Damien Besancenot

Radu Vranceanu

MIGRATORY POLICY IN DEVELOPING COUNTRIES: HOW TO BRING BEST PEOPLE BACK?

Keywords: Temporary migration, Return migrants, Hybrid Bayesian Equilibrium, Signaling model. JEL Classification: J61, F22, D82

This paper analyzes the decision of a migrant to return or stay within the framework of a signaling model with exogenous migratory costs. If employers have only imperfect information about the type of a worker and good workers migrate, bad workers might copy their strategy in order to get the same high wage as the good workers. Employers will therefore reduce the wage they pay to migrants and good workers incur a loss compared to the perfect information setup. In one hybrid equilibrium of the game, the more bad workers migrate, the higher the incentive for good workers to come back. Policy implications follow.

Introduction

Compared to the other dimensions of economic globalization --international trade and movements of capital --international flows of labour used to be traditionally small. The existence of a system of double barriers, with restrictions on people to leave poor countries and restrictions to reach the developed ones, can explain both why immigration flows were small and why most of the migrants were of the permanent type. However, in the last few years, subject to an aging population and shortages of labor in some sectors, many governments in the developed countries become more favorable to open their countries to foreign workers. True, political support to immigration of skilled workers is much stronger than for the non skilled. To the contrary, in the developing countries, many policymakers are upset about the "brain-drain" phenomenon, where skilled workers are attracted by the perspective of higher living standards in the developed countries, thus depleting what seems to be a very scarce resource in these areas. Migration of the unskilled workforce is seen as a less harmful phenomenon, especially if unemployment in the developing country is high.

In 2005-2007, twelve relatively poor Eastern European countries joined the European Union. This created an exceptional opportunity for workers form the East to live and work in the West (Mansour and Quillin, 2006). According to the European Commission's latest report on crossborder labor mobility (European Commission, 2008), more and more workers from the EU's 12 newest member states have been relocating to the Western regions of Europe since being allowed to move freely. However, their numbers remain small as compared to the population of the host countries. For instance, the number of Bulgarians and Romanians who found a job in one of the EU's 15 older member states grew to 1.6 million in 2007, that is 0.4 percent of the total population of the EU's richest nations, up from 1.3 million in 2006 --when those two countries

were not yet part of the EU. Citizens from Poland and from the other nine countries that joined the EU in 2004 now account for 0.5 percent of the resident population of the EU's richest nations, up from 0.3 percent in 2004.

Another element to be taken into account when analyzing international movements of labor pertains to the distinction between temporary and permanent migration. At difference with permanent migrants, temporary migrants work abroad for a pre-determined time period, then return. It may be argued that workers hired under a fixed-term contract have no other choice but to return, so there is no genuine economic decision to analyze. However, in many cases, workers who go abroad on temporary contracts can stay longer should they really want it.

In general, temporary migration is seen as bringing a net positive contribution to the development of the sending country. True, migration, even if it is temporary, diminishes the amount of available labor at home. If there are labor shortages, this first impact of temporary migration is negative; yet, if the unemployment in the developing country is high, temporary migration can alleviate the pressure on the welfare system. Other benefits stem from the improvement in human capital connected to short spells of work abroad and the investment in the origin country of the migrants' savings [START_REF] Ruhs | The potential of temporary migration programmes in the future international migration policy[END_REF]. It turns out that a non-negligible proportion of the new East-West European migrants are temporary. From the outset, they do not commit on staying forever in the host country. They consider the case for working for some time in the West, then to come back [START_REF] Dustmann | Temporary migration and economic assimilation[END_REF][START_REF] Dustmann | Return migration: theory and empirical evidence from the UK[END_REF]. 3 For example, data for the UK and Ireland suggest that around half of the EU-8 citizens who have come to work in the UK since 2004 may have already left the country again (European Commission, 2008).

Several economists aimed at explaining the migrant's decision to return. In general, in these analyses, critical elements are the costs and benefits of migration. Benefits are most often interpreted as a better wage, given that in general migrants come from less developed regions (Hicks, 1932). Costs to migration measure some form of disutility connected to living far from one's friends and family. A standard model of return migration was worked out by [START_REF] Borjas | Who leaves? The outmigration of the foreign-born[END_REF], building on the methodology introduced by [START_REF] Borjas | Self-selection and the earnings of immigrants[END_REF] in an early paper. 4The basic framework is a self-selection model with stochastic income. From the outset, skill distribution of the migrants dominates the skill distribution of the stayers. Upon arriving in the rich country, the migrant makes a draw from a known distribution of wages; if the wage is low enough, he will return, if not he will stay. If he returns, he will make again a draw from a random distribution. The decision to migrate, then to return is driven by a comparison between the actual and expected gains. One important conclusion is that if both cost of migrating and the cost of returning increase, the frequency of return migrants declines. 5 Stark (1995) works out a migration model with asymmetric information about the type of the migrants, where migrants' true skills are revealed with a lag. When low-skilled workers are no longer pooled with high-skilled workers, they return. [START_REF] Dustmann | Return migration: theory and empirical evidence from the UK[END_REF] analyze the worker's decision to migrate, stay or return within a continuous time model with investment in human capital; the worker decides on the optimal migration and return moments such as to maximize intertemporal utility, given that he must weight the cost of living among foreigners against the benefit of enhancing his skills by working in a more demanding environment. Indeed, migrants might take advantage of their stay abroad to foster their productivity by acquiring new skills, learning new production and organization methods, foreign languages, or join useful professional networks. 6Some economists have unveiled from the data that return migrants tend to find a job easier, and that sometimes they get better wages than similar left-home workers. For instance, [START_REF] Iara | Skill diffusion by temporary migration? Returns to Western European working experience in the EU accesssion countries, Working Paper #210[END_REF] found that young Eastern European man who moved and came back tend to claim in average 30% higher earnings than equivalent left-home workers. [START_REF] Hazans | Post-enlargement return migrants' earnings premium: Evidence from Latvia[END_REF] shows that Latvian return migrants command an earnings' premium as high as 15% compared to identical stayers. 7 Such empirical evidence is consistent with both the self-selection cum bad luck model or the human capital accumulation model.

This paper develops an original, complementary explanation that is consistent with these stylized facts. Migration is analyzed as a signal for hidden productivity along the lines of the classical model by [START_REF] Spence | Job market signaling[END_REF]. Our analysis is cast as a simple two-period game, with two types of workers --highly productive and less productive, under imperfect information of the employers about the type of the worker. To neutralize one important cause of migration, we assume that worker's productivity is the same in the sending and the destination country. Firms pay the worker a wage identical to the expected productivity. Migration can have signaling virtues only if migratory costs are larger for bad workers than for good workers. In this paper, we build on a cost structure where all good workers migrate at the first period. At the second period, they can either return back home or and stay in the host country. Despite bearing higher migratory costs, bad workers can choose to migrate in order to be perceived as good workers and get a better wage. If they migrate, they can either return or stay. The game presents various equilibria depending on parameter values. In some cases, equilibria are multiple: which one actually materializes depends on firm equilibrium beliefs. The most interesting case is a complete hybrid equilibrium where both good and bad workers implement mixed strategies: some but not all good workers return, some but not all bad workers migrate and return.

Besides the productivity differential, an important variable in our analysis is the migratory cost. Such a cost has a strong psychological component, related to the energy that the migrant needs to spend in order to adapt to a different environment. Other components such as traveling costs have a "transaction cost" nature, and can be easier converted into monetary expenses. Governments in the origin country can have an impact on such costs. They may for instance provide free language courses for prospective emigrants, subsidize transport (for instance, subsidize a discount journey every year), set up cultural centers in the destination country, provide good consular services, and so on. By choosing their country of destination, and the "distance" between home and destination, good workers can also have a bearing on migratory costs. In this context, the concept of distance is not only geographical, but also cultural. Notice that when migration is used as a signaling device, bad workers can only follow good workers, they have no word to say about choosing the destination.

The logic of strategic signaling has already been applied to the one-way migration decision by Katz and Stark (1987). In their model, employers in the rich country know only the distribution of immigrants' skills and pay them the average expected productivity. Imperfect information is responsible for inefficient migration patterns as compared to the perfect information setup. They show that the highly skilled workers may want to pay the price of signalling, should such a signalling device exist.

Our analysis builds on several simplifying assumptions: at difference with [START_REF] Borjas | Who leaves? The outmigration of the foreign-born[END_REF] and other companion papers, in our setup wages are deterministic (there is no human capital accumulation is weak. 7 The test controls for demographic characteristics, education, foreign and unemployment experience of family members. random shock to be discovered by the migrant once abroad). We also rule out the possibility of investing in human capital, which was an essential element in the paper by [START_REF] Dustmann | Return migration: theory and empirical evidence from the UK[END_REF]. These simplifications will allow us to focus on the pure signaling virtues of migratory strategies. As in the permanent migration analysis of Katz and Stark (1987), a migrant's wage depends not only on his own characteristics, hard work and luck, but also on the various strategies of the other workers.

The paper is organized as follows. The next section introduces the model. Section 3 presents the pure strategy equilibria of the game. Hybrid equilibria are analyzed in section 4. Policy issues are introduced in section 4. The last section presents the conclusion.

