Weak Axiom of Revealed Preference Theory - YouTube

Today, let’s move on to the weak axiom of revealed preference. We use the revealed preference

theory on the assumption that the consumer has preferences and chooses the best bundle

of goods she can afford. But a question arises that how can we tell that the consumer is

maximising? Or what kind of observation would lead us to conclude that the consumer is not

maximising? Have a look at this figure, and let’s interpret

it according to the logic of revealed preference. When are in budget line L1, suppose the consumer

prefers the bundle (y1,y2). This implies that when the consumer’s budget line is L1, the

optimal bundle chosen is (y1,y2), though the bundle (x1,x2) lies inside of the budget line

L1 and is affordable. Then, it means that (y1,y2) is revealed preferred to (x1,x2),

because (y1,y2) is preferred by the consumer when (x1,x2) is affordable. And suppose that

the relative prices changes, and the consumer moves to budget line L2. Then (x1,x2) becomes

more expensive than (y1,y2), as (x1,x2) lies on the budget line, whereas, (y1,y2) lies

inside of it. It has to be noted that the bundles (x1,x2) and (y1,y2) are affordable

when the consumer is at both of these budget lines. Now, suppose (x1,x2) is chosen when

the consumer is at budget line L2. This allows us to conclude two things: (1) (x1,x2) is

preferred to (y1,y2) when the consumer is at budget line L2; and (2) (y1,y2) is preferred

to (x1,x2) when the consumer is at budget line L1, which is clearly absurd. The consumer

has chosen (x1, x2) when she could have chosen (y1, y2), indicating that (x1, x2) was preferred

to (y1, y2), but then she chose (y1, y2) when she could have chosen (x1,x2)—indicating

the opposite! So this consumer cannot be a maximizing consumer. 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 like his tastes and preferences.

The theory of consumer choice implies that such observations will not occur.

The weak axiom of revealed preference states that, if (x1, x2) is directly revealed preferred

to (y1,y2), and the two bundles are not the same, then it cannot happen that (y1, y2)

is directly revealed preferred to (x1, x2). Suppose (x1,x2) is purchased at prices (p1,p2)

and bundle (y1, y2) is purchased at prices (q1, q2).

As we explained earlier, suppose when the consumer is at budget line L2, he purchases

the bundle (x1,x2). Then, according to the theory of revealed

preference, p1x1 + p2x2 ≥ p1y1 + p2y2. That is, the total expenditure of the bundle (x1,x2)

must be greater or equal to the bundle (y1,y2) at prices (p1,p2) for the theory of RP to

hold. This essentially means that the bundle (y1,y2) must be affordable to the consumer

when the consumer opts to consume (x1,x2). Given this, suppose the consumer moves to

budget line L1 and the prices change to (q1,q2). And then, it must not be the case that

q1y1 + q2y2 ≥ q1x1 + q2x2. That is, here, the prices change to (q1,q2). So the consumer

can purchase (y1,y2) here only if (x1,x2) isn’t affordable. Since the consumer purchased

the bundle (x1,x2) when he was at budget line L2, he won’t be consistent with his preferences

if he purchases the bundle (y1,y2) at budget line L1. But he will still be consistent with

his preferences if he purchased (y1,y2) when the bundle (x1,x2) wasn’t affordable. So

it mustn’t be the case that the total expenditure on (y1,y2) is greater than or equal to the

total expenditure on (x1,x2) at prices (q1,q2). This is because the bundle (x1,x2) was preferred

to bundle (y1,y2) in equation 1, even though (y1,y2) was affordable. So in case of change

in prices to (q1,q2) in equation 2, if the consumer is purchasing the bundle (y1,y2)

when (x1,x2) is available, then it means that the weak axiom of the revealed preference

isn’t satisfied. This example represents this violation because, as you can see at

budget line L1, the bundle (x1, x2) was available as it lies inside the budget line, but still

the consumer chooses (y1,y2). Have a look at this figure, where the consumer

purchases (x1,x2) at budget line L1. Here, the other bundle (y1,y2) wasn’t affordable

as it lies outside the budget line L1. Suppose the consumer moves to budget line L2, and

then, as a result of this movement he chooses the bundle (y1,y2). Here, there doesn’t

arise the problem of consistency, as the other bundle (x1,x2) lies outside the budget line.

