Course: Microeconomics Text: Varian’s Intermediate Microeconomics 1

 Economists assume that consumers choose the best bundle of goods they can afford.  This chapter first specifies in detail what consumer can afford : the budget constraint or the consumption possibility set.  What is best for consumer, or the preference on the possible consumption bundles, will be discussed in the next chapter. 2

 A consumption bundle containing x 1 units of commodity 1, x 2 units of commodity 2 and so on up to x n units of commodity n is denoted by the vector (x 1, x 2, …, x n ).  Commodity prices are p 1, p 2, …, p n. 3

 Q: When is a consumption bundle (x 1, …, x n ) affordable at prices p 1, …, p n ? 4

 A: When total expenditure is smaller than income: p 1 x 1 + … + p n x n  m where m is the consumer’s (disposable) income.  That is, one’s total expenditure is smaller than or equal to one’s income. 5

 The bundles that are only just affordable by the consumer is one’s budget constraint. This is the set { (x 1,…,x n ) | x 1  0, …, x n  and p 1 x 1 + … + p n x n  m }. 6

 The consumer’s budget set is the set of all affordable bundles; B(p 1, …, p n, m ) = { (x 1, …, x n ) | x 1  0, …, x n  0 and p 1 x 1 + … + p n x n  m }  The budget constraint is the upper boundary of the budget set. 7

x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 m /p 2 8

x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 2 m /p 1 9

x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 Just affordable m /p 2 10

x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 Just affordable Not affordable m /p 2 11

x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 Affordable Just affordable Not affordable m /p 2 12

x2x2 x1x1 Budget constraint (budget line) is p 1 x 1 + p 2 x 2 = m. m /p 1 Budget Set the collection of all affordable bundles. m /p 2 13

x2x2 x1x1 p 1 x 1 + p 2 x 2 = m is x 2 = -(p 1 /p 2 )x 1 + m /p 2 so slope is -p 1 /p 2. m /p 1 Budget Set m /p 2 14

 For n = 2 and x 1 on the horizontal axis, the constraint’s slope is -p 1 /p 2. What does it mean?  Note: For a linear line: y=mx+c, m is the slope of the line, while c is the y-intercept. 15

 For n = 2 and x 1 on the horizontal axis, the constraint’s slope is -p 1 /p 2. What does it mean?  To hold income m constant, increasing x 1 by 1 must reduce x 2 by p 1 /p 2. 16

x2x2 x1x1 Slope is -p 1 /p p 1 /p 2 17

x2x2 x1x1 +1 -p 1 /p 2 Opp. cost of an extra unit of commodity 1 is p 1 /p 2 units foregone of commodity 2. 18

x2x2 x1x1 The opp. cost of an extra unit of commodity 2 is p 2 /p 1 units foregone of commodity 1. -p 2 /p

 The budget constraint and budget set depend upon prices and income. What happens as prices or income change? 20

Original budget set x2x2 x1x1 21

Original budget set New affordable consumption choices x2x2 x1x1 Original and new budget constraints are parallel (same slope). 22

Original budget set x2x2 x1x1 23

x2x2 x1x1 New, smaller budget set Consumption bundles that are no longer affordable. Old and new constraints are parallel. 24

 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.  The slope –p 1 / p 2 does not change. 25

 When income increases, NO original choice is lost and new choices are added, so higher income cannot make a consumer worse off.  When income decreases, the consumer may (typically will) be worse off, as one can no longer afford some of the bundles anymore. 26

 What happens if just one price decreases?  Suppose p 1 decreases. 27

Original budget set x2x2 x1x1 m/p 2 m/p 1 ’ m/p 1 ” -p 1 ’/p 2 28

Original budget set x2x2 x1x1 m/p 2 m/p 1 ’ m/p 1 ” New affordable choices -p 1 ’/p 2 29

Original budget set x2x2 x1x1 m/p 2 m/p 1 ’ m/p 1 ” New affordable choices Budget constraint pivots; slope flattens from -p 1 ’/p 2 to -p 1 ”/p 2 -p 1 ’/p 2 -p 1 ”/p 2 30

 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.  Similarly, increasing one price pivots the constraint inwards (consider a price change from p 1 ” to p 1 ’ ), reduces choice and may (typically will) make the consumer worse off. 31

 Quantity/per-unit tax: price increases from p to p+t.  Quantity/per-unit subsidy: price decreases from p to p-s.  Ad valorem/value tax: price increases from p to (1+t)p  Ad valorem/value subsidy: price decreases from p to (1-s)p 32

