Lecture 1 Consistency

Lecture 1 Consistency - Economics 101, UCLA Jernej Copic...

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Unformatted text preview: Economics 101, UCLA Jernej Copic Lecture 1. Rationality in decisions and expectations over lotteries. Bob interrupted Number One and Alan in their conversation. Indeed, more of an instruction than a conversation: "When you make it to the W coast, make sure to first contact Squitty. He will give you some necessary training. Also, here is the stash of greenbacks in 1-dollar bills, and our depot of flower pots is at your disposal, should you need to temporarily borrow some for your training; just make sure that there are no flower pots missing when you are done. Whatever cash you are left with shall be your reward, and you don't need to pay any extra fees to Squitty, just enjoy the training, hehe... So? Why are you still here, staring at me? Start walking already, or you will never make it there!" Accompanied by these warm words of fatherly farewell Alan walked through the door of the little flower shop and embarked on his mission. As soon as he closed the door behind him, Bob jumped: "What is it that Squitty will train him in?" - "Rational behavior," was the cold reply. "Huh?" "Well," continued Number One, "I know I can be pretty sure that Alan will not behave rationally, unless he gets some training." "Why is that?" "He seems to not choose consistently. For example, when he is lounging in his apartment on Sunday afternoons above the 5th Ave, figuring out what to do, he sometimes has a choice between watching TV, hanging out with you guys, and sleeping. When it is not too noisy to sleep, that is. Of course, when it is too noisy to sleep, he only has the choice between watching TV and hanging out. Now, my point is that he chooses hanging out with you guys in the latter case, and he decides to watch TV in the former case." Bob was still confused: "I still don't understand why that would be inconsistent. Perhaps he just likes it that way, or maybe his mood changes, or maybe he has enough of one or the other and wants to do something different for a change?" "First, Alan is a very monotonous guy and a man of habit. He would never try something 1 different than what his impulses guide him to do, and as we all know, his impulses are not particularly interesting. He has no moods, and he never does anything to change himself, such as for example reading a book, or attending a lecture. One could say he is a pretty simple man. So much so that he even influences the climate in the vicinity of his apartment so even the weather down there never changes. Still, Alan never likes two things just equally well - he is never quite indifferent." Bob insisted. "But does that make him inconsistent?" "Of course not - His choices do. See, let's describe his available choices, or his choice set by {x, y, z} in the first case, and {x, y} in the second case. You follow?" "You mean x would be `TV', y would be `hanging out', and z would be `sleep' ?" "Good, you are a genius! I hope you understand that the advantage of this description is that we can in this way describe many other situations that Alan may face..." - Bob's face lit up. "Oh, cool, you are right!" "Ok, so now things are very simple. Alan's choices are {x, y} y; {x, y, z} x, and that is inconsistent. The reason is that Alan chooses y when x is available, so that means he prefers y. But then, he chooses x when y is still available, so now he suddenly prefers x, and that is inconsistent!" "Oh, I see," said Bob, "so if instead, his choice were like mine, which is {x, y, z} z, would he then be consistent?" "Well, in that case, we would not know for sure quite yet, but things would be more promising. It would depend on his choice when faced with other possibilities. But we could just go by a simple principle, called the Weak Axiom of Revealed Preference (WARP). It says that whenever someone chooses x when y is available, they cannot choose y when x is still available. And see, Alan clearly doesn't satisfy that." Bob pondered for a while, and then replied. "Okay, so perhaps Alan is not consistent, but why should that be problematic?" "For one thing, he is unpredictable. If he chose as you suggested, {x, y} y, and {x, y, z} 2 z, and assuming that he satisfied WARP with all his other choices as well, then we could deduce that he liked z more than y, and y more than x. To be concise, we would then normally write z y x. The symbol is called a strict preference relation, because Alan is never indifferent. So whenever we would see him facing a decision between some of these possibilities, we could with certainty predict what he would do. But this guy is inconsistent, so we can't say that he likes one more than the other, and so we can't predict what he will do." "I see. I am just wondering if perhaps this could be advantageous for him." "Yeah, right... Huh! - the opposite! And that's exactly why I sent Alan Squitty's way, who will give him some rationality training. Squitty has precise instructions from me about how to do that, and it is also why I gave Alan access to our flower pot depot." - "What do you mean?" Asked Bob. "First, remember that there are two types of flower pots. The big green ones, and the smaller but