Question

For each of the following utility functions, draw an indifference map with 3 indifference curves. Be sure to label your axes, and label your curves as IC1, IC2, and IC3, where U1 < U2 < U3. (5 points each) a. ?(?, ?) = 3? + 5? b. ?(?, ?) = ? 2 + ? 2 c. ?(?, ?) = −? 2 + ln(?) d. ?(?, ?) = min(3?, 5?)

Answer 1

a)

The utility function is given as

$\small U=3x&plus;5y$

This is an equation of a straight line. The goods are a perfect substitute, and hence the ICs are a straight line. The higher the utility, the further away the ICs are from the origin. The three ICs with their respective utilities are:

1. $\small U1=10=3x&plus;5y$
2. $\small U2=12=3x&plus;5y$
3. $\small U3=15=3x&plus;5y$

The indifference map represents the above utility functions for different utilities described above:

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b)

The utility function is given as

$\small U=x^2&plus;y^2$

This is an equation of a circle. In the positive quadrant, it is concave to the origin. The goods are bad to the consumer, and hence the ICs are concave to the origin. The higher the utility, the further away the ICs are from the origin. The three ICs with their respective utilities are:

1. $\small U1=1.5=x^2&plus;y^2$
2. $\small U2=9=x^2&plus;y^2$
3. $\small U3=25=x^2&plus;y^2$

The indifference map represents the above utility functions for different utilities described above:

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c)

The utility function is given as

$\small U=-x^2&plus;\ln(y)$

This is an equation of a parabola. In the positive quadrant, it is positively sloped and concave. This implies one of the goods is bad to the consumer, and hence the ICs are concave to the origin. The higher the utility, the further away the ICs are from the origin. The three ICs with their respective utilities are:

1. $\small U1=0.001=x^2&plus;\ln (y)$
2. $\small U2=0.9=x^2&plus;\ln (y)$
3. $\small U3=1.5=x^2&plus;\ln (y)$

The indifference map represents the above utility functions for different utilities described above:

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d)

The utility function is given as

$\small U=min \left \{3x,5y}{ \right \}$

This is an equation of a right-angled curve. This implies one of the goods are perfect complement and used in a fixed proportion, and hence the ICs have a kink. The higher the utility, the further away the ICs are from the origin. The three ICs with their respective utilities are:

1. $\small U1=4=min\left \{ 3x,5y \right \}$
2. $\small U2=8=min\left \{ 3x,5y \right \}$
3. $\small U3=12=min\left \{ 3x,5y \right \}$

The indifference map represents the above utility functions for different utilities described above: