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Midterm Exam 1B for Intermediate Microeconomics Analysis | ECON 173, Exams of Microeconomics

Drake UniversityMicroeconomics
Prof. William M. Boal

Material Type: Exam; Professor: Boal; Class: INTERMED MICROECON ANALYSIS; Subject: Economics; University: Drake University; Term: Fall 2008;

Typology: Exams

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Uploaded on 07/30/2009

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Intermediate Microeconomic Analysis (Econ 173) Signature:
Drake University, Fall 2008
William M. Boal Printed name:
MIDTERM EXAMINATION #1 VERSION B
“Mathematical Tools”
September 9, 2008
INSTRUCTIONS: This exam is closed-book, closed-notes, and calculators are NOT permitted.
Point values for each question are noted in brackets. As usual in this course, “exp(x)” denotes
the exponential function (also written ex ) while “ln(x)” denotes the natural logarithm function
(logarithm to base e).
I. MULTIPLE CHOICE: Circle the one best answer to each question. Feel free to use
margins for scratch work. [3 pts each—36 pts total]
(1) Suppose y = 2x2 – 3x + 3. Then the
derivative of y with respect to x is given
by
a. dy/dx = 3 .
b. dy/dx = 2 .
c. dy/dx = 2x – 3x .
d. dy/dx = 2x2 – 3x + 3 .
e. dy/dx = 4x – 3 .
f. dy/dx = – 3x + 3 .
(2) Suppose y = 3(x–1)2. Then the
derivative of y with respect to x is given
by
a. dy/dx = 3 .
b. dy/dx = 2 .
c. dy/dx = 6(x–1) .
d. dy/dx = 2(x–1) .
e. dy/dx = 3(x–1)2 .
f. dy/dx = 3(x–1) .
(3) Consider the following functional forms,
where a, b, and c denote constants.
Which form has constant slope (or
derivative)?
a. y = ln (ax) .
b. y = exp (ax) .
c. y = a + b x .
d. y = a + (b/x) .
e. y = a + b x + c x2 .
f. y = a xb .
(4) If x increases by 1 percent, then ln(x)
increases by about
a. 0.01 percent.
b. 1 percent.
c. ln(1), or about 0 units.
d. 0.01 units.
e. 1 units.
The next question refers to the following
graph of y = f(x) .
(5) In this graph, the derivative of y with
respect to x (that is, df/dx) equals zero at
a. no point on the graph.
b. one point on the graph.
c. two points on the graph.
d. three points on the graph.
e. four points on the graph.
f. more than four points on the graph.
(6) Suppose we wish to maximize the
function y = f(x), which is continuously
x
y=f(x)
pf3
pf4
pf5

Partial preview of the text

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Intermediate Microeconomic Analysis (Econ 173) Signature: Drake University, Fall 2008 William M. Boal Printed name:

MIDTERM EXAMINATION #1 VERSION B

“Mathematical Tools”

September 9, 2008

INSTRUCTIONS: This exam is closed-book, closed-notes, and calculators are NOT permitted. Point values for each question are noted in brackets. As usual in this course, “ exp(x) ” denotes the exponential function (also written ex^ ) while “ ln(x) ” denotes the natural logarithm function (logarithm to base e ). I. MULTIPLE CHOICE: Circle the one best answer to each question. Feel free to use margins for scratch work. [3 pts each—36 pts total] (1) Suppose y = 2x^2 – 3x + 3. Then the derivative of y with respect to x is given by a. dy/dx = 3. b. dy/dx = 2. c. dy/dx = 2x – 3x (^). d. dy/dx = 2x^2 – 3x + 3. e. dy/dx = 4x – 3. f. dy/dx = – 3x + 3. (2) Suppose y = 3(x–1)^2. Then the derivative of y with respect to x is given by a. dy/dx = 3. b. dy/dx = 2. c. dy/dx = 6(x–1). d. dy/dx = 2(x–1). e. dy/dx = 3(x–1)^2. f. dy/dx = 3(x–1). (3) Consider the following functional forms, where a, b, and c denote constants. Which form has constant slope (or derivative)? a. y = ln (ax). b. y = exp (ax). c. y = a + b x. d. y = a + (b/x). e. y = a + b x + c x^2. f. y = a xb^. (4) If x increases by 1 percent, then ln(x) increases by about a. 0.01 percent. b. 1 percent. c. ln(1), or about 0 units. d. 0.01 units. e. 1 units. The next question refers to the following graph of y = f(x). (5) In this graph, the derivative of y with respect to x (that is, df/dx) equals zero at a. no point on the graph. b. one point on the graph. c. two points on the graph. d. three points on the graph. e. four points on the graph. f. more than four points on the graph. (6) Suppose we wish to maximize the function y = f(x), which is continuously x y=f(x)

