Policy and Complex Systems
the expected measures of d and v can each be expressed as a yet-undefined mean of its lower and upper interval bound , consistent with the intermediate value theorem . 8
The geometric mean of x 1 and x 2 can be expressed in one of the following forms , where g (·) is a geometric mean function :
g ( x 1 , x 2 ) = xx1xx2 = exp [ ! ! ( logx 1 + logx 2 )]. ( 2 )
Note that the second expression of g ( x 1 , x 2
) in equation ( 2 ) is the anti-log of the arithmetic mean of the log-transformed values of x 1 and
x 2
. This will become important later .
Expected value , as the term is used here , means simply the determination of the measure of an expected outcome . For this paper , the expected outcome is the underreporting rate . The expected value is , therefore , the determination of the measure of the tax underreporting rate .
IV - Preliminary Claims
The following are claims and respective
proofs necessary for deriving the expected values of the detection and underreporting rates .
Claim 1 The value hierarchy a < d v < z always exists if the tax authority has audit selection criteria that perform more efficiently than a random number generator at predicting which returns might contain underreported tax .
Proof . If we set v = z ( i . e ., the underreporting rate equal to the audit success rate ) it would mean that the tax authority ’ s audit selection criteria are equivalent to a random number generator . If the tax authority randomly selects returns for audit in large enough numbers , the proportion of audited returns containing underreported tax would be equivalent to the proportion of all returns in population R containing underreported tax .
Yet , assuming the tax authority ’ s audit selection algorithms are even marginally more efficient that a random number generator , the proportion of audited returns containing underreported tax will always be greater than the proportion of returns in population R containing underreported tax . 9 Therefore , we can conclude that v < z . Given equation ( 1 ), if v < z , then a < d and d < v . Consequently , we can assume the value hierarchy a < d < v < z always exists if the tax authority has audit selection criteria that perform more efficiently than a random number generator at predicting which returns might contain underreported tax .
Claim 2 The measures of d and v converge at the value √m ; meaning d cannot have a measure greater than √m , and v cannot have a measure less than √m .
Proof . Per equation ( 1 ), as the value of d changes by a factor of n , the value of v changes by a factor of 1 / n , or dn = v / n . It follows from this that :
n = vv / dd
( 3 )
8
The intermediate value theorem states that given an interval I = [ h , k ] in the real numbers R and a continuous function f : I → R , if w is a number between f ( h ) and f ( k ), then there exists a position q , where q ( h , k ) such that f ( q ) = w . Where w = 0 , this is also known as the theorem of Bolzano ( Russ , 1980 ).
9
An example of a tax authority ’ s audit selection algorithms is the Internal Revenue Service ’ s Discriminant Inventory Function System ( DIF ) score . According to the I . R . S . ( 2012 ), “[ DIF ] is a mathematical technique used to score income tax returns for [ audit ] potential . . . . Each return measured under DIF receives a DIF score . Generally , the higher the score , the greater the audit potential .” Further , the agency ( 2013 ) holds that “[ i ] f your return is selected because of a high score under the DIF system , the potential is high that an examination of your return will result in a change to your income tax liability .”
9