A hyperbolic discounter with a yearly beta=0.5 and delta=1 has 10,000 to invest for retirement, which is 30 years from today. Her utility function for wealth at retirement is u(w)=w. The agent cares about her wealth at retirement and her effort in finding the right investment. The money is currently sitting in account A, earning 0% interest. She can costlessly transfer to account B in which it would earn an interest of 1% per year. Alternatively, she can exert some effort and find an investment C in which her money would earn an interest of 3% per year. The one-off effort cost of finding this account is equivalent to e=400 (this is not a finder fee that comes out of her wealth, instead it is a mental cost associated with the hassle of finding the account). Every year, the agent decides whether to transfer the whole amount, and, if so, to which account. Any money transferred to accounts B or C stays there until retirement.

1) Write down the utility function of this agent as of the first year depending on the decision she takes in the first year.

2) Assuming the agent is nave, describe what the agent will end up doing over the 30 years (will she keep her wealth in account A, or transfer it to account B or C, and if so when?). Be sure to explain your reasoning/show your calculations.

3) If the agent is sophisticated instead of naive, describe what the agent will end up doing over the 30 years (will she keep her wealth in account A, or transfer it to account B or C, and if so when?).