budgetary and other constraints on choice

1

Budgetary and Other Constraints on Choice

2

Consumption Choice Sets

A consumption choice set is the collection of all consumption choices available to the consumer.

What constrains consumption choice?

Budgetary, time and other resource limitations.

3

Budget Constraints

A consumption bundle containing x1 units of commodity 1, x2 units of commodity 2 and so on up to xn units of commodity n is denoted by the vector (x1, x2, … , xn).

Commodity prices are p1, p2, … , pn.

4

Budget Constraints

Q: When is a consumption bundle (x1, … , xn) affordable at given prices p1, … , pn?

5

Budget Constraints Q: When is a bundle (x1, … , xn)

affordable at prices p1, … , pn?

A: When p1x1 + … + pnxn mwhere m is the consumer’s (disposable) income.

6

Budget Set and Constraint for Two Commoditiesx2

x1

Budget constraint isp1x1 + p2x2 = m.

m /p1

m /p2

7


x1

Budget constraint isp1x1 + p2x2 = m.m /p2

m /p1

8


x1


m /p1

Just affordable

m /p2

9


x1


m /p1

Just affordableNot affordable

m /p2

10


x1


m /p1

AffordableJust affordable

Not affordable

m /p2

11


x1


m /p1

BudgetSet

the collection of all affordable bundles.

m /p2

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x1

p1x1 + p2x2 = m is x2 = -(p1/p2)x1 + m/p2

so slope is -p1/p2.

m /p1

BudgetSet

m /p2

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Budget Constraints For n = 2 and x1 on the horizontal

axis, the constraint’s slope is -p1/p2. What does it mean?

x pp

x mp2

121

2

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Budget Constraints For n = 2 and x1 on the horizontal

axis, the constraint’s slope is -p1/p2. What does it mean?

Increasing x1 by 1 must reduce x2 by p1/p2.

See Notes

x pp

x mp2

121

2

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Budget Sets & Constraints; Income and Price Changes

The budget constraint and budget set depend upon prices and income. What happens as prices or income change?

16

How do the budget set and budget constraint change as income m

increases?

Originalbudget set

x2

x1

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Higher income gives more choice

Originalbudget set

New affordable consumptionchoices

x2

x1

Original andnew budgetconstraints areparallel (sameslope).

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decreases?

Originalbudget set

x2

x1

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decreases?x2

x1

New, smallerbudget set

Consumption bundlesthat are no longeraffordable.Old and new

constraintsare parallel.

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Budget Constraints - Income Changes

Increases in income m shift the constraint outward in a parallel manner, thereby enlarging the budget set and improving choice.

Decreases in income m shift the constraint inward in a parallel manner, thereby shrinking the budget set and reducing choice.

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Budget Constraints - Income Changes

No original choice is lost and new choices are added when income increases, so higher income cannot make a consumer worse off.

An income decrease may (typically will) make the consumer worse off.

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Budget Constraints - Price Changes

What happens if just one price decreases?

Suppose p1 decreases.

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How do the budget set and budget constraint change as p1 decreases

from p1’ to p1”?

Originalbudget set

x2

x1

m/p2

m/p1’ m/p1”

-p1’/p2

24



Originalbudget set

x2

x1

m/p2

m/p1’ m/p1”

New affordable choices

-p1’/p2

25



Originalbudget set

x2

x1

m/p2

m/p1’ m/p1”

New affordable choices

Budget constraint pivots; slope flattens from -p1’/p2 to -p1”/p2

-p1’/p2

-p1”/p2

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Reducing the price of one commodity pivots the constraint outward. No old choice is lost and new choices are added, so reducing one price cannot make the consumer worse off.

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Similarly, increasing one price pivots the constraint inwards, reduces choice and may (typically will) make the consumer worse off.

The rate of exchange chance (see an example).

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Uniform Ad Valorem Sales Taxes A uniform sales tax levied at rate t

changes the constraint from p1x1 + p2x2 = mto (1+t)p1x1 + (1+t)p2x2 = m

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Uniform Ad Valorem Sales Taxes A uniform sales tax levied at rate t changes

the constraint from p1x1 + p2x2 = mto (1+t)p1x1 + (1+t)p2x2 = mi.e. p1x1 + p2x2 = m/(1+t).

