Choice. Economic Rationality u The principal behavioral postulate is that a decisionmaker chooses...

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.

Economic Rationality

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

The answer is simple: the consumer chooses the bundle within the choice set that is on the indifference curve that provides the highest level of utility.

Rational Constrained Choice

x1

x2

x1*

x2*


x1

x2

x1*

x2*

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


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.(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*).

Rational Constrained Choice For well-behaved (monotonic and

convex) preferences, tangency is a necessary and sufficient condition for optimality.

At the optimum the consumer substitutes one good for the other at a rate identical to the market’s.

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

12 1 1 2 2

11 2 1 2 1

/

/

a b

a b

dx U x ax x axMRS

dx U x bx x bx

Computing Ordinary Demands - a Cobb-Douglas Example.

So the MRS is

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

12 1 1 2 2

11 2 1 2 1

/

/

a b

a b

dx U x ax x axMRS

dx U x bx x bx

** *2 1 12 1*

1 2 2

ax p bpx x

bx p ap (A)


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

* *1 1 2 2p x p x m (B)


So now we know that

* *12 1

2

bpx x

ap (A)

* *1 1 2 2p x p x m (B)


So now we know that

xbpap

x21

21

* * (A)

* *1 1 2 2p x p x m (B)

Substitute


So now we know that

xbpap

x21

21

* * (A)

* *1 1 2 2p x p x m (B)

* *11 1 2 1

2

bpp x p x m

ap

Substitute

and get

This simplifies to ….


*1

1( )

amx

a b p


*2

2( )

bmx

a b p

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

* *

then gives

*1

1( )

amx

a b p


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 * *1 2

1 2

( , ) ,( ) ( )

am bmx x

a b p a b p


x1

x2

*1

1( )

amx

a b p

*2

2( )

bmx

a b p

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* = m

(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

*2

2

mx

p

x1 0*

MRS = -1

Slope = -p1/p2 with p1 > p2


x1

x2

*1

1

mx

p

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

* *1 2

1

( , ) ,0m

x xp

and

* *1 2

2

( , ) 0,m

x xp

if p1 < p2

if p1 > p2


x1

x2

MRS = -1

Slope = - p1/p2 with p1= p2

1

m

p

2

m

p


x1

x2

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

2

m

p

1

m

p

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

x1

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

x2 = ax1


x1

x2

MRS = -

MRS = 0

MRS is undefined

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

x2 = ax1


x1


x2 = ax1

Which is the mostpreferred affordable bundle?


x1


x2 = ax1

The most preferredaffordable bundle


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

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

* *1 2

1 2 1 2

,m am

x xp ap p ap

Application: choosing a tax

Which is less harmful, a quantity tax or an income tax, both allowing the same revenue?

1. We can show that an income tax is always better in the sense that given any quantity tax, there is an income tax that leaves the consumer at a higher indifference curve.


2. Outline of the argument:

a) original budget constraint: p1x1 + p2x2 = m

b) budget constraint with quantity tax:

(p1 + t)x1+p2x2 = m

c) optimal choice with tax: (p1+t)x1* + p2x2

* = m

d) revenue raised is tx1*


e) income tax (lump-sum) that raises the same amount of revenue leads to budget constraint: p1x1 + p2x2 = m - tx1*i) this line has the same slope as the original budget line, but lies inwards


ii) also passes through (x1*;x2*)

iii) proof: p1x1*+ p2x2* = m - tx1*

iv) this means that (x1*; x2* ) is affordable under the income tax, so the optimal choice under the income tax must be even better than (x1 *; x2*)

Choice. Economic Rationality u The principal behavioral postulate is that a decisionmaker chooses...

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