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

x 2 x 1

Rational Constrained Choice

Utility x 2 x 1

Rational Constrained Choice

Utility x 2 x 1

Rational Constrained Choice

Utility x 2 x 1

Rational Constrained Choice

Utility x 1 x 2

Rational Constrained Choice

Utility x 1 x 2

Rational Constrained Choice

Utility x 2 x 1

Rational Constrained Choice

Utility x 2 x 1

Rational Constrained Choice

Utility x 2 Affordable, but not the most preferred affordable bundle.

x 1

Rational Constrained Choice

Utility x 2 The most preferred of the affordable bundles.

Affordable, but not the most preferred affordable bundle.

x 1

Rational Constrained Choice

Utility x 2 x 1

Rational Constrained Choice

Utility x 2 x 1

Rational Constrained Choice

x 2 Utility x 1

Rational Constrained Choice

x 2 Utility x 1

Rational Constrained Choice

x 2 x 1

Rational Constrained Choice

x 2 Affordable bundles x 1

Rational Constrained Choice

x 2 Affordable bundles x 1

Rational Constrained Choice

x 2 More preferred bundles Affordable bundles x 1

Rational Constrained Choice

x 2 More preferred bundles Affordable bundles x 1

Rational Constrained Choice

x 2 x 2 * x 1 * x 1

Rational Constrained Choice

x 2 (x 1 *,x 2 *) is the most preferred affordable bundle.

x 2 * x 1 * x 1

Rational Constrained Choice 

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



Ordinary demands will be denoted by x 1 *(p 1 ,p 2 ,m) and x 2 *(p 1 ,p 2 ,m).

Rational Constrained Choice 

When x 1 * > 0 and x 2 * > 0 the demanded bundle is INTERIOR .



If buying (x 1 *,x 2 *) costs $m then the budget is exhausted.

Rational Constrained Choice

x 2 (x 1 *,x 2 *) is interior.

(x 1 *,x 2 *) exhausts the budget.

x 2 * x 1 * x 1

Rational Constrained Choice

x 2 (x 1 *,x 2 *) is interior.

(a) (x 1 *,x 2 *) exhausts the budget; p 1 x 1 * + p 2 x 2 * = m.

x 2 * x 1 * x 1

Rational Constrained Choice

x 2 (x 1 *,x 2 *) is interior .

(b) The slope of the indiff.

curve at (x 1 *,x 2 *) equals the slope of the budget constraint.

x 2 * x 1 * x 1

Rational Constrained Choice 

(x 1 *,x 2 *) satisfies two conditions:



(a) the budget is exhausted; p 1 x 1 * + p 2 x 2 * = m



(b) the slope of the budget constraint, -p 1 /p 2 , and the slope of the indifference curve containing (x 1 *,x 2 *) are equal at (x 1 *,x 2 *).

Computing Ordinary Demands 

How can this information be used to locate (x 1 *,x 2 *) for given p 1 , p 2 m?

and

Computing Ordinary Demands a Cobb-Douglas Example.



Suppose that the consumer has Cobb-Douglas preferences.

2 )



Computing Ordinary Demands a Cobb-Douglas Example.



Suppose that the consumer has Cobb-Douglas preferences.



Then MU 1



2 )

 

U x 1

 

ax a 1



1 x b 2 MU 2

  

U x 2

 

1

Computing Ordinary Demands a Cobb-Douglas Example.



So the MRS is MRS



dx 2 dx 1

     

1 2

 

ax a 1



1 x



b 2 1

 

ax 2 bx 1 .

Computing Ordinary Demands a Cobb-Douglas Example.



So the MRS is MRS



dx 2 dx 1

     

1 2

 

ax a 1



1 x



b 2 1

 

ax 2 bx 1 .



At (x 1 *,x 2 *), MRS = -p 1 /p 2 so

Computing Ordinary Demands a Cobb-Douglas Example.



So the MRS is MRS



dx 2 dx 1

     

1 2

 

ax a 1



1 x



b 2 1

 

ax 2 bx 1 .



At (x 1 *,x 2 *), MRS = -p 1 /p 2 so



ax * 2 bx * 1

 

p 1 p 2



x * 2



bp 1 ap 2 x * 1 .

(A)

Computing Ordinary Demands a Cobb-Douglas Example.



(x 1 *,x 2 *) also exhausts the budget so p x *



p x *



m .

