The Australian National University
ECON8025: Semester One, 2023
Tutorial 4 Questions
Dr Damien S. Eldridge
22 March 2023
Tutorial Assignment 2
This assignment involves submitting answers for each of the tutorial ques-
tions, but not for the additional practice questions, that are contained on
the tutorial 4 questions sheet (this document). You should submit your
answers on the Turnitin submissions link for Tutorial Assignment 2 that
is available on the Wattle site for this course (under the “Assessments”
block) by no later than 08:00:00 am on Monday 20 March 2023. If you
have trouble accessing the Wattle site for this course or the Turnitin sub-
mission link, please submit your assignment to the course email address
(ECON8025@anu.edu.au). One of the tutorial questions will be selected for
grading and your mark for this tutorial assignment will be based on the
quality and accuracy of your answer to that question. The identity of the
question that is selected for grading will not be revealed to students un-
til some point in time after the due date and time for submission of this
assignment.
A Note on Sources
These questions do not originate with me. They have either been influenced
by, or directly drawn from, other sources.
Key Concepts
Budget-Constrained Expenditure Maximisation, Uncompensated (or Mar-
shallian, or Walrasian, or Ordinary) Demand Functions (or Correspon-
dences), Indirect Utility Functions, Utility-Constrained Expenditure Min-
1
imisation, Compensated (or Hicksian) Demand Functions (or Correspon-
dences), Expenditure Functions, The Lagrangean Approach to Constrained
Optimisation (Maximisation or Minimisation) Problems, Interior Solutions,
Corner Solutions, The First-Order Conditions for am Interior Solution at
which the Budget Constraint Binds, The Kuhn-Tucker First-Order Condi-
tions, Choice Functions or Correspondences (“Arg Max”s and “Arg Min”s),
Value Functions (Maximum Value Functions and Minimum Value Func-
tions).
Tutorial Questions
Tutorial Question 1
Consider an individual whose preferences are defined over bundles of non-
negative amounts of each of two commodities. Suppose that this individ-
ual’s preferences can be represented by a utility function U : R2+ −→ R of
the form U (x1, x2) = ln (x1 + 1) + 2
√
x2, where x1 denotes the individual’s
consumption of commodity one, and x2 denotes the individual’s consump-
tion of commodity two. This individual is a price taker in both commodity
markets. The price of commodity one is p1 > 0, and the price of commodity
two is p2 > 0. This individual is endowed with an income of y > 0.
1. Does this individual have quasi-linear preferences? Justify your an-
swer.
2. Are this individual’s preferences locally non-satiated? Justify your
answer.
3. What is this individual’s budget-constrained utility maximisation
problem?
4. Suppose that the individual will optimally consume strictly positive
amounts of both commodities. What is the individual’s optimal con-
sumption bundle in this case? Under what circumstances, if any, will
this case occur?
5. Can it ever be optimal for this individual to choose to consume zero
units of commodity one? If so, what would be his or her optimal
consumption of commodity two? Under what circumstances, if any,
will this case occur?
6. Can it ever be optimal for this individual to choose to consume zero
units of commodity two? If so, what would be his or her optimal
consumption of commodity one? Under what circumstances, if any,
will this case occur?
2
7. What are the Marshallian demand functions (or possibly correspon-
dences) for commodity one and commodity two for this individual?1
8. What is this individual’s indirect utility function?
Tutorial Question 2
Consider a consumer that faces the following budget-constrained utility
maximisation problem:
Maximise U (q1, q2, q3) = q1 + ln (q2q3)
subject to p1q1 + p2q2 + p3q3 6 y, q1 > 0, q2 > 0, and q3 > 0,
where p1 is the price of commodity one, p2 is the price of commodity two,
p3 is the price of commodity three, and y is the consumer’s income. The
choice variables in this problem are q1, q2, and q3. You may assume that this
consumer will optimally choose to exhaust his or her budget throughout this
question. You may also assume that 0 < p1 < 2p1 < y < ∞, 0 < p2 < ∞,
and 0 < p3 < ∞. Note that these inequality restrictions on the economic
parameters of this problem ensure that it has a unique solution that involves
the consumer choosing to purchase strictly positive amounts of all three
commodities.
1. What conditions characterise an optimal consumption bundle for this
consumer? Justify your answer.
2. What are the consumer’s Marshallian demand functions (or corre-
spondences) for each of the three commodities?2 Justify your answer.
3. What is the consumer’s indirect utility function? Justify your an-
swer?
Tutorial Question 3
Consider a consumer whose preferences over bundles of strictly positive
amounts of each of three distinct commodities can be represented by a
utility function U : R3+ −→ R of the form
U (q1, q2, q3) = q1 +
√
q2q3.
1Marshallian demands are also known as Walrasian demands, ordinary demands, and
uncompensated demands.
2Marshallian demands are also known as Walrasian demands, ordinary demands, and
uncompensated demands.
3
The constant per unit price of commodity one is p1 ∈ (0,∞), the constant
per unit price of commodity two is p2 ∈ (0,∞), and the constant per unit
price of commodity three is p3 ∈ (0,∞). The consumer has an income of
y ∈ (0,∞).
1. What is the consumer’s budget-constrained utility maximisation prob-
lem?
2. What is the Lagrangean function for the consumer’s budget-constrained
utility maximisation problem?
3. (What are the first-order conditions for the consumer’s budget-constrained
utility maximisation problem (assuming that a strictly positive amount
of each commodity will optimally be purchased)?
4. What are the consumer’s Marshallian demands3 for each of the com-
modities (assuming that a strictly positive amount of each commod-
ity is optimally purchased)? In the case of each commodity, identify
whether the Marshallian demand mapping is a function or a cor-
respondence. (You may assume that appropriate second-order con-
ditions for a maximum and constraint qualification conditions are
satisfied.)
5. What is the consumer’s indirect utility function (assuming that a
strictly positive amount of each commodity is optimally purchased)?
Tutorial Question 4
Consider a consumer whose preferences over consumption bundles of non-
negative amounts of each of three distinct commodities can be represented
by a modified Stone-Geary utility function of the form
U (q1, q2, q3) =

(q1 − γ1)α1 (q2 − γ2)α2 (q3 − γ3)1−α1−α2 if (q1, q2, q3) ∈ Q̂,
−∞ if (q1, q2, q3) 6∈ Q̂,
where q1 is the quantity of commodity one, q2 is the quantity of commodity
two, q3 is the quantity of commodity three, and
Q̂ =
{
(q1, q2, q3) ∈ R3+ : q1 > γ1, q2 > γ2, q3 > γ3
}
.
Note that q1, q2, and q3 are choice variables, while γ1, γ2, γ3, α1 and α2
are fixed preference parameters. Typically, we would assume that γ1 > 0,
3Marshallian demands are also known as Walrasian demands, ordinary demands, and
uncompensated demands.
