Intermediate Microeconomic Analysis (Econ 173)
Drake University, Fall 2005
William M. Boal
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MIDTERM EXAMINATION #1 VERSION C
“Mathematical Tools”
September 6, 2005
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—75 pts total]
(1) Suppose the derivative of the function
y = f(x) equals 0.5 at a particular value of x
. At that point, the graph of the function is
a. upward-sloping.
b. downward-sloping.
c. vertical.
d. horizontal.
e. cannot be determined from the
information given.
(2) Suppose the derivative of the function
y = f(x) equals 2 at a particular value of x .
If x then increases by 0.5, y will
a. decrease by approximately 1.0.
b. decrease by approximately 0.5
c. increase by approximately 1.0.
d. increase by approximately 1.5.
e. increase by approximately 2.0.
(3) Suppose the derivative of the function
y = f(x) equals -4 at a particular value of x .
If x then increases by 0.2, y will
a. decrease by approximately 4.0.
b. decrease by approximately 3.8.
c. decrease by approximately 0.8.
d. decrease by approximately 0.2.
e. increase by approximately 0.8.
(4) Suppose y = 5/x. Then the derivative of
y with respect to x is given by
a. dy/dx = -5.
b. dy/dx = 5x.
c. dy/dx = 5/x.
d. dy/dx = -5/x.
e. dy/dx = -5/x2.
(5) Suppose y = 2x + 5x2. Then the
derivative of y with respect to x is given
by
a. dy/dx = 5.
b. dy/dx = 2 + 5x.
c. dy/dx = 2 + 10x.
d. dy/dx = 2x + 10x2.
e. dy/dx = x2 + (5/3)x3.
(6) Consider the following functional forms,
where a, b, and c denote constants.
Which form has constant slope (or
derivative)?
a. y = exp (ax).
b. y = ln (ax).
c. y = a + b x.
d. y = a xb.
e. y = a + b x + c x2.
Intermediate Microeconomic Analysis (Econ 173)
Drake University, Fall 2005
(7) If x increases by 5 percent, then ln(x)
increases by about
a. ln(5), or about 1.6 units.
b. 0.05 percent.
c. 0.05 units.
d. 5 percent.
e. 5 units.
The next question refers to the following
graph of y = f(x) .
y=f(x)
x
(8) 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. more than three points on the graph.
(9) 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. d2f/dx2 = 0, if x = x*.
b. df/dx = 0, if x = x*.
c. f(x*) = 0.
d. x* = 0.
e. All of the above.
Midterm Examination #1 Version C
Page 2 of 6
(10) Suppose the elasticity of the function
y = f(x) equals 0.5 at a particular value of
x. If x increases by 3 percent, then y will
a. remain constant.
b. increase by about 0.5 percent.
c. increase by about 1.5 percent.
d. increase by about 3 percent.
e. increase by about 3.5 percent.
(11) Suppose the elasticity of the function
y = f(x) equals -2 at a particular value of x.
If x increases by 3 percent, then y will
a. decrease by about 6 percent.
b. decrease by about 3 percent.
c. decrease by about 2 percent.
d. decrease by about 1 percent.
e. increase by about 2 percent.
(12) Consider the following functional
forms, where a, b, and c denote
constants. Which form has constant
elasticity?
a. y = exp (ax).
b. y = ln (ax).
c. y = a + b x.
d. y = a xb.
e. y = a + b x + c x2.
The next two questions refer to the
following graph of a level curve, or contour,
of the function y = f(x1,x2) .
x1
x2
Intermediate Microeconomic Analysis (Econ 173)
Drake University, Fall 2005
(13) By definition, all points along the curve
in this graph have identical values of
a. y .
b. the marginal rate of substitution
c. x1 .
d. x2 .
e. all of the above.
(14) According to this graph, if x2
decreases 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.
(15) Consider the function y=f(x1,x2) .
Suppose at a particular point, y/x1=3, and
y/x2=1. If x1 increases by 0.3 and
simultaneously x2 increases by 0.2 , then y
will
a. remain constant.
b. increase by about 0.5 .
c. increase by about 1.1 .
d. increase by about 4.0 .
e. increase by about 4.5 .
(16) Consider the function y = f(x1,x2) .
Suppose at a particular point, y/x1=4, and
y/x2=2. If x1 increases by 1, then y will
remain constant only if x2
a. decreases by about 0.5 .
b. decreases by about 1 .
c. decreases by about 2 .
d. decreases by about 3 .
e. decreases by about 4 .
