© 2010 W. W. Norton & Company, Inc. 2 Budgetary and Other Constraints on Choice

© 2010 W. W. Norton & Company, Inc. 2 Consumption Choice Sets u A consumption choice set is the collection of all consumption choices available to the consumer. u What constrains consumption choice? –Budgetary, time and other resource limitations.

© 2010 W. W. Norton & Company, Inc. 3 Budget Constraints u 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 ). u Commodity prices are p 1, p 2, …, p n.

© 2010 W. W. Norton & Company, Inc. 4 Budget Constraints u Q: When is a consumption bundle (x 1, …, x n ) affordable at given prices p 1, …, p n ?

© 2010 W. W. Norton & Company, Inc. 5 Budget Constraints u Q: When is a bundle (x 1, …, x n ) affordable at prices p 1, …, p n ?  A: When p 1 x 1 + … + p n x n  m where m is the consumer’s (disposable) income.

© 2010 W. W. Norton & Company, Inc. 6 Budget Constraints  The bundles that are only just affordable form the consumer’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 }.

© 2010 W. W. Norton & Company, Inc. 7 Budget Constraints  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 } u The budget constraint is the upper boundary of the budget set.

© 2010 W. W. Norton & Company, Inc. 8 Budget Set and Constraint for Two Commodities x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 m /p 2

© 2010 W. W. Norton & Company, Inc. 9 Budget Set and Constraint for Two Commodities x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 2 m /p 1

© 2010 W. W. Norton & Company, Inc. 10 Budget Set and Constraint for Two Commodities x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 Just affordable m /p 2

© 2010 W. W. Norton & Company, Inc. 11 Budget Set and Constraint for Two Commodities x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 Just affordable Not affordable m /p 2

© 2010 W. W. Norton & Company, Inc. 12 Budget Set and Constraint for Two Commodities 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

© 2010 W. W. Norton & Company, Inc. 13 Budget Set and Constraint for Two Commodities x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 Budget Set the collection of all affordable bundles. m /p 2

© 2010 W. W. Norton & Company, Inc. 14 Budget Set and Constraint for Two Commodities 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

© 2010 W. W. Norton & Company, Inc. 15 Budget Constraints u If n = 3 what do the budget constraint and the budget set look like?

© 2010 W. W. Norton & Company, Inc. 16 Budget Constraint for Three Commodities x2x2 x1x1 x3x3 m /p 2 m /p 1 m /p 3 p 1 x 1 + p 2 x 2 + p 3 x 3 = m

© 2010 W. W. Norton & Company, Inc. 17 Budget Set for Three Commodities x2x2 x1x1 x3x3 m /p 2 m /p 1 m /p 3 { (x 1,x 2,x 3 ) | x 1  0, x 2  0, x 3   0 and p 1 x 1 + p 2 x 2 + p 3 x 3  m}

© 2010 W. W. Norton & Company, Inc. 18 Budget Constraints u For n = 2 and x 1 on the horizontal axis, the constraint’s slope is -p 1 /p 2. What does it mean?

© 2010 W. W. Norton & Company, Inc. 19 Budget Constraints u For n = 2 and x 1 on the horizontal axis, the constraint’s slope is -p 1 /p 2. What does it mean? u Increasing x 1 by 1 must reduce x 2 by p 1 /p 2.

© 2010 W. W. Norton & Company, Inc. 20 Budget Constraints x2x2 x1x1 Slope is -p 1 /p p 1 /p 2

© 2010 W. W. Norton & Company, Inc. 21 Budget Constraints 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.

© 2010 W. W. Norton & Company, Inc. 22 Budget Constraints x2x2 x1x1 Opp. cost of an extra unit of commodity 1 is p 1 /p 2 units foregone of commodity 2. And the opp. cost of an extra unit of commodity 2 is p 2 /p 1 units foregone of commodity 1. -p 2 /p 1 +1

© 2010 W. W. Norton & Company, Inc. 23 Budget Sets & Constraints; Income and Price Changes u The budget constraint and budget set depend upon prices and income. What happens as prices or income change?

© 2010 W. W. Norton & Company, Inc. 24 How do the budget set and budget constraint change as income m increases? Original budget set x2x2 x1x1

© 2010 W. W. Norton & Company, Inc. 25 Higher income gives more choice Original budget set New affordable consumption choices x2x2 x1x1 Original and new budget constraints are parallel (same slope).

© 2010 W. W. Norton & Company, Inc. 26 How do the budget set and budget constraint change as income m decreases? Original budget set x2x2 x1x1

© 2010 W. W. Norton & Company, Inc. 27 How do the budget set and budget constraint change as income m decreases? x2x2 x1x1 New, smaller budget set Consumption bundles that are no longer affordable. Old and new constraints are parallel.

