1. Chapter 2 Budget Constraint

2. 2 Consumption Theory Economists assume that consumers choose the best bundle of goods they can afford. In this chapter, we examine how to describe what a consumer can afford: the budget constraint, and budget set. What is best for a consumer, or the preference on the possible consumption bundles, will be discussed in the next chapter.

3. 3 Budget Constraints A consumption bundle containing x 1 units of commodity 1, x 2 units of commodity 2 and so on up to x n units of commodity n is denoted by the vector (x 1, x 2, …, x n ). Commodity prices are p 1, p 2, …, p n.

4. 4 Budget Constraints Q: When is a consumption bundle (x 1, …, x n ) affordable at prices p 1, …, p n ? A: When total expenditure is smaller than income: p 1 x 1 + … + p n x n  m where m is the consumer’s (disposable) income. That is, the consumer’s affordable consumption bundles are those that don’t cost more than m.

5. 5 Budget Constraints The bundles that are only just affordable form the consumer’s budget constraint. This is the set { (x 1,…,x n ) | x 1  0, …, x n  and p 1 x 1 + … + p n x n  m }.

6. 6 Budget Constraints The consumer’s budget set is the set of all affordable bundles; B(p 1, …, p n, m ) = { (x 1, …, x n ) | x 1  0, …, x n  0 and p 1 x 1 + … + p n x n  m } The budget constraint is the upper boundary of the budget set.

7. 7 Budget Set and Constraint for Two Commodities x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 m /p 2

8. 8 Budget Set and Constraint for Two Commodities x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 2 m /p 1

9. 9 Budget Set and Constraint for Two Commodities x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 Just affordable m /p 2

10. 10 Budget Set and Constraint for Two Commodities x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 Just affordable Not affordable m /p 2

11. 11 Budget Set and Constraint for Two Commodities x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 Affordable Just affordable Not affordable m /p 2

12. 12 Budget Set and Constraint for Two Commodities x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 Budget Set the collection of all affordable bundles. m /p 2

13. 13 Budget Set and Constraint for Two Commodities x2x2 x1x1 p 1 x 1 + p 2 x 2 = m is x 2 = -(p 1 /p 2 )x 1 + m/p 2 so slope is -p 1 /p 2. m /p 1 Budget Set m /p 2

14. 14 Budget Constraint for Three Commodities x2x2 x1x1 x3x3 m /p 2 m /p 1 m /p 3 p 1 x 1 + p 2 x 2 + p 3 x 3 = m

15. 15 Budget Set for Three Commodities x2x2 x1x1 x3x3 m /p 2 m /p 1 m /p 3 { (x 1,x 2,x 3 ) | x 1  0, x 2  0, x 3   0 and p 1 x 1 + p 2 x 2 + p 3 x 3  m}

16. 16 Budget Constraints For n = 2 and x 1 on the horizontal axis, the constraint’s slope is -p 1 /p 2. What does it mean? Holding income m constant, increasing x 1 by 1 unit must reduce x 2 by p 1 /p 2 units.

17. 17 Budget Constraints x2x2 x1x1 Slope is -p 1 /p 2 +1 -p 1 /p 2

18. 18 Budget Constraints x2x2 x1x1 +1 -p 1 /p 2 Opp. cost of an extra unit of commodity 1 is p 1 /p 2 units foregone of commodity 2.

19. 19 Budget Constraints x2x2 x1x1 The opp. cost of an extra unit of commodity 2 is p 2 /p 1 units foregone of commodity 1. -p 2 /p 1 +1

20. 20 Budget Sets & Constraints as Income or Price Changes The budget constraint and budget set depend upon prices and income. When prices and incomes change, the set of goods that a consumer can afford changes as well. How do these changes affect the budget constraint and budget set?

21. 21 How do the budget set and budget constraint change as income m increases? Original budget set x2x2 x1x1

22. 22 Higher income gives more choice Original budget set New affordable consumption choices x2x2 x1x1 Original and new budget constraints are parallel (same slope).

23. 23 How do the budget set and budget constraint change as income m decreases? Original budget set x2x2 x1x1

24. 24 How do the budget set and budget constraint change as income m decreases? x2x2 x1x1 New, smaller budget set Consumption bundles that are no longer affordable. Old and new constraints are parallel.

25. 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 –p 1 / p 2 does not change.

26. 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, a consumer may (typically will) be worse off, as one can no longer afford some of the bundles anymore.

27. 27 Budget Constraints - Price Changes What happens to the budget constraint and budget set if one of the prices changes? For example, consider a situation in which p 1 decreases while p 2 and income remain unchanged.

28. 28 How do the budget set and budget constraint change as p 1 decreases? Original budget set x2x2 x1x1 m/p 2 m/p 1 ’ m/p 1 ” -p 1 ’/p 2

29. 29 How do the budget set and budget constraint change as p 1 decreases? Original budget set x2x2 x1x1 m/p 2 m/p 1 ’ m/p 1 ” New affordable choices -p 1 ’/p 2

30. 30 How do the budget set and budget constraint change as p 1 decreases? Original budget set x2x2 x1x1 m/p 2 m/p 1 ’ m/p 1 ” New affordable choices Budget constraint pivots; slope flattens from -p 1 ’/p 2 to -p 1 ”/p 2 -p 1 ’/p 2 -p 1 ”/p 2

31. 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, reduces choice and may (typically will) make the consumer worse off.

