Proble Set 2: Solutions ECON 30: Interediate Microeconoics Prof. Marek Weretka Proble (Marginal Rate of Substitution) (a) For the third colun, recall that by definition MRS(x, x 2 ) = ( ) U x ( U ). x 2 Utility Function U x U x 2 MRS(x, x 2 ) MRS(2,3) (i) U(x, x 2 ) = x x 2 x 2 x x 2 x 3 2 (ii) U(x, x 2 ) = x 3 x 5 2 3x 2 x 5 2 5x 3 x 4 2 3x 2 5x 9 0 (iv) U(x, x 2 ) = 3 ln x + 5 ln x 2 3 x 5 x 2 3x 2 5x 9 0 (b) MRS(2, 3) = 9/0 for utility function U(x, x 2 ) = x 3 x 5 2 has the following interpretation: At bundle (2, 3), to reain indifferent about the change (i.e., reain at the sae utility level), a consuer is willing to give up 9/0 of x 2 for one additional unit of x. (Or, after losing one unit of x, he ust receive 9/0 of a unit of x 2 to be as well off as he was at bundle (2, 3).) So at the point (2, 3), good two is ore valuable since he needs to get less of it than he lost of the other good to reain as satisfied. If 0.0000 of good one is taken away, he would have to receive approxiately 0.0000 ( 9 ) = 0.000009 units of good two 0 to reain indifferent to the change. (c) The two utility functions share the sae MRS functions because U(x, x 2 ) = 3 ln x + 5 ln x 2 is a onotonic transforation of U(x, x 2 ) = x 3 x 5 2. To see this, let f(u) = ln(u) (f(u) is a onotonic function). Then letting u = x 3 x 5 2, we have that f(u) = ln(x 3 x 5 2) = 3 ln x + 5 ln x 2. If one function is a onotonic transforation of another, the two describe the sae preferences since they will they rank bundles in the sae way. (They assign different values to the bundle, but we do not use these cardinal nubers in deterining the utility-axiizing choices we only care about ordinal coparisons.) Proble 2 (Well-Behaved Preferences) (a) Instead of using utility function U(x, x 2 ) = x 3 x 2, we can use a onotonic transforation instead: U(x, x 2 ) = 3 ln x + ln x 2. (To get this, let f(u) = ln(u). Then f(u) = ln(x 3 x 2 ) = 3 ln x + ln x 2. Again, even though these are not the sae utility func-
tions, they ll give the sae MRS and thus the sae results.) Using U(x, x 2 ) = 3 ln x + ln x 2, we get than MU = U x = 3 x and MU 2 = U MRS(x, x 2 ) = MU MU 2 = ( ) U x ( U x 2 ), we have here that MRS(x, x 2 ) = 3x 2 x 2 = x 2. Since x. This is the MRS for any bundle (x, x 2 ), which is also the slope of the indifference curve passing through that point. (b) Using our answer in (a), we get that MRS(, ) = 3 = 3. This tells us that the slope of the indifference curve passing through the point (, ) is 3: CDs, x 2 MRS = 3 DVDs, x At (, ), good one is locally ore valued since, to copensate for a loss of 3 units of good two (the CDs), Alicia only needs unit of good one (the DVDs) to aintain the initial level of happiness. (c) The two secrets of happiness for well-behaved preferences are: () x + x 2 = (Since ore is preferred to less, spend all of your incoe.) (2) MRS = (Marginal utility per dollar spent is equalized.) Note: An equivalent way of writing this is MU MU 2 = (using the definition of MRS) or MU. All three ways are exactly the sae. = MU 2 Graphically, we re finding the bundle for which the budget line is tangent to an indifference curve: 2
CDs, x 2 = 40 0 5 = 20 DVDs, x Given that = 40, = 20, and = 800, we can rewrite these two equations as () 40x + 20x 2 = 800 (2) 3x 2 x = 40 20 = x 2 = 2 3 x (d) To find Alicia s optial bundle, we just use the two equations above to solve for our two unknowns, which are x and x 2. (So there s no econoics here, only Algebra.) You can just take x 2 = 2 3 x fro equation (2) and plug it into () to get 40x + 20( 2 3 x ) = 800 = x = 5. Plug x = 5 into either equation to find that x 2 = 0. Alicia s optial bundle, given these prices and her incoe, is (5, 0), which is interior (she s consuing non-zero aounts of both). This is shown in the figure above. Proble 3 (Perfect Copleents) (a) The indifference curves passing through (5, ), (0, 0), and (5, 4) are shown below. The vertices all fall along the dotted line along which x 2 = 5x. (It shows the cobinations for which Trevor consues five ties as any strawberries (x 2 ) as he does units of ilk (x )). The MRS at each of these points (without using any forulas and only looking at the graph) is zero: MRS(5, ) = MRS(0, 0) = MRS(5, 4) = 0. Strawberries, x 2 x 2 =5x (0, 0) (5, 4) (5, ) Milk, x 3
