Public Sector Economics

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1 Public Sector Economics Munich, 4 April 08 Preference Divergence between the Electorate and their Elected Representatives Didier Laussel and Ngo Van Long

2 Preference Divergence Between the Electorate and Their Elected Representatives Didier Laussel y Aix-Marseille University Ngo Van Long z McGill University 3 February 08 Correspondence: y Aix-Marseille University, CNRS, EHESS, Centrale Marseille, AMSE. didier.laussel@outlook.fr z Department of Economics, McGill University. ngo.long@mcgill.ca

3 Preference Divergence Between the Electorate and Their Elected Representatives Abstract: The citizen candidates models of democracy assume that politicians have their own preferences that are not fully revealed at the time of elections. We study the optimal delegation problem which arises between the median voter (the writer of the constitution) and the (future) incumbent politician when not only the state of the world and but also the politician s type (preferred policy) are the policy-maker s private information. We show that it is optimal to tie the hands of the politician by imposing both a policy oor and a policy cap and delegating him/her the policy choice only in between the cap and the oor. The delegation interval is shown to be the smaller the greater is the uncertainty about the politician s type.these results are also applicable to settings outside the speci c problem that our model addresses. JEL-Classi cation: D8;H0 Keywords:Representative democracy; optimal delegation; political uncertainty; policy caps; policy oors; citizen candidates.

4 Introduction Representative democracy may be best de ned as a political system in which the power to choose public policies is delegated to elected representatives (which we will call here politicians for short). It is traditionally opposed to direct democracy, a system in which the general public determines itself the laws and policies. The evaluation of the costs and bene ts of delegation is then central in the old debate between the two systems. The main alleged advantage of representative democracy is that representatives are able to specialize in policy-making. This allows them to devote more time to study the state of the world than ordinary citizens and thus to make better informed decisions. Benjamin Constant (89, p.6) put it very neatly when he wrote: The representative system is nothing but an organization by means of which a nation charges a few individuals to do what it cannot or does not wish to do herself. Poor men look after their own business; rich men hire stewards. This is the history of ancient and modern nations. The representative system is a proxy given to a certain number of men by the mass of the people who wish their interests to be defended and who nevertheless do not have the time to defend them themselves. Edmund Burke (854) said the same when writing that it is the duty of the representative to sacri ce his repose, his pleasures, his satisfactions, to those of the constituents. The debate on the relative merits of delegating decisions to politicians versus direct democracy has recently been revived after the Brexit referendum, and questions are raised concerning (i) whether it is a good thing to accept as binding the verdict of a relatively narrow majority of voters (most of whom were not well informed about the question on which they were voting), and more generally, (ii) to what extent should the The idea that elections may be a screening process allowing to delegate the policy choices to policymakers who are deemed more competent than the general population is more controversial. Le système representatif n est autre chose qu une organisation à l aide de laquelle une nation se décharge sur quelques individus de ce qu elle ne peut ou ne veut pas faire elle-même. Les individus pauvres font euxmêmes leurs a aires: les hommes riches prennent des intendants. C est l histoire des nations anciennes et des nations modernes. Le système représentatif est une procuration donnée à un certain nombre d hommes par la masse du peuple, qui veut que ses intérêts soient defendus, et qui néanmoins n a pas le temps de les défendre toujours lui-même. The above English translation appeared in Fontana (988). 3

5 citizens of a democracy be able to make decisions directly? (See, for example, Peter Singer (06), among others). Given the number and the complexity of public decisions to be taken in a modern society, the advantages of representative democracy are so important that in one way or another all modern democracies are basically representative democracies, with sometimes some elements of direct or participative democracy like referendums and/or popular initiatives. It has however long been recognized that this system has costs of its own. The main drawback of representative democracy is that representatives are free, once elected, to promote their own interests. This was expressed forcefully by Jean-Jacques Rousseau (76) who thought that even if it is possible that there exists a coincidence between what the constituency and their representatives want at the time the former delegates their power to the latter, there is nothing to ensure that this coincidence will continue. 3 When legislators are citizen candidates, their actions in o ce follow their own preferences. The sources of divergence between the preferred policies of the median voter and those of the elected representatives are indeed numerous, even when they have the same information about the state of the world. In the rst place, representatives cannot be bound by a binding mandate; indeed they cannot commit to implement speci c policies once elected because it is impossible to write complete contracts describing what the representatives should do in each of the multitude of circumstances that could occur during their term of o ce. In the second place, even if the representatives could commit, as assumed for instance by Downs (957) and Stokes (963), this does not eliminate ex ante political uncertainty, i.e., the uncertainty (prior to and up to the time of the election) about the elected politician s type. On one hand, it is indeed well known that even if the election is a single issue one (one-dimensional policy space) there may be multiple voting equilibria as soon as there are more than two candidates. On the other hand, when the policy space is multi-dimensional, the voters have to choose between 3 Le souverain peut bien dire : Je veux actuellement ce que veut un tel homme, ou du moins ce qu il dit vouloir ; mais il ne peut pas dire : Ce que cet homme voudra demain, je le voudrai encore. Du Contrat Social, chapter VII,. 4

