Download Microeconomics Theory, Lecture Notes - Economics - 3 and more Study notes Economics in PDF only on Docsity!
Chapter 3
The Traditional Approach to
Consumer Theory
In the previous section, we considered consumer behavior from a choice-based point of view. That is, we assumed that consumers made choices about which consumption bundle to choose from a set of feasible alternatives, and, using some rather mild restrictions on choices (homogeneity of degree zero, Walras’ law, and WARP), were able make predictions about consumer behavior. Notice that our predictions were entirely based on consumer behavior. In particular, we never said anything about why consumers behave the way they do. We only hold that the way they behave should be consistent in certain ways. The traditional approach to consumer behavior is to assume that the consumer has well-defined preferences over all of the alternative bundles and that the consumer attempts to select the most preferred bundle from among those bundles that are available. The nice thing about this approach is that it allows us to build into our model of consumer behavior how the consumer feels about trading off one commodity against another. Because of this, we are able to make more precise predictions about behavior. However, at some point people started to wonder whether the predictions derived from the preference-based model were in keeping with the idea that consumers make consistent choices, or whether there could be consistent choice-based behavior that was not derived from the maximization of well-defined preferences. It turns out that if we define consistent choice making as homogeneity of degree zero, Walras’ law, and WARP, then there are consistent choices that cannot be derived from the preference-based model. But, if we replace WARP with a slightly stronger but still reasonable condition, called the Strong Axiom of Revealed Preference (SARP),
then any behavior consistent with these principles can be derived from the maximization of rational preferences. Next, we take up the traditional approach to consumer theory, often called “neoclassical” con- sumer theory.
3.1 Basics of Preference Relations
We’ll continue to assume that the consumer chooses from among L commodities and that the commodity space is given by X ⊂ RL +. The basic idea of the preference approach is that given any two bundles, we can say whether the first is “at least as good as” the second. The “at-least-as- good-as” relation is denoted by the curvy greater-than-or-equal-to sign: º. So, if we write x º y, that means that “x is at least as good as y.” Using º, we can also derive some other preference relations. For example, if x º y, we could also write y ¹ x, where ¹ is the “no better than” relation. If x º y and y º x, we say that a consumer is “indifferent between x and y,” or symbolically, that x ∼ y. The indifference relation is important in economics, since frequently we will be concerned with indifference sets. The indifference curve Iy is defined as the set of all bundles that are indifferent to y. That is, Iy = {x ∈ X|y ∼ x}. Indifference sets will be very important as we move forward, and we will spend a great deal of time and effort trying to figure out what they look like, since the indifference sets capture the trade-offs the consumer is willing to make among the various commodities. The final preference relation we will use is the “strictly better than” relation. If x is at least as good as y and y is not at least as good as x, i.e., x º y and not y º x (which we could write y ² x), we say that x  y, or x is strictly better than (or strictly preferred to) y. Our preference relations are all examples of mathematical objects called binary relations. A binary relation compares two objects, in this case, two bundles. For instance, another binary relation is “less-than-or-equal-to,” ≤. There are all sorts of properties that binary relations can have. The first two we will be interested in are called completeness and transitivity. A binary relation is complete if, for any two elements x and y in X, either x º y or y º x. That is, any two elements can be compared. A binary relation is transitive if x º y and y º z imply x º z. That is, if x is at least as good as y, and y is at least as good as z, then x must be at least as good as z. The requirements of completeness and transitivity seem like basic properties that we would like any person’s preferences to obey. This is true. In fact, they are so basic that they form economists’
prefer less of them (like garbage or noise). However, it is not possible for all goods to always be bads if preferences are non-satiated. (Why?) It’s time for a brief discussion about the practice of economic theory. Recall that the object of doing economic theory is to derive testable implications about how real people will behave. But, as we noted earlier, in order to derive testable implications, it is necessary to impose some restrictions on (make assumptions about) the type of behavior we allow. For example, suppose we are interested in the way people react to wealth changes. We could simply assume that people’s behavior satisfies Walras’ Law, as we did earlier. This allows us to derive testable implications. However, it provides little insight into why they satisfy Walras’ Law. Another option would be to assume monotonicity — that people prefer more to less. Monotonicity implies that people will satisfy Walras’ Law. But, it rules out certain types of behavior. In particular, it rules out the situation where people prefer less of an object to more of it. But, introspection tells us that sometimes we do prefer less of something. So, we ask ourselves if there is a weaker assumption that allows people to prefer less to