Deriving the Expected Value of the Tax Underreporting Rate
There is no evidence that the tax authority will detect underreporting on every return it audits . In other words , audits are less than perfect detectors of underreported tax . For this reason , one can assume that the value of z is a lower bound of the probability of an audited return containing underreporting . The upper bound , arguably , is 1.00 , since it is possible that 100 percent of returns selected for audit actually contain underreporting , even if the underreported tax eludes the tax authority upon examination .
For this reason , z also takes on an interval value , where I z
= [(| S 1
|/| A |), 1 ]. It stands to reason that the fulcrum on the I z interval “ seesaw ” is also the geometric mean of the lower and upper bounds . Denote the lower bound as z and the upper bound as z '. Denote the expected value of the measure of v using z as E ( v ), as seen in equation ( 14 ). Denote the expected value of the measure of v using z ' as E '( v ). Denote the geometric mean of the two as v *.
Because v *=
it follows that
( 16 )
Let ’ s look at the results for v * using IRS enforcement data and Phillips ’ s approximation of v = 0.325 based on his manual count of audited returns as a benchmark . For calendar years 2007 through 2013 , the average overall individual income tax audit rate a is 0.0103 . The average lower bound calculation of z for the same period is 0.8269 . Employing equation ( 14 ), this yields
while employing equation ( 16 ) yields
It appears the use of v * and not E ( v ) to approximate the measure of v produces results closer to those found in the real world per Phillips . While this does not confirm the accuracy of v * over E ( v ), it does suggest that the former is perhaps a more accurate approximation method than the latter , as would be theoretically expected in finding the fulcrum point of the “ seesaw ” between z and z '. Under Law ’ s definition of ABM validity in § 1 , this suggests that ABMs using v * would be more “ valid ” than those using ( v ).
VI - Conclusion
Some complexity science researchers
have found that models such as
ABMs can help discover the iterative dynamics of tax compliance in a self-report / audit system . The validity of these models depend on accurate input assumptions , which include the audit and underreporting rates . Although audit rates are typically available from tax authority enforcement statistics , underreporting rates remain hidden . Thus , researchers require a method to approximate the underreporting rate given available enforcement data .
This paper attempts a derivation of the underreporting rate given only enforcement data . This approximation should move decisions regarding tax compliance policy from the chaotic sector to its proper home in the political sector of the Stacey Matrix . Hopefully researchers using ABMs and other models will more accurately discover the dynamics and emergent properties of tax compliance as a complex system . Such discoveries can better inform tax policymakers as to the laws and rules that best meet the dual goals of a tax authority ; namely , maximized revenue collection and enhanced general welfare .
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