Separating the Accountability and Competence Effects of Mayors on Municipal Spending

A.1 The General Structure of the Electoral Game

In any given term, there are three kinds of players: a representative voter V , a pair of incumbent politicians P (the mayor and the deputy-mayor), and a pair of challengers in ticket C (one for mayorship and one for deputy-mayorship), belonging to the opposition party. The sequence of the events is depicted in Figure A.1.

Figure A.1:

The general structure of the electoral game.

At the beginning of the term, the incumbent politicians can be either ‘competent’ types θ _C , with probability μ′, or ‘incompetent’ types θ _I , with probability 1 − μ′. For ‘debutant’ politicians, at their first term of office, the type is determined by nature N with probability μ′ ≡ μ ₀ ∈ (0, 1). For ‘career’ politicians, μ′ represents the voter’s belief about the types in office, based on the observation of their past policy performance. The voter knows that debutant politicians are competent with probability μ ₀ but she does not observe the actual type.^[14]

To implement policy, the incumbent politicians take a binary choice about their level of effort: either low effort a ̲ or high effort a ̄ . Also the policy outcome is binary: either a low (bad) outcome L or a high (good) outcome H. Incompetent types always achieve outcome L, no matter the effort exerted. Instead, competent types exerting high effort always achieve outcome H, while competent types exerting low effort achieve outcome H with probability γ ∈ (0, 1) and outcome L with probability 1 − γ. However, as represented in Figure A.1, for competent types there is no direct link between effort and outcome, since with probability ɛ ^t > 0, ɛ ∈ (0, 1), competent types can become incompetent after choosing the level of effort (with ɛ ^t independent of effort), where t ≥ 1 is a measure of ‘experience’ (e.g. the number of terms in office). The probability ɛ ^t is decreasing in t to capture the fact that experience in policy making helps to retain competence.

The cost of exerting low effort is normalized to zero, while that of exerting high effort is c ≥ 0, the value of which is drawn by Nature at the beginning of the game from a given cumulative distribution F(c). The realization of c is private information but the distribution F is common knowledge. For the politicians, the benefits of reelection are equal to B > 0, exogenously given, irrespective of effort exerted and competence status. Politicians are assumed to maximize the expected flow of present and future rents of office, net of the cost of effort, discounted at rate δ ∈ 0,1 over the relevant time span.

At the end of the term, the voter observes the outcome and then the election is held, in which the incumbent politicians P run against a ticket of challengers C . Since the challengers are drawn by Nature, the voter’s belief that the challengers are competent types is μ ₀. As for the incumbents, depending on the observed outcome O (H or L), the belief a about the level of effort taken ( a ̲ or a ̄ ), and the prior belief μ′ that they were competent at the beginning of the term, the voter forms her posterior belief μ ( O , a , μ ′ , P ) that the incumbents, if reelected, will enter the following term as competent types. The voter is assumed to care only about policy performance in the following political term. Therefore, on the basis of beliefs μ ₀ and μ ( O , a , μ ′ , P ) , she casts her ballot for the ticket of candidates that maximizes the probability of achieving a high outcome in the ensuing term. It is also assumed that two politicians in ticket who lose an election are both out of politics forever. The policy game is solved for Perfect Bayesian Equilibria in pure strategies.

Following Alt, Bueno de Mesquita, and Rose (2011), we assume that γ > μ ₀. This assumption implies that the voter prefers a pair of politicians who exert low effort but are competent with certainty (for whom the probability of achieving a good outcome H is γ) to a randomly drawn pair of new politicians who exert high effort with certainty (for whom the probability of outcome H is μ ₀). Without imposing such an assumption, the voter would end up always preferring to vote for the challengers, thus making impossible to re-elect competent incumbents. In our electoral game, we introduce the additional restriction that μ 0 > μ ̃ 0 > 0 , which ensures that the voter never reelects a pair of politicians after observing a bad policy outcome L (see Assumption 2 below for the definition of the threshold probability μ ̃ 0 ).^[15]

Given the general structure of the policy game illustrated above, we now describe the specific features of first and second terms of office.