Drake University, Fall 2008 Page 2 of 7 differentiable. Assuming there are no restrictions on the possible values of x, the maximizing value x* must satisfy a. f(x) = 0. b. x = 0. c. d^2 f/dx^2 = 0, if x = x. d. df/dx = 0, if x = x. e. All of the above. (7) Consider the following functional forms, where a, b, and c denote constants. Which form has constant elasticity? a. y = ln (ax). b. y = exp (ax). c. y = a + b x. d. y = a + (b/x). e. y = a + b x + c x^2. f. y = a xb. The next two questions refer to the following graph of a level curve, or contour, of the function y = f(x 1 ,x 2 ). (8) By definition, all points along the curve in this graph have identical values of a. x 1. b. x 2. c. y. d. both x 1 , and x 2. e. the marginal rate of substitution f. all of the above. (9) According to this graph, if x 2 increases and y is to be held constant, then x 1 must a. equal zero. b. increase. c. decrease. d. remain constant. e. cannot be determined from the information given. (10) Suppose y = (x 1 +5)^2 (x 2 –4)^3. Then the partial derivative of y with respect to x 2 is given by the formula a. y/x 2 = 2(x 1 +5) (x 2 –4)^3

  • (x 1 +5)^2 3(x 2 –4)^2. b. y/x 2 = 2(x 1 +5) 3(x 2 –4)^2. c. y/x 2 = 2(x 1 +5). d. y/x 2 = 3(x 2 –4)^2. e. y/x 2 = (x 1 +5)^2 3(x 2 –4)^2. f. y/x 2 = 2(x 1 +5) (x 2 –4)^3. (11) Consider the following functional forms for y = f(x 1 ,x 2 ), where a, b, c, d, and e denote constants. Which form has constant partial derivatives (y/x 1 and y/ x 2 )? a. y = ax 1 + bx 2 + c (x 1 x 2 )1/^. b. y = a x 1 b^ x 2 c^. c. y = a (x 1 –b)c^ (x 2 –d)e^. d. y = a + b x 1 –1^ + c x 2 –^. e. y = a + b x 1 1/2^ + c x 2 1/^. f. y = a + b x 1 + c x 2. (12) Consider the following functional forms, where a, b, c, and d denote constants. Which form has constant partial elasticities ( 1 and  2 )? a. y = ax 1 + bx 2 + c (x 1 x 2 )1/^. b. y = a x 1 b^ x 2 c^. c. y = a (x 1 –b)c^ (x 2 –d)e^. d. y = a + b x 1 –1^ + c x 2 –^. e. y = a + b x 1 1/2^ + c x 2 1/^. f. y = a + b x 1 + c x 2. x 2 x 1

Drake University, Fall 2008 Page 4 of 7 (7) [4 pts] Consider the function y = f(x 1 ,x 2 ). Suppose at a particular point, the partial elasticity of this function with respect to x 1 equals 0.5, and the partial elasticity with respect to x 2 equals 1.2. Further suppose x 1 increases by 2 percent and simultaneously x 2 increases by 3 percent. a. Will y increase or decrease?

b. By about how much? %

(8) [2 pts] Consider the function y = f(x 1 ,x 2 ). Suppose at a particular point, the y/x 1 = 5, and y/x 2 = 2. Then the value of the marginal rate of substitution of x 2 for x 1 (that is, the | slope| of the level curve with x 1 on the vertical axis and x 2 on the horizontal axis) equals III. PROBLEMS: Please write your answers in the boxes on this question sheet. Show your work and circle your final answers. (1) [Optimization: 12 pts] Consider the function y = f(x) = x^2  5 x + 7. Assume that x cannot be negative. a. Find a formula (in terms of x ) for the derivative of y with respect to x ( dy/dx ). b. Compute the value x* that maximizes this function, subject to the restriction that x cannot be negative. [Hint: sketch the curve.] c. Compute the maximum value y* = f(x*), subject to the restriction that x cannot be negative.

Drake University, Fall 2008 Page 5 of 7 (2) [Marginal rate of substitution: 8 pts] Consider the following three functions. (i) y = 7 x 1 0.4^ x 2 0.^. (ii) y = 2 x 1 + 3 x 2. (iii) y = x 12 x 23. a. Which two functions have exactly the same formula, in terms of x 1 and x 2 , for the marginal rate of substitution of x 2 for x 1? (Recall that the MRS of x 2 for x 1 is the | slope| of the level curve with x 1 on the vertical axis and x 2 on the horizontal axis.) b. What is that formula?

Drake University, Fall 2008 Page 7 of 7 IV. CRITICAL THINKING: Answer just one of the questions below (your choice). [6 pts] (1) Suppose the partial derivatives of y = f(x 1 ,x 2 ) are both positive. Do the level curves of this function slope up or down? Explain your reasoning. (2) Suppose y is inversely proportional to x. What must be the elasticity of y with respect to x? Explain your reasoning. Which question are you answering? _______. Please write your answer below. Full credit requires good grammar, accurate spelling, and correct reasoning. [end of exam]