Strong Axiom of Revealed Preference Theory (Varian) - YouTube

I have already discussed what the weak axiom of revealed preference is. Today we will have

a look at the Strong Axiom of Revealed Preference Theory.

The Weak Axiom of Revealed Preference requires that if X is directly revealed preferred to

Y, then we should never observe Y being directly revealed preferred to X. I will now explain

how it is different from the strong axiom of revealed preference. The Strong Axiom of

Revealed Preference (SARP) requires that the same sort of condition hold for indirect revealed

preference. I will explain this in detail. We have already studied in the theory of RP

that, if a bundle of goods X is revealed preferred to a bundle Y , and Y is in turn revealed

preferred to a bundle Z, then X must in fact be preferred to Z. This is nothing but the

theory of transitivity. And here X is said to be indirectly revealed preferred to Z.

In a more formalised manner we can define strong axiom as that, if (x1,x2) is revealed

preferred to (y1,y2) (either directly or indirectly) and (y1,y2) is different from (x1, x2), then

(y1, y2) cannot be directly or indirectly revealed preferred to (x1, x2). In the weak

axiom, the focus was only for direct revealed preference between two bundles of goods.

Since the underlying preferences of the consumer must be transitive, it follows that the revealed

preferences of the consumer must be transitive. That is, since the assumption of transitivity

is important for choosing well behaved preferences, the SARP is a necessary implication of optimising

behaviour. This means that if a consumer is always choosing the best things that he can

afford, then his observed behaviour must satisfy SARP. There is also another side to the same

coin. If the observed choices satisfy SARP, we can construct nice, well-behaved preferences

from those observed choices. In this sense SARP is a sufficient condition for optimising

behaviour. Thus SARP is both a necessary and a sufficient condition for observed choices

to be compatible with the economic model of consumer choice.

Now, we will learn as to how to check for SARP.

The table represents the various sets of prices 1,2 and 3 and bundles 1,2 and 3. The entries

in the table represents the total cost of various bundles at various sets of prices.

For eg: The entry in Row 1, Column 1, ie, 20 represents the total cost of bundle 1 at

prices 1, and entry in Row 2, Column 1, ie, 21 represents the total cost of bundle 1 at

prices 2 and so on. At the same time, the entry in Row 1, Column 2, that is 10 represents

the total cost of bundle 2 at prices 1. We assume here that it is the diagonal elements

which are preferred by the consumer. The diagonal elements are 20, 20 and 10, as marked by red

lines. This means the consumer selects bundle 1, which costs 20 at prices 1, and bundle

2 at prices 2 which costs 20 and bundle 3 at prices 3 which costs 10. At prices 1, the

bundle which is assumed to be selected is the first one, that is, 20 as I mentioned

before. Now, let’s focus solely on prices 1 or the first row. If you have a look at

bundle 2, it costs 10. But, we selected bundle 1, which costs 20. We can see that bundle

2 was also affordable to the consumer at prices 1 since it costs less than 20. Remember we

are talking solely about prices 1, and if you look at bundle 3, it costs 22. The cost

of bundle 3 is greater than bundle 1. Here, we can say that the bundle 1 is revealed preferred

to bundle 2 at prices 1. That is because, bundle 2 was also affordable when the consumer

selected bundle 1 (because bundle 2 costs less than bundle 1). So let’s put a star

in bundle 2, keeping in mind that bundle 2 wasn’t selected by the consumer even though

it was affordable. Note that at prices 1, which we are talking about, bundle 3 isn’t

affordable as it costs more than bundle 1. Ignore the star in the parenthesis for a moment,

I will explain that further. At prices 2, the bundle selected is bundle 2 as we assumed.

Let’s talk about prices 2 now. So the bundle selected is 2, which costs 20. Here, we can

see that the bundle 3 was also affordable which costs 15, but the consumer didn’t

select it. So we can say that bundle 2 is revealed preferred to bundle 3 here. Now,

have a look at bundle 1, it costs 21, which is more than the cost of bundle 2 which is

selected. That means it isn’t affordable. Let’s put a star in bundle 3 now, since

bundle 3 was affordable to the consumer, but wasn’t preferred. Now if we look at first

two rows, that is prices 1 and 2, we get the result that bundle 1 is preferred to bundle

2 at prices 1. And bundle 2 is preferred to bundle 3 at prices 2. That means that bundle

1 is revealed preferred to bundle 2 and bundle 2 is revealed preferred to bundle 3. From

the assumption of transitivity, we get the result that bundle 1 is indirectly revealed

preferred to bundle 3. So let’s put a star with a parenthesis in bundle 3 at prices 1

to represent this indirect revealed preference. Now, let’s look at prices 3. At prices 3,

the bundle which is assumed to be selected is 3. It costs 10, and if we look at bundle

1 and 2, we can see that those bundles costs more than 10. This means that the bundles

1 and 2 weren’t affordable to the consumer at prices 3. Remember we got a result that

bundle 1 is indirectly revealed preferred to bundle 3, and we can’t disprove it at

prices 3. This is because at prices 3, when the consumer selects bundle 3, the bundle

1 isn’t affordable. So we can conclude that the SARP holds here.