The Marginal Rate of Substitution can be defined as the rate at which a consumer is willing to forgo a number of units good X for one more of good Y at the same utility. The Marginal Rate of Substitution is used to analyze the indifference curve.

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## Marginal Rate of Substitution

Suppose you and your friend is playing Scrabble. Both of you decided on some new rules for the game. You suggested that if any of you want a new piece of the alphabet you have to do any of the three things.

Either you have to skip a turn or you have to exchange with any other alphabet you have or you have to cut down your earned points. Suppose you want to get a new piece of an alphabet.

You decided to cut your points. How much will you cut down your points to get a new piece of the alphabet?

In the real world and in the business world we come across this type of situation. We have to settle for some commodity instead of others. We have to substitute one service instead of others.

This is the concept of the marginal rate of substitution. Let us get ourselves ready to know them.

**Indifference Curve**

It is a curve that shows the combination of goods which gives the same level of satisfaction to the consumers so that an individual is indifferent. In other words, the consumer gives equal preference to all such combinations.

It is a graph that gives a consumer equal satisfaction, making the consumer indifferent. An indifference curve shows the combination of services which a consumer can prefer over the other.

For any consumer, utility function (U) is a function of the quantities of goods. Suppose there are two commodities x_{1} and x_{2}. Then

U = f (x_{1}, x_{2}) = constant = U_{0}.

On the indifference curve, the quality consumed of one commodity is compensated by the increase in the quantity consumed of the other commodity.

**Assumption**

- Utility is cardinal.
- Consumer is rational.
- Goods consumed are substitutable.
- Availability of more goods is always better.
- A consumer will have transitivity in his choice. Suppose a consumer prefer item ‘A’ over ‘B’. Also, he chooses item ‘B’ over ‘C’. Then it must prefer item ‘A’ over ‘C’.

**Properties of Indifference Curve**

- The difference curve has a negative slope.
- Indifferent curves do not intersect.
- They are convex from below, i.e., convex to the origin.
- An indifference curve that lies to the right of another, yields more utility.

## Marginal Rate of substitution

It is the rate at which the consumer is willing to give up commodity ‘X’ for one more unit of commodity ‘Y’. He tries to maintain the same level of satisfaction.

In simple words, it is the same as the utility gained for good Y as the utility lost for good X. One can calculate the marginal rate of substitution as

M.R.S. _{Y X} = Δ X / Δ Y, on any point on the indifference curve.

**Derivation of Formula Marginal Rate of Substitution**

For any consumer, utility function (U) is a function of the quantities of goods. Suppose there are two commodities x_{1} and x_{2}. Then

U = f (x_{1}, x_{2}) = constant = U_{0}.

Taking total differential, we get

d U = ∂ f / ∂ x_{1} . d x_{1} + ∂ f / ∂ x2 . d x_{2} = d U_{0} = 0

⇒ f _{x1} d _{x1} + f _{x2} d x_{2}

⇒ − d x_{2} / d x_{1} = f _{x1 }/ f _{x2} = (∂ U / ∂ x_{1} ) ÷ (∂ U / ∂ x_{2}) = MU x_{1} / MU x_{2}

The slope (d x_{2} / d x_{1}) of the tangent at any point on an indifference curve is the rate at which x_{1} must be substituted for x_{2} or vice versa.

The negative of the slope (− d x_{2} / d x_{1}) is the marginal rate of substitution of x_{1} for x_{2}.

(source – econ 150)

**Assumptions**

- The consumer is logical and knowledgeable to consume every unit of goods.
- Goods are equal in size and shape.
- No time gap between consumption.
- No change in income, preference, taste, and fashion.
- Utility is cardinal.
- Marginal unit of money is constant.

**Limitations**

This law doesn’t apply to

- Dissimilar units.
- Unreasonable quantity.
- Unsuitable time period.
- Rare collections like coins, stamps etc.
- Change in taste and fashion of the consumer.
- Abnormal person.
- Changing the income of the consumer.
- Habitual goods.
- Durable and valuable goods.

**Types of Marginal Rate of Substitution**

**Diminishing**

The marginal rate of substitution is diminishing. One can obtain it if the consumer is willing to give up less and less unit of good Y for every additional unit of good X.

**Constant**

The marginal rate of substitution is constant also. One can obtain this if, for one more unit of Y, only one unit of X is given up. It is constant for perfect substitution.

**Increasing**

Suppose a consumer substitutes a commodity X for the other commodity Y at an increasing rate to maintain the same level of satisfaction. In this case, one can obtain an increasing marginal rate of substitution.

## Solved Example on Marginal Rate of Substitution

**Problem: Indifference curves are convex to the origin because(choose the correct choice)**

- Two goods are perfect substitutes.
- The two goods are complementary goods.
- Two goods are imperfect substitutes.
- None of the above.

Solution: iii. Two goods are imperfect substitutes.

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