Maxime Menuet
email: maxime.menuet@univ-orleans.frpatrick.villieu@univ-orleans.fr

Patrick Villieu

WHY ARE REFORMS INCOMPLETE? REPUTATION VERSUS THE "NEED FOR ENEMIES"

Keywords: D7, D8, E61, C72

Why do Politicians not solve social problems? One reason may be that such problems are difficult to solve, or that Politicians are incompetent. But there is another reason: in representative democracies, competent Politicians sometimes lack the incentive to solve problems to keep their enemies alive, in order to conserve an electoral advantage. This paper shows that reputational strengths can, to some extent, circumvent Politicians incentives not to address the problems. If the reputation of an incumbent Politician depends on the amount of reforms he implements, and positively affects his probability of being reelected, the trade-off between reputation and the need for enemies leads to an incomplete set of reforms, which can handle only a part of the problems. This mechanism might contribute to the explanation of the high degree of persistence of some social or economic diseases such as, specifically, public indebtedness.

INTRODUCTION

The fact that a society can create social cohesion through identifying a common enemy is a leitmotiv of political science and has been described by a number of works in psychology, sociology, history or social anthropology (see e.g. [START_REF] Bailey | The Need for Enemies : a bestiary of political forms[END_REF]). The idea that enemies serve a function for societies and for individuals comes from psychological fundamentals such that children or adolescents construct their identity through a need for opposition and that everybody needs "to identify some people as allies and others as enemies" [START_REF] Volkan | The need to have enemies and allies: A developmental approach[END_REF]. In his "beloved enemies", [START_REF] Barash | Beloved enemies: Our need for opponents[END_REF] even affirms that it is in human nature to need to create opponents. Such arguments have been applied to sociology and politics notably in the context of the cold war [START_REF] Wolfe | The rise and fall of the Soviet threat: Domestic sources of the cold war consensus[END_REF]. More generally, following [START_REF] Finlay | Enemies in politics[END_REF], "it seems that we have always needed enemies and scapegoats; if they have not been readily available, we have created them". [START_REF] Murray | Do People Need Foreign Enemies? American Leaders' Beliefs after the Soviet Demise[END_REF] distinguish two typical explanations for why people need enemies. The first one is that people "psychologically need enemies as suitable targets for the displacement of their personal fears and hostilities"; the second one is that political leaders can create enemies "to mobilize the nation around common aims".

In this paper, we consider the possibility that this second motivation to the "need for enemies" might be extended to the fight against macroeconomic diseases whose liquidation is a goal shared by the community and which can be manipulated by Politicians in power. There are numerous examples showing that macroeconomic variables may have been diverted from their stated objectives.

The objective of "fighting inflation" during the 1980s might have been an instrument for imposing reforms in the job market (for ending wage indexation), and the persistence of an inflation threat might have helped Governments to maintain a pressure on wages. In some European countries, the persistence of high levels of unemployment might have exerted a threat to workers for them to accept low working conditions or might have been used as a signal that the Government was tough and determined to fight inflation [START_REF] Drazen | Credibility of Policies vs Credibility of Policymakers[END_REF], while the declared goal of "fighting unemployment" ensured a sufficient level of social cohesion. More recently, the crisis of sovereign debts in European Monetary Union gave rise to a novel enemy, namely the public debt, whose persistence exerts pressures leading populations to accept austerity measures.

In all of these examples, the macroeconomic "enemy" can be used to create a "unifying myth" (according to [START_REF] Bailey | The Need for Enemies : a bestiary of political forms[END_REF]) for other purposes, possibly welfare enhancing. 1 But such myths can also be used by Politicians interested in their reelection, in the same way than our fear of foreign 1 Notice that mythmaking is not necessarily counterproductive from a social welfare perspective. A society can take advantage to tying its minds on a unifying myth, if such a myth facilitates overcoming some particular interests, for example.

enemies. The "need for enemies" has often been mentioned to describe Politicians' attempts to justify continuing failed policies, or to create some scapegoats in order to escape from internal reforms. 2 But not implementing internal reforms may also be the consequence of our need for keeping the enemy alive. Indeed, the "need for enemies" will be looked on in quite a different light if, instead of being regarded as a factor which comes from the incapacity of Politicians to undertake reforms, it is seen as a factor which leads Politicians not to undertake reforms. [START_REF] Fergusson | The Need for Enemies[END_REF] develops an interesting application of this mechanism to the fight against guerilla groups like the FARC in Columbia, suggesting that president Uribes incentive to eliminate the guerilleros would have been mitigated by the fact his electoral advantage would have been removed if he had eliminated them too quickly. As [START_REF] Fergusson | The Need for Enemies[END_REF] states the problem arises when Politicians are elected because they are "the person for the job": once the job is over, the Politicians may be replaced even if (and because) they have successfully completed the job for which they were elected. A classic example in history (also cited by [START_REF] Fergusson | The Need for Enemies[END_REF] is Winston Churchill who led Britain to victory in the Second World War as prime minister, but was immediately removed by electors as soon as the war was won in 1945. Another example, more recent and more closely linked to economics, might be related to the episode of Claudio Monti in Italy, where the initial massive popular support for the centrist coalition has collapsed once the reforms implemented. 3 For sure, austerity and unpopular reforms may have led to a desire to sanction the coalition, but the fact that reforms were already launched probably may have removed the electoral advantage of Monti because the threat of the enemy was pushed aside.

These examples also show that the "need for enemies" story is only one part of the piece. Generally, one can find mechanisms that ensure that reforms are undertaken, at least partially. In some extreme circumstances, "the person for the job" is designated, by an election or not, and she does the job, because she is the "right" person (she has a great sense of community that goes beyond his electoral interest) or because this is the "right" time (she takes advantage of favorable conditions).

In other circumstances, the society may delegate some objectives to an independent agent -such that a "conservative" central banker à la [START_REF] Rogoff | The Optimal Degree of Commitment to an Intermediate Target[END_REF] to fight inflation, for example. However, such a delegation of decisions to an "independent" agent or to a "providential" person, even if she 2 One of the most influential examples of this point of view is given in the Report entitled The First Global Revolution by the Council of the Club of Rome (1991): "The need for enemies seems to be a common historical factor. Some states have striven to overcome domestic failure and internal contradictions by blaming external enemies. "

3 In November 2011, Monti accepted to form a Government that would remain in office until the next scheduled general elections. He became Prime Minister of Italy and formed a "technocratic" cabinet composed of unelected professionals. On this occasion, he received the largest support ever acquired in a confidence vote in the Italian Parliament. However, after having introduced emergency austerity measures, restored financial stability and implemented structural reforms in the labor market, Montis coalition was only able to come fourth at the election of February 2013, obtaining 10.5% of the vote.

does not become a dictator (benevolent or not), might raise a problem of democratic control. 4 The other possibility, more usual in democracies where the "rules of the game" are based on elections, is to resort to reputational strengths to circumvent Politicians incentives not to reform.

Effectively, in general, electors do not perfectly perceive the "competence" 5 of Politicians, either incumbent Politicians or their opponents. However, by launching reforms, the incumbent Politician can attempt to signal his competence and to improve his reputation in order to maximize his chances of reelection, while his opponent cannot, since he is not at the helm of power. Such a signaling mechanism gives rise to a trade-off between "not to reform" in order to keep the enemy alive and "to reform" in order to maximize reputation.

The basic mechanism driving our analysis can be described by the following example of an electoral competition in a municipality. Suppose that there are two candidates. The incumbent defends the construction of a tennis hall and his challenger a library. Suppose also that the incumbent candidate has an advantage in public finance management, possibly because she is in office and has access to more information. If she leaves a healthy financial position at the end of his term, the election will be played on the relative popularity of the opponents and on the "ideological" preferences of electors ("sportsmen" against "intellectuals"). However, the incumbent candidate can attempt to take advantage of his competence in public finance by leaving a deteriorated financial situation in order to induce some intellectuals not to support his challenger, who is less competent to solve this problem. The same argument can be applied to national elections: a Government in office, who is "skilled" in solving a problem, can be induced not to provide much effort to fight this problem in order to keeping alive an electoral advantage, beyond voters ideological preferences or popularity shocks hitting his campaign. But not to solve the problem can damage his reputation to be competent, thus the Politician in office can use the choice of reform effort as a strategic variable in the voting process.