6 packages: there is nothing to ensure that on each issue the elected representative s proposed policies are the ones preferred by a majority. Two recent examples are relevant. In France, although President Macron was recently elected by a large majority, the elimination of the wealth tax (Impôt de Solidarité sur la Fortune), although present in his election platform, is, according to all polls, opposed by a majority of French voters. Similarly, US President Trump s project of repealing Obamacare is opposed by a majority of US voters. More generally, McKelvey (976, p. 47) has shown that in the case of multi-dimensional policy spaces it is theoretically possible to design voting procedures which, starting from any given point, will end up at any other point in the space of alternatives, even at Pareto dominated ones. The purpose of the present paper is to study the optimal design of a representative democracy system as a speci c delegation problem which belongs to a class of delegation problems dealing with settings in which, according to Amador and Bagwell (03) a principal faces an informed but biased agent, and contingent transfers between the principal and the agent are infeasible. 4 Indeed, political settings are highly relevant examples of this kind of problems where legal rules limit or even completely forbid transfers to elected representatives. Compared to the existing theoretical models of delegation, the type of delegation problem we analyze in our paper has a novel feature. While our model and the existing models share a common feature, namely, the agent (the incumbent politician) has private information about the state of the world, and her preference is generally biased with respect to that of the principal s 5, we depart from the existing models by assuming that the principal does not know the direction and magnitude of the agent s bias. The representative s type as determined by the electoral process is a random variable. 6 Throughout this paper we restrict attention 4 The bias is also referred to as the preference divergence between the agent and the principal; see Alonso and Matouschek (008). An example of preference divergence is Angela Merkel s unwillingness to put a ceiling on the number of refugees to be admitted into Germany. 5 The principal is here the writer of the constitution. Throughout the paper we call him/her the median voter though this coincidence may be true only initially. 6 We do not model the electoral process here; we simply treat it a black box, in order to focus on the delegation problem. 5

7 to the case where there is more uncertainty about the state of the world than about the politician s preferences. 7 For the sake of simplicity, we suppose that the voting procedure is unbiased, in the sense that the expected representative s type is the median voter s type. Before analyzing in full our more general model, we consider two benchmark cases. In the rst benchmark, we assume that the median voter, not knowing the direction and magnitude of bias in the politician s preference, must choose one of the following alternatives: (i) giving the elected politician the full freedom to choose policy, versus (ii) requiring that policy must be decided by voters (who are uninformed about the state of the world). We show that in this case, as long as the median voter s uncertainty about the politician s bias is less than the uncertainty about the state of the world, the median voter s expected utility is higher under (i) than under (ii). In our second benchmark scenario, we assume that the politician s bias is known. In this case we obtain results which are in line with Amador and Bagwell (03): the principal sets a policy cap when the politician is known to have a rightist bias, and a policy oor is set when the politician is known to have a leftist bias. We obtain novel results in the more general case where interval delegation is possible, in a scenario where the representative s bias is random while the electoral process is unbiased. We show that the policy choice should then be delegated to the elected representative only within bounds: an upper bound (policy cap) and a lower bound (policy oor). Cap and oor are imposed to avoid extreme policy choices under delegation. Our main result is that the writer of the constitution always nds it optimal to tie the hands of future politicians in this way. Furthermore, we show that the interval of parameter values over which delegation occurs shrinks when the political uncertainty increases. There are several strands of literature on policy choice that are related to our model. Starting from Osborne and Slivinski (996) and Besley and Coate (997), the economic literature has developed an alternative theory of policy choice in representative democra- 7 As will be shown below, this assumption implies that the median voter strictly prefers unfettered representative democracy (i.e. full delegation to the politician) to direct democracy. 6

8 cies, 8 i.e., models in which citizen candidates run for election 9. These models depart from the traditional Hotelling model of electoral competition (Downs (957)) in two important respects: (i) the citizens-candidates have policy preferences instead of being exclusively o ce-motivated, 0 and (ii) they cannot commit to implement any policy other than their most preferred one. Under free entry, the number and types of the candidates are determined at equilibrium. Whereas the initial models assumed complete information, they have been subsequently extended to a setting where each candidate s ideal point is his/her private information. While this literature focuses on the electoral process - which in our model is simply treated as a black box with random outcomes- its conclusions con rm the existence of political uncertainty: there are generally multiple equilibria and the type of the elected representative cannot be predicted with certainty. Indeed, this literature shows that the elected politician s type almost always di ers from the median voter s type. 3 Our model is closer to the political agency models, as surveyed in Besley (006), which also deal with the relationship between citizens/voters (the principals) and the politicians/government (the agent). The agents are generally better informed than the former. This literature encompasses moral hazard models where politicians may use their o ce to extract rents (Ferejohn, 986; Hillman, 08; Persson and Tabellini, 000), and adverse selection models where the issue is to select the good politicians (Besley and Prat, 004). There are also models which combine moral hazard and adverse selection (Besley and Case,995; Coate and Morris,995; 8 Besley and Coate (997, p. 85). 9 See for instance Hamlin and Hjortlund (000) for the proportional representation case, and Mattozzi and Snowberg (08) for an application of the citizen candidates model to taxation and redistribution. 0 Angela Merkel s preference regarding the number of refugees to be admitted is arguably di erent from that of the median voter. She might have been motivated by ethical considerations, or possibly by the prospect of a Nobel Peace Prize. See Peace-Prize-opening-country-s-doors-desperate-refugees.html In the models of Osborne and Slivinski (996) and Besley and Coate (997), candidates are completely informed about the ideal policies of all others. See Großer and Palfrey (009, 04) for theoretical analyses. For an experimental study, see Großer and Palfrey (07). 3 Großer and Palfrey (07, p. ) report that, both theoretically and experimentally, candidate entry is from the extremes of the policy space. 7

