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Demand and Comparative Statics Demand Marshallian Demand Graphically Demand • Using the tools we have developed up to this point, we can now determine demand for an individual consumer Demand and Comparative Statics • We seek demand functions * – x1 = x1( p1, p2, m) and * – x2 = x2( p1, p2, m) ECON 370: Microeconomic Theory • From our previous example Summer 2004 – Rice University – Preferences of the form: u = xαy1 – α – And a budget constraint: pxx + pyy ≤ m Stanley Gilbert – Result in demands: * • x = αm / px * • y = (1 – α)m / py Econ 370 - Ordinal Utility 2 Marshallian Demand Graphically • In general, we are interested in tracing out x2 Marshallian Demand Curves. •A Marshallian Demand Curve describes how demand for a good changes: Preferences – As its own price changes, and 3 2 1 x1 – Holding all other prices and income constant p1 p1 p1 P p 3 • Functionally, that means graphing 1 * – x1 = x1( p1, p2, m) 2 p1 – Versus p1 1 p1 – And holding p2 and m constant Demand for 1 Good Q Econ 370 - Ordinal Utility 3 Econ 370 - Ordinal Utility 4 1 Aggregating Demand Aggregating Demand (cont) • Ultimately, we want to look at market demand • Consumer i’s ordinary demand function for good j is – x i( p , p , mi) • That is, we want to see how the demands for many j 1 2 – (notice that we make allowance for different consumers people add together to make up market demand having different income) • If we have n people • Then aggregate demand for good xj is: – And let i = 1 … n 1 n n i i – Xj( p1, p2, m , …, m ) = Σi=1 xj ( p1, p2, m ) • Two goods, x , and x 1 2 … • If all consumers are identical, then – Xj( p1, p2, M) = n × xj( p1, p2, m) – Where M = n × m Econ 370 - Ordinal Utility 5 Econ 370 - Ordinal Utility 6 Aggregating Demand Graphically Aggregating Demand Graphically • Graphically, we add demand curves holding price • Graphically, we add demand curves holding price constant constant – That is, horizontally – That is, horizontally Consumer 1 Consumer 2 Market Consumer 1 Consumer 2 Market P P P P P P Q1 Q2 Qm Q1 Q2 Qm Econ 370 - Ordinal Utility 7 Econ 370 - Ordinal Utility 8 2 The “Law” of Demand The “Law” of Demand (cont) • What, in Economics is called the “Law” of • The “Law” of Demand is not a law in the sense Demand says that it is required by Economic theory – Demand curves “always” slope downward • A good that does not obey the law of demand is – That is, Demand for a good “always” decreases as the theoretically possible and is called a Giffen Good price for that good increases (holding everything else constant) – (A good that obeys the law of demand is called an Ordinary Good) • This statement is different from what is in your book—which I will discuss in due time—but it is • However, so far, no one has ever identified a the more usual statement of the law of demand Giffen good • So the “Law” of demand is a good description of empirical reality Econ 370 - Ordinal Utility 9 Econ 370 - Ordinal Utility 10 Elasticity of Demand Calculating Elasticity • One question we are interested in is how responsive • There are two types of Elasticity: demand is to changes is price dx x p dx • We use something called Elasticity to measure it – Point Elasticity: ε = i i = i i dpi pi xi dpi • In general, Elasticity is the ratio of the percent change of 1 2 2 1 (here) demand to percent change in price pi ∆xi pi + pi xi − xi – Arc Elasticity: ε = = 1 2 2 1 xi ∆pi x + x p − p • That is: i i i i • We will almost always use Point Elasticity %∆xi εi = %∆pi Econ 370 - Ordinal Utility 11 Econ 370 - Ordinal Utility 12 3 Elasticity in General Observations on Demand Elasticity • There are many different Elasticities • Own-Price Elasticity may be different at different points on the demand curve • Own-Price Elasticity measures how demand • Since demand generally