Off-the-peak preferences over government size^* by Francisco Martínez-Mora M. Socorro Puy^** Documento de Trabajo 2010-05

Serie Capital Humano y Empleo CÁTEDRA Fedea – Santander

February 2010

The authors thank Subir Bose, Gianni De Fraja, John Duggan, Mark Fey, Michel Le Breton, Miltos Makris, Ludovic Renou, Luigi Siciliani and Al Slivinsky for their helpful comments. Financial assistance from Ministerio de Ciencia e Innovación under the project SECO2008-03674/ECON, and Junta de Andalucía under the project SEJ1645 is gratefully acknowledged.

** University of Leicester and FEDEA

*** Universidad de Málaga.

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Francisco MartÌnez-Mora University of Leicester and FEDEA

M. Socorro Puy Universidad de M·laga

January 22, 2010

Abstract

We show that preferences-bias towards overprovision or underprovision can explain the asymmetric location of electoral candidates with respect to the median voter. We analyze the determinants of preferences o§-the-peak and Önd that: (i) The sign of the third derivative of the policy-induced utility function indicates whether preferences are bias towards overprovision (positive) or underprovision (negative). (ii) The analog of Kimballís coe¢ cient of prudence can be used to measure the asymmetry of preferences. (iii) Consumersírisk aversion and government corruption (in the form of decreasing e§ectiveness producing public good) induce votersí preferences to be more intense towards underprovision.

Key-words: Single-peaked preferences, preferences-bias, coe¢cient of prudence, di§erentiated platforms, risk-aversion. JEL classiÖcation numbers: D72, H31,H5.

The authors thank Subir Bose, Gianni De Fraja, John Duggan, Mark Fey, Michel Le Breton, Miltos Makris, Ludovic Renou, Luigi Siciliani and Al Slivinsky for their helpful comments. Financial assistance from Ministerio de Ciencia e InnovaciÛn under the project SECO2008-03674/ECON, and Junta de AndalucÌa under the project SEJ1645 is gratefully acknowledged.

1 Introduction

Many models of political competition assume that preferences of voters over policies are single-peaked and symmetric around the peak. In fact, the assumption that voters possess quadratic preferences over policies is the most standard one in models that study the strategic location of candidates. There are however notable exceptions, such as the classical downsian model of political competition (Black, 1948; Downs, 1957), whose only restriction is singlepeakedness.¹ In a well-known result, this model predicts that the two competing platforms converge in equilibrium to the median voterís bliss point, which obviously renders the non-symmetry of votersípreferences innocuous. Nevertheless, and despite its profound ináuence, the median voter theorem does not fully satisfy political scientists because that prediction is not in line with what we observe in real-world electoral competition.²

In the last decades, a number of articles have presented models of electoral competition where equilibrium platforms are di§erentiated. For instance, Calvert (1985) and Wittman (1983) derive separation of platforms from policy-motivated candidates with uncertainty on the distribution of voters. Krasa and Polborn (2009) derive this separation from the parties di§erent abilities in producing public good. Likewise, Osborne and Slivinski (1996) and Besley and Coate (1997) propose the citizen-candidate model with a class of two-candidate equilibria where candidates locate symmetrically around the median (if preferences of the median are symmetric).³

In this paper, we show that in models where candidatesí platforms do not converge, the non-symmetry of policy preferences has a relevant impact on the equilibrium location of platforms (and consequently, in the electoral outcome). To illustrate this point, we follow the citizen-candidate approach, where the median plays the following central role: while she is never offered her most preferred choice, equilibrium platforms exactly counteract

¹Another exception is given by the directional voting model due to Rabinowitz and Macdonald (1989), in which preferences are asymmetric.

²Another critic that has been put forward is that the downsian model does not account for preferences of politicians over policies (e.g. Roemer, 2001).

³Two-candidate electoral competition is empirically relevant (see, for instance Jones, 1999; Wright and Riker,1989) and theoretically important: not only many models assume the exogenous existence of two political contestants, also the citizen-candidate model with sincere voters predicts the existence of just two-candidate equilibria for wide regions of the parameter set (Osborne and Slivinski, 1996).

each other to make her indi§erent.⁴

We introduce two natural alternatives to symmetric preferences, asymmetric towards overprovision ñwhich we name wasteful policy preferencesñ and asymmetric towards underprovision ñ which we name scrooge policy preferencesñ. We show that, when preferences of the median voter are wasteful, the candidate proposing a larger government size becomes less moderate than her rival. However, when preferences of the median voter are scrooge, the candidate proposing a smaller government size becomes less moderate than her rival.

We characterize the proposed types of preferences. In doing so, we show that the third derivative of the utility representation of policy preferences with respect to the policy variable indicates the direction of the nonsymmetry of preferences, with positive sign indicating wasteful preferences, and a negative one implying scrooge preferences. When the third derivative equals zero, in turn, preferences are symmetric.

The sign of the third derivative of the relevant utility function has already been shown to have important implications in other economic settings. For example, in the theory of precautionary saving due to Kimball (1990), the sign of the third derivative of the utility function over consumption governs the presence or absence of a precautionary saving motive (where precautionary saving means to forego consumption this period when there is uncertainty about future periods).⁵ In fact, following the theory of risk aversion (Pratt, 1964), we show that Kimballís coe¢cient of prudence measures the asymmetry of preferences.

We next analyze what factors drive preferences to be non-symmetric. We provide several examples of scrooge and wasteful preferences, and discuss what types of preferences emerge in terms of di§erent redistributive programs and votersí primitive preferences over private and public good. For the widely used case where primitive preferences over private and public good are quasi-linear, we show that a positive coe¢cient of prudence (which implies DARA or CARA speciÖcations of votersírisk aversion over private consumption), as well as a government with constant or decreasing-e§ectiveness (or corruption), drive preferences of voters to be scrooge. Our analysis also reveals that the symmetric property of policy preferences holds only under quite stringent conditions.

