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The logic of revealed preference can be stated in terms of two axioms- the weak axiom and the strong axiom. Fig. 10.4 enables us to predict two things- (1) (x_{1}, x_{2}) is preferred to (y_{1}, y_{2}); and (2) (y_{1}, y_{2}) is preferred to (x_{1}, x_{2}).

In this case the consumer has apparently chosen (x_{1}, x_{2}), when he could have chosen (y_{1}, y_{2}). This means that (x_{1},x_{2}) was preferred to (y_{1}, y_{2}). But then he had chosen (y_{1}, y_{2}) when he could have chosen (x_{1}, x_{2}). Since these two things cannot happen at the same time there is inconsistency or logical contradiction.

In fact, such things do not happen. If a consumer is choosing the best things he can afford, then the things that are affordable, but not chosen, must be worse than what is chosen.

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**This basic point is brought into focus by the following axiom: **

**Statement of**** the Axiom:**

If (x_{1}, x_{2}) is directly revealed preferred to (y_{1}, y_{2}) and the two bundles are not the same, then (y_{1}, y_{2}) cannot, at the same time, be directly revealed preferred to (x_{1}, x_{2}).

**Algebraic Statement****:**

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If a bundle (x_{1}, x_{2}) is demanded at prices (p_{1}, p_{2}) and another bundle (y_{1}, y_{2}) is purchased at prices (p’_{1}, p’_{2}), then if;

p_{1}x_{1} + P_{2}x_{2} __>__ p_{1}y_{1} + p_{2}y_{2 }

**Then the following inequality cannot hold:**

p’_{1}y_{1} + p’_{2}y_{2} > p’_{1}x_{1} + p’_{2}x_{2}

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**Verbal Statement:**

This means that 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.

This means that if the Y-bundle is affordable when the X-bundle is purchased then when the X-bundle is purchased, the Y-bundle must not be affordable.

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Thus the WARP has been violated Fig. 10.4 by choosing both (x_{1}, x_{2}) and (y_{1}, y_{2}). This type of behaviour is not rational (maximising) behaviour. There is no set of indifference curves that could be drawn in Fig. 10.4 which could make the both bundles optimal ones at the same time.

In contrast, the consumer in Fig. 10.5 is behaving consistently. In this case it is possible to derive (locate) indifference curves for which the behaviour is optimal.

**Example 1:**

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A consumer purchases x_{1} = 20, x_{2} = 10, at prices p_{1} = 2, p_{2} = 6.

She is also observed to purchase x’_{1} = 18, x’_{2} = 4 at prices p’_{1} = 3, p’_{2} = 5. Is her behaviour consistent with the axioms of revealed preference theory?

**Solution:**

**The first purchase at original prices is:**

p_{1}x_{1} = 2 x 20 = 40 and p_{2}x_{2} = 6 x 10 = 60.

So the total expenditure (TE) is 100.

**The second purchase at original prices is: **

p_{1}x’_{1} = 2 x 18 = 36 and p_{2}x’_{2} = 6 x 4 = 24.

Thus his TE is 60.

**The first purchase at new prices is: **

P’_{1}x_{1} = 3 x 20 = 60 and p’_{2}x_{2} = 5 x 10 = 50.

So TE = 110. The second purchase at new prices is p’_{1}x’_{1} = 3 x 18 = 54 and p’_{2}x’_{2} = 5 x 4 = 20. Thus TE = 74. Here X^{0} = (20, 10), X^{1} = (18, 4), P^{0} = (2, 6) and P^{1} = (3, 5). So P^{0}X^{0} = 2 x 20 + 6 x 10 = 100 and P^{0}X^{1} = 2 x 18 + 4 x 6 = 60. So P^{0}X^{0} > P^{0}X^{1}.

Therefore is revealed preferred to X^{1}. Here P^{1}X^{1} = 3 X 18 + 5 X 4 = 74 and P^{1}X^{0} = 3 x 20 + 5 x 10 = 110. So P^{1}X^{1} < P^{1}X^{0}, or X^{1} is not revealed preferred to X^{0}. Thus WARP is fulfilled and the behaviour of the consumer is consistent with the WARP.

**Checking WARP****:**

We know that the WARP is the condition which must be satisfied by the consumer who always chooses the best things he can afford. Suppose we observe three choices of bundles of goods at three prices. Let us look at the data in Table 10.2.

On the basis of the above data, we can compute the consumer’s cost of purchasing each bundle of goods at three different sets of prices, as shown in Table 10.3. For example, the element in row 3, column 1 indicates how much money the consumer will have to spend at the third set of prices to buy the first bundle of goods.

The diagonal elements (20, 20, 16) measure how much money the consumer is spending at each choice. The off-diagonal elements in each row measure how much he would have spent if he had purchased a different bundle.

Thus it is possible to say whether bundle 3 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 m row 3, column 3 (the actual expenditure the consumer incurs at the third set of prices to buy the third bundle, i.e., Rs. 16).

Since in this case bundle 1 was affordable when bundle 3 was purchased, bundle 3 is revealed preferred to bundle 1. Thus we put a star in row 3, column of Table 10.3.

In this Table row 1, column 2 contains a star and row 2, column 1 also contains a star. This means that observation 2 could have been chosen when the consumer had actually chosen observation 1 and vice versa. This is a violation of WARP. So the conclusion is that the data depicted Tables 10.2 and 10.3 are not relevant for a consumer with stable preferences if we assumed that he is always choosing the best things he can afford.

**Example 2:**

Suppose there are only three goods X, Y and Z with associated prices given by P_{x} P_{y} and P_{z} respectively. Consider the following bundles of these goods chosen with the associated prices.

(i) How would the weak axiom of revealed preference rank these three bundles?

(ii) Would that ranking obey the principle of transitivity?

**Solution:**

We have three bundles A, B and C with three price situations P_{A}, P_{B} and P_{c}, respectively.

A = (3, 4, 4); P_{A} = (2, 2, 2)

B = (2, 1, 4); P_{B} = (2, 1, 4)

C = (1, 3, 3); P_{c}= (5, 2, 2)

**The weak axiom of revealed preference will rank these three bundles in the following way:**

(i) A is revealed to be preferred to B.

(ii) B is revealed to be preferred to C.

(iii) A is revealed to be preferred to C.

**Proof of (i):**

According to the weak axiom of revealed preference, A is revealed to be preferred to B if and only if.

The existence of these two inequalities proves that according to WARP, bundle A is revealed to be preferred to bundle B.

This proves statement (ii).

**Proof of (iii): **

Similarly we can show that

P_{A}A __>__ P_{A}C [which implies that P_{C}A> P_{C}C.]

**The ranking is also obeying the principle of transitivity since the following statement holds:**

If A is revealed to be preferred to B and B is revealed to be preferred to C, then C must never be revealed to be preferred to A.

The statement is true since statements (ii) and (iii) hold.

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