Case for tax quantity Case for tax on both sales tax and income

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Uniform Ad Valorem Sales Taxesx2

x1

mp2

mp1

p1x1 + p2x2 = m

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x1

mp2

mp1

p1x1 + p2x2 = m

p1x1 + p2x2 = m/(1+t)

mt p( )1 1

mt p( )1 2

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x1

mt p( )1 2

mp2

mt p( )1 1

mp1

Equivalent income lossis

m mt

tt

m

1 1

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Uniform Ad Valorem Sales Taxes Case for tax quantity

Case for sales tax and income tax Quantity subsidy (see notes) Value Subsidy (see notes) Lump-sum tax or subsidy (see notes)

Budget set with rationing (see notes)

34

Preferences

35

Rationality in Economics

Behavioral Postulate:A decisionmaker always chooses its most preferred alternative from its set of available alternatives.

So to model choice we must model decisionmakers’ preferences.

36

Preference Relations Comparing two different consumption

bundles, x and y: strict preference: x is more preferred

than is y. weak preference: x is as at least as

preferred as is y. indifference: x is exactly as preferred as

is y.

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Preference Relations

Strict preference, weak preference and indifference are all preference relations.

Particularly, they are ordinal relations; i.e. they state only the order in which bundles are preferred.

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denotes strict preference; x y means that bundle x is preferred strictly to bundle y.

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denotes strict preference so x y means that bundle x is preferred strictly to bundle y.

denotes indifference; x y means x and y are equally preferred.

denotes weak preference;x y means x is preferred at least as much as is y.

~~

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Assumptions about Preference Relations

Completeness: For any two bundles x and y it is always possible to make the statement that either x y or y x.

~

~

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Reflexivity: Any bundle x is always at least as preferred as itself; i.e.

x x.~

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Transitivity: Ifx is at least as preferred as y, andy is at least as preferred as z, thenx is at least as preferred as z; i.e.

x y and y z x z.~ ~ ~

43

Indifference Curves

Take a reference bundle x’. The set of all bundles equally preferred to x’ is the indifference curve containing x’; the set of all bundles y x’.

44

Indifference Curves

xx22

xx11

x”x”

x”’x”’

x’ x’ x” x” x”’ x”’x’

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Indifference Curves

xx22

xx11

zz xx yy

x

y

z

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Indifference Curves

x2

x1

xAll bundles in I1 arestrictly preferred to all in I2.

y

z

All bundles in I2 are strictly preferred to all in I3.

I1

I2

I3

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Indifference Curves

x2

x1

I(x’)

x

I(x)

WP(x), the set of bundles weakly preferred to x.

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Indifference Curves

x2

x1

WP(x), the set of bundles weakly preferred to x.

WP(x) includes I(x).

x

I(x)

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Indifference Curves

x2

x1

SP(x), the set of bundles strictly preferred to x, does not include I(x).

x

I(x)

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Indifference Curves Cannot Intersect

xx22

xx11

xxyy

zz

II11

I2 From IFrom I11, x , x y. From I y. From I22, x , x z. z.Therefore y Therefore y z. z.

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Indifference Curves Cannot Intersect

xx22

xx11

xxyy

zz

II11

I2 From IFrom I11, x , x y. From I y. From I22, x , x z. z.Therefore y Therefore y z. But from I z. But from I11 and Iand I22 we see y z, a we see y z, a contradiction. contradiction.

52

Slopes of Indifference Curves

When more of a commodity is always preferred, the commodity is a good.

If every commodity is a good then indifference curves are negatively sloped.

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BetterBetter

Worse

Worse

Good 2Good 2

Good 1Good 1

Two goodsTwo goodsa negatively sloped a negatively sloped indifference curve.indifference curve.

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If less of a commodity is always preferred then the commodity is a bad.

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Better

Better

WorseWorse

Good 2Good 2

Bad 1Bad 1

One good and oneOne good and onebad a bad a positively sloped positively sloped indifference curve.indifference curve.

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Utility Functions

A preference relation that is complete, reflexive, transitive and continuous can be represented by a continuous utility function.

