14.3 Apportionment Methods 935 Whenever we use Jefferson’s method, we first round the modified quota to the nearest hundredth and then determine the modified lower quota. In Example 4, we used a modified divisor of 34,900 to obtain the desired sum of 250, which is the number of legislative seats to be apportioned. There are an infinite number of modified divisors (in a small range) that could be used to obtain the modified lower quotas that we obtained. In fact, any modified divisor from about 34,758 to about 34,901 would result in the modified lower quotas we obtained in Table 14.33. If you selected a modified divisor lower than about 34,758, you would obtain modified lower quotas for which the sum of the seats is too high. For example, if you selected a modified divisor of 34,750, you would obtain modified lower quotas of 28, 35, 143, 23, and 22. The sum of these seats is 251, which is above the number of seats to be allocated. If you selected a modified divisor greater than about 34,901, you would obtain modified lower quotas for which the sum of the seats is too low. For example, if you selected a modified divisor of 34,925, you would obtain modified lower quotas of 28, 35, 142, 23, and 21. The sum of these seats is 249, which is below the number of seats to be allocated. With Jefferson’s method, we will always use a modified divisor that is less than the standard divisor used with Hamilton’s method. Recall from the discussion of Hamilton’s method that if an apportionment method assigns either the lower or upper quota, it is said to satisfy the quota rule. Hamilton’s method satisfies the quota rule, since it uses either the lower quota or upper quota of the standard quota. Notice that the standard quota for state C from Example 3 is 141.78. According to the quota rule, state C should receive either 141 or 142 seats. Using Jefferson’s method, state C received 143 seats (see Table 14.33). Therefore, Jefferson’s method violates the quota rule. The first case in the House of Representatives in which Jefferson’s method led to a violation of the quota rule occurred in 1832. New York had a standard quota of 38.59 but was awarded 40 seats using Jefferson’s method. Daniel Webster argued that this result was unconstitutional and suggested a compromise between Hamilton’s method and Jefferson’s method, leading to Webster’s method. Let’s discuss how Webster proposed to apportion the seats in the House of Representatives. Table 14.33 shows the modified quotas for all the states. Now try Exercise 13 Table 14.33 Republic of Hydrangea Population Using a Modified Divisor, d 34,900 = State A B C D E Total Population 1,003,200 1,228,600 4,990,700 813,000 764,500 8,800,000 Standard quota 28.50 34.90 141.78 23.10 21.72 Modified quota 28.74 35.20 143.00 23.30 21.91 Modified lower quota 28 35 143 23 21 250 The sum of the modified lower quotas is now 250, our desired sum. Each state is awarded the number of legislative seats listed in Table 14.33 under the category of modified lower quota. 7
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