fedea

Fundación de Estudios de Economía Aplicada

Asymmetric single-peaked preferences and their utility representation by Francisco Martínez-Mora M. Socorro Puy ** Documento de Trabajo 2011-03

SERIE: Capital Humano y Empleo CÁTEDRA Fedea-Santander

January 2011

* University of Leicester and FEDEA. ** Universidad de Málaga.

Francisco MartÌnez-Moray University of Leicester and FEDEA

M. Socorro Puyz Universidad de M·laga

June 10, 2011

Abstract

The symmetry of single-peaked preferences is a widely used assumption. When the space of alternatives has a meaningful metric, such restriction is only justiÖed for its analytical convenience. This paper analyzes how to deal with asymmetric preferences in a tractable way. First, we introduce two types of asymmetric preferences (shortfall and excess avoidance), provide su¢cient conditions for preferences to be of one type or the other and a coe¢ cient that measures the degree of asymmetry. Second, we deÖne the family of generalized distancemetric utility functions that represents any asymmetric preferences maintaining analytical tractability and we compare it to previous proposals.

Key-words: Single-peaked preferences, asymmetric preferences, quadratic preferences, linex loss function, prudence. JEL classiÖcation numbers: D72, H31, H5.

The authors thank Subir Bose, Luis CorchÛn, Gianni De Fraja, Juanjo Dolado, Mark Fey, Sidartha Gordon, Michel Le Breton, Miltos Makris, Jordi MassÛ, Antonio NicolÛ, Clara Ponsati, Ludovic Renou, Luigi Siciliani, Al Slivinsky and, especially, John Duggan for their helpful comments. Financial assistance from Ministerio de Ciencia e InnovaciÛn under the project SECO2008-03674/ECON is gratefully acknowledged.

yE-mail: fmm14@leicester.ac.uk. Website: http://www.le.ac.uk/ec/fmm14/

zE-mail: mps@uma.es. Website: http://sites.google.com/site/socorropuy/

1 Introduction

Models of political economics commonly assume that preferences of voters and policy-makers are single-peaked with respect to a linear order of the policy alternatives. In other words, they assume that agents have a single bliss point and that deviations from it in any direction generate increasing utility losses. Single-peaked preferences have usually been restricted to be symmetric about the bliss point. That is to say, utility losses from symmetric deviations above and below the peak have been assumed identical. In fact, endowing agents with quadratic or euclidean preferences over policy alternatives is the standard assumption in models of political competition, monetary policy making or in the literature on Öscal response to foreign-aid.¹ That is not problematic when the space of policy alternatives does not have a speciÖc and meaningful metric. In other cases, for example when the policy alternatives correspond to levels of spending, taxation or interest rates, analytical convenience is the only justiÖcation for the symmetry assumption.

There are indeed important reasons that call such simpliÖcation into question in the context of economic policy analysis. On the one hand, it has been shown that symmetric single-peaked preferences cannot be deduced from a standard utility-maximization problem with a well-behaved primitive utility function and a linear constraint (Milyo, 2000). On the other hand, an increasing body of theoretical and empirical literature points towards the relevance of relaxing the symmetry restriction in a number of research areas. In the context of monetary policy, Blinder (1997) notes that: "Academic macroeconomists tend to use quadratic loss functions for reason ofmathematical convenience, without thinking much about their substantive implications. The assumption is not innocuous, [...] practical central bankers and academics would beneÖt from more serious thinking about the functional form of the loss function".

This paper o§ers a formal analysis of asymmetric single-peaked preferences. We believe that our results will be useful in the analysis of economic policy-making and, in general, in any setting where the policy space has a speciÖc metric and single-peakedness is an adequate modelling assumption. Within the family of single-peaked preferences, we introduce two basic types of asymmetric preferences: those that show shortfall avoidance and those displaying excess avoidance.² We say that preferences exhibit shortfall avoidance when they favour alternatives above the peak over their symmetric counterparts; in the opposite case, we say that they show excess avoidance.

¹Among the political economy models that consider symmetric preferences: Enelow and Hinich (1982), Ansolabehere and Snyder (2000), Groseclose (2001), AragonÈs and Palfrey (2002) who analyze the e§ect of voters preferences on candidatesípersonal characteristics; Palfrey (1983), Osborne and Slivinsky (1996) who study the the strategic entry of candidates; Calvert (1985), Bernhardt, Duggan and Squintani (2007) who account for candidatesíuncertainty about the location of the median voter.

Our contribution is double-fold. On the one hand, we provide su¢cient conditions for the underlying preferences to show shortfall avoidance or excess avoidance, and identify others that allow the comparison of degrees of asymmetry. Interestingly, we Önd that the coe¢ cient of prudence proposed by Kimball (deÖned by the third derivative divided by the second derivative of the utility speciÖcation) provides a measure of the asymmetry of preferences. On the other hand, we deÖne a rich family of utility functions, the generalized distance-metric utility functions, that can not only represent the two basic types of preference-bias, accommodating any degree of asymmetry, but also more complex asymmetries.³ Most importantly, our speciÖcation inherits the analytical tractability of the distance-functions, including smoothness, for which we believe it could be fruitfully exploited in future theoretical and empirical research. We then compare our proposal to other utility functions that have been used in the literature to represent asymmetric preferences. In particular, we consider the linex function proposed by Varian (1974), and further studied by Zellner (1986), and the piecewise asymmetric function proposed by Waud (1976). We show that, while these proposals are convenient in that the direction and degree of asymmetry depend on a single parameter, they are limited with respect to the asymmetries they can represent.

