On the feasibility of unpopular policies under re-election concerns.

Chiu, Y. Stephen

1. Introduction

A common concern about political decision making is that re-election motives lead incumbent politicians to select policies that, although popular among the electorate, are inferior to available, less popular alternatives. This concern, reflected in the notions such as demagogy and mob rule, implies that politicians may be penalized for choosing policies that they believe to be the best, but ironically are rewarded for choosing popular policies that they do not necessarily believe to be the best. In this paper, I study this common concern through a series of models. (1)

For the main model that I will study, politicians differ only in utility from holding office. The main result is that strong re-election motives on the part of the incumbent politicians in general do not render efficient and unpopular policies infeasible. In fact I will show that, under very general conditions, voters' voting decisions do not depend on whether the incumbents have chosen popular policy options, nor are incumbents' policy decisions responsive to policy popularity. We may call this result the irrelevance or neutrality of policy popularity. Note that voters only have limited information, and there is a possibility that the unpopular policy may be superior to its alternatives. Voters would take their policy opinions as tentative and be willing to revise them upon the arrival of new information. Then an incumbent's re-election chances will not deteriorate simply because of her choice of unpopular policies. Foreseeing this, incumbents with even strong re-election motives will not find it beneficial to choose popular but inefficient polices. (2)

To convey the above ideas, I will examine an infinite horizon political agency model, along the line pioneered by Barro (1973). There are two important ingredients in this model. First, politicians are of two types: strong and weak. Both types of politicians are concerned about both social welfare and the utility gained from holding office, whereas the strong type puts a smaller weight on the latter. Each politician's type is the politician's private information. If the politicians were homogeneous, the choice between different incumbents would be inconsequential. If their heterogeneity were public information, voters would not need the politicians' past records to infer their preferred future policies. In either case, the question of policy manipulation would no longer exist. (In section 6, I will introduce a third type of politicians.)

The second ingredient is a proper modelling of "a policy that could be unpopular but potentially superior." A policy is popular when more voters think that the policy will yield a welfare greater than those of its alternatives. Such a policy's "popularity" reflects its ex ante efficiency as perceived by the electorate. The incumbent politician, nevertheless, observes an additional imperfect-information signal about the likely efficacy of the policy. (3) Therefore, a popular policy might be interim inefficient, whereas an unpopular policy might be interim efficient.

One crucial assumption for the policy popularity irrelevance result is that the only difference among politicians is their utility derived from holding office. This suggests that for policy popularity to be relevant, other sorts of heterogeneity among politicians are needed. Suppose, for instance, that the public believes with some probability that the incumbent is ignorant in the sense that she knows about different policy options even less than the public does. Then a choice of an unpopular policy might signal the incumbent's ignorance; an incumbent who is informed but with a strong re-election motive might then reject the unpopular policy even if it is interim efficient.

The rest of the paper is organized as follows. Section 2 introduces the model. Section 3 prepares preliminaries for later sections. Section 4 establishes the efficient perfect Bayesian equilibrium under the assumption that voters vote for the candidate who has a greater probability of being strong. It also discusses the issue of equilibrium uniqueness. Section 5 argues that the basic efficiency result holds in several extensions of the basic model. Section 6 discusses an extension in which there are three types of politicians. This section qualifies my basic efficiency result. Section 7 contains some concluding remarks.

2. Model

I use a political agency approach pioneered by Barro (1973) and Ferejohn (1986) and later developed by Austen-Smith and Banks (1989), Banks and Sundaran (1993), and Coate and Morris (1995). In particular, the model presented here closely resembles that of Coate and Morris in the way that informational characteristics about the policy are modeled. I use an infinite horizon model: In each period, the incumbent politician must decide whether to implement a policy or to maintain the status quo. The policy's outcome is uncertain and voters have only limited information about the policy's efficacy. At the end of the period, upon the revelation of the policy outcome, an election between the incumbent and a challenger is held. The winner will become the next period's office holder and will face a policy decision and an election as did the immediate predecessor.

Voters

There is a continuum of infinitely living citizens among whom is a median voter, whose interest coincides with social welfare. As will subsequently become clear, the median voter's preferences dictate the election outcome. (4) Among the citizens is also an incumbent. All agents in the model are risk neutral. In each period t, the current period social welfare, [w.sub.t], is defined as the sum of the income of citizens. Normalizing the measure of citizens to one and assuming all agents discount future payoffs with the common discount factor [delta], the discounted social welfare (the same as the median voter's discounted payoff) measured at period t is thus [[sigma].sup.[infinity].sub.t'=t] [[delta].sup.t'-t][w.sub.t'].

Policy Choices

In each period t, an observable choice between the status quo and an alternative policy, which can differ from period to period, is required for the incumbent politician. The status quo payoff is deterministic and equals zero, whereas the alternative policy payoff, which is revealed immediately before the election, is stochastic and takes the value of either [W.sub.tG] (for a good outcome) or [W.sub.tB] (for a bad outcome), where [W.sub.tG] > 0 > [W.sub.tB]. At the beginning of period t, nature chooses between a high-yield state and a low-yield state with probabilities [[PHI].sub.t] and 1 - [[PHI].sub.t], respectively. In case of the high-yield (low-yield) state, nature chooses between the good outcome and bad outcome with probabilities [[theta].sub.tH] and 1 - [[theta].sub.tH] (probabilities [[theta].sub.tL] and 1 - [[theta].sub.tL], where [[theta].sub.tH] > [[theta].sub.tL]), respectively. Although these probabilities and the values of [W.sub.tg] and [W.sub.tB] are commonly known, the realized state is know n only to the incumbent. Because of this, voters are never certain whether the incumbent has chosen efficiently. Define [W.sub.tH] [equivalent to] [[theta].sub.tH][W.sub.tG] + (1 - [[theta].sub.tH])[W.sub.tB] and [W.sub.tL] [equivalent to] [[theta].sub.tL][W.sub.tG] + (1 - [[theta].sub.tL])[W.sub.tB] as the expected payoff from the policy given the state is high yield and low yield, respectively. To be economically interesting. I assume that [W.sub.tH] > 0 and [W.sub.tL] < 0 for all t.

DEFINITION 2.1. The policy at time t is popular if [[PHI].sub.t] [greater than or equal to] [[PHI].sub.t] where [[PHI].sub.t] is defined by [[PHI].sub.t][W.sub.tH] + (1 - [[PHI].sub.t])[W.sub.tL] = 0. In other words, the policy is popular if and only if it is ex ante efficient. The policy is unpopular if it is not popular. In addition, the policy is interim efficient if the economic state is high yield.

Therefore, a policy can be popular but interim inefficient, and can be unpopular but interim efficient.

Politicians

Each politician can be either strong or weak, and the type is the politician's private information. The current period utility of a type i politician at period t is [w.sub.t] if she is not in office, and [w.sub.t] + [k.sub.i] if she is in office, i = S. W. In the above stipulation, [w.sub.t] is the social welfare of period t and [k.sub.i] is the politician's utility derived from holding office where 0 [less than or equal to] [k.sub.s] < [k.sub.w]. Therefore, the objective of a politician of type i, i S, W, at time t is to maximize the expected value of [summation over ([infinity]/t'=t)] [[delta].sup.t'-t][[p.sub.t'][k.sub.i] + [w.sub.t'] where [p.sub.t'] is the (endogenously determined) probability of being in office at time t'. The preferences of each type of politician are commonly known. Denote by [[lambda].sub.t] the prior probability that the incumbent in period t is strong. The parameter [[lambda].sub.1] is chosen by nature, whereas the determination of [[lambda].sub.t], t = 2, 3, ... will be explained later.

Election

At the end of each period t, an election takes place in which the incumbent is matched with a random challenger whose probability of being strong is drawn from a cumulative function [M.sub.t]([[micro].sub.t]). Each voter chooses either the incumbent or the challenger. The one who receives more votes will be the incumbent in the next period. Since there are two candidates only, the single peakedness condition of preferences (defined over the two candidates) trivially holds. Therefore, the median voter's preferences will dictate the election outcome. I assume that the median voter will use the voting strategy that the politician with a greater probability of being strong is selected. (5) This voting strategy is natural since the median voter's preferences coincide with that of social welfare, and a strong politician cares relatively more about social welfare than a weak politician does. Hereafter, unless otherwise stated, I will use "reputation" and "probability of being strong" interchangeably. I also assume t hat an incumbent, once defeated, has no chance of returning to office. This assumption will be relaxed in section 5. The efficiency of alternative voting strategies is also discussed in section 5.

