Chapter 2: Budget Constraint PDF

Chapter 2: Budget Constraint Mohsen Bakhshi-Moghaddam Queen’s University Microeconomic Theory I Chapter 2 1 / 20 Consumer Theory Consumers choose the best bundle of goods they can afford. What we...

Chapter 2: Budget Constraint Mohsen Bakhshi-Moghaddam Queen’s University Microeconomic Theory I Chapter 2 1 / 20 Consumer Theory Consumers choose the best bundle of goods they can afford. What we mean by ”best”: will be discussed in the next chapter What we mean by ”can afford”: budget constraint Microeconomic Theory I Chapter 2 2 / 20 The Budget Constraint (I) The consumer’s consumption bundle of goods: X = (x1 , x2 ) x1 : the amount of good 1 that the consumer chooses to consume x2 : the amount of good 2 that the consumer chooses to consume Prices: (p1 , p2 ) The amount of money the consumer can spend: m Budget constraint p1 x1 + p2 x2 ≤ m Microeconomic Theory I Chapter 2 3 / 20 The Budget Constraint (II) p1 x1 + p2 x2 ≤ m p1 x1 : the amount of money spent on good 1 p2 x2 : the amount of money spent on good 2 Interpretation: the amount of money spent on the two goods should be no more than the total amount of money available. This set of affordable consumption bundles at prices (p1 , p2 ) and income m is called the budget set of the consumer. Microeconomic Theory I Chapter 2 4 / 20 Why Two-Good? Example: x1 : the amount of milk consumed; x2 : everything else the consumer might want to consume; under this interpretation, we can think of good 2 as being the dollars that the consumer can use to spend on all other goods. ⇒ p2 = 1 (the price of one dollar is one dollar!) Budget constraint: p1 x1 + x2 ≤ m Good 2: a composite good such a composite good is invariably measured in dollars to be spent on goods other than good 1. Microeconomic Theory I Chapter 2 5 / 20 Graph the Budget Constraint (I) Budget line: the set of bundles that cost exactly m =⇒ p1 x1 + p2 x2 = m Rearrange: x2 = pm2 − pp12 x1 =⇒ slope: − pp12 ; vertical intercept: pm2 ; horizontal intercept: pm1 find the two intercepts and connect them with a straight line: x2 m p2 slope:− pp12 budget set m x1 p1 Microeconomic Theory I Chapter 2 6 / 20 Graph the Budget Constraint (II) Economic interpretation of the slope: the rate at which the market is willing to substitute good 1 for good 2; If consumption of good 1 increases by ∆x1 , how much will consumption of good 2 change to satisfy the budget constraint? Note: if the consumer satisfies her budget constraint before and after making the change, she must satisfy: p 1 x1 + p 2 x2 = m (1) and p1 (x1 + ∆x1 ) + p2 (x2 + ∆x2 ) = m (2) subtracting Eq.(1) from (2) gives: ∆x2 p1 p1 ∆x1 + p2 ∆x2 = 0 =⇒ =− ∆x1 p2 Slope: measures the opportunity cost of consuming good 1. Microeconomic Theory I Chapter 2 7 / 20 Changes in Budget Constraint (I) (1) Changes in income: from m to m0 0 p1 New budget line: x2 = mp2 − p2 x1 → m0 > m: parallel shift outward; m0 < m: parallel shift inward x2 m p2 m x1 p1 Microeconomic Theory I Chapter 2 8 / 20 Changes in Budget Constraint (I) (1) Changes in income: from m to m0 0 p1 New budget line: x2 = mp2 − p2 x1 → m0 > m: parallel shift outward; m0 < m: parallel shift inward x2 m0 p2 m p2 m m0 x1 p1 p1 Microeconomic Theory I Chapter 2 8 / 20 Changes in Budget Constraint (II) (2) Changes in the price of one good while holding the price of another good and income fixed: Increasing p1 to p10 m p10 New budget line: x2 = p2 − p2 x 1 (different slope!) with the two points: (0, pm2 ), ( pm0 , 0) 1 x2 m p2 p slope:− p1 2 m x1 p1 Microeconomic Theory I Chapter 2 9 / 20 Changes in Budget Constraint (II) (2) Changes in