The Marginal Rate of Substitution (MRS) is an important concept in microeconomics that shows the rate at which a consumer is willing to trade off one good for another while maintaining the same level of utility. To derive the MRS step by step, we follow a process based on a consumer's utility function and indifference curves.

Here’s a detailed step-by-step explanation of the derivation:

Step 1: Understand the Utility Function

First, we need a utility function. A utility function, $U(x_1, x_2)$, represents the consumer's preference over two goods, $x_1$ and $x_2$. For example, let’s assume the utility function is:

$U(x_1, x_2) = x_1 \cdot x_2$

This is a simple Cobb-Douglas utility function, where the consumer derives utility from consuming goods $x_1$ and $x_2$.

Step 2: Define the Indifference Curve

An indifference curve shows all combinations of goods $x_1$ and $x_2$ that give the consumer the same level of utility. For a given utility level $U_0$, the indifference curve is the set of all points where:

$U(x_1, x_2) = U_0$

For our utility function, if the utility level is $U_0$, the indifference curve is:

$x_1 \cdot x_2 = U_0$

Step 3: Partial Derivatives and Marginal Utilities

To compute the Marginal Rate of Substitution, we need the marginal utilities of both goods. The marginal utility of a good is the partial derivative of the utility function with respect to that good.

• Marginal utility of $x_1$ ($MU_1$):

$MU_1 = \frac{\partial U(x_1, x_2)}{\partial x_1} = x_2$

• Marginal utility of $x_2$ ($MU_2$):

$MU_2 = \frac{\partial U(x_1, x_2)}{\partial x_2} = x_1$

Step 4: Definition of MRS

The Marginal Rate of Substitution is defined as the rate at which a consumer is willing to give up some amount of one good in exchange for another good, while keeping the utility constant. Mathematically, the MRS is given by the ratio of the marginal utilities:

$MRS = - \frac{MU_1}{MU_2}$

In our example:

$MRS = - \frac{x_2}{x_1}$

This means that the consumer is willing to give up $\frac{x_2}{x_1}$ units of good 2 for an additional unit of good 1 to maintain the same level of utility.

Step 5: Interpretation

The MRS decreases as the consumer consumes more of good $x_1$ (and less of good $x_2$), which reflects the diminishing marginal utility—i.e., the more you have of a good, the less you're willing to give up other goods to get more of it.

Example

Let’s assume a consumer has the following amounts of two goods: $x_1 = 4$ and $x_2 = 8$.

From Step 4, we know that:

$MRS = - \frac{x_2}{x_1} = - \frac{8}{4} = -2$

This means the consumer is willing to give up 2 units of $x_2$ for an additional unit of $x_1$ while maintaining the same utility level.

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Key Insights

• The MRS measures the slope of the indifference curve.
• The negative sign indicates the trade-off: as you consume more of one good, you consume less of the other.
• The magnitude of the MRS tells you how much of one good a consumer is willing to trade for another.

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Would you like more details or have any questions on this topic?

Related Questions:

1. How does the diminishing MRS affect consumer behavior?
2. How do perfect substitutes affect the MRS between two goods?
3. Can you explain the difference between MRS and the marginal rate of technical substitution (MRTS)?
4. How do income and substitution effects relate to changes in MRS?
5. How does MRS relate to the concept of equilibrium in consumer choice?

Tip:

Remember that the MRS is different for different points on the indifference curve, reflecting the changing willingness to substitute goods based on relative consumption levels.