Median voter theorem - continuous choice€¦ · A is the Condorcet winner although voter 3’s...

Intro Prefs & Voting Electoral comp. Voter Turnout Agency GIP SIP Rent seeking Partisans

Median voter theorem - continuous choice

In most economic applications voters are asked to make anon-discrete choice - e.g. choosing taxes. In theseapplications the condition of single-peakedness is relatedto the curvature properties of the political preferencefunction. If it is quasi-concave in the political choicevariable (e.g. the tax rate), then the preference functionhas a "single peak" and the median voter theorem statedabove readily applies.It is important to note that the condition has to be satisfiedby the political preference function.

Median voter theorem - continuous choice

Is strict concavity (or more mildly quasi-concavity) ofpolitical preferences guaranteed by strict concavity (orquasi-concavity) of the underlying preference function(direct utility function)? Not necessarily! The reason is thatthe latter distinguishes from the direct utility function by thefact that is also reflects optimal individual choices (e.g.labor supply and savings decision) and general equilibriumeffects.

Continuous choice - An Example

ui = c + θib(g) with b′> 0 and b

′′< 0.

c = I − T (I (exogenously given))θi measures the preference for public consumption g.Taxes finance public consumption. With a continuum ofindividuals whose size is normalized at unity the publicbudget constraint reads g = T . The political preferenceover the tax rate is ui = I − T + θib(T ).FOC: −1 + θib

′=0.

SOC: θib′′< 0(by strict concavity of b(g)). ⇒ Here, strict

concavity of the direct utility function ensures strictconcavity (and thus single-peakedness) of politicalpreferences.

Continuous choice - An Example - Modified

Individuals also derive utility from consuming leisure lui = c + h(l) + θib(g) with h′ > 0 and h′′ < 0.Time endowment is normalized to unity. L + l = 1.Private consumption is c = w(1− t)L.Each individual chooses labor supply L = 1− l such thatui = w(1− t)L + h(1− L) + θib(g) is maximized taking thetax rate and the level of public consumption as given.FOC: w(1− t)− h′ = 0⇒ Labour supply functionL∗(w(1− t)). ((dL∗)/(dt)) = (w/(h′′)) < 0.

Continuous choice - An Example - Modified

The most preferred tax rate follows from maximizingu∗ = w(1− t)L∗ + h(1− L∗) + θib(wtL∗).FOC: −wL∗ + θib′(wL∗ + wt((dL∗)/(dt))) = 0.SOC: −w((dL∗)/(dt)) + θib′′(wL∗ + wt((dL∗)/(dt)))2 +θib′(2w((dL∗)/(dt)) + wt((d2L∗)/(dt2))) ≷ 0 where((d2L∗)/(dt2)) = ((wh′′′((dL∗)/(dt)))/((h′′)2)).Note: The sign of h′′′ is not predeterminedAlthough the direct utility function exhibits strong regularityproperties, single-peakedness of the induced politicalpreferences is not guaranteed.

Median voter theorem with single-crossing

Single-peakedness is only a sufficient condition for votingcycles not to arise as illustrated in example 2.2.Example 2.2:

1 2 31st A A C. B B A3rd C C B

Table 3: Preferences III

A vs. B: 3:0; A vs. C: 2:1⇒ A winsA is the Condorcet winner although voter 3’s preferences arenot single-peaked (see Figure on next slide).

Median voter theorem with single-crossingFig. 2.3

Figure 3: Preferences III

A second, more general condition which rules out voting cyclesis the condition of single-crossing.

Definition (single-crossing)Preferences are single-crossing if for any two voters i and j(i < j) and for any two alternatives q′ and q′′ with q′ < q′′, wehave

uj(q′) > uj(q′′)⇒ ui(q′) > ui(q′′) andui(q′′) > ui(q′)⇒ uj(q′′) > uj(q′).

The key difference to the notion of single-peakedness isthat voters (instead of policy alternatives) are rankedaccording to their ”ideology“.

Example 2.2 (contd.): Assume A < B < C. Are thesepreferences single-crossing?

Voters 1 and 3 (1 is the left voter); alternatives A and C (Ais the ”left“ alternative)→ no contradiction possibleVoters 1 and 3 (1 is the left voter); alternatives B and C (Bis the ”left“ alternative)→ no contradiction possibleVoters 1 and 3 (1 is the left voter); alternatives A and B (Ais the ”left“ alternative)⇒ u3(A) > u3(B)⇒ u1(A) > u1(B)

(equivalently for voter 2)

Example 2.3: Assume A < B < C.

1 2 31st A B C. B A B3rd C C A

Table 4: Preferences IV

Are these preferences single-crossing?

Fig. 2.4

Figure 4: Preferences IV

Voters 1 and 2 (1 is the left voter); alternatives B and C (Bis the ”left“ alternative)→ u2(B) > u2(C)⇒ u1(B) > u1(C)X

Voters 1 and 2 (1 is the left voter); alternatives A and C (Ais the ”left“ alternative)→ u2(A) > u2(C)⇒ u1(A) > u1(C)X

⇒No contradiction can be constructed for any othercombination of voters and policy alternatives.⇒ preferences are single-crossing (and single-peaked): B isthe Condorcet winner.

The concept of single-crossing allows us to state a 2nd versionof the Median voter theorem:

Theorem (Median voter theorem (single-crossingversion))If there is an odd number of voters, individual preferences aresingle-crossing and the policy space is one-dimensional, thenthe option most preferred by the voter with a median innatecharacteristic is the Condorcet winner.

Some final remarks:

The order i < j is meant to be an invariant order reflectinginnate characteristics (such as the political ideology,exogenous productivity or taste for public goods).Both concepts (single-peakedness and single-crossing)are logically independent.The concept of single-crossing is an ordinal concept. Itonly requires political preferences to be monotone in thetype of the voter.

Some final remarks (cont’d):With single-crossing the politically decisive voter is the onewho is the median of the invariant order of types.Single-peakedness characterizes the politically decisivevoter as the one whose preferred alternative is the medianof the distribution of the most preferred alternatives.The two conditions are only sufficient conditions for aCondorcet winner to exist. It is still possible that aCondorcet winner exists although individual preferences donot satisfy either of both conditions (see example 2.4).

Example 2.4: Assume A < B < C.

1 2 31st A B C. B A A3rd C C B

Table 5: Preferences V

Voters 2 and 3 (2 is the left voter); alternatives A and B (A is the”left“ alternative)⇒ u3(A) > u3(B)⇒ u2(A) > u2(B)⇒wrong=⇒ Preferences are neither single-crossing nor single-peaked(see Figure on the next slide). But a Condorcet winner exists: Avs. B: 2:1; A vs. C: 2:1

Median voter theorem with single-crossingFig. 2.5 3 2

Figure 5: Preferences V

Single-crossing - Continuous choice

If majority voting involves a non-discrete choice, it sufficesin many economic applications to check that the marginalrates of substitution are monotone in the voters’ type.Whether preferences are single-crossing can be easilyverified in the case of an effectively one-dimensional policyspace -> see a basic finding in Milgrom (1994) whoserelevance for majority voting is illustrated in Gans andSmart (1996): With an effectively one-dimensional policyspace political preferences are single-crossing if and only ifthey satisfy the Spence-Mirrlees condition.

Single-crossing - Continuous choice: Spence-MirrleesCondition

Assume ui(x , y , θ) where θ is an individual-specificpreference parameter.The Spence-Mirrlees condition requires the marginal rateof substitution between x and y to vary monotonically withthe individual’s type θFormally, ((dx)/(dy)) = −((δui)/(δy))/((δui)/(δx)) mustbe either increasing or decreasing in θ for any combination(x , y).

Single-crossing - Continuous choice: An Example

The political preference function is defined over the twopolicy variables T and g, i.e. ui = I − T + θib(g).Effectively, the policy problem is one-dimensional since Tand g are uniquely linked via the public budget constraints,g = T . Thus, g is a function of T , i.e. g(T ).MRS(T ,g) = ((dT )/(dg)) = −((δui)/(δg))/((δui)/(δT )) =θib′(g)

MRS(T ,g) is increasing in the preference type θi whichguarantees single-crossing .

Single-crossing - Continuous choice: An Example -contd.

Political preferences are defined asu∗ = w(1− t)L∗ + h(1− L∗) + θib(g).Slope of the indifference curve in (g, t) space is((dT )/(dg)) = −((δui)/(δg))/((δui)/(δt)) =((θib′(g))/(wL∗))

Since b′ and L∗ are independent of θi , MRS(T ,g) is strictlyincreasing in the voter’s type θi .Note, as shown above the political preference function maynot be single-peaked. However, it is single-crossing.As a consequence, the single-crossing version of theMedian Voter Theorem can be invoked in characterizingthe political equilibrium.

