EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Choice and Constraints on Choice The BUDGET Set…Feasible Choice Economic Rationality The principal behavioral postulate is that a decision maker chooses its most preferred alternative from those available. The available choices constitute the choice set. 2 1 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Consumption Choice Sets A consumption choice set is the collection of all consumption choices available to the consumer. What constrains consumption choice? Budgetary, time and other resource limitations. 3 Budget Constraints 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 (x1, x2, … , xn). Commodity prices are p1, p2, … , pn. 4 2 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Budget Constraints Q: When is a consumption bundle (x1, … , xn) affordable at prices p1, … , pn? A: When p1x1 + … + pnxn m where m is the consumer’s (disposable) income. 5 Budget Constraints The bundles that are only just affordable form the consumer’s budget constraint. This is the set { (x1,…,xn) | x1 0, …, xn 0 and p1x1 + … + pnxn = m } These are the bundles ON, rather than INSIDE the budget constraint 6 3 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Budget Constraints The consumer’s budget set is the set of all affordable bundles; B(p1, … , pn, m) = { (x1, … , xn) | x1 0, … , xn 0 and p1x1 + … + pnxn m } The budget constraint is the upper boundary of the budget set. 7 Budget Set and Constraint for Two Commodities x2 m /p2 p2x2 = m Budget constraint is p1x1 + p2x2 = m p1x1 = m m /p1 x1 8 4 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Budget Set and Constraint for Two Commodities x2 Budget constraint is p1x1 + p2x2 = m. m /p2 Not affordable Just affordable Affordable m /p1 x1 9 Budget Set and Constraint for Two Commodities x2 Budget constraint is p1x1 + p2x2 = m. m /p2 the collection of all affordable bundles. Budget Set m /p1 x1 10 5 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Budget Set and Constraint for Two Commodities x2 p1x1 + p2x2 = m m /p2 p2x2 = - p1x1 + m x2 = - (p1/p2)x1 + m/p2 so slope is -p1/p2 Budget Set m /p1 x1 11 Budget Constraints x2 Opp. cost of an extra unit of commodity 1 is p1/p2 units foregone of commodity 2. -p1/p2 +1 x1 12 6 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Budget Constraints x2 Opp. cost of an extra unit of commodity 1 is p1/p2 units foregone of commodity 2. And the opp. cost of an extra +1 unit of commodity 2 is -p2/p1 p2/p1 units foregone of commodity 1. x1 13 Budget Constraints If n = 3 what do the budget constraint and the budget set look like? 14 7 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Budget Constraint for Three Commodities x2 m /p2 p1x1 + p2x2 + p3x3 = m m /p3 m /p1 x3 x1 15 Budget Set for Three Commodities x2 m /p2 { (x1,x2,x3) | x1 0, x2 0, x3 0 and p1x1 + p2x2 + p3x3 m} m /p3 m /p1 x3 x1 16 8 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Budget Sets & Constraints; Income and Price Changes The budget constraint and budget set depend upon prices and income. What happens as prices or income change? 17 Budget set change: income m increases x2 Original budget set x1 18 9 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Higher income gives more choice x2 New affordable consumption choices Original and new budget constraints are parallel (same slope). Original budget set x1 19 Budget set change: income m decreases x2 Original budget set x1 20 10 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Budget set change: income m decreases x2 Consumption bundles that are no longer affordable. New, smaller budget set Old and new constraints are parallel. x1 21 Budget Constraints - Income Changes 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. 22 11 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Budget Constraints - Income Changes No original choice is lost and new choices are added when income increases, so higher income cannot make a consumer worse off. An income decrease may (typically will) make the consumer worse off. 23 Budget Constraints - Price Changes What happens if just one price decreases? Suppose p1 decreases. 24 12 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets p1 decreases from p1' to p1" x2 m/p2 -p1'/p2 Maximum affordable amount of x1 rises (budget buys more x1 at smaller p1) Original budget set m/p1' m/p1" x1 25 p1 decreases from p1' to p1" x2 m/p2 New affordable choices -p1'/p2 Original budget set m/p1' m/p1" x1 26 13 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets p1 decreases from p1' to p1" x2 m/p2 New affordable choices -p1'/p2 Original budget set Budget constraint pivots; slope flattens from -p1'/p2 to -p1"/p2 -p "/p 1 m/p1' 2 m/p1" x1 27 Budget Constraints - Price Changes 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. 