more, at least sometimes, that still implies Walras’ Law. It turns out that local nonsatiation is just such an assumption. It allows for people to prefer less to more — even to prefer less of everything — the only requirement is that, no matter which bundle the consumer currently selects, there is always a feasible bundle nearby that she would rather have. By selecting the weakest assumption that leads to a particular result, we accomplish two tasks. First, the weaker the assumptions used to derive a result, the more “robust” it is, in the sense that a greater variety of initial conditions all lead to the same conclusion. Second, finding the weakest possible condition that leads to a particular conclusion isolates just what is needed to bring about the conclusion. So, all that is really needed for consumers to satisfy Walras’ Law is for preferences to be locally nonsatiated — but not necessarily monotonic or strongly monotonic. The assumptions of monotonicity or local nonsatiation will have important implications for the way indifference sets look. In particular, they ensure that Ix = {y ∈ X|y ∼ x} are downward sloping and “thin.” That is, they must look like Figure 3.1. If the indifference curves were thick, as in Figure 3.2, then there would be points such as x, where in a neighborhood of x (the dotted circle) all points are indifferent to x. Since there is no strictly preferred point in this region, it is a violation of local-nonsatiation (or monotonicity). In addition to the indifference set Ix defined earlier, we can also define upper-level sets and lower-level sets. The upper level set of x is the set of all points that are at least as good as x, Ux = {y ∈ X|y º x}. Similarly, the lower level set of x is the set of all points that are no
x 2
x 1
I (^) x
Figure 3.1: Thin Indifferent Sets
x 2
x 1
Ix x
Figure 3.2: Thick Indifference Sets
that prevents flat regions from appearing on indifference curves. However, there are reasons why we want to rule out indifference curves with flat regions. Because of this, we strengthen the convexity assumption with the concept of strict convexity. A preference relation is strictly convex if for any distinct bundles y and z (y 6 = z) such that y º x and z º x, ty + (1 − t) z  x. Thus imposing strict convexity on preferences strengthens the requirement of convexity (which actually means that averages are at least as good as extremes) to say that averages are strictly better than extremes.
3.2 From Preferences to Utility
In the last section, we said a lot about preferences. Unfortunately, all of that stuff is not very useful in analyzing consumer behavior, unless you want to do it one bundle at a time. However, if we could somehow describe preferences using mathematical formulas, we could use math techniques to analyze preferences, and, by extension, consumer behavior. The tool we will use to do this is called a utility function. A utility function is a function U (x) that assigns a number to every consumption bundle x ∈ X. Utility function U () represents preference relation º if for any x and y, U (x) ≥ U (y) if and only if x º y. That is, function U assigns a number to x that is at least as large as the number it assigns to y if and only if x is at least as good as y. The nice thing about utility functions is that if you know the utility function that represents a consumer’s preferences, you can analyze these preferences by deriving properties of the utility function. And, since math is basically designed to derive properties of functions, it can help us say a lot about preferences. Consider a typical indifference curve map, and assume that preferences are rational. We also need to make a technical assumption, that preferences are continuous. For our purposes, it isn’t worth derailing things in order to explain why this is necessary. But, you should look at the example of lexicographic preferences in MWG to see why the assumption is necessary and what can go wrong if it is not satisfied. The line drawn in Figure 3.3 is the line x 2 = x 1 , but any straight line would do as well. Notice that we could identify the indifference curve Ix by the distance along the line x 2 = x 1 you have to travel before intersecting Ix. Since indifference curves are downward sloping, each Ix will only intersect this line once, so each indifference curve will have a unique number associated with it. Further, since preferences are convex, if x  y, Ix will lay above and to the right of Iy (i.e. inside Iy), and so Ix will have a higher number associated with it than Iy.
x 2
x 1
a Ix
Figure 3.3: Ranking Indifference Curves
We will call the number associated with Ix the utility of x. Formally, we can define a function u (x 1 , x 2 ) such that u (x 1 , x 2 ) is the number associated with the indifference curve on which (x 1 , x 2 ) lies. It turns out that in order to ensure that there is a utility function corresponding to a particular preference relation, you need to assume that preferences are rational and continuous. In fact, this is enough to guarantee that the utility function is a continuous function. The assumption that preferences are rational agrees with how we think consumers should behave, so it is no problem. The assumption that preferences are continuous is what we like to call a technical assumption, by which we mean that is that it is needed for the arguments to be mathematically rigorous (read: true), but it imposes no real restrictions on consumer behavior. Indeed, the problems associated with preferences that are not continuous arise only if we assume that all commodities are infinitely divisible (or come in infinite quantities). Since neither of these is true of real commodities, we do not really harm our model by assuming continuous preferences.
