Chapter 8: Slutsky Decomposition

Transcription

1 Econ 33 Microeconomic Analysis Chapter : Slutsky Decomposition Instructor: Hiroki Watanabe Spring 13 Watanabe Econ 33 Slutsky Decomposition 1 / 59 1 Introduction Decomposing Effects 3 Giffen Is Income-Inferior Examples 5 Hicks Now We Know Watanabe Econ 33 Slutsky Decomposition / 59 1 Introduction Overview Background Decomposing Effects 3 Giffen Is Income-Inferior Examples 5 Hicks Now We Know Watanabe Econ 33 Slutsky Decomposition 3 / 59

2 Overview Chpt : Categories and definitions. Chpt : Price change demand. Chpt 1: Price change welfare. Watanabe Econ 33 Slutsky Decomposition / 59 Overview Discussion 1.1 (Orange vs Tomato) Listen to NPR News Clip. Orange juice is not a required staple for people to live. So they will turn to other things and that will bring down prices eventually. When the price of OJ goes up, do people 1 increase consumption of tomato juice to substitute away from OJ, or can they possibly consume less tomato juice? Watanabe Econ 33 Slutsky Decomposition 5 / 59 Overview Exogenous Variable φ C (p, m) φ C (p, m) m normal income inferior p C Giffen LOD p T substitute complement Watanabe Econ 33 Slutsky Decomposition / 59

3 Overview Δ C denotes change in C. If C = 5 and then C becomes after the price change (or income change), Δ C = 3. C Δ denotes change in. T If T = 3 and then T becomes after the price change (or income change), 3 Δ =. 5 Watanabe Econ 33 Slutsky Decomposition 7 / 59 Overview Split the price change in three progressive stages: 1 Original: O Transitional: T 3 Final: F Watanabe Econ 33 Slutsky Decomposition / 59 Background Consider (p O C, p T) = (1, 1) and (p F C, p T) = (1, 1). Δ C has two components: 1 Income effect, C : some money left after purchasing O C Substitution effect, C : Liz has to give up only one slice for a cup of tea. Watanabe Econ 33 Slutsky Decomposition 9 / 59

4 Background T responds to the change in p C as well: 1 Income effect, T Substitution effect, T In total, T or? Watanabe Econ 33 Slutsky Decomposition 1 / 59 Background Definition 1. (Income & Substitution Effect) 1 Income effect is a change in demand Δ due to having more purchasing power. Substitution effect is a change in demand Δ S due to change in relative price. 3 Total effect Δ is the sum of the above: Δ := Δ + Δ S. Watanabe Econ 33 Slutsky Decomposition 11 / 59 Background To resolve ambiguity with tomato juice consumption in Discussion 1.1 we want to separate two effects from each other. Recall the change in parameters: 1 Income change: Price change: Watanabe Econ 33 Slutsky Decomposition 1 / 59

5 1 Introduction Decomposing Effects Eliminating Income Effect Slutsky Decomposition Example 1: Cobb-Douglas 3 Giffen Is Income-Inferior Examples 5 Hicks Now We Know Watanabe Econ 33 Slutsky Decomposition 13 / 59 Eliminating Income Effect Goal: Eliminate the income effect (change in purchasing power) and isolate the substitution effect. Idea: You can compensate the income to suppress the change in purchasing power. Then let Liz solve her UMP with compensated income. Watanabe Econ 33 Slutsky Decomposition 1 / 59 Eliminating Income Effect 1 1 Tea x T (cups) Cheese x C (slices) Watanabe Econ 33 Slutsky Decomposition 15 / 59

6 Eliminating Income Effect 1 1 Tea x T (cups) Cheese x C (slices) Watanabe Econ 33 Slutsky Decomposition / 59 Eliminating Income Effect 1 1 Tea x T (cups) Cheese x C (slices) Watanabe Econ 33 Slutsky Decomposition 17 / 59 Eliminating Income Effect 1 1 Tea x T (cups) Cheese x C (slices) Watanabe Econ 33 Slutsky Decomposition 1 / 59

7 Eliminating Income Effect We ll take away m m T dollars from m. Liz: I don t feel any change in my purchasing power. With income adjustment, I still use up my income to purchase O regardless of the price. Or equivalently, change (p O C, p T, m) (p F C, p T, m T ) reserves Liz s purchasing power since she can afford O in both environments. Setting income at m T suppresses the income effect and isolates the substitution effect. Watanabe Econ 33 Slutsky Decomposition 19 / 59 Eliminating Income Effect We solve two problems and compare the result: 1 UMP O : max ( C, T ) s.t. p O C C + p T T = m φ C (p O with the solution O C =, p T, m) φ T (p O C, p. T, m) Transition: UMP T 3 UMP F : max ( C, T ) s.t. p F C C + p T T = m φ C (p F with the solution F C =, p T, m) φ T (p F C, p. T, m) Watanabe Econ 33 Slutsky Decomposition / 59 Eliminating Income Effect Problem.1 (UMP O ) max ( C, T) s.t. p O C + pt T = m C O = (φ C (p O C, pt, m), φt (p O, pt, m)). C pure substitution effect Δ S Problem. (UMP T ) max ( C, T) s.t. p F C C + pt T = mt. a T = (φ C (p F C, pt, mt ), φ T (p F C, pt, mt )), a m T := p F C O C + pt O T. pure income effect Δ Problem.3 (UMP F ) max ( C, T) s.t. p F C + pt T = m C F = (φ C (p F C, pt, m), φt (p F, pt, m)) C Watanabe Econ 33 Slutsky Decomposition 1 / 59

