chapter 5 choice

Chapter 5Choice

2

Economic Rationality

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

The available choices constitute the choice set.

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

3

Rational Constrained Choice

x1

x2

4


x1

x2

Affordablebundles

5


x1

x2

Affordablebundles

More preferredbundles

6


Affordablebundles

x1

x2

More preferredbundles

7


x1

x2

x1*

x2*

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

8

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 byx1*(p1,p2,m) and x2*(p1,p2,m).

9


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

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

10


x1

x2

x1*

x2*

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

11


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.

12

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*).

13

Rational Constrained Choice Condition (b) can be written as:

That is,

Therefore, at the optimal choice, the ratio of marginal utilities of the two commodities must be equal to their price ratio.

2 1 1

1 2 2

dx MU pMRS

dx MU p

1 1

2 2

MU p

MU p

14

Mathematical Treatment

The consumer’s optimization problem can be formulated mathematically as:

One way to solve it is to use Lagrangian method:

15

Mathematical Treatment Assuming the utility function is

differentiable, and there exists an interior solution.

The first order conditions are:

16

Mathematical Treatment

So, for interior solution, we can solve for the optimal bundle using the two conditions:

1 1

2 2

1 1 2 2

,

.

MU p

MU p

p x p x m

17

Computing Ordinary Demands - a Cobb-Douglas Example. 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

18

Computing Ordinary Demands - a Cobb-Douglas Example. 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)

19

Computing Ordinary Demands - a Cobb-Douglas Example. So now we have

Solving these two equations, we have

xbpap

x21

21

* * (A)

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

xam

a b p11

*

( ).

x

bma b p2

2

*

( ).

20

Computing Ordinary Demands - a Cobb-Douglas Example. So we have discovered that the most

preferred affordable bundle for a consumer with Cobb-Douglas preferences

is

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

( , )( )

,( )

.* * ( )x xam

a b pbm

a b p1 21 2

21

A Cobb-Douglas Example

x1

x2

xam

a b p11

*

( )

x

bma b p

2

2

*

( )

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

22

A Cobb-Douglas Example Note that with the optimal bundle:

The expenditures on each good:

It is a property of Cobb-Douglas Utility function that expenditure share on a particular good is a constant.

( , )( )

,( )

.* * ( )x xam

a b pbm

a b p1 21 2

.)(

,)(

),( *22

*11

mba

bm

ba

axpxp

23

Special Cases

24

Special Cases

25

Some Notes

For interior optimum, tangency condition is only necessary but not sufficient.

There can be more than one optimum.

Strict convexity implies unique solution.

26

Corner Solution

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.

27

Examples of Corner Solutions -- the Perfect Substitutes Case

x1

x2

MRS = -1

28


x1

x2

MRS = -1

Slope = -p1/p2 with p1 > p2.

29


x1

x2

x1 0*

MRS = -1

Slope = -p1/p2 with p1 > p2.

2

*2 p

mx

30


x1

x2

x2 0*

MRS = -1

Slope = -p1/p2 with p1 < p2.

1

*1 p

mx

31

Examples of Corner Solutions -- the Perfect Substitutes Case So when U(x1,x2) = x1 + x2, the most

preferred affordable bundle is (x1*,x2*) where

and

0,),(

1

*2

*1 p

mxx if p1 < p2

2

*2

*1 ,0),(

p

mxx if p1 > p2

32


x1

x2

MRS = -1

Slope = -p1/p2 with p1 = p2.

1p

m

2p

m

33


x1

x2

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

1p

m

2p

m

34

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

x1

x2Better

35


x1

x2

36


x1

x2Which is the most preferredaffordable bundle?

37


x1

x2

The most preferredaffordable bundle

38


x1

x2


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

39

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

x1

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

x2 = ax1

40


x1

x2

MRS = -

MRS = 0

MRS is undefined

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

x2 = ax1

41


x1

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

x2 = ax1

42


x1


x2 = ax1

Which is the mostpreferred affordable bundle?

43


x1


x2 = ax1


44


x1

x2

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

x2 = ax1

x1*

x2*

45


x1


x2 = ax1

x1*

x2*

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

46


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

Solving the two equations, we have

.app

amx;

appm

x21

*2

21

*1

47


x1

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

x2 = ax1

xm

p ap11 2

*

x

amp ap

2

1 2

*

chapter 5 choice

Documents

ordinary demand x1

rational constrained

preferred affordable

property of cobb

budget constraint

optimal bundle

interior solution

douglas utility function