Continuity means that small changes to a consumption bundle cause only small changes to the preference level.

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Utility Functions

A utility function U(x) represents a preference relation if and only if:

x’ x” U(x’) > U(x”)

x’ x” U(x’) < U(x”)

x’ x” U(x’) = U(x”).

~

58

Utility Functions

Utility is an ordinal (i.e. ordering) concept.

E.g. if U(x) = 6 and U(y) = 2 then bundle x is strictly preferred to bundle y.

59

Utility Functions & Indiff. Curves Consider the bundles (4,1), (2,3) and

(2,2). Suppose (2,3) (4,1) (2,2). Assign to these bundles any numbers

that preserve the preference ordering;e.g. U(2,3) = 6 > U(4,1) = U(2,2) = 4.

Call these numbers utility levels.

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Utility Functions & Indiff. Curves

An indifference curve contains equally preferred bundles.

Equal preference same utility level. Therefore, all bundles in an

indifference curve have the same utility level.

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Utility Functions & Indiff. Curves So the bundles (4,1) and (2,2) are in

the indiff. curve with utility level U

But the bundle (2,3) is in the indiff. curve with utility level U 6.

On an indifference curve diagram, this preference information looks as follows:

62


U 6U 4

(2,3) (2,2) (4,1)

x1

x2

63


Another way to visualize this same information is to plot the utility level on a vertical axis.

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U(2,3) = 6

U(2,2) = 4 U(4,1) = 4

Utility Functions & Indiff. Curves3D plot of consumption & utility levels for 3 bundles

x1

x2

Utility

65


This 3D visualization of preferences can be made more informative by adding into it the two indifference curves.

66


U

U

Higher indifferencecurves containmore preferredbundles.

Utility

x2

x1

67


Comparing more bundles will create a larger collection of all indifference curves and a better description of the consumer’s preferences.

68


U 6U 4U 2

x1

x2

69


As before, this can be visualized in 3D by plotting each indifference curve at the height of its utility index.

70


U 6U 5U 4U 3U 2U 1

x1

x2

Utility

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The collection of all indifference curves for a given preference relation is an indifference map.

An indifference map is equivalent to a utility function; each is the other.

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Utility Functions

There is no unique utility function representation of a preference relation.

Suppose U(x1,x2) = x1x2 represents a preference relation.

Again consider the bundles (4,1),(2,3) and (2,2).

73

Utility Functions U(x1,x2) = x1x2, so

U(2,3) = 6 > U(4,1) = U(2,2) = 4;

that is, (2,3) (4,1) (2,2). Definir transformación monótona

(ver notas)

74

Utility Functions U(x1,x2) = x1x2 (2,3) (4,1) (2,2). Define V = U2.

75

Utility Functions U(x1,x2) = x1x2 (2,3) (4,1) (2,2). Define V = U2. Then V(x1,x2) = x1

2x22 and

V(2,3) = 36 > V(4,1) = V(2,2) = 16so again(2,3) (4,1) (2,2).

V preserves the same order as U and so represents the same preferences.

76

Goods, Bads and Neutrals A good is a commodity unit which

increases utility (gives a more preferred bundle).

A bad is a commodity unit which decreases utility (gives a less preferred bundle).

A neutral is a commodity unit which does not change utility (gives an equally preferred bundle).

77

Goods, Bads and NeutralsUtility

Waterx’

Units ofwater aregoods

Units ofwater arebads

Around x’ units, a little extra water is a neutral.

Utilityfunction

78

Some Other Utility Functions and Their Indifference Curves

Instead of U(x1,x2) = x1x2 considerV(x1,x2) = x1 + x2. What do the indifference curves for

this “perfect substitution” utility function look like?

79

Perfect Substitution Indifference Curves

5

5

9

9

13

13

x1

x2

x1 + x2 = 5

x1 + x2 = 9

x1 + x2 = 13

V(x1,x2) = x1 + x2.

80

Perfect Substitution Indifference Curves

5

5

9

9

13

13

x1

x2

x1 + x2 = 5

x1 + x2 = 9

x1 + x2 = 13

All are linear and parallel.

V(x1,x2) = x1 + x2.

budgetary and other constraints on choice

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