4
γ2 > 0, γ3 > 0, 0 < α1 < 1, 0 < α2 < 1, and α1 +α2 < 1. You should make
these typical assumptions when answering this question.
Suppose that this consumer is a price taker in all markets, faces a price
vector (p1, p2, p3) ∈ R3++, and is endowed with a monetary income of
y > p1γ1 + p2γ2 + p3γ3 > 0. You may assume that the budget constraint
binds and any required constraint qualification conditions and second-order
conditions for a maximum are satisfied by any solution to this consumer’s
budget-constrained utility maximisation problem.
1. What is this consumer’s budget-constrained utility maximisation prob-
lem?
2. What is the Lagrangean function for this consumer’s budget-constrained
utility maximisation problem?
3. What are the first-order conditions for this consumer’s budget-constrained
utility maximisation problem?
4. Find the Marshallian demand functions (or correspondences) for this
consumer.4 Be sure to show all of your working.
5. What is the point income elasticity of demand for each of the three
commodities?5
Additional Practice Questions
Additional Practice Question 1
Suppose that a consumer has perfect complements, or Leontief, preferences
over bundles of non-negative amounts of each of two commodities. The
consumer’s consumption set is R2+. The consumer’s preferences can be
represented by a utility function of the form U(x1, x2) = min(x1, x2).
1. Illustrate the consumer’s weak preference set for an arbitrary (but
fixed) utility level U .
2. Illustrate a representative iso-expenditure line for the consumer.
4Marshallian demands are also known as Walrasian demands, ordinary demands, and
uncompensated demands.
5The point income elasticity of the Marshallian demand for commodity k, when
the vector of prices and income is (p, y) = (p1, p2, · · · , pn, y), is given by εyk (p, y) =
∂ ln(qdk(p,y))
∂ ln(y) =
(
y
qdk(p,y)
)(
∂qdk(p,y)
∂y
)
.
5
3. Illustrate the consumer’s utility-constrained expenditure minimisa-
tion problem.
4. Illustrate the derivation of the consumer’s compensated (or Hicksian)
demand curve for commodity one.
5. Find the algebraic expression for the consumer’s compensated (or
Hicksian) demand functions for commodity one and commodity two?
6. Find an algebraic expression for the consumer’s expenditure function.
Additional Practice Question 2
Suppose that a consumer has perfect substitutes preferences over bundles
of non-negative amounts of each of two commodities. The consumer’s con-
sumption set is R2+. The consumer’s preferences can be represented by a
utility function of the form U(x1, x2) = x1 + x2.
1. Illustrate the consumer’s weak preference set for an arbitrary (but
fixed) utility level U .
2. Illustrate a representative iso-expenditure line for the consumer.
3. Illustrate the consumer’s utility-constrained expenditure minimisa-
tion problem.
4. Illustrate the derivation of the consumer’s compensated (or Hicksian)
demand curve for commodity one.
5. Find the algebraic expression for the consumer’s compensated (or
Hicksian) demand functions for commodity one and commodity two?
6. Find an algebraic expression for the consumer’s expenditure function.
Additional Practice Question 3
Suppose that a consumer has preferences over bundles of non-negative
amounts of each two goods, x1 and x2, that can be represented by a Cobb-
Douglas utility function of the form
U (x1, x2) = x
α
1x
1−α
2 ,
where 0 < α < 1. The consumer is a price taker who faces a price per
unit of good one that is equal to $p1 and a price per unit of good two that
is equal to $p2. Answer each of the following questions. To keep things
relatively simple, focus only on interior solutions in which positive amounts
of both commodities are consumed.
6
1. What is the consumer’s (utility-constrained) expenditure minimisa-
tion problem?
2. What conditions characterise the consumer’s optimal choice of con-
sumption bundle for this problem?
3. What are the consumer’s compensated (Hicksian) demand functions
for good one and good two?
4. Illustrate a representative compensated demand curve for each com-
modity.
5. What is the consumer’s optimal expenditure level, given the minimum-
utility constraint? (In other words, what is the consumer’s expendi-
ture function?)
Additional Practice Question 4
Suppose that a consumer has preferences over bundles of non-negative
amounts of each two goods, x1 and x2, that can be represented by a quasi-
linear utility function of the form
U (x1, x2) = x1 +
√
x2.
The consumer is a price taker who faces a price per unit of good one that is
equal to $p1 and a price per unit of good two that is equal to $p2. Answer
each of the following questions. To keep things relatively simple, focus
only on interior solutions in which positive amounts of both commodities
are consumed.
1. What is the consumer’s (utility-constrained) expenditure minimisa-
tion problem?
2. What conditions characterise the consumer’s optimal choice of con-
sumption bundle for this problem?
3. What are the consumer’s compensated (Hicksian) demand functions
for good one and good two?
4. Illustrate a representative compensated demand curve for each com-
modity.
5. What is the consumer’s optimal expenditure level, given the minimum-
utility constraint? (In other words, what is the consumer’s expendi-
ture function?)
7
Additional Practice Question 5
Suppose that a consumer has preferences over bundles of non-negative
amounts of each two goods, x1 and x2, that can be represented by a con-
stant elasticity of substitution (CES) utility function of the form
U (x1, x2) = (x
ρ
1 + x
ρ
2)
1
ρ .
The consumer is a price taker who faces a price per unit of good one that is
equal to $p1 and a price per unit of good two that is equal to $p2. Answer
each of the following questions. To keep things relatively simple, focus
only on interior solutions in which positive amounts of both commodities
are consumed.
1. What is the consumer’s (utility-constrained) expenditure minimisa-
tion problem?
2. What conditions characterise the consumer’s optimal choice of con-
sumption bundle for this problem?
3. What are the consumer’s compensated (Hicksian) demand functions
for good one and good two?
4. Illustrate a representative compensated demand curve for each com-
modity.
5. What is the consumer’s optimal expenditure level, given the minimum-
utility constraint? (In other words, what is the consumer’s expendi-
ture function?)
Answers for the Tutorial Questions
Answer for Tutorial Question 1
Part 1
In order for preferences over consumption bundles that contain non-negative
amounts of each of two commodities to be quasi-linear in commodity one,
they must be able to represented by a utility function of the form U (x1, x2) =
ax1 + β (x2). Note, however, that this does not require that every utility
function representation of these preferences take this form. Recall that a
utility function representation of ordinal preferences is only unique up to a
strictly increasing transformation. Regardless of the particular admissible
utility function that is used to represent a particular preference ordering,
the marginal-rate-of-substitution of commodity one for commodity two will
8
be the same (for valid utility function representations of that preference or-
dering that are at least once continuously differentiable). This means that,
if preferences are quasi-linear in commodity one, then we must have
MRS1,2 (x1, x2) =
MU1 (x1, x2)
MU1 (x1, x2)
=
a
β′ (x2)
= g (x2) .
