The Australian National University
ECON8025: Semester One, 2023
Tutorial 2 Answers
Dr Damien S. Eldridge
9 March 2023
Tutorial Assignment 1
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 2 questions sheet (this document). You should submit your
answers on the Turnitin submissions link for Tutorial Assignment 1 that is
available on the Wattle site for this course (under the “Assessments Items
1: Tutorial Assignments” block) by no later than 08:00:00 am on Monday 6
March 2023. If you have trouble accessing the Wattle site for this course or
the Turnitin submission 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 until 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
Consumption Set, Preferences, Weak Preference, Strict Preference, Indif-
ference, Preference Properties, Weak Completeness, Reflexivity, Strong
1
Completeness, Transitivity, Rationality, Local Non-Satiation, Monotonic-
ity, Strong Monotonicity, Convexity, Strict Convexity, Regularity Condi-
tions, Utility Functions, Ordinality and Ranking, Non-Uniqueness of Util-
ity Functions, Utility Functions Only Unique Up To a Strictly Increas-
ing Transformation, Marginal Rate of Substitution as the Slope of an In-
difference Curve, Diminishing MRS and Downward Sloping Indifference
Curves that are Bowed-In Towards the Origin, Indifference Curves, Weak-
Preference Sets, Indifference Curve Maps, Commodities that are “Goods”,
Commodities that are “Bads”, Commodities that are “Neutrals”, Com-
modities that are “Goods” up to a point and then become “Bads”, Bliss
Points, Perfect Substitutes, Perfect Complements, Cobb-Douglas Prefer-
ences, Stone-Geary Preferences, Constant-Elasticity-of-Substitution Pref-
erences, Lexicographic Preferences, Preference Properties, Utility Func-
tions, Indifference Curves (Utility Level Sets, Iso-Utility Curves),“At Least
As Good As” (Utility Upper Contour) Sets.
Tutorial Questions
Tutorial Question 1
Suppose that a consumer has preferences defined over the consumption set
R2+ = {(x1, x2) : x1 ∈ [0,∞) , x2 ∈ [0,∞)}. In other words, this consumer
has preferences defined over the set of all bundles (combinations) of non-
negative quantities of each of two commodities. Suppose that these pref-
erences can be represented by a utility function of the form U : R2+ −→ R.
Consider the following example of such a utility function.
Perfect Substitutes: U(x1, x2) = x1 + x2.
Complete the following exercises for this utility function.
1. Find the equation of a representative indifference curve. (The equa-
tion should express x2 as a function of x1 and the fixed utility level
U).
2. Draw the representative indifference curve.
3. Are the preferences rational (that is, are they both strongly complete
and transitive)? Justify your answer.
4. Are the preferences locally non-satiated? What about monotone?
What about strongly monotone? Justify your answers.
5. Are the preferences convex? What about strictly convex? Justify
your answer.
2
Tutorial Question 2
Repeat Tutorial Question 1 for the case in which the utility function is
Leontief (Perfect Complements): U(x1, x2) = min (x1, x2) .
Tutorial Question 3
Repeat Tutorial Question 1 for the case in which the utility function is
Cobb-Douglas Special Case: U(x1, x2) = x
1
2
1 x
1
2
2 .
Tutorial Question 4
Repeat Tutorial Question 1 for the case in which the utility function is
Quasi-Linear Special Case 1: U(x1, x2) = x1 + ln (x2) .
In this particular case, you may assume that the consumption set is
R+ × R++ = {(x1, x2) : x1 ∈ [0,∞) , x2 ∈ (0,∞)} .
Tutorial Question 5
Repeat Tutorial Question 1 for the case in which the utility function is
Quasi-Linear Special Case 2: U(x1, x2) = x1 +
√
x2.
Tutorial Question 6
Repeat Tutorial Question 1 for the case in which the utility function is
Scarf-Shapley-Shubik Special Case:
U(x1, x2) = max (min (x1, 2x2) ,min (2x1, x2)) .
Tutorial Question 7
Repeat Tutorial Question 1 for the case in which the utility function is
Bliss Point Special Case: U(x1, x2) = −
√
(x1 − 5)2 + (x2 − 5)2.
3
Additional Practice Questions
Additional Practice Question 1
Lexicographic Preferences on R2+: Suppose that a consumer has lex-
icographic preferences over bundles of non-negative amounts of each of
two commodities. The consumer’s consumption set is R2+. The consumer
weakly prefers bundle a = (a1, a2) over bundle b = (b1, b2) if either (i)
a1 > b1, or (ii) both a1 = b1 and a2 > b2. In any other circumstance, the
consumer does not weakly prefer bundle a to bundle b. (Note that these
preferences are not continuous. Furthermore, they cannot be represented
by a utility function.)
1. Under what circumstances will bundle c = (c1, c2) be strictly pre-
ferred to bundle d = (d1, d2)?
2. Under what circumstances will bundle c = (c1, c2) be indifferent to
bundle d = (d1, d2)?
3. Are these preferences weakly complete? Explain why.
4. Are these preferences reflexive? Explain why.
5. Are these preferences strongly complete? Explain why.
6. Are these preferences transitive? Explain why.
7. Are these preferences rational? Explain why.
8. Show that these preferences are strongly monotone. (Note that this
means that they are also monotone and locally non-satiated.)
9. Show that these preferences are strictly convex. (Note that this means
that they are also convex.)
10. Show that these preferences are not continuous.
Additional Practice Question 2
Judith loves dogs, hates cats, and doesn’t care one way or the other about
birds.
1. Illustrate her indifference curves map (including the direction of in-
creasing utility) over bundles of dogs and cats.
2. Illustrate her indifference curves map (including the direction of in-
creasing utility) over bundles of dogs and birds.
3. Illustrate her indifference curves map (including the direction of in-
creasing utility) over bundles of cats and birds.
4
Additional Practice Question 3
Suppose that a consumer has preferences over bundles of non-negative
amounts of each of two commodities that involve each commodity being
good up to a point and then becoming bad. The consumer’s consumption
set is R2+.
1. Illustrate the indifference curve map for the consumer.
2. Indicate the direction of increasing utility for the consumer.
Additional Practice Question 4
Different Indifference Curves Should Not Cross: If the weak pref-
erence ordering is “rational”, then indifference curves cannot cross (or,
more generally, indifference sets cannot intersect). In some undergraduate
presentations of a result along these lines1, either monotonicity or strong
monotonicity (some form of “more is preferred to less” property) is also as-
sumed. The reason for this is, presumably, that it simplifies the explanation
of the result. Nonetheless, from a theoretical perspective, this approach is
undesirable because it involves making a superfluous assumption. Attempt
to establish the validity of this result without assuming anything other than
the “rationality” of the weak preference ordering.
