off-the-peak preferences and the outcome of elections francisco martínez-mora university of...

Off-the-peak preferences and the outcome of elections

Francisco Martínez-MoraUniversity of Leicester and FEDEA

&M. Socorro Puy

Universidad de Málaga

Moscow, 22nd of July, 2010

1. Introduction

Some models of two-party political competition (e.g. the downsian model) predict the convergence of policy platforms at the median voter's peak, a very sharp prediction for a very complicated problem.

In that context, if preferences are single-peaked, a unique aspect of the preference relation of voters is relevant for the outcome of elections: the location of the median's bliss point.

Platform convergence is, notwithstanding, at odds with the empirical evidence and with the complexity of real-world political competition.

Accordingly, in recent decades, a number of articles have put forward models where platform convergence does not characterize equilibrium.

1. Introduction

Political equilibria with separation of candidates:

Calvert (1985), Wittman (1983): differentiated platforms result in a model with policy-motivated candidates who have uncertainty on the location of the median voter.

Caplin & Nalebuff (1997): Membership-based equilibrium (no member of a party can

improve by changing to the rival party); Two internally consistent parties (the platform of each party is

located at the the median of its members’ bliss points.

Palfrey (1984), Osborne & Slivinsky (1996), Besley & Coate (1996): Separation of platforms derives from the endogenous entry of candidates.

1. Introduction

Making predictions about the outcome of elections in that context is, of course, much less straightforward, as it depends on the interplay between at least two elements:

1) Parties choice of platforms, which could in turn depend on within-party interactions among partisan and opportunistic members.

2) The median voter's preferred option between the proposed alternatives.

Without platform convergence, the median voter is no longer offered his most preferred option and, hence, must choose between policy proposals that do not fully satisfy her.

Hence, another aspect of the preference relation, the shape of the median voter's policy preferences off-the-peak becomes crucial.

1. Introduction

But, what do we know about the shape of policy preferences off-the-peak?

Models have usually assumed that policy preferences are symmetric about the peak, meaning that the voter in question is indifferent between symmetric deviations with respect to her peak.

Ansolabehere & Snyder (2000), Groseclose (2001), Aragonés & Palfrey (2002).

Palfrey (1984), Osborne & Slivinsky (1996).

Calvert (1985), Bernhardt, Duggan & Squintani (2007)

In this paper, we study the determinants of voters preferences off-the-peak, as well as the political consequences implied by different 'types' of preferences off-the-peak.

1. Introduction

Other economic contexts where off-the-peak(/target/reference point) preferences are relevant, and where our results may be of use:

Monetary policy with asymmetric preferences over deviations from the inflation target (Ruge-Murcia, 2003).

Fiscal response to foreign aid, where policy-makers have asymmetric preferences over deviations with respect to target spending or tax revenues (Heller, 1975; Feeny, 2006).

Prospect theory and loss aversion also introduces asymmetric preferences in the sense that agents are more sensitive to losses than to gains with respect to a reference point (Kahneman and Tversky, 1991).

1. Introduction

We define three types of preferences “off-the-peak”: symmetric, wasteful and scrooge.

We obtain that, across these types, systematic differences result in:

a raw measure of parties' electoral power, and (within the context of the citizen-candidate model) in

the location of equilibrium platforms, or the expected policy outcome.

A positive sign of the third derivative of the indirect utility function with respect to the policy variable indicates wasteful preferences, while a negative sign indicates scrooge preferences.

Roadmap

2 A general policy problem

3 Off-the-peak preferences, definitions

4 Off-the-peak preferences, results

5 Determinants of preferences off-the-peak

6 Conclusions.

2. A general policy problem

A jurisdiction populated by a continuum of households.

Households (indexed with i) differ by income yi, median income: ym, average (and total) income Y.

Two goods: Private consumption, xi, publicly expenditure, e.

Units of quality of the publicly provided good are normalized to be measured in units of the numeraire.


Preferences represented by a generic utility function U(xi, e) increasing in the both of its arguments.

Public expenditure financed through taxation; tax-bill function: τ(e,yi ).

The policy-induced utility function is thus given by:

V(e, yi) ≡ U(yi – τ(e,yi ),e)

Notation: V’, V’’, V’’’ denote the first, second and third derivative of the policy-induced utility function with respect to public spending.