We consider an incumbent Government who is simply seeking reelection. Before the election, he tries to manipulate the vote by introducing "reforms" in the public finance stance, which result in "liquidating" a part of public debt. 6 However, the success of reforms depends not only on his 4 In modern history, a salient example of issues raised by such a problem of democratic control following a radical change in the "rules of the game" has been the questioning about the implementation of the Constitution of 1958 in France, following the failure of the Fourth Republic to reform the country. Thus, on May 19, 1958, General de Gaulle asserted at a press conference that he was "at the disposition of the country". In response to a journalist who feared that he would violate civil liberties, De Gaulle retorted vehemently: "Have I ever done that? Does anyone believe that, at age 67, I am going to launch a career as a dictator? " For an interesting anthropological analysis of the "rules of the political game" in France during the period, see [START_REF] Bailey | Stratagem and Spoils. A Social Anthropology of Politics[END_REF], chapter 9.

5 In our paper, "competence" may describe either the Politician's ability or his concern for reforms. Electors do not know if the Politicians are very committed to the reforms (for example, they do not know at the time of the election, the exact composition of the future team in power). Thus, "competence" describes the expected comparative advantage of the Politicians in the problem to be solved. 6 We consider reforms that Politicians may deal directly and which are acceptable by the great majority of people, like, reform effort, and on his competence, but also on the "macroeconomic or political context", which symbolizes the difficulty to reform and that we define as a random shock hitting the efficiency of reforms. Electors do not perceive the random shocks that affect the efficiency of reforms, nor the effort of Government, but only the global outcome of the reforms, namely the amount of public debt "liquidation". Knowing this outcome, they attempt to infer the degree of competence of the incumbent Government by a Bayesian procedure. It follows that a higher amount of reforms will raise the reputation of the incumbent Politician, i.e. the probability that he is competent. Finally, the chance of reelection of the incumbent Government depends both on his "reputation", and on the "need to keep the enemy alive". This results in a trade-off which, generally, gives rise to incomplete reforms because it is the interest of the incumbent Government to reform, but only partially.

The amount of reforms that will be implemented is the outcome of a political equilibrium resulting from the confrontation between the temptation of the Government to take advantage of his competence and the control of his reputation by the voters. In equilibrium, this amount will positively depend on the probability that the incumbent Government is competent (which enhances the benefits of "reputation") and will negatively depend on the competence gap between the incumbent Government and his challenger (which strengthens the "need for keeping public debt alive"). In addition, our model exhibits a "Goldilocks theorem" for the probability of reelection of an incumbent Government: to maximize his chances of success, the macroeconomic context must be neither "too bad" nor "too good". Effectively, if a "too bad" economic context generates a low amount of reforms which reduces the incumbent Politician's reputation, a "too good" economic context, ceteris paribus, may give rise to an amount of reforms that weakens the electoral advantage of a competent Politician.

Therefore, our model offers three types of explanations for why reforms are incomplete: because macroeconomic conditions are difficult, because the incumbent Government is incompetent or because he is competent but he acts strategically.

This paper is connected with a large number of analyses in political economy literature focusing on "positive" approaches of reforms and that Besley and Coate (1998) qualifies as "sources of inefficiency" in representative democracies. With a closely concern to ours, but using a very different specification, [START_REF] Alesina | Why are Stabilizations Delayed ?[END_REF] and [START_REF] Drazen | The Benefit of Crisis for Economic Reforms[END_REF] are interested in the optimal especially, reforms that improve public finances, by reducing "wasteful" expenditures or by generating exceptional receipts.

Of course, a number of reforms involve conflicts between social groups to defend specific interests or jeopardize some acquired positions. In such conditions, reforms can be abandoned because they are too politically costly or because they do not correspond to the preferences of voters supporting the incumbent party. Although these ingredients can be easily introduced in our model (see Appendix D) we want to develop another argument for why Governments do not reform even if electors recognize that what is costly is the fact not to reforming. For this reason, we consider that the Government in office does not attempt to maximize some partisan objective (such that his electors' welfare), but simply tries to maximize his chances to be reelected (net from the cost of reform effort).

delay in reforms following a "war of attrition" and the benefits of crisis to implement reforms more rapidly. In our framework, reforms are not only delayed but remain optimally incomplete. In a model with asymmetric information like ours, [START_REF] Cukierman | When Does It Take a Nixon to Go to China ?[END_REF] also presents a framework where the Politician who cares most about doing something is the least likely to do it (like President Nixon, who opened the door to the international legitimization of the People's Republic of China in the early 1970s, in spite of his strong anti-Communist convictions). But the argument of [START_REF] Cukierman | When Does It Take a Nixon to Go to China ?[END_REF] rests on the fact that voters are imperfectly informed about the ideology of policymakers, while in our model they are imperfectly informed about their competencies. More specifically, two cases of strategic behavior of a Politician facing rational forward-looking voters have been extensively studied in the literature. The first case focuses on uncertainty about Governments preferences (as in, e.g., [START_REF] Alesina | A Positive Theory of Fiscal Deficits and Government Debt[END_REF]), or about Governments abilities [START_REF] Cukierman | A positive theory of discretionary policy, the cost of democratic government and the benefits of a constitution[END_REF], [START_REF] Rogoff | Elections and Macroeconomic Policy Cycles[END_REF] and [START_REF] Rogoff | Equilibrium Political Budget Cycles[END_REF], among others). In these frameworks, the incumbent Politician acts strategically in order to "signal" his preferences or abilities to voters. In the second case, studied in particular by [START_REF] Persson | Why a Stubborn Conservative Would Run A Deficit: Policy with Time-Inconsistent Preferences[END_REF], [START_REF] Alesina | A Positive Theory of Fiscal Deficits and Government Debt[END_REF] and [START_REF] Milesi-Ferretti | How Cynical Can an Incumbent Be ? Strategic Policy in a Model of Government Spending[END_REF] among others, the incumbent Politician uses a state variable (typically public debt) to manipulate the choices of voters or of future policymakers.7 Our model can be viewed as joining these two strands of literature. Effectively, by choosing his reform effort, an incumbent Government can affect election outcomes both through signaling and through the residual burden which he will bequeath to his possible, incompetent, successors.

The rest of the paper is organized as follows. Section 1 presents the baselines of the model, Section 2 describes the way Households compute the reputation of the incumbent Government, following a Bayesian approach, and Section 3 presents the resulting political equilibrium. Finally, Section 4 produces some numerical results and Section 5 concludes.

THE MODEL

We are interested in a very general set of problems which focuses on an Agent who is hired by a Principal to carry out a specific task, namely to "liquidate" a "problem". There are two periods, and the Agent can remove a part of the "problem" at each period. At the end of the first period, the Principal has to decide whether to renew or not the Agent in office. In this game, the Agent can act strategically: even if he is "competent", i.e. he can solve the problem in the first period, he has no interest to do so, because he would not be renewed once the problem solved. Thus, he is induced not to liquidate in the first period, in order to maximize his chances to be renewed. But of course, the fact that the Agent does not liquidate in the first period may affect the probability to be renewed, because it's a bad signaling (zero liquidation might signal that the Agent is incompetent or is a shirker and affect his reputation).

In this paper, we attempt to solve the trade-off between these two conflicting forces: the need to keep enemies alive and the need to preserve reputation. We model this trade-off in a public finance set-up, with the Principal the Electors, the Agent a Politician or a (local or central) Government.

Effectively, any incumbent Government has at least one comparative advantage relative to his opponents for reforming some problem, that we regard to be the fiscal stance.

Agents' preferences

We consider one economy with N districts indexed by n ∈ {1, • • • , N }, each populated by a continuum of Households (or Citizens) with measure normalized to unity. In addition, there are two Politicians (or parties) denoted by D and R respectively, where i ∈ {D, R} represents the ideological bias.

There are two periods. In period 1, a Politician i ∈ {D, R} holds power, and at the end of period 1 there is an election for deciding who will be in power in period 2. In the first period, the Politician initially in power distributes to Households the public good g 1 = N n=1 g 1n . To finance this public good, he can resort to public debt or distorsive taxation. In addition, at each period t ∈ {1, 2}, he can launch reforms that lead to liquidate a part l t , 0 ≤ l t ≤ 1, of his financial needs. The part l t corresponds to the share of public expenditure that can be financed without detrimental effect on Household's welfare, namely cuts in wasteful spending (such as unnecessary "operating" expenditures arising from bureaucracy, corruption or X-inefficiencies), flat taxes, or extraordinary resources (donations, ability to withdraw funds from international institutions), all elements that we do not model explicitly, but th at depend on the ability of the Politician in office to generate "painless" resources. At the end of the first period, the residual amount of public spending is financed by issuing public debt (d 1 ), so that the budget constraint is simply: d 1 = (1 -l 1 )g 1 . Then, the election takes place and the Government i, i ∈ {D, R}, is renewed or his challenger takes office.