Additional Practice Question 5
Consider a consumer that has preferences defined over bundles of non-
negative amounts of each of two commodities. The consumer’s consump-
tion set is R2+. Suppose that the consumer’s preferences can be represented
by a utility function, U(x1, x2). We could imagine a three dimensional
graph in which the “base” axes are the quantity of commodity one avail-
able to the consumer (q1) and the quantity of commodity two available to
the consumer (q2) respectively. The third axis will be the “height” axis.
This will represent the value taken by the utility function at each combina-
tion of q1 and q2. An indifference curve map for this consumer is essentially
the view that you would get of this graph if you were looking down on it
from directly above, so that your line of sight is parallel with the “utility”
axis. It will be a two-dimensional diagram that looks like a topographical
map that people might use when they are hiking. The indifference curves
play the role of contour lines. They indicate the locus of commodity bun-
dles that yield the same utility level. Explain why it might be a good idea
1See, for example, Pindyck, RS, and DL Rubinfeld (2013), Microeconomics (eighth
edition), Pearson, USA (pp. 72–73), and Borjas, GJ (2016), Labor economics (seventh
edition), McGraw-Hill, USA (pp. 28–29).
5
to indicate the direction (or directions) in which utility is increasing on the
consumer’s indifference curve map.
Additional Practice Question 6
Cobb-Douglas Preferences: Cobb-Douglas preferences on the consump-
tion set R2+ can be represented by a utility function of the form
U (q1, q2) = Aq
α
1 q
β
2 ,
where A > 0, α ∈ (0, 1), and β ∈ (0, 1) are fixed parameters.
1. If we assume that preferences are ordinal, explain why these precise
preferences are also represented by the utility function
U (q1, q2) = q
γ
1q
1−γ
2 ,
where γ = α
(α+β)
. Is γ ∈ (0, 1)?
2. If we assume that preferences are ordinal and restrict attention to the
consumption set R2++, explain why these precise preferences are also
represented by the utility function
U (q1, q2) = γ ln (q1) + (1− γ) ln (q2) .
Why did we need to restrict the consumption set to R2++ for this part
of this question?
Additional Practice Question 7
Cobb-Douglas Preferences Specific Example: Consider the utility
function U(x, y) = x0.5y0.5 that is defined on the consumption set R2+.
1. Find the equation of the indifference curve that corresponds to U =
40.
2. What is the slope of the indifference curve for U = 40 for any given
value of x?
3. What is the equation of an arbitrary indifference curve for this utility
function?
4. What is the slope of an arbitrary indifference curve for this utility
function at any given value of x?
6
Additional Practice Question 8
Marginal Rates of Substitution: Calculate the marginal rate of sub-
stitution for an arbitrary commodity bundle of the form (x, y) >> (0, 0)
(that is, where x > 0 and y > 0) for each of the following utility functions.
1. Quasi-Linear Preferences Example 1: U (x, y) = x+
√
y.
2. Quasi-Linear Preferences Example 2: U (x, y) = x+ ln (y).
3. Stone-Geary Preferences: U (x, y) = (x− x0)α (y − y0)1−α, where
x0 > 0, y0 > 0, and α ∈ (0, 1) are fixed parameters.
4. Constant-Elasticity-of-Substitution (CES) Preferences:
U (x, y) = (αxρ + βyρ)
1
ρ , where x0 > 0, y0 > 0, and α ∈ (0, 1) are
fixed parameters
7
Appendices
Appendix A: Some Remarks
Remark 1 It can be shown that perfect substitutes preferences, perfect
complements preferences, and Cobb-Douglas preferences are special cases
of CES preferences.
Remark 2 It can be shown that Cobb-Douglas preferences are a special
case of Stone-Geary preferences.
Appendix B: Some Notation
• The set of real numbers is denoted by R.
• The set of non-negative real numbers is denoted by R+. It is simply
the collection of all real numbers that are greater-than-or-equal-to
zero.
• The set of positive real numbers is denoted by R++. It is simply the
collection of all real numbers that are strictly-greater-than zero.
• The set of natural numbers is denoted by N. It is the set of counting
numbers, {1, 2, 3, · · · }. Some people include zero in this set, while
others do not. My preference is not to include zero in the set of
natural numbers.
• If we want to form a set that includes both the number zero and
every natural number, but nothing else, then we can form the set
of non-negative integers. This is the set {0, 1, 2, 3, · · · }. This set is
denoted by Z+.
• The set R2 is the Cartesian product of the set of real numbers with
itself. This is the set of all ordered-pairs of real numbers. In other
words, it is the two-dimensional rectangular coordinate plane that
you should be familiar with from high-school. It is formally defined
to be
R2 = R× R = {(x, y) : x ∈ R, y ∈ R} .
• The non-negative quadrant of R2 is denoted by R2+.
• The strictly positive quadrant of R2 is denoted by R2++.
8
Answers for the Tutorial Questions
Answer for Tutorial Question 1
Consider the utility function U(x1, x2) = x1 + x2. A representative indif-
ference curve for this utility function satisfies the equation x1 + x2 = U ,
which can be rearranged to obtain x2 = −x1 +U . This is the equation of a
straight-line in (x1, x2)–space with a slope of (−1) and an x2–intercept at
the point (0, U).
Since these preferences are represented by a utility function, we know
that they must be rational. It can be shown that a necessary (but not
always sufficient) condition for the existence of a utility function represen-
tation of preferences is that those preferences be rational.
These preferences are strongly monotone (and hence also monotone and
locally non-satiated). This can be seen by picking an arbitrary point on the
representative indifference curve and noting that every point that is either
north, or east, or north-east of that point belongs to a different indifference
curve with a higher utility level than that of the representative indifference
curve.
These preferences are convex but not strictly convex. This can be seen
by choosing two distinct points that both belong to the representative indif-
ference curve. Every point on the straight line between them also belongs
to the representative indifference curve. These points are all indifferent,
and hence weakly preferred, to the original two points, but they are not
strictly preferred to those points.
Answer for Tutorial Question 2
Consider the utility function U(x1, x2) = min(x1, x2). A representative in-
difference curve for this utility function satisfies the equation min(x1, x2) =
U , which is equivalent to
x2

= U if x1 > U ,
∈ [U,∞) if x1 = U ,
is undefined if x1 < U .
This is the equation of an L–shaped right-angled curve in (x1, x2)–space
with the kink occurring on the line x2 = x1 at the point (U,U).
Since these preferences are represented by a utility function, we know
that they must be rational. It can be shown that a necessary (but not
always sufficient) condition for the existence of a utility function represen-
tation of preferences is that those preferences be rational.
These preferences are monotone (and hence also locally non-satiated),
but not strongly monotone. This can be seen by picking an arbitrary point
9
on the representative indifference curve such that x2 > x1 = U and noting
that every point that is directly north of that point belongs to the same
indifference curve and hence is not strictly preferred to the initially selected
point.
These preferences are convex but not strictly convex. This can be seen
by picking an arbitrary point on the representative indifference curve such
that x2 > x1 = U and noting that every point that is directly north of
that point belongs to the same indifference curve and hence is not strictly
preferred to the initially selected point.