(B)

Computing Ordinary Demands a Cobb-Douglas Example.



So now we know that x * 2



bp 1 ap 2 x * 1 p x *



p x *



m .

(A) (B)

Computing Ordinary Demands a Cobb-Douglas Example.



So now we know that Substitute x * 2



bp 1 ap 2 x * 1 p x *



p x *



m .

(A) (B)

Computing Ordinary Demands a Cobb-Douglas Example.



So now we know that Substitute x * 2



bp 1 ap 2 x * 1 p x *



p x *



m .

and get p x *



p 2 bp 1 ap 2 This simplifies to ….

x * 1



m .

(A) (B)

Computing Ordinary Demands a Cobb-Douglas Example.

x * 1



( a am



) 1 .

Computing Ordinary Demands a Cobb-Douglas Example.

x * 1



( a am



) 1 .

Substituting for x 1 * in p x *



p x *



m then gives x * 2



( a bm



2 .

Computing Ordinary Demands a Cobb-Douglas Example.

So we have discovered that the most preferred affordable bundle for a consumer with Cobb-Douglas preferences 2 )



is ( * 1 * 2 )



( ( a am



b p 1 , ( a bm



2 ) .

Computing Ordinary Demands a Cobb-Douglas Example.

x 2 2 )



x * 2



( a bm



2 x * 1



( a am



) 1 x 1

Rational Constrained Choice 

When x 1 * > 0 and x 2 * > 0 and (x 1 *,x 2 *) exhausts the budget, and indifference curves have no ‘kinks’, the ordinary demands are obtained by solving:



(a) p 1 x 1 * + p 2 x 2 * = y



(b) the slopes of the budget constraint, -p 1 /p 2 , and of the indifference curve containing (x 1 *,x 2 *) are equal at (x 1 *,x 2 *).

Rational Constrained Choice 

But what if x 1 * = 0?



Or if x 2 * = 0?



If either x 1 * = 0 or x 2 * = 0 then the ordinary demand (x 1 *,x 2 *) is at a corner solution to the problem of maximizing utility subject to a budget constraint.

Examples of Corner Solutions - the Perfect Substitutes Case

x 2 MRS = -1 x 1

Examples of Corner Solutions - the Perfect Substitutes Case

x 2 MRS = -1 Slope = -p 1 /p 2 with p 1 > p 2 .

x 1

Examples of Corner Solutions - the Perfect Substitutes Case

x 2 MRS = -1 Slope = -p 1 /p 2 with p 1 > p 2 .

x 1

Examples of Corner Solutions - the Perfect Substitutes Case

x 2 x * 2



y p 2 MRS = -1 Slope = -p 1 /p 2 with p 1 > p 2 .

x * 1



0 x 1

Examples of Corner Solutions - the Perfect Substitutes Case

x 2 MRS = -1 x * 2



0 Slope = -p 1 /p 2 with p 1 < p 2 .

x * 1



y p 1 x 1

Examples of Corner Solutions - the Perfect Substitutes Case

So when U(x 1 ,x 2 ) = x 1 + x 2 , the most preferred affordable bundle is (x 1 *,x 2 *) where ( x * 1 , x * 2 )

   

y p 1 , 0

  

if p 1 < p 2 and ( x * 1 , x * 2 )

  

y 0 , p 2

 

if p 1 > p 2 .

Examples of Corner Solutions - the Perfect Substitutes Case

x 2 y p 2 MRS = -1 Slope = -p 1 /p 2 with p 1 = p 2 .

y p 1 x 1

Examples of Corner Solutions - the Perfect Substitutes Case

x 2 y p 2 All the bundles in the constraint are equally the most preferred affordable when p 1 = p 2 .

y p 1 x 1

Examples of Corner Solutions - the Non-Convex Preferences Case

x 2 x 1

Examples of Corner Solutions - the Non-Convex Preferences Case

x 2 x 1

Examples of Corner Solutions - the Non-Convex Preferences Case

x 2 Which is the most preferred affordable bundle?

x 1

Examples of Corner Solutions - the Non-Convex Preferences Case

x 2 The most preferred affordable bundle x 1

Examples of Corner Solutions - the Non-Convex Preferences Case

Notice that the “tangency solution” x 2 is not the most preferred affordable bundle.