Information

The sequence of moves of the game is summarized in Figure 1. The only information asymmetry between voters and politicians concerns the actual state of economy and the actual type of each politician, as explained earlier. The structure, parameters, and payoffs of the game, and the preferences of the two types of politicians, are commonly known.

REMARK 1. In particular, the reputation of the first period incumbent, [[lambda].sub.1], the discount factor [delta], the parameters regarding policy efficacy [[PHI].sub.t], [[theta].sub.tH], [[theta].sub.tL], [w.sub.tG], [w.sub.tB], t = 1, 2,... , and the cumulative functions of the challenger's reputation [M.sub.t]([[micro].sub.t]), t = 1, 2, ... are fixed and known to all agents in the model throughout the game.

3. Preliminaries

Because the initial reputation of the first period incumbent [[lambda].sub.1], parameters of policy efficacy [[PHI].sub.t], [[theta].sub.tH], [[theta].sub.tL], [w.sub.tG], [w.sub.tB], t = 1, 2,..., and cumulative functions [M.sub.t]([[micro].sub.t]), t = 1, 2, ... are data of the model, I will not explicitly state them as arguments when specifying strategies. Let [h.sub.t] be a history at the beginning of period t that includes all publicly observed information. Let [H.sub.t] [h.sub.t] be the set of all such histories at the beginning of period t. Specifically, [h.sub.1] = [[lambda].sub.1] and [H.sub.1] = {[h.sub.1]}, I define recursively [h.sub.t] = ([h.sub.t-1], [d.sub.t-1], [w.sub.t-1], [[lambda].sup.U.sub.t-1], [[micro].sub.t-1], [e.sub.t-1]), t = 2, 3, ... where [d.sub.t-1] [member of] {A, R} is the policy choice in period t - 1 with [d.sub.t-1] = A denoting an acceptance of the policy and [d.sub.t-1] = R a rejection; [w.sub.t-1] [member of] {[w.sub.t-1,G], [w.sub.t-1,B], 0} is the economic outcome in p eriod t - 1 with [w.sub.t-1] = [w.sub.t-1,G] denoting a good outcome payoff, [w.sub.t-1] = [w.sub.t-1,B] a bad outcome payoff, and 0 the status quo payoff; [[lambda].sub.U.sub.t-1] [member of] [0, 1] is the updated probability at the end of period t - 1 that the incumbent politician is strong; [[micro].sub.t-1] [member of] [0, 1] is the probability that the challenger in period t - 1 is strong; and [e.sub.t-1] [member of] {0, 1} is the incumbent's re-election outcome in period t - 1 with 0 denoting a failure and 1 a success. Hence, [H.sub.t] [equivalent to] [H.sub.t-1] X {A, R} X {[w.sub.tG], [w.sub.tB], 0} X [0, 1] x [0, 1] X {0, 1}, t = 2,3, ...

REMARK 2. Defining histories to contain beliefs, as I have done here, is somewhat unusual. My justification is as follows. For [h.sub.1] (which equals [[lambda].sub.1]), [[lambda].sub.1] is the probability chosen by nature and is taken by all voters as a datum on which their decisions are based. Hence it should be viewed as part of [h.sub.t]. Similarly, for any history [h.sub.t], t = 2, 3, ... all the beliefs contained in the history are in fact data being used by voters in previous periods to make voting decisions. It is in this sense that they are part of the histories.

I denote the mixed (behavioral) strategy of the incumbent i at period t by [[sigma].sub.ti]: [H.sub.t] [right arrow] [0, 1] as the probability of accepting the policy where i = SH, SL, WH, WL indicates the type of the incumbent (strong S or weak W) and the state of the economy (high-yield state H, or low-yield state L). Define [[sigma].sub.t] [equivalent to] ([[sigma].sub.tSH], [[sigma].sub.tSL], [[sigma].sub.tWH], [[sigma].sub.tWL): [H.sub.t] [right arrow] [[0, 1].sup.4]. Let [[lambda].sup.U.sub.t]: [H.sub.t] X {A, R} X {[w.sub.tG], [w.sub.tB], 0} [right arrow] [0, 1] be a mapping from histories [H.sub.t], current policy decisions, and current outcomes to the posterior probability about the incumbent's being strong at the end of period t. Then [([[sigma].sub.t]).sub.t=1,2,...] and [([[lambda].sup.U.sub.t]).sub.t=1,2,...], together with the voting strategy that the politician with the greater reputation is elected, constitute a perfect Bayesian equilibrium (PBE, hereafter) if (i) under any circumstance, given the beliefs, [([[lambda].sup.U.sub.t]).sub.t=1,2,...], no agent can gain by unilaterally deviating from his or her prescribed strategy and (ii) the beliefs [([[lambda].sup.U.sub.t]).sub.t=1,2,...] are updated via Bayes rule whenever applicable. Note that the first condition applies to both politicians and voters, and hence the proof of a PBE involves the optimality of not only politicians' strategies but also voters' voting strategies. I have assumed that voters will simply vote for the one with a greater reputation. To economize the notation, I will not state this voting strategy in a PBE. The optimality of this voting strategy will be discussed later. Another remark is that the voting strategy implies [[lambda].sub.t] = max {[[lambda].sup.U.sub.t-1], [[micro].sub.t-1]}, Together with the realization of the reputation of the challenger in each period, I know how [[lambda].sub.t] evolves across time in a PBE.

Consider the moment when the incumbent in period t facing a history [h.sub.t] [member of] [H.sub.t] has learned the state [[xi].sub.t] [member of] {H, L}, but has yet to make a policy choice. Given a PBE, I define [V.sup.S*.sub.t]([h.sub.t], [[xi].sub.t]) as a politican's discounted utility or value function measured at time t when she is strong and is the incumbent of period t, [V.sup.W*.sub.t]([h.sub.t], [[xi].sub.t]) when she is weak and is the incumbent of period t, [Y.sup.S*.sub.t]([h.sub.t], [[xi].sub.t]) when she (being either strong or weak) is no longer the incumbent and the actual incumbent of period t is strong, and [Y.sup.W*.sub.t]([h.sub.t], [[xi].sub.t]) when she (being either strong or weak) is no longer the incumbent and the actual incumbent of period t is weak. (6) (Note that the prior reputation of the incumbent for period t is contained in h1 through the relation [[lambda].sub.t] = max {[[lambda].sup.U.sub.t-1], [[micro].sub.-1]}.) I define [V.sup.j.sub.t]([h.sub.t], [[xi]sub.t], A) and [V.sup.j.sub.t]([h.sub.t], [[xi].sub.t], R]) as a j-type (j = S, W) incumbent's discounted utility earned by accepting the policy and rejecting the policy, respectively with when facing history [h.sub.t] and knowing state [[xi].sub.t], given that all other simultaneous and future simultaneous and future strategies stipulation in the PBE will be implemented.

REMARK 3. Whenever the expectation notation [E.sub.t] or [E.sub.t+1] is used, the information set is that available to the agent at the beginning of the period (t or t+1) before the economic state is revealed to the incumbent.

