Chapter 6. Demand Relationships Among Goods

Transcription

1 Chapter 6 Demand Relatonshps Among Goods Up to ths pont, we have held the pre of other goods onstant. Now we onsder how hanges n p affet n a two-good world.

2 I p I p I p I p p p ( ) ( ) then I p then ( ) U 2 U ( ) U 2 U Gross Complements: p. For eample, butter and toast. p Gross Substtutes: p. For eample, Peps and Coke. p I p

3 Slutsk-tpe Equaton an be adapted to ross effets. For a Two Good World wth dmnshng MRS : If p hanges If p hanges Normal Gross substtutes f s nferor. Gross omplements (substtutes) f s normal and the rossnome effet outweghs (does not outwegh) the ross-substtuton effet. Inferor p p + or - + p p + U U Cross-substtuton effet alwas (+) n 2-good world. Not the ase n n-good world. I - I + - ; e ; e,p,p Elastt form e,p s Elastt form,p s e,i (+) for normal good. (-) for nferor good. Cross-nome effet s (-) for a normal good (the usual ase); (+) for an nferor good. It equals the hange n from a ver small hange n I tmes the quantt of onsumed. e e,i

4 It s also lear that the Gross Substtutes and Gross Complements ategores are the result of the rossnome effet as well as the ross-substtuton effet. The ombnaton of the two effets ma lead to wrong onluson about whether goods are substtutes or omplements (strtl speakng). For more than one good, the Cross-Slutsk Equaton would be: Can be < n n-good world. p p U I or e e s e,,,, I Changes n the pre of an good or hanges n nome ma affet the onsumpton of all goods.

5 Gross Effets for a Delne n p I p p p 2 p Gross omplements: p p or p Gross omp Gross sub I p U U U I 2 p Gross substtutes: p or p p The ndfferene urves represent two onsumers wth dfferent preferenes (utlt funtons) for and ; both have U n ommon, but U and U are dfferent. Remember, ndfferene urves for the same utlt funton annot ross.

6 Substtuton Effet Normal goods an be ether gross substtutes or gross omplements. Here we wll dsuss the dfferene between gross substtutes and omplements and net substtutes and omplements. Cross-Sub effet p Own-Sub effet Gross omplements U normal good Gross substtutes nferor good Own-substtuton effet s negatve, U U, p but ross-substtuton effet s postve n a 2- good world (not n n-good world), Cross-nome effet for normal good s negatve,. I p U U I.

7 The ross-substtuton effet wll be postve n a two-good world f the ndfferene urves are strtl onve (MRS dmnshes). p p then U and p U. As before, the own-substtuton effet s negatve, but the rosssubsttuton effet s postve n two-good world (but not n an n- good world). U

8 Gross Versus Net Substtutes and Net Complements Gross refers to the ombnaton of the nome and substtuton effets as a rteron for lassfaton. Most real world stuatons onl onsder these effets (Unompensated or Marshallan demand urves used n the defnton). Gross Subs: X p X Gross Comps: p

9 Net refers to the substtuton effet alone as a rteron for lassfaton. Ths lassfaton uses the Hksan (ompensated) demand urve n the defnton. Net Subs: p U Net Comps: U p In a two-good world wth dmnshng MRS, the two goods annot be net omplements. Not true n n-good world.

10 Hks Seond Law of Demand The ompensated demand funton for a partular good s: (p,..., p n, U) Ths ompensated demand funton s homogeneous of degree zero n pres (U onstant). If pres double, the pont of tangen wth the ndfferene urve does not hange beause the pre rato does not hange. Applng Euler s Theorem to ths funton: p p2... pn p p p Dvde ths equaton b to get: If MRS s dmnshng, e In other words, for good 2, e n and 2 e e 3 e e 2... e.... n. Good s a net substtute for all other goods n general. e n

11 Gross defntons are not smmetr. For eample, an be a gross substtute for 2 and at the same tme 2 an be a gross omplement for. Ths results from the nome effet. If one good s mportant n total onsumpton (hgh proporton of ependtures), t ma have a large nome effet, and f the other good s unmportant n total onsumpton (small proporton of ependtures), t ma have a small nome effet. Tpall, p p 2 2 And sgns ould be dfferent.

12 Net effets are not ambguous. The onsder onl the shape of the Indfferene urve. The effet of an nrease n p on onsumpton of 2 s the same as the effet of an nrease n p 2 on onsumpton of beause onl the pre rato hanges. Onl p p 2 hanges when a pre hanges. Net effets are alwas smmetr n magntude and sgn. p U p U

13 In the two-good ase, wth strtl onve ndfferene urves, the own-substtuton effet must be negatve, and thus, b Hks Seond Law of Demand the ross-substtuton effet must be postve. Almost all ndvdual goods turn out to be net substtutes n the real world f the are related at all. In addton, the nome effets are small for ndvdual goods, so most ndvdual goods appear to be gross substtutes. Fnall, smmetr of the ross-substtuton effets, and negatvt of the own-substtuton effets are two of the maor results of the theor of ndvdual hoe.

