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Microeconomics Comparative Statics Microeconomics Comparative statics Harald Wiese Leipzig University Harald Wiese (Leipzig University) Comparative statics 1 / 42 Structure Introduction Household theory Budget Preferences, indifference curves, and utility functions Household optimum Comparative statics Decisions on labor supply and saving Uncertainty Market demand and revenue Theory of the firm Perfect competition and welfare theory Types of markets External effects and public goods Pareto-optimal review Harald Wiese (Leipzig University) Comparative statics 2 / 42 Introduction Comparative statics, parameters and variables Parameters: describe the economic situation (input of economic models), e.g., preferences of a household Variables: are the result of economic models (after application of the equilibrium concept), e.g., profit-maximizing prices Comparative: Comparison of equilibria resulting from alternative parameter values Static: Adaption processes are not analyzed Harald Wiese (Leipzig University) Comparative statics 3 / 42 Introduction Comparative Statics in household theory The demand for good 1 for money budget: G G x1 = x1 (p1, p2, m), for endowment budget: A A x1 = x1 (p1, p2, w1, w2). How does demand for good 1 change due to a change in prices p1, p2, income m, endowment w1, w2. Harald Wiese (Leipzig University) Comparative statics 4 / 42 Comparative statics Introduction Impact of the own price price consumption curve and demand curve for money budget price consumption curve and demand curve for endowment budget price elasticity of demand Impact of the other good's price Impact of income Slutsky equations Harald Wiese (Leipzig University) Comparative statics 5 / 42 Impact of the own price ∗ ∗ Assign an optimum (x1 (p1) , x2 (p1)) 2 to every p1! 2 1 1 2 Price consumption curve: Locus of these household optima price consumption curve expressed as a function x = h (x )! 2 1 1 1 1 Demand curve: 1 1 1 1 1 ∗ Locus of (x1 (p1) , p1) 1 expressed as a function demand curve ∗ x = f (p1)! 1 1 1 1 1 1 1 Harald Wiese (Leipzig University) Comparative statics 6 / 42 Impact of the own price Cobb-Douglas utility function 1 2 3 3 U(x1, x2) = x1 x2 with the household optimum ∗ 1 m ∗ 2 m x1 = , x2 = 3 p1 3 p2 Demand curve for good 1: ∗ 1 m x1 = f (p1) = 3 p1 Price consumption curve: 2 m x2 = h (x1) = 3 p2 Problem What about perfect complements? Harald Wiese (Leipzig University) Comparative statics 7 / 42 Impact of the own price Perfect complements 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Harald Wiese (Leipzig University) Comparative statics 8 / 42 Impact of the own price Demand curves for money budget Definition (ordinary goods) ¶xG ¶xA 1 < 0 or 1 < 0 ¶p1 ¶p1 Definition (non-ordinary goods) ¶xG ¶xA 1 > 0 or 1 > 0 ¶p1 ¶p1 Giffen goods if the budget is a money budget Problem Sketch a price consumption curve for a Giffen good! Harald Wiese (Leipzig University) Comparative statics 9 / 42 Impact of the own price Price consumption curve and demand curve for endowment budget Initial situation: budget line 2 price increase for good 1 through B. Is good 1 ordinary for money budget? for endowment budget? endowment 1 Harald Wiese (Leipzig University) Comparative statics 10 / 42 Impact of the own price Price consumption curve and demand curve for endowment budget For a Cobb-Douglas utility function with parameter a, the demand for good 1 can be obtained as follows ∗ m x1 = a p1 p w + p w = 1 1 2 2 p1 p2 = w1 + w2 p1 Good 1 is ordinary for money budget and also for endowment budget. Harald Wiese (Leipzig University) Comparative statics 11 / 42 Impact of the own price Price consumption curve and demand curve for endowment budget ¶xG ¶xA 1 6= 1 ? ¶p1 ¶p1 Take the derivative of A G x1 (p1, p2, w1, w2) = x1 (p1, p2, p1w1 + p2w2). with respect to p1: A G G G G ¶x1 ¶x1 ¶x1 ¶ (p1w1 + p2w2) ¶x1 ¶x1 = + · = + w1. ¶p1 ¶p1 ¶m ¶p1 ¶p1 ¶m | {z } p1 " by one unit increases value of endowment G ¶x1 Endowment income effect: ¶m w1 Harald Wiese (Leipzig University) Comparative statics 12 / 42 Impact of the own price Price elasticity of demand Definition relative change of effect [%] elasticity = relative change of cause [%] Elasticities for demand cause: change in price of the same good price of the other good income effect: change in demand Harald Wiese (Leipzig University) Comparative statics 13 / 42 Impact of the own price Price elasticity of demand dx1 x1 ¶x1 p1 #x1,p1 = dp = , 1 ¶p1 x1 p1 ∗ 1 −1 for Cobb-Douglas utility function x1 = f (p1) = 3 mp1 1 −2 p1 #x1,p1 = (−1) mp1 3 x1 1 p = (−1) mp−2 1 3 1 1 −1 3 mp1 = −1. 