Answer for Tutorial Question 3
Consider the utility function U(x1, x2) = x
1
2
1 x
1
2
2 . A representative indiffer-
ence curve for this utility function satisfies the equation x
1
2
1 x
1
2
2 = U , which
can be rearranged to obtain x2 =
U
2
x1
. This is the equation of a rectangular
hyperbola in (x1, x2)–space.
Since these preferences are represented by a utility function, we know that
they must be rational. It can be shown that a necessary (but not always
sufficient) condition for the existence of a utility function representation of
preferences is that those preferences be rational.
These preferences are monotone (and hence locally non-satiated), but not
strongly monotone, on R2+. They are strongly monotone (and hence also
monotone and locally non-satiated) on R2++. The indifference curve cor-
responding to a utility level of zero is capital-L shaped and runs along
the horizontal and vertical axes, with the kink occurring at the origin.
Any indifference curve that corresponds to a strictly positive utility level is
smooth, downward sloping, and bowed-in towards the origin. This can be
seen by picking an arbitrary point on the representative indifference curve
and noting that every point that is either north, or east, or north-east of
that point belongs to a different indifference curve with a higher utility
level than that of the representative indifference curve.
These preferences are strictly convex (and hence also convex) if we restrict
attention to consumption bundles that contain strictly positive amounts
of both commodities. If we allow for the possibility that some consump-
tion bundles might contain a zero amount of either or both commodities,
then these preferences are convex, but not strictly convex. (I would like to
thank one of the students in the Semester One 2022 instance of the class
for pointing out that the restriction to consumption bundles that contain
strictly positive amounts of both commodities is required for these pref-
10
erences to be strictly convex.) This can be seen by choosing two distinct
points that both belong to the representative indifference curve. If the rep-
resentative indifference curve involves a strictly positive utility level, then
all bundles that lie on it must contain strictly positive amounts of both
commodities. In such cases, every point on the straight line between the
two (distinct) chosen points on the representative indifference curve belongs
to an indifference curve that has a higher level of utility than the repre-
sentative indifference curve. We will now proceed to discuss the following
three propositions.
1. If preferences on R2+ can be represented by the utility function
U(x1, x2) = x
1
2
1 x
1
2
2 , then those preferences are convex.
2. If preferences on R2+ can be represented by the utility function
U(x1, x2) = x
1
2
1 x
1
2
2 , then those preferences are not strictly convex.
3. If preferences on R2++ can be represented by the utility function
U(x1, x2) = x
1
2
1 x
1
2
2 , then those preferences are strictly convex.
First, we will pride a rather formal proof of the first proposition. Second,
we will provide a counter-example that could be used as part of a proof of
the third proposition by the method of ”proof by contradiction”. Third, we
will provide an indication of the reason that the restriction to consumption
bundles that contain strictly positive amounts of both commodities might
allow us to modify the proof of the first proposition to establish the strict
convexity, rather than just the convexity, of these preferences.
A Discussion of Proposition One
Claim 3 If preferences on R2+ can be represented by the utility function
U(q1, q2) = q
1
2
1 q
1
2
2 , then those preferences are convex.
Proof. Consider two distinct consumption bundles, x = (x1, x2) ∈ R2+
and y = (y1, y2) ∈ R2+, such that U (x1, x2) > U and U (y1, y2) > U . This
requires that
U (x1, x2) = x
1
2
1 x
1
2
2 =
√
x1x2 > U
and
U (y1, y2) = y
1
2
1 y
1
2
2 =
√
y1y2 > U .
Noting that
U(q1, q2) = q
1
2
1 q
1
2
2 =
√
q1q2 > 0
for all (q1, q2) ∈ R2+, we can assume that U > 0 without loss of gen-
erality. Furthermore, it must also be that case that U (x1, x2) > 0 and
11
U (y1, y2) > 0. Thus we must have x1x2 > U
2
and y1y2 > U
2
.
Recall that x and y are distinct consumption bundles. This means that
x 6= y, which requires that either (i) x1 6= y1 and x2 = y2, or (ii) x1 = y1
and x2 6= y2, or (iii) x1 6= y1 and x2 6= y2. Construct a consumption bundle
zλ =
(
zλ1 , z
λ
2
) ∈ R2+ as z = λx + (1− λ) y, where 0 < λ < 1. (In other
words, we have constructed the consumption bundle zλ by taking some
convex combination of the consumption bundles x and y.) Note that
zλ =
(
zλ1 , z
λ
2
)
= λx+ (1− λ) y
= λ (x1, x2) + (1− λ) (y1, y2)
= (λx1, λx2) + ((1− λ) y1, (1− λ) y2)
= (λx1 + (1− λ) y1, λx2 + (1− λ) y2) .
Since x 6= y and 0 < λ < 1, we know that both zλ 6= x and zλ 6= y. (In
other words, we know that x, y, and zλ are distinct consumption bundles.)
We need to show that
U
(
zλ1 , z
λ
2
)
=
(
zλ1
) 1
2
(
zλ2
) 1
2 =
√
zλ1 z
λ
2 > U .
The fact that
U(q1, q2) = q
1
2
1 q
1
2
2 =
√
q1q2 > 0
for all (q1, q2) ∈ R2+ means that U
(
zλ1 , z
λ
2
)
> 0, and it also allows us to
assume that U > 0 without loss of generality. This means that showing
that √
zλ1 z
λ
2 > U
is equivalent to showing that
zλ1 z
λ
2 > U
2
.
Note that
zλ1 z
λ
2 = (λx1 + (1− λ) y1) (λx2 + (1− λ) y2)
= λ2x1x2 + λ (1− λ) (x1y2 + y1x2) + (1− λ)2 y1y2.
Recall that x1x2 > U
2
and y1y2 > U
2
. This means that
zλ1 z
λ
2 = λ
2x1x2 + λ (1− λ) (x1y2 + y1x2) + (1− λ)2 y1y2
> λ2U2 + λ (1− λ) (x1y2 + y1x2) + (1− λ)2 U2.
12
If it is the case that x1y2 + y1x2 > 2U
2
, then we obtain
zλ1 z
λ
2 > λ2U
2
+ λ (1− λ) (x1y2 + y1x2) + (1− λ)2 U2
> λ2U2 + λ (1− λ)
(
2U
2
)
+ (1− λ)2 U2
= λ2U
2
+ 2λ (1− λ)U2 + (1− λ)2 U2
=
{
λ2 + 2λ (1− λ) + (1− λ)2}U2
= {λ+ (1− λ)}2 U2
= 12U
2
= 1U
2
= U
2
.
This would mean that
zλ1 z
λ
2 > U
2
,
as required.
Consider the following (more than) exhaustive list of possibilities:
1. x1 > y1 and x2 > y2;
2. x1 < y1 and x2 < y2;
3. x1 > y1 and x2 < y2; and
4. x1 < y1 and x2 > y2.
