LECTURE NOTE 2 UTILITY

LECTURE NOTE 2 UTILITY W & L INTERMEDIATE MICROECONOMICS PROFESSOR A. JOSEPH GUSE In the last note Preferences we introduced the mathematical device of a preference relation. Let it be said first that the introduction of utility functions is not meant to replace nor is it an alternative to that model of consumers tastes. Utility functions are only meant to represent preference relations. Economists use utility functions as a matter of analytical and computation convenience. (And convenient they are!) Definition 1. A Utility Function u : X R represents a preference relation if for all pairs of consumption bundles {x,y} X X, x y u(x) u(y). Here X is meant to denote the set of all consumption bundles. Lower case x X and lower case y X then are particular bundles. x, for example might represent (5 haircuts, 2 tickets to see Led Zeppelin, and 4 bananas) while y might be (1 haircut, 0 tickets, and 16 bananas). The definition of representations simply says that if ranks the first bundle above the second then the utility function must assign the first bundle a higher number. The numbers themselves have meaning only in so far as they order the consumption bundles. Example 1. Suppose that John s preferences given by (which we take to be rational) ranks the alternatives x, y (as described above) and z = (8 haircuts, 1 ticket and 13 bananas) as follows: x y z. In words, John prefers x to y and y to z. Any of the following utility functions satisfy the requirements of representation. bundle u û ψ v x 3 30 400001 87 y 2 20 400000 3.1415 z 1 10-6 2 Note that the magnitude of the numbers assigned to bundles does not matter. Furthermore, the relative magnitude of the differences doesn t matter either. For example, one might read ψ to mean that John likes y a lot more than z and John likes x only a little more than y. Meanwhile, applying a similar interpretive rule to the function v, we might conclude the opposite. SUCH INTERPRETATIONS ARE UNFOUNDED. John preferences are what they are; ψ and v are both perfectly good representations. We say that we only care about the ordinal properties of a utility function, not its cardinal properties. One problem with this statement is that people often make statements about their tastes which sound cardinal in nature. Sally might say, I like licorice jelly beans only a little more than cinnamon, but I like cinnamon 3 times more than I like mint. Is the ordinal model of human preferences missing something important about the way the human mind experiences or imagines different choices? The answer is no. 1

2 W & L INTERMEDIATE MICROECONOMICS PROFESSOR A. JOSEPH GUSE Statements such as the one about preferences for jelly bean flavor can only have real meaning if quantities like a little more and three times refer to something real. Perhaps Sally is only willing to give up a little bit more time and effort to get a licorine bean instead of a cinnamon, but would work three times as long for a given amount of cinnamon beans than she would for the same number of mint ones. In other words, when pressed, the quantities implicit in Sally s original statement must refer to some unnamed hypothetically sacrificed good such as leisure time in order for them to have any meaning. Sally s feeling about jelly beans are then easily captured by expanding the commodity space to include her leisure time and using an ordinal preference relation to compare combinations different flavored beans and leisure time. 1. Monotonic Transformations Definition 2. Let S R. A function f : S R is said to be strictly increasing everywhere or monotone increasing if for all x S and all δ > 0 we have f(x + δ) > f(x). Monotone or monotonic when applied to functions means that they always move in the same direction. A function can be monotone increasing or monotone decreasing. We are particularly interested in the increasing type. A monotone increasing function which is differentiable everywhere would have a strictly positive first derivative at all values in the domain. Sometimes we may want think of our monotonic tranformation as defined on some restricted domain. For example, the function f : [0, ) R with f(x) = x 2 is monotone increasing only if we are careful to restrict the domain to the set of all non-negative real numbers. The notion that it does not matter how a utility labels indifference curves as long as it assigns higher numbers to more preferred bundles is captured mathematically as follows. Definition 3. If the utility function u : X R represents preferences given by some relation, than u : X R with u (x) = f(u(x)) also represents if f is strictly increasing for all values in the image of u. When f is strictly increasing as required to preserve the representation, we say that the new utility function u is a monotonic transformation of the original u. 1 It is important to keep in mind that we are using the term monotonic in two different ways - one to describe preferences and this new usage to describe transformations of utility functions. Don t get confused. When we say that preferences are monotonic, we mean that the consumer can always be made better off by increasing the quantity of each good. (e.g. moving to a consumption bundles that has more pizza and more beer.) A monotonic transformation, on the other hand, refers to building a new function by wrapping it in a strictly increasing function. Suppose for example that some utility function u represents preferences which are not monotonic. For example u(z,b) = b (z 5) 2 represents preferences where sometimes less pizza is better and hence not monotonic. Now consider the utility functions u (b,z) = tanh ( b (z 5) 2). Since tanh is a strictly increasing function, this new utility function is a monotonic transformation of the original and it still represents preferences which are NOT monotonic. Got it? 1 This is quite remarkable at some level because one implication is that continuity of preferences are preserved even under non-continuous monotonic transformations. Check this.

