Unformatted text preview: Suggested Solutions – HW3 Instructors – Dror Brenner and Ryan Miller
1. If the utility function is convex (i.e. preferences are “wellbehaved”), and if the MRS is welldefined (i.e. the goods are not perfect complements), then the optimal consumption bundle is an interior solution. We will have the following two equations in and : • • =− + = These can be solved as a system of 2 equations in 2 unknowns to obtain the optimal consumption bundle for the consumer, ∗ , ∗ (a) Calculate the MRS: =− Solve the system: − 2 3 4 =− , 6
∗ =− 2 3 =− 2 3 4
∗ +6 = 200 , = 20,20 (b) Calculate the MRS: =− Solve the system: − 3 1+ =− , +3 2
∗ =− 1+ +3 3 +2 = 97 , ∗ = 15,26 (c) This is a case of preferences over two goods which are perfect complements. Hence, the MRS for these two goods is not welldefined: however, as we have discussed in section, the consumer will optimize by choosing ∗ and ∗ such that the two terms in the min ⋅ function are equal. (Note: though this is an abuse of notation, it is sort of as if the MRS = −∞ for the vertical portion of the indifference curve, and MRS=0 for the horizontal portion, so the kink is the only place where the consumer could ever optimize). So, set: 2 Solve the system: 2 = 3
∗ = 3 , ,
∗ 4 + = 50 = 5,30 1 (d) Calculate the MRS: =− Solve the system: −
∗ =− 1 ⁄2 3
∗ 6
⁄ 1 ⁄2
∗ 6
⁄ 3 =− , 2 , +2
∗ = 154 , = 4927 1 , or 96 64 ≅ 51.32, 0.02 2. For this question, it would be easiest to look at part (b) first, and then solve part (a) (a) Bob has a concave utility function. Therefore, his optimal consumption bundle is a corner solution, i.e. ⁄ , 0 o r 0, ⁄ gives him higher utility. See the answer to part (b) for more whichever of discussion. In this case, the solution is:
∗ , ∗ = 0,1000 Mark has a convex utility function. Therefore, his optimal consumption bundle is interior, and we can find it in a similar way to Question 1. Calculate the MRS: =− Solve the system:
⁄ 1 ⁄2 1 ⁄2 ⁄ ⁄ ⁄ =− ⁄ − ⁄ 4 =− , 1
∗ 4 + = 1000 , ∗ = 50,800 (b) Bob’s utility function is not convex. We can see this by thinking about what one of his indifference curves would look like: his utility function is the equation for a circle. Hence, Bob will choose to spend all of his income on one good. We can also see this graphically: think about the highest indifference curve (i.e. in this case, the largest circle) we could draw for Bob, that would contain his budget line. It would in fact be the circle that touches one of the points where Bob spends all his money on one good, i.e. one of the bundles on the axis or the axis! If we were to try to simply set = − ⁄ , note that we would, in fact, be finding the *smallest* possible utility for Bob, given his budget constraint. This is why it is always important to check whether a utility function is convex. On the other hand, Mark’s utility function is convex (his preferences are “wellbehaved”). Therefore, the optimal bundle for him will be an interior solution. 2 3. (a) These goods are perfect substitutes. Thus, the MRS is constant: 1 = − = −1 1 We want to compute the Engel curve and the inverse demand for , but since the MRS is constant, we cannot find an interior solution. Since , = + and the goods are perfect substitutes, the or . That is, consumer will reach her highest indifference curve by spending all her income on either her optimal consumption bundle is either
∗ , ∗ = 1000 ,0 Or (since = 2):
∗ , ∗ = 0, 500 Then she will only ever purchase at all if her utility from doing so is highest: 1000 ,0 = 1000 ⇔ > 500 = <2 0, 500 Since = 1 to begin, this condition is satisfied, so as we vary her income , the consumer will spend all of it on . Thus the Engel curve is (in this case) her budget equation, when = 0: =
M slope = p1 = 1 X1 The Engel curve is always positively sloped; thus, Now, holding = 1000 and letting is a normal good (not inferior). is: vary, the ordinary demand for
∗ = 1000/ , 0,500 , 0, <2 =2 >2 3 X1 X1= 500 p1 p1= 2 Since the demand curve does not have any positively sloped part, is an ordinary good (not Giffen). The inverse demand is obtained by simply expressing as a function of ∗ . Visually, “inverting” the demand function looks like reflecting it about the 45° line in the positive quadrant. 2, ∞ , 2, = 1000/ ∗ ,
p1 =0 0 < ∗ < 500 ∗ > 500 ∗ p1= 2 X1 X1= 500 4 (b) These goods are perfect complements. Thus, the MRS is not well defined. However, as we have discussed in section and in class, we know that since , = min , , the consumer’s optimal bundle for any income will be one where = . Combined with the budget constraint + = , the optimal bundle is:
∗ , ∗ = + , + (it will Holding = 1 and = 2 fixed and solving for lets us find the Engel curve for coincidentally look identical, in this case, to that for ): =
M + ∗ =3 ∗ slope = p1 + p2 = 3 X1 Now, holding = 1000 and letting vary, the ordinary demand for
∗ is: = 1000 +2 X1 X1= 500 p1 5 Again, the inverse demand is obtained by simply expressing
∗ as a function of = 1000
∗ ∗ . = 1000 ⇒ +2 ∗ + 2 = 1000 ⇒ −2 p1 X1 X1= 500 (Note: if ∗ > 500 then simply using the above equation would give us some negative price < 0. So, we should be careful to specify that the inverse demand function is not defined for these values of ∗ .) (c) These preferences are quasilinear with only depend on : appearing linearly in the utility function. Hence, the MRS will 1 ⁄2
⁄ =− We can find the optimal consumption bundle by setting = − ⁄ , and adding the budget constraint + = . Of course, since the preferences are quasilinear, the equation = − ⁄ alone is enough to let us solve for the optimal quantity ∗ of : − 1 2 1 = − ⇒ 2
∗ 1 =− 1 2 =1 To sketch the Engel curve for , we should fix = 1 and = 2 and let vary in the budget equation, + = . However, if the solution is interior, then we cannot solve this. The Engel curve will be vertical, since ∗ = 1 no matter how large is. If the solution is not interior (the problem does not ask for this, but I’ll include it for fun), then intuitively the consumer will maximize utility by trying to consume as close to ∗ = 1 as possible. That is, the consumer will spend all his income on . 6 M M=1 X1 X1 = 1 The inverse demand for is found by setting in terms of ∗ . Thus, fixing = 2, − 1 2 =− 2 =− ⁄ (for the interior solution) and solving for 1
∗ ⇒ = p1 p1 = 1 X1 X1 = 1 7 4. (a) Let be the price of “all other goods” in 1970. Then the budget equations for the two years are: 1970: 2 + = 1000 1971: 2.42 + 1.1 = 1133 All other goods X'2 = 1030/p2 X2 = 1000/p2 1971 1970 Coconuts X'1 = 468.2 X1 = 500 (b) No, we cannot say this, because the price of coconuts also increased. The drop in consumption could be due to a substitution effect which outweighs the income effect, as illustrated by the indifference curves and budget sets in the following (nottoscale) diagram:
All other goods Coconuts 8 ...
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 Spring '08
 Mehta
 Utility

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