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帮我解决图片中的题 Advance Micro I Homework 5 Instructor: Erya Yang September 26, 2024 ...
Oct 6, 2024
帮我解决图片中的题
1 Answer
a
Utility Representation: Chris's utility function can be represented as u(x)=−e−αx u(x) = -e^{-\alpha x} . Given that he is indifferent between the lottery L L and receiving $40 for sure, we have: \( u(40) = 0.5 \cdot u(100) + 0.5 \cdot u(0) \)
b
Certainty Equivalents: For lotteries L1,L2,L3 L_1, L_2, L_3 , we calculate the certainty equivalents by solving u(CE)=0.5⋅u(1000)+0.5⋅u(0) u(CE) = 0.5 \cdot u(1000) + 0.5 \cdot u(0) , u(CE)=0.5⋅u(10000)+0.5⋅u(0) u(CE) = 0.5 \cdot u(10000) + 0.5 \cdot u(0) , and u(CE)=0.5⋅u(100000)+0.5⋅u(0) u(CE) = 0.5 \cdot u(100000) + 0.5 \cdot u(0)
c
Intuitive Reasonableness: The certainty equivalents should decrease as the lottery amounts increase, reflecting Chris's risk aversion
Answer
Chris's utility function is u(x)=−e−αx u(x) = -e^{-\alpha x} and his certainty equivalents for the lotteries are calculated based on this function.
Key Concept
Expected Utility Theory
Explanation
Chris's preferences illustrate how expected utility theory accounts for risk aversion in decision-making under uncertainty.
2 Answer
a
Completeness of Preferences: If ⪰ \succeq is complete, then for any x x and y y , either x⪰y x \succeq y or y⪰x y \succeq x holds. Thus, x≻y x \succ y implies y⪰x y \succeq x does not hold
b
Uniqueness of ≻ \succ : The strict preference ≻ \succ uniquely determines ⪰ \succeq since if x≻y x \succ y , then y⪰x y \succeq x cannot be true
c
Example of Incomplete Rankings: Consider ⪰ \succeq as {a⪰b,b⪰c} \{a \succeq b, b \succeq c\} and ⪰′ \succeq' as {a⪰′b,b⪰′c} \{a \succeq' b, b \succeq' c\} where a≻b≻c a \succ b \succ c holds in both but ⪰≠⪰′ \succeq \neq \succeq'
Answer
The strict preference ≻ \succ uniquely determines the weak preference ⪰ \succeq , and distinct incomplete rankings can share the same strict part.
Key Concept
Completeness and Strict Preferences
Explanation
Completeness ensures that preferences can be ranked, while strict preferences provide a unique ordering that can differ in weak preferences.
3 Answer
a
Weak Preference Completeness: The person’s choice of the second cheapest wine implies a consistent weak preference, which is complete and transitive within smaller menus
b
Violations of WARP: In larger menus, the choice may violate the weak axiom of revealed preference (WARP) due to the complexity of options leading to inconsistent choices
c
No Contradiction: This situation does not contradict representation results because WARP violations can occur without negating the underlying preference structure
Answer
The observed weak preference is complete and transitive in small menus but may violate WARP in larger ones without contradicting representation results.
Key Concept
Weak Axiom of Revealed Preference (WARP)
Explanation
WARP allows for inconsistencies in choices when faced with larger sets of options, reflecting the complexity of decision-making.
4 Answer
a
Revealed Preferences: The choice function indicates that C(1,2,3)={1,3} C(1,2,3) = \{1,3\} suggests that 1 and 3 are preferred over 2, and similar logic applies to other combinations
b
Preference Relations: From the choices, we can derive that 1≻2 1 \succ 2 , 3≻2 3 \succ 2 , 3≻4 3 \succ 4 , 5≻4 5 \succ 4 , etc., leading to a preference ordering
c
Summary of Preferences: The revealed preferences can be summarized as 1≻2 1 \succ 2 , 3≻2 3 \succ 2 , 3≻4 3 \succ 4 , 5≻4 5 \succ 4 , 5≻6 5 \succ 6 , 7≻6 7 \succ 6 , 9≻8 9 \succ 8 , etc
Answer
The revealed preferences indicate a clear ordering based on the choice function, showing preferences among the options.
Key Concept
Revealed Preference Theory
Explanation
Revealed preference theory allows us to infer preferences based on observed choices, providing insights into consumer behavior.
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