Let me evaluate the value of [V.sup.j.sub.t]([h.sub.t], [[xi].sub.t], A), j = S, W. First of all, it contains a current payoff of [w.sub.t[xi]] + [k.sub.j]. To compute future payoffs, first consider the case in which a good outcome results. The incumbent either wins or loses. If the incumbent wins, then her discounted payoff starting from the beginning of period t + 1 will be

[E.sub.t+1] [V.sup.j*.sub.t+1]{[[h.sub.t], A, [w.sub.tG], [[lambda].sup.U.sub.t](G), [[micro].sub.t], 1], [[xi].sub.t+1]} [equivalent to] [[PHI].sub.t+1] [V.sup.j*.sub.t+1]{[[h.sub.t], A, [w.sub.tG], [[lambda].sup.U.sub.t](G), [[micro].sub.t], 1], H} + (1 - [[PHI].sub.t+1])[V.sup.j*.sub.t+1]{[[h.sub.t], A, [w.sub.tG], [[lambda].sup.U.sub.t](G), [[micro].sub.t], 1], L} (3.1)

where [[micro].sub.t] < [[lambda].sup.U.sup.t](G), which is the incumbent's posterior reputation given that the accepted policy yields a good outcome. (7) If the incumbent loses, provided that the challenger's type is i, i = S, W, her discounted payoff starting from the beginning of period t + 1 will be

[E.sub.t+1][Y.sup.i*.sub.t+1]{[[h.sub.t], A, [w.sub.tG], [[lambda].sup.U.sub.t](G), [[micro].sub.t], 0], [[xi].sub.t+1] [equivalent to] [[PHI].sub.t+1][Y.sup.i*.sub.t+1]{[[h.sub.t], A, [w.sub.tG], [[lambda].sup.U.sub.t](G), [[micro].sub.t], 0], H} + (1 - [[PHI].sub.t+1])[Y.sup.i*.sub.t+1]{[[h.sub.t], A, [w.sub.tG], [[lambda].sup.U.sub.t](G), [[micro].sub.t], 0], L} (3.2)

where the new incumbent's reputation is [[lambda].sub.t+1] = [[micro].sub.t] [greater than or equal to] [[lambda].sup.U.sub.t](G). Likewise, the payoffs when a bad outcome results can be calculated. Putting all these together, I have

[V.sup.j.sub.t]([h.sub.t], [[xi].sub.t], A) = [w.sub.t[xi]] + [k.sub.j] + [summation over (i=G,B)] [delta][[theta].sup.i.sub.t[xi]]{[[integral].sup.[[lambda].sup.U.sub. t](i).sub.0] [E.sub.t+1] [V.sup.j*.sub.t+1]{[[h.sub.t], A, [w.sub.ti], [[lambda].sup.U.sub.t](i), [[micro].sub.t], 1], [[xi].sub.t+1]} [dM.sub.t]([[micro].sub.t]) + [[integral].sup.1.sub.[[lambda].sup.U.sub.t](i)] [[summation over (t=S, W)] [[micro].sup.l.sub.t][E.sub.t+1][Y.sup.l*.sub.t+1]{[[h.sub.t], A, [w.sub.ti], [[lambda].sup.U.sub.t](i), [[micro].sub.t], 0], [[xi].sub.t+1]}] [dM.sub.t]([[micro].sub.t])} (3.3)

where [[theta].sup.G.sub.t[xi]] = [[theta].sub.t[xi]], [[theta].sup.B.sub.t[xi]] = 1 - [[theta].sub.t[xi]], [[micro].sup.S.sub.t] = [[micro].sub.t], [[micro].sup.W.sub.t] = 1 - [[micro].sub.t], and [[lambda].sup.U.sub.t](B) is the incumbent's posterior reputation given that the accepted policy yields a bad outcome. In the same fashion, I evaluate the value of [V.sup.j.sub.t]([h.sub.t], [[xi].sub.t], R) as follows:

[V.sup.j.sub.t]([h.sub.t], [[xi].sub.t], R) = [k.sub.j] + [delta]{[[integral].sup.[[lambda].sup.U.sub.t](R).sub.0] [E.sub.t+1][V.sup.j*.sub.t+1]{[[h.sub.t], R, 0, [[lambda].sup.U.sub.t](R), [[micro].sub.t], 1], [[xi].sub.t+1]} [dM.sub.t]([[micro].sub.t]) + [[integral].sup.1.sub.[[lambda].sup.U.sub.t](R)] [[summation over (l=S,W)] [[micro].sup.l.sub.t][E.sub.t+1][Y.sup.l*.sub.t+l]{[[h.sub.t], R, 0, [[lambda].sup.U.sub.t](R), [[micro].sub.t], 0], [[xi].sub.t+1]}] [dM.sub.t]([[micro].sub.t])} (3.4)

where [[lambda].sup.U.sub.t](R) is the incumbent's posterior reputation given that the policy is rejected.

4. Efficient Equilibrium

With the preliminaries in the last section, I now show the existence of a PBE where [[sigma].sub.tSH]([h.sub.t]) = [[sigma].sub.tWH]([h.sub.t]) = 1, and [[sigma].sub.tSL]([h.sub.t]) = [[sigma].sub.tWL]([h.sub.t]) = 0 for all [h.sub.t] [member of] [H.sub.t], t = 1, 2, . . . . Intuitively, such a description states that regardless of history, policy popularity, and incumbent type, the policy is accepted when the economy is of a high-yield state and the status quo is maintained otherwise. It is straightforward to establish the following lemma (proof omitted), which is key to my main result.

LEMMA 4.1. Given any history [h.sub.t]. Suppose [[sigma].sub.tSH]([h.sub.t]) = [[sigma].sub.tWH]([h.sub.t]) = 1, and [[sigma].sub.tSL]([h.sub.t]) = [[sigma].sub.tWL]([h.sub.t])= 0. Let [[lambda].sub.t] and [[lambda].sup.U.sub.t](i) be the prior and posterior reputation of the incumbent in period t where i = G (acceptance with a good outcome), B (acceptance with a bad outcome), R (rejection). We have [[lambda].sup.U.sub.t](G) = [[lambda].sup.U.sub.t](B) = [[lambda].sup.U.sub.t](R) = [[lambda].sub.t].

In other words, an incumbent's updated reputation is exactly the same as her prior reputation, regardless of policy choice and outcome. The idea is in fact very simple. Since both the strong and the weak behave exactly the same, the policy choice and corresponding policy outcome do not provide further useful information to distinguish the type of the incumbent, and therefore, the prior belief is retained. I also have the following lemma:

LEMMA 4.2. Consider any pair of [h.sub.t], [h'.sub.t] [member of] [H.sub.t] that are consistent with the same [[lambda].sub.t]. Suppose [[sigma].sub.[tau]SH]([h.sub.[tau]]) = [[sigma].sub.[tau]WH]([h.sub.[tau]]) = 1 and [[sigma].sub.[tau]SL]([h.sub.[tau]]) = [[sigma].sub.[tau]WL]([h.sub.[tau]]) = 0 for all [h.sub.[tau]] [member of] [H.sub.[tau]], [tau] = t, t + 1, . . . . I have:

(i) [E.sub.t][V.sup.j*.sub.t]([h.sub.t], [[xi].sub.t]) = [E.sub.t][V.sup.j*.sub.t]([h'.sub.t], [[xi].sub.t]), j = S, W and

(ii) [E.sub.t][Y.sup.S*.sub.t]([h.sub.t], [[xi].sub.t]) = [E.sub.t][Y.sup.S*.sub.t]([h'.sub.t], [[xi].sub.t]) = [E.sub.t][Y.sup.W*.sub.t]([h.sub.t], [[xi].sub.t]) = [E.sub.t][Y.sup.W*.sub.t]([h.sub.t], [[xi].sub.t].

PROOF. See the Appendix.

The lemma states that all aspects of any history [h.sub.t] other than [[lambda].sub.t], are pay-off irrelevant. The reason is that these other aspects affect neither incumbent's strategy nor the re-election chance of period t's incumbent.

To show that the proposed strategies and beliefs constitute a PBE, I first show that given any arbitrary history [h.sub.t], no i [member of] { SH, SL, WH, WL } will have an incentive to deviate unilaterally once and then conform to the prescription thereafter. Consider period t's incumbent i's payoff by accepting the policy and rejecting the policy, respectively, where i = SH, SL, WH, WL. Substituting [[lambda].sup.U](G) = [[lambda].sup.U](B) = [[lambda].sup.U](R) = [[lambda].sub.t] (Lemma 4.1) into Equations 3.3 and 3.4, I have

[V.sup.j.sub.t]([h.sub.t], [[xi].sub.t], A) = [w.sub.t[xi]] + [k.sub.j] + [summation over (i=G,B)] [delta][[theta].sup.i.sub.t[xi]]{[[integral].sup.[[lambda].sub.t].sub .0] [E.sub.t+1][V.sup.j*.sub.t+1][([h.sub.t], A, [w.sub.ti], [[lambda].sub.t], [[micro].sub.t], 1), [[xi].sub.t+1]] [dM.sub.t]([[micro].sub.t]) + [[integral].sup.1.sub.[[lambda].sub.t]] [[summation over (l=S,W)] [[micro].sup.l.sub.t][E.sub.t+1][Y.sup.l*.sub.t+1][([h.sub.t], A, [w.sub.ti], [[lambda].sub.t], [[micro].sub.t], 0), [[xi].sub.t+1]] [dM.sub.t]([[micro].sub.t])} (4.1)

where [[theta].sup.G.sub.t[xi]] = [[theta].sub.t[xi]], [[theta].sup.B.sub.t[xi]] = 1 - [[theta].sub.t[xi]], [[micro].sup.S.sub.t] = [[micro].sub.t], [[micro].sup.W.sub.t] = 1 - [[micro].sub.t], [[xi].sub.t] = H, L, and

