1Demand and Comparative StaticsECON 370: Microeconomic TheorySummer 2004 – Rice UniversityStanley GilbertEcon 370 - Ordinal Utility 2Demand• Using the tools we have developed up to this point, we can now determine demand for an individual consumer• We seek demand functions– x1*= x1( p1, p2, m) and– x2*= x2( p1, p2, m)• From our previous example– Preferences of the form: u = xαy1 – α– And a budget constraint: pxx + pyy ≤ m– Result in demands: • x*= αm / px• y*= (1 – α)m / pyEcon 370 - Ordinal Utility 3Marshallian Demand• In general, we are interested in tracing out Marshallian Demand Curves.•A Marshallian Demand Curve describes how demand for a good changes:– As its own price changes, and– Holding all other prices and income constant• Functionally, that means graphing– x1*= x1( p1, p2, m)– Versus p1– And holding p2and m constantEcon 370 - Ordinal Utility 4Graphicallyx2x1PQDemand for Good 1Preferencesp13p13p12p12p11p112Econ 370 - Ordinal Utility 5Aggregating Demand• Ultimately, we want to look at market demand• That is, we want to see how the demands for many people add together to make up market demand• If we have n people– And let i = 1 … n• Two goods, x1, and x2 …Econ 370 - Ordinal Utility 6Aggregating Demand (cont)• Consumer i’s ordinary demand function for good j is– xji( p1, p2, mi)– (notice that we make allowance for different consumers having different income)• Then aggregate demand for good xjis:– Xj( p1, p2, m1, …, mn) = Σi=1nxji( p1, p2, mi)• If all consumers are identical, then– Xj( p1, p2, M) = n × xj( p1, p2, m)– Where M = n × mEcon 370 - Ordinal Utility 7Aggregating Demand Graphically• Graphically, we add demand curves holding price constant– That is, horizontallyPQ1PQ2PQmConsumer 1 Consumer 2 MarketEcon 370 - Ordinal Utility 8Aggregating Demand Graphically• Graphically, we add demand curves holding price constant– That is, horizontallyPQ1PQ2PQmConsumer 1 Consumer 2 Market3Econ 370 - Ordinal Utility 9The “Law” of Demand• What, in Economics is called the “Law” of Demand says– Demand curves “always” slope downward– That is, Demand for a good “always” decreases as the price for that good increases (holding everything else constant)• This statement is different from what is in your book—which I will discuss in due time—but it is the more usual statement of the law of demandEcon 370 - Ordinal Utility 10The “Law” of Demand (cont)• The “Law” of Demand is not a law in the sense that it is required by Economic theory• A good that does not obey the law of demand is theoretically possible and is called a Giffen Good– (A good that obeys the law of demand is called an Ordinary Good)• However, so far, no one has ever identified a Giffen good• So the “Law” of demand is a good description of empirical realityEcon 370 - Ordinal Utility 11Elasticity of Demand• One question we are interested in is how responsive demand is to changes is price• We use something called Elasticity to measure it• In general, Elasticity is the ratio of the percent change of (here) demand to percent change in price• That is:iiipx∆∆=%%εEcon 370 - Ordinal Utility 12Calculating Elasticity• There are two types of Elasticity:12122121iiiiiiiiiiiippxxxxpppxxp−−++=∆∆=ε– Arc Elasticity:iiiiiiiidpdxxppdpxdx==ε– Point Elasticity:• We will almost always use Point Elasticity4Econ 370 - Ordinal Utility 13Elasticity in General• There are many different Elasticities• Own-Price Elasticity measures how demand changes as its own price changes• Income Elasticity measures how demand changes as income changes• Cross-Price Elasticities measure how demand changes as the prices of other goods changeEcon 370 - Ordinal Utility 14Observations on Demand Elasticity• Own-Price Elasticity may be different at different points on the demand curve• Since demand generally decreases as price increases, we have ε < 0– But usually for convenience we work with | ε |.• We call demand Elastic if | ε | > 1• We call demand Inelastic if | ε | < 1• In general, – when demand is elastic, it responds strongly to changes in price– when demand is inelastic, it responds weakly to changes in priceEcon 370 - Ordinal Utility 15Example: Linear Demand Curve• Suppose inverse demand is linear– pi= a – bxi– xi= a/b – pi/ b• We calculate dxi/dpi= -1 / b• Own-price elasticity of demand isiiiidpdxxp=εiipap−−=()⎟⎠⎞⎜⎝⎛−−=bbpapii1Example: Points of Interestpi= a – bxipixiaa / biipap−−=εObserve thatpi= 0000=−−=aε⇒ε = 0pi= a−∞=−−=aaaε⇒ε = -∞122−=−−=aaaε⇒2api=ε = -1a / 2a / 2b5Example: Summarypixiaa/bε=−1ε=0a/2a/2bε=−∞own-price elasticown-price inelasticown-price unit elasticEcon 370 - Ordinal Utility 18Important Special Cases• What is the elasticity of a horizontal demand curve?• What is the elasticity of a vertical demand curveEcon 370 - Ordinal Utility 19Income Elasticity of Demand• We are also interested in demand changes as income changes• We use Income Elasticity to measure it, where– Income Elasticity for Good i is defined as:dmdxxmiiim=εEcon 370 - Ordinal Utility 20Income Effects Graphicallyx1x2m1m2x1mm1m2m3m3Engle CurvePreferences6Econ 370 - Ordinal Utility 21Graphical Income Elasticity (cont)x1x2m1m2m3Engle CurvePreferencesIncome OfferCurvex1mm1m2m3Econ 370 - Ordinal Utility 22Normal and Inferior Goods• For most goods demand increases as income increases– That is εim> 0– And the Engle curve is sloped forward– Such goods are called Normal Goods• However, demand for some goods decreases as income increases– That is εim< 0– And the Engle curve is sloped backward– Such goods are called Inferior Goods• Note that some goods may be both Normal and Inferior in different price rangesEcon 370 - Ordinal Utility 23Inferior to What?• What exactly is an inferior good, and what is it inferior to?• Example: the Yugo– Very cheap car: Sold for ~ $4,000 in the US in 1986– Poor quality and very small– Sold moderately well in the US– Still in production in Serbia– Who do you think bought the cars?– What do you think happened when their income increased?• Inferior to What?– To other goods availableEcon 370 - Ordinal Utility 24Normal, Inferior and Giffen Goods• A Normal Good cannot be a Giffen Good• -or- All Giffen Goods are Inferior goods• (Can you