Choice Theory PDF

2 Choice Theory People make choices all the time for a variety of reasons. In economics, we explore decisionmaking from the perspective of “rational” individuals. In these notes, we will explore exactly what the assumption of rationality entails and its implications...

2 Choice Theory People make choices all the time for a variety of reasons. In economics, we explore decisionmaking from the perspective of “rational” individuals. In these notes, we will explore exactly what the assumption of rationality entails and its implications for consumer choice. Concepts Covered Axioms of Rational Choice Utility Functions Marginal Rate of Substitution Indifference Curves 2.1 Axioms of Rational Choice Although rationality in everyday language can mean many things, in economics it has a very specific (and weaker than you might think) definition. In particular, it does not necessarily mean that an individual always makes the “right” choice or that they never make mistakes. Essentially, all it means is that an individual’s choices are internally consistent. We now turn to a more formal definition of rationality in choice. Notation To introduce the idea of rational choice, we need some new notation. Imagine a consumer is making a decision between two choices A and B. These can be goods (deciding between getting pizza or a burger) but they could also be more abstract choices (deciding whether to come to class or take a nap). We use the following notation to say A is strictly preferred to B AB If a consumer is indifferent between A and B we will write A∼B If A is weakly preferred to B (either preferred or indifferent), we write AB 1 Three Axioms A rational consumer in economics have preferences that satisfy the following three assumptions 1. Completeness: For any two possible choices A and B, either A  B or B  A The completeness axiom implies that consumers must be able to state their preference over any two pairs of goods. Note that the axiom does not rule out indifference. Completeness allows a consumer to be indifferent between two choices as long as they are aware of their indifference. It rules out cases where a consumer has no idea whether they prefer one good to another. 2. Transitivity: For any three possible choices A, B, and C, if A  B and B  C then A  C You are probably familiar with the transitivity property from math classes and the idea here is similar. If I like blue t-shirts more than green, and green more than pink, then it would be strange if I liked pink more than blue. Transitivity ensures a rational consumer would not. 3. Continuity: For any 2 choices A and B, if A  B then anything sufficiently close to A should also be preferred to B The continuity axiom is the most technical and the definition I have given is not the most precise (it is also the least important for our purposes and is sometimes left out of the definition). As an example, assume a consumer strictly prefers a 250 mL can of Coke to a 250 mL can of Sprite. Continuity says they should also prefer a 249.9 mL can of Coke to a 250 mL can of sprite. 2.2 Utility Functions The reason we use the three axioms of choice presented on the previous page is that they allow us to move from a relatively abstract concept of preferences into more familiar mathematical territory. In particular, it can be shown that if the three axioms of choice hold, it is always possible to describe a consumer’s preference using a mathematical function. Definition A utility function is a function U such that whenever A  B, U (A) ≥ U (B) (and whenever A  B, U (A) > U (B)). In other words, for any choice (remember these choices can be anything), we can assign a real number value that describes how much happiness, or utility that choice brings to the consumer. This transformation from preference relations to a real valued function is extremely important because it allows us to use mathematical tools to explore consumer choice. 2 Non-Uniqueness of Utility Note the the theorem mentioned above does not say that there is only one way to map a consumers preferences into a utility function. In fact, a consumer’s preference can be described by infinitely many utility functions as long as they preserve the same ordering of choices. In more technical terms, we say that the number produced by a utility function is ordinal, meaning that it can only represent the order of a consumer’s preference, but not the magnitude of those preferences. Alternatively, a cardinal function gives numbers that represent some magnitude. For example, a function that described distances would be a cardinal function. 2 miles is not only longer than 1 mile, it is exactly twice as long. On the other hand, if eating an ice cream cone gives 10 utility and eating a slice of pizza gives 5, it does not mean that the individual likes ice cream cones twice as much (or that they would like 2 ice cream cones as much as one slice of pizza). All we could say in this case is that one ice cream cone is preferred to eating one slice of pizza. Often it is mathematically convenient to transform a utility function into a different function that represents the same preference. It turns out that if we want to preserve the same preference ordering we can apply any transformation to a utility function as long as it is monotonically increasing. This idea is easiest to see through an example. Example Let’s assume that a consumer has four hours of free time and they are trying to decide how to allocate it between watching Netflix and playing video games. Assume the consumer’s preferences over the two activities can be described by the following utility function U (N, V ) = N V 2 Where N is the number of hours spent watching Netflix and V is the number of hours spent playing video games. Compare the following possible combinations of (N, V ): A : (4, 0), B : (3, 1), C : (2, 2), D : (1, 3), E : (0, 4). Show that a consumer with the utility function above has the following preference ordering DCBA∼E Now look at the following utility functions. Which ones preserve this preference ordering? Which do not? 1. U (N, V ) = N V 2 + 10 2. U (N, V ) = ln(N ) + 2 ln(V ) 3. U (N, V ) = N + V 2 4. U (N, V ) = N 1/2 V 5. U (N, V ) = −N V 2 6. U (N, V ) = N 2 V 3 Hopefully you found that 1,2, and 4 preserve the same ordering of preferences while 3, 5, 6 do not. What’s different about these transformations? In each case, we can find a monotonically increasing transformation of the original utility function that turns it into the new function. For number 1, the transformation is f (x) = x+10 for 2 it is f (x) = ln(x) and for 4 it is f (x) = x1/2 , which are each increasing functions. Note that while I only used 5 different combinations as an example, the ordering would be preserved for any combination of N and V. For choices 3, 5, and 6, we cannot find such a function. There is no function that maps the original function into numbers 3 or 6 and number 5 needs a decreasing function (f (x) = −x, which would exactly flip the preference ordering. 