Intermediate Microeconomics – Revealed Preferences

- underlying preferences - whatever they may be = know to be strictly convex
--> unique demanded bundle at each budget
- assumption = not necessary for theory of revealed preference, but the exposition = simpler w/ it

- depicted consumer's demanded bundle (x1, x2) / another arbitrary bundle (y1, y2), beneath consumer's BL
- willing to postulate that consumer = optimizing consumer of sort that we have been studying\- bundle (y1, y2) = certainly affordable purchase at given budget - consumer could have bought it if he/she wanted to, would even have had money left over
- since (x1, x2) = optimal bundles, must be better than anything else that consumer could afford --> in particular must be better than (y1, y2)
- same arg holds for any bundle on/underneath BL other than D bundle
- since it could have been bought at given budget but wasn't --> bought must be better
- here is where we use assumption that there is unique demanded bundle for each budget
- all of bundles in shaded area underneath BL = revealed worse than demanded bundle (x1, x2)
- This is b/c they could have been chosen, but were rejected in favor of (x1, x2)

- Let (x1, x2) be bundle purchased at p's (p1, p2) when consumer has income m
- What does it mean to say that (y1, y2) = affordable at those p's / m?
- It simply means that (y1, y2) satisfies BC p1y1 + p2y2 ≤ m
- Since (x1, x2) = bought at given budget, it must satisfy BC w/ equality p1 x1 + p2x2 = m
- Putting 2 equations together, fact that (y1, y2) is affordable at budget (p1, p2, m) => p1x1 + p2x2 ≥ p1y1 + p2y2

- if above inequality = satisfied/ (y1, y2) = actually diff. bundle from (x1, x2), say that (x1, x2) = directly revealed preferred to (y1, y2)
- left-hand side of inequality = expenditure on bundle that is actually chosen at p's (p1, p2) --> revealed preference = relation that holds b/t bundle that is actually demanded at some budget/bundles that could have been demanded at that budget

- so that IC have flat spots, it may be that some bundles that are on BL might be just as good as D bundle
- complication can be handles w/o too much difficulty, but easier to just assume it away

- misleading b/c does not inherently have anything to do w/ prefs, although we've seen above that if consumer = making optimal choices, 2 ideas = closely related
- instead of saying "X = revealed preferred to Y", would be better to say "X is chosen over Y"
- when sya that X = revealed preferred to Y, all are claiming that X = chosen when Y could have been chosen, that is that p1x1 + p2x2 ≥ p1y1 + p2y2

- It follows from our model of consumer behavior—that people are choosing the best things they can
afford—that the choices they make are preferred to the choices that they could have made
- Or, in the terminology of the last section, if (x1, x2) is directly revealed preferred to (y 1, y 2), then (x1, x2) is in fact preferred to (y 1, y 2). Let us state this principle more formally

Let (x1, x2) be the chosen bundle when prices are (p 1, p 2), and let (y 1, y 2) be some other bundle such that p 1 x1 + p 2 x2 ≥p 1y 1 + p 2y 2. Then if the consumer is choosing the most preferred bundle she can afford, we must have (x1, x2) >- (y 1, y 2)

- if we observe that one bundle is chosen when another one is affordable, then we have learned something about the preferences between the two bundles: namely, that the first is preferred to the second
- q 1y 1 + q 2y 2 ≥ q 1z 1 + q 2z 2. Then we know that (x1, x2) >- (y 1, y 2) and that (y 1, y 2) >- (z 1, z 2). From the transitivity assumption we can conclude that (x1, x2) >- (z 1, z 2

- Revealed preference / transitivity tell us that (x1, x2) must be better than (z1, z2) for consumer who made illustrated choices
- It is natural to say that in this case (x1, x2) is indirectly revealed preferred to (z 1, z 2)
*- Of course "chain" of observed choices may be longer than just three: if bundle A is directly revealed preferred to B, and B to C, and C to D, ... all the way to M, say, then bundle A is still indirectly revealed preferred to M
- The chain of direct comparisons can be of any length

- If a bundle is either directly or indirectly revealed preferred to another bundle, we will say that the first bundle = RP to 2nd

- can conclude from these observations that since (x1, x2) is revealed preferred, either directly or indirectly, to all of the bundles in the shaded area, (x1, x2) is in fact preferred to those bundles by the consumer who made these choices. Another way to say this is to note that the true indifference curve through (x1, x2), whatever it is, must lie above the shaded region

- By observing choices made by consumer, we can learn about his/her preferences
- As we observe more and more choices, we can get a better and better estimate of what consumer's preferences are like
- Most economic policy involves trading off some goods for others: if we put a tax on shoes and subsidize clothing, we'll probably end up having more clothes and fewer shoes. In order to evaluate the desirability of such a policy, it is important to have some idea of what consumer preferences between clothes and shoes look like. By examining consumer choices, we can extract such information through the use of revealed preference and related techniques

