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The Theory of the Consumer Budget Constraints Two Good Case Budget

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The Theory of the Consumer Budget Constraints Two Good Case Budget Professor Jay Bhattacharya Spring 2001 The Theory of the Consumer Budget Constraints • A consumer must choose among bundles of • Model of individual choice: goods: (x1, x2,…xn) example: (fish, beef, – “Consumers choose the best bundle of goods milk,CDs,books). and services that they can afford.” • Each good has a price: (p , p …p ). •1)“afford”: depends upon opportunities / 1 2 n budget constraints • The consumer has income I to spend on •2)“best”: depends upon preferences goods. •3)“choose”: assumes optimizing (goal oriented) behavior Spring 2001 Econ 11--Lecture 2 1 Spring 2001 Econ 11--Lecture 2 2 Budget Set Two Good Case x2 I p p • Consider the case of 2 goods (x , x )(e.g.., 2 slope = 1 1 2 p video games, baby food). Let’s say the 2 price of these goods are p1, p2. • A bundle (x1,x2) is affordable (in the ≤ budget set) if and only if p1x1+ p2x2 I. x • The set of affordable bundles is the budget 1 ≤ I p1x1 + p2x2 I set. p1 Spring 2001 Econ 11--Lecture 2 3 Spring 2001 Econ 11--Lecture 2 4 Budget Line x2 Budget Line ‘Facts’ I p p slope = 1 • The budget line intersects the x1 axis at I/p1 2 p2 • The budget line intersects the x2 axis at I/p2 • The slope of the budget line is -p1/p2 – How do you show this? • p1x1 + p2x2 = I • x = I/p -(p /p ) x x1 2 2 1 2 1 I p x + p x ≤ I • The consumer can buy only positive 1 1 2 2 p1 ≥ ≥ amounts of goods, so x1 0andx2 0 Spring 2001 Econ 11--Lecture 2 5 Spring 2001 Econ 11--Lecture 2 6 Econ 11--Lecture 2 1 Professor Jay Bhattacharya Spring 2001 Example: Determining the Budget Line What Does the Budget Line Tell Us? • The ‘opportunity cost’ of consuming an additional • x1 = houses (not in L.A.!) • x2 =BMWs unit of good 1 in terms of lost consumption of • p1 = $35,000 • p2 = $70,000 • I = $140,000 good 2. BMWs – Q; In this example, what is the opportunity cost of a BMW? slope = − 1 2 2 – A: Two houses. – Why? The answer is determined by the shape of the budget line. The slope of the budget line is equal to the 1 price of good 1 in terms of good 2. This slope tells us how much good 1 we must give up to get an additional unit of good 2. 123 4houses Spring 2001 Econ 11--Lecture 2 7 Spring 2001 Econ 11--Lecture 2 8 Budget Constraints with More than Two The Effect of a Price Change on the Goods Budget Line x • We can expand the bundle of goods to three 2 x2 or more goods – Budget line: p1x1+p2x2+p3x3 = I N = – N goods: å pi xi I i=1 • Often, we define good 2 as a composite good (i.e., all other goods) x1 x1 –e.g.: x1=gradschool, x2= all other goods Price of x2 falls Price of x1 rises Spring 2001 Econ 11--Lecture 2 9 Spring 2001 Econ 11--Lecture 2 10 The Effect of a Change in Income on the What Happens if the Prices of Both Budget Line Goods Double? x2 x2 • The effect is the same as if income were cut in half. ()+ ()= x1 2 p1 x2 2p2 I I x p + x p = 1 1 2 2 2 x x Income Rises 1 Income Falls 1 • What would happen if both prices and income double? Spring 2001 Econ 11--Lecture 2 11 Spring 2001 Econ 11--Lecture 2 12 Econ 11--Lecture 2 2 Professor Jay Bhattacharya Spring 2001 Budget ‘Lines’ Can Be Nonlinear Volume Discounts •Thefirstd units of x cost p • Up to now, we have only considered budget lines 1 1 when there are fixed and non-variable prices. • Any units after d cost p1/2 • Insomeapplications,suchanassumptionmaynot x2 apply. •Examples: – Progressive income taxes – Volume discounts – Food stamps (and other welfare programs) d x1 Spring 2001 Econ 11--Lecture 2 13 Spring 2001 Econ 11--Lecture 2 14 Food Stamps Food Stamps Budget Constraint housing • 2 goods, food and housing, with prices $1 and $2 respectively. 25 units • Income is equal to $50 • The consumer has a coupon (which can’t be sold) worth $10 of food – draw budget constraint – how much does it cost to trade food for housing (with and without the coupon) 10 units 50 units 60 units food Spring 2001 Econ 11--Lecture 2 15 Spring 2001 Econ 11--Lecture 2 16 Revealed Preference Axiom of Revealed Preference • A careful analysis of budget constraints can • Intuitive explanation of the axiom of lead to powerful predictions about revealed preference: consumer behavior. – Given price and income, if two bundles of • A good example of this is revealed goods (say A and B) are available to a preference analysis. consumer and he chooses A, then A will never be chosen over B no matter what prices and • Using just budget constraints and observed income. choices, we can prove that demand curves – If the consumer chooses B, then A must not be slope downward. affordable, given prices and income. Spring 2001 Econ 11--Lecture 2 17 Spring 2001 Econ 11--Lecture 2 18 Econ 11--Lecture 2 3 Professor Jay Bhattacharya Spring 2001 Graphical Demonstration of Graphical Demonstration (II) Revealed Preference x2 x2 Now, facing constraint I2,the consumer still prefers A to B. C Presumably he would prefer any The consumer facing budget bundle between C and D to A, line I1 picks good bundle A D since those bundles have more of A over good bundle B. A both x1 and x2. B B I 1 I2 x1 x1 Spring 2001 Econ 11--Lecture 2 19 Spring 2001 Econ 11--Lecture 2 20 Graphical Demonstration (III) Downward Sloping Demand x 2 • Suppose we observe that a consumer is Finally, facing constraint I3,the indifferent between two bundles of goods, C consumer might choose bundle and D. The goods are X and Y. B. But that is only because A bundle A is not available. • Suppose that C is chosen when prices are: ( C C ) pX , pY B • Suppose that D is chosen when prices are: I3 ( D D ) pX , pY x1 Spring 2001 Econ 11--Lecture 2 21 Spring 2001 Econ 11--Lecture 2 22 Downward Sloping Demand (II) Downward Sloping Demand (III) • Adding these two equations together and • Since the consumer is indifferent between C combining terms yields: and D, when C is chosen, D must cost at (pC − pD )()X − X + (pC − p D )()Y −Y ≤ 0 least as much (and perhaps more) than C: X X C D Y Y C D C + C ≤ C + C • If the price of Y is fixed then: pX X C pY YC pX X D pY YD (pC − pD )()X − X ≤ 0 • Similarly, when D is chosen, C must cost at X X C D least as much (and perhaps more) than D: • This means that, holding all else except pX fixed, price and quantity move in opposite p D X + p DY ≤ p D X + p DY X D Y D X C Y C directions—downward sloping demand. Spring 2001 Econ 11--Lecture 2 23 Spring 2001 Econ 11--Lecture 2 24 Econ 11--Lecture 2 4.
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