There are a number of research areas where asymmetric preferences have been found to play critical roles. For example, following Blinderís suggestion, Ruge-Murcia (2003) develops a theoretical model of monetary policy-making which endows the central bank with an asymmetric loss function and Önds that some important results (such as the linear relation between unemployment and the average ináation deviation from the target) do not extend to the asymmetric case. Moreover, he provides empirical evidence supporting the existence of a non-symmetric objective function. Along similar lines, Dolado et al. (2004) show that asymmetric preferences are theoretically important, as they imply the existence of a non-linear monetary policy rule, and provide empirical evidence of such non-linearities in the US. Exploiting the same implication of asymmetric preferences, Surico (2007) o§ers evidence supporting that the preferences of the Fed were asymmetric with respect to target values before 1979, when output contractions triggered stronger responses than output expansions. Surico stated that " [...] potential evidence of asymmetries in the central bank objective may be interpreted as evidence of asymmetries in the representative agentís utility."

²This terminology is adapted from the one introduced by Cukierman and Muscatelli (2008). These authors refer to ináation avoidance and recession avoidance preferences.

³For instance, our proposal can capture cases where preferences display excess avoidance in some parts of the policy space and shortfall avoidance in others.

Likewise, within the literature on Öscal response to foreign-aid, Heller (1975) and Feeny (2006) highlight the relevance of policymakersíasymmetric single-peaked preferences over deviations with respect to target spending or tax revenues. We conclude this brief and non-exhaustive review of the literature by recalling the work of Waud (1976) who, in a study of economic policy-making under uncertainty, revealed that the optimal response of the policy-maker to changes in uncertainty crucially hinges on the symmetry or asymmetry of preferences.⁴

The remaining of the paper is organized as follows. The next section presents the environment and deÖnitions. Section 3 studies conditions on the utility representation that guarantee the emergence of asymmetric singlepeaked preferences and compares preferences in terms of their degrees of asymmetry. Section 4 proposes the generalized distance-metric utility representation. Section 5 analyzes other utility speciÖcations. Section 6 concludes.

2 The environment and deÖnitions

An agent has preferences deÖned over alternatives in R.⁵ The space of alternatives we consider has a speciÖc and meaningful metric; thus, it is not possible to alter the spatial location of the alternatives.⁶ The preference R of the agent on the set of alternatives is a complete preorder. The set of complete preorders on the set of alternatives is .

⁴Prospect theory also accounts for asymmetric preferences, though in a di§erent sense. Following empirical and experimental evidence, this literature assumes that agents are more sensitive to losses than to gains with respect to a reference point (Kahneman and Tversky, 1991; Benartzi and Thaer 1995). In contrast to our analysis, the reference point is not a maximizer.

⁵ All our results extend to cases where the set of alternatives is a subset of R. See Appendix.

⁶See Eguia (2010) for an analysis that endogeneizes the spatial representation of the set of alternatives.

The strict and indi§erence preference relations induced by R are denoted by and I respectively. Given , the peak of when it exists, is an alternative strictly preferred to any other alternative. Let denote the peak of :

DeÖnition 1 A preference satisÖes single-peakedness (SP) if there exists a peak of R; and for all such that and

The SP property of preferences requires that, at each side of the peak, alternatives located closer to the peak are preferred to those located further away from it. The set of complete preorders satisfying on the set of alternatives is denoted by . We refer to a generic element of as a SP preference.

Every alternative below the peak can be associated to another alternative above the peak according to the indi§erence preference relation. In what follows, we interpret as a deviation o§ the peak. We next deÖne a function that assigns to every deviation below the peak, a deviation above the peak according to the indi§erence preference relation.

DeÖnition 2 The preference-bias function associated to assigns to every deviation the corresponding deviation for which

Observe that every preference-bias function associated to a SP preference is a strictly increasing function. We propose three basic types of SP preferences. These are the symmetric SP preferences, the shortfall avoidance preferences and the excess avoidance preferences.

DeÖnition 3 We say that the preference is symmetric when d for all d. We say that the preference shows shortfall avoidance when for all d: We say that the preference shows excess avoidance when for all d.

