(a) On the same axes, sketch Alice's budget set and some contours u₁(x, y) = c for positive values of c. By referring to your sketch explain why the method of Lagrange multipliers can be used to find the maximum value of u₁(x, y) subject to her budget constraint. Hence use the method of Lagrange multipliers to find Alice's optimal bundle. (b) On the same axes, sketch Bob's budget set and some contours up (x, y) = c For positive values of c.

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Chapter1: Making Economics Decisions
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Problem 1. Two consumers, Alice and Bob, have utility functions given by
u₁(x, y) = x¹/²y¹/4 and ug (x, y) = 4x² + y²,
respectively, where x and y are the number of apples and bananas respectively.
Suppose that apples and bananas cost p₁ and p2 dollars each respectively and that both
consumers have M dollars to spend on apples and bananas.
(a) On the same axes, sketch Alice's budget set and some contours
u₁(x, y) = c for positive values of c.
By referring to your sketch explain why the method of Lagrange multipliers can be used to find the
maximum value of u₁(x, y) subject to her budget constraint.
Hence use the method of Lagrange multipliers to find Alice's optimal bundle.
(b) On the same axes, sketch Bob's budget set and some contours up (x, y) = c
For positive values of c.
By referring to your sketch, find Bob's optimal bundle.
Transcribed Image Text:Problem 1. Two consumers, Alice and Bob, have utility functions given by u₁(x, y) = x¹/²y¹/4 and ug (x, y) = 4x² + y², respectively, where x and y are the number of apples and bananas respectively. Suppose that apples and bananas cost p₁ and p2 dollars each respectively and that both consumers have M dollars to spend on apples and bananas. (a) On the same axes, sketch Alice's budget set and some contours u₁(x, y) = c for positive values of c. By referring to your sketch explain why the method of Lagrange multipliers can be used to find the maximum value of u₁(x, y) subject to her budget constraint. Hence use the method of Lagrange multipliers to find Alice's optimal bundle. (b) On the same axes, sketch Bob's budget set and some contours up (x, y) = c For positive values of c. By referring to your sketch, find Bob's optimal bundle.
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