The model

Main assumptions

There are two countries, the sending and the destination country. The internationally mobile labor force of the sending country is normalized to 1. A worker has the same productivity in the sending and the destination country. In the sending country there are  highly productive workers, with a productivity  g (good type) and 1 - workers with a low productivity  b (bad type) , with  g   b . Worker productivity is a deterministic variable; it is the same in the sending and the host country. Firms make zero expected profits: they pay wages identical to the expected productivity of each worker.

Migratory costs per period are c g for the type g and c b for the type b, with c g   g and c b   b . Furthermore, migration can be a useful signaling device only if c g  c b . We assume that this condition holds hereafter. 8Workers live for two periods. At the first period, workers can migrate ( M ) or not ( M ̄); at the second period, those who migrated at the first period can return home ( R, or stay in the host country ( S. In order to keep the model as simple as possible, we restrict the range of parameters such as to rule out the cases where good workers have no incentive migrate even if this action allows them to signal their type. This implies that the migratory cost of the good workers is low enough, more precisely that c g  21 - g - b .9

We define by w 1 (and w 2 ) the wage expected by a worker at the end of period 1 (and respectively period 2), depending on the history of the game (i.e., the observed decisions).

We denote by  the proportion of good migrants who choose to return and by 1 - the proportion of good migrants who choose to stay.

Bad workers can either migrate or not; if they migrate they can either stay or return. The proportion of bad workers who leave and stay is q, the proportion of workers who leave and return is p, thus the proportion of bad workers who do not migrate is 1p -q.

The basic sequence of decisions is summarized by a decision tree in Figure 1.

Type g

θ g Type b θ b α 1-α Payoffs M M M _ M _ Stay Return Stay Return w 1 (M)+w 2 (M,S) -2c g w 1 (M)+w 2 (M,R) -c g w 1 (M)+w 2 (M,R) -c b w 1 (M)+w 2 (M,S) -2c b w 1 (M)+w 2 (M,M) _ _ 1-π π p q 1-p-q _ Type g θ g Type b θ b α 1-α Payoffs M M M _ M _ Stay Return Stay Return w 1 (M)+w 2 (M,S) -2c g w 1 (M)+w 2 (M,R) -c g w 1 (M)+w 2 (M,R) -c b w 1 (M)+w 2 (M,S) -2c b w 1 (M)+w 2 (M,M) _ _ 1-π π p q 1-p-q _ Figure 1.

Decision tree

At the beginning of the game, Nature decides on the type of worker, b or g. At time t  1, good workers migrate, bad workers have the choice between migrating or not; at t  2 all migrants, bad or good, have the choice between returning home or staying in the host country. There are three main strategies: M ̄, M ̄ -do not migrate, M, S --migrate and stay and M, R --migrate and return. Only the first one reveals the type of worker. The two others can be implemented by both good and bad workers, hence the dotted lines that connects the similar strategies.

Strategies, beliefs, payoffs

We define the strategy of the g -type worker by a linear combination of the two pure strategies: M, R and M, S . Let us define by  s the conditional probability that the worker is good given his past strategy s . Under our assumptions, employers' beliefs are: a) After the first period:

 M  Pr g |M  PrM| g  Pr g  PrM  PrM| g  Pr g  PrM| g  Pr g   PrM| b  Pr b      1 -p  q  M ̄ Pr g |M ̄  0 b) After the second period:  MR  Pr g |M, R  PrM, R| g  Pr g  PrM, R| g  Pr g   PrM, R| b  Pr b      1 -p  MS  Pr g |M, S  PrM, S| g  Pr g  PrM, S| g  Pr g   PrM, S| b  Pr b   1 - 1 -  1 -q  M ̄M ̄ Pr g |M ̄, M ̄  0
Since firms pay wages identical to the worker's expected productivity, workers' equilibrium payoffs contingent upon their strategy are:

U b M ̄, M ̄  w 1 M ̄  w 2 M ̄  2 b U i M, R  w 1 M -c i  w 2 M, R   g  1 -p  q b   1 -p  q -c i   g  1 -p b   1 -p U i M, S  w 1 M  w 2 M, S -2c i   g  1 -p  q b   1 -p  q  1 - g  1 -q b 1 -  1 -q -2c i
where i ∈ g, b .

A Nash equilibrium of this game is defined as a situation where workers choose their optimal migratory strategy given employers' beliefs, and employers' beliefs are correct given workers' optimal migratory strategies. We can distinguish between three types of equilibria: a separating configuration --where each type of worker chooses a specific migratory strategy, a pooling configuration --where all workers carry out the same migratory strategy, and a hybrid configuration --where at least one type of worker is randomizing between pure strategies.

The list of all logically possible combinations of strategies can be drawn easily. Hereafter we will analyze only on the feasible equilibria. 10 We firstly analyze the pure strategy equilibria of the game then turn to mixed strategy equilibria.

Pure strategy equilibria

Proposition 1. The game presents a socially efficient separating equilibrium where bad workers do not migrate M ̄, M ̄ and good workers migrate and return M, R , under the sufficient and necessary condition:

 g - b   c b 2 . Proof See Appendix A
If the migratory cost of bad workers is too high, they do not migrate and migration is a strategy specific to the good worker. The latter has no incentive to stay for good, since migration signals him as a good worker (and he can spare the migratory cost at the second period).

In this equilibrium, firms in the origin country can use the migratory track record of an individual as an efficient screening device. Furthermore, because high productivity workers come back, overall output in the origin country edges up.

Proposition 2. The game presents a socially inefficient separating equilibrium where bad workers do not migrate M ̄, M ̄ and good workers migrate and stay M, S , under the sufficient and necessary condition:

c g   g - b   c b .
10 The proof of inexistence of the other equilibria can be provided on request.

Proof See Appendix A

In this equilibrium, employers in the sending country do not expect good workers to come back. Equilibrium beliefs are such that the return strategy is associated to a bad worker. Hence, such an equilibrium can exist only if the migratory cost of good workers is not too high. To the opposite, the migratory cost c b of the bad workers must be large enough to prevent them from copying the strategy of the good workers.

Compared to the former efficient separating equilibrium, at time t  2 the economy-wide output is lower because all the highly productive workers have left for good. The only left home workers are the low productive ones.

Proposition 3. The game presents a pooling equilibrium where all workers migrate and stay, under the sufficient and necessary condition:

c b    g - b . Proof See Appendix A.
Here, the migratory cost is so low that bad workers would leave too, in order to mimic the good ones. Furthermore, if a worker does not migrate, he will be perceived as a bad worker. The same rationale prompts them to stay abroad, since the decision to return is associated to a bad worker. The good workers are therefore "trapped" into an inefficient strategy.

Proposition 4. The game presents a pooling equilibrium where all workers migrate and return, under the sufficient and necessary condition:

c b 2   g - b  Proof See Appendix A.
In this equilibrium the good worker will return, but will not benefit from the signalling effect. However, he has no other choice but to migrate, since if he stays he will be perceived as a bad one. Despite the high migratory cost, btype workers also migrate in a first step, because the large frequency of good workers allow them to benefit of a high wage. 11

Hybrid equilibria

Probably the most interesting situation is that where both types of workers play Nash mixed strategies. Indeed, in real life, patterns of migration seldom display pure strategies. Before analyzing these equilibria, we should state a proposition that allows to simplify the list of possible actions. 11 Appendix A shows that this equilibrium can occur for

c b 2   g - b  if firms beliefs are Prg|MS  0. If Prg|MS  1, the equilibrium exists only for c b 2   g - b   c g 1- .
Proposition 5. If good workers are indifferent between returning or staying  ∈0, 1 , all bad workers who have migrated at the first period prefer to return q  0 .

Proof. Starting from the indifference condition:

U g M, R  U g M, S w 1 M -c g  w 2 M, R  w 1 M  w 2 M, S -2c g w 1 M  w 2 M, R  w 1 M  w 2 M, S -c g
given that c b  c g , we can write:

w 1 M  w 2 M, R  w 1 M  w 2 M, S -c b w 1 M  w 2 M, R -c b  w 1 M  w 2 M, S -2c b
The later inequality is equivalent to U b M, R  U b M, S, thus q  0.

We have thus established that  ∈0, 1 q  0. We can now analyze the equilibria.

4.1. Full Hybrid Equilibrium. Some good workers return, the other stay. Some bad workers migrate and return, the other do not migrate.

In this equilibrium,  ∈0, 1 , p ∈0, 1 and q  0.

