Chapter 5Chapter 5 Choice - Lancaster University...Chapter 5Chapter 5 Choice Economic...

Chapter 5Chapter 5

ChoiceChoice

Economic RationalityEconomic Rationality

The principal behavioral postulate is that a decisionmaker chooses itsthat a decisionmaker chooses its most preferred alternative from those available to itavailable to it.

The available choices constitute the choice set.

How is the most preferred bundle in How is the most preferred bundle in the choice set located?

Rational Constrained ChoiceRational Constrained Choice

x2

xx1


UtilitUtility

x1


UtilitUtility x2

x1


Utility

x2

x1


Utility

x2

x1

x2


Utility

Affordable, but not the most preferred affordable bundle.

x2

x1

x2


Utility The most preferredof the affordable

Affordable, but not bundles.

the most preferred affordable bundle.

x2

x1

x2


Utility

x2

x1

x2


Utility

x22

x1


x2

UtilityUtility

x1


x2

UtilityUtilityx1


x22

x1


x22

Affordable

x1

bundles


x22

More preferredMore preferredbundles

Affordable

x1

bundles

Rational Constrained ChoiceRational Constrained Choicex2

More preferredbundles

Affordablebundles

xx1

Rational Constrained ChoiceRational Constrained Choicex2

x2*

xx * x1x1*

Rational Constrained ChoiceRational Constrained Choicex2 (x1* x2*) is the most(x1 ,x2 ) is the most

preferred affordablebundlebundle.

x2*

xx * x1x1*


The most preferred affordable bundle is called the consumer’s ORDINARYis called the consumer s ORDINARY DEMAND at the given prices and budgetbudget.

Ordinary demands will be denoted byy yx1*(p1,p2,m) and x2*(p1,p2,m).


When x1* > 0 and x2* > 0 the demanded bundle is INTERIORdemanded bundle is INTERIOR.

If buying (x1*,x2*) costs $m then the b d i h dbudget is exhausted.

Rational Constrained ChoiceRational Constrained Choicex2 (x1* x2*) is interior(x1 ,x2 ) is interior.

(x1*,x2*) exhausts thebudget.

x2*

g

xx * x1x1*

Rational Constrained ChoiceRational Constrained Choicex2 (x1* x2*) is interior(x1 ,x2 ) is interior.

(a) (x1*,x2*) exhausts thebudget; p x * + p x * = mbudget; p1x1* + p2x2* = m.

x2*

xx * x1x1*

Rational Constrained ChoiceRational Constrained Choicex2 (x1* x2*) is interior(x1 ,x2 ) is interior .

(b) The slope of the indiff.curve at (x * x *) equalscurve at (x1*,x2*) equals

the slope of the budget

x2*constraint.

xx * x1x1*

Rational Constrained ChoiceRational Constrained Choice (x1*,x2*) satisfies two conditions:( 1 , 2 ) (a) the budget is exhausted;

p x * + p x * = mp1x1* + p2x2* = m (b) the slope of the budget constraint,

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

Computing Ordinary DemandsComputing Ordinary Demands

How can this information be used to locate (x1* x2*) for given p1 p2 andlocate (x1 ,x2 ) for given p1, p2 and m?

Computing Ordinary Demands -a Cobb-Douglas Example.

Suppose that the consumer has Cobb-Douglas preferencesCobb Douglas preferences.

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


Suppose that the consumer has Cobb-Douglas preferencesCobb Douglas preferences.

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

Then

( , )1 2 1 2

MU U ax xa b1 1

12

x11

1 2

UMU Ux

bx xa b2

21 2

1


So the MRS is

MRS dx U x ax x axa b

2 1 1

12 2 /MRS

dx U x bx x bxa b 1 2 1 21 1 /

.


So the MRS is


2 1 1

12 2 /MRS

dx U x bx x bxa b 1 2 1 21 1 /

.

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


So the MRS is


2 1 1

12 2 /MRS

dx U x bx x bxa b 1 2 1 21 1 /

.

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

b* ax

bxpp

x bpap

x2

1

12

212

1** *. (A)

bx p p1 2 2


(x1*,x2*) also exhausts the budget so

p x p x m1 1 2 2* * . (B)


So now we know thatbp1* *x bpap

x212

1* * (A)

p x p x m1 1 2 2* * . (B)



x212

1* * (A)

Substitutep x p x m1 1 2 2

* * . (B)Substitute



x212

1* * (A)

Substitutep x p x m1 1 2 2

* * . (B)Substitute

d tp x p bp

apx m1 1 2

11

* * . and get

ap2This simplifies to ….


x ama b p1

1

*( )

.a b p1( )


x ama b p1

1

*( )

.

