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

Rational Constrained ChoiceRational Constrained Choice

UtilitUtility

x1

Rational Constrained ChoiceRational Constrained Choice

UtilitUtility x2

x1

Rational Constrained ChoiceRational Constrained Choice

Utility

x2

x1

Rational Constrained ChoiceRational Constrained Choice

Utility

x2

x1

Rational Constrained ChoiceRational Constrained Choice

Utility

x2

x1

Rational Constrained ChoiceRational Constrained Choice

Utility

x2

x1

Rational Constrained ChoiceRational Constrained Choice

Utility

x2

x1

x2

Rational Constrained ChoiceRational Constrained Choice

Utility

Affordable, but not the most preferred affordable bundle.

x2

x1

x2

Rational Constrained ChoiceRational Constrained Choice

Utility The most preferredof the affordable

Affordable, but not bundles.

the most preferred affordable bundle.

x2

x1

x2

Rational Constrained ChoiceRational Constrained Choice

Utility

x2

x1

x2

Rational Constrained ChoiceRational Constrained Choice

Utility

x22

x1

Rational Constrained ChoiceRational Constrained Choice

x2

UtilityUtility

x1

Rational Constrained ChoiceRational Constrained Choice

x2

UtilityUtilityx1

Rational Constrained ChoiceRational Constrained Choice

x22

x1

Rational Constrained ChoiceRational Constrained Choice

x22

Affordable

x1

bundles

Rational Constrained ChoiceRational Constrained Choice

x22

Affordable

x1

bundles

Rational Constrained ChoiceRational Constrained Choice

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*

Rational Constrained ChoiceRational Constrained Choice

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).

Rational Constrained ChoiceRational Constrained Choice

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

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

Then

( , )1 2 1 2

MU U ax xa b1 1

12

x11

1 2

UMU Ux

bx xa b2

21 2

1

Computing Ordinary Demands -a Cobb-Douglas Example.

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 /

.

Computing Ordinary Demands -a Cobb-Douglas Example.

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 /

.

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

Computing Ordinary Demands -a Cobb-Douglas Example.

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 /

.

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

b* ax

bxpp

x bpap

x2

1

12

212

1** *. (A)

bx p p1 2 2

Computing Ordinary Demands -a Cobb-Douglas Example.

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

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

Computing Ordinary Demands -a Cobb-Douglas Example.

So now we know thatbp1* *x bpap

x212

1* * (A)

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

Computing Ordinary Demands -a Cobb-Douglas Example.

So now we know thatbp1* *x bpap

x212

1* * (A)

Substitutep x p x m1 1 2 2

* * . (B)Substitute

Computing Ordinary Demands -a Cobb-Douglas Example.

So now we know thatbp1* *x bpap

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 ….

Computing Ordinary Demands -a Cobb-Douglas Example.

x ama b p1

1

*( )

.a b p1( )

Computing Ordinary Demands -a Cobb-Douglas Example.

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( ).

Computing Ordinary Demands -a Cobb-Douglas Example.

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*).

Rational Constrained ChoiceRational Constrained Choice

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

Examples of Corner Solutions --the Perfect Substitutes Case

x2MRS = -1MRS 1

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

x11

Examples of Corner Solutions --the Perfect Substitutes Case

x2MRS = -1MRS 1

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

x11

Examples of Corner Solutions --the Perfect Substitutes Case

x2MRS = -1

x yp2

2

* MRS 1

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

x1x1 0* 1x1 0

Examples of Corner Solutions --the Perfect Substitutes Case

x2MRS = -1MRS 1

Slope = p /p with p < p

*

Slope = -p1/p2 with p1 < p2.

x1x y*

x2 0

1x yp1

1

Examples of Corner Solutions --the Perfect Substitutes Case

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.

Examples of Corner Solutions --the Perfect Substitutes Case

x2MRS = -1MRS 1

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

p2

x1y 1p1

Examples of Corner Solutions --the Perfect Substitutes Case

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

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

x11

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

Which is the most preferredWhich is the most preferredaffordable bundle?

x11

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

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

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

x2 = ax1MRS = 0

x11

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

MRS = -

x2 = ax1MRS = 0

x11

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

MRS = - MRS is undefinedMRS is undefined

x2 = ax1MRS = 0

x11

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

x2 = ax1

x11

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

Which is the mostpreferred affordable bundle?

x2 = ax1

x11

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

The most preferredaffordable bundle

x2 = ax1

x11

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

x2 = ax1

x2*

x1x1* 1

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

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

x2 = ax1

x2*

x1x1* 1

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

(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*.

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* = m(a) gives p1x1 p2ax1 m

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

1 appmx

211 app

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* = m(a) gives p1x1 p2ax1 mwhich gives .

appamx;

appmx *

2*1

appapp 212

211

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* = 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.

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

x2 = ax1x2

*

amp ap1 2

x1x m1* 1p ap1

1 2