Proposition 6. The game presents an equilibrium where a fraction  ∈0, 1 of good migrants decide to return, the other 1 - good workers stay in the host country; a fraction p ∈0, 1 of bad workers migrate and return, the other 1 -p do not migrate, with:

 *   g - b -c g  c g 2 g - b  -c g  c b  c g  c b  - g - b  p *   1 - 2 g - b  -c g  c b  c g  c b  - g - b  provided that: c g  c b  2   g - b   min c g  1 2 c b , c g  c b 1  
Proof With  ∈0, 1, p ∈0, 1, (and q  0, workers' payoffs become:

U b M ̄, M ̄  w 1 M ̄  w 2 M ̄, M ̄  2 b U i M, R  w 1 M -c i  w 2 M, R   g  1 -p b   1 -p -c i   g  1 -p b   1 -p U i M, S  w 1 M  w 2 M, S -2c i   g  1 -p b   1 -p   g -2c i
A first equilibrium condition:

U g M, R  U g M, S  g  1 -p b   1 -p -c g   g  1 -p b   1 -p   g  1 -p b   1 -p   g -2c g c g  1 -p g - b  -c g 
allows to define the probability  as a function of p and various parameters:

  1 -   g - b -c g  c g p
Notice the necessary condition for existence:

 g - b -c g   0  c g   g - b .
The second equilibrium condition defines p according to parameters and  :

U b M, R  U b M ̄, M ̄  g  1 -p b   1 -p -c b   g  1 -p b   1 -p  2 b
The equilibrium probability p * can be determined if we substitute  by its former expression :

 g  1 -p b   1 -p -c b   1-   g - b -c g c g p g  1 -p b  1-   g - b -c g c g p  1 -p  2 b p *   1 - 2 g - b  -c g  c b  c g  c b  - g - b 
We then get :

 *  1 -   g - b -c g  c g p *   g - b -c g  c g 2 g - b  -c g  c b  c g  c b  - g - b 
This equilibrium exists if both p * ∈ 0, 1 and  * ∈ 0, 1. We check first the conditions for p *  0 and  *  0.

We must have either

2 g - b  -c g -c b   0 and c g  c b  - g - b   0 or 2 g - b  -c g -c b   0 and c g  c b  - g - b   0.
Only the former case is possible; it entails that:

c g  c b  2   g - b   c g  c b  with p *  0 and  *  0 for  g - b   c g c b  2 .
Then condition  *  1 requires that:

 g - b -c g  c g 2 g - b  -c g  c b  c g  c b  - g - b   1   g - b   c g  1 2 c b
and condition p *  1 needs:

 1 - 2 g - b  -c g  c b  c g  c b  - g - b   1   g - b   1 1   c g  c b 
All these conditions can be written in a compact form :

c g  c b  2   g - b   min c g  1 2 c b , c g  c b 1  
We can easily see that

c g c b  2  min c g  1 2 c b , c g c b 1
, . thus there are always some values of  g - b  such as this equilibrium can exist. We can also remark that the efficient separating equilibrium is not consistent with this hybrid equilibrium. Indeed, the separating

equilibrium is possible if 2 g - b   c b which is in stark contradiction with c g  c b   2 g - b  .
In this equilibrium, the probability that the good worker comes back  * is a decreasing function in both c g and c g . If policymakers aim at increasing the proportion of good workers that return, they should reduce migratory costs. This policy would also lead to an increase in the proportion of bad workers that migrate and return.

Bad workers are indifferent between "not migrate" and "migrate-return". Good workers migrate and return.

Proposition 7. There is a partial Hybrid equilibrium 1 where all good workers migrate and return   1 , a fraction p ∈0, 1 of the bad workers migrate and return, no bad worker migrates and stays q  0 , with

p   1 - 2 g - b  -c b c b provided that: c b 2   g - b   min c g  1 2 c b , c b 2 .
Proof Equilibrium conditions are:

U b M, R  U b M ̄, M, U b M, R  U b M, S and U g M, R  U g M, S.
Assume that good workers migrate and return so (   1 . Bad workers are indifferent between "not migrate" and "migrate-return". Thus, we can write :

U b M, R  U b M ̄, M ̄  g  1 -p b   1 -p -c b   g  1 -p b   1 -p  2 b p   1 - 2 g - b  -c b  c b
Here p can be seen as an increasing function in  g - b . An equilibrium exists only if:

0  p  1  1 2 c b   g - b   1 2 c b Indeed, 2 g - b  -c b  0  c b 2   g - b  a n d  1- 2  g - b -c b c b  1   g - b   c b
2 . Given the structure of the full Hybrid equilibrium, should one migrant decide to stay, then employers believe that he is of the good type:

 MS  Pr g |M, S  1
Thus the payoff of a worker who stays in the host country is:

U i M, S  w 1 M  w 2 M, S -2c i   g  1 -p b   1 -p   g -2c i , with i ∈ b, g.
Given these beliefs, the strategy M, S is dominated for the b -worker if :

U b M, R  U b M, S  g  1 -p b   1 -p -c b   g  1 -p b   1 -p   g  1 -p b   1 -p   g -2c b p   1 - c b  g - b  -c b
We replace p by its equilibrium expression:

 1 - 2 g - b  -c b c b   1 - c b  g - b  -c b   g - b   3 2 c b
Consider now the g -workers. For them, this equilibrium implies that M, R is a dominant strategy given the previous beliefs :

U g M, R  U g M, S w 1 M -c g  w 2 M, R  w 1 M  w 2 M, S -2c g  g  1 -p b   1 -p   g -c g p   1 - c g  g - b -c g 
For the equilibrium p , this condition is tantamount to:

2 g - b  -c b c b  c g  g - b -c g    g - b   c g  1 2 c b Given that c g  1 2 c b  c b  1 2 c b
, the necessary condition for this equilibrium is:

c b 2   g - b   min c g  1 2 c b , c b 2 .
Notice that the "efficient separating equilibrium", requiring  g - b   c b 2 , is mutually exclusive with this equilibrium.

Good workers are indifferent between "return or stay". Bad workers migrate and return.

Proposition 8. There is a partial Hybrid equilibrium 2 where all bad workers migrate and return ( p  1 ), no bad worker migrants and stays q  0 and a fraction  of the good workers migrate and return, and the other 1 - good workers migrate and stay, with

  1 -   g - b -c g  c g
provided that:

c g  c b 1     g - b   c g 1 - Proof Equilibrium conditions are: U g M, R  U g M, S, U b M, R  U b M, S and U b M, R  U b M ̄, M. We have show in Proposition 5 that  ∈ 0, 1  q  0, that is U g M, R  U g M, S  U b M, R  U b M, S.
For  ∈ 0, 1 and p  1, we have:

U b M ̄, M ̄  w 1 M ̄  w 2 M ̄, M ̄  2 b U i M, R  w 1 M -c i  w 2 M, R   g  1 - b -c i   g  1 - b   1 - U i M, S  w 1 M  w 2 M, S -2c i   g  1 - b   g -2c i
First condition: good workers play a mixed strategy if: U g M, R  U g M, S. With p  1 , this implies:

 g - b 1 -  c g   1 -   1 -   g - b -c g  c g
We know that this equilibrium exists for 0    1 which is tantamount to:

0  1 -   g - b -c g  c g  1.
This imposes additional constraints on parameters:

 g - b -c g  0  c g   g - b 1 -   g - b -c g  c g  1   g - b   c g 1 -
In a compact form, the necessary conditions can be written:

c g   g - b   c g 1 - Second condition, U b M, R  U b M ̄, M ̄, is true for:  g  1 - b -c b   1-   g - b -c g c g  g  1 - b  1-   g - b -c g c g  1 -  2 b c g  c b 1     g - b .
So, in a compact form, the necessary conditions for this equilibrium to exist are:

max c g , c g  c b 1     g - b   c g 1 - But because c g  c b , we have c g  c g c b 1
. So the necessary and sufficient condition is:

c g  c b 1     g - b   c g 1 -
This equilibrium can exist only if if the proportion of good workers in the total population is

large enough: 1 -c g  c b   c g 1      c b 2c g c b .

A synthesis and policy implications

The analysis of temporary migration has been developed in the case where good workers have an incentive to migrate, more precisely if their migratory cost is relatively low:

c g  21 - g - b    g - b   c g 21 - .
In this context, we have shown that the game presents several equilibria, that are summarized in the Table 1.

EQ.

 * p * q * Necessary and sufficient conditions Sep. "efficient"

1 0 0  g - b   c b 2 Sep. "inefficient" 0 0 0 c g   g - b   c b Pooling MS 0 0 1 c b    g - b  Pooling MR 1 1 0 c b 2   g - b  Hybrid Full 0   *  1 0   *  1 0 c g c b  2   g - b   min c g  1 2 c b , c g c b 1 Hybrid 1 1 0   *  1 0 c b 2   g - b   min c g  1 2 c b , c b 2 Hybrid 2 0   *  1 1 0 c g c b 1   g - b   c g 1- , if   c b 2c g c b Table 1. A

summary of the equilibria

The most complex configuration of this game appears for   c b 2c g c b . In the opposite case, the Hybrid 2 equilibrium does not exist. In the following, we will develop more in detail the complex picture.

