Msc Micro I exam. Lecturer: Todd Kaplan.
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1 Msc Micro I exam. Lecturer: Todd Kaplan. Please answer exactly 5 questions. Answer one question from each of sections: A, B, C, and D and answer one additional question from any of the sections A, B, C, or D. For instance, answering, 4, 6, 7, 8 is valid. Answering, 3, 4, 5, 7 is also valid. Answering, 2, 3, 4, 5 is not valid. GOOD LUCK! Section A.. Does strong transitivity and completeness imply weak transitivity? Explain. No. Take the example of where someone is indifferent between prices that are different by 50 cents or less but care about differences larger than this. Strong transitivity holds since if p b > p a +50 and p c > p b + 50, we have p c > p a However, weak transitivity does not hold since indifference transitivity does not hold. More formally, we may have x y and y z yet x z but this does not violate strong transitivity since we don t have x y, y z, etc. This does violate weak transitivity since we have z y and y x but not z x. 2. We showed on the homework that if independence (and weak transitivity hold) then the following Property holds. Property P: for all L, L, L, L and x we have if L L and L L, then xl + ( x)l xl + ( x)l. Now show that if P and Independence hold, then weak transitivity holds. Say L L 2 and L 2 L 3. If Property P holds, then we can choose L = L, L = L 2, L = L 2, L = L 3 and x = /2. Thus, L 2 + L 2 2 L L 2 3. By Independence, we have L L 3. Section B. 3. Ari likes to have similar things. He prefers bundle x to y if and only if min{ x2+, x+ y2+, y+ }. For the following provide both an x+ x2+ y+ y2+ answer and an argument supporting your answer. (i) Draw the indifference curves. (ii) Does Ari s preferences satisfy transitivity? (iii) Does Ari s preferences satisfy convexity? (iv) Does Ari s preferences satisfy weak monotonicity? (v) Does Ari s preferences satisfy continuity?
2 (i) all indifference curves are two rays symmetric around the 45 degree line. For instance, y = 2x + and y = (x )/2 form one indifference curve. The slope increases for the one above the 45 degree line as they start further away from the origin. (ii) yes, x y and y z implies min{ x2+ and min{ y2+ z2+, y+ y+ y2+, x+ x+ x2+ y2+ y+, y+ y2+ }, z+ x2+ } = min{, x+ z+ z2+ x+ x2+ x z. (iii) Yes. Intuitively closer to the 45 degree line is better. A convex combination of two points cannot be further away. It is easy to show this with a picture on the indifference curves. (Note they don t need to formally show this, it will be too messy). (iv) No, we have (, 2) (, 3). (v). Yes,B(z) and W (z) are closed sets. You can see this again from the indifference curves. 4. Crazy Chris has strictly monotonic preferences and chose a sure chance of $3800 over a 80% chance of $6000. He also chose a 40% chance of $5500 over a 50% chance of $4000. Show how Crazy Chris cannot have vnm expected utility. Show how by using different weights of probability (with a w(p) function) can explain his choice. By choosing a sure chance of $3800 over a 80% chance of $6000, if Chris had vnm preferences, then u(3800) 0.8u(6000) + 0.2u(0). By choosing a 40% chance of $5500 over a 50% chance of $4000, if Chris had vnm preferences, then 0.4u(5500) + 0.6u(0) 0.5u(4000) + 0.5u(0). Multiplying this latter inequality by 2 on each side and combining the u(0) terms yields 0.8u(5500) + 0.2u(0) u(4000). However we have u(4000) > u(3800), so 0.8u(5500) + 0.2u(0) u(4000) > u(3800) 0.8u(6000) + 0.2u(0). z2+ z+, z+ z2+ } = This implies that u(5500) > u(6000), which can t happen with monotonic preferences. With a w(p) function this can be consistent. Take utility u(x) = x and w(0.5)=0.5. The two choices become 3800 w(0.8)6000 and 2
3 w(0.4) If w(0.8) = 0.6 and w(0.4) = 0.45 works (overweighting small probabilities and underweighting high probabilities). Section C. 5. Take an uncertain environment with three possible outcomes:, 2, and 3. Utility for these outcomes is as follows: u() = 4, u(2) =, u(3) =. Utility over a lottery is vnm expected utility except when one probability is strictly less than /4 on an outcome, it is treated as 0. For instance, if there is a 5/6 chance of outcome, the utility is 5u() = What does the indifference curve going through the point (,, ) look like when represented in a triangle? Explain your answer. We as long as all probability is greater than /4, we have p + p2 + p3 = and 4p +p2 +p3 = 2. Eliminating p3 yields 4p+p2+-p-p2=2 or 3p=. So it is a line parallel to the 2-3 side going through the middle of the triangle. When it reaches the point of p2=2/3-/4=5/2 and p3=/4, it jumps to the line 4p+p2=2. This has points of (/3,2/3,0) and moves toward the point (/2,0,/2) until it hits the point where p3=/4. Similar for the other side. 6. Sam has vnm preferences. He is playing a game show where one of two suitcases will be opened. In suitcase A, there is a /2 chance of million dollars and /2 chance of nothing. In suitcase B, there is a 3/4 chance of million dollars and a /4 chance of nothing. Show that if he has a p chance of suitcase A and a (-p) chance of suitcase B, his utility is p U(A) + ( p) U(B) (using only the definition of vnm and that with vnm the utility of a complex lottery is equal to its utility of the reduced form lottery). According to the definition of vnm U(p A + ( p) B) = (p + 2 ( p) 3)u( million) + (p + ( p) )u(0) Rearranging yields p( u( million) + u(0)) + ( p)( 3 u( million) u(0)). Again according to the definition of vnm u( million)+ u(0) = U(A) and 3u( million) + u(0) = U(B). Thus, we have the answer. 4 4 Section D. 7. Auctions: You are bidding in a first-price auction for a painting by Monet against two other bidders. Each of you has a private, independent value v i for such a painting that is drawn uniformly from [0, ] (in billions of shekels). There is a probability of /2 that the painting is a fake and worthless to all. What is your equilibrium bid function? If this was 3
4 instead for charity and an all-pay auction was used, what would be your equilibrium bid function? Each bidder would wish to choose b to maximize his expected profits P robwin(b) ( 2 v i b). Assuming the others bid b(v) = α v, we have expected profits equals The first-order condition is ( b a )2 ( 2 v i b). 2( b a )( a ) ( 2 v i b) ( b a )2 = 0 This simplies to or b a = 2 a ( 2 v i b) b = v i 3 For the all-pay auction, a bidder solves max ṽ 2 v ṽ 2 b(ṽ). The FOC is v 2 = b (v). Integrating and setting the constant to zero yields b(v) = v Game Theory: Consider the following prisoners dilemma in simple payoffs. \2 C D 5 8 C D 8 2 Utility over payoffs for each player i is u i (x i, x i ) = x i a max{x i x i, 0} b max{x i x i, 0} where a, b 0 and a b. For what values of a and b is (C, C) a Nash equilibrium? For which of these values is (D, D) also a Nash equilibria? Explain. 4
5 (C, C) will be a Nash Equilibrium whenever u i (5, 5) u i (8, 0) (since there would not be an incentive to deviate). We have u i (5, 5) = 5 and u i (8, 0) = 8 b 8. Thus, for (C, C) to be an equilibrium we must have 5 8 b b 3. (D, D) will be an equilibrium whenever u i (2, 2) u i (0, 8). We have u i (2, 2) = 2 and u i (8, 0) = a 8. The inequality u i (2, 2) u i (0, 8) is always satisfied. 5
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