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Utility Functions

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Attached are 2 utility functions that I'm having trouble with.

Problem 3

For each of the following utility functions, draw indifference curves for utility level 12 and 16. Indicate three bundles on each of these indifference curves by specifying the coordinates of each bundle. Shade the weakly preferred set of bundles you indicated on the indifference curves of utility 16. For each of the utility functions, determine and explain whether the preferences they represent are convex, concave, or non-convex preferences.

a) U_3 (x_1,x_2 )=?x_1?^2-?2x?_2;
b) U_4 (x_1,x_2 )=Min{4x_1-4x_2,2x_1-x_2}.

Solution Summary

The solution includes drawings of the requested utility function, as well as a full justification of why they are drawn that way. Additionally, example bundles are given for each of the functions.

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Problem 3

For each of the following utility functions, draw indifference curves for utility level 12 and 16. Indicate three bundles on each of these indifference curves by specifying the coordinates of each bundle. Shade the weakly preferred set of bundles you indicated on the indifference curves of utility 16. For each of the utility functions, determine and explain whether the preferences they represent are convex, concave, or non-convex preferences.

a) U_3 (x_1,x_2 )=?x_1?^2-?2x?_2;
b) U_4 (x_1,x_2 )=Min{4x_1-4x_2,2x_1-x_2}.

In order to complete this problem, the first step is to obtain x2 as a function of x1. This will allow us to draw the indifference curves. This can be done as follows. For the utility function in (a):

Where U is any given utility level (in this particular case, the utility levels of interest are 12 and 16). Note that the final equation above ...

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