32. 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. 33 Uniform Ad Valorem Sales Taxes A uniform sales tax levied at rate t on all goods changes the constraint from p 1 x 1 + p 2 x 2 = m to (1+ t )p 1 x 1 + (1+ t )p 2 x 2 = m i.e. p 1 x 1 + p 2 x 2 = m /(1+ t ).

34. 34 Uniform Ad Valorem Sales Taxes x2x2 x1x1 p 1 x 1 + p 2 x 2 = m

35. 35 Uniform Ad Valorem Sales Taxes x2x2 x1x1 p 1 x 1 + p 2 x 2 = m p 1 x 1 + p 2 x 2 = m/(1+t)

36. 36 Uniform Ad Valorem Sales Taxes x2x2 x1x1 Equivalent income loss is

37. 37 Uniform Ad Valorem Sales Taxes x2x2 x1x1 A uniform ad valorem sales tax levied at rate t is equivalent to an income tax levied at rate

38. 38 Lump-Sum 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.

39. 39 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.

40. 40 The Food Stamp Program Suppose m = $100, p F = $1 and the price of “other goods” is p G = $1. The budget constraint is then F + G =100.

41. 41 The Food Stamp Program G F 100 F + G = 100: before stamps.

42. 42 The Food Stamp Program G F 100 F + G = 100: before stamps. Budget set after 40 food stamps issued. 14040

43. 43 The Food Stamp Program G F 100 F + G = 100: before stamps. Budget set after 40 food stamps issued. 140 The family’s budget set is enlarged. 40

44. 44 The Food Stamp Program What if food stamps can be traded on a black market for $0.50 each?

45. 45 The Food Stamp Program G F 100 F + G = 100: before stamps. Budget constraint after 40 food stamps issued. 140 120 Black market trading makes the budget set larger again. 40

46. 46 Budget Constraints - Relative Prices How does the unit of account affect the budget constraint and budget set? Suppose prices and income are measured in dollars. Say p 1 =$2, p 2 =$3, m = $12. Then the constraint is 2x 1 + 3x 2 = 12.

47. 47 Budget Constraints - Relative Prices If prices and income are measured in cents, then p 1 =200, p 2 =300, m =1200 and the constraint is 200x 1 + 300x 2 = 1200, the same as 2x 1 + 3x 2 = 12. Changing the unit of account changes neither the budget constraint nor the budget set.

48. 48 Budget Constraints - Relative Prices The constraint for p 1 =2, p 2 =3, m =12 2x 1 + 3x 2 = 12 can also be written as 1x 1 + (3/2)x 2 = 6, the constraint for p 1 =1, p 2 =3/2, m =6. Setting p 1 =1 makes commodity 1 the numeraire and defines all prices relative to p 1. For example, 3/2 is the price of commodity 2 relative to the price of commodity 1.

49. 49 Budget Constraints - Relative Prices Multiplying all prices and income by any constant k does not change the budget constraint: kp 1 x 1 +kp 2 x 2 =km. Any commodity can be chosen as the numeraire without changing the budget set or the budget constraint. It is also clear from the graph that, only the ratios p 1 /p 2, m/p 1 and m/p 2 are relevant to the budget line and budget set.

50. 50 Shapes of Budget Constraints Q: What makes a budget constraint a straight line? A: A straight line must have a constant slope. The constraint is p 1 x 1 + p 2 x 2 = m. So if prices are constants, the constraint is a straight line.

51. 51 Shapes of Budget Constraints But what if prices are not constants? For example, bulk buying discounts, or price penalties for buying “too much”. Then constraints will be curved or have kinks.

52. 52 Shapes of Budget Constraints - Quantity Discounts Suppose p 2 is constant at $1 but that p 1 =$2 for 0  x 1  20 and p 1 =$1 for x 1 >20.

53. 53 Shapes of Budget Constraints - Quantity Discounts Suppose p 2 is constant at $1 but that p 1 =$2 for 0  x 1  20 and p 1 =$1 for x 1 >20. Then the constraint’s slope is - 2, for 0  x 1  20 - p 1 /p 2 = - 1, for x 1 > 20 Also assume m=100. {

54. 54 Shapes of Budget Constraints with a Quantity Discount m = $100 50 100 20 Slope = - 2 / 1 = - 2 (p 1 =2, p 2 =1) Slope = - 1/ 1 = - 1 (p 1 =1, p 2 =1) 80 x2x2 x1x1

55. 55 Shapes of Budget Constraints with a Quantity Discount m = $100 50 100 20 Slope = - 2 / 1 = - 2 (p 1 =2, p 2 =1) Slope = - 1/ 1 = - 1 (p 1 =1, p 2 =1) 80 x2x2 x1x1

56. 56 Shapes of Budget Constraints with a Quantity Discount m = $100 50 100 20 80 x2x2 x1x1 Budget Set Budget Constraint

57. 57 Shapes of Budget Constraints with a Quantity Penalty x2x2 x1x1 Budget Set Budget Constraint

58. 58 What’s Next? The budget set describes what consumption bundles are affordable to the consumers. 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.