Consider the tax evasion decision problem:

Question 1 (20 pts.) Consider the tax evasion decision problem discussed in class. Y is the taxpayer's actual income, which is equal to 160. X is the reported income, and t - the proportional income tax. The taxpayer's preferences are given by the utility function U(Y,) = In Y,, where Y, is income in state j E {c, nc], Ync = Y - tX and Ye = (1 -t)Y - Ft(Y - X). The probability of being caught, state c, is p = 0.25. The fine, F = 0.5, is levied in state c. a) (2 pts.) Write down the Lagrangean corresponding to the taxpayer's optimization problem. b) (3 pts.) Write down the first order conditions. c) (1 pt.) Briefly interpret the first order condition with respect to Ync. d) (3 pts.) Calculate and briefly interpret MRSc,nc. e) (4 pts.) Calculate the individually optimal quantities Y", Yrc, X* as a function of any parameters. f) (2 pts.) Show that the individually optimal share of underreported income, , is independent of income, Y. g) (3 pts.) Calculate the range of the tax rate, t, for which a taxpayer evades a portion of their income. h) (2 pts.) Briefly interpret your result in part g)