A.2 First Term Policy Games (FTPG)

The two categories of FTPG that fit our empirical data are illustrated in Figure A.2. In a FTPG, the incumbent ticket is composed of a ‘debutant’ mayor and a ‘debutant’ deputy. However, mayors can be of two classes: either the deputy of the former mayor, or a ‘truly debutant’ politician who held no role in the previous administration. We label as P 1 e the former, and as P 1 n e the latter, class of politicians, where the subscript 1 stands for ‘first-term incumbents’ and the superscripts e/ne stand for ‘mayor with previous experience’ and ‘mayor without previous experience’, respectively.^[16]

Figure A.2:

First term policy games (FTPG).

The incumbents are of class P 1 e if the last election was won by the deputy mayor of the incumbent party running as candidate for mayorship, with the term-limited mayor passing on the torch to her deputy. Instead, the incumbents are of class P 1 n e if the last election was won either by the challengers of the opposition party, or by a ‘debutant’ candidate of the incumbent party, since the deputy of the former term-limited mayor did not run for elections.

Apart from the possibility of having the two described classes of incumbent politicians, a FTPG evolves as described in the previous section. The only difference is that, if the incumbents are of class P 1 n e , then their probability of entering as competent types is equal to μ 1 ′ P 1 n e = μ 0 (Nature’s draw). If, instead, they are of class P 1 e , then the voter holds a prior belief μ 1 ′ P 1 e = μ 2 O , a , μ 2 ′ , P 2 d about them being competent types, based on the observed policy outcome O, the belief about exerted effort a in the previous term (a second term, as policy makers of class P 2 d , see below), and the prior belief μ 2 ′ of being competent types in the previous term.

Note, from Figure A.2, that the transition probability of a pair of competent politicians to incompetent types is equal to ɛ for both classes P 1 e and P 1 n e of policy makers. That is, we assume that the relevant period determining the transition probability is the number of terms jointly served by the mayor and the deputy, which is one term for both classes of first term policy makers, irrespective of whether the mayor served previously as a deputy.^[17]

Notice finally that, at the end of the political term, at the election stage (see the bottom part of Figure A.2), the game moves on to a second term policy game if the voter reelects the incumbents (where policy makers are of class P 2 d or P 2 n d , see below), while it moves to another FTPG if the voter elects the challengers (where policy makers are of class P 1 n e ).

A.3 Second Term Policy Games (STPG)

The categories of STPG that fit our empirical data are illustrated in Figure A.3. As described in the previous section, the first term politicians who enter their second term as incumbents can be of two classes: either P 1 e or P 1 n e . Both classes share the common feature that while the incumbent mayor is term limited, the deputy mayor is not. It is not therefore precluded to the latter to take over the leadership and run for mayorship at the following election.

Figure A.3:

Second term policy games (STPG).

We model the process by which a lame duck mayor passes on the torch to her/his deputy in a simple way. If the incumbent politicians are competent types, their party Π ‘promotes’ with given probability π ∈ (0, 1) the deputy by allowing her/him to run for mayorship at the following election. Instead, if the politicians are incompetent types, their party Π appoints a ticket of debutant candidates.^[18] In the former case, the second term incumbents with the deputy running for mayorship are denoted by P 2 d . In the latter case, the second term incumbents with the deputy not running for mayorship are denoted by P 2 n d . Once the party’s decision has been taken, and irrespective of whether the incumbents are of class P 2 d or P 2 n d , the political game unfolds – choice of effort, random transition from competent to incompetent types, determination of the policy outcome – in the same way as illustrated in the previous sections. The only relevant difference concerns the transition probability of a pair of competent politicians to incompetent types, which is now equal to ɛ ² for both classes P 2 d or P 2 n d of policy makers, since in both cases the number of terms jointly served by the mayor and the deputy is two.^[19]

At the final stage of the STPG, elections are held. If the election is between policy makers of class P 2 d and a ticket of debutant challengers C of the opposition party (see the bottom-left part of Figure A.3), then the voter draws her ballot on the basis of her posterior beliefs μ 2 O , a , μ 2 ′ , P 2 d about competence of the incumbents versus that, equal to μ ₀, of the challengers. If the winners are the former, then the game moves to a FTPG in which the incumbent politicians are of class P 1 e , whereas, if the winners are the latter, then it moves to a FTPG in which the incumbents are of class P 1 n e . If, instead, the second term incumbent deputy has not been appointed for mayorship, then (see the bottom-right part of Figure A.3) the electoral competitors are identical ‘debutant’ politicians C from both political parties, and therefore the voter casts her ballot by flipping a coin.