[V.sup.j.sub.t]([h.sub.t], [[xi].sub.t], R]) = [k.sub.j] + [delta]{[[integral].sup.[[lambda].sub.t].sub.0] [E.sub.t+1][V.sup.j*.sub.t+1][([h.sub.t], R, 0, [[lambda].sub.t], [[micro].sub.t], 1), [[xi].sub.t+1]] [dM.sub.t]([[micro].sub.t]) + [[integral].sup.l.sub.[[lambda].sub.t]] [summation over (l=S,W)] [[micro].sup.l.sub.t][E.sub.t+1][Y.sup.l*.sub.t+1][([h.sub.t], R, 0, [[lambda].sub.t], [[micro].sub.t], 0), [[xi].sub.t+1]] [dM.sub.t]([[micro].sub.t])} (4.2)

where [[xi].sub.t] = H, L. Note that the proposed strategies depend only on the state of the economy, but not on the history itself. Given that all agents are to follow the prescription onwards from period t + 1, using Lemma 4.2, [E.sub.t+l][V.sup.j*.sub.t+1]([h.sub.t+1], [[xi].sub.t+1]) is independent of [h.sub.t+1] as long as [[lambda].sub.t+1] (which is contained in [h.sub.t+1]) is the same. So are the values of [E.sub.t+l][Y.sup.S*.sub.t+1]([h.sub.t+1], [[xi].sub.t+1]), and [E.sub.t+l][Y.sup.W*.sub.t+1]([h.sub.t+1], [[xi].sub.t+1]). Therefore, by subtracting Equation 4.2 from Equation 4.1, I obtain: [V.sup.j.sub.t]([h.sub.t], [[xi].sub.t], A) - [V.sup.j.sub.t]([h.sub.t], [[xi].sub.t], R) = [W.sub.t[xi]] > 0 if [xi] = H and <0 if [xi] = L. This suggests that given the stated prescription and beliefs, at time t no incumbent i [member of] {SH, SL, WH, WL} will have an incentive to deviate unilaterally from her prescribed strategy at time t and conform to it thereafter. Using the one-stage-deviation principle in s ubgame perfection (Theorem 4.2 in Fudenberg and Tirole 1991, p. 110), I establish that nobody will gain from unilaterally deviating more than once. I therefore establish the following proposition:

PROPOSITION 4.3. There exists a PBE ([sigma], [[lambda].sup.U]) in which the incumbent always accepts the policy if the economy is of a high-yield state, and always rejects the policy if the economy is of a low-yield state. Formally, in that PBE ([sigma], [[lambda].sup.U]), [[sigma].sub.tSH]([h.sub.t]) = [[sigma].sub.tWH]([h.sub.t]) = 1, and [[sigma].sub.tSL]([h.sub.t]) = [[sigma].sub.tWL]([h.sub.t]) = 0 for all [h.sub.t] [member of] [H.sub.t], t = 1, 2,... and voters have beliefs [[lambda].sup.U.sub.t](G) = [[lambda].sup.U.sub.t](B) = [[lambda].sup.U.sub.t](R) = [[lambda].sub.t], t = 1,2,....

The result is fairly intuitive. Given that voters view the incumbent equally strong regardless of policy choice and outcome, the incumbent cannot gain any future benefit from manipulating policy choice to influence re-election chances. Since manipulating policy choice will bring an immediate loss in the current period, the incumbent simply makes the efficient decision, which better maximizes her objective.

The proposition can be alternatively understood as follows. Since voters would like to vote for strong politicians, weak incumbents would like to mimic strong incumbents. Note that, a priori, weak incumbents do not have preferences on the policy that are different from those of strong incumbents. Therefore, as long as strong incumbents are prescribed to act efficiently, weak incumbents will follow suit. This accounts for our pooling, but efficient, equilibrium. The insight differs from that revealed in many signaling games (e.g., Spence 1973) in which signaling is more costly for weak-type agents to make.(8)

Salmon (1993) argues that reputation may play a role in making unpopular but efficient policies feasible. The idea is that although adopting unpopular but efficient policy now may hamper re-election chances, it may also build up a reputation in the long run. Such a reputation is useful, of course, only when the incumbent can be re-elected once defeated. The analysis here, however, does not utilize such a reputation argument. I will modify the model to allow losing incumbents to return for election in section 5.

One final observation about the equilibrium is that the reputation of office holders must be monotonically nondecreasing, with the only changes occurring when an incumbent is replaced by a challenger with a better reputation. Since [[lambda].sup.U.sub.t] = [[lambda].sub.t], the incumbent in period t + 1 will have a greater reputation if and only if the incumbent in period t is replaced. Once I assume that incumbents may have a fixed chance of not running for re-election, the monotonicity of the incumbents' reputation is no longer guaranteed.

Although the equilibrium I have identified has a quite simple structure, I do not expect it to be unique. One cause of multiple equilibria in incomplete information games is that there are few restrictions on off-the-equilibrium-path beliefs. To avoid multiple equilibria of this sort, I confine my focus to equilibria where both acceptance and rejection are observed with positive probability along the equilibrium path. In addition, two more conditions are required. (i) In equilibrium, [[lambda].sup.U*.sub.t](G) [greater than or equal to] [[lambda].sup.U*.sub.t](B) for all [h.sub.t], t = 1, 2,... and (ii) in equilibrium, [[sigma].sub.tSH] [greater than or equal to] [[sigma].sub.tWH] for all [h.sub.t], t = 1, 2,... The first condition states that voters consider an incumbent more likely to be strong when the policy outcome is good instead of bad. The second condition states that, provided the economy is of high yield, the strong incumbent is more likely to accept the policy than the weak incumbent.

PROPOSITION 4.4. (Unique equilibrium outcome) For every PBE in which (i) both acceptance and rejection are observed with positive probability, (ii) [[lambda].sup.U*.sub.t](G) [greater than or equal to] [[lambda].sup.U*.sub.t](B), and (iii) [[sigma].sub.tSH] [greater than or equal to] [[sigma].sub.tWH] for all histories [h.sub.t], t = 1, 2,..., it must be true that [[sigma].sup.*.sub.tSH]([h.sub.t]) = [[sigma].sup.*.sub.tWH]([h.sub.t]) = 1, and [[sigma].sup.*.sub.tSL]([h.sub.t]) = [[sigma].sup.*.sub.tWL]([h.sub.t]) = 0, t = 1, 2,...

PROOF. See the Appendix.

Since in the PBE studied both strong and weak politicians make decisions efficiently, the reader may wonder why the strong type is preferred in the first place. In addition to the justification provided in section 2, it would still be helpful to think of a different consideration--moral hazard. Suppose that occasionally the incumbent may be required to put in an unobservable effort to achieve a task, the outcome of which promotes voters' welfare in an expected value sense. Since the strong incumbent is concerned relatively more about social welfare than the weak, it is reasonable to assume that her level of effort would be more appropriate, and hence voters should prefer the strong incumbent to the weak. In fact, this argument does not require that the extra benefits be substantial or the incumbent need make an effort in every period. As formally putting this moral hazard issue into the model will significantly complicate the analysis, I am content to simply point out the issue here.

5. Extensions

In this section, I maintain the voting strategy stipulated in section 2 to study a few extensions, while leaving the extension involving three types of politicians to the next section.

Policy Outcome Revealed after the Election

I argue here that an efficient PBE similar to that stipulated in section 4 exists even in the case when the policy outcome is revealed after the election. Suppose, specifically, the policy outcome is revealed at the beginning of the next period. Although an exact analysis for this case is even more involved (which is part of the reason why I chose to present a model with contemporary outcomes), the basic insight is very similar. Suppose that each incumbent is thought to behave efficiently hereafter. Thus, the incumbent's posterior reputation will be the same as the prior one, whatever policy outcome is revealed. This suggests that by deviating in the current period the incumbent gains neither greater re-election chances for the current election nor greater chances in the next period. However, by choosing an inefficient policy, the incumbent is expected to receive a smaller gain at the beginning of the next period. The incumbent thus has a strict incentive to act efficiently. The belief that the incumbent choo ses efficiently is thus supported.