8 Slutsky Decomposition Substitution effect Δ S is given by Δ S = φ(p F C, p T, m T ) φ(p O C, p T, m). Income effect Δ is given by Δ = φ(p F C, p T, m) φ(p F C, p T, m T ). Total effect Δ is given by Δ = φ(p F C, p T, m) φ(p O C, p T, m) = Δ + Δ S. Watanabe Econ 33 Slutsky Decomposition / 59 Slutsky Decomposition Definition. (Slutsky Decomposition) Slutsky decomposition splits the total effect into two parts: Δ = Δ + Δ S. Watanabe Econ 33 Slutsky Decomposition 3 / 59 Example 1: Cobb-Douglas Example.5 (Cobb-Douglas Utility Function) Suppose Liz consumes cheesecakes C and tea T. Initial price of p O C = was slashed in half to pf C = 1. p T = 1 and m = throughout. Her preference is represented by ( C, T ) = C T, whose MRS at ( C, T ) is T C. What are Δ S and Δ? Watanabe Econ 33 Slutsky Decomposition / 59

9 Example 1: Cobb-Douglas Steps: 1 UMP O : Find O. Find m T := p F C O C + p T T. 3 UMP T : Find T. UMP F : Find F. 5 Compute Δ := Δ + Δ S. Watanabe Econ 33 Slutsky Decomposition 5 / 59 Example 1: Cobb-Douglas 1 UMP O : max ( C, T ) = C T s.t. p O C C + T = m. MRS at ( C, T ) is T C. O = (, ). Watanabe Econ 33 Slutsky Decomposition / 59 Example 1: Cobb-Douglas Tea x T (cups) Cheese x C (slices) Indifference Curves Watanabe Econ 33 Slutsky Decomposition 7 / 59

10 Example 1: Cobb-Douglas Find m T. O = (, ). m T := p F C + = 1. 3 UMP T : max ( C, T ) = C T s.t. p F C C + T = m T. T = (, ). Δ S = T O = =. Watanabe Econ 33 Slutsky Decomposition / 59 Example 1: Cobb-Douglas Tea x T (cups) Cheese x C (slices) Indifference Curves Watanabe Econ 33 Slutsky Decomposition 9 / 59 Example 1: Cobb-Douglas UMP F : max ( C, T ) = C T s.t. p F C C + T = m. F = (, ). Δ = F T = =. 5 Slutsky decomposition: Δ = Δ + Δ S = + =. Watanabe Econ 33 Slutsky Decomposition 3 / 59

11 Example 1: Cobb-Douglas Tea x T (cups) Cheese x C (slices) Indifference Curves Watanabe Econ 33 Slutsky Decomposition 31 / 59 Example 1: Cobb-Douglas Proposition. (Cobb-Douglas Utility Function and Cross-Price Effect) Cross-price effect is absent for Cobb-Douglas utility function. Proof. Income effect and substitution effects cancel each other out (see above). Watanabe Econ 33 Slutsky Decomposition 3 / 59 1 Introduction Decomposing Effects 3 Giffen Is Income-Inferior Fact Normal Good Income-Inferior Good Examples 5 Hicks Now We Know Watanabe Econ 33 Slutsky Decomposition 33 / 59

12 Fact Fact 3.1 (Sign of Substitution Effect) Substitution effect Δ S C against p C is positive if preferences are convex. If purchasing power remains the same, decrease in p C results in increase in C. 1 1 Note LOD says p C implies C. The fact above says p C implies C if the purchasing power remains the same. Watanabe Econ 33 Slutsky Decomposition 3 / 59 Normal Good For a normal good with convexity, Watanabe Econ 33 Slutsky Decomposition 35 / 59 Normal Good 1 1 Tea x T (cups) Cheese x C (slices) Watanabe Econ 33 Slutsky Decomposition 3 / 59

13 Normal Good Suppose p C. Slutsky decomposition: Δ C = Δ + Δ S. C C }{{}}{{} + by normality + by Fact 3.1 Conclude: p C Δ C. Conclude: a normal good satisfies LOD. Watanabe Econ 33 Slutsky Decomposition 37 / 59 Income-Inferior Good For an income-inferior good with convexity, Watanabe Econ 33 Slutsky Decomposition 3 / 59 Income-Inferior Good 1 1 Tea x T (cups) Cheese x C (slices) Watanabe Econ 33 Slutsky Decomposition 39 / 59