1 1 and for U (x1, x2) = 3 ln x1 + 2 ln x2? Harald Wiese (Leipzig University) Comparative statics 14 / 42 Impact of the own price Price elasticity of demand 0 ordinary goods non-ordinary goods 1,1 1 1 For ordinary goods: #x1,p1 > −1 , j#x1,p1 j < 1 Definition Demand is called inelastic if j#x1,p1 j < 1 holds. Harald Wiese (Leipzig University) Comparative statics 15 / 42 Acquisitive crime j#x,pj < 1 ) expenditures increase with respect to price: d (px (p)) dx = x + p dp dp p dx = x 1 + x dp = x (1 + #x,p) = x (1 − j#x,pj) > 0. Drug addicts have inelastic demand. Higher taxes or criminalization increase prices and hence increase expenditures for drugs leading to a potential increase in acquisitive crime. Harald Wiese (Leipzig University) Comparative statics 16 / 42 Impact of the other good's price If the price of butter increases, many consumers buy margarine instead. Hence, ¶x margarine > 0 ¶pbutter Examples Substitutes: butter and margarine car and bike Complements: cinema and popcorn left shoe and right shoe Harald Wiese (Leipzig University) Comparative statics 17 / 42 Impact of the other good's price Problem Sketch a household diagram with x-axis: heroin y-axis: methadone The price of heroin increases and hence ... (K 69) Problem Sketch demand curves with x-axis: number of visits to cinema y-axis: price of visits to cinema Price of popcorn increases and hence ... (K 70) Price of theater increases and hence ... (K 71) Harald Wiese (Leipzig University) Comparative statics 18 / 42 Impact of the other good's price Cross price elasticity of demand is given by dx1 x1 ¶x1 p2 #x1,p2 = dp = 2 ¶p2 x1 p2 and for a Cobb-Douglas utility function we have x∗ = 1 m p0 and 1 3 p1 2 hence ¶x1 p2 p2 #x1,p2 = = 0 · = 0 ¶p2 x1 x1 0 complements substitutes 1,2 1 2 Harald Wiese (Leipzig University) Comparative statics 19 / 42 Impact of income 2 income consumption curve 1 1 1 1 Engel curve (normal good) 1 1 1 1 Harald Wiese (Leipzig University) Comparative statics 20 / 42 Impact of income Cobb-Douglas utility function 1 2 3 3 U(x1, x2) = x1 x2 with household optimum ∗ 1 m ∗ 2 m x1 = , x2 = 3 p1 3 p2 Engel curve for good 1: ∗ 1 m x1 = q (m) = 3 p1 Income consumption curve: ∗ ∗ 2 m 2 3p1x1 p1 ∗ ∗ x2 = = = 2 x1 = g (x1 ) 3 p2 3 p2 p2 Problem And for U (x1, x2) = min (x1, 2x2)? Harald Wiese (Leipzig University) Comparative statics 21 / 42 Impact of income Definition (normal goods) ¶x 1 > 0 ¶m Definition (inferior goods) ¶x 1 < 0 ¶m Problem Sketch the Engel curve for an inferior good! Problem (K 73) Harald Wiese (Leipzig University) Comparative statics 22 / 42 Impact of income Income elasticity is given by dx1 x ¶x1 m # = 1 = x1,m dm m x m ¶ 1 and for a Cobb-Douglas utility function x∗ = 1 m and hence 1 3p1 ¶x1 m 1 m #x1,m = = 1 = 1 ¶m x1 3p1 m 3p1 inferior goods normal goods 0 1 necessary luxury 1, goods goods Harald Wiese (Leipzig University) Comparative statics 23 / 42 Impact of income Monotonic preferences ) all income is spend on consumption of goods. m single good ) x (p, m) = p , ¶x m 1 m #x,m = = m = 1 ¶m x p p p1x1 p2x2 several goods ) s1 := m and s2 := m : s1#x1,m + s2#x2,m = 1. (proof in the book) Harald Wiese (Leipzig University) Comparative statics 24 / 42 Impact of income Good 1 is called normal if: ¶xA ¶xA ¶xG 1 > 0, 1 > 0 oder 1 > 0. ¶w1 ¶w2 ¶m Normality is equivalent for money budget and endowment budget. Harald Wiese (Leipzig University) Comparative statics 25 / 42 Slutsky equations An intuitive explanation for the three effects 1 Substitution effect or opportunity-cost effect: p1 " ) p1/p2 " ) x1 # and x2 " 2 Consumption income effect (monetary e.): p1 " ) overall consumption possibilities decrease ) x1 # if 1 is a normal good 3 Endowment income effect: p1 " ) value of endowment increases ) x1 " if 1 is a normal good Harald Wiese (Leipzig University) Comparative statics 26 / 42 Slutsky equation Substitution effect 2 1 substitution effect absolute substitution effect in the figure S S relative substitution effect Dx1 or dx1 Dp1 dp1 Harald Wiese (Leipzig University) Comparative statics 27 / 42 Slutsky equation Absolute substitution effect for perfect complements 2 1 Harald Wiese (Leipzig University) Comparative statics 28 / 42 Slutsky equation Relative substitution effect is never positive Starting point 2 old household optimum X and p1 # Impossible: W optimal or S Dx1 > 0 Dp1 because of W affordable before 1 for strict monotonicity X % V W Harald Wiese (Leipzig University) Comparative statics 29 / 42 Slutsky equation Income effects Endowment income effect: G G ¶x1 ¶p (p1w1 + p2w2) ¶x1 · = w1 ¶m ¶p1 ¶m | {z } p1 " by one unit increases value of endowment Consumption income effect: G ¶x1 (−x1) ¶m | {z } p1 " by one unit increases expenditures and thereby decreases disposable income Harald Wiese (Leipzig University) Comparative statics 30 / 42 Slutsky equation for money budget G S G ¶x1 ¶x1 ¶x1 B = − x1 .
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