Preferences and Transformation
- III.B.5: Explain the meaning of and solve for the marginal rate of substitution.
- III.B.6: Determine whether a utility function satisfies the following properties: monotonocity, quasiconcavity, homotheticiy, essentiality, and quasilinearity.
- III.B.7: Recognize and apply positive monotonic transformations of utility functions.
- III.B.8: Determine whether two utility functions represent the same preferences.
Use the graph above to answer the questions below.
Alma’s preferences are represented by the function of the following form \[u_1(x_1,x_2) = x_1^ax_2^b\] Suppose $a = 2$ and $b=3$.
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Derive Alma’s Marginal Rate of Substitution (algebraically). In words, explain what this means.
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Is this a quasiconcave utility function? (Hint: Think about indifference curves and upper contour sets. You should not need to do any algebra.)
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What does your answer from (b) indicate about Alma’s preferences? Are they convex?
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Are both goods essential? Remember that a good is essential if it is not possible to have more utility than u(0,0) without a positive amount of that good.
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Are these preferences monotone? Remember that monotonic preferences mean that a bundle with more of both goods is preferred to a bundle with less of both goods.
Bailey’s preferences are represented by the following function: \[u_2(x_1,x_2) = 2 \ln x_1 + 3 \ln x_2\]
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Calculate the utilities that Alma and Bailey each receive at $x_1=16$ and $x_2=16$. Do the same for the bundle $x_1=128$ and $x_2=4$. Is it possible that Alma and Bailey have the same preferences?
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What is are the Marginal Rates of Substitution for Alma and Bailey for these two bundles? If Alma and Bailey had the same preferences, would they have to have the same MRS’s for each bundle?
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Prove that Alma and Bailey do or do not have the same preferences.