1 Budget Constraint Topics • Budget constraint. Budget set. Budget line. Numéraire. Non-linear budget constraints. Taxes, subsidies, and rationing. Modeling composite goods. Budget constraint with endowment. Reading § Varian [V], ch 2; Вэриан [В], гл 2 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. Budget Constraints • A bundle (𝑥1, … , 𝑥𝑛) affordable at prices 𝑝1, … , 𝑝𝑛 when 𝑝1𝑥1 + ⋯ + 𝑝𝑛𝑥𝑛 ≤ 𝑚 where 𝑚 is the consumer’s (disposable) income. Budget Constraints • The bundles that are only just affordable form the consumer’s budget constraint or budget set. This is the set {(𝑥1, … , 𝑥𝑛)|𝑥1 ≥ 0, … , 𝑥𝑛 ≥ 0 and 𝑝1𝑥1 + ⋯ + 𝑝𝑛𝑥𝑛 ≤ 𝑚} • Since most of our economic insights can be derived by confining attention to the case of two goods {(𝑥1, 𝑥2)|𝑥1 ≥ 0, 𝑥2 ≥ 0 and 𝑝1𝑥1 + 𝑝2𝑥2 ≤ 𝑚} Sometimes we will refer to the goods as 𝑥 and 𝑦 instead of 1 and 2. In that case, per-unit prices will be written as 𝑝𝑥 and 𝑝𝑦 Budget Constraints • The consumer’s budget set is the set of all affordable bundles; 𝐵(𝑝! , 𝑝" , 𝑚) = {(𝑥! , 𝑥" )|𝑥1 ≥ 0, 𝑥" ≥ 0 and 𝑝! 𝑥! + 𝑝" 𝑥" ≤ 𝑚} which expresses the idea that the expenditure on good 1 (𝑝1𝑥1) and the expenditure on good 2 (𝑝2𝑥2) should not exceed the consumer’s income. • The upper boundary of the budget set is a budget line: {(𝑥! , 𝑥" ) ∈ ℝ"# |𝑝! 𝑥! + 𝑝" 𝑥" = 𝑚} Budget Set for Two Commodities 𝒙𝟐 Vertical intercept 𝑚/𝑝2 𝑝1𝑥1 + 𝑝2𝑥2 = 𝑚 is 𝑥2 = −(𝑝1/𝑝2)𝑥1 + 𝑚/𝑝2 so slope of budget line is −𝑝1/𝑝2 Budget line Budget set 𝑚/𝑝1 𝒙𝟏 Horizontal intercept Budget Constraints • The budget line’s slope is −𝑝1/𝑝2. What does it mean? 𝑝1 𝑚 𝑥" = − 𝑥! + 𝑝2 𝑝2 • Increasing 𝑥1 by 1 must reduce 𝑥2 by 𝑝1/𝑝2 Budget Constraints 𝒙𝟐 Slope is -p1/p2 −𝑝1/𝑝2 +1 𝒙𝟏 Budget Constraints 𝒙𝟐 Opportunity cost of an extra unit of commodity 1 is p1/p2 units foregone of commodity 2. −𝑝1/𝑝2 +1 𝒙𝟏 Budget Constraints 𝒙𝟐 +1 −𝑝2/𝑝1 And the opportunity cost of an extra unit of commodity 2 is p2/p1 units foregone of commodity 1. 𝒙𝟏 Higher income gives more choice 𝒙𝟐 New affordable consumption choices Original budget set Original and new budget constraints are parallel (same slope). 𝒙𝟏 How do the budget set and budget constraint change as income m decreases? 𝒙𝟐 Consumption bundles that are no longer affordable. New, smaller budget set Old and new constraints are parallel. 𝒙𝟏 How do the budget set and budget constraint change as 𝒑𝟏 decreases from 𝒑%𝟏 to 𝒑%% 𝟏? 𝒙𝟐 𝑚/𝑝2 New affordable choices −𝑝() /𝑝2 Original budget set Budget constraint pivots; slope flattens from −𝑝() /𝑝2 to −𝑝()) /𝑝2 −𝑝()) /𝑝2 𝑚/𝑝!" 𝑚/𝑝!"" 