Economic essay

Assignment 2:

The Median Voter Model

1 Introduction

Microeconomics tools are not only used to explain typically economic phenomena, like

wages, interest rates, stock prices, marketing strategies, and so on. They are also used

to explain politics. Nowadays, political economics is an important field in social sciences.

In this paper, we describe probably the best known political-economic model, the median

voter model. It was developed by Downs (1957),1 who was heavily inspired by Hotelling

(1929).2 The Hotelling model is discussed in Chapter 14 of Frank and Cartwright (2016).3

The median voter model describes the consequences of electoral competition for policy

outcomes. The model is an economic model in the sense that each actor - in the present

model, each party and each voter - makes decisions on the basis of cost-benefit analyses.

As in any economic model, assumptions are made about each actor’s objective function.

We now describe the model.

2 The Model

Two parties, party L and party R, run for office. The election revolves around policy x,

say government spending. Before the election, each party chooses a platform. We denote

by xaL the platform chosen by party L, and by x a R the platform chosen by party R. After

the election, the platform of the party that has won the election is implemented. So,

parties always do what they have promised.

Parties are assumed to care only about winning the election. The utility function of

party P ∈{L, R} can therefore be represented by

UP = IP

1Downs, A., 1957, An Economic Theory of Democracy, New York Harper. 2Hotelling, H., 1929, Stability in Competition, Economic Journal, 39, 41-57. 3Frank, R. and E. Cartwright, 2016, Microeconomics and Behaviour, 2nd European Edition, McGraw-

Hill Education.

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where IP = 1 if party P has won the election, and IP = 0 if party P has lost it.

The volting population is made of an infinite number of voters. Each voter either votes

for party L or for party R. There is no abstention. Let vi reflects voter i’s vote decision.

It denotes the probability with which i votes for party L. Thus, vi = 1, means that i

votes with probability one for party L, and vi = 0 means that i votes with probability

one for party R. Voters have different preferences over government spending. Let

Ui (x) = −(x−xi) 2

(1)

describe voter i’s preferences about x. In Equation (1), xi denotes the most preferred

value of x from voter i’s perspective. It is called voter i’s bliss point. Clearly, if i could

choose x, he would choose x = xi. This value of x maximizes Ui (x). The problem is that

voters have different bliss points. For example, John wants the government to spend more

than Ann, xJohn = 70 > xAnn = 40. Figure 1 depicts their utility functions. The dashed

Figure 1: Utility function of Ann and John

line depicts Ann’s utility function, while the solid line depicts John’s utility function.

Figure 1 illustrates that the closer is x to i’s bliss point, the higher is i’s utility.

Question 1: Are John and Ann risk-averse or risk-loving?

As mentioned above, the electorate consists of an infinite number of citizens. Some

of these citizens have bliss points that are even lower than that of Ann. Some have bliss

points higher than that of John. Finally, some have bliss points between xAnn and xJohn.

For simplicity, we initially assume that citizens’ bliss points are uniformly distributed on

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the interval [20, 110], meaning that no citizen wants public spending to be lower than 20,

no citizen wants public spending to be higher than 110, and the same number of citizens

wants x to be equal to y as to z, provided that 20 ≤ y ≤ 110 and 20 ≤ z ≤ 110. The timing of the model is as follows. First, parties simultaneously offer platforms,

xaL and x a R. Next, after having observed x

a L and x

a R, voters simultaneously maker their

vote decisions. Finally, the party that obtains the majority of votes wins the election. Its

platform is implemented.

3 Solving the Model

The model is solved by backward induction. We first determine the optimal decision of

each voter i. Each voter observes xaL and x a R. Suppose that x

a R ≥ x

a L. The analysis

for xaR ≤ x a L is identical. To determine voter i’s optimal strategy, it is convenient to

distinguish four cases.

1. xi > x a R > x

a L . Then,

−(xaR −xi) 2 > −(xaL −xi)

2 ,

implying that it is a dominant strategy for voter i to vote for party R. The intuition

is straightforward. xi > x a R > x

a L means that both parties propose to spend less

than voter i wants the government to spend. As a result, voter i prefers the platform

that leads to the highest amount of spending, vi = 0.

2. xi < x a L < x

a R . Then,

−(xaL −xi) 2 > −(xaR −xi)

2 ,

implying that it is a dominant strategy for voter i to vote for party L, vi = 1 This

case is the opposite of the first one.

3. xaR > xi > x a L. In the third case, voter i faces a dilemma. Party R proposes to

spend more than he wants, while party L proposes to spend less than he wants. In

this case voter i votes for party L if

−(xaL −xi) 2

> −(xaR −xi) 2

xi −xaL > x a R −xi (2)

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4. If xaR = x a L, each voter is indifferent between both parties. We assume that in this

case each voter casts his ballot for party L with probability one half, vi = 1 2

The analysis above shows that voter i votes for party L if his bliss point is closer to

xaL than to x a R. This property of voters’ voting strategies has an important implication.

Consider voter i and voter j. Suppose that xi > xj, x a R > x

a L, and that voter j votes for

party R. Then, voter i also votes for party R. Likewise, if xi < xj, x a R > x

a L, and voter j

votes for party L, then voter i also votes for party L.

Question 2: Show graphically that if xi < xj, x a R > x

a L and vi = 0, then vj = 0.

We have now arrived at one of the main insights of the median voter model. If in our

model the median voter strictly prefers party L to party R, party L wins the election.

Analogously, if the median voter strictly prefers party R to party L, party R wins the

election. Earlier, we have assumed that xi is uniformly distributed on the interval [20, 110].

Under this assumption, the median voter’s bliss point is xm = 65. Our analysis thus shows

that if xL is closer to 65 than xR, L wins the election. Otherwise, R wins.

We now turn to the analysis of parties’ platform decisions, xL and xR. Parties antici-

pate how voters will respond to their platforms. Most importantly, each party knows that

if it gets the support of the median voter, it will win the election. As is usual in economic

models, in equilibrium actors do not have an incentive to deviate. It is now easy to see

that in equilibrium each party chooses a platform that coincides with the median voter’s

bliss point. If not, one party has an incentive to deviate. As an example, suppose that

xaL = 60 and x a R = 80. In this case, party L wins the election. However, by choosing

xaR ∈ (60, 70) party R can win the election. It therefore has an incentive to deviate. This incentive to deviate for one party always exists unless xaL = x

a R = xm. Hence, the median

voter model predicts that parties choose platforms that coincide with the median voter’s

bliss point.

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Your Essay

The median voter model discussed above is based on many assumptions. For example,

1. Political parties only care about office.

2. The bliss points of voters are uniformly distributed.

In your essay you should first analyze how the outcomes of the median voter model

are affected by the assumption that the bliss points of voters are uniformly distributed.

To this end, assume instead that the bliss points of voters are distributed as in Figure 2

below.

0 100

0.0075

0.02

20

Density

x_i

Figure 2: Alternative distribution of voter bliss points

Next, you should show how the outcomes of the median voter model are affected by

replacing the assumption that political parties only care about winning the election by the

assumption that political parties are ideological, that is they care about policy outcomes,

like voters do. Assume that parties have different ideologies. Assume again that the bliss

points of voters are uniformly distributed on [20, 110] and that parties must implement

what they promise in case they get elected.

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