Symmetric SP preferences induce indi§erence between alternatives located symmetrically around the peak. When a preference shows shortfall avoidance, the comparison between two alternatives symmetrically located at each side of the peak is such that the alternative located above the peak is higher in the preference ordering than the alternative located below the peak. Of course, it is the opposite when a preference shows excess avoidance. The proposed types of SP preferences do not fully classify the set . Our aim, however, is to capture two natural and meaningful types of asymmetries in the set of SP preferences.⁷

Because the preference-bias function is independent of the location of the peak, one can compare degrees of asymmetry between pairs of preferences even when their respective peaks do not coincide. The following deÖnition establishes the binary relation more avoidance than on

DeÖnition 4 Let with and denoting their respective preference-bias functions and peaks. We say that shows more shortfall avoidance than when for all d (or equivalently, shows more excess avoidance than

Thus, according to this deÖnition, degrees of asymmetry are comparable across pairs of preference relations when their associated preference-bias functions are such that one is always above the other, i.e., for all Observe, therefore, that the proposed binary relations are partial preorders (transitive but not complete) on the sets of SP preferences which show shortfall avoidance or excess avoidance.⁸

3 Conditions on the utility representation

Every preference can be represented by a strictly quasi-concave utility function which has a maximizer. The maximizer (or peak of satisÖes:

\[e ^ {*} = \arg \max V (e)\tag{1}\]

In what follows, we consider di§erentiable utility representations so that the peak of R satisÖes

SP

.

⁷In particular, these deÖnitions do not cover preferences which show shortfall avoidance in some ranges of the domain and excess avoidance in others. It would of course be possible to generalize them to account for any SP preference in

⁸Again, the deÖnition could be generalized to compare degrees of asymmetry in speciÖc intervals of the domain of d.

3.1 Conditions for asymmetric SP preferences

In terms of the utility representation, when the SP preference shows shortfall avoidance, then for all and when the preference shows excess avoidance, in turn, holds for all d. Figure 1 depicts two examples of utility functions representing shortfall avoidance and excess avoidance respectively.

For SP preferences, holds for alternatives below the peak, whereas is true for alternatives above the peak.

A characterization of symmetric single-peaked preferences in terms of the properties of the utility representation V is straightforward to establish. Preferences are symmetric if and only if for all That is to say, if and only if marginal utility at every pair of symmetric deviations above and below the peak coincide. Similarly, certain properties of the slope of guarantee that preferences show shortfall avoidance or excess avoidance.

Proposition 1 Let V be the utility representation of

Properties (1a) or (2a) guarantee that R shows shortfall avoidance:

for all d

(2a) V0 strictly convex for all

where

Properties guarantee that R shows excess avoidance:

for all d

(2b) V0 strictly concave for all

where

Proof. The following claims prove our statement.¹⁰

Claim 1: for all d:

Proof of Claim 1: Utility loss derived from reducing to is measured by , whereas utility loss from increasing to d is measured by : By (1a), for all Solving for the integral and simplifying for all d, i.e., preferences show shortfall avoidance.

⁹ This result is derived by di§erentiating with respect to d.

¹⁰The analogous statements on preferences showing excess avoidance can be proved following a similar reasoning (that we omit in the interest of brevity).

Claim 2:

Proof of Claim 2: By strict convexity of we have for all d. By SP, ; and substituting in the inequality yields for all d.

By conditions (1a) and (1b), the comparison of the slopes of the utility representation of SP preferences at every symmetric deviation with respect to the peak reveals the direction of the asymmetry. Conditions and (2b) also reveal the direction of the asymmetry by checking whether the marginal utility function is strictly concave or strictly conve x.¹¹

3.2 Conditions for comparing degrees of asymmetry

Figure 2 depicts two utility representations of preferences where (represented by shows more excess avoidance than (represented by . We put together both utility representations around their respective peaks in order to compare their associated preference-biased function.

Next, we show that the comparison between the slopes of di§erent utility representations indicates the strength of the asymmetry across di§erent preferences. For each pair of preferences ; we denote by their respective peaks.

Proposition 2 Let be two utility representations of respectively. If for all ; then shows more excess avoidance than equivalently, shows more shortfall avoidance than :

Proof. Condition for all d implies that the utility loss derived from reducing to which is measured by 2 in comparison to the utility loss derived from reducing to which is measured by ; is such that . By deÖnition of the preference-bias function, there is for which , at the same time, and given that for all

(2a)

(2b)

¹¹For a concrete application of conditions and see Cukierman and Muscatelli (2008) where central banksípreferences show excess avoidance over ináation and shortfall avoidance over output. As shown by these authors, this asymmetry of preferences is not innocuous as it induces a non-linear Taylor rule.

\[\int_ {e _ {2} ^ {*}} ^ {e _ {2} ^ {*} + \delta_ {1}} V _ {2} ^ {\prime} (e) d e \geq A.\tag{2}\]

By deÖnition of the preference-bias function there is for which

\[\int_ {e _ {2} ^ {*}} ^ {e _ {2} ^ {*} + \delta_ {2}} V _ {2} ^ {\prime} (e) d e = B.\tag{3}\]

Because ; conditions (2) and (3) imply from where we derive that for all d:

In other terms, the proposed su¢cient condition for to show more excess avoidance than implies that and , for all , below the peak, equal-distance deviations generate more disutility with than with , and above the peak, equal-distance deviations generate more disutility with than with

The degree of asymmetry can also be compared using the degree of concavity or convexity of the marginal utility function. For this, one of the marginal utility speciÖcations must be obtained as an increasing transformation of the other.