⁴In a similar vein, and according to the membership-based equilibrium due to Caplin and Nalebu§ (1997), if the platform of each party is the mean (or the median) of its members and no member of the party can improve by moving to the rival party, then in equilibrium an agent (not necessarily the median) is indi§erent between the two parties platforms.

⁵ Groseclose (2001) shows that the sign of the third derivative of the distant function between the ideal policy of voters and the implemented policy indicates whether swing voters prefer moderate movements form high-valence than low-valence candidates.

The remaining of the paper is organized as follows. The next section presents a general model of public good provision where the political process follows the citizen-candidate approach. Section 3 introduces three types of preferences over public expenditure o§-the-peak: symmetric, wasteful and scrooge, and inquires about their implications for the outcome of the electoral process. In that section we also characterize and compare the proposed types of preferences. Section 4 analyzes the determinants of policy preferences o§- the-peak in terms of the primitives of the model. Section 5 concludes.

2 A general policy problem

Consider a jurisdiction populated by a continuum of households with mass normalized to unity. Households are heterogeneous with respect to exogenously given income . Income is distributed across the population according to some given distribution function with strictly positive density function on the support : The income of household i is denoted by . The mean and median income of the overall population are denoted by and respectively.

Preferences of household i are deÖned over bundles of private consumption , and public expenditure Units of quality of the publicly provided good are normalized to be measured in units of the numeraire. Preferences have continuous utility representation , with strictly positive marginal utility in both arguments.

Public expenditures are Önanced through taxation levied on every household. Each household tax bill depends on the level of public expenditure and income. Household iís tax bill is described by a continuous function , strictly increasing in the level of public expenditure. The tax bill is constrained to balanced government budget, so that every function satisÖes

\[\int_ {\underline {{y}}} ^ {\bar {y}} \tau (e, y) f (y) d y = e.\tag{1}\]

As a feasibility constraint, we require for all 6

Households choose private consumption in a decentralized way. From strictly increasing marginal utility in private consumption, Induced preferences over public expenditure are derived from the indirect utility function (or policy-induced utility) of household i which can be written as

\[V (e, y _ {i}) \equiv U (y _ {i} - \tau (e, y _ {i}), e).\tag{2}\]

While the utility representation U is identical across households, heterogeneity in income induces heterogenous preferences over public expenditure. We use V0; V 00; V 000 to denote the Örst, second and third derivatives of V with respect to e. We assume that V is a function in argument e.

The following conditions on V are useful in the analysis that follows. The two Örst conditions on single-peakedness and single-crossing are standard. The third condition is a novel condition that, as we show below, provides additional insights into the nature of non-symmetric preferences over public expenditure.

Single-Peaked (SP): V has a maximizer and is strictly quasi-concave in e. Strict Single-Crossing (SSC): is strictly monotonic in .

Monotonic Satiation (MS): is strictly monotonic in

By SP, every household i has an ideal policy (or peak), denoted by such that

\[e _ {i} ^ {p} = \underset {e} {\arg \max} V (e, y _ {i}).\tag{3}\]

Every household is worse o§ as the distance between a policy e and their ideal policy increases. Condition SP characterizes the entire domain of single peaked preference relations.⁸ When the ideal policy of a household i is derived as an interior solution to (3), we have . The peak of the median is denoted by

x_i = y_i ty_i

⁶This constraint imposes an upper bound on the total amount of public expenditure. ⁷For instance, under proportional income taxation, optimal private consumption is where t is the marginal tax rate. According to (1) we have t = ^ey~ ely where y~ denotes mean income, from where (e; y_i) = eyi_y~ e yi

V (e; y_i) > V (e0; y_i)

e0 < e < e^p_i

e; e0

e^p_i > e > e0.

⁸Of course, this characterization accounts for a convex policy space and preferences with utility representation. Preferences over policies are single-peaked when the most preferred policy of household i is a singleto and for every pair of policies we have n e^p_i ; V (e, yi) > V (e', yi) when e' < e < e, or e > e > e′.

In turn, SSC is a su¢ cient condition that guarantees that majority preferences and preferences of the household with median income coincide.⁹ By this condition, can be either strictly increasing or strictly decreasing in income. In the former case, rich households prefer more public expenditure than poor ones (e.g. public education). In the latter, the opposite holds, i.e., rich households prefer less public expenditure than poor ones (e.g. income redistribution).

Finally, MS is a regularity condition that restricts changes in the marginal utility of public expenditure to be strictly monotonic. According to MS, the marginal utility derived from public expenditure is either a strictly concave or a strictly convex function. Strict concavity of the marginal utility function implies that the principle of diminishing marginal utility (which basically captures the process of satiation) applies with increasing intensity. Thus, marginal utility derived from public expenditure falls faster at higher level of public expenditure. Strict convexity of course implies the opposite, that marginal utility falls more slowly at higher level of public expenditure.¹⁰

The electoral process that determines the level of public expenditure in the jurisdiction follows the citizen-candidate pattern. This process has three stages. In stage 1, households simultaneously decide whether or not to become candidates. In doing so, they must weigh, on the one hand, some positive entry costs, , against, on the other hand, some positive beneÖts derived from holding o¢ce, , and from choosing or a§ecting policy. In stage 2, each household casts a vote. We consider sincere voting, i.e., each household votes for the candidate whose platform better matches their own policy preferences.¹¹ Finally, in stage the candidate that achieves more votes becomes the winning candidate and implements her ideal policy denoted by . If two candidates tie, we account for the expected policy between the two winning candidates and we interpret it as a measure of the asymmetric location of candidates with respect to the peak of the median.¹²

⁹ In fact, this is the one-dimensional version of the Spence-Mirrlees condition that characterizes the domain of strict single crossing preferences (see Milgrom, 1994; Gans and Smart, 1996). In our model, preferences over policies are strict single-crossing when for all and for all then ^e0 6⁼ e; y_j > y_i; if V (e; y_i) > V (e0; y_i) V (e; y_j ) > V (e0; y_j ):

¹⁰Monotonic satiation is not such a stringent assumption as it may in principle seem. For example, CARA or DARA preferences for private consumption entail i.e., u000 > 0, that its marginal utility falls slower at high levels of consumption (e.g. Kimball, 1990).

