Question

Frank's utility function for two goods, \(X\) and \(Y\), is given by \(U=X Y\). Find Frank's indifference curves, when utility is 10,20 and 30 . Plot these indifference curves. How should Frank compare the following two bundles: \((X=1, Y=10)\) and \((X=\) \(5, Y=2) ?\)

Short answer

The indifference curves for utility levels 10, 20, and 30 are given by \( XY = 10 \), \( XY = 20 \), and \( XY = 30 \) respectively. The bundles \((X=1, Y=10)\) and \((X=5, Y=2)\) are equally preferred as they both give a utility of 10.

Step-by-step solution

Understanding Utility Function

Frank's utility function is given by \( U = XY \), where \( X \) and \( Y \) are quantities of goods \( X \) and \( Y \) respectively. The utility values describe the indifference curves, where for a particular utility level, any combination of \( X \) and \( Y \) giving that utility represents a point on the curve.

Deriving Indifference Curves Equations

To find the indifference curves, set the utility \( U \) to specific values: 10, 20, and 30. The equations are:1. For \( U = 10 \), the indifference curve is \( XY = 10 \).2. For \( U = 20 \), the indifference curve is \( XY = 20 \).3. For \( U = 30 \), the indifference curve is \( XY = 30 \).These equations describe the curves on a graph in the \( X \)-\( Y \) plane.

Graphical Representation of Indifference Curves

Plot each indifference curve on the \( X \)-\( Y \) graph:- For \( U = 10 \), plot points like (1,10), (2,5), (5,2), (10,1) creating a hyperbola.- For \( U = 20 \), plot points where \( XY = 20 \).- For \( U = 30 \), plot points where \( XY = 30 \).The curves do not intersect and each is a distinct hyperbola.

Calculating Utility for Given Bundles

Compute the utility for the bundles:- Bundle \((X=1, Y=10)\) gives \(U = 1 \times 10 = 10\).- Bundle \((X=5, Y=2)\) gives \(U = 5 \times 2 = 10\).

Comparing Bundles

Since both bundles \((X=1, Y=10)\) and \((X=5, Y=2)\) yield the same utility of 10, they are on the same indifference curve. Therefore, from Frank's perspective, they are equally preferred.

Key concepts

Utility Function

The idea of a utility function is central to understanding consumer preferences in microeconomic analysis. A utility function, like Frank's, typically maps combinations of goods to a numerical value which stands for the consumer's level of satisfaction, or 'utility.'
For example, Frank's utility with two goods is given by the mathematical representation: \[ U = XY \] Here, 'U' denotes the utility, and 'X' and 'Y' represent the quantities of two respective goods. The role of this function is to encapsulate Frank's preferences by assigning a higher value to more preferred combinations.
Utility functions guide the drawing of indifference curves, which visually illustrate combinations of goods providing the same utility level, suggesting Frank feels 'indifferent' towards any choosing along that curve.

Bundle Comparison

Bundle comparison examines how different combinations of goods satisfy a consumer’s preferences—important in determining how goods choices vary with shifts in economic conditions.
To compare bundles, like these in Frank's case:

• Bundle 1: \((X=1, Y=10)\)
• Bundle 2: \((X=5, Y=2)\)

We calculate the utility each bundle provides using Frank's utility function, where for Bundle 1: \[ U = 1 \times 10 = 10 \] And for Bundle 2: \[ U = 5 \times 2 = 10 \] Even though the composition is different, both bundles yield the same utility score of 10.
This equivalence means Frank sees no preference between these two bundles he would be equally content with either.

Microeconomic Analysis

Microeconomic analysis uses tools like utility functions and indifference curves to study how individuals make choices based on available resources and preferences.
Indifference curves are a primary focus in this type of analysis as they detail all the combinations of goods that offer the consumer the same satisfaction level.

• Derived from utility functions, these curves help illustrate consumer behavior. Frank's curves, formed from the utility expression \( XY = U \), indicate levels \( U = 10, 20, \) and \( 30 \).
• Each curve is unique to its utility value and doesn’t intersect with others.

Analysts use these insights to predict how changes in price or income can affect consumer choices. Through systematic evaluation of indifference curves, economists get a visual and mathematical grasp of consumer welfare and market behavior. This analysis further helps optimize resource allocation and improve overall economic policies.