Chapter 2 Budget Constraint

1. Chapter 2 Budget Constraint Course: Microeconomics Text: Varian’s Intermediate Microeconomics

2. Consumer Theory • 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.

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. Budget Constraints • Q: When is a consumption bundle (x1, … , xn) affordable at prices p1, … , pn?

5. Budget Constraints • Q: When is a consumption bundle (x1, … , xn) affordable at prices p1, … , pn? • A: When total expenditure is smaller than income: p1x1 + … + pnxn£mwhere m is the consumer’s (disposable) income. • That is, one’s total expenditure is smaller than or equal to one’s income.

6. Budget Constraints • The bundles that are only just affordable by the consumer is one’s budget constraint. This is the set{ (x1,…,xn) | x1 ³ 0, …, xn³ 0 and p1x1 + … + pnxn=m }.

7. Budget Constraints • The consumer’s budget setis the set of all affordable bundles;B(p1, … , pn, m) ={ (x1, … , xn) | x1³ 0, … , xn³ 0 and p1x1 + … + pnxn£m } • The budget constraint is the upper boundary of the budget set.

8. Budget Set and Constraint for Two Commodities x2 Budget constraint is p1x1 + p2x2 = m. m /p2 x1 m /p1

9. Budget Set and Constraint for Two Commodities x2 Budget constraint is p1x1 + p2x2 = m. m /p2 x1 m /p1

10. Budget Set and Constraint for Two Commodities x2 Budget constraint is p1x1 + p2x2 = m. m /p2 Just affordable x1 m /p1

11. Budget Set and Constraint for Two Commodities x2 Budget constraint is p1x1 + p2x2 = m. m /p2 Not affordable Just affordable x1 m /p1

12. Budget Set and Constraint for Two Commodities x2 Budget constraint is p1x1 + p2x2 = m. m /p2 Not affordable Just affordable Affordable x1 m /p1

13. Budget Set and Constraint for Two Commodities x2 Budget constraint (budget line) is p1x1 + p2x2 = m. m /p2 the collection of all affordable bundles. Budget Set x1 m /p1

14. Budget Set and Constraint for Two Commodities x2 p1x1 + p2x2 = m is x2 = -(p1/p2)x1 + m/p2 so slope is -p1/p2. m /p2 Budget Set x1 m /p1

15. Budget Constraints • For n = 2 and x1 on the horizontal axis, the constraint’s slope is -p1/p2. What does it mean? • Note: For a linear line: y=mx+c, m is the slope of the line, while c is the y-intercept.

16. Budget Constraints • For n = 2 and x1 on the horizontal axis, the constraint’s slope is -p1/p2. What does it mean? • To hold income m constant, increasing x1 by 1 must reduce x2 by p1/p2 .

17. Budget Constraints x2 Slope is -p1/p2 -p1/p2 +1 x1

18. Budget Constraints x2 Opp. cost of an extra unit of commodity 1 is p1/p2 units foregone of commodity 2. -p1/p2 +1 x1

19. Budget Constraints x2 The opp. cost of an extra unit of commodity 2 is p2/p1 units foregone of commodity 1. +1 -p2/p1 x1

20. 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?

21. How do the budget set and budget constraint change as income m increases? x2 Original budget set x1

22. Higher income gives more choice x2 New affordable consumptionchoices Original and new budget constraints are parallel (same slope). Original budget set x1

23. How do the budget set and budget constraint change as income m decreases? x2 Original budget set x1

24. How do the budget set and budget constraint change as income m decreases? x2 Consumption bundles that are no longer affordable. Old and new constraints are parallel. New, smaller budget set x1

25. 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. • The slope –p1 / p2 does not change.

26. Budget Constraints - Income Changes • 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.

27. Budget Constraints - Price Changes • What happens if just one price decreases? • Suppose p1 decreases.

28. How do the budget set and budget constraint change as p1decreases from p1’ to p1”? x2 m/p2 -p1’/p2 Original budget set m/p1’ x1 m/p1”

29. How do the budget set and budget constraint change as p1decreases from p1’ to p1”? x2 m/p2 New affordable choices -p1’/p2 Original budget set m/p1’ x1 m/p1”

30. How do the budget set and budget constraint change as p1decreases from p1’ to p1”? x2 m/p2 New affordable choices Budget constraint pivots; slope flattens from -p1’/p2to -p1”/p2 -p1’/p2 Original budget set -p1”/p2 m/p1’ x1 m/p1”

31. 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. • 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.

32. Tax and Subsidy • 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

33. Uniform Ad Valorem Sales Taxes • A uniform sales tax levied at rate t on all goods changes the constraint from p1x1 + p2x2 = mto (1+t)p1x1 + (1+t)p2x2 = mi.e. p1x1 + p2x2 = m/(1+t).

34. Uniform Ad Valorem Sales Taxes x2 Equivalent income lossis x1

35. Uniform Ad Valorem Sales Taxes x2 A uniform ad valoremsales tax levied at rate tis equivalent to an incometax levied at rate x1

36. Tax and Subsidy • 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.

37. 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? • Here we assume one of the two goods is food.

38. 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.

39. The Food Stamp Program G F + G = 100: before stamps. 100 F 100

40. The Food Stamp Program G F + G = 100: before stamps. 100 Budget set after $40 foodstamps issued. F 40 100 140

41. The Food Stamp Program G F + G = 100: before stamps. 100 Budget set after $40 foodstamps issued. The family’s budgetset is enlarged. F 40 100 140

42. Budget Constraints - Relative Prices • 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.

43. 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,upon simplification, it is the same as 2x1 + 3x2 = 12. • Changing the unit of account changes neither the budget constraint nor the budget set.

44. Budget Constraints - Relative Prices • 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.

45. Budget Constraints - Relative Prices • 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.

46. 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 = mso if prices are constants then a constraint is a straight line.

47. Shapes of Budget Constraints • 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.

48. 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.

49. 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 • Also assume m=100. {

50. Shapes of Budget Constraints with a Quantity Discount x2 m = $100 Slope = - 2 / 1 = - 2 (p1=2, p2=1) 100 Slope = - 1/ 1 = - 1 (p1=1, p2=1) x1 80 20 50