Explain the law of the marginal rate of substitution.

The law of the marginal rate of substitution states that as a consumer substitutes one good for another while keeping the total utility constant, the rate at which they are willing to exchange one good for another will diminish. In other words, as a consumer consumes more of one good and less of another, the amount of the second good they are willing to give up for an additional unit of the first good decreases.

This law is based on the principle of diminishing marginal utility, which states that as a consumer consumes more of a good, the additional satisfaction or utility they derive from each additional unit of that good decreases. As a result, the consumer will be less willing to give up a larger quantity of the second good in exchange for a smaller quantity of the first good.

The law of the marginal rate of substitution is important in understanding consumer behavior and preferences, as it helps to explain how consumers make choices and allocate their resources in order to maximize their utility.

Law of Marginal Rate of Substitution (MRS)

The Law of Marginal Rate of Substitution states that in a situation where two goods (X and Y) are interchangeable, the consumer will reach equilibrium when the marginal utility of one good divided by its price is equal to the marginal utility of the other good divided by its price:

MRS (X, Y) = MUx / Px = MUy / Py

where:

- MRS (X, Y) is the Marginal Rate of Substitution of Good X for Good Y
- MUx is the Marginal Utility of Good X
- Px is the Price of Good X
- MUy is the Marginal Utility of Good Y
- Py is the Price of Good Y

Interpretation:

This law implies that at the point of equilibrium, the consumer derives the same additional satisfaction (marginal utility) from consuming a unit of either good X or Y, taking into account their respective prices. In other words, the consumer is allocating their budget optimally to maximize their total utility.

Example:

Suppose a consumer has $100 to spend on two goods: apples and oranges. The marginal utility of each additional apple is 10 utils, and its price is $1 per apple. The marginal utility of each additional orange is 15 utils, and its price is $2 per orange.

Calculate MRS (Apples, Oranges):

MRS (Apples, Oranges) = MUapples / Papples = MUoranges / Poranges

= 10 utils / $1 = 15 utils / $2

= 10/1 = 15/2

= 5

Interpretation:

At this point of consumption, the consumer derives the same additional satisfaction from consuming 5 apples as they do from consuming 3 oranges. They have reached equilibrium and are allocating their budget optimally to maximize their total utility.