Mid-exam review class

1. Mid-exam review class

2. Consumer theory • A typical consumer will satisfy himself as much as possible with limited resources. • First I should use proper tools to describe the “limited resources” and “satisfaction”

3. Budget Set • Two important definitions: • The budget set consists of all bundles that are affordable at the given prices and income. • The budget lineconsists of all bundles that just exhaust the consumer’s income. • You should describe the above two definitions in arithmetical and geometrical ways!

4. Preference • Def.1 Consumption set is the set which consists of all consumption bundles, or say, consumption plans which the typical consumer desires, regardless whether they are feasible or not. • Def.2 If the binary relation defined on the consumption set satisfies Axiom 1,2 and 3, then this binary relation is said to be the Preference relation.

5. Preference Axiom 1 Completeness: x is w.p. to y or y is w.p. to x for any pair of x and y. Axiom 2 Reflexivity: x is w.p. to x for any bundle x. Axiom 3 Transitivity: If x is w.p. to y and y is w.p. to z, then x is w.p. to z.

6. Strict preference relation andindifference relation • Def.3 If the binary relation defined on the consumption set satisfies and then we say ,which is called the strict preference relation. Def.4 If the binary relation defined on the consumption set satisfies and then we say ,which is called the indifference relation,

7. Indifference curves • Def. 5 An indifference curve is a set of consumption bundles with the same desire level of a representative consumer. • Now we can conclude that for a specific preference, there’s a unique shape of indifference curves corresponding to it. • We can use a cluster of IC to describe a specific preference.

8. Well-behaved preference • Def.6 A given preference is called well-behaved preference if it satisfies Axiom 1,2,3,4 and 5. • Axiom 4monotonic (meaning more is better) • Axiom 5convex (meaning average are preferred to extremes).

9. Utility function • Utility function is a way to describe preference. • Def. 7 A mapping u: is called the utility function which stands for the preference relation if the mapping satisfies u ( x ) ≥ u ( y ) if and only if bundle x is w.p. to bundle y.

10. Positive Monotonic transformation • Lemma: Ordinal utility holds that the size of the utility difference between any two consumption bundles doesn’t matter. • So what we care is only the ordinal represented by the amount of utility function. • Positive Monotonic transformation of the utility function represents the same preference as the original utility function.

11. Positive Monotonic transformation • For a given preference, there’s at least one utility function to describe it.

12. 2 A utility function u • Take a slice at given utility level • Project down to get contours U(x1,x2) The indifference curve x2 0 x1

13. Demand • Three factors of demand: • price, income and preference • Now we can obtain the demand function, and carry out the comparative statics.

14. Demand function • Key point: • Lagrangian method • Two properties: • 1) • 2)

15. Demand function • Warning: the two arithmetical properties only hold for the tangency case. • In order to obtain the demand, we can also use the geometrical tools. • Find the highest IC in the budget set! • Interior solution: Tangency case and kinky case • Boundary solution: • Some examples

16. Comparative statics(1) • How does the demand of one good change corresponding to the change in income with the prices constant? • Def. income offer curve and Engle curve • For different types of goods • For different types of preferences (H.M)

17. Comparative statics(1) • Def. normal goods(p97) and inferior goods(p98) • Def. luxury goods and necessary goods(p101) • To make the def. more clearly, we can impose the def. of income elasticity of demand.

18. Comparative statics(1) • Arithmetical form: • Geometrical meaning: • The ratio of the slope of the ray and the slope of the tangent line of some point on the Engle curve.

19. Comparative statics(1) • Strict def. • Normal goods: • Inferior goods: • Luxury goods: • Necessary goods:

20. Comparative statics(2) • How does the demand of one good change corresponding to the change in its price with the income and price of the other good constant? • Def. price offer curve and personal demand curve • For different types of goods • For different types of preferences (H.M)

21. Comparative statics(2) • Ordinary goods • Giffen goods • Substitutes • Complements • Distinguish the ordinary and Giffen goods in the figures of price offer curve and demand curve

22. 画四种曲线的思路 • 总原则：直观切入，代数辅助。 • 1、为了有好的引导，应先求出两种商品的需求函数； • 2、根据定义和函数形式开始直观作图； • 3、对比函数形式，检验图像的正确性； • 4、最后要检查曲线（尤其是价格提供曲线和需求曲线）的起点、终点和渐近线。

