finance 30210: managerial economics consumer demand analysis
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Finance 30210: Managerial Economics
Consumer Demand Analysis
We can begin our representation of the consumer with a demand function…
csd PPIPDQ ,,,
Quantity demanded
Is a function of…
Price IncomePrices of Substitutes
Prices of Compliments
Quantity
10,40,100 cs PPID
Price
$20
40
10,40,100,2040 D
Given a demand function we can characterize the behavior of demand with elasticity..
Quantity
cs PPID ,,
Price
P
Qdp
%
%
$20
$15
40 60
%25% P
%50% dQ
225
50
p
Price elasticity will always be a negative number
Quantity
100ID
Price
$20
40 56
410
40I
I
QdI
%
%
Income elasticity will generally be a positive number
Given a demand function we can characterize the behavior of demand with elasticity..
110ID
%40% dQ
%10% I
Quantity
40sPD
Price
$20
40 44
5.20
10
sP
s
dP P
Qs
%
%
Cross price elasticity will be a positive number for substitutes
Given a demand function we can characterize the behavior of demand with elasticity..
48sPD
%10% dQ
%20% sP
Quantity
14cPD
Price
$20
32 40
5.40
20
I
c
dP P
Qc
%
%
Cross price elasticity will be a negative number for compliments
Given a demand function we can characterize the behavior of demand with elasticity..
10sPD
%20% dQ
%40% sP
Note that if the demand relationship is linear, elasticity is not constant
csd ePdPcIbPaQ
Q
Pb
Q
P
P
Q
P
Q
p
p
dp
%
%
Q
Ic
Q
I
I
Q
I
Q
I
p
dp
%
%
Q
Pd s
Ps
Q
Pe c
Pc
For example….
IPQd 0025.260
Quantity
000,40ID
Price
$30
100
1100
000,400025.
6.100
302
I
p
$80
Let’s try something a little more complicated…a non-linear demand relationship
22
1
02.2100 IPQd
Quantity
20ID
Price
$400
29.
68
400 2
12
1
2
1
Q
P
Q
PP
Q
P
P
Qp
$2916
68
24.
68
2004.04.04.
22
Q
I
Q
II
Q
I
I
QI
Sometimes we use demand functions that are linear in logs…
Quantity
40ID
Price
$12
40
IPd eQ 012.15.150
LN(Quantity)
40ID
Price
$12
3.7
IPd eLNQLN 012.15.150
.15 .012150 P IdLN Q LN LN e
IPQLN d 01215.5
Quantity
40ID
Price
$12
40
IPd eQ 012.15.150
8.11215.15.150
150)15(.15.150
012.15.
012.15.012.15.
P
e
Pe
Q
Pe
Q
P
P
QIP
IPIP
p
48.40012.012.150
150)012(.012.150
012.15.
012.15.012.15.
I
e
Ie
Q
Ie
Q
I
P
QIP
IPIP
I
A little math trick…recall the derivative of the natural log
xx
x 1ln
x
xx
ln
This just says that the difference in logs is a percentage change
Therefore, if we start with elasticity
PP
Q
P
Q
P
Qp
ln
%
ln
%
%
LN(Quantity)
40ID
Price
$12
3.7
IPQLN d 01215.5
8.11215.15.ln
%
%
PPP
Q
P
Qp
48.40012.012.ln
%
%
III
Q
I
QI
Sometimes we use demand functions that are linear in logs…
Sometimes we use demand functions that are linear in logs…
Quantity
40ID
Price
$10
3
IQ ePe d 045.65.16
Quantity
40ID
LN(Price)
2.3
3
IQ ePLNeLN d 045.65.16
Id eLNPLNLNQ 045.65.16
IPLNQd 045.65.7.2
Sometimes we use demand functions that are linear in logs…
Quantity
40ID
LN(Price)
2.3
3
IPLNQd 045.65.7.2
22.3
65.165.
1
ln%
%
QQP
Q
P
Qp
6.3
40045.045.
%
%
Q
I
Q
I
I
Q
I
QI
Sometimes we use demand functions that are linear in logs…
Quantity
40ID
Price
$5
8.8
4.12.135. IPQd
LN(Quantity)
40ID
LN(Price)
1.6
2.2
4.12.135. IPLNQLN d
4.12.135. ILNPLNLNQLN d
ILNPLNQLN d 4.12.104.1
Sometimes we use demand functions that are linear in logs…
LN(Quantity)
40ID
LN(Price)
1.6
2.2
ILNPLNQLN d 4.12.104.1
2.1ln
ln
%
%
P
Q
P
Qp
4.1ln
ln
%
%
I
Q
I
QI
Log linear demand curves have constant elasticities!