Proposition 3 Let be strictly concave representations of are strictly concave and for all where , then shows more excess avoidance than

Proof. By strict concavity of and ; the functions and are strictly decreasing functions and they can be related by a strictly increasing transformation such that where . This implies that when Di§erentiating the expression,

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

V0₁ ; V 0₂

e R,

R₂

R₁.

¹² Similarly, if are strictly convex and for all then shows more shortfall avoidance than

13If e2 = e1

^e1 ^{= 0;} − e*

^{V0001 (e)}_V00(e) to V1"(e) V1(e)

V000(e) ²_V00(e) . V2′(e)

¹³ If e₂ = e₁, then = e₂ and it is possible to compare degrees of asymmetry just by comparing

From where

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

By strict concavity of and and . Then, implies concave). By deÖnition of ; it follows that ; or equivalently, : Substituting function

\[0 = \int_ {e _ {2} ^ {*} - d} ^ {e _ {2} ^ {*} + \delta_ {2} (d)} V _ {2} ^ {\prime} (e) d e = \int_ {e _ {2} ^ {*} - d} ^ {e _ {2} ^ {*} + \delta_ {2} (d)} g (V _ {1} ^ {\prime} (e - \Lambda)) d e.\]

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

This proposition reveals an analogy between our analysis and the theories developed by Arrow (1971), Pratt (1964) and Kimball (1990). In Arrow-Prattís theory of risk aversion, concavity of the utility function over consumption indicates the presence of risk aversion, while according to Kimballís theory of precautionary savings, concavity of the marginal utility function entails precautionary saving behavior. In each case, the degree of concavity of the utility function or the degree of concavity of the marginal utility function measures risk aversion or precautionary savings respectively.¹⁴ These behavioral traits become thus comparable across pairs of concave functions such that one is a concave transformation of the other. In our context, as long as is strictly concave, the curvature of the marginal utility function determines the degree of asymmetry. We can therefore apply the coe¢ cient of prudence proposed by Kimball (1990) to measure whether preferences show more shortfall avoidance for the more convex marginal utility representation, and more excess avoidance for the more concave marginal utility representation.

u000 u′" u11 u00

¹⁴ Arrow-Prattís coe¢ cient of risk aversion is deÖned by , whereas Kimballís coe¢ - cient of prudence is deÖned by where u measures utility over private consumption.

DeÖnition 5 Let V be strictly concave utility representation of is strictly concave, we refer to the ratio as the coe¢cient of excess avoidance. If is strictly convex, we refer to the ratio as the coe¢cient of shortfall avoidance.

The proposed coe¢cients of asymmetry can be classiÖed into three types: constant, increasing or decreasing in Thus, we say that V satisÖes constant excess avoidance when is constant for all , and we say that satisÖes increasing (decreasing) excess avoidance when it is increasing (decreasing) in Likewise, we say that V satisÖes constant shortfall avoidance when is constant for all ; and we say that V satisÖes increasing (decreasing) shortfall avoidance when it is increasing (decreasing) in e.

4 Asymmetric SP utility representation

In this section, we show that a generalization of any distance-metric utility function allows for the utility representation of any asymmetric SP preference relation.

A distance-metric utility function is deÖned by where is a continuous and strictly increasing distance function between the peak and the alternative e. Particular examples of are the quadratic function, in which ; or the distance function induced by a norm, in which ; or any function where

Given a preference , the preference-bias function associated to R assigns to each deviation below the peak, the deviation above the peak for which the agent is indi§erent. Let be the inverse of the preference-bias function.¹⁵ The generalized distance-metric utility function is deÖned by

\[V (e) = \left\{ \begin{array}{c l} - f (e - e ^ {*}) & \text {when} e \leq e ^ {*} \\ - f (\delta^ {- 1} (e - e ^ {*})) & \text {when} e > e ^ {*} \end{array} \right..\]

Below the peak, the proposed utility function coincides with the distancemetric utility function. In order to capture the preference-bias, the level of utility derived from any alternative above the peak is equal to the corresponding distance-metric utility value at its indi§erent alternative below the peak. If the SP preference relation is symmetric, i.e., the generalized distance-metric utility function collapses to the distance-metric utility function. In the Appendix we specify the generalized distance-metric utility function for the case in which the set of alternatives is bounded.

SP

8

¹⁵Because the preference satisÖes the SP condition, the preference-bias function is biyective and it has an inverse.

Arguably, an advantage of the generalized distance-metric utility function is that it is derived from the set of indi§erent alternatives of the primitive preference relation. Another is that it maintains the tractability of the distance functions (including smoothness if the distance function is smooth). Moreover, it can accommodate every SP preference, including those which satisfy our deÖnition of shortfall and excess avoidance.