𝒙𝟏 Budget Constraint with 3 Goods 𝒙𝟐 𝑚/𝑝2 𝑝( 𝑥( + 𝑝* 𝑥* + 𝑝+ 𝑥+ = 𝑚 𝑚/𝑝3 𝑚/𝑝1 𝒙𝟏 𝒙𝟑 Budget Constraint with 3 Goods 𝒙𝟐 𝑚/𝑝2 {(𝑥1, 𝑥* , 𝑥+ )|𝑥1 ≥ 0, 𝑥* ≥ 0, 𝑥+ ≥ 0 and 𝑝( 𝑥( + 𝑝* 𝑥* + 𝑝+ 𝑥+ ≤ 𝑚} 𝑚/𝑝3 𝑚/𝑝1 𝒙𝟏 𝒙𝟑 Composite Good • We can often interpret one of the goods as representing everything else the consumer might want to consume. • For example, we study a consumer’s demand for good 1 (𝑥! ). We can then let 𝑥" stand for everything else the consumer might want to consume. • It is convenient to think of good 2 as being the dollars that the consumer can use to spend on other goods. Composite Good • Under this interpretation the price of good 2 will automatically be 1, since the price of one dollar is one dollar. Thus the budget constraint is 𝑝! 𝑥! + 𝑥" ≤ 𝑚 • So, good 2 represents a composite good that stands for everything else that the consumer might want to consume other than good 1. Such a composite good is invariably measured in dollars to be spent on goods other than good 1. Numéraire • “Numéraire” means “unit of account”. • Suppose prices and income are measured in rubles. Say, 𝑝! = 2, 𝑝" = 3, 𝑚 = 12. Then the constraint is 2𝑥! + 3𝑥" = 12 Numéraire • If prices and income are measured in kopeks, then 𝑝! = 200, 𝑝" = 300, m=1200 and the constraint is 200𝑥! + 300𝑥" = 1200, the same as 2𝑥! + 3𝑥" = 12. • Changing the numeraire changes neither the budget constraint nor the budget set. Numéraire • The constraint for 𝑝! = 2, 𝑝" = 3, 𝑚 = 12 2𝑥! + 3𝑥" = 12 is also 𝑥! + (3/2)𝑥" = 6, the constraint for 𝑝! = 1, 𝑝" = 3/2, 𝑚 = 6. • Setting 𝑝! = 1 makes good 1 the numeraire and defines all prices relative to 𝑝! ; e.g. 3/2 is the price of good 2 relative to the price of good 1. Numéraire • Any good can be chosen as the numeraire without changing the budget set or the budget constraint. • Suppose three goods with prices 𝑝! = 2, 𝑝" = 3, and 𝑝& = 6 Þ – price of good 2 relative to good 1 is 3/2, – price of good 3 relative to good 1 is 3. • Relative prices are the rates of exchange of goods 2 and 3 for units of good 1. Taxes, Subsidies, and Rationing • Governments often use policies that affect the shape of the budget constraint, in particular, taxes, subsidies and rationing. • Quantity tax – A certain amount of tax should be paid to the government for each unit of the good purchased – 𝑡: quantity tax per unit of good 𝑖 – 𝑝# → 𝑝# + 𝑡 (higher price) • Value tax (Ad Valorem tax) – Tax on the value (i.e. price) of the good – 𝜏: tax rate – 𝑝# → 𝑝# (1 + 𝜏) (higher price) Taxes, Subsidies, and Rationing • Subsidy – Government gives an amount of money to the consumer that depends on the amount of the good purchased – the opposite of a tax – 𝑠: quantity subsidy per unit of good 𝑖 – 𝑝# → 𝑝# − 𝑠 (lower price) – 𝜎: subsidy rate – 𝑝# → 𝑝# (1 − 𝜎) (lower price) Taxes, Subsidies, and RaAoning • Lump-sum tax or subsidy – Government