- If we are willing to add more assumptions about consumer preferences, we can get more precise estimates about the shape of indifference curves
- ex. suppose we observe two bundles Y and Z that are revealed preferred to X, as in Figure 7.3, and that we are willing to postulate preferences are convex --> know that all of the weighted averages of Y and Z are preferred to X as well. If we are willing to assume that preferences are monotonic, then all the bundles that have more of both goods than X, Y , and Z—or any of their weighted averages—are also preferred to X

- conclude that all of the bundles in the upper shaded area are better than X, and that all of the bundles in the lower shaded area are worse than X, according to the preferences of the consumer who made the choices. The true indifference curve through X must lie somewhere between the two shaded sets. We've managed to trap the indifference curve quite tightly simply by an intelligent application of the idea of revealed preference and a few simple assumptions about preferences

- All of the above relies on the assumption that the consumer has preferences and that she is always choosing the best bundle of goods she can afford. If the consumer is not behaving this way, the "estimates" of the indifference curves that we constructed above have no meaning
- If (x1, x2) is directly revealed preferred to (y 1, y 2), and the two bundles are not the same, then it cannot happen that (y 1, y 2) is directly revealed preferred to (x1, x2). In other words, if a bundle (x1, x2) is purchased at prices (p 1, p 2) and a different bundle (y 1, y 2) is purchased at prices (q 1, q 2), then if p 1 x1 + p 2 x2 ≥p 1y 1 + p 2y 2 it must not be the case that q 1y 1 + q 2y 2 ≥ q 1 x1 + q 2 x2
- In English: if the y-bundle is affordable when the x-bundle is purchased, then when the y-bundle is purchased, the x-bundle must not be affordable

- (1) (x1, x2) is preferred to (y 1, y 2); and (2) (y 1, y 2) is preferred to (x1, x2)
- consumer has apparently chosen (x1, x2) when she could have chosen (y 1, y 2), indicating that (x1, x2) was preferred to (y 1, y 2), but then she chose (y 1, y 2) when she could have chosen (x1, x2)—indicating the opposite!
- Either the consumer is not choosing the best bundle she can afford, or there is some other aspect of the choice problem that has changed that we have not observed. Perhaps the consumer's tastes or some other aspect of her economic environment have changed. In any event, a violation of this sort is not consistent with the model of consumer choice in an unchanged environment. The theory of consumer choice implies that such observations will not occur. If the consumers are choosing the best things they can afford, then things that are affordable, but not chosen, must be worse than what is chosen

- condition that must be satisfied by a consumer who is always choosing the best things he or she can afford
- WARP = logical implication
- of that model and can therefore be used to check whether or not a particular consumer, or an economic entity that we might want to model as a consumer, is consistent with our economic model

- Let's consider how we would go about systematically testing WARP in practice. Suppose that we observe several choices of bundles of goods at different prices. Let us use (p t 1 , p t 2 ) to denote the tth observation of prices and (x t 1 , x t 2 ) to denote the t th observation of choices. To use a specific example, let's take the data in Table 7.1.
- Given these data, we can compute how much it would cost the consumer to purchase each bundle of goods at each different set of prices, as we've done in table 7.2
- other entries in each row measure how much she would have spent if she had purchased a different bundle. Thus we can see whether bundle 3, say, is revealed preferred to bundle 1, by seeing if the entry in row 3, column 1 (how much the consumer would have to spend at the third set of prices to purchase the first bundle) is less than the entry in row 3, column 3 (how much the consumer actually spent at the third set of prices to purchase the third bundle). In this particular case, bundle 1 was affordable when bundle 3 was purchased, which means that bundle 3 is revealed preferred to bundle 1. Thus we put a star in row 3, column 1, of the table. From a mathematical point of view, we simply put a star in the entry in row s, column t, if the number in that entry is less than the number in row s, column s

- In this framework, a violation of WARP consists of two observations t and s such that row t, column s, contains a star and row s, column t, contains a star. For this would mean that the bundle purchased at s is revealed preferred to the bundle purchased at t and vice versa. We can use a computer (or a research assistant) to check and see whether there are any pairs of observations like these in the observed choices. If there are, the choices are inconsistent with the economic theory of the consumer. Either the theory is wrong for this particular consumer, or something else has changed in the consumer's environment that we have not controlled for. Thus the Weak Axiom of Revealed Preference gives us an easily checkable condition for whether some observed choices are consistent with the economic theory of the consumer

- observe that row 1, column 2, contains a star and row 2, column 1, contains a star. This means that observation 2 could have been chosen when the consumer actually chose observation 1 and vice versa
- We can conclude that the data depicted in Tables 7.1 and 7.2 could not be generated by a consumer with stable preferences who was always choosing the best things he or she could afford