf (k * 3 ) βf (k * 3 )
, then the steady state capital stock equals k * 3 which does not depend on the wariness level. The intuition behind this is that when households really care about the worst consumption over two periods of life, they should choose the same level of consumption when young and old.

Notice that k * 3 is lower than the steady state k s of the economy without wariness if and only if the interest rate of the economy without wariness is high (in the sense that βf (k s ) > 1).

Proposition 4 leads to a direct consequence which compares the steady states of the economies with and without wariness.

Corollary 4.

1. k * ≥ k s if and only if one of the two following conditions hold:

(1) f (k * 3 ) < 1 γ+β ; (2) 1+γ β ≥ f (k * 3 ) ≥ 1 γ+β and βf (k s ) ≤ 1.

k * < k s if and only if one of the two following conditions hold

: (1) f (k * 3 ) > 1+γ β ; (2) 1+γ β ≥ f (k * 3 ) ≥ 1 γ+β and βf (k s ) > 1.
Condition βf (k s ) > 1 (respectively, βf (k s ) < 1) can be named as "high (respectively, low) real interest rate condition". Indeed, we can write β = 1 1+ρ where ρ is the subjective interest rate, and f (k s ) = 1 + r s where r s represents the real capital return at the steady state of the economy without wariness. By consequence, βf (k s ) > 1 becomes 1+rs 1+ρ > 1, or equivalently, r s > ρ.

Tractable cases

In this section, we consider some tractable cases where we can explicitly find the equilibrium capital path and compute the steady state. Denote

σ 1 ≡ 1 1+β+γ , σ 2 ≡ 1+γ 1+β+γ .
Observe that σ 1 is decreasing but σ 2 is increasing in γ. When there is no wariness γ = 0, we have σ 1 = σ 2 = 1 1+β .

Corollary 5. Assume that f (k) = Ak α and u(c) = ln c. At equilibrium, the dynamics of capital is characterized as follows:

nk t+1 =    (1 -σ 2 )(1 -α)Ak α t if k t < a 2 (1 -σ 1 )(1 -α)Ak α t if k t > a 1 (16) nk t+1 + nαAk α t+1 = (1 -α)Ak α t if k t ∈ [a 2 , a 1 ] ( 17
)
where

a i ≡ n αA(1-σ i ) σ i 1 1-α (1 -σ i )(1 -α)A 1 α for i = 1, 2. 5
The unique steady state is determined by

k * =                    k * 1 ≡ (1-σ 1 )(1-α)A n 1 1-α if σ 1 ≥ αn 1-α , i.e., 1-α αn ≥ 1 + β + γ k * 3 ≡ (1-α-nα)A n 1 1-α if σ 1 > αn 1-α > σ 2 , i.e., 1 + β 1+γ < 1-α αn < 1 + β + γ k * 2 ≡ (1-σ 2 )(1-α)A n 1 1-α if σ 2 ≤ αn 1-α , i.e., 1-α αn ≤ 1 + β 1+γ . ( 18
)
Proof. This is just a direct consequence of Lemma 4. See Appendix A for detailed computations.

The following graphic illustrates the relationship k t+1 = G(k t ). When there is no wariness (i.e., γ = 0), we have σ 1 = σ 2 = σ = 1 1+β and hence we recover the standard system nk t+1 = β 1+β (1 -α)Ak α t . In this case, we have lim

t→∞ k t = k s ≡ β(1 -α)A n(1 + β) 1 1-α . ( 19
)
We can verify that condition k * 3 < k s , i.e., 1-α αn < 1+β is equivalent to βf (k s ) > 1.6 Proposition 5 (comparative statics). Assume that f (k) = Ak α and u(c) = ln c.

When γ is low

: γ ≤ 1-α αn -(1 + β), or, equivalently, βf (k s ) ≤ 1+β 1+β+γ then the capital path (k t ) converges to k * 1 which is higher than k s . Notice that βf (k s ) ≤ 1+β
1+β+γ requires a low real interest rate, i.e., βf (k s ) < 1.

2. When 1-α αn ≤ 1 + 1+γ , or, equivalently, βf (k s ) ≥ (1+β)(1+γ) 1+β+γ , the capital path (k t ) converges to k * 2 which is lower than k s . Here, we need that βf (k s ) > 1.

