week 01 - part b

THE BUDGET CONSTRAINT

FEASIBILITY IN A MARKET SETTING

INTRODUCTION TO DECISION MAKING BY CONSUMERS

We want to study how individual agents (consumers and then firms) make optimal decisions

Two elements of any decision making problem what is feasible what is desirable

Today we address the first element in a market setting The consumption choice set is the collection of all

alternatives available to the consumer availability here is in the broadest sense we are interested in the collection of feasible alternatives

ROAD MAP

Its not just about the money the budget constraint and its determinants

Everything is relative changing prices and income

Carrots and sticks in your purse uniform sales taxes and the food stamp program

On sale for a limited time only quantity discounts in the budget constraint

BUDGET CONSTRAINTS What constrains consumption choices?

budgetary, time and other resource limitations focus on budgetary restrictions first

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 x = (x1, x2, , xn)

Commodity prices are p = (p1, p2, , pn)

When is a given consumption bundle x = (x1, x2, , xn) affordable at prices p = (p1, p2, , pn)

BUDGET CONSTRAINTS For bundle x to be affordable at prices p it must be that

p1x1 + + pnxn m where m is the consumers (disposable) income

The bundles that are affordable given prices p and income m form the consumers budget set

B(p,m) = { (x1,,xn) | x1 0, , xn 0 and p1x1 + + pnxn m }

The bundles that are just affordable belong to the consumer budget constraint the budget constraint is the upper boundary of the budget set

BUDGET SET AND CONSTRAINT FOR TWO COMMODITIES x2

x1

Budget constraint is p1x1 + p2x2 = m

m/p1

m/p2


x1 m/p1

m/p2 Budget constraint is

p1x1 + p2x2 = m


x1 m/p1

m/p2

Just affordable Not affordable



x1 m/p1

m/p2

Just affordable Not affordable

Affordable



x1 m/p1

m/p2

Budget set



x1 m/p1

m/p2

We can write the budget constraint as x2 = - (p1/p2) x1 + m/p2

Budget set



x1 m/p1

m/p2

We can write the budget constraint as x2 = - (p1/p2) x1 + m/p2

Opportunity cost of an extra unit of good 1 is p1/p2 units foregone of commodity 2

+1

-p1/p2

BUDGET CONSTRAINT FOR THREE COMMODITIES

x2

x1

x3

m/p2

m/p1

m/p3

p1x1 + p2x2 + p3x3 = m

BUDGET CONSTRAINT FOR THREE COMMODITIES

x2

x1

x3

m/p2

m/p1

m/p3

{(x1,x2,x3) | xi 0, and p1x1 + p2x2 + p3x3 m}

BUDGET SETS, INCOME AND PRICE CHANGES The budget constraint and budget set depend upon prices

and income we want to understand what happens as prices or income

change

A change in income is easily understood there is no move in relative prices of goods, so the budget

constraint moves parallel to the original budget constraint outwards if income increases making the consumer

better off or inwards if income decreases making the consumer

worse off

HIGHER INCOME GIVES MORE CHOICE

Original budget set

New affordable consumption choices

x2

x1

Original and new budget constraints are parallel

CHANGES AS PRICE OF GOOD 1 DECREASES

Original budget set

x2

x1

m/p2

m/p1 m/p1

-p1/p2


Original budget set

x2

x1

m/p2

m/p1 m/p1

New affordable choices

-p1/p2


Original budget set

x2

x1

m/p2

m/p1 m/p1

New affordable choices

Budget constraint pivots and slope flattens from - p1/p2 to - p1/p2

-p1/p2

-p1/p2

BUDGET CONSTRAINTS - PRICE CHANGES Reducing the price of one commodity pivots the budget

constraint outward no initial feasible bundle is lost and new bundles are

affordable now, so reducing one price cannot make the consumer worse off

Similarly, increasing one price pivots the budget constraint inwards

some consumption bundles are no longer affordable, so the choices of the consumer are reduced

this makes her (typically) worse off

UNIFORM AD VALOREM SALES TAXES

An ad valorem sales tax levied at a rate of 5% for commodity k increases its price from pk to (1 + 0.05)pk = 1.05pk