We can notice that

  c b 2c g c b  c g  1 2 c b  c g c b 1 and c g  1 2 c b  c b 2 . So, for   c b 2c g c b , min c g  1 2 c b , c g c b 1  c g c b 1 and min c g  1 2 c b , c b 2  c b 2 . We also have for   c b 2c g c b  c g c b 1  c b 2 . The efficient separating equilibrium exists if  g - b   c b
2 . Yet the model has been developed for the case where good workers do migrate, i.e.

 g - b   c g 21- . So the efficient separating case can be feasible only if:

c g 21 -   g - b   c b 2 .
There is a non empty set for  g - b  only if:

c g 1 -  c b  c g  c b 1 -    c b -c g c b
On the other hand, we argued that the full configuration of equilibria requires   c b 2c g c b . Hence, the most general picture obtains only for:

c b 2c g  c b    c b -c g c b
There is a non empty interval for  if:

c b 2c g  c b  c b -c g c b  c g  c b /2
Figure 2 provides a graphic representation of the regionning of equilibria in this case. To bring some intuition about the thresholds, we choose c b  2 , c g  0. 5 and   0. 7 ( 0. 66 . Regionning of the equilibria Some policy implications can be inferred from this simple model. We can notice that, depending on parameters, the game can present multiple equilibria. For instance, for

 2 12    2-0.5 2  0. 75. α + + 1 g b c c 2 g b c c + 1 α - 1
c b 2   g - b   c b c g
1 , either the Pooling MR or the Inefficient Separating equilibrium or the Full Hybrid equilibrium can occur. Which one actually materializes depends on firms' equilibrium beliefs. In turn, if the economic model is prone for multiple equilibria, it is very difficult for policymakers to work out effective policies, since they cannot infer what is the type of the prevailing equilibrium.

From the origin country policymakers' point of view, the most desired equilibrium is the "efficient separating one". For a given productivity differential, they might be tempted to reach it by rising migratory costs of the bad workers. This policy might not always be efficient. For instance, in this configuration of the game the efficient separating equilibrium always comes with the inefficient separating one. Unfortunately policymakers have no method to impose their preferred situation. Policymakers may also try to manipulate the productivity differential, by improving training programs, such as to narrow the gap between good and bad workers along the hidden but valuable characteristics.

The full hybrid equilibrium is an interesting situation, insofar as both good and bad workers play mixed strategies. Contrary to what intuition would suggest, if this equilibrium prevails, more good workers will return if migratory costs are small. Indeed, if migratory costs are low, more bad workers will migrate, and the wage of the good workers will decline. Hence, they will have a stronger incentive to come back and avoid the migratory cost.

In all the hybrid equilibria good workers are penalized as compared to separating equilibria because, due to bad migrants, their wage is They may try to signal themselves by choosing such a migratory destination that the migratory cost c g (and implicitly c b , because by assumption c b  c g  becomes so high that the bad workers have no incentive to follow them (in other words, they might push costs so high that only the separating equilibria does prevail).12 If this logic holds, one can expect to see migration flowing toward "hard" places, even if such a tough experience has no impact on the migrant's productivity.

Conclusion

Temporary migration called the attention of policymakers and economists in the recent period. In particular, temporary migration seems to be the dominant contemporary pattern of migration between Eastern and Western European regions. In general, in developed countries temporary migrants benefit of a more favorable public opinion than permanent migrants, since they are seen as filling a well identified labor shortage (mainly in the hospitality sector, cleaning, agriculture, food processing, etc.). From the developing country perspective, whatever workers' qualifications, temporary migration is often seen as a source of additional income, since temporary migrants use to reinvest most of their earnings in the origin country. Furthermore, there is a strong belief that working abroad might help improving a worker's human capital.

Many factors drive an individual's decision to migrate then return. Existing literature has emphasized two of them: investment in human capital --individuals work abroad for some time in order to learn and improve their skills then come back home, and errors --workers go abroad hoping to get a better wage; some of them have bad luck, get low wages, thus they prefer to come back. In this paper we put forward another possible motive, which can be matched with both the former explanations. In our model, good workers always migrate at the first period. They do it because their migratory cost is rather low and because by migrating they can get a better wage than they could get at home. At the second period, some of them return, the other don't. Whether good workers return or not depends on the strategy of the bad workers. If bad workers do not migrate, the mere decision to migrate signals a worker as being of the high-productivity type and allows him to claim a high wage. If at least some bad workers mimic the behavior of the good workers and migrate as well, migration becomes a less efficient signaling strategy. Rational employers would adjust wages accordingly, thus penalizing good workers as compared to the perfect information setup. Notice that the signaling effect of migration can be obtained independently of any improvement in migrants' human capital.

From the point of view of the sending country, the most appealing configuration corresponds to the separating equilibrium with return migration. In this configuration, migration is a useful signaling device: firms can use the migratory track record to screen individuals. Stayers and migrants are paid to their true productivity; furthermore, the latter come back and contribute to output growth. Unfortunately, this equilibrium might not be single. One important contribution of this paper is to emphasize the scope for multiple equilibria. In turn, this makes efficient return policies hard to design, since policymakers have no reliable means to detect the type of the prevailing equilibrium. Policy recommendations differ depending on the equilibrium. In general, migratory costs must be raised in order to obtain a separating equilibrium. Yet, if the full hybrid equilibrium is at work, a developing country that aims at bringing back its best people must reduce migratory costs.

If migrants can control migratory costs, for instance by choosing their destination, good migrants might implement a policy of over-signaling: they can choose such a remote location that bad workers cannot follow. In this case, the separating equilibrium prevails, at the expense of good migrants who bear abnormal migratory costs. This is the price that good workers must pay in order to remove the informational asymmetry. In this context, any policy that helps suppressing imperfect information would be welfare improving.

Case 1. Out-of-equilibrium beliefs:

 MS  Pr g |M, S  0 U g M, R  U g M, S 2 g -c g   g -2c g   b  g - b  -c g is true.
The two other conditions are

U b M ̄, M ̄  U b M, S  2 b   g -2c b   b  c b  1 2  g - b  U b M ̄, M ̄  U b M, R  2 b  2 g -c b  c b  2 g - b 
Notice that the latter encompasses the former. Case 2 : Out-of-equilibrium beliefs:  MS  Pr g |M, S  1 .

U b M ̄, M ̄  w 1 M ̄  w 2 M ̄  2 b U i M, R  w 1 M -c i  w 2 M, R  2 g -c i U i M, S  w 1 M  w 2 M, S -2c i  2 g -2c i
Condition U g M, R  U g M, S  2 gc g  2 g -2c g is true. The other two conditions are

U b M ̄, M ̄  U b M, S  2 b  2 g -2c b  c b   g - b U b M ̄, M ̄  U b M, R  2 b  2 g -c b  c b  2 g - b 
The latter encompasses the former. Hence, the sufficient and necessary condition is:

 g - b   c b /2.

A.2. Pooling equilibria A. 2.1All workers migrate and stay M, S

In this equilibrium,   0, q  1 and p  0 Equilibrium conditions:

U b M, S  U b M, R, U b M, S  U b M ̄, M ̄, U g M, S  U g M, R.
The payoffs:

U b M ̄, M ̄  w 1 M ̄  w 2 M ̄  2 b U i M, S  w 1 M  w 2 M, S -2c i  2 g  1 - b  -2c i
Case 1 : beliefs :  MR  Pr g |M, R  1.

U i M, R  w 1 M -c i  w 2 M, R   g  1 - b -c i   g
Equilibrium conditions:

U b M, S  U b M, R 2 g  1 - b  -2c b   g  1 - b -c b   g  g  1 - b  -c b   g
Impossible.

Case 2 : beliefs :  MR  Pr g |M, R  0. Thus:

U i M, R  w 1 M -c i  w 2 M, R   g  1 - b -c i   b
Equilibrium conditions:

U b M, S  U b M, R  2 g  1 - b  -2c b   g  1 - b -c b   b   g - b   c b U b M, S  U b M ̄, M ̄  2 g  1 - b  -2c b  2 b   g - b   c b U g M, S  U g M, R2 g  1 - b  -2c g   g  1 - b -c g   b   g - b   c g
Necessary and sufficient condition:

c b    g - b .
A. 2.2.All workers migrate and return M, R Equilibrium probabilities   1 and p  1. Equilibrium conditions: U b M, R  U b M, S, U b M, R  U b M ̄, M ̄, U g M, R  U g M, S.

Payoffs:

U b M ̄, M ̄  w 1 M ̄  w 2 M ̄  2 b U i M, R  w 1 M -c i  w 2 M, R  2 g  1 - b  -c i
Case 1. Let us consider that the firm beliefs are such as  MS  Pr g |M, S  0. It follows that:

U i M, S  w 1 M  w 2 M, S -2c i   g  1 - b   b -2c i
Equilibrium conditions are:

U b M, R  U b M ̄, M ̄   g - b   c b 2 U b M, R  U b M, S   g - b   -c b
which is always true and

U g M, R  U g M, S   g - b   -c b
Which is true as well. So this equilibrium exists for these beliefs for  g - b   c b 2 . Case 2. Let us consider now the alternative beliefs:  MS  Pr g |M, S  1. In this case, the payoff connected to the strategy MS is:

U i M, S  w 1 M  w 2 M, S -2c i   g  1 - b   g -2c i
Equilibrium conditions are:

U b M, R  U b M, S  c b 1 -   g - b  U b M, R  U b M ̄, M ̄   g - b   c b 2
The two conditions are jointly fulfilled if:

c b 2   g - b   c b 1 -
The third condition is:

U g M, R  U g M, S  2 g  1 - b  -c g   g  1 - b   g -2c g   g - b   c g 1 -
But c g  c b . So the two former conditions entail the sufficient and necessary condition:

c b 2   g - b   c g 1 -
In the text, we refer to the broader domain of existence of this equilibrium, more precisely  g - b   c b 2 , which builds on the former set of beliefs.

  s g  play M, R with probability  play M, S with probability 1 - The strategy of the b -type worker is a linear combination of the three pure strategies: M ̄, M ̄, M, R and M, S : s b  play M, R with probability p play M, S with probability q play M ̄, M ̄ with probability 1p -q

  Figure 2. Regionning of the equilibria

CEPN and University Paris 13, rue Jean-Baptiste Clément, Villetaneuse, France. E-mail: besancenot@univ-paris13.fr

ESSEC Business School, BP 50105, 95021 Cergy,

Temporary migration can also be observed between Latin and North America, and within the East-Asian region.