(This list of possibilities is more than exhaustive because we know that
x 6= y.) In case (1) we have
x1y2 > y1y2 > U
2
and
y1x2 > y1y2 > U
2
,
which means that
x1y2 + y1x2 > U
2
.
In case (2) we have
x1y2 > x1x2 > U
2
13
and
y1x2 > x1x2 > U
2
,
which means that
x1y2 + y1x2 > U
2
.
Establishing the required result for cases (3) and(4) is not quite as straight-
forward as it is for case (1) and case (2).
Consider case (3). Recall that this involves x1 > y1 and x2 < y2. Note that
x1y2 + y1x2 = x1y1 + y1x2 + 0 + 0
= x1y2 + y1x2 + (x1x2 − x1x2) + (y1y2 − y1y2)
= x1y2 + y1x2 + x1x2 − x1x2 + y1y2 − y1y2
= x1x2 + y1y2 + {x1y2 − x1x2 − y1y2 + y1x2}
= x1x2 + y1y2 + {x1 (y2 − x2)− y1 (y2 − x2)}
= x1x2 + y1y2 + (x1 − y1) (y2 − x2) .
Recall that x1x2 > U
2
and y1y2 > U
2
. The fact that x1 > y1 in this case
means that (x1 − y1) > 0. The fact that x2 < y2 in this case means that
(y2 − x2) > 0. This means that (x1 − y1) (y2 − x2) > 0 in this case. Thus
we have
x1y2 + y1x2 = x1x2 + y1y2 + (x1 − y1) (y2 − x2)
> U2 + U2 + 0
= 2U
2
.
As such, we know that
x1y2 + y1x2 > 2U
2
in case (3).
Consider case (4). Recall that this involves x1 < y1 and x2 > y2. We
have already established that
x1y2 + y1x2 = x1x2 + y1y2 + (x1 − y1) (y2 − x2) .
This can be rearranged to obtain
14
x1y2 + y1x2 = x1x2 + y1y2 + (x1 − y1) (y2 − x2)
= x1x2 + y1y2 + (1) (x1 − y1) (y2 − x2)
= x1x2 + y1y2 + (−1)2 (x1 − y1) (y2 − x2)
= x1x2 + y1y2 + {(−1) (x1 − y1)} {(−1) (y2 − x2)}
= x1x2 + y1y2 + (y1 − x1) (x2 − y2) .
Recall that x1x2 > U
2
and y1y2 > U
2
. The fact that x1 < y1 in this case
means that (y1 − x1) > 0. The fact that x2 > y2 in this case means that
(x2 − y2) > 0. This means that (y1 − x1) (x2 − y2) > 0 in this case. Thus
we have
x1y2 + y1x2 = x1x2 + y1y2 + (y1 − x1) (x2 − y2)
> U2 + U2 + 0
= 2U
2
.
As such, we know that
x1y2 + y1x2 > 2U
2
in case (4).
A Discussion of Proposition Two
Suppose that we want to do not restrict our attention to consumption bun-
dles that have a strictly positive amount of each commodity. Instead, we
only require that they contain a non-negative amount of each commodity.
Consider the consumption bundle x = (0, 1) ∈ R2+, the consumption bundle
y = (0, 3) ∈ R2+, and the consumption bundle z = (0, 2) ∈ R2+. We have
U (x) = U (0, 1) =
√
(0) (1) =
√
0 = 0,
U (y) = U (0, 3) =
√
(0) (3) =
√
0 = 0,
and
U (z) = U (0, 2) =
√
(0) (2) =
√
0 = 0.
Note that z is a (non-trivial) convex combination of x and y, since
15
z = (0, 2)
=
(
0 + 0, 1
2
+ 3
2
)
=
(
0, 1
2
)
+
(
0, 3
2
)
=
(
1
2
)
(0, 1) +
(
1
2
)
(0, 3)
=
(
1
2
)
x+
(
1
2
)
y
= λx+ (1− λ) y,
where λ = 1
2
∈ (0, 1). Consider the reference utility level given by U = 0.
Clearly we have U (x) = 0 > 0 and U (y) = 0 > 0. Since z is a (non-
trivial) convex combination of x and y, we would have U (z) > 0 if these
preferences were strictly convex on R2+. The fact that U (z) = 0 6> 0 means
that these preferences are not strictly convex on R2+.
A Discussion of Proposition Three
Suppose now that we restrict attention to consumption bundles that con-
tain strictly positive amounts of both commodities.
Recall that both x1x2 > U
2
and y1y2 > U
2
. Note that if either x1x2 > U
2
,
or y1y2 > U
2
, or both x1x2 > U
2
and y1y2 > U
2
, then we can replace U
with k = min
{√
x1x2,
√
y1y2
}
everywhere in the proof of Proposition One
and still have that proof remain valid. As such, we know that if either
x1x2 > U
2
, or y1y2 > U
2
, or both x1x2 > U
2
and y1y2 > U
2
, then we
have U
(
zλ1 , z
λ
2
)
> k > U . This means that any case that prevents these
preferences from being strictly convex must involve both x1x2 = U
2
and
y1y2 = U
2
.
Suppose that both x1x2 = U
2
and y1y2 = U
2
. Since both x ∈ R2++ and
y ∈ R2++, we know that x1 > 0, x2 > 0, y1 > 0, and y2 > 0. This means
that U
2
> 0 and hence that U > 0. Recall that x 6= y. This means that
either x1 6= y1, or x2 6= y2, or both x1 6= y1 and x2 6= y2. Since x 6= y and
x1x2 = y1y2 = U
2
> 0, we know that we must have either (i) x1 > y1 and
x2 < y2, or (ii) x1 < y1 and x2 > y2.
If x1 > y1 and x2 < y2, then we are in case (3) from the proof of Proposition
16
One, and have (x1 − y1) (y2 − x2) > 0, which means that
x1y2 + y1x2 = x1x2 + y1y2 + (x1 − y1) (y2 − x2)
= U
2
+ U
2
+ (x1 − y1) (y2 − x2)
= 2U
2
+ (x1 − y1) (y2 − x2)
> U
2
+ U
2
+ 0
= 2U
2
.
As such, we know that
x1y2 + y1x2 > 2U
2
in this case.
If x1 < y1 and x2 > y2, then we are in case (4) from the proof of Proposition
One, and have (y1 − x1) (x2 − y2) > 0, which means that
x1y2 + y1x2 = x1x2 + y1y2 + (y1 − x1) (x2 − y2)
= U
2
+ U
2
+ (y1 − x1) (x2 − y2)
= 2U
2
+ (y1 − x1) (x2 − y2)
> U
2
+ U
2
+ 0
= 2U
2
.
As such, we know that
x1y2 + y1x2 > 2U
2
in this case.
