Lecture # 07-a Consumer Choice (conclusion) Lecturer: Martin Paredes

1. Lecture # 07-a Consumer Choice (conclusion) Lecturer: Martin Paredes

2. Outline Consumer Choice (conclusion) Duality Composite Goods Some Applications

3. Consumer Choice Example: Perfect Complements • Suppose U(X,Y) = min {X,Y} I = € 1000 PX = € 50 PY = € 100 • Which is the optimal choice for the consumer?

4. Example: Corner Solution – Perfect Complements Y BL: 50X + 100Y = 1000 10 X 0 20

5. Example: Corner Solution – Perfect Complements Y BL 10 U = min{X,Y} X 0 20

6. Example: Corner Solution – Perfect Complements Y BL 10 6.6 U = min{X,Y} X 0 6.6 20

7. Duality • The mirror image of the original (primal) constrained optimisation problem is called the dual problem. • Min PX . X + PY . Y subject to: U(X,Y) = U0 • X,Y where U0 is a target level of utility.

8. Duality Y U0 X 0

9. Duality Y U0 E1 = PXX + PYY X 0

10. Duality Y U0 E1 E2 X 0

11. Duality Y Optimal choice (interior solution) at point A • A U0 E1 E* E2 X 0

12. Duality • Suppose U0 = U*, which is the level of utility that solves the primal problem • Then an interior optimum, if it exists, of the dual problem also solves the primal problem.

13. Duality Y Optimal choice (interior solution) • Y* U = U* PXX + PYY = E* X 0 X*

14. Example: • Suppose U(X,Y) = XY PX = € 50 PY = € 100 • Which is the basket that minimizes the expenditure necessary to attain a utility level of U0 = 50?

15. We have to solve a system of two equations for two unknowns: 1. MRSX,Y = PX PY 2. XY = 50

16. MRSX,Y = MUX = Y MUY X • PX = 50 = 1 PY 100 2 • So X = 2Y

17. Utility level: XY = U0 = 50 • Then: (2Y) * Y = 50 Y2 = 25 => Y* = 5 => X* = 10

18. Example: Expenditure Minimization Y E* = 50X + 100Y = 1000 • 5 U = XY = 50 X 0 10

19. Composite Good • Consumers usually purchase more than two goods. • Economists often want to focus on the consumer’s selection of a particular good. • What to do? • Use a composite good in the vertical axis, that represents the amount spent on all other goods combined. • By convention, the price of a unit of the composite good equals 1. (Pm = 1)

20. Composite Good m Preference directions • I/Pm= I • A -PX/Pm = -PX IC X I/PX

21. Some Applications • Borrowing and Lending • Consider a consumer that lives for two periods. • Suppose the consumer has an income of I1 in period 1, and I2 in period 2 • To represent the consumption choice in each period, define the composite goods • C1: consumption in period 1 (in €) • C2: consumption in period 2 (in €)

22. Borrowing and Lending • If the consumer cannot borrow or lend, he will spend I1 in period 1, and I2 in period 2

23. Example: Borrowing and Lending C2 C1

24. Example: Borrowing and Lending C2 • A C2* = I2 C1 C1* = I1

25. Example: Borrowing and Lending C2 Preference direction • A C2* = I2 IC0 C1 C1* = I1

26. Suppose the consumer can put money in the bank and earn an interest rate r. • If he decreases his consumption in period 1 by € X (so C1 = -X), he will increase his consumption in period 2 by C2 = (1+r) * X • Slope of budget line: dC2 = - (1+r) dC1 • He will be able to spend up to I1 * (1+r) + I2 in period 2, while consuming nothing in period 1

27. Suppose the consumer can borrow money from the bank at the same interest rate r. • If he increases his consumption in period 1 by borrowing € X (so C1 = X), he will have to pay back € (1+r) * X. Hence his consumption in period 2 decreases by C2 = - (1+r) * X • Slope of budget line: dC2 = - (1+r) dC1 • He will be able to borrow up to I1 + I2 / (1+r) in period 1, while consuming nothing in period 2

28. Example: Borrowing and Lending C2 • I2+ I1(1+r) • A I2 C1 I1

29. Example: Borrowing and Lending C2 • I2+ I1(1+r) • A I2 • C1 I1 I1+I2/(1+r)

30. Example: Borrowing and Lending C2 • I2+ I1(1+r) • A I2 • C1 I1 I1+I2/(1+r)

31. Example: Borrowing and Lending C2 • I2+ I1(1+r) • A I2 Slope = -(1+r) • C1 I1 I1+I2/(1+r)

32. Example: Borrowing and Lending C2 • I2+ I1(1+r) Case 1: Borrowing in period 1 • A I2 IC0 Slope = -(1+r) • C1 I1 I1+I2/(1+r)

33. Example: Borrowing and Lending C2 • I2+ I1(1+r) Case 1: Borrowing in period 1 • A I2 • C2B B IC1 IC0 Slope = -(1+r) • C1 C1B I1 I1+I2/(1+r)

34. Example: Borrowing and Lending C2 • I2+ I1(1+r) Case 2: Lending in period 1 • A I2 IC0 Slope = -(1+r) • C1 I1 I1+I2/(1+r)

35. Example: Borrowing and Lending C2 • I2+ I1(1+r) Case 2: Lending in period 1 • D C2D IC2 • A I2 IC0 Slope = -(1+r) • C1 C1D I1 I1+I2/(1+r)

36. Some Applications 2. Quantity Discounts • Suppose a consumer spends his income of € 550 on electricity and other goods • Suppose the power company sells electricity at a price of € 11 per unit.

37. Composite Good Example: Quantity Discounts 0 Electricity

38. Composite Good Example: Quantity Discounts 550 Slope = -PE = -11 0 40 Electricity

39. Composite Good Example: Quantity Discounts 550 • A IC0 Slope = -PE = -11 0 18 40 Electricity

40. Suppose the power company offers the following quantity discount: • € 11 per unit for the first 10 units • € 8 per unit for additional units.

41. Composite Good Example: Quantity Discounts 550 • A Slope = -PE = -11 0 10 18 40 Electricity

42. Composite Good Example: Quantity Discounts 550 • A Slope = -PE = -8 Slope = -PE = -11 0 10 18 40 75 Electricity

43. Composite Good Example: Quantity Discounts 550 • A IC0 Slope = -PE = -8 Slope = -PE = -11 0 10 18 40 75 Electricity

44. Composite Good Example: Quantity Discounts 550 • A B IC1 Slope = -PE = -8 Slope = -PE = -11 0 10 18 30 40 75 Electricity

45. Summary • The budget line represents the set of all baskets that the consumer can buy if he spends all of his income • The budget line rotates as prices change and shifts when income changes. • If the consumer chooses the consumption bundle by maximizing utility given his budget constraint, the optimal consumption basket will lie at a tangency between an indifference curve and the budget line or at a corner point.