Itself an extenssion of the classical Roy model[START_REF] Roy | Some thoughts on the distribution of earnings[END_REF].

See alsoDe Coulon and Piracha (2002) for an empirical test of a variant of this model with Albanian data.

Mesnard (2004) uses data from a Tunisia survey to show that the evidence about this process of

To keep the model as simple as possible, we assume that migratory costs are time-invariant. In a more powerful model, migratory cost could change in time. However, the direction of change is not clear, some cost components should go down (migrants adapt to the new environment), some cost components rise (migrants miss their friends and family).

The proof is straightforward if we compare the two-period payoff of a good worker that firstly migrates then comes back 2 gc g  with the payoff of a good worker who decides to fond into the mass of workers 2 g  1 - b . It can be shown that this condition also rules out a hybrid equilibrium where good workers are indifferent between migrating or not.

Of course, it should be checked that this strategy is rational, i.e., that good workers' payoff is larger with such a high cost than in the hybrid case.

Appendix A. Proofs of existence of the pure strategy equilibria

A 1. Separating equilibria A 1.1. Bad workers do not migrate M ̄, M ̄. Good workers migrate and stay abroad M, S .

Equilibrium probabilities are   0 and q  p  0. Equilibrium conditions are:

In this equilibrium, the payoffs become:

U i M, R depends on beleifs. There can be investigated two situations:

Sufficient and necessary condition:

A.1. 2. Bad workers do not migrate M ̄, M ̄. Good workers migrate and return M, R .

In this case   1, and q  p  0. The payoffs are:


Gilles Dolfo

Jacques Vigué
email: jacques.vigue@irsamc.ups-tlse.fr

A pendulum with a broken cylindrical symmetry: normal modes and beats

Keywords:

A pendulum, which has the cylindrical symmetry around its suspension wire, has two degenerate normal modes. We break this symmetry by putting a small magnet in the pendulum body and by running the pendulum in the presence of an inhomogeneous magnetic field. The force due to the magnetic field gradient modifies the equilibrium position and the spatial derivatives of this force define the normal modes of the pendulum with two different oscillation frequencies. We describe this system theoretically and we test these results by a series of experiments. This system provides a classical analogue of the degenerate perturbation theory in quantum mechanics.

Introduction

Since the discovery of the isochronism of small oscillations by Galileo Galilei, the pendulum has been used for an extraordinary variety of physical and technical applications [START_REF] Baker | The pendulum. A case study in physics[END_REF][START_REF]The pendulum: Scientific, Historical, Philosophical and Educational Perspectives[END_REF][START_REF] Pook | Understanding Pendulums. A Brief Introduction[END_REF]. In the present paper, we analyze the normal modes of a pendulum with a slightly broken cylindrical symmetry. When this symmetry is perfect, the normal modes are degenerate: the pendulum can oscillate in any direction with the same frequency and any combination of two orthogonal oscillations, which represents an elliptic oscillation, is also a solution of Newton's equations of motion. If a perturbation breaks the cylindrical symmetry, the frequency degeneracy of the normal modes is lifted. If the perturbation is conserved by time reversal, the normal modes correspond to two orthogonal directions of oscillation: this behavior is illustrated by the Blackburn pendulum also called the Y-suspended pendulum [START_REF] Crowell | Motion of the Earth as viewed from the moon and the Y-suspended pendulum[END_REF][START_REF] Whitaker | A note on the Blackburn pendulum[END_REF] and by the experiment described in the present paper. If the perturbation changes sign by time reversal, the normal modes are circular oscillations in opposite sense of rotation as illustrated by the Foucault pendulum [START_REF] Foucault | Démonstration physique du mouvement de rotation de la terre au moyen du pendule[END_REF]: for this pendulum, the mode degeneracy is lifted by the Coriolis force and this pendulum provides a classical effect of a geometric phase [START_REF] Hannay | Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian[END_REF][START_REF] Bergmann | Foucault pendulum through basic geometry[END_REF].

Finally, the lifting of an eigenstate degeneracy is well known in perturbation theory in quantum mechanics and the present study provides a perfect analogue in classical mechanics. In both cases, starting from a degenerate situation, the perturbation defines the new eigenstates or the new normal modes which are obtained by the diagonalization of the perturbation matrix expressed in the initial basis describing the degenerate normal modes.

We have built a pendulum made of sphere supported a piano string. This suspension makes that the pendulum is a good example of 2D quasi-harmonic oscillator (the usual suspension by a spring used in clocks or the use of knives limit the pendulum motion to in-plane oscillations and the pendulum is then described as a 1D quasiharmonic oscillator [START_REF] Baker | The pendulum. A case study in physics[END_REF]). A small magnet was placed near the sphere center in order to excite the pendulum oscillation in a controlled manner, thanks to the force due to a magnetic field gradient. If the laboratory magnetic field is inhomogeneous, the pendulum cylindrical symmetry is broken. The pendulum equilibrium position is then displaced, proportionally to the first derivatives of the field, and the frequency degeneracy of the normal modes is lifted, the new normal modes corresponding to linear oscillations. The rigidity with respect to torsion of the piano string is large so that the rotation of the magnet in the horizontal plane is negligible.

We develop the theory of this effect in the general case and in the practical case of the magnetic field produced by a magnetic dipole. If the two normal modes are excited simultaneously with equal oscillation amplitudes, one observes a beat pattern, with the oscillation varying from linear in one direction to linear in the perpendicular direction, with intermediate periods of elliptic and even circular oscillations. All the calculations are feasible by undergraduate students and the experiment is easy to build. This paper is organized as follows: we first develop the theory of a pendulum carrying a magnet in the presence of an inhomogeneous magnetic field; we then express the magnetic field of a dipole in cartesian coordinates; its first and second spatial derivatives, which are presented in an appendix, are used to calculate the magnetic field effect on the pendulum motion ; we describe our experimental set-up and some experimental tests of the theory. We present some concluding remarks.

Theory

The pendulum

We consider a pendulum having a perfect cylindrical symmetry around the vertical axis except that the pendulum body carries a small magnet, of magnetic moment m P , which interacts with the laboratory magnetic field B. We consider only small amplitude oscillations and we neglect the torsion of the suspension wire so that the direction of the magnetic moment m P is fixed in space.

Newton's equations in zero magnetic field

We describe the pendulum as a point mass M at the end of a wire of length l. The equilibrium position chosen as the origin O of the coordinate system with the x and y axes in the horizontal plane. The z-axis is vertical, oriented upward. Because of the wire, the mass M is on the sphere of radius l centered at x = y = 0, z = l and, neglecting fourth order terms in x, y, its altitude is z ≈ (x 2 + y 2 ) / (2l). The gravitational potential energy is given by

E g = M gz ≈ M g x 2 + y 2 2l , (1)
where g is the local acceleration of gravity. From E g , we deduce the restoring force F g = -∇E g and Newton's equations of motion

M d 2 x dt 2 ≈ -k g x, M d 2 y dt 2 ≈ -k g y. (2)
We note k g = M g/l the gravitational restoring force constant. The motion is harmonic in the x and y directions with the same angular frequency ω = k g /M = g/l. The two normal modes, corresponding to oscillations along the x and y axes, are degenerate and any other choice of orthogonal axes is equivalent. The general solution of Newton's equations is

x = a x cos (ωt + ϕ x ) , y = a y cos (ωt + ϕ y ) . (3)
These equations describe any linear or elliptic harmonic oscillation of angular frequency ω.

Newton's equations in the presence of a magnetic field

If the magnetic moment m P interacts with the external magnetic field B, there is a supplementary potential energy E m

E m = -m P • B = - 3 i=1 m P,i B i , (4)
where the indices i = 1, 2, 3 are used to label the x,y,z coordinates and the cartesian components of the vectors. The magnetic force F m exerted on the magnet is given by F m = -∇E m whose components are given by

F m,j = 3 i=1 m P,i ∂B i ∂x j (5)
If the magnetic force is very small with respect to the weight M g, the sole effect of the vertical component of the force is to slightly correct the weight, a negligible effect. The horizontal components of the magnetic force displace the equilibrium position to the point where the total force vanishes. The coordinates x e,j of the new equilibrium position are

x e,j = 3 i=1 m P,i k g ∂B i ∂x j for j = 1, 2 (6)
The pendulum oscillation is sensitive to the second derivatives of the potential energy calculated in the x-y plane at the new equilibrium position. We consider small horizontal displacements i from the new equilibrium position and we limit the total potential energy E pot = E g + E m to its second order Taylor expansion

E pot = E pot (x e , y e ) + 1 2 i,j [k g δ ij + k m,i,j ] i j with k m,i,j = k m,j,i = ∂ 2 E m ∂x i ∂x j = - 3 k=1 m P,k ∂ 2 B k ∂x i ∂x j (7)
where δ ij is the Kronecker symbol. We derive Newton's equations of motion