Different Definitions of Strong and Weak Incumbents

Coate and Morris (1995) assume that a politician, either strong or weak, cares about social welfare only when she is the incumbent. A plausible explanation of such a utility function is that the politician, whenever not the incumbent, may have a reservation price that is not directly related to the current economic situation, but, whenever in office, gains from a more prosperous economy that confers her with a greater material or psychological benefit.

PROPOSITION 5.1. Consider a model with the same specification as that stipulated in section 2 but with the following new definitions of politician: A politician is strong if her current period utility equals w + [k.sub.s] when she is the incumbent and equals 0 otherwise, where w is the social welfare of the current period and [k.sub.s] is the utility that she derives from holding office; a politician is weak if her current period utility equals w + [k.sub.w] when she is the incumbent and equals 0 otherwise, where w is the social welfare of the current period and [k.sub.w] is the utility that she derives from holding office (0 [less than or equal to] [k.sub.s] < [k.sub.w]). Then there exists a PBE in which the incumbent always accepts the policy if the economy is high yield and always rejects the policy if the economy is low yield.

The proof is again achieved by the one-stage-deviation principle. The intuition behind the result is similar to that I have already emphasized.

Losers Can Come Back for Re-election

So far I have assumed that an incumbent who once loses is unable to stand for re-election in the future. Here I argue that the relaxation of this assumption does not nullify the efficient equilibrium. The basic insight of the previous sections holds true. Given that the updated probabilities that voters have about an incumbent being strong do not alter with the incumbent's decision, policy manipulation affects neither the chances of re-election nor the expected future payoffs conditional on the election outcome; the only concern is the direct effect of the policy decision. The efficient equilibrium is thus supported. (9)

Bi-Party Competition

Another interesting extension posits a partisan model in which two parties compete and dominate the political arena. Suppose that each party's leadership can be either strong or weak, with the weak putting a greater weight on the utility gained from holding office. Suppose the reputation of each party evolves in the following way: During its incumbency each party's leadership remains unchanged and so does its type; but once the incumbent party loses in an election, I assume that there will be a possibility of leadership change in that party. The latter assumption gives rise to uncertainty about the type of the leadership of the challenging party, serving the role of [M.sub.t]([[micro].sub.t]), and thus replicates the basic model in section 2, with another modification that the losing party can return for re-election. Putting aside the issue of ideology and party's policy preferences, it can be shown that the insight prevailing in previous models also works in this case.

Alternative Voting Strategies

So far I have found that there exists a PBE in which the incumbent always acts efficiently when voters use the voting strategy that elects the politician with a greater reputation. When alternative voting strategies are used that explicitly penalize (reward) the choice of unpopular policy but reward (penalize) the choice of popular policy, the efficient equilibrium cannot, in general, be supported. Those that explicitly penalize (reward) the choice of unpopular policy will force incumbents to reject (accept) unpopular policies that are interim efficient (inefficient). Such voting strategies are not optimal, even if all politicians are of the same type. There are circumstances in which efficient policy making dictates the adoption of an unpopular policy while the precommitted voting strategy penalizes such a policy choice. In this case, the incumbent may act inefficiently to enhance her chance of re-election. Thus, the voting strategy described in this paper is justified.

6. Three Types of Politicians

The previous discussion suggests that re-election motives need not lead to the infeasibility of efficient but unpopular policies. A crucial assumption is that politicians differ solely in the utility that they gain from holding office. Once other sorts of heterogeneity are introduced, however, the result may be different. Suppose, for instance, that there is an additional type of politician who is stupid or dumb in the sense that she is less familiar with a policy than the public. That is, the dumb incumbent in period t does not know [[phi].sub.t]; nor does she observe the state of economy. I assume that she simply randomizes with equal probability between the policy and the status quo. (10) Consequently, to avoid being viewed as dumb, weak incumbents may accept popular policies that are interim inefficient or reject unpopular policies that are interim efficient.

To examine this possibility, I focus on a simplified, two-period version of my model. I denote a politician's type by i, where i = S, W, D, when she is strong, weak, and dumb, respectively. I assume that the second period will be solved so that the expected second period welfare resulting from a type i second-period incumbent will be [w.sub.2i], where i = S, W, D, and that these values are commonly known and agreed upon by all parties at the outset of the game. It is natural to assume that [w.sub.2S] > [w.sub.2W] > [w.sub.2D].

Denote the prior probability that the first-period incumbent is of type i by [[lambda].sub.1i], where i = S, W, D. At the end of the first period, upon observing policy outcome I, voters update their beliefs about the incumbent's type, coming up to [[lambda].sup.U.sub.1](i\j), the posterior probability that the incumbent is of type i, where i = S, W, D and j = G, B, R. If the incumbent is retained, then the median voter foresees an expected payoff of [[sigma].sub.i=S,W,D][[lambda].sup.U.sub.1](i\j)[w.sub.2i] conditional on the observation of policy outcome j, where j = G, B, R. The incumbent is retained as long as that expected payoff is not less than the expected payoff resulting from the challenger, [w.sup.C]. The probabilities associated with the challenger's true type are random variables and are not realized until immediately before the election. Once these probabilities are realized, voters can compute [w.sup.C] to make their election choices. Let N([w.sup.C]) be the resulting cumulative function of [w.sup.C] perceived by the public at the beginning of the game. I assume that the random variable [w.sup.C] has a support of [[w.sub.2S], [w.sub.2D]]. I further assume that the parameters of the game are set such that the strong incumbent's dominant strategy is always to choose efficiently.

I call the strategy profile that [[sigma].sub.1SH], = 1, [[sigma].sub.1SL] = 0, [[sigma].sub.1WH] = 1, and [[sigma].sub.1WL] = 0 the efficient strategy profile, and call the resulting outcome the efficient outcome. With the additional information that [[sigma].sub.1DH] = [[sigma].sub.1DL] = 0.5, the updated probabilities of the incumbent's type in the first penod, after some computation, are as follows:

[[lambda].sup.U.sub.1](i\R) = [[lambda].sub.i]/(1 - [[lambda].sub.D]) + [[lambda].sub.D]/[2(1 - [[PHI].sub.1])],

[[lambda].sup.U.sub.1](i\G) = [[lambda].sub.i]/(1 - [[lambda].sub.D]) + [[lambda].sub.d]{1 + [(1 - [[PHI].sub.1])/[[PHI].sub.1]]([[theta].sub.L]/[[theta].sub.H])}/2,

[[lambda].sup.U.sub.1](i\B) = [[lambda].sub.i]/(1 - [[lambda].sub.D]) + [[lambda].sub.D]{1 + [(1 - [[PHI].sub.1])/[[PHI].sub.1]][(1 - [[theta].sub.L])/(1 - [[theta].sub.H])]}/2, (6.1)

where i = S, W, and [[lambda].sup.U.sub.1](D\j) = 1 - [[lambda].sup.U.sub.1](S\j) - [[lambda].sup.U.sub.1](W\j) where j = R, G, B. Given the efficient strategy prescription, I can compute, as in sections 3 and 4, the payoff to the incumbent of each type (strong and weak) under each state (high, low) for each policy choice (acceptance or rejection) and check if any type under any circumstance has a unilateral incentive to deviate. The following proposition is obtained (proof omitted).

PROPOSITION 6.1. Suppose [k.sub.w] is sufficiently large. Consider the efficient strategy profile and corresponding updated beliefs. (i) Given [[lambda].sub.1] = ([[lambda].sub.1S], [[lambda].sub.1W], [[lambda].sub.1D]), as [[PHI].sub.1] decreases (increases), the weak incumbent facing a high (low) state is more likely to have a unilateral incentive to deviate from her strategy prescription. (ii) Given [[PHI].sub.1], for [[lambda].sub.1D] = 0, the efficient outcome is always feasible. (iii) Given [[PHI].sub.1], for sufficiently large [[lambda].sub.1D], the efficient outcome is always feasible.

