# 辅导案例-ECON 2101

THE AUSTRALIAN NATIONAL UNIVERSITY
Semester Two, 2019: Final Examination – 1 November 2019
Microeconomics 2
ECON 2101
Writing Time: Three Hours.
Permitted Materials: Non-programmable Calculator.
Page 1 of 6 – Microeconomics 2 (ECON 2101)
Instructions
1. The exam consists of four questions, each of which is worth twenty-five marks. As
such, there are a total of one-hundred marks available on this exam.
2. The division of marks between parts of a question is indicated within each question
itself.
3. Please attempt as many questions as you can in the allotted time for this exam.
4. Answer each question in the script book (or books) provided.
5. Please start each question on a new page of the script book.
6. A copy of the “Some useful formulae” document that was provided during this
course can be found at the end of this exam.
7. Good luck!
Page 2 of 6 – Microeconomics 2 (ECON 2101)
Question 1 (25 marks)
1. Ted describes his preferences over combinations of hamburgers and glasses of scotch
as follows. “I like both hamburgers and scotch up to a point, but after that point
further consumption of either of them makes me feel increasingly ill.” Draw a family
of representative indi↵erence curves that illustrate Ted’s preferences. Indicate the
direction(s) of increasing satisfaction for Ted in your diagram. Are Ted’s prefer-
ences locally non-satiated? You should include a brief explanation that provides a
2. “If a consumer doesn’t consume any snails but does consume Big Macs, then his
marginal rate of substitution between snails and Big Macs when his snail consump-
tion is zero must be equal to the ratio of the price of snails to the price of Big Macs.”
Is this claim true, false or ambiguous? Justify your answer. (5 marks.)
3. Is the following claim true, false, or ambiguous? Justify your answer.
“A consumer has the utility function U (x1, x2) = x1 + 2
p
x2

. The price of com-
modity one is \$2 and the price of commodity two is \$1. The consumer’s income is
\$20. If the price of commodity two rises to \$2, then entire change in demand for
commodity two is due to the substitution e↵ect.” (7 marks.)
4. Describe and illustrate the derivation of the optimal choice of health status, health
care, and the composite “all other consumption” commodity in the Wagsta↵ model
of the demand for health care.1 Analyse the impact of an increase in the price of
health care on the optimal choice of health status, health care, and the composite
“all other consumption” commodity in the Wagsta↵ model. (8 marks.)
1Wagsta↵, A (1986), “The demand for health: Theory and applications”, The Journal of Epidemiology
and Community Health 40(1), March, pp. 1–11.
Page 3 of 6 – Microeconomics 2 (ECON 2101)
Question 2 (25 marks)
1. Herbie’s utility function is
U (x1, x2) = x1 + 8x2 x
2
2
2
,
where x1 is the amount of co↵ee he consumes per week and x2 is the amount of tea
he consumes per week. Herbie has \$200 a week to spend. The price of co↵ee is \$1
per cup. The price of tea is currently \$4 per cup. Herbie has received an invitation
to join a club devoted to the consumption of tea. If he joins the club, Herbie can
get a discount on the purchase of tea. If he belonged to the club, he could buy tea
for \$1 per cup. What is the largest membership fee that Herbie would be willing to
pay to join this club? (8 marks.)
2. Bernice’s preferences can be represented by the utility function
U (x1, x2) = min (x1, x2). The price of commodity one is one dollar, the price of
commodity two is two dollars, and her income is twelve dollars. Suppose that an
economic shock results in the price of commodity one rising to three dollars and the
price of commodity two falling to one dollar. What is the the compensating varia-
tion measure of the impact of this shock on her welfare? What is the the equivalent
variation measure of the impact of this shock on her welfare? Illustrate your answers
using diagrams involving indi↵erence curves and budget lines. (8 marks.)
3. Rosalie is a von Neumann-Morgenstern expected utility maximiser whose initial
wealth is \$1, 000. She is o↵ered the chance to invest all of this wealth in a project
which has a fifty percent chance of making \$200 and a fifty percent chance of losing
\$100. The utility that she derives from various wealth levels is shown in the table
below. Will Rosalie invest in the project? What is the risk premium that Rosalie