In the second (and last) period, the Government newly elected can once again launch reforms and he has to repay public debt and interests (namely: (1 + r)d 1 , where r is the constant real interest rate) and there is no other spending. To finance the debt burden, the Government must levies taxes from Households τ 2 = N n=1 τ 2n , since he cannot borrow in the last period (d 2 = 0).

Taking into account reforms launched in the second period (l 2 ), the Government intertemporal budget constraint is simply given by ( 1)

τ 2 = (1 + r)d 1 -l 2 (1 -l 1 )g 1 = (1 -l 1 )(1 + r -l 2 )g 1 .
Household n (thereafter we identify district n to Household n) derives utility from consumption in the two periods: c 1n and c 2n , and from the public good g 1n , namely, assuming a log-linear utility function for simplicity

(2)

U n := u(c 1n , c 2n , g 1n ) = log(c 1n ) + 1 1 + ρ log(c 2n ) + λ log(g 1n ),
where ρ is the rate of discount and λ is a positive parameter reflecting the preference for the public good. Household n receives a constant level of income y n in the first period and saves for consumption and tax payments during second period, hence, defining by s n Household n' saving, the following budget constraints hold: s n = y n -c 1n and, c 2n = (1 + r)s n -τ 2n . Thus, the intertemporal budget constraint is given by:

c 1n + c 2n /(1 + r) = y n -τ 2n /(1 + r). Consequently,
first order conditions conduct to the following usual relationships

c * 1n = 1 + ρ 2 + ρ (y n - 1 1 + r τ 2n ), and c * 2n = 1 + r 1 + ρ c * 1n .
And the Household n 's utility becomes8

(3)

U n = u 0n - τ 2n (2 + ρ) y n (1 + ρ)(1 + r) ,
where

u 0n := 1 1+ρ {(2 + ρ) log(y n 1+ρ 2+ρ ) + log( 1+r 2+ρ )} + λ log(g 1n ).
Finally, anticipating on resolution, the symmetric equilibrium implies that: τ 2n = τ 2 /N , and

g 1n = g 1 /N . Therefore, by introducing (1) in (3) and defining γ n := (2 + ρ)/y n (1 + ρ)(1 + r)N , we
obtain a reduced form for Household n's utility (4)

U n = u 0n -γ n g 1 (1 -l 1 )(1 + r -l 2 ).
Consequently, Households' utility positively depends on the amount of liquidation in each period (which is the part of public spending which will not lead to an increase in public debt or in taxes).

Besides, from (4), since l 2 ≤ 1, Citizen n's welfare is maximum when l 1 = 1.

Households as voters

The key assumption in our model is that the amount of liquidation is not totally a control variable for the Government, but depends on his effort in implementing reforms (his control variable e), on his degree of competence (q), but also on the economic or social context, which may be more or less favorable to the success of reforms, and that we model as a random shock (ε). This shock represents the state of the power relationship between pro and con-reforms in the society and symbolizes either, social resistance to reforms (strikes, demonstrations, lobbying,...) or economic climate, which may be more or less conducive to reforms (good times allow providing compensation to the losers).

Each Politician can be competent (g-type), or incompetent (b-type). Thus, we define the success probability of the Government i, i ∈ {D, R}, of competence j, j ∈ {b, g}, in attempting to reform in the first period by: l j 1i = ε 1 q j i e j 1i , where q j i ≥ 0, e j 1i ≥ 0, and

ε t ≥ 0, t ∈ {1, 2}, with mean E(ε t ) =: α ≤ 1.
In each period the Government in office produces an exogenous average amount of effort in reforming (say e). However, in the period preceding the election (t = 1) the incumbent Politician is induced to act strategically by choosing an amount of reform effort (possibly) different from e, namely: e j 1i = e + z j i ≥ 0, where z j i ≥ -e denotes the extra reform effort (positive or negative) that a "Machiavellian" Politician of type i and competence j will undertake to manipulate Electors. In the second period, the newly elected Government has no short-run electoral concern, thus: e j 2i = e, so that the amount of reforms is simply: l j 2i = ε 2 q j i e. To save notations, we define: ε 1 =: ε, and e j 1i =: e j i , which we consider to be the choice variable of Government i of competence j (since the choice of e j i is equivalent to the choice of z j i ). Obviously, a competent Government i has a higher probability of success in reforming than an incompetent one (i.e. q g i > q b i , i ∈ {D, R}). In addition, two Politicians may differ in their preferences or in their abilities, generating, for some problem, a comparative advantage in favor of the incumbent Politician. In this way, without loss of generality, we suppose that Politician R is, on average, more competent than Politician D in public finance reforms. An alternative interpretation is that the abilities of Politicians are the same, but Politician R has a bias in favor of reforms.9

Households know the ideological bias of the Government in place (in what follows, we will consider, unless otherwise and without loss of generality, that the Government initially in office is of type R).

They also know the amount of liquidation that arises in the first period l 1R . However, they cannot observe the reform effort of the Politician in office (e j R ), because they don't know his competence q j R nor the random shock ε that hits the success probability of liquidation, which are private information for the incumbent Government. We will show below how Households can infer the competence of Government in office from the signal l 1R . After the election, Government R is reelected or a new Government of type D takes place. From equation (4) in the case of a Government i ∈ {D, R} and competence j ∈ {b, g} is elected, Household n's expected utility is (5)

E j i [U n ] =    u 0n -γ n g 1 (1 -l 1R )(1 + r -l j 2D ) if i = D, u 0n -γ n g 1 (1 -l 1R )(1 + r -l j 2R ) + θ n + ξ if i = R.
In addition to utility derived from consumption, Households have preferences over ideology and other characteristics of Politicians which we call "popularity". Thus, Household n feels an additional expected utility (θ n + ξ) if the Politician R is in power. In order to avoid generating a deterministic result of the election, this term includes two random components: θ n is isidiosyncratic and ξ is common to all voters. On the one hand, following the probabilistic voting models of [START_REF] Lindbeck | Balanced-budget redistribution as the outcome of political competition[END_REF], and [START_REF] Persson | Political Economics: Explaining Economic Policy[END_REF], each Household n has an ideological preference θ n in favor of Politician R. Therefore, θ 1 , • • • , θ N are independent random variables, constant over time, and uniformly distributed on [-1 2s , 1 2s ], with density s > 0. 10 On the other hand, the random variable ξ reflects the general popularity of Politician R, and is uniformly distributed on [-1 2k , 1 2k ], with density k > 0. 11 Finally, even if Households don't know the precise degree of Politicians' competence in reforming, they may have some prior information on this competence, possibly based on past behavior of Politicians. In what follows, we suppose that Households assign some prior probability δ, 0 ≤ δ ≤ 1, to the fact that a Politician is competent (i.e. g-type). Therefore, we introduce a random variable

X i which represents Politician i's competence, i ∈ {D, R}. If Politician i is skilled (respectively unskilled), then X i = g (respectively X i = b), with (6) P{X i = g} = 1 -P{X i = b} = δ.
10 s is a measure of voters' responsiveness to Government decisions. Indeed, as s increases, Households care more about Government policy relative to ideology, see equation ( 5).

11 We could easily introduce the case where the popularity shock (ξ) negatively depends on the amount of reforms l 1R . Effectively, reforms can involve "political" costs because they face special interests (such as e.g. agents who collect taxes, who might have took benefits from corruption) and give rise to a punishment vote, even if Citizens recognize that reforms are, globally, desirable. In other words, the drug may be good for the patient but hard to swallow, like a visit to the dentist. Appendix D shows how this feature can be introduced within our framework without change in our results.

However, there is an asymmetry in our model, since the Government in office can send a signal to increase this prior probability by launching reforms. In other words, he can attempt to increase its reputation to be competent by liquidating a high part of public debt (high level of l 1R ). On the contrary, his opponent, who is not in office, cannot send such a signal, so that his reputation remains at the initial level δ.

Timing of events

The timing of events can be depicted as follows.

1. The random shock ε on the success probability of reforms is revealed to the Government in office. The Government R of competence j ∈ {b, g} decides the effort in implementing reforms e j R , in order to maximize his chances of reelection. Thus, the amount of effective liquidation of public spending will be: l j 1R = εq j i e j i .

