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Advanced Microeconomics Comparative Statics and Duality Theory Advanced Microeconomics Comparative statics and duality theory Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 62 Part B. Household theory and theory of the …rm 1 The household optimum 2 Comparative statics and duality theory 3 Production theory 4 Cost minimization and pro…t maximization Harald Wiese (University of Leipzig) Advanced Microeconomics 2 / 62 Comparative statics and duality theory Overview 1 The duality approach 2 Shephard’slemma 3 The Hicksian law of demand 4 Slutsky equations 5 Compensating and equivalent variations Harald Wiese (University of Leipzig) Advanced Microeconomics 3 / 62 Maximization and minimization problem Maximization problem: Minimization problem: Find the bundle that Find the bundle that maximizes the utility for a minimizes the expenditure given budget line. needed to achieve a given utility level. x2 indifference curve with utility level U A C B budget line with income level m x1 Harald Wiese (University of Leipzig) Advanced Microeconomics 4 / 62 The expenditure function and the Hicksian demand function I Expenditure function: e : R` R R, ! (p, U¯ ) e (p, U¯ ) := min px 7! x with U (x ) U¯ The solution to the minimization problem is called the Hicksian demand function: χ : R` R R` , ! + (p, U¯ ) χ (p, U¯ ) := arg min px 7! x with U (x ) U¯ Harald Wiese (University of Leipzig) Advanced Microeconomics 5 / 62 The expenditure function and the Hicksian demand function II Problem Express e in terms of χ and V in terms of the household optima! Lemma For any α > 0: χ (αp, U¯ ) = χ (p, U¯ ) and e (αp, U¯ ) = αe (p, U¯ ) . Obvious?! Harald Wiese (University of Leipzig) Advanced Microeconomics 6 / 62 The expenditure function and the Hicksian demand function III Problem Determine the expenditure function and the Hicksian demand function for a 1 a the Cobb-Douglas utility function U (x1, x2) = x1 x2 with 0 < a < 1! Hint: We know that the indirect utility function V is given by: V (p, m) = U (x (p, m)) a 1 a m m = a (1 a) p1 p2 a 1 a a 1 a = m. p p 1 2 Harald Wiese (University of Leipzig) Advanced Microeconomics 7 / 62 Hicksian demand and the expenditure function utility expenditure maximization minimization objective function utility expenditure parameters prices p, income m prices p, utility U¯ notation for x (p, m) χ (p, U¯ ) best bundle(s) name of Marshallian Hicksian demand function value of V (p, m) e (p, U¯ ) objective function = U (x (p, m)) = p χ (p, U¯ ) Harald Wiese (University of Leipzig) Advanced Microeconomics 8 / 62 Applying the Lagrange method (recipe) ` ¯ L (x, µ) = ∑ pg xg + µ [U U (x)] g =1 with: U – strictly quasi-concave and strictly monotonic utility function; prices p >> 0; µ > 0 – Lagrange multiplier - translates a utility surplus (in case of U (x) > U¯ ) into expenditure reduction Increasing consumption has a positive direct e¤ect on expenditure, but a negative indirect e¤ect via a utility surplus (scope for expenditure reduction) and µ Harald Wiese (University of Leipzig) Advanced Microeconomics 9 / 62 Applying the Lagrange method (recipe) Di¤erentiate L with respect to xg : ∂L (x1, x2, ..., µ) ∂U (x1, x2, ..., x`) ! ∂U (x1, x2, ..., x`) ! pg = pg µ = 0 or = ∂xg ∂xg ∂xg µ and hence, for two goods g and k ∂U (x1,x2,...,x`) x p ∂ g =! g or MRS =! MOC U (x ,x ,...,x ) ∂ 1 2 ` pk ∂xk Note also: ∂L(x,µ) = U¯ U (x) =! 0 ∂µ Harald Wiese (University of Leipzig) Advanced Microeconomics 10 / 62 Applying the Lagrange method Comparing the Lagrange multipliers λ – the shadow price for utility maximization (translates additional ∂V income m into higher utility: λ = ∂m ); µ – the shadow price for expenditure minimization (translates ¯ additional utility U¯ into higher expenditure: = ∂e(p,U ) ). µ ∂U¯ We note without proof 1 µ = λ Harald Wiese (University of Leipzig) Advanced Microeconomics 11 / 62 The duality theorem here, duality does work x2 indifference curve with utility level U At the budget line through point C A (budget m), the household optimum C is at point B and B budget line with income level m the utility is U (B) = V (p, m) . The expenditure x1 needed to obtain B’sutility level or A’sutility level is equal to m : e (p, V (p, m)) = m. Harald Wiese (University of Leipzig) Advanced Microeconomics 12 / 62 The duality theorem here, duality does work x2 indifference curve with utility level U The minimal expenditure for the A indi¤erence curve passing through A C (utility level U¯ ) is B budget line with income level m denoted by e (p, U¯ ) and achieved by bundle B. x1 With income e (p, U¯ ) (at point C), the highest achievable utility is V (p, e (p, U¯ )) = U¯ . Harald Wiese (University of Leipzig) Advanced Microeconomics 13 / 62 The duality theorem Conditions for duality Theorem ` Let U : R+ R be a continuous utility function that obeys local nonsatiation! and let p >> 0 be a price vector. ) If x (p, m) is the household optimum for m > 0: χ (p, V (p, m)) = x (p, m) e (p, V (p, m)) = m. If χ (p, U¯ ) is the expenditure-minimizing bundle for U¯ > U (0): x (p, e (p, U¯ )) = χ (p, U¯ ) V (p, e (p, U¯ )) = U¯ . Harald Wiese (University of Leipzig) Advanced Microeconomics 14 / 62 The duality theorem here, duality does not work x2 At (p, m) , the household optimum is at the bliss point with V (p, m) = 9. A B 9 The expenditure 7 needed to obtain 6 this utility level is smaller than m : x1 e (p, V (p, m)) < m. Here, local nonsatiation is violated. Harald Wiese (University of Leipzig) Advanced Microeconomics 15 / 62 Main results Theorem Consider a household with a continuous utility function U, Hicksian demand function χ and expenditure function e. Shephard’slemma: The price increase of good g by one small unit increases the expenditure necessary to uphold the utility level by χg . Roy’sidentity: A price increase of good g by one small unit decreases the budget available for the other goods by χg and indirect utility by ∂V the product of the marginal utility of income ∂m and χg . Hicksian law of demand: If the price of a good g increases, the Hicksian demand χg does not increase. ∂χ (p,U¯ ) ¯ The Hicksian cross demands are symmetric: g = ∂χk (p,U ) . ∂pk ∂pg Slutsky equations see next slide Harald Wiese (University of Leipzig) Advanced Microeconomics 16 / 62 Main results Theorem Money-budget Slutsky equation: ∂xg ∂χg ∂xg = χg . ∂pg ∂pg ∂m Endowment Slutsky equation: endowment money ∂xg ∂χg ∂xg = + ωg χg . ∂pg ∂pg ∂m Harald Wiese (University of Leipzig) Advanced Microeconomics 17 / 62 Shephard’slemma Overview 1 The duality approach 2 Shephard’slemma 3 The Hicksian law of demand 4 Slutsky equations 5 Compensating and equivalent variations Harald Wiese (University of Leipzig) Advanced Microeconomics 18 / 62 Shephard’slemma Assume a price increase for a good g by one small unit. To achieve the same utility level, expenditure must be increased by at most ∂e χg ∂pg Shephard’sLemma (see manuscript): ∂e = χg ∂pg Harald Wiese (University of Leipzig) Advanced Microeconomics 19 / 62 Roy’sidentity Duality equation: U¯ = V (p, e (p, U¯ )) . Di¤erentiating with respect to pg : ∂V ∂V ∂e 0 = + ∂pg ∂m ∂pg ∂V ∂V = + χg (Shephard’slemma) ∂pg ∂m Roy’sidentity: ∂V = ∂V χ . ∂pg ∂m g Harald Wiese (University of Leipzig) Advanced Microeconomics 20 / 62 The Hicksian law of demand Overview 1 The duality approach 2 Shephard’slemma 3 The Hicksian law of demand 4 Slutsky equations 5 Compensating and equivalent variations Harald Wiese (University of Leipzig) Advanced Microeconomics 21 / 62 Compensated (Hicksian) law of demand Assume: ` p and p0 – price vectors from R ; ` ` χ (p, U¯ ) R+ and χ (p0, U¯ ) R+ – expenditure-minimizing bundles necessary2 to achieve a utility of2 at least U¯ . p0 χ U¯ , p0 p0 χ (U¯ , p) . expenditure- expenditure- minimizing| {z } minimizing| {z } bundle at p0 bundle at p ... (see the manuscript) If the price of one good increases, the Hicksian demand for that good ) cannot increase: ∂χg 0. ∂pg Harald Wiese (University of Leipzig) Advanced Microeconomics 22 / 62 Concavity and the Hesse matrix concavity De…nition Let f : M R be a function on a convex domain M R`. ! ) f is concave if f (kx + (1 k) y) kf (x) + (1 k) f (y) for all x, y M and for all k [0, 1] (for – convex). 2 2 f is strictly concave if f (kx + (1 k) y) > kf (x) + (1 k) f (y) holds for all x, y M with x = y and for all k (0, 1) (for < – strictly convex). 2 6 2 Harald Wiese (University of Leipzig) Advanced Microeconomics 23 / 62 Hesse matrix De…nition Let f : R` R be a function. ! The second-order partial derivative of f with respect to xi and xj (if it exists) is given by ∂ ∂f (x ) 2 ∂xi ∂ f (x) fij (x) := = . ∂xj ∂xi ∂xj If all the second-order partial derivatives exist, the Hesse matrix of f is given by f11 (x) f12 (x) f1` (x) f21 (x) f22 (x) Hf (x) = 0 1 . B f (x) f (x) f (x) C B `1 n2 `` C @ A Problem Determine the Hesse matrix for f (x, y) = x2y + y 2. Harald Wiese (University of Leipzig) Advanced Microeconomics 24 / 62 Hesse matrix Believe me (or check in the script) that Lemma (diagonal entries) If a function f : R` R is concave (strictly concave), the diagonal entries of its Hesse matrix! are non-positive (negative).
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