decreases as price increases, we changes as its own price changes have ε < 0 • Income Elasticity measures how demand changes – But usually for convenience we work with | ε |. as income changes • We call demand Elastic if | ε | > 1 • Cross-Price Elasticities measure how demand • We call demand Inelastic if | ε | < 1 changes as the prices of other goods change • In general, – when demand is elastic, it responds strongly to changes in price – when demand is inelastic, it responds weakly to changes in price Econ 370 - Ordinal Utility 13 Econ 370 - Ordinal Utility 14 Example: Linear Demand Curve Example: Points of Interest • Suppose inverse demand is linear pi pi = a – bxi ε = − a − pi – pi = a – bxi 0 – xi = a/b – pi / b Observe that p = 0 ⇒ ε = − = 0 pi i a − 0 • We calculate dxi /dpi= -1 / b ε = -∞ a a 2 a pi = ⇒ ε = − = −1 • Own-price elasticity of demand is 2 a − a 2 a ε = − = −∞ ε = -1 pi = a ⇒ pi dxi pi ⎛ 1⎞ pi a / 2 a − a ε = = ⎜− ⎟ = − xi dpi ()a − pi b ⎝ b⎠ a − pi ε = 0 a / 2b a / b xi Econ 370 - Ordinal Utility 15 4 Example: Summary Important Special Cases • What is the elasticity of a horizontal demand curve? pi ε =−∞ a own-price elastic a/2 ε =−1 own-price unit elastic • What is the elasticity of a vertical demand curve own-price inelastic ε = 0 a/2b a/b xi Econ 370 - Ordinal Utility 18 Income Elasticity of Demand Income Effects Graphically • We are also interested in demand changes as income changes Preferences Engle Curve x2 m • We use Income Elasticity to measure it, where m3 – Income Elasticity for Good i is defined as: m2 m3 m dx ε = i m2 im x dm i m1 m1 x1 x1 Econ 370 - Ordinal Utility 19 Econ 370 - Ordinal Utility 20 5 Graphical Income Elasticity (cont) Normal and Inferior Goods • For most goods demand increases as income Preferences Engle Curve increases x2 m – That is εim > 0 Income Offer – And the Engle curve is sloped forward Curve – Such goods are called Normal Goods m3 m • However, demand for some goods decreases as 2 income increases m1 – That is εim < 0 – And the Engle curve is sloped backward m 3 – Such goods are called Inferior Goods m m 1 2 x1 x1 • Note that some goods may be both Normal and Inferior in different price ranges Econ 370 - Ordinal Utility 21 Econ 370 - Ordinal Utility 22 Inferior to What? Normal, Inferior and Giffen Goods • What exactly is an inferior good, and what is it • A Normal Good cannot be a Giffen Good inferior to? • -or- All Giffen Goods are Inferior goods • Example: the Yugo – Very cheap car: Sold for ~ $4,000 in the US in 1986 • (Can you see why?) – Poor quality and very small • However, most Inferior Goods are not Giffen – Sold moderately well in the US Goods – 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 available Econ 370 - Ordinal Utility 23 Econ 370 - Ordinal Utility 24 6 Can all goods be inferior? Cross-Price Elasticity of Demand • All goods cannot be Inferior, that is • Finally, we are interested in how demand changes as other prices change • At least one good must be Normal • We use Cross-Price Elasticity to measure it, where •Why? x2 – Cross-Price Elasticity for Good i relative to the price of good j is defined as: p j dxi εij = xi dp j m2 m1 x1 • Are there any preferences that will result in consumption decreasing for both goods when income increases? Econ 370 - Ordinal Utility 25 Econ 370 - Ordinal Utility 26 Substitutes and Complements Examples • If consumers see two goods as close substitutes for • Perfect Complements/Substitutes each other, what sign will the cross-price elasticity • Homothetic Preferences of demand have? ε – εij > 0 •x = Ap – What makes this interesting? • If consumers see two goods as complements for each other, what sign will the cross-price elasticity of demand have? – εij < 0 Econ 370 - Ordinal Utility 27 Econ 370 - Ordinal Utility 28 7.
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