This means that if x ∈ R2++, y ∈ R2++, x1x2 = U2, y1y2 = U2, and x 6= y,
then we must have U
(
zλ1 , z
λ
2
)
> U .
Combing these results, we can conclude that if we restrict attention to
consumption bundles that contain strictly positive amounts of both com-
modities, then these preferences are strictly convex.
Answer for Tutorial Question 4
Consider the utility function U(x1, x2) = x1+ln(x2). A representative indif-
ference curve for this utility function satisfies the equation x1 +ln(x2) = U ,
17
which can be rearranged to obtain x2 =
eU
ex1
. The x2–intercept for this
representative indifference curve occurs at the point (0, eU). The represen-
tative indifference curve is downward sloping from this point and bowed-in
towards the origin. There is no x1–intercept for this representative indif-
ference curve, Instead, x1 will approach (positive) infinity as x2 decreases
towards zero.
Since these preferences are represented by a utility function, we know
that they must be rational. It can be shown that a necessary (but not
always sufficient) condition for the existence of a utility function represen-
tation of preferences is that those preferences be rational.
These preferences are strongly monotone (and hence also monotone and
locally non-satiated). This can be seen by picking an arbitrary point on the
representative indifference curve and noting that every point that is either
north, or east, or north-east of that point belongs to a different indifference
curve with a higher utility level than that of the representative indifference
curve.
These preferences are strictly convex (and hence also convex). This can
be seen by choosing two distinct points that both belong to the represen-
tative indifference curve. Every point on the straight line between them
belongs to an indifference curve that has a higher level of utility than the
representative indifference curve.
Answer for Tutorial Question 5
Consider the utility function U(x1, x2) = x1 +
√
x2. A representative indif-
ference curve for this utility function satisfies the equation x1 +
√
x2 = U ,
which can be rearranged to obtain x2 =
(
U − x1
)2
. Ignoring any points
that occur to the right of the x1–intercept of this representative indif-
ference curve, it will downward sloping and bowed-in towards the ori-
gin in (x1, x2)–space. The x1–intercept occurs at the point (U, 0). The
x2–intercept occurs at the point (0, U
2
).
Since these preferences are represented by a utility function, we know
that they must be rational. It can be shown that a necessary (but not
always sufficient) condition for the existence of a utility function represen-
tation of preferences is that those preferences be rational.
These preferences are strongly monotone (and hence also monotone and
locally non-satiated). This can be seen by picking an arbitrary point on the
representative indifference curve and noting that every point that is either
north, or east, or north-east of that point belongs to a different indifference
curve with a higher utility level than that of the representative indifference
curve.
These preferences are strictly convex (and hence also convex). This can
be seen by choosing two distinct points that both belong to the represen-
18
tative indifference curve. Every point on the straight line between them
belongs to an indifference curve that has a higher level of utility than the
representative indifference curve.
Remark 4 The reason that you do not include points on an indifference-
curves for the quasi-linear utility function of the form x1 +
√
x2 that lie to
the right of its x1-intercept is that they implicitly involve negative values
for the
√
x2. But by convention, the
√
x2 is usually taken to mean the
positive square-root. If we want to have the negative square-root function,
then we should specify
(−√x2). If we want to allow both the positive and
the negative square-root, then we should specify ±√x2, where ± means
“plus or minus”, but note that this would then be a correspondence and not
a function. Since we want to work with utility functions here, we could
choose to use either x1 +
√
x2, or x1 − √x2, but not both of them. If
we choose to use x1 +
√
x2, then both commodities are “goods” (in the
sense that “more of either commodity, with no change in the amount of the
other commodity, is preferred to less”. If we choose to use x1 −√x2, then
commodity one is a “good” (in the sense that “more of commodity one,
with no change in the amount of commodity two, is preferred to less”),
but commodity two is a “bad” (in the sense that “less of commodity two,
with no change in the amount of commodity one, is preferred to more”).
Following the standard convention, we will interpret x1 +
√
x2 to mean
x1 + (the positive square-root of x2). (The x1-intercept for the indifference
curve in which U = k occurs when x1 = k. Points that belong to the
graph of the equation x2 = (k − x1)2 that lie to the right of the vertical line
x1 = k must have x1 > k. In order for U = k, this would in turn require
that
√
x2 < 0. As such, we will not include such points in the set of points
that belong to that indifference curve.)
19
Answer for Tutorial Question 6
20
Remark 5 The special case of the Scarf-Shapley-Shubik utility function
that is considered in this question is
U (x1, x2) = max (min (x1, 2x2) ,min (2x1, x2)) .
This utility function can be expressed in an alternative “multiple-case” form
as
U (x1, x2) =

2x1 if 0 6 2x1 6 x2,
x2 if 0 6 x1 6 x2 6 2x1,
x1 if 0 6
(
1
2
)
x1 6 x2 6 x1,
2x2 if 0 6 x2 6
(
1
2
)
x1.
A useful way to approach the derivation of the “multiple-case” representa-
tion of this utility function is as follows. First, note that there are at most
four regions of the non-negative quadrant of Euclidean two-space (that is,
of R2+) that are relevant. This is the case because, ignoring threshold cases
in which the two relevant components are equal to each other, (i) either
min (x1, 2x2) = x1 or min (x1, 2x2) = 2x2 (this is the first primary sub-
case), and (ii) either min (2x1, x2) = 2x1 or min (2x1, x2) = x2 (this is
the second primary sub-case). This means that, again ignoring threshold
cases in which the two relevant components are equal to each other, there
are four possible intermediate sub-cases for the utility function. These are:
max (x1, 2x1), max (x1, x2), max (2x2, 2x1), and max (2x2, x1). These four
intermediate sub-cases can be used to obtain the three ”threshold lines” that
partition R2+ into four the four sub-regions that appear in the “multiple-
case” representation of this utility function. These threshold lines are given
by the equations x2 =
(
1
2
)
x1, x2 = x1, and x2 = 2x1. Following this, you
would then need to determine which of x1, 2x2, 2x1, and x2 is the largest
in each of the four regions. (As can be seen from the “multiple-case” form
of this utility function that is provided above, the answer to this question
is different in the different regions, except for points that fall on the three
threshold lines).
A nice discussion of an example of Scarf-Shapley-Shubik preferences that
is very similar, although not quite the same, as the one considered in this
tutorial question can be found in Takayama, A (1985), Mathematical eco-
nomics (second edition), Cambridge University Press, USA, pp. 211–213.
It is on the basis of this discussion in Takayama that I use the term “Scarf-
Shapley-Shubik preferences” to describe preferences that have a similar form
to the one considered in this tutorial question. The utility function in the
21
example that is considered in the discussion in Takayama is
U (x, y) =

x if x 6 y
2
,
y
2
if y
2
< x < y,
x
2
if x = y,
x
2
if y < x < 2y,
y if x > 2y.
=

x if x 6 y
2
,
y
2
if y
2
< x < y,
x
2
if y 6 x < 2y,
y if x > 2y.