A.4 Political Equilibria and Policy Outcomes

The Perfect Bayesian Equilibrium, in pure strategies, of the two-term limit political game set up in the previous sections is formally characterized in Proposition 1 in Section A.7. In this section, we describe its general features.

Provided that μ ̃ 0 < μ 0 < γ , with μ ̃ 0 a function of (γ, ɛ, π), in any equilibrium of the policy game, at the end of a first term the voter reelects the incumbent politicians (either of class P 1 n e or of class P 1 e ) only after observing a good policy outcome H. Otherwise, after observing a bad outcome L, the voter elects the candidates of the opposition party (the challengers C ). At the end of a second term, provided that the deputy-mayor is running as a candidate for mayor (class P 2 d politicians), and again only after observing a good outcome H, the voter elects the deputy-mayor; otherwise, after observing a bad outcome L, the voter elects the challengers. If, instead, at end of a second term the deputy is not running for mayorship (class P 2 n d politicians), the voter, regardless of the outcome, randomly elects with equal probability one of the two pairs of (identical, because both ‘debutants’) candidates in ticket.

As for competence, given the above equilibrium voting strategies, and given that, by assumption, only competent politicians can obtain a good outcome H, all policy makers elected by the voter observing a good outcome enter the following term as competent types with certainty, and they remain competent on the basis of their seniority in office. Hence, as shown in Figure A.4, the probability Pr(θ _C ) that, after choosing the effort level, the policy makers remain competent is equal to 1 − ɛ for class P 1 e (since the mayor and the deputy jointly stayed in office for one term), and to 1 − ɛ ² for classes P 2 d and P 2 n d (since the mayor and the deputy jointly stayed in office for two terms). As for politicians of class P 1 n e , randomly drawn as competent types by Nature with probability μ ₀, they remain competent during their first term of office with probability μ ₀(1 − ɛ).

Figure A.4:

Illustration of proposition 1.

As for accountability, the choice of effort by competent policy makers (recall that, since incompetent types can never obtain a good outcome H, they always choose low effort a ̲ ) is then determined by the randomly realized cost c of high effort (recall that the cost of low effort is normalized to zero) and by the future prospects of staying in office. In this respect, Figure A.4 shows that (competent) policy makers P 2 n d always choose low effort a ̲ , since neither the mayor (term limited) nor the deputy (not running for mayorship) have a direct interest in the following political term. Instead, exerting high effort can be worth the cost for the other three classes of policy makers. Specifically, first-term policy makers P 1 n e and P 1 e exert high effort a ̄ only if c < c ̂ 1 , whereas second-term policy makers P 2 d exert high effort a ̄ only if c < c ̂ 2 d . Note that the probability, F c ̂ 2 d , that politicians P 2 d exert high effort is higher than the probability, F ( c ̂ 1 ) , that either P 1 n e or P 1 e politicians exert high effort, since c ̂ 2 d > c ̂ 1 . The reason is that, while all are accountable politicians, the probability of obtaining a good outcome by P 2 d is higher than that of both P 1 n e and P 1 e , because of the seniority effect of the politicians in ticket, which in turn gives stronger incentives to exert high effort.

A.5 Disentangling Accountability and Competence

Following Alt, Bueno de Mesquita, and Rose (2011), the equilibrium probabilities about competence and effort are used to define the policy makers’ expected performance as the probability of achieving a good outcome H, which is equal to

Using the probabilities Pr(θ _C ) and the thresholds c ̂ 1 < c ̂ 2 d shown in Figure A.4, and letting F ̂ 1 ≡ F ( c ̂ 1 ) , F ̂ 2 d ≡ F c ̂ 2 d , the expected performances for the four classes of politicians are shown in Table A.1.

