Intermediate Microeconomic Analysis (Econ 173)
Drake University, Fall 2009
William M. Boal
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MIDTERM EXAMINATION #1 VERSION A
“Mathematical Tools”
September 9, 2009
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. Use margins for scratch
work. [3 pts each—39 pts total]
(1) Suppose y = 3x2 + 5x + 2. Then the
derivative of y with respect to x is
a. dy/dx = 3 .
b. dy/dx = 5 .
c. dy/dx = 5x + 2 .
d. dy/dx = 6x + 5 .
e. dy/dx = 3x + 5 .
f. dy/dx = 3x2 + 5x + 2 .
(4) If x increases by 3 percent, then ln(x)
increases by about
a. ln(3), or about 1.1 units.
b. 0.03 percent.
c. 3 percent.
d. 3 units.
e. 0.03 units.
(2) Suppose y = 3(2x+3) –1/2. Then the
derivative of y with respect to x is
a. dy/dx = 3 .
b. dy/dx = 6 .
c. dy/dx = (2x+3)-3/2 .
d. dy/dx = -3 (2x+3) –3/2 .
e. dy/dx = -(3/2) (2x+3) –3/2 .
f. dy/dx = 3(2x+3)-1/2 .
(5) Suppose we wish to maximize the
function y = f(x), which is continuously
differentiable. Assuming there are no
restrictions on the possible values of x, the
maximizing value x* must satisfy
a. d2y/dx2 = 0, if x = x*.
b. dy/dx = 0, if x = x*.
c. f(x*) = 0.
d. x* = 0.
e. All of the above.
(3) Consider the following functional forms.
Which form has constant slope (or
derivative)?
a. y = 5 + 2 x .
b. y = 7 + (3/x) .
c. y = 5 + 6 x + (1/2) x2 .
d. y = 4 x –2 .
e. y = ln (2x) .
f. y = exp (5x) .
Intermediate Microeconomic Analysis (Econ 173)
Drake University, Fall 2009
The next question refers to the following
graph of y = f(x) .
Midterm Examination #1 Version A
Page 2 of 8
The next two questions refer to the
following graph of a level curve, or contour,
of the function y = f(x1,x2) .
y=f(x)
x1
x
x2
(6) 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.
(7) Suppose y and x are strictly positive
variables. If the derivative dy/dx is positive,
then the elasticity of y with respect to x
a. must be positive.
b. must be negative.
c. must be zero.
d. can be positive or negative but not zero.
e. can be positive, negative, or zero.
(8) Consider the following functional forms.
Which form has constant elasticity?
a. y = 5 + 2 x .
b. y = 7 + (3/x) .
c. y = 5 + 6 x + (1/2) x2 .
d. y = 4 x –2 .
e. y = ln (2x) .
f. y = exp (5x) .
(9) By definition, all points along the curve
in this graph have identical values of
a. y .
b. x1 .
c. x2 .
d. both x1, and x2 .
e. the marginal rate of substitution.
f. all of the above.
(10) According to this graph, if x2 increases
and y is to be held constant, then x1 must
a. equal zero.
b. increase.
c. decrease.
d. remain constant.
e. cannot be determined from the
information given.
(11) Suppose y = (x1–2)3 (x2–5)2 . Then the
partial derivative of y with respect to x1 is
given by the formula
a. y/x1 = 3(x1–2)2.
b. y/x1 = 2(x2–5) .
c. y/x1 = 3(x1–2)2 2(x2–5).
d. y/x1 = 3(x1–2)2 (x2–5)2 .
e. y/x1 = (x1–2)3 2(x2–5) .
f. y/x1 = 3(x1–2)2 (x2–5)2
+ (x1–2)3 2(x2–5) .
Intermediate Microeconomic Analysis (Econ 173)
Drake University, Fall 2009
(12) Consider the following functional
forms for y = f(x1,x2). Which form has
constant partial derivatives (y/x1 and
y/x2)?
a. y = 8 + 3 x1 + 5 x2 .
b. y = 4x1 + 5x2 + 3 (x1x2)1/2 .
c. y = 3 x13 x27 .
d. y = 9 (x1–3)2 (x2–5)4 .
e. y = 15 + 3 x1–1 + 7 x2–1 .
f. y = 14 + 5 x11/2 + 4 x21/2 .
Midterm Examination #1 Version A
Page 3 of 8
(13) Consider the following functional
forms for y = f(x1,x2). Which form has
constant partial elasticities (1 and 2) ?
a. y = 8 + 3 x1 + 5 x2 .
b. y = 4x1 + 5x2 + 3 (x1x2)1/2 .
c. y = 3 x13 x27 .
d. y = 9 (x1–3)2 (x2–5)4 .
e. y = 15 + 3 x1–1 + 7 x2–1 .
f. y = 14 + 5 x11/2 + 4 x21/2 .
