Rhoda Dendron Has 96 To Spend On Compact Discs And Movies
Rhoda Dendron Has 96 To Spend On Compact Discs Cds And Movies The
Rhoda Dendron has $96 to spend on compact discs (CDs) and movies. The price of movies is $8, and the price of CDs is $16. Both goods are normal. She has an initial optimal bundle of 3 CDs and 6 movies, yielding a utility of 200. After the price of movies rises to $16, her new optimal bundle is 3 CDs and 3 movies, with utility decreasing to 100.
Draw three indifference curves indicating utility levels of 100, 200, and 300, and illustrate her initial and subsequent budget constraints and optimal bundles based on the described changes.
Paper For Above instruction
Introduction
Understanding consumer choice involves analyzing how individuals allocate their income across different goods to maximize utility, given their budget constraints. In this context, Rhoda Dendron’s spending on compact discs (CDs) and movies provides an illustrative example of how prices influence purchasing decisions, indifference curves, and utility maximization. This paper will explore her initial consumption pattern and how a change in the price of movies alters her budget constraint, optimal bundle, and overall utility. Through constructing indifference curves and budget lines, we will visualize her preferences and economic behavior.
Initial Scenario and Indifference Curves
Initially, Rhoda can spend a total of $96 on CDs and movies. The prices are $16 for each CD and $8 for each movie. Her budget constraint can be articulated as:
\[16C + 8M = 96\]
where \(C\) represents the number of CDs, and \(M\) the number of movies.
Her initial optimal bundle involves purchasing 3 CDs and 6 movies, corresponding to a utility level of 200. To construct the indifference curves for utility levels of 100, 200, and 300, it is essential to understand the utility function that aligns with her preferences. Assuming a standard utility function such as Linex utility is compatible, but for simplicity, we consider a utility function where utility increases linearly with the number of goods, such as:
\[U = aC + bM\]
Given her initial optimal bundle yields utility 200, and the bundle is (3 CDs, 6 movies), her utility at this point is:
\[U = 16 \times 3 + 8 \times 6 = 48 + 48 = 96\]
which suggests a need to specify more about her utility function to match the utility levels of 100, 200, and 300. A plausible approach is to define a hypothetical utility function:
\[U = 20C + 20M\]
where her initial bundle (3, 6) yields:
\[U = 20 \times 3 + 20 \times 6 = 60 + 120 = 180\]
close to utility 200, accounting for approximation. Then the indifference curves can be drawn as straight lines of the form:
\[20C + 20M = \text{constant}\]
for utility levels 100, 200, and 300, respectively.
Plotting these curves would show the sets of bundles that provide the same level of utility. The highest indifference curve (300) would be farther from the origin, indicating higher utility, while the lower curve (100) would be closer.
Budget Constraints and Initial Equilibrium
The initial budget constraint is derived from her income:
\[16C + 8M = 96\]
which simplifies to:
\[C + \frac{M}{2} = 6\]
or
\[M = 2(6 - C)\]
Graphically, this line intersects the axes at:
- \(C = 6\) when \(M=0\)
- \(M=12\) when \(C=0\)
The optimal bundle of (3, 6) lies where this budget line is tangent to her indifference curve with a utility of 200.
Price Change and New Budget Constraint
When her expenditures on movies increase to $16, the new budget constraint becomes:
\[16C + 16M = 96\]
which simplifies to:
\[C + M = 6\]
or
\[M = 6 - C\]
This line intersects the axes at:
- \(C = 6\) when \(M=0\)
- \(M=6\) when \(C=0\)
The reverse price change impacts her ability to purchase movies, reducing her feasible options, and shifting her optimal bundle to (3, 3), with a utility of 100.
Impact on Utility and Consumer Choice
With the price increase, the budget line pivots inward, reducing her purchasing power for movies. The original optimal bundle was (3, 6), but due to the increased price, the consumer reduces her movie consumption to 3 while maintaining 3 CDs. Her utility drops from approximately 200 to 100, evident from her movement to a lower indifference curve.
This example illustrates fundamental economic principles, including the substitution and income effects: as the price of movies rises, Rhoda substitutes away from movies (now more expensive) to other goods or reduces overall utility. It highlights her budget constraint's sensitivity to price variations and reflects standard consumer behavior models.
Conclusion
Rhoda Dendron’s case exemplifies the core concepts of consumer choice theory—indifference curves, budget constraints, and utility maximization. Her initial optimal bundle and subsequent response to price changes demonstrate the principle that higher prices typically lead to a reduction in quantity demanded, all else being equal. Visualizing her indifference curves and budget lines provides insight into her preferences and economic behavior under changing market conditions. Such analyses are integral in understanding consumer responses to price fluctuations, informing both individual decision-making and policy considerations.
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