Clearly, any comparison between first and second terms politicians includes both a competence and an accountability effect. For instance, in the comparison between P 1 e and P 2 d , the former are not only less accountable, but also less likely to be competent, than the latter. Our empirical strategy to disentangle accountability and competence effects in the elections of Italian municipalities is thus the following.

By comparing the performances of first term policy makers, P 1 n e and P 1 e , we can identify a competence effect, since, while equally accountable (the expected effort is the same), the politicians with previous experience P 1 e are more likely to be competent types than those without experience. Formally, the competence effect is equal to

By comparing the performances of second term policy makers, P 2 n d and P 2 d , we can identify an accountability effect, since, while all having the same probability of being competent types, the politicians P 2 d whose deputy is running for mayor are accountable while lame duck policy makers P 2 n d are not. Formally, the accountability effect is equal to

These comparisons in the performance of first- and second-term policy makers are at the heart of our empirical analysis in Section 3 of the paper.

A.6 Theoretical Interpretation of Empirical Results

In terms of the theoretical model, our main empirical results can be expressed as Z 1 n e < Z 1 e ≈ Z 2 d ≈ Z 2 n d , where the performance index Z is minus per capita municipal expenditure, with lower expenditure implying better performance.

If we look at Eq. (A.1), we see that a significant gap Z 1 e − Z 1 n e > 0 can be due, ceteris paribus, to a low value of μ ₀, or to a low value of ɛ. However, a low value of ɛ also implies, from Eq. (A.2), a significant gap Z 2 d − Z 2 n d > 0 , which is not supported by our data. On the other hand, a negligible gap Z 2 d − Z 2 n d can be the result of a large value of γ. And a large value of γ can also explain the non-significant gap Z 1 e − Z 2 d emerging from our data. In sum, our empirical findings are coherent with an institutional setup in which first term politicians without previous experience are competent with a low probability μ ₀, and competent politicians obtain a good outcome with a large probability γ when exerting low effort.

A.7 Proposition 1: Statement and Proof

In the First Term Policy Game (FTPG), and in the Second Term Policy Game (STPG), respectively represented in Figures A.2 and A.3, μ j ′ ( P ) denotes the voter’s prior belief that in term j, j = 1, 2, policy makers of class P are competent types before choosing their effort level, and μ j ( O , a , μ j ′ ( P ) , P ) denotes the posterior belief that policy makers of class P – who obtained outcome O by exerting effort a – are competent types.

The following proposition characterizes the pure strategy Perfect Bayesian Equilibria of the policy game.

A.7.1 Actions Strategies (Parts A, B, C)

The equilibrium actions profiles – the decision to exert high or low effort – are characterized only for competent policy makers, since incompetent politicians, being unable to obtain a good outcome, always exert low effort.

Given the equilibrium voting strategies defined in Part D of the proposition, and the equilibrium probabilities of competent types defined in Part E, the present value V ₁ of the payoffs accruing to first-term policy makers, either P 1 n e or P 1 e , and the present value V 2 d of the payoffs accruing to second-term policy makers with deputy-candidates, P 2 d , are characterized by the following two-equation system:

(A.5) V 1 = B − C ( a 1 ) + ( 1 − ε ) δ g ( a 1 ) π V 2 d + ( 1 − π ) B ,

(A.6) V 2 d = B − C a 2 d + ( 1 − ε 2 ) δ g a 2 d V 1 ,

where C ( a ̲ 1 ) = 0 , C ( a ̄ 1 ) = c , g ( a ̲ 1 ) = γ , g ( a ̄ 1 ) = 1 , C a ̲ 2 d = 0 , C a ̄ 2 d = c , g a ̲ 2 d = γ , g a ̄ 2 d = 1 . Equation (A.5) shows that the flow of payoffs for politicians P 1 n e and P 1 e is given by current (first-term) payoff B − C(a ₁), plus the expected payoff from a second term in office, π V 2 d + ( 1 − π ) B (where the payoff accruing to lame-duck policy makers P 2 n d is simply B), weighted by the probability 1 − ɛ of retaining competence, the probability g(a ₁) of obtaining outcome H, and the discount factor δ. Equation (A.6) shows that the flow of payoffs for politicians P 2 d is given by current (second-term) payoff B − C a 2 d , plus the expected payoff from a first term in office by the deputy-candidate, V ₁, weighted by the probability 1 − ɛ ² of retaining competence, the probability g a 2 d of obtaining outcome H, and the discount factor δ.