When

1 + β 1+γ < 1-α αn < 1 + β + γ, or, equivalently, 1 + β 1 + β + γ < βf (k s ) < (1 + β)(1 + γ) 1 + β + γ (20) the capital path (k t ) converges to k * 3 . Moreover, k * 3 ≥ k s ⇔ αn 1 -α ≤ 1 1 + β ⇔ βf (k s ) ≤ 1. ( 21
)
Note that we need

β 1+γ -β = β γ 1+γ < 1-α αn -(1 + β) < γ. Proof. k * 2 is decreasing in γ because it is decreasing in σ 2 and σ 2 is increasing in γ. k * 1
is increasing in γ because it is decreasing in σ 1 and σ 1 is decreasing in γ.

Figure 2 shows the capital path for different values of the wariness level.

(1) γ ≤ 1-α αn -(1 + β); (2) 1 + β 1+γ < 1-α αn < 1 + β + γ and 1-α αn ≥ 1 + β. Condition (2) is equivalent to 1 + β ≤ 1-α αn < 1 + β + γ.
By consequence, we obtain the following result.

Corollary 6. Assume that f (k) = Ak α and u(c) = ln c. We have that:

k * ≥ k s if and only if 1 + β ≤ 1-α
αn , or equivalently βf (k s ) ≤ 1. Figure 3 illustrates how the physical capital at the steady state depends on the wariness level γ. On the right hand side, we have 1 + β > 1-α αn and k * < k s , ∀γ > 0. On the left hand side, we have 1 + β < 1-α αn and k * > k s , ∀γ > 0. When we work with a CRRA utility function, we have the following result which is a direct consequence of Propositions 3 and 4. 1-σ with 0 < σ = 1, then at equilibrium, the dynamics of capital is characterized as follows

nk t+1 = s γ (w t , R t+1 ) =                (1-α)Ak α t 1+(αAk α-1 t+1 ) -1 σ +1 β -1 σ 1 if k t > a 1 (1-α)Ak α t 1+αAk α-1 t+1 if a 2 ≤ k t ≤ a 1 (1-α)Ak α t 1+(αAk α-1 t+1 ) -1 σ +1 β -1 σ 2 if k t < a 2 . ( 22
)
where

a i ≡ n αA(1-σ i ) σ i 1 1-α (1 -σ i )(1 -α)A 1 α for i = 1, 2.
The unique steady state is determined by

k * =            k * 1 if 1-α αn > 1 + β + γ k * 3 ≡ (1-α-nα)A n 1 1-α if 1 + β 1+γ ≤ 1-α αn ≤ 1 + β + γ k * 2 if 1-α αn ≤ 1 + β 1+γ . ( 23
)
where k * i is the positive solution of the equation nk

* i = (1-α)A(k * i ) α 1+(αA(k * i ) α-1 ) -1 σ +1 β -1 σ i , for i = 1, 2. 1. When σ 1 ≡ 1 1+β+γ > αn 1-α , i.e., γ is low: γ < 1-α αn -(1 + β), then the capital path (k t ) converges to k * 1 which is higher than k s . 2. When σ 2 ≤ αn 1-α , i.e., 1-α αn ≤ 1 + β 1+γ , the capital path (k t ) converges to k * 2 which is lower than k s . 3. When σ 1 < αn 1-α < σ 2 , i.e., 1 + β 1+γ < 1-α αn < 1 + β + γ, the capital path (k t ) converges to k * 3 . Moreover, k * 3 ≥ k s ⇔ βf (k s ) ≤ 1 ⇔ αn 1 -α ≤ 1 1 + β . ( 24
)
By combining 3 cases, we observe that: k * ≥ k s if and only if 1 + β ≤ 1-α αn , or equivalently βf (k s ) ≤ 1.

Effect of wariness on dynamic efficiency

Following [START_REF] Malinvaud | Capital accumulation and efficient allocation of resources[END_REF] and [START_REF] Cass | On capital overaccumulation in the aggregative, neoclassical model of economic growth: A complete characterization[END_REF], we define the notion of dynamic efficiency.

Definition 4. 1. A sequence of capital stock

(k t ) is a feasible path if f (k t ) -nk t+1 ≥ 0, ∀t ≥ 0.
2. A feasible path (k t ) is efficient if there does not exist another feasible path

(k t ) satisfying f (k t ) -nk t+1 ≥ f (k t ) -nk t+1 , ∀t and there exists at least t 0 ≥ 0 with f (k t 0 ) -nk t 0 +1 > f (k t 0 ) -nk t 0 +1 .
3. An intertemporal equilibrium is dynamically efficient if its capital path is efficient.

According to the consumption good market clearing condition nk t+1 +c t + dt n = f (k t ), any equilibrium capital path is feasible. An interesting question is whether it is efficient.

