Consider your tastes for five dollar bills and ten dollar bills (and suppose that you could have partial $10 and $5 bills).
A: Suppose that all you care about is how much money you have, but you don’t care whether a particular amount comes in more or fewer bills.
(a) With the number of five dollar bills on the horizontal axis and the number of ten dollar bills on the vertical, illustrate three indifference curves from your indifference map.
(b) What is your marginal rate of substitution of ten dollar bills for five dollar bills?
(c) What is the marginal rate of substitution of five dollar bills for ten dollar bills?
(d) Are averages strictly better than extremes? How does this relate to whether your tastes exhibit diminishing marginal rates of substitution?
(e) Are these tastes homothetic? Are they quasilinear?
(f) Are either of the goods on your axes “essential”?
B: Continue with the assumption that you care only about the total amount of money in your wallet, and let five dollar bills be denoted x1 and ten dollar bills be denoted x2.
(a) Write down a utility function that represents the tastes you graphed in A(a). Can you think of a second utility function that also represents these tastes?
(b) Calculate the marginal rate of substitution from the utility functions you wrote down in B(a) and compare it to your intuitive answer in A(b).
(c) Can these tastes 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?
(d) 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?
(e) 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?