23. 注意： • 原点一般来说都会在收入提供曲线和恩格尔曲线上面 • 在分析价格提供曲线和个人需求曲线时，我们要尤其注意起点、终点和渐近线。而起点和终点分别是价格趋向于无穷大和零的情形。对此李杰老师认为这两个极限点可取可不取；而我认为对于两个极限情形我们都取空心点，毕竟极限情形是无法达到的。

24. Deeper sight • Two problems about price and income we should think about seriously: • We have known How the demand of one good changes corresponding to the change in its price with the income and price of the other good constant, but do you know the mechanism that forms the ordinary good or Giffen good? (Ch 8) • Do you know the source of income? (Ch 9)

25. Chapter 8 • Slutsky and Hicks decomposition: • The common ground: the process of the change in demand corresponding to the change in the relative price can be decomposed into substitution effect and income effect.

26. Chapter 8 • Def. of substitution effect : the difference between the new optimal choice and the original optimal choice with the real income unchanged and the relative price adjusted. • Def. of income effect : the difference between the final optimal choice and the optimal choice with the real income unchanged and the relative price adjusted.

27. Chapter 8 • Slutsky and Hicks decomposition: • The differentia: the def. of “unchanged real income” • Slutsky: • After the change in the relative price, the typical consumer is still able to buy the original optimal consumption bundle. (Keep the purchasing power constant before and after the change in the relative price.)

28. Chapter 8 • Hicks: • After the change in the relative price, the typical consumer is still able to reach the same level of satisfaction as the original optimal consumption bundle. (Keep the utility constant before and after the change in the relative price.) • Figures.

29. Slutsky decomposition

30. Hicks decomposition

31. Chapter 8 • Warning : • Subs. Effect is non-positive, while the sign of the income effect isn’t unique, depending on the type of the good. • Can we compare the subs. Effects? • Giffen goods must be the inferior goods, but not vice versa. • The law of demand. • Some examples.

32. Chapter 8

33. Chapter 8

34. Chapter 8

35. Chapter 8

36. Chapter 9 • To explain the source of “income”, we impose a def. of “endowment”, which means the initial holding of different goods. • Budget constraint: • We can see the endowment is always on the budget line, independent of the relative price.

37. Chapter 9

38. Chapter 9 • Figure: Gross demand, net demand or net supply • Now in the equilibrium, we should find the optimal choice (gross demand) and make it clear that the decision the typical consumer makes (net demand or net supply).

39. Figure 1

40. Figure 2

41. Chapter 9 • Comparative statics: • Income(1 and 2) figure • Price :figure

42. Chapter 9 • How to obtain the price offer curve? • Endowment point must be on the price offer curve, and the line through w and a, which is another point on the price offer curve is the budget line corresponding to a. • The area which the price offer curve must be in.

44. Chapter 9 • From personal gross demand curve to net demand or supply curve (Def.) (figure) • Two examples: (figures) • Impose a quantity tax on the consumption of good 1 • Analysis of the labor supply curve (from the price offer curve to the labor supply curve)

47. Applications of two-good modelSummary： • 在分析具体背景之后，要找准应该把那些因素抽象为两种商品。这个是分析问题的突破口。 • 确定主旨句：这个经济主体要在一定的约束条件下追求自身利益的最大化。 • 接下来就做翻译工作，刻画好“约束”和“自身利益”，正如前面所述，这就是你们分析的出发点。 • 展开相应的几何分析或代数推导，建议两者联合运用。 • 最后得到我们想要的“最优选择”，有需要还要进行比较静态分析。

48. Outlines • Applications of two-good model with endowment point • Intertemporal choices (Fisher) • Insurance market (Stiglitz)

49. Intertemporal choices • The key sentence: • A typical and forward-looking consumer will make a consumption decision to maximize his lifetime utility with the intertemporal budget constraint. • How to describe the intertemporal budget constraint and the preference?

50. Stiglitz: how does the Insurance market operates? • You’re the risk averser and have a risky asset: • You can go into the insurance market and spread your risk: • Questions:1 how to fix the premium? • 2 what level of insured amount should you choose?