Substituting for x1* in

a b p1( )

Substituting for x1 in p x p x m1 1 2 2

* *

bm*then gives

x bma b p2

2( ).


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

a bU x x x xa b( , )1 2 1 2

is( , ) , .* * ( )x x am bm

1 2 ( , )( )

,( )

.( )x xa b p a b p1 2

1 2

Computing Ordinary Demands -C bb D l E la Cobb-Douglas Example.

x2 U x x x xa b( )U x x x xa b( , )1 2 1 2

x2*

bma b p2( )a b p2( )

xam* x1x ama b p1

1

*( )

Rational Constrained ChoiceRational Constrained Choice When x1* > 0 and x2* > 0 1 2

and (x1*,x2*) exhausts the budget,and indifference curves have noand indifference curves have no

‘kinks’, the ordinary demands are obtained by solving:are obtained by solving:

(a) p1x1* + p2x2* = y (b) the slopes of the budget constraint,

-p /p and of the indifference curve-p1/p2, and of the indifference curve containing (x1*,x2*) are equal at (x1*,x2*).


But what if x1* = 0? Or if x * = 0? Or if x2* = 0? If either x1* = 0 or x2* = 0 then the

ordinary demand (x1*,x2*) is at a corner solution to the problem ofcorner solution to the problem of maximizing utility subject to a budget constraintconstraint.

Examples of Corner Solutions --the Perfect Substitutes Case

x2MRS = -1MRS 1

x11


x2MRS = -1MRS 1

Slope = p /p with p > pSlope = -p1/p2 with p1 > p2.

x11


x2MRS = -1

x yp2

2

* MRS 1

Slope = p /p with p > pSlope = -p1/p2 with p1 > p2.

x1x1 0* 1x1 0


x2MRS = -1MRS 1

Slope = p /p with p < p

*

Slope = -p1/p2 with p1 < p2.

x1x y*

x2 0

1x yp1

1


So when U(x x ) = x + x the mostSo when U(x1,x2) = x1 + x2, the mostpreferred affordable bundle is (x1*,x2*)where

0,y)x,x( *2

*1 if p1 < p2

0,p

)x,x(1

21

and

if p1 < p2

and

** y0)xx( if p > p

2

21 py,0)x,x( if p1 > p2.


x2MRS = -1MRS 1

Slope = -p1/p2 with p1 = p2.y

p2

x1y 1p1


x2All the bundles in theAll the bundles in the constraint are equally the

most preferred affordable

yp2

most preferred affordablewhen p1 = p2.

x1y 1p1

Examples of Corner Solutions --the Non-Convex Preferences Casex2

x11


Which is the most preferredWhich is the most preferredaffordable bundle?

x11


The most preferredThe most preferredaffordable bundle

x11

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

N ti th t th “t l ti ”x2Notice that the “tangency solution”is not the most preferred affordable

The most preferredbundle.

The most preferredaffordable bundle

x11

Examples of ‘Kinky’ Solutions -- the Perfect Complements Casex2 U(x1,x2) = min{ax1,x2}

x2 = ax1

x11


x2 = ax1MRS = 0

x11


MRS = -

x2 = ax1MRS = 0

x11


MRS = - MRS is undefinedMRS is undefined

x2 = ax1MRS = 0

x11


x2 = ax1

x11


Which is the mostpreferred affordable bundle?

x2 = ax1

x11


The most preferredaffordable bundle

x2 = ax1

x11


x2 = ax1

x2*

x1x1* 1


(a) p x * + p x * = m(a) p1x1* + p2x2* = m

x2 = ax1

x2*

x1x1* 1


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

x2 = ax1

x2*

x1x1* 1

Examples of ‘Kinky’ Solutions -- the Perfect Complements Case

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


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

Substitution from (b) for x2* in (a) gives p1x1* + p2ax1* = m(a) gives p1x1 p2ax1 m


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

Substitution from (b) for x2* in (a) gives p1x1* + p2ax1* = m(a) gives p1x1 p2ax1 mwhich gives *

1 appmx

211 app


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

Substitution from (b) for x2* in (a) gives p1x1* + p2ax1* = m(a) gives p1x1 p2ax1 mwhich gives .

appamx;

appmx *

2*1

appapp 212

211


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

Substitution from (b) for x2* in (a) gives p1x1* + p2ax1* = m(a) gives p1x1 p2ax1 mwhich gives .

appamx;

appmx *

2*1

A bundle of 1 commodity 1 unit andappapp 21

221

1 y

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


x2 = ax1x2

*

amp ap1 2

x1x m1* 1p ap1

1 2

Chapter 5Chapter 5 Choice - Lancaster University...Chapter 5Chapter 5 Choice Economic...

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