9 Fearon,999; among others). The basic element of these models is the idea that politicians may be made accountable 4 to the citizens via electoral competition. 5 While a politician s concern for reelection does act as a discipline device, these models nd that it does not ensure full accountability. Indeed, Fearon (999) nds that voters are better o using elections to select types, rather than disciplining incumbents, and Besley (006, p. ) concludes that the reelection mechanism is imperfect since it is unable to always eliminate dissonant politicians. 6 This is clearly in line with our assumptions about the existence and persistence of political uncertainty. Finally, our paper has much in common with the papers on optimal delegation problem (e.g., Holmström,977 and 984) that focus on interval delegation, i.e., the set over which the policy choice is delegated to the agent is a single interval. 7 Other papers have provided conditions for the optimality of interval delegation under various settings: Amador, Werning and Angeletos (006) when money burning is allowed; Alonso and Matouschek (008) when money burning is not allowed; Ambrus and Egorov (009) for uniform distributions and quadratic utility functions. A recent paper by Amador and Bagwell (03) provides necessary and su cient conditions which encompass previous results as special cases. Our paper departs from all this literature by analyzing the case when the agent s bias is a random variable whose realized value is private information of the agent. 8 We show that when the uncertainty about the state of the world is greater than the political uncertainty and when 4 On political accountability, see Przeworski et al. (999), For a review of recent theoretical and empirical research on electoral accountability, see Ashworth (0). Of note is the array of various alternative approaches to explaining why an incumbent might have incentives to act contrary to voters interests. These include signaling models and multi-task models. In contrast to the two-period models surveyed by Ashworth, there is also a strand of literature that assumes repeated electoral competition: see, for example, Duggan and Feys (006) and Banks and Duggan (008). 5 The voters hold the incumbent politician responsible for the consequences of his/her actions while in o ce, and the politician takes this into account. 6 In the same way, Coate and Morris (995) nd that political competition does not prevent ine cient methods of redistribution to be employed. 7 Interval delegation models are di erent from strategic delegation models, which explore strategic motives for voters to elect politicians di erent from themselves. For strategic delegation motives in election, see Harstad (00) and Christiansen (03). Besley and Coate (997) study the desirability of limiting the set of politician types to whom voters can strategically delegate. 8 Koessler and Martimort (0) have instead studied the case when the decision space is two-dimensional. They assume quadratic utility functions and uniform distribution of the agent s type. 8

10 the political process is not biased, interval delegation holds with both a policy cap and a policy oor. A further result is that the delegation interval gets smaller when the political uncertainty increases. These results clearly apply to many other contexts, and not just to the speci c problem studied in our model. The rest of this paper is organized as follows. Section outlines the model. Section 3 analyzes two benchmark cases. Section 4 characterizes the equilibria. Section 5 concludes. The Model A policy x has to be chosen from a set X = [x L ; x H ] which is a closed interval of the real line. There is a continuum of voters, uniformly distributed along the real interval [ ; ]; we refer to t [ ; ] as the type of voter. The population mass is normalized to unity. The utility of the voter of type t is (B + + t)x where B is a technology parameter, and is the state of the world representing random deviation from B. We assume that the realization is unknown to the voter at the time when a decision on x must be made. To x ideas, one can think of x as the amount of a public good to be provided (e.g., defence expenditure), B + is the productivity (or e ectiveness) of the public good, t is the voter s preference parameter, and (=)x is the per capita cost of nancing the public good (including deadweight losses due to nancing it by means of distortionary taxes 9 ). The voter with t = 0 is called the median voter. Denote by () the probability density function for. Assume that () is strictly positive only for belonging to the real interval [ "; "]. Since voters do not have information about the realized value of, under direct democracy, the median voter would choose x to maximize his expected utility, Bx x, that is, he would vote for x = B. Following the insight of Benjamin Constant, we assume that x politicians, thanks to their specialization, are informed about. 9 In this paper, we do not model these costs. 9

11 We consider a problem where the median voter is the principal and the politician as the agent. At the time when the decision on x must be taken, is not known to the voters, but it is known to the incumbent politician. A politician of type t [ ; ] has the utility function (B + + t)x x Following the political economy approach (see, e.g. Hillman s book, 08), we suppose that politicians are motivated only by self-interest. If the incumbent politician were unconstrained in her decision making, then, given that she has private information about the realized, she would choose x to maximize her own utility, knowing her type is t. This means that she would choose x = B + + t () while the interest of the median voter is best served if x = B +. If t 6= 0, we say that the politician is biased (relative to the median voter s preference). 0 A positive t means the politician s bias is toward the right (e.g., she would like to spend more on defence than the median voter would want). Similarly, a negative t means that the politician s bias is toward the left.. The median voter s problem The politician knows her type t, but the voters do not: they only know that t is distributed with a probability density function (t), where t [ ; ]. At the time when the decision on x must be made, + t is the politician s private information. De ne = + t Then the density function for, denoted by f(), is the convolution ( )(), i.e., f() = ( )() () ( )d () 0 For example, when a politician is of type t = (the most extreme right-wing type), upon observing that = ", she will choose the defence spending level x = B + " + if she is unconstrained in her choice. But any spending level greater than B + " would never be in the best interest of the median voter. 0

