### What Is the Marginal Rate of Substitution (MRS)?

In economics, the marginal rate of substitution (MRS) is the amount of a good that a consumer is willing to give up for another good, as long as 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, displaying a frontier of equal utility for each combination of "good A" and "good B."

### Key Takeaways

- The marginal rate of substitution (MRS) is the amount of a good that a consumer is willing to give up for another good, as long as the new good is equally satisfying.
- It forms a downward sloping curve, called the indifference curve.
- At any given point along an indifference curve, the MRS is the slope of the indifference curve at that point.

### The Formula for MRS Is

$\begin{aligned} &|MRS_{xy}| = \frac{dy}{dx} = \frac{MU_x}{MU_y} \\ &\textbf{where:}\\ &x, y=\text{two different goods}\\ &\frac{dy}{dx}=\text{derivative of y with respect to x}\\ &MU=\text{marginal utility of good x, y}\\ \end{aligned}$

#### Marginal Rate of Substitution

### What Does the Marginal Rate of Substitution Tell You?

The marginal rate of substitution is an economics term that refers to the point at which one good is substitutable for another. It forms a downward sloping curve, called the indifference curve, where each point along it represents quantities of good X and good Y that you would be happy substituting for one another. It is always changing for a given point on the curve, and mathematically represents the slope of the curve at that point.

At any given point along an indifference curve, the MRS is the slope of the indifference curve **at that point**. Note that most indifference curves are actually curves, so their slopes are changing as you move along them. If the marginal rate of substitution of X for Y or Y for X is diminishing, the indifference’ curve must be convex to the origin. If it is constant, the indifference curve will be a straight line sloping downwards to the right at a 45° angle to either axis. If the marginal rate of substitution is increasing, the indifference curve will be concave to the origin.

The law of diminishing marginal rates of substitution states that MRS* decreases *as one moves down the standard convex-shaped curve, which is the indifference curve.

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

For example, 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 the consumer is willing to give up for more hamburgers. 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 in order to obtain 1 extra hamburger.

### Limitations of Marginal Rate of Substitution

The marginal rate of substitution does not examine a combination of goods that a consumer would prefer more or less than another combination but examines which combinations of goods the consumer would prefer just as much. It also does not examine marginal utility – how much better or worse off a consumer would be with one combination of goods rather than another – because all combinations of goods along the indifference curve are valued the same by the consumer.