4. UTILITY
1. A monotonic transformation is one that preserves the order of numbers such that
u>v implies f(u)>f(v). We are given that raising a number to an odd power is a
monotonic transformation. Suppose the function raises a number to an even power.
This is a monotonic transformation for all positive numbers including 0. However, it
is not true for negative numbers. For example ½ > -1 but ¼ < 1.
2. Transformation 1 is a monotonic transformation. Transformation 2 is not a
monotonic transformation. Transformation 3 is not a monotonic transformation.
Transformation 4 is a monotonic transformation. Transformation 5 is a monotonic
transformation. Transformation 6 is not a monotonic transformation.
Transformation 7 is a monotonic transformation. Transformation 8 is not a
monotonic transformation.
3. Suppose a diagonal line through the origin intersected an indifference curve twice
with points of intersection A and B. Without loss of generality assume A is closer to
the origin and B is further from the origin along the diagonal. This means the
individual gets more of both goods at A than at point B. However, the fact that both
points are on the same indifference curve indicates that the individual is indifferent
between the two points, hence contradicting the claim that they have monotonic
preferences.
4. In the case of the first utility function, the individual has concave preferences. In the
case of the second utility function, the goods are perfect substitutes.
5. The first utility function is a Cobb-Douglas utility function. The second utility function
is not a monotonic transformation due to the even exponent.
6. The first utility function represents Cobb-Douglas preferences. Both subsequent
utility functions represent monotonic preferences.
7. The marginal rate of substitution involves the ratio of the 2 marginal utilities of the 2
respective goods. A monotonic transformation does not change the ordering of
preferred bundles and hence does not affect the ratios.
1. A monotonic transformation is one that preserves the order of numbers such that
u>v implies f(u)>f(v). We are given that raising a number to an odd power is a
monotonic transformation. Suppose the function raises a number to an even power.
This is a monotonic transformation for all positive numbers including 0. However, it
is not true for negative numbers. For example ½ > -1 but ¼ < 1.
2. Transformation 1 is a monotonic transformation. Transformation 2 is not a
monotonic transformation. Transformation 3 is not a monotonic transformation.
Transformation 4 is a monotonic transformation. Transformation 5 is a monotonic
transformation. Transformation 6 is not a monotonic transformation.
Transformation 7 is a monotonic transformation. Transformation 8 is not a
monotonic transformation.
3. Suppose a diagonal line through the origin intersected an indifference curve twice
with points of intersection A and B. Without loss of generality assume A is closer to
the origin and B is further from the origin along the diagonal. This means the
individual gets more of both goods at A than at point B. However, the fact that both
points are on the same indifference curve indicates that the individual is indifferent
between the two points, hence contradicting the claim that they have monotonic
preferences.
4. In the case of the first utility function, the individual has concave preferences. In the
case of the second utility function, the goods are perfect substitutes.
5. The first utility function is a Cobb-Douglas utility function. The second utility function
is not a monotonic transformation due to the even exponent.
6. The first utility function represents Cobb-Douglas preferences. Both subsequent
utility functions represent monotonic preferences.
7. The marginal rate of substitution involves the ratio of the 2 marginal utilities of the 2
respective goods. A monotonic transformation does not change the ordering of
preferred bundles and hence does not affect the ratios.