12 In what follows, we assume that and are uniform density functions, i.e., 8 < 0 8 < " () = 8 [ "; "] : " 0 8 > " 8 < (t) = : 0 8t < 8t [ ; ] 0 8t > Given the uniform density functions and, the convolution () can be computed as follows. For any given [ " ; " + ], let us denote by D() the set of values that are consistent with t [ ; ], i.e., D() f [ "; "] j g : (3) Then f() = D() d (4) 4" Using (4), we nd that has the following trapezoid-shaped density function 8 "++ < ; 8 [ " ; "] 4" f() = ; 8 [ "; " ] ; (5) : " "+ 8 [" ; " + ] 4" and consequently, the corresponding cumulative distribution function is 8 >< F () = >: ("++) ; 8 [ " ; "] 8" +" ; 8 [ "; " ] : (6) " +(+") " +6" ; 8 [" ; " + ] 8" Given any prescribed schedule x(:) that associates to each + t an action x(), the expected utility of the median voter is W M = " " (B + )x ( + t) (x ( + t)) " d dt (7) This expected utility may be conveniently rewritten as W M = E [(B + )x] E x (8)

13 where and E x "+ x() f()d x() f()d (9) " E [(B + )x] where D() is given by eq. (3). = "+ " " (B + )x ( + t) " " d dt (0) (B + ) d d 4" x() D() Since is the politician s private information, the median voter s problem of expected utility maximization is a principal-agent problem. The median voter must design a schedule x(:) that is incentive-compatible to the agent. Applying the revelation principle, the median voter s optimization problem can be formulated as follows. Problem : Choose a function x(:) that maximizes (8), subject to the incentivecompatibility constraint: an agent that has private information will choose action x() in preference to any other action x(b), i.e., = arg max (B + )x(b) b (x(b)) (). Properties of incentive-compatible schedules in the absence of transfers Before fully characterizing the solution of Problem, we must state a number of properties that any incentive-compatible scheme x(:) must satisfy, given that transfers are not feasible. The median voter, as principal, o ers the elected politician a schedule x(). To x ideas, suppose the median voter writes a constitution which in e ect tells the elected politician the following message: Here is the schedule (the function) x(:) de ned over the set of possible values of [ " ; " + ]. You must report a value b belonging to this set. If your report Unlike the standard (text-book) principal agent model with transfers between the principal and the agent (e.g., La ont and Martimore, 00), here the principal does not receive any transfer from the agent. Her only payo is her personal expected utility derived from x. This is in line with the observation made by Amador and Bagwell (03), that contingent transfers between the principal and the agent are often infeasible.

14 is b, you will be required to take action x(b). Clearly, a schedule x(:) is able to induce the agent to report truthfully if and only if under that schedule, the agent cannot obtain a better payo by reporting a false value b 6= : Let A (b; ) denote the agent s payo, where is the true value and b is the reported value: A (b; ) = (B + )x(b) x(b) () A schedule x(:) induces truth telling if and only if for each ; we have W A () A (; ) A (b; ) for all b [ " ; " + ] : That is, W A () = (B + )x() x() (B + )x(b) x(b) (3) By a standard revealed preference argument, any incentive-compatible schedule x() is nondecreasing and almost everywhere di erentiable for all [ " ; " + ]. In addition, by Berge s maximum theorem, we know that W A () is a continuous function. Over any interval ( ; ) where x() is di erentiable, since A (b; ) is maximized at b =, the following rst order condition must hold, where x(b) is evaluated at b =, [(B + ) x ()] dx d = 0 (4) that is, either x() B = or dx=d = 0 on ( ; ). We report this result as Fact : Fact : Schedules that are incentive-compatible to the politician must be non-decreasing and almost everywhere di erentiable. Over any interval of di erentiability [ 0 ; ], the graph of x() B against is either horizontal (i.e., x() B = constant), or coincides with a segment of the 45 degree ray through the origin, i.e., x() B = for all [ 0 ; ]. Applying the envelope theorem to (), we nd that dw A d (b; It follows from (5) that the following two equations hold: W A () = W A () + = x(b) where b = (5) dw A ( 0 ) d 0 d 0 = W A () + x( 0 )d 0 (6) A politician that observes a higher will want to choose a higher x. Therefore any schedule that is decreasing in over some range of will not be able to induce the politician to report the true. 3

15 W A () = W A () dw A ( 0 ) d 0 = W A () d 0 x( 0 )d 0 (7) where ", and " +. It is important to note that we cannot treat W A () and W A () as constant. We will determine these values endogenously, as part of the optimization problem of the median voter. 3 In general, any incentive-compatible schedule x(:) B, while being non-decreasing and a.e. di erentiable, may exhibit an upward jump discontinuity. At any point e ( " ; "+) where there is an upward jump, we will use the following notation: lim x() x (e) < x + (e) lim x() (8) "e #e Since W A () is continuous, the following relationship holds W A (e) = (B + e)x (e) x (e) = (B + e)x + (e) x+ (e) It follows that h x + (e) x (e) i = (B + e) x + (e) From this equation, we deduce the following Fact: x (e) Fact : At any point ea of upward jump discontinuity of an incentive-compatible schedule x() B, it holds that e = (x+ (e) B) + (x (e) B) : (9) As a consequence of Fact and Fact, we deduce Fact 3: Fact 3: Under incentive compatibility, if a schedule x() B has an upward jump of discontinuity at some e (from the value x (e) B to the value x + (e) B) it must hold that x() B = x (e) B for all ( 0 ; e), and x() B = x + (e) B for all (e; 00 ), where 0 x (e) B and 00 x + (e) B. In other words, a jump must be from one horizontal segment to another. It follows from Fact, Fact, and Fact 3 that there are potentially four di erent types of incentive-compatible schedules: 3 See the proof of Proposition. 4

