Problem Set 2: Preferences∗Econ 11 Fall 2015Due Date: October 6, 20151. Tony Stark (aka Iron Man) has different preferences over weapons andgoods.(a) For each of the following pairs of goods find the marginal rate ofsubstitution and based on its value draw a representative indifferencecurve1.i. Tony can always substitute 3 telescoping super-cavitation swordsfor 1 flamethrower, independent of the number he has of each.ii. Tony really enjoys the generator in his chest that can shut downelectrical devices in a 600 mile radius, and eyelash curlers don’tgive him any utility. Assume the generator is the X good.iii. Tony always buys 1 mini-gun with 888 uranium rounds.iv. Tony’s preferences over repulsor rays and pulse bolts are givenby the following utility function: U (RR, P B) = RR1/2P B−1/2.v. Tony’s preferences over superpower inhibiting nanobot (SPIN)darts and grenades can be described by the following utility func-tion: U (G, SP ) = (G1/2+ P1/2)2. However, after 100 grenadeshe runs the risk of blowing himself up, so his utility is given by:U(G, SP ) = (20 − G1/2+ SP1/2)2.(b) Pepper Potts has been trying to explain to Tony the difference be-tween ordinal and cardinal rankings. In particular, Pepper arguesthat Tony’s preferences over telescoping super-cavitation swords andflamethrowers can also be represented by the utility functionU0(S, F ) = 2S + 6FIs Pepper right?∗For errors or corrections, please email me to [email protected] is not necessary to graph exactly the function, it is enough to determine the generalshape implied by the MRS.12. Captain America’s preferences over shields with silver stars (X) and redand blue spandex suits (Y ) are given by: U (X, Y ) = X1/4Y3/4.(a) Graph a typical indifference curve.(b) Obtain the marginal utility of X (M Ux) and the marginal utility ofY (MUy).(c) Obtain the marginal rate of substitution.(d) Determine if it is increasing or decreasing when X increases.(e) Compare the MRS with the following functions:• U(X, Y ) =X1/4Y3/42• U(X, Y ) =X1/4Y3/41/2• U(X, Y ) =14ln(X) +34ln(Y )3. The Hulk and Loki only consume radioactive green hamburgers (H) andf˚arik˚al (mutton stew with cabbage) (F ). Their respective marginal rateof substitution are given by the following equations:MRSH=H3FMRSL=3HF(a) Assume both have three units of each dish. For their preferences tobe consistent with the axioms of utility representation, which goodshould be graphed on the X axis?(b) Plot a graph of their indifference curves corresponding to their cur-rent bundle. In general terms, which good do each prefer more? Thatis, which good gives them a higher utility when consuming one moreunit?(c) Propose a trade that would make both of them happier. How canyou use your knowledge of the MRS to show that the proposed trademakes them better-off?4. Natalia Romanova (Black Widow) enjoys spending time with Clint Barton(Hawkeye), but the more time she spends with him, the less she likes it.Although Natalia prefers spending 4 hours than 3 hours with Clint, sheprefers spending 4 hours than 5 hours with him. Moreover, she derives thesame utility from spending 3 or 5 hours with Clint. Determine whether thefollowing statements are true or false, and provide a proper justificationfor your answer.(a) The marginal utility of the fourth hour is zero.(b) Every hour spent with Clint before the fourth one is a good for Na-talia, and every hour after the fourth is a bad.(c) Natalia’s preferences are not transitive.2(d) Since Natalia is indifferent between spending 3 or 5 hours with Clint,her preferences are not complete.(e) Natalia’s preferences are convex.5. Thor’s preferences over hammers (H) and red capes (R) can be describedby the following equation:U(H, R) =2Hδ+ Rδ1/δ(a) Calculate the MRS.(b) Using the MRS, draw a representative indifference curve for the fol-lowing cases:• δ = 1• δ → −∞• δ = 0(c) Which axiom would be violated if δ >