M d 2 dt 2 x y = - k g + k m,x,x k m,x,y k m,x,y k g + k m,y,y x y . (8)
As this matrix is symmetric, it can be diagonalized by a rotation of the coordinate system to new axes u and v. Newton's equations are then uncoupled

M d 2 dt 2 u v = - k g + k m,u,u 0 0 k g + k m,v,v u v . (9)
Provided that (k g + k m,u,u ) and (k g + k m,v,v ) are both positive (in the opposite case, the new equilibrium position is unstable), the normal modes describe harmonic oscillations along the new axes with the angular frequencies

ω u = k g + k m,u,u M ≈ ω 1 + k m,u,u 2k g , ω v = k g + k m,v,v M ≈ ω 1 + k m,v,v 2k g (10)
where the approximate forms are valid if k m,u,u , k m,v,v k g and then, the frequency difference is equal to

ω u -ω v ≈ ω k m,u,u -k m,v,v 2k g (11)

Motion of the pendulum

In the general case, the excitation of the pendulum produces an oscillation which is a superposition of the two normal modes with amplitudes u 0 and v 0 and initial phases ϕ u and ϕ v

u(t) = u 0 cos ψ u (t) and v(t) = v 0 cos ψ v (t) with ψ u (t) = ω u t + ϕ u and ψ v (t) = ω v t + ϕ v (12)
If the x,y axes are deduced from the u,v axes by a rotation of angle θ, the pendulum position is given by

x (t) y (t) = cos θ sin θ -sin θ cos θ u(t) v(t) . (13)
We note x u = u 0 cos θ, x v = v 0 sin θ, y u = -u 0 sin θ and y v = v 0 cos θ. We thus get

x (t) = [x u + x v ] cos ψ -cos ψ + + [-x u + x v ] sin ψ -sin ψ + y (t) = [y u + y v ] cos ψ -cos ψ + + [-y u + y v ] sin ψ -sin ψ + (14) with ψ + (t) = [ψ u (t) + ψ v (t)] /2 and ψ -(t) = [ψ u (t) -ψ v (t)] /2. ψ + (t) describes a fast oscillation at the normal mode mean frequency ω + = (ω u + ω v ) /2 and ψ -(t) a slow oscillation at the difference frequency ω -= (ω u -ω v ) /2.
The modulus of the amplitudes of the fast oscillation are given by

˜ xω + (t) = x 2 u + x 2 v + 2x u x v cos (2ψ -(t)) exp (-t/τ ) , ˜ y ω + (t) = y 2 u + y 2 v + 2y u y v cos (2ψ -(t)) exp (-t/τ ) . (15)
We have taken into account the damping of the oscillation by the factor exp (-t/τ ). We assume that the damping forces are equal for the two normal modes, as a consequence of the cylindrical symmetry of the pendulum (a possible exception may occur if the pendulum magnet induces eddy currents in nearby conductors). The amplitudes ˜ xω + (t) and ˜ y ω + (t) present beats at the frequency

ω beat = |ω u -ω v | ≈ ω |k m,u,u -k m,v,v | 2k g (16)
The total (kinetic and potential) energy E(t) is proportional to

E(t) = (k g + k m,u,u ) u 2 0 + (k g + k m,v,v ) v 2 0 2 exp (-2t/τ ) . (17)
E(t) decays because of damping but without any oscillation. The energy is partly transferred between the oscillations along the x and y-directions but the total energy is conserved, if damping can be neglected. The energy transfer is complete in the particular case u 0 = ±v 0 and θ = ±π/4.

Toward an experiment

We consider that the inhomogeneous part of the magnetic field B is produced by a magnetic dipole and that we can neglect the inhomogeneous part of the magnetic field due to other sources. The pendulum equilibrium position is at the origin of the x, y system of axes while the magnetic moment m F , represented by a red arrow, is at the origin of the X, Y system of axes. Φ is the angle between X and x. The pendulum, represented by the red dashed circle, carries the magnetic moment m P parallel to the x-axis and represented by the blue arrow. The distance between the two magnetic moments is noted r.

Calculation of the magnetic potential energy

Figure 1 represents the geometrical arrangement of our experiments. The magnetic moment m P carried by the pendulum and the fixed magnetic moment m F are in the same horizontal plane. The line going through the center of the fixed magnet and through the equilibrium position O of the pendulum is parallel to m F . This line is chosen as the X-axis of a second system of axes with its origin at the center of the magnetic moment m F . The Y-axis is in the horizontal plane, perpendicular to the Xaxis, and the Z-axis is in the vertical direction. We note Φ the angle between the x axis and the X axis. The potential energy of the magnetic moment m P is given by

E m = -m P (B X cos Φ + B Y sin Φ) (18)
The values of B X and B Y , their first and second derivatives with respect to X and Y are given in the Appendix. From now on, we assume that the displacement of the equilibrium position of the pendulum due to the presence of the magnetic moment m F is very small with respect to the distance r between the two magnetic moments and we calculate the field derivatives at the equilibrium position in the absence of m F . Using eq. ( 6) written in the X, Y coordinate system, we deduce the new equilibrium position

X e -r r = - k m 4k g cos Φ and Y e r = + k m 8k g sin Φ with k m = 24µ 0 m P m F 4πr 5 ( 19
)
k m is a restoring force constant. We also rewrite eq. ( 7) in the X, Y coordinate system

E pot (X e + ε X , Y e + ε Y ) = k g 2 ε 2 X + ε 2 Y + k m 2 -ε 2 X cos Φ + ε X ε Y sin Φ + ε 2 Y 2 cos Φ , (20)
where the potential energy E pot (X e , Y e ) has been omitted. To calculate the normal modes, we must give a diagonal form to the potential energy term. This calculation is easy but the general expressions are not very simple and we discuss only two particular cases in the next subsection.

Two particular positions of the magnet m F

In the case Φ = 0, the potential energy E pot is given by

E pot = 1 2 k g ε 2 X + ε 2 Y + k m -ε 2 X + ε 2 Y 2 ( 21
)
The matrix is already diagonal in the X, Y basis and the normal modes correspond to oscillations along the X and Y directions with the angular frequencies

ω X = k g -k m M and ω Y = k g + (k m /2) M (22)
The beat frequency is given by

ω beat = |ω X -ω Y | ≈ ω 3k m 4k g . ( 23
)
In the case Φ = π/2, the potential energy E pot is given by

E pot = 1 2 k g ε 2 X + ε 2 Y + k m ε X ε Y (24)
The symmetry by exchange of X and Y proves that the potential is diagonal in the U , V basis with U = (X + Y )/ √ 2 and V = (-X + Y )/ √ 2 i.e. in a basis rotated by an angle equal to π/4. In this basis, E pot is given by

E pot = 1 2 k g ε 2 U + ε 2 V + k m 2 ε 2 U -ε 2 V . ( 25
)
The normal modes correspond to the oscillations along the U and V axes with the angular frequencies

ω U = k g + (k m /2) M and ω V = k g -(k m /2) M (26)
and the beat frequency is given by

ω beat = |ω V -ω U | ≈ ω k m 2k g . ( 27
)
4. The experiment

The set-up

The pendulum is placed in a box to protect it from air currents. Its suspension wire is a piano string (diameter 0.49 ± 0.01mm, length 404 ± 0.5 mm) which is rigid against rotation, so that the magnet rotation due to the torque exerted by the magnetic field is negligible and the angle Φ is constant. This pendulum was used to test the Stokes drag force on a sphere [START_REF] Dolfo | Experimental test of unsteady Stokes' drag force on a sphere[END_REF]. It is made of a plastic sphere (radius a ≈ 15 mm) containing an aluminium alloy cylinder holding the pendulum magnet which is fixed to the piano string. Because the piano string wire is also rigid with respect to flexion, the restoring force is partly due to elastic forces. We have published the theoretical description of this type of pendulum [START_REF] Dolfo | A more accurate theory of a flexible-beam pendulum[END_REF]. We note k ef f = M ω 2 the effective restoring force constant which replaces k g . The fixed magnet is cubic (side length 20 mm) and the pendulum magnet is a parallelepiped (2×3×10 mm 3 ): both magnets are made of various grades of sintered NdFeB alloys. Following Petruska and Abbott [START_REF] Petruska | Optimal Permanent-Magnet Geometries for Dipole Field Approximation[END_REF], for a cube-shaped magnet with uniform magnetization, the magnetic field of a cube-shaped magnet is very close to the field of a magnetic dipole even at distances as small as twice the cube edge length. We have measured this field as a function of the distance along the direction of the magnetic moment and we have thus deduced the magnetic moments m P and m F : their measured values agree with the value calculated with the equation B r V m /µ 0 (V m being the magnet volume), using a remanent field B r ≈ 1.3 T, very close to the value given by the company which produced the magnets. The pendulum oscillation is resonantly excited by the force exerted by a magnetic gradient on the pendulum magnet. To produce this magnetic field gradient, we use a flat coil with about 1000 turns and a mean radius R ≈ 15 mm, located out of the box enclosing the pendulum, at a distance equal to ∼ 160 mm from the pendulum magnet. This coil is powered by a sine signal at the pendulum frequency. This signal is produced by the computer and amplified by a power amplifier. With a peak voltage equal to 6 V applied for about 20 seconds, we obtain oscillations amplitudes ∼ 0.5 mm. The parameters (diameter, number of turns) of this coil have not been optimized: in particular, the maximum gradient force of a flat coil occurs at a distance equal to half its radius and a considerably more efficient excitation is obviously possible.