Midterm Examination #1 Version C
Page 3 of 6
(17) Suppose y = (x1-2)3 (x2-5)2 . Then the
partial derivative of y with respect to x2 is
given by the formula
a. y/x2 = 2(x1-2)3 (x2-5) .
b. y/x2 = (x1-2)3 (x2-5)2 .
c. y/x2 = 6(x1-2)2 (x2-5).
d. y/x2 = 3(x1-2)2 (x2-5)2.
e. y/x2 = 2(x2-5).
(18) Consider the following functional forms
for y = f(x1,x2), where a, b, c, d, and e
denote constants. Which form has constant
partial derivatives (y/x1 and y/x2)?
a. y = a (x1-b)c (x2-d)e .
b. y = a + b x1-1 + c x2-1 .
c. y = a + b x1 + c x2 .
d. y = ax1 + bx2 + c (x1x2)1/2 .
e. y = a x1b x2c .
(19) Revenue equals price times quantity
sold. If price increases by 6 percent and the
quantity sold decreases by 3 percent, then
revenue
a. increases by about 1 percent.
b. increases by about 2 percent.
c. increases by about 3 percent.
d. increases by about 5 percent.
e. increases by about 6 percent.
(20) Average cost (sometimes called "unit
cost") equals total cost divided by the
number of units produced. If total cost
increases by 3 percent and the number of
units produced increases by 4 percent, then
average cost
a. decreases by about 0.75 percent.
b. decreases by about 1 percent.
c. increases by about 3 percent.
d. decreases by about 4 percent.
e. increases by about 7 percent.
Intermediate Microeconomic Analysis (Econ 173)
Drake University, Fall 2005
Midterm Examination #1 Version C
Page 4 of 6
(21) Consider the function y = f(x1,x2) .
Suppose at a particular point, the partial
elasticity of this function with respect to x1
equals 1.5, and the partial elasticity with
respect to x2 equals 0.5. If x1 increases by
2 percent and simultaneously x2 increases
by 1 percent , then y will
a. increase by about 0.5 percent.
b. increase by about 1.5 percent.
c. increase by about 2.0 percent.
d. increase by about 3.5 percent.
e. increase by about 6.0 percent.
(24) Consider the function y = f(x1,x2) .
Suppose at a particular point, 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 0.5. If x1 increases by 3 units and
we wish to keep y constant, then x2 must
a. decrease by about 2/3 unit.
b. decrease by about 1.5 units.
c. decrease by about 2 units.
d. decrease by about 3 units.
e. decrease by about 6 units.
(22) Consider the following functional
forms, where a, b, c, and d denote
constants. Which form has constant partial
elasticities?
a. y = a (x1-b)c (x2-d)e .
b. y = a + b x1-1 + c x2-1 .
c. y = a + b x1 + c x2 .
d. y = ax1 + bx2 + c (x1x2)1/2 .
e. y = a x1b x2c .
The next question refers to the following
graph of a level curve, or contour, of the
function y = f(x1,x2) .
(23) Consider the function y = f(x1,x2) .
Suppose at a particular point, the y/x1=3,
and y/x2=6. Then 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
a. 0.5 .
b. 2.0 .
c. 3.0 .
d. 6.0 .
e. 9.0 .
x1
x2
(25) Along this level curve, as we move
down and to the right, 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) is
a. diminishing.
b. constant and equal to zero.
c. constant and equal to one.
d. increasing.
e. infinite.
II. PROBLEMS: Please write your answers in the boxes on this question sheet. Show your
work and circle your final answers.
Intermediate Microeconomic Analysis (Econ 173)
Drake University, Fall 2005
Midterm Examination #1 Version C
Page 5 of 6
(1) [12 pts] Consider the function y = f(x) = x2 + 6x + 8 . 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.
c. Compute the maximum value y* = f(x*), subject to the restriction that x cannot be
negative.
(2) [8 pts] Consider the following three functions.
(i)
y = 1/x1  1/x2 .
(ii)
y = [(1/x1) + (1/x2)]-1 .
(iii) y = x1 + x2 .
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 (that 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 2005
Midterm Examination #1 Version C
Page 6 of 6
III. CRITICAL THINKING: Answer just one of the questions below (your choice). Full
credit requires good grammar, accurate spelling, and correct reasoning. [5 pts]
(1) Suppose y is strictly proportional to x. What must be the elasticity of y with respect to x?
Explain your reasoning.
(2) Suppose the partial derivatives of y = f(x1,x2) are both positive. Do the level curves of y
slope up or down? Explain your reasoning.
Which question are you answering? _______. Please write your answer below.
[end of exam]