Theorem 1: Every preference can be represented by the generalized distance-metric utility function. Furthermore, this utility speciÖcation can be used to compare pairs of preferences such that one of them shows more shortfall avoidance (or less excess avoidance) than the other.

Proof. First, we show that every preference relation is represented by the generalized distance-metric utility function. The preference ordering across alternatives located at the same side of the peak is captured by the distance-metric utility function below the peak, and above the peak, by a function that is strictly decreasing in distance (given that is a strictly increasing function in all its domain). The preference ordering of pairs of alternatives located at opposite sides of the peak can be deduced by identifying the pairs of alternatives yielding equal utility. Thus, and simplifying : Since , we deduce that for all d. Second, we show that the generalized distance-metric utility function can be used to compare degrees of asymmetries across pairs of preferences. Suppose that for all Then, because is always a strictly increasing function, implies that . Plugging this inequality into the generalized distance-metric utility function we obtain that for all In addition, for all We deduce, using Proposition 2, that shows more shortfall avoidance than .

For instance, if we take a linear preference bias-function with

¹⁶Observe that, o§ the peak, all the proposed distance functions are di§erentiable.

; the corresponding generalized distance-metric utility function is

\[V (e) = \left\{ \begin{array}{c l} - f (e - e ^ {*}) & \text {when} e \leq e ^ {*} \\ - f (\frac {e - e ^ {*}}{k}) & \text {when} e > e ^ {*} \end{array} \right..\]

Under this utility speciÖcation, represents a particular class of preferences that show shortfall avoidance, while corresponds to another class of SP preferences that exhibit excess avoidance.

Of course, the preference-bias function does not need to be linear; we can think of diverse functional forms yielding di§erent asymmetric SP preferences. Thus, it is possible to represent more complex asymmetries than the basic types previously deÖned. For instance, we can represent preferences that display excess avoidance for some values of and shortfall avoidance for others.¹⁷

Our proposal accommodates every continuous and strictly increasing distance metric function. For instance, a rich family of utility speciÖcations are given by:

\[V (e) = \left\{ \begin{array}{c l} - | e - e ^ {*} | ^ {\lambda} & \text {when} e \leq e ^ {*} \\ - \left| \delta^ {- 1} (e - e ^ {*}) \right| ^ {\lambda} & \text {when} e > e ^ {*} \end{array} \right.\]

where di§erent values of yield di§erent utility functions. It is straightforward to check that these functions are smooth, i.e. that they are continuously di§erentiable in all the domain, including the peak.

5 Other asymmetric SP utility representations

In this section we analyze other utility representations of asymmetric SP preferences in the literature and compare them to our proposal. We study the linex loss function proposed by Varian (1974) and further studied by Zellner (1986), and a simple piecewise asymmetric function proposed by Waud (1976).

(d) = [d (1 a)],

a [0; 1), > 0

= 1;

< 1,

(d) d, d d

d

> 1,

(d) < d, d > d

¹⁷A preference-bias function that creates such kind of asymmetry is where and determines whether preferences exhibit excess avoidance near the peak and shortfall avoidance from some distance onwards, or viceversa. In particular, if then and so that preferences show shortfall avoidance below and excess avoidance above d. The opposite case occurs d* when which implies that , and . The threshold ^{(d) d} 8^{d d (d) > d} 8^{d > d} distance is given by ^{d = (1 a) 1} β

5.1 The linex loss function

The linex (for linear-exponential) loss function has been used to represent policy makersípreferences over ináation and output gap (e.g. Ruge-Murcia, 2003; Dolado et al, 2004; Surico, 2007) as well as in Bayesian econometrics (Zellner, 1986) and optimal forecasting (Christo§ersen and Diebold, 1997). According to our notation, this utility speciÖcation is deÖned by:

\[V (e) = - \exp (\alpha [ e ^ {*} - e ]) + \alpha [ e ^ {*} - e ] + 1\]

where . Observe that this speciÖcation yields

Solving for the Örst, second and third derivatives we obtain:

\[\begin{array}{r c l} {V ^ {\prime}} & = & {\alpha \exp (\alpha [ e ^ {*} - e ]) - \alpha} \\ {V ^ {\prime \prime}} & = & {- \alpha^ {2} \exp (\alpha [ e ^ {*} - e ])} \\ {V ^ {\prime \prime \prime}} & = & {\alpha^ {3} \exp (\alpha [ e ^ {*} - e ])} \end{array}\]

The second derivative guarantees concavity of the utility speciÖcation. By Proposition 1, the sign of the third derivative implies that the linex utility function represents shortfall avoidance when and excess avoidance when : Additionally, using Proposition 3, we can obtain the coe¢cient of asymmetry, which is equal to when there is excess avoidance and to when there is shortfall avoidance. Note that the coe¢cient of asymmetry of the linex function is constant.

Therefore, according to our results, the linex function is able to represent preferences with the two types of asymmetry (shortfall and excess avoidance). Moreover, given that its coe¢cient of asymmetry is constant, it is easy to represent and compare di§erent degrees of asymmetry just by changing or comparing the value of . This utility representation, however, cannot represent every preference . For instance, it cannot represent preferences which show excess avoidance for some values of d; and shortfall avoidance for others. Likewise, it cannot represent preferences with increasing or decreasing avoidance.

