Report

Course: Microeconomics Text: Varian’s Intermediate Microeconomics 1 Economists assume that consumers choose the best bundle of goods they can afford. This chapter first specifies in detail what consumer can afford: the budget constraint or the consumption possibility set. What is best for consumer, or the preference on the possible consumption bundles, will be discussed in the next chapter. 2 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. 3 Q: When is a consumption bundle (x1, … , xn) affordable at prices p1, … , pn? 4 Q: When is a consumption bundle (x1, … , xn) affordable at prices p1, … , pn? A: When total expenditure is smaller than income: p1x1 + … + pnxn m where m is the consumer’s (disposable) income. That is, one’s total expenditure is smaller than or equal to one’s income. 5 The bundles that are only just affordable by the consumer is one’s budget constraint. This is the set { (x1,…,xn) | x1 0, …, xn and p1x1 + … + pnxn = m }. 6 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 x2 m /p2 Budget constraint is p1x1 + p2x2 = m. m /p1 x1 8 x2 m /p2 Budget constraint is p1x1 + p2x2 = m. m /p1 x1 9 x2 m /p2 Budget constraint is p1x1 + p2x2 = m. Just affordable m /p1 x1 10 x2 m /p2 Budget constraint is p1x1 + p2x2 = m. Not affordable Just affordable m /p1 x1 11 x2 m /p2 Budget constraint is p1x1 + p2x2 = m. Not affordable Just affordable Affordable m /p1 x1 12 x2 m /p2 Budget constraint (budget line) is p1x1 + p2x2 = m. the collection of all affordable bundles. Budget Set m /p1 x1 13 x2 m /p2 p1x1 + p2x2 = m is x2 = -(p1/p2)x1 + m/p2 so slope is -p1/p2. Budget Set m /p1 x1 14 For n = 2 and x1 on the horizontal axis, the constraint’s slope is -p1/p2. What does it mean? x2 = p1 p2 x1 m p2 Note: For a linear line: y=mx+c, m is the slope of the line, while c is the y-intercept. 15 For n = 2 and x1 on the horizontal axis, the constraint’s slope is -p1/p2. What does it mean? x2 = p1 p2 x1 m p2 To hold income m constant, increasing x1 by 1 must reduce x2 by p1/p2 . 16 x2 Slope is -p1/p2 -p1/p2 +1 x1 17 x2 Opp. cost of an extra unit of commodity 1 is p1/p2 units foregone of commodity 2. -p1/p2 +1 x1 18 x2 The opp. cost of an extra unit of commodity 2 is p2/p1 units foregone of commodity 1. +1 -p2/p1 x1 19 The budget constraint and budget set depend upon prices and income. What happens as prices or income change? 20 x2 Original budget set x1 21 x2 New affordable consumption choices Original and new budget constraints are parallel (same slope). Original budget set x1 22 x2 Original budget set x1 23 x2 Consumption bundles that are no longer affordable. New, smaller budget set Old and new constraints are parallel. x1 24 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. The slope –p1 / p2 does not change. 25 When income increases, NO original choice is lost and new choices are added, so higher income cannot make a consumer worse off. When income decreases, the consumer may (typically will) be worse off, as one can no longer afford some of the bundles anymore. 26 What happens if just one price decreases? Suppose p1 decreases. 27 x2 m/p2 -p1’/p2 Original budget set m/p1’ m/p1” x1 28 x2 m/p2 New affordable choices -p1’/p2 Original budget set m/p1’ m/p1” x1 29 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 30 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. Similarly, increasing one price pivots the constraint inwards (consider a price change from p1” to p1’), reduces choice and may (typically will) make the consumer worse off. 31 Quantity/per-unit tax: price increases from p to p+t. Quantity/per-unit subsidy: price decreases from p to p-s. Ad valorem/value tax: price increases from p to (1+t)p Ad valorem/value subsidy: price decreases from p to (1-s)p 32 A uniform sales tax levied at rate t on all goods changes the constraint from p1x1 + p2x2 = m to (1+t)p1x1 + (1+t)p2x2 = m i.e. p1x1 + p2x2 = m/(1+t). 33 x2 m p2 m ( 1 t ) p2 Equivalent income loss is m m 1 t = t 1 t m m ( 1 t ) p1 p1 m x1 34 x2 m p2 m ( 1 t ) p2 A uniform ad valorem sales tax levied at rate t is equivalent to an income tax levied at rate t . 1 t m m ( 1 t ) p1 p1 x1 35 Lump-sum tax: government tax a fixed sum of money, T, regardless of individual’s behavior. This is equivalent to a decrease in income by T, implying an inward parallel shift of budget line. Similarly, lump-sum subsidy S implies an outward parallel shift of budget line corresponding to an amount S. 36 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? Here we assume one of the two goods is food. 37 Suppose m = $100, pF = $1 and the price of “other goods” is pG = $1. The budget constraint is then F + G =100. 38 G F + G = 100: before stamps. 100 100 F 39 G F + G = 100: before stamps. 100 Budget set after $40 food stamps issued. 40 100 140 F 40 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 41 How does the unit of account affect the budget constraints and budget set? Suppose prices and income are measured in dollars. Say p1=$2, p2=$3, m = $12. Then the constraint is 2x1 + 3x2 = 12. 42 If prices and income are measured in cents, then p1=200, p2=300, m=1200 and the constraint is 200x1 + 300x2 = 1200, upon simplification, it is the same as 2x1 + 3x2 = 12. Changing the unit of account changes neither the budget constraint nor the budget set. 43 The constraint for p1=2, p2=3, m=12 2x1 + 3x2 = 12 is also 1x1 + (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; e.g. 3/2 is the price of commodity 2 relative to the price of commodity 1. 44 Multiplying all prices and income by any constant k does not change the budget constraint. kp1x1 +kp2x2 =km Any commodity can be chosen as the numeraire (by taking k=1/pi) without changing the budget set or the budget constraint. It is also clear from the graph that, only the ratios p1/p2, m/p1 and m/p2 are relevant to the budget line and budget set. 45 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. 46 But what if prices are not constants? E.g. bulk buying discounts, or price penalties (or tax) for buying “too much”. Then constraints will be curved or have kinks. 47 Suppose p2 is constant at $1 but that p1=$2 for 0 x1 20 and p1=$1 for x1>20. 48 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 { Also assume m=100. 49 x2 Slope = - 2 / 1 = - 2 (p1=2, p2=1) 100 m = $100 Slope = - 1/ 1 = - 1 (p1=1, p2=1) 20 50 80 x1 50 x2 Slope = - 2 / 1 = - 2 (p1=2, p2=1) 100 m = $100 Slope = - 1/ 1 = - 1 (p1=1, p2=1) 20 50 80 x1 51 x2 m = $100 100 Budget Constraint Budget Set 20 50 80 x1 52 x2 Budget Constraint Budget Set x1 53 The budget set describes what consumption bundles are affordable to the consumers. The budget constraint is typically described by p1 x1 + p2 x2 = m, which is a straight line when prices are constant. When income increases, budget set shifts outward, enlarging the budget set. When prices increases, the slope of budget line changes, and the it shrinks the budget set. 54 The next chapter will introduce preference, which describes the ordering of what a consumer likes among the consumption bundles. Then we can combine both preference and budget constraint to analyze consumer’s choice. 55