How to Calculate Marginal Rate of Substitution

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There are a number of economic principles that are important to learn when operating a business. One of these is the marginal rate of substitution, or MRS. While you can find a marginal rate of substitution calculator when you need one, you will be better served in the long run to learn how to calculate MRS yourself. Fortunately, the marginal rate of substitution formula isn't difficult so long as you know the values of the items being substituted.

Defining the Marginal Rate of Substitution

To calculate the marginal rate of substitution, the following formula is used:

|MRS(X, Y)| = ΔY/ΔX

In this formula, both X and Y represent the values of two items believed to be of equivalent quality. The ΔY and ΔX in a fraction represent the derivative of Y in respect to X, giving us an idea of the rate at which the two values can change. By taking this derivative, you can find the marginal utility of the two goods, which tells you how much of item X is equivalent to a specified amount of item Y or vice versa. Note that the MRS is always a positive number, because you solve for an absolute value (indicated by the | | around the MRS in the formula.)

Finding Perfect Substitutes

In some cases, the perceived values of two items will be equivalent regardless of the quantities involved. These items are referred to as perfect substitutes, as consumers do not appear to have a preference between them and will freely choose one or the other based on availability. This is often useful from the consumer's point of view, but it may not be as good for your company depending on the circumstances of the goods in question.

Having perfect substitutes is beneficial to a business if you produce both of the goods involved and they carry the same price and production margins; it won't matter whether consumers buy item A or item B because your company makes the same profit off of both purchases. Perfect substitution becomes troublesome if one item has a higher margin than the other or if one of the items is produced by a competitor, however. In these cases, your company may need to take steps to increase the perceived value of the product you most wish consumers to prefer.

Limitations of MRS

The marginal rate of substitution cannot be used to determine consumer preference, though some companies try to use it in this manner. The formula doesn't take into account if the consumer has a preference for one of the goods over the other; instead, it assumes that both goods are seen as equally valued by the consumer and the consumer likes both an equivalent amount. It also does not take into account whether consumers would be better off with a different combination of goods. While it is a useful tool, it's important not to try and use it as any sort of a gauge of consumer intent or product preferences.

What Is the Marginal Rate of Technical Substitution?

Occasionally, you may hear reference to the marginal rate of technical substitution, or MRTS. This concept is very similar to the marginal rate of substitution, though it is typically used in terms of labor. The MRTS determines the rate at which one labor input can be substituted for another without affecting the overall output of the system. The primary difference between MRS and MRTS is that the marginal rate of substitution focuses on finding equilibrium on the consumer side, whereas the marginal rate of technical substitution is focused on finding equilibrium on the producer side.

To calculate a marginal rate of technical substitution, use the formula MRTS(L,K) = - ΔK/ ΔL, with K representing cost and L representing labor input. Note that while this looks significantly like the marginal rate of substitution formula, the value is multiplied by -1 (indicated by the negative sign in front of the division). This multiplication by a negative number does not occur when calculating MRS.