Wealth (\$) Bernoulli-Cramer Utility
900 200
950 210
1,000 214
1,010 214.5
1,100 218.5
1,200 220

Page 4 of 6 – Microeconomics 2 (ECON 2101)
Question 3 (25 marks)
Consider a two-person (A and B), two-commodity (1 and 2) pure exchange economy.
Suppose that individual A’s preferences can be represented by the utility function
UA

qA1 , q
A
2

=

qA1
1
3

qA2
2
3 ,
where qA1 is individual A’s consumption of commodity one and q
A
2 is individual A’s con-
sumption of commodity two. Similarly, suppose that individual B’s preferences can be
represented by the utility function
UB

qB1 , q
B
2

=

qB1
1
3

qB2
2
3 ,
where qB1 is individual B’s consumption of commodity one and q
B
2 is individual B’s con-
sumption of commodity two. Individual A is endowed with one unit of commodity one and
two units of commodity two, while individual B is endowed with two units of commodity
one and one unit of commodity two.
1. Find the set of Pareto ecient allocations for this economy. (5 marks.)
2. Find the set of individually rational allocations for this economy. (5 marks.)
3. Find the contract curve for this economy. (You should use the definition of the
contract curve that was specified in class rather than the one that is used in the
textbook for this course.) (5 marks.)
4. Assuming that both consumers are price takers, find the competitive equilibrium for
this economy. Be sure to specify the equilibrium consumption of each commodity
by each player and the equilibrium price ratio (p = p1p2 ). (5 marks.)
5. Illustrate this economy and your answers to the earlier parts of this question using
an Edgeworth-Bowley box diagram. (5 marks.)
Page 5 of 6 – Microeconomics 2 (ECON 2101)
Question 4 (25 marks)
1. A firm that produces a single homogeneous product has two factories. One factory
has the cost function C1 (y1) = 2y21+90 and the other has the cost function C2 (y2) =
6y22 + 40. If the firm wishes to produce a total of thirty-two units of its product as
cheaply as possible, how many units will be produced in the second factory? Justify
2. Suppose that the short-run marginal cost of producing a good is \$40 for the first
200 units and \$50 for each additional unit beyond 200. How much output should a
profit-maximising firm in this situation produce in the short-run if the market price
3. “A public good will always be under-provided in a free market.” Is this claim true,
4. “If there are no transaction costs, then an ecient outcome can be achieved in the
presence of an externality if the government completely allocates tradable property
rights over the externality generating activity. Furthermore, the final outcome will
be invariant to the specific initial allocation of property rights that is chosen by
the government.” Is this claim true, false or ambiguous? Justify your answer.
(8 marks.)
——— End of Examination ———
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Page 6 of 6 – Microeconomics 2 (ECON 2101)
The Australian National University
ECON2021: Microeconomics 2
Semester Two, 2019
Some Useful Formulae
Dr Damien S. Eldridge
26 August 2019
Partial Derivatives of Multivariate Functions are a Special Case of
Derivatives of Univariate Functions
Consider the multivariate real-valued function y = f(x1, x2). Fix the value
taken by x2 at x2 = k. The multivariate function f can now be viewed as a
univariate function g of the form y = g(x1) = f(x1; k). The partial derivative
of the multivariate function f with respect to the variable x1 is simply the
derivative of the univariate function g with respect to x1. In other words,
@y
@x1
=
@f(x1, x2)
@x1
=
df(x1; k)
dx1
=
dg(x1)
dx1
= g0(x1).
Useful Rules for Di↵erentiation
If appropriate regularity conditions are satisfied, then each of the following
di↵erentiation rules is valid.
• The product rule: If f(x) = u(x)v(x), then
df(x)
dx
=

du(x)
dx

v(x) +

dv(x)
dx

u(x) = u0(x)v(x) + v0(x)u(x).
• The quotient rule: If f(x) = u(x)v(x) (which requires that v(x) 6= 0),
then
df(x)
dx
=