14 Composte Goods The number of goods n the real world s too large to handle analtall. Thus, we an ombne man related goods nto a omposte good and treat them as one good both n theoretal onstrutons and n real world analss. For eample, foods, meats, vegetables, dar produts, energ, transportaton, eduaton, housng, and possbl all other goods an be omposte goods! We an legtmatel assume a group of goods s one omposte good so long as all pres wthn the omposte move together proportonatel, e., all pres nrease or derease b the same perentage. If ths happens, the goods are eellent substtutes; therefore, taken together, the an be vewed as one good. Ths makes our two-dmensonal graph of ndfferene urves vald n man rumstanes.

15 Household Produton Model Indvduals do not reeve utlt dretl from man of the goods purhased. These goods are onl nputs nto a produton proess that results n utlt. Man nputs are nvolved n some ases. For eample: Raw steak, haroal, lghter flud, a grll, and labor are all nputs n produng utlt from an outdoor barbeque. Bat, boat, gasolne, transportaton to lake, old drnks, e, rod, reel, lne, labor are used to produe utlt from a fshng trp.

16 Therefore, the onsumer an be vewed as a frm that bus nputs and ombnes them to produe somethng that gves hm/her utlt. Defne produton funtons that desrbe ths proess as: G = f (X, X 2, X 3 ) fnal good G 2 = f 2 (X, X 2, X 3 ) fnal good 2 Ths produton ours subet to a budget onstrant I = P X + P 2 X 2 + P 3 X 3. You an vsualze the trade-off between produng G and G 2. For eample, what are the opportunt osts (shadow pres or mplt pres) of produng eah of the fnal goods?

17 Attrbutes Model Based upon the dea that onsumers do not reeve utlt dretl from goods, but rather from the attrbutes of those goods. For eample, do ou reeve utlt from: A new ar or from the transportaton, speed, ar ondtonng, red olor, mage or status, CD plaer that t provdes? Spnah or from the taste, teture, ron, gut fll, vtamns, fber, et., that t provdes? Assume two attrbutes (a and a 2 ) and three goods that provde dfferent amounts of eah attrbute: ax X ax X2 ax X3 2 ax X ax X2 ax X3 a 2 3 a 2 3 a = fnanal seurt, a 2 = urrent nome, X = savngs, X 2 = stoks, X 3 = real estate. These are produton funtons for gettng attrbutes from the three goods. The onsumer s obetve (utlt) s related onl to a, not to X eept ndretl. The onsumer wll hoose amounts of X, X 2, and X 3 that wll gve utlt mamzng amounts of a and a 2, subet to a budget onstrant.

18 Current nome (alores) Indfferene urves represent three dfferent ndvduals. a 2 * X a * 2 a * X Savngs (re) X Stoks (wheat) 2 * X 2 * X 3 Bundles that ould be purhased Fnanal seurt (vtamns) X 3Real estate (spnah) a The ndfferene urves are for a and a 2. X, X 2, X 3 are onl was to get a and a 2 and provde no other utlt. The ras show the amounts of a omng from addtonal unts of X. The budget onstrant lmts amounts of eah X that an be purhased. If all nome were spent on X, the onsumer ould get a * and a 2* b onsumng X * of X, and so forth. The lnes onnetng the ras reflet ombnatons of X that are possble. The ndfferene urves and resultng tangenes show the optmal ombnatons of the X s, wth the optmal amounts of eah attrbute on the aes. The loseness of the optmal pont to the X ra ndates the porton of the budget spent on X. Assumng the onsumer s mamzng utlt, he/she would never onsume all three goods at one. The onsumer annot onsume more goods than the number of attrbutes. For eample, n our two-attrbute ase, the onsumer an onsume at most two goods. In these models, orner solutons are lkel. ( eg., a a2) The opportunt ost of a n terms of a 2 opportunt ost s the mplt pre of a n terms of a 2. s smpl the slope of the onstrant lnes. Ths These mplt pres are alled hedon pres of a and a 2 beause a and a 2 are not real goods wth $ pres. If the onl dfferene between X and X 2 s the dfferene n amounts of a and a 2 that eah ontans, then the slope of the onstrant shows these hedon pres. Could atuall alulate the mplt pres for a and a 2 from the pres of X, X 2, and X 3 and the funtons that produe a and a 2 from X, X 2, and X 3.