(b) His preferences can be represented by the utility function U(x, x 2 ) = in{5x, x 2 }. In general, if preferences are perfect copleents where a of x ust be consued for every b of x 2, the utility function can be expressed as U(x, x 2 ) = in{ a x, b x 2}, and the line along which all of the vertices of those L-shaped indifference curves lie is a x = b x 2. So using this forula directly U(x, x 2 ) = in{x, 5 x 2} but, ultiplying everything through by 5 (which would be a onotonic transforation!) we get U(x, x 2 ) = in{5x, x 2 }. To find the level of utility associated with the indifference curves passing through (5, ), (0, 0), and (5, 4), we use this utility function to find that: U(5, ) = in{5 5, } = in{25, } = U(0, 0) = in{5 0, 0} = in{50, 0} = 0 U(5, 4) = in{5 5, 4} = in{75, 4} = 4 Strawberries, x 2 x 2 =5x (0, 0) u = 0 (5, ) (5, 4) u =4 u = Milk, x Notice that if you would have used utility function U(x, x 2 ) = in{x, 5 x 2}, you would get: U(5, ) = in{5, 5 } = in{5, 5 } = 5 U(0, 0) = in{0, 0} = in{0, 2} = 2 5 U(5, 4) = in{5, 5 4} = in{5, 4 5 } = 4 5 Either way, we see that (0, 0) is the ost preferred (i.e., gives the highest utility aong the three), followed by (5, 4) and then (5, ). (c) Multiplying our utility function by ten and adding two is equivalent to taking a onotonic transforation f(u) = 0u + 2. If we take our utility U(x, x 2 ) = in{5x, x 2 }, we get U trans (x, x 2 ) = 0 in{5x, x 2 } + 2. Then U(5, ) = 0 in{25, } + 2 = 0 + 2 = 2 U(0, 0) = 0 in{50, 0} + 2 = 0 0 + 2 = 02 U(5, 4) = 0 in{75, 4} + 2 = 0 4 + 2 = 42 4
(New) Strawberries, x 2 x 2 =2x Again, the indifference curves do not ove and the preference ranking aong the bundles is preserved, we just have the above levels of utility attached to each of the indifference curves. (d) Letting =, =, and = 00, the two secrets of happiness for perfect copleents are () x + x 2 = = x + x 2 = 00 (Trevor spends all of his incoe.) (2) x 2 = 5x (He consues only to optial proportions along the dotted line along which x 2 = 5x.) Note: These types of preferences are not well behaved like Cobb Douglas preferences are, so we use a different second secret of happiness for these preferences. We can no longer use MRS = since the MRS of the indifference curve is not defined at the kink. To find the deand for both ilk (x ) and strawberries (x 2 ) we solve the equations in () and (2): Plug x 2 = 5x into equation () for x 2, so x + (5x ) = 00 = x = 00/6. Plug this into either equation to solve for x 2 and get x 2 = 500/6. This is interior since Trevor is consuing non-zero aounts of both goods (i.e., x > 0 and x 2 > 0). (e) With larger strawberries, the new optial proportion of ilk (x ) and strawberries (x 2 ) is two strawberries for every unit of ilk, or x 2 = 2x. Our indifference curves are the sae shape as they were before, but now the vertices of these L-shaped indifference curves (which will be the optial bundles) lie along x 2 = 2x, as seen below. Milk, x These new preferences can be represented by utility function U(x, x 2 ) = in{2x, x 2 }. 5
Proble 4 (Perfect Substitutes) (a) When two goods are perfect substitutes, we know the indifference curves are linear and downward-sloping, in this case having a constant slope of. The indifference curves passing through points (3, 2) and (3, 3) are shown below: Jonagold, x 2 3 2 (3, 3) (3, 2) 3 Red Delicious, x (b) Soe utility functions that could represent these functions: (i) U(x, x 2 ) = x + x 2 (ii) U(x, x 2 ) = ln(x + x 2 ) (iii) U(x, x 2 ) = (x + x 2 ) 2 (iv) U(x, x 2 ) = 8 x + 8 x 2 (v) U(x, x 2 ) = 6x + 6x 2 Each of these five utility functions represents a onotonic transforation of any of the others; they all represent the sae underlying perfect-substitute preferences over Red Delicious (x ) and Jonagold (x 2 ) apples and give indifference curves having the sae MRS = (you can verify this). (c) MRS(x, x 2 ) = for any (x, x 2 ). This eans that starting fro any (x, x 2 ) bundle, Kate is always willing to give up one Red Delicious (x ) apple to get one additional Jonagold (x 2 ) (or vice versa). This eans, as noted above, that the indifferences curves will be linear everywhere. (d) Letting = 2, =, and = 00 we can first turn to the coodity space to see how to go about funding the optial bundle. We know that the first secret of happiness (spending all of one s incoe) will always hold, so the optial choice is along the budget line. But the budget line is not tangent to any indifference curve here: The budget line has a slope of = 2 and we just deterined that MRS(x, x 2 ) = everywhere, so 6
MRS anywhere. Jonagold, x 2 = 00 (0, 00) = 50 Red Delicious, x We can see that at the indifference curve furthest out fro the origin (the one with highest utility) but still touching the budget line, Kate is consuing only Jonagolds (x 2 ) and no Red Delicious (x ): x = 0, x 2 = = 00 = 00. This is not an interior solution (since x = 0) but rather is a corner solution. As a general rule, when MRS < (the indifference curves are ore flat than the budget line), the consuer chooses to consue only x 2. Here, MRS = < = 2. This akes sense intuitively here: If the two types of apples are : perfect substitutes for Kate, she should just consue the one that is cheaper. (e) Now we have that MRS = > = and Kate consues only the cheaper apple, the 2 Red Delicious (x ), with x = = 00 = 00, x 2 = 0. Jonagold, x 2 = 50 (00, 0) = 00 Red Delicious, x (f) With this price change, MRS = everywhere, so all of the cobinations along the budget line lie along the sae indifference curve, and thus any bundle is equally as good as 7
any other bundle on the budget line. Kate can choose any (x, x 2 ) cobination such that x + x 2 = or x + x 2 = 00: Jonagold, x 2 = 00 all optial bundles along here = 00 Red Delicious, x 8