© 2010 W. W. Norton & Company, Inc. 28 Budget Constraints - Income Changes u Increases in income m shift the constraint outward in a parallel manner, thereby enlarging the budget set and improving choice.

© 2010 W. W. Norton & Company, Inc. 29 Budget Constraints - Income Changes u Increases in income m shift the constraint outward in a parallel manner, thereby enlarging the budget set and improving choice. u Decreases in income m shift the constraint inward in a parallel manner, thereby shrinking the budget set and reducing choice.

© 2010 W. W. Norton & Company, Inc. 30 Budget Constraints - Income Changes u No original choice is lost and new choices are added when income increases, so higher income cannot make a consumer worse off. u An income decrease may (typically will) make the consumer worse off.

© 2010 W. W. Norton & Company, Inc. 31 Budget Constraints - Price Changes u What happens if just one price decreases? u Suppose p 1 decreases.

© 2010 W. W. Norton & Company, Inc. 32 How do the budget set and budget constraint change as p 1 decreases from p 1 ’ to p 1 ”? Original budget set x2x2 x1x1 m/p 2 m/p 1 ’ m/p 1 ” -p 1 ’/p 2

© 2010 W. W. Norton & Company, Inc. 33 How do the budget set and budget constraint change as p 1 decreases from p 1 ’ to p 1 ”? Original budget set x2x2 x1x1 m/p 2 m/p 1 ’ m/p 1 ” New affordable choices -p 1 ’/p 2

© 2010 W. W. Norton & Company, Inc. 34 How do the budget set and budget constraint change as p 1 decreases from p 1 ’ to p 1 ”? 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

© 2010 W. W. Norton & Company, Inc. 35 Budget Constraints - Price Changes u 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.

© 2010 W. W. Norton & Company, Inc. 36 Budget Constraints - Price Changes u Similarly, increasing one price pivots the constraint inwards, reduces choice and may (typically will) make the consumer worse off.

© 2010 W. W. Norton & Company, Inc. 37 Uniform Ad Valorem Sales Taxes  An ad valorem sales tax levied at a rate of 5% increases all prices by 5%, from p to (1+0  05)p = 1  05p. u An ad valorem sales tax levied at a rate of t increases all prices by tp from p to (1+t)p. u A uniform sales tax is applied uniformly to all commodities.

© 2010 W. W. Norton & Company, Inc. 38 Uniform Ad Valorem Sales Taxes u A uniform sales tax levied at rate t 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

© 2010 W. W. Norton & Company, Inc. 39 Uniform Ad Valorem Sales Taxes u A uniform sales tax levied at rate t 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).

© 2010 W. W. Norton & Company, Inc. 40 Uniform Ad Valorem Sales Taxes x2x2 x1x1 p 1 x 1 + p 2 x 2 = m

© 2010 W. W. Norton & Company, Inc. 41 Uniform Ad Valorem Sales Taxes x2x2 x1x1 p 1 x 1 + p 2 x 2 = m p 1 x 1 + p 2 x 2 = m/(1+t)

© 2010 W. W. Norton & Company, Inc. 42 Uniform Ad Valorem Sales Taxes x2x2 x1x1 Equivalent income loss is

© 2010 W. W. Norton & Company, Inc. 43 Uniform Ad Valorem Sales Taxes x2x2 x1x1 A uniform ad valorem sales tax levied at rate t is equivalent to an income tax levied at rate

© 2010 W. W. Norton & Company, Inc. 44 The Food Stamp Program u Food stamps are coupons that can be legally exchanged only for food. u How does a commodity-specific gift such as a food stamp alter a family’s budget constraint?

© 2010 W. W. Norton & Company, Inc. 45 The Food Stamp Program u Suppose m = $100, p F = $1 and the price of “other goods” is p G = $1. u The budget constraint is then F + G =100.

© 2010 W. W. Norton & Company, Inc. 46 The Food Stamp Program G F 100 F + G = 100; before stamps.

© 2010 W. W. Norton & Company, Inc. 47 The Food Stamp Program G F 100 F + G = 100: before stamps.

© 2010 W. W. Norton & Company, Inc. 48 The Food Stamp Program G F 100 F + G = 100: before stamps. Budget set after 40 food stamps issued

© 2010 W. W. Norton & Company, Inc. 49 The Food Stamp Program G F 100 F + G = 100: before stamps. Budget set after 40 food stamps issued. 140 The family’s budget set is enlarged. 40

© 2010 W. W. Norton & Company, Inc. 50 The Food Stamp Program u What if food stamps can be traded on a black market for $0.50 each?

© 2010 W. W. Norton & Company, Inc. 51 The Food Stamp Program G F 100 F + G = 100: before stamps. Budget constraint after 40 food stamps issued Budget constraint with black market trading. 40

© 2010 W. W. Norton & Company, Inc. 52 The Food Stamp Program G F 100 F + G = 100: before stamps. Budget constraint after 40 food stamps issued Black market trading makes the budget set larger again. 40

© 2010 W. W. Norton & Company, Inc. 53 Budget Constraints - Relative Prices u “Numeraire” means “unit of account”. u 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 = 12.