By solving the equation system (A.5) and (A.6) in the unknowns V ₁ and V 2 d , we obtain the equilibrium payoffs as a function of the actions about effort chosen by the policy makers, as follows:

(A.7) V 1 a 1 , a 2 d = B − C ( a 1 ) + ( 1 − ε ) δ g ( a 1 ) B − π C a 2 d 1 − ( 1 − ε ) ( 1 − ε 2 ) δ 2 g ( a 1 ) g a 2 d π ,

V 2 d a 1 , a 2 d = B − C a 2 d + ( 1 − ε 2 ) δ g a 2 d B − π C ( a 1 ) 1 − ( 1 − ε ) ( 1 − ε 2 ) δ 2 g ( a 1 ) g a 2 d π

(A.8) + ( 1 − ε ) ( 1 − ε 2 ) δ 2 g ( a 1 ) g a 2 d ( 1 − π ) B 1 − ( 1 − ε ) ( 1 − ε 2 ) δ 2 g ( a 1 ) g a 2 d π .

The policy game admits four possible configuration strategies by policy makers P 1 n e / P 1 e and P 2 d (term-limited policy makers P 2 n d always set low effort a ̲ 2 n d ):

A = a ̄ 1 , a ̄ 2 d , a ̄ 1 , a ̲ 2 d , a ̲ 1 , a ̄ 2 d , a ̲ 1 , a ̲ 2 d .

For any strategy profile a 1 , a 2 d ∈ A , the payoffs defined in Eqs. (A.7) and (A.8) can be expressed as non-increasing linear functions of the cost of effort c, as follows:

(A.9) V 1 a 1 , a 2 d , c = α 1 a 1 , a 2 d − β 1 a 1 , a 2 d × c ,

(A.10) V 2 d a 1 , a 2 d , c = α 2 d a 1 , a 2 d − β 2 d a 1 , a 2 d × c ,

where α > 0 denotes the intercept term and β ≥ 0 the slope coefficient.

Under the four configuration strategies, the coefficients of V 1 a 1 , a 2 d , c are equal to

α 1 a ̄ 1 , a ̄ 2 d = 1 + δ ( 1 − ε ) 1 − δ 2 ( 1 − ε ) ( 1 − ε 2 ) π B , β 1 a ̄ 1 , a ̄ 2 d = 1 + δ ( 1 − ε ) π 1 − δ 2 ( 1 − ε ) ( 1 − ε 2 ) π , α 1 a ̄ 1 , a ̲ 2 d = 1 + δ ( 1 − ε ) 1 − δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ π B , β 1 a ̄ 1 , a ̲ 2 d = 1 1 − δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ π , α 1 a ̲ 1 , a ̄ 2 d = 1 + δ ( 1 − ε ) γ 1 − δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ π B , β 1 a ̲ 1 , a ̄ 2 d = δ ( 1 − ε ) γ π 1 − δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ π , α 1 a ̲ 1 , a ̲ 2 d = 1 + δ ( 1 − ε ) γ 1 − δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ 2 π B , β 1 a ̲ 1 , a ̲ 2 d = 0 ,