LECTURE NOTE 2 UTILITY 3 2. Are All Rational Preferences Representable? (Technical) A rational preference relation will consistently rank any subset of consumption bundles from most preferred to least preferred. Shouldn t it be the case that any such relation may be represented by a utility function? We just need to assign higher number to more preferred bundles. There s plenty of numbers, right? Surprising, it it turns out, that there are not always enough numbers. The most commonly sighted example is the case of lexicographic preferences. Example 2. Let be defined on R 2 + as follows. For any two bundles (x 1,x 2 ) and (y 1,y 2 ) we have (1) x 1 > y 1 x y (2) x 1 = y 1 and x 2 > y 2 x y (3) x 1 = y 1 and x 2 = y 2 x y The lexicographic ordering ranks one bundle over another if it contains more of the first commodity. In cases where two bundles have exactly the same quantity of the first good, the ranking is decided by comparing how much of the second good the bundles have. Ties are only possible when the two consumption bundles have the same quantity of both goods (i.e. they are the same bundle). It is easy to verify that the lexicographic ordering is rational. (try proving this as a problem.) The key observation for understanding why such a preference relation cannot be represented by a utility function is that indifference sets are singletons. A utility function, if it were to represent this relation, it would be charged with the difficult task of assigning a unique number to every point in the positive quadrant of two-dimensional space. That, in itself is actually possible mathematically (see space-filling curves, Cantor, Hilbert, Peano). However, the task is subtlely worse than that. It requires those numbers to be assigned in such a way that there is a gap between the utilities levels of (x,0) and (x, 1) and that there be such a gap for all x. Despite that fact that there are uncountable infinite real numbers, there simply are not enough numbers to do that! The complete proof is more technical, but that is the basic intuition. (Try proving this rigorously as a problem. Hint the proof relies on the impossbility of constructing a one-to-one mapping from an uncountable set to a countable set.) It is worth noting how strange perfectly lexicographic preference would be. Suppose someone has such preferences for cocaine and food. When presented with any two bundles, this person looks first to the cocaine. If one bundle has more, he picks it. He picks even if the other bundles has a lot more food and just a tiny bit less cocaine. He only compares bundles on food, when the amount of cocaine is exactly tied. This means, that even though, food is a good, this person is never willing to substitute it for cocaine at any rate. This sounds a little bit like the behavior of the MRS in the case of perfect complements, but it is actually quite different. In that case, someone with 5 right shoes and 1 left shoe is not willing to give up any left shoes for another right shoe and if they did get another right shoe, they would be on the same indifference curve. In this case, the consumer is not willing to give up any cocaine for another pound of food and yet, if they got another pound of food they would report being better off! Is the existence of preference relations which cannot be represented by a utility function such as the lexicographic ordering a big problem for our theory? Not really. If we believe that everything has its price, then certainly it is not a problem. But even if there are some things on which a consumer may be

4 W & L INTERMEDIATE MICROECONOMICS PROFESSOR A. JOSEPH GUSE unwilling to compromise, such preferences can still be representable as long as those uncompromising goods come in discrete quantities. The problem with cocaine in the previous example was not so much that the consumer was willing to give up all his food (not matter how much he has) to get the smallest portion of addition cocaine you can imagine. The problem was that there were an uncountably infinite number of levels of cocaine from which he was willing to make such a trade. While it is easy to imagine a good for which a consumer might sacrifice any amount of other consumption (say the life of a cherished family member), it is difficult to imagine uncountably infinite levels of such goods. For example, a parent might be willing to sacrifice any amount for their child s health if the choice is between good health and very poor health or death, but if health is thought of as a continuum, would that parent really give up everything so that the child experiences one less sneeze over a lifetime? or lives a fraction of a second longer? It that if adding another member to our collection of nice assumptions called continuity make the issue go away. A utility function representing a continuous (and rational) preference relation is guaranteed to exist. 3. M.R.S. versus Marginal Utility Marginal utility is defined at the rate of change in utility as one of the goods in a consumption bundle increases. MU Z = u(x z,x b x z ) What good is this notion? We don t care about the cardinal properties of utilities, so telling me a rate of change in it is, on its face, a useless exercise. Afterall, a we could apply a monotonic transformation to u and the give the marginal utility of the new utility function a completely different shape. Where under the original utility function it may have appeared that utility increases at a decreasing rate, we could easily find a tranformation that makes it look like utility increases at a decreasing rate. Both would be correct as long as they both represent the underlying preference relation. However, it turns out that taking the ratio of two marginal utility functions can tell us something very useful: the marginal rate of substitution. Intuitively the reason we can take two measures - which when taken separately are useless - and make something useful out of them is that when you take the ratio of two marginal utilities the cardinal properties in each cancel each other out! In fact, the ratio of marginal utilities measured at a particular consumption bundle is the same no matter which utility function is used to represent the underlying preferences. To see this, suppose that u represents and f is a strictly increasing function. Then v() = f(u()) also represents.

LECTURE NOTE 2 UTILITY 5 v(x z,x b )/ x z v(x z,x b )/ x b = f(u(x z,x b ))/ x z f(u(x z,x b ))/ x b f u u x = z f u u x b = u(x z,x b )/ x z u(x z,x b )/ x b