¹¹Thus, according to the citizen-candidate, votersíonly concern is the platform of the candidates and not other personal characteristics.

¹²Since we are interested in the equilibrium location of candidates, the expected policy

We study equilibria with two competing candidates, which we name Low (L) and High (H). Their platforms specify an amount of public expenditure, and are denoted by and respectively, where . A two-candidate equilibrium is a (pure strategy) Nash equilibrium at the entry stage with the particularity that just two voters become candidates.

DeÖnition: A two-candidate equilibrium is given by a pair of platforms such that no candidate improves withdrawing from the contest, and no other voter improves by becoming candidate.

For every possible distribution of income across households, we guarantee existence of two-candidate equilibria. To prove this statement, we extend the proof of Osborne and Slivinski (1996) to preferences over policies satisfying SP and SSC.¹³ The proof is in the Appendix.

Proposition 1 Assume that policy preferences satisfy SP and SSC. 2c, there always exist two-candidate equilibria. Furthermore, in every twocandidate equilibrium, candidates tie.

In every two-candidate equilibrium, candidates gather the same proportion of votes. This equilibrium property implies that the median voter must be indi§erent between the equilibrium platforms. By SSC, the median voter coincides with the median income household, so that, in every equilibrium . By SP, the equilibrium platforms must be located at di§erent sides of the peak of the median, i.e.,

3 Preferences o§-the-peak

In this section, we introduce two alternatives to symmetry ñwhich we name wasteful and scrooge policy preferencesñand we inquire about their implications in terms of the location of equilibrium platforms. We also characterize the proposed types of preferences and provide a coe¢ cient that measures the degree of scroogeness or wastefulness.

is more informative than any tie-breaking rule. In this point we deviate from the citizencandidate that assumes that ties are dealt with an equal-probability rule. Our alternative assumption, however, does not entail any substantial variation in terms of equilibrium location of candidates.

¹³Osborne and Slivinski assume that preferences of voters over policies are symmetric around the peak and that voters only di§er in their peak. Under their assumption, SSC is obviously satisÖed.

3.1 DeÖnitions of preferences

Given household iís peak and letting d be a positive amount of public expenditure, we say that policy preferences of household i are symmetric (around the peak) when

\[V (e _ {i} ^ {p} - d, y _ {i}) = V (e _ {i} ^ {p} + d, y _ {i}) \text { for all } d \in (0, e _ {i} ^ {p}).\]

In that case, household i is indi§erent between any two policies that are equidistant to their peak.¹⁴

We consider two natural alternatives to symmetry. Figure 1 illustrates these notions. These alternatives capture preference-bias towards overprovision and underprovision with respect to their peak.

(a)

(b)

Figure 1: (a) Symmetric preferences (b) Wasteful Preferences (c) Scrooge preferences

Suppose household i is given a choice between two symmetric deviations from their peak, and , we say that their preferences are wasteful whenever

\[V (e _ {i} ^ {p} - d, y _ {i}) < V (e _ {i} ^ {p} + d, y _ {i}) \text { for all } d \in (0, e _ {i} ^ {p}).\]

Thus, o§-the-peak, that household prefers an excess in public expenditure over its symmetric defect. In the opposite case, that is whenever

\[V (e _ {i} ^ {p} - d, y _ {i}) > V (e _ {i} ^ {p} + d, y _ {i}) \text { for all } d \in (0, e _ {i} ^ {p}),\]

2e^p_i < e:

e^p_i

¹⁴Let e be the upper bound in public expenditure. We implicity assume that Note also that this deÖnition, together with the subsequent deÖnitions of wasteful and scrooge preferences, only apply when is an interior solution.

we say that preferences of household i are scrooge,¹⁵ which implies that household i prefers a defect from the peak over its symmetric excess.

3.2 Implications of wasteful and scrooge preferences

As it follows from Proposition 1, in every two-candidate equilibrium condition holds. As a consequence, if preferences of the median are symmetric, then the expected policy is , however, if preferences of the median are wasteful or scrooge, the equilibrium platforms cannot be equidistant to the peak of the median. Thus, when preferences of the median are wasteful, and when preferences of the median are scrooge, (Figure 2 illustrates this point). The proposed arguments prove our next result.

Figure 2: Illustration of Proposition 2, median with scrooge preferences

Proposition 2 Assume that policy preferences satisfy SP and SSC, then the expected policy in every two-candidate equilibrium:

- coincides with the peak of the median when her preferences are symmetric,

- is above the peak of the median when her preferences are wasteful,

- is below the peak of the median when her preferences are scrooge.

Thus, the two proposed alternatives to symmetric preferences entail an asymmetric location of equilibrium platforms with respect to the peak of the median. It is important to stress that only the shape of the median voter preferences o§-the-peak matters to derive the above result. In turn, by SSC, the only relevant aspect of preferences of non-decisive voters is the location of their peak (whether it is to the right or to the left of the median).