while the coefficients of V 2 d a 1 , a 2 d , c are equal to

α 2 d a ̄ 1 , a ̄ 2 d = 1 + δ ( 1 − ε 2 ) + δ 2 ( 1 − ε ) ( 1 − ε 2 ) ( 1 − π ) 1 − δ 2 ( 1 − ε ) ( 1 − ε 2 ) π B , β 2 d a ̄ 1 , a ̄ 2 d = 1 + δ ( 1 − ε 2 ) π 1 − δ 2 ( 1 − ε ) ( 1 − ε 2 ) π , α 2 d a ̄ 1 , a ̲ 2 d = 1 + δ ( 1 − ε 2 ) γ + δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ ( 1 − π ) 1 − δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ π B , β 2 d a ̄ 1 , a ̲ 2 d = δ ( 1 − ε 2 ) γ 1 − δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ π , α 2 d a ̲ 1 , a ̄ 2 d = 1 + δ ( 1 − ε 2 ) + δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ ( 1 − π ) 1 − δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ π B , β 2 d a ̲ 1 , a ̄ 2 d = 1 1 − δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ π , α 2 d a ̲ 1 , a ̲ 2 d = 1 + δ ( 1 − ε 2 ) γ + δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ 2 ( 1 − π ) 1 − δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ 2 π B , β 2 d a ̲ 1 , a ̲ 2 d = 0 .

It is then immediate to see that, for V 1 a 1 , a 2 d , c ,

(A.11) α 1 a ̄ 1 , a ̄ 2 d > α 1 a ̄ 1 , a ̲ 2 d > α 1 a ̲ 1 , a ̄ 2 d > α 1 a ̲ 1 , a ̲ 2 d > 0 ,

(A.12) β 1 a ̄ 1 , a ̄ 2 d > β 1 a ̄ 1 , a ̲ 2 d > β 1 a ̲ 1 , a ̄ 2 d > β 1 a ̲ 1 , a ̲ 2 d = 0 ,

that is, the intercept decreases, and the slope coefficient decreases in absolute value, as the strategy profile changes from a ̄ 1 , a ̄ 2 d to a ̄ 1 , a ̲ 2 d , to a ̲ 1 , a ̄ 2 d , to a ̲ 1 , a ̲ 2 d .

As for V 2 d a 1 , a 2 d , c ,

(A.13) α 2 d a ̄ 1 , a ̄ 2 d > α 2 d a ̲ 1 , a ̄ 2 d > α 2 d a ̄ 1 , a ̲ 2 d > α 2 d a ̲ 1 , a ̲ 2 d > 0 ,

(A.14) β 2 d a ̄ 1 , a ̄ 2 d > β 2 d a ̲ 1 , a ̄ 2 d > β 2 d a ̄ 1 , a ̲ 2 d > β 2 d a ̲ 1 , a ̲ 2 d = 0 ,

that is, the intercept decreases, and the slope coefficient decreases in absolute value, as the strategy profile changes from a ̄ 1 , a ̄ 2 d to a ̲ 1 , a ̄ 2 d , to a ̄ 1 , a ̲ 2 d , to a ̲ 1 , a ̲ 2 d .

For given cost of high effort c, policy makers choose the strategy profile a 1 , a 2 d such that

(A.15) a 1 , a 2 d ∈ A = arg max V 1 a 1 , a 2 d , c ,

(A.16) a 1 , a 2 d ∈ A = arg max V 2 d a 1 , a 2 d , c .

The typical solution is represented in Figure A.5. Denote with V 1 * ( . ) the payoff function associated to the solution of problem (A.15). Panel (1) of Figure A.5 then shows that V 1 * a ̄ 1 , a ̄ 2 d , c if 0 ≤ c < c ̂ 1 , V 1 * a ̲ 1 , a ̄ 2 d , c if c ̂ 1 < c < c ̂ 2 d , and V 1 * a ̲ 1 , a ̲ 2 d , c if c > c ̂ 2 d , where

(A.17) c ̂ 1 = c  such that  V 1 a ̄ 1 , a ̄ 2 d , c = V 1 a ̲ 1 , a ̄ 2 d , c ,

(A.18) c ̂ 2 d = c  such that  V 1 a ̲ 1 , a ̄ 2 d , c = V 1 a ̲ 1 , a ̲ 2 d , c ,

where c ̂ 1 and c ̂ 2 d are defined in Eqs. (A.3) and (A.4), respectively.

Figure A.5:

Politicians payoffs as a function of the cost of effort.