Result (i) confirms my earlier intuition that the weak incumbent's policy choice is responsive to policy popularity. When [[PHI].sub.1] approaches zero, it follows from Equation 6.1 that [[lambda].sup.U.sub.1](i\R) [right arrow] 2[[lambda].sub.i]/(2 - [[lambda].sub.D]), [[lambda].sup.D.sub.1](i\G) [right arrow] 0, and [[lambda].sup.D.sub.1](i\B) [right arrow] 0, where i = S, W. That is, a weak incumbent at the high state, by following the prescribed strategy, will be regarded as a dumb politician with certainty, which implies a zero probability of being re-elected. However, by unilaterally deviating, she still has some chance of being re-elected. Therefore, the efficient outcome is less likely to be feasible when [[PHI].sub.1] approaches zero, and likewise when [[PHI].sub.1] approaches one.

Result (ii) corresponds to the efficiency result for the main model of this paper, where there are only two types of politicians. Recall that voters are aware that the incumbent's information about efficacy is superior to the voters' prior information. When [[lambda].sub.1D] is negligible, voters consider policy popularity as useless information. However, when [[lambda].sub.1D] becomes larger, unwise decisions on the part of the incumbent become probable, and foreseeing this, voters consider policy popularity informative. Therefore, weak incumbents will condition their decisions on policy popularity, even though from their viewpoint policy popularity is completely uninformative. However, their responsiveness to policy popularity is not monotone in [[lambda].sub.1D]. Result (iii) states that, for a very large [[lambda].sub.1D], any updated beliefs must assign the incumbent with a very large probability of being dumb. From the viewpoint of the weak incumbent, the increase in re-election chances from a unilatera l deviation is too little to justify the deviation. (According to Equation 6.1, as [[PHI].sub.1] [right arrow] 1, lim [[lambda].sup.U.sub.1](D\j) [right arrow] 1 where j = R, G, B.)

To further the understanding of the proposition, I now report some simulation results. The following parameterization is used: [w.sub.2S] = 5, [w.sub.2W] = 4, [w.sub.2D] = 0, [w.sub.1G] = 10, [w.sub.1B] = - 10, [w.sub.1R] = 0, [[theta].sub.H] = 0.8, [[theta].sub.L] = 0.2, [delta] = 0.9, [k.sub.S] = 0, [k.sub.W] = 30, and the random variable [w.sup.C] is uniformly distributed in [[w.sub.2S], [w.sub.2D]]. With this specification, a policy is popular if and only if [[PHI].sub.1] > 0.5.

Some simulation results are depicted in the four panels of Figure 2 for different values of [[PHI].sub.1]. For each panel, an equilateral triangle (unit simplex) is drawn to represent the space of [[lambda].sub.1]. Each side of the triangle has a length of 2/[square root of (3)] so that the sum of the distance of any point, say A in Figure 2b, in the triangle to each side always equals unity. Therefore, each point in the triangle uniquely determines a 3-tuple [[lambda].sub.1] where the distance from the point to the bottom side is [[lamabda].sub.1S], to the top left side [[lambda].sub.1w], and to the top right side [[lambda].sub.1D]. Conversely, each 3-tuple [[lambda].sub.1] can be uniquely represented by one and only one point in the space. (If the 3-tuple has one element equal to zero, then it will be located on a side of the triangle. In case it has two elements equal to zero, it will be located on a vertex of the triangle.)

My simulation shows that the efficient outcome is always feasible for all [[lambda].sub.1] [equivalent to] ([[lambda].sub.1S], [[lambda].sub.1W], [[lambda].sub.1D]) for [[PHI].sub.1] [member of] [0.186, 0.776]. However, this is not true for more extreme values of [[PHI].sub.1]. For [[PHI].sub.1] <0.186, there are 3-tuples [[lambda].sub.1], that, given the efficient strategy profile and corresponding beliefs, the weak type in the high state finds it beneficial to unilaterally deviate by rejecting the policy. The set of 3-tuples [[lambda].sub.1] where such a unilateral deviation occurs is depicted as a shaded region in Figure 2a and b where [[PHI].sub.1] = 0.1 and 0.15, respectively. First, note that the shaded region in Figure 2a is much larger than that in Figure 2b. This suggests that when the policy is getting more unpopular, the unilateral tendency of the weak incumbent to reject an efficient policy is more profound. This is consistent with part (i) of Proposition 6.1. Second, the shaded region will expand t oward but never reach the top left side and the bottom left vertex as [[phi].sub.1] approaches zero. This is consistent with parts (ii) and (iii) of the proposition.

For [[PHI].sub.1] > 0.776, the simulation shows that there are 3-tuples [[lambda].sub.1] that, given the efficient strategy profile and corresponding beliefs, the weak type might have a unilateral incentive to deviate. But unlike the case where [[PHI].sub.1] < 0.186, now it is in the low state that the weak type will deviate. The set of 3-tuples [[lambda].sub.1] where such a unilateral deviation occurs is depicted as a shaded region in Figure 2c and d where [[PHI].sub.1] = 0.8 and 0.9, respectively. That the shaded region in Figure 2d is much larger than that in Figure 2c is consistent with part (i) of Proposition 6.1. That the shaded region will expand toward but never reach the top right side and the bottom left vertex as [[PHI].sub.1] approaches unity is consistent with parts (ii) and (iii) of the proposition.

My 3-type 2-period model shows that re-election concerns may lead to policy manipulation; the irrelevance or neutrality of policy popularity breaks down once other sorts of heterogeneity exist among politicians in addition to the differing utility gained from holding office. Although I have defined this third type of politician as one who is completely ignorant of the economy, she could also be driven by corruption, for example, or simply by interests that diverge in various ways from the interests of the median voter (such as special interest lobbying). In that way, it is not hard to see that the policy choices of weak politicians will still be responsive to policy popularity. However, the exact relationship between policy popularity and the weak incumbents' policy choices is, in general, model specific. Suppose the third type of politician is dumb in the sense that she always makes the popular choices. Then to avoid being viewed as dumb, weak incumbents may reject popular policies even if they are interim e fficient, or accept unpopular policies even if they are interim inefficient.

It is worthwhile relating these results to those in a recent paper by Mon-is (2001) about political correctness, which contains a model very similar to the model in this section. His model is similar in the sense that one gives bad advice to avoid looking like someone who is biased in a certain way. Therefore, the "politically correct" advisor in his paper is a lot like the weak politician when she is trying to avoid looking like a dumb politician. Incidentally, the Morris model suggests that even a politician who is only concerned about social welfare may try to increase her re-election chances by choosing popular but suboptimal policies. If a strong politician thinks that she might get replaced by a bad politician once she loses re-election, this may pressure her to choose popular policies to assure good policies in the future. I do not have such a result because I have explicitly assumed that acting efficiently is the strong politician's dominant strategy. But presumably the result is feasible once the lat ter assumption is relaxed. Whereas Morris is interested in a detailed look at the pattern of inefficiency, here I am interested in investigating when efficiency is still feasible.

7. Concluding Remarks

We often hear complaints about the interference of political objectives on good policy making. A particular form of these complaints is political decision making: Re-election concerns lead incumbent politicians to select policies that, although popular among the electorate, are inferior to available, less popular alternatives. One contribution of this paper is to pose and analyze the question in a formal way.

In my core model and its various extensions with two types of politicians, I have shown that the efficient outcome can generally be supported as an equilibrium outcome. (11) All these models assume that politicians differ only in the utility gained from holding office. In the model studied in section 6, where I introduced a third type of politician who differs from other types in the knowledge about the economy, I found that the weak incumbent starts to condition her policy decisions according to policy popularity, which she knows contains no extra information about policy efficacy. The contrast between the results from section 6 and those from earlier sections shows that for policy manipulation to occur, a difference in re-election concerns (i.e., the difference in utility gained from holding office) per se is in general not sufficient, and other sorts of heterogeneity among politicians are needed. I have also related the model in section 6 with Morris's (2001) interesting work about political correctness.

Plausible reasons why political decision making may be inefficient have been raised in the literature. One particular reason is that voters and politicians have different time preferences. The mortality of agents and the time-to-build nature of investment presumably create a different time preference between voters and the incumbent. This line of argument has been put forward by Glazer (1989), Tabellini and Alesina (1990), and Garfinkel (1994). Another source of inefficiency is due to private interest lobbying. For the influence of private interests on policy making, see the seminal paper by Stigler (1971). Since self-interested politicians also derive utility from the private interest that lobbies for the available policy (or the status quo), it is not hard to imagine that the motivation of the self-interested incumbent's policy choice would be distorted. Although clearly interesting, these studies differ from this one in an important aspect: Inefficient policy making will arise even in the absence of re-ele ction concerns. Moreover, their studies do not explore the very issue of whether re-election concerns per se might lead office-seeking politicians to choose popular but inefficient policies.

