Question

Economists use utility functions to describe consumers' relative preference for two or more commodities (for example, vanilla vs. chocolate ice cream or leisure time vs. material goods). The Cobb-Douglas family of utility functions has the form \(U(x, y)=x^{a} y^{1-a},\) where \(x\) and \(y\) are the amounts of two commodities and \(0

Short answer

Answer: The Marginal Rate of Substitution (MRS) for the given utility function at the point \((x, y) = (8, 12)\) is -3.

Step-by-step solution

Compute the Marginal Utilities

To compute the marginal utilities, find the partial derivatives of the utility function with respect to x and y. The given utility function is \(U(x, y) = x^{a}y^{1-a}\). \(\frac{\partial U}{\partial x} = a x^{a-1} y^{1-a}\) \(\frac{\partial U}{\partial y} = (1-a) x^{a} y^{-a}\) Thus, the marginal utilities for the given utility function are: \(\frac{\partial U}{\partial x} = a x^{a-1} y^{1-a}\) \(\frac{\partial U}{\partial y} = (1-a) x^{a} y^{-a}\)

Show the MRS formula

To show that the MRS is equal to \(-\frac{a}{1-a}\frac{y}{x}\), remember that the Marginal Rate of Substitution (MRS) is the ratio of the marginal utilities: \(MRS = -\frac{\partial U/\partial x}{\partial U/\partial y}\) Now, substitute the expressions for the partial derivatives we found in Step 1: \(MRS = -\frac{a x^{a-1} y^{1-a}}{(1-a) x^{a} y^{-a}}\) Now, simplify the equation: \(MRS = -\frac{a}{1-a} \frac{y}{x}\) Thus, we have shown that the MRS for the Cobb-Douglas utility function is equal to \(-\frac{a}{1-a}\frac{y}{x}\).

Find the MRS for the given utility function and point

Now, we want to find the MRS for \(U(x, y) = x^{0.4}y^{0.6}\) at the point \((x, y) = (8, 12)\). First, we need to identify the values of a, x, and y: \(a = 0.4, x = 8, y = 12\) Now, plug these values into the MRS formula we derived in Step 2: \(MRS = -\frac{0.4}{1-0.4} \frac{12}{8} = -\frac{0.4}{0.6} \frac{12}{8}\) Simplify the equation: \(MRS = -\frac{2}{1} \times \frac{3}{2} = -3\) Therefore, the MRS for the utility function \(U(x, y) = x^{0.4}y^{0.6}\) at the point \((x, y) = (8, 12)\) is equal to \(-3\).

Key concepts

Cobb-Douglas Utility Function

In the realm of economics, the Cobb-Douglas utility function serves as a fundamental tool for depicting a consumer's preferences for two types of commodities. This function is particularly useful because it provides a straightforward representation of how consumers derive satisfaction, or 'utility', from different combinations of goods. The general expression of the Cobb-Douglas utility function is given by \(U(x, y) = x^{a} y^{1-a}\). Here, \(x\) and \(y\) are quantities of two commodities and \(a\) is a parameter that must lie between zero and one.
The parameter \(a\) signifies how much weight or importance a consumer assigns to one commodity compared to the other. This flexibility makes it possible to model a variety of consumer preferences. Key features of this utility function include constant proportional trade-offs between goods, making it easier to study consumer behavior. It is often visualized using indifference curves which represent levels of satisfaction across different combinations of goods.

Marginal Rate of Substitution

Imagine you're deciding how to divide your budget between ice cream and cake. The decision often involves giving up some of one to gain more of the other. The rate at which you are willing to make that trade is called the Marginal Rate of Substitution (MRS). In mathematical terms, MRS is defined as the negative ratio of the marginal utilities of two goods.
For the Cobb-Douglas utility function, the MRS can be calculated using the formula \( MRS = -\frac{a}{1-a} \frac{y}{x} \). This formula shows how the ratio of the quantities \(y\) and \(x\) changes as \(a\) varies, reflecting the shift in preferences. The MRS diminishes as more of \(x\) is consumed, highlighting the principle of diminishing marginal utility, where more of a good decreases its extra satisfaction relative to others.

Partial Derivatives

In calculus, partial derivatives play a key role in understanding changes in multivariable functions, especially in utility functions like Cobb-Douglas. Here, partial derivatives show the rate of change of utility with respect to a change in one of the goods, while keeping the other constant. For a Cobb-Douglas utility \(U(x, y) = x^{a} y^{1-a}\), the partial derivatives are:

• \(\frac{\partial U}{\partial x} = a x^{a-1} y^{1-a}\)
• \(\frac{\partial U}{\partial y} = (1-a) x^{a} y^{-a}\)

These derivatives, often referred to as marginal utilities, reveal how the utility shifts as more of either \(x\) or \(y\) is consumed. The concept is crucial for deriving the MRS, which helps understand consumer trade-offs and optimize their satisfaction.

Indifference Curves

Indifference curves are a vital concept in microeconomics, representing various combinations of two goods that provide the same level of utility to the consumer. For the Cobb-Douglas utility function, each curve reflects a different utility level and typically has a convex shape. This convexity indicates that as a consumer substitutes one good for another, they require increasing amounts of the second good to maintain the same level of satisfaction.
The slope of the indifference curve at any point is the Marginal Rate of Substitution (MRS), which shows the trade-off ratio the consumer is willing to accept. The fact that these curves never intersect reflects the consistency of consumer preferences. Analyzing these curves helps economists understand how consumers allocate their resources and how changes in prices and income affect their choices.