The most preferred affordable bundle x 1

Examples of ‘Kinky’ Solutions - the Perfect Complements Case

x 2 U(x 1 ,x 2 ) = min{ax 1 ,x 2 } x 2 = ax 1 x 1

Examples of ‘Kinky’ Solutions - the Perfect Complements Case

x 2 U(x 1 ,x 2 ) = min{ax 1 ,x 2 } x 2 = ax 1 MRS = 0 x 1

Examples of ‘Kinky’ Solutions - the Perfect Complements Case

x 2 U(x 1 ,x 2 ) = min{ax 1 ,x 2 } MRS = -



x 2 = ax 1 MRS = 0 x 1

Examples of ‘Kinky’ Solutions - the Perfect Complements Case

x 2 U(x 1 ,x 2 ) = min{ax 1 ,x 2 } MRS = -



MRS is undefined x 2 = ax 1 MRS = 0 x 1

Examples of ‘Kinky’ Solutions - the Perfect Complements Case

x 2 U(x 1 ,x 2 ) = min{ax 1 ,x 2 } x 2 = ax 1 x 1

Examples of ‘Kinky’ Solutions - the Perfect Complements Case

x 2 U(x 1 ,x 2 ) = min{ax 1 ,x 2 } Which is the most preferred affordable bundle?

x 2 = ax 1 x 1

Examples of ‘Kinky’ Solutions - the Perfect Complements Case

x 2 U(x 1 ,x 2 ) = min{ax 1 ,x 2 } The most preferred affordable bundle x 2 = ax 1 x 1

Examples of ‘Kinky’ Solutions - the Perfect Complements Case

x 2 U(x 1 ,x 2 ) = min{ax 1 ,x 2 } x 2 * x 1 * x 2 = ax 1 x 1

Examples of ‘Kinky’ Solutions - the Perfect Complements Case

x 2 U(x 1 ,x 2 ) = min{ax 1 ,x 2 } (a) p 1 x 1 * + p 2 x 2 * = m x 2 = ax 1 x 2 * x 1 * x 1

Examples of ‘Kinky’ Solutions - the Perfect Complements Case

x 2 U(x 1 ,x 2 ) = min{ax 1 ,x 2 } (a) p 1 x 1 * + p 2 x 2 * = m (b) x 2 * = ax 1 * x 2 = ax 1 x 2 * x 1 * x 1

Examples of ‘Kinky’ Solutions - the Perfect Complements Case

(a) p 1 x 1 * + p 2 x 2 * = m; (b) x 2 * = ax 1 *.

Examples of ‘Kinky’ Solutions - the Perfect Complements Case

(a) p 1 x 1 * + p 2 x 2 * = m; (b) x 2 * = ax 1 *.

Substitution from (b) for x 2 * in (a) gives p 1 x 1 * + p 2 ax 1 * = m

Examples of ‘Kinky’ Solutions - the Perfect Complements Case

(a) p 1 x 1 * + p 2 x 2 * = m; (b) x 2 * = ax 1 *.

Substitution from (b) for x 2 * in (a) gives p 1 x 1 * + p 2 ax 1 * = m which gives x * 1



p 1



m ap 2

Examples of ‘Kinky’ Solutions - the Perfect Complements Case

(a) p 1 x 1 * + p 2 x 2 * = m; (b) x 2 * = ax 1 *.

Substitution from (b) for x 2 * in (a) gives p 1 x 1 * + p 2 ax 1 * = m which gives x * 1



p 1



m ap 2 ; x * 2



p 1 am



ap 2 .

Examples of ‘Kinky’ Solutions - the Perfect Complements Case

(a) p 1 x 1 * + p 2 x 2 * = m; (b) x 2 * = ax 1 *.

Substitution from (b) for x 2 * in (a) gives p 1 x 1 * + p 2 ax 1 * = m which gives x * 1



p 1



m ap 2 ; x * 2



p A bundle of 1 commodity 1 unit and 1 am



ap 2 .

a commodity 2 units costs p 1 m/(p 1 + ap 2 + ap 2 ; ) such bundles are affordable.

Examples of ‘Kinky’ Solutions - the Perfect Complements Case

x 2 U(x 1 ,x 2 ) = min{ax 1 ,x 2 } x * 2



p 1 am



ap 2 x 2 = ax 1 x * 1



p 1



m ap 2 x 1