In other words, if preferences are quasi-linear in commodity one (and ap-
propriate differentiability assumptions are satisfied), then it must be the
case that the marginal-rate-of-substitution of commodity one for commod-
ity two is only a function of both the consumption of commodity two, and
not of the consumption of commodity one.
In order for preferences over consumption bundles that contain non-negative
amounts of each of two commodities to be quasi-linear in commodity two,
they must be able to represented by a utility function of the form U (x1, x2) =
α (x1) + bx2. Note, however, that this does not require that every utility
function representation of these preferences take this form. Recall that a
utility function representation of ordinal preferences is only unique up to a
strictly increasing transformation. Regardless of the particular admissible
utility function that is used to represent a particular preference ordering,
the marginal-rate-of-substitution of commodity one for commodity two will
be the same (for valid utility function representations of that preference or-
dering that are at least once continuously differentiable). This means that,
if preferences are quasi-linear in commodity two, then we must have
MRS1,2 (x1, x2) =
MU1 (x1, x2)
MU1 (x1, x2)
=
α′ (x1)
b
= h (x1) .
In other words, if preferences are quasi-linear in commodity two (and ap-
propriate differentiability assumptions are satisfied), then it must be the
case that the marginal-rate-of-substitution of commodity one for commod-
ity two is only a function of the consumption of commodity one, and not
of the consumption of commodity two.
We are told that an individual’s preferences can be represented by a utility
function U : R2+ −→ R of the form U (x1, x2) = ln (x1 + 1)+(2)
√
x2, where
x1 denotes the individual’s consumption of commodity one, and x2 denotes
the individual’s consumption of commodity two. Note that
MU1 (x1, x2) =
∂
∂x1
U (x1, x2) =
1
x1 + 1
+ 0 =
1
x1 + 1
,
and
MU2 (x1, x2) =
∂
∂x2
U (x1, x2) = 0 +
(
1
2
)
(2)x
( 12)−1
2 = (1)x
−( 12)
2 =
1√
x2
.
9
This means that
MRS1,2 (x1, x2) =
MU1 (x1, x2)
MU1 (x1, x2)
=
(
1
x1+1
)
(
1√
x2
) = √x2
x1 + 1
.
Note that the marginal-rate-of-substitution of commodity one for com-
modity two for these preferences is a function of both the consumption
of commodity one and the consumption of commodity two. As such, these
preferences cannot be quasi-linear in either commodity one or commodity
two.
Part 2
We are told that an individual’s preferences can be represented by a utility
function U : R2+ −→ R of the form U (x1, x2) = ln (x1 + 1)+(2)
√
x2, where
x1 denotes the individual’s consumption of commodity one, and x2 denotes
the individual’s consumption of commodity two. Note that
MU1 (x1, x2) =
∂
∂x1
U (x1, x2) =
1
x1 + 1
+ 0 =
1
x1 + 1
> 0
for all (x1, x2) ∈ R2+. Note also that
MU2 (x1, x2) =
∂
∂x2
U (x1, x2) = 0 +
(
1
2
)
(2)x
( 12)−1
2 = (1)x
−( 12)
2 =
1√
x2
.
for all (x1, x2) ∈ R+×R++ =
{
(x1, x2) ∈ R2+ : x2 6= 0
}
. (The partial deriva-
tive
∂
∂x2
U (x1, x2) is not defined when x2 = 0.) Finally, note that, for any fixed
x1 ∈ R+, we have
U (x1, ε) = ln (x1 + 1) + (2)
√
ε > ln (x1 + 1) = U (x1, 0)
for all ε > 0. As such, we can conclude that this individual’s preferences
are strongly monotone. Since this individual’s preferences are strongly
monotone, they must be locally non-satiated.
Part 3
The individual’s budget-constrained utility maximisation problem is to
choose x1 > 0 and x2 > 0 to maximise U (x1, x2) = ln (x1 + 1) + (2)
√
x2
subject to the constraint that p1x1 + p2x2 6 y.
10
Part 4
Since this individual’s preferences are locally non-satiated, we know that
he will optimally choose to exhaust his budget. This means that any opti-
mal consumption bundle for this individual that contains strictly positive
amounts of both commodities must satisfy the following two conditions:
MRS1,2 (x1, x2) =
p1
p2
,
and
p1x1 + p2x2 = y.
Note that
MRS1,2 (x1, x2) =
p1
p2
⇐⇒
√
x2
x1 + 1
=
p1
p2
⇐⇒ √x2 =
(
p1
p2
)
(x1 + 1)
⇐⇒ x2 =
(
p1
p2
)2
(x1 + 1)
2 ⇐⇒ x2 =
(
p21
p22
)(
x21 + 2x1 + 1
)
.
Upon substituting this expression into the (binding) budget constraint, we
obtain
p1x1 + p2x2 (x1) = y
⇐⇒ p1x1 + p2
(
p21
p22
)
(x21 + 2x1 + 1) = y
⇐⇒ p1x1 +
(
p21
p2
)
(x21 + 2x1 + 1) = y
⇐⇒ p1x1 +
(
p21
p2
)
x21 + 2
(
p21
p2
)
x1 +
(
p21
p2
)
= y
⇐⇒ p1p2x1 + p21x21 + 2p21x1 + p21 = p2y
⇐⇒ p21x21 + (p1p2 + 2p21)x1 + (p21 − p2y) = 0
⇐⇒ ax21 + bx1 + c = 0,
11
where a = p21, b = (p1p2 + 2p
2
1), and c = (p
2
1 − p2y). Upon applying the
quadratic formula to this quadratic equation, we obtain
x1(1), x1(2) =
−b±√b2−4ac
2a
=
−(p1p2+2p21)±
√
(p1p2+2p21)
2−4p21(p21−p2y)
2p21
=
−p1(p2+2p1)±
√
p21p
2
2+4p
3
1p2+4p
4
1−4p41+4p21p2y
2p21
=
−p1(p2+2p1)±
√
p21p2(p2+4p1+4y)
2p21
=
−p1(p2+2p1)±
√
p21
√
p2(p2+4p1+4y)
2p21
=
−p1(p2+2p1)±(p1)
√
p2(p2+4p1+4y)
2p21
=
−(p2+2p1)±
√
p2(p2+4p1+4y)
2p1
=
−(p2+2p1)+
√
p2(p2+4p1+4y)
2p1
,
−(p2+2p1)−
√
p2(p2+4p1+4y)
2p1
=
−p2−2p1+
√
p2(p2+4p1+4y)
2p1
,
−p2−2p1−
√
p2(p2+4p1+4y)
2p1
.
Since p1 > 0, p2 > 0, and y > 0, we know that there are two distinct real
solutions for this quadratic equation. These solutions are
x1(1) =
−p2 − 2p1 +
√
p2 (p2 + 4p1 + 4y)
2p1
,
and
x1(2) =
−p2 − 2p1 −
√
p2 (p2 + 4p1 + 4y)
2p1
.
