Consider a consumer whose preferences can be represented by a utilityfunction, U : R2+ −→ R, of the form U (x1, x2) = −x21 − x22. Suppose thatthis consumer’s budget set is given byB (p1, p2, y) = {(x1, x2) ∈ R2+ : p1x1 + p2x2 6 y} ,where (p1, p2, y) ∈ R3++ are arbitrary “prices and income parameters”.1. Find this consumer’s Marshallian demand function or correspondence.Justify your answer.2. Find this consumer’s indirect utility function. Justify your answer.
Question
Consider a consumer whose preferences can be represented by a utilityfunction, U : R2+ −→ R, of the form U (x1, x2) = −x21 − x22. Suppose thatthis consumer’s budget set is given byB (p1, p2, y) = {(x1, x2) ∈ R2+ : p1x1 + p2x2 6 y} ,where (p1, p2, y) ∈ R3++ are arbitrary “prices and income parameters”.1. Find this consumer’s Marshallian demand function or correspondence.Justify your answer.2. Find this consumer’s indirect utility function. Justify your answer.
Solution
考虑一个消费者，其偏好可以由一个效用函数U : R2+ −→ R表示，形式为U (x1, x2) = −x21 − x22。假设这个消费者的预算集由B (p1, p2, y) = {(x1, x2) ∈ R2+ : p1x1 + p2x2 ≤ y}给出，其中(p1, p2, y) ∈ R3++是任意的“价格和收入参数”。
- 找到这个消费者的马歇尔需求函数或对应关系。证明你的答案。
首先，我们需要最大化消费者的效用函数，即最小化-x1^2 - x2^2，同时满足预算约束p1x1 + p2x2 ≤ y。这是一个二次规划问题，可以通过拉格朗日乘数法求解。设拉格朗日函数为L = -x1^2 - x2^2 + λ(y - p1x1 - p2x2)，对x1, x2, λ分别求偏导并令其等于0，可以得到马歇尔需求函数。
- 找到这个消费者的间接效用函数。证明你的答案。
间接效用函数是在给定价格和收入下，消费者可以获得的最大效用。在这个问题中，我们已经找到了马歇尔需求函数，将其代入效用函数U (x1, x2) = −x21 − x22，就可以得到间接效用函数。
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