Advanced Microeconomics Preliminary Remarks Ronald Wendner Department of Economics University of Graz, Austria Course # 320.911
Todays program Organization of course Important principles of notation we gonna use Preference and choice R. Wendner (U Graz, Austria) Microeconomics 2 / 15
Organization Organization of our course Essential prerequisites 320.901 Mathematics 320.902 Game Theory Familiarity with the basic concepts in microeconomics as offered by undergraduate microeconomics textbooks such as Nicholson, Microeconomic Theory R. Wendner (U Graz, Austria) Microeconomics 3 / 15
Organization Required reading Mas-Colell, A., M.D. Whinston, J.R. Green (1995), Microeconomic Theory, New York, Oxford: Oxford University Press My presentation + additional materials: My website Teaching Advanced Microeconomics my materials are no substitute R. Wendner (U Graz, Austria) Microeconomics 4 / 15
Organization Further references Sydsaeter, K., P. Hammond (2012, 4th ed.), Essential Mathematics for Economic Analysis, Harlow: Pearson Education Ltd. Corbae, D., M.B. Stinchcombe, J. Zeman (2009), An Introduction to Mathematical Analysis for Economic Theory and Econometrics, Princeton et al.: Princeton University Press. Sydsaeter, K., P. Hammond, A. Seierstad, A. Strom (2008, 2nd ed.), Further Mathematics for Economic Analysis, Harlow: Pearson Education Ltd. Novshek, W. (1993), Mathematics for Economists, San Diego et al.: Academic Press Inc. Velleman, D.J. (2006), How to Prove it, Cambridge et al.: Cambridge University Press Dixit, A. (1990), Optimization in Economic Theory, New York: Oxford University Press. put on reserve ( Semesterhandapparat ) in FB. R. Wendner (U Graz, Austria) Microeconomics 5 / 15
Organization Topics Notation Preference and choice Consumer choice Classical demand theory Aggregate demand Production Equilibrium and basic welfare properties R. Wendner (U Graz, Austria) Microeconomics 6 / 15
Organization Typical agenda & workload recap (last classes main results) lecture or problem sets required reading for following class R. Wendner (U Graz, Austria) Microeconomics 7 / 15
Organization Typical agenda & workload recap (last classes main results) lecture or problem sets required reading for following class workload for weeks with 1 class per week: 6 ECTS = 6 x 25 hours = 150h = 10h per week = 8h per week in addition to class R. Wendner (U Graz, Austria) Microeconomics 7 / 15
Organization Grading based on percentage grades in class participation: 15 % midterm exam, 24 April, 2018: 40% final exam, 26 June, 2018: 45% R. Wendner (U Graz, Austria) Microeconomics 8 / 15
Organization Grading based on percentage grades in class participation: 15 % midterm exam, 24 April, 2018: 40% final exam, 26 June, 2018: 45% letter grades: 86% - 100%: Sehr gut (A); 73% - 85%: Gut (B); 60% - 72%: Befriedigend (C); 50% - 59%: Genügend (D); 0% - 49%: Nicht genügend (F). R. Wendner (U Graz, Austria) Microeconomics 8 / 15
Vector- and Matrix Notation Mathematical ingredients in microeconomics sets functions, (binary) relations, correspondences scalars, vectors, matrices R. Wendner (U Graz, Austria) Microeconomics 9 / 15
Vector- and Matrix Notation Mathematical ingredients in microeconomics sets functions, (binary) relations, correspondences scalars, vectors, matrices frequently R N N i=1 R = R R... R (= N-fold Cartesian product) x R N = x 1 x 2. x N is a column (!) vector with transpose x T = (x 1, x 2,..., x N ) R. Wendner (U Graz, Austria) Microeconomics 9 / 15
Consider vectors x, y R N inner product x y x T y R. Wendner (U Graz, Austria) Microeconomics 10 / 15
Consider vectors x, y R N inner product x y x T y x 0 x i 0, i = 1,...N x R N + x = 0 x i = 0, i = 1,...N R. Wendner (U Graz, Austria) Microeconomics 10 / 15
Consider vectors x, y R N inner product x y x T y x 0 x i 0, i = 1,...N x R N + x = 0 x i = 0, i = 1,...N x 0 x i > 0, i = 1,...N x R N ++ R. Wendner (U Graz, Austria) Microeconomics 10 / 15
Consider vectors x, y R N inner product x y x T y x 0 x i 0, i = 1,...N x R N + x = 0 x i = 0, i = 1,...N x 0 x i > 0, i = 1,...N x R N ++ single- and vector valued functions single valued function f(x) : R N R, with x R N, N 1 R. Wendner (U Graz, Austria) Microeconomics 10 / 15
