ECON 186 Class Notes: Optimization Part 2
|
|
- Reynard Curtis
- 5 years ago
- Views:
Transcription
1 ECON 186 Class Notes: Optimization Part 2 Jijian Fan Jijian Fan ECON / 26
2 Hessians The Hessian matrix is a matrix of all partial derivatives of a function. Given the function f (x 1,x 2,...,x n ), where all partial derivatives exist and are continuous, the Hessian of f is 2 f 2 f x 2 x 1 1 x f x 1 x n 2 f 2 f x H(f ) = 2 x f x2 2 x 2 x n f 2 f x n x 1 x n x f xn 2 Given the quadratic form d 2 z = f xx dx 2 + 2f xy dxdy + f yy dy 2, the Hessian determinant (sometimes called the Hessian) is H = f xx f xy f yx f yy Jijian Fan (UC Santa Cruz) ECON / 26
3 Examples Is q = 5u 2 + 3uv + 2v 2 either positive or negative definite? The discriminant of q is , with first leading principal minor 5 > 0 and second leading principal minor = = 7.75 > 0 So, q is positive definite. Given f xx = 2, f xy = 1, and f yy = 1 at a certain point on a function z = f (x,y), does d 2 z have a definite sign at that point? The discriminant of the quadratic form d 2 z is , which has leading principal minors 2 < 0 and = 1 > 0, so d 2 z is negative definite, which means the point in question is a local maximum. Jijian Fan (UC Santa Cruz) ECON / 26
4 Three-variable Quadratic Forms Similar conditions can analogously be obtained for a function of three or more variables. Consider a quadratic form q with three variables u 1, u 2, and u 3. Then: q(u 1,u 2,u 3 ) = d 11 (u 2 1) + d 12 (u 1 u 2 ) + d 13 (u 1 u 3 ) + d 21 (u 2 u 1 ) + d 22 (u 2 2) +d 23 (u 2 u 3 ) + d 31 (u 3 u 1 ) + d 32 (u 3 u 2 ) + d 33 (u 2 3) = = [ u 1 u 2 ] u 3 d 11 d 12 d 13 d 21 d 22 d 23 d 31 d 32 d 33 u 1 u 2 u i=1 j=1 u Du d ij u i u j Jijian Fan (UC Santa Cruz) ECON / 26
5 Three-variable Quadratic Forms Now, there are three leading principal minors: D 1 d 11 D 2 d 11 d 12 d 21 d 22 D 3 d 11 d 12 d 13 d 21 d 22 d 23 d 31 d 32 d 33 The sufficient condition for positive definiteness (local minimum) is that D 1 > 0, D 2 > 0, and D 3 > 0. The sufficient condition for negative definiteness (local maximum) is that D 1 < 0, D 2 > 0, and D 3 < 0. Jijian Fan (UC Santa Cruz) ECON / 26
6 Examples Find and classify the critical points of the function f (x,y,z) = x 2 + y 2 + 7z 2 + xy + 3yz. fx = 2x + y, f y = 2y + x + 3z, f z = 14z + 3y. It is easy to see that the only critical point is (0,0,0). f xx = 2, f yy = 2, f zz = 14, f xy = f yx = 1, f xz = f zx = 0, f yz = f zy = 3. We then compute the Hessian: f xx f yx f zx f yx f yy f yz f zx f zy f zz = D1 = 2 > 0, D 2 = = 4 1 = 3 > 0, D = = = 2(28 9) 14 = 24 > 0 So, since the Hessian is positive definite, the only critical point (0,0,0) is a local minimum. Jijian Fan (UC Santa Cruz) ECON / 26
7 Examples Find the extreme values of f (x 1,x 2,x 3 ) = z = x x 1x 3 + 2x 2 x 2 2 3x 2 3 f1 = 3x x 3, f 2 = 2 2x 2, f 3 = 3x 1 6x 3 So, we have a system of three equations: 3x x 3 = 0 x 3 = x 2 1 x 3 = ( 1 2 ) 2 = x 2 = 0 x 2 = 1 3x 1 6x 3 = 0 x 1 = 2x 3 x 1 = 2x 2 1 x 1 = 1 2 Additionally, since x 1 2x 3 = 0, (0,1,0) must also be a solution, so the two roots are (0,1,0) and ( 1 2,1, 1 4). Jijian Fan (UC Santa Cruz) ECON / 26
8 Examples f 11 = 6x 1, f 22 = 2, f 33 = 6, f 12 = f 21 = 0, f 13 = f 31 = 3, f 23 = f 32 = 0 So, the Hessian is 6x D 1 (0,1,0) = 0, so we already know that the point (0,1,0) is indefinite, and in fact is not an extremum at all. D 1 ( 1 2,1, 1 4 ) = 3 < 0, D 2 ( 1 2,1, 1 4 ) = = 6 > 0, D 3 ( 1 2,1, 1 4 ) = = = = 18 < 0 The Hessian is negative definite, so the point ( 1 2,1, 1 4) is a maximum. Jijian Fan (UC Santa Cruz) ECON / 26
9 Profit Maximization Example Consider a competitive firm with the following profit function: π = R C = PQ wl rk where P is price, Q is output, L is labor, K is capital, w is wage, r is the rental rate of capital. Since the firm is in a competitive market, P, w, and r are exogenous, while L, K, and Q are endogenous. However, Q is also function of K and L via the Cobb-Douglas production function Q = Q(K,L) = L α K β Assume that there are decreasing returns to scale where α = β < 1 2. Substituting in, the objective function becomes π(k,l) = PL α K α wl rk Jijian Fan (UC Santa Cruz) ECON / 26