This could also be written as
U (x, y) = max
(
min
(x
2
, y
)
,min
(
x,
y
2
))
.
I obtained the precise form of the utility function considered in this question
from Question E on the August 1999 Microeconomic Theory Comprehen-
sive Exam from the University of Texas at Austin. I think that exam was
written by Prof. Dale Stahl and Prof. Max Stinchcombe. In that question,
the utility function is written as
U1 (x1, y1) = max {min (2x1, y1) ,min (x1, 2y1)} ,
which is equivalent to the one that is used in this tutorial question.
Remark 6 Recall that utility function representations of preferences are
only unique up to a strictly increasing transformation assuming that we
have an ordinal focus, so that only the induced ranking of consumption
bundles matters). While the Takayama (1985, p. 211) version of the utility
function for a special case of Scarf-Shapley-Shubik preferences is different
that the utility function that is employed in this question, they both actually
represent the same underlying preferences. In other words, they are two
different utility function representations for a common underlying special
case of Scarf-Shapley-Shubik preferences. The reason for this is that each
22
of these utility functions can be obtained from the other one by applying a
strictly increasing transformation. If we denote the Takayama version of
the utility function by T (x1, x2) and the version that is employed in this
question by U (x1, x2), then we have
T (x1, x2) = max
(
min
(x1
2
, x2
)
,min
(
x1,
x2
2
))
and
U (x1, x2) = max (min (x1, 2x2) ,min (2x1, x2)) .
Note that
U (x1, x2) = 2T (x1, x2) = f (T (x1, x2)) ,
where f (x) = 2x is a strictly increasing function. Note also that
T (x1, x2) =
(
1
2
)
U (x1, x2) = g (U (x1, x2)) ,
where g (x) =
(
1
2
)
x is a strictly increasing function.
23
Answer for Tutorial Question 7
24
Answers for the Additional Practice Ques-
tions
Answer for the Additional Practice Question 1
Consider a lexicographic weak preference relation defined on the consump-
tion set R2+ (or, more precisely, on R2+ × R2+).
Claim 7 If preferences are lexicographic, c = (c1, c2) ∈ R2+, d = (d1, d2) ∈
R2+, and c d, then either (i)c1 > d1, or (ii) both c1 = d1 and c2 > d2.
Proof. Recall that c d if and only if both c % d and d 6% c. Since c % d,
we know that either (i) c1 > d1, or (ii) both c1 = d1 and c2 > d2. Since
d 6% c, we know that either (iii) d1 < c1, or (iv) both d1 = c1 and d2 6> c2.
Combining these four cases, we can only have c d if either (a) c1 > d1,
or (b) both c1 = d1 and c2 > d2.
Claim 8 If preferences are lexicographic, c = (c1, c2) ∈ R2+, d = (d1, d2) ∈
R2+, and c ∼ d, then both c1 = d1 and c2 = d2.
Proof. Recall that c ∼ d if and only if both c % d and d % c. Since c % d,
we know that either (i) c1 > d1, or (ii) both c1 = d1 and c2 > d2. Since
d % c, we know that either (iii) d1 > c1, or (iv) both d1 = c1 and d2 > c2.
The only way that both of these situations can be simultaneously true is if
both c1 = d1 and c2 = d2.
Claim 9 % is weakly complete over R2+.
Proof. Let x = (x1, x2) ∈ R2+ and y = (y1, y2) ∈ R2+ such that x 6= y.
This means that either x1 6= y1 or x2 6= y2 or both. If x1 6= y1, then either
x1 > y1, in which case x % y, or y1 > x1, in which case y % x. If x1 = y1,
then it must be the case that x2 6= y2. In such cases, we must have either
x1 = y1 and x2 > y2, in which case x % y, or x1 = y1 and y2 > x2, in
which case y % x. As such, we can conclude that, in every case where
x = (x1, x2) ∈ R2+ and y = (y1, y2) ∈ R2+ such that x 6= y, we have either
x % y or y % x. Thus we know that % is weakly complete over R2+.
Claim 10 % is reflexive over R2+.
25
Proof. Let x = (x1, x2) ∈ R2+. Clearly we have x1 = x1 and x2 = x2. This
means that x1 = x1 and x2 ≥ x2. As such, we can conclude that x % x.
Thus we know that % is reflexive over R2+.
Claim 11 % is strongly complete over R2+.
Proof. Let x = (x1, x2) ∈ R2+ and y = (y1, y2) ∈ R2+. Clearly, it must be
the case that either x = y or x 6= y. If x = y, then we know that both x % y
and y % x because we have already shown that % is reflexive over R2+. If
x 6= y, then we know that either x % y or y % x because we have already
shown that % is weakly complete over R2+. As such, we can conclude that,
if x = (x1, x2) ∈ R2+ and y = (y1, y2) ∈ R2+, then either % y or y % x or
both. This we know that % is strongly complete over R2+.
Claim 12 % is transitive over R2+.
Proof. Let x = (x1, x2) ∈ R2+, y = (y1, y2) ∈ R2+ and x = (z1, z2) ∈ R2+
such that x % y and y % z. Since x % y, we know that either (i) “x1 > y1”
or (ii) “x1 = y1 and x2 ≥ y2”. Since y % z, we know that either (iii) “y1 >
z1” or (iv) “y1 = z1 and y2 ≥ z2”. There are four possible combinations to
consider. These are:
• (a) case (i) and case (iii);
• (b) case (i) and case (iv);
• (c) case (ii) and case (iii); and
• (d) case (ii) and case (iv).
Consider combination (a). Here we have x1 > y1 and y1 > z1. Note
that x1 ∈ R, y1 ∈ R and z1 ∈ R. Since the binary “strictly greater than”
relation, >, is transitive over R, we know that x1 > z1. As such, we can
conclude that x % z.
Consider combination (b). Here we have x1 > y1 and y1 = z1. This
means that x1 > z1. As such, we can conclude that x % z.
Consider combination (c). Here we have x1 = y1 and y1 > z1. This
means that x1 > z1. As such, we can conclude that x % z.
Consider combination (d). Here we have x1 = y1 and y1 = z1. This
means that x1 = z1. We also have x2 ≥ y2 and y2 ≥ z2. Note that x2 ∈ R,
26
y2 ∈ R and z2 ∈ R. Since the binary “greater than or equal to” relation,
≥, is transitive over R, we know that x2 ≥ z2. As such, we can conclude
that x % z.
We have shown that x % z in all four of the possible cases that can occur
when x % y and y % z. Thus we can conclude that % is transitive over R2+.
Claim 13 % is rational over R2+.
Proof. We have already shown that % is both strongly complete and
transitive over R2+. Thus we can conclude that % is rational over R2+.
Claim 14 % is strongly monotone over R2+.
