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Problem Set 2: SolutionsECON 301: Intermediate MicroeconomicsProf. Marek WeretkaProblem 1 (Marginal Rate of Substitution)(a)For the third column, recall that by definitionMRS(x1, x2) =−(∂U∂x1)(∂U∂x2).Utility Function∂U∂x1∂U∂x2MRS(x1, x2)MRS(2,3)(i)U(x1, x2) =x1x2x2x1−x2x1−32(ii)U(x1, x2) =x31x523x21x525x31x42−3x25x1−910(iv)U(x1, x2) = 3 lnx1+ 5 lnx23x15x2−3x25x1−910(b)MRS(2,3) =−9/10 for utility functionU(x1, x2) =x31x52has the following interpre-tation: At bundle (2,3), to remain indifferent about the change (i.e., remain at the sameutility level), a consumer is willing to give up 9/10 ofx2for one additional unit ofx1. (Or,after losing one unit ofx1, he must receive 9/10 of a unit ofx2to be as well off as he wasat bundle (2,3).) Soat the point(2,3), good two is more valuable since he needs to get lessof it than he lost of the other good to remain as satisfied. If 0.00001 of good one is takenaway, he would have to receive approximately 0.00001×(910) = 0.000009 units of good twoto remain indifferent to the change.(c)The two utility functions share the same MRS functions becauseU(x1, x2) = 3 lnx1+5 lnx2is a monotonic transformation ofU(x1, x2) =x31x52. To see this, letf(u) = ln(u)(f(u) is a monotonic function). Then lettingu=x31x52, we have thatf(u) = ln(x31x52) =3 lnx1+ 5 lnx2. If one function is a monotonic transformation of another, the two describethe same preferences since they will they rank bundles in the same way. (They assign dif-ferent values to the bundle, but we do not use thesecardinalnumbers in determining theutility-maximizing choices—we only care aboutordinalcomparisons.)Problem 2 (Well-Behaved Preferences)(a)Instead of using utility functionU(x1, x2) =x31x12, we can use a monotonic trans-formation instead:U(x1, x2) = 3 lnx1+ lnx2. (To get this, letf(u) = ln(u). Thenf(u) = ln(x31x2) = 3 lnx1+ lnx2. Again, even though these are not the same utility func-1
tions, they’ll give the same MRS and thus the same results.)UsingU(x1, x2) = 3 lnx1+ lnx2, we get thanM U1=∂U∂x1=3x1andM U2=∂U∂x2=1x2. SinceMRS(x1, x2) =−MU1MU2=−(∂U∂x1)(∂U∂x2), we have here thatMRS(x1, x2) =−3x2x1. This is the MRSfor any bundle (x1, x2), which is also the slope of the indifference curve passing through thatpoint.(b)Using our answer in(a), we get thatMRS(1,1) =−3·11=−3. This tells us that theslope of the indifference curve passing through the point (1,1) is−3:DVDs,x1CDs,x211MRS=
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Problem Set 2: SolutionsECON 301: Intermediate MicroeconomicsProf. Marek WeretkaProblem 1 (Marginal Rate of Substitution)(a)For the third column, recall that by definitionMRS(x1, x2) =−(∂U∂x1)(∂U∂x2).Utility Function∂U∂x1∂U∂x2MRS(x1, x2)MRS(2,3)(i)U(x1, x2) =x1x2x2x1−x2x1−32(ii)U(x1, x2) =x31x523x21x525x31x42−3x25x1−910(iv)U(x1, x2) = 3 lnx1+ 5 lnx23x15x2−3x25x1−910(b)MRS(2,3) =−9/10 for utility functionU(x1, x2) =x31x52has the following interpre-tation: At bundle (2,3), to remain indifferent about the change (i.e., remain at the sameutility level), a consumer is willing to give up 9/10 ofx2for one additional unit ofx1. (Or,after losing one unit ofx1, he must receive 9/10 of a unit ofx2to be as well off as he wasat bundle (2,3).) Soat the point(2,3), good two is more valuable since he needs to get lessof it than he lost of the other good to remain as satisfied. If 0.00001 of good one is takenaway, he would have to receive approximately 0.00001×(910) = 0.000009 units of good twoto remain indifferent to the change.(c)The two utility functions share the same MRS functions becauseU(x1, x2) = 3 lnx1+5 lnx2is a monotonic transformation ofU(x1, x2) =x31x52. To see this, letf(u) = ln(u)(f(u) is a monotonic function). Then lettingu=x31x52, we have thatf(u) = ln(x31x52) =3 lnx1+ 5 lnx2. If one function is a monotonic transformation of another, the two describethe same preferences since they will they rank bundles in the same way. (They assign dif-ferent values to the bundle, but we do not use thesecardinalnumbers in determining theutility-maximizing choices—we only care aboutordinalcomparisons.)Problem 2 (Well-Behaved Preferences)(a)Instead of using utility functionU(x1, x2) =x31x12, we can use a monotonic trans-formation instead:U(x1, x2) = 3 lnx1+ lnx2. (To get this, letf(u) = ln(u). Thenf(u) = ln(x31x2) = 3 lnx1+ lnx2. Again, even though these are not the same utility func-1
tions, they’ll give the same MRS and thus the same results.)UsingU(x1, x2) = 3 lnx1+ lnx2, we get thanM U1=∂U∂x1=3x1andM U2=∂U∂x2=1x2. SinceMRS(x1, x2) =−MU1MU2=−(∂U∂x1)(∂U∂x2), we have here thatMRS(x1, x2) =−3x2x1. This is the MRSfor any bundle (x1, x2), which is also the slope of the indifference curve passing through thatpoint.(b)Using our answer in(a), we get thatMRS(1,1) =−3·11=−3. This tells us that theslope of the indifference curve passing through the point (1,1) is−3:DVDs,x1CDs,x211MRS=
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