2. Households observe the amount of liquidation l 1R . They don't know the shock ε nor Politicians' competence.

3. Households revise their prior probability that the incumbent Government is competent δ, knowing the signal l 1R .

4. The general popularity shock ξ is revealed and Households vote.

5. The Politician i ∈ {D, R} who got the most votes takes power in period 2.

6. If the initial set of reforms have been unsuccessful (l 1R < 1), a second set of reforms is launched by the newly elected Government, and the game ends.

As usual, we look for the pure subgame perfect equilibrium, and we solve the model by backwards induction. The two crucial stages in our model are steps 1 (choice of effort by the incumbent Government) and 3 (Households revise their probability that the incumbent Government is competent after having received the signal l 1R ). Let us depict these stages in the two following Sections.

GOVERNMENT'S REPUTATION AND SIGNALING

Households don't know the incumbent Government's competence, but they observe the actual liquidation of public debt l 1R , which they use as a signal to revise their beliefs about Government reputation δ. To this end, Households adopt a Bayesian procedure to revise the probability that the Government is competent. According to the fact that the incumbent Government is of type R, the signal l 1R can be obtained only in two cases:

l 1R =      l g 1R = εq g R e g R if the Government is competent (g-type), l b 1R = εq b R e b R if the Government is incompetent (b-type).
Households guess that the amount of liquidation l j 1R is an increasing function of εq j R , denoted by h(•), and which is given by the following definition.

DEFINITION 1: (Households guess) Households guess is given by the function h(•) continuous, increasing, positive, such that:

(7) l j 1R = h(εq j R ).
This means that the elasticity of effort in implementing reforms with respect to εq j R must be less than unity (in absolute value), which will be the case in equilibrium (see Eq. ( 31) in the following Section). Using the method of undetermined coefficients, this guess will be verified as a rational expectation equilibrium, as shown in Section 4. Now we define the probability that the incumbent Government is competent knowing the signal l 1R , that we note p R and whose value is established in the following Theorem.

THEOREM 1: (Bayesian revision of probability) Given the signal l 1R , the probability that the incumbent Government is competent is

(8) p R = δ + δ(1 -δ)Λ 1 + δΛ , where Λ := P{ε = ε g (l 1R )} P{ε = ε b (l 1R )} -1, ∀l 1R ∈ [0; 1].
PROOF: The bayesian revision of probability takes place in two stages.

Step 1. The signal received by Households can take any particular value l 1R only in two cases: if the random shock ε takes the value ε b or ε g respectively, where ε b and ε g are (9)

l j 1R = h(εq j R ) = l 1R ⇒ ε = 1 q j R h -1 (l 1R ) =: ε j (l 1R ), ∀j ∈ {b, g}.
To reduce the notations, we note ε j (l 1R ) =: ε j , j ∈ {b, g}. Of course ε b and ε g depend on the signal l 1R , more precisely: dε j /dl 1R ≥ 0, with ε g < ε b . Indeed, if the observed amount of liquidation is high, it's likely that the shock that caused this amount is also high. But, since a competent Government has more chances to succeed in reforms, the required shock associated to a skilled Government to verify the observed level of liquidation (ε g ) is less than the one required if the Government is unskilled (ε b ).

Step 2. Let l ∈ L 2 ([0; 1]) be the signal received by Households in period 1. The probability that the incumbent Government is competent knowing the signal l = l 1R is, according to Bayes rule:

(10) p R = P{X R = g|l = l 1R } = P{l = l 1R |X R = g}P{X R = g} P{l = l 1R } .
Yet the unconditional probability that the Government is competent is the initial value of his "reputation" δ, thus, P{X R = g} = δ. Besides, in accordance with the law of total probability,

P{l = l 1R } = P{l = l 1R |X R = g}P{X R = g} + P{l = l 1R |X R = b}P{X R = b}. Furthermore, according to Step 1, P{l = l 1R |X R = j} = P{ε = ε j (l 1R )}, j ∈ {b, g}. Therefore, (10) becomes (11) p R = δP{ε = ε g (l 1R )} δP{ε = ε g (l 1R )} + (1 -δ)P{ε = ε b (l 1R )} .
Finally, after rewriting, we obtain the above Eq. ( 8).

COROLLARY 1: The probability p R is increasing in Λ.

PROOF: By (8), ∂p R /∂Λ = δ(1 -δ)/(1 + δΛ) 2 ≥ 0.
Theorem 1 shows that the reputation of the incumbent Government p R depends on the signal received by Households. Effectively, since ε g (•) and ε b (•) positively depends on the amount of observed liquidation l 1R , Λ is also a function of the signal l 1R in Eq. ( 8).

The next step is to precise the probability distribution of the shock ε in order to explicit the linkage between l 1R and p R . For this purpose, we introduce the probability density function f of the shock ε, defined on [β, +∞[, β ≥ 0. In addition, we suppose f is a Normal density N(α, σ 2 ), as in Figure 1, and is given by

(12) f : s → 1 κ √ 2Πσ 2 exp{- (s -α) 2 2σ 2 }1 [β,+∞[ (s),
where 1 is the indicator function, κ is the density parameter given by: κ

:= [1 -G(β)], with G(•)
is the c.d.f. of the Normal distribution N(α, σ 2 ).

With such a density function we assume that intermediate shocks are more probable than "high"

or "low" ones. Therefore, the opportunities to reduce public debt are distributed in such a manner that "moderate" liquidations are more probable. Another interpretation might be that ε is a shock on Politician competences, and that there is higher occurrence of moderately competent Politicians. In what follows, we define q i := q b i +q g i and, qi := q g i -q b i , hence q b i = (q i -q i )/2, and q g i = (q i +q i )/2, i ∈ {D, R}. Furthermore, in our simulations below, we will suppose that the differential qi is small.

f (ε) ε • • 0 x(l 1R ) α x(l 1R ) ε g (l 1R ) ε b (l 1R ) β x(l 1R )
The following Proposition establishes the link between the signal l 1R and the " reputation of the incumbent Politician.

THEOREM 2: (The Government's reputation) The probability that the incumbent Politician is competent according to the signal l 1R is given by

(13) p R = δ + Π(l 1R ), ∀l 1R ∈ [0; 1],
where the signal function Π(•) ∈ C 1 (R).

COROLLARY 2: For the Gaussian distribution (12), there is β > 0 such that Π ′ (•) ≥ 0.

PROOF: We prove the Theorem 2 in two Steps. In the first Step we replace the probability p R with a density relation, which is given by the following Lemma, and in the second Step we compute the signal function Π(•) and we prove Corollary 2.

LEMMA 1: Given the density of probability f , the probability p R in Eq. ( 8) can be written as

(14) p R = δf (ε g (l 1R )) δf (ε g (l 1R )) + (1 -δ)f (ε b (l 1R ))
.

PROOF: See Appendix A.

Equation ( 14) can be rewritten as: p R = δ + Π(l 1R ), where

(15) Π(l 1R ) = δ(1 -δ) Λ(l 1R ) 1 + δ Λ(l 1R )
, and Λ(l

1R ) := f (ε g (l 1R )) f (ε b (l 1R )) -1.
Step 2. By (9), and our previous definition of q R and qR , we can write:

(16) ε b (•) = 2h -1 (•) q R -qR , and ε g (•) = 2h -1 (•) q R + qR .
Hence, with the Gaussian distribution (12):

Λ(•) = exp{ 1 2σ 2 [ε b (•) -ε g (•)][ε b (•) + ε g (•) -2α]} -1.
Now, let us define the function x(•) by:

x(•) = 2q R h -1 (•)/(q R -qR )(q R + qR ), so that ε b (•) + ε g (•) = 2x(•), and ε b (•) -ε g (•) = 2q R x(•)/q R . Thus, we obtain (17) Λ(•) = exp{ 2q R x(•) q R σ 2 [x(•) -α]} -1. We can remark that Π(l 1R ) ≥ 0 if Λ(l 1R ) ≥ 0, namely if x(l 1R ) ≥ α, or equivalently if l 1R ≥ l 1R := h(α[q R -qR ][q R + qR ]/2q R ).
We can easily compute:

Π ′ (l 1R ) = δ(1 -δ) [1 + δ Λ(l 1R )] 2 Λ′ (l 1R ) = δ(1 -δ)[1 + Λ(l 1R )] [1 + δ Λ(l 1R )] 2 2q R x ′ (l 1R ) q R σ 2 [2x(l 1R ) -α].
Notice that:

x ′ (•) ≥ 0, since h ′ (•) ≥ 0; hence Π ′ (l 1R ) ≥ 0 if 2x(l 1R ) ≥ α, namely if l 1R ≥ l 1R
, where:

l 1R := h(α[q R -qR ][q R + qR ]/4q R ). Finally, we define β = x(l 1R ) = α/2.
From equations ( 15) and ( 17) it is clear that the reputation of the incumbent Government p R can be positively or negatively affected by the "signal" of liquidation. Effectively, if the liquidation l 1R is "small", Π(l 1R ) is negative and the incumbent Politician suffers from a loss of reputation (p R < δ). If the amount of liquidation is "high", on the contrary, Π(l 1R ) becomes positive and the reputation of the incumbent Politician improves (p R > δ). With the Gaussian distribution (12),

Π(l 1R ) ≥ 0 if l 1R ≥ l 1R , and Π(l 1R ) ≤ 0 if l 1R ≤ l 1R ≤ l 1R .
However, in both cases, the reputation of the incumbent Government is increasing with l 1R , as Corollary 2 ensures. Thus, the Politician on power has an incentive to reform in order to strengthen its reputation as soon as l 1R ≥ l 1R , which corresponds to ε ≥ β = α/2. Nevertheless, the actual amount of liquidation is not directly chosen by the Government in power, who can only choose his level of effort in reforming. Therefore, in bad economic or political contexts (low values of ε) it is not an advantage to hold power, because the absence of reform will lead voters to degrade the reputation of the incumbent Politician.

