Tecnologías para aire acondicionado

June 18, 2011

1

Preferences and Observed Behavior

Rather than starting from an abstract utility function, let us consider starting instead from the actual, observed behavior of the consumer.

Consider the budget line shown in figure 1. Suppose that this consumer chooses bundle X. Now, it is clear that the consumer could have afforded bundle Y. In this case, we can say that bundle X is revealed preferred to bundle Y. In general, any time a bundle X is chosen when another bundle Y is affordable, then we say that bundle X isrevealed preferred to bundle Y. This is writtenX RP Y or XY.

To emphasize, this concept is based on real-life observable behavior. If you bought 2 hamburgers and 1 milkshake at Burger King when you could have afforded 1 hamburger and 3 milkshakes, then you have revealed through your observed choices that you prefer the first option.

We can use this idea to build up indifference curves. In figure 2, the consumer chooses bundle X onBCX,

bundle Y onBCY and bundle Z onBCZ.

Now, bundle X was affordable when bundle Y was chosen, meaning thatY X. Similarly, bundle X was affordable when bundle Z was chosen, meaning thatZX. Thus, whatever indifference curve we construct that includes bundle X, it must be drawn such that bundle Y and bundle Z are on higher indifference curves. A possible indifference curve is sketched in figure 2. Notice that, by getting more observations of this sort, we can continue to bound in and get a more precise idea of how the indifference curve is shaped.

2

Weak Axiom of Revealed Preference

TheWeak Axiom of Revealed Preference (WARP) states that if bundle X is revealed preferred to bundle Y, then bundle Y cannot be revealed preferred to bundle X.

The choices shown in figure 3 violate WARP. On BCX, when bundle X was chosen, bundle Y would

have been affordable. So X Y. Also, on BCY, when bundle Y was chosen, bundle X would have been

affordable, soY X. SinceX Y andY X both, this consumer’s behavior violates WARP.

The choices shown in figure 4 satisfy WARP. Bundle X was not affordable when bundle Y was chosen, and vice versa, so the consumer has not revealed anything about his preferences over X and Y.

3

Checking the Weak Axiom: Example 1

A consumer purchases two goods: Good 1 and Good 2. We make two observations of his purchases on days when the prices are different. We will call the consumer’s observed purchases bundle X and bundle Y.

• Observation X: WhenP1= 1 and P2= 3, the consumer buysx1= 4 and x2= 2

• Observation Y: WhenP1= 3 and P2= 5, the consumer buysy1= 3 andy2= 1

Figure 1: Bundle X revealed preferred to Bundle Y

Figure 3: Consumer choices that violate WARP

Bundle X Bundle Y Prices X 10 6 Prices Y 22 14

At price vector X, bundle X was purchased. However, bundle Y would have been affordable (bundle X cost 10 at X prices, but bundle Y would have cost only 6 at X prices), soXY.

At price vector Y, bundle Y was purchased. Notice that bundle X would not have been affordable (bundle Y cost 14 at Y prices, but bundle X would have cost 22 at Y prices). Therefore, we do not have Y revealed preferred to X.

This consumer’s choices satisfy WARP.

4

Checking the Weak Axiom: Example 2

Again a consumer purchases two goods: Good 1 and Good 2. We make two observations of his purchases on days when the prices are different. As before, we call the consumer’s observed purchases bundle X and bundle Y.

• Observation X: WhenP1= 1 and P2= 2, the consumer buysx1= 1 and x2= 2

• Observation Y: WhenP1= 2 and P2= 1, the consumer buysy1= 2 andy2= 1

Now, make a table showing how much each observed bundle X and Y costs at each of the price vectors.

Bundle X Bundle Y

Prices X 5 4

Prices Y 4 5

At price vector X, bundle X was purchased. However, bundle Y would have been affordable (bundle X cost 5 at X prices, but bundle Y would have cost only 4 at X prices), soX Y.

At price vector Y, bundle Y was purchased. However, bundle X would have been affordable (bundle Y cost 5 at Y prices, but bundle X would have cost only 4 at Y prices), soY X.

This consumer’s choices violate WARP since XY andY X both.

5

Strong Axiom of Revealed Preference

Consider a consumer who purchases three goods: Good 1, Good 2 and Good 3. We make three different observations of his purchases on days when the prices are different. We will call the consumer’s observed purchases bundle X, bundle Y and bundle Z.

• Observation X: WhenP1= 1, P2= 2 andP3= 8, the consumer buysx1= 2, x2= 1 andx3= 3

• Observation Y: WhenP1= 4, P2= 1 andP3= 8, the consumer buysx1= 3, x2= 4 andx3= 2

• Observation Z: When P1= 3, P2= 1 andP3= 2, the consumer buysx1= 2, x2= 6 andx3= 2

As before, make a table showing how much each observed bundle X, Y and Z costs at each of the price vectors.

Bundle X Bundle Y Bundle Z

Prices X 28 27 30

Prices Y 33 32 30

Bundle X was purchased at X prices, but bundle Y would have been affordable, so XY. Bundle Y was purchased at Y prices, but bundle Z would have been affordable, so Y Z. Bundle Z was purchased at Z prices, but bundle X would have been affordable, soZ X.

Notice that this consumer’s behavior satisfies WARP. There are no bundles A and B where AB and BAboth. However, the revealed preferences are intransitive: XY andY Z, butZX.

We say that a consumer’s choices satisfy theStrong Axiom of Revealed Preference(SARP) if they satisfy WARPandthe preference ordering is transitive. So, for the example above, the consumer’s behavior satisfies WARP, but it violates SARP.

6

Afriat’s Theorem

The reason that revealed preference is so important is that it establishes a link between observed real-life behavior of consumers and the theoretical, abstract utility-maximization framework that we developed in earlier sections. Afriat’s Theorem states that, as long as a consumer’s behavior satisfies SARP, then we can find a utility function that the consumer is obeying.

In other words, even though people in the real world might not sit down and think about maximizing a utility function, as long as their observed behavior satisfies SARP (which is a fairly minimal condition) then we can find a utility function that the person acts as if he’s maximizing. So, using the utility-maximization framework makes sense as a logical construct for observed choices.