Chapter 6Gross ComplementsGross SubstitutesA Mathematical TreatmentSubstitutes and ComplementsSlide 6Gross Substitutes and ComplementsAsymmetry of the Gross DefinitionsSlide 9Slide 10Slide 11Net Substitutes and ComplementsSlide 13Slide 14Composite CommoditiesComposite Commodity TheoremSlide 17Slide 18Slide 19Composite CommodityExample: Composite CommoditySlide 22Slide 23Slide 24Slide 25Slide 26Slide 27Household Production ModelSlide 29Slide 30Slide 31The Linear Attributes ModelSlide 33Slide 34Slide 35Slide 36Slide 37Slide 38Important Points to Note:Slide 40Slide 41Slide 42Chapter 6DEMAND RELATIONSHIPS AMONG GOODSCopyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.MICROECONOMIC THEORYBASIC PRINCIPLES AND EXTENSIONSEIGHTH EDITIONWALTER NICHOLSONGross ComplementsQuantity of Good XQuantity of Good YX1X0Y1Y0U1U0When the price of Y falls, the substitution effect may be so small that the consumer purchases more X and more YIn this case, we call X and Y gross complementsX/PY < 0Gross SubstitutesQuantity of Good XQuantity of Good YIn this case, we call X and Y gross substitutesX1X0Y1Y0U0When the price of Y falls, the substitution effect may be so large that the consumer purchases less X and more YU1X/PY > 0A Mathematical Treatment•The change in X caused by changes in PY can be shown by a Slutsky-type equationIXYPXPdPXUYYXY constantsubstitutioneffect (+)income effect(-) if X is normalcombined effect(ambiguous)Substitutes and Complements•For the case of many goods, we can generalize the Slutsky analysisIiUjijijiXXPXPdPXjconstant for any i or j–This implies that the change in the price of any good induces income and substitution effects that may change the quantities of every good demandedSubstitutes and Complements•Two goods are substitutes if one good may replace the other in use–examples: tea & coffee, butter & margarine•Two goods are complements if they are used together–examples: coffee & cream, fish & chipsGross Substitutes and Complements•The concepts of gross substitutes and complements include both substitution and income effects•Two goods are gross substitutes if Xi /Pj > 0•Two goods are gross complements ifXi /Pj < 0Asymmetry of the Gross Definitions•One undesirable characteristic of the gross definitions of substitutes and complements is that they are not symmetric•It is possible for X1 to be a substitute for X2 and at the same time for X2 to be a complement of X1Asymmetry of the Gross Definitions•Suppose that the utility function for two goods is given byU(X,Y) = ln X + Y•Setting up the LagrangianL = ln X + Y + (I – PXX – PYY)Asymmetry of the Gross Definitions gives us the following first-order conditions:L/X = 1/X - PX = 0L/Y = 1 - PY = 0L/ = I - PXX - PYY = 0•Manipulating the first two equations, we getPXX = PYAsymmetry of the Gross Definitions•Inserting this into the budget constraint, we can find the Marshallian demand for YPYY = I – PY–An increase in PY causes a decline in spending on Y–Since PX and I are unchanged, spending on X must rise ( X and Y are gross substitutes)–But spending on Y is independent of PX ( X and Y are independent of one another)Net Substitutes and Complements•The concepts of net substitutes and complements focuses on only substitution effects•Two goods are net substitutes if 0constantUjiPX• Two goods are net complements if0constantUjiPXNet Substitutes and Complements•This definition is intuitively appealing–it looks only at the shape of the indifference curve•This definition is also theoretically desirable–it is unambiguous because the definitions are perfectly symmetricconstantconstant UijUjiPXPXGross ComplementsQuantity of Good XQuantity of Good YX1X0Y1Y0U1U0Even though X and Y are gross complements, they are net substitutesSince MRS is diminishing, the own-price substitution effect must be negative so the cross-price substitution effect must be positiveComposite Commodities•In the most general case, an individual who consumes n goods will have demand functions that reflect n(n+1)/2 different substitution effects•It is often convenient to group goods into larger aggregates–examples: food, clothing, “all other goods”Composite Commodity Theorem•Suppose that consumers choose among n goods•The demand for X1 will depend on the prices of the other n-1 commodities•If all of these prices move together, it may make sense to lump them into a single composite commodityComposite Commodity Theorem•Let P20…Pn0 represent the initial prices of these other commodities–Assume that they all vary together (so that the relative prices of X2…Xn do not change)•Define the composite commodity to be total expenditures on X2…Xn at the initial pricesY = P20X2 + P30X3 +…+ Pn0XnComposite Commodity Theorem•The individual’s budget constraint isI = P1X1 + P20X2 +…+ Pn0Xn = P1X1 + Y•If we assume that all of the prices P20…Pn0 change by the same factor (t) then the budget constraint becomesI = P1X1 + tP20X2 +…+ tPn0Xn = P1X1 + tY–Changes in P1 or t induce substitution effectsComposite Commodity Theorem•As long as P20…Pn0 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 X20…Xn0 behave–only focuses on total spending on X20…Xn0Composite Commodity•A composite commodity is a group of goods for which all prices move together•These goods can be treated as a single commodity–The individual behaves as if he or she is choosing between other goods and spending on this entire composite groupExample: Composite Commodity•Suppose that an individual receives utility from three goods:–food (X)–housing services (Y), measured in hundreds of square feet–household operations (Z), measured by electricity use•Assume a CES utility functionExample: Composite Commodity•The Lagrangian technique can be used to derive demand functions ZYXZYXU111 ),,( utility ZXYXXPPPPPXIZYXYYPPPPPYIYZXZZPPPPPZIExample: Composite Commodity•If initially I = 100, PX = 1, PY = 4, and PZ = 1, then•X* = 25, Y* = 12.5, Z* = 25–$25 is spent on food and $75 is spent on housing-related needsExample: Composite Commodity•If we