ECON 134A Midterm: ECON 134A UCSB Test1A 2012Summer

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turquoisegnu128

31 Jan 2019

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Find all values of the constant c that make the function f (x) continuous on ( , ). f (x) = ( c2x2 3x 1, 3c cos(x 1), if x < 1 if x 1: (10 pts. ) Find the linear approximation of the function f (x) = sin (2x) at a = Find the derivative of the function f (x) = 3x 1 using the limit de - nition of the derivative. No credit will be given if limit de nition is not used: (10 pts. ) Find an equation of the tangent line to the curve at the point (1, 1). 4q3x y = x y + 9: (25 pts; 5 pts. each) find the limit, if it exists. If the limit does not exist, explain why. All work must be shown. (a) lim x 2+ (b) lim x . |2 x| x2 7x + 10 sin (5x) x (c) lim x (d) lim x 0.

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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) = x^ay^b (Cobb-Douglas utility function)

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

John M. Keynes: U(x,y) = x + y^b (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 p_x and p_y respectively.

d. Now assume that a=1/2, b = 1/4, I = 600, p_x = 10 and p_y = 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, s_x, is defined as

(p_x 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.

aquamarinescorpion997

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.

limejackal967

Need assistance with these problems!

1. The following is a linear programming formulation of a labor planning problem. There are four overlapping shifts, and management must decide how many employees to schedule to start work on each shift. The objective is to minimize the total number of employees required while the constraints stipulate how many employees are required at each time of day. The variables X₁ - X₄ represent the number of employees starting work on each shift (shift 1 through shift 4).
Minimize X₁ + X₂ + X₃ + X₄
Subject to: X₁ + X₄%u2265 12 (shift 1)
X₁ + X₂%u2265 15 (shift 2)
X₂ + X₃%u2265 16 (shift 3)
X₃ + X₄%u2265 14 (shift 4)
all variables %u2265 0

Find the optimal solution using QM.
How many workers would be assigned to shift 1? (Points : 3) 12
13
0
none of the above