By using the same argument in Proposition 2.4 in de la Croix and Michel ( 2002), we obtain the following result.

Lemma 7. Under conditions in Proposition 3, an intertemporal equilibrium is dynamically

efficient if f (k * ) > n, and dynamically inefficient if f (k * ) < n.
When f is strictly decreasing and f (0) > n > f (∞), there exists a unique so-called golden rule capital stock k = k GR determined by f (k GR ) = n (note that k GR does not depend on γ). In such a case, the conditions f (k * ) > n and f (k * ) < n are respectively equivalent to k * < k GR and k * > k GR . Hence, we can investigate dynamic efficiency by comparing the steady state k * with the golden rule capital stock.

The following result is a direct consequence of Proposition 4 and Lemma 7.

Proposition 6. Let assumptions in Proposition 3 be satisfied.

1. If f (k * 3 ) < 1 γ+β and Assumption 3 holds, then the intertemporal equilibrium is

dynamically efficient if f (k * 1 ) > n and dynamically inefficient if f (k * 1 ) < n. By consequence, if f (k * 3 ) < 1 γ+β < n, then intertemporal equilibrium is dynamically inefficient. 2. If f (k * 3 ) > 1+γ β and Assumption 3 holds, then the intertemporal equilibrium is dynamically efficient if f (k * 2 ) > n and dynamically inefficient if f (k * 2 ) < n. By consequence, if f (k * 3 ) > 1+γ β > n, then intertemporal equilibrium is dynamically efficient. 3. If 1+γ β ≥ f (k * 3 ) ≥ 1 γ+β , the intertemporal equilibrium is dynamically efficient if f (k * 3 ) > n and dynamically inefficient if f (k * 3 ) > n.
Proposition's point 3 indicates that when the capital return f (k * 3 ) and the wariness have a middle value, the equilibrium is dynamically efficient (inefficient) if there is under-accumulation (over-accumulation) at k * 3 . Points 1 and 2 are more interesting because k * 1 and k * 2 depend on the wariness level γ 2 . Proposition's point 1 suggests that when the capital return f (k *

3 ) is lower than 1 γ+β , the equilibrium is dynamically efficient if there is under-accumulation at k * 1 , i.e., k * 1 < k GR . According to Proposition 4, k * 1 is strictly increasing in the wariness level γ.

So, in this case, the equilibrium is more likely to be dynamically efficient if the wariness is low enough. By contrast, when the capital return f (k * 3 ) is higher than (1 + γ)/β, then the equilibrium is more likely to be dynamically efficient if the wariness is quite high.

When f (k) = Ak α , we find that k GR = αA n 1 1-α . Therefore, we can explicitly find conditions for the dynamic efficiency.

Corollary 8. Assume that f (k) = Ak α , u(c) = u 1-σ
1-σ , ∀0 < σ = 1, and u(c) = ln(c), for σ = 1. We have that:

1. If 1-α nα > 1 + γ + β, then the inter-temporal equilibrium is dynamically efficient if 1-α α < 1 + n 1-1 σ ( β 1+γ ) -1 σ and dynamically inefficient if 1-α α > 1 + n 1-1 σ ( β 1+γ ) -1 σ . 2. If 1-α n < 1 + β 1+γ , then the inter-temporal equilibrium is dynamically efficient if 1-α α < 1+n 1-1 σ (β +γ) -1 σ and dynamically inefficient if 1-α α > 1+n 1-1 σ (β +γ) -1 σ . 3. If 1 + β 1+γ ≤ 1-α n ≤ 1 + γ + β, then the inter-temporal equilibrium is dynamically efficient if α > 1 n+2 and inefficient if α < 1 n+2 . Proof. See Appendix A.

Wariness and endogenous growth

We now introduce endogenous growth à la [START_REF] Romer | Increasing returns and long-run growth[END_REF]. Assume that the production function at date t is f t (k) = A t f (k), where the productivity A t is taken as given by the firm. But, in equilibrium, due to a positive externality, A t is a function of the aggregate capital k t : A t = A(k t ). By consequence, we obtain the dynamical system of (k t )

