3. Consider an agent who evaluates utility delayed by k periods with a discount factor of...

 

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3. Consider an agent who evaluates utility delayed by k periods with a discount factor of 8k. Time is discrete and indexed by t € {0,1,2,...}. This individual has to complete a project (which only takes one period to complete) before or during period T, where the (undiscounted) utility cost of completing the project in period t is t a) Explain the difference between an exponential discounter, a naïve hyperbolic discounter and a sophisticated hyperbolic discounter. b) Suppose the individual is an exponential discounter with = 1 and 8 = 1. When will the project be completed? c) Now suppose the individual is a naïve hyperbolic discounter with ß = 2 and 8 = 1. Calculate when this individual will plan on completing the project, and when it will actually be completed. d) Now consider the behaviour of a sophisticated hyperbolic discounter with ß = 2/1/2₁ and 8 1. Prove that if T is even, then the individual will finish the project in period 0, whereas if T is odd the project will be completed in period 1. [Hint: start by considering how the individual will behave in period T-1, and then work your way backwards.] 3. Consider an agent who evaluates utility delayed by k periods with a discount factor of 8k. Time is discrete and indexed by t € {0,1,2,...}. This individual has to complete a project (which only takes one period to complete) before or during period T, where the (undiscounted) utility cost of completing the project in period t is t a) Explain the difference between an exponential discounter, a naïve hyperbolic discounter and a sophisticated hyperbolic discounter. b) Suppose the individual is an exponential discounter with = 1 and 8 = 1. When will the project be completed? c) Now suppose the individual is a naïve hyperbolic discounter with ß = 2 and 8 = 1. Calculate when this individual will plan on completing the project, and when it will actually be completed. d) Now consider the behaviour of a sophisticated hyperbolic discounter with ß = 2/1/2₁ and 8 1. Prove that if T is even, then the individual will finish the project in period 0, whereas if T is odd the project will be completed in period 1. [Hint: start by considering how the individual will behave in period T-1, and then work your way backwards.]