# The Median Voter Theorem

Imagine that the income tax schedule is linear and exogenously given. The tax rate is endogenously determined by voting. Note that the tax rate and the size of government spending is a one-to-one relationship since income is exogenously given. We may use both variables for the same meaning. If the demand for public goods differs among voters, how is the size of spending determined? A plausible mechanism is majority rule by voting.

Consider the typical majority voting system where each voter has one vote. Voters behave honestly with respect to their preferences. In this regard, the outcome of majority voting may reflect the interests of the median voter. Let us explain this.

Compare Y and an increase in Y, *AY +* Y. Which would each voter choose? As shown in Fig. 12.2, the relationship between welfare and Y is normally concave and has a unimodal shape. If voters pay significant attention to Y and think that Y is too small, they vote for an increase in Y, AY + Y. When Y = 0, all voters choose AY + Y. Then, as Y increases, some voters turn against AY + Y. Namely, if the income effect of public goods is zero and tax increases with income, richer voters begin to vote against an increase in Y, AY + Y, as Y increases. If we arrange voters according to income, the median voter with median income attains her or his optimal public goods, YM, with majority voting. If Y increases more than YM, most voters are against it.

The above argument implies the median voter theorem:

Public spending is determined by the median income voter.

In the foregoing three-person model, person 2 is the median voter and Y_{M} is the outcome of majority voting. If we compare Y_{M} and Y_{L}, persons 2 and 3 vote for Y_{M}

Fig. 12.2 **The median voter theorem**

and person 1 votes for Y_{L}. Y_{M} wins. If we compare Y_{M} and Y_{R}, persons 1 and 2 vote for Y_{M} and person 3 votes for Y_{R}. Again Y_{M} wins. If voters are identical with respect to income but different with respect to preferences, a similar argument applies. In such an instance, public spending is determined by the median preference voter.

This is the median voter theorem, which has an important policy meaning. The size of government is determined by the median voter; hence, the size is not too large or small. Many empirical studies have examined whether spending is determined by the median income voter. For example, a simple test is to regress government spending on the after-tax income of the median voter. However, it is hard to estimate this proposition directly. Empirical results so far seem ambiguous.