Proof. Recall that if % is the lexicographic preference ordering over R2+
and x, y ∈ R2+, then x % y if and if either (a) x1 > y1 (so that x1 ≥ y1 but
y1 x1) or (b) x1 = y1 (so that x1 ≥ y1 and y1 ≥ x1) and x2 ≥ y2.
Consider two commodity bundles, x = (x1, x2) ∈ R2+ and y = (y1, y2) ∈
R2+, such that x ≥ y. This means that one of the following must be true:
(a) x1 > y1 and x2 > y2; or (b) x1 > y1 and x2 = y2; or (c) x1 = y1 and
x2 > y2. In both case (a) and case (b), we have x1 > y1. Thus we know
that x % y in both of these cases. Furthermore, in case (c), we have x1 = y1
and x2 > y2. Thus we know that x % y in this case as well. Hence we
can conclude that, if x ≥ y, then x % y. Thus the lexicographic preference
ordering over R2+ is strongly monotone.
Corollary 15 Since % is strongly monotone over R2+, we know that it is
also both monotone and locally non-satiated over R2+.
Claim 16 % is strictly convex over R2+.
Proof. Recall that if % is the lexicographic preference ordering over R2+
and x, y ∈ R2+, then x % y if and if either (a) x1 > y1 (so that x1 ≥ y1 but
y1 x1) or (b) x1 = y1 (so that x1 ≥ y1 and y1 ≥ x1) and x2 ≥ y2.
Consider two distinct bundles, x = (x1, x2) ∈ R2+ and y = (y1, y2) ∈ R2+.
The fact that these bundles are distinct means that x 6= y. Without loss
of generality, we can assume that x % y. Since x 6= y, the assumption that
x % y means that either (a) x1 > y1 or (b) x1 = y1 and x2 > y2. Construct
a third bundle, zλ = (zλ1 , z
λ
2 ) by taking a convex combination of x and y.
This means that zλ = λx+(1−λ)y, where λ ∈ (0, 1). In order to show that
27
the lexicographic preference ordering is strictly convex, we need to show
that zλ y for all λ ∈ (0, 1).
Note that we have zλ1 = λx1 + (1 − λ)y1 and zλ2 = λx1 + (1 − λ)y2.
Recall that we have two cases to consider. These are (a) x1 > y1 or (b)
both x1 = y1 and x2 > y2.
Under case (a), we have
zλ1 = λx1 + (1− λ)y1
> λy1 + (1− λ)y1 (because x1 > y1 and λ ∈ (0, 1))
= y1.
Since zλ1 > y1, we know that z
λ y for all λ ∈ (0, 1) in this case.
Under case (b), we have
zλ1 = λx1 + (1− λ)y1
= λy1 + (1− λ)y1 (because x1 = y1)
= y1.
Since zλ1 = y1, we need to consider the second commodity components of
the consumption bundles. Note that in case (b) we have
zλ2 = λx2 + (1− λ)y2
> λy2 + (1− λ)y2 (because x2 > y2 and λ ∈ (0, 1))
= y2.
Since zλ1 = y1 and z
λ
2 > y2, we know that z
λ y for all λ ∈ (0, 1) in this
case.
Combining the two cases, we can conclude that if x % y, x 6= y and
zλ = λx + (1− λ)y, then zλ y for all λ ∈ (0, 1). Thus the lexicographic
preference ordering over R2+ is strictly convex.
Corollary 17 Since % is strictly convex over R2+, we know that it is also
convex over R2+.
Claim 18 The lexicographic weak preference relation is not continuous on
the consumption set R2+.
28
Proof. Consider the sequence of bundles {xn}n∈N where xn = (xn1 , xn2 ) =(
1
n
, 0
)
. Consider also the sequence of bundles {yn}n∈N where yn = (yn1 , yn2 ) =
(0, 1). Note that xn1 =
1
n
> 0 = yn1 for all n ∈ N. Thus we know that
xn % yn for all n ∈ N. However, note also that x∞1 = limn−→∞ xn1 =
limn−→∞ 1n = 0. Thus we have x
∞
1 = 0 = y
∞
1 and x
∞
2 = 0 < 1 = y
∞
2 . In
other words, x∞1 = y
∞
1 and x
∞
2 6> y∞2 . Thus we know that x∞ 6% y∞. Since
there are two convergent sequences of bundles, {xn}n∈N and {yn}n∈N, such
that both (i) xn % yn for all n ∈ N, and (ii) x∞ 6% y∞, we know that this
preference ordering is not continuous.
Remark 19 It can be shown that the lexicographic weak preference relation
on the consumption set R2+ cannot be represented by a utility function. If,
however, the consumption set is restricted to Z2+, or some subset of it, then
the lexicographic preference relation can be represented by a utility function.
29
Answer for the Additional Practice Question 2
Part 1
30
Part 2
Part 3
31
Answer for the Additional Practice Question 3
Answer for the Additional Practice Question 4
Claim 20 If a consumer’s weak preference relation % is rational on his
consumption set C, then his indifference sets must be mutually disjoint (that
is, any pair of different indifference sets must have an empty intersection).
(In other words, his indifference curves cannot cross if his preferences are
rational.)
Proof. (By contradiction.) Let % be a rational weak preference relation
on the consumption set C. Suppose that it is possible for two different
indifference sets for this weak preference relation to have a non-empty in-
tersection. Let I1 and I2 be such a pair of indifference sets.
First, consider the case in which both I1 \ (I1 ∩ I2) 6= ∅ and I1 \
(I1 ∩ I2) 6= ∅. This requires that there exist two distinct consumption
bundles, a ∈ C and b ∈ C such that both (i) a ∈ I1 but a /∈ I2 and (ii)
b ∈ I2 but b /∈ I1. Since these indifference sets are not disjoint, we know
32
that there exists some bundle c ∈ I1 ∩ I2. We have both a ∈ I1 and c ∈ I1.
This requires that a ∼ c, which in turn requires that both a % c and c % a.
We also have both b ∈ I2 and c ∈ I2. This requires that b ∼ c, which in
turn requires that both b % c and c % b. Since the weak preference relation
is rational, we know that it is transitive. Transitivity, along with the fact
that both b % c and c % a, means that b % a. Transitivity, along with
the fact that both a % c and c % b, means that a % b. Since both a % b
and b % a, we must have a ∼ b. But this requires that a ∈ I2 and b ∈ I1,
contradicting the claim that these are distinct indifference sets. Thus we
know that if the weak preference relation is rational, then any pair of dif-
ferent indifference curves must be disjoint if they each contain at least one
element that does not belong to the other indifference set.
Second, consider the case in which one of the two (non-empty) indif-
ference sets is a proper subset of the other indifference set. Without loss
of generality, suppose that I1 ⊂ I2, so that there exists some consumption
bundle a ∈ C such that a ∈ I2 but a /∈ I1. Since these indifference sets are
not disjoint, we know that there exists some bundle b ∈ I1 ∩ I2. We have
both a ∈ I2 and b ∈ I2. This requires that a ∼ b, which in turn requires
that both a % b and b % a. We have b ∈ I1 and a 6 inI1. This requires
that either (i) b % a but a 6% b, or (ii) a % b but b 6% a. In case (i), we
obtain the contradiction that a % b and a 6% b. In case (ii), we obtain the
contradiction that b % a but b 6% a. Thus we can conclude that if the weak
preference relation is rational, then one non-empty indifference set cannot
be a proper subset of another indifference set.
