A taxpayer can be of two types: they may have

A taxpayer can be of two types: they may have income the IRS doesn’t know of prior to fíling or they may not. Call them Tu and Tn, for a taxpayer with or without unreported income, respectively. Tu must decide whether or not to declare the unreported income on their return. Tn will certainly NOT declare and pay taxes on income they don’t have. Suppose the probability that a taxpayer is Tu is .8, and the probability that she is Tn is .2.

If Tn declares the income, she pays taxes of $1000 on it, while the IRS receives $1000. If there is no declaration, the IRS must decide whether to audit the taxpayer or not. If she is Tu and audited, the taxpayer pays 2000 – 1000 in taxes plus a 1000 penalty– while the IRS gets 1000 – 600= 400, where 600 is the cost of auditing. If she is Tn and audited, she gets 0 while the IRS gets a payoff of -600, the cost of the audit. If there is a declaration and no audit, the taxpayer and the IRS both get nothing.

Explain why neither of the taxpayer’s pure strategies – pooling on declare, or separating with Tu declaring and Tn not declaring – can be part of a Bayesian Perfect Equilibrium. Then find a semi- separating equilibrium in which:

Tn never declares, Tu declares with probability r, while the IRS audits if no declaration with probability s.

Find r and s. Make sure you specify the equilibrium belief of the IRS when no income is declared: the probability, conditional on no declaration, that they are dealing with Tu.