5.2 The piecewise asymmetric function of Waud

Waud (1976) proposes another utility representation of asymmetric preferences. Using our notation, this utility speciÖcation is deÖned by:

\[V (e) = \left\{ \begin{array}{l l} - \lambda f (e ^ {*} - e) & \mathrm{when} e \leq e ^ {*} \\ - f (e - e ^ {*}) & \mathrm{when} e > e ^ {*} \end{array} \right.\tag{4}\]

where f is a strictly increasing and convex function, and Observe that represents preferences showing shortfall avoidance, while corresponds to preferences showing excess avoidance. Additionally, by Proposition when , the higher the coe¢cient ; the more excess avoidance the preferences show (likewise, when , the smaller the coe¢ cient ; the more shortfall avoidance the preferences display). Thus, the function proposed by Waud can represent preferences of the two basic asymmetric types (shortfall and excess avoidance). Furthermore, it allows for comparisons of degrees of asymmetry just by checking the value of . Nevertheless, this utility representation cannot represent every preference

By the deÖnition of the preference-bias function, for all which implies for all Given a convex function that condition imposes a restriction on the preference-bias function and thereby on the preferences it can represent. To illustrate such restriction, let be the quadratic function; then, the preference-bias function must satisfy i.e., has to be linear in This example illustrates that this utility speciÖcation cannot represent every preference : Additionally, there are preference-bias functions for which no function exists such that Expression (4) represents these preferences (consider again the case of preferences which show excess avoidance for some values of and shortfall avoidance for others).

6 Conclusion

Symmetric single-peaked preferences are usually represented by distancemetric utility functions. Our analysis revealed that such utility speciÖcations can be easily transformed to accommodate single-peaked preferences with any direction and degree of asymmetry, including the two basic types (shortfall and excess avoidance) as well as more complex asymmetries. Our utility speciÖcation, which we named the generalized distance-metric utility function, maintains the analytical tractability and smoothness of the distance functions. Another advantage is that it is directly derived from the set of indi§erent alternatives of the primitive preference relation. Our proposal can be used to capture the preferences of policy-makers, investors, politicians, the media or, in general, in any context where the policy space has a meaningful metric and where single-peakedness is a relevant modelling assumption. This is for instance the case in the models studied by Blinder (1997), Ruge-Murcia (2003), Feeny (2006) or Dolado et al. (2004) among others. We believe our proposal could also be fruitfully exploited in Bayesian econometrics and in optimal forecasting, as it would allow to incorporate richer asymmetries in the loss function.

On the other hand, we identiÖed su¢cient conditions on standard utility representations that reveal the direction (if any) of the asymmetry of preferences and another that allows the comparison of degrees of asymmetry across di§erent preference relations. We found that an analogous of Kimballís coe¢cient of prudence can be used to measure the degree of asymmetry of preferences and, hence, to compare degrees of shortfall or excess avoidance. Finally, we analyzed two proposals of asymmetric utility representations ñ the linex function and the piecewise asymmetric function of Waud (1976). We explained that the sign and degree of asymmetry of these utility speci-Öcations depend on a single parameter, for which they are able to represent preferences exhibiting di§erent degrees of shortfall or excess avoidance. For the same reason and in contrast to our proposal, however, these utility (or loss) functions are limited in the range of single-peaked preferences they can represent.

Appendix

If the set of alternatives were bounded, as for instance it is the case when ; we could also deÖne the distance-metric utility function. For that, suppose that we take e su¢ciently large as to guarantee that there exists an alternative such that i.e., such that the agent is indi§erent between 0 and e~. This implies that : The preferencebias function is then deÖned so that The inverse of the preference-bias function is then bounded above by ， which corresponds to the indi§erence relation We can extend the domain of to every in a strictly increasing way. Thus, the generalized distance-metric utility function is deÖned by

\[V (e) = \left\{ \begin{array}{c l} - f (e - e ^ {*}) & \text {when} e \in [ 0, e ^ {*} ] \\ - f (\delta^ {- 1} (e - e ^ {*})) & \text {when} e \in (e ^ {*}, \overline {{e}} ] \end{array} \right..\]

¹⁸The assumption that 0 is made only to simplify the exposition. Alternatively, if there exists an alternative such that then the preference-bias function is deÖned on band the analysis remains unchanged.

