Law of Substitution: Law of Equi-Marginal Utility

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The Law of Substitution (Law of Equi-Marginal Utility)

The law of substitution explains consumer equilibrium about multi-commodity case.

In real life, the consumers consume various goods and services in their daily life.

They have limited budget to purchase goods and services.

So, they allocate their scarce resources to purchase various goods in order to get maximum satisfaction.

 

The concept of the law of substitution was developed by H.H. Gossen in 1854 AD.

So, this law is also called second law of Gossen.

However, this law was further developed and completed by Alfred Marshall.

 

Definition

According to Alfred Marshall, “In order to maximize the utility, the consumer allocates his income on various goods in such a way that the ratio of marginal utility to price of each goods be equal to marginal utility of money.”

 

Assumptions

The law of equi-marginal utility is based on the following important assumptions:

Rational consumer

Diminishing marginal utility

Utilities are independent

Money income of the consumer is given and fixed

Prices of the goods are given.

 

A consumer consumes many goods and services in real life.

But for our simplicity, suppose consumer consumes only two goods namely X and Y.

For a rational consumer to be in equilibrium he should purchase that combination of commodities (X and Y) for which the marginal utility obtained from X should equal to the price paid to obtain the last unit of X and marginal utility obtained from Y should equal to the price paid to obtain the last unit Y under the given budget limitations.

 

Therefore, the consumer will be in equilibrium at that point where the last rupees spent on each commodity gives the same amount marginal utility.

 

The second condition reminds us that consumer’s budget (money, income, M) set aside for consumption of goods and services must be fully spent; no part should remain unspent.

In order to maximize his utility, a rational consumer should equalizer the ratios of marginal utility and price of each commodity.

 

The equality law is also known as the law of equi-marginal utility.

When the consumer substitutes one commodity for other until the ratios are equal; the law is also known as law of substitution.

When the consumer equalizes these ratios, his utility will be maximized; sometimes the law is also known as law of utility maximization.

 

Now, a detail illustration will help to explain this rule.

For simplicity, we limit our discussion to just two commodities namely X and Y.

However, the analysis can readily be extended to any number of commodities.

 

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Let us suppose:

Here, Amount = Rs = $ = £ = € = ₹ = Af = ৳ = Nu = Rf = රු = Currency of your country 

The money income (M) of the consumer is $/₹/Rs 5; which he spends on two goods X and Y.

The price per unit of X and Y be $1 each, that is, P_X = $1 = P_Y

Now, we use table to discuss the law of equi-marginal utility or law of substitution.

 

Units of commodity	Marginal utility from X	Marginal utility from Y
	Per $/₹/Rs marginal utility of X (MUX/PX)	Per $/₹/Rs marginal utility of Y (MUY/PY)
First	10	12
Second	8	10
Third	6	8
Fourth	4	6
Fifth	2	4
Total	30	40

 

Table summarizes the choice of rational consumer.

Column 2 shows the amount of extra or marginal utility obtained from commodity X.

Column 3 shows the marginal utility from commodity Y.

In each case the relationship between the numbers of units of the commodity obtained and the corresponding marginal utility reflect the law of diminishing marginal utility.

 

Suppose a consumer is trying to decide which combination of two commodities X and Y; he should purchase to maximize his utility.

Obviously price of commodities and his money income are the basic determining factor of the choice.

 

If we divide marginal utility of each commodity by their respective price then we will get marginal utility on per unit basis.

In our example, since price is 1 marginal utility on unit and per $/₹/Rs basis are the same.

Now, we have marginal utilities on unit and per rupee basis and price tags of X and Y.

In what order should he allocate his money on units of X and Y to achieve the highest degree of utility within the limit imposed by his money income and prices?

What combination of X and Y will he have obtained at the time he exhausted his $5?

 

Looking on columns 2 and 3 of the table, we find that the consumer should first spend $1 on the first unit of Y, because its marginal utility per rupee is 12 utils is higher than X’s 10 utils,

But, the consumer finds he is indifferent about whether he should buy a second unit of Y or first unit of X.

 

If consumer buys both of them, now he has 1 unit of X and 2 units of Y.

 

Does this combination of X and Y represent the maximum amount of utility which the consumer can obtain?

The answer is “No”.

 

This collection of goods only costs $/₹/Rs 3 [(1 × 1) + (2 × 1)], he has $/₹/Rs 2 of income remaining, which he can spend to achieve a still higher level of total utility.

 

Examining columns 2 and 3 again, we find that the consumer is indifferent between third units of Y and two units of X.

Marginal utility per rupee spent on each now the same at 8 units for the last rupee spent on each product and his money income of Rs 5 is fully spent, that is [(2 × $1) + (3 × $1) = 5].

The utility maximizing combination of commodities attainable by the consumer is 2 units of X and 3 units of Y.

The consumer is obtaining 18 (10 + 8) utils of satisfaction from 2 units of X and 30 (12 + 10 + 8) utils from the 3 units of Y.

His $5 of income is optimally spent from which he derives 48 (18 + 30) utils of satisfaction.

 

It is noticeable that there are other combinations of X and Y which are obtainable with $5.

But none of these will yield a level of total utility as high as do 2 units of X and 3 units of Y.

For example 1 units of X and 4 units of Y can be obtained from $5.

However, this combination violates utility maximizing rule; total utility here is 46 utils, clearly inferior to the utility 48 utils yielded by 2 of X and 3 of Y.

 

Furthermore, there are other combination of X and Y (such as 3 of X and 4 of Y or 1 of X and 2 of Y) wherein the marginal utility of the last rupee is the same for both X and Y.

 

But such combinations are either unobtainable with the consumer’s limited money income (as 3 of X and 4 of Y) or fail to exhaust her money income (as 1 of X and 2 of Y) and therefore, do not yield him the maximum utility attainable.

 

In the figure the equilibrium combination of commodities X and Y is X₁Y₁ where the consumer is obtaining equal marginal utilities (X₁E₁ = Y₁E₂) from X and Y.

 

Any other combination is not equilibrium, which violates the utility maximizing rule.

If the consumer desires to consume one addition unit of X he will obtain X₁X₂EE₁ additional utility; because of limited income to consume more unit of X the consumer must sacrifice some unit of Y and he will loss Y₁Y₂E₂E₃ utility.

 

Consumer loss in utility because of sacrifice in the consumption of Y is greater than the gain of utility from the consumption of more X.

 

This is because of the operation of diminishing marginal utility in the consumption process.

The same conclusion will be derived if the consumer desires to increase the consumption of more of Y and less of X.

 

 

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