Question

Suppose that the world is populated entirely with rational but amoral individuals that only get utility from wealth. An individual’s utility function is U(W)=ln(W+1) where W is the wealth of the individual. Some individuals start with 0 initial wealth while other individuals start with a wealth of w>0. Everyone can decide whether or not to steal x units of wealth. The individual know their wealth when they take the decision to steal or not. Stealing has two consequences: first the individual gains x$ of wealth up to 2$ of wealth. Second, with probability p=0.6 the individual gets caught and has to pay back 2x$ , that is twice the amount stolen. The penalty is capped by the individual’s wealth: the individual can never get less than 0$ of wealth in that world.

a) How much should the individual steal if they have an initial wealth of w>>0?

b) How much should the individual steal if they started with no initial wealth?

c) Now suppose that there is no limit to the amount that can be stolen and that the penalty when caught is simply to pay back the amount that was stolen. What amount should an individual with a large intial wealth steal? What about an individual that starts with w=0?