¹⁵Our choice of the term íscroogeíis motivated by Ebenezer Scrooge, a mean character in Charles DickensíA Christmas Carol.

The shape of preferences o§-the-peak generate di§erent conÖgurations of two-candidate equilibria.¹⁶ When preferences are symmetric, every twocandidate equilibrium entails an equivalent prediction in terms of the expected policy. When preferences are non-symmetric, predictions concerning the expected policy may vary across di§erent two-candidate equilibria. In particular, we next show that under our proposed conditions, it is possible to order the multiplicity of equilibria since higher polarization of the candidatesí platforms is translated into more distant between the peak of the median and the expected policy. The proof is in the Appendix.

Proposition 3 Assume that policy preferences satisfy SP, SSC, MS and that preferences of the median are either wasteful or scrooge. Then, the more polarized the two equilibrium platforms are, the larger the wedge between the expected policy and the peak of the median.

Thus, our result reveals that within the proposed conditions, polarization of equilibrium platforms and asymmetry with respect to the median come together.

3.3 On the characterization of preferences

We have shown that the non-symmetry of preferences has relevant policy implications, as illustrated by the citizen-candidate approach. The following characterization results, in turn, do not depend on a speciÖc electoral procedure. We Örst provide a characterization of preferences that satisfy SP and that are symmetric (around the peak).

Theorem 1: Assume that policy preferences satisfy SP, then preferences are symmetric if and only if for all

¹⁶Policy preferences o§-the-peak are also crucial when equilibrium candidates do not tie (e.g. Caplin and Nalebu§, 1997) since, in this case, the median voterís o§-the-peak preferences determine which party wins. More generally, scrooge (wasteful) preferences of the median voter provide candidates proposing low (high) spending with some electoral advantage. That advantage may or not be decisive depending on the speciÖc location of the platforms in equilibrium.

Proof. By SP, when V is symmetric for all . This implies that at any point ; the e§ect on V derived from increasing in any amount, is equal to the e§ect on V derived from decreasing by the same amount. In particular, when modifying and by an amount that approaches zero, we have

Conversely, let V satisfy SP and for all , and suppose, by contradiction, that V is non-symmetric. Then, the utility loss derived from decreasing public expenditure from to cannot equal the utility loss derived from increasing public expenditure from to . Without loss of generality, consider the case where ; that expression is equivalent to ; which is in contradiction with for all

Hence, symmetric preferences require the slope of the policy-induced utility function to be equal in absolute terms at every pair of symmetric deviations from the peak. Clearly, when this condition does not hold, preferences are non-symmetric. Theorem 1 indicates that, apart from cases where the marginal function falls at a constant rate , whenever , symmetry requires a somewhat farfetched behavior of the indirect utility function in the public expenditure direction.¹⁷

Following the above result, we want to provide a characterization of wasteful and scrooge preferences. By restricting the set of preferences according to MS, we show that the sign of the third derivative of characterizes the proposed types of preferences.

Theorem 2: Assume that policy preferences satisfy SP and MS then,

- preferences are wasteful if and only if V0 is strictly convex in

- preferences are scrooge if and only if is strictly concave in

Proof. If is strictly convex , equivalently, in , by Jensensíinequality, the expected value of in the interval is above the value of in the mean of for all . Thus,

¹⁷ To see this, note that for any concave interval to the right of the peak, the corresponding (symmetric) interval to the left of the peak must be the exact symmetric but convex.

\[\int_ {e _ {i} ^ {p} - d} ^ {e _ {i} ^ {p} + d} \frac {V ^ {\prime} (e , y _ {i})}{2 d} d e > V ^ {\prime} (e _ {i} ^ {p}, y _ {i}).\tag{4}\]

Solving for the integral, By continuity of and SP, ; and for all , preferences are wasteful. If is strictly concave, the sign of Expression (4) is reversed and this proves that preferences are scrooge. Conversely, let preferences be wasteful and suppose that is not strictly convex. Then, by MS, must be strictly concave, which implies that preferences are scrooge, contradicting that preferences are wasteful. Following a similar line of reasoning, it is straightforward to prove the case where is strictly concave.

The above proof reveals that the condition, alongside strict concavity of are su¢ cient conditions for scrooge preferences. Likewise, the SP condition together with strict convexity of are su¢cient conditions for wasteful policy preferences. Only under MS, the strict concavity or convexity of become necessary conditions for scrooge and wasteful preferences, respectively.

As illustrated in Figure 3, when is strictly concave, the marginal utility of public spending falls faster at higher levels of public expenditure. In that case, the utility loss emerging from receiving too much of public expenditure (an excess over the peak equal to is larger than the utility loss derived from too low a level of provision (a shortfall over the peak equal to . Strict convexity of obviously implies the opposite: that marginal utility falls (and satiation occurs) more slowly at higher levels of public expenditure. In that case, it is more important to avoid a shortfall of spending with respect to the peak than to incur in an excess.

3.4 Degrees of wastefulness and scroogeness

There is an analogy between our result concerning the characterization of wasteful and scrooge preferences, and the theories developed by Pratt (1964) and Kimball (1990). According to Prattís theory of risk aversion, concavity of a utility function over consumption indicates the presence of risk aversion, while according to Kimballís theory of precautionary savings, concavity of the marginal utility of second period consumption entails precautionary savings. In both cases, the degree of concavity of the relevant function measures risk aversion or precautionary savings. These behavioral traits become thus comparable across pairs of concave functions such that one is a concave transformation of the other.