Assumptions on preferences:

Single–Peaked SP: V has a unique maximizer (i.e. it is strictly quasi-concave in e).

Strict Single–Crossing SSC: V’ is strictly monotonic in y.

Monotonic–Satiation MS: V’’ is strictly monotonic in e.

Ideal policy of voter with income yi: epi = argmax V(e, yi)

e

3. Off-the-peak preferences: definitions

Consider a two-party (L and H) election and any pair of feasible electoral platforms: (eL, eH).

Restrict attention to cases where eL < epm < eH, and rewrite

platforms as deviations from the median's peak:

eL= epm - dL eH = ep

m + dH

Next, define the space of electoral platforms, as that of all feasible pairs of policy platforms in terms of deviations from ep

m:

dL

epm

Y

dH

dH

= dL

45º


Let the locus of indifference δ(dL, ym) provide, for each distance to the left of the peak ep

m, dL, the corresponding distance to the right of the peak that leaves households with median income indifferent.

It is implicitly defined by:

V(epm-d, ym) = V(ep

m+δ(d, ym), ym)

Points on the locus of indifference δ(dL, ym) constitute pairs of platforms that leave the median indifferent and, hence, under SSC, gather equal support from the electorate.


Depending on whether the median voter favours or not an excess of spending over a symmetric shortfall with respect to her peak, preferences will be defined as wasteful or scrooge.

e

Symmetric preferences: δ(d)=d

V

d d

ep-d ep ep+δ(d)

Wasteful preferences: δ(d)>d

e

V

ep-d ep ep+δ(d)

d d

ep-d ep+δ(d)Scrooge preferences: δ(d)<d

e

d d

V


Consider two different policy-induced utility functions Vi; i=1,2 satisfying SP and MS, and corresponding to households with income yi, (y1≠ y2), and with peaks ep

i.

Suppose V1 and V2 are scrooge (δ(d)<d): we say that V2 is more scrooge than V1 when:

δ2(d) < δ1(d) 0<d ≤ minep1,e

p2

Suppose V1 and V2 are wasteful (δ(d)>d): we say that V2 is more wasteful than V1 when:

δ2(d) > δ1(d) 0<d ≤ min ep1,e

p2 .

3. Off-the-peak preferences: results

Proposition 1: Under SP and SSC

The space of electoral platforms is partitioned by the locus of indifference into two regions.

The region above the locus of indifference δ(dL, ym) contains pairs of platforms such that the median (and a majority of the population) favours the proposal of party L.

The region below the locus of indifference contains pairs of platforms such that the median (and a majority of the population) favours the proposal of party H.

dL

epm

Y

dH

δ(dL, ym)

L wins

H wins


Define the relative size of the area of the space of electoral platforms within which a certain party wins as a raw measure of its electoral power.

Proposition 2:

Electoral power of party L is greater (smaller) the flatter (steeper) the slope of the locus of indifference. This is defined by:

That is to say, the electoral advantage of one or the other party hinges on the relative speed at which the median's utility falls at each side of her peak.

)y,(eV')yd,-(eV'

mPm

mPm

mLd y,d'


Corollary 1: Under SP, preferences are symmetric iff :

V’(epi – d,yi) = - V’(ep

i + d, yi) d.

That is to say, symmetry of policy preferences is a very restrictive assumption, as it requires V’ to be linear or a farfetched shape (More on this later on).

To derive the next results, we impose MS:

Proposition 3: Under SP and MS,

i) preferences are wasteful iff V’’’> 0 (or V’ strictly convex),

ii) preferences are scrooge iff V’’’< 0 (or V’ strictly concave). .


e

V , V ’

d d

Utility loss from a defeat of d over the peak

Utility loss from an excess of d over the peak

V ’

ep


We next show an analogy with the theories developed by Arrow (1971), Pratt (1964) and Kimball (1990):

According to Arrow-Pratt's theory of risk aversion, concavity of a utility function over consumption indicates the presence of risk aversion.

According to Kimball's 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 the stregth of the precautionary motive for saving.

Here,we prove that, under MS and SP, the curvature of the marginal policy-induced utility function determines the degree of scroogeness or wastefulness of preferences and the electoral power of each party.


