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ECON 7030 Microeconomic Analysis
创建日期：2024-05-27 16:49:53
所属分类：经济Economics
标签： ECON 7030
Microeconomic Analysis

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ECON 7030 Microeconomic Analysis

Lecture 6: Demand
University of Queensland
Semester 1, 2022
Last year. . .
• We have seen how we can find utility maximising and
expenditure minimising bundles
• You may think: Let’s practise calculating the optimal bundle!
• That’s not a bad idea for assessments, but is of limited use in
economic studies
Finding Optimal Bundles is Just the Beginning
• Utility is a theoretical construct, it is not observable
• In applications, we can rarely get the “real” utility function of
an individual and calculate his/her optimal consumption
bundle
• Also, since utility is not observable, if I am looking at a single
choice, I can always come up with an ad-hoc utility function
that makes that choice optimal
Goal: Testable Predictions
• The purpose of utility maximisation and expenditure
minimisation is to be able to give predictions on consumers’
behaviours
• We also want these predictions to be testable — that is,
empirical observations or data can (at least potentially)
differentiate our predictions against alternative predictions.
• One way to do it is to look at how choices change when
observable variables change
• This is also useful for applications, as while we may not have
a “real” utility function, we may still predict the effect of
certain policies
What Are We After?
• Recall that the condition for a utility maximising bundle is
given by
Ux(x , y)
Uy (x , y)
=
Px
Py
Pxx + Pyy = M
• There are two kinds of variables
Endogenous Variables (Choice Variables): x , y
Exogenous Variables (Parameters): Px , Py , M
• What we want: To be able to say how the endogenous
variables change when an exogenous variable change
(This is known as Comparative Statics)
Utility Maximising Bundle as a Function of the
Parameters
• We have two equations in two unknowns:
Ux(x
∗, y∗)
Uy (x∗, y∗)
=
Px
Py
Pxx
∗ + Pyy∗ = M
• Unless we are really unlucky, a solution exists — we will call it
(x∗, y∗)
• Notice that x∗ and y∗ depends on Px , Py and M — when the
value(s) of Px , Py and M change(s), the values of x
∗ and y∗
change
• In other words, x∗ and y∗ are functions of Px , Py and M
(Marshallian) Demand
• We have
x∗ = x∗(Px ,Py ,M)
y∗ = y∗(Px ,Py ,M)
• These functions express the quantity of x and y the consumer
is willing and able to purchase given the prices and income
• Or, these function express the quantity demanded for Good X
and Good Y given the prices and income
• In other words, the functions x∗ and y∗ are our (Marshallian)
demand functions
Comparative Statics
x∗ = xm(Px ,Py ,M)
y∗ = ym(Px ,Py ,M)
• There are three changes (in x∗) that we are interested in
Own Price Effect Change in x∗ due to a change in Px
Cross Price Effect Change in x∗ due to a change in Py
Income Effect Change in x∗ due to a change in M
• Associated with these effects are three elasticities:
Own Price Elasticity % change in x∗ / % change in Px
Cross Price Elasticity % change in x∗ / % change in Py
Income Elasticity % change in x∗ / % change in M
• We are more interested in elasticities than just changes
because elasticity is unit free — even if you change the unit of
measurements, you are not going to change the elasticity
Elasticity: In General
• Suppose I have two variables, v and w , and suppose
v = v(w , α), where α is some other parameters
• The elasticity of v with respect to w is
% change in v
% change in w
=
∆v
v × 100
∆w
w × 100
=
w
v
∆v
∆w
=
w
v
∂v
∂w
A Little Trick for Those Interested
• If both v and w are always positive,
w
v
∂v
∂w
=
∂ ln v
∂ lnw
• Given the Cobb-Douglas Utility, U(x , y) = xαy1−α
We can solve for the utility-maximising x∗ and get
x∗ = α
M
Px
Hence
ln x∗ = lnα + lnM − lnPx
• Which gives
Own Price Elasticity =
∂ ln x∗
∂ lnPx
= −1
Income Elasticity =
∂ ln x∗
∂ lnM
= 1
Digression: A Little Trick for Econometric Estimations
• Due to the little trick, when you run your regression in
econometrics, you can estimate
ln x∗ = β0 +β1 lnPx +β2 lnPy +β3 lnM + other controls, etc.
• Then β1, β2 and β3 will give you the own-price elasticity,
cross-price elasticity and income elasticity, respectively
Rest of This Lecture
1 Change in Income
2 Change in Price
3 Featured Example
Part I
Change in Income
Change in Income
x
y
U0
M
M ′
xA
yA A
B
C
D
• Question: Where should the new bundle be? Like B, C , or D?
Answer:
Both Goods are Normal
x
y
U0
U1
M
M ′
xA
yA
xC
yC
A
C
Both X and Y are normal goods
• The consumption of a normal good increases when income
increases
X is Inferior and Y is Normal
x
y
U0
U1
M
M ′
xA
yA
xB
yB
A
B
X is an inferior good
Y is a normal good
• The consumption of an inferior good decreases when income
increases
X is Normal and Y is Inferior
x
y
U0
U1
M
M ′
xA
yA
xD
yD
A
D
X is a normal good
Y is an inferior good
Can Both Goods Be Inferior?
x
y
U0
M
M ′
xA
yA
xE
yE
A
E
• If consumption of both goods falls, the consumer will not
exhaust his/her budget
Budget Balancedness
• In fact, the prediction that the consumer will exhaust his/her
budget is one that we can test
• We know
Pxx
∗(Px ,Py ,M) + Pyy∗(Px ,Py ,M) = M
When there is a change in M,
Pxx
∗
M
M
x∗
∆x∗
∆M
+
Pyy
∗
M
M
y∗
∆y∗
∆M
= 1
sxηx + syηy = 1
where sx and sy are the expenditure shares of x and y ,
respectively
and ηx and ηy are the income elasticities of x and y ,
respectively
Budget Balancedness
• The equation
sxηx + syηy = 1
says that the weighted average (by expenditure shares) of
income elasticities of all goods has to be 1
• This equation extends to more than 2 goods as well
• This is something that we can test with empirical data
• Also this rules out some possibilities
Some Terminology
Income Elasticity Category Sub-category
Negative Inferior Good
Positive, < 1
Normal Good
Necessity
Positive, ≥ 1 Luxury
• Since the weighted average of income elasticities has to be 1,
there must be at least one luxury good
Engel Curve
• If we trace the path of consumption bundles as income
changes, we get an Engel Curve
0
x
y
Engel CurveM1
U1
xA
yA
M2
U2
xB
yB
M3
U3
xC
yC
Engel Curve can bend
• A good may be normal at some income range but inferior at
another
0
x
y
Engel Curve
M1
U1
xA
yA
M2
U2
xB
yB
M3
U3
xC
yC
Going from Engel Curve to Demand Curves
0
x
y
Engel Curve
xA
yA
xB
yB
xC
yC
0
x
Px
Px
D1 D2 D3
When Income Increases: Summary
Normal Goods Inferior Goods
η Positive
Negative
Demand shifts. . . Right
Left
Engel Curve slope upward* Downward**
*Both goods are normal.
**One good is inferior and another good is normal.
Part II
Change in Price
Change in Price
0
x
y
M1
slope=− px
py
U1
A
xA
yA
slope=− p′x
py

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ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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