Which of these candidate values for x1 is the appropriate one? Clearly
x1(1) > x1(2). This means that either (i) 0 < x1(2) < x1(1), (ii) 0 = x1(2) <
x1(1), (iii) x1(2) < 0 < x1(1), (iv) x1(2) < 0 = x1(1), or (v) x1(2) < x1(1) < 0.
Since we require both x1 and x2 to be strictly positive in this part of this
question, we can restrict our attention to case (i), case (ii), and case (iii).
In case (ii) and case (iii), the only choice of x1 that is strictly positive is
x∗1 = x1(1) =
−p2 − 2p1 +
√
p2 (p2 + 4p1 + 4y)
2p1
.
12
In case (i), we have 0 < x1(2) < x1(1), which means that
0 < x2(2) =
(
p1
p2
)2 (
x1(2) + 1
)2
<
(
p1
p2
)2 (
x1(1) + 1
)2
= x2(2).
Since this individual’s preferences are strongly monotone, the fact that
(0, 0) <<
(
x1(2), x2(2)
)
<<
(
x1(1), x2(1)
)
in case (i) means that
U
(
x1(1), x2(1)
)
> U
(
x1(2), x2(2)
)
in case (i). As such, we know that the appropriate choice for for x1 in case
(i) is
x∗1 = x1(1) =
−p2 − 2p1 +
√
p2 (p2 + 4p1 + 4y)
2p1
.
Thus we can conclude that the optimal choice of x1 when an optimal con-
sumption bundle for this individual must strictly positive amounts of both
commodities is
x∗1 =
−p2 − 2p1 +
√
p2 (p2 + 4p1 + 4y)
2p1
.
This means that the optimal choice of x2 when an optimal consumption
bundle for this individual must strictly positive amounts of both commodi-
ties is
x∗2 =
(
p1
p2
)2
(x∗1 + 1)
2
=
(
p21
p22
)(−p2−2p1+√p2(p2+4p1+4y)
2p1
+ 1
)2
=
(
p21
p22
)(−p2−2p1+√p2(p2+4p1+4y)
2p1
+ 2p1
2p1
)2
=
(
p21
p22
)(−p2+√p2(p2+4p1+4y)
2p1
)2
=
(
p21
p22
)((−p2+√p2(p2+4p1+4y))2
4p21
)
=
(
−p2+
√
p2(p2+4p1+4y)
)2
4p22
=
(
−p2+
√
p2(p2+4p1+4y)
2p2
)2
.
13
Note that both x∗1 > 0 and x
∗
2 > 0 if x
∗
1 > 0. This will be the case if
x∗1 > 0⇐⇒
−p2 − 2p1 +
√
p2 (p2 + 4p1 + 4y)
2p1
> 0
⇐⇒ −p2−2p1+
√
p2 (p2 + 4p1 + 4y) > 0⇐⇒
√
p2 (p2 + 4p1 + 4y) > p2+2p1
⇐⇒ p2 (p2 + 4p1 + 4y) > (p2 + 2p1)2 ⇐⇒ p22+4p1p2+4p2y > p22+4p1p2+4p21
⇐⇒ 4p2y > 4p21 ⇐⇒ y >
p21
p2
⇐⇒ p
2
1
p2
< y.
Thus we can conclude that, if
p21
p2
< y, then the optimal consumption bundle
for this individual is
(x∗1, x
∗
2) =
(
−p2−2p1+
√
p2(p2+4p1+4y)
2p1
,
(
−p2+
√
p2(p2+4p1+4y)
2p2
)2)
.
An alternative approach to answering Part (4), Part (5), and Part (6) of this
question to that employed here would involve the use of the Kuhn-Tucker
first-order conditions for this inequality-constrained non-linear program-
ming problem.
Part 5
Approach 1
Recall that this individual’s preferences are locally non-satiated. This
means that he will optimally exhaust his budget. As such, if x∗1 = 0 is
ever possible, then the optimal consumption bundle in such cases must be
(x∗1, x
∗
2) =
(
0,
y
p2
)
.
Such cases will occur when the non-negativity constraint on x∗1 from the
answer for Part (4) of this question is binding. In other words, it will occur
when
x∗1|Part (4) = x1(1) =
−p2 − 2p1 +
√
p2 (p2 + 4p1 + 4y)
2p1
6 0.
We established in the answer for Part (4) of this question that
x∗1|Part (4) > 0⇐⇒
p21
p2
< y,
14
which means that
x∗1|Part (4) 6 0⇐⇒
p21
p2
> y.
Thus we can conclude that, if
p21
p2
> y, then the optimal consumption bundle
for this individual is
(x∗1, x
∗
2) =
(
0,
y
p2
)
.
Approach 2
Recall that this individual’s preferences are locally non-satiated. This
means that he will optimally exhaust his budget. As such, if x∗1 = 0 is
ever possible, then the optimal consumption bundle in such cases must be
(x∗1, x
∗
2) =
(
0,
y
p2
)
.
Such cases will occur if the indifference curve for this individual that passes
through the point (x∗1, x
∗
2) =
(
0, y
p2
)
does not intersect the budget set for
this individual at any other point. (Recall that the budget set includes
non-negativity constraints on both commodities, as well as the budget
constraint.) In a “budget-line and indifference curves” graph on which
consumption of commodity one appears on the horizontal axis and con-
sumption of commodity two appears on the vertical axis, a sufficient (but
not necessary) condition for this to occur is that the slope of this individ-
ual’s indifference curve that passes through the point (x∗1, x
∗
2) =
(
0, y
p2
)
is
never steeper than the budget line at any point of the form (x1, x2) ∈ R2+
that belongs to that indifference curve.
This individual’s preferences are sufficiently well behaved that a necessary
and sufficient condition for (x∗1, x
∗
2) =
(
0, y
p2
)
to be the optimal consump-
tion bundle is that, at that point, the slope of the indifference curve that
passes through that point is no steeper than the slope of the budget line.
Since both the relevant indifference curve (for this individual, because his
preferences are well behaved) and the budget line are downward sloping,
they will both have negative slopes. The steepness of a negative slope
at any any given point will be determined by the magnitude, or absolute
15
value, of that slope at that point. Note that∣∣∣IC Slope at (x∗1, x∗2) = (0, yp2)∣∣∣ 6 |Budget Line Slope|
⇐⇒
∣∣∣∣−MRS1,2 (x1, x2)|(x1,x2)=(0, yp2 )
∣∣∣∣ 6 ∣∣∣−(p1p2)∣∣∣
⇐⇒ MRS1,2 (x1, x2)|(x1,x2)=(0, yp2 ) 6
p1
p2
⇐⇒
√
x2
x1+1
∣∣∣
(x1,x2)=
(
0, y
p2
) 6 p1
p2
⇐⇒
√
y
p2
0+1
6 p1
p2
⇐⇒
√
y
p2
1
6 p1
p2
⇐⇒
√
y
p2
6 p1
p2
⇐⇒ y
p2
6 p
2
1
p22
⇐⇒ y 6 p21
p2
⇐⇒ p21
p2
> y.