nk t+1 = s t =          s β 1 (w t , R t+1 ) if R t+1 < 1 γ+β wt 1+R t+1 if 1 γ+β ≤ R t+1 ≤ 1+γ β s β 2 (w t , R t+1 ) if R t+1 > 1+γ β =                s β 1 A(k t ) f (k t ) -k t f (k t ) , A(k t+1 )f (k t+1 ) if A(k t+1 )f (k t+1 ) < 1 γ+β A(kt) f (kt)-ktf (kt) 1+A(k t+1 )f (k t+1 ) if 1 γ+β ≤ A(k t+1 )f (k t+1 ) ≤ 1+γ β s β 2 A(k t ) f (k t ) -k t f (k t ) , A(k t+1 )f (k t+1 ) if A(k t+1 )f (k t+1 ) > 1+γ β .
For the sake of simplicity, we consider the logarithmic utility function u(c) = ln(c) and the Cobb-Douglas production function f (k) = k α . In this case, the wage equals w t = (1 -α)A t k α and the capital return equals R t = αA t k α-1 t . According to Lemma 2, the saving of household is given by

s t =          β 1 wt 1+β 1 if R t+1 < 1 γ+β wt 1+R t+1 if 1 γ+β ≤ R t+1 ≤ 1+γ β β 2 wt 1+β 2 if R t+1 > 1+γ β
where recall that β 1 ≡ β + γ, β 2 ≡ β 1+γ . Assume that the endogenous productivity A t is determined by A t = Ak 1-α t , where A > 0 represents the exogenous productivity of our model. Hence, the wage is

w t = (1 -α)Ak 1-α t k α t = (1 -α)Ak t while the capital return equals R t = αAk 1-α t k α-1 t
= αA, ∀t. Therefore, we obtain the following result. Proposition 7. Consider an endogenous growth model described above. The rate of growth of the capital stock per capita equals

k t+1 k t = g(γ) ≡          β 1 (1-α)A n(1+β 1 ) if αA < 1 γ+β (1-α)A n(1+αA) if 1 γ+β ≤ αA ≤ 1+γ β β 2 (1-α)A n(1+β 2 ) if αA > 1+γ β . ( 25
)
Notice that when there is no wariness (γ = 0), we recover the standard saving s t = βwt 1+β and the rate of capital growth equals g 0 ≡ β(1-α)A n(1+β) . So, our result leads to interesting implications.

1. If (γ + β)αA < 1 (the wariness and productivity are quite low), then the growth rate of physical per capital g(γ) is increasing in the wariness level γ and g(γ) > g 0 .

2. If αA > 1+γ β (the wariness is quite low but the productivity is quite high), then the growth rate of physical per capital g(γ) is decreasing in the wariness level γ and g(γ) < g 0 .

If 1

γ+β ≤ αA ≤ 1+γ β (the productivity is moderate and the wariness is high), then g(γ) ≥ g 0 if and only if αβA ≤ 1 (i.e., the capital return is low).

Observe that g(γ) ≤ g 0 if and only if one of the following conditions holds: (1) αA > 1+γ β ;

(2) (1) The growth rate of capital stock per capita in the economy with wariness is less than that in the economy without wariness, i.e., g(γ) ≤ g 0 if and only if the capital return is high in the sense that αβA ≥ 1.

(2) g(γ) > g 0 if and only if αβA < 1 (the capital return is low).

Conclusion

We have investigated the effects of wariness on the economic dynamics of the economy in OLG models. Under standard assumptions of utility and production functions, we have proved the capital stock of intertemporal equilibrium, in the exogenous growth economy, converges to a steady state. We have also studied how this steady state depends on the wariness. We have shown that whether the effects are positive or negative depend on the structure of the economy, specially the interest rate of the economy without wariness. This insight holds not only for the exogenous growth model but also for the endogenous growth framework à la [START_REF] Romer | Increasing returns and long-run growth[END_REF].

A promising avenue for future research involves extending our work to encompass a stochastic economy with incomplete markets.

A Appendix

Proof of Proposition 1. For the sake of simplicity, we remove the subscript and consider the problem

P (w, R) : max (c,d,s) U (c, d) ≡ u(c) + βu(d) + γ min(u(c), u(d)) (26) c + s ≤ w, d ≤ Rs, c, d, s ≥ 0, (27)
Denote U * the maximum value of the problem P (w, R). We also observe that

U * = max max c+s≤w, d≤Rs c≥d u(c) + (β + γ)u(d) , max c+s≤w, d≤Rs c≤d (1 + γ) u(c) + β 1 + γ u(d) . ( 28
)
Denote

β 1 ≡ β + γ, β 2 ≡ β (1+γ) . We have β 1 > β > β 2 . Consider the first problem (P 1 ) : max (c,d,s) u(c) + β 1 u(d) (29) c + s ≤ w, d ≤ Rs, c ≥ d ≥ 0 (30)
In optimal, we must have c + s = w and d = Rs. So, solving (P 1 ) is equivalent to solving the following problem

