Home Economics Marginal Rate of Substitution (MRS): Formula & Definition

# Marginal Rate of Substitution (MRS): Formula & Definition

## What is marginal rate of substitution?

The marginal rate of substitution is basically referred to as the rate at which a consumer is willing to sacrifice somewhat quantity of Good 2 or good Y (which we called as good X2 or good Y) in return of good 1 or good X (which we called as good X1 or good X) and remains equally satisfied as he was with good X1 or good X.

## Define marginal rate of substitution?

The marginal rate of substitution definition can be stated as the amount at which a consumer is keen to abandon a number of units’ good X for one more of good Y at the alike efficacy. So, it is mainly used to analyze the indifference curve that helps to study the behavior of consumers. As we place the 2 goods i.e. good X and good Y or say good 1 and good 2 on the indifference curve and also place the utility and need on the same curve and study their behavior.

## What Is MRS In Economics?

MRS economics can be referred to as the utility gained for good Y while you lose utility of good X. Therefore, MRS of Y and X is the change in X divided by change in Y at any point on the indifference curve.

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## Marginal Rate Of Substitution Formula

The (MRS) marginal rate of substitution formula can be stated as follows:

∣MRSxy​∣ = dx / dy ​ = MUy / ​MUx​​

Where in the above formula,

• x, y = two different goods
• dx dy = derivative of y with respect to x
• MU = marginal utility of good x, y​

Or you can also write down this formula as follows,

Marginal Rate of Substitution of Good X for Good Y (MRSxy) = ∆ Y / ∆ X

Although, you should note that  and  are not the same, as they are the reciprocal of each other, as we represent  of them as,

MRSyx = 1  ∕  MRSxy

### How to calculate marginal rate of substitution from a utility function?

You can derive the Formula of Marginal Rate of Substitution as we are given the quantities of goods involve utility function (U) for any consumer. So, here we assume that there are two commodities x1 and x2. Then

U = f (x1, x2) = constant = U0

Taking total differential, we get

d U = ∂ f / ∂ x1 . d x1 + ∂ f / ∂ x2 . d x2 = d U0 = 0

⇒ f x1 d x1 + f x2 d x2

⇒ – d x2 / d x1 = f x1 / f x2 = (∂ U / ∂ x1 ) ÷ (∂ U / ∂ x2) = MU x1 / MU x2

Here, we have the indifference curve slope as (d x2 / d x1) that also tells us the rate at which x1 must be substituted for x2 or vice versa.

Also, here The negative of the slope (- d x2 / d x1) is known to be as the marginal rate of substitution of x1 for x2.

### What Is the Marginal Rate of Technical Substitution (MRTS) ?

The marginal rate of technical substitution (MRTS) is basically an economic theory that tells us the rate at which one factor must decrease to achieve the same level of satisfaction while we decrease the rate at which we consume another factor.

So, MRTS basically gives and take relationship, by which the firm can maintain constant output.

### Marginal rate of technical substitution formula

The (MRTS) marginal rate of technical substitution formula can be stated as follows:

MRTS (L, K) = – Δ L / Δ K​ = MPK / ​MPL

Where in the above formula:

• K = Capital
• L = Labor
• MP = Marginal products of each input
• Δ L Δ K = amount of capital that can be reduced when labor is increased (typically by one unit)​

## Indifference Curve

The Indifference Curve represents the graph of 2 commodities i.e. commodity X and commodity Y that provides satisfaction to the consumer.

### Properties of Indifference Curve

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

## Types of Marginal Rate of Substitution

Diminishing MRS

Constant MRS

Increasing MRS

• ### Diminishing MRS

Diminishing MRS occurs when the consumer is willing to give up a less and lesser amount of good Y in exchange for good X than the MRS obtained is diminishing one.

• ### Constant MRS

Constant MRS occurs when the consumer gives up 1 more unit of good Y to get one more unit of good X, so here exists perfect substitution resulting in constant MRS.

• ### Increasing MRS

Increasing MRS occurs when the consumer is willing to give up additional unit of good X at an increasing rate o that he can achieve the same level of satisfaction. Thus, here one can attain an increasing marginal rate of substitution.

## Marginal Rate Of Substitution Example

By the use of the following examples, you will be able to understand calculations relating to Substitution of Marginal rate.

### Example 1:

#### Graph representation of MRS In reference to the above graphical representation of image and indifference curve, we have 2 commodities of goods with us as, Hamburger and Hotdogs as good X and good Y or can also denote them by good X1 and good X2.

Here, the consumer must choose between Hotdogs and hamburgers. And in order to get the marginal rate of substitution the consumers choose between both the goods i.e. Hotdogs and Hamburger to provide them an equal level of satisfaction.

So, with these combinations of both the goods the curve line is negative, which shows us a diminishing rate of substitution.

