Consumption Choice Sets
• A consumption choice set is the collection of all consumption choices available to the Chapter Two consumer. • What constrains consumption choice? Budgetary and Other Constraints – Budgetary, time and other resource limitations. on Choice
Budget Constraints Budget Constraints
• A consumption bundle containing x 1 units • Q: When is a consumption bundle of commodity 1, x 2 units of commodity 2 (x 1, … , x n) affordable at given prices and so on up to x n units of commodity n is p1, … , p n? denoted by the vector (x 1, x 2, … , x n).
• Commodity prices are p 1, p 2, … , p n.
1 Budget Constraints Budget Constraints
• Q: When is a bundle (x 1, … , x n) affordable • The bundles that are only just affordable at prices p 1, … , p n? form the consumer’s budget constraint. • A: When This is the set p x + … + p x ≤ m 1 1 n n ≥ ≥ 0 where m is the consumer’s (disposable) { (x 1,…,x n) | x 1 0, …, x n and = income. p1x1 + … + p nxn m }.
Budget Set and Constraint for Budget Constraints Two Commodities x2 • The consumer’s budget set is the set of all Budget constraint is affordable bundles; m /p 2 p1x1 + p 2x2 = m. B(p 1, … , p n, m) = ≥ ≥ Not affordable { (x 1, … , x n) | x 1 0, … , x n 0 and ≤ Just affordable p1x1 + … + p nxn m } • The budget constraint is the upper Affordable boundary of the budget set.
m /p 1 x1
2 Budget Set and Constraint for Budget Set and Constraint for Two Commodities Two Commodities x2 x2
Budget constraint is p1x1 + p 2x2 = m is m /p 2 m /p 2 p1x1 + p 2x2 = m. x2 = -(p 1/p 2)x 1 + m/p 2 so slope is -p1/p 2. the collection of all affordable bundles. Budget Budget Set Set
m /p 1 x1 m /p 1 x1
Budget Constraint for Three Budget Constraints Commodities
• If n = 3 what do the budget constraint and x2 the budget set look like? p1x1 + p 2x2 + p 3x3 = m m /p 2
m /p 3 x3
m /p 1 x1
3 Budget Set for Three Budget Constraints Commodities • For n = 2 and x 1 on the horizontal axis, x ≥≥≥ ≥≥≥ ≥≥≥ the constraint’s slope is -p /p . What 2 { (x 1,x 2,x 3) | x 1 0, x 2 0, x 3 0 and 1 2 ≤≤≤ does it mean? m /p 2 p1x1 + p 2x2 + p 3x3 m} p m x ===−−− 1 x +++ 2 p 1 p m /p 3 2 2 x3 • Increasing x 1 by 1 must reduce x 2 by p1/p 2. m /p 1 x1
Budget Constraints Budget Constraints x x 2 2 Opp. cost of an extra unit of Slope is -p1/p 2 commodity 1 is p 1/p 2 units foregone of commodity 2. And the opp. cost of an extra -p1/p 2 +1 unit of commodity 2 is +1 -p /p 2 1 p2/p 1 units foregone of commodity 1.
x1 x1
4 Budget Sets & Constraints; Higher income gives more Income and Price Changes choice x2 New affordable consumption • The budget constraint and budget set choices depend upon prices and income. What Original and happens as prices or income change? new budget constraints are parallel (same Original slope). budget set
x1
How do the budget set and Budget Constraints - Income budget constraint change as Changes x2 income m decreases? • Increases in income m shift the constraint Consumption bundles outward in a parallel manner, thereby that are no longer enlarging the budget set and improving affordable. choice. Old and new • Decreases in income m shift the New, smaller constraints constraint inward in a parallel manner, budget set are parallel. thereby shrinking the budget set and reducing choice. x1
5 Budget Constraints - Income Budget Constraints - Price Changes Changes • No original choice is lost and new choices • What happens if just one price decreases? are added when income increases, so • Suppose p 1 decreases. higher income cannot make a consumer worse off. • An income decrease may (typically will) make the consumer worse off.
How do the budget set and Budget Constraints - Price budget constraint change as p 1 Changes x2 decreases from p ’ to p ”? 1 1 • Reducing the price of one commodity m/p 2 New affordable choices pivots the constraint outward. No old choice is lost and new choices are added, Budget constraint so reducing one price cannot make the -p ’/p pivots; slope flattens 1 2 consumer worse off. from -p1’/p 2 to Original -p ”/p -p1”/p 2 budget set 1 2
m/p 1’ m/p 1” x1
6 Budget Constraints - Price Uniform Ad Valorem Sales Changes Taxes • Similarly, increasing one price pivots the • An ad valorem sales tax levied at a rate of constraint inwards, reduces choice and 5% increases all prices by 5%, from p to may (typically will) make the consumer (1+0 .05)p = 1 .05p. worse off. • An ad valorem sales tax levied at a rate of t increases all prices by tp from p to (1+ t)p. • A uniform sales tax is applied uniformly to all commodities.
