Court-ordered redistricting and the law of 1/n

More specifically, each of n legislative districts internalizes only a fraction (1/n) of the cost of all projects. See Lee (2015) for a more detailed discussion of the law of 1/n.

Some recent studies also report inconsistent findings at the local government levels (e.g., MacDonald 2008; Jordahl and Liang 2010; Petterson-Lidbom 2012).

Suppose that a winning coalition requires two-thirds of the lower house districts nested within half of the upper house districts. The share of the lower house districts needed to win the simple majority of the upper house is 1 / 3 (that is, 2 / 3 \(\times\) 1 / 2). Thus, a simple majority of the lower house is sufficient to win the upper house.

The assumption about House districts embedded in Senate districts follows the previous literature (e.g., Ansolabehere et al. 2003; Chen and Malhotra 2007. In some states, House district lines cut across the Senate district lines, but that is not critical for the main results.

Although many states had multi-member districts in the 1960s and 1970s, single-member districts became popular gradually as the courts ruled increasingly that multi-member districts hurt minority representation. See Crain (1977) for related discussions.

This setup allows us to better identify the effect of redistricting in both chambers.

We assume a closed rule in which legislators are not allowed to offer amendments once a proposal has been made.

We assume that \(A > 1\) to guarantee interior solutions.

This assumption rules out weakly dominated strategies.

Remark The basic model assumes that each of m Senate districts is divided into an equal number of House districts. If that assumption is relaxed, the jth Senate district (\(j \in \{1,\ldots , m \}\)) would be divided into \(k_{j}\) House districts, where \(k_{1}\le \cdots \le k_{m}.\) Then, \(n=\sum _{j=1}^{m}k_{j}\) is the total number of House districts. In that case, we still obtain qualitatively similar results. Proofs are available upon request to the authors.

A winning coalition must include more than half of the House districts in more than half of the Senate districts. The number of House legislators required for a Senate majority is then \(\frac{1}{4} n\) (i.e., \(\frac{1}{2} m\) senators times \(\frac{1}{2} k\) representatives). A simple majority of House districts is sufficient for winning the Senate.

In equilibrium, the proposer induces a coalition member to agree on the bill with the default payoff of 0.

By totally differentiating (3), we obtain \(dg/dG= \frac{1}{n} \cdot \frac{1}{u^{\prime }(g) - \frac{{\widetilde{d}}(n)-1}{n}}\), which is greater than 0 because \(u^{\prime }(g) > \frac{{\widetilde{d}}(n)-1}{n}\) (as the proposer selects g at less than the optimal level for coalition members). Noting that \(\frac{{\widetilde{d}}(n)- 1}{n} \simeq 1/2\), dg / dG also declines in n.

Notice that malapportionment is an issue for both upper chambers (between the mth Senate district and other Senate districts) and lower chambers (between House districts located in the mth Senate district and other House districts).

Specifically, the payoff for a representative located within the 1st through mth Senate district is given by \(u(g) - \tau\). On the other hand, the payoff for a representative located within the mth Senate district is given by \(\left[ u(g^{\prime }) - \tau \right] + \left[ u(g^{\prime \prime }) - \tau \right]\) where \(g^{\prime }\) and \(g^{\prime \prime }\) denote the amount of spending for each segment of population.

During the 1962–1972 period, for instance, the size of the lower chamber increased in 11 states and decreased in 14 states, and the size of the upper chamber increased in 16 states and decreased in 5 states.

In (5) and (6), \(({\widetilde{d}}(n)-1)\) refers to the number of House coalition districts, and \(n+k\) refers to the number of voters (i.e., population).

After mergers, each House district has a population of 2.

The idea behind Proposition 2 can be illustrated with a simple example. Suppose that a state legislature consists of 10 districts: 9 districts with a population of 1 and the 10th district with a population of 3. The total population is then 12. Before redistricting, the 10th district is excluded from the coalition because that district has higher project costs. Thus, the winning coalition consists of five districts (assuming that 50% of a legislature will pass the bill), but less than half of the total population. More specifically, the coalition has a vote share of 0.5 with the population share of 0.42. In case of division, the 10th district is divided into 3 districts with populations of 1. With 12 districts of equal populations, the winning coalition needs to attract one additional district. In the case of mergers, 10 districts are combined into six districts each with a population of 2. The winning coalition now consists of three districts with a total population of 6. Although the number of districts declined because of mergers, the coalition must provide projects to one additional citizen. In both cases, the extant coalition faces higher costs of proposing spending projects owing to the inclusion of additional members.

To see this, note that \(u(g)=\frac{Y}{n+k}\) both before and after redistricting (because a coalition member’s payoff is 0), implying that if g falls, Y also must fall.

More generally, the populations of more than \({\widetilde{d}}(n)\) House districts are 1, and other House districts may have populations greater than 1. In addition, more than half of the House districts located in more than \({\widetilde{d}}(m)\) Senate districts have populations of 1.

Proofs are available upon request to the authors. Note that if mergers of districts follow population growth in those districts, the 1 / n effect could be negative—that is, between \(t=1\) and \(t=3\), total spending increases whereas the number of legislative districts declines. The relationship between seats and spending is then negative. If mergers of districts follow population declines, on the contrary, both total spending and the number of legislative districts fall. Because mergers of districts typically take place when those districts experience outflows of residents, we expect that the relationship between seats and spending is positive in the post-Baker period.

Alaska and Hawaii’s expenditures per capita are considered outliers, and Nebraska’s legislature is unicameral.

It would be interesting to see if the theory also holds in the pre-Baker period (i.e., before 1962). We did not examine the pre-Baker period, however, because disaggregated data on state governments before 1962 are rare and available mainly for abnormal years, such as the Great Depression and World War II.

The Citizen Ideology Index was obtained from the revised 1960–2008 Citizen Ideology Series calculated according to Berry et al. (1998).

We ignore the potential multicollinearity between \(L_{it}\) and \(U_{it}\) as the correlation between the two is not high (0.19 for the total sample period; 0.21 for the Baker period; 0.19 for the post-Baker period).

Some states amended their constitutions to alter the sizes of their legislatures, but for reasons encompassing all legislative matters, not just those related to budget allocation (Lee 2015).

On a similar note, Malhotra (2006) showed that larger government size can lead to an increase in legislative professionalism while legislature size remains fixed.

The Chow test (\(p \hbox {value} < 0.005\)) indicates a structural break between the Baker and post-Baker periods—that is, the coefficients on Lower and Upper are statistically different between the two periods at the 1% level.

Data refer to the average of 1978 and 2008.

Note that \({\widetilde{d}}(m){\widetilde{d}}(k)\le {\widetilde{d}}(n)\) as long as \(m\ge 3\) and \(k\ge 3.\)

Technically there are many equilibria but all have the same outcomes in terms of project sizes.

argmax \(\left[ A\log (1+g)-g\right]>0\Leftrightarrow A>1.\)