# Marginal Rate of Substitution – MRS Economics Definition – Example

## What Is the Marginal Rate of Substitution (MRS)? The Marginal Rate of Substitution is the rate at which a consumer is willing to exchange units of good X for one more unit of good Y assuming both have the same utility.

In economics, the MRS is the amount of a good that a consumer is willing to consume in relation to another good.  Of course, this assumes that the new good is equally satisfying. It’s used in indifference theory to analyze consumer behavior. The marginal rate of substitution is calculated between two goods placed on an indifference curve.  The curve displays a frontier of utility for each combination of item X and item Y.

An indifference curve is usually downward sloping and convex. The MRS is the slope of the indifference curve at any given point along the curve. When the law of diminishing marginal rates of substitution is in effect, the marginal rate of substitution forms a downward, negative sloping, convex curve.  This curve is a graphic representation showing how more consumption of one good relates to another.

## Marginal Rate of Substitution – A Closer Look

MRS is used to analyze consumer behaviors for a variety of purposes. The marginal rate of substitution is an economic term that refers to the amount of one good a consumer might substitute for another. MRS economics involves a sloping curve, called the indifference curve.  Each point along it represents quantities of good X and good Y that consumers would be willing to substitute one for another.

The slope of the indifference curve is essential to the marginal rate of substitution analysis. At any given point along an indifference curve, the MRS is the slope of the indifference curve at that point. Most indifference curves are actually curves, so the slopes are changing as you move along them. Most indifference curves are also usually convex.  This is because as you consume more of one good you will consume less of the other. Indifference curves can be straight lines if a slope is constant.  It results in an indifference curve represented by a downward-sloping straight line.

If the marginal rate of substitution is increasing, the indifference curve will be concave to the origin. This is typically not common.  It means a consumer would consume more of X for the increased consumption of Y and vice versa. Usually, marginal substitution is diminishing.  This means a consumer chooses the substitute in place of another good rather than simultaneously consuming more. The law of diminishing marginal rates of substitution states that MRS decreases as one moves down a standard convex-shaped indifference curve.  (Source: investopedia.com)

## Marginal Rate of Substitution and Indifference Curves

The MRS is linked with indifference curves because the slope of this curve is the MRS.

#### Diminishing

When the MRS is diminishing, it means a consumer is willing to give up fewer units of good Y for every additional unit of good X he consumes.  It is generally used for defining the utility of consumption for a given economic agent.  It has an MRS that constantly changes along the curve.

• A concave curve – tends toward zero along the x-axis when diminishing the quantity of y.  The curve approaches infinity along the y-axis when diminishing the quantity of x.
• A convex curve – A person whose preferences are convex always prefers mixtures of goods to extremes of either good. If we draw a line between two points on the same indifference curve, then each point on the line is a mixture of the two end-points. When the indifference curves are convex, all points on the line between the end-points give higher utility than the end-points.

#### Constant

The marginal rate of substitution can also be constant. It means that for one more unit of Y, only one unit of X is given up. Constant tends to mean there is perfect substitution.  Both goods are perfect substitutes since the lines are parallel and the MRS = 1.  The slope has an angle of 45º with each axis. For perfect substitutes, the MRS will remain constant.

#### Increasing

Suppose a consumer substitutes a commodity X for the other commodity Y at an increasing rate to maintain the same level of satisfaction. It implies an increasing marginal rate of substitution.  In this case, the horizontal fragment of each indifference curve has an MRS approaching zero and the vertical fractions an MRS approach infinity. (Source: policonomics.com)

## Calculating the MRS Formula

The marginal rate of substitution (MRS) formula is:

### |MRSx,y| = dy/dx = MUx/MUy

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

## Example of How to Use the Marginal Rate of Substitution

For example, assume a consumer must choose between hamburgers and hot dogs. In order to determine the marginal rate of substitution, the consumer is asked what combinations of hamburgers and hot dogs provide the same level of satisfaction.

When these combinations are graphed, the slope of the resulting line is negative. This means that the consumer faces a diminishing marginal rate of substitution.  The more hamburgers they have relative to hot dogs, the fewer hot dogs they are willing to consume. If the marginal rate of substitution of hamburgers for hot dogs is -2, then the individual would be willing to give up 2 hot dogs for every additional hamburger he consumes.  (Source: investopedia.com)

## Marginal Rate of Substitution Example

To illustrate an example, use the following table as points on an indifference curve. This table is known as the indifference schedule.

Going from point A to point B, we can see that the consumer is ready to exchange three units of Good Y for one additional unit of Good X. Therefore, at this stage, the consumer’s Marginal Rate of Substitution of X for Y is three.

The MRS of X for Y represents the amount of Y which the consumer has to give up for the gain of one additional unit of X so that his or her level of his or her utility (satisfaction) remains the same. We assume that any of the five combinations in the table have the same level of utility. For example, if the consumer goes from D to E, then the marginal rate of substitution becomes 1. (Source: intelligenteconomist.com)

### The Principle of Diminishing Marginal Rate of Substitution

The MRS of Good X for Good Y diminishes as more and more of Good X is substituted for Good Y. In other words, as the consumer has more and more of good X he is prepared to give up less and less of Good Y. The rate at which the consumer substitutes Good X for Good Y is greater at the beginning. But, as he continues the substitution process, the rate of substitution begins to fall.  (Source: ibid)

## Limitations of Marginal Rate of Substitution

MRS analysis is generally limited to only two variables.  It does not examine combinations of goods that a consumer would prefer more or less over another combination. Also, MRS does not necessarily examine marginal utility.  It treats the utility of both comparable goods equally.  However, in actuality, they may have varying utility.  Also, this principle doesn’t apply to:

• Dissimilar units.
• 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.

## Up Next: What Is FF&E – Furniture, Fixtures, and Equipment?

FF&E is an acronym for furniture, fixtures, and equipment.  It refers to movable furniture, fixtures, or other equipment that have no permanent connection to the structure of a building. These items include desks, chairs, computers, electronic equipment, tables, bookcases, and partitions.  FF&E items typically depreciate substantially over their long-term use.  But, they are nevertheless important costs to consider when valuing a company.  These items are sometimes referred to as FF&A which stands for furniture, fixtures, and accessories.