Uniform Ad Valorem Sales Uniform Ad Valorem Sales Taxes x2 Taxes • A uniform sales tax levied at rate t m changes the constraint from p1x1 + p 2x2 = m p1x1 + p 2x2 = m p2 to m p1x1 + p 2x2 = m/(1+ t) + (1+ t)p 1x1 + (1+ t)p 2x2 = m ()1++ t p2 i.e.
p1x1 + p 2x2 = m/(1+ t).
m m x + 1 ()1 ++ t p1 p1
7 Uniform Ad Valorem Sales Uniform Ad Valorem Sales Taxes x Taxes x 2 2 A uniform ad valorem m m sales tax levied at rate t p Equivalent income loss p is equivalent to an income 2 is 2 t m m tax levied at rate . −−− m === t +++ + m m + 1 t ()1++ t p2 1+++ t 1+++ t ()1++ t p2
m m x m m x + 1 + 1 ()1 ++ t p1 p1 ()1 ++ t p1 p1
Other Taxes (Subsidies) The Food Stamp Program
• Quantity tax: p1+t • Food stamps are coupons that can be • Lump sum tax : m-t legally exchanged only for food. • How does a commodity-specific gift such • Rationing: x1 8 The Food Stamp Program The Food Stamp Program G F + G = 100: before stamps. • Suppose m = $100, p F = $1 and the price of “other goods” is p = $1. G 100 Budget set after 40 food • The budget constraint is then stamps issued. F + G =100. The family’s budget set is enlarged. 40 100 140 F The Food Stamp Program The Food Stamp Program G F + G = 100: before stamps. • What if food stamps can be traded on a 120 black market for $0.50 each? 100 Budget constraint after 40 food stamps issued. Budget constraint with black market trading. 40 100 140 F 9 Budget Constraints - Relative The Food Stamp Program G Prices F + G = 100: before stamps. 120 • “Numeraire” means “unit of account”. 100 Budget constraint after 40 • Suppose prices and income are measured food stamps issued. in dollars. Say p 1=$2, p 2=$3, m = $12. Black market trading Then the constraint is makes the budget 2x 1 + 3x 2 = 12. set larger again. 40 100 140 F Budget Constraints - Relative Budget Constraints - Relative Prices Prices • If prices and income are measured in • The constraint for p 1=2, p 2=3, m=12 2x + 3x = 12 cents, then p 1=200, p 2=300, m=1200 and 1 2 the constraint is is also 1x 1 + (3/2)x 2 = 6, the constraint for p =1, p =3/2, m=6. 200x 1 + 300x 2 = 1200, 1 2 the same as Setting p 1=1 makes commodity 1 the numeraire and defines all prices relative 2x 1 + 3x 2 = 12. • Changing the numeraire changes neither to p1; e.g. 3/2 is the price of commodity the budget constraint nor the budget set. 2 relative to the price of commodity 1. 10 Budget Constraints - Relative Budget Constraints - Relative Prices Prices • p =2, p =3 and p =6 ⇒ • Any commodity can be chosen as the 1 2 3 numeraire without changing the budget set • price of commodity 2 relative to or the budget constraint. commodity 1 is 3/2, • price of commodity 3 relative to commodity 1 is 3. • Relative prices are the rates of exchange of commodities 2 and 3 for units of commodity 1. Shapes of Budget Constraints Shapes of Budget Constraints • Q: What makes a budget constraint a • But what if prices are not constants? straight line? • E.g. bulk buying discounts, or price • A: A straight line has a constant slope and penalties for buying “too much”. the constraint is • Then constraints will be curved. p1x1 + … + p nxn = m so if prices are constants then a constraint is a straight line. 11 Shapes of Budget Constraints - Shapes of Budget Constraints Quantity Discounts with a Quantity Discount x • Suppose p is constant at $1 but that 2 m = $100 2 Slope = - 2 / 1 = - 2 p =$2 for 0 ≤ x ≤ 20 and p =$1 for x >20. 100 1 1 1 1 (p =2, p =1) Then the constraint’s slope is 1 2 ≤ ≤ - 2, for 0 x1 20 -p /p = Slope = - 1/ 1 = - 1 1 2 { - 1, for x 1 > 20 (p 1=1, p 2=1) and the constraint is 20 50 80 x1 Shapes of Budget Constraints Shapes of Budget Constraints with a Quantity Discount with a Quantity Penalty x2 m = $100 x2 100 Budget Budget Constraint Constraint Budget Set Budget Set 20 50 80 x1 x1 12 Shapes of Budget Constraints - Shapes of Budget Constraints - One Price Negative One Price Negative • Commodity 1 is stinky garbage. You x2 x2 = 2x 1 + 10 are paid $2 per unit to accept it; i.e. p1 = - $2. p 2 = $1. Income, other than from Budget constraint’s slope is accepting commodity 1, is m = $10. -p1/p 2 = -(-2)/1 = +2 • Then the constraint is - 2x + x = 10 or x = 2x + 10. 1 2 2 1 10 x1 Shapes of Budget Constraints - More General Choice Sets One Price Negative x2 • Choices are usually constrained by more Budget set is than a budget; e.g. time constraints and all bundles for other resources constraints. ≥≥≥ which x 1 0, ≥≥≥ • A bundle is available only if it meets every x2 0 and ≤≤≤ constraint. x2 2x 1 + 10. 10 x1 13 More General Choice Sets More General Choice Sets Other Stuff Other Stuff Choice is also budget constrained. At least 10 units of food must be eaten to survive Budget Set 10 Food 10 Food More General Choice Sets More General Choice Sets Other Stuff Choice is further restricted by a time constraint. So what is the choice set? 10 Food 14 More General Choice Sets More General Choice Sets Other Stuff Other Stuff The choice set is the intersection of all of the constraint sets. 10 Food 10 Food 15