Here Is The Background Information A Consumer Spends Her Inc

Here Is The Background Informationa Consumer Spends Her Income On

Here is the background information: A consumer spends her income on nuts (n) and fruits (f) both of which are measured in units in pounds. The total utility is shown below: #of units consumed TU(n) TU(f) she has a budget of $12 and P(n)= $2, and P(f) = $1

Question: Would Q(n) = 5 and Q(f) = 2 be utility maximizing? Explain not in reference to TU but MU or marginal utility. I said yes, because the marginal utility for both n and f is the same (I used the marginal utility rule), but I don't know if that is correct, help?

Paper For Above instruction

To determine whether consuming 5 pounds of nuts (n) and 2 pounds of fruits (f) constitutes a utility-maximizing point for the consumer, it is essential to analyze the marginal utility per dollar spent on each good. Marginal Utility (MU) and the Marginal Utility per dollar (MU/P) serve as the keystones for such an analysis, based on the utility maximization rule that consumers allocate their income in a way that equates the marginal utility per dollar across all goods.

Given the consumer’s budget of $12, and the prices of nuts and fruits being $2 and $1 respectively, the total expenditure for purchasing 5 pounds of nuts and 2 pounds of fruits amounts to:

• Cost of nuts: 5 pounds × $2 = $10
• Cost of fruits: 2 pounds × $1 = $2

Total expenditure: $10 + $2 = $12, which exhausts the consumer’s entire budget, indicating that this bundle is affordable.

However, affordability alone does not ensure utility maximization. To evaluate whether this combination is optimal, we analyze the marginal utility per dollar for each good. If we denote MU_n as the marginal utility of nuts and MU_f as that of fruits, the consumer maximizes utility by equating the ratio of marginal utility to price for both goods:

• MU_n / P_n = MU_f / P_f

According to the previous statement, the consumer believed that the marginal utility for both nuts and fruits is the same, implying:

• MU_n = MU_f

Given the prices:

• P_n = $2
• P_f = $1

The marginal utility per dollar for nuts: MU_n/2, and for fruits: MU_f/1. Since MU_n = MU_f, the ratios become:

• MU_n/2 and MU_f/1

If MU_n = MU_f, then:

• MU_n/2 < MU_f/1, if MU_n < 2× MU_f

Thus, unless the marginal utilities are proportional to the prices (i.e., MU_n/2 = MU_f/1), the ratio of marginal utility per dollar spent on nuts and fruits will not be equal.

In the absence of explicit numerical values for the marginal utilities, we only know that if MU_n and MU_f are equal, then the marginal utility per dollar spent on fruits is higher than that for nuts because:

• MU_f/1 = MU_f
• MU_n/2 = (MU_f)/2 < MU_f

Therefore, if the marginal utility of nuts is equal to that of fruits, the consumer may not be maximizing utility by choosing this bundle. Instead, she could increase her total utility by reallocating her spending toward the good with higher marginal utility per dollar, which, in this case, seems to be fruits.

In conclusion, the assumption that MU_n = MU_f does not necessarily equalize the marginal utility per dollar across goods unless the prices are proportionally aligned to the marginal utilities. Without detailed marginal utility values, it remains inconclusive whether the bundle (Q(n) = 5, Q(f) = 2) is utility-maximizing. Nonetheless, based solely on the principle that at the optimum MU_n/P_n = MU_f/P_f, this bundle would likely not be optimal unless the marginal utilities satisfy this proportionality criterion.

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