An ad valorem sales tax levied at a rate of t increases the price of good k from pk to (1 + t)pk

A uniform sales tax is applied uniformly to all commodities levied at rate t changes the constraint from

p1x1 + p2x2 = m to (1+t)p1x1 + (1+t)p2x2 = m


An ad valorem sales tax levied at a rate of 5% for commodity k increases its price from pk to (1 + 0.05)pk = 1.05pk

An ad valorem sales tax levied at a rate of t increases the price of good k from pk to (1 + t)pk

A uniform sales tax is applied uniformly to all commodities levied at rate t changes the constraint from

p1x1 + p2x2 = m to p1x1 + p2x2 = m/(1+t)


x2

x1

mp2

mp1

p1x1 + p2x2 = m

p1x1 + p2x2 = m/(1+t)

m(1+ t)p1

mt p( )1 2+


x2

x1

mp2

mp1

p1x1 + p2x2 = m

p1x1 + p2x2 = m/(1+t)

m(1+ t)p1

mt p( )1 2+

Equivalent income loss is

m t/(1+t) of the consumers

income


x2

x1

mp2

mp1

m(1+ t)p1

mt p( )1 2+

A uniform ad valorem sales tax levied at rate t is equivalent to an income tax levied at rate t /(1+t)

Equivalent income loss is

m t/(1+t) of the consumers

income

THE FOOD STAMP PROGRAM Food stamps are coupons that can be legally exchanged

only for food (and other basic products) We want to understand how a given number of food

stamps ($40) alter a familys budget constraint first scenario: no secondary market for food stamps second scenario: secondary market available

Two things to keep in mind actual effects of the food stamp program onto budget set (and

consumers welfare) what is missing in this simple analysis

Suppose m = $100, pF = $1 and the price of other goods is pG = $1; the budget constraint is then F + G = 100

THE FOOD STAMP PROGRAM W/O BLACK MARKET

G

F 100

100

Original budget set F + G = 100

40


G

F 100

100


40 140


G

F 100

100


Budget set after 40 food stamps issued

140 40

THE FOOD STAMP PROGRAM WITH BLACK MARKET

G

F 100

100


Budget set after 40 food stamps issued

140 40

Budget constraint with black market trading at 0.50$ per food stamp

120

SHAPES OF BUDGET CONSTRAINTS In the 2-good case, the budget constraint is a straight line

the slope is constant (and equal to - p1/p2) the opportunity cost of one good in terms of the other is always

the same In the n-good case, the budget constraint is a hyperplane

mathematically, the budget constraint is an affine function This is the case whenever prices are posted (each additional

unit of a good costs the same) What if prices change

bulk-buying discounts (home products) price penalties for buying too much (mobile plans)

SHAPES OF BUDGET CONSTRAINTS - QUANTITY DISCOUNTS Suppose p2 is constant at $1 Suppose p1 = $2 for 0 x1 20 but then p1 = $1 for x1 > 20 Then the constraints slope is

- 2, for 0 x1 20 - p1/p2 = - 1, for x1 > 20

Assume the consumer has $100 Three key points to understand the budget constraint

intercept with the vertical axis (buying only good 2) the bundle at which the slope changes the intercept with the horizontal axis (buying only good 1)

{

SHAPES OF BUDGET CONSTRAINTS - QUANTITY DISCOUNTS

50

100

20

slope is - 2 / 1 = - 2 (p1 = 2, p2 = 1)

slope = - 1/ 1 = - 1 (p1 = 1, p2 = 1)

80

x2

x1

SHAPES OF BUDGET CONSTRAINTS - QUANTITY DISCOUNTS

50

100

20 80

x2

x1 Budget Set

Budget Constraint

SHAPES OF BUDGET CONSTRAINTS - QUANTITY PENALTY

x2

x1

Budget Constraint

Budget Set

MORE GENERAL FEASIBLE SETS Choices are usually constrained by more than a budget

restriction time constraints and other resources constraints

A consumption bundle is feasible/affordable only if it meets every constraint

So the feasible set (the generalized budget set) is the intersection of all these constraints

week 01 - part b

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