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3. Slutsky Equations Slutsky 方程式 Y Part 2C. Individual Demand Functions 3. Slutsky Equations Slutsky 方程式 Own-Price Effects A Slutsky Decomposition Cross-Price Effects Duality and the Demand Concepts 2014.11.20 1 Own-Price Effects Q: Wha t h appens t o purch ases of good x chhhange when px changes? x/px Differentiation of the FOCsF.O.Cs from utility maximization could be used. HHthiowever, this approachhi is cumb ersome and provides little economic insight. 2 The Identity b/w Marshallian & Hicksian Demands: * Since x = x(px, py, I) = hx(px, py, U) Replacing I by the EF, e(px, py, U), and U by gives x(px, py, e(px, py, )) = hx(px, py, ) Differenti ati on ab ove equati on w.r.t. px, we have xxehx x pepp x x xx p x U = constant I = hx = x x xx x pp I xxU = constant 3 x xx x pp I xxU = constant S.E. I.E. ( – ) ( ?) The S. E. is always negative h x 0 as long as MRS is diminishing. px The Law of Deman d hldholds x x as long as x is a normal good. 00 Ipx xx If x is a Giffen good , 00 hen x must be an inferior good. pIx E/px = hx = x A$1iA $1 increase in px raiditises necessary expenditures by x dollars. 4 Compensated Demand Elasticities The compensated demand function: hx(px, py, U) Compensated OwnOwn--PricePrice Elasticity of Demand dhx hhp e x xx hpx , x dp x px hx px Compensated CrossCross--PricePrice Elasticity of Demand dhx hh p e xxy hpxy, dp y phyx p x 5 OwnOwn--PricePrice Elasticity form of the Slutsky Equation x h x x x ppxx I x php xI p xxx x x pxxx p x IxI ee se x,,pxxxhIh pxx ,I px where s x Expenditure share on x. x I The Slutsky equation shows that the compensatdted an d uncompensa tdted pri ce elasticities will be similar if the share of income devoted to x is small . the income elasticity of x is small. 6 A Slutsky Decomposition Example: Cobb-Douglas utility function U(x,y) = x0.5y0.5 1 I 1 I The Marshallian Demands: x y 2 px 2 py 050.50 05.5 11II I The IUF: (,ppIxy ,) 0.5 0.5 22ppppxx 2 xy The EF: 0.5 0.5 ep(,xy p ,)2 p xy p The Hicksian Demands: e p0.5 e p0.5 y h x hx 0.5 y 0.5 pp ppyy xx 7 The Slutskyyp Decomposition: xI1 TE.. 2 0 ppxx2 0.5 0.5 hx 11ppyyII 1 SE..1.5 1.5 0.5 0.5 2 0 ppx 2224xxxyx pppp xI1111II I IE.. x 2 0 Ippp22xx 4 x 8 Numerical Example: CobbCobb--DouDouggylas utility function U(x,y) = x0.5y0.5 Let px = $1, py = $4, I = $8 1 I 1 I The Marshallian Demands: x 4 y 1 2 px 2 py 0.5 0.5 The IUF: ()412( ppIxy,,)4 1 2 The EF: ep(,xy p ,) I 8 The Hicksian Demand for x: p0.5 40.5 p0.5 10.5 y h x 21 hx 0.5 0.5 24 y 0.5 0.5 px 1 py 4 9 Suppose that px : $1 $4 18 18 The Marshallian Demands: x '1 y '1 24 24 0.5 0.5 The IUF: (,ppIxy ,)1 1 1 050.50 05.5 Thlihe real income: eepp'(,,')241216 xy The Hicksian Demand for x: 40.5 40.5 hx 22 h 22 40.5 y 40.5 The Slutsky Decomposition: TE..: x 1 4 3 SE..: hx 2 4 2 IE.. TE .. SE .. (3)(2) 1 10 Figure: The Slutsky Decomposition px : $1 $4 y TE..: x 1 4 3 SE..: hx 24 2 4 IE.. TE .. SE .. (3)(2) 1 I I = –2y 2 1 IC IC1 0 1 2 4 8 x I.E. S.E. 11 Figure: The Slutsky Decomposition px : $1 $4 px TE..: x 1 4 3 SE..: hx 24 2 4 IE.. TE .. SE .. (3)(2) 1 1 hx x 1 2 4 x I.E. S.E. 12 Crossross--PiPrice EffEfftects The identity b/w Marshallian & Hicksian Demands: x(px, py, e(px, py, )) = hx(px, py, ) Differenti ati on ab ove equati on w.r.t. py, we have xxehx x pepp x yyy p y U = constant I = hy = y x xx y ppyy I U = constant 13 CrossCross--PricePrice Elasticity form of the Slutsky Equation xxh x y ppyy I x ppph xI yyyx y pppyyx p xIxI ee se x,,phpyxIyxy , py where s y Expenditure share on y. y I 14 Definition: Gross Substitutes Two goods are (gross) substitutes if one good may replace the other in use. i.e., if x i 0 p j e.g, tea & coffee, butter & margarine Definition: Gross Complements Two goods are (gross) complements if they are used together. i.e., if x i 0 p j e.g., coffee & cream, fish & chips 15 FigureFigure:: Gross Substitutes When the price of y falls, the substitution effect may be so large y that the consumer purchases less x and more y. In this case , we call x and y gross substitutes. y1 y x/p > 0 0 U1 y U0 x1 x0 x 16 FigureFigure:: Gross Complements When the price of y falls, the substitution effect may be so small that y the consumer purchases more x and more y. In this case , we call x and y gross complements. y1 y0 x/py < 0 U1 U0 x x 0 1 x 17 Definition: Net Substitutes Two goods are net substitutes if hi x 0 or i 0 p j p j U constant Definition: Net Complements Two goods are net complements if h x i 0 or i 0 p j p j U constant Note: The concepts of net substitutes and complements fllfocuses solely on subs titu tion eff ect ss. 18 x xx y pp I yyU = constant S.E. I.E. ( + ) ( ?) h The S .E . is always positive x 0 if DMRS and n = 2. py If x is a normaldl good, IEI.E. < 0 . The combined effect is ambiguous. x 0 S.E. > |I.E.| Gross Substitutes py x SES.E. <|IE|< |I.E.| Gross Complements 0 py If x is an inferior good, both S . E. > 0 x and I.E. >0 Gross Substitutes 0 py 19 Case of Manyy( Goods (nn>> 2) The Generalized Slutsky Equation is: xx x ii x i ppj I jjU =constant When n > 2, hi/pj can be negative. i.e., xi and xj can be net complementscomplements. If the utility function is quasi-concave, then the crosscross--netnet--substitutionsubstitution effects are symmetricmmetric. i.e., h h i j pp Proof: ji 2 h e p e e p j hj i i ppppppj jiji i 20 Asyyymmetry of the Gross CrossCross--PricePrice Effects The gross definitions of substitutes and complements are not symmetric. It is possible for xi to be a substitute for xj and at the same time for xj to be a complement of xi. 21 DDlituality and the Deman d Concep ts “Dual” Problem UMP EMP Slutsky Equation xxh x x* p p I x* xp(, p ,) I xx hppU(, ,) xy xxh xxy x y* pyypI Roy ’s Iden tity Shephard’s Lemma x(,ppIxy ,) xppI(,xy ,) hpp xxy (, ,(, ppI xy ,)) hppUxxy(, ,) px hppUxxy(, ,) xppeppU (, xy ,(, xy ,)) e I px * * UppI(,xy ,) eeppU (,xy ,) * ee ((())ppxy,,( pp xy,,II)) * UppeppUU(,xy ,(, xy ,)) 22.
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