THE POLITICAL EQUILIBRIUM

To describe the political equilibrium, we first characterize the electoral process, before studying the trade-off between reputation and the need for enemies, and determining the optimal reform effort of the incumbent Government.

The determination of Citizens' vote

At the end of period 1 the election takes place and Citizens vote for the candidate which gives them the highest expected utility according to the signal l 1R . There are two candidates: the incumbent Politician, of type-R and his type-D challenger. On the one hand, if Politician R is reelected, Citizen n's welfare is

(18) E R [U n ] = p R E g R [U n ] + (1 -p R )E b R [U n ],
with p R = δ + Π(l 1R ) the probability that the incumbent Politician is competent. According to definition (5), Citizen n's expected utility if R reelected is given by

(19) E j R [U n ] = u 0n -γ n g 1 (1 -l 1R )(1 + r -l j 2R ) + θ n + ξ, j ∈ {b, g}.
By ( 18) and ( 19) we obtain12

(20) E R [U n ] = u 0n -γ n g 1 (1 -l 1R ){1 + r -αe[p R q g R + (1 -p R )q b R ]} + θ n + ξ.
On the other hand, if the opponent D is elected, Citizen n's welfare is

(21) E D [U n ] = p D E g D [U n ] + (1 -p D )E b D [U n ],
where, according to definition (6), the probability that the Politician D is competent is simply:

p D = δ, since the challenger cannot enhance his reputation by signaling. Therefore, Household n's expected utility becomes

(22) E D [U n ] = u 0n -γ n g 1 (1 -l 1R ){1 + r -αe[δq g D + (1 -δ)q b D ]}.
Household n will support the incumbent Government R if the expected differential of welfare W n,R is positive, i.e.

(23)

W n,R := E R [U n ] -E D [U n ] ≥ 0.
By ( 20) and ( 22), it follows:

(24)

W n,R = αeγ n g 1 (1 -l 1R )[∆ R + qR Π(l 1R )] + θ n + ξ,
where, ∆ R := δ(q g R -q g D ) + (1 -δ)(q b R -q b D ) ≥ 0 represents the initial (namely, before signaling) average gap of reputation between type-R and type-D Politicians. In what follows, we will consider that this gap remains positive after signaling, namely that: ∆ R ≥ -q R Π(•).

The term (1-l 1R ) in the RHS of (24) reflects the incentive to "keep the enemies alive". Effectively, since ∆ R + qR Π(l 1R ) ≥ 0, the incumbent Government (of type R) is reputed to be more competent than his challenger. If he liquidates all public expenditure before the election (l 1R = 1), electoral choices will be subject to ideological preferences or random shocks on popularity only (θ n + ξ). But the incumbent Politician can take advantage of his competence gap by keeping the problem alive, thus he has an incentive not to liquidate public debt in the first period. 13 In the absence of any reputational concern (namely, if Π(•) = 0), the Government in office would not reform (l * 1R = 0). But this incentive can be offset by the term qR Π(l 1R ) in the RHS of ( 24), which reflects the gain of reputation obtained by signaling: the more the incumbent Government liquidates, the more he increases the probability to be competent on the eyes of the voters. Finally, we define by S n,R Household n 's vote for party R. Given our assumptions about the distribution of ideological preferences, S n,R can be expressed as:

S n,R := P θ {W n,R ≥ 0} = P θ {θ n ≥ -θ} = 1/2s -θ s dx = 1 2 + sθ, where θ := αeγ n g 1 (1 -l 1R )[∆ R + qR Π(l 1R ] + ξ.
Clearly, Household n's vote for party D is given by 1 -S n,R . From both candidates' point of view, S n,i , i ∈ {D, R}, is a random variable, since it is a transformation of the random variable ξ capturing party R 's average popularity.

Let us consider a majoritarian rule in which the party having obtained at less 50% of votes wins

13 Households' welfare is maximized for l 1R = 1, but if l 1R < 1, the electors will suffer more damages if the Politician D is elected, than if R is reelected. Thus, even if the incumbent Government was able to finance the public good g 1 entirely by painless resources (namely if he was able to fix l 1R = 1), he is induced not to do so in order to preserve his electoral advantage.

the election. Under this electoral rule, µ R , which denotes the reelection probability of Politician R, is given by ( 25)

µ R := P ξ { N n=1 S n,R ≥ 1 2 N },
where the probability refers to the random variable ξ. By simplification, we suppose that Households have the same income y and thus γ n =: γ for all n ∈ {1, • • • , N }. Therefore, by the definition of S n,R and our previous assumption that ξ is uniformly distributed on [-1 2k ; 1 2k ], we have:

(26) µ R = P ξ {ξ ≥ -ξ} = 1/2k -ξ k dx = 1 2 + kξ =: µ R (l 1R ),
where

ξ := αeγg 1 (1 -l 1R )[∆ R + qR Π(l 1R )].
The expression of the probability of reelection µ R in equation ( 26) points out the cross-relation between "reputation" and the "need for enemies".

The trade-off between reputation and the "need for enemies"

From relation ( 26), Households' vote depends on the amount of liquidation that the incumbent Government implements before the election l 1R . This gives the incumbent Politician of competence j ∈ {b, g} the possibility to manipulate the probability of reelection by setting an adequate amount of liquidation l j 1R . In this Subsection, we first describe how the "economic or political context" ε influences the probability of reelection of the incumbent Government, and then what would be the optimal choice of liquidation to maximize this probability.

PROPOSITION 1: ("Goldilocks Theorem") Ceteris paribus, to maximize the reelection probability of the incumbent Government, the "economic or political context" must be neither "too bad" nor "too good".

PROOF: From equation ( 26), the reelection probability of a Politician of competence q j R who provides an effort e j R is, using µ 0 := kαeγg 1 :

(27) µ j R - 1 2 = µ 0 (1 -εq j R e j R )[∆ R + qR Π(εq j R e j R )],
The first term in the RHS of ( 27) is decreasing in ε, while the second term is increasing in ε.

Thus, there is, in general, a critical value of the shock ε that maximizes µ j R . This value (ε) is such that dµ j R /dε = 0 and d 2 µ j R /dε 2 ≤ 0 and is implicitly determined by the following relation:

(

1 -εq j R e j R )Π ′ (εq j R e j R ) = ∆ R qR + Π(εq j R e j R )
, where, in accordance with our previous assumptions:

Π ′ (•) ≥ 0 and ∆ R qR + Π(•) ≥ 0. µ j R ε • ε 0 Figure 2. A Goldilocks Theorem
A "good" economic or political context (namely, ε > ε) gives rise to a "involuntary" high amount of reform that weakens the electoral advantage of the incumbent Politician (for an unchanged level of reform effort e j R ). Conversely, a "bad" context (ε < ε) generates a low amount of reforms which reduces the "reputation" of the incumbent Politician. In both cases, the probability of reelection will be low. Thus the chances of the Government in office are maximized when the shock on the possibility to reform is neither "too bad" nor "too good".

