Please HELP There are three categories of income tax filers

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There are three categories of income tax filers in the USA: those who never evade taxes in a year, those who sometimes do it in a year, and those who always do it in a year. An examination of audited tax returns from one year to the next shows that of those who did not evade tax last years, 95% continue in the same category this year, 4% move to the \"sometimes\" category, and the others move to the \"always\" category. For those who sometimes evade taxes, 6% move to \"never\", 90% stay in the same category, and 4% move to \"\"always\". As for the \"always\" evaders, the respective percentages Are 0% (\"never\"), 10% (\"sometimes\"), and 90% (\"always\").

a) Define the stage, the states, and the matrix of transition probabilities of the Markov chain.

b) In the long run, what would be the percentages of the \"never\", \"always\" and \"sometimes\" categories?

c) Statistics show that a taxpayer in the \"sometimes\" category evades taxes on about $5000 per tax return and that one in the \"always\" category does it for about $12000. Assuming an average income tax rate of 12% and a filers population of 70 million, determine the annual reduction in collected taxes due to evasion

Solution

With the given information we we form the markov chain as follows. We take as states the category of tax evaders N(Never), S(sometimes) A(always). from this information we determine the transition probabilities as represented below.

Let P represent this transition matrix, and let V be the probability vector We want to find V such that

                                                            VP=V

where v=(N, S,A)

.95N+.06S+0A=N

.04N+.9S+.01A=S

.01N+.04S+.9A=A

Also recall that V is a probability vector hence N+S+A=1

solving the above equations for N, S, A we have

N=44%

S=36.7%

A=19.1%

answeer to c.

36.7*70million*$5000= 160440 Million

19.1*70million*$12000=128450 Million

Total reduction in tax collection=.12*(160440+128450)= $34666.8 Million