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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 \(0 < a < 1\) is a parameter. Level curves on which the utility function is constant are called indifference curves; the utility is the same for all combinations of \(x\) and \(y\) along an indifference curve (see figure). a. The marginal utilities of the commodities \(x\) and \(y\) are defined to be \(\partial U / \partial x\) and \(\partial U / \partial y,\) respectively. Compute the marginal utilities for the utility function \(U(x, y)=x^{a} y^{1-a}\) b. The marginal rate of substitution (MRS) is the slope of the indifference curve at the point \((x, y) .\) Use the Chain Rule to show that for \(U(x, y)=x^{a} y^{1-a},\) the MRS is \(-\frac{a}{1-a} \frac{y}{x}\) c. Find the MRS for the utility function \(U(x, y)=x^{0.4} y^{0.6}\) at \((x, y)=(8,12)\)

Step by step solution

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a. Marginal Utilities

To compute the marginal utilities, we need to take the partial derivatives of the utility function with respect to \(x\) and \(y\): $$ \frac{\partial U}{\partial x} = \frac{\partial}{\partial x}(x^a y^{1-a}) $$ $$ \frac{\partial U}{\partial y} = \frac{\partial}{\partial y}(x^a y^{1-a}) $$ Differentiating, we get: $$ \frac{\partial U}{\partial x} = ax^{a-1} y^{1-a} $$ $$ \frac{\partial U}{\partial y} = (1-a)x^a y^{-a} $$ b. Chain Rule to find MRS

02

b. Marginal Rate of Substitution (MRS)

The MRS is defined as the ratio of marginal utilities: $$ \text{MRS} = -\frac{dy}{dx} = -\frac{\partial U / \partial x}{\partial U / \partial y} $$ From our results above, we have: $$ \text{MRS} = -\frac{ax^{a-1} y^{1-a}}{(1-a)x^a y^{-a}} $$ Simplifying, we get: $$ \text{MRS} = -\frac{a}{1-a} \frac{y}{x} $$ c. Find the MRS for a specific utility function

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c. Calculate MRS at \((x, y)=(8, 12)\)

We are given that the utility function is \(U(x, y)=x^{0.4} y^{0.6}\), and want to find the MRS at \((x, y)=(8, 12)\). First, note that the parameter \(a\) is 0.4. Using the formula we got for MRS from part b, we can plug in the values of \(x\), \(y\), and \(a\): $$ \text{MRS} = -\frac{a}{1-a} \frac{y}{x} = -\frac{0.4}{1-0.4} \frac{12}{8} $$ Simplifying, we find the MRS at the given point: $$ \text{MRS} = -\frac{2}{3}

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Utility Functions in Economics

Utility functions are a cornerstone concept in the study of economics, particularly in the analysis of consumer behavior. They represent a way to quantify how consumers value different goods and services. The utility a person derives from these goods can then be used to predict decision-making and consumption patterns.

Imagine you're at the store deciding between vanilla and chocolate ice cream. Your choice will depend on which flavor yields more satisfaction or 'utility'. If a utility function of the form \(U(x, y) = x^a y^{1-a}\) was used to model your preferences, \(x\) and \(y\) would encode the amounts of vanilla and chocolate ice cream, and the parameter \(a\) would reflect the weight you give to vanilla ice cream in your utility calculation. The crucial point is that this equation simplifies how we understand decisions; instead of considering a whole narrative about why you like vanilla more, economists use the utility function's parameters to ascribe numeric values to your preferences.

Partial Derivatives

Fundamentals of Partial Derivatives

When dealing with functions that involve more than one variable, like utility functions, partial derivatives become essential tools. The partial derivative of a function with respect to one of its variables is simply the derivative taking all other variables as constants.

For instance, if you're trying to understand how your utility changes as you consume more vanilla ice cream while keeping your chocolate ice cream consumption constant, you would look at the partial derivative of your utility function with respect to vanilla, notated as \( \frac{\partial U}{\partial x} \). This measure helps economists realize the additional utility (marginal utility) gained from a tiny increase in consumption of one good, holding everything else constant. Partial derivatives underpin the idea of marginal analysis, which is used throughout economics to make comparisons and optimizations.

Indifference Curves

Exploring Indifference Curves

Indifference curves are graphical representations that illustrate combinations of two goods that give a consumer the same level of satisfaction. If we plot different combinations of vanilla and chocolate ice cream consumption on a graph, every point on the same indifference curve means equally preferable choices that provide the same utility.

One of the beautiful aspects of indifference curves is that they help visualize the concept of the marginal rate of substitution (MRS). This rate tells us how willing a person is to trade off one good for another, while remaining equally satisfied, hence 'indifferent'. When we say the MRS is the slope of the indifference curve at any point, we’re essentially finding out how much of one ice cream flavor a person is willing to give up for an extra serving of the other flavor, without changing their overall level of happiness.