 A uniform sales tax levied at rate t on all goods changes the constraint from p 1 x 1 + p 2 x 2 = m to (1+ t )p 1 x 1 + (1+ t )p 2 x 2 = m i.e. p 1 x 1 + p 2 x 2 = m /(1+ t ). 33

x2x2 x1x1 Equivalent income loss is 34

x2x2 x1x1 A uniform ad valorem sales tax levied at rate t is equivalent to an income tax levied at rate 35

 Lump-sum tax: government tax a fixed sum of money, T, regardless of individual’s behavior.  This is equivalent to a decrease in income by T, implying an inward parallel shift of budget line.  Similarly, lump-sum subsidy S implies an outward parallel shift of budget line corresponding to an amount S. 36

 Food stamps are coupons that can be legally exchanged only for food.  How does a commodity-specific gift such as a food stamp alter a family’s budget constraint?  Here we assume one of the two goods is food. 37

 Suppose m = $100, p F = $1 and the price of “other goods” is p G = $1.  The budget constraint is then F + G =

G F 100 F + G = 100: before stamps. 39

G F 100 F + G = 100: before stamps. Budget set after $40 food stamps issued

G F 100 F + G = 100: before stamps. Budget set after $40 food stamps issued. 140 The family’s budget set is enlarged

 How does the unit of account affect the budget constraints and budget set?  Suppose prices and income are measured in dollars. Say p 1 =$2, p 2 =$3, m = $12. Then the constraint is 2x 1 + 3x 2 =

 If prices and income are measured in cents, then p 1 =200, p 2 =300, m =1200 and the constraint is 200x x 2 = 1200, upon simplification, it is the same as 2x 1 + 3x 2 = 12.  Changing the unit of account changes neither the budget constraint nor the budget set. 43

 The constraint for p 1 =2, p 2 =3, m =12 2x 1 + 3x 2 = 12 is also 1x 1 + (3/2)x 2 = 6, the constraint for p 1 =1, p 2 =3/2, m =6.  Setting p 1 =1 makes commodity 1 the numeraire and defines all prices relative to p 1 ;  e.g. 3/2 is the price of commodity 2 relative to the price of commodity 1. 44

 Multiplying all prices and income by any constant k does not change the budget constraint. kp 1 x 1 +kp 2 x 2 =km  Any commodity can be chosen as the numeraire (by taking k=1/p i ) without changing the budget set or the budget constraint.  It is also clear from the graph that, only the ratios p 1 /p 2, m/p 1 and m/p 2 are relevant to the budget line and budget set. 45

 Q: What makes a budget constraint a straight line?  A: A straight line has a constant slope and the constraint is p 1 x 1 + … + p n x n = m so if prices are constants then a constraint is a straight line. 46

 But what if prices are not constants?  E.g. bulk buying discounts, or price penalties (or tax) for buying “too much”.  Then constraints will be curved or have kinks. 47

 Suppose p 2 is constant at $1 but that p 1 =$2 for 0  x 1  20 and p 1 =$1 for x 1 >20. 48

 Suppose p 2 is constant at $1 but that p 1 =$2 for 0  x 1  20 and p 1 =$1 for x 1 >20.  Then the constraint’s slope is - 2, for 0  x 1  20 -p 1 /p 2 = - 1, for x 1 > 20  Also assume m=100. { 49

m = $ Slope = - 2 / 1 = - 2 (p 1 =2, p 2 =1) Slope = - 1/ 1 = - 1 (p 1 =1, p 2 =1) 80 x2x2 x1x1 50

m = $ Slope = - 2 / 1 = - 2 (p 1 =2, p 2 =1) Slope = - 1/ 1 = - 1 (p 1 =1, p 2 =1) 80 x2x2 x1x1 51

m = $ x2x2 x1x1 Budget Set Budget Constraint 52

x2x2 x1x1 Budget Set Budget Constraint 53

 The budget set describes what consumption bundles are affordable to the consumers.  The budget constraint is typically described by p 1 x 1 + p 2 x 2 = m, which is a straight line when prices are constant.  When income increases, budget set shifts outward, enlarging the budget set.  When prices increases, the slope of budget line changes, and the it shrinks the budget set. 54

 The next chapter will introduce preference, which describes the ordering of what a consumer likes among the consumption bundles.  Then we can combine both preference and budget constraint to analyze consumer’s choice. 55