fancier looking blue ones. For some inexplicable reason, on all odd days in a month, Alan prefers the green flower pots, and on even days, he prefers the blue ones. The first thing that Squitty will do is that he will figure out how much Alan prefers the green ones on the odd days. Just to make things simple, lets use x and y again: x =`green flower pot', and y =`blue flower pot'. Now what Squitty will do is that he will offer different payments to Alan for the pots until he figures out what is the money-equivalent difference between the two pots for Alan. For example, he may test Alan on two choice sets {x + $2, y}, and then {x + $2.01, y}. Suppose it turns out that on even days, in the first case, Alan chooses y, and in the second case, he chooses x+$2.01. On odd days, Squitty offers Alan choices {x, y +$1}, and {x, y + $1.01}, and suppose Alan's choice is {x, y + $1} x, {x, y + $1.01} y. By this system Squitty would figure out that Alan likes x over y on odd days by 1$. On even days, Squitty would find out that Alan likes y over x by $2. Finally, by offering Alan a choice between not doing anything at all and buying a y for some amount Squitty will be able to also find out how much Alan likes the blue flower pots in terms of money, not only the monetary difference that Alan assigns to the different flower pots. For example, Squitty could give Alan a choice like {y - $1, 0}, which would represent buying y for $1 or doing nothing. Suppose that in this manner Squitty finds out that in fact, on odd days, Alan 3 doesn't care for blue pots at all, which means that Alan on odd days cares for blue pots as much as for $0. On even days, Squitty finds find out that Alan doesn't care for green pots. You following?" Bob nodded silently. "The point is that by such a procedure, Squitty can find out how much Alan likes the two pots in terms of money." "Oh, I get it! I have heard of something similar! That is exactly how economists measure consumers preferences over groceries: they observe at what prices the consumers still buy some groceries and at what prices they don't buy them anymore. Well, I guess, things must be more complicated there because the choice sets are much bigger, and the preferences over goods may in some way interact... But the principle is still the same, right?" "Yep, but don't interrupt my thread - you know I am old and can easily forget what I have been talking about - show some respect! Now, under the above Alan's preferences, the plan for teaching Alan some rationality is straightforward. On day one, Squitty will offer Alan the choice between Alan doing nothing, or giving Squitty x and getting $0.99 or giving y and getting $0.05. Since Alan here gives flower pots, let's write that choice set as {-x + $0.99, -y + $0.05, 0}. Clearly, on day 1, Alan's choice will be -y + $0.05. On day 2, Squitty will present Alan with the choice of {x - $0.01, y - $1.95, 0}. Since Alan on even days likes x as much as $0 and y as much as $2, he will decide to take y and pay Squitty $1.95. In this way, Alan will after 2 days have exactly the same amount of flower pots as at the beginning, so he won't have to worry about me, but will be $1.90 down cash-wise, hehe. Moderately painful, I should say. Then, on days 1,3,5,..., Squitty will give Alan the choice {-x + $0.99, -y + $0.05, 0}, and on days 2,4,6,..., the choice between {x - $0.01, y - $1.95, 0}. Until Alan learns to be consistent, he will be losing money. And when Alan learns to be consistent, he will every day attach the same payoff to the same possibility. This payoff can then be called Alan's utility function." "Oh, I understand. If Alan is inconsistent, then it is like there are two Alans who are in conflict with one another and that's why Alan can be ripped off!" Exclaimed Bob. "I have a question though: is it necessary to think of his payoff in terms of money? Can't we 4 think of something else?" "True, you are right. Let's go back a step. When Alan is consistent, we can figure out a unique ranking of the different alternatives that describes his choices, which is described by the prefence relation . Then, instead of thinking of the ranking, we can think of some numbers which describe his satisfaction from these different alternatives in the right way so that the alternative he likes most gets the highest number. For example, suppose he ranked sleep hangout T V . Then we could ask him how much he liked each one of these, on the scale of 10. And maybe he would say sleep was 9, hang out was 7 and TV was 6. These numbers are then a particular representation of Alan's utility function u, so that u(z) = 9, u(y) = 7, and u(x) = 6. See, now it has nothing to do with money. But I kind of like to simplify my life by thinking in terms of money, and that is also easy to measure to figure out how much you like something in terms of money, I will just check out what you do when you get different prices for that object, and then see when you buy it and when you don't. Besides, as good old Methuselah used to say, `money' is the greas of the univers, my dear." "Okay, okay, so what you are saying is that we can assign these utility numbers in many different ways, as long as we describe the order in which a consistent person ranks his alternatives, and one possibility, which is easy to implement in practice when we have money is through assigning monetary values to the alternatives. But now I am again confused about why we would want to do that and not just simply stick with ranking?" "I'll give you a good reason. So you like gambling, correct?" "Oh, yeaaah!" Bob's eyes got a weird fervent shine. "See, when you play the roulette, how do you figure out what you should be doing?" "I mean, I just sort of do it by the feeling, if you know what I mean..." "Alright, but see, the thing is the following. If you have 1 green, what are your odds of winning when you bet on a red?" "Oh, comon man, that is super easy - since there are 18 reds and 18 blacks and one green, the odds are 18 ." 