du(x)
dx

v(x)

dv(x)
dx

u(x)
(v(x))2
=
u0(x)v(x) v0(x)u(x)
(v(x))2
.
1
• The chain rule: If f(x) = g (h(x)) and y = h(x), then
df(x)
dx
=

dg (h(x))
dh(x)
◆✓
dh(x)
dx

=

dg(y)
dy
◆✓
dh(x)
dx

,
evaluated at y = h(x). (In other word, the only variable in the final
derivative that you obtain should be x. The variable y should not
appear in that derivative.)
• The inverse function rule: Suppose that the inverse function of f
is g. This means that, if y = f(x), then x = g(y). If appropriate
regularity conditions are satisfied, then df(x)dx =
1
( dg(y)dy )
when dg(y)dy is
evaluated at the point y = f(x).
Some Useful Derivatives
If appropriate regularity conditions are satisfied, then each of the following
derivative results is valid.
• Derivative of a constant function: If b is a constant real number
and f(x) = b, then df(x)dx = 0.
• Derivative of a linear function: If a is a constant real number and
f(x) = ax, then df(x)dx = a.
• Derivative of a sum: If f(x) = g(x)+h(x), then df(x)dx = dg(x)dx + dh(x)dx .
• Derivative of an ane function: If a and b are constant real num-
bers and f(x) = ax+ b, then df(x)dx = a.
• Derivative of a power function 1: If c 6= 0 is a constant real number
and f(x) = xc, then df(x)dx = cx
c1.
• Derivative of a multiple of a function: If a 6= 0 is a constant real
number and f(x) = ag(x), then df(x)dx = a
dg(x)
dx .
• Derivative of a power function 2: If a and c 6= 0 are constant real
numbers and f(x) = axc, then df(x)dx = acx
c1.
• Derivative of a polynomial function: Suppose that n 2 {1, 2, 3, · · · }.
If a0, a1, · · · , an are all constant real numbers and f(x) = anxn +
2
an1xn1 + · · ·+ a1x+ a0, then
df(x)
dx =
danxn
dx +
danxn
dx + · · ·+ da1xdx + da0dx
= annxn1 + an1(n 1)xn2 + · · ·+ a1(1)x0 + 0
= annxn1 + an1(n 1)xn2 + · · ·+ a1,
assuming that x 6= 0.
• Derivative of an exponential function 1: If e is the real number
that is known as Euler’s constant and f(x) = ex, then df(x)dx = e
x.
• Derivative of an exponential function 2: If a > 0 is a constant
real number such that a 6= 1, and f(x) = ax, then df(x)dx = ax ln (a).
• Derivative of an exponential function 3: If e is the real num-
ber that is known as Euler’s constant and f(x) = eg(x), then df(x)dx =
eg(x)

dg(x)
dx

.
• Derivative of an exponential function 4: If a > 0 is a con-
stant real number such that a 6= 1, and f(x) = ag(x), then df(x)dx =
ag(x) ln (a)

dg(x)
dx

.
• Derivative of a logarithmic function 1: If e is the real number
that is known as Euler’s constant and f(x) = loge(x) = ln(x), then
df(x)
dx =
1
x .
• Derivative of a logarithmic function 2: If a > 0 is a constant real
number such that a 6= 1, and f(x) = loga (x), then df(x)dx = 1x ln(a) .
• Derivative of a logarithmic function 3: If e is the real number
that is known as Euler’s constant and f(x) = loge(g(x)) = ln(g(x)),
then df(x)dx =

1
g(x)
⌘⇣
dg(x)
dx

= g
0(x)
g(x) .
• Derivative of a logarithmic function 4: If a > 0 is a constant real
number such that a 6= 1, and f(x) = loga (g(x)), then
df(x)
dx
=