(P 1 ) : max s u(w -s) + β 1 u(Rs) (31) w -s ≥ Rs ≥ 0 ⇔ 0 ≤ s ≤ w 1 + R (32) Denote G(s) ≡ u(w -s) + β 1 u(Rs). We have G (s) = -u (w -s) + β 1 Ru (Rs) and G (s) = u (w -s) + β 1 R 2 u (Rs) < 0. Observe that G (0) = +∞ and G ( w 1+R ) = u ( Rw 1+R )(β 1 R -1)
. By consequence, we obtain that:

1. If β 1 R ≥ 1, then c = d = w R 1+R and s = w 1+R . 2. If β 1 R < 1, then s = s β 1 which is a unique solution to the equation G (s) = 0. Consider the second problem (P 2 ) : max (c,d,s) u(c) + β 2 u(d), ( 33
) c + s ≤ w, d ≤ Rs, 0 ≤ c ≤ d. ( 34
)
By using the similar argument, the solution of (P 2 ) is given by

1. If β 2 R ≤ 1, then c = d = w R 1+R and s = w 1+R . 2. If β 2 R > 1, then s = s β 2 which is a unique solution to the equation -u (w -s) + β 2 Ru (Rs) = 0.
We now come back to the original problem.

U * = max max c+s≤w, d≤Rs c≥d u(c) + (β + γ)u(d) , max c+s≤w, d≤Rs c≤d (1 + γ) u(c) + β 1 + γ u(d) . ( 35
)
We consider three cases.

1. If β 1 R < 1, then the solution of (P 1 ) satisfies s = s β 1 . Since β 1 > β 2 , we have β 2 R < 1 which implies that the solution of (P 2 ) satisfies c = d = w R 1+R and s = w 1+R . Note that the allocation c = d = w R 1+R and s = w 1+R is feasible for the problem (P 1 ). This follows that the maximum value of the problem (P 1 ) is higher

than (1 + γ) u(w R 1+R ) + β 1+γ u(w R 1+R ) = (1 + γ + β)u( Rw 1+R
). So, the solution of the problem (P (w, R)) satisfies s = s β 1 .

2. If β 2 R > 1, then by using the same argument, we obtain that s = s β 2 .

If β

1 R ≥ 1 ≥ β 2 R, then c = d = w R 1+R and s = w 1+R . Proof of Proposition 2. Given γ, denote β 1 (γ) ≡ β + γ, β 2 (γ) ≡ β 1+γ . Since 0 < γ 1 < γ 2 , we have 1 γ 2 +β < 1 γ 1 +β < 1+γ 1 β < 1+γ 2 β . So, we consider 5 cases. 1. If R t+1 < 1 γ 2 +β then R t+1 < 1 γ 1 +β . According to Proposition 1, we have s t (γ 1 ) = s β 1 (γ 1 ) (ω t , R t+1 ) and s t (γ 2 ) = s β 1 (γ 2 ) (ω t , R t+1 ). Thanks to Lemma 1, we have s β 1 (γ 1 ) (ω t , R t+1 ) < s β 1 (γ 2 ) (ω t , R t+1 ). Hence s t (γ 1 ) < s t (γ 2 ). 2. If 1 γ 2 +β ≤ R t+1 < 1 γ 1 +β then we have s t (γ 1 ) = s β 1 (γ 1 ) (ω t , R t+1 ) < wt 1+R t+1 . Since 1 γ 2 +β ≤ R t+1 < 1+γ 2 γ , Proposition 1 implies that s t (γ 2 ) = wt 1+R t+1 . So s t (γ 1 ) < s t (γ 2 ). 3. It is a direct consequence of Proposition 1. 4. If 1+γ 1 β < R t+1 ≤ 1+γ 2 β , we have s t (γ 1 ) = s β 2 (γ 1 ) (ω t , R t+1 ) and s t (γ 2 ) = wt 1+R t+1 . So, s t (γ 1 ) > s t (γ 2 ). 5. If R t+1 > 1+γ 2
β , by using the similar argument as point 1, we get that s

t (γ 1 ) > s t (γ 2 ). 4. If f (k * 2 ) > 1+γ β , then s (2) (k * 2 ) = nk * 2 > w(k * 2 ) 1+f (k * 2 ) . So, n = ω(k * 3 ) k * 3 (1+f (k * 3 )) > w(k * 2 ) k * 2 (1+f (k * 2 )) . Thus, k * 2 > k * 3 , f (k * 2 ) < f (k * 3 ). 5. If f (k * 2 ) < 1+γ β , then s (2) (k * 2 ) = nk * 2 < w(k * 2 ) 1+f (k * 2 ) . So, n = ω(k * 3 ) k * 3 (1+f (k * 3 )) < w(k * 2 ) k * 2 (1+f (k * 2 )) . Thus, k * 3 > k * 2 , f (k * 3 ) < f (k * 2 ) < 1+γ β . 6. If f (k * 2 ) = 1+γ β , then s (2) (k * 2 ) = nk * 2 = w(k * 2 ) 1+f (k * 2 ) . So, k * 2 = k * 3 .
Proof of Corollary 1. (1) Let βR t+1 < 1. We consider two cases:

