Order the answer to: In this exercise, you will…
|Question||Consider a 2-good world, with goods x and x represented on the two axes in any graphs you draw. (a) Draw your two axes and pick some arbitrary bundle that contains at least some of each good. (b) Draw the 45-degree line in your graph â?? this is a ray that represents all bundles that have equal amounts of x and x in them. (c) Pick a second bundle B = such that In other words, pick B such that it has equal amounts of x and x and such that it has more of x and x than A. (d) Is A more or less preferred than the bundle (0,0)? Is B more or less preferred than A? (e) Now imagine moving along the 45-degree line from(0,0) toward B. Can you use the continuity property of tastes we have assumed to conclude that there exists some bundle C between (0,0) and B such that the consumer is indifferent between A and C? (f) Does the same logic imply that there exists such an indifferent bundle along any ray from the origin and not just along the 45-degree line? (g) How does what you have just done demonstrate the existence of a well-defined indifference map? Next we show that the same logic implies that there exists a utility function that represents these tastes. (a) If you have not already done so, illustrate A(a)-(e). (b) Denote the distance from (0,0) to C on the 45-degree line as and assign the value t to the bundle A. (c) Imagine the same procedure for labeling each bundle in your graph â?? i.e. for each bundle, determine what bundle on the 45-degree line is indifferent and label the bundle with the distance on the 45-degree line from (0,0) to the indifferent bundle. The result is a function u(x ,x ) that assigns to every bundle a number. Can you explain how this function meets our definition of a utility function? (d) Can you see how the same method of proof would work to prove the existence of a utility function when there are more than 2 goods (and when tastes satisfy rationality, continuity and monotonicity)? (e) Could we have picked a ray other than the 45-degree line to construct the utility values associated with each bundle?|
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