16 (i) First, the fully exible schedule, x() B = for all [ "; + "]. That is, the politician is unconstrained in her choice of x. (Being unconstrained, the politician will choose x B = because this maximizes her utility.) (ii) Second, semi- exible schedules : x() B = for all in some interval [ 0 ; ], with a policy cap (i.e., x B is not permitted to exceed < " + ), or a policy oor (i.e., x B is not permitted to be below the level 0 > " ), or both a cap and a oor. (iii) Third, step schedules : x() is a step function. There are possibly n intervals, [ 0 ; ), [ ; ),..., [ n ; n ] and over each interval i, x is a constant, x i, where x < x < ::: < x n = x n : (iv) Fourth, hybrid schedules : a hybrid schedule may include several exible segments, and several step-wise discontinuities. Due to the incentive-compatibility constraint () and the presence of the double integral (0), the solution of Problem is not straightforward. We must proceed using a number of steps before reaching our main results. 4 Before solving Problem, in order to aid our intuition, it will be instructive to consider two simpler scenarios, which also serve as useful benchmarks for comparison with our main results. 3 Two benchmark scenarios In this section, we consider two benchmark scenarios. 3. The rst benchmark: choice between unfettered representative democracy and direct democracy In the rst benchmark case, we assume that the median voter, while facing uncertainty about both and t, is restricted to making a choice between two extreme alternatives: (i) Giving 4 Our main results are that (i) it is optimal to impose restrictions on the politician s range of actions, in the form of a policy cap x and a policy oor, x, and (ii) the greater is the uncertainty about the politician s type, as measured by the length of the support of the (uniform) distribution of t, the smaller is the length of the interval [x; x]. 5

17 the politician complete freedom to choose any action x she wants (this extreme scenario is called the unfettered representative democracy case); and (ii) Direct democracy : the voters decide on the policy level x by voting, even though they are completely uninformed about the realized state of the world. Under alternative (i), the elected politician, having observed + t, is free to choose any x. She will then choose the policy level that maximizes her own self-interest, i.e., x( + t) = B + + t. In this case, the median voter s resulting expected utility can be computed as follows: " (B + )(B + + t) " (B + + t) " d d dt = B + 6 " Under alternative (ii), the decision on x is made under direct democracy (the voters directly decide on x). Since voters do not observe, and since E[] = 0, the median voter will choose x to maximize Bx x, resulting in the constant policy, i.e. x = B always. His expected utility is B. Thus we have obtained the following result for our rst benchmark scenario: Result (choice between unfettered representative democracy and direct democracy) Unfettered representative democracy would give rise to a higher welfare level to the median voter, as compared with direct democracy, if and only if < " (i.e., i the uncertainty about the politician s type is smaller than the uncertainty about the state of the world, ). In choosing between unfettered representative democracy (with an informed, but possibly biased politician) and uninformed direct democracy, the median voter faces the trade-o between, on one hand, the bene ts of relying on a politician who has access to information about, and on the other hand, the costs of (i) letting the politician set an x that may exceed the upper bound on what the median voter would ever want, B + " (as the politician may have a rightist bias, and may choose x > B + "), and (ii) letting the politician set an x that falls below what the median voter would ever want (as the politician may have a leftist bias). If < ", these costs are not too large relative to the bene ts, and therefore unfettered 6

18 representative democracy is superior to direct democracy. Remark : In considering the rst benchmark scenario, we have assumed that the expected value of t is zero, i.e., on average, the politician s bias is zero. It is easy to consider the modi ed assumption that t is uniformly distributed over some interval [ + ; + ], where 6= 0. Then the expected value of the politician bias is. In this case even though we maintain the assumption that < ", it can be shown that direct democracy is better than unfettered representative democracy if the politician s bias is expected to be su ciently large, in the sense that 3 > (" ). 3. The second benchmark scenario: the politician s bias is known We now turn to the second benchmark scenario. Here, we assume that the bias t is known, but it is not too large. Speci cally, we suppose that " < t < ". This assumption rules out (i) the existence of a politician with a right-wing bias t > "; such a politician would choose an action x > B + " if > 0, while the median voter would never want x > B + ", whatever the realization of [ "; "], and (ii) the existence of a politician with a left-wing bias t < ". Under this benchmark scenario, the optimal delegation in our model is simply a special case of a more general model of delegation under a known bias, which has been considered in Amador and Bagwell (03). Since the distribution of is uniform over [ "; "], it follows that, given that t is a known constant, the distribution of is uniform over [ " + t; " + t]. With only one source of randomness, the solution of Problem can be characterized relatively easily. We can prove the following results, for the case of a rightist bias, i.e., t > 0. (A symmetric argument applies for the case of a leftist bias.) Result : Given a known rightist bias t > 0, the optimal incentive-compatible schedule x() B has the properties that: (i) for all < " t, the politician is given the freedom to choose her self-interest-maximizing choice, namely, x() B =, and (ii) for all " t (i.e., for all " t), x() B must be equal to the cap value " t. 7

