Demand Part I Demand Functions - Stanford University

Professor Jay Bhattacharya

Spring 2001

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.

X2

The consumer's choice of (X1,X2) (i.e. demand for X1 and X2) depends upon:

? prices (p1, p2) ? income (I) ? preferences--U(X1,X2)

X1

Spring 2001

Econ 11--Lecture 5

1

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

X1 = f ( p1, p2 , I ) X 2 = g( p1, p2 , I )

Spring 2001

Econ 11--Lecture 5

2

Comparative Statics

? What happens to demand when prices or income changes?

?e.g., if prices

I

double and

p2

income doubles,

what happens to

demand?

I

2 p2

I

I

2 p1

p1

Spring 2001

Econ 11--Lecture 5

3

Zero Degree Homogeneity of Demand

?

A function degree k if

f(fx(1tx,x1,2t,x...n ,.x..nt)xnis)

homogenous of

= t k f (x1, xn ,...xn )

? Marshallian demand functions are homogenous of degree zero. This fact is consistent with the absence of "money illusion."

X 1( p1, p2 , I ) = X1 (2 p1,2 p2 ,2)

Spring 2001

Econ 11--Lecture 5

4

What happens to demand when income changes?

? Budget constraint shifts in/out. Slope of budget constraint does not change.

x2 Increasing income

Spring 2001

Econ 11--Lecture 5

x1

5

Income Expansion Path

(Income-Offer Curve)

x2

I2 p2

Prices are fixed along the income

I1

expansion path

p2

I0

p2

Spring 2001

I0

I1

I2

p1

p1

p1

Econ 11--Lecture 5

x1

6

Econ 11--Lecture 5

1

Professor Jay Bhattacharya

Spring 2001

Engel Curves

? Engel curve relates income to quantity

demanded.

x1

A "Normal" Good when income rises, the consumer buys more of

x1

? But what if the IEP or Engel curve looks like this? An increase in income leads to more x2 but less x1.

x2 ? x1 is an "inferior" good.

x1

Spring 2001

Econ 11--Lecture 5

Income

7

IEP

Spring 2001

x1

Econ 11--Lecture 5

Engel Curve Income

8

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.

X > 0.

? Inferior Good:Demand for a good x

decreases with income

? This implies that the slope of the Engel curve is

negative.

X < 0.

Spring 2001

Econ 11--Lecture 5

9

Examples

Normal Goods Beef Cars Haircuts at a salon

Inferior Goods Potatoes Bus tickets Haircuts by your mother

Spring 2001

Econ 11--Lecture 5

10

All Goods Can't Be Inferior

? "Proof" #1: If income expands, the IEP cannot

point toward the origin.

x2

x2

x2

IEP

IEP

IEP

x1

Both Normal

x1 Inferior

x1 x1 Normal

x1

x2 Normal

x2 Inferior

Spring 2001

Econ 11--Lecture 5

11

All Goods Can't Be Inferior

? Proof #2: use budget constraint.

p1x1 + p2 x2 =

p1

x1

+

p2

x2

=1

Both x1 and x2 can't be negative.

Thus, both x1 and x2 can't be inferior goods.

Spring 2001

Econ 11--Lecture 5

12

Econ 11--Lecture 5

2

Professor Jay Bhattacharya

Spring 2001

A Good Can't be Inferior at all Income Levels

? Why not? Start with zero income. As income increases, if you ever consume that good, it is normal (at that income level).

? In order for a consumer to purchase less of a good as income increases, he must once have consumed some of it.

x1

Engel Curve

normal

inferior

Spring 2001

Income

Econ 11--Lecture 5

13

Consider 2 Engel Curves

x

x

Demand increases "slower" than income.

Demand increases "faster" than income.

I % change in x < % change in I

Necessity

I % change in x > % change in I

Luxury

Spring 2001

Econ 11--Lecture 5

14

Elasticity

? The elasticity of y with respect to x is defined as the percentage change in y induced by a small percentage change in x.

? Why do we need this concept? It is unit free.

? e.g. How much coffee are you willing to trade for a bagel? Determined by the slope of indifference curve = MRS.

? The value of the MRS depends upon the "units" of coffee we are using.

Spring 2001

Econ 11--Lecture 5

15

Elasticity

? Elasticity of y with respect to x is defined as the percentage change in y induced by a small percentage change in X.

y,x

=

dy dx

x y

y y

/

x x

Y,X

=

d ln y d ln x

since

d ln y = dy y

Spring 2001

Econ 11--Lecture 5

16

Income elasticity of demand

? Elasticity of a good, x, with respect to

income.

dx d

x

ex,I

Definitions

ex,I 0

Inferior Normal

ex,I 1

Necessity Luxury

Spring 2001

Econ 11--Lecture 5

17

? Engel curves for luxuries have a convex

shape:

x

slope

=

x* I*

slope = dx dI ( ) x*,I *

x*

Spring 2001

dx

> x*

dI ( ) x*,I* I *

I*

? I

I* x*

dx dI

( ) x*,I *

eI,x

>1

Econ 11--Lecture 5

18

Econ 11--Lecture 5

3

Professor Jay Bhattacharya

Spring 2001

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

19

? 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

Sx

=

xpx

? I will prove that if x is a necessity:

dS x < 0 d

Spring 2001

Econ 11--Lecture 5

20

log Sx = log x + log px - log

Totally differentiate this log "share" equation:

d log Sx = d log x + d log px - log Hold prices constant, i.e., set d log px = 0

? d log Sx = d log x - d log

? d log S x = d log x -1 d log d log

? I dS x = I dx - 1 S x dI x dI

Spring 2001

Econ 11--Lecture 5

21

I dS x = I dx - 1 S x dI x dI

?

I Sx

dS x dI

= eI,x

-1

but for necessities, eI ,x < 1

? I dS x < 0 ? dS x < 0

S x dI

dI

so the data confirm Engel's Law

Spring 2001

Econ 11--Lecture 5

22

? The expenditure weighted sum of income elasticities is equal to 1.

? Thus, all goods cannot be necessities. Nor can all goods be luxuries.

S1eI ,1 + S 2eI ,2 = 1

where

S1

=

x1 p1

S2

=

x2 p1

Spring 2001

Econ 11--Lecture 5

23

? Proof: Start with the budget constraint

p1x1 + p2 x2 =

?

p1

dx1 d

+

p2

dx2 d

=1

?

x1

x1

p1

dx1 d

+

x2

x2

p2

dx2 d

=1

?

?? ?

x1 p1

???????

x1

dx1 d

????

+

?? ?

x2 p2

???????

x2

dx2 d

???? = 1

? S1eI ,1 + S2eI ,2 = 1

Spring 2001

Econ 11--Lecture 5

24

Econ 11--Lecture 5

4

Professor Jay Bhattacharya

Spring 2001

What happens to demand when price changes?

x2 "Price-Consumption Curve" or "Price-Offer" Curve

slope = - p1 p2

Spring 2001

Econ 11--Lecture 5

slope = - p1 p2

x1

25

"Marshallian" Demand Curve

(Demand Curve)

? In the graph, we hold constant income and the prices of all other goods.

p1

p1

p1

Spring 2001

Econ 11--Lecture 5

x1 26

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."

Spring 2001

Econ 11--Lecture 5

27

Econ 11--Lecture 5

5

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download