28 14 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Budget Constraints - Price Changes Similarly, increasing one price pivots the constraint inwards, reduces choice and may (typically will) make the consumer worse off. 29 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. 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. 30 15 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Uniform Ad Valorem Sales Taxes A uniform sales tax levied at rate t changes the constraint from p1x1 + p2x2 = m to (1+t)p1x1 + (1+t)p2x2 = m i.e. p1x1 + p2x2 = m/(1+t). 31 Uniform Ad Valorem Sales Taxes x2 m p2 p1x1 + p2x2 = m m p1 x1 32 16 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Uniform Ad Valorem Sales Taxes x2 m p2 m ( 1 t ) p2 p1x1 + p2x2 = m p1x1 + p2x2 = m/(1+t) m ( 1 t ) p1 m p1 x1 33 Uniform Ad Valorem Sales Taxes x2 m p2 m ( 1 t ) p2 Equivalent income loss is m m 1 t m = m 1 t 1 t 1 t m mt m = 1 t t = m 1+ t m ( 1 t ) p1 m p1 x1 34 17 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Uniform Ad Valorem Sales Taxes x2 m p2 m ( 1 t ) p2 A uniform ad valorem sales tax levied at rate t is equivalent to a uniform income tax levied at rate t 1+ t Equivalent income loss t m 1 + t m ( 1 t ) p1 m p1 x1 35 The Food Stamp Program 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? 36 18 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets The Food Stamp Program Suppose m = $100, pF = $1 and the price of “other goods” is pG = $1. The budget constraint is then F + G =100. 37 The Food Stamp Program G F + G = 100: before stamps. 100 100 F 38 19 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets The Food Stamp Program G F + G = 100: before stamps. 100 Budget set after 40 food stamps issued. 40 100 140 F 39 The Food Stamp Program G F + G = 100: before stamps. 100 Budget set after 40 food stamps issued. The family’s budget set is enlarged. 40 100 140 F 40 20 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets The Food Stamp Program What if food stamps can be traded on a black market for $0.50 each? 41 The Food Stamp Program G 120 100 F + G = 100: before stamps. Budget constraint after 40 food stamps issued. Budget constraint with black market trading. Black market trading makes the budget set larger again. 40 100 140 F 42 21 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets If the government gave you a subsidy of $200 per month that you had to spend on housing and if you could spend the remainder of your income in any way you wished… …the effect of the subsidy would differ from the effect of a $200 per month unrestricted increase in your income only if A. housing were an inferior good for you. B. housing were a normal good for you. C. you would spend less than $200 per month on housing when you received the unrestricted $200 monthly increase in your income. D. you would spend more than $200 per month on housing when you received the unrestricted $200 monthly increase in your income. Other Goods 43 If consumer buys fewer than $200/PH units of housing, then optimum occurs somewhere here [only if unrestricted grant] If consumer buys at least $200/PH units of housing, then consume a combination of Housing and Other Goods somewhere here, with unrestricted grant or $200 tied grant. $200/pH Housing 44 22 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets If the government gave you a subsidy of $200 per month that you had to spend on housing and if you could spend the remainder of your income in any way you wished… …the effect of the subsidy would differ from the effect of a $200 per month unrestricted increase in your income only if A. housing were an inferior good for you. B. housing were a normal good for you. C. you would spend less than $200 per month on housing when you received the unrestricted $200 monthly increase in your income. D. you would spend more than $200 per month on housing when you received the unrestricted $200 monthly increase in your income. 45 Budget Constraints - Relative Prices “Numeraire” means “unit of account”. Suppose prices and income are measured in dollars. Say p1=$2, p2=$3, m = $12. Then the constraint is 2x1 + 3x2 = 12. 46 23 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Budget Constraints - Relative Prices If prices and income are measured in cents, then p1=200, p2=300, m=1200 and the constraint is 200x1 + 300x2 = 1200, the same as 2x1 + 3x2 = 12. Changing the numeraire changes neither the budget constraint nor the budget set. 47 Budget Constraints - Relative Prices The constraint for p1=2, p2=3, m=12 2x1 + 3x2 = 12 is also 1×x1 + (3/2)x2 = 6, the constraint for p1=1, p2=3/2, m=6. Setting p1=1 makes commodity 1 the numeraire and defines all prices relative to p1; i.e. 3/2 is the price of commodity 2 relative to the price of commodity 1. 