10 Profit Maximization Example First order conditions: π ( w 1 L = PαLα 1 K α w = 0 K = Pα L1 α) α π K = PαLα K α 1 r = 0 Jijian Fan (UC Santa Cruz) ECON / 26
11 Profit Maximization Example Before we continue, let s make sure that these equations for L and K do actually give us a maximum. H = π LL π LK = Pα(α 1)L α 2 K α Pα 2 L α 1 K α 1 Pα 2 L α 1 K α 1 Pα(α 1)L α K α 2 π KL π KK = P 2 α 2 (α 1) 2 L 2α 2 K 2α 2 P 2 α 4 L 2α 2 K 2α 2 = P 2 α 2 (α 2 2α + 1)L 2α 2 K 2α 2 P 2 α 4 L 2α 2 K 2α 2 = P 2 α 2 L 2α 2 K 2α 2 (1 2α) + P 2 α 4 L 2α 2 K 2α 2 P 2 α 4 L 2α 2 K 2α 2 = P 2 α 2 L 2α 2 K 2α 2 (1 2α) H 1 = Pα(α 1)L α 2 K α < 0 and H > 0, so the hessian is negative definite, so L and K as defined by the FOC s represents the optimal quantities that will maximize profit. Jijian Fan (UC Santa Cruz) ECON / 26
12 Profit Maximization Example Plugging in the FOC for L into the FOC for K, we get [ ( ] w 1 α 1 PαL α K α 1 r = PαL α Pα L1 α) α r = 0 PαL α [ ( w Pα ) 1 ] α 1 α ( L 1 α w ) α 1 α α = Pα Pα = P α 1 α +1 α α 1 = (Pα) α 1 L 2α 1 α w α 1 α α +1 L α2 +2α 1+α 2 α w α 1 α L (1 α)(α 1) α +α r r r = 0 (Pα) α 1 L 2α 1 α w α 1 α = r (Pα) α 1 w α 1 α r 1 = L 2α 1 α = L 1 2α α L = ( Pαw α 1 r α) 1 2α 1 Jijian Fan (UC Santa Cruz) ECON / 26
13 Profit Maximization Example Similarly, we can find that K = ( Pαr α 1 w α) 1 1 2α Then, we can find the optimal quantity expressed only as a function of the exogenous parameters: Q = (L ) α (K ) α = ( Pαw α 1 r α) α 1 2α ( Pαr α 1 w α) α 1 2α = ( α 2 P 2 = wr ) α 1 2α Jijian Fan (UC Santa Cruz) ECON / 26
14 Constrained Optimization Up to this point, we have considered only problems of unconstrained optimization, that is, where an economic entity chooses the values of some variables to optimize a dependent variable with no restriction. However, consider a firm which seeks to maximizes profits with the production of two goods, but faces a production quota where Q 1 + Q 2 = 950. In this case, the choice variables are not only simultaneous, but also dependent. The solving of this problem is called constrained optimization. As another example, consider that a consumer wants to maximize their utility, given by U = x 1 x 2 + 2x 1 However, the consumer does not have an infinite amount of money, so they cannot buy an infinite amount of goods as would maximize their utility. Instead, the individual only has $60 to spend and x 1 costs $4 and x 2 costs $2, so their budget constraint is 4x 1 + 2x 2 = 60 Jijian Fan (UC Santa Cruz) ECON / 26
15 Constrained Optimization So, the individual s optimization problem can be stated as max U = x 1 x 2 + 2x 1 subject to 4x 1 + 2x 2 = 60 We call this constraint a budget constraint and it restricts the domain of the utility function, and as a result, the range of the objective function. In an unconstrained setting, x 1 and x 2 could take any value 0, but now the pair (x 1,x 2 ) must lie on the budget line. Jijian Fan (UC Santa Cruz) ECON / 26
16 Constrained Optimization The first method of solving constrained optimization is that of substitution. In the above example, we can take the budget constraint and find: x 2 = 60 4x 1 = 30 2x 1 2 Plug into the utility function to get: Also, we can easily see that du2 dx1 2 constrained maximum of U. U = x 1 (30 2x 1 ) + 2x 1 = 32x 1 2x 2 1 U x 1 = 32 4x 1 = 0 x 1 = 8 x 2 = 30 2(8) = 14 U = 8(14) + 2(8) = 128 = 4 < 0, so x 1 = 8 represents a Jijian Fan (UC Santa Cruz) ECON / 26
17 Constrained Optimization Another method, which is generally much more useful, especially for more complex and more than one constraint is called the Lagrange-multiplier method. The Lagrangian function for the previous example is: L = x 1 x 2 + 2x 1 + λ (60 4x 1 2x 2 ) λ is called the Lagrange multiplier (which we will discuss later). To solve the Lagrangian, we treat λ as a choice variable, so that the derivative with respect to λ will automatically satisfy the constraint. The first order conditions are: L = x λ = 0 λ = x x 1 4 L = x 1 2λ = 0 λ = x 1 x 2 2 L λ = 60 4x 1 2x 2 = 0 Jijian Fan (UC Santa Cruz) ECON / 26
