1. relation between exp. function and IUF
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1 Dualty Dualty n consumer theory II. relaton between exp. functon and IUF - straghtforward: have m( p, u mn'd value of expendture requred to attan a gven level of utlty, gven a prce vector; u ( p, M max'd value of utlty attanable at a gven ncome, gven a prce vector. - exp functon s monotoncally ncreasng n u, and IUF s monotoncally ncreasng n M; therefore can nvert these functons - m(p,u(p,mm, & u(p,m(p,uu - snce both IUF and exp. f'n nhert propertes of utlty functon, can use them to descrbe preferences as well as utlty f'n. Page of 9
2 Dualty Example: CD utlty: ux (, x xα x β. gven { p, p, M }, Marshallan demands are x D(, M p M α, ( α+ β p x D ( p, M β M ( α+ β p. IUF: u ( αm α ( βm ( α+ β p ( α+ β p β 3. Ths s ugly: go back to orgnal utlty, α+ β and create vxx ( ( ux (, x ; defne γ α α + β - then get demands x D(, M p M γ, x D ( p, M ( γ M p p and IUF γm γ ( γ M ( γ u v( x, x ( ( p p γ γ ( γ γ p γ p γ M 4. gven { p, p, u }, Hcksan demands are p x H (, ( p u u ( γ γ p γ x H ( p, u u( p ( γ γ p γ Page of 9
3 Dualty p and m(p,u ( p up [ ( γ γ ( ( γ + p γ ] p γ p γ up ( ( γ p γ γ γ p 5. Now: set u u u( D( p, M, D ( p, M γ ( γ γ γ γ ( γ p p M Substtutng ths nto exp. fn above, and smplfyng, yelds m, u p M.. Relaton between Marshallan and Hcksan demand functons. - from above, Hcksan demand for good one p s x H (, ( p u u ( γ γ p γ - let u γ ( γ γ γ u ( p, M γ ( γ p p M (IUF for same prces, and gven ncome - sub'g n value for u, and smplfyng, gves x H (, (, M (, p u p M γ D p M p - Hcksan and Marshallan demand are the same (have the same value when utlty or M values are chosen approprately (also works when M m( p, u Page 3 of 9
4 Dualty What does ths gve us? - Slutsky equaton: ncome and substtuton effects of a prce change. - start wth dentty x H (, p u D(, p m(, p u (notce ths equalty refers to levels - dfferentate wrt prce of good, keepng utlty constant - so adustng M m( p, u as necessary- yelds H D D m + p p M p ( m( p, u - from Shephard's Lemma, p H x - substtutng ths nto (, and rearrangng, gves Slutsky equaton: D H D x p p M So what? H p gves substtuton effect, and last term gves ncome effect. Page 4 of 9
5 Dualty. suppose, so equaton s D H D x p p M LHS: slope of Marshallan demand curve for good. Is ths +ve or -ve? -frst term s slope of Hcksan demand curve, so s -ve: the substtuton effect; -nd term s the ncome effect: slope of Engel curve multpled by -quantty; ths s -ve f normal good, +ve f s nferor: the ncome effect. -demand curve for normal good has negatve slope; -demand curve for nferor good can have postve slope, f good s strongly nferor, so D / M s large; and/or sgnfcant part of expendtures (f x large. Page 5 of 9
6 Dualty. Slutsky equaton n elastcty form: a own prce: (multply through by p / x,clever use of multplyng by p D p H p M D x becomes x p x p x M M ε σ s η b cross-prce: D H D x p p M ε σ s η becomes Def'n of substtutes and complements? Notce H (, m u m(, u H p p p p p p p p (why? Young's theorem: order of dfferentaton doesn't matter So: f H / p >(<0, usng Hcksan demand we classfy goods as substtutes (complements no matter how we measure t; - same s not true for Marshallan demands: possble for goods and to be substtutes wrt change n one prce, complements wrt change n other prce. Page 6 of 9
7 Dualty - why do we care? Competton polcy frequently looks at cross-prce elastctes n defnng markets. 3. Slutsky matrx: nxn matrx of dervatves of Hcksan demand wrt prces. Ths matrx s symmetrc, negatve sem-defnte; can test theory. 4. Propertes of Marshallan demand curves: a Cournot aggregaton b Engel aggregaton Both are restrctons theory mposes on systems of demand functons; gve crossequaton restrctons whch can be used n emprcal testng. Both start wth budget constrant, px M, whch s equalty n equlbrum.. Take the total dervatve of the budget equaton (so dfferentate wrt M and all prces, and then set changesn all except M and one prce equal to zero Page 7 of 9
8 Dualty D D [ x + ] ( 0 p dp + p dm dm p M a Cournot aggregaton: effects of change n sngle prce must sum to zero (snce M s constant. - n expresson above, set dm0, dvde both sdes by dp : D x + p 0 p n elastcty form: + 0 s sε b Engel aggregaton: effects of change n expendture from a change n ncome must equal the change n ncome: Page 8 of 9
9 Dualty - n expresson for total dfferental, set prce changes equal to zero, dvde both sdes by dm n D p M or n sη Page 9 of 9
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