Relevance of the Median Voter Theorem

Gerber and Lewis (2004)"Beyond the Median: VoterPreferences, District Heterogeneity, and PoliticalRepresentation", Journal of Political Economy, 112:analyze to what extent the median voter theorem canexplain real-world political choices.In a nutshell, they approach the question whether alegislator’s behavior is tied to the median preference of thedistrict the legislator represents.

Gerber and Lewis (2004): Data and Methodology

The authors estimate preferences of Los Angeles Countyvoters based on a variety of elections in 1992.A second source of data are voter choices revealed in anumber of state-wide and local ballot measures on varioustopics ranging from taxation of candies, property taxexemption of home of person who dies while on activemilitary service, to the introduction of congressional termlimits.The voting record used in the analysis contains a completeenumeration of all the vote choices made by a given voter,as well as identifying information about the legislativedistrict in which the ballot was cast

Gerber and Lewis (2004): Data and Methodology

Voters are grouped according to his/her ideology. Voterswho support three times the republican candidate out ofthe four races for legislative office are classified asrepublicans (similar for democrats). The remaining set ofvoters are classified as independent votersFor each of these subgroups the authors compute the voterpreference using data from the state-wide ballot measures.The median preference exhibits some interestingproperties. It is more to the left when the share oflow-income households and the share of high incomehouseholds becomes larger. The median voter is alsomore to the left the higher the educational attainment ofthe population is - see Table 3 GL

Gerber and Lewis (2004)

Figure 6: Analysis of Median Preferences

Gerber and Lewis (2004): Results

The legislators behavior is inferred from their roll call votes,i.e. from the record of how the district’s legislator voted ona piece of legislation.If the median voter theorem is valid, the legislator’sbehavior must follow from the median voter preference.To formally test for it the authors first regress the medianpreference on the legislator behavior. The validity of themedian voter reasoning can be inferred from two sources:

1 The median preference and legislator behavior shouldexhibit a certain degree of co-movement, i.e. the coefficienthas to be positive and statistically significant. This is afundamental condition for the median voter theorem to beempirically relevant.

2 The share of legislative behavior explained by the medianpreference (measured by R2) should be sufficiently high.

Gerber and Lewis (2004): Results contd.

1st regression: The coefficient of median preference ispositive. The R2 is 0.37 leaving a significant amount ofvariation in legislator behavior unexplained.2nd regression: party ideology is included as anexplanatory variable. The sign of the coefficient is positive.The fit of the regression increases to R2 = 0.92 with theconsequence of rendering the effect of median preferenceinsignificant.3rd regression: Interaction between median preferenceand the variance within each district included. The R2 isslightly increased to 0.93. Representing the main result ofthe paper, the interaction term has a negative impact oflegislators behavior while the median preference has onceagain a positive and significant effect on legislator’sbehavior.

Gerber and Lewis (2004): Regression Results

Figure 7: Regression Results

Gerber and Lewis (2004): Regression Results contd.

In heterogenous districts legislator’s behavior appears to beless related to the median voter preference.Possible explanations:

Lobbying (policy for campaign contributions; incentives tolobby in heterogenous districts tend to be larger).Party loyalty (party re-election concerns can be moreeasily traded-off against the median preference if voters donot easily detect such a deviation).

2.3 Preferences — Multi-dimensional voting

Figure 8: Separate voting I

2.3 Preferences — Multi-dimensional voting

The example gives rise to two questions:

1 Which additional conditions are necessary for voting not tobe cyclic?

2 If voting cycles do not arise how can one characterize thepolicy outcome?

⇒ The plan of this part is to introduce three voting conceptswhich exclude voting cycles. These are respectively:structure-induced equilibrium, intermediate preferences andprobabilistic voting.

Structure-induced equilibrium

Idea:Partitioning of the voting process into a number ofuni-dimensional voting stages.If conditions underlying either of the two the Median votertheorems are satisfied, the alternative preferred by themedian voter is the Condorcet winner along each policydimension.Let’s take an example...