The similarity between our approach and theirs stems from the fact that, as we show next, under MS and as long as preferences satisfy a stronger version of SP (strict concavity of , the curvature of the marginal policyinduced utility function determines the degree of non-symmetry of preferences. We can therefore apply the coe¢cient of prudence that Kimball (1990) proposes, to measure the level of scroogeness or wastefulness.¹⁸

In order to state and prove this result, we need a deÖnition of the degrees of wastefulness and scroogeness of preferences: Consider two di§erent utility functions satisfying SP and which peaks are respectively. Each of these functions correspond to two di§erent households with income and y₂ respectively. The functions and are implicitly deÖned by each of the following functions

\[\begin{array}{r} V _ {1} (e _ {1} ^ {p} - d, y _ {1}) = V _ {1} (e _ {1} ^ {p} + \delta_ {1} (d), y _ {1}) \\ V _ {2} (e _ {2} ^ {p} - d, y _ {2}) = V _ {2} (e _ {2} ^ {p} + \delta_ {2} (d), y _ {2}), \end{array}\]

i.e., each of the households is indi§erent between two deviations from their peak and . We say that is more scrooge than when for all min . Likewise, we say that is more wasteful than when for all min

¹⁸Recall that Kimballís coe¢cient of prudence is deÖned by

Theorem 3: Let be policy-induced utility functions satisfying strict concavity in , and MS. Let 2

are scrooge and for all 2 then is more scrooge than ; and

are wasteful and for all , then is more wasteful than

Proof. Consider the case in which and are scrooge (an analogous argument proves the result for wasteful preferences). By assumption, and are strictly decreasing functions of and can be related by a strictly increasing transformation such that where (we omit argument from and in the interest of clarity). is obtained as an increasing transformation of with the particularity that the argument of is normalized to . This implies that when : We di§erentiate the expression,

\[\begin{array}{c} {V _ {2} ^ {\prime \prime} (e) = g ^ {\prime} (V _ {1} ^ {\prime} (e - \Delta)) V _ {1} ^ {\prime \prime} (e - \Delta)} \\ {V _ {2} ^ {\prime \prime \prime} (e) = g ^ {\prime \prime} (V _ {1} ^ {\prime} (e - \Delta)) [ V _ {1} ^ {\prime \prime} (e - \Delta) ] ^ {2} + g ^ {\prime} (V _ {1} ^ {\prime} (e - \Delta)) V _ {1} ^ {\prime \prime \prime} (e - \Delta).} \end{array}\]

From where

\[- \frac {V _ {2} ^ {\prime \prime \prime} (e)}{V _ {2} ^ {\prime \prime} (e)} = - \frac {V _ {1} ^ {\prime \prime \prime} (e - \Delta)}{V _ {1} ^ {\prime \prime} (e - \Delta)} - \frac {g ^ {\prime \prime} (V _ {1} ^ {\prime} (e - \Delta)) V _ {1} ^ {\prime \prime} (e - \Delta)}{g ^ {\prime} (V _ {1} ^ {\prime} (e - \Delta))}.\]

By MS, and . Then, implies strictly concave). By deÖnition of ; we have ; or equivalently, : Substituting function

\[0 = \int_ {e _ {2} ^ {p} - d} ^ {e _ {2} ^ {p} + \delta_ {2} (d)} V _ {2} ^ {\prime} (e) d e = \int_ {e _ {2} ^ {p} - d} ^ {e _ {2} ^ {p} + \delta_ {2} (d)} g (V _ {1} ^ {\prime} (e - \Delta)) d e.\tag{5}\]

By strict concavity of ， where : Since g is strictly increasing and ; then for all d: Since for all ; and by deÖnition of we deduce that for all min

According to this result, policy preferences are more scrooge for the "more concave" marginal utility representation, and more wasteful for the "more convex" marginal utility representation. The advantage of using this coe¢cient to measure scroogeness and wastefulness is that it can be easily calculated for speciÖc applications.¹⁹

When the peak derived from two di§erent policy-induced utility function coincides, then ; and it is possible to compare degrees of non-symmetry just by comparing

\[- \frac {V _ {1} ^ {\prime \prime \prime} (e , y _ {1})}{V _ {1} ^ {\prime \prime} (e , y _ {1})} \text {to} - \frac {V _ {2} ^ {\prime \prime \prime} (e , y _ {2})}{V _ {2} ^ {\prime \prime} (e , y _ {2})} \text {for every} e < 2 e _ {1} ^ {p}.\]

Because it is the curvature around the peak that matters, in cases where the peak is not reached at the same level of public expenditure, the comparison must be made around the corresponding peak. That is to say, when 2 we shall compare

\[- \frac {V _ {1} ^ {\prime \prime \prime} (e - \Delta , y _ {1})}{V _ {1} ^ {\prime \prime} (e - \Delta , y _ {1})} \text {and} - \frac {V _ {2} ^ {\prime \prime \prime} (e , y _ {2})}{V _ {2} ^ {\prime \prime} (e , y _ {2})} \text {for every} e < \min \left\{2 e _ {1} ^ {p}, 2 e _ {2} ^ {p} \right\}.\]

In this case, when , then i.e., preferences are compared around their respective peaks. Note that, as it occurs in Pratt (1964) and Kimball (1990), the ordering of preferences according to the level of scroogeness or wastefulness is a partial ordering (transitive but not complete).

In the Appendix we provide some numerical examples with plausible parameters. We compare preferences of a median voter with di§erent coe¢- cients of risk aversion. In the proposed examples we show, using our result in Theorem 3, that more risk aversion makes preferences "more scrooge". As a consequence, when the median voter is more risk averse, the candidate proposing a smaller government size is always less moderate than her counterpart.

V000 V00

000 V V

¹⁹We have exactly considered Kimballís coe¢ cient. Another alternative is to deÖne two separated coe¢cients, a coe¢cient of scroogeness and a coe¢cient of wastefulness (that coincides with Kimball). In this way, both coe¢ cients are positive and the higher the coe¢ cient, the more scrooge or more wasteful preferences are.

