THE AUSTRALIAN NATIONAL UNIVERSITY

Semester Two, 2019: Final Examination – 1 November 2019

Microeconomics 2

ECON 2101

Reading Time: Fifteen Minutes.

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

justification for each of your answers. (5 marks.)

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

associates with the project? Justify your answers. (9 marks.)

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

your answer. (5 marks.)

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

of output is $46? Justify your answer. (5 marks.)

3. “A public good will always be under-provided in a free market.” Is this claim true,

false or ambiguous? Justify your answer. (7 marks.)

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 @[email protected] = 1 and @[email protected] = 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 @[email protected] = 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 @[email protected] = 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 @[email protected] =

1

2

p

x2

.

• Quasi-Linear Example 2: Suppose that x2 > 0. If f(x1, x2) =

x1 + ln (x2), then

@f

@x1

= 1 and @[email protected] =

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.

5