Structure-induced equilibriumTwo-Dimensional Voting – Separate Voting

fig. 3.2

Figure 9: Separate voting II

Our example could describe a two-chamber parliamentarysystem, e.g. Bundestag and a State parliament.Due to the separating structure imposed on the votingproblem, the equilibrium is referred to as astructure-induced equilibrium.

Our example has featured circular indifference curvesimplying the existence of ”political“ best responses whichare independent of the political choices taken in the otherpolicy dimension (“political” dominant strategies).

The structure-induced equilibrium features two properties:1 The median voter in each dimension does not change with

the choice taken in the other dimension.2 The Condorcet winner in each dimension is the level of x

(or y ) which is contained in the bliss-point of the respectivemedian voter.

⇒ Both neat properties will break away if indifference curvesare not circular (see the example on the next slide).

Figure 10: Separate voting III

Generally, the concept of a structure-induced equilibriumcomes in two forms:

1 Simultaneous voting: Each parliament takes thedecisions of the other parliament as given.

2 Sequential voting: One parliament moves first. Thereby, itanticipates the other parliament’s best-response to its ownchoice. The other parliament moves afterwards taking thepolitical choice of the “leader” as given.

⇒ In contrast to ”circular“ preferences, the voting outcomediffers wrt timing for ”non-circular“ preferences! (See figures onnext two slides)

Figure 11: Separate voting IV

Figure 12: Separate voting V

Intermediate preferences

Idea:Put more restriction on the preferences instead of on thevoting process.Conflict over a multi-dimensional policy should berepresented as a uni-dimensional conflict.

More formally, consider preferences of an individual areW (q, αi), where q denotes a policy vector and αi gives thepreference parameter of individual i . If individual preferencescan be rewritten such that

W (q, αi) = J(q) + K (αi)H(q),

where K (αi) is monotonic in αi , then the individual is said tohave intermediate preferences.

Denote the median of the distribution of preferences typesby αm and the policy choice most preferred by this voterq(αm).

If this policy bundle is pitched against any other policybundle, q(αm) receives at least half of the votes.The fact that q(αm) is the Condorcet winner is due to themonotonicity of K (αi).

Since q(αm) is the optimal choice for an αm-type voter

J(q(αm)) + K (αm)H(q(αm)) ≥ J(q) + K (αm)H(q)

is satisfied for any q 6= q(αm).

When H(q(αm))− H(q) > 0, the inequality can equivalently bewritten as

K (αm) ≥ J(q)− J(q(αm))

H(q(αm))− H(q).

⇒ If K (αi) is strictly increasing (decreasing), the inequalityholds for all voters with αi > (<) αm which guarantees thatq(αm) receives at least half of the votes. An analogousreasoning applies when H(q(αm))− H(q) < 0.

Example: Consider preferences areui = U(c) + αiG(q1) + (1− αi)F (q2), where

c is private consumptionq1 and q2 are two types of public expenditures

Gross income is equal to 1 and τ are tax revenues such thatc = 1− τ . With a population normalized to unity the publicbudget constraint is τ = q1+ q2.

Example (cont’d): The preferences satisfy the intermediatepreference condition with

J(q) = U(1− q1 − q2) + F (q2),K (αi) = αi andH(q) = G(q1)− F (q2).

Probabilistic voting

Another way of solving the non-existence problem is tointroduce uncertainty from the candidates’ viewpointsThis ensures stable equilibria in both one-dimensionalpolicy space (e.g. if preferences are neither single-peakednor single-crossing) and multi-dimensional policy spaceUncertainty overcomes discontinuities (which causecycling)We relegate a thorough discussion of this model to the endof the next section

References

Gerber and Lewis (2004),Beyond the Median: VoterPreferences, District Heterogeneity, and PoliticalRepresentation, Journal of Political Economy, 112,1364-1383Gans, J.S. and M. Smart (1996), Majority Voting withsingle-crossing preferences, Journal of Public Economics,59, 219 - 237

3. Electoral competition-based on the script by Marko Köthenbürger and Tobias Seidel-

We want to study electoral competition along the lines ofthe policy question how the size of government spending isdetermined

...and maintain two key assumptions:

1 Politicians are opportunistic (no partisan preferences)2 Politicians always implement the announced policy (no

commitment problem)

⇒ Politicians only maximize their probability of holding officeand not a social welfare function!

Median voter theorem - continuous choice€¦ · A is the Condorcet winner although voter 3’s...

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