1
g(x) ln (a)
◆✓
dg(x)
dx

=
g0(x)
g(x) ln (a)
if g(x) 6= 0.
3
Partial Derivatives of Some Common Functional Forms for Utility
Functions and Production Functions
• Perfect Substitutes: If f(x1, x2) = x1+x2, then @f@x1 = 1 and @f@x2 = 1.
• Generalised Perfect Substitutes: Let a > 0 and b > 0 be constant
real numbers. If f(x1, x2) = ax1 + bx2, then
@f
@x1
= a and @f@x2 = b.
• Cobb-Douglas: Suppose that both x1 > 0 and x2 > 0. If f(x1, x2) =
Axb1x
c
2, then
@f
@x1
= Abxb11 x
c
2 and
@f
@x2
= Acxb1x
c1
2 .
• Stone-Geary: Let 1 and 2 be constant real numbers. Suppose that
both x1 > 1 and x2 > 2. If f(x1, x2) = A (x1 1)b (x2 2)c, then
@f
@x1
= Ab (x1 1)b1 (x2 2)c and @f@x2 = Ac (x1 1)
b (x2 2)c1.
• Quasi-Linear Example 1: Suppose that x2 > 0. If f(x1, x2) =
x1 +
p
x2, then
@f
@x1
= 1 and @f@x2 =
1
2
p
x2
.
• Quasi-Linear Example 2: Suppose that x2 > 0. If f(x1, x2) =
x1 + ln (x2), then
@f
@x1
= 1 and @f@x2 =
1
x2
.
• Constant Elasticity of Substitution Example 1: Suppose that
both x1 > 0 and x2 > 0. Let 1 < ⇢ <1 be a constant real number.
If
f(x1, x2) = (x

1 + x

2)
1
⇢ ,
then
@f
@x1
=

1

(x⇢1 + x

2)
1
⇢1 ⇢x⇢11 = x⇢11 (x⇢1 + x⇢2) 1⇢⇢
and
@f
@x2
=

1

(x⇢1 + x

2)
1
⇢1 ⇢x⇢12 = x⇢12 (x⇢1 + x⇢2) 1⇢⇢ .
• Constant Elasticity of Substitution Example 2: Suppose that
both x1 > 0 and x2 > 0. Let 1 < ⇢ < 1, a > 0, and b > 0 be
constant real numbers. If
f(x1, x2) = (ax

1 + bx

2)
1
⇢ ,
then
@f
@x1
=

1

(ax⇢1 + bx

2)
1
⇢1 ⇢ax⇢11 = ax⇢11 (ax⇢1 + bx⇢2) 1⇢⇢
and
@f
@x2
=

1

(ax⇢1 + bx

2)
1
⇢1 ⇢bx⇢12 = bx⇢12 (ax⇢1 + bx⇢2) 1⇢⇢ .
4
Point Elasticities
If y = f(x) is a real-valued univariate function that satisfies appropriate
regularity conditions, then the point elasticity of y with respect to x is given
by
"yx =
% change in y
% change in x
= % change in f(x)% change in x
=

x
f(x)
⌘⇣
df(x)
dx

= d ln(f(x))d ln(x) .
Reciprocals and Surds as Power Functions
• The reciprocal function f(x) = 1x can be expressed in power function
form as f(x) = x1.
• The reciprocal power function f(x) = 1xa , where a > 0 is a strictly
positive constant real number, can be expressed in power function form
as f(x) = xa.
• The second degree surd function is the square-root function (f(x) =p
x). It can be written in power form as f(x) = x
1
2 .
• The third degree surd function is the cube-root function (f(x) = 3px).
It can be written in power form as f(x) = x
1
3 .
• The nth degree surd function f(x) = npx, where n 2 {2, 3, 4, · · · }, can
be written in power function form as f(x) = x
1
n .
A Brief Comment on Terminology
In this document, a function has been very loosely (and perhaps even slop-
pily) described as taking the form “y = f(x)”. Strictly speaking, this is the
equation that describes the graph of the function f(x). All coordinates of
the form (x, y) that satisfy the equation y = f(x) belong to the graph of the
function f(x), with x being the independent variable and y the associated
dependent variable.
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