1.Dr. Claw is hatching a plan to steal all of the cherry blossoms in Japan, cornering the market on haiku inspiration. He likes his henchmen to be big, fast, and obedient. Lou the Lunk weighs 340 pounds, runs slowly, and is somewhat obedient. Turncoat Tom weighs 240 pounds, runs very fast, and is very disobedient. Brainwashed Bobby weighs 150 pounds, has average speed, and is extremely obedient.
(a) Suppose Dr. Claw prefers henchman A to henchman B if A is superior to B in at least two of these dimensions, otherwise he is indifferent. Are Dr. Clawâs preferences complete? Are they transitive? Explain your answer.
(b) Now suppose Dr. Claw only prefers henchman A to henchman B if A is superior to B in all three dimensions. If A and B are identical in all three traits, then he is indifferent. Otherwise, Dr. Claw cannot compare the two henchmen. Are Dr. Clawâs preferences complete in this case? Are they transitive? Explain.
2. In this exercise, I will show you how to plot some indifference curves for the utility function u(x, y) = xy. Carefully read and follow these steps.
(a) For each of the values x = 1, 2, 3, 4, 5, find the corresponding value of y such that u(x, y) = 5. Plot these points on a graph of the x, y coordinate plane.
(b) Find the formula for the slope of the indifference curve passing through any point (x, y). Calculate these slopes for each point from part (a).
(c) Plot the slopes from part (b) on the same graph by drawing a short line segment at each point with its corresponding slope. For example, at (1, y1), draw a small line segment with the slope of the indifference curve through (1, y1).
(d) On the same graph, approximate the indifference curve corresponding to u(x, y) = 5 by sketching a curve that passes through each of the plotted points from (a), matching the slope of the line segments from (c) at each point.
(e) Repeat parts (a)-(d), but now for u(x, y) = 10, using the same graph.
1.Dr. Claw is hatching a plan to steal all of the cherry blossoms in Japan, cornering the market on haiku inspiration. He likes his henchmen to be big, fast, and obedient. Lou the Lunk weighs 340 pounds, runs slowly, and is somewhat obedient. Turncoat Tom weighs 240 pounds, runs very fast, and is very disobedient. Brainwashed Bobby weighs 150 pounds, has average speed, and is extremely obedient.
(a) Suppose Dr. Claw prefers henchman A to henchman B if A is superior to B in at least two of these dimensions, otherwise he is indifferent. Are Dr. Clawâs preferences complete? Are they transitive? Explain your answer.
(b) Now suppose Dr. Claw only prefers henchman A to henchman B if A is superior to B in all three dimensions. If A and B are identical in all three traits, then he is indifferent. Otherwise, Dr. Claw cannot compare the two henchmen. Are Dr. Clawâs preferences complete in this case? Are they transitive? Explain.
2. In this exercise, I will show you how to plot some indifference curves for the utility function u(x, y) = xy. Carefully read and follow these steps.
(a) For each of the values x = 1, 2, 3, 4, 5, find the corresponding value of y such that u(x, y) = 5. Plot these points on a graph of the x, y coordinate plane.
(b) Find the formula for the slope of the indifference curve passing through any point (x, y). Calculate these slopes for each point from part (a).
(c) Plot the slopes from part (b) on the same graph by drawing a short line segment at each point with its corresponding slope. For example, at (1, y1), draw a small line segment with the slope of the indifference curve through (1, y1).
(d) On the same graph, approximate the indifference curve corresponding to u(x, y) = 5 by sketching a curve that passes through each of the plotted points from (a), matching the slope of the line segments from (c) at each point.
(e) Repeat parts (a)-(d), but now for u(x, y) = 10, using the same graph.
