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ECON7030 MICROECONOMIC ANALYSIS

创建日期：2024-05-27 15:37:03

所属分类：经济Economics

标签： ECON7030

MICROECONOMIC ANALYSIS

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ECON7030 MICROECONOMIC ANALYSIS

MOCK PROBLEM SET 2
Instruction:
This is an open-book problem set. You are NOT ALLOWED to discuss the problem
set and/ or the answers with anyone else. You can do your own research to find the
answers. If your answers are based on a reference, you MUST properly CITE the source
of such reference. The use of AI to develop responses is prohibited and may constitute
student misconduct under the Student Code of Conduct.
You have 24 hours to complete AND submit your answers by the due time. Hand-
written as well as typed answers are welcome. Make sure that you upload a single PDF
file containing multiple pages. Submit on time, ideally at least 1 hour in advance of the
deadline. It shows competence and professionalism, and it will ensure that you avoid late
penalties.
Answer all questions. Justify all your answers. Every graph (figure or diagram) in
your answers has to be well-labelled. For functions that intersect the axis, the points
where they intersect them must be identified (providing the corresponding numbers).
All quantities of goods can be treated as continuous variables unless explicitly stated
otherwise. Show your work.
Question 1 True, False, or Uncertain? Justify your answers.
(a) (5 marks) Inflation and the rising cost of living are currently a big problem. Suppose
that the prices of all the goods in your consumption bundle increase at the same rate
of 5%. The Union negotiated a nominal salary increase of the same rate, 5%. You
are equally well off.
Solution: True.
When the prices of all the goods in your consumption bundle increase at the same
rate (no substitution effect), an increasing in income by the same rate will move you
back to the original bundle that yield the same level of utility.
(b) (5 marks) Tom consumes only two goods, X and Y . He spends 40% of his income
in X. Tom’s income elasticity of demand for X is β, where β > 0. Tom’s (Marshal-
lian) demand for X has a price elasticity equal to −0.4β. X and Y are perfect
complements for Tom. [Hint: Slutsky equation.]
1
Solution:True.
Based on the Slutsky Equation:
mxx =
h
xx − sxηx
We know that sx = 0.4, ηx = β. Hence −sxηx = −0.4β. If mxx = −0.4β = −sxηx,
then hxx = 0 or the substitution effect is nil. As a result, good X and Y can only be
perfect complements.
(c) (5 marks) John has monotonic preferences for two goods, X and Y . The price of
Y is one dollar per unit. The following diagram illustrates one of his indifference
curves, labelled U0. It also shows two budget lines corresponding to income levels
M and M ′, respectively, while the prices of X and Y remain constant. A = (xA, yA)
is the optimal bundle when the income is M . When the income changes to M ′,
ceteris paribus, John’s new optimal bundle(s) must be located along the
blue section.
x
y
U0
M
M ′
xA
yA A
B
C
D
Solution: False or Uncertain
• If both x and y are normal goods and income increases from M to M ′, then he
will consume more of both x and y. Hence, the new optimal bundle would be
along the blue section such as C.
• If x is normal but y is inferior and income increases from M to M ′, then he will
consume more x but less y. Hence, the new optimal bundle would be along the
green section such as D.
• If x is inferior but y is normal and income increases from M to M ′, then he will
consume less x but more y. Hence, the new optimal bundle would be along the
red section such as B.
(d) (5 marks) Bob consumes only two goods, X and Y . Consider the price combinations
A, B and C, in the following table. Bob’s chosen bundle in each situation is also
shown in the table. Bob’s preference satisfies both WARP and SARP.
2
Situation Px Py X Y
A $2 $4 6 2
B $5 $5 3 5
C $8 $2 5 4
Solution: False.
Bob’s preference satisfies WARP but violates SARP. Let’s compute the costs of each
bundle given different situation
Cost of bundle A Cost of bundle B Cost of bundle C Revealed Pref.
A 2(6)+4(2)=20 2(3)+4(5)=26 2(5)+4(4)=26 A R B, A R C
B 5(6)+5(2)=40 5(3)+5(5)=40 5(5)+5(4)=45 B R A, B R C