Observe that the utility derived from alternatives is below and that the utility function is strictly decreasing in . If we take a linear preference bias-function with ; the corresponding generalized distance-metric utility function would be

\[V (e) = \left\{ \begin{array}{c l} - f (e - e ^ {*}) & \text {when} e \in [ 0, e ^ {*} ] \\ - f (\frac {e - e ^ {*}}{k}) & \text {when} e \in (e ^ {*}, \overline {{e}} ] \end{array} \right..\]

Note that this function extends, in a natural way, the domain of to alternatives in the interval

References

1. [1] Ansolabehere S. and J.M. Snyder (2000), Valence Politics and Equilibrium in Spatial Election Models, Public Choice 103: 327ñ336.
2. [2] AragonÈs E. and T. R. Palfrey (2002), Mixed Equilibrium in a Downsian Model with a Favored Candidate, Journal of Economic Theory 103: 131- 161.
3. [3] Arrow K. (1971), Essays in the Theory of Risk-Bearing. Chicago, IL: Markham Publishing Company.
4. [4] Benartzi S. and R.H. Thaler (1995), Myopic Loss Aversion and the Equity Premium Puzzle, Quarterly Journal of Economics 110: 73-92.
5. [5] Bernhardt D., J. Duggan and F. Squintani (2009), The Case of Responsible Parties, American Political Science Review 103: 570ñ587.
6. [6] Blinder A. (1997), Distinguished lecture on economics and government: what central bankers could learn from academics and vice versa, Journal of Economic Perspectives 11: 3ñ19.
7. [7] Calvert R. (1985), Robustness of the Multidimensional Voting Model: Candidate Motivations, Uncertainty, and Convergence, American Journal of Political Science 29: 69-95.
8. [8] Christo§ersen, P. F. and F. X. Diebold (1997), Optimal Prediction under Asymmetric Loss, Econometric theory 13: 808-817.
9. [9] Cukierman A. and A. Muscatelli (2008), Nonlinear Taylor Rules and Asymmetric Preferences in Central Banking: Evidence from the United Kingdom and the United States, The B.E. Journal of Macroeconomics 8, Article 7.
10. [10] Dolado, J. J., R. Maria-Dolores and F. Ruge-Murcia (2004), Nonlinear Monetary Policy Rules: Some New Evidence for the U.S, Studies in Nonlinear Dynamics and Econometrics 8: 1-32.
11. [11] Eeckhout L. and H. Schlesinger (2006), Putting Risk in Its Proper Place, American Economic Review 96: 280-289.
12. [12] Eguia J. X. (2011), Foundations of Spatial Preferences, Journal of Mathematical Economics 47: 200-205.
13. [13] Enelow J. M. and M. J. Hinich (1982), Nonspatial Candidate Characteristics and Electoral Competition, Journal of Politics 44: 115-130.
14. [14] Feeny S. (2006), Policy Preferences in Fiscal Response Studies, Journal of International Development 18: 1167-1175.
15. [15] Groseclose T. (2001), A Model of Candidate Location When One Candidate Has a Valence Advantage, American Journal of Political Science 45: 862-886.
16. [16] Heller P. S. (1975), A model of Public Fiscal Behavior in Developing Countries: Aid, Investment, and Taxation, American Economic Review 65: 429-445.
17. [17] Kahneman D. and A. Tversky (1991), Loss Aversion and Riskless Choice: a Reference Dependent Model, Quarterly Journal of Economics 106: 1039-1061.
18. [18] Kimball M. S. (1990), Precautionary Saving in the Small and in the Large, Econometrica 58: 53-73.
19. [19] Milyo J. (2000), A Problem with Euclidean Preferences in Spatial Models of Politics, Economics Letters 66, 179-182.
20. [20] Osborne M. J. and A. Slivinski (1996), A Model of Political Competition with Citizen-Candidates, Quarterly Journal of Economics 111: 65-96.
21. [21] Palfrey T. R. (1984), Spatial Equilibrium with Entry, Review of Economic Studies 51: 139-156.
22. [22] Pratt J. W. (1964), Risk Aversion in the Small and in the Large, Econometrica 32: 122-136.
23. [23] Pratt J. W. and R. J. Zeckhauser (1987) Proper Risk Aversion, Econometrica 55: 143-154.
24. [24] Ruge-Murcia F. J. (2003), Ináation Targeting under Asymmetric Preferences, Journal of Money, Credit and Banking 35: 763-785.
25. [25] Surico P. (2007), The Fedís Monetary Policy Rule and U.S. Ináation: The Case of Asymmetric Preferences, Journal of Economic Dynamics and Control 31: 305-324.
26. [26] Varian, H. (1974), A Bayesian Approach to Real State Assessment, in Feinberg, F. E. and Zellner, A. (eds.) Studies in Bayesian Economics in honour of L. J. Savage. North-Holland. Amsterdam.
27. [27] Waud, R. N. (1976), Asymmetric Policymaker Utility Functions and Optimal Policy Under Uncertainty, Econometrica 44: 53-66.