3.2.1 Utility is an Ordinal Concept
Notice that the numbers assigned to the indifference curves in defining the utility function were essentially arbitrary. Any assignment of numbers would do, as long as the order of the numbers assigned to various bundles is not disturbed. Thus if we were to multiply all of the numbers by 2 , or add 6 to them, or take the square root, the numbers assigned to the indifference curves after the transformation would still represent the same preferences. Since the crucial characteristic of a utility function is the order of the numbers assigned to various bundles, but not the bundles themselves, we say that utility is an ordinal concept.
0 2 4 6 8 10
x 1 0
2
4
6
8
10
x 2
0
1
2
3
ux
0 2 4 6 8 10
x 1
Figure 3.4: Function u (x)
Figure 3.5: Level sets of u (x)
Notice the curvature of the surface. Now, consider Figure 3.5, which shows the level sets (Ix) for various utility levels. Notice that the indifference curves of this utility function are convex. Now, pick an indifference curve. Points offering more utility are located above and to the right of it. Notice how the contour map corresponds to the 3D utility map. As you move up and to the right, you move “uphill” on the 3D graph. Quasiconcavity is a weaker condition than concavity. Concavity is an assumption about how the numbers assigned to indifference curves change as you move outward from the origin. It says that the increase in utility associated with an increase in the consumption bundle decreases as you move away from the origin. As such, it is a cardinal concept. Quasiconcavity is an ordinal concept. It talks only about the shape of indifference curves, not the numbers assigned to them. It can be shown that concavity implies quasiconcavity but a function can be quasiconcave without being concave (can you draw one in two dimensions). It turns out that for the results on utility
Figure 3.7: Level sets of v (x).
sets.^6 To see why v (x) is quasiconcave, let’s look at the level sets of v (x) in Figure 3.7. Even though v(x) is curved in the other direction, the level sets of v (x) are still convex. Hence v (x) is quasiconcave. The important point to take away here is that quasiconcavity is about the shape of level sets, not about the curvature of the 3D graph of the function. Before going on, let’s do one more thing. Recall u (x) = x (^14) 1 x
(^14) 2 and^ v^ (x) =^ x^
(^32) 1 x
(^32) 2.^ Now, consider the monotonic transformation f (u) = u^6. We can rewrite v (x) = x (^64) 1 x^
(^64) 2 =
μ x (^14) 1 x^
(^14) 2
= f (u (x)). Hence utility functions u (x) and v (x) actually represent the same preferences! Thus we see that utility and preferences have to do with the shape of indifference curves, not the numbers assigned to them. Again, utility is an ordinal, not cardinal, concept. Now, here’s an example of a function that is not quasiconcave.
h (x) = ¡x^2 + y^2 ¢^ 14 μ 2 +^14
sin
8 arctan
³ (^) y x
(^6) That isn’t a proof, just an illustration.
20 40 60 80 100
x 1
20
40
60
80
100
x 2
0
10
20 utility
20 40 60 80 100
x 1
Figure 3.8: Function h(x)
Don’t worry about where this comes from. Figure 3.8 shows the 3D plot of h (x). Figure 3.9 shows the isoquants for this utility function. Notice that the level sets are not convex. Hence, function h (x) is not quasiconcave. After looking at the mathematical analysis of the consumer’s problem in the next section, we’ll come back to why it is so hard to analyze utility functions that look like h (x).