The same type of argument can be found for Government's incentive to reform. Let's suppose that the Government can directly choose the amount of liquidation l j 1R (this is not true, because it can only choose his effort e j 1R , but the mechanisms giving rise to the trade-off between reputation and the need for enemies will be similar, as we will show below). The impact of reforms (which result in an effective amount of liquidation l j 1R ) on the probability of reelection of the incumbent Government can be obtained by the following derivative of relation ( 27):

(28) 1 µ 0 dµ j R dl j 1R = qR (1 -l j 1R )Π ′ (l j 1R ) -[∆ R + qR Π(l j 1R )].
The first term in the RHS of relation ( 28) represents the marginal gain of "reputation" on the probability to be renewed: R(l j 1R ) := qR (1 -l j 1R )Π ′ (l j 1R ) ≥ 0. This gain positively depends on the marginal effect of reforms on the probability that agents assign to the fact that the Government is competent (Π ′ (l j 1R )) and to the skill gap of the incumbent Politician qR . 14 In addition, this gain is the higher the lower liquidation has been in the first period. Effectively, if the amount of reforms before the election has been important, public debt will be low, and the interest burden that can be alleviated by reforms in the second period will be relatively small. In this case, "reputation" to be very competent in reforming will not significantly improve the chances of success in the election.

The second term in the RHS of relation ( 28) represents the marginal gain of "keeping the enemy alive": N (l j 1R ) := [∆ R + qR Π(l j 1R )] ≥ 0. By not reforming in the first period, the incumbent Government takes advantage of his initial reputation to be, in average, more competent than his challenger (∆ R ). But this gain also positively depends on the amount of liquidation undertaken before the election, since the signal of liquidation improves the reputation of the incumbent Government: the more the Government is perceived as competent, the higher the benefits of "keeping the public debt alive" will be. This "need for public debt" is positively related to the competence gap between a skilled and a unskilled incumbent Government (q R ). Figure 3 illustrates the trade-off between the "need for enemies" and "reputation".

µ j R l j 1R lj 1R lj 1D 1 N (l j 1R ) N (l j 1D ) R(l j 1R )
Marginal gain of keeping the enemy alive Marginal gain of reputation

• • 0 Figure 3.
The trade-off between "reputation" and "the need for enemies"

To maximize the probability of reelection, the Politician in office should choose the amount of reform lj R which cancels relation (28), at the intersection of relations R(l j 1R ) and N (l j 1R ). Since N (l j 1R ) is increasing in l j 1R while R(l j 1R ) is decreasing 15 in l j 1R , there is, in general one interior solution lj R ∈ [0, 1]. So far, we have assumed that the incumbent Government is of type R. If the Government in office is of type D, with a challenger of type R, the analysis is unchanged, with the subscript D replacing the subscript R. The only change in is that the initial gap of reputation becomes:

15 Under the sufficient condition that qR → 0, see Appendix B.

∆ D = -∆ R ≤ 0. Thus, for a given gap of competence of the incumbent Government (namely: qD = qR ), the marginal benefit of "keeping the enemy alive" is lesser for a type-D incumbent Government than for a type-R one (N (l j 1D ) ≤ N (l j 1R )), while the marginal benefit of reputation is unchanged (R(l j 1D ) = R(l j 1R )). Therefore, the amount of reforms that maximizes the probability of reelection of a type-D Government is higher than those of a type-R Government: if the Politician in office is less competent than his challenger, he will implement more reform ( lj D ≥ lj R ) to maximize the probability of reelection (see Figure 3). 16However, the incumbent Government cannot choose the actual amount of liquidation l j 1i , which depends on the random shock that hits the probability of success of reforms. He can only choose the reform effort e j i . Let us now turn our attention to the optimal choice of effort.

The optimal choice of reform effort

In what follows, we suppose that the reform effort is costly for the incumbent Government.

Indeed, reforms require embarking on a negotiation process that generates costs, whether the increased workload of the Government, or psychological or political costs. Thus, for a liquidation effort e j i , we define the cost function c(e j i ), where c(•) is an increasing and strictly convex function, and we suppose that the incumbent Politician R seeks to maximize his probability of reelection net of the cost of effort. Consequently, the optimal effort of liquidation should satisfy in equilibrium: [START_REF] Persson | Why a Stubborn Conservative Would Run A Deficit: Policy with Time-Inconsistent Preferences[END_REF] e j * R = argmax

e j R {µ j R -c(e j R )}.
The solution of this program is characterized in the following Proposition and Corollary.

PROPOSITION 2: (Optimal reform effort) The optimal amount of reform l j * R corresponding to the optimal choice of effort e j * R of the incumbent Government R of competence j, j ∈ {b, g}, satisfies the following relation:

(30) R(l j * 1R ) = N (l j * 1R ) + 1 µ 0 εq j R c ′ ( l j * 1R εq j R
).

PROOF: See Appendix B.

COROLLARY 3: For (l j * 1R , εq j R ) verifying the optimal condition (30), there is a function ϕ ∈ C 1 (R + * ), defined on R + * → R + * , such that the optimal amount of reforms can be written:

(31) l j * 1R = ϕ(εq j R ), where ϕ ′ (•) ≥ 0. PROOF: See Appendix B.
In relation ( 30), the optimal decision on liquidation comes from the trade-off between "reputation" and the "need for enemies" on the reelection probability that we have analyzed in Subsection 3.B, but also on the cost of reforming. Relation (30) shows that the result of the optimal liquidation positively depends on the "economic or political context" (ε) and on the competence of the incumbent Government (q j R ). It is also positively associated to the skill-gap of type-R Politicians (q R ) and negatively associated to the average initial reputation gap between type-R and type-D Politicians (∆ R ). 17

When the competence gap between a skilled and an unskilled incumbent Government qR increases, the marginal gain of reputation R(l j 1R ) increases while the marginal gain of the "need for enemies" N (l j 1R ) may increase or decrease, depending on the sign of Π(l j 1R ). In any case, as we can verify in Eq. ( 28), R(l j 1R ) rises more than proportionally with respect to N (l j 1R ), thus inducing the incumbent Government to further liquidate public spending. This effect is independent of the incumbent Politician's intrinsic competence, j ∈ {b, g}. Indeed, even an unskilled Politician can take advantage of the fact that if he launches more reforms, his reputation will rise.

Conversely, the higher the initial reputation-gap between type-R and type-D Politicians (∆ R ), the higher the success probability of the incumbent Government, independently of his actions to enhance his reputation. Therefore, when ∆ R increases, it is in the interest of the Government in office to liquidate less in order to take advantage of the "need for enemies" mechanism.

Finally, to establish the existence of a rational expectation equilibrium, we have to identify the function ϕ(εq j R ) in ( 31) and the function h(εq j R ) in (7).

IDENTIFICATION AND SOME NUMERICAL RESULTS

In order to establish formal results, we consider from now that Household's guess (h(•)) is a linear increasing function of the facility to reform εq j R , namely

(32) h(εq j R ) = a + bεq j R ,
17 Effectively, from the implicit function theorem, it is straightforward to show from equation ( 30) that: l j * 1R = ϕ(εq j R , ∆ R , qR ), where: ∂ 1 ϕ ≥ 0, ∂ 2 ϕ ≤ 0, and

∂ 3 ϕ ≥ 0. See Appendix B.
where a and b ≥ 0 are parameters that have to be identified in equilibrium.

To proceed to identification, we suppose a simple quadratic function of effort, such that: c(e j R ) = c 0 (e j R ) 2 , and we consider linear approximations of equations ( 13) and ( 30). The strategy of linearization is established in the following Lemma.

LEMMA 2: (Linearization) For ε ≥ β, if the variance of ε (σ 2 ) is small: i. The signal function Π(•) can be approximate by

(33) Π(l 1R ) ≈ φ 0 + φ 1 l 1R ,
where φ 0 := φ 0 (a, b), and φ 1 := φ 1 (b) > 0.

ii. The optimal value of liquidation is, for small value of qR :

(34)

l j * 1R = A (a, b) + B (a, b) εq j R ,
where A(•) and B(•) > 0 are parameters depending on a and b.

PROOF: See Appendix C.

The final step of the resolution is to identify parameters of relations ( 32) and (34) to prove the existence of a rational expectation equilibrium verifying the initial guess of Households on the form of Government signal.

PROPOSITION 3: (Identification) The rational expectation equilibrium is established for the couple (a * , b * ) that identifies relations (32) and (34). Thus, the two following relations ensure identification:

A(a * , b * ) = a * , (35) B(a * , b * ) = b * . ( 36
)
In general one can find one couple (a * , b * ) that verifies these two relations.

PROOF: See Appendix C.

Let us now present some numerical results. Simulations are operated from the following benchmark calibration: α = δ = 0.5, q R = µ 0 = 1, qR = ∆ R = 0.1, σ 2 = 0.05 and c 0 = 0.01. For this calibration, a * = 0.0068 and b * = 0.5771. Results are robust to changes in parameters, as we will show.

