The following figure depicts possible indifference curves of Alan, whose utility depends on his and/or his friend Beatrice’s levels of consumption. Alan’s consumption is along the horizontal axis while Beatrice’s consumption is along the vertical axis. Look at the figure and select the letter(s) that could represent altruistic preferences.多项选择题
A
(a)
B
(b)
C
(c)
D
(d)

Question

The following figure depicts possible indifference curves of Alan, whose utility depends on his and/or his friend Beatrice’s levels of consumption. Alan’s consumption is along the horizontal axis while Beatrice’s consumption is along the vertical axis. Look at the figure and select the letter(s) that could represent altruistic preferences.多项选择题

A

(a)

B

(b)

C

(c)

D

(d)

BlackTom AI · Accepted Answer

Altruistic preferences mean Alan’s utility increases when either his own consumption (A) or Beatrice’s consumption (B) increases; mathematically u_A > 0 and u_B > 0. 1. Relevant concepts - Marginal utilities: u_A is marginal utility of Alan’s consumption, u_B of Beatrice’s. - Indifference curve slope: along u(A,B) = constant, du = u_A dA + u_B dB = 0, so dB/dA = −u_A/u_B. - Monotonicity: if both partials are positive, higher A or higher B (holding the other fixed) increases utility, so indifference curves must slope downward (dB/dA < 0) and higher indifference curves lie to the northeast. 2. Step-by-step reasoning - Start with du = u_A dA + u_B dB = 0. - Solve for slope: dB/dA = −u_A/u_B. - Since altruism implies u_A > 0 and u_B > 0, their ratio is positive, so dB/dA is negative. - Therefore, indifference curves for altruistic preferences must slope downward: an increase in B must be compensated by a decrease in A to keep utility constant. 3. Intermediate calculation shown - du = 0 => u_A dA + u_B dB = 0. - dB/dA = −u_A/u_B. - With u_A > 0 and u_B > 0, −u_A/u_B < 0. 4. Why (c) is correct - Option (c) shows indifference curves that slope downward (and typically are monotone and convex), consistent with both partial derivatives being positive. That shape indicates Alan values increases in Beatrice’s consumption (altruism): raising B allows lowering A while keeping utility unchanged. 5. Why other options are incorrect - (a) depicts vertical indifference curves (utility depends only on A): u_B = 0, not altruistic. - (b) depicts upward-sloping curves (utility falls when B rises): u_B < 0, implying spite or jealousy, the opposite of altruism. - (d) shows non-monotonic or kinked/ambiguous shapes that do not reflect strictly positive marginal utility for B across the domain (e.g., utility might not increase with B everywhere), so it fails the definition of pure altruism. 6. Conclusion Because altruism requires u_A > 0 and u_B > 0, yielding downward-sloping indifference curves, option (c) is the only choice consistent with altruistic preferences. This matches the provided answer: (c).

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