3.3 The Utility Maximization Problem (UMP)
Now that we have defined a utility function, we are prepared to develop the model in which consumers choose the bundle they most prefer from among those available to them. 7 In order to (^7) Notice that in the choice model, we never said why consumers make the choices they do. We only said that those choices must appear to satisfy homogeneity of degree zero, Walras’ law, and WARP. Now, we say that the consumer acts to maximize utility with certain properties.
ensure that the problem is “well-behaved,” we will assume that preferences are rational, continuous, convex, and locally nonsatiated. These assumptions imply that the consumer has a continuous utility function u (x), and the consumer’s choices will satisfy Walras’ Law. In order to use calculus techniques, we will assume that u () is differentiable in each of its arguments. Thus, in other words, we assume indifference curves have no “kinks.” The consumer’s problem is to choose the bundle that maximizes utility from among those avail- able. The set of available bundles is given by the Walrasian budget set Bp,w = {x ∈ X|p · x ≤ w}. We will assume that all prices are strictly positive (written p >> 0 ) and that wealth is strictly positive as well. The consumer’s problem can be written as:
max x≥ 0 u (x) s.t. : p · x ≤ w. The first question we should ask is: Does this problem have a solution? Since u (x) is a continuous function and Bp,w is a closed and bounded (i.e., compact) set, the answer is yes by the Weierstrass theorem - a continuous function on a compact set achieves its maximum. How do we find the solution? Since this is a constrained maximization problem, we can use Lagrangian methods. The Lagrangian can be written as:
L = u (x) + λ (w − p · x)
Which implies Kuhn-Tucker first-order conditions (FOC’s):
ui (x∗) − λ∗pi ≤ 0 and xi (ui (x∗) − λ∗pi) = 0 for i = (1, ..., L) w − p · x∗^ ≥ 0 and λ∗^ (w − p · x∗) = 0
Note that the optimal solution is denoted with an asterisk. This is because the first-order conditions don’t hold everywhere, only at the optimum. Also, note that the value of the Lagrange multiplier λ is also derived as part of the solution to this problem. Now, we have a system with L + 1 unknowns. So, we need L + 1 equations in order to solve for the optimum. Since preferences are locally non-satiated, we know that the consumer will choose a consumption bundle that is on the boundary of the budget set. Thus the constraint must bind. This gives us one equation. The conditions on xi are complicated because we must allow for the possibility that the consumer chooses to consume x∗ i = 0 for some i at the optimum. This will happen, for example, if the relative
price of good i is very high. While this is certainly a possibility, “corner solutions” such as these are not the focus of the course, so we will assume that x∗ i > 0 for all i for most of our discussion. But, you should be aware of the fact that corner solutions are possible, and if you come across a corner solution, it may appear to behave strangely. Generally speaking, we will just assume that solutions are interior. That is, that x∗ i > 0 for all commodities i. In this case, the optimality condition becomes
ui (x∗ i ) − λ∗pi = 0. (3.1)
Solving this equation for λ∗^ and doing the same for good j yields:
− ui^ (x
∗ i ) uj
x∗ j
´ = − (^) ppi j for all i, j ∈ { 1 , ..., L}.
This turns out to be an important condition in economics. The condition on the right is the slope of the budget line, projected into the i and j dimensions. For example, if there are two commodities, then the budget line can be written x 2 = − p p^12 x 1 + (^) pw 2. The left side, on the other hand, is the slope of the utility indifference curve (also called an isoquant or isoutility curve). To see why − ui(x ∗ i ) uj (x∗ j ) is the slope of the isoquant, consider the following identity:^ u^ (x^1 , x^2 (x^1 ))^ ≡^ k, where k is an arbitrary utility level and x 2 (x 1 ) is defined as the level of x 2 needed to guarantee the consumer utility k when the level of commodity 1 consumed is x 1. Differentiate this identity with respect to x 1 :^8
u 1 + u 2 dx dx^21 = 0 dx 2 dx 1 =^ −^
u 1 u 2
So, at any point (x 1 , x 2 ), − u u^12 ((xx^11 ,x,x^22 )) is the slope of the implicitly defined curve x 2 (x 1 ). But, this curve is exactly the set of points that give the consumer utility k, which is just the indifference curve. As mentioned earlier, we call the slope of the indifference curve the marginal rate of substitution (MRS): M RS = − u u^12. Thus the optimality condition is that at the optimal consumption bundle, the MRS (the rate that the consumer is willing to trade good x 2 for good x 1 , holding utility constant) must equal the ratio of the prices of the two goods. In other words, the slope of the utility isoquant is the same as the slope of the budget line. Combine this with the requirement that the optimal bundle be on (^8) Here, we adopt the common practice of using subscripts to denote partial derivatives, ∂u ∂x(xi )= ui.