Proposition 4: Let V1; V2 be two policy-induced utility functions satisfying strict concavity in e, and MS. Let Δ=ep

2-ep

1:

If V1; V2 are scrooge and:

then V2 is more scrooge than V1.

If V1; V2 are wasteful and:

then V2 is more wasteful than V1.

P2

P1

11

11

22

22 2e,2emine )y(e,''V)y(e,'''V

)y(e,''V)y(e,'''V

0;

P2

P1

11

11

22

22 2e,2emine )y(e,''V)y(e,'''V

)y(e,''V)y(e,'''V

0;

4. An application: two-candidate equilibria in citizen-candidate model

Citizen-Candidate model (Besley &Coate, 1997; Osborne &Slivinsky, 1996).

Each household has a vote.

Three stages:

Stage 1: Each voter strategically decides whether or not to become candidate (if a voter becomes candidate, her platform is her ideal policy).

c = cost of becoming candidate; b = benefits of holding office.

Stage 2: Voters vote sincerely. Plurality rule. Stage 3: The implemented policy is the platform of the winning

candidate. If there is a tie, each candidate wins with probability a half.

4. An application: two-candidate equilibria in citizen-candidate model

A two-candidate equilibrium is a pair (eL, eH) such that:

i) no candidate improves by withdrawing from the contest, ii) no voter improves becoming candidate.

Proposition 5: (location of platforms). Under SP and SSC:

1) (eL, eH) are located symmetrically around epm when

preferences of the median are symmetric;

2) eL is more moderate with respect to epm than eL when

preferences of the median are wasteful;

3) eH is more moderate than eH if the median’s preferences are scrooge.

5. Determinants of preferencesoff-the-peak

Consider the case of quasi-linear preferences:

V(e,yi) ≡ u(yi – τ(e,yi )) + e.

Third derivative of the policy-induced utility function:

V’’’= – u’’’(τ’)3 +3u’’ τ’ τ’’ – u’ τ’’’

Factors that influence the shape of preferences over e:

Risk aversion, u’’ < 0; CARA, DARA or CRRA (or positive coefficient of prudence) u’’’>0.

Government effectiveness: Constant effectiveness τ’’ = 0; decreasing effectiveness τ’’ > 0; τ’’’ > 0 (due to decreasing returns to scale, or to increasing corruption).

5. Determinants of preferencesoff-the-peak

Remark 1: Uncommon conditions for symmetric preferences.

Remark 2: Government decreasing effectiveness induce scrooge preferences.

Remark 3: Risk averse households with CARA, DARA or CRRA induce scrooge preferences.

u’’= 0 u’’< 0u’’’= 0

u’’< 0u’’’> 0

τ’’ = 0 Symmetric Scrooge

τ’’ > 0τ’’’ = 0

Symmetric Scrooge Scrooge

τ’’ > 0τ’’’ > 0

Scrooge Scrooge Scrooge

6. Concluding remarks

We have shown that, whenever political equilibrium is characterised by separation of platforms, the shape of voters preferences off-the-peak has a relevant impact on the outcome of two-party electoral competition.

Taken as given parties' platforms, whether preferences are symmetric, wasteful or scrooge, determines the electoral advantage of each party.

For instance, wasteful preferences imply that the party offering higher spending has an advantage in the sense that its platform can be located at a greater distance from the median's peak and still win.

The more wasteful are preferences, the greater this advantage for party H.

In the context of the citizen-candidate model, we showed that there is an impact as well in the choice of platforms: when the median has scrooge preferences the platform of the L party is less moderate than its counterpart, and the expected policy is below the median's peak.


Symmetric preferences require very stringent conditions in economic policy problems.

The ratio -V'''/V’’ measures the degree of scroogeness (if negative) or wastefulness (if positive) of specific policy preferences.

Scrooge preferences emerge from:

CARA, DARA, CRRA specifications of risk (or a positive coefficient of prudence).

Government decreasing effectiveness (due to tax distortions, corruption…).


On-going research:

Off-the-peak party preferences and the choice of platforms.

The design of the tax system and the probability of reelection of an incumbent party.

off-the-peak preferences and the outcome of elections francisco martínez-mora university of...

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