NATIONAL UNIVERSITY OF SINGAPORE
EC2104 Quantitative Methods for Economic Analysis
Semester 2, AY2019-2020
Midterm Examination
March 6, 2020
Time Allowed: 70 minutes
INSTRUCTIONS TO CANDIDATES
1. This examination paper comprises (12) printed pages, including this page.
2. This is a CLOSED book examination.
3. You may use a scientific (non-graphing) calculator.
4. There are fifteen (15) multiple-choice questions.
5. Each correct answer is worth 1 point. No marks will be deducted for a wrong answer.
6. Use Form FAS-10 (the bubble form) to submit your answers. Use Form FAS-10
strictly according to the important instructions contained therein. Use 2B pencils
only.
7. You are required to turn in Form FAS-10 only. Answers from the question booklet
would not be accepted.
Do not turn over until you are told to start writing
Problem 1: Utility refers to the amount of satisfaction derived from consuming goods and
services. Consider the utility function given by:
1 − x2
U (x) = ln
1 + x2
Find the marginal utility of consumption.
2x
1 − x2
2x
(B)
1 − x2
4x
(C) −
1 − x4
4x
(D)
1 − x4
(E) None of the above
(A) −
Problem 2: The Coefficient of Relative Risk Aversion is defined as follows:
γ=−
V 00 (x)
x
V 0 (x)
Find the Coefficient of Relative Risk Aversion for a consumer with the following utility
function:
V (x) =
1
x1−θ
1−θ
1
θ
(B) θ
(A)
(C) 1 − θ
(D) 1 + θ .
(E) None of the above.
2
Problem 3: Given the following utility function, determine the output level (x) that
corresponds to the minimum and maximum utility for the consumer.
W (x) = x3 − 6x2 + 9x + 1,
1
x ∈ [ , 5]
2
1
, Maximum: 5
2
(B) Minimum: 1, Maximum: 5
(A) Minimum:
(C) Minimum: 3, Maximum: 5
(D) Minimum: 5, Maximum: 1
(E) None of the above
Problem 4: Determine the interval that the utility function is convex given the following
marginal utility:
U 0 (x) = U (3 − U ),
x>0
(A) 0 < U < 1.5
(B) 1.5 < U < 3
(C) 3 < U < 6
(D) U > 6
(E) None of the above
3
Problem 5: The marginal propensity to consume is often used to quantify induced consumption. It is given that national income (y) and consumption (x) is related via the following
equation:
x
x = ey
Find an expression for
dy
(i.e the marginal propensity to consume).
dx
(For clarity, note that the right hand side of the equation refers to the exponential function, e raised to the power of xy .)
x−y
x ln x
xy
(B)
x ln x
x
(C)
y
(A)
(D) xy
(E) None of the above.
Problem 6: Given the following function f (x), find the values of p and q for which the
derivative f 0 (1) exists.
f (x) =

 x2 + 3x + p
 qx + 2
if x ≤ 1
if x > 1
(A) p = 3, q = 5
(B) p = 5, q = 3
(C) p = 3, q = 3
(D) p = 5, q = 5
(E) No value of p and q makes it possible for f to be differentiable at x = 1.
4
Problem 7: Determine the value of an asset that is determined from the following integral.
Z 0
x2 ex dx
−∞
(A) 1
(B) 2
(C) 3
(D) 4
(E) None of the above
Problem 8: Given that p is a constant parameter, find the area of the region bounded by
the following 2 curves:
y 2 = 2px
x2 = 2py
(A) p2
2
(B) p2
3
4
(C) p2
3
5 2
(D) p + 2
3
(E) None of the above
5
Problem 9: Let X(t) denote the total value of investments. The growth rate of the investments has the following relationship over time:
Ẋ(t)
= tX(t)
X(t)
Find X(t) as a function of t.
(A) −2t2 − 2
(B) 2t2 + 2
−2
(C)
t
2
t +2
(D)
t
(E) None of the above
Problem 10: The dimensions of a rectangular box are measured to be 15cm, 20cm and
30cm. Each measurement is correct to within 0.1 cm. Use differentials to estimate the largest
possible error (in cm3 ) when the volume of the box is calculated from these measurements.
(A) 100
(B) 125
(C) 135
(D) 625
(E) None of the above
6
Problem 11: Consider the following production function: f (x, y) = ln(x5 y 2 ) + ln(xy 4 ).
Which of the following statements is true.
(A) f (x, y) is homogenous and homothetic.
(B) f (x, y) is neither homogenous nor homothetic.
(C) f (x, y) is homogenous but not homothetic.
(D) f (x, y) is homothetic but not homogenous.
(E) None of the above
Problem 12: Consider the following production function:
z(x1 , x2 , ...xn ) =
n
X
ai ln(xi )
i=1
where a1 , . . . , an are constants. Find the partial elasticities of z with respect to xi
(A) ai
(B)
n
X
ai
i=1
ai
xi
ai
(D)
z
(E) None of the above
(C)
7
Problem 13: Find the value of
and y(t) = 2t.
df
at t = 3 for the function f (x, y) = xy, with x(t) = 2t
dt
(A) 48
(B) 48 ln 2 + 16
(C) 64
(D) 64 ln 2 + 2
(E) None of the above
Problem 14: Consider a country with the following tax rate on wealth w :
t(w) = a +
(bw + c)p
w
Given that a, b, c, p are all constant parameters that are strictly positive, find the level(s) of
wealth w such that t0 (w) = 0
(A) 0
(B) c + b
(C) c(p − b)
c
(D)
p−b
(E) None of the above
8
Problem 15: A consumer has the following utility over goods x1 and x2 :
U (x1 , x2 ) = xa1 x1−a
2 ,
0<a<1
Find the elasticity of substitution of x1 and x2 .
(A) 1
(B) a
(C) 1 − a
(D) 1 + a
(E) None of the above
End of Paper
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