The problem with this procedure is that it could potentially take a very long time to find the optimal point. The calculus approach allows us to do it much faster by finding the point along the budget line that has the same slope as the indifference curve. This is a much easier task, but it turns out that it is really just a shortcut for the procedure outlined above.
3.3.1 Walrasian Demand Functions and Their Properties
So, suppose that we have found the utility maximizing point, x∗. What have we really found? Notice that if the prices and wealth were different, the utility maximizing point would have been different. For this reason, we will write the endogenous variable x∗^ as a function of prices and wealth, x (p, w) = (x 1 (p, w) , x 2 (p, w) ..., xL (p, w)). This function gives the utility maximizing bundle for any values of p and w. We will call x (p, w) the consumer’s Walrasian demand function, although it is also sometimes called the Marshallian or ordinary demand function. This is to distinguish it from another type of demand function that we will study later. As a consequence of what we have done, we can immediately derive some properties of the Walrasian demand function:
- Walras’ Law: p · x (p, w) ≡ w for all p and w. This follows from local nonsatiation. Recall the definition of local non-satiation: For any x ∈ X and ε > 0 there exists a y ∈ X such that ||x − y|| < ε and y  x. Thus the only way for x to be the most preferred bundle is if there the nearby point that is better is not in the budget set. But, this can only happen if x satisfies p · x (p, w) ≡ w.
- Homogeneity of degree zero in (p, w). The definition of homogeneity is the same as always. x (αp, αw) = x (p, w) for all p, w and α > 0. Just as in the choice based approach, the budget set does not change: Bp,w = Bαp,αw. Now consider the first-order condition:
− ui^ (x
∗ i ) uj
x∗ j
´ (^) = − p pi j^ for all^ i, j^ ∈^ {^1 , ..., L}^. Suppose we multiply all prices by α > 0. This makes the right hand side − αp αpij = − p pij , which is just the same as before. So, since neither the budget constraint nor the optimality condition are changed, the optimal solution must not change either.
- Convexity of x (p, w). Up until now we have been assuming that x (p, w) is a unique point. However, it need not be. For example, if preferences are convex but not strictly convex, the
isoquants will have flat parts. If the budget line has the same slope as the flat part, an entire region may be optimal. However, we can say that if preferences are convex, the optimal region will be a convex set. Further, we can add that if preferences are strictly convex, so that u () is strictly quasiconcave, then x (p, w) will be a single point for any (p, w). This is because strict quasiconcavity rules out flat parts on the indifference curve.
A Note on Optimization: Necessary Conditions and Sufficient Conditions
Notice that we derived the first-order conditions for an optimum above. However, while these conditions are necessary for an optimum, they are not generally sufficient - there may be points that satisfy them that are not maxima. This is a technical problem that we don’t really want to worry about here. To get around it, we will assume that utility is quasiconcave and monotone (and some other technical conditions that I won’t even mention). In this case we know that the first-order conditions are sufficient for a maximum. In most courses in microeconomic theory, you would be very worried about making sure that the point that satisfies the first-order conditions is actually a maximizer. In order to do this you need to check the second order conditions (make sure the function is “curved down”). This is a long and tedious process, and, fortunately, the standard assumptions we will make, strict quasiconcavity and monotonicity, are enough to make sure that any point that satisfies the first-order conditions is a maximizer (at least when the constraint is linear). Still, you should be aware that there is such things as second-order conditions, and that you either need to check to make sure they are satisfied or make assumptions to ensure that they are satisfied. We will do the latter, and leave the former to people who are going to be doing research in microeconomic theory.
A Word on Nonconvexities
It is worthwhile to spend another moment on what can happen if preferences are not convex, i.e. utility is not quasiconcave. We already mentioned that with nonconvex preferences it becomes necessary to check second-order conditions to determine if a point satisfying the first-order condi- tions is really a maximizer. There can also be other complications. Consider a utility function where the isoquants are not convex, shown in Figure 3.11.
When the budget line is given by line 1, the optimal point will be near x. When the budget line is line 2, the optimal points will be either x or y. But, none of the points between x and y on line