Finally, some remarks about two pieces of related work are in order. Wittman (1995) maintains, as I do here, that voters will take the possibility of policy manipulation into consideration. However, he does not spell out the effect of choosing unpopular policies on re-election chances. It is unclear whether he also agrees with the common concern that choosing unpopular policies may lead to lower chances of re-election. Salmon (1993) maintains, as I do here, that voters are aware of the limitation of their information and hence do not tend to pressure the incumbent on policy choices. However, he seems to admit that mob rule does exist, at least in the short run. Specifically, he argues that to gain reputation politicians might be willing to sacrifice short-term re-election chances. Another difference is that no formal model is provided in Wittman (1995) and Salmon (1993).

Appendix A: Proof of Lemma 4.2

PROOF. Note that each politician derives utility from two sources: the economy and holding office. Recall that a politician cannot run for election once losing her office. Her utility in this case derives solely from the economy. When all future policies are efficiently made, the per period social welfare is [[phi].sub.[tau]][W.sub.[tau]H], [tau] = t, t + 1,... Hence a not-in-office politician's utility, denoted by [EY.sup.J*.sub.t]([h.sub.t],[[xi].sub.t] = [summation over ([infinity]/i=0)] [[delta].sup.i][[phi].sub.t+1][W.sub.t+i,H], is independent of [h.sub.t], as long as [h.sub.t] is such that the politician is no longer an incumbent at period t. Note that [EY.sup.j*.sub.t]([h.sub.t], [[xi].sub.t]) = [EY.sup.i*.sub.t]([h.sub.t], [[xi].sub.t]) where j = S, W, i = S, W, i [not equal to] j. This proves part (ii). For part (i), it is clear that [EV.sup.j*.sub.t]([h.sub.t], [[xi].sub.t]) = [summation over ([infinity/i=0)] [[delta].sup.i][[phi].sub.t+i][W.sub.t+i,H] + [summation over ([infinity]/i=0)] [p.sub.t+i][[delta].sup.i][k.sub.j], where [p.sub.t+i] is the expected probability that the incumbent at time t will still be in incumbency at time t + i. Clearly, [p.sub.t] = 1 and [p.sub.t+i] = [[PHI].sup.i-1.sub.l=0] [M.sub.t+l]([[lambda].sub.t+l], i = 1, 2,... Thus I have [EV.sup.j*.sub.t]([h.sub.t], [[xi].sub.t]) = [EY.sup.j*.sub.t]([h.sub.t], [[xi].sub.t]) + {1 + [summation over ([infinity]/i=1)][[[PHI].sup.i-1.sub.l=0] [M.sub.t+l]([[lambda].sub.t+l][[delta].sub.i]]} [k.sub.j], which is independent of [h.sub.t], as long as [[lambda].sub.t] is fixed. This proves (i). QED.

Appendix B: Proof of Proposition 4.4.

PROOF. I divide the proof into several parts.

CLAIM 1. [[lambda].sup.U.sub.t](G) [greater than or equal to] [[lambda].sup.U.sub.t](B) implies that in equilibrium, (i) [[sigma].sub.tSH] < 1 [right arrow] [[sigma].sub.tSL] = 0 and (ii) [[sigma].sub.tWH] < 1 [right arrow] [[sigma].sub.tWL] = 0.

PROOF. Consider (i). [[sigma].sub.tSH] < 1 implies that [V.sub.S.sup.t]([h.sub.t], H, A) [less than or equal to] [V.sup.S.sub.t]([h.sub.t], H, R). But I have (a) [V.sup.S.sub.t]([h.sup.t, H, A) > [V.sup.S.sub.t]([h.sub.t], L, A) by [[lambda].sup.U.sub.t](G) [greater than or equal to] [[lambda].sup.U.sub.t](B), and (b) [V.sup.S.sub.t]([h.sub.t], H, R) = [V.sup.S.sub.t]([h.sup.t], L, R). Altogether, I have [V.sup.S.sub.t]([h.sub.t], L, A) < [V.sup.S.sub.t]([h.sub.t], L, R). Hence the strictly optimal strategy of SL is [[sigma].sub.tSL] = 0. The proof for (ii) is exactly the same and is omitted. QED.

Strategies that are admitted by Claim 1 are as follows:

strong incumbents' strategies weak incumbents' strategies

a. [[sigma].sub.tSH] < 1, i. [[sigma].sub.tWH] < 1,
 [[sigma].sub.tSL] = 0 [[sigma].sub.tWL] = 0

b. [[sigma].sub.tSH] = 1, ii. [[sigma].sub.tWH] = 1,
 [[sigma].sub.tSL] < 1 [[sigma].sub.tWL] < 0

c. [[sigma].sub.tSH] = 1, iii. [[sigma].sub.tWH] = 1,
 [[sigma].sub.tSL] = 1 [[sigma].sub.tWL] = 1

CLAIM 2. Refer to Table Al. The left-hand side gives the only strategies that are permitted by [[lambda].sup.U.sub.t](G) [greater than or equal to] [[lambda].sup.U.sub.t](B). The right-hand side gives the beliefs consistent with each set of strategies on the left-hand side.

Table A1.

Strategies Corresponding beliefs by Bayes rule

a.i. subcase [alpha] where [[lambda].sup.U.sub.t](R) <
 [[sigma].sub.tSH] > [[lambda].sup.U.sub.t](G) =
 [[sigma].sub.tWH] [[lambda].sup.U.sub.t](B)
a.i. subcase [beta] where [[lambda].sup.U.sub.t](R) >
 [[sigma].sub.tSH] < [[lambda].sup.U.sub.t](G) =
 [[sigma].sub.tWH] [[lambda].sup.U.sub.t](B)
a.i. subcase [gamma] where [[lambda].sup.U.sub.t](R) =
 [[sigma].sub.tSH] = [[lambda].sup.U.sub.t](G) =
 [[sigma].sub.tWH] [not equal [[lambda].sup.U.sub.t](B)
 to] 0
a.i. subcase [delta] where [[lambda].sup.U.sub.t](R) =
 [[sigma].sub.tSH] = [lambda.sub.t],
 [[sigma].sub.tWH] = 0 [[lambda].sup.U.sub.t](G) and
 [[lambda].sup.U.sub.t](B) cannot
 be determined by Bayes rule
a.ii [[lambda].sup.U.sub.t](R) >
 [[lambda].sup.U.sub.t](G) [greater
 than or equal to]
 [[lambda].sup.U.sub.t](B)
a.iii [[lambda].sup.U.sub.t](R) >
 [[lambda].sup.U.sub.t](G)>
 [[lambda].sup.U.sub.t](B)
b.i where [[sigma].sub.tWH] = 0 [[lambda].sup.U.sub.t](R) <
 or [[sigma].sub.tSL] = 0 [[lambda].sup.U.sub.t](G) =
 [[lambda].sup.U.sub.t](B)
b.ii where [[sigma].sub.tSL] [less [[lambda].sup.U.sub.t](R) [greater
 than or equal to] than or equla to]
 [[sigma].sub.tWL] [[lambda].sup.U.sub.t](G) =
 [greater than or equal to]
 [[lambda].sup.U.sub.t](B)
b.iii [[lambda].sup.U.sub.t](R) >
 [[lambda].sup.U.sub.t](G) >
 [[lambda].sup.U.sub.t](B)
c.i where [[sigma].sub.tWH] = 0 0 = [[lambda].sup.U.sub.t](R) <
 [[lambda].sup.U.sub.t](G) =
 [[lambda].sup.U.sub.t](B) = 1
c.iii [[lambda].sup.U.sub.t](G) =
 [[lambda].sup.U.sub.t](B) =
 [[lambda].sub.t], but
 [[lambda].sup.U.sub.t](R) cannot
 be determined by Bayes rule

PROOF. The claim is easy to verify. One remark is about the strategies of b.i. With these strategies, with some manipulation, I have updated beliefs

[lambda](G) = 1/1 + [(1 - [lambda])/[lambda]][[sigma].sub.WH]/{1 + [[sigma].sub.SL][(1 - [phi])/[phi]]([[theta].sub.L]/[[theta].sub.H])} and

[lambda](B) = 1/1 + [(1 - [lambda])/[lambda]]/[[sigma].sub.WH]/{1 + [[sigma].sub.SL][(1 - [phi])/[phi]][(1 - [[theta].sub.L])/(1 - [[theta].sub.H])}

so that [lambda](G) < [lambda](B) if [[sigma].sub.WH] > 0 and [[sigma].sub.SL] > 0. QED.