Thus we can conclude that, if
p21
p2
> y, then the optimal consumption bundle
for this individual is
(x∗1, x
∗
2) =
(
0,
y
p2
)
.
Part 6
Recall that this individual’s preferences are locally non-satiated. This
means that he will optimally exhaust his budget. As such, if x∗2 = 0 is
ever possible, then the optimal consumption bundle in such cases must be
(x∗1, x
∗
2) =
(
y
p1
, 0
)
.
Such cases will occur if the indifference curve for this individual that passes
through the point (x∗1, x
∗
2) =
(
y
p1
, 0
)
does not intersect the budget set for
this individual at any other point. (Recall that the budget set includes
16
non-negativity constraints on both commodities, as well as the budget
constraint.) In a “budget-line and indifference curves” graph on which
consumption of commodity one appears on the horizontal axis and con-
sumption of commodity two appears on the vertical axis, a sufficient (but
not necessary) condition for this to occur is that the slope of this individ-
ual’s indifference curve that passes through the point (x∗1, x
∗
2) =
(
y
p1
, 0
)
is
never flatter than the budget line at any point of the form (x1, x2) ∈ R2+
that belongs to that indifference curve.
This individual’s preferences are sufficiently well behaved that a necessary
and sufficient condition for (x∗1, x
∗
2) =
(
y
p1
, 0
)
to be the optimal consump-
tion bundle is that, at that point, the slope of the indifference curve that
passes through that point is no flatter than the slope of the budget line.
Since both the relevant indifference curve (for this individual, because his
preferences are well behaved) and the budget line are downward sloping,
they will both have negative (or, at least, non-positive) slopes. The steep-
ness of a negative slope (or, for that matter, any slope, be it positive, zero,
or negative) at any any given point will be determined by the magnitude,
or absolute value, of that slope at that point. Note that∣∣∣IC Slope at (x∗1, x∗2) = ( yp1 , 0)∣∣∣ > |Budget Line Slope|
⇐⇒
∣∣∣∣−MRS1,2 (x1, x2)|(x1,x2)=( yp1 ,0)
∣∣∣∣ > ∣∣∣−(p1p2)∣∣∣
⇐⇒ MRS1,2 (x1, x2)|(x1,x2)=( yp1 ,0) >
p1
p2
⇐⇒
√
x2
x1+1
∣∣∣
(x1,x2)=
(
y
p1
,0
) > p1
p2
⇐⇒
√
0
y
p1
+1
> p1
p2
⇐⇒ 0y
p1
+1
> p1
p2
⇐⇒ 0 > p1
p2
,
which is impossible, because p1
p2
> 0 (since both p1 > 0 and p2 > 0). Thus
we can conclude that (x∗1, x
∗
2) =
(
y
p1
, 0
)
is never an optimal bundle for this
consumer (when y > 0). Furthermore, we can also conclude that x∗2 6= 0
for this consumer (when y > 0).
17
Part 7
Upon combining the answers for Part (4), Part (5), and Part (6) of this
question, we obtain the ordinary demands for commodity one and com-
modity two for this individual. The ordinary demand for commodity one
for this individual is
x1 (p1, p2, y) =

0 if y 6 p
2
1
p2
,
−p2−2p1+
√
p2(p2+4p1+4y)
2p1
if y >
p21
p2
.
The ordinary demand for commodity two for this individual is
x2 (p1, p2, y) =

y
p2
if y 6 p
2
1
p2
,
(
−p2+
√
p2(p2+4p1+4y)
2p2
)2
if y >
p21
p2
.
18
Part 8
The indirect utility function for this individual is
V (p1, p2, y)
= max {U (x1, x2) : p1x1 + p2x2 6 y, x1 > 0, x2 > 0}
= max
{
ln (x1 + 1) + (2)
√
x2 : p1x1 + p2x2 6 y, x1 > 0, x2 > 0
}
= U (x1 (p1, p2, y) , x2 (p1, p2, y))
= ln (x1 (p1, p2, y) + 1) + (2)
√
x2 (p1, p2, y)
=

ln (0 + 1) + (2)
√
y
p2
if y 6 p
2
1
p2
,
ln
(
−p2−2p1+
√
p2(p2+4p1+4y)
2p1
+ 1
)
+ (2)
√(
−p2+
√
p2(p2+4p1+4y)
2p2
)2
if y >
p21
p2
,
=

ln (1) + (2)
√
y
p2
if y 6 p
2
1
p2
,
ln
(
−p2−2p1+
√
p2(p2+4p1+4y)
2p1
+ 2p1
2p1
)
+ 2
(
−p2+
√
p2(p2+4p1+4y)
2p2
)
if y >
p21
p2
,
=

0 +
√
4y
p2
if y 6 p
2
1
p2
,
ln
(
−p2+
√
p2(p2+4p1+4y)
2p1
)
+
(
−p2+
√
p2(p2+4p1+4y)
p2
)
if y >
p21
p2
,
=

√
4y
p2
if y 6 p
2
1
p2
,
ln
(
−p2+
√
p2(p2+4p1+4y)
2p1
)
+
(
−p2+
√
p2(p2+4p1+4y)
p2
)
if y >
p21
p2
.
19
Answer for Tutorial Question 2
Part (1)
The Lagrangean function for the individual’s budget-constrained utility
maximisation problem is
L (q1, q2, q3, λ) = U (q1, q2, q3) + λ [y − p1q1 − p2q2 − p3q3]
= q1 + ln (q2q3) + λ [y − p1q1 − p2q2 − p3q3] .
The first-order conditions that characterise the individual’s optimal con-
sumption bundle are
∂L
∂q1
= 1− λp1 = 0⇐⇒ λ = 1
p1
,
∂L
∂q2
=
1
q2
− λp2 = 0⇐⇒ λ = 1
p2q2
,
∂L
∂q3
=
1
q3
− λp3 = 0⇐⇒ λ = 1
p3q3
,
and
∂L
∂λ
= y − p1q1 − p2q2 − p3q3 = 0⇐⇒ p1q1 + p2q2 + p3q3 = y.
Upon employing the first of these first-order conditions to eliminate the
Lagrange multiplier λ from the the second and third of these first-order
conditions, can conclude that the individual’s optimal consumption bundle
(q∗1, q
∗
2, q
∗
3) is characterised by the following three equations:
1
p2q2
=
1
p1
,
1
p3q3
=
1
p1
,
and
p1q1 + p2q2 + p3q3 = y.
Part (2)
We can rearrange the first two of the three conditions that characterise the
individual’s optimal consumption bundle (q∗1, q
∗
2, q
∗
3) to obtain
q∗2 =
p1
p2
, and
20
q∗3 =
p1
p3
.
