MRS in Economics: What It Is and the Formula for Calculating It

Marginal Rate of Substitution

Investopedia / Madelyn Goodnight

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 consume compared to another good, as long as the new good is equally satisfying.

MRS is used in indifference theory to analyze consumer behavior. When someone is indifferent to substituting one item for another, their marginal utility for substitution is zero since they neither gain nor lose any satisfaction from the trade.

Key Takeaways

  • The marginal rate of substitution (MRS) is the willingness of a consumer to replace one good for another good, as long as the new good is equally satisfying.
  • The marginal rate of substitution is the slope of the indifference curve at any given point along the curve and displays a frontier of utility for each combination of "good X" and "good Y."
  • When the law of diminishing MRS is in effect, the MRS forms a downward, negative sloping, convex curve showing more consumption of one good in place of another.
  • MRS may not inform analysts of true utility as it assumes both products can be exchanged for the same utility.
  • MRS is also limited in that it only considered two items; it does not consider how additional units may factor into different consumption preferences.
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What Is an Indifference Curve?

Formula and Calculation of the Marginal Rate of Substitution (MRS)

The marginal rate of substitution (MRS) formula is:

M R S x y = d y d x = M U x M U y where: x , y = two different goods d y d x = derivative of y with respect to x M U = marginal utility of good x, y \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} MRSxy=dxdy=MUyMUxwhere:x,y=two different goodsdxdy=derivative of y with respect to xMU=marginal utility of good x, y

What the MRS Can Tell You

The marginal rate of substitution is a term used in economics that refers to the amount of one good that is substitutable for another and is used to analyze consumer behaviors for a variety of purposes. MRS is calculated between two goods placed on an indifference curve, displaying a frontier of utility for each combination of "good X" and "good Y." The slope of this curve represents quantities of good X and good Y that you would be happy substituting for one another.

MRS is a critical component for businesses to understand when analyzing consumption trends or for government entities to understand when setting public policy. Consider an example of a government wanting to analyze how offering electric vehicle incentives may spur more environmentally-friendly purchases. Understanding how MRS is impacted before and after a tax incentive can allow for the government to analyze the financial implications of the plan.

MRS and the Indifference Curve

The slope of the indifference curve is critical to the marginal rate of substitution analysis. MRS is the slope of the indifference curve at any single point along the curve. The slope will often be different as one moves along an indifference curve.

Most indifference curves are usually convex 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, resulting 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 since it means a consumer would consume more of X for the increased consumption of Y (and vice versa). Usually, marginal substitution is diminishing, meaning 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 curve, which is the indifference curve.

Example of MRS

For example, a consumer must choose between hamburgers and hot dogs. 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 consumption. 

Marginal Rate of Substitution
Investopedia/Julie Bang

Limitations of the MRS

The marginal rate of substitution has a few limitations. The main drawback is that it does not examine a combination of goods that a consumer would prefer more or less than another combination. This generally limits the analysis of MRS to two variables. As this is most often graphically depicted using only x and y variables, other variables that may still factor consumption may not be appropriately considered.

MRS does not necessarily examine marginal utility since it treats the utility of both comparable goods equally, though in actuality they may have varying utility. In the example above, consider how the utility of a hamburger (with it's potential lettuce, onion, or other vegetable dressings) may vary from that of a plain hot dog.

MRS vs. MRT

Marginal rate of substitution is tied to the marginal rate of transformation (MRT). Whereas MRS focuses on the consumer demand side, MRT focuses on the manufacturing production side.

Often, the two concepts are intertwined and drive the other. For example, consider a global shortage of flour. A manufacturer may be more inclined to bake less cakes and more bread as bread is a more efficient product to make based on material constraints.

As a result, consumers may find cake shortages result in much higher prices. This may in turn result in a stronger MRS between cake and bread as consumers may be enticed by lower costs of the over-produced item. On the other hand, if consumers don't prove to have any reason to substitute bread for cake, a manufacturer may be handcuffed into producing a less-efficient good to meet market demand.

What Is the Relationship Between Indifference Curve and MRS?

Essentially, MRS is the slope of the indifference curve at any single point along the curve. Most indifference curves are usually convex because as you consume more of one good you will consume less of the other. So, MRS will decrease as one moves down the indifference curve.

This is known as the law of diminishing marginal rate of substitution. If the marginal rate of substitution is increasing, the indifference curve will be concave, which means that a consumer would consume more of X for the increased consumption of Y and vice versa, but this is not common.

What are the Drawbacks of Marginal Rate of Substitution?

The marginal rate of substitution has a few limitations. The main drawback is that it does not examine a combination of goods that a consumer would prefer more or less than another combination. This generally limits the analysis of MRS to two variables. Also, MRS does not necessarily examine marginal utility because it treats the utility of both comparable goods equally though in actuality they may have varying utility. 

What Is Indifference Curve Analysis?

Indifference curve analysis operates on a simple two-dimensional graph. Each axis represents one type of economic good. The consumer is indifferent between any of the combinations of goods represented by points on the indifference curve because these combinations provide the same level of utility to the consumer. Indifference curves are heuristic devices used in contemporary microeconomics to demonstrate consumer preference and the limitations of a budget.

The Bottom Line

For economic and financial planning reasons, it's critical that various entities understand how consumers may substitute one good for other. This concept called marginal rate of substitution, measures the relationship between two products and how likely a consumer is to buy one in the place of the other. This information is useful in setting manufacturing levels or gauging public policy.

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