18 Constrained Optimization λ = λ x ( x = x 1 2 Marginal rate of substitution x 1 = x ) 2x 2 = x 2 4 = 0 4x 2 = 56 x 2 = x 1 28 = 0 4x 1 = 32 x 1 = 8 Jijian Fan (UC Santa Cruz) ECON / 26
19 Lagrangian Method Given an objective function z = f (x,y) subject to g(x,y) = c we can write the Lagrangian function as L = f (x,y) + λ [c g(x,y)] Then, the first order conditions are L λ : c g(x,y) = 0 L x : f x λg x = 0 L y : f y λg y = 0 Jijian Fan (UC Santa Cruz) ECON / 26
20 Lagrangian Example A firm s production function is y = x + z and input prices are w x and w z. Find the quantities of x and z that minimize cost subject to the production function. L = w x x + w z z λ( x + z y) L x : w x 1 2 λx 1 2 = 0 wx = 1 2 λx 1 2 λ = 2wx x 1 2 L z : w z 1 2 λz 1 2 = 0 wz = 1 2 λz 1 2 λ = 2wz z 1 2 2w x x 1 2 = 2wz z 1 2 x 1 2 = z 1 2 w z w x x = z w 2 z w 2 x Jijian Fan (UC Santa Cruz) ECON / 26
21 Lagrangian Example y = x + z = z 2 1 w 1 z + z 1 z 2 w z + z 1 2 w x 2 = w x w x z 1 2 = w x y z = w x 2 y 2 w x + w z (w x + w z ) 2 = z 1 2 (w x + w z ) w x Plugging back into the marginal rate of technical substitution, ( w x 2 = x y 2 ) w 2 z (w x + w z ) 2 wx 2 = w 2 z y 2 (w x + w z ) 2 Jijian Fan (UC Santa Cruz) ECON / 26
22 Lagrangian Multiplier Interpretation The optimal value of L depends on λ (c), x (c), y (c), so L = f (x,y ) + λ [c g (x,y )] dl dc = (f x λ g x ) dx dc + (f y λ g y ) dy dc + [c g (x,y )] dλ dc + λ However, the first order conditions tell us that c = g (x,y ), f x = λ g x, and f y = λ g y, so the first three terms on the right hand side drop out and we are left with dl dc = λ So, the value of the Lagrange multiplier at the solution of the problem is a measure of the effect of a change in the constraint via the parameter c on the optimal value of the objective function. Jijian Fan (UC Santa Cruz) ECON / 26
23 Lagrangian-Method with Multiple Constraints The Lagrange-multiplier method is equally applicable when there is more than one constraint, we just need a Lagrange-multiplier for each constraint. Consider the function f (x 1,x 2,...,x n ) subject to two constraints: g (x 1,x 2,...,x n ) = c and h (x 1,x 2,...,x n ) = d. Then, the Lagrangian function can be written as: L = f (x 1,x 2,...,x n ) + λ [c g (x 1,x 2,...,x n )] + µ [d h (x 1,x 2,...,x n )] Then, the first-order conditions will consist of the following (n + 2) simultaneous equations: L λ = c g (x 1,x 2,...,x n ) = 0 L µ = d h (x 1,x 2,...,x n ) = 0 L i = f i λg i µh i = 0 (i = 1,2,...,n) Jijian Fan (UC Santa Cruz) ECON / 26
24 Multi-Constraint Lagrangian Example Find the maximum and minimum of f (x,y,z) = 4y 2z subject to 2x y z = 2 and x 2 + y 2 = 1. The Lagrangian function is: L = 4y 2z + λ (2 2x + y + z) + µ ( 1 x 2 y 2) The first order conditions are: L λ : 2 2x + y + z = 0 (1) L µ : 1 x 2 y 2 = 0 (2) L x : 2λ 2x µ = 0 (3) L y : 4 + λ 2y µ = 0 (4) L z : 2 + λ = 0 (5) Jijian Fan (UC Santa Cruz) ECON / 26
25 Multi-Constraint Lagrangian Example Plug in λ = 2 to (3) and (4): (5) λ = 2 ( 3) 2(2) 2x µ = 0 2x µ = 4 x = 2 µ (6) (4) y µ = 0 6 = 2y µ y = 3 µ (7) Plug in (6) and (7) to (2): ( 1 2 ) 2 ( ) 3 2 = 0 1 = 13 µ µ µ 2 µ = ± 13 Jijian Fan (UC Santa Cruz) ECON / 26
26 Multi-Constraint Lagrangian Example So there are two possible solutions, where µ = 13 and µ = 13. Case 1: µ = 13 Plugging back in to (6), (7), and then (2), we get x = 2 13, y = 3 13, and 0 = 2 + 2( 2 13 ) z z = Case 2: µ = 13 Plugging back in to (6), (7), and then (2), we get x = 2 13, y = 3 13, and 0 = 2 2( 2 13 ) z z = These are both potential optimum. To confirm, we must use the second order conditions we will learn in the next lecture. Jijian Fan (UC Santa Cruz) ECON / 26
Constrained optimization.
