拔高尼科爾森系列課件ch6-demand relationships among goods

1、Chapter 6DEMAND RELATIONSHIPS AMONG GOODSContentsThe two-good caseSubstitutes and ComplementsNet Substitutes and ComplementsComposite CommoditiesHousehold Production ModelThe two-good caseThe Two-Good CaseThe types of relationships that can occur when there are only two goods are limitedBut this cas

2、e can be illustrated with two-dimensional graphsGross ComplementsQuantity of xQuantity of yx1x0y1y0U1U0When the price of y falls, the substitution effect may be so small that the consumer purchases more x and more y ( e effect is larger)In this case, we call x and y gross complementsx/py 0A Mathemat

3、ical TreatmentThe change in x caused by changes in py can be shown by a Slutsky-type equationsubstitutioneffect (+)Convex e effect(-) if x is normalcombined effect(ambiguous)Substitutes and ComplementsSubstitutes and ComplementsFor the case of many goods, we can generalize the Slutsky analysis for a

4、ny i or jthis implies that the change in the price of any good induces e and substitution effects that may change the quantity of every good demandedSubstitutes and ComplementsTwo goods are substitutes if one good may replace the other in useexamples: tea & coffee, butter & margarineTwo goods are co

5、mplements if they are used togetherexamples: coffee & cream, fish & chipsGross Substitutes and ComplementsThe concepts of gross substitutes and complements include both substitution and e effectstwo goods are gross substitutes if xi /pj 0two goods are gross complements ifxi /pj 0y/px = 0Net Substitu

6、tes and ComplementsNet Substitutes and ComplementsThe concepts of net substitutes and complements focuses solely on substitution effectstwo goods are net substitutes if two goods are net complements if Net Substitutes and ComplementsThis definition looks only at the shape of the indifference curveTh

7、is definition is unambiguous because the definitions are perfectly symmetric（prove： Shephards lemma and youngs theorem)Gross ComplementsQuantity of xQuantity of yx1x0y1y0U1U0Even though x and y are gross complements, they are net substitutesSince MRS is diminishing, the own-price substitution effect

8、 must be negative so the cross-price substitution effect must be positiveSubstitutability with Many GoodsOnce the utility-maximizing model is extended to may goods, a wide variety of demand patterns e possibleAccording to Hicks second law of demand, “most” goods must be substitutesSubstitutability w

9、ith Many GoodsTo prove this, we can start with the compensated demand functionxc(p1,pn,V) (homogeneous degree zero in all prices)Applying Eulers theorem yieldsSubstitutability with Many GoodsIn elasticity terms, we getSince the negativity of the substitution effect implies that eiic 0, it must be th

10、e case that“most” goods are net substitutesComposite CommoditiesComposite CommoditiesIn the most general case, an individual who consumes n goods will have demand functions that reflect n(n+1)/2 different substitution effectsIt is often convenient to group goods into larger aggregatesexamples: food,

11、 clothing, “all other goods”Composite Commodity TheoremSuppose that consumers choose among n goodsThe demand for x1 will depend on the prices of the other n-1 commoditiesIf all of these prices move together, it may make sense to lump them into a single composite commodity (y)Composite Commodity Theo

12、remLet p20pn0 represent the initial prices of these other commoditiesassume that they all vary together (so that the relative prices of x2xn do not change)Define the composite commodity y to be total expenditures on x2xn at the initial pricesy = p20 x2 + p30 x3 + pn0 xnComposite Commodity TheoremThe

13、 individuals budget constraint isI = p1x1 + p20 x2 + pn0 xn = p1x1 + yIf we assume that all of the prices p20pn0 change by the same factor (t 0) then the budget constraint esI = p1x1 + tp20 x2 + tpn0 xn = p1x1 + tychanges in p1 or t induce substitution effectsComposite Commodity TheoremAs long as p2

14、0pn0 move together, we can confine our examination of demand to choices between buying x1 and “everything else”The theorem makes no prediction about how choices of x2xn behaveonly focuses on total spending on x2xnComposite CommodityA composite commodity is a group of goods for which all prices move

15、togetherThese goods can be treated as a single commoditythe individual behaves as if he is choosing between other goods and spending on this entire composite groupHousehold Production ModelHousehold Production ModelAssume that there are three goods that a person might want to purchase in the market:

16、 x, y, and zthese goods provide no direct utilitythese goods can be combined by the individual to produce either of two home-produced goods: a1 or a2the technology of this household production can be represented by a production functionHousehold Production ModelThe individuals goal is to choose x,y,

17、 and z so as to maximize utility utility = U(a1,a2) subject to the production functionsa1 = f1(x,y,z)a2 = f2(x,y,z) and a financial budget constraintpxx + pyy + pzz = IHousehold Production ModelTwo important insights from this general model can be drawnbecause the production functions are measurable

18、, households can be treated as “multi-product” firmsbecause consuming more a1 requires more use of x, y, and z, this activity has an opportunity cost in terms of the amount of a2 that can be producedThe Linear Attributes ModelIn this model, it is the attributes of goods that provide utility to indiv

19、idualsEach good has a fixed set of attributesThe model assumes that the production equations for a1 and a2 have the forma1 = ax1x + ay1y + az1za2 = ax2x + ay2y + az2zThe Linear Attributes Modela1a20 xThe ray 0 x shows the combinations of a1 and a2available from successively larger amounts of good xy

20、The ray 0y shows the combinations ofa1 and a2 available from successivelylarger amounts of good yzThe ray 0z shows thecombinations of a1 anda2 available fromsuccessively largeramounts of good zThe Linear Attributes ModelIf the individual spends all of his or her e on good xx* = I/pxThat will yielda1

21、* = ax1x* = (ax1I)/pxa2* = ax2x* = (ax2I)/pxThe Linear Attributes Modela1a20 xyzx* is the combination of a1 and a2 that would beobtained if all e was spent on xx*y*y* is the combination of a1 and a2 thatwould be obtained if all e wasspent on yZ*z* is the combination ofa1 and a2 that would beobtained

22、 if all e wasspent on zThe Linear Attributes Modela1a20 xyzx*y*z*All possible combinations from mixing thethree market goods are represented bythe shaded triangular area x*y*z*A utility-maximizing individual would neverconsume positive quantities of all threegoodsThe Linear Attributes Modela1a20 xyz

23、Individuals with a preference towarda1 will have indifference curves similarto U0 and will consume only y and zU0Individuals with a preferencetoward a0 will haveindifference curves similarto U1 and will consume onlyx and yU1The Linear Attributes ModelThe model predicts that corner solutions (where i

24、ndividuals consume zero amounts of some commodities) will be relatively commonespecially in cases where individuals attach value to fewer attributes than there are market goods to choose fromConsumption patterns may change abruptly if e, prices, or preferences changeContentsThe two-good caseSubstitu

25、tes and ComplementsNet Substitutes and ComplementsComposite CommoditiesHousehold Production ModelImportant Points to Note:When there are only two goods, the e and substitution effects from the change in the price of one good (py) on the demand for another good (x) usually work in opposite directionsthe sign of x/py is ambiguousthe substitution effect is positivethe e effect is negativeImportant Points to Note:In cases of more than two goods, demand relationships can be specified in two waystwo goods are gross substitutes if xi /pj 0 and gross complements if xi /pj 0 and net complements