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Exam 4: Tastes and Indifference Curves

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Question 1

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Suppose bundle A is better than bundle B for a consumer,and bundle C is an average of bundles A and B. a.Use the continuity,convexity and monotonicity assumptions to formally prove that this implies that bundle C is better than bundle B. b.Did you also -- implicitly or explicitly -- use the rationality axioms?

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a.Start with bundle A and imagine slowly removing some of each good from the bundle.Continuity and monotonicity then implies that eventually we will get to a bundle \(\bar { A }\) that is just as good (but no better)than B,i.e.. \(\bar { A } \sim B\) . Next,take the average bundle between \(\bar { A }\) and B -- and call it \(\bar { C }\) .Convexity implies that \(\bar { C }\) .is at least as good as B. Finally,note that C contains more of every good than \(\bar { C }\) .Thus,monotonicity implies that C \(\succ\) \(\bar { C }\) --- and,since \(\bar { C }\) is at least as good as B,this implies that C \(\succ\) B. b.Yes,both the completeness and the transitivity assumptions were used.Completeness was used implicitly because we created bundles and just assumed that the consumer is able to compare them to other bundles.Transitivity was used in the last step where we concluded that we know C is better than B because we C is better than \(\bar { C }\) and \(\bar { C }\) is at least as good as B.

Question 2

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Consider a worker who dislikes working end enjoys consuming a composite good.With labor hours on the horizontal and the composite consumption good on the vertical axis,which of the following statements are true. a. If the worker's tastes are convex,the slope of indifference curves increases as we move to the right in the graph. b. The worker becomes better off as we move to the northwest in the graph. c. A tax on wage income does not change this worker's indifference map. d. All of the above. e. None of the above.

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D The graph below shows an indifference curve that satisfies convexity here -- A and B are indifferent,and C is the average of the two.C has more consumption (that the worker likes)and less labor (which the worker does not like)then D --- and thus C is preferred to D.Since D is indifferent to A and B,we therefore have the average C being preferred to the extremes A and B.We also see that the more preferred bundles lie to the northwest -- with more consumption and less labor.Finally,a tax on wages has nothing to do with how a worker feels about consumption and working --- a tax on wages affects the worker's budget,not his tastes. 11ea78ab_dc8e_ad00_8379_4ded1625f3ab_TB2157_00

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The number of units of the good on the horizontal axis that we are willing to give up to get one more unit of the good on the vertical axis is equal to the absolute value of the slope of the indifference curve.

You like bundle A better than bundle B,and bundle C is an average between A and B.If your tastes satisfy convexity,then C is at least as good as A and as B.

Prove formally that the rationality axioms alone rule out the possibility of indifference curves crossing.

Explain the following statement: Individuals with different tastes might have the same tastes at the margin at their current consumption bundles.

Bundle A is worse than bundle B,and bundle C is an average of bundles A and B.Then our usual assumptions about tastes imply that bundle B is at least as good as bundle C.

If you observe me choosing bundle A over bundle B on Monday,bundle B over bundle C on Tuesday and bundle C over bundle A on Wednesday,it must be that my tastes violate transitivity.

Suppose tastes are NOT monotonic anywhere.Then diminishing MRS is not consistent with convexity of tastes.

Explain the following statement: For the same individual,tastes over goods may vary at the margin as we move from one bundle to another.

When the price of beer goes up,our model of tastes would typically require tastes to change.

You like bundle A better than bundle B,and bundle C is an average of bundles A and B.Which of the following is correct if your tastes satisfy our usual assumptions? a. Bundle C is at least as good as bundle B. b. Bundle A is at least as good as bundle C. c. Both (a)and (b). d. None of the above. e. There is not enough information to tell.

If the marginal rate of substitution is not diminishing,it must mean that tastes violate convexity (assuming that our other assumptions about tastes hold).

Complete tastes are tastes that make people desire at least some of every good.

Consider the utility function $u \left( x _ { 1 } , x _ { 2 } \right) = \ln x _ { 1 } + x _ { 2 } ^ { 1 / 2 }$ .(Explain all your answers.) a.Derive the function for the marginal rate of substitution. b.Do the tastes represented by this utility function satisfy diminishing MRS? c.The marginal utility of a good is defined as the change in utility from additional consumption of that good (holding all else constant).Derive the marginal utility of $x _ { 1 }$ and $x _ { 2 }$ . d.Why does an ordinal approach to utility theory not any attention to what you derived in (c)? e.Why does an ordinal approach to utility not treat the marginal rate of substitution the way it treats the marginal utility concept?