Week3 Handout ECON2101

7/29/2019 Week3 Handout ECON2101

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Choice

Ch 5: Choice

Ch 6: Demand (sec. 2,5,6,8 & appendix)

Ch 15: Market Demand (sec. 1& 2)

2

Rational Choice

The principal behavioral postulate isthat a decisionmaker chooses itsmost preferred alternative from thoseavailable to it.

In terms of our model, this meanschoosing a bundle from the highestindifference curve that can bereached without exceeding thebudget set.

3

Rational Constrained Choice

x1

x2

Affordablebundles


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4


Affordablebundles

x1

x2

More preferred

bundles

5


x1

x2

x1*

x2*

(x1*,x2*) is the most

preferred affordablebundle.

6


The most preferred affordable bundleis called the consumers ORDINARYDEMAND at the given prices andbudget.

Ordinary demands will be denoted byx1*(p1,p2,m) and x2*(p1,p2,m).


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7

Choice with monotonic preferences

Supposed preferences are monotonic,i.e. more is better;

Then the consumer will alwayschoose a bundle that exhausts thebudget.

The chosen bundle is interior if itcontains strictly positive quantities ofboth goods.

8


When preferences are monotonic,indifference curves are smoothlyconvex and the chosen bundle isinterior, (x1*,x2*) satisfies twoconditions: (a) the budget is exhausted;

p1x1* + p2x2* = m (b) the slope of the budget constraint, -

p1/p2, and the slope of the indifferencecurve containing (x1*,x2*) are equal at(x1*,x2*).

9

Choice: The canonical case

x1

x2

x1*

x2*

(x1*,x2*) is interior .(a) (x1*,x2*) exhausts thebudget; p1x1* + p2x2* = m.(b) The slope of the indiff.curve at (x1*,x2*) equals

the slope of the budgetconstraint.


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10

Solving for the optimum bundle

If the budget is exhausted, then theoptimum bundle must satisfy the

budget constraint with equality: p1x1* + p2x2* = m

If the optimum bundle is interior,then it must be a point at which theslope of the budget line equals theslope of the indifference curve, i.e.:

- p1/p2 = MRS

We can use these two equations tosolve for the two variables x1*and x2*.

11

Computing Ordinary Demands -a Cobb-Douglas Example.

Suppose that the consumer hasCobb-Douglas preferences.

Then

U x x x xa b( , )1 2 1 2

MU Ux

ax xa b11

1 1 2

MUU

xbx xa b2

21 2

1

12


So the MRS is

At (x1*,x2*), MRS = -p1/p2 so

MRSdx

dx

U x

U x

ax x

bx x

ax

bx

a b

a b2

1

1

2

11

2

1 21

2

1

/

/.

ax

bx

p

p

xbp

ap

x2

1

1

22

1

21

*

*

* *. (A)


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13

Computing Ordinary Demands -a Cobb-Douglas Example. So now we know that

MRS = slope of budget line:

budget constraint is satisfied with

equality

xbp

apx2

1

21

* *(A)

p x p x m1 1 2 2* *

. (B)

14


xbp

apx2

1

21

* * (A)

p x p x m1 1 2 2

* *.

(B)

p x pbp

apx m1 1 2

1

21

* *.

Substitute

and get

This simplifies to .

15


xbm

a b p2

2

*

( ).

Substituting for x1* in

p x p x m1 1 2 2* *

then gives

xam

a b p1

1

*

( ).


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16


So we have discovered that the mostpreferred affordable bundle for a consumerwith Cobb-Douglas preferences

U x x x xa b( , )1 2 1 2

is( , )

( ),

( ).

* *x xam

a b p

bm

a b p1 2

1 2

17

Computing Ordinary Demands- a Cobb-Douglas Example.

x1

x2

xam

a b p1

1

*

( )

x

bm

a b p

2

2

*

( )

U x x x xa b( , )1 2 1 2

18

Computing Ordinary Demands - a Cobb-Douglas Example.

Try this with numbers instead of symbols:

Let a=1/3, b=2/3, m=75

Let prices be p1 and p2 for now

Then we get

The consumer spends 1/3 of her income on good1, and 2/3 of her income on good 2. Once weknow the prices, we can compute the actualquantities.

(This is true only for this class of utility functions.)

.50

,25

),(21

*

2

*

1

ppxx


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19


When x1* > 0 and x2* > 0

and (x1*,x2*) exhausts the budget,

and indifference curves have nokinks, the ordinary demands are

obtained by solving:

(a) p1x1* + p2x2* = y

(b) the slopes of the budget constraint, -p1/p2, and of the indifference curve

containing (x1*,x2*) are equal at (x1*,x2*).

20

Another way to look at optimisation

Pick any level of x1, and supposeconsumer buys this much x1.