• For γ ∈ (0, 1 R t+1 -β), Proposition 1 implies that s t (γ) = s β 1 (γ) which is increasing in γ. • For 1 R t+1 -β ≤ γ, we have 1 β+γ ≤ R t+1 < 1 β < 1+γ β .
Then Proposition 1 implies that s t = wt 1+R t+1 which does not depend on γ.

(2) Let βR t+1 > 1. We consider two cases:

• For γ < βR t+1 -1, i.e., 1+γ β < R t+1 , Proposition 1 implies that s t (γ) = s β 2 (γ) which is decreasing in γ.

• For γ ≥ βR t+1 -1, we have 1 β < R t+1 < 1+γ β . Then Proposition 1 implies that s t = wt 1+R t+1 which does not depend on γ.

Proof of Corollary 5. First we verify that the conditions Lemma 4 hold when u(c) = ln c and f (k) = Ak α . In fact, we have ω

(k) = (1 -α)Ak α . Thus the function h(k) = ω(k) k = (1 -α)Ak α-1 is decreasing in k and then ω(k) k(1+f (k)
) is also decreasing in k. The function f (k)u (nkf (k)) = 1 nk is also decreasing in k. The rate of return R t+1 = f (k t+1 ) = αAk α-1 t+1 . From Lemma 4, the dynamic of capital at equilibrim satisfies the following equation

nk t+1 =          β 1 ω(kt) 1+β 1 = β 1 (1-α)Ak α t 1+β 1 = (1 -σ 1 )(1 -α)Ak α t if αAk α-1 t+1 < 1 γ+β (1-α)Ak α 1+αAk α-1 t+1 . if 1 γ+β ≤ αAk α-1 t+1 ≤ 1+γ β (1 -σ 2 )(1 -α)Ak α t if αAk α-1 t+1 > 1+γ β .
When αAk α-1 t+1 < 1 γ+β , we have k t+1 = G(k t ) with G(k) = G 1 (k) = 1 n (1-σ 1 )(1-α)Ak α . By consequence, we can check that condition αAk α-1 t+1 < 1 γ+β becomes k 1 > a 1 . Similarly, αAk α-1 t+1 > 1+γ β becomes k t < a 2 while 1 γ+β ≤ αAk α-1 t+1 ≤ 1+γ β becomes a 2 ≤ k t ≤ a 1 . As a consequence, we obtain ( 16) and ( 17).

Proof of Corollary 7. Remind that k * 1 , k s and k * 2 are respectively the solution of the following equations

g 1 (k) ≡ (1 -α)Ak α-1 1 + (αAk α-1 ) -1 σ +1 β -1 σ 1 = n g s (k) ≡ (1 -α)Ak α-1 1 + (αAk α-1 ) -1 σ +1 β -1 σ = n g 2 (k) ≡ (1 -α)Ak α-1 1 + (αAk α-1 ) -1 σ +1 β -1 σ 2 = n
Since the function αAk α-1 is strictly decreasing in k and the function

x 1+β -1 σ x 1-1 σ = 1 x -1 +β -1 σ x -1
σ is strictly increasing in x > 0, then the function g 1 (k), g s (k), g 2 (k) are strictly decreasing in k. Since β 1 = β + γ and β 2 = β 1+γ , we obtain that k * 1 is strictly decreasing in γ and k * 2 is strictly increasing in γ. Moreover,

k * 1 > k s > k * 2 .
Proof of Corollary 8. We firstly consider the case u(c) = ln c.

1. If 1-α nα > 1 + γ + β then from Corollary 5, we have

k * = k * 1 = (1-σ 1 )(1-α)A n 1 1-α . The condition k GR > k * is equivalent to (1 -σ 1 )(1 -α)A n < αA n .
Using the fact that σ 1 = 1 1+γ+β , we obtain the equivalent condition 1-α α < 1-σ 1 = 1 + 1 β+γ . Hence k * < k GR if and only if 1-α α < 1 + 1 β+γ . Inversely, we see that k * > k GR if and only if 1-α α > 1 + 1 β+γ .