takes away (or gives) some fixed amount of money, regardless of individual’s behavior – Impact on the income – 𝑡: lump-sum tax – 𝑚 → 𝑚 − 𝑡 (lower income) – 𝑠: lump-sum subsidy – 𝑚 → 𝑚 + 𝑠 (higher income) Taxes, Subsidies, and Rationing • Rationing – Governments sometime impose rationing constraints that limit the consumption of some good to a specific amount. – If good 𝑥 is rationed so that no more than 𝑥̅ could be consumed, then the budget line becomes vertical at that point 𝑥̅ Non-Linear Budget Constraint 𝑥2 ℝ&' |5𝑥% {(𝑥% , 𝑥& ) ∈ + 5𝑥& ≤ 100} or {(𝑥% , 𝑥& ) ∈ ℝ&' |𝑥& ≤ 20 − 𝑥% } 𝑥2 𝑥1 {(𝑥% , 𝑥& ) ∈ ℝ&' |𝑥& ≤ 24 − 𝑥% for 𝑥% > 4 and 𝑥& = 20 for 0 ≤ 𝑥% ≤ 4} 𝑥1 Example (Food Stamp): A person can purchase food (good 1) or clothing (good 2) at prices 𝑝! = $5 and 𝑝" = $5 per unit with her income of 𝑚 = $100. The government gives her an endowment of 4 food stamps where each stamp entitles her to one unit of food. We will assume that food, clothing and food stamps are divisible. Non-Linear Budget Constraint Example (Quantity Discount): A person can buy the (divisible) goods 1 and 2 with her income of 120 rub. The price of good 2 is fixed at 𝑝$ = 6 rub per unit. However, she receives a price discount for purchases in excess of 6 units of good 1: the price of good 1 is 𝑝% =10 rub per unit up to 6 units, and 6 rub per unit for each subsequent unit of good 1 or fraction thereof. 𝑥2 𝑠𝑙𝑜𝑝𝑒 = −5/3 {(𝑥% , 𝑥& ) ∈ ℝ&' |10𝑥% + 2𝑥& ≤ 120} 𝑠𝑙𝑜𝑝𝑒 = −1 𝑥1 Non-Linear Budget Constraint 𝑥2 Example: A person’s income is still 120 rub and 𝑝$ remains at 6 rub per unit, but now 𝑝% is 10 rub per unit if she buys fewer than 6 units of good 1 and 6 rub per unit if she buys 6 units or more, i.e., 10 if 𝑥% < 6 𝑝% = $ 6 if 𝑥% ≥ 6 𝑠𝑙𝑜𝑝𝑒 = −5/3 𝑠𝑙𝑜𝑝𝑒 = −1 In other words, she receives a discounted per-unit price for good 1 when she buys a sufficiently large quantity, 𝑥1 and this discount applies for all units of good 1 purchased. Endowment* [V, ch 9 “Buying and Selling”] • No money income • But consumer is endowed 𝜔 = 𝜔! , 𝜔" *The idea of budget constraint with endowment will be discussed in details later Budget Constraint: Endowment • So, given 𝑝1 and 𝑝2, the budget constraint for a consumer with an endowment 𝜔 = 𝜔! , 𝜔" is 𝑝1𝑥1 + 𝑝" 𝑥" = 𝑝1𝜔1 + 𝑝" 𝜔" • The budget set is {(𝑥1, 𝑥2)|𝑝1𝑥1 + 𝑝" 𝑥" ≤ 𝑝1𝜔1 + 𝑝" 𝜔" , 𝑥! ≥ 0, 𝑥" ≥ 0} {(𝑥1, 𝑥2) ∈ ℝ"# |𝑝1𝑥1 + 𝑝" 𝑥" ≤ 𝑝1𝜔1 + 𝑝" 𝜔" } Budget Constraint: Endowment 𝒙𝟐 𝑝1𝑥1 + 𝑝* 𝑥* = 𝑝1𝜔1 + 𝑝* 𝜔* 𝜔" 𝜔1 𝒙𝟏 Budget Constraint: Endowment 𝒙𝟐 𝑝1𝑥1 + 𝑝* 𝑥* = 𝑝1𝜔1 + 𝑝* 𝜔* 𝜔" Budget set {(𝑥1, 𝑥2)|𝑝1𝑥1 + 𝑝* 𝑥* ≤ 𝑝1𝜔1 + 𝑝* 𝜔* , 𝑥( ≥ 0, 𝑥* ≥ 0} 𝜔1 𝒙𝟏 Budget Constraint: Endowment 𝒙𝟐 𝑝1𝑥1 + 𝑝* 𝑥* = 𝑝1𝜔1 + 𝑝* 𝜔* 𝜔" 𝑝() 𝑥1 + 𝑝*) 𝑥* = 𝑝() 𝜔1 + 𝑝*) 𝜔* 𝜔1 𝒙𝟏