ch5

Chapter Five

Choice

Economic Rationality

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

The available choices constitute the choice set.

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

Rational Constrained Choice

x1

x2


x1

x2Utility


Utility x2

x1


x1

x2

Utility


Utility

x1

x2


Utility

x1

x2

Affordable, but not the most preferred affordable bundle.


x1

x2

Utility

Affordable, but not the most preferred affordable bundle.

The most preferredof the affordablebundles.


x1

x2

Utility


Utility

x1

x2


Utilityx1

x2


x1

x2


x1

x2

Affordablebundles


x1

x2

Affordablebundles

More preferredbundles


Affordablebundles

x1

x2

More preferredbundles


x1

x2

x1*

x2*


x1

x2

x1*

x2*

(x1*,x2*) is the mostpreferred affordablebundle.


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

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


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

If buying (x1*,x2*) costs $m then the budget is exhausted.


x1

x2

x1*

x2*

(x1*,x2*) is interior.

(x1*,x2*) exhausts thebudget.


x1

x2

x1*

x2*

(x1*,x2*) is interior.(a) (x1*,x2*) exhausts thebudget; p1x1* + p2x2* = m.


x1

x2

x1*

x2*

(x1*,x2*) is interior .(b) The slope of the indiff.curve at (x1*,x2*) equals the slope of the budget constraint.

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

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

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

Computing Ordinary Demands

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

Computing Ordinary Demands - a Cobb-Douglas Example.

Suppose that the consumer has Cobb-Douglas preferences.

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


Suppose that the consumer has Cobb-Douglas preferences.

Then

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

MUUx

ax xa b1

1112

MUUx

bx xa b2

21 2

1


So the MRS is

MRSdxdx

U xU x

ax x

bx x

axbx

a b

a b

2

1

1

2

112

1 21

2

1

//

.


So the MRS is

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

MRSdxdx

U xU x

ax x

bx x

axbx

a b

a b

2

1

1

2

112

1 21

2

1

//

.


So the MRS is

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

MRSdxdx

U xU x

ax x

bx x

axbx

a b

a b

2

1

1

2

112

1 21

2

1

//

.

ax

bx

pp

xbpap

x2

1

1

22

1

21

*

** *. (A)


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

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


So now we know that

xbpap

x21

21

* * (A)

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


So now we know that

xbpap

x21

21

* * (A)

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

Substitute


So now we know that

xbpap

x21

21

* * (A)

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

p x pbpap

x m1 1 21

21

* * .

Substitute

and get

This simplifies to ….


xam

a b p11

*

( ).


xbm

a b p22

*

( ).

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

* *

then gives

xam

a b p11

*

( ).


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 pbm

a b p1 21 2


x1

x2

xam

a b p11

*

( )

x

bma b p

2

2

*

( )

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

Rational Constrained Choice When x1* > 0 and x2* > 0

and (x1*,x2*) exhausts the budget,and indifference curves have no ‘kinks’, 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*).


But what if x1* = 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 of maximizing utility subject to a budget constraint.

Examples of Corner Solutions -- the Perfect Substitutes Case

x1

x2

MRS = -1


x1

x2

MRS = -1

Slope = -p1/p2 with p1 > p2.


x1

x2

xy

p22

*

x1 0*

MRS = -1

Slope = -p1/p2 with p1 > p2.


x1

x2

xyp11

*

x2 0*

MRS = -1

Slope = -p1/p2 with p1 < p2.


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

0,py

)x,x(1

*2

*1

and

2

*2

*1 p

y,0)x,x(

if p1 < p2

if p1 > p2.


x1

x2

MRS = -1

Slope = -p1/p2 with p1 = p2.

yp1

yp2


x1

x2

All the bundles in the constraint are equally the most preferred affordable when p1 = p2.

yp2

yp1

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

x1

x2Better


x1

x2


x1

x2

Which is the most preferredaffordable bundle?


x1

x2

The most preferredaffordable bundle


x1

x2


Notice that the “tangency solution”is not the most preferred affordablebundle.

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

x1

x2U(x1,x2) = min{ax1,x2}

x2 = ax1


x1

x2

MRS = 0

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

x2 = ax1


x1

x2

MRS = -

MRS = 0


x2 = ax1


x1

x2

MRS = -

MRS = 0

MRS is undefined


x2 = ax1


x1


x2 = ax1


x1


x2 = ax1

Which is the mostpreferred affordable bundle?


x1


x2 = ax1



x1


x2 = ax1

x1*

x2*


x1


x2 = ax1

x1*

x2*

(a) p1x1* + p2x2* = m


x1


x2 = ax1

x1*

x2*

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


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

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

21

*1 app

mx


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

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

appam

x;app

mx

21

*2

21

*1


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

.app

amx;

appm

x21

*2

21

*1


x1


x2 = ax1

xm

p ap11 2

*

x

amp ap

2

1 2

*

ch5

Education

rational constrained

ordinary demand x1

u x1 ax1 x

x2a b u x1

ordinary demands

cobbdouglas preferences

cobbdouglas example

dx1 u x