the price of one good while holding the price of another good and income fixed: Increasing p1 to p10 m p10 New budget line: x2 = p2 − p2 x 1 (different slope!) with the two points: (0, pm2 ), ( pm0 , 0) 1 x2 m p2 p p0 slope:− p1 slope:− p1 2 2 m m x1 p10 p1 Microeconomic Theory I Chapter 2 9 / 20 Changes in Budget Constraint (III) (3) Changes in the price of both goods: Multiplying p1 and p2 by t: m New budget line: tp1 x1 + tp2 x2 = m =⇒ p1 x1 + p2 x2 = t Equivalent to dividing income by t t > 1: shift inward; t < 1: shift outward Microeconomic Theory I Chapter 2 10 / 20 Changes in Budget Constraint (IV) (4) Changes in the price of both goods and income together: Example: both prices go up and income goes down, then: m m The intercepts p1 and p2 must decrease =⇒ budget line shifts inward; What about the slope? If price 2 increases more than price 1 ⇒ − pp12 decreases (in absolute value) ⇒ the budget line becomes flatter; If price 2 increases less than price 1 ⇒ the budget line becomes steeper. Microeconomic Theory I Chapter 2 11 / 20 The Numeraire (I) The budget line is defined by two prices & one income, but one of these variables is redundant → peg one of them to some fixed value and describe the same budget set Consider p1 x1 + p2 x2 = m Want p2 to be 1 p1 m Divide by p2 on both sides: p2 x1 + x2 = p2 The budget set does not change. Good 2 is the numeraire. p1 p2 : relative price of good 1 m p2 : relative income Microeconomic Theory I Chapter 2 12 / 20 The Numeraire (II) Example p1 = 2, p2 = 3, m = 20 Budget line: 2x1 + 3x2 = 20 2 20 Divide by 3: 3 x1 + x2 = 3 2/3 is the price of good 1, relative to the price of good 2. 20/3 is the income of the consumer, relative to the price of good 2. Microeconomic Theory I Chapter 2 13 / 20 Tax, Subsidies and Rationing (I) I Taxes Quantity tax: e.g. pay t dollars for each unit of good 1 purchased. → Change in price of good 1: from p1 to p1 + t → New budget line (steeper): (p1 + t)x1 + p2 x2 = m Value tax (ad valorem tax): a tax on the value (the price) of a good. → Usually in percentage terms, e.g. a sales tax of 13% → Change in price of good 1: from p1 to (1 + τ )p1 → New budget line: (1 + τ )p1 x1 + p2 x2 = m A tax increases the price to the consumer. Microeconomic Theory I Chapter 2 14 / 20 Tax, Subsidies and Rationing (II) I Subsidies Quantity subsidy: the government gives an amount to the consumer that depends on the amount of goods purchased. → E.g., a subsidy is s dollars per unit of consumption of good 1 → Change in price of good 1: from p1 to p1 − s → New budget line (flatter): (p1 − s)x1 + p2 x2 = m Ad valorem subsidy: a subsidy based on the price of the good being subsidized. → E.g., price of good 1 is subject to an ad valorem subsidy at rate σ → Change in price of good 1: from p1 to (1 − σ)p1 → New budget line: (1 − σ)p1 x1 + p2 x2 = m A subsidy decreases the price to the consumer. Microeconomic Theory I Chapter 2 15 / 20 Tax, Subsidies and Rationing(III) I Lump-sum tax or subsidy Lump-sum tax of T : the government takes away T dollars from the consumer, regardless of her behavior. → New budget line: p1 x1 + p2 x2 = m − T → Budget line shifts inward Lump-sum subsidy of S: giving the consumer S dollars. → New budget line: p1 x1 + p2 x2 = m + S → Budget line shifts outward Microeconomic Theory I Chapter 2 16 / 20 Tax, Subsidies and Rationing (IV) I Rationing Consumption limit on a good. E.g., good 1 were rationed so that no more than x̄1 could be consumed by a given consumer. Budget constraint: p1 x1 + p2 x2 ≤ m when 0 ≤ x1 ≤ x̄1 Assume x̄1 < pm1 x2 m p2 budget set x̄1 m x1 p1 Microeconomic Theory I Chapter 2 17 / 20 Tax, Subsidies and Rationing (V) I Combine taxes, subsidies and rationing E.g., a consumer can consume good 1 at price p1 up to some level x̄1 ; then pay a tax t on all consumption in excess of x̄1. What is the budget constraint here? Microeconomic Theory I Chapter 2 18 / 20 Tax, Subsidies and Rationing (V) I Combine taxes, subsidies and rationing E.g., a consumer can consume good 1 at price p1 up to some level x̄1 ; then pay a tax t on all consumption in excess of x̄1. What is the budget constraint here? If x1 ≤ x̄1 , the budget line is p1 x1 + p2 x2 = m ⇒ slope = − pp12 If x1 > x̄1 , the budget line is p1 x̄1 + (p1 + t)(x1 − x̄1 ) + p2 x2 = m Microeconomic Theory I Chapter 2 18 / 20 Tax, Subsidies and Rationing (V) I Combine taxes, subsidies and rationing E.g., a consumer can consume good 1 at price p1 up to some level x̄1 ; then pay a tax t on all consumption in excess of x̄1. What is the budget constraint here? If x1 ≤ x̄1 , the budget line is p1 x1 + p2 x2 = m ⇒ slope = − pp12 If x1 > x̄1 , the budget line is p1 x̄1 + (p1 + t)(x1 − x̄1 ) + p2 x2 = m → Rearrange: x2 = m−p p2 1 x̄1 − p1p+t 2 (x1 − x̄1 ) ⇒ slope = − p1p+t 2 Microeconomic Theory I Chapter 2 18 / 20 Tax, Subsidies and Rationing (V) I Combine taxes, subsidies and rationing E.g., a consumer can consume good 1 at price p1 up to some level x̄1 ; then pay a tax t on all consumption in excess of x̄1. What is the budget constraint here? If x1 ≤ x̄1 , the budget line is p1 x1 + p2 x2 = m ⇒ slope = − pp12 If x1 > x̄1 , the budget line is p1 x̄1 + (p1 + t)(x1 − x̄1 ) + p2 x2 = m → Rearrange: x2 = m−p p2 1 x̄1 − p1p+t 2 (x1 − x̄1 ) ⇒ slope = − p1p+t 2 m Assume x̄1 < p1 Microeconomic Theory I Chapter 2 18 / 20 Tax, Subsidies and Rationing (V) I Combine taxes, subsidies and rationing E.g., a consumer can consume good 1 at price p1 up to some level x̄1 ; then pay a tax t on all consumption in excess of x̄1. What is the budget constraint here? If x1 ≤ x̄1 , the budget line is p1 x1 + p2 x2 = m ⇒ slope = − pp12 If x1 > x̄1 , the budget line is p1 x̄1 + (p1 + t)(x1 − x̄1 ) + p2 x2 = m → Rearrange: x2 = m−p p2 1 x̄1 − p1p+t 2 (x1 − x̄1 ) ⇒ slope = − p1p+t 2 m Assume x̄1 < p1 x2 m p2 budget set x̄1 m+t x̄1 m x1 p1 +t p1 Microeconomic Theory I Chapter 2 18 / 20 Summary 1. Consumption bundle (x1 , x2 ): how much of each good is consumed (p1 , p2 ): prices of the two goods m: money the consumer has to spend budget constraint: p1 x1 + p2 x2 ≤ m all (x1 , x2 ) that satisfy this constraint make up the budget set. 2. Two goods can think of good 2 as a composite good, representing money spent on other goods budget constraint becomes p1 x1 + x2 ≤ m 3. Budget line p1 x1 + p2 x2 = m also written as x2 = m/p2 − (p1 /p2 )x1 set x1 = 0 to find vertical intercept (m/p2 ); set x2 = 0 to find horizontal intercept (m/p1 ) slope measures opportunity cost of good 1 Microeconomic Theory I Chapter 2 19 / 20 Summary 4. Changes in budget line increasing m makes parallel shift out increasing p1 makes budget line steeper increasing p2 makes budget line flatter multiplying all prices by t is like dividing income by t multiplying all prices and income by t doesn’t change budget line 5. The numeraire can assign one price a value of 1 and measure another price and income relative to that. 6. Taxes, subsidies, and rationing quantity tax (levied on units bought): p1 + t value tax (levied on dollars spent): (1 + τ )p1 subsidies (opposite of a tax): (a) p1 − s; (b) (1 − σ)p1 lump sum tax or subsidy: independent of the consumer’s choices rationing: can’t consume more than a certain amount of some good Microeconomic Theory I Chapter 2 20 / 20