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notes_4 - Consumer Theory Part II 1 Problem Set 2 Solution...

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Unformatted text preview: Consumer Theory: Part II February 19, 2009 1 Problem Set 2: Solution Execise 3.13 Let’s put Peanut Butter on the vertical axes and Jelly on the horizontal axes. 1. 1-Peanut Butter is the same as 2-Jelly. So I can always substitute Peanut Butter for 2 ounces of Jelly. Therefore the MRS is fixed and equal to 0.5: these goods are perfect substitutes MRS = 1 2 = MU J MU PB so, up to a multiplicative constant, MU J = 1 and MU PB = 2, giving the equation U = 2 PB + J 2. Jelly doesn’t affect utility. 3. Jelly is a bad 4. Peanut Butter and Jelly must be consumed together and in fixed pro- portions: they are perfect complements. I want 2-PB for every 1-Jelly: therefore PB must always be double the amount of Jelly. PB = 2 J This is the line where the corners are aligned Exercise 3.17 The utility function is U ( x,y ) = xy + x 1. Is the assumption ”the more the better” satisfied for both goods ? If this is the case then the utility must be increasing with respect to 1 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 jel y Peanut U = 2 peanut + 1 jel y MRS= 1/2 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 jel y Peanut utility doens't increase by increasing Jel y 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 jel y Peanut 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 jel y Peanut PB = 2 J Figure 1: Exercise 3.13 both goods; and a function is increasing if its partial derivatives are positive. MU x = y + 1 > MU y = x > So, the assumption is satisfied for both goods 2. The marginal utility for x remains constant as x changes: x doesn’t appear into MU x , therefore a change in x will not modify the marginal utility. ∂MU x ∂x = 0. 3. The marginal rate of substitution is MRS = y + 1 x 4. The marginal rate is decreasing in x : mathematically, ∂MRS ∂x =- y + 1 x 2 < Exercise 4.6 Good MU Price HotDog 5 1 Soda 3 . 5 2 Dave is not maximizing: in fact, at the optimum, it must be that MRS = p x /P y . Another way to write this is MU x p x = MU y p y At the optimum the marginal utility in dollar value must be equalized across good. This is not the case here Good MU Price MU/P HotDog 5 1 5 Soda 3 . 5 6 the marginal utility in dollar value for Soda (6) is higher than that for Hot Dog (5): therefore Dave should consume more Soda and less Hot Dogs. Exercise 4.11 The utility is U = √ H + √ M (the text implies that H is the good on the horizontal axes). 1. the marginal rate of substitution MRS H,M = MU H MU M = r M H this is decreasing in H : ∂MRS ∂H =- . 5 r M H 3 < 2. The indifference curves intersect the axes U = √ H + √ M = ⇒ M =0 U = √ H = ⇒ H = U 2 3. Income is 24, P H = 2 and P M = 1. At the optimum, MRS = P H P M MRS = r M H = 2 1 = ⇒ M = 4 H substitute this condition into the budget constrain P H H + P M M = I 2 H + (4 H ) = 24 = ⇒ H * = 4 and M * = 16 3 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 H M U = 3 U = 2 2 Constrained Maximization: the Lagrange Mul- tiplier Method [OPTIONAL] Suppose you face the problem of maximizing a certain function given a...
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notes_4 - Consumer Theory Part II 1 Problem Set 2 Solution...

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