Homework 1: Preferences, Lotteries, Expected Value and Expected Utility

Transcription

1 Homework 1: Preferences, Lotteries, Expected Value and Expected Utility Solution Guide 1. Study your own preferences (a) In each case, indicate if you prefer one of the goods over the other or if you are indifferent: i. cheese and eggs ii. bread and butter iii. milk and sugar iv. cheese and potatoes v. ham and cheese vi. ham and eggs vii. potatoes and eggs viii. bread and milk ix. sugar and cheese (b) Check if your preferences are transitive. [SOLUTION] (a) The answer to this question depends on each one s preferences. Typically, different people will produce different answers. (b) Based on your answers on (a) you must check that your preferences are transitive. That is, you must check that whenever the alternative A is preferred to the alternative B, and alternative B is preferred to alternative C, then A should also be preferred to C (for whatever alternatives A, B, and C in cheese, eggs, bread, butter, milk, etc.) For instance, if you find that cheese is preferred to eggs, eggs preferred to ham, and ham preferred to cheese, then your preferences are NOT transitive. Preferences are transitive whenever such cycles do not occur. 2. Represent the following situation. I am considering going for a vacation during the Easter break, but I am not quite sure which is the best choice. Usually it rains (with probability 0.7), in which case going to the mountains might be a good choice (it gives me utility of 100), but if the weather is sunny I will not enjoy the mountains so much (my utility will be 30). If contrary to the tradition the weather is sunny and hot (with probability 0.3), the best choice will be the beach (utility of 120), but spending my holiday on the beach when it is raining is horrible (utility -20). 1

2 [SOLUTION] My actions (the alternatives I have to choose from) are : Mountain and Beach. That is, I must decide whether go to the mountains or to the beach. These determine the action branches at the beginning of the three. Then, after I have made my choice (Mountain or Beach), the weather may be Rainy (70% chances) or Sunny (30% chances). These define the chance branches after each one of the choice branches The sequence of the tree is important. The tree must represent the real problem: I have to decide were to go before knowing the weather. The problem is represented in the following tree: Rainy (0.7) 100 Mountain Beach Sunny (0.3) Rainy (0.7) Sunny (0.3) There are 3 cards on the table, you are unable to see the colors, but you know that there is one blue, one yellow and one red. If you take the blue one you have a probability of 70% of winning a prize of $100, if you take the yellow one the probability is 50%, and the red one gives you a probability of 30%. (a) Represent the lottery you are playing. (Note: The probability to withdraw any of the cards is 1/3) (b) What is the set of final outcomes? (c) Represent the reduced form of this lottery. (d) Are you willing to play such gamble? 2

3 [SOLUTION] (a) In this case there are no choice branches, you have nothing to choose (well, at the end you must decide if you want to play the gamble or not, but this is not part of the tree). The gamble proposed is just a lottery where everything is random. The sequence of the gamble is that first you choose a card (without knowing its color) and then you play a lottery whose probabilities of winning depend on the color of the card you chose in the first place The problem is represented in the following tree (b) The set of final outcomes in this case is 0 and 100. These are the only possible results. Thus, what matters at the end is what are the probabilities of obtaining each of these two outcomes. We find them in the reduced lottery in the next item (c) We compute now the probabilities of earning 100 and of earning 0 To earn 100, the possible events are: (blue card and win) or (yellow card and win) or (red card and win) In terms of probabilities: 3

4 p(100) = p(blue) p(win)+p(yellow) p(win)+p(red) p(win) = = 1 3 (0.7)+ 1 3 (0.5)+ 1 (0.3) = Thus, we have that p(100) = 0.5 and therefore, p(0) = 0.5 as in the following tree (d) I would play the gamble since I have nothing to lose! (But different people might have a different view) A different question would be: How much are you willing to pay to play the gamble? The answer to that question would depend on each one s risk attitude 4. John Smith has in mind 3 investment opportunities: to invest in the wood industry, water energy, and solar energy. The outcome of each of this options depends on the energetic policy of his country. If the government adopts a sustainable growth economic policy (this can happened with probability of 0.5) the payoffs will be different compared to the case of an open market policy, as in the table below: Ecologist Open Market Wood Water Solar (a) Represent his decision problem. (b) Assume that he is risk neutral (that is, he values lotteries according to their expected value). What is his choice? (c) Assume now that he is risk averse and his utility function is u(x) = x. What is his choice now? 10 (d) Assume now that he is risk lover and his utility function is u(x) = 10x 2. What is his choice? 4

5 [SOLUTION] (a) My actions in this case (the alternatives I have to choose from) are : Wood, Water and Solar. These define the action branches at the beginning of the three. Then, after I have made my choice, the energetic policy may be Ecological (50% chances) or Open Market (50% chances). These define the chance branches after each one of the choice branches Again, the sequence of the tree is important. The tree must represent the real problem: I have to decide my investment before knowing the energetic policy. The problem is represented in the following tree 5

6 (b) If John Smith is risk neutral we have that u(x) = x. Hence, E(Wood) = (0.5) 25+(0.5) 59 = 42 E(Water) = (0.5) 49+(0.5) 36 = 42.5 E(Solar) = (0.5) 64+(0.5) 36 = 50 Hence, a risk neutral John Smith would choose Solar as it provides the highest Expected Utility (c) If John Smith is risk averse we have that u(x) = x 10. Hence, E(Wood) = (0.5) 10 +(0.5) 10 = E(Water) = (0.5) 10 +(0.5) 10 = E(Solar) = (0.5) 10 +(0.5) 10 = 0.7 Hence, a risk averse John Smith would also choose Solar as it provides the highest Expected Utility (d) If John Smith is risk lover we have that u(x) = 10x 2. Hence, E(Wood) = (0.5) 10(25) 2 +(0.5) 10(59) 2 = E(Water) = (0.5) 10(49) 2 +(0.5) 10(36) 2 = E(Solar) = (0.5) 10(64) 2 +(0.5) 10(36) 2 = Hence, a risk lover John Smith would also choose Solar as it provides the highest Expected Utility 6