C 8(6)+2(2)=52 8(3)+2(5)=34 8(5)+2(4)=48 C R B, C R A
WARP:
(i) We have B R A and A R B.
(ii) We have C R B and B R C.
Hence, WARP is satisfied.
SARP: There is only one indirect revealed preference relation which can be identified
as the following:
C R B and B R A ⇒ C IR A
(i) We have B R A and A R B.
(ii) We have C R B and B R C.
(iii) We have C IR A and A R C
(iv) We have C IR A and C R A. This case does not satisfy SARP.
Hence, SARP is violated.
(e) (5 marks) Let
p: per-unit price of insurance,
pi: probability of accident,
L: loss wealth from the accident
X: amount of purchased insurance.
An insurance purchaser is an expected utility maximiser. The insurance is unfair
in favour of the insurance company; that is, the per-unit premium is higher than
the probability of insurance payout, p > pi. The insurance model suggests that
risk-averse consumers choose to be fully insured, i.e, X = L.
3
Solution: False. The risk-averse consumer will be under-insured.
Let c1 be a consumption with accident and c2 be a consumption without accident.
p > pi ⇒ p
1− p >
pi
1− pi
The slope of the BL is steeper than the slope of th IC at the certainty line. The
risk-averse utility maximiser will choose the bundle (c1, c2) such that
MRSc1,c2 =
piu′(c1)
(1− pi)u′(c2) =
p
1− p
For the above equation to hold and this consumer is risk-averse meaning u′(·) > 0
and u′′(·) < 0, thus
u′(c1)
u′(c2)
> 1
u′(c1) > u′(c2)
c1 < c2
Given that c1 = w − L + (1− p)X and c2 = w − pX where L is the loss if accident
occurs and X is the amount of insurance purchased.
The consumption without accident is greater than the consumption with accident,
c1 > c2 implies that this consumer is under-insured, i.e., X < L. In particular,
c1 < c2
w − L+ (1− p)X < w − pX
w − pX + (X − L) < w − pX
∴ X − L < 0
X < L
Question 2 The 2022 Australia Federal election is just around the corner. The compe-
tition between the two candidates, Scoot Morris and Antone Alba, is fierce.
As a consumer (and Australian voter), you only care about public transportation and
child care. Let public transportation be good X and child care be good Y . Each trip in
the public transportation costs $10 and each day of child care service costs $100. Your
after-tax income is $500 per week. Assume that you always consume a positive amount
of both X and Y (also known as interior bundles). Your utility function is given by:
U(x, y) = αxβ + 2y + 1,
4
xy
U1
U2
Figure 1. Vertically Parallel ICs
where α > 0 and 0 < β < 1.
(a) (5 marks) Write down the MRS in terms of α, β, x, and y.
Solution:
MRSxy =
MUx
MUy
=
αβxβ−1
2
=
αβ
2x1−β
MRSxy does not depend on y. This utility function is quasi-linear and it is strictly
concave in x. In particular,
U ′x = αβx
β−1 > 0
U ′′x = αβ(β − 1)xβ−2 < 0
(b) (5 marks) Draw two indifference curves. Do the indifference curves have the same
slope along any vertical line (when x is held constant)? Illustrate and explain.
Solution:Yes, the ICs have the same slope along any vertical line when x is held
constant. This is because the slope of the ICS, or MRSxy, does not depend on y.
Any bundles with the same amount of x, regardless of the amount of y, would have
the same MRSxy.
(c) (2 marks) Is it possible to establish whether Y is a normal or inferior good? Explain.
Solution: Yes, it is possible. Since there is no income effect on X, i.e, the income
elasticity of X is zero, X is neither normal nor inferior. Therefore, Y must be a
normal good. Recall that when analysing the 2-good situation, at least one good
must be a normal good.
5
(d) (5 marks) Based on your answer in part (c), is the price elasticity of Hicksian demand
for Y greater, smaller, or equal to that of Marshallian demand? Explain. DO NOT
derive the Hicksian and Marshallian demand functions for Y .
Solution: Recall that the price elasticity of demand for y is determined by:
y =
%∆y
%∆Py
When Py decreases by 10%, the substitution effect causes y
H to increases. Since y
is a normal good, the income effect reinforces the substitution effect and causes y to
increase even further. yM , consisting of both income and substitution effects, will
therefore increase more than yH . In particular, %∆yH < %∆yM .As a result,
Hy =
%∆yH
%∆Py
<
%∆yM
%∆Py
= My