4 Determinants of preferences o§-the-peak

In this section we analyze what may drive preferences to be symmetric, wasteful, or scrooge. For expositional purposes, hereafter we analyze the case of quasi-linear preferences where , yielding the following indirect utility function

\[V (e, y _ {i}) \equiv u \left(y _ {i} - \tau (e, y _ {i})\right) + e,\tag{6}\]

with The proposed conditions guarantee SP, with V strictly concave. In addition, if the cross derivative , we can also guarantee SSC.²¹

As we show below, attitudes toward risk and precautionary savings, as well as the form of the tax bill plays a central role in shaping the preferences of households over public expenditure. We measure di§erent degrees of risk aversion according to the Arrow-Prattís coe¢cient of absolute risk aversion Increasing absolute risk aversion (IARA) holds when Likewise, decreasing absolute risk aversion (DARA) and constant absolute risk aversion (CARA) correspond to and respectively, where , and both cases imply

The generic tax bill function can capture di§erent Önance schemes, as well as di§erent degrees of e§ectiveness in transforming tax revenues into public expenditure. We say that the government is constant-e§ective when the scheme is linear in . Particular instances of constante§ective governments are given by the lump-sum tax, where or the proportional tax, where . A progressive or regressive tax scheme can also be constant-e§ective. Thus, for instance, the tax bill function with strictly increasing in income is progressive whenever and regressive when When is strictly convex in e we say that government is decreasing-e§ective, . Convexity of in e can represent congestions in government ability to transform tax revenue into public expenditure. Other interpretations are tax distortions, or corruption of public o¢cials, in which case convexity of in reáects the deviation of tax revenues to other purposes di§erent from public expenditure.²³ Examples of convex under lump-sum or proportional income taxation are given by and (with respectively.

u00 = 00 = 0

²⁰We should exclude since the ideal policy of the household cannot be derived as an interior solution to their utility maximization problem.

21To

V "y = −u"(1 − τ y)τ′ + u'τ "y,

u00 = 0; 00_ey > 0

u″ < 0, τey ≥ 0

taxation, (e; y_i) = ey_i y~ y

^{21 To} see this, the cross derivative ^V00ey ^{= u00(1 0}y^{)0 + u000}ey where ^{u00 < 0; 00}ey ⁰ or > 0 are su¢ cient conditions for SSC. For instance, under proportional income , we have 00 = ¹y~ 1y which implies that rich households demand more ey public expenditure.

ρ

^y_y (y)f(y)dy = 1:

²²For these functions to qualify as tax bill functions, the government budget must be balanced. This implies that must satisfy A particular example of a non-proportional taxation that satisÖes these requirements is the education Önance scheme studied by BÈnabou (2002).

Solving for the derivatives of (6) with respect to

\[V ^ {\prime} = - u ^ {\prime} \tau^ {\prime} + 1\tag{7}\]

\[V ^ {\prime \prime} = u ^ {\prime \prime} \left[ \tau^ {\prime} \right] ^ {2} - u ^ {\prime} \tau^ {\prime \prime}\tag{8}\]

\[V ^ {\prime \prime \prime} = - u ^ {\prime \prime \prime} \left[ \tau^ {\prime} \right] ^ {3} + 3 u ^ {\prime \prime} \tau^ {\prime} \tau^ {\prime \prime} - u ^ {\prime} \tau^ {\prime \prime \prime}.\tag{9}\]

Following the result in Theorem 2, the sign of (9) determines whether preferences satisfy the symmetric, scrooge or wasteful requirement. The following table summarizes in which cases the primitives of the problem u and yield each type of policy preferences. We cover most, if not all, the commonly used utility functions, as well as all the examples of tax bill functions we have mentioned.

Table 1: Determinants of the shape of preferences o§-the-peak

²³See, for instance, the model on career concerns proposed by Persson and Tabellini (2000, chapter 4).

²⁴ In this case, V000 = u0[ 3r_A000 000].

According to Table 1, we can describe, in terms of the primitives of the policy problem, the conditions for symmetric preferences.

Remark 1: Symmetric preferences follow from:

i) risk neutral households and a decreasing-e§ective government with marginal tax bill increasing at a constant speed with public expenditure ， ii) risk averse households with a speciÖc form of where , and a constant-e§ective government.

Two particular examples of tax bill functions satisfying condition i) are given by with and by with . Note, however, that preferences are not symmetric for As far as we know, no commonly used utility function satisÖes condition ii).²⁶ Remark 1 thus suggests that symmetric policy preferences require placing quite stringent restrictions on the primitives of the policy problem. It is worth mentioning that there are more cases, not considered in the table, where preferences are symmetric. To identify all these cases, we can check, according to our result in Theorem 1, the necessary and su¢cient condition for symmetry, which can be written as

\[u ^ {\prime} \left(e _ {i} ^ {p} - d\right) \tau^ {\prime} \left(e _ {i} ^ {p} - d, y _ {i}\right) + u ^ {\prime} \left(e _ {i} ^ {p} + d\right) \tau^ {\prime} \left(e _ {i} ^ {p} + d, y _ {i}\right) = 2 \text {for all} d \in \left(0, e _ {i} ^ {p}\right).\tag{10}\]

Thus, if and but , preferences will be symmetric as long as is symmetric around ; or if and but 2 preferences will be symmetric as long as is symmetric around . We disregard these last two cases since, a priori, there is no implicit reason, but a mathematical argument, for either or to satisfy such symmetry requirements. Furthermore, such cases are not robust to small variations of and

Remark 2: If either i) or ii), then preferences are scrooge:

i) risk neutral households and a decreasing-e§ective government with ii) risk averse households according to the CARA or DARA condition, and either a constant-e§ective government or a decreasing-e§ective government with

u′″ = 0

(u′ > 0, u″ < 0, u" >

u00 < 0; 00 > 0;

g(e; y_i) = u0(e)0(e; y_i)

e^p_i :

²⁶ In fact, most of them show derivatives that alternate in sign (u0 > 0; u00 < 0; u000 > 0).This statement is justiÖed by Pratt and Zeckhauser (1987).