Figure 4 presents the equilibrium amount of reform effort implemented by a skilled (continuous line) or an unskilled (dashed line) Government, and the associated levels of liquidation and probabilities of reelection, as functions of shocks on the "macroeconomic or political context" (ε). To this end, we conduct 10 random draws of the shock ε around values of 0.5, 1 and 1.5 respectively (first plot of Figure 4). The optimal liquidation process closely follows the stochastic behavior of ε, but, of course, for the same shock ε the liquidation implemented by a skilled Government is always higher than the one implemented by an unskilled Politician (third plot of Figure 4).

The fourth plot in Figure 4 illustrates the "Goldilocks Theorem" of Section 3.B, but generalizes this Theorem to the endogenous choice of reform effort, while the latter was considered as exogenous in Section 3.B. Effectively, the optimal probability of reelection is maximum for a range of shocks close to ε ≈ 1, and decreases for worse or better macroeconomic or political contexts. Thus it is not an advantage to beneficiate to "exceptionally favorable" conditions. Does it pay to be relatively 18 incompetent? Figure 4 shows that, beyond some threshold for exogenous shocks (close to unity in our simulation), the chances of success of an incompetent type-R Government are higher than the chances of a competent one. Effectively, for very high shocks, the optimal liquidation (the one that maximizes the probability of reelection minus the cost of effort) implemented by a competent Government is so strong that the enemy (the unliquidated part of public debt) becomes very low. Hence, a less competent Politician would be able to keep the enemy more alive, thus increasing his chances in the election.

Nevertheless, the incompetent Politician needs to deliver a higher effort than the competent Politician (see the second chart of Figure 4), thus his gain (the probability of reelection minus the cost of effort) is always lower the one of a competent Politician. Furthermore, with the Gaussian distribution (12) most of the shocks are close to α = 1/2, as we have shown. Therefore, cases in which the probability of reelection is higher for an incompetent Government than for a competent one are very unusual.

To implement some robustness checks, we depict the effect of changes in qR and ∆ R on the equilibrium amount of liquidation, respectively in Figure 5 and Figure 6, for a given shock ε. 19 If the competence-gap between the type-R Politicians (q R ) increases, coefficient a * increases while coefficient b * increases and then decreases, as Figure 5 shows. The optimal amount of liquidation increases with qR both for a skilled and for an unskilled Government, but in a much more significant 18 Remember that the type-R Government is more competent, on average, than his type-D challenger. 19 In Figures 5 and6, ε = 0.5. extent if the Government is skilled. As a result, the reputation of the skilled Government rises quite rapidly, while the reputation of the unskilled Government declines since the observed liquidation is lower than it should be, given the competence-gap qR .

Essentially, these results can be explained in the same manner as in the previous Section. When the skill gap increases, the incumbent Government, if he is the skilled one, must take advantage of this gap, by undertaking many reforms for enhancing his reputation. The same is true for the unskilled Government, but he will reform only to a lesser extent, precisely because of its incompetence, so he cannot benefit from reputational gains.

On the contrary, if the competence-gap between type-R and type-D Governments (∆ R ) increases, the incumbent Government is induced to reduce the amount of reform efforts and the resulting liquidated share of public debt decreases (see Figure 6) because the benefits of keeping the enemy alive are higher. This is even more true that the reputation does not change with ∆ R , as shows Figure 6. Effectively, reputation depends on the term h -1 (l j * 1R ) in relation ( 14). Yet, in equilibrium: h

-1 l j * 1R = (l j * 1R -a * )/b * = q R q j R ε is independent of ∆ R .
In other words, rational voters know that a change in liquidation associated to a change in the competence gap between type-R Politicians and type-D Politicians does not affect the probability that the incumbent type-R Government is competent or not.

CONCLUSION

In this paper we have shown that the trade-off between "Reputation" and the "Need for enemies" may generate an incomplete set of reforms, because, in our set up, the liquidation of the public debt is made endogenous within the political process. One important question raised by this perspective is whether Governments use reforms to manipulate voters, and especially if they are Machiavellian enough to let a problem to persist in spite of their competence to liquidate this problem. In this respect, our paper joins a number of results in political economy literature in which Governments use public debt strategically to manipulate the choices of voters or of future policymakers.20 Furthermore, in our framework, the need for enemies is only part of the story.

Beyond the need for keeping the enemy alive which leads competent Governments to undertake only few reform effort, reforms may also be incomplete because Governments are incompetent or because of adverse shocks hitting the economy, out of the control of Politicians.

Therefore even a strategic Government who chooses the level of effort in order to maximize his chances of reelection can be removed if the economy is hit by strongly positive or negative shock ("Goldilocks Theorem"). Indeed, the exercise of power is a risky game: if the context is "too bad", a strategic incumbent Government will be perceived as incompetent, even if his effort is strong.

Definitely, there are many examples showing that bad economic or political contexts are detrimental for the election of incumbent Politicians. Finding circumstances such that a good context has been prejudicial is a more complex task (in conformity with our model, such circumstances are rare). This might be the case of the US mid-term election in 1994 for example, which was more devastating to the Presidents party than any midterm since 1946, despite relatively good economic performances.21 Besides, the "state of the economy" midterm loss indicator suggested much smaller losses, and the results of this election were very enigmatic in relation to all traditional explanatory factors (see [START_REF] Campbell | The Presidential Pulse and the 1994 Midterm Congressional Election[END_REF]. Another prominent example may be the failure of Prime Minister Jospin in France, who was ousted of the second round of presidential election in 2002, while the economic situation had improved during his term (rather strong economic growth, decline in unemployment and in public debt): everything happened as though the improvement of the economic climate had made him lose its comparative advantage in the election, to the benefit of his challengers. The United Kingdom general election of 1997 may also symbolize the enemy-loss syndrome. In this occasion in effect, despite falling unemployment and a strong economic recovery following the early 1990s recession, Prime Minister John Major suffered from the Conservatives' worst defeat since 1906, with their lowest percentage share of the vote since 1832.

Our setup may lead to interesting prospects for future research. First, the trade-off between "Reputation" and the "Need for enemies" might be studied in other contexts, specifically microeconomic ones, in order to examine how to implement optimally incentives in a job contract, for example. Second, our model, like much of existent literature, describes a non-repeated game.

Besides, introducing the trade-off between "Reputation" and the "Need for enemies" in a repeatedgame framework might be interesting. Effectively, in an intertemporal framework, an incumbent Government who has a good chance of losing the election can seek to worsen the situation of his successor, to increase its chances in a subsequent election. By bequeathing a high debt burden to his possible successor, the incumbent Government can force its newly elected challenger to "pay the bills" [START_REF] Alesina | A Positive Theory of Fiscal Deficits and Government Debt[END_REF]. Thus, a bad electoral context would be an additional inducement to not to reform, if the incumbent Politician has a chance to return to power in the future. It would be particularly interesting to study the interplay between the electoral process and Citizens welfare in such intertemporel frameworks, in the context of Dynamic Stochastic General Equilibrium Models augmented with electoral cycles, in the lines of [START_REF] Battaglini | A Dynamic Theory of Public Spending, Taxation and Debt[END_REF], for example. Finally, it would be interesting to use opinion surveys to empirically test the "need for enemies" mechanism in order to show how the incumbent politicians may lose their electoral advantage when the economic or political context tends to alleviate the problem for which they are relatively competent. Π(l 1R ) in the neighborhood of l 1R = l 1R (namely, x(l 1R ) = x(l 1R )), where x(l 1R ) = α: Π (l 1R ) = Π l 1R + Π ′ l 1R (x(l 1R ) -α) + o(|x(l 1R ) -α| 2 ). 24By substituting x(l 1R ) with its value, as Π(l 1R ) = 0, and Π ′ (l 1R ) = αω, we obtain:

Π (l 1R ) ≈ 2αωq R h -1 (l 1R ) (q R -qR )(q R + qR )
-α 2 ω.

Since h -1 (l 1R ) linearly depends on l 1R in the guess function (32), we can write:

Π (l 1R ) ≈ φ 0 + φ 1 l 1R .
Notice that φ 0 and φ 1 depends on the parameters a and b of the guess function in relation (32), namely:    φ 1 = αωq b(q -q)(q + q) =: φ 1 (b),

φ 0 = -(aφ 1 + α 2 ω) =: φ 0 (a, b),
where q := q R /2 is the Government R's average competence, and q := qR /2.