To measure simultaneously the two displacements ε x (t) and ε y (t) of the pendulum, we use two shadow detectors described in our previous paper [START_REF] Dolfo | Experimental test of unsteady Stokes' drag force on a sphere[END_REF]. With a laser beam radius w ≈ 1 mm, the sensitivity is about 1 -2 × 10 3 V/m and the deviations from linearity are below 5% for a displacement smaller than ±0.5 mm, as measured using the shadow of a knife edge placed on a translation stage. The shadow detector signals are digitized at 40 Hz and we thus get the measurements of the displacements x (t) and y (t) as a function of the time t.

In another set of experiments, we have used a magnetic induction detector made of a pair of similar coils of solenoid valves [START_REF] Dolfo | Damping mechanisms of a pendulum[END_REF]: this detector is less sensitive than the shadow detectors. We compensate this reduced sensitivity by increasing the pendulum magnet volume to 1 cm 3 and the oscillation amplitude to a few millimeters. We thus obtain very nice signals. We have then used a 1 cm 3 magnet as the fixed magnet so that the magnetic moment product m P m F keeps a similar value. The advantages of the magnetic induction detector are that it does not require any fine alignment and that its linearity range is considerably larger. However, as we have only one such detector, we have recorded only one displacement, x (t): this explains why we do not present the results of this second set of experiments. In order to record the two displacements with two such detectors, the coils should be placed on the axes (x ± y) / √ 2.

The experiments

With the excitation coil axis and the pendulum magnet aligned with the x-axis, we excite the oscillation in this direction. We have fixed the angle Φ = π/2 because, in this case, the normal modes are theoretically excited with the same amplitude, leading to the maximum beat contrast. Moreover, when Φ = π/2, the displacement of the pendulum is along the Y-axis (see eqs; 19) and the modification of the distance r, being of the Table 1. This table gives the mass M of the pendulum body, its oscillation frequency ω/(2π), the effective restoring force constant k ef f = M ω 2 , the dimensions and magnetic moments m P and m P of the two magnets.

M (g) 20.9 ω/(2π) (Hz) 0.844

k ef f (N/m) 0.588 m P (A.m 2 ) 0.060 ± 0.003 m F (A.m 2 )
8.5 ± 0.2 second order in the displacement, is negligible. We have measured the beat pattern for various values of the distance r between the centers of the two magnets.

Experimental protocol

We first measure the shadow detector signal during 200 s with a numerical oscilloscope and, using its FFT function, we measure the two pendulum frequencies, if they are resolved or their mean value in the opposite case. Then, we choose the excitation frequency as the mean of the two frequencies in order to excite equally the two normal modes. After the excitation period, the pendulum oscillates freely for a duration equal to 4000 s before a new excitation. Because of the sensitivity of the shadow detectors, any adjustment of pendulum or of the shadow detectors put the pendulum in oscillation and the damping of this oscillation takes a long time. Because of this uncontrolled excitation, the first excitation period usually occurs while the pendulum is already oscillating and the initial conditions are ill-defined while at the end of a recording period, the oscillation amplitudes are negligibly small and the excitation puts the pendulum in a well-defined state. We show in figure 2 an example of a small part of the recorded signals. The beat patterns are obvious and they exhibit a very good contrast.

Data analysis

We extract the oscillation amplitudes of the fast oscillation ˜ xω + (t) and ˜ y ω + (t) by a sliding Fast Fourier Transform over 256 points i.e. over 6.4 s. The measured amplitudes are fitted by the following equation

˜ ω + (t) = A + B cos 2πt T beat + φ exp (-t/τ ) , (28)
which is equivalent to eq. (15). Figure 3 presents an example of these amplitudes and the squared amplitudes with their fits by eq. (28). The quantities A i and B i (i = x or i = y) would be equal in the ideal case and the beat contrast is then maximum. However, any misalignment of the magnetic moments m P and m F and of the direction of the excitation modifies the relative amplitudes of these modes. Because of the long duration of the experiments, we have not made much effort to improve the beat contrast. We have measured the beat period T beat for various values of the distance r between the magnet centers, covering the 140 -220 mm range by 20 mm steps. Equation (27) relates this period to the pendulum period T pend = 2π/ω by

T beat T pend = 2k ef f k m = Cr 5 with C = 4πk ef f 12µ 0 m P m F ( 29
)
where the dependence in r 5 is deduced from the dependence of k m given by equation ( 19). This behavior is tested in fig. 4. The best fit to the measured beat period gives C = (4.17 ± 0.012) × 10 5 m -5 while the value of C deduced from the measured magnetic moments is C = (4.8 ± 0.4) 10 5 m -5 . With Φ = π/2, we deduce from eq. ( 19) that X e = 0 and

Y e r = k m 8k ef f = T pend 8T beat . (30)
In our experiments, when r increases from 140 to 220 mm, the ratio T pend /T beat varies from about 0.045 down to 0.0047 and the ratio Y e /r decreases from 1.1% to 0.12% and the modification of the distance due to this displacement is fully negligible.

Direct observation of the pendulum motion

If the pendulum starts oscillating along the x-axis at t = 0, its oscillation becomes elliptic and at t = T beat /4, the oscillation is circular in one sense (depending of the sense of the two magnetic moment); then after a period of elliptic oscillation, the oscillation becomes linear along the y-axis at t = T beat /2; at t = 3T beat /4, the oscillation is circular in the opposite sense and at t = T beat , the oscillation is back to linear along the x-axis exactly as in t = 0. Figure 5 illustrates this behavior.

We have also made a movie of the pendulum motion in the configuration Φ = π/2, with an excitation along the x-axis at t = 0. The pendulum oscillation follows our theoretical predictions: some minor deviations are due to the large amplitude used in this experiment which makes that the pendulum explore a large spatial region while our calculation is valid in the small amplitude limit [13].

Concluding remarks

We have studied the motion of a pendulum having the cylindrical symmetry, with two normal modes which are degenerate. We have shown that, if the pendulum T /T beat = 0 1/4 1/2 3/4 carries a magnetic moment, an inhomogeneous magnetic field displaces the equilibrium position and lifts the mode degeneracy. The resulting normal modes are defined by the symmetry of the perturbation. In the present case, the perturbation is conserved by time reversal, because, although a magnetic dipole changes sign by time-reversal, the perturbation involves the product of two magnetic dipoles: the normal modes correspond to linear oscillations. The Foucault's pendulum is the most famous case in which the perturbation, due to Coriolis effect, is not conserved by time reversal and, in this case, the normal modes are circular oscillations. We have shown that, when the mode degeneracy is lifted, the pendulum oscillation measured along a given direction exhibit beats and, with a correct choice of parameters, the energy transfer from one direction of oscillation to the perpendicular direction can be complete in theory and almost complete in our experiments. We have also verified that the beat period varies as r 5 , where r is the distance between the magnet centers, in good agreement with theory.

Acknowledgments

We want to thank D. Castex, E. Panader, S. Faure, L. Polizzi and W. Volondat for their contributions to the experiment. Financial supports from CNRS INP, CNRS MI DEFI and université P. Sabatier are gratefully acknowledged.

Appendix I: Magnetic field of a permanent magnet

The magnetic field produced by a magnet is given by its multipolar expansion and, because ∇ • B = 0, one can prove that this expansion contains only odd power terms [START_REF] Petruska | Optimal Permanent-Magnet Geometries for Dipole Field Approximation[END_REF]. The field due to the 2 2n+1 -pole decreases like r -(2n+1) where r is the distance to the magnet center: the first non-vanishing terms is the dipolar term with B ∝ r -3 and for the next non-vanishing term B ∝ r -5 . If r is large with respect to the magnet dimensions, the magnetic field B of a compact magnet with a uniform magnetization is very well approximated by the dipolar term [START_REF] Petruska | Optimal Permanent-Magnet Geometries for Dipole Field Approximation[END_REF] and the magnetic moment m is equal to

µ 0 m = V m B r (31)
where V m is the magnet volume and B r is the remanent field of the magnet.

The magnetic field of a magnetic moment m are usually expressed in polar coordinates. Here, we write the components of the magnetic field in a cartesian coordinate system X, Y and Z, with the X-axis parallel to the magnetic moment m and with the origin at the center of the magnet

B X = µ 0 m F 4π × 2X 2 -Y 2 -Z 2 (X 2 + Y 2 + Z 2 ) 5/2 B Y = µ 0 m F 4π × 3XY (X 2 + Y 2 + Z 2 ) 5/2 B Z = µ 0 m F 4π × 3XZ (X 2 + Y 2 + Z 2 ) 5/2 (32)
We need the first and second derivatives of B X and B Y with respect to X and Y for Z = 0. Their calculation is simple but tedious and we have used Mathematica to verify our results. The first derivatives are equal to

∂B X ∂X = µ 0 m F 4π × 3X (-2X 2 + 3Y 2 ) (X 2 + Y 2 ) 7/2 ∂B X ∂Y = ∂B Y ∂X = µ 0 m F 4π × 3Y (-4X 2 + Y 2 ) (X 2 + Y 2 ) 7/2 ∂B Y ∂Y = µ 0 m F 4π × 3X (X 2 -4Y 2 ) (X 2 + Y 2 ) 7/2 (33)
At the pendulum equilibrium position in the absence of the magnet, X = r with r > 0, Y = 0, Z = 0, the first derivatives of B X and B Y are given by ∂B X ∂X = -6µ 0 m F 4πr 4 , ∂B X ∂Y = 0.

