Consider the tax evasion decision problem discussed in class. Y is the taxpayers actual income, X the reported income, and t the proportional income tax. The taxpayers preferences are given by the utility function U(Yj)=lnYj , where Yi is income in state j{c, nc} , Ync=Y-tX and Yc=1-tY-Ft(Y-X) . The probability of being caught, state c , is p .

1. (8 pts.) Calculate the individually optimal quantities Yc*, Ync*,X* as a function of the parameters.
2. (2 pts.) Show that the individually optimal share of underreported income, X*Y , is independent of income, Y .
3. (2 pts.) If p=0.5, t=0.5, calculate the range of the fine F for which a taxpayer evades a portion of their income.
4. (2 pts.) Briefly interpret your result in i).
5. (2 pts.) If F=0.1, t=0.1 , calculate the range of the probability of audit p for which a taxpayer does not evade any income.
6. (2 pts.) Briefly interpret your result in 5