We would expect the price elasticity of Hicksian demand for Y to be smaller than
than of Marshallian demand.
(e) Scoot Morris promises an income tax reduction that increases you after-tax weekly
income by 10%, from $500 to $550.
(i) (2 mark) Without solving the utility maximisation problem, establish how your
consumption of X and Y will change. Justify.
Solution: Since the ICs are parallel when x is constant, the increased income
will not affect the optimal x but will increase the optimal y.
(ii) (4 marks) Find the optimal consumption bundles with and without this tax
reduction.
Solution: Assume interior solution.
Tangency Condition:
MRSxy =
Px
Py
αβ
2x1−β
=
Px
Py
x1−β =
αβPy
2Px
x∗ =
(
αβPy
2Px
) 1
1−β
(1)
6
Budget Equation:
Pxx+ Pyy = M
Px
(
αβPy
2Px
) 1
1−β
+ Pyy = M
y∗ =
M − Px
(
αβPy
2Px
) 1
1−β
Py
(2)
Plug in M = 500, Py = 100, and Px = 10 into eq. (1) and (2), we get
x∗0 =
(
αβ × 100
20
) 1
1−β
= (5αβ)
1
1−β
y∗0 =
500− 10 (5αβ) 11−β
100
= 5− 0.1 (5αβ) 11−β
Next, Scoot’s campaign would increase M by 10% or M1 = 550. Therefore,
x∗1 =
(
αβ × 100
20
) 1
1−β
= (5αβ)
1
1−β
= x∗0
y∗1 =
550− 10 (5αβ) 11−β
100
= 5.5− 0.1 (5αβ) 11−β
> y∗0
In conclusion, Scoot’s campaign will not affect the amount of optimal X you consume
but you would increase Y consumption by 0.5 unit. Since your preference is strongly
monotone (you can easily verify this!), you would strictly prefer the new bundle with
more Y and the same amount X.
(f) Now consider Antone Alba’s campaign instead. He promises to reduce the child care
cost by 10%.
(i) (2 mark) Without solving the utility maximisation problem, establish how your
consumption of X and Y will change. Justify.
Solution: The totol (price) effect of the decrease in py includes both substitution
and income effect.
The effect of the decrease in py on X:
- Substitution effect: X decreases.
- Income effect: X remains unchanged*.
7
*There is no income effect on X due to the nature of the quasi-linear utility
function. The effect of the decrease in py on Y :
- Substitution effect: Y increases.
- Income effect: Y increases**.
**Since X is not a normal good, then Y must be a normal good.
Therefore, we could expect that the X consumption will decrease and the Y
consumption will increase.
(ii) (2 marks) Find the optimal consumption bundle with the reduced cost of child
care.
Solution:Antone’s campaign affects Py whereas Px and M remain the same. In
particular, P ′y = 90. Hence,
x∗2 =
(
αβ × 90
20
) 1
1−β
= (4.5αβ)
1
1−β
< x∗0 = x
∗
1
y∗2 =
500− 10 (4.5αβ) 11−β
90
=
50− (4.5αβ) 11−β
9
(iii) (5 marks) With a well-labeled diagram, illustrate the Hicksian decomposition of
the price effect of Antone’s campaign into the substitution effect and the income
effect. Do not forget to explain and label the diagram.
Solution:
8
y child
care)
A-→B : 5.E .
B→c : I. E. Co☒= 0)
5900-0=5.5-6
5
YE - - - - - -
--B&i•
'
,
!A
✗ ( Bus trip )
✗ 50
(g) (5 marks) Now assume that α = 1 and β = 0.2, who should win your vote, Scoot
Morris or Antone Alba? If you are indifferent between the two, choose a “silent”
vote. Show and explain only within the context of your previous answers and the
given information available. Please do not assert your personal political views here.
9
Solution: Let α = 1 and β = 0.2, then
x∗1 = (5αβ)
1
1−β
= (5× 1× 0.2) 10.8
= 1
y∗1 = 5.5− 0.1 (5αβ)
1
1−β
= 5.5− 0.1(5× 1× 0.2) 10.8
= 5.4
x∗2 = (4.5αβ)
1
1−β
= (4.5× 1× 0.2) 10.8
= 0.919
y∗2 =
50− (4.5αβ) 11−β
9
=
50− (4.5× 1× 0.2) 10.8
9
= 5.453
U(x, y) = αxβ + 2y + 1
= x0.2 + 2y + 1
U(x∗1, y
∗
1) = 1
0.2 + 2(5.4) + 1
= 12.800
U(x∗2, y
∗
2) = 0.919
0.2 + 2(5.453) + 1
= 12.889
Since U(x∗1, y
∗
1) < U(x
∗
2, y
∗
2), you should vote for Antone.
(h) Your sister, Josephine, is also an eligible voter who faces the same dilemma. She
earns the same as you. However, we do not know her utility function.
(i) (5 marks) During the family gathering on ANZAC day, Josephine indicated
that, if Scoot Morris wins, she will choose the bundle S = (3, 5.2). Assume
that Josephine’s preferences satisfy WARP. Draw the set of possible bundles
that Josephine may choose from if Antone wins. Call this set W . REDRAW
the diagram only with the budget lines under Scoot’s and Antone’s campaigns.
Illustrate the set W in this new diagram.

