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With the following condition: UxxUx^2-2UxyYxUy+uyyUy^2<0, check the convexity of the indifference curves for each of the following functions and describe the precise relationship between diminishing marginal utility and quasi-concavity for each case.

A. U(x,y)=3x+y

B. U(x,y)=(x.y)^1/2

C. U(x,y)=(x)^1/2+y

D. U(x,y)=(x^2-y^2)^1/2

E. U(x,y)=(xy)/(x+y)

(^1/2 used inplace for square root)

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Related questions

PROBLEM 1 (This one is designed to give you lots of math practice!)

Consider the following individual utility functions:

Adam Smith: U(x,y) = xy

Jeremy Bentham: U(x,y) = xayb (Cobb-Douglas utility function)

Alfred Marshall: U(x,y) = ln x + ln y

John M. Keynes: U(x,y) = x + yb (Quasi-linear utility function)

Joan Robinson: U(x,y) = aX + bY (linear; perfect substitutes)

a. Solve for the MRS(XY) for each of the gentlemen (i.e., all but Joan Robinson). Show that Mr. Smith, Mr. Bentham, and Mr. Marshall all have diminishing MRS(XY), but that if a=b=2, their preferences exhibit constant, increasing, and decreasing marginal utility respectively. (Fortunately, diminishing MRS is the condition that guarantees convex indifference curves, so we can solve for their optimal consumption!)

 

b. What is the general formula for Mrs. Robinsons (constant) MRS?

c. Find demand functions for x and y for Smith, Bentham, Marshall, and Keynes. Assume that each man has income I and that the prices of x and y are px and py respectively.

d. Now assume that a=1/2, b = 1/4, I = 600, px = 10 and py = 20. Use these values to find the utility-maximizing quantities of x and y consumed by each man.

e. Find an expression for the share of income Mr. Bentham and Mr. Marshall each devote to the consumption of good x, where this share, sx, is defined as

(px x)/I.

f. Find expressions for the income elasticity of demand for good x for Mr. Bentham and Mr. Marshall. Find expressions for the (own) price elasticity of demand for good y for these two men.

5. For each of the following utility functions, draw the three indierence curves that correspond to the three stated utility levels, labeling each curve with the corresponding utility

level. Label any intercepts or \kinks" in the line to fully identify the position of the curve.

(a) U(X; Y ) = minf3X; Y g; utility levels = 3; 6; 9

(b) U(X; Y ) = minf3X; 2Y g; utility levels = 6; 12; 18.

(c) U(X; Y ) = 5X + 2Y ; utility levels = 10; 15; 20.




6. For each of the following utility functions, derive MUX; MUY , and the MRS. Show your

steps.

(a) U(X; Y ) = X2Y

(b) U(X; Y ) = X(1/3) Y(2/3)

(c) U(X; Y ) = 4rootXY

(d) U(X; Y ) = XY + 3X + 2Y

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. Claw’s 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. Claw’s 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.

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