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1) Al Einstein has a utility function that we can describe by u(x1, x2) = x 2/1 + 2x1x2 + x 2/2

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1) Al Einstein has a utility function that we can describe by u(x1,

x2) = x 2/1 + 2x1x2 + x 2/2 . Al's wife, El Einstein, has a utility function v(x1, x2) = x2 + x1.
(a) Calculate Al's marginal rate of substitution between x1 and x2.
(b) What is El's marginal rate of substitution between x1 and x2?
(c) Do Al's and El's utility functions u(x1, x2) and v(x1, x2) represent the same preferences?
(d) Is El's utility function a monotonic transformation of Al's?


2)Marie Curie has a utility function U(x, y) = max{x, 2y}
(a) Draw a graph, and plot the lines given by the equations x = 10, and 2y = 10.
(b) What's the value of U(x, y) when x = 10 and 2y < 10?
(c) What's the value of U(x, y) when x < 10 and 2y = 10?
(d) On the same chart, plot the indifference curve along which U(x, y) = 10.
(e) Are Marie's preferences convex?


3)Convert these marketing pitches into mathematical utility functions:
(a) Everything goes with Biscuits
(b) Banks Brewery asks that you drink responsibly.
(c) Sunflower margarine tastes just like butter!


4)Suppose each individual in a society gets utility from varying degrees of a specific policy P, which can be between 0 and 1 so that 0 < P < 1. Individual 1's utility function is given by u1(P) = P 0.5 , while individual 2's utility function is given by u2(P) = 0.3P, individual 3's utility function is given by u3(P) = P −0.5 , and individual 4's utility function is given by u4 = u1(P) + u2(P) + u3(P). Assuming these are cardinal utility functions:
(a) In words, describe how the policy affects each individual.
(b) Calculate the utility of each individual for each value of P from 0.05 to 0.95 in increments of 0.5.
(c) There is no correct answer to this question: how would you choose a value of P if you were the government? Why? What value of P would you choose?
(d) Why is there no correct answer to the question above?
(e) Which one of these individuals would you elect as your leader and why?


5)Here are some utility functions. Find a) the marginal utility with respect to x1, b) the marginal utility with respect to x2, c) the marginal rate of substitution between x1 and x2.
(a) u(x1, x2) = 2x1 + 3x2
(b) u(x1, x2) = 4x1 + 6x2
(c) u(x1, x2) = ax1 + bx2
(d) u(x1, x2) = 2√ x1 + x2
(e) u(x1, x2) = ln(x1) + x2
(f) u(x1, x2) = v(x1) + x2
(g) u(x1, x2) = x1x2
(h) u(x1, x2) = x a 1x b 2
(i) u(x1, x2) = (x1 + 2)(x2 + b)
(j) u(x1, x2) = (x1 + a)(x2 + b)
(k) u(x1, x2) = x a 1 + x a 2


6)Tom Edison is eating 5 avocados and 20 sugar apples. The marginal rate of substitution is the rate at which Tom is willing to trade sugar apples for an extra avocado. At his current consumption bundle of 5 avocados and 20 sugar apples, his marginal rate of substitution is 4. Compared to his current
(a) Will Tom be better off if he instead consumes 6 avocados and 20 sugar apples?
(b) Will Tom be better off if he instead consumes 6 avocados and 15 sugar apples?
(c) Will Tom be better off if he instead consumes 7 avocados and 16 sugar apples?
(d) Will Tom be better off if he instead consumes 6 avocados and 16 sugar apples?


7)Give three examples of utility functions that give exactly the same preferences as u(x, y) = xy. Derive the marginal rate of substitution in each case and check whether they are equal.