2.3 Marginal Rate of Substitution Definition An important tool in relating a consumer’s preference over various goods is the marginal rate of substitution (MRS). If we have a utility function over two goods, U (x, y), the MRS is defined as ∂U ∂x M RS = ∂U ∂y Let’s think about what the MRS tells us. On top we have what we call the marginal utility of x (the partial derivative of utility with respect to good x). It is the increase in utility that results from increasing the amount of good x by a small amount. On the bottom we have the same for y, the marginal utility of y. The MRS is therefore the ratio of the marginal utilities of x and y. Example Let’s assume utility is linear in x and y U (x, y) = 2x + 4y With this utility function, the marginal utility of x is 2 and the marginal utility of y is 4 so the MRS is 2/4 = 1/2. We can interpret this value as the amount of y the consumer would need to receive in order to make up for losing one unit of x. For example, let’s say the consumer started out with 5 units of x and 5 units of y. We can see that this basket would give the consumer a utility of 30. But there are other combinations that would also give us 30 utility. The MRS tells us how to find these other baskets. It tells us that if x goes down by 1 then y has to go up by 1/2. So we know that (4, 5.5) will also give us the same utility (plug it into the utility function to prove it to yourself). In this example, MRS is constant so the consumer always receives twice as much utility from y as they do from x. In general, MRS will not be a constant, but a function of x and y. For example, if we had a utility function U (x, y) = x1/2 y 1/2 , we would get an MRS that changes depending on how much x and y the consumer currently has Ux (1/2)x−1/2 y 1/2 y M RS = = 1/2 −1/2 = Uy (1/2)x y x 4 2.4 Indifference Curves Definition The previous example showed how the consumer could achieve the same utility with a different combination of goods. An indifference curve plots all values that give the consumer a given utility level. To find an indifference curve, we can set the utility function equal to some arbitrary utility level, which we will denote Ū U (x, y) = Ū Note here that U (x, y) is a function, while Ū is a number. For any utility function, there are infinitely many indifference curves (one for each value of Ū Example Let’s return to our example from above with U = 2x + 4y and create a graph that plots all combinations of x and y that give this consumer exactly 30 utility (so we are setting Ū = 30 30 = 2x + 4y =⇒ y = 7.5 − 0.5x Notice that this function is a line with y intercept at 7.5 and a slope of -1/2, which is (the negative of) our MRS. As we increase the amount of x we have, moving right along the curve, we need less y to maintain the same utility level. The MRS tells us how much less. y 7.5 (5, 5) x 0 15 Again, this indifference curve plots every combination of x and y that give the consumer exactly 30 utility. Although we only plotted one indifference curve, we could draw one for every possible level of utility. 5  When we move away from linearity, it gets a bit trickier. Return to the second example above U (x, y) = x1/2 y 1/2 Now let’s set an arbitrary fixed utility level Ū and solve for y as a function of this utility and x Ū 2 Ū = x1/2 y 1/2 =⇒ y = x Plugging in various values for Ū will give us our indifference curves. Some examples are plotted below (it would be good practice to sketch an indifference curve by hand by plugging in values for x and y) 10 8 6 y 4 Ū = 4 2 Ū = 3 Ū = 2 0 Ū = 1 0 1 2 3 4 5 x Along each of these curves the consumer receives the same utility. As we move up and to the right (more x and more y), we increase utility. Down and to the left decreases utility. Again, the big difference between this utility function and the example on the previous page is that marginal utilities (and therefore MRS) are no longer constant. We calculated above that this utility function has an MRS equal to y/x. The MRS in this case again represents the negative slope (derivative) of an indifference curve, but now that slope is changing at each point on the curve. Convexity In the example above, we can see that as we increase the amount of x we have, the increase in utility relative to y falls and vice versa. For example, let’s say the consumer initially has 10 units of x and 10 units of y, which gives us an MRS of 1. The consumer is indifferent between receiving one more unit of x vs one more unit of y. However, as we increase the amount of x relative to y (moving to 11 and 9), the MRS starts to fall, meaning that the marginal utility of x falls relative to y. Now the consumer relatively prefers receiving more y relative to x. In other words, with the utility function given above, the more of a good a consumer receives, the less they want additional units of that good (relative to the other) Any utility function that has this property will have convex indifference curves like those in the graph above. We will return to the idea of convexity in the future. 6 Appendix: Proof MRS is slope of indifference curve Above we said that the MRS is the slope of an indifference curve. This fact was easy to see in the linear example, but it isn’t as obvious in general. To prove it, let’s consider a utility function U (x, y). An indifference curve sets utility equal to some fixed constant U (x, y) = Ū Now let’s take the total differential of this equation (see math review notes) with respect to x (note the derivative of the constant on the right is just 0) dU ∂U dx ∂U dy = 0 =⇒ + =0 dx ∂x dx ∂y dx dy Solving this equation for dx ∂U dy ∂x = − ∂U dx ∂y Which says that the slope of the indifference curve is the (negative) MRS. 7