The theory of revealed preference allows us to use information about consumer choices to inter how the consumer must rank bundles if he is maximizing utility with budget constraints. Alternatively, the revealed preference model explains that how the consumer spends his income with given prices of different commodities. The change in price allows consumer to buy of more or less commodities. But increase in income also helps to buy of more one commodity or buy more of other commodity. Such model is based on certain assumptions.

**Assumptions****: **

**Revealed preference theory is based on following assumptions: **

1. Consumer spends his entire income on two commodities only.

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2. The consumer chooses one commodity for each price vector p and income situation.

3. There is one and only one price p and income combination at which bundle x is chosen by consumer.

4. The consumer choices are consistent. The X^{0} with price p^{0} and if different bundle x^{1} is chosen then x^{0} will be no longer feasible alternative.

**Model****: **

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Let us assume that P^{0} will be the price vector at which x^{0} can be purchased. The consumer chooses x^{1} when X^{0} was chosen. The cost for consumer of consumption bundle is p^{0}x^{1 }> p^{0}x^{0}. The consumer chooses x^{0} when p^{0}x^{0 }= M_{0}. Now the price changes then it increases from p^{0 }to p^{1}. In the economy, such rise in prices is always observed of commodities. At new price level, p^{1}x^{0 }consumer is not happy because he/has to pay more in terms of money.

But his choices are changing and he prefer P^{1}x^{0} > p^{1}x^{1} then consumer is still better off. Similarly P^{1}x^{0 }≥ p^{0}x^{1} implies p^{1}x^{1}< p^{1}x^{0}. At new adjustment, the previous bundle of consumption gives more satisfaction to consumer. It also means that P^{0}x^{0 }≥ p^{0}x^{1} → P^{1}x^{1}≤ P^{1}x^{0}.

**The Weak Axiom of Revealed Preference (WARP)****: **

In the Weak Axiom of Revealed Preference (WARP), we have assumed two commodities. Now x^{0} is chosen at p^{0}, M^{0 }where B_{0} is at equilibrium. Now, X^{1} is chosen at p^{1}m^{1}. The price effect shows a choice structure, it satisfies the weak axiom of revealed preference. There may be a rational preference relation consistent with these choices.

**In figure 2.4: **

B_{0} = It is a consumers’ budget line defined by P_{0} and M_{0 }

X_{0} = The initial bundle chosen by consumer on B_{0 }

B_{1} = It is a budget line after the fall in p_{1} with

X^{1} = The new bundle chosen on B_{1}

In the figure, the line B_{1} shifts to B_{2}. The bundle x^{2} is just right to x^{0}. Therefore x^{0} and x^{2} is the substitution effect. Both goods are alternatively purchased. The x^{0} and x^{1} is income effect which is shown in diagram. It is because of fall in price p_{1}. The P^{0}x^{0} is the price vector and consumption vector. The P^{1}x^{1} is the new price vector and consumption vector. The consumer’s income is adjusted until at m_{2}x^{0}. It can be purchased at the new prices p^{1}, so that p^{1}x^{0} = M_{2}. The price vector p1 and the compensated money income is M_{2}. The consumer chooses x^{2}, it is because all income is spent. We have

**The compensating change in M ensures that: **

Now x^{2} is chosen because x^{0} is still available to consumers. In routine life, consumer expects to buy new commodities when old commodities exist. It means likeness and preferences changes.

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From the equation (10), we have

Here, X^{2} is not purchased by consumer but x^{0} is purchased. It may be because new choice of commodity is not suitable to him.

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Now we consider equation (11)

Now, equation (12) can be written as

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Subtracting equation (14) from (13) gives the following equation,

And multiplying above equation by (-1) we will have following equation,

This prediction applies irrespective of the number and direction of price changes. In case of a change in the j^{th} price, only p^{1} and p^{0} differ in p** _{j}**.

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**Therefore equation (15) can be written as: **

We can also derive Slutsky equation from the behavioral assumption. Therefore M_{2 }= p^{1}x^{0} and M_{0 }= p^{0}x^{0} the compensating reduction in M is

**The case of Δ in P _{i} only we have: **

Above equation explains that the change in price is equivalent to the change in money income. Government offset the effect of price rise with dearness allowance to public sector workers. Such dearness allowance increases the money income up to the new rise in price level.

The price effect of p_{i} on X** _{j}** is and this can be partitioned into substitution effect (x

^{2}

**-x**

_{j}**) and income effect (x**

_{j}^{1}

**-x**

_{j}**). We divide price substitution and income effect by D Pi.**

_{j}ADVERTISEMENTS:

**It is as follows: **

But from equation (18) we have ΔM = ⎯ Δp_{i}x^{0}_{j}

**. ^{.}.** Δp

_{i}= ⎯ ΔM/X

^{0}

_{i}substituting this into second term of equation (19), we have on the right hand side

Above equation helps us to show the utility maximizing theory of the consumer. The revealed preference theory and utility maximization are equal in their nature.