(a) State whether True or False and provide an explanation (or an example): If preferences are...

(a) State whether True or False and provide an explanation (or an example): If preferences are convex then the utility function must also always be convex.

(b) Is U=(5x1+10x2)2 a monotonic transformation of U=x1+x2? Explain.

Expert Solution

a) TRUE: Convex preferences means that averages (rather than extremes) are preferred, If preferences are convex then indifference curves are convex to the origin.

Suppose consumption bundles x and y lie on an indifference curve.

By convexity of preferences, [tx+ (1−t)y] (an average of the two bundles x and y) lies on a higher indifference curve for t ∈ [0,1].

By monotonicity, this higher indifference curve lies to the northeast of the original indifference curve. Hence the indifference curve is convex, as shown below:

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b) FALSE: Not a monotonic transformation; easy grounds for falsification is that MRS should be constant under a monotonic transformation.

MRS of U=x1+x2 -----> (-1) (they are perfect substitutes in 1:1 ratio)

whereas MRS of U=(5x1+10x2)2 -----> - MUx2 / MUx1 = - 5 / (5/2) = -2 (perfect substitutes but not in ratio 1:1)

i.e. MRS IS NOT THE SAME AFTER TRANSFORMATION, so not a monotonic transformation
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Rahul Sunny answered 1 year ago

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