∂B Y ∂X = 0, ∂B Y ∂Y = 3µ 0 m F 4πr 4 (34)
The second derivatives are equal to

∂ 2 B X ∂X 2 = µ 0 m F 4π × 3 (8X 4 -24X 2 Y 2 + 3Y 4 ) (X 2 + Y 2 ) 9/2 ∂ 2 B X ∂X∂Y = µ 0 m F 4π × 15XY (4X 2 -3Y 2 ) (X 2 + Y 2 ) 7/2 = ∂ 2 B Y ∂X 2 ∂ 2 B X ∂Y 2 = µ 0 m F 4π × 3 (-4X 4 + 27X 2 Y 2 -4Y 4 ) (X 2 + Y 2 ) 9/2 = ∂ 2 B Y ∂X∂Y ∂ 2 B Y ∂Y 2 = µ 0 m F 4π × 15XY (-3X 2 + 4Y 2 ) (X 2 + Y 2 ) 9/2 (35)
The presence of equalities among these derivatives is not surprising. As ∆B = 0 and ∇ × B = 0 in free space, one can prove that the 27 second derivatives ∂ 2 B i /∂X j ∂X k can be expressed as a functions of 7 of them only. At the equilibrium position in the absence of the magnet, X = r, Y = 0, Z = 0, the second derivatives B X and B Y are given by

∂ 2 B X ∂X 2 = 24µ 0 m F 4πr 5 , ∂ 2 B X ∂X∂Y = 0 , ∂ 2 B X ∂Y 2 = - 12µ 0 m F 4πr 5 , ∂ 2 B Y ∂X 2 = 0 , ∂ 2 B Y ∂X∂Y = - 12µ 0 m F 4πr 5 , ∂ 2 B Y ∂Y 2 = 0. (36)

Figure 1 .

 1 Figure1. Schematic drawing of a top-view of the arrangement. The pendulum equilibrium position is at the origin of the x, y system of axes while the magnetic moment m F , represented by a red arrow, is at the origin of the X, Y system of axes. Φ is the angle between X and x. The pendulum, represented by the red dashed circle, carries the magnetic moment m P parallel to the x-axis and represented by the blue arrow. The distance between the two magnetic moments is noted r.

Figure 2 .

 2 Figure 2. The recorded signals x (t) (blue line) and y (t) (red line) converted in millimeters, to the measured sensitivities of the shadow detectors. The experiment is done with a distance r = 140 ± 2 mm and the beat period is T beat = 26.47 ± 0.02 s.

Figure 3 .

 3 Figure 3. The amplitudes of the fast oscillations converted in millimeters thanks to the measured sensitivities of the shadow detectors. Upper panel: the amplitudes ˜ x,ω+ (t) (blue full line) and ˜ y,ω+ (t) (red full line). Lower panel : the squares of these amplitudes showing the almost complete energy exchange between the oscillation along the x and y directions. The experiment is done with a distance r = 180±2 mm leading to a beat period T beat = 91.94 ± 0.04 s.

Figure 4 .

 4 Figure 4. The beat period T beat (s) is plotted as a function of the distance r (mm) between the centers of the two magnets in a loglog plot. The squares represent the experimental results and the dashed straight line is the best fit following T beat ∝ r 5 .

Figure 5 .

 5 Figure 5. Plot of the calculated motion of the pendulum in the u,v plane, assuming an equal excitation of the two normal modes in the case T beat = 100T pend . Each top panel describes the motion during a time interval equal to the pendulum period, chosen at multiples of T beat /4. The bottom panel describes the motion during one full beat period T beat , exhibiting a Lissajous curve.


Corinne Rousse

Nicolas Garnier

Gaetano Benčić

Davor Munda

Wine production in the Santa Marina villa (Istria -Croatia)

The villa is more than 100 m long and 50 m wide. It is built on a sloping ground, with two overlaying terraces facing the sea. The residential rooms seem to be situated mainly on the low terrace, around a large courtyard.

Service buildings are identi ed on the upper terrace, including a set of rooms for oil -and probably wine -processing, corresponding to the pars fructuaria of the villa.

Oil production took place in the main room (11 x11m). The remains of two oil mills (mola olearia type) and two large presses have been evidenced. Despite the destruction, signi cant elements of the machinery can be recognized, such as the rectangular press-beds built with a masonry foundation and one monolithic bloc used as a frame for the wooden uprights (arbores) on which the lever was xed. We suppose that the presses could reach a lengh of 10/11m with the use of a winch to power the lever.

On the south side of the room, a rectangular basin over 10 m long seems to have been connected to the presses by a pipe. In a later phase, the basin was divided by a small transverse masonry wall which rested on imbrices allowing the liquid to be ltered. To the west of the basin, another room located below, which has not been studied yet , could have been used as storage.

These structures are very similar to those observed in the villas of Brioni island (Val Catena / Uvala Verige, Monte Collisi / Kolči and Val Madona / Uvala Dobrika), in which the presence of great cellars, that could host up to 60 /80 dolia, associated with rectangular basins of great dimensions suggest principally a wine production, as disclosed by J. P. Brun (BRUN 2004).

Originally designed to produce wine, the equipment could then have been reused or transformed to process oil, in accordance with the new requirements of the owners of the villas, as the transformations of the basin seems to show in the case of Santa Marina.

To test this hypothesis, four samples were taken from the long basin of the villa of Santa Marina: one from the mortar of the northern wall (under the trace left by the pipe); two from the plaster covering the pavement, made in opus spicatum ; one directly from the pavement (spica ).

Biochemical analyzis, carried out by N. Garnier (2020), con rm that the long basin of the villa of Santa Marina contained grape juice, with the presence in all the samples of malic and tartaric acid. Two samples from the bottom of the basin reveal very clear traces of grapes and another fruit rich in malic acid (apple, pear or quince). They also contain traces of acids from alcoholic fermentation (malonic, maleic, succinic, fumaric and pyruvic acids), indicating a fermented drink, in this case red wine.

Only the sample from the north wall of the basin bears traces of a strong impregnation with fatty acids, suggesting that it was in contact with vegetable oil. During the survey, we also noticed traces caused by oleic acid on the mortar.

These results con rm the existence of a wine production in the villa of Santa Marina undoubtedly intended for local consumption, together with the production of oil, well known in Istria. They substanciate the palynological, iconographic and archaeological data already identi ed (including the production of wine amphoras in the Loron workshop).

The wine presses and the cellar are perhaps to be found in other rooms of the villa, not yet excavated. It also seems that the basin was then reused to decant the oil in the last phases of production of the villa.

  Wine production in the Santa Marina villa (Istria -Croatia) Corinne Rousse Aix Marseille université, CNRS, Centre Camille Jullian, Aix-en-Provence, France Nicolas Garnier SAS Laboratoire N. Garnier, Vic-le-Comte / AoRoc UMR CNRS 8046 Ecole Normale Supérieure de Paris Gaetano Benčić Zavičajni muzej Poreštine, Poreč, researches carried out by the Museum of the territory of Poreč (ZMP) and the Centre Camille Jullian have focused on a large roman villa facing the bay of Santa Marina, about 10 km north to the colony of Parentium (Istria, Croatia). The residence was part of an aristocratic estate that also included a big glina, on the site of Loron, connected to the villa and mainly dedicated to the production of oil amphoras. The stamps give the name of famous owners, beginning with T. Statilius Taurus Sisenna (cos 16 A.D.) who built the complex around 10.A.D. Then, the villa became an imperial property, from the time of Domitian (81A.D.) to Hadrian at least. During the IVth c. A.D., the buildings are spoliated and reused, before being completely abandoned in the Vth c. A.D.

  P. Brun, Archéologie du vin et de l'huile dans l'Empire romain, Paris, 2004. R. Matijašić, Gospodarstvo antičke Istre, Pula, 1998. C. Rousse, D. Munda, G. Benčić, O. Bourgeon, P. Maggi, V. Dumas, K. Bartolić, E. Botte, F. Welc : Loron / Santa Marina (Tar-Vabriga, Poreč, Croatie). La villa de Santa Marina. L'huilerie. Campagne 2019, Chronique des activités archéologiques de l'École française de Rome. Balkans, 2020. https://doi.org/10.4000/cefr.4862 Contacts : corinne.rousse@univ-amu.fr labo.nicolasgarnier@free.fr gaetano.bencic@muzejporec.hr davor.munda@muzejporec.hr The villa in context a) in Istria; b) facing the bay of Santa Marina; c) near the Loron site (the villa is on the north shore, Loron on the south) D. Kaniewski & al. 2019 : Reconstruction of the agropastoral activities from the Mirna region, next to Santa Marina and Loron Garnier 2020 (left). Biochemical anlysis of the pavement of Santa Marina basin.