Suppose that you setting prices for US Air. You know that you face the following demand curve for the New York/Chicago Shuttle…
Quantity
Price
$200
400
PQ 41200
$80,000 If you wanted to increase revenues, should you increase or decrease your price?D
Quantity
Price
$199
400
PQ 41200
$80,396
D
PQTR
PPTR 41200
241200 PPTR
PP
TR81200
40020081200
P
TR
If you wanted to increase revenues, should you increase or decrease your price?
You should decrease price. A $1 price decrease will raise revenues by $400
$200
404
Why didn’t revenue go up by $400?
Quantity
Price
$150
400
PQ 41200
$90,000D
241200 PPTR
081200
PP
TR
150$P
Suppose that you wanted to maximize revenues?
$200
600
Now, let’s go about this a little differently…
Quantity
Price
$200
400
PQ 41200
$80,000
2400
2004
p
PQTR
QPTR %%%
PPTR p %%%
PTR p %1%
PTR %%
Every 1% drop in price will raise revenues by 1%
D
Using the point elasticity gives you the same answer…
Quantity
Price
$200
400
PQ 41200
$86,400
$180
480
%8100*000,80
000,80400,86
Why didn’t revenues go up by 10%?
5.1480
1804
p
PTR %5.%
D
We could also maximize revenues using elasticity…
Quantity
D
Price
$150
600
PQ 41200 PTR p %1%
When the elasticity hits -1, revenues stop growing when you lower your price.
14
Q
Pp
141200
4
P
P
PP 412004 150$P
$90,000
Quantity
D
Price
$150
600
PQ 41200
Quantity
D
Price
$200
600
$250
$140
$50
$90,000 2.0p
875.p
2p
5p
PDQ
Obviously, the more information you have, the better decisions you will make…
The best pricing decisions would come from a demand curve
However, knowing a few elasticities is quite helpful as well!
Revenue maximizing price Revenue maximizing price
Suppose that you setting prices for US Air. You know that you face the following demand curve for the New York/Chicago Shuttle…
IPQ 241000 Suppose that median income is equal to $50,000.
Income in thousands
Quantity
D
Price
$125
600
$75,000
Suppose that a recession causes a 20% drop in income. How much would we have to lower our price if we wanted to keep sales constant?
40241000600 P
Quantity
D
Price
$125
600
$72,000
Suppose that a recession causes a 20% drop in income. How much would we have to lower our price if we wanted to keep sales constant?
Now income is $40,000
IPQ 241000
120P
$120
20102 Q
2054 Q
Now, again with elasticity…
Suppose that median income is equal to $50,000.
Quantity
D
Price
$125
600
$75,000
83.600
1254
p
17.600
502
I
Suppose that a recession causes a 20% drop in income. How much would we have to lower our price if we wanted to keep sales constant?
IPQ 241000
Quantity
D
Price
$125
600
4.32017.% Q
4.31.483.% Q
IPQ Ip %%%
2017.%83.0 P
$123
1.4% P
1.4% P
Suppose that a recession causes a 20% drop in income. How much would we have to lower our price if we wanted to keep sales constant?
83.p
17.I
$73,800
Quantity
D
Price
$125
600
$75,000
Quantity
D
Price
$125
600
$75,000
83.p
17.I
IPQ 241000 IPDQ ,
Again, the more information you have, the better decisions you will make…
The best pricing decisions would come from a demand curve
However, knowing a few elasticities is quite helpful as well!
Now, suppose that you realized that you actually faced two types of customers : Recreational and business travelers. Could you do better?
Quantity
D
Price
$150
500Quantity
D
Price
$150
100
PQB 2800
Business
PQR 2400
Recreational
$400
$200
$75,000
$15,000
6.p
3p
Recall the aggregate demand curve we had previously….what we had was actually a piece of what aggregate demand really looked like
Quantity
D
Price
$200
600
PQB 2800
$400
$90,000
PQR 2400
At a price above $200, recreational travelers don’t fly. The only demand is coming from business travelers.
At a price below $200, we now have two types of demanders flying.
+
PQ 41200 $150
400
Could we do better?
PQ 2800
Suppose that we decided to ignore recreational travelers and cater to business clients…
Quantity
D
Price
$200
600
$400
$80,000PQ 41200
400
No…that’s not a good idea!
PQ 2800 22800 PPPQTR 04800
PP
TR
200$P
Why don’t we just charge them different prices?
Quantity
D
Price
$200
400Quantity
D
Price
$100
200
200$
4800
2800
28002
P
PP
TR
PPTR
PQB
Business Recreational
$400
$200
$80,000
$20,000
100$
4400
2400
24002
P
PP
TR
PPTR
PQR
1p1p
Quantity
Price
$200
$400
The real question is: Is this feasible to do empirically?