© 2010 W. W. Norton & Company, Inc. 54 Budget Constraints - Relative Prices u 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, the same as 2x 1 + 3x 2 = 12. u Changing the numeraire changes neither the budget constraint nor the budget set.

© 2010 W. W. Norton & Company, Inc. 55 Budget Constraints - Relative Prices u The constraint for p 1 =2, p 2 =3, m=12 2x 1 + 3x 2 = 12 is also 1.x 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.

© 2010 W. W. Norton & Company, Inc. 56 Budget Constraints - Relative Prices u Any commodity can be chosen as the numeraire without changing the budget set or the budget constraint.

© 2010 W. W. Norton & Company, Inc. 57 Budget Constraints - Relative Prices u p 1 =2, p 2 =3 and p 3 =6  u price of commodity 2 relative to commodity 1 is 3/2, u price of commodity 3 relative to commodity 1 is 3. u Relative prices are the rates of exchange of commodities 2 and 3 for units of commodity 1.

© 2010 W. W. Norton & Company, Inc. 58 Shapes of Budget Constraints u Q: What makes a budget constraint a straight line? u 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.

© 2010 W. W. Norton & Company, Inc. 59 Shapes of Budget Constraints u But what if prices are not constants? u E.g. bulk buying discounts, or price penalties for buying “too much”. u Then constraints will be curved.

© 2010 W. W. Norton & Company, Inc. 60 Shapes of Budget Constraints - Quantity Discounts  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.

© 2010 W. W. Norton & Company, Inc. 61 Shapes of Budget Constraints - Quantity Discounts  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 and the constraint is {

© 2010 W. W. Norton & Company, Inc. 62 Shapes of Budget Constraints with a Quantity Discount m = $ Slope = - 2 / 1 = - 2 (p 1 =2, p 2 =1) Slope = - 1/ 1 = - 1 (p 1 =1, p 2 =1) 80 x2x2 x1x1

© 2010 W. W. Norton & Company, Inc. 63 Shapes of Budget Constraints with a Quantity Discount m = $ Slope = - 2 / 1 = - 2 (p 1 =2, p 2 =1) Slope = - 1/ 1 = - 1 (p 1 =1, p 2 =1) 80 x2x2 x1x1

© 2010 W. W. Norton & Company, Inc. 64 Shapes of Budget Constraints with a Quantity Discount m = $ x2x2 x1x1 Budget Set Budget Constraint

© 2010 W. W. Norton & Company, Inc. 65 Shapes of Budget Constraints with a Quantity Penalty x2x2 x1x1 Budget Set Budget Constraint

© 2010 W. W. Norton & Company, Inc. 66 Shapes of Budget Constraints - One Price Negative u Commodity 1 is stinky garbage. You are paid $2 per unit to accept it; i.e. p 1 = - $2. p 2 = $1. Income, other than from accepting commodity 1, is m = $10. u Then the constraint is - 2x 1 + x 2 = 10 or x 2 = 2x

© 2010 W. W. Norton & Company, Inc. 67 Shapes of Budget Constraints - One Price Negative 10 Budget constraint’s slope is -p 1 /p 2 = -(-2)/1 = +2 x2x2 x1x1 x 2 = 2x

© 2010 W. W. Norton & Company, Inc. 68 Shapes of Budget Constraints - One Price Negative 10 x2x2 x1x1 Budget set is all bundles for which x 1  0, x 2  0 and x 2  2x

© 2010 W. W. Norton & Company, Inc. 69 More General Choice Sets u Choices are usually constrained by more than a budget; e.g. time constraints and other resources constraints. u A bundle is available only if it meets every constraint.

© 2010 W. W. Norton & Company, Inc. 70 More General Choice Sets Food Other Stuff 10 At least 10 units of food must be eaten to survive

© 2010 W. W. Norton & Company, Inc. 71 More General Choice Sets Food Other Stuff 10 Budget Set Choice is also budget constrained.

© 2010 W. W. Norton & Company, Inc. 72 More General Choice Sets Food Other Stuff 10 Choice is further restricted by a time constraint.

© 2010 W. W. Norton & Company, Inc. 73 More General Choice Sets So what is the choice set?

© 2010 W. W. Norton & Company, Inc. 74 More General Choice Sets Food Other Stuff 10

© 2010 W. W. Norton & Company, Inc. 75 More General Choice Sets Food Other Stuff 10

© 2010 W. W. Norton & Company, Inc. 76 More General Choice Sets Food Other Stuff 10 The choice set is the intersection of all of the constraint sets.