Download Econ 11 Lecture 5: Demand and Its Properties by Professor Jay Bhattacharya and more Schemes and Mind Maps Statics in PDF only on Docsity!
Spring 2001 Econ 11--Lecture 5 1
Demand Part I
- Recap: The Consumer’s Optimization Problem
- The budget constraint and the tangency condition determine the amount of each good the consumer will purchase.
The consumer’s choice of ( X 1 , X 2 ) (i.e. demand for X 1 and X 2 ) depends upon:
- prices ( p 1 , p 2 )
- income ( I )
- preferences—U( X 1 , X 2 ) X 1
X 2
Spring 2001 Econ 11--Lecture 5 2
Demand Functions
- A Marshallian demand function relates the quantity demanded of a good to prices and income
- Demand depends on all prices
- Preferences and constraints together determine the shape of demand
2 1 2
1 1 2
X g p p I
X f p p I
Spring 2001 Econ 11--Lecture 5 3
Comparative Statics
- What happens to demand when prices or income changes?
p 1
I 2 p 1
I
2 p 2
I
p 2
I
–e.g., if prices double and income doubles, what happens to demand?
Spring 2001 Econ 11--Lecture 5 4
- A function f ( x 1 , x 2 ,… xn ) is homogenous of degree k if
- Marshallian demand functions are homogenous of degree zero. This fact is consistent with the absence of “money illusion.”
X 1 ( p 1 , p 2 , I )= X 1 ( 2 p 1 , 2 p 2 , 2 Ι)
Zero Degree Homogeneity of
Demand
f ( tx 1 , txn ,... txn ) = tkf ( x 1 , xn ,... xn )
Spring 2001 Econ 11--Lecture 5 5
What happens to demand when income
changes?
- Budget constraint shifts in/out. Slope of budget constraint does not change.
Increasing income
x 1
x 2
Spring 2001 Econ 11--Lecture 5 6
Income Expansion Path
(Income-Offer Curve)
1
2 p
I 1
1 p
I 1
0 p
I
2
0 p
I
2
1 p
I
2
2 p
I Prices are fixed along the income expansion path
x 1
x 2
Spring 2001 Econ 11--Lecture 5 7
Engel Curves
- Engel curve relates income to quantity demanded. A “Normal” Good when income rises, the consumer buys more of x 1
x 1
Income (^) Spring 2001 Econ 11--Lecture 5 8
- But what if the IEP or Engel curve looks like this? An increase in income leads to more x 2 but less x 1. - x 1 is an “inferior” good.
IEP x^1 Engel Curve
x 2 x 1
Income
Spring 2001 Econ 11--Lecture 5 9
Normal and Inferior Goods
- Normal Good :Demand for a good x increases with income - This implies that the slope of the Engel curve is positive.
- Inferior Good :Demand for a good x decreases with income - This implies that the slope of the Engel curve is negative. < 0. ∂Ι
∂ X
∂Ι
∂ X
Spring 2001 Econ 11--Lecture 5 10
Examples
Normal Goods Beef Cars Haircuts at a salon
Inferior Goods Potatoes Bus tickets Haircuts by your mother
Spring 2001 Econ 11--Lecture 5 11
All Goods Can’t Be Inferior
Both Normal x 1 Inferior x 2 Normal
x 1 Normal x 2 Inferior
x 2
x 1
x 2
x 1
x 2
x 1
- “Proof” #1: If income expands, the IEP cannot point toward the origin.
IEP IEP IEP
Spring 2001 Econ 11--Lecture 5 12
All Goods Can’t Be Inferior
- Proof #2: use budget constraint.
11 22
∂Ι
x p
x p
px px
Both and ∂^ x^2 can’t be negative.
Thus, both x 1 and x 2 can’t be inferior goods.
∂ x 1
Spring 2001 Econ 11--Lecture 5 19
Engel’s Law
- Engel’s Law: “Food is a necessity”
- Expenditure on Food / Income
- 1935-1939 35.4%
- 1952 32.2%
- 1963 25.2%
- 1998 19%
Spring 2001 Econ 11--Lecture 5 20
- If x is a necessity, then as income increases, the share of income spent on x decreases: - Define the share of income spent on x as Sx - I will prove that if x is a necessity:
x =^ x
xp S
d Ι
dS (^) x
Spring 2001 Econ 11--Lecture 5 21
log Sx = log x +log px −log Ι
Totally differentiate this log “share” equation:
d log Sx = d log x + d log px −∆log Ι
Hold prices constant, i.e., set d log px = 0
Þ d log Sx = d log x − d log Ι
1 log
log log
log − Ι
Þ
d
d x d
d Sx
Þ = − 1
dI
dx x
I
dI
dS S
I (^) x x Spring 2001 Econ 11--Lecture 5 22
dI
dx x
I
dI
dS S
I (^) x x
Þ x = I , x − 1 x
e dI
dS S
I
but for necessities, eI , x <^1
Þ < 0 Þ < 0 dI
dS dI
dS S
I (^) x x x
so the data confirm Engel’s Law
Spring 2001 Econ 11--Lecture 5 23
- The expenditure weighted sum of income elasticities is equal to 1.
- Thus, all goods cannot be necessities. Nor can all goods be luxuries.
S 1 eI , 1 + S 2 eI , 2 = 1
1 =^11
x p
S
2 =^21
x p
S
where
Spring 2001 Econ 11--Lecture 5 24
- Proof: Start with the budget constraint
11 22
Ι
Þ
d
p dx d
pdx
px px
1
1
2 2
1 2 2 1
11
2 2 2 1 1 2 1
1
÷÷^ = ø
ö ççè
æ Ι
Ι ÷ ø ç ö è
æ Ι ÷÷+ ø
ö ççè
æ Ι
Ι ÷ ø ç ö è
æ Ι Þ
= Ι
Ι Ι
Ι
Ι Ι Þ
d
dx x
xp d
dx x
xp
d pdx x
x d pdx x
x
Þ S 1 eI , 1 + S 2 eI , 2 = 1
Spring 2001 Econ 11--Lecture 5 25
What happens to demand when price
changes?
x 2
x 1
“Price-Consumption Curve” or “Price-Offer” Curve
2
1 p slope =− p^2
1 p slope =− p ′
Spring 2001 Econ 11--Lecture 5 26
“Marshallian” Demand Curve
(Demand Curve)
- In the graph, we hold constant income and the prices of all other goods.
x 1
p 1 ′
p 1
p 1
Spring 2001 Econ 11--Lecture 5 27
The Law of Demand
- The ‘Marshallian” demand curve slopes downward (usually). - The “weak” law of demand. - It is theoretically possible for the Marshallian demand curve to slope upward.
- The “Marshallian demand curve is the demand curve that we most often use. Thus, we often just call it the “demand curve.”