This is “the Truth”. Two types of individuals making purchasing decisions. Individual decisions added up across all individuals create an aggregate demand curve.
PQ 2400 PQ 2800
$200 ,41200
200$ ,2800
PP
PPQ
Quantity
Price
$200
$400
If you do a linear estimation, what you end up with something like this…not really what we want
bPaQ
$200 ,41200
200$ ,2800
PP
PPQ
bPaQ
However, if you put a dummy variable in the regression for business/recreational traveler, you could get at the truth…
Quantity
Price
$200
$400
PQ 2400 PQ 2800
cDbPaQ bPaQ
cDbPaQ
Business ,1
Recreation ,0D
YES! We can do this!
Now, lets change it up a little….
QuantityD
Price
$150
450Quantity
D
Price
$150
150
PQB 600
Business
PQR 3600
Recreational
$600
$200
$67,500
$22,500
33.p
3p
Now, if we charge different prices….
QuantityD
Price
$300
300Quantity
D
Price
$100
300
$600
$200
$90,000
$30,000
300$
2600
600
6002
P
PP
TR
PPTR
PQB
Business
1p
100$
6600
3600
36002
P
PP
TR
PPTR
PQB
Recreational
1p
Quantity
Price
$200
$600
Again, we have to ask: Is this feasible to do empirically?
This is “the Truth”. Two types of individuals making purchasing decisions. Individual decisions added up across all individuals create an aggregate demand curve.
PQR 3600
PQB 600
PQ 41200
$300
PQB 600
PQR 3600 +
Quantity
Price
$200
$600
PQR 3600
PQB 600
bPaQ
If you do a linear estimation, what you end up with something like this…
bPaQ
Quantity
Price
$200
$600
PQR 3600
PQB 600
cDbPaQ
What if we tried the dummy variable trick…
cDbPaQ
Business ,1
Recreation ,0D
bPaQ
We’re Screwed! Time to try something else.
Here’s the process that takes place in the economy…
Individual consumers have preferences over a variety of goods…they have limited incomes and face market prices. Consumers make choices on what to buy
P
QD
P
QD
P
QD
P
QD
Individual decisions can be represented by individual demand curves
P
QD
The market aggregates those decisions into a market demand curve
Here is the problem we face…
P
QD
We can estimate a market demand curve…
P
QD
P
QD
P
QD
P
QD
The problem is that the market demand often tells us very little about what is happening in the background…
So, here is how we attack the problem…
We see if we can come up with a numerical representation of preferences…
P
QD
P
QD
P
QD
P
D
We then derive the resulting demand curves that come from the consumer choice problem…
P
QD
We then aggregate the individual demand curves to get a market demand. This we can compare to aggregate data to see if we are correct
How do we get an understanding of consumer preferences? We observe what they do!
Television ACost = $500
Television BCost = $2500
Suppose you walk into the store with a choice between two TVs.
Suppose that you choose Television A
Either you prefer Television A or you can’t afford Television B
Suppose that you choose Television B You must prefer Television B
Suppose that you observed the following consumer behavior
P(Bananas) = $4/lb.
P(Apples) = $2/Lb.
Q(Bananas) = 10lbs
Q(Apples) = 20lbs
P(Bananas) = $3/lb.
P(Apples) = $3/Lb.
Q(Bananas) = 15lbs
Q(Apples) = 15lbs
What can you say about this consumer?
Is strictly preferred to
Choice A
Choice B
Choice B Choice A
How do we know this?
Consumers reveal their preferences through their observed choices!
P(Bananas) = $4/lb.
P(Apples) = $2/Lb.
Q(Bananas) = 10lbs
Q(Apples) = 20lbs
P(Bananas) = $3/lb.
P(Apples) = $3/Lb.
Q(Bananas) = 15lbs
Q(Apples) = 15lbs
Cost = $80 Cost = $90
Cost = $90 Cost = $90
B Was chosen even though A was the same price!
Choice A Choice B
What about this choice?
P(Bananas) = $2/lb.
P(Apples) = $4/Lb.
Q(Bananas) = 25lbs
Q(Apples) = 10lbs
Q(Bananas) = 10lbs
Q(Apples) = 20lbs
Cost = $90
Q(Bananas) = 15lbs
Q(Apples) = 15lbsCost = $90
Cost = $100
Is strictly preferred to Choice C Choice B
Choice C
Is choice C preferred to choice A?
