# There are three little pigs, Damien, Bruno, and Raja. Each of the three pigs lives for five periods, 1, 2, 3, 4, and 5, and is a hyperbolic…

There are three little pigs, Damien, Bruno, and Raja. Each of the three pigs lives for five periods, 1, 2, 3, 4, and 5, and is a hyperbolic discounter as introduced in class. In period 1, Damien and Bruno have to decide whether to build a house of bricks or of straw (the market is out of sticks). Building a house of straw has no instantaneous utility cost (it takes no effort), but building a house of bricks has an instantaneous utility cost of 3 (it requires some effort). Every day the three little pigs get utility from rolling around in the mud near the house they live in. This includes the first period, in which they roll around in the mud after houses have been built. But how much utility they get depends on whether they are safe from the big bad wolf, and this in turn depends on what their house is made of. If a pig lives in a brick house, he gets an instantaneous utility of 5 each period from rolling around in the mud. But if a pig lives in a straw house, he knows that the big bad wolf could come and huff and puff and blow his house down, and so he only gets instantaneous utility of 4 each period from rolling around in the mud.

(a) Damien has a long-run discount factor δ =0.9 and a short-run discount factor β =0.9. Write out his discounted utilities with each of the two possible houses. Which type of house will he build in the first period?

(b) Bruno instead has δ =0.9 and β =0.5. Which type of house will he build in the first period?

(c) Raja has a cold and Mama Pig lets him stay in her brick house for the first period and roll around in the mud near her house. Hence, Raja only needs to build his house in the second period. Like Bruno, he has δ =0.9 and β =0.5. Write out Raja’s discounted utilities from the two types of houses from the perspective of period 1. (Remember to write Raja’s discounted utility beginning with the period in which he is planning, the first period, which is before the period he will build.) From the perspective of period 1, which type of house would he like to build in the second period?

(d) Without commitment, which type of house will Raja end up building in the second period?

(e) Give a verbal example of a partial (i.e. not full) commitment device that Mama Pig could offer that will help Raja stick to his period-1 plan.

(f) Show how your commitment device could work in terms of utilities. Specifically, give an example of how the commitment device might modify Raja’s instantaneous utilities in the different periods, and show that this is sufficient for him to stick with his period-1 plan for which house to build.