Book cover for Intermediate Microeconomics: A Modern Approach

Intermediate Microeconomics: A Modern Approach

Hal R. Varian

ISBN #9780393927023

7th Edition

224 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter covered the fundamental concept of utility in economics, emphasizing how utility functions are constructed and used to analyze consumer behavior. Key forms of utility functions include perfect substitutes, perfect complements, quasilinear, and Cobb-Douglas, each with unique characteristics. Additionally, the chapter detailed the importance of marginal utility and the marginal rate of substitution (MRS) as tools for understanding consumer choice. Real-world applications such as commuting illustrate how these theoretical constructs are used in everyday decision-making.

Learning Objectives

1

Understand the concept of utility in economics and how it is used to analyze consumer behavior.

2

Differentiate between various forms of utility functions such as perfect substitutes, perfect complements, quasilinear, and Cobb-Douglas.

3

Explain the roles of marginal utility and the marginal rate of substitution (MRS) in consumer decision-making.

4

Apply the construction of utility functions to real-world scenarios, such as commuting decisions.

Key Concepts

CONCEPT

DEFINITION

Utility

A measure of satisfaction or preference that a consumer derives from consuming goods and services.

Cardinal Utility

A concept that assumes utility can be measured and assigned numerical values, allowing for meaningful comparisons of satisfaction levels.

Utility Function

A mathematical representation of a consumer's preferences, which assigns a number to each possible bundle of goods that reflects the level of satisfaction obtained.

Perfect Substitutes

A type of utility function where a consumer is willing to substitute one good for another at a constant rate.

Perfect Complements

A type of utility function where goods are consumed in fixed proportions and one good has no value without the other.

Quasilinear Preferences

A utility function where one good is taken in a linear fashion, implying that the marginal utility of money is constant.

Cobb-Douglas Preferences

A common utility function form characterized by a multiplicative relationship between goods, reflecting diminishing marginal utility and constant elasticity of substitution.

Marginal Utility

The additional satisfaction gained from consuming an extra unit of a good or service.

Marginal Rate of Substitution (MRS)

The rate at which a consumer is willing to give up one good in exchange for an additional unit of another good while maintaining the same level of utility.

Example Problems

Example 1

The text said that raising a number to an odd power was a monotonic transformation. What about raising a number to an even power? Is this a monotonic transformation? (Hint: consider the case $f(u)=u^{2}$.)

Example 2

Which of the following are monotonic transformations? (1) $u=2 v-13$ (2) $u=-1 / v^{2}$ (3) $u=1 / v^{2}$ (4) $u=\ln v$ (5) $u=-e^{-v}$ $(6) \quad u=v^{2}$ (7) $u=v^{2}$ for $v>0 ;$ (8) $u=v^{2}$ for $v<0$.

Example 3

We claimed in the text that if preferences were monotonic, then a diagonal line through the origin would intersect each indifference curve exactly once. Can you prove this rigorously? (Hint: what would happen if it intersected some indifference curve twice?)

Example 4

What kind of preferences are represented by a utility function of the form $u\left(x_{1}, x_{2}\right)=\sqrt{x_{1}+x_{2}} ?$ What about the utility function $v\left(x_{1}, x_{2}\right)=$ $13 x_{1}+13 x_{2} ?$

Example 5

What kind of preferences are represented by a utility function of the form $u\left(x_{1}, x_{2}\right)=x_{1}+\sqrt{x_{2}} ?$ Is the utility function $v\left(x_{1}, x_{2}\right)=x_{1}^{2}+2 x_{1} \sqrt{x_{2}}+x_{2}$ a monotonic transformation of $u\left(x_{1}, x_{2}\right) ?$
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Step-by-Step Explanations

QUESTION

How do you calculate the marginal utility of a good from a given utility function?

STEP-BY-STEP ANSWER:

Step 1: Identify the utility function, U(x, y), where x and y are quantities of two goods.
Step 2: To calculate the marginal utility of good x (MUx), take the partial derivative of U with respect to x, i.e., MUx = ∂U/∂x.
Step 3: Similarly, calculate the marginal utility of good y (MUy) by taking the partial derivative of U with respect to y, i.e., MUy = ∂U/∂y.
Step 4: Interpret the results as the additional satisfaction received from consuming an extra unit of x or y, respectively.
Final Answer: Calculate ∂U/∂x and ∂U/∂y to determine the marginal utilities of goods x and y.

Calculating Marginal Utility

QUESTION

How do you determine the MRS from a utility function?

STEP-BY-STEP ANSWER:

Step 1: Begin with the utility function U(x, y).
Step 2: Calculate the marginal utilities MUx = ∂U/∂x and MUy = ∂U/∂y.
Step 3: The MRS is given by the ratio of the marginal utilities: MRS = MUx / MUy.
Step 4: This ratio represents the rate at which the consumer is willing to substitute good y for good x while maintaining the same level of utility.
Final Answer: Use MRS = (∂U/∂x) / (∂U/∂y) to quantify the trade-off between goods x and y.

Determining the Marginal Rate of Substitution (MRS)

QUESTION

How do you construct different types of utility functions such as Cobb-Douglas or perfect complements?

STEP-BY-STEP ANSWER:

Step 1: Identify the consumer's preferences and the relationship between the goods.
Step 2: For a Cobb-Douglas utility function, use a multiplicative form U(x, y) = x^a * y^b where a and b reflect the relative importance of each good.
Step 3: For perfect substitutes, use a linear form U(x, y) = ax + by, where the constants a and b represent the rate of substitution.
Step 4: For perfect complements, use a minimum function such as U(x, y) = min{ax, by} to represent fixed consumption ratios.
Step 5: For quasilinear preferences, create a function of the form U(x, y) = v(x) + y where one good enters linearly.
Final Answer: Tailor the functional form of the utility function based on the nature of the goods and consumer preferences.

Constructing a Utility Function

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Common Mistakes

  • Confusing the different types of utility functions and applying the wrong functional form to a given set of preferences.
  • Overlooking the role of marginal utility fluctuations when small changes in consumption occur.
  • Miscomputing MRS by neglecting the correct application of partial derivatives in utility functions.
  • Assuming that all consumers have identical utility functions, thereby ignoring individual differences in preferences.