19 Similarly, given a known leftist bias t < 0, the optimal incentive-compatible schedule x() B has the properties that (i) for all > " t, the politician is given the freedom to choose her self-interest-maximizing choice, namely, x() B =, and (ii) for all " t (i.e., < " t), x() B must equal the oor value " t > ". In the case of a known rightist bias, imposing a policy cap serves to rule out extreme-right policies. The greater is the bias, the lower is the optimal cap. Similarly, in the case of a known leftist bias, imposing a policy oor rules out extreme left policies. The greater is the bias, the higher is the oor. Remark : Result above may be generalized to the case where the distribution of is not necessarily uniform. Assume simply a strictly positive continuous density function f() over the set [ "; "]. It may be shown, using the approach of Amador and Bagwell (03), that if F () + tf() is non-decreasing, 5 then: (a) In the case of a known rightist bias t > 0, the optimal schedule x() B consists of (i) a exible policy x() B = for all < y (i.e., all y t), and (ii) for all y (i.e., all y t), x() B must equal the cap y, where y is the solution of the following equation 6 " y ( y)f()d = 0: t (b) In the case of a known leftist bias t < 0, the optimal schedule x() B consists of (i) a exible policy x() B = for all z t, (ii) for all z t, x() B must equal the oor value z, where z is the solution of the following equation z " t ( z)f()d = 0: 5 This is the condition (c), of Amador and Bagwell (03, p. 550). An alternative and simpler proof is possible, along the lines of Laussel and Resende (08). 6 For the uniform distribution, we can see that y = " t. 8

20 4 Optimal cap and oor: The case where the politician s bias is private information We now solve for the optimal schedule x() B when voters do not know the politician s bias, t. We assume that the variance of the distribution of t is smaller than the variance of the distribution of the state of the world,, i.e., we assume < ". Proposition : It is optimal for the median voter to set both a policy cap and a policy oor, and to delegate the policy choice to the politician only for intermediate values of : The cap is x = B + ", that is, x() B = " for all " ; " +. The oor is x() = B " + for all " ; " +. And, for all " + ; ", the politician is free to choose her x, and her choice is x() = B +. The length of this delegation interval is ". Thus, the greater is the uncertainty about political bias, the smaller is the delegation interval. Before proving Proposition, it is useful to state the following Lemma: Lemma : For any incentive-compatible schedule x() B, the median voter s expected utility is given by W M = W A () + or, equivalently, by W M = W A () + "+ " "+ " x() [() (B + ) + h()] f()d (0) x() [() (B + ) ()] f()d () where = ", = " +, and where (i) () is the expected value of + B conditional on the sum + t being equal to, where t [ ; ] : It is given by () f() D() ( + B) d 4" (ii) h() [ F ()] =f(), and () F ()= f() where f() and F () are given by equations (5) and (6). Note that h() is the inverse of the hazard rate. 9

21 Proofs of Lemma and Proposition : See the Appendix. According to Proposition, for a given level " of the state-of-the-world uncertainty, the length of the delegation interval (over which the policy choice is fully delegated to a politician) is " + ; ". This length is decreasing in the level of political uncertainty,. Full delegation, i.e. unfettered representative democracy, is optimal i there is no political uncertainty at all ( = 0 ). The optimal policy is pictured in Figure, where we set B = 0; " = and = : The cap x() = :5 applies for [:5; 3] and the oor x() = :5 applies for [ 3; :5]: The exible policy x() = is preferred when [ :5; :5] : The dashed line indicates what would be a exible policy outside the range in which it is optimal. 5 Conclusion In this paper we have analyzed a novel optimal delegation game between the median voter (the writer of the constitution) and the elected representative (the incumbent politician) in which the latter is privately informed of the state of the world and the former is uninformed about both the state of the world and the politician s type. The politician s type is seen as the exogenous random outcome of an electoral process which is not analyzed here. Our game may be considered to take place before the politician is elected, for instance when the constitution or, more generally the fundamental laws, are written. The main result is that, in that setting, it is optimal for the writer of the constitution, presumably the median voter, to impose an upper and a lower bound to the policies which the incumbent politician is allowed to implement, i.e. to tie the hands of future politicians. This interval delegation result was already obtained in delegation models with known agent s type but is new in a model where the politician s type is private information. Moreover, the form taken here by the interval delegation, with an upper and a lower limit, is itself new compared with the results obtained in applied delegation models with known agent s type. The most prominent example is optimal trade agreements between two governments 0