Upon substituting these two expressions into the third of the conditions
that characterise the individual’s optimal consumption bundle (q∗1, q
∗
2, q
∗
3),
we obtain
p1q1 + p2q
∗
2 + p3q
∗
3 = y
⇐⇒ p1q1 + p2
(
p1
p2
)
+ p3
(
p1
p3
)
= y
⇐⇒ p1q1 + p1 + p1 = y
⇐⇒ p1q1 + 2p1 = y
⇐⇒ p1 (q1 + 2) = y
⇐⇒ q1 + 2 = yp1
⇐⇒ q∗1 = yp1 − 2.
Thus we can conclude that the individual’s optimal consumption bundle is
(q∗1, q
∗
2, q
∗
3) =
(
y
p1
− 2, p1
p2
,
p1
p3
)
.
This means that the individual’s Marshallian demand function is
qD (p1, p2, p3, y) =
(
qD1 (p1, p2, p3, y) , q
D
2 (p1, p2, p3, y) , q
D
3 (p1, p2, p3, y)
)
=
(
y
p1
− 2, p1
p2
, p1
p3
)
.
21
Part (3)
The individual’s indirect utility function is
V (p1, p2, p3, y) = max {U (q1, q2, q3) : p1q1 + p2q2 + p3q3 = y}
= max {q1 + ln (q2q3) : p1q1 + p2q2 + p3q3 = y}
= U
(
qD1 (p1, p2, p3, y) , q
D
2 (p1, p2, p3, y) , q
D
3 (p1, p2, p3, y)
)
= qD1 (p1, p2, p3, y) + ln
((
qD2 (p1, p2, p3, y)
) (
qD3 (p1, p2, p3, y)
))
= y
p1
− 2 + ln
((
p1
p2
)(
p1
p3
))
= y
p1
− 2 + ln
(
p21
p2p3
)
= y
p1
− 2 + ln (p21)− ln (p1p3)
= y
p1
− 2 + 2 ln (p1)− (ln (p2) + ln (p2))
= y
p1
− 2 + 2 ln (p1)− ln (p2)− ln (p2) .
Answer for Tutorial Question 3
Part 1
The consumer’s budget-constrained utility maximisation problem is to choose
q1 > 0, q2 > 0, and q3 > 0, to maximise his utility function,
U (q1, q2, q3) = q1 +
√
q2q3,
subject to his budget constraint,
p1q1 + p2q2 + p3q3 6 y.
Part 2
The Lagrangian function for this budget-constrained utility maximisation
problem is
L (q1, q2, q3, λ) = q1 +√q2q3 + λ [y − p1q1 − p2q2 − p3q3]
= q1 +
√
q2
√
q3 + λ [y − p1q1 − p2q2 − p3q3]
= q1 + q
1
2
2 q
1
2
3 + λ [y − p1q1 − p2q2 − p3q3] .
22
Part 3
Assuming that a strictly positive amount of each commodity will optimally
be purchased and that the consumer will optimally spend all of his income,
the first-order conditions for this budget-constrained utility maximisation
problem are
L1 = 1− λp1 = 0⇐⇒ λ = 1
p1
,
L2 =
√
q3
2
√
q2
− λp2 = 0⇐⇒ λ = 1
2p2
√
q3
q2
,
L3 =
√
q2
2
√
q3
− λp3 = 0⇐⇒ λ = 1
2p3
√
q2
q3
,
and
Lλ = y − p1q1 − p2q2 − p3q3 = 0⇐⇒ p1q1 + p2q2 + p3q3 = y.
Part 4
We have
λ = λ
⇐⇒ 1
2p2
√
q3
q2
= 1
2p3
√
q2
q3
⇐⇒ 1
2p2
√
q3√
q2
= 1
2p3
√
q2√
q3
⇐⇒ 2p3q3 = 2p2q2
⇐⇒ p3q3 = p2q2
⇐⇒ q3 =
(
p2
p3
)
q2.
This means that, if this consumer will optimally choose to consume strictly
positive amounts of all three commodities, then he will optimally choose to
spend an identical amount on commodity three as he does on commodity
23
two. Note that
λ = λ
⇐⇒ 1
p1
= 1
2p2
√
q3
q2
⇐⇒
√
q3
q2
= 2p2
p1
⇐⇒ q3
q2
=
4p22
p21
⇐⇒
(
p2
p3
)
q2
q2
=
4p22
p21
⇐⇒
(
p2
p3
)
1
=
4p22
p21
⇐⇒ p2
p3
=
4p22
p21
⇐⇒ p21 = 4p
2
2p3
p2
⇐⇒ p21 = 4p2p3.
This means that the consumer will optimally choose to consume strictly
positive amounts of all three commodities only if p21 = 4p2p3. Note that
we cannot pin down a specific value for either q2 or q3 when the consumer
optimally chooses to consume strictly positive amounts of all three com-
modities. If the consumer chooses to purchase α units of commodity two,
then he will also choose to purchase
(
p2
p3
)
α units of commodity three. We
know, from the budget constraint, that in such circumstances, he will also
choose to purchase
q1 =
y − p2α− p3
(
p2
p3
)
α
p1
=
y − p2α− p2α
p1
=
y − 2p2α
p1
units of commodity one.
Recall that all three commodities are being consumed in strictly positive
quantities here. In order to ensure that both q2 > 0 and q3 > 0, we need
α > 0. In order to ensure that q1 > 0, we need
y − 2p2α
p1
> 0⇐⇒ y − 2p2α > 0⇐⇒ y > 2p2α⇐⇒ α < y
2p2
.
As such, we can conclude that if the consumer optimally choose to consume
strictly positive amounts of all three commodities (which requires that p21 =
24
4p2p3), then his Marshallian demand correspondences for each of the three
commodities are
q1 (p1, p2, p3, y) =
{
γ ∈ R+ : γ = y − 2p2α
p1
, 0 < α <
y
2p2
}
,
q2 (p1, p2, p3, y) =
{
α ∈ R+ : 0 < α < y
2p2
}
,
and
q3 (p1, p2, p3, y) =
{
β ∈ R+ : β =
(
p2
p3
)
α, 0 < α <
y
2p2
}
.
Note that each of these Marshallian demands are set-valued, rather than
single-valued, when the consumer optimally wants to consume strictly pos-
itive amounts of all three commodities. As such, if we restrict attention to
the case in which the consumer optimally wants to consume strictly pos-
itive amounts of all three commodities, we can conclude that all three of
the Marshallian demands are correspondences rather than functions.