37 "Right, so then, when you bet $1 on red, what is your expected payoff in terms of money?" 5 "Well, that's pretty simple too it's 18 19 1+ (-1). 37 37 But I don't really care about that. I just care about what happens when I bet. I am sure I can win!" "Indeed, you are a genius! I guess you don't exactly play roulette once, right? So if you keep betting, then after a while, you will find out that on average, you are getting your expected payoff in every game." Bob now seemed immersed in thought, deep as that could be. After a while, he emerged from his trance, somewhat wiser, but also somewhat worried. "Oh man! Now I get it why I always lose money on the roulette - my average payoff is negative oh, those bastards! Ah, I still kinda enjoy gambling - it's not all about the money..." "Even in general, you see," Number One continued with his lecture entirely undisturbed, "whenever you are faced with a lottery over different things, when you can compute the expectation of this lottery, then you can figure out what you want to do in a very informed way. Like, I had this crazy uncle, who knew me damn well, and he really was a brute. So he had this fantastic muscle car collection. Once upon a time, he cornered me and wouldn't let me go before I made a choice. The choice he gave me was between three different lotteries. He had a box of ten-sided dice, by the way. The first lottery was him giving me a `69 chevy with probability 0.1 (that is if the roll of a die was 10), a `70 mustang with a probability 0.3 (when the roll of a die was 7, or 8, or 9), and me cleaning his toilet for the rest of his life with probability 0.6 (if the roll was 6 or less). The second lottery was between me buying him cigarettes with probability 0.5, and he giving me a collection of his disgusting stamps with probability 0.5. And the third possibility was that I just go free and he gives me a kick in my hind (no lottery there)." "I get it now! So by knowing your utility values of different alternatives, chevy, mustang, toilet cleaning, cigarettes, ugly stamps, and a kick in the but, you can actually figure out what is your best choice in expectation! So what happened?" "Ugh, I should have just taken that kick... What the hell you think I was doing on monday mornings before school? That 6 old bastard sure died of pure malice..." APPENDIX. Choice sets, e.g., {x, y}, {x, y, z} represent the possibilities available to an individual at some economic decision; x, y, z are alternatives, or goods. Consistent choices satisfy WARP: if x is chosen when y is available, then y cannot be chosen when x is still available. If choices of an individual are consistent, then they can be rationalized by a preference relation . A preference relation can be represented by a utility function u. When there is money, then a convenient utility function measures by how much an individual likes different alternatives in terms of money. This can be elucidated by presenting individual with different choice sets, where prices, or relative prices, for the goods are varied in a given choice set. If an individual's choices are not consistent then he can be exploited; Consistency is equivalent to rationality. When a choice set consists of lotteries, a rational individual will compute the expected value of each lottery, and then decide accordingly. A lottery is given by probabilities of different alternatives, e.g., L = (px , py , pz ), where px is the probability of alternative x. The individual's expected utility from lottery L is then, EL u = px u(x) + py u(y) + pz uz . An individual prefers lottery L over lottery L , if, EL u > EL u. 7 EXERCISES. 1. Is an indiviual with the following choices consistent? {x, y, z} x, {x, y} y 2. Is an indiviual with the following choices consistent? Explain. {x, y, z} z, {x, y} x 3. Suppose Bob likes a Rip Curl T-shirt. How would you find out how much he likes the T-shirt in terms of money? 4. Suppose Alan can buy either a cup of tea or a cup of coffee. How would you fidn out by hoe much he prefers one over the other? 5. No One has the following utility over wine (x), cheese (y), dried meat (z), and olives (w): u(x) = 2, u(y) = -1, u(z) = 1, u(w) = 3. Which of the following lotteries does he like most? L1 = ( 1 , 1 , 1 , 1 ), L2 = ( 1 , 0, 1 , 1 ), L3 = ( 1 , 1 , 0, 1 ). 4 4 4 4 3 3 3 4 4 2 8 ...
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This note was uploaded on 02/06/2012 for the course ECON 101 taught by Professor Buddin during the Spring '08 term at UCLA.

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