²⁵ Note that u000 = 0 implies that

²⁷Not to mention the case where and where symmetry requires the product function to be symmetric around

²⁸See the numerical example in the Appendix for a particular instance of these conditions.

Risk neutrality, as well as risk aversion with DARA or CARA speciÖcations of risk, are the assumptions more generally invoked in economic applications.²⁹ Likewise, a constant-e§ective government or a decreasing-e§ective government with a marginal tax bill that increases faster at higher levels of public expenditure , seems to us the appropriate form according to economic intuition. Thus, Remark 2 covers most commonly used utility functions as well as most of the widely applicable tax bill functions.

It is intuitive that consumers exhibiting DARA or CARA speciÖcations of risk are more afraid of too low private consumption (or equivalently, too high levels of public expenditure) and therefore, they possess scrooge preferences. Likewise, when the cost of raising public funds (or corruption) increases faster at higher levels of public expenditure , preferences for underprovision become more intense.

An alternative utility representation (not included in Table 1) is given by the quasi-linear (money-metric) speciÖcation of policy preferences : This speciÖcation yields the following indirect utility function

\[V (e, y _ {i}) \equiv y _ {i} - \tau (e, y _ {i}) + h (e),\tag{11}\]

in which h can be interpreted as a human capital production function, provided that e measures public expenditure in education. Alternatively, h could be a health status production function, provided that e measures public expenditure in health care. In both cases, positive and decreasing marginal returns in production emerge as natural assumptions, i.e., These conditions together with and guarantee SP and SSC. The third derivative of (11) is

\[V ^ {\prime \prime \prime} = - \tau^ {\prime \prime \prime} + h ^ {\prime \prime \prime}.\tag{12}\]

This reveals that, convex marginal returns contributes towards wastefulness.³⁰

²⁹On the one hand, recent empirical evidence presented by Chiappori and Paiella (2008) does not give support to the IARA condition. On the other hand, DARA is widely considered to be reasonable (see Arrow 1971), and as postulated by Pratt (1964): DARA is implied by such behavior of investing in risky securities as one becomes richer.

U(xi, e) = xi + 2αe2

V000 > 0

y~ y

³⁰ For instance, Roemer (2001) considers the utility representation U(x_i; e) = x_i + 2e ¹2 . Under proportional income taxation, V = y_i e yi + 2e ¹2 , and it is straightforward to show that and hence, preferences are wasteful.

5 Concluding Remarks

The symmetry assumption of policy preferences is widely spread in the political economics literature. While this condition seems reasonable when the policy issue at stake is purely ideological, our analysis revealed that it requires placing very stringent restrictions on the primitives of an economic policy problem.

We showed that the non-symmetry of policy preferences has relevant electoral implications. In particular, it can explain, in a simple and intuitive way, why competing candidates may not adopt equally moderated policies with respect to the median voter. Likewise, it can explain why polarization of candidatesíplatforms encompasses more asymmetry with respect to the median voter.

Our main Önding regarding the shape of preferences o§-the-peak is that the sign of the third derivative of the policy-induced utility function with respect to the policy variable indicates the direction of the bias of preferences towards overprovision (when V0 is convex) or underprovision (when V0 is concave). This feature of the policy-induced utility function is straightforward to check, and the result is applicable to a large body of policy problems, as long as preferences over policies are single-peaked. Even if we analyze a multidimensional policy space, and provided that preferences over policies are separable in each issue dimension, we can also check whether preferences over each issue shape the proposed forms (symmetric, wasteful, or scrooge).

Interestingly, the techniques used by Pratt to compare degrees of risk aversion, and by Kimball to compare degrees of prudence, can be used to compare degrees of non-symmetry of preferences. As we showed, Kimballís coe¢cient deÖnes a partial order of preferences in terms of the bias towards overprovision (wastefulness) or towards underprovision (scroogeness). This measure provides a tool that can be used, in future work, to analyze the impact of speciÖc political reforms on policy preferences (where both, the e§ect over the peak, and o§-the-peak are relevant to predict the electoral implications).³¹

³¹There are papers that analyze the impact of government spending and corruption on votersíbelieves over the ability of the incumbent government (see, for instance, Persson and Tabellini, 2000; Gavazza and Lizzeri, 2009). In contrast with this literature, and within the context of perfect information, we analyze the mechanism through which di§erent spending programs have an impact over policy-preferences and, consequently over the valuation of candidatesíplatforms.

Appendix

Proof of Proposition 1:

By SP, there always exist a pair of policies such that where DeÖne and note that, given the deÖnition of , that function satisÖes . By either : In the former case, , and households with income below strictly prefer over e ; whereas those with income above strictly prefer over : When ; preferences of households over with respect to are reversed. Let us Örst show that the proposed policies qualify as two-candidate equilibrium. If two households with preferred policies and respectively, become candidates, then there is a tie and If , then and ; so that both High and Low are better-o§ running the election; that is to say, no candidate has incentives to withdraw. Next, we show that no other household improves by entering the race: If a candidate with preferred policy where or ; entered the race, she would give the victory to the candidate whose platform is furthest from hers in the policy space. Likewise, a candidate with preferred policy that entered the race, could not win provided the two other candidates are su¢ciently close to each other. A candidate with no chance of winning an election may still want to enter the race, as that may modify the equilibrium policy to her beneÖt. A necessary condition to enter in this case is (when the preferred policy of this additional candidate is in the interval , or (when the preferred policy of this additional candidate is in the interval ). In both cases, however, candidates with platforms and can be located su¢ciently close to each other to guarantee that whatever the entry cost, the above conditions do not hold. Finally, suppose that there is an equilibrium where candidates do not tie. Then, the candidate that loses the election can improve by withdrawing from the contest, in contradiction with the assumption that this is an equilibrium.