Then he spends p1x1, and has m-p1x1left to spend on the other good

with which he can buy [m -p1x1]/p2units of good 2

This gives him U(x1, [m -p1x1]/p2)amount of utility

His problem is to pick x1 so as tomaximise U(x1, [m -p1x1]/p2)

21

Another way to look at optimisation

Apply this to the case U(x1,x2)=x1x2.

To maximise we take a first derivativeand equate it to zero, to give

and substitute back in the

budget constraint to get

U xm p x

px

m p x

p

mx

p

p x

p( , ) [ ]1

1 1

2

1

1 1

2

1

2

1 1

2

2

m

p

p x

px

m

p2

1 1

2

1

1

20

2

xm

p2

22


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22

Examples of Corner Solutions --the Perfect Substitutes Case

x1

x2 MRS = -1

Budget line:Slope = -p1/p2 with p1 > p2.

23

Examples of Corner Solutions --the Non-Convex Preferences Case

x1

x2Which is the most preferred

affordable bundle?

24

Examples of Corner Solutions --the Non-Convex Preferences Case

x1

x2

The most preferredaffordable bundle


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28

Examples of Kinky Solutions -- thePerfect Complements Case

x1

x2

U(x1,x2) = min{ax1,x2}

x2 = ax1

x1*

x2*

(a) p1x1* + p2x2* = m(b) x2* = ax1*

29

Examples of Kinky Solutions -- the Perfect

Complements Case

(a) p1x1* + p2x2* = m; (b) x2* = ax1*.

Substitution from (b) for x2* in

(a) gives p1x1* + p2ax1* = mwhich gives

A bundle of 1 commodity 1 unit and

acommodity 2 units costs p1 + ap2;m/(p1 + ap2) such bundles are affordable.

.app

amx;app

mx

21

*2

21

*1

30

Examples of Kinky Solutions --the Perfect Complements Case

x1

x2 U(x1,x2) = min{ax1,x2}

x2 = ax1

xm

p ap1 1 2*

x

am

p ap

2

1 2

*


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Demand

Demand is the pattern of the consumers(or markets) consumption behaviour asprices and income change.

Demand curves trace these patternsassuming that only one price changes,while income and other prices remainconstant.

32

Demand

Here we trace how the quantity of agood in the consumers chosenbundle changes in response tochanges in

the price of that good

the price of the other good

the consumers income

In each case holding all the otherparameters constant

33

Properties of Demand Functions

Comparative statics analysis ofordinary demand functions -- thestudy of how ordinary demandsx1*(p1,p2,y) and x2*(p1,p2,y) changeas prices p1, p2 and income y change.


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34

Own-Price Changes

How does x1*(p1,p2,y) change as p1changes, holding p2 and y constant?

Suppose only p1 increases, from p1 top1 and then to p1.

35

Own-Price Changes

x1

x2

p1= p1

p1 = p1

Fixed p2 and y.

p1x1 + p2x2 = y

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Own-Price Changes

x1

x2

p1= p1p1=p1

Fixed p2 and y.

p1 = p1

p1x1 + p2x2 = y


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37

x2

x1x1*(p1)

Own-Price Changes

p1 = p1

Fixed p2 and y.

38

x2

x1x1*(p1)

p1

x1*(p1)

p1

x1*

Own-Price ChangesFixed p2 and y.

p1 = p1

Here we plot thequantity demanded ofx1 (from the horizontalaxis) against the pricep1 implicit in thebudget line.

39

x2

x1

x1*(p1) x1*(p1)x1*(p1)

p1

x1*(p1)

x1*(p1)

p1

p1

p1 = p1

x1*



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40

x2

x1x1*(p1) x1*(p1)

x1*(p1)

p1

x1*(p1)x1*(p1)

x1*(p1)

p1

p1

p1

x1*


41

x2

x1x1*(p1) x1*(p1)

x1*(p1)

p1

x1*(p1)x1*(p1)

x1*(p1)

p1

p1

p1

x1*

Own-Price Changes Ordinarydemand curvefor commodity 1Fixed p2 and y.

42

x2

x1

x1*(p1) x1*(p1)x1*(p1)

p1

x1*(p1)x1*(p1)

x1*(p1)

p1

p1

p1

x1*

Own-Price Changes Ordinarydemand curvefor commodity 1Fixed p2 and y.


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x2

x1x1*(p1) x1*(p1)

x1*(p1)

p1

x1*(p1)x1*(p1)

x1*(p1)

p1

p1

p1

x1*

Own-Price Changes Ordinarydemand curve

for commodity 1

p1 priceoffer

curve

Fixed p2 and y.

44

Own-Price Changes

The curve containing all the utility-maximizing bundles traced out as p1changes, with p2 and y constant, isthe p1- price offer curve.

The plot of the x1-coordinate of thep1- price offer curve against p1 is the

ordinary demand curve forcommodity 1.

