If Anu's marginal rate of substitution of X for Y is 5 and the per unit price of X is 9 and of Y is 2, is she optimizing her consumption? Explain your answer. Do you suggest a chan... If Anu's marginal rate of substitution of X for Y is 5 and the per unit price of X is 9 and of Y is 2, is she optimizing her consumption? Explain your answer. Do you suggest a change in her consumption bundle to maximize her satisfaction?
Understand the Problem
The question is asking whether Anu is optimizing her consumption based on her marginal rate of substitution and the prices of goods X and Y. We need to determine if her consumption is aligned with the principle of utility maximization, which states that the ratio of the marginal utility to price for both goods should be equal. The marginal rate of substitution indicates how much of good Y Anu is willing to give up for an additional unit of good X, and we need to compare this to the price ratio.
Answer
Anu is optimizing her consumption if \( \frac{MU_X}{P_X} = \frac{MU_Y}{P_Y} \) and \( \frac{MU_X}{MU_Y} = \frac{P_X}{P_Y} \).
Answer for screen readers
Anu is optimizing her consumption if ( \frac{MU_X}{P_X} = \frac{MU_Y}{P_Y} ) and ( \frac{MU_X}{MU_Y} = \frac{P_X}{P_Y} ).
Steps to Solve
- Identify the Marginal Utility Ratios
We need to determine the ratios of marginal utility to price for both goods X and Y. Let's denote the marginal utility of good X as (MU_X), the marginal utility of good Y as (MU_Y), the price of good X as (P_X), and the price of good Y as (P_Y).
The ratios will be: $$\frac{MU_X}{P_X}$$ for good X and $$\frac{MU_Y}{P_Y}$$ for good Y.
- Set Up the Optimization Condition
According to the utility maximization principle, for Anu to be optimizing her consumption of goods X and Y, the following condition must hold: $$\frac{MU_X}{P_X} = \frac{MU_Y}{P_Y}$$
- Compare the Marginal Rate of Substitution with Price Ratio
Next, we compare the marginal rate of substitution (MRS), which is defined as: $$MRS = \frac{MU_X}{MU_Y}$$
Then, we need to compute the price ratio: $$\frac{P_X}{P_Y}$$
If (MRS) equals the price ratio, then Anu is optimizing her consumption.
- Conclusion from Comparison
If the conditions established in the previous steps are met, conclude whether Anu is maximizing her utility based on her consumption of goods X and Y.
Anu is optimizing her consumption if ( \frac{MU_X}{P_X} = \frac{MU_Y}{P_Y} ) and ( \frac{MU_X}{MU_Y} = \frac{P_X}{P_Y} ).
More Information
This analysis is based on the concept of utility maximization, which is fundamental in economics. It reflects how consumers allocate their resources to maximize satisfaction based on their preferences and the prices of goods.
Tips
One common mistake is to ignore the price ratios when calculating the marginal rate of substitution. Remember to always compare both ratios to determine if consumption is optimized.
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