In the above figure, when the consumer consumes 5 units of Hotdogs he gets 31 units of Hamburger, and 31 units of Hamburger. And in order to have additional units of Hotdogs he has to give up Hamburger. Thus, in order to consume 10 units of Hotdogs, the consumer gives up some units of Hamburger, and the consumer only 16 units of Hamburger.

#### Case of additional units of Hamburger

• At Point A: The consumer consumes 15 units of Hotdogs and 10 units of Hamburger, so to consume additional units of Hamburger the consumer gives up Hotdogs and consumes only 10 units of Hotdogs and gets additional units of Hamburger as 16 units instead of 10. So, he shifts at point B.
• At Point B: The consumer consumes 10 units of Hotdogs and 1 unit of Hamburger, so to consume additional units of Hamburger the consumer gives up Hotdogs and consumes only 5 units of Hotdogs and gets additional units of HAmburger as 31 units instead of 16. So, he shifts at point C.
• At Point C: The consumer consumes 5 units of Hotdogs and 31 units of Hamburger, so to consume additional units of Hamburger the consumer has to give up Hotdogs and will be getting additional units of Hamburger that will provide him the same level of satisfaction.

### Example 2:

#### Tabular representation of MRS

Here, we have presented the following table showing different combinations of good X and good Y, to provide equal satisfaction to the consumer, as:

CombinationGood XGood Y
A210
B37
C45
D53
E62

The above table shows us that the consumer is willing to give up three units of Good Y in order to get one additional unit of Good X. Therefore, at Point A in the above table the MRS of X for Y for the consumer is 3.

The amount of Y that consumer is willing to give up to get additional units or one additional unit of X in order to get the same utility and satisfaction is termed as MRS of X for Y. Here, while calculating MRS we assume that the consumer gets the same level of utility at all points.

#### Key Points to notice

• It involves 2 goods that the consumer uses or consumes and they provide equal satisfaction to the consumer.
• It also involves substituting one good for the other good to provide equal MRS utility to the consumer i.e. giving one good for the consumption of the other.
• It is the slope of the indifference curve that we notice at any point of the IC curve.
• An indifference curve is a convex curve, showing consuming more than 1 commodity for another.

#### Assumptions Involved

The following assumptions are applied to MRS as the utility of consumer changes, which are as follows:

• The consumer is rational and conversant to consume every unit of goods.
• All the goods are equal in size and shape.
• There is no time gap between consumption.
• There is no change in income, preference, taste, and fashion.
• The utility is cardinal.
• Marginal unit of money is constant.

#### Limitations to MRS formula

This law of MRS doesn’t apply to the following conditions:

• When there exist dissimilar units.
• There exists an unreasonable quantity of goods.
• There is the unsuitable time period.
• In the case of rare collections like coins, stamps, etc.
• There is some change in the taste and fashion of the consumer.
• In the case of an abnormal person.
• When the income of the consumer is changing or fluctuating.
• In the case of habitual goods.
• In the case of durable and valuable goods.

#### Marginal rate of substitution calculator

Here, we have mentioned marginal rate of substitution calculator that you can use to calculate the MRS of goods with the final and an initial number of units consumed and the initial and final total utility in order to find out the marginal utility of goods. So, you can use marginal rate calculator that will assist you to calculate marginal rate and save your time as well.

## Frequently Asked Questions

• ### How to find the marginal rate of substitution?

You can find the marginal rate of substitution by using the MRS formula as follows:

∣MRSxy​∣ = dx  ∕  dy ​ = MUy  ∕  ​MUx

Where in the above formula we have,

• x, y = two different goods
• dx dy = derivative of y with respect to x
• MU = marginal utility of good x, y​

Or you can also write down this formula as follows,

Marginal Rate of Substitution of Good X for Good Y (MRSxy) =ΔY/ΔX

• ### Is marginal rate of substitution constant?

Yes, the marginal rate of substitution can be constant too other than increasing and decreasing. It is constant when one unit of X is given up to get the one more unit of Y, where there exists perfect substitution.

• ### Why does MRS decrease?

MRS decreases when the consumer is willing to give up a less and lesser amount of good Y in exchange of good X than the MRS obtained is diminishing one.

• ### Why are indifference curves bowed inward?

Indifference curves are convex in nature or say are bowed inward due to the marginal rate of substitution (MRS) which tells us that the marginal utility of consuming good decreases as the supply of the good increases and vice versa.

## Conclusion

MRS is stated to as the utility gained for good Y while you lose utility of good X. Therefore, MRS of Y and X is the change in X divided by change in Y at any point on the indifference curve.

Therefore, the formula for the same is as . And also we have studies the different types of it as ∣MRSxy​∣ = dx / dy ​ = MUy / ​MUx​​. diminishing MRS, increasing MRS, and constant MRS that exist when the consumer is willing to sacrifice more or less of goods, that in all helps us to study marginal rate of substitution thoroughly.

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