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Changing m Elasticity Examples Changing own price Changing other good’s price Analyzing demand Intermediate Micro Lecture 6 Chapter 6 of Varian Examples Changing m Elasticity Examples Changing own price Changing other good’s price Analyzing demand Can model utility function and decisions I I Even for p, m we don’t observe Can use demand functions to model comparative statics I I I I Comparative statics: Studying the effect on the equilibrium outcome due to a change in parameters. How does demand change as income increases? What are the effects on demand of a change in prices? Categorize goods based on comparative statics Examples Changing m Elasticity Examples Changing own price Changing other good’s price Review: Demand function Start with basic consumer decision problem: maxx1 ,x2 u(x1 , x2 ) s.t.p1 x1 + p2 x2 = m Leave p1 , p2 , m as parameters, and obtain demand functions x1 (p1 , p2 , m) x2 (p1 , p2 , m) Examples Changing m Elasticity Examples Changing own price Changing other good’s price Examples Changing m x1 (p1 , p2 , m) I Consider effect of ↑ m I Easiest measure: I Normal good: x1 is a normal good (at (p1 , p2 , m)) if I Inferior good: x1 is an inferior good (at (p1 , p2 , m)) dx1 dm (= d dm x1 (p1 , p2 , m)) dx1 dm > 0 1 if dx dm < 0 Changing m Elasticity Examples Changing own price Changing other good’s price Normal vs Inferior x1 is normal x2 is normal x1 is inferior x2 is normal Can both goods be inferior? Examples Changing m Elasticity Examples Changing own price Income offer curve I Income offer curve: A graph of all optimal bundles for a given p1 , p2 , for all values of m I m varies I p1 , p2 stay constant I Plug various values of m into demand functions, plot results I If both goods are normal, income offer curve is upward sloping (%) Changing other good’s price Examples Changing m Elasticity Examples Changing own price Income offer curve I Income offer curve: A graph of all optimal bundles for a given p1 , p2 , for all values of m I m varies I p1 , p2 stay constant I Plug various values of m into demand functions, plot results I If both goods are normal, income offer curve is upward sloping (%) Changing other good’s price Examples Changing m Elasticity Examples Changing own price Changing other good’s price Examples Engel curve I Engel curve: A graph of the demand for one of the goods, for all values of m, holding constant p1 , p2 I m varies I p1 , p2 stay constant I Plug various values of m into demand function, plot results I If the good is normal, Engel curve is upward sloping (%) I The Engel curve never slopes downward Changing m Elasticity Examples Changing own price Changing other good’s price Income offer curve axes: x1 , x2 Engel curves ← axes: xi , m ↑ Examples Changing m Elasticity Examples Changing own price Changing other good’s price Elasticity - Not in book! dxi dm ∗ m xi I Income elasticity of demand: xi ,m = I This formula is called point (income) elasticity Percent change in xi relative to the percent change in m I I I ∆x/x Non-calculus formula: ∆m/m Called arc (income) elasticity I Formula for (instantaneous) percent growth of y due to z: d dz ln(y (z)) I xi ,m = d ln(x1 (p1 ,p2 ,m)) dm d ln(m) dm Examples Changing m Elasticity Examples Changing own price Changing other good’s price Why use elasticity - Not in book! I dxi : dm change in xi due to increase in m dxi dm dx i I dm I I > 0: increasing in m < 0: decreasing in m Scale? I dxi ∗ m (income elasticity of demand): Same sign dm xi I Note that elasticity (and slope!) can vary with m as dxi dm Examples Changing m Elasticity Examples Changing own price Changing other good’s price Elasticity-based definitions- Not in book! I Unit elasticity: when xi ,m = 1 I xi grows at same rate as m I Any ray through the origin has unit elasticity Engel curve with unit elasticity Examples Changing m Elasticity Examples Changing own price Changing other good’s price Homothetic preferences I I Homothetic preferences: A set of preferences with the property that, if (x1 , x2 ) ∼ (y1 , y2 ), then (tx1 , tx2 ) ∼ (ty1 , ty2 ), ∀t ≥ 0 Equivalent properties: I I I Income offer curves are straight lines through the origin, for any (p1 , p2 ) Engel curves are straight lines through the origin, for any (p1 , p2 ) xi ,m for any (p1 , p2 , m), for any good i Examples Changing m Elasticity Examples Changing own price Changing other good’s price Elasticity-based definitions- Not in book! I Luxury good: xi for which xi ,m > 1 I xi grows at faster rate than m To identify on Engel curve I 1. Draw ray from origin to point 2. If curve crosses ray from left to right, good is luxury at this (p1 , p2 , m) Engel curve for Ikea furniture Examples Changing m Elasticity