Deriving the Expected Value of the Tax Underreporting Rate
contains underreported tax . In ÃŨ , a return does not experience audit and does not contain underreported tax . Therefore , the set of all possible outcomes is [ AU , AŨ , ÃU , ÃŨ ]. This set establishes the sample space .
The magnitude |·| is the number of returns in a type or state . The magnitude of type A is | AU | + | AŨ |. The magnitude of type à is | ÃU | + | ÃŨ |. The magnitude of type U is | AU | + | ÃU |. The magnitude of type Ũ is | AŨ | + | ÃŨ |. Since the total number of returns filed for a given taxable year is large , the ratio of the magnitude of a type or state to the magnitude of another type or the overall population of returns is sufficiently equivalent to the probability of the type / state occurring so as to make the ratio and the probability measure synonymous . 5 In general , the relative frequency of an event type t that is a subset of the larger event type T approximates the probability Pr ( t ) of type t , or | t |/| T | ≈ Pr ( t ). As | T | gets larger , the relative frequency better approximates the true probability of type t occurring . When | T | is very large , the relative frequency becomes equivalent to the true probability of the type occurring , or | t |/| T | Pr ( t ).
Since | R | is large in a country such as the United States where almost 150,000,000 individual tax returns are filed every year , it is sufficient for purpose of this discussion to hold that the proportion of returns experiencing an event type to the total number of returns that make up the larger set under consideration ( the ” given ” in conditional probability ) is equivalent to the true probability of returns experiencing that event type . 6 For simplicity , the paper employs the following shorthand notation for the most significant probability measures
involved in this inquiry .
• Let a denote the overall probability of A where Pr ( A ) | A |/| R | ( audit rate ).
• Let d denote the conditional probability of A given U where Pr ( A | U ) | AU |/| U | ( detection rate ).
• Let v denote the overall probability of U where Pr ( U ) | U |/| R | ( underreporting rate ).
• Let z denote the conditional probability of U given A where Pr ( U | A ) | AU |/| A | ( audit success rate ).
• Finally , let m denote the overall probability of state AU where Pr ( S 1
)
| AU |/| R | ( mixed rate ).
Most tax authorities maintain data on the tax returns they receive . From these data , one can usually calculate the measures of states AU and AŨ . Because one can know the measures of AU and AŨ , one can calculate the measures a and z . Because a and z are known , by definition m is known . States ÃU and ÃŨ traditionally have unknown measures . Accordingly , measures d and v are usually unknown .
Previous work ( Manhire , 2014 ) showed that the relationship between a , d , v , and z is identical to the theorem of Bayes , 7 which states that :
m = az = dv .
( 1 )
The current paper assumes that any proposition for measuring the expected value of the underreporting rate must be consistent with the theorem of Bayes ; that is , a proposition cannot be valid if it violates equation ( 1 ). The paper further assumes that
5
E . g ., taxpayers subject to U . S . internal revenue laws filed 145,236,429 individual income tax returns in calendar year 2013 . ( Internal Revenue Service , 2015 ). This makes | R | sufficiently large .
6
This same equivalence assumption might not be possible in countries with a low number of filed tax returns .
7
The theorem states that Pr ( H K ) = Pr ( H ) Pr ( K | H ) = Pr ( K ) Pr ( H | K ).
8