Choice B
Choice A
Is strictly preferred to Choice B Choice A
Is strictly preferred to Choice C Choice B
Is strictly preferred to Choice C Choice A
Rational preferences exhibit transitivity
C > B > A
Consumer theory begins with the assumption that every consumer has preferences over various combinations of consumer goods. Its usually convenient to represent these preferences with a utility function
BAU :
A BU
Set of possible consumption choices “Utility Value”
Q(Bananas) = 25lbs
Q(Apples) = 10lbs
Q(Bananas) = 10lbs
Q(Apples) = 20lbs
Q(Bananas) = 15lbs
Q(Apples) = 15lbs
Choice C
Choice A
Choice B
Using the previous example (Recall, C > B > A)
)20,10()15,15()10,25( UUU
We require that utility functions satisfy a few basic properties
20),( yxUx
y
A
B
C
BUCU
BUAU
AUCU
)()(
There is a definite ranking of all choices (i.e. transitivity)
25),( yxU
This tells us that indifference curves can’t cross on another
20),( yxU
x
y
A
B
C BUCU
BUAU
AUCU
)()(
We run into a problem!
25),( yxU
Suppose that indifference curves did cross…
)()( CUAU
20),( yxUx
y
A
B
C
More is always better!
)()( AUCU
We require that utility functions satisfy a few basic properties
20),( yxUx
y
A
B
C
People Prefer Moderation!
)()( AUCU
15
5 15
5
10
10
25),( yxU
We require that utility functions satisfy a few basic properties
Note: This is a result of diminishing marginal utility…this guarantees that demand curves slope down!
20),( yxUx
yA
BC
People prefer extremes!
)()( AUCU
25),( yxU
What if we didn’t have decreasing marginal utility?
Increasing marginal utility produces some weird decisions!!
x
y
*y
*x
20),( yxU
We can characterize preferences with a few statistics. First, how does a consumer prefer one good relative to another.
x
yMarginal Utility of Y
Marginal Utility of X
),(
),(**
**
yxU
yxUMRS
x
y
y
x
The marginal rate of substitution (MRS) measures the amount of Y you need to be get in order to give up a little of X
The marginal rate of substitution (MRS) measures the amount of Y you require to give up a little of X
x
y
*y
*x
20),( yxU
)','(),( ** yxMRSyxMRS
'y
'x
If you have a lot of X relative to Y, then X is much less valuable than Y MRS is low!
The elasticity of substitution measures the curvature of the indifference curve
x
y'
x
y
x
y
MRSxy
%
%
Elasticity of substitution measures the degree to which your valuation of X depends on your holdings of X
y
x
small is
y
x
large is
The elasticity of substitution measures the curvature of the indifference curve
If the elasticity of substitution is small, then small changes in x and y cause large changes in the MRS
If the elasticity of substitution is large, then large changes in x and y cause small changes in the MRS
MRSxy
%
%
X
Y
X
Y
Elasticity of Substitution measures the degree in which you can alter the mix of goods. Consider a couple extreme cases:
Perfect substitutes can always be can always be traded off in a constant ratio
Perfect compliments have no substitutability and must me used in fixed ratios
Elasticity is 0Elasticity is Infinite
Consumers solve a constrained maximization – maximize utility subject to an income constraint.
),(max,
I ypx ptosubject
yxU
yx
yx
As before, set up the lagrangian…
)(),(),( ypxpIyxUyx yx
)(),(),( ypxpIyxUyx yx
First Order Necessary Conditions
0),( xx pyxU
ypxpI yx
0),( yy pyxU
y
x
y
x
P
P
yxU
yxU
),(
),(
x
x
y
y
p
yxU
p
yxU ),(),(
y
x
),(max0,0
I ypx ptosubject
yxU
yx
yx
xp
I
yp
I
*y
*x
y
x
y
x
P
P
yxU
yxU
),(
),(
ypxpI yx
y
x
Individual demand Curves represent a summary of the individual consumer choice problem
x
xp
*x
xp
*x
ypID ,
yp
I
xp
I
y
x x
y
x x
small is MRS
large is MRS
xp
xp
Willingness to pay is low
Willingness to pay is high
The marginal rate of substitution controls the height of the demand curve
ypID ,
ypID ,
y
x*x
The elasticity of substitution will effect the price elasticity of the demand curve
x
xp
'x *x
xp
xp'
D
MRSxy
%
%
xx p
x
%
%
yp
I
xp
I
xp
I
''x
y
x x
xp
small is small is x
y
x x
xp
large is large is x
Elasticity of Substitution vs. Price Elasticity
y
x x
xp
0 0x
y
x x
xp
x
Perfect Complements vs. Perfect Substitutes
An Example: Cobb-Douglas Utility
yxyxU ),(
yxyxU x1),(
1),( yxyxU y
x
y
yx
yx
yxU
yxU
y
x
1
1
),(
),(
),(
),(**
**
yxU
yxUMRS
y
x Recall, Marginal Rate of Substitution measures the relative value of x. This will determine the intercept of the demand curve
1
x
yMRS
xy
MRS
MRSxy
MRSxy
*%
% 1
xy
xy
xy
MRS
MRSxy
%
%
Recall, elasticity of substitution is measuring the degree of flexibility in the consumption of X and Y and will determine the slope of the demand curve
max 5.5.