22 that face private political pressures as studied by Bagwell and Staiger (005) or Amador and Bagwell ((03), Section 4) where the main result is a simple tari cap. 7 Another example is the existence of a simple oor (minimum) equilibrium output in public intrinsic common agency games (Martimort, Semenov and Stole (06)). More generally, Amador and Bagwell (03, Proposition 3) specialize their results on interval delegation to the case where the agent s bias is not a ected by the state of the world, and they give su cient conditions for the interval allocation to take the form of a simple cap. In contrast, from the present model, where not only the state of the world but also the agent s type are private information, it seems that the interval allocation is taking the form of a delegation interval between a oor and a cap. We are rather con dent that these results would extend to the case of more general distribution functions. More speci cally, we conjecture that three natural assumptions may be su cient to entail the optimality of an interval delegation with a cap and a oor: (i) a nonincreasing hazard rate; h 0 () 0; (ii) a derivative 0 () and (iii) a symmetric density function f() that is non-decreasing between " and 0 and non-increasing between 0 and " + : Obviously, these conditions are satis ed when the distributions of and t are uniform as assumed here. We hope to establish a proof of this more general result in a future technical note. Are our results likely to be substantially changed when the incumbent must periodically run for reelection? Clearly not if politicians are primarily policy-motivated: a pure screening e ect of reelection reduces but does remove ex ante political uncertainty. In the opposite polar case, if the incumbent rst and foremost wants to be reelected, the writer of the constitution has to explicitly account for this constraint in the incumbent s program. 8 This will be the subject of further research. Finally, it would certainly be interesting to consider the case where the political process 7 See also Beshkar, Bond and Ro (0), and Amador and Bagwell (0). 8 More precisely the incumbent will never report a value b such that, conditional on this observation, the median voter would prefer to pick randomly a next politician.

23 is not only random but also biased, i.e. when the writer of the constitution rationally expects the politician s type to di er from his/her own one for a variety of reasons. 9 Very likely, the symmetry between the oor and the cap would be lost, the writer of the constitution willing to tie more the hands of future politicians in the direction in which they are more likely to be biased. 9 A possible reason is an external reward (e.g., the prospect of a Nobel Peace Prize). Other reasons include the in uence of pressure groups, and the possibly greater in uence of a uent people on the politicians decisions. See for instance Gilens (04), and Gilens and Page (08).

24 Appendix Proof of Result The proof is given for the case of a rightist bias (t > 0). A similar argument applies in the case of a leftist bias. The proof proceeds in three steps. Step : With a known bias t, it is never optimal for the median voter to set a discontinuous schedule x() B involving a step function over an interval [ 0 ; ] [ " + t; " t]. Proof: In view of Facts, and 3, without loss of generality, consider a schedule x() B that has a discontinuity at point e = ( 0 + )=, with x (e) B = 0 and x + (e) B =. Then x() B = 0 for all [ 0 ; e) and x() B = for all [e; ]. We now show that we can construct an alternative schedule x # () B that dominates x() B. Let x # () B = x() B for all outside [ 0 ; ], and x # () B = for all [ 0 ; ]. Then the median voter s expected utility under schedule x # () B exceeds his expected utility under x() B by the amount given below: 30 " Write 0 ( t) d " e 0 ( t) 0 0 d ( t) " e d x = 0 ; u = e = x + y ; y = > x The terms that have t as a multiplicative factor cancel each other out. The remaining terms are " u x ( + x x)d + " y u ( + y y)d = 48" (y x)3 > 0 Step : If t > 0, any schedule x() B with a oor, i.e., x() B = 0 for all [ " + t; 0 ] is not optimal. Proof: Consider an alternative schedule x # () which is identical to x() outside the interval [ " + t; 0 ], and x # () B = for all in the interval [ " + t; 0 ] : 30 The expression below does not contain B because it can be veri ed that the terms involving B in the three integrals cancel each other out. 3

25 The excess of the median voter s expected utility under x # (:) over his expected utility under x(:) is = " 0 "+t ( t) d 0 ( t) 0 " "+t 0 d Let x = " + t and y = 0 > " + t. Then y ( t) d = y Thus " = y t x x ( t)y y y d = y x y x 3 6 y t x y y x y yt (y x) x x 3 6 yt (y x) y (y x) y (y x) = 6 (y x) (3t x + y) = 6 ( 0 + " t) (t + " + 0 ) > 0 if t > 0: Step 3: From Step and Step, the only remaining possibilities are (i) a fully exible schedule, and (ii) a semi- exible schedule with a cap that applies for all [ ; " + t], for some " + t. Note that a fully exible policy is equivalent to a semi- exible policy, with a cap that applies only at = " + t. Therefore, to prove Result, it su ces to show that the optimal cap must be operative for all " t, and at = " t, x(" t) = " t. To prove this, we compute the median voter s expected utility under a semi- exible policy and show that it attains its maximum at = " t. The median voter s expected utility is computed as follows. Let x = B " + t and v = B + " + t. Let y =. Then the median voter seeks to maximize the sum of following integrals, by choosing y y v (B + t)(b + ) (B + ) d + (B + t)(b + y) x y (B + y) d Di erentiating the above integrals with respect to y, using Leibnitz s rule, we obtain the rst order condition for a maximum: (v y) (t v + y) = 0 4

26 The solution y = v t satis es both the FOC and SOC. 3 Proof of Lemma From (3) and (6), and de ning = " and = " + x() = (B + )x() + W A () = (B + )x() + W A () + (B + )x() + W A () + K() x( 0 )d 0 with K 0 () x(). Then x() f()d = W A () (B + )x()f()d + K()f()d Integrating K()f() by parts gives Then K()f()d = [K()F () K()F ()] = x()d x()h()f()d x() f()d = W A () + x()f ()d = x()f ()d x() F () f()d f() x() [h() (B + )] f()d (A.) Using (8), (9), (0) and (A.), the expected utility of the median voter can be written as W M = x(a) [() + h() (B + )] f()d + W A () where () f() D() (B + ) d: 4" This establishes (0). A similar argument establishes (). As a veri cation, note that if we subtract equation () from (0), we get 0 = W () W () + "+ " x()d = W () W () + "+ " dw d d 3 The other solution to the FOC does not satisfy the SOC, and corresponds to a local minimum. 5