Proof of ii. By (30), and according to the linearization of Π(•), the optimal amount of reform l j * 1R becomes: punishment vote, even if voters recognize that reform are, globally, desirable. With such an extension, our results are unchanged to within a constant. Regarding Equation ( 28), the marginal gain of "reputation" (R(l j 1R )) is identical, and the marginal gain of "keeping the enemy alive" (N (l j 1R )) becomes: Ñ (l j 1R ) = ∆ R + qR Π(l j 1R ) + η 0 , where η 0 = η/γeαx 0 , which represents the marginal loss of popularity when the amount of liquidation increases. Therefore, by (30), the optimal amount of reform l j * 1R satisfies the new following relation: ).

l j * 1R = 1 2 [ qR (φ 1 -φ 0 ) -∆ R qR φ 1 + c 0 µ 0 (εq j R ) 2
ξ • • • • 1 2k 1 2k -ηl 1R -1 2k -1 2k -ηl 1R
Since η 0 ≥ 0, the optimal reform effort e j * R is lower. Indeed, if η > 0, an adverse shock of popularity is more likely to happen, and the Government will choose a lower effort to weaken the loss of popularity. Finally, linearization and identification procedures are unchanged except for a constant η 0 .

Figure 1 .

 1 Figure 1. The density function of ε

Figure 4 .

 4 Figure 4. Optimal reform effort, liquidation and probability of reelection as functions of shocks on the "economic or political context".

Figure 5 .Figure 6 .

 56 Figure 5. Effect of changes in the competence gap of the incumbent Government.

]

  =: W (εq j R ).By linearizing equation (32) in the neighborhood of εq j R = αq, we obtain:l j * 1R ≈ A (a, b) + B (a, b) εq j R , where A(a, b) = W (αq) -αqW ′ (αq), and B(a, b) = W ′ (αq). Hence, using c 1 := c 0 /2µ 0 (αq) 2 :A(a, b) := [2q(φ 1 -φ 0 ) -∆ R ](qφ 1 -c 1 ) 4(qφ 1 + c 1 ) 2 , B(a, b) := c 1 αq [2q(φ 1 -φ 0 ) -∆ R ] 2(qφ 1 + c 1 ) 2 .

k 0 Figure 7 .

 07 Figure 7. The density of popularity shock (ξ)

[START_REF] Aghion | Government Domestic Debt and the Risk of Default: A Political-Economic Model of the Strategic Role of Government Debt[END_REF] present an interesting model in which public debt can be used as a political instrument to ensure reelection. In the incumbent Politician (typically a conservative one) can more credibly commit not to default than his opponent, he has an incentive to excessive accumulation of public debt in order to increase the number of bond-holders (which will vote for the most "competent candidate, i.e. the one which is less likely to default).

We use log(1 -τ 2n yn(1+r) ) ≈ -τ 2n yn(1+r) for "small" taxes' rate τ 2n /yn.

In our model "competence" may also reflect the degree of Government involvement in pursing reforms. The type R-Government is strongly committed to this goal, while the type-D Government may have interest in (or may be more "competent" for) other objectives. Considering a multi-objective framework would not fundamentally change the model, the main point being that the Government of type R has an incentive to keep alive his comparative advantage.

We use α := Eε 2 .

In this Subsection, we consider that the functional form Π(•) is invariant to changes in parameters. This hypothesis will be relaxed in the following Section.

This property can be likened to the argument of[START_REF] Cukierman | When Does It Take a Nixon to Go to China ?[END_REF], that Governments do not always do what is expected of them.

See, e.g.[START_REF] Persson | Why a Stubborn Conservative Would Run A Deficit: Policy with Time-Inconsistent Preferences[END_REF],[START_REF] Alesina | A Positive Theory of Fiscal Deficits and Government Debt[END_REF],[START_REF] Aghion | Government Domestic Debt and the Risk of Default: A Political-Economic Model of the Strategic Role of Government Debt[END_REF][START_REF] Aghion | Government Domestic Debt and the Risk of Default: A Political-Economic Model of the Strategic Role of Government Debt[END_REF][START_REF] Milesi-Ferretti | How Cynical Can an Incumbent Be ? Strategic Policy in a Model of Government Spending[END_REF][START_REF] Milesi-Ferretti | How Cynical Can an Incumbent Be ? Strategic Policy in a Model of Government Spending[END_REF].

The economy was healthier than in most midterms economies, as shown by the annual economic growth, which was about 2.5% in 1994, compared to an average midterm growth rate of only 1.4% from 1946 to 1994, for example.

Where o(•) is a function such as: o(|x(l 1R ) -α| 2 ) → 0 when l 1R → l 1R .

APPENDIX A: PROOF OF LEMMA 1

Step 1. We prove the following result:

where for simplicity ε j := ε j (l 1R ), ∀l 1R ≥ 0.

To this end, we define the functions' sequence (ψ j n ) n≥1 by: (A.2)

Yet, f is continuous. Thus, there is ζ ∈ [ε j -1/n; ε j + 1/n], such that:

Furthermore, ε j -1/n ≤ ζ ≤ ε j + 1/n. So, by taking the limit:

Finally we obtain equation (A.1):

Step 2. We can write:

Therefore, by (A.3) and ( 11), we can write:

Finally, by (A.1), lim n→+∞ (n/2)P{ψ b n (ε)} = f (ε j ), and multiplying (A.4) by n/2, we obtain:

The first order condition for the maximization of Government's program ( 29) implies, from ( 27):

Since εq j R e j R = l j R , this equation immediately results in (30). The second order condition implies:

Effectively, by hypothesis: c ′ (•) ≥ 0, c ′′ (•) > 0, and Π ′ (•) ≥ 0 according to Corollary 2. Thus a sufficient but unnecessary condition for the SOC to be verified is that: qR Π ′′ (l j 1R ) → 0, which is the case either if Π ′′ (l j 1R ) = 0 (as in our simulations in Section 5), or if the differential of competence is small (q R → 0), which we have supposed. Consequently, the amount of liquidation l j 1R verifying the first order condition (30) is a maximum and solution of the problem [START_REF] Persson | Why a Stubborn Conservative Would Run A Deficit: Policy with Time-Inconsistent Preferences[END_REF], which we will note

The optimal amount of liquidation l j * 1R is defined by the relation Ψ(l j * 1R , εq j R ) = 0, where, by (30):

Hence,

Consequently, since c ′ (•) ≥ 0 and, c ′′ (•) > 0, we have either if Π ′′ (l j * 1R ) = 0, or qR → 0: ∂ 1 Ψ < 0, and ∂ 2 Ψ ≥ 0. Therefore:

In addition, according to the implicit function theorem, there is an application ϕ ∈ C 1 , defined on

Proof of i. By ( 15) and ( 16), when qR → 0:

the signal function can be approximate by: Π(l 1R ) = δ(1 -δ) Λ(l 1R ). In addition, by ( 17), when

where

.

By choosing a sufficiently small value for the variance of shocks σ 2 , most of random shocks ε will be very "close" to their mean α. 23 Therefore, we consider the following linear approximation of 23 Indeed, if σ 2 → 0, ε is degenerate random variable, thus: ε → Eε = α.

PROOF of PROPOSITION 3.

Replacing φ 0 and φ 1 with their values in Lemma 2, identification restrictions ( 35)-( 36) become:

By introducing (C.2) in (C.1), we get:

, where ν := ωαq (q -q)(q + q)

.

By introducing this value in (C.2), we obtain:

3) is a second degree polynomial, on the form: φ 2 1 -w 1 φ 1 -w 2 = 0, where w 2 > 0, and w 1 > 0, since: 5qν -2αω = 3ωαq 2 + 2αω q2 (q -q)(q + q) > 0.

Therefore, this polynomial has only one positive root, namely:

Finally, we obtain: a * = αqν(qφ * 1 -c 1 )/2φ * 1 c 1 , and b * = ν/φ * 1 .

APPENDIX D: THE POPULARITY SHOCK DEPENDING ON REFORMS One possible extension of our model is to introduce the case where the popularity shock (ξ) negatively depends on the amount of reforms l 1R , with sensitivity η, η ≥ 0. Therefore, ξ is uniformly distributed on [-1 2k -ηl 1R ; 1 2k -ηl 1R ] with density k > 0, as Figure D1 shows. 25 Effectively, the liquidation of public debt, even if it is considered by Citizens as an acceptable goal for the reform of Government finance, can involve "political" cost because it faces special interests. In such a way, electors will not base their votes uniquely on the future policies of candidates but also on the current behavior of the incumbent Politician: too much reform currently can give rise to a 25 This analysis in the main text corresponds to the particular case where η = 0.