Examples Changing own price Changing other good’s price Elasticity-based definitions- Not in book! I Necessary good: xi for which xi ,m < 1 I xi grows at slower rate than m To identify on Engel curve I 1. Draw ray from origin to point 2. If curve crosses ray from right to left, good is ncessary at this (p1 , p2 , m) Engel curve for Ikea furniture Examples Changing m Elasticity Examples Changing own price Changing other good’s price Perfect substitutes u(x1 , x2 ) = 2x1 + 3x2 m = x1 + 2x2 x1 (1, 2, m) = m, x2 (1, 2, m) = 0 Income offer curve Engel curve for x1 Examples Changing m Elasticity Examples Changing own price Changing other good’s price Cobb Douglas u(x1 , x2 ) = x10.4 x20.6 m = 0.5x1 + 1.5x2 x1 (0.5, 1.5, m) = 0.8m, x2 (0.5, 1.5, m) = 0.4m Income offer curve Engel curve for x2 Examples Changing m Elasticity Examples Changing own price Changing other good’s price Examples Quasilinear u(x1 , x2 ) = ln(x1 ) + 0.25x2 m= x1 + x2 m if m < 4 0 if m < 4 x1 (1, 1, m) = , x2 (1, 1, m) = 4 if m ≥ 4 m − 4 if m ≥ 4 Income offer curve Engel curve for x1 Changing m Elasticity Examples Changing own price Changing p1 : effect on x1 x1 (p1 , p2 , m) I Consider effect of ↑ pi on xi I Derivative: I dxi dpi I dxi dpi < 0 for all known goods Giffin good: A good for which I No documented examples! dxi dpi >0 Changing other good’s price Examples Changing m Elasticity Examples Changing own price Changing other good’s price Own-price elasticity - Not in book! I I dxi Own-price elasticity of demand: xi ,pi = − dp ∗ i % ↓ in xi relative to % ↑ in p1 xi ,pi xi ,pi = 0 0 < xi ,pi < 1 xi ,pi = 1 1 < xi ,pi < ∞ xi ,pi = ∞ Description Perfectly inelastic Inelastic Unit elastic Elastic Perfectly elastic pi xi Interpretation xi does not change when pi does xi changes less than pi does xi changes by same % pi does xi changes more than pi does ↑ pi ⇒ xi = 0, ↓ pi ⇒ xi = ∞ Examples Changing m Elasticity Examples Changing own price Changing other good’s price Inverse demand x1 (p1 , p2 , m) I Take m, p2 as fixed I Rewrite demand function as x1 (p1 ) Can find inverse demand function: p1 (x1 ) I I I Gives p1 that causes x1 to be optimal Only exists if each value x1 optimal only for one p1 Examples Changing m Elasticity Examples Changing own price Changing other good’s price Implications of own-price elasticity - Not in book! I I Expenditure on good 1 = p1 x1 If xi ,pi > (=, <)1 I ↑ p1 causes ↓ ( no change, ↑) in p1 x1 xi ,pi xi ,pi = 0 0 < xi ,pi < 1 xi ,pi = 1 1 < xi ,pi < ∞ xi ,pi = ∞ Description Perfectly inelastic Inelastic Unit elastic Elastic Perfectly elastic Interpretation xi does not change when pi does xi changes less than pi does xi changes by same % pi does xi changes more than pi does ↑ pi ⇒ xi = 0, ↓ pi ⇒ xi = ∞ Examples Changing m Elasticity Examples Changing own price Changing other good’s price Changing p1 : effect on x2 - Not in book! x2 = I m p1 x1 − p2 p2 Suppose ↑ p1 I dx2 dp1 > (=, <)0 when xi ,pi > (=, <)1 I Complements: Two goods for which I Substitutes: Two goods for which dx2 dp1 dx2 dp1 <0 >0 Examples Changing m Elasticity Examples Changing own price Price offer curve I Price offer curve: A graph of all optimal bundles for a given m, p2 , for all values of p1 I p1 varies, m, p2 constant I Plug values of p1 into demand functions, plot I Complements: POC upward sloping (%) I Substitutes: POC downward sloping (&) Changing other good’s price Examples Changing m Elasticity Examples Changing own price Price offer curve I Price offer curve: A graph of all optimal bundles for a given m, p2 , for all values of p1 I p1 varies, m, p2 constant I Plug values of p1 into demand functions, plot I Complements: POC upward sloping (%) I Substitutes: POC downward sloping (&) Changing other good’s price Examples Changing m Elasticity Examples Changing own price Demand curve I I Demand curve: A graph of the demand for good i, for all values of pi , holding constant m, pnot i Non-Giffin goods: downward-sloping or flat Changing other good’s price Examples Changing m Elasticity Examples Changing own price Changing other good’s price Examples Perfect substitutes u(x1 , x2 ) = x1 + x2 10 =p1 x1 + x2 10 if p < 1 if p1 < 1 0 p1 1 10 − x1 if p1 = 1 , x (p , 1, 10) = x1 (p1 , 1, 10) = [0, 10] if p1 = 1 2 1 10 if p1 > 1 0 if p1 > 1 Price offer curve Demand curve for x1 Changing m Elasticity Examples Changing own price Changing other good’s price Cobb-Douglas u(x1 , x2 ) = x10.75 x20.25 40 = 2x1 + p2 x2 x1 (2, p2 , 40) = 15, x2 (2, p2 , 40) = Price offer curve 10 p2 Demand curve for x2 Examples Changing m Elasticity Examples Changing own price Changing other good’s price Quasilinear u(x1 , x2 ) = ln(x1 ) + x2 10 = p1 x1 + x2 x1 (p1 , 1, 10) = p11 , x2 (p1 , 1, 10) = 9 Price offer curve Demand curve for x1 Examples