So from this figure, we can observe that there wasn’t a violation of WARP as one bundle

wasn’t affordable when the other bundle was chosen.

Now, let’s numerically check how a violation of WARP can take place. Have a look at this

table, where there are three observations, i.e., 1,2 and 3 at prices p1 and p2, and with

bundles x1 and x2. Given these data, let’s compute how much it would cost the consumer

to purchase each bundle of goods at each different set of prices.

See the entry, in row 1, column 1, which we obtain with the prices under observation 1

and bundle 1. As we can see in the first table, the first set of prices i.e., p1, p2 is 1

and 2. And the first bundle is also given. Here, X1 is 1 and x2 is 2 under observation

1. Row 1 column 1 represents the total cost of bundle 1 at first set of prices. It will

be 1.1 ie (p1x1) +2.2 ie (p2x2) which we get as 5. Now, let’s move on to row 2, column

1. This represents the bundle 1, ie (x1,x2) with prices under the 2nd set of observation,

which is 2.1+1.2 which is 2+2 which is 4. The third row, first column represents the

bundle 1 with prices under 3rd set of observation, so 1.1+1.2 which equals 1+2 which is 3. Now

let’s move on to column 2, which represents the second bundle i.e., (2,1) at various set

of prices. So Row 1, Column 2 will be 1.2+2.1 which is 2+2 which is 4. And Row 2, Column

2 will be 2.2+1.1 which equals 4+1 which is 5. And Row 3, Column 2 will be 1.2+1.1 which

is 2+1 which is 3. Now let’s move on to Column 3, which measures the bundle (2,2)

at various sets of prices. Row 1, Column 3 will be 1.2+2.2 which is 2+4 which equals

6. Row 2, Column 3 will be 2.2+1.2 which is 4+2 which equals 6. R3, C3 will be 1.2 +2.1

which is 2+2 which is 4. So this is same as the previous column, with

just the total cost being given. Now, suppose the diagonal terms in the table

measures how much money the consumer is spending at each choice. As given by the red lines

on top, 5, 5 and 4 represents the diagonal elements. First, let’s look at the prices

under observation 1. So this means that at prices 1, the optimal bundle which the consumer

chose is assumed to be bundle 1, for which the price is 5. The consumer could have purchased

bundle 2 also, as the price is only 4. But he chose 5. But at the first set of prices,

the consumer couldn’t have chosen bundle 3, as the price of the bundle 3 exceeds the

price of the optimal bundle which is bundle 1. Since bundle 2 was affordable to the consumer,

let’s put a star there. Now at second set of prices, the consumer’s optimal bundle

is assumed to be bundle 2. The price of bundle 2 is 5 and since the bundle 1 was also affordable

to the consumer but he didn’t select it, let’s put a star there. Note that, here,

at second set of prices, bundle 3 isn’t affordable as it’s price is 6. Now in case

of third set of prices, here the consumer selects the bundle 3, for which the price

is 4. As we can see, here in the 3rd set of prices, bundle 1 and 2 were affordable but

the consumer didn’t select those. So we are putting a star in those entries.

Now, remember the theory of RP. At 1st set of prices, bundle 1 can be said to be revealed

preferred to bundle 2. That is because bundle 2 was affordable, but the consumer didn’t

choose the same. At second set of prices, bundle 2 can be said to be revealed preferred

to bundle 1. Here, the bundle 1 was affordable, but he didn’t choose it. So from the 1st

and 2nd set of prices itself, we can see the violation of WARP. Though both bundles were

affordable at 1st and 2nd set of prices, the consumer didn’t show consistency in preferences.

At the same time, bundle 3 wasn’t affordable in both of these cases. At the third set of

prices, the consumer chooses bundle 3 when bundle 1 and 2 were available. But this doesn’t

violate WARP because bundle 3 wasn’t affordable at 1st and 2nd set of prices. But the table

can be seen to be clearly violating WARP as there isn’t a consistency in bundle 1 and

2 at 1st and 2nd set of prices.