1. Given the utility function U = 4, where a, /30, prove that the indifference curves are convex...

Question

1. Given the utility function U = 4, where a, /3>0, prove that the indifference curves are convex towards the origin.
2. Solve the problems of: (i) maximizing U = (x,x2)3 + ,x2 subject to pi, +p2>2 = y, and (ii) maximizing U = en, + en, subject to pi, +p2>2 = and explain the relationship between the solutions.

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Robert · Accepted Answer

1. We have U=x1αx2β.
Given there are only two goods, to show that the utility function is convex to the origin it is sufficient to show that it exhibits diminishing marginal rate of substitution.
Totally differentiating the utility function gives us,
dU= (αx1α-1x2β)dx1 + (βx1αx2β-1)dx2.
Along an indifference curve the utility remains unchanged, i.e. dU=0,
Therefore,

1. Given the utility function U = 4, where a, /3>0, prove that the indifference curves are convex towards the origin. 2. Solve the problems of: (i) maximizing U = (x,x2)3 + ,x2 subject to pi, +p2>2 =...

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