assuming there are only 2 loan options, small and large, a person may take out the large loan because thy maximise their utility try spending slightly more than what the small loan gives them (taking interest rates into account)

Consider an agent who wants to take out a loan. The agent can take out either a small loan, or a large loan. If she takes out the small loan, she can spend the money in period 1 on something she wants now and repay the loan in the second period. If she takes out the large loan, she can choose to spend it in two ways: A) Spend some of the loan money on something she wants in the first period, some of the money on something she wants in the second period and pay back the loan in the second period. B) Spend all the loan money in the first period on something she wants + something she 'doesn't want as much' and pay all the loan back in the second period. The table below summarises the utilities Utility in period 2 -2. Loan Small Loan Large Loan (Scenario A) Large Loan (Scenario B) Utility in Period 1 2.5 3 6 -2 -7 For simplicity assume 8 =1. Analyse the optimal consumption decisions of three types of agents: Exponential discounting; naive hyperbolic discounting ( B= 1/2); and sophisticated hyperbolic discounting ( B= 1/2). Answer the following questions for each agent type. a) At Period 0, what loan do you expect each agent to choose? What will their utility be from making this choice? If they choose the large loan, will they expect to follow scenario A or B? Which Scenario will they actually follow? Explain your reasoning. (30 points) b) A friend of the sophisticated agent offers to stop them spending all their money in one period, for a fee. Would the sophisticated agent be willing to pay their friend for this service today, and if so how much would they be willing to pay? (5 points) Consider an agent who wants to take out a loan. The agent can take out either a small loan, or a large loan. If she takes out the small loan, she can spend the money in period 1 on something she wants now and repay the loan in the second period. If she takes out the large loan, she can choose to spend it in two ways: A) Spend some of the loan money on something she wants in the first period, some of the money on something she wants in the second period and pay back the loan in the second period. B) Spend all the loan money in the first period on something she wants + something she 'doesn't want as much' and pay all the loan back in the second period. The table below summarises the utilities Utility in period 2 -2. Loan Small Loan Large Loan (Scenario A) Large Loan (Scenario B) Utility in Period 1 2.5 3 6 -2 -7 For simplicity assume 8 =1. Analyse the optimal consumption decisions of three types of agents: Exponential discounting; naive hyperbolic discounting ( B= 1/2); and sophisticated hyperbolic discounting ( B= 1/2). Answer the following questions for each agent type. a) At Period 0, what loan do you expect each agent to choose? What will their utility be from making this choice? If they choose the large loan, will they expect to follow scenario A or B? Which Scenario will they actually follow? Explain your reasoning. (30 points) b) A friend of the sophisticated agent offers to stop them spending all their money in one period, for a fee. Would the sophisticated agent be willing to pay their friend for this service today, and if so how much would they be willing to pay? (5 points)