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Apportioning Representatives in the United States Congress - The Quota Rule

Author(s): 
Michael J. Caulfield (Gannon University)

The fact that the affected states in the discrepancy just mentioned are Virginia and Delaware is no coincidence. In general, Jefferson’s method is biased in favor of larger states and against smaller ones. It violates what is called the “Quota Rule:

Quota

Here is a table that compares the populations and Representatives of Virginia and Delaware from 1790 through 1840:

Year

VA Pop.

Quota

Rep’s

DE Pop.

Quota

Rep’s

1790

630,560

18.31036

19

55,540

1.61278

1

1800

747,362

21.55048

22

61,812

1.78237

1

1810

817,615

22.47609

23

71,004

1.95188

2

1820

895,303

21.25999

22

70,943

1.68462

1

1830

1,023,503

20.58844

21

75,432

1.51736

1

1840

1,060,202

14.86167

15

77,043

1.07997

1

Totals

119.04703

122

9.62898

7

By 1810, New York had overtaken Virginia as the most populous state in the Union. If we look at its numbers instead of Virginia’s, the discrepancy between that large state and Delaware is even more pronounced:

Year

NY Pop.

Quota

Rep’s

DE Pop.

Quota

Rep’s

1790

331,589

9.628765

10

55,540

1.61278

1

1800

577,805

16.66124

17

61,812

1.78237

1

1810

953,043

26.19898

27

71,004

1.95188

2

1820

1,368,775

32.50313

34

70,943

1.68462

1

1830

1,918,578

38.59347

40

75,432

1.51736

1

1840

2,428,919

34.04803

35

77,043

1.07997

1

Totals

157.6336

163

9.62898

7

Using Jefferson’s method, New York always had its quota rounded up. On two occasions, its quota was rounded up more than one whole unit, in violation of the Quota Rule. In contrast, Delaware’s quota was rounded up only once. More striking are the cumulative results, showing New York well above and Delaware well below their expected totals.

There is a simple explanation for Jefferson’s bias toward the large states. The method works by lowering the divisor D by some d until the rounding fits the specified number of seats. But lowering the divisor causes the quotient to grow at a faster rate if the dividend is higher. For example,

formula1

Lowering the divisor by 7,000 in each case raises the quotient by more than 2.2 in the case of the large dividend but only by 0.75 in the case of the small one.

Michael J. Caulfield (Gannon University), "Apportioning Representatives in the United States Congress - The Quota Rule," Loci (November 2010), DOI:10.4169/loci003163

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