Proof of Proposition 3:

We can write and as a function of so that and

b=2.

³² In such case, we consider that the beneÖts each candidate derives from holding o¢ ce reduce to

where

\[V (e _ {m} ^ {p} - d, y _ {m}) = V (e _ {m} ^ {p} + \delta (d), y _ {m}).\]

By the implicit function theorem, . Also, in equilibrium, the utility loss from too low a level of provision must equal the utility loss from too high a level of provision so that,

\[\int_ {e _ {m} ^ {p} - d} ^ {e _ {m} ^ {p}} V ^ {\prime} (e, y _ {m}) d e = \int_ {e _ {m} ^ {p}} ^ {e _ {m} ^ {p} + \delta (d)} V ^ {\prime} (e, y _ {m}) d e.\tag{13}\]

By SP, is decreasing and . Scrooge preferences entail and, by MS, is strictly concave in : Thus, for condition (13) to hold, it must be the case that . According to ; scrooge preferences imply : Wasteful preferences entail and, by MS, is strictly convex in e: Following the same reasoning, under wasteful preferences, . After straightforward manipulation, the outcome of two-candidate equilibria can be expressed as

\[e ^ {*} (d) = e _ {m} ^ {p} + \frac {\delta (d) - d}{2}.\tag{14}\]

Taking derivatives

\[e ^ {* \prime} (d) = \frac {\delta^ {\prime} (d) - 1}{2},\tag{15}\]

where when preferences are scrooge, and when preferences are wasteful.

Numerical Examples:

We present two numerical examples to further illustrate our results. We quantify the impact of the shape of policy preferences o§-the-peak for utility conÖgurations with the same median voterís peak.

We adopt a lognormal income distribution that matches the 2006 mean ($66,570) and median ($48,201) income in the US (in thousands). This requires setting the mean parameter at and the variance parameter at . We follow a particular form of decreasing-e§ective government proposed by Hansen and Kessler (2001). These authors consider proportional income taxation in addition to a cost associated to raising public funds. Convexity of the function that captures the cost of raising public funds is translated into convexity of in According to their speciÖcation, y~ where is the cost of raising public funds.

Policy preferences are derived from the following primitive utility,

\[U (x _ {i}, e) = \frac {1}{1 - \alpha} x _ {i} ^ {1 - \alpha} + \beta e\]

which displays DARA. Such speciÖcation yields the following policy-induced utility function:

\[V (e, y) = \frac {1}{1 - \alpha} \left[ \frac {y _ {i}}{\widetilde {y}} \left[ \widetilde {y} ^ {2} - 2 \widetilde {y} e \right] ^ {\frac {1}{2}} \right] ^ {1 - \alpha} + \beta e.\]

The induced policy preferences satisfy SP, SSC and MS and the peak is increasing in household income.³³ Both the presence of risk aversion (with DARA and thus and of public sector distortions (with generate scrooge preferences.

The taste parameters are chosen so that the median voterís peak matches US spending in K-12 education per pupil in 2006 ($9,138). We compare two cases with di§erent coe¢ cients of risk aversion (but the same peak): in example 1, and , whilst in example 2, and . We follow the citizen-candidate approach in which costs from becoming candidate and beneÖts from holding o¢ ce are set at and respectively.

Table 2: Numerical examples

Table 2 contains the results for three of the continuum of equilibria that emerge in each example: the Örst column presents results of the equilibrium with closest-to-median candidates, the second one an intermediate case, and the third one the equilibrium with the most polarized candidates. The Örst two rows contain the policy proposals of candidates Low and High. We use and to denote the income level of candidates with platforms and respectively. Finally, the last row indicates the wedge between the expected policy and the peak of the median voter in percentage rates. The examples show that, with scrooge preferences, the candidate proposing a smaller government size is less moderate than her rival . Furthermore, with more polarization of candidates the magnitude of di§erences between the median voterís most preferred policy and the expected policy can be signiÖcant (up to 12.37% in example 1).

τ = tyi

0 =

^yi ^{y~2 2~ye 12 > 0; 00 = y}i^{y~ y~2 2~ye 32 > 0; 000 = 3y}i^{y~2 y~2 2~ye 52 > 0;} yi (y2 − 2ye)−1 > 0, τ" = yiy (y2 − 2ye)−x > 0, τ" = 3yiy2 (y2 − 2ye)−2 > 0, and

³³Solving for t we have and since = ty , it follows that 00_ey = y~² 2~ye ² > 0 Tey = (g2 − 2¯e)− > 0

Figure 4: Comparison of the coefficient of scroogeness

Figure 4 depicts Kimballís coe¢ cient (or, equivalently, the coe¢ cient of scroogeness). Let and be the policy-induced utility function for examples 1 and respectively. Taking into account that the peak is the same in both speciÖcations, Figure 4 illustrates that in the whole domain which, according to Theorem implies that preferences of the median voter in example 1 (with greater absolute risk aversion), are more scrooge than those of the median voter in example 2. As a consequence, in example 1 the impact of polarization is more intense, and, in particular, the candidate proposing a smaller government size becomes less moderate with respect to the peak of the median.

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