Upon combining these two cases, we have established the validity of the
claim.
Answer for the Additional Practice Question 5
It is a good idea to indicate the direction of increasing satisfaction (or
utility) on an indifference curve map for the same reason that is useful
to indicate the direction of increasing height on a topographical (contour)
map. It is helpful to know whether one is ascending a hill or descending a
hill.
Answer for the Additional Practice Question 6
We will work through this question as an example in one or more of the
lectures for this class, if I remember to do so. (If I do not remember to do
so, then please remind me.)
33
Answer for the Additional Practice Question 7
Part 1
The utility function is
U (x, y) = U(x, y) = x0.5y0.5 =
√
x
√
y =
√
xy.
The indifference curve corresponding to an amount of utility equal to 40
“utils” is given by the condition
U (x, y) = 40⇐⇒ √xy = 40⇐⇒ xy = 1, 600⇐⇒ y = 1, 600
x
.
Thus the equation of the indifference curve corresponding to an amount of
utility equal to 40 “utils” is
y =
1, 600
x
.
Note that the graph of this indifference curve will be a rectangular hyper-
bola.
Part 2
The slope of the indifference curve corresponding to an amount of utility
equal to 40 “utils” is given by
dy
dx
=
d
dx
(
1, 600
x
)
=
d
dx
1, 600x−1 = −1, 600x−2 = −
(
1, 600
x2
)
.
We can show that the marginal rate of substitution of x for y for this utility
function is given by
MRSx,y (x, y) =
y
x
.
(Exercise: Show this.) At first glance, it might appear that
dy
dx
6= −MRSx,y (x, y)
in this case (when U = 40). Oh ye of little faith! Recall that the equation
for the indifference curve corresponding to a utility level of forty “utils”
was
y =
1, 600
x
.
Upon substituting this into the marginal rate of substitution expression,
we obtain
MRSx,y (x, y) =
y
x
=
(
1,600
x
)
x
=
1, 600
x2
,
so that we do indeed have
dy
dx
=
1, 600
x2
= −MRSx,y (x, y) .
34
Part 3
The utility function is
U (x, y) = U(x, y) = x0.5y0.5 =
√
x
√
y =
√
xy.
The indifference curve corresponding to an arbitrary, but given, amount of
utility, say k > 0, is given by the condition
U (x, y) = k ⇐⇒ √xy = k ⇐⇒ xy = k2 ⇐⇒ y = k
2
x
.
Thus the equation of the indifference curve corresponding to a utility level
of k > 0 is
y =
k2
x
.
Note that the graph of any such indifference curve (when the utility level
is k > 0) will be a rectangular hyperbola.
Part 4
The slope of the indifference curve corresponding to a utility level of k > 0
is given by
dy
dx
=
d
dx
(
k2
x
)
=
d
dx
k2x−1 = −k2x−2 = −
(
k2
x2
)
.
We can show that the marginal rate of substitution of x for y for this utility
function is given by
MRSx,y (x, y) =
y
x
.
(Exercise: Show this.) At first glance, it might appear that
dy
dx
6= −MRSx,y (x, y)
in this case (when U = k > 0). Oh ye of little faith! Recall that the
equation for the indifference curve corresponding to a utility level of k > 0
was
y =
k2
x
.
Upon substituting this into the marginal rate of substitution expression,
we obtain
MRSx,y (x, y) =
y
x
=
(
k2
x
)
x
=
k2
x2
,
so that we do indeed have
dy
dx
=
k2
x2
= −MRSx,y (x, y) .
35
Answer for the Additional Practice Question 8
Part 1
The utility function is
U (x, y) = x+
√
y = x+ y
1
2 .
The marginal utility of commodity x is
MUx (x, y) =
∂U (x, y)
∂x
= 1.
The marginal utility of commodity y is
MUy (x, y) =
∂U (x, y)
∂y
=
(
1
2
)
y−(
1
2) =
1
2
√
y
.
The marginal rate of substitution of commodity x for commodity y is
MRSx,y (x, y) =
MUx (x, y)
MUy (x, y)
=
1(
1
2
√
y
) = 2√y.
Part 2
The utility function is
U (x, y) = x+ ln y.
The marginal utility of commodity x is
MUx (x, y) =
∂U (x, y)
∂x
= 1.
The marginal utility of commodity y is
MUy (x, y) =
∂U (x, y)
∂y
=
1
y
.
The marginal rate of substitution of commodity x for commodity y is
MRSx,y (x, y) =
MUx (x, y)
MUy (x, y)
=
1(
1
y
) = y.
Part 3
The utility function is
U (x, y) = (x− x0)α (y − y0)1−α .
36
The marginal utility of commodity x is
MUx (x, y) =
∂U (x, y)
∂x
= α (x− x0)α−1 (y − y0)1−α .
The marginal utility of commodity y is
MUy (x, y) =
∂U (x, y)
∂y
= (1− α) (x− x0)α (y − y0)1−α−1
= (1− α) (x− x0)α (y − y0)−α .
The marginal rate of substitution of commodity x for commodity y is
MRSx,y (x, y) =
MUx (x, y)
MUy (x, y)
=
α (x− x0)α−1 (y − y0)1−α
(1− α) (x− x0)α (y − y0)−α
=
α (y − y0)
(1− α) (x− x0) .
Part 4
The utility function is
U (x, y) = (αxρ + βyρ)
1
ρ .
The marginal utility of commodity x is
MUx (x, y) =
∂U (x, y)
∂x
=
(
1
ρ
)
(αxρ + βyρ)(
1
ρ)−1 ραxρ−1
= αxρ−1 (αxρ + βyρ)(
1
ρ)−1 = αxρ−1 (αxρ + βyρ)
1−ρ
ρ .
The marginal utility of commodity y is
MUy (x, y) =
∂U (x, y)
∂y
=
(
1
ρ
)
(αxρ + βyρ)(
1
ρ)−1 ρβyρ−1
= βyρ−1 (αxρ + βyρ)(
1
ρ)−1 = βyρ−1 (αxρ + βyρ)
1−ρ
ρ .
The marginal rate of substitution of commodity x for commodity y is
MRSx,y (x, y) =
MUx (x, y)
MUy (x, y)
=
αxρ−1 (αxρ + βyρ)
1−ρ
ρ
βyρ−1 (αxρ + βyρ)
1−ρ
ρ
=
αxρ−1
βyρ−1
=
αy1−ρ
βx1−ρ
=
(
α
β
)(y
x
)1−ρ
.
37