Week 01 - Part B
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Transcript of Week 01 - Part B
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THE BUDGET CONSTRAINT
FEASIBILITY IN A MARKET SETTING
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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
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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
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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)
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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
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BUDGET SET AND CONSTRAINT FOR TWO COMMODITIES x2
x1
Budget constraint is p1x1 + p2x2 = m
m/p1
m/p2
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BUDGET SET AND CONSTRAINT FOR TWO COMMODITIES x2
x1 m/p1
m/p2 Budget constraint is
p1x1 + p2x2 = m
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BUDGET SET AND CONSTRAINT FOR TWO COMMODITIES x2
x1 m/p1
m/p2
Just affordable Not affordable
Budget constraint is p1x1 + p2x2 = m
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BUDGET SET AND CONSTRAINT FOR TWO COMMODITIES x2
x1 m/p1
m/p2
Just affordable Not affordable
Affordable
Budget constraint is p1x1 + p2x2 = m
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BUDGET SET AND CONSTRAINT FOR TWO COMMODITIES x2
x1 m/p1
m/p2
Budget set
Budget constraint is p1x1 + p2x2 = m
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BUDGET SET AND CONSTRAINT FOR TWO COMMODITIES x2
x1 m/p1
m/p2
We can write the budget constraint as x2 = - (p1/p2) x1 + m/p2
Budget set
Budget constraint is p1x1 + p2x2 = m
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BUDGET SET AND CONSTRAINT FOR TWO COMMODITIES x2
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
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BUDGET CONSTRAINT FOR THREE COMMODITIES
x2
x1
x3
m/p2
m/p1
m/p3
p1x1 + p2x2 + p3x3 = m
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BUDGET CONSTRAINT FOR THREE COMMODITIES
x2
x1
x3
m/p2
m/p1
m/p3
{(x1,x2,x3) | xi 0, and p1x1 + p2x2 + p3x3 m}
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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
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HIGHER INCOME GIVES MORE CHOICE
Original budget set
New affordable consumption choices
x2
x1
Original and new budget constraints are parallel
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CHANGES AS PRICE OF GOOD 1 DECREASES
Original budget set
x2
x1
m/p2
m/p1 m/p1
-p1/p2
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CHANGES AS PRICE OF GOOD 1 DECREASES
Original budget set
x2
x1
m/p2
m/p1 m/p1
New affordable choices
-p1/p2
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CHANGES AS PRICE OF GOOD 1 DECREASES
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
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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
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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
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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 p1x1 + p2x2 = m/(1+t)
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UNIFORM AD VALOREM SALES TAXES
x2
x1
mp2
mp1
p1x1 + p2x2 = m
p1x1 + p2x2 = m/(1+t)
m(1+ t)p1
mt p( )1 2+
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UNIFORM AD VALOREM SALES TAXES
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
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UNIFORM AD VALOREM SALES TAXES
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
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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 = $1; the budget constraint is then F + G = 100
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THE FOOD STAMP PROGRAM W/O BLACK MARKET
G
F 100
100
Original budget set F + G = 100
40
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THE FOOD STAMP PROGRAM W/O BLACK MARKET
G
F 100
100
Original budget set F + G = 100
40 140
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THE FOOD STAMP PROGRAM W/O BLACK MARKET
G
F 100
100
Original budget set F + G = 100
Budget set after 40 food stamps issued
140 40
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THE FOOD STAMP PROGRAM WITH BLACK MARKET
G
F 100
100
Original budget set F + G = 100
Budget set after 40 food stamps issued
140 40
Budget constraint with black market trading at 0.50$ per food stamp
120
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THE FOOD STAMP PROGRAM WITH BLACK MARKET
G
F 100
100
Original budget set F + G = 100
Budget set after 40 food stamps issued
140 40
Budget constraint with black market trading at 0.50$ per food stamp
120
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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)
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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)
{
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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
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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
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SHAPES OF BUDGET CONSTRAINTS - QUANTITY DISCOUNTS
50
100
20 80
x2
x1 Budget Set
Budget Constraint
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SHAPES OF BUDGET CONSTRAINTS - QUANTITY PENALTY
x2
x1
Budget Constraint
Budget Set
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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