Consider vectors x, y R N inner product x y x T y x 0 x i 0, i = 1,...N x R N + x = 0 x i = 0, i = 1,...N x 0 x i > 0, i = 1,...N x R N ++ single- and vector valued functions single valued function f(x) : R N R, with x R N, N 1 vector valued function f(x) : R N R M, with x R N, N 1, M 1 vector of M functions, each of which is defined on the domain R N R. Wendner (U Graz, Austria) Microeconomics 10 / 15
Consider (single function) f(x) : R N R, with x R N, N 1 Gradient at x: f( x) R N (column vector) f( x) f( x)/ x 1 f( x)/ x 2. f( x)/ x N R. Wendner (U Graz, Austria) Microeconomics 11 / 15
Consider (single function) f(x) : R N R, with x R N, N 1 Gradient at x: f( x) R N (column vector) f( x) f( x)/ x 1 f( x)/ x 2. f( x)/ x N What is the gradient of a utility/production function? R. Wendner (U Graz, Austria) Microeconomics 11 / 15
Consider (the vector-valued function) f(x) : R N R M, with x R N, N 1, M 1, and f(x) = f 1 (x) f 2 (x). f M (x) R. Wendner (U Graz, Austria) Microeconomics 12 / 15
Consider (the vector-valued function) f(x) : R N R M, with x R N, N 1, M 1, and f(x) = f 1 (x) f 2 (x). f M (x) Jacobian matrix at x: Df( x) is M N matrix of FO partial derivatives Df( x) f 1 ( x)/ x 1, f 1 ( x)/ x 2,..., f 1 ( x)/ x N f 2 ( x)/ x 1, f 2 ( x)/ x 2,..., f 2 ( x)/ x N. f M ( x)/ x 1, f M ( x)/ x 2,..., f M ( x)/ x N R. Wendner (U Graz, Austria) Microeconomics 12 / 15
Consider (the vector-valued function) f(x) : R N R M, with x R N, N 1, M 1, and f(x) = f 1 (x) f 2 (x). f M (x) Jacobian matrix at x: Df( x) is M N matrix of FO partial derivatives Df( x) f 1 ( x)/ x 1, f 1 ( x)/ x 2,..., f 1 ( x)/ x N f 2 ( x)/ x 1, f 2 ( x)/ x 2,..., f 2 ( x)/ x N. f M ( x)/ x 1, f M ( x)/ x 2,..., f M ( x)/ x N for M = 1: Df(x) = [ f(x)] T R. Wendner (U Graz, Austria) Microeconomics 12 / 15
Restrictions on the Jacobian consider f(x, y) with x R K, y R L so that f(x, y) : R K+L R M with y R L being constant R. Wendner (U Graz, Austria) Microeconomics 13 / 15
Restrictions on the Jacobian consider f(x, y) with x R K, y R L so that f(x, y) : R K+L R M with y R L being constant D x f(x, y) is a Jacobian of dimension M K R. Wendner (U Graz, Austria) Microeconomics 13 / 15
Restrictions on the Jacobian consider f(x, y) with x R K, y R L so that f(x, y) : R K+L R M with y R L being constant D x f(x, y) is a Jacobian of dimension M K Consider single function g(x) : R N R Hessian matrix D 2 g(x) is the symmetric N N matrix R. Wendner (U Graz, Austria) Microeconomics 13 / 15
Restrictions on the Jacobian consider f(x, y) with x R K, y R L so that f(x, y) : R K+L R M with y R L being constant D x f(x, y) is a Jacobian of dimension M K Consider single function g(x) : R N R Hessian matrix D 2 g(x) is the symmetric N N matrix D 2 g(x) = D( g(x)) = g(x)/( x 1 x 1), g(x)/( x 1 x 2),..., g(x)/( x 1 x N ) g(x)/( x 2 x 1), g(x)/( x 2 x 2),..., g(x)/( x 2 x N ). g(x)/( x N x 1), g(x)/( x N x 2),..., g(x)/( x N x N ) R. Wendner (U Graz, Austria) Microeconomics 13 / 15
The beauty of matrix notation remember the chain rule for M=N=1: g(x) : R R, and f(.) : R R so that f(g(x)) : R R then: [f(g(x))]/ x = f (g(x)).g (x) R. Wendner (U Graz, Austria) Microeconomics 14 / 15
The beauty of matrix notation remember the chain rule for M=N=1: g(x) : R R, and f(.) : R R so that f(g(x)) : R R then: [f(g(x))]/ x = f (g(x)).g (x) or using our differential operator D: D xf(g(x)) = Df(x).Dg(x) R. Wendner (U Graz, Austria) Microeconomics 14 / 15
The beauty of matrix notation remember the chain rule for M=N=1: g(x) : R R, and f(.) : R R so that f(g(x)) : R R then: [f(g(x))]/ x = f (g(x)).g (x) or using our differential operator D: D xf(g(x)) = Df(x).Dg(x) lets go fully general... R. Wendner (U Graz, Austria) Microeconomics 14 / 15
Chain rule for M 1, N 1 consider f(g(x)), where f : R K R M, g : R N R K, x R N : R. Wendner (U Graz, Austria) Microeconomics 15 / 15
Chain rule for M 1, N 1 consider f(g(x)), where f : R K R M, g : R N R K, x R N : f(g(x)) = f 1 (g 1(x),..., g K(x)) f 2 (g 1(x),..., g K(x)). f M (g 1(x),..., g K(x)) what is D xf(g(x))? R. Wendner (U Graz, Austria) Microeconomics 15 / 15
Chain rule for M 1, N 1 consider f(g(x)), where f : R K R M, g : R N R K, x R N : f(g(x)) = f 1 (g 1(x),..., g K(x)) f 2 (g 1(x),..., g K(x)). f M (g 1(x),..., g K(x)) what is D xf(g(x))? D xf(g(x)) = Df(g(x)) Dg(x) dimensions: D xf(g(x)) [M N] = Df(g(x)) [M K] Dg(x) [K N] R. Wendner (U Graz, Austria) Microeconomics 15 / 15