14 Income-Inferior Good Suppose p C. Slutsky decomposition: Δ C = Δ + Δ S. C C }{{}}{{} - by inferiority + by Fact 3.1 Can not conclude: p C Δ C. Can not conclude: an income-inferior good satisfies LOD. Watanabe Econ 33 Slutsky Decomposition / 59 Income-Inferior Good In case of extreme income-inferiority, income effect may be larger in magnitude than the substitution effect, causing C even with p C. Conclude: an income-inferior cheesecake may be Giffen. Recall the definition: Giffen good: p C C. Watanabe Econ 33 Slutsky Decomposition 1 / 59 Income-Inferior Good Conversely, if a cheesecake is Giffen, Δ C Δ }{{} C = Δ + Δ S. C C }{{}}{{}? + by Fact 3.1 has to be negative. Conclude: Giffen cheesecake has to be income-inferior. Watanabe Econ 33 Slutsky Decomposition / 59

15 Income-Inferior Good Proposition 3. (Giffen Is Income Inferior) m-wise C p C -wise normal LOD income-inferior Giffen Watanabe Econ 33 Slutsky Decomposition 3 / 59 Income-Inferior Good Proof. See above. Remark There is no such thing as a normal Giffen cheesecake. Watanabe Econ 33 Slutsky Decomposition / 59 1 Introduction Decomposing Effects 3 Giffen Is Income-Inferior Examples Example : Perfect Complements Example 3: Perfect Substitutes Example : Quasi-Linear Preferences 5 Hicks Now We Know Watanabe Econ 33 Slutsky Decomposition 5 / 59

16 Example : Perfect Complements Right Shoes x R Indifference Curves Left Shoes x L Watanabe Econ 33 Slutsky Decomposition / 59 Example : Perfect Complements Rotation does not change : Δ S = T O =. Δ = Δ +. Watanabe Econ 33 Slutsky Decomposition 7 / 59 Example 3: Perfect Substitutes 1 1 Indifference Curves Bottles x Six Packs x Watanabe Econ 33 Slutsky Decomposition / 59

17 Example 3: Perfect Substitutes T is already F. Δ = + Δ S (all the change is due to substitution effect). Watanabe Econ 33 Slutsky Decomposition 9 / 59 Example : Quasi-Linear Preferences For quasi-linear utility function ƒ ( C, T ) = C + ( T ), composite good picks up all the income effect. Indifference curves are parallel to each other. So, if you parallel shift the budget line, C grows in direct proportion, whereas T stays the same.... why? Watanabe Econ 33 Slutsky Decomposition 5 / 59 Example : Quasi-Linear Preferences 1 Indifference Curves 15 Tea x T Composite Goods x C Watanabe Econ 33 Slutsky Decomposition 51 /

18 Example : Quasi-Linear Preferences MRS is constant at any given T : MRS(1, ) = MRS(3, ) = MRS( , ). i.e., one basket is worth the same amount of tea regardless of the number of baskets. Baskets Liz s income (less expenditure on tea) She won t get saturated by cash regardless of the amount of cash. Watanabe Econ 33 Slutsky Decomposition 5 / 59 Example : Quasi-Linear Preferences Parallel shifts cannot change T. She is willing to give up the same amount of tea (MRS cups) no matter how rich or poor she is. Therefore, income effect for tea is absent for quasi-linear preferences. Δ T = + Δ S T. Compare this, for example, to Cobb-Douglass. MRS grows smaller as cheesecake consumption. Watanabe Econ 33 Slutsky Decomposition 53 / 59 1 Introduction Decomposing Effects 3 Giffen Is Income-Inferior Examples 5 Hicks Now We Know Watanabe Econ 33 Slutsky Decomposition 5 / 59

19 An alternative way to compensate (adjust) income level for Slutsky decomposition. Slutsky: adjust m to guarantee that O is available under p F C. Hicks: adjust m in the way that guarantees ( O ) under p F C. When Δp C is small, the Slutsky substitution effect is almost the same as the Hicksian substitution effect. Comes in handy because it does not depend on the value of utility level. We ll use this in Chapter 1. Watanabe Econ 33 Slutsky Decomposition 55 / 59 Tea x T (cups) Cheese x C (slices) Indifference Curves (Hicks) (Slutsky) Watanabe Econ 33 Slutsky Decomposition 5 / 59 1 Introduction Decomposing Effects 3 Giffen Is Income-Inferior Examples 5 Hicks Now We Know Watanabe Econ 33 Slutsky Decomposition 57 / 59

20 Decomposing total change in quantity demanded. Property of Giffen goods. Alternative compensation. Watanabe Econ 33 Slutsky Decomposition 5 / 59 Index Cobb-Douglas Utility Function, 5 convex, 35 cross-price effect, 33 Δ, see total effect Δ, see income effect Δ S, see substitution effect Giffen good, income effect, 9 11, income-inferior good, 39 normal good, 3 perfect complements, 7 perfect substitutes, 9 purchasing power, 11 quasi-linear preferences, 53 Slutsky decomposition,, 55 substitution effect, 9 11, Hicksian, 55 total effect, 11 Watanabe Econ 33 Slutsky Decomposition 59 / 59