0,0
I ypx ptosubject
yx
yx
yx
)(),( 5.5. ypxpIyxyx yx
y
x
y
x
p
p
yx
yx
yxU
yxU
5.5.
5.5.
5.
5.
),(
),(x
p
py
y
x
An Example: Cobb-Douglas Utility
Marginal Rate of Substitution
max 5.5.
0,0
I ypx ptosubject
yx
yx
yx
Ixp
ppxp
y
xyx
Iypxp yx
y
xp
Ix
2
yp
Iy
2
An Example: Cobb-Douglas Utility
x
xp
*x
xp
x
p
p
x
p
x x
xxx
%
%
xp
Ix
2
, 5.5. yxyxU
22 xx p
I
p
x
1
2
2 2
x
x
xx
pI
p
p
I
Constant price elasticity
max 5.5.
0,0
I ypx ptosubject
yx
yx
yx
xp
Ix
2 0
%
%
x
p
p
x
p
x y
yyy
Zero cross price elasticity
max 5.5.
0,0
I ypx ptosubject
yx
yx
yx
xp
Ix
2
x
I
I
x
I
xI
%
%
1
2
2
1
x
x
pI
I
p
Constant income elasticity
max 5.5.
0,0
I ypx ptosubject
yx
yx
yx
xp
Ix
2
yp
Iy
2
Ipx x lnln2
1lnln
Ipy y lnln2
1lnln
Log-linear demand curves…
Returning to our airline example…suppose we used a customer survey or some other method and came up with preferences of airline travelers
Business Recreational
, 95.05. yAyAU
, 98.02. yAyAU
Airline Travel Everything Else
95.
05.
y
AMRS
1
98.
02.
y
AMRS
1
Different intercepts but the same slope…
Returning to our airline example…suppose we used a customer survey or some other method and came up with preferences of airline travelers
Business Recreational
, 95.05. yAyAU
, 98.02. yAyAU
Airline Travel Everything Else
IpA lnln05.lnln
p
IA
05.
IpA lnln02.lnln
p
IA
02.
LN(Quantity)
LN(Price)
IpAB lnln05.lnln
IpAR lnln02.lnln +
IpA ln2ln202.ln05.lnln
We end up with something like this…
BARA A
Business
Recreation
Aggregate
We could estimate a log linear demand curve with a dummy variable to check this
dDIcpbaA lnlnln
Suppose that we wanted different slopes…
1,1
yxyxU
The Cobb-Douglas utility function is a special case of the CES (Constant elasticity of substitution) utility functions…
1
1
x
yMRS
1
1
The parameter alpha will govern the marginal rate of substitution which influences the intercept of the demand curve
The parameter rho will govern the elasticity of substitution which influences the slope of the demand curve
Suppose that we wanted different slopes…Suppose that we wanted different slopes…
The Cobb-Douglas utility function is a special case of the CES (Constant elasticity of substitution) utility functions…
The resulting demand curve looks like this…
PIpx x lnlnlnlnln
A price index comprised of both good’s prices
1,1
yxyxU
Returning to our airline example…suppose we used a customer survey or some other method and came up with preferences of airline travelers
Business Recreational
95.
05.8.
y
AMRS
25.1
98.
02.2.
y
AMRS
5
95.05., 2.
12.2. yxyxU
98.02., 8.
18.8. yxyxU
Different intercepts and different slopes…
Returning to our airline example…suppose we used a customer survey or some other method and came up with preferences of airline travelers
PIpA lnlnln2.05.ln2.ln
Business Recreational
95.05., 2.
12.2. yxyxU
98.02., 8.
18.8. yxyxU
PIpA lnlnln8.02.ln8.ln
Quantity
Price
Now, we have something like this…
PIpAB lnlnln2.05.ln2.ln
PIpAR lnlnln8.02.ln8.ln +
PIpA ln2ln2ln02.ln8.05.ln2.ln
Business
Recreation
Aggregate
We could estimate a log linear demand curve to check this
PdIcpbaA lnlnlnln
RABA
A
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