Denote with V 2 d * ( . ) the payoff function associated to the solution of problem (A.16). Panel (2) of Figure A.5 then shows that V 2 d * a ̄ 1 , a ̄ 2 d , c if 0 ≤ c < c ̂ 1 , V 2 d * a ̲ 1 , a ̄ 2 d , c if c ̂ 1 < c < c ̂ 2 d , and V d 2 * a ̲ 1 , a ̲ 2 d , c if c > c ̂ 2 d .

To complete the proof, we have to show that the patterns represented in panels (1) and (2) of Figure A.5 are the only possible ones. To this end, we have to show that (i) c ̂ 2 d > c ̂ 1 , (ii) V 1 a ̄ 1 , a ̲ 2 d , c < V 1 * ( . ) for all c, (iii) V 2 d a ̄ 1 , a ̲ 2 d , c < V 2 d * ( . ) for all c.

As for point (i), it is immediate to see that

c ̂ 2 d − c ̂ 1 = ( 1 − γ ) [ 1 − δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ π ] [ δ ( 1 − ε ) ( 1 − ε 2 ) γ ( 1 − π ) + ε ( 1 − ε ) ] B [ 1 + δ ( 1 − ε ) ( 1 − γ ) π − δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ π ] [ 1 − δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ 2 π ] > 0 .

As for point (ii), let

k 0 = c  such that  V 1 a ̄ 1 , a ̲ 2 d , c = V 1 a ̲ 1 , a ̄ 2 d , c , k 00 = c  such that  V 1 a ̄ 1 , a ̲ 2 d , c = V 1 a ̄ 1 , a ̄ 2 d , c ,

where

(A.19) k 0 ≡ δ ( 1 − ε ) ( 1 − γ ) B 1 − δ ( 1 − ε ) γ π ,

(A.20) k 00 ≡ δ ( 1 − ε 2 ) ( 1 − γ ) 1 + δ ( 1 − ε ) B 1 + δ ( 1 − ε 2 ) ( 1 − γ ) − δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ π .

That V 1 a ̄ 1 , a ̲ 2 d , c < V 1 * ( . ) for all c follows from the fact that

k 00 − k 0 = δ ( 1 − γ ) [ δ ( 1 − ε ) ( 1 − ε 2 ) γ ( 1 − π ) + ε ( 1 − ε ) ] B [ 1 − δ ( 1 − ε ) γ π ] [ 1 + δ ( 1 − ε 2 ) ( 1 − γ ) − δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ π ] > 0 .

As for point (iii), let

k 00 = c  such that  V 2 d a ̄ 1 , a ̲ 2 d , c = V 2 d a ̄ 1 , a ̄ 2 d , c , k 000 = c  such that  V 2 d a ̄ 1 , a ̲ 2 d , c = V 2 d a ̲ 1 , a ̄ 2 d , c ,

where k ₀₀ is defined in Eq. (A.20) and

(A.21) k 000 ≡ δ ( 1 − ε 2 ) ( 1 − γ ) B 1 − δ ( 1 − ε 2 ) γ .

That V 2 d a ̄ 1 , a ̲ 2 d , c < V 2 d * ( . ) for all c follows from the fact that

k 00 − c ̂ 1 = δ ( 1 − γ ) [ 1 − δ 2 ( 1 − ε ) ( 1 − ε 2 ) π ] [ 1 + δ ( 1 − ε 2 ) ( 1 − γ ) − δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ π ] × [ δ ( 1 − ε ) ( 1 − ε 2 ) γ ( 1 − π ) + ε ( 1 − ε ) ] B [ 1 + δ ( 1 − ε ) ( 1 − γ ) π − δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ π ] > 0 , k 000 − c ̂ 2 d = δ 2 ( 1 − ε 2 ) γ ( 1 − γ ) ( 1 − ε 2 ) [ 1 + δ ( 1 − ε ) γ ( 1 − π ) ] + 1 − ε B [ 1 − δ ( 1 − ε 2 ) γ ] [ 1 − δ 2 ( 1 − ε ) ( 1 − ε 2 ) γ 2 π ] > 0 . 

This ends the proof of parts A, B and C of the proposition.