27 which is correct. Proof of Proposition The proof consists of three steps. The rst is to show that a exible policy is always better than any discontinuous incentive-compatible policy when " ; " + ;. The second one is to prove that (i) it is better to have some policy cap (resp. policy oor) and that (ii) the optimal policy cap (resp. policy oor) is such that x() B = " " for all ; " + (resp. such that x() B = " + for all " ; " + ). Then, in step 3, we prove that discontinuous policies are also dominated at the two extremes of the interval. Step : Consider any incentive-compatible schedule x() B that has a jump discontinuity at some e, where e = 0+ and " + < 0 < < ". We will show that such a schedule is dominated by a similar schedule that does not have a jump. In fact, let x (e) B = 0 and x + (e) B =. Then x() B = 0 for all [ 0 ; e) and x() B = for all [e; ]. We now show that we can construct an alternative schedule x # () B that dominates x() B. Let x # () B be the same as x() B except over the interval [ 0 ; ] ; where x # () B =. Note that x() B is a step function over the interval [ 0 ; ] and the discontinuity occurs at e = ( 0 + )=, with B + 0 = lim "e x() x (e) < x + (e) lim #e x() = B + > B + e We must show that the excess of the median voter s expected utility under the schedule x # (:) B over his expected utility under the schedule x(:) B is strictly positive. Using (0), and de ning g() = [() (B + ) + h()] f(), this excess is given ( 0 ; ) = = 0 g()d e 0 0 g()d + e g()d 0 g()d f [G( ) G(e)] + 0 [G(e) G( 0 )]g where G() "+ g()d 6

28 Notice that ( 0 ; 0 ) = 0. Thus, if we can show 0 ; )=@ > 0 then we can conclude that ( 0 ; ) > 0 or all > 0. To show 0 ; )=@ > 0, note 0 ; )=@ = G(e) G( ) + g(e)( e) (A.) If g 0 () < 0 (which implies that G(:) is a strictly concave function), then clearly the RHS of (A.) is indeed strictly positive,. We now prove that g 0 () < 0. Recall that by de nition, g() [ F ()] (B + )f() + D() ( + B) d 4" and f() and F () are given by (5) and (6). Then direct computation shows that 8 < g() = : +(+") 3"+" ; 8 [ " ; "] 4" " ; 8 [ "; " ] " ; 8 [" ; " + ] 4" (" )("+ ) It is easy to see that g 0 () < 0; 8 ( " ; " + ): : (A.3) Step : Having shown that it is not optimal to have a discontinuous schedule x() B, we now show that, among all the continuous semi- exible schedules x() B, setting a policy cap is always optimal, and, moreover, the best policy cap is x() B = " " ; " +. for all Let us consider an arbitrary incentive-compatible schedule x() with a policy cap x B = for some, i.e., x() B = for all [ ; " + ]. Using (0), we compute the di erence between the median voter s expected utility obtained from this schedule, and his/her expected utility obtained from an alternative schedule x y () such that (i) x y () B is identical 3 to x() B for all [ " ; ] and (ii) for all [ ; " + ] we have x y () B = (i.e., it is exible in this interval). The di erence in expected utility levels obtained from x() and x y () is W ( ) = "+ ( )g()d; (A.4) where g() is given by (A.3). Direct computation shows that 3 This implies, in particular, that x() = x y (), so that W A () has the same value under the two schedules. 7

29 8 >< " 3 (++ ) 3 ( " ) < 0; if [ " ; "] 6 48" W ( ) = ( ")( ")( + ") 0; if " [ "; " ] : >: ( 48" + ")( + " ) 3 ; if [" ; " + ] Thus, W ( ) is equal to 0 at the value = " + and also at the value = " ; and it is strictly positive for all (" ; " + ) and strictly negative if < ". In particular, W ( ) attains its maximum value at = " cap is to set the cap x = B + " : It follows that the best policy ; that is, x() = B + " for all " ; " + : With the optimal cap, we nd that the endogenously determined W A () is given by W A () =. (B + ) B + " B + " Step 3: We now show that any schedule x() B that has a discontinuous jump to a cap, where the discontinuity occurs at some e in the interior of the interval " ; " + with the cap being set such that x B = where " + > e (i.e., x() B = for all (e; " + ]) is dominated by the schedule x # () B which is identical to x() B for all 0 where + 0 = e, and which has a cap x # B = 0, i.e. x # () B = 0 for all [ 0 ; " + ]. Indeed the expected median voter s utility under x # (:) exceeds the one under x(:) by "+ 0 + ( 0 ) g()d = 96" ( 0 )( 0 + " + )( + 0 (" + )) > 0: Remark: By an argument symmetrical to Step, a policy oor is always optimal. In fact, using equation () to compute the di erence between the median voter s expected utility obtained from a schedule x() B with a oor, and an alternative schedule x y () B without a oor, we can show that the best policy oor is x() = B " + for all " ; " + : By an argument symmetrical to Step 3, one can show that any schedule with a discontinuity at some e in the interior of the interval " ; " + is dominated by a continuous schedule. With the optimal oor, we nd that the endogenously determined W A () is given by W A () = (B + ) B " + B " +. 8