2. If 1-α αn ≤ 1 + β 1+γ , then from Corollary 5, we have k * = k * 2 = (1-σ 2 )(1-α)A n 1 1-α . So, we find

f (k * ) = αA (1 -σ 2 )(1 -α)A n α-1 1-α = αn (1 -σ 2 )(1 -α)
.

By consequence, condition f (k * ) > n is equivalent to 1-α α < 1 1-σ 2 . Recall that σ 2 ≡ 1+γ 1+β+γ . So, 1 1-σ 2 = 1+β+γ β = 1 + 1+γ β . Hence, f (k * ) > n becomes

1-α α < 1 + 1+γ β while f (k * ) < n becomes 1-α α > 1 + 1+γ β . 3. If 1+ β 1+γ ≤ 1-α n ≤ 1+γ +β then k * = k * 3 = (1-α-nα)A n 1 1-α .
Hence the condition k * < k GR satisfies if and only if (1-α-nα)A n < αA n which is equivalent to α > 1 n+2 . So, we obtain our result.

We now consider the case u(c) = u 1-σ 1-σ , ∀0 < σ = 1.

1. If 1-α n > 1 + γ + β then k * = k * 1 is the solution of the equation g 1 (k) ≡

(1-α)Ak α-1

1+(αAk α-1 ) -1 σ +1 β -1 σ 1 = n.
Here, recall that β 1 ≡ β + γ, β 2 ≡ β 1+γ .

We can verify that g 1 (k) is a strictly decreasing function and

g 1 (k GR ) = 1-α α n 1+n 1-1 σ β -1 σ 1 < n = g 1 (k * 1 ) if and only if 1-α α 1+n 1-1 σ β -1 σ 1 < 1. Hence k *
1 < k GR and then the inter-temporal equilibrium is dynamically efficient if 1-α α < 1 + n 1-1 σ β -1 σ 1 . Inversely, the inter-temporal equilibrium is inefficient.

If 1-α

n < 1 + β 1+γ then k * = k * 2 is the solution of the equation

g 2 (k) ≡ (1 -α)Ak α-1 1 + (αAk α-1 ) -1 σ +1 β -1 σ 2 = n.
We can verify that g 2 (k) is a strictly decreasing function and then we have

g 2 (k GR ) = n 1+n 1-1 σ β -1 σ 2 1-α α < n = g 2 (k *
2 ) if and only if

(1-α)

α 1+n 1-1 σ β -1 σ 2 < 1 .
Hence k * 2 < k GR and then the inter-temporal equilibrium is dynamically efficient if

1-α α < 1 + n 1-1 σ β -1 σ 2 and inefficient if 1-α α > 1 + n 1-1 σ β -1 σ 2 .
3. We use the same argument in the proof of part 3 of the case u(c) = ln(c).

Proposition 4 .

 4 Let assumptions in Proposition 3 be satisfied. 1. If f (k * 3 ) < 1 γ+β and Assumption 3 holds, then the capital path converges to k * = k * 1 which is the unique solution to the equation nk = s β 1 ω(k), f (k) . Moreover, k * 1 is strictly increasing in the wariness level γ and k * 1 > k s . 2. If f (k * 3 ) > 1+γ β and Assumption 3 holds, then the capital path converges to k * = k * 2 which is the unique solution to the equation nk = s β 2 ω(k), f (k) . Moreover, k * 2 is strictly decreasing in the wariness level γ and k

Figure 1 :

 1 Figure 1: G(k) with different wariness levels for A = 8, α = 0.3, β = 0.7, n = 1.3.

Figure 2 :

 2 Figure 2: Dynamic capital with different wariness levels γ for β = 0.7, α = 0.3, n = 1.3, A = 8.

Figure 3 :

 3 Figure 3: The capital at the steady state versus On the left: A = 5, n = 1.2, β = 0.6, α = 0.3. On the right: A = 5, n = 1.6, β = 0.9, α = 0.3.

Definition 1. An

  intertemporal equilibrium is a positive sequence (R t , w t , c t , d t+1 , s t , K t+1 , L t ) t≥0 which satisfies the following conditions:(1) given the sequence (R t , w t ), the allocation (K t , L t ) is a solution to the problem (P f,t ) and the allocation (c t , s t , d t+1 ) is a solution to the problem (P t ),

See de la Croix and Michel (2002) for an introduction of OLG models.

