Well, I've been working on this problem, but I can't get the right path to the solution. 1. The problem statement, all variables and given/known data "Consider the following version of the Ehrlich economic model of crime where an individual has the expected utility function: U = p ln(Iu) + (1 - p) ln (Is). p = objective probability of being caught Iu= Criminal's income if caught Is= Criminal's income if not caught The criminal's initial endowment of time is T hours a day and should be divided into "time for crime" (tc) and "time for legal work" (tl). So T = tc + tl. If the criminal choose to work legally, he gets an income of wl> 0 per unit of time; while engaged in crime he gets an income of wc> wl per unit of time. If the individual is caught committing crimes, he get a penalty f > wc - wl per unit of time. So, the income of an individual who commits a crime but is not arrested is Is = wltl + wctc while if he get arrested Iu = wltl + wctc - f tc . 2. Relevant equations Then, the criminal divides his time between crime and work legally, an his problem is given by: max U = p ln(Iu) + (1 - p) ln (Is). with tc, tl[itex]\geq[/itex]0 s.t. Is= wltl + wctc Iu = wltl + wctc - f tc T = tc + tl a) For solving the problem in ( Iu,Is), rewrite the budget constraint as a linear equation: Is = a - bIu . Find a and b. b) Graph the problem in ( Iu,Is). Solve the problem in ( Iu,Is) when wl=1 , wc= 2, f= 1.5 , T= 3. Then find the corresponding solutions for (tc,tl)." 3. The attempt at a solution Using differentiation and the Kuhn - Tucker conditions I've concluded that a= ((1- p) + p (Is))/(1-p) b= p (Is)/(1-p)Iu But I don't believe that's correct. Hope you could help. Thank you for reading.