Suppose p1 = 11, p2 = 3 and M = 63. What quantity of good 2 will Qin demand?

Question

BlackTom AI · Accepted Answer

Question restatement: The problem gives p1 = 11, p2 = 3, and M = 63, and asks for the quantity of good 2 that Qin will demand.

Key point to start with: to determine a demand for good 2, we need Qin's preferences (the form of the utility function or the underlying demand system). Without specifying how Qin values goods 1 and 2 relative to each other, there is no unique solution for x2.

Reasoning under common benchmark models:
- If Qin faces a Cobb-Douglas utility u(x1, x2) = x1^a x2^b with a + b = 1 (a and b are the expenditure shares on goods 1 and 2), the demand for good 2 is x2 = (b * M) / p2. If we assume a = b = 0.5 (equal shares), then x2 = (0.5 * 63) / 3 = 31.5 / 3 = 10.5. In this benchmark, Qin would demand 10.5 units of good 2.
- If Qin has preferences of perfect substitutes with u = x1 or u = x2 (i.e., the agent cares only about one good and the cheaper good dominates), Qin would spend all income on the cheaper good 2 since p2 = 3 < p1 = 11. Then x2 = M / p2 = 63 / 3 = 21, a clearly different outcome from 10.5 or any other particular value.
- If Qin has perfect complements, such as u = min{x1, x2}, the optimal bundle under a linear budget constraint 11x1 + 3x2 ≤ 63 would typically satisfy x1 = x2, giving 11x + 3x = 63 → 14x = 63 → x = 4.5, so x2 = 4.5 in that case.

Observations:
- In each of these standard specifications, the resulting x2 is 10.5, 21, or 4.5 depending on the assumed utility form. None of these straightforward benchmark cases yield x2 = 6.
- The absence of Qin’s utility function or any additional constraints means the exact quantity of good 2 cannot be uniquely determined from p1, p2, and M alone.

Potential reasons for the reported answer (6):
- It is possible the dataset assumes a specific utility specification not stated in the prompt (for example, a nonstandard Cobb-Douglas with a particular share b that yields x2 = (b * 63) / 3 = 6, which would require b = 18/63 ≈ 0.2857, an unusual but possible preference weight).
- Alternatively, there could be a misprint or mismatch between the question data and the provided answer.

Summary in plain terms: Without knowing Qin’s utility function or preference structure, the quantity of good 2 demanded cannot be uniquely determined from p1, p2, and M alone. Under common models, the predicted x2 values would vary (e.g., 10.5 under equal Cobb-Douglas shares, 21 under pure substitution favoring good 2, or 4.5 under a perfect-complements setup). The fact that the listed answer is 6 suggests either a nonstandard or unstated preference specification, or a data error. The key lesson is that demand quantities hinge critically on the assumed preferences, not just prices and income.

Suppose p1 = 11, p2 = 3 and M = 63. What quantity of good 2 will Qin demand?Numerical

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