II. SHORT ANSWER: Please write your answers in the boxes on this question sheet. Use
margins for scratch work.
(1) [4 pts] Suppose the derivative of the function y = f(x) equals –2 at a particular value of x .
Moreover, the elasticity of y with respect to x equals –1.2. Further suppose that x increases
by 0.4.
a. Will y increase or decrease?
b. By about how much?
units
(2) [4 pts] Suppose the derivative of the function y = f(x) equals 5 at a particular value of x .
Moreover, the elasticity of y with respect to x equals 1.5. Further suppose that x increases
by 4 percent.
a. Will y increase or decrease?
b. By about how much?
%
(3) [4 pts] Consider the function y = f(x1,x2) . Suppose at a particular point, y/x1 = 5, and
y/x2 = 6, and that the partial elasticities are 1 = 1 and 2 = 1.5. Further suppose that x1
increases by 0.4 and simultaneously x2 increases by 0.5.
a. Will y increase or decrease?
b. By about how much?
units
Intermediate Microeconomic Analysis (Econ 173)
Drake University, Fall 2009
Midterm Examination #1 Version A
Page 4 of 8
(4) [4 pts] Consider the function y = f(x1,x2) . Suppose at a particular point, y/x1 = 3, and
y/x2 = 1.5, and that the partial elasticities are 1 = 0.5 and 2 = 0.7. Further suppose x2
increases by 2, but suppose we want to keep the value of y constant.
a. Must x1 increase or decrease?
b. By about how much?
units
(5) [4 pts] Revenue equals price times quantity sold. Suppose price decreases by 3 percent and
the quantity sold increases by 5 percent.
a. Will revenue increase or decrease?
b. By about how much?
%
(6) [4 pts] The capital-labor ratio equals the quantity of capital divided by the quantity of labor.
Suppose the quantity of capital increases by 5 percent and the quantity of labor increases by 2
percent.
a. Will the capital-labor ratio increase or decrease?
b. By about how much?
%
(7) [4 pts] Consider the function y = f(x1,x2) , where y denotes output, x1 denotes capital
input, and x2 denotes labor input. Suppose at a particular point, the partial elasticity of output
with respect to capital equals 0.4, and the partial elasticity with respect to labor equals 0.7.
Further suppose capital increases by 4 percent and simultaneously labor increases by 2 percent.
a. Will output increase or decrease?
b. By about how much?
%
(8) [2 pts] Consider the function y = f(x1,x2) . Suppose at a particular point, the y/x1 = 2, and
y/x2 = 7. Then the value of the marginal rate of substitution of x2 for x1 (that is, the |slope|
of the level curve with x1 on the vertical axis and x2 on the horizontal axis) equals
Intermediate Microeconomic Analysis (Econ 173)
Drake University, Fall 2009
Midterm Examination #1 Version A
Page 5 of 8
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) = 2 x2 + 20 x + 2. 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.
Intermediate Microeconomic Analysis (Econ 173)
Drake University, Fall 2009
Midterm Examination #1 Version A
Page 6 of 8
(2) [Marginal rate of substitution: 8 pts] Consider the following three functions.
(i)
(ii)
(iii)
y = 5 x11/2 x21/2 .
y = 4 x13 x23 .
y = 5 ( x11/2 + x21/2 ) .
a. Which two functions have exactly the same formula, in terms of x1 and x2, for the
marginal rate of substitution of x2 for x1 ? (Recall that the MRS of x2 for x1 is the
|slope| of the level curve with x1 on the vertical axis and x2 on the horizontal axis.)
b. What is that formula?
Intermediate Microeconomic Analysis (Econ 173)
Drake University, Fall 2009
Midterm Examination #1 Version A
Page 7 of 8
(3) [Partial elasticities: 8 pts] Suppose y = x13 (x2–2)5 .
a. Find a formula for the partial elasticity of y with respect to x1. Express your answer in
terms of x1 and x2 alone, not y.
b. Find a formula for the partial elasticity of y with respect to x2. Express your answer in
terms of x1 and x2 alone, not y.
Intermediate Microeconomic Analysis (Econ 173)
Drake University, Fall 2009
Midterm Examination #1 Version A
Page 8 of 8
IV. CRITICAL THINKING: Answer just one of the questions below (your choice). [3 pts]
(1) Suppose y = f(x1,x2). Further suppose that the sum of the partial elasticity of y with respect
to x1 and the partial elasticity of y with respect to x2 equals 1. That is, 1 + 2 = 1. If x1 and
x2 both simultaneously increase by 3 percent, does y increase by more than 3 percent, exactly 3
percent, or less than 3 percent? Justify your answer.
(2) Suppose y = f(x1,x2). Further suppose y/x1 is positive, but y/x2 is negative. Do the
level curves of f(x1,x2) slope up or down? Justify your answer.
Circle the question you are answering. Please write your answer below. Full credit requires
good grammar, accurate spelling, and correct reasoning.
[end of exam]