48 24 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Budget Constraints - Relative Prices Any commodity can be chosen as the numeraire without changing the budget set or the budget constraint. 49 Budget Constraints - Relative Prices p1=2, p2=3 and p3=6 price of commodity 2 relative to 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. 50 25 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Will spends his entire income on 8 sacks of acorns and 8 crates of butternuts. The price of acorns is 9 dollars per sack and his income is 96 dollars. He can just afford a commodity bundle with A sacks of acorns and B crates of butternuts that satisfies the budget equation A. 9A + 6B = 96 B. 18A + 6B = 192 C. 8A + 8B = 96 D. 9A + 6B = 192 51 Will spends his entire income on 8 sacks of acorns and 8 crates of butternuts. The price of acorns is 9 dollars per sack and his income is 96 dollars. He can just afford a commodity bundle with A sacks of acorns and B crates of butternuts that satisfies the budget equation Budget Eqn: 9A +PBB = 96 Spends 9×8 = $72 on acorns Left with $96 – $72 = $24 to spend on 8 crates butternuts PB must be $24/8 = $3 Budget const: 9A + 3B = 96 Equivalent to 18A + 6B = 192 A. 9A + 6B = 96 B. 18A + 6B = 192 C. 8A + 8B = 96 D. 9A + 6B = 192 52 26 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Shapes of Budget Constraints Q: What makes a budget constraint a straight line? A: A straight line has a constant slope and the constraint is p1x1 + … + pnxn = m so if prices are constants then a constraint is a straight line. 53 Shapes of Budget Constraints But what if prices are not constants? E.g. bulk buying discounts, or price penalties for buying “too much”. Then constraints will be curved. 54 27 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Shapes of Budget Constraints Quantity Discounts Suppose p2 is constant at $1 but that p1=$2 for 0 x1 20 and p1=$1 for x1>20. Then the constraint’s slope is - 2, for 0 x1 20 -p1/p2 = - 1, for x1 > 20 { and the constraint is 55 x2 100 Shapes of Budget Constraints Quantity Discounts m = $100 Slope = - 2 / 1 = - 2 (p1=2, p2=1) Slope = - 1/ 1 = - 1 (p1=1, p2=1) 20 50 80 x1 56 28 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets x2 100 Shapes of Budget Constraints Quantity Discounts m = $100 Slope = - 2 / 1 = - 2 (p1=2, p2=1) Slope = - 1/ 1 = - 1 (p1=1, p2=1) 20 50 x1 80 57 x2 Shapes of Budget Constraints Quantity Discounts m = $100 100 Budget Constraint Budget Set 20 50 80 x1 58 29 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Shape of Budget Constraint with a Quantity Penalty x2 Budget Constraint Budget Set x1 59 Application: Budget Constraint with Quantity Penalty x2 Suppose x1 is consumption($) NOW, x2 is consumption($) in FUTURE, Consumer/economic agent has income $m1 NOW and $m2 in FUTURE m2 Can lend at interest rate rL, but must pay rH (>rL) to borrow Intertemporal (over time) budget set looks like this m1 x1 60 30 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Shapes of Budget Constraints One Price Negative Commodity 1 is stinky garbage. You are paid $2 per unit to accept it; i.e. p1 = − $2. p2 = $1. Income, other than from accepting commodity 1, is m = $10. Then the constraint is − 2x1 + x2 = 10 or x2 = 2x1 + 10. 61 Shapes of Budget Constraints One Price Negative x2 x2 = 2x1 + 10 Budget constraint’s slope is -p1/p2 = -(-2)/1 = +2 10 x1 62 31 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets Shapes of Budget Constraints One Price Negative x2 Budget set is all bundles for which x1 0, x2 0 and x2 2x1 + 10. 10 x1 63 More General Choice Sets Choices are usually constrained by more than a budget; e.g. time constraints and other resources constraints. A bundle is available only if it meets every constraint. 64 32 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets More General Choice Sets Other Stuff At least 10 units of food must be eaten to survive Food 10 65 More General Choice Sets Other Stuff Choice is also budget constrained. Budget Set 10 Food 66 33 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets More General Choice Sets Other Stuff Choice is further restricted by a time constraint. 10 Food 67 More General Choice Sets So what is the choice set? 68 34 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets More General Choice Sets Other Stuff 10 Food 69 More General Choice Sets Other Stuff 10 Food 70 35 EC270 Lecture Notes Consumer Budget Constraints, Budget Sets More General Choice Sets Other Stuff The choice set is the intersection of all of the constraint sets. 10 Food 71 36