Question

There are three categories of income tax filters in Mexico: those that never evade taxes, those that sometimes do, and those that always do. An examination of audited tax returns from one year to the next shows that of those who did not evade taxes last year, 95% will continue in the same category this year; 4% will move to the "sometimes" category, and the rest will move to the "always" category. For those who sometimes evade taxes, 6% will move to "never", 90% will remain the same, and 4% will move to "always". As regards the "always" evaders, the respective percentages are 0, 10 and 90%.

1. Express the problem as a Markov chain.
2. In the long run, what would be the percentages of the “never”, “sometimes” and “always” tax evasion categories?

3. Statistics show that a taxpayer in the “sometimes” category evades taxes totaling approximately $ 5,000 per return and in the “always” category totaling approximately $ 12,000. Assuming the taxpayer population is 70 million and the average income tax rate is 12%, determine the annual reduction in taxes collected due to evasion.

Answer 1

solution: from the given data......

(1) The second matrix is the is the 3 x 3 matrix we start with

The states are never, sometime, and always. Where we are now is the vertical states

Examples: To transition from never to never we use the 0.95% transition probability. To find the odds of going from Sometimes this year to never next year we read the probability in the middle row of the left column.?? ?To find the same odds for sometimes to sometimes, we read the probability in the middle row of the middle column. To find the same odds for sometimes to always, we read the probability in the middle row of the right column.

(2)  To find the long term probability we multiply the matrix many times. (see your book or websites on how to multiply 3x3 matrices.

after many multiplications (I made an Excel spreadsheet for this)

we find the top matrix. This shows us that no matter where the average tax payers start, after many years 44.1% of them will never cheat, 36.8% of them will sometimes cheat, and 19.1% of them will always cheat.

(3)

OK for the sometimes people, we have 70,000,000*$5,000*0.12*0.39 = ???$15,441,176,471

for the always people we have ?70,000,000*$12,000*0.12*0.19 = $19,270,588,235

all together this is about $35,000,000,000 ($35 billion) please rate my answer, thank you.