Optimal Choice and Tangency in Consumer Decision Making

1. Chapter 5 Choice • Key Concept: Optimal choice, where a consumer chooses the best he can afford. • Tangency is neither necessary nor sufficient. • The necessary and sufficient condition: The set of consumption bundles strictly preferred to A cannot intersect with the budget set. (月亮形區域為空)

2. Chapter 5 Choice • budget set + preference → choice • Optimal choice: the best one can afford.

3. Fig. 5.1

4. Suppose the consumer chooses bundle A. • A is optimal A w B for any B in the budget set ↔ The set of consumption bundles strictly preferred to A cannot intersect with the budget set. (月亮形區域為空)

5. A

6. sufficient optimum necessary

7. typhoon → rain → wet sufficient optimum necessary

8. A optimal↔月亮形區域為空. • A optimal →月亮形區域為空? • (necessary) • If not, then 月亮形區域不為空. • That means there exists a bundle B such that B s A and B is in the budget set. • Then A is not optimal.

9. A optimal↔月亮形區域為空. • A optimal ←月亮形區域為空? • (sufficient) • All B such that B s A is not affordable, so for all B in the budget set, we must have A w B. • Hence A is optimal.

10. The indifference curve tangent to the budget line is neither necessary nor sufficient for optimality. sufficient optimum necessary

11. necessary: optimal → tangent • optimal not tangent • kinked preferences • tangent is not defined • (intuition)

12. Fig. 5.2

13. necessary: optimal → tangent • optimal not tangent • corner solution角解 • vs. interior solution內解 • (intuition)

14. Fig. 5.3

15. sufficient: tangent → optimal • tangent not optimal • satiation

16. Fig. 3.7

17. sufficient: tangent → optimal • tangent not optimal • convexity is violated

18. Fig. 5.4

19. Despite all of these, the usual tangent condition MRS1, 2= -p1/ p2 has a nice interpretation. • So we still need to get the economic intuition.

20. The MRS is the rate the consumer is willing to pay for an additional unit of good 1 in terms of good 2. • The relative price ratio is the rate the market asks a consumer to pay for an additional unit of good 1 in terms of good 2. • At optimum, these two rates are equal. (主觀相對價格 vs. 客觀相對價格)

21. If they are not equal, we will get: • when |MRS1, 2| > p1/ p2 • should buy more of 1 • when |MRS1, 2| < p1/ p2 • should buy less of 1

22. We now know what the optimal choice is, let us turn to demand since they are related.

23. The optimal choice of goods at some price and income is the consumer’s demanded bundle. • A demand function gives you the optimal amount of each good as a function of prices and income faced by the consumer.

24. Denote the demand function x1 (p1, p2, m). • At p1, p2, m, the consumer demands x1 • Let us work out some examples.

25. Perfect substitutes • u(x1, x2) = x1 + x2 • p1> p2: x1 = 0, x2 = m/ p2 • p1= p2: x1 belongs to [0, m/ p1] and x2 = (m- p1 x1)/p2 • p1< p2: x1 = m/ p1, x2 = 0

26. Fig. 5.5

27. Perfect complements • u(x1, x2) = min{x1, x2} • x1 = x2 = m/ (p1+ p2)

28. Fig. 5.6

29. Neutrals or bads • Why spend money on them?

30. Discrete goods • Just compare.

31. Fig. 5.7

32. Non convex preferences • Probably a corner solution

33. Fig. 5.8

34. Cobb-Douglas • u(x1, x2) = x1ax21-a • u(x1, x2) = a lnx1 + (1-a) lnx2 • |MRS1, 2| = p1/ p2 • (a/x1)/[(1-a)/x2] = p1/ p2

35. |MRS1, 2| = p1/ p2 • (a/x1)/[(1-a)/x2] = p1/ p2 • a/(1-a) = p1x1/ p2x2 • x1 = am/ p1 and x2 = (1-a)m/ p2 • This is useful if when we are estimating utility functions, we find that the expenditure share is fixed.

36. Table 5.1

37. Implication of the MRS condition: at equilibrium, we don’t need to know the preferences of each individual, we can infer that their MRS’ are the same. • This has an useful implication for Pareto efficiency because if their MRS’ are different, they can trade and Pareto improve.

38. One small example: butter (price:2) and milk (price: 1) • A new technology that will turn 3 units of milk into 1 unit of butter. Will this be profitable? • Another new tech that will turn 1 unit of butter into 3 units of milk. Will this be profitable?

39. quantity tax and income tax • If the government wants to raise a certain amount of revenue, is it better to raise it via a quantity tax or income tax?

40. Suppose gov imposes a quantity tax of t dollars per unit of x1. • budget constraint: (p1+t) x1 + p2 x2= m • at optimum: (x1*, x2*) so that • (p1+t) x1* + p2 x2* = m • tax revenue is t x1* • to compare, we want income tax R* to raise the same revenue • R* = t x1*

41. (x1’, x2’) is the optimum with income tax. • p1 x1’+ p2 x2’ = m - R* but we know that • (p1+t) x1* + p2 x2* = m • p1 x1* + p2 x2* = m - t x1* • p1 x1* + p2 x2* = m - R* • so (x1*, x2*) is on the income tax budget line • Hence, (x1’, x2’) w (x1*, x2*).

42. Fig. 5.9

43. Income tax better than quantity tax? • For any consumer, we can find an income tax raising the same tax revenue and the consumer is better off. • Yet the amount of income tax will differ from consumer to consumer. • uniform income tax vs. quantity tax (consider those who do not consume good 1)

44. Income tax might discourage earning. • We ignore supply side

45. Chapter 5 Choice • Key Concept: Optimal choice, where a consumer chooses the best he can afford. • Tangency is neither necessary nor sufficient. • The necessary and sufficient condition: The set of consumption bundles strictly preferred to A cannot intersect with the budget set. (月亮形區域為空)