Part 5
Assuming that the consumer optimally wants to consume strictly positive
amounts of all three commodities, his indirect utility function is
V (p1, p2, p3, y) = max {U (q1, q2, q3) : p1q1 + p2q2 + p3q3 = y, q1 > 0, q2 > 0, q3 > 0}
= U (q1 (p1, p2, p3, y) , q2 (p1, p2, p3, y) , q3 (p1, p2, p3, y))
= q1 (p1, p2, p3, y) +
√
q2 (p1, p2, p3, y) q3 (p1, p2, p3, y)
= y−2p2α
p1
+
√
α
(
p2
p3
)
α
= y−2p2α
p1
+
√(
p2
p3
)
α2
= y
p1
−
(
2p2
p1
)
α +
(√
p2
p3
)
α
= y
p1
−
(√
4p22
p21
)
α +
(√
p2
p3
)
α
= y
p1
+
(√
p2
p3
−
√
4p22
p21
)
α.
Recall that a necessary condition for the consumer to optimally want to
consume strictly positive amounts of all three commodities is that p21 =
25
4p2p3. Upon substituting this into the above expression for the consumer’s
indirect utility function, we obtain
V (p1, p2, p3, y) =
y
p1
+
(√
p2
p3
− 4p22
p21
)
α
= y
p1
+
(√
p2
p3
−
√
4p22
4p2p3
)
α
= y
p1
+
(√
p2
p3
−
√
p2
p3
)
α
= y
p1
+ (0)α
= y
p1
+ 0
= y
p1
.
Answer for Tutorial Question 4
Please see the handwritten answer that is attached to the end of this doc-
ument. (It is titled “ECON8025 — Semester One, 2021 Final Exam
Question Four Answer Key”.)
26
Answers for the Additional Practice Ques-
tions
Answer for Additional Practice Question 1
U
U
U
U
U
0
0
Q1
Q1
Q2
P1
Q1 = Q2
( P1
0U + P2
0U ) / P1
0 ( P1
1U + P2
0U ) / P1
1 ( P1
2U + P2
0U ) / P1
2
P1
0
P1
1
P1
2
h1
-1 ( P1 ; P2
0 , U )
If both prices are strictly positive, the consumer will always want to
choose the kink-point on the minimum-utility-constraint indifference curve.
Thus the consumer’s Hicksian demands must satisfy the following two equa-
tions: q1 = q2 (Equation 1) and min (q1, q2) = U (Equation 2), where U is
the required minimum utility level. The solution to this system of equa-
tions is q∗1 = q
∗
2 = U . Thus the consumer’s compensated (or Hicksian)
demand functions for the two commodities are
h1 (p1, p2, U) = U
27
and
h2 (p1, p2, U) = U .
The consumer’s expenditure function is
e (p1, p2, U) = p1h1 (p1, p2, U) + p2h2 (p1, p2, U) = p1U + p2U = (p1 + p2)U .
Answer for Additional Practice Question 2
U = ( P1
0 U ) / P1
0
= ( P1
1 U ) / P1
1
= ( P2
0 U ) / P1
1
U
U
0
0
Q1
Q1
Q2
P1
P1
0
P1
1 = P2
0
P1
2
( P2
0 U ) / P1
2
h1
-1 ( P1 ; P2
0 , U )
Both the minimum-utility-constraint indifference curve and the family
of iso-expenditure curves are straight lines in the case of perfect substitutes.
If the iso-expenditure lines are steeper than the minimum utility indiffer-
ence curve (that is, if p1
p2
> 1 ⇐⇒ p1 > p2), then the consumer would
optimally choose the corner solution in which only commodity two (the
cheaper commodity) is purchased. If the iso-expenditure lines are flatter
28
than the minimum utility indifference curve (that is, if p1
p2
< 1⇐⇒ p1 < p2),
then the consumer would optimally choose the corner solution in which
only commodity one (the cheaper commodity) is purchased. In order to
reach the minimum required utility level, the consumer will need to pur-
chase U units of commodity one in such a case. If the iso-expenditure
lines have the same slope as the minimum utility indifference curve (that
is, if p1
p2
= 1 ⇐⇒ p1 = p2), then the consumer will be indifferent between
purchasing any combination of the two commodities that sums to the min-
imum required utility level. There is a family of potential solutions in such
a case. This is the reason that the compensated (or Hicksian) demands for
two commodities are correspondences, rather than functions, when the con-
sumer views the two commodities as perfect substitutes. The consumer’s
compensated (or Hicksian) demand correspondences for the two commodi-
ties are
(h1 (p1, p2, U) , h2 (p1, p2, U))

= (U, 0) if p1 < p2,
∈ {(γ, U − γ) : γ ∈ [0, U ]} if p1 = p2,
= (0, U) if p1 > p2.
The consumer’s expenditure function is
e (p1, p2, U) = p1h1 (p1, p2, U) + p2h2 (p1, p2, U)
=

p1U if p1 < p2,
p1γ + p2 (U − γ) if p1 = p2,
p2U if p1 > p2,
=

p1U if p1 6 p2,
p2U if p1 > p2,
= min (p1, p2)U .
Answer for Additional Practice Question 3
The consumer’s expenditure minimisation problem is
min
{
p1q1 + p2q2 : q
α
1 q
1−α
2 = U
}
.
The two equations (first-order conditions) that characterise the optimal
consumption bundle for this problem are
MRS1,2 =
p1
p2
29
and
qα1 q
1−α
2 = U .
The marginal rate of substitution for this version of the Cobb-Douglas
utility function is
MRS1,2 =
αq2
(1− α) q1 .
The first of the two first-order conditions can be rewritten as
q2 =
(
1− α
α
)(
p1
p2
)
q1.
Upon substituting this expression for q2 into the second of the first-order
conditions, we obtain
qα1
[(
1− α
α
)(
p1
p2
)
q1
]1−α
= U
⇐⇒ qα1
(
1− α
α
)1−α(
p1
p2
)1−α
q1−α1 = U
⇐⇒ q1
(
1− α
α
)1−α(
p1
p2
)1−α
= U
⇐⇒ q∗1 =
U(
1−α
α
)1−α (p1
p2
)1−α
⇐⇒ q∗1 =
(
α
1− α
)1−α(
p2
p1
)1−α
U .
Upon substituting this expression for q1 into the earlier expression for q2,
we obtain
q∗2 =
(
1− α
α
)(
p1
p2
)
q∗1
⇐⇒ q∗2 =
(
1− α
α
)(
p1
p2
)(
α
1− α
)1−α(
p2
p1
)1−α
U
⇐⇒ q∗2 =
(
1− α
α
)(
p1
p2
)(
1− α
α
)α−1(
p1
p2
)α−1
U
⇐⇒ q∗2 =
(
1− α
α
)α(
p1
p2
)α
U .
Thus the compensated (or Hicksian) demands for the two commodities are
h1 (p1, p2, U) =
(
α
1− α
)1−α(
p2
p1
)1−α
U
30
and
h2 (p1, p2, U) =
(
1− α
α
)α(
p1
p2
)α
U .