45

Own-Price Changes

What does a p1 price-offer curve looklike for Cobb-Douglas preferences?

Take

Then the ordinary demand functionsfor commodities 1 and 2 are

U x x x xa b( , ) .1 2 1 2


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46

Own-Price Changes

x p p ya

a b

y

p

1 1 2

1

*( , , )

x p p yb

a b

y

p2 1 2

2

*( , , ) .

and

Notice that x2* does not vary with p1 so the

p1 price offer curve is flat and the ordinarydemand curve for commodity 1 is a

rectangular hyperbola.

47

x1*(p1) x1*(p1)

x1*(p1)

x2

x1

p1

x1*

Own-Price Changes Ordinarydemand curvefor commodity 1

is

Fixed p2 and y.

x

by

a b p

2

2

*

( )

xay

a b p1

1

*

( )

xay

a b p1

1

*

( )

48

Own-Price Changes

Usually we ask Given the price forcommodity 1 what is the quantitydemanded of commodity 1?

But we could also ask the inversequestion At what price forcommodity 1 would a given quantityof commodity 1 be demanded?


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49

Own-Price Changes

p1

x1*

p1

x1

Given p1, what quantity is

demanded of commodity 1?Answer: x1 units.

The inverse question is:

Given x1 units aredemanded, what is the

price ofcommodity 1?Answer: p1

50

Own-Price Changes

Taking quantity demanded as givenand then asking what must be pricedescribes the inverse demandfunction of a commodity.

51

Own-Price Changes

A Cobb-Douglas example:

xay

a b p1

1

*

( )

is the ordinary demand function and

pay

a b x1

1( )*

is the inverse demand function.


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52

Own-Price Changes

A perfect-complements example:

xy

p p1

1 2

*

is the ordinary demand function and

py

xp1

1

2*

is the inverse demand function.

53

Income Changes

How does the value of x1*(p1,p2,y)change as y changes, holding both p1and p2 constant?

54

x2

x1

Income ChangesFixed p1 and p2.

y < y < y


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55

x2

x1


y < y < y

x1x1

x1

x2x2x2

56

x2

x1


y < y < y

x1x1

x1

x2x2x2

Income

offer curve

57

Income Changes

A plot of the demand bundles atdifferent incomes is called theincome offer curve.

A plot ofquantity demanded of onegood against income is called anEngel curve.


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x2

x1


y < y < y

x1x1

x1

x2x2x2

Income

offer curve

x1*

y

x1x1

x1

yy

y

Note that the horizontalaxis in the right hand

diagram is x1, but thevertical axis is income, y.

59

x2

x1


y < y < y

x1x1

x1

x2x2x2

Income

offer curve

x1*

y

x1x1

x1

yyy Engel

curve;good 1

60

x2

x1


y < y < y

x1x1

x1

x2x2x2

Income

offer curve x2*

y

x2x2

x2

yy

y

Engel

curve;good 2


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61

Income Changes and Cobb-Douglas Preferences An example of computing the

equations of Engel curves; the Cobb-Douglas case.

The ordinary demand equations are

U x x x xa b( , ) .1 2 1 2

xay

a b px

by

a b p1

12

2

* *

( );

( ).

62

Income Changes and Cobb-Douglas Preferences

xay

a b px

by

a b p1

12

2

* *

( );

( ).

Rearranged to isolate y, these are:

y

a b p

a x

ya b p

bx

( )

( )

*

*

11

22

Engel curve for good 1

Engel curve for good 2

63

Income Changes and Cobb-Douglas Preferences

y

yx1*

x2*

ya b p

ax

( ) *11

Engel curvefor good 1

ya b p

bx

( ) *22

Engel curve

for good 2


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64

Income Changes

In the Cobb-Douglas example theEngel curves were straight lines.

Q: Is this true in general? A: No. Engel curves are straight lines

if the consumers preferences arehomothetic.

That is, the consumers MRS is thesame anywhere on a straight linedrawn from the origin.

Or, the indifference curves are blownup versions of each other, projectingout from the origin.

65

Income Effects -- ANonhomothetic Example

Quasilinear preferences are nothomothetic.

For example,

U x x f x x( , ) ( ) .1 2 1 2

U x x x x( , ) .1 2 1 2

66

Quasi-linear Indifference Curves

x2

x1

Each curve is a vertically shifted

copy of the others.

Each curve intersects

both axes.


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67

Income Changes; Quasilinear

Utilityx2

x1x1~

x1*

y

x1~

Engel

curvefor

good 1

68

Income Changes; Quasilinear

Utilityx2

x1x1~

x2*

y Engel

curvefor

good 2


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69

Additionalmaterial forself-study

70

More examples of demand curvesderived from known preferences:

perfect-complements utility function

perfect-substitutes utility function

71

Own-Price Changes

What does a p1 price-offer curve look like

for a perfect-complements utility function?