Given these preliminaries, I can prove Proposition 4.4 now. The plan of attack is to examine all eases listed in Claim 2 and to knock off all of them save the case b.ii in which [[sigma].sub.tSL] = [[sigma].sub.tWL] = 0.

a.i. subcase [alpha]: The strategy [[sigma].sub.tSH] < 1 suggests that [V.sup.S.sub.t]([h.sub.t], H, A) [less than or equal to] [V.sup.S.sub.t]([h.sub.t], H, R). But given the updated belief that [[lambda].sup.U.sub.t](R) < [[lambda].sup.U.sub.t](G) = [[lambda].sup.U.sub.t](B),SH can obtain a strictly greater payoff by accepting than rejecting the policy. (Accepting the policy gives her a greater payoff now, as well as greater future payoffs since re-election chances are greater.) This is a contradiction.

a.i. subcase [beta]: The strategies such that [[sigma].sub.tSH] < [[sigma].sub.tWH] are not allowed.

a.i. subcase [gamma]: Given that [[lambda].sup.U.sub.t](R) = [[lambda].sup.U.sub.t](G) = [[lambda].sup.U.sub.t](B), SH should have found it optimal to adopt [[sigma].sub.tSH] = 1. A contradiction.

a.i. subcase [delta]: In this case, acceptance of the policy will not be observed with positive probability. This case is ruled out explicitly.

a.ii and a.iii: That [[sigma].sub.tSH] < [[sigma].sub.tWH] is not allowed according to the third condition in the proposition.

b.i: Given the beliefs that [[lambda].sup.U.sub.t](R) < [[lambda].sup.U.sub.t](G) = [[lambda].sup.U.sub.t](B), WH would have played [[lambda].sub.tWH] = 1 to increase both the current payoff and the re-election chances. However, [[sigma].sub.tWH] 1 is ruled explicitly in this subcase. A contradiction.

b.ii: Suppose [[sigma].sub.tSL] < [[sigma].sub.tWL], which does not equal zero. It follows that [[lambda].sup.U.sub.t](R) > [[lambda].sup.U.sub.t](G) > [[lambda].sup.U.sub.t](B). Foreseeing these updated beliefs, WL would have chosen a strategy of [[sigma].sub.tWL] = 0. (This could have avoided an expected loss of welfare in the current period and could have brought greater re-election chances too.) Hence a contradiction. Suppose [[sigma].sub.tSL] = [[sigma].sub.tWL]. It follows that [[lambda].sup.U.sub.t](R) = [[lambda].sup.U.sub.t](G) = [[lambda].sup.U.sub.t](B). Foreseeing these beliefs, SL would choose [[sigma].sub.tSL] = 0. So would WL choose [[sigma].sub.tWL] = 0 too.

b.iii: Given the updated beliefs that [[lambda].sup.U.sub.t](R) > [[lambda].sup.U.sub.t](G) > [[lambda].sup.U.sub.t](B), the said strategy [[sigma].sub.tWL] = 1 could not be optimal for WL. She should have used [[sigma].sub.tWL] = 0 to avoid making current period loss and to obtain greater re-election chances.

c.i: Given the stated updated beliefs, WH would have played [[sigma].sub.tWH] = 1 to increase both the current payoff and the re-election choice. A contradiction.

c.iii: Since rejection of the policy is not observed with positive probability, this case is ruled out. QED.

Received October 1999; accepted July 2001.

(1.) Although there is a vast literature on the policy making of rational incumbents (see, e.g., Alesina 1987; Bernhardt and Ingberman 1985; Rogoff and Silbert 1988; Rogoff 1990; to name a few), only Salmon (1993) and Wittman (1989, 1995) have touched on this issue so far as I have been able to determine. Harrington (1993) touches upon the issue tangentially. However, his approach is different from most economic writings by assuming uncommon beliefs among agents, and allows the result that politicians are penalized for choosing policies that they consider the best.

(2.) Similar insights can be found in Salmon (1993) and Wittman (1989, 1995). See section 7 for a comparison.

(3.) When the policy is simply a public project, like building a new airport or a new conference center, it is likely that the incumbent knows more about its efficacy than the public does.

(4.) Allowing noncoincidence of the median voter's interest with social welfare introduces another force in the model. Since this force is not the point on which I shall focus, it will not be pursued further. For the role of uncertainty in identifying the median voter, see Glazer (1989), Tabellini and Alesina (1990), and Garfinkel (1994).

(5.) The tie-breaking rule is inconsequential, as the probability of having a tie is zero.

(6.) Note that given [h.sub.t] and [[xi].sub.t] a politician, regardless of type, receives the same [Y.sup.S*.sub.t]([h.sub.t], [[xi].sub.t]) when she is no longer the incumbent and the actual incumbent of period t is strong. The reason is that a politician, regardless of type, receives the same per period utility when not in incumbency. The same argument works for the same [Y.sup.W*.sub.t]([h.sub.t], [[xi].sub.t]) as well.

(7.) To abuse the notation a little bit, I omit the term [h.sub.t], in [[lambda].sup.U.sub.t](G). Likewise, I will omit it in [[lambda].sup.U.sub.t](B) and [[lambda].sup.U.sub.t](R), which I will define later.

(8.) This point also helps clarify the difference in insights between the current paper and that of Coate and Morris (1995). Coate and Morris assume that the bad incumbent, unlike the good incumbent, has a financial tie to a special interest group, which in turn benefits from government transfers and implementation of a policy. This implies that it is more costly for the bad incumbent to adopt decisions that hurt the interest group. A semiseparating equilibrium thus results.

(9.) Presumably, the loser's possibility of running for election makes the cumulative function [M.sub.t]([[micro].sub.t]), at least partly, endogenized, and this presumably alters the expected future payoffs through changes in the value functions [V.sup.S*.sub.t+1]([h.sub.t+1], [[xi].sub.t+1]), [V.sup.W*.sub.t+1]([h.sub.t+1], [[xi].sub.t+1]), [Y.sup.S*.sub.t+1]([h.sub.t+1], [[xi].sub.t+1]), and [Y.sup.W*.sub.t+1]([h.sub.t+1], [[xi].sub.t+1]). However, as these terms enter both [V.sup.j.sub.t]([h.sub.t+1], [[xi].sub.t+1], A) and [V.sup.j.sub.t]([h.sub.t+1], [[xi].sub.t+1], R), j = G, B, and what matters in the incumbent's policy decision is the comparison of the two, cancellation of those terms under the equilibrium belief makes such a modification inconsequential. It is worth pointing out that in this model as well as the efficient equilibrium described in the last section. an incumbent, once she has lost office, would never be able to return. This is because she has been replaced by an opponent with a greater reputation, and the opponent's reputation never changes once elected.

(10.) Alternative definitions also will lead to similar inefficiency results, though the exact conditions under which inefficiency occurs will differ.

(11.) In the calculation of social welfare, because the incumbent (and the set of politicians) is of measure zero, I do not take the incumbent's welfare into account. This is consistent with the general practice that a planner's welfare is not viewed as part of social welfare. However, this may not be appropriate if there are a finite number of members, among whom is an incumbent, in the economy, as in the citizen-candidates model formulated by Besley and Coate (1997), who explicitly make the latter comment. See also Osborne and Slivinski (1996), who pioneered the citizen-candidates approach.

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Y. Stephen Chiu (*)

(*.) Department of Decision Sciences and Managerial Economics, The Chinese University of Hong Kong, Shatin, Hong Kong. E-mail ywchiu@baf.msmail.cuhk.edu.hk.

I thank Jonathan Hamilton (the editor), two anonymous referees, and seminar participants at The Chinese University of Hong Kong for helpful Comments. Helpful research assistance from Timothy Ng is acknowledged. Any remaining errors are mine.