Consider your tastes for right and left shoes.
A: Suppose you, like most of us, are the kind of person that is rather picky about having the shoes you wear on your right foot be designed for right feet and the shoes you wear on your left foot be designed for left feet. In fact you are so picky that you would never wear a left shoe on your right foot or a right shoe on your left foot — nor would you ever choose (if you can help it) not to wear shoes on one of your feet.
(a) In a graph with the number of right shoes on the horizontal axis and the number of left shoes on the vertical, illustrate three indifference curves that are part of your indifference map.
(b) Now suppose you hurt your left leg and have to wear a cast (which means you cannot wear shoes on your left foot) for 6 months. Illustrate how the indifference curves you have drawn would change for this period. Can you think of why goods such as left shows in this case are called neutral goods?
(c) Suppose you hurt your right foot instead. How would this change your answer to part (b).
(d) Are any of the tastes you have graphed homothetic? Are any quasilinear?
(e) In the three different tastes that you graphed, are any of the goods ever “essential”? Are any not essential?
B: Continue with the description of your tastes given in part A above and let x1 represent right shoes and let x2 represent left shoes.
(a) Write down a utility function that represents your tastes as illustrated in A(a). Can you think of a second utility function that also represents these tastes?
(b) Write down a utility function that represents your tastes as graphed in A(b).
(c) Write down a utility function that represents your tastes as drawn in A(c).
(d) Can any of the tastes you have graphed in part A be represented by a utility function that is homogeneous of degree 1? If so, can they also be represented by a utility function that is not homogeneous?
(e) Refer to end-of-chapter exercise 4.13 where the concepts of “strong monotonicity,” “weakmono- tonicity” and “local non-satiation” were defined. Which of these are satisfied by the tastes you have graphed in this exercise?
(f) Refer again to end-of-chapter exercise 4.13 where the concepts of “strong convexity” and “weak convexity” were defined. Which of these are satisfied by the tastes you have graphed in this exercise?