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CS 333 CP1406 KIT308 Economics 701 ECON3740 ECE 5759 BFC3340 FINC2012 COMP4500 MATH3871 FIT3080 18-819C Economic ECO202Y5Y PHY100 FINC-780 Econ 100B EC602 IE582 EE450 COMP9517 WRDS 150B ECON 570 LECTURE 2 ECA5101 ECE 4420 ADAD9311 FIT3031 CS 544 MF 703 CS 6140 KIT501 ENGSCI 344 CSC171 IE 332 FINM8007 21S2 COSC 1114 FE545 E471 SIT102 CSC108 MAT301 IAE 101 COP3503 STAT2004 EC 303 CS1026 STAT1203 MATH3171 ISOM5160 IFN657 FIT2014 ACCT2102 DSCI550 STATS 330 ELEC421 COMP3670 QBUS6840 MEES:698B MAST30027 MIET1077 Economics IFT 6758 FIT5122 BUSN5101 CptS 111 HRMT5502 Math2018 PP 312 INFS3200 CZ3005 CSSE1001 SENG 2011 MKTG 553Q BUFN 640 COMP507 ETF5952 DATA 311 ECE4043 BANK3011 finance M3H623544 HW1 INFS 7410 AI6103 STAT3888 INFS3208 ACCT7103 ECOS3013 PSYCH 20B F79MA STAT3006 ENGG952 2X MGSC 7000 ECE4132 MSF 526 MATH 3033 CDS 511 EEEE4120 EEET2465 ECA5103 MATH2070 ELEC3104 BU51019 EEE8147 ENGR20003 COSC 1107 ETF3200 FNCE 435 MSFI 435 ISyE6420 GENG4405 HW ETC5523 IE 7315 MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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