Question

There are three little pigs, Luke, Kethan, and Arjun. 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, Luke and Kethan 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 get to 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.

1. Luke has a long-run discount factor δ = 0.8 and a short-run discount factor β = 0.7. Write out his discounted utilities with each of the two possible houses. Which type of house will he build in the first period?

2. Kethan instead has δ = 0.8 and β = 0.3. Which type of house will he build in the first period?

3. Arjun 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, Arjun only needs to build his house in the second period. Like Kethan, he has δ = 0.8 and β = 0.3. Write out Arjun’s discounted utilities from the two types of houses from the perspective of period 1. (Remember to write Arjun’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?

4. Without commitment, which type of house will Arjun end up building in the second period? 1

5. Give a verbal example of a commitment device that Mama Pig could offer that will help Arjun stick to his period-1 plan.

6. Show how your commitment device could work in terms of utilities. Specifically, give an example of how the commitment device might modify Arjun’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. Use numbers to illustrate this.

Answer 1

Answer 1 :-

For Luke :-

The brick house will give him :-

(5-3)/(1.07)¹ =1.87 utility

While the straw house will give him :-

4/(1.07)¹= 3.73 utility

Therefore Luke will choose the straw house .

Answer 2 :-

In case of Kethan ,

the straw house will give him :-

4/1.03=3.88 utility

While ,

the brick house will give him ,

2/1.03 = 1.94 utility

Thus Kethan will choose straw house .

Answer 3 :-

In case of arjun who is living with Mama Pig for first period :-

In brick house :-

Utility = 5/(1.03)¹+((5-3)/(1.03)²= 4.85+1.89

=6.74

While in straw house he will get ,

Utility = 5/(1.03)¹+4/(1.03)²=4.85+3.77=8.62

The first period utility after reducing the costs of making a house is greater than one which implies the benefit of constructing brick house is providing him with additional benefit .(4.85-3)>1(additional utility from brick house construction).

Therefore Arjun will choose to construct brick house .

Answer 4 :-

In this case Arjun is not living with Mumma Pig in first period ,

Now his utility for brick house ,

4/(1.03)¹+2/(1.03)²

^{= 3.88+1.88}

^=5.76

Now for stick house ,

Utility = 4/(1.03)+4/(1.03)²

=3.88+3.77

=7.65

In this case if we reduce construction cost from period one utility then ,

(3.88-3)<1 .

Thus he has no motivation to build the brick house in second period and will choose to construct straw house .

Answer 5 :-

The verbal commitment of mama pig that she will keep Arjun in her brick house for the first period will motivate Arjun to construct a brick house in in second period .

Answer 6 :-

Consolidated effect on utility in long run for Arjun :-

Period	Brick house	Non Brick house
1	5	4
2	(5-3)=2	4
3	5	4
4	5	4
5	5	4
Total	22	20
Discounting	22/(1.08)5=14.99 utility	20/(1.08)5=13.61 utility