U x x x x( , ) min , .1 2 1 2

Then the ordinary demand functionsfor commodities 1 and 2 are

x p p y x p p yy

p p1 1 2 2 1 2

1 2

* *( , , ) ( , , ) .

With p2 and y fixed, higher p1 causessmaller x1* and x2*.

72

p1

x1*

Fixed p2 and y.

x

y

p p

2

1 2

*

xy

p p1

1 2

*

Own-Price Changes

x1

x2

p1

xy

p p1

1 2

*

p1 = p1

y/p2

73

p1

x1*

Fixed p2 and y.

x

y

p p

2

1 2

*

x yp p

11 2

*

Own-Price Changes

x1

x2

p1

p1

p1 = p1

x

y

p p1

1 2

*

y/p2

74

p1

x1*

Fixed p2 and y.

x

y

p p

2

1 2

*

x yp p

11 2

*

Own-Price Changes

x1

x2

p1

p1

p1

xy

p p1

1 2

*

p1 = p1

y/p2


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75

p1

x1*

Ordinarydemand curve

for commodity 1is

Fixed p2 and y.

x

y

p p

2

1 2

*

xy

p p1

1 2

*

x yp p

11 2

* .

Own-Price Changes

x1

x2

p1

p1

p1

y

p2

y/p2

76

Own-Price Changes

What does a p1 price-offer curve looklike for a perfect-substitutes utility

function?

U x x x x( , ) .1 2 1 2

Then the ordinary demand functions

for commodities 1 and 2 are

77

Own-Price Changes

x p p yif p p

y p if p p1 1 2

1 2

1 1 2

0*( , , )

,

/ ,

x p p yif p p

y p if p p2 1 2

1 2

2 1 2

0*( , , )

,

/ , .

and

78

Fixed p2 and y.

Own-Price Changes

x2

x1

p1

x1*

Fixed p2 and y.

p1

p2 = p1

p1

xy

p1

1

*

0 12

xy

p

*

y

p2

p1 price

offer

curve

Ordinarydemand curve

for commodity 1

79

The remaining slidesremind you of material youhave seen in micro 1.

It may be useful toremember this materialnow, and relate it to thematerial you have

encountered in this course.80

Cross-Price Effects

If an increase in p2 increases demand for commodity 1 then

commodity 1 is a gross substitute for

commodity 2.

reduces demand for commodity 1 thencommodity 1 is a gross complement for

commodity 2.


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81

Cross-Price Effects

A perfect-complements example:

xy

p p1

1 2

*

x

p

y

p p

1

2 1 22

0*

.

so

Therefore commodity 2 is a gross

complement for commodity 1.

82

Cross-Price Effects

p1

x1*

p1

p1

p1

y

p2

Increase the price of

good 2 from p2 to p2and

83

Cross-Price Effects

p1

x1*

p1

p1

p1

y

p2

Increase the price ofgood 2 from p2 to p2and the demand curve

for good 1 shifts inwards

-- good 2 is acomplement for good 1.

84

Cross-Price Effects

A Cobb- Douglas example:

xby

a b p2

2

*

( )so

85

Cross-Price Effects

A Cobb- Douglas example:

xby

a b p2

2

*

( )

x

p2

1

0*

.

so

Therefore commodity 1 is neither a gross

complement nor a gross substitute forcommodity 2.86

Income Effects

A good for which quantity demandedrises with income is called normal.

Therefore a normal goods Engelcurve is positively sloped.


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87

Income Effects

A good for which quantity demandedfalls as income increases is called

income inferior. Therefore an income inferior goods

Engel curve is negatively sloped.

88

x2

x1

Income Changes; Goods

1 & 2 Normal

x1x1

x1

x2x2x2

Income

offer curve

x1*

x2*

y

y

x1x1

x1

x2x2

x2

yy

y

yy

y

Engel

curve;good 2

Engel

curve;good 1

89

Income Changes; Good 2 Is Normal, Good 1

Becomes Inferior

x2

x1

Income

offer curve

90

x2

x1 x1*

x2*

y

y



Income Changes; Good 2 Is Normal, Good 1

Becomes Inferior

91

Ordinary Goods and Giffen Goods

A good is called ordinary if thequantity demanded of it alwaysincreases as its own price decreases.

If, for some values of its own price,the quantity demanded of a goodrises as its own-price increases thenthe good is called Giffen.

92

Ordinary Goods

Fixed p2 and y.

x1

x2

p1 priceoffer

curve

x1*

Downward-sloping

demand curve

Good 1 is

ordinary

p1


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93

Giffen Goods

Fixed p2 and y.

x1

x2p1 price offercurve

x1*

Demand curve has

a positively

sloped part

Good 1 is

Giffen

p1

Documents

Week3 Handout ECON2101