Part 2C. 3. Slutsky Equations Slutsky Slutsky Own-Price Effects

Transcription

1 Part 2C. Individual Demand Functions 3. Slutsk Equations Slutsk 方程式 Own-Price Effects A Slutsk Decomposition Cross-Price Effects Dualit and the Demand Concepts

2 Own-Price Effects Q: What happens to purchases of good change when p changes? /p Differentiation of the FOCsfrom F.O.Cs utilit maimization could be used. However, this approach his cumbersome and provides little economic insight. 2

3 The Identit b/w Marshallian & Hicksian Demands: Since * = (p, p, I) = h (p, p, U) Replacing I b the EF, e(p, p, U), and U b gives (p, p, e(p, p, )) = h (p, p, ) Differentiation above equation w.r.t. p, we have I e h h p e p p = h = p p I U = constant p p U = constant 3

4 p p I U = constant S.E. ( ) I.E. (?) The S.E. is alwas negative h 0 as long as MRS is diminishing. p The Law of Demand holds as long as is a normal good. 0 0 I p If is a Giffen good, hen must be an inferior good. p 0 0 I E/p = h = A$1i increase in p raises necessar ependitures b dollars. 4

5 Compensated Demand Elasticities The compensated demand function: h (p, p, U) Compensated Own-Price Elasticit of Demand dh h h p e h, p dp p h p Compensated Cross-Price Elasticit of Demand e h, p dh h p h dp p h p 5

6 Own-Price Elasticit form of the Slutsk Equation h p p I p h p p I p p I I e e s e p s I, p h, p, I where Ependiture share on. The Slutsk equation shows that the compensated tdand uncompensated tdprice elasticities will be similar if the share of income devoted to is small. the income elasticit of is small. 6

7 A Slutsk Decomposition Eample: Cobb-Douglas utilit function U(,) = I 1 I The Marshallian Demands: p p The IUF: ( p, p, I) The EF: ep (, p, ) 2p p I 1 I I 2 p 2 p 2 p p The Hicksian Demands: 0.5 e e p h 0.5 p p h e e p p p

8 The Slutsk Decomposition: 1 I TE p 2 p h 1 p 1 p I 1 I SE.. 0 pp 2 p 2 p 2 p p 4 p I I IE I 2 p 2 p 4 p 8

9 Numerical Eample: Cobb-Douglas utilit function U(,) = Let p = $1, p = $4, I = $8 1 I The Marshallian Demands: 4 p The IUF: ( p, p, I ) The EF: ep (, p, ) I 8 The Hicksian Demand for : 1 I p 2 1 h p p 1 h p 1 p p 9

10 Suppose that p : $1 $4 1 8 The Marshallian Demands: ' The IUF: ( p, p, I) The real lincome: e e p p 18 ' ' (,, ') The Hicksian Demand for : h h The Slutsk Decomposition: TE..: 143 SE..: h 242 IE.. TE.. SE.. ( 3) ( 2) 1 10

11 Figure: The Slutsk Decomposition p : $1 $4 p I 4 2 TE..: 143 SE..: h IE.. TE.. SE.. ( 3) ( 2) 1 I = 2 1 IC 1 IC I.E. S.E. 11

12 Figure: The Slutsk Decomposition p : $1 $4 p p 4 TE..: 143 SE..: h IE.. TE.. SE.. ( 3) ( 2) I.E. S.E. h 12

13 Cross-Price Effects The identit b/w Marshallian & Hicksian Demands: (p, p, e(p, p, )) = h (p, p, ) Differentiation above equation w.r.t. p, we have I e h h p e p p = h = p p I U = constant p p U = constant 13

14 Cross-Price Elasticit form of the Slutsk Equation h p p I p h p p I p p I I e e s e, p h, p, I s where Ependiture share on. p I 14

15 Definition: Gross Substitutes Two goods are (gross) substitutes if one good ma replace the other in use. i.e., if i 0 pp j e.g, tea & coffee, butter & margarine Definition: Gross Complements Two goods are (gross) complements if the are used together. i.e., if i 0 p j e.g., coffee & cream, fish & chips 15

16 Figure: Gross Substitutes 1 When the price of falls, the substitution effect ma be so large that the consumer purchases less and more. In this case, we call and gross substitutes. 0 U 1 /p > 0 U

17 Figure: Gross Complements 1 When the price of falls, the substitution effect ma be so small that the consumer purchases more and more. In this case, we call and gross complements. 0 U 0 U1 /p <

18 Definition: Net Substitutes Two goods are net substitutes if h h p i j 0 i or 0 p j U constant Definition: Net Complements Two goods are net complements if h hi 0 i or 0 p p j j U constant Note: The concepts of net substitutes and complements focuses solel l on substitution effects. 18

19 p p I U = constant S.E. ( + ) I.E. (?) The S.E. is alwas positive h 0 if DMRS and n = 2. p If is a normal good, IE I.E. < 0. The combined effect is ambiguous. 0 S.E. > I.E. Gross Substitutes p SE S.E. < IE I.E. Gross Complements 0 p If is an inferior good, both S.E. > 0 and I.E. >0 Gross Substitutes 0 p 19

20 Case of Man Goods (n > 2) ) The Generalized Slutsk Equation is: j p p I i i i j j U =constant When n > 2, h i /p j can be negative. i.e., i and j can be net complements. If the utilit function is quasi-concave, then the cross-net-substitution effects are smmetric. i.e., h h i j p j p i Proof: 2 h e p e j h i i e p j pj pj pipj pi pi 20

21 Asmmetr of the Gross Cross-Price Effects The gross definitions of substitutes and complements are not smmetric. It is possible for i to be a substitute for j and at the same time for j to be a complement of i. 21

22 Dualit and the Demand Concepts UMP Dual Problem EMP Slutsk Equation * h * p p I ( p, p, I) h( p, p, U) h * p p I Ro s Identit Shephard s Lemma ( p, p, I) p I ( p, p, I) h ( p, p, ( p, p, I)) h ( p, p, U) ( p, p, e( p, p, U)) h( p, p, U) e p * U p p I (,, ) * e e ( p, p, ( p, p, I)) I * U ( p, p, e( p, p, U)) U * e e p p U (,, ) 22