28. [28] Wittman D. (1983), Candidate Motivation: A Synthesis of Alternative Theories, American Political Science Review 77: 142-157.
29. [29] Zellner, A. (1986), Bayesian Estimation and Prediction using Asymmetric Loss Functions, Journal of the American Statistical Association 81: 446-451.
30. 2011-03: “Asymmetric Single-Peaked Preferences and their Utility Representation”, Francisco Martínez-Mora, M. Socorro Puy.
31. 2011-02: “Strategic Behaviour of Exporting and Importing Countries of a Non-Renewable Natural Resource: Taxation and Capturing Rents”, Emilio Cerdá Tena y Xiral López Otero.
32. 2011-01: “Politicians' Luck of the Draw: Evidence from the Spanish Christmas Lottery”, Manuel Bagues y Berta Esteve-Volart.
33. 2010-31: “Risk Aversion and the Effect of Family Background on Student Effort”, Pedro Landeras.
34. 2010-29: “Executive Pay with Observable Decision”, Marco Celentaniy, Rosa Loveira y Pablo Ruiz-Verdú.
35. 2010-28: “Incentives, Resources and the Organization of the School System”, Facundo Albornoz, Samuel Berlinski y Antonio Cabrales.
36. 2010-27: “Retirement Incentives, Individual Heterogeneity and Labour Transitions of Employed and Unemployed Workers”, J. Ignacio García Pérez, Sergi Jimenez-Martín y Alfonso R. Sánchez-Martín.
37. 2010-26: “Social Security and the Job Search Behavior of Workers Approaching Retirement”, J. Ignacio García Pérez y Alfonso R. Sánchez Martín.
38. 2010-25: “A Double Sample Selection Model for Unmet Needs, Formal Care and Informal Caregiving Hours of Dependent People in Spain”, Sergi Jiménez-Martín y Cristina Vilaplana Prieto.
39. 2010-24: “Health, Disability and Pathways into Retirement in Spain”, Pilar García-Gómez, Sergi Jiménez-Martín y Judit Vall Castelló.
40. 2010-23: Do we agree? Measuring the Cohesiveness of Preferences”, Jorge Alcalde-Unzu y Marc Vorsatz.
41. 2010-22: “The Weight of the Crisis: Evidence from Newborns in Argentina”, Carlos Bozzoli y Climent Quintana-Domeneque.
42. 2010-21: “Exclusive Content and the Next Generation Networks”·, Juan José Ganuza and María Fernanda Viecens.
43. 2010-20: “The Determinants of Success in Primary Education in Spain”, Brindusa Anghel y Antonio Cabrales.
44. 2010-19: “Explaining the Fall of the Skill Wage Premium in Spain”, Florentino Felgueroso, Manuel Hidalgo y Sergi Jiménez-Martín.
45. 2010-18: “Some Students are Bigger than Others, Some Students’ Peers are Bigger than Other Students’ Peers”, Toni Mora y Joan Gil.
46. 2010-17: “Electricity Generation Cost in Isolated System: the Complementarities of Natural Gas and Renewables in the Canary Islands”, Gustavo A. Marrero y Francisco Javier Ramos-Real.
47. 2010-16: “Killing by Lung Cancer or by Diabetes? The Trade-off Between Smoking and Obesity”, Federico A. Todeschini, José María Labeaga y Sergi Jiménez-Martín.
48. 2010-15: “Does Gender Matter for Academic Promotion? Evidence from a Randomized Natural Experiment”, Natalia Zinovyeva y Manuel F. Bagues.
49. 2010-14: “Spain, Japan, and the Dangers of Early Fiscal Tightening”, R. Anton Braun y Javier Díaz-Giménez.
50. 2010-13: “Competition and Horizontal Integration in Maritime Freight Transport”, Pedro Cantos-Sanchez, Rafael Moner-Colonques, Jose Sempere-Monerris y Óscar Alvarez.
51. 2010-12: “Eppur si Muove! Spain: Growing without a Model”, Michele Boldrin, J. Ignacio Conde-Ruiz y Javier Díaz Gimenez.
52. 2010-11: “The Spanish Business Bankruptcy Puzzle and the Crisis”, Marco Celentani, Miguel García-Posada y Fernando Gómez.
53. 2010-10: “Promoting Employment of Disabled Women in Spain; Evaluating a Policy”, Judit Vall Castello.
54. 2010-09: “The Role of Construction in the Housing Boom and Bust in Spain”, Carlos Garriga.
55. 2010-08: “Did Good Cajas Extend Bad Loans? Governance, Human Capital and Loan Portfolios”, Vicente Cuñat y Luis Garicano.
56. 2010-07: “Unemployment and Temporary Jobs in the Crisis: Comparing France and Spain”, Samuel Bentolila, Pierre Cahuc, Juan J. Dolado y Thomas Le Barbanchon.
57. 2010-06: “La Subida del Impuesto Sobre el Valor Añadido en España: Demasiado Cara y Demasiado Pronto”, Juan Carlos Conesa , Javier Díaz-Giménez, Julián Díaz-Saavedra y Josep Pijoan-Mas.
58. 2010-05: “Off-the-peak Preferences Over Government Size”, Francisco Martínez-Mora y M. Socorro Puy.
59. 2010:04: “Green Shoots? Where, when and how?”, Maximo Camacho, Gabriel Perez-Quiros y Pilar Poncela.
60. 2010-03: “The Spanish Crisis from a Global Perspective”, Jesús Fernández-Villaverde y Lee Ohanian.