The consumer’s expenditure function is
e (p1, p2, U) = p1h1 (p1, p2, U) + p2h2 (p1, p2, U)
= p1
(
α
1−α
)1−α (p2
p1
)1−α
U + p2
(
1−α
α
)α (p1
p2
)α
U
=
[
p1
(
α
1−α
)1−α (p2
p1
)1−α
+ p2
(
1−α
α
)α (p1
p2
)α]
U
=
[(
α
1−α
)1−α
pα1p
1−α
2 +
(
1−α
α
)α
pα1p
1−α
2
]
U
=
[(
α
1−α
)1−α
+
(
1−α
α
)α]
pα1p
1−α
2 U
=
[
α1−ααα+(1−α)α(1−α)1−α
αα(1−α)1−α
]
pα1p
1−α
2 U
=
[
α+(1−α)
αα(1−α)1−α
]
pα1p
1−α
2 U
=
[
1
αα(1−α)1−α
]
pα1p
1−α
2 U
= α−α (1− α)α−1 pα1p1−α2 U .
Answer for Additional Practice Question 4
The consumer’s expenditure minimisation problem is
min {p1q1 + p2q2 : q1 +√q2 = U} .
Assuming an interior solution, the two equations (first-order conditions)
that characterise the optimal consumption bundle for this problem are
MRS1,2 =
p1
p2
and
q1 +
√
q2 = U .
The marginal rate of substitution for this version of a quasi-linear utility
function is
MRS1,2 = 2
√
q2.
31
The first of the two first-order conditions can be rewritten as
q∗2 =
p21
4p22
.
Upon substituting this expression for q∗2 into the second of the first-order
conditions, we obtain
q∗1 +
√
q∗2 = U ⇐⇒ q∗1 +
√
p21
4p22
= U ⇐⇒ q∗1 +
p1
2p2
= U ⇐⇒ q∗1 = U −
p1
2p2
.
Thus the compensated (or Hicksian) demands for the two commodities are
h1 (p1, p2, U) = U − p1
2p2
and
h2 (p1, p2, U) =
p21
4p22
.
The consumer’s expenditure function is
e (p1, p2, U) = p1h1 (p1, p2, U) + p2h2 (p1, p2, U)
= p1
(
U − p1
2p2
)
+ p2
(
p21
4p22
)
= p1U − p
2
1
2p2
+
p21
4p2
= p1U − 2p
2
1
4p2
+
p21
4p2
= p1U − p
2
1
4p2
.
Answer for Additional Practice Question 5
The consumer’s expenditure minimisation problem is
min
{
p1q1 + p2q2 : (q
ρ
1 + q
ρ
2)
1
ρ = U
}
.
The two equations (first-order conditions) that characterise the optimal
consumption bundle for this problem are
MRS1,2 =
p1
p2
and
(qρ1 + q
ρ
2)
1
ρ = U .
32
The marginal rate of substitution for this version of the constant-elasticity-
of-substitution utility function is
MRS1,2 =
(
q1
q2
)ρ−1
.
The first of the two first-order conditions can be rewritten as(
q1
q2
)ρ−1
=
p1
p2
⇐⇒ q1
q2
=
(
p1
p2
) 1
ρ−1
⇐⇒ q1 =
(
p1
p2
) 1
ρ−1
q2.
Upon substituting this expression for q1 into the second of the first-order
conditions, we obtain(((
p1
p2
) 1
ρ−1
q2
)ρ
+ qρ2
) 1
ρ
= U ⇐⇒
((
p1
p2
) ρ
ρ−1
qρ2 + q
ρ
2
) 1
ρ
= U
⇐⇒
(
p1
p2
) ρ
ρ−1
qρ2 + q
ρ
2 = U
ρ ⇐⇒
((
p1
p2
) ρ
ρ−1
+ 1
)
qρ2 = U
ρ
⇐⇒
(
p
ρ
ρ−1
1 + p
ρ
ρ−1
2
p
ρ
ρ−1
2
)
qρ2 = U
ρ ⇐⇒ qρ2 =
(
p
ρ
ρ−1
2
p
ρ
ρ−1
1 + p
ρ
ρ−1
2
)
U
⇐⇒ q∗2 =
p
1
ρ−1
2 U(
p
ρ
ρ−1
1 + p
ρ
ρ−1
2
) 1
ρ
.
Upon substituting this expression for q2 into the earlier expression for q1,
we obtain
q∗1 =
(
p1
p2
) 1
ρ−1
q∗2 =
p 11−ρ1
p
1
1−ρ
2
 p 1ρ−12 U(
p
ρ
ρ−1
1 + p
ρ
ρ−1
2
) 1
ρ
=
p
1
ρ−1
1 U(
p
ρ
ρ−1
1 + p
ρ
ρ−1
2
) 1
ρ
.
Thus the compensated (or Hicksian) demands for the two commodities are
h1 (p1, p2, U) =
p
1
ρ−1
1 U(
p
ρ
ρ−1
1 + p
ρ
ρ−1
2
) 1
ρ
and
h2 (p1, p2, U) =
p
1
ρ−1
2 U(
p
ρ
ρ−1
1 + p
ρ
ρ−1
2
) 1
ρ
.
33
The consumer’s expenditure function is
e (p1, p2, U) = p1h1 (p1, p2, U) + p2h2 (p1, p2, U)
= p1
 p 1ρ−11 U(
p
ρ
ρ−1
1 +p
ρ
ρ−1
2
) 1
ρ
+ p2
 p 1ρ−12 U(
p
ρ
ρ−1
1 +p
ρ
ρ−1
2
) 1
ρ

= p
ρ−1
ρ−1
1
 p 1ρ−11 U(
p
ρ
ρ−1
1 +p
ρ
ρ−1
2
) 1
ρ
+ p ρ−1ρ−12
 p 1ρ−12 U(
p
ρ
ρ−1
1 +p
ρ
ρ−1
2
) 1
ρ

=
p
ρ−1+1
ρ−1
1 U(
p
ρ
ρ−1
1 +p
ρ
ρ−1
2
) 1
ρ
+
p
ρ−1+1
ρ−1
2 U(
p
ρ
ρ−1
1 +p
ρ
ρ−1
2
) 1
ρ
=
p
ρ
ρ−1
1 U(
p
ρ
ρ−1
1 +p
ρ
ρ−1
2
) 1
ρ
+
p
ρ
ρ−1
2 U(
p
ρ
ρ−1
1 +p
ρ
ρ−1
2
) 1
ρ
=

(
p
ρ
ρ−1
1 +p
ρ
ρ−1
2
)
(
p
ρ
ρ−1
1 +p
ρ
ρ−1
2
) 1
ρ
U
=
(
p
ρ
ρ−1
1 + p
ρ
ρ−1
2
)1− 1
ρ
U
=
(
p
ρ
ρ−1
1 + p
ρ
ρ−1
2
) ρ
ρ
− 1
ρ
U
=
(
p
ρ
ρ−1
1 + p
ρ
ρ−1
2
) ρ−1
ρ
U .
34