We see that a i is decreasing in σ i , for i = 1, 2. Therefore, a 1 is increasing but a 2 is decreasing in γ.

Indeed, we observe that f (k s ) = αA(k s ) α-1 = αA n(1+β) β(1-α)A = n α(1+β) β(1-α) . Thus, βf (k s ) = n(1+β)α (1-α) .

Proof of Lemma 5. For the sake of simplicity, we write s (i)

is strictky decreasing for any k > 0, then we get that

2. By using the same argument, we have that:


Pascale Desrumaux

Eric Dose

Pascal Malola

Christine Jeoffrion

Nathalie Leroy-Frémont

may come

Moral harassment and lack of support from colleagues and hierarchy...

Harcèlement moral et manque de soutien des collègues et de la hiérarchie… Se soumettre ou se démettre ? Des besoins frustrés à l'intention de quitter Pascale DESRUMAUX, Professeur de Psychologie du travail et des organisations Univ. Lille, EA 4072 -PSITEC -Psychologie : Interactions, Temps, Émotions, Cognition, F-59000 Lille, France. pascale.desrumaux@univ-lille3.fr Eric DOSE, Univ. Lille, EA 4072 -PSITEC -Psychologie : Interactions, Temps, Émotions, Cognition, F-59000 Lille, France. Mail : eric.dose@outlook.com Pascal MALOLA, Univ. Lille, EA 4072 -PSITEC -Psychologie : Interactions, Temps, Émotions, Cognition, F-59000 Lille, France. pascalmalola@yahoo.fr Nathalie LEROY, IUT de Valenciennes, Responsable Mobilité et développement Qualité, FORMASUP, France. nathalie.leroy@univ-valenciennes.fr Mots Clés : harcèlement moral, soutien social des collègues et de la hiérarchie, frustration des besoins, intention de quitter 1. Problématique : Se développant dans les organisations en raison de techniques de management qui nient les besoins psychologiques (BP) et donnent priorité à une gestion centrée sur des objectifs financiers, le harcèlement moral au travail (HMT) [START_REF] Einarsen | Workplace bullying: Development in theory, research and practice[END_REF] concentre une synergie d'agissements et de techniques mises en oeuvre (Hirigoyen, 1998 ;Leyman, 1990) visant à réduire les solidarités, à isoler les personnes et à les maltraiter ou les terroriser psychologiquement. Ces pratiques, induites par la recherche de gains organisationnels, détruisent les personnes en niant leurs BP et génèrent perte de sens, émotions (Desrumaux, et al, 2018) ainsi que des risques de passage à l'acte (Nicolas, et al., 2016). La détérioration des climats limitent le soutien et entrainent une fuite des ressources humaines (brain drain) (Notelaers & De Witte, 2003 ;Poilpot-Rocaboy et al, 2011). 2. Méthodologie :Partant des modèles instanciant les effets du harcèlement moral au travail (Desrumaux & Malola, 2014), sont évalués d'une part, les effets du harcèlement moral au travail et de la frustration des BP (autonomie, compétence et affiliation sociale) sur l'intention de quitter l'organisation et d'autre part, les effets médiateurs du soutien social reçu des collègues et de la hiérarchie entre le harcèlement moral au travail et la frustration des BP et l'intention de départ. Un questionnaire a été rempli par des salariés (N = 266) de la région Hauts de France. 3. Résultats : Les analyses de régression montrent un effet médiateur partiel du soutien des collègues entre le HMT et l'intention de départ alors que le soutien des collègues ne médiatise pas le lien entre la frustration des BP et l'intention de quitter. Le soutien de la hiérarchie exerce un effet médiateur partiel dans la relation entre le harcèlement moral au travail et l'intention de départ. Un effet partiel du soutien de la hiérarchie est observé entre la frustration des besoins psychologiques et l'intention de quitter. 4. Discussion : Le harcèlement moral au travail et la frustration des besoins psychologiques participent aux intentions de départ particulièrement en l'absence de soutien. Les politiques organisationnelles pourraient mettre en place des plans de préventions contre le harcèlement moral au travail alertant sur le caractère déshumanisant et non éthique des modes de management financiers et développer des mesures visant à se centrer sur les BP des salariés et sur les conduites citoyennes au travail. Desrumaux, P., Jeoffrion, C., Bouterfas, N., De Bosscher, S., & Boundenghan, M. (2018). Workplace bullying: how do bystanders' emotions and the type of bullying influence their willingness to help-giving? Nordic Psychology, doi.org/10.1080Psychology, doi.org/10. /19012276.2018.1430610 .1430610