ams/econ 11b supplementary notes ucsc Constrained optimization. c 2016, Yonatan Katznelson 1. Constraints In many of the optimization problems that arise in economics, there are restrictions on the values
More informationECON2285: Mathematical Economics
ECON2285: Mathematical Economics Yulei Luo Economics, HKU September 17, 2018 Luo, Y. (Economics, HKU) ME September 17, 2018 1 / 46 Static Optimization and Extreme Values In this topic, we will study goal
More informationLecture Notes for Chapter 12
Lecture Notes for Chapter 12 Kevin Wainwright April 26, 2014 1 Constrained Optimization Consider the following Utility Max problem: Max x 1, x 2 U = U(x 1, x 2 ) (1) Subject to: Re-write Eq. 2 B = P 1
More informationTutorial 3: Optimisation
Tutorial : Optimisation ECO411F 011 1. Find and classify the extrema of the cubic cost function C = C (Q) = Q 5Q +.. Find and classify the extreme values of the following functions (a) y = x 1 + x x 1x
More informationOptimization, constrained optimization and applications of integrals.
ams 11b Study Guide econ 11b Optimization, constrained optimization and applications of integrals. (*) In all the constrained optimization problems below, you may assume that the critical values you find
More informationPartial derivatives, linear approximation and optimization
ams 11b Study Guide 4 econ 11b Partial derivatives, linear approximation and optimization 1. Find the indicated partial derivatives of the functions below. a. z = 3x 2 + 4xy 5y 2 4x + 7y 2, z x = 6x +
More informationMaximum Value Functions and the Envelope Theorem
Lecture Notes for ECON 40 Kevin Wainwright Maximum Value Functions and the Envelope Theorem A maximum (or minimum) value function is an objective function where the choice variables have been assigned
More informationEC /11. Math for Microeconomics September Course, Part II Lecture Notes. Course Outline
LONDON SCHOOL OF ECONOMICS Professor Leonardo Felli Department of Economics S.478; x7525 EC400 20010/11 Math for Microeconomics September Course, Part II Lecture Notes Course Outline Lecture 1: Tools for
More informationSolutions. ams 11b Study Guide 9 econ 11b
ams 11b Study Guide 9 econ 11b Solutions 1. A monopolistic firm sells one product in two markets, A and B. The daily demand equations for the firm s product in these markets are given by Q A = 100 0.4P
More informationz = f (x; y) f (x ; y ) f (x; y) f (x; y )
BEEM0 Optimization Techiniques for Economists Lecture Week 4 Dieter Balkenborg Departments of Economics University of Exeter Since the fabric of the universe is most perfect, and is the work of a most
More informationCHAPTER 4: HIGHER ORDER DERIVATIVES. Likewise, we may define the higher order derivatives. f(x, y, z) = xy 2 + e zx. y = 2xy.
April 15, 2009 CHAPTER 4: HIGHER ORDER DERIVATIVES In this chapter D denotes an open subset of R n. 1. Introduction Definition 1.1. Given a function f : D R we define the second partial derivatives as
More informationSolutions. F x = 2x 3λ = 0 F y = 2y 5λ = 0. λ = 2x 3 = 2y 5 = x = 3y 5. 2y 1/3 z 1/6 x 1/2 = 5x1/2 z 1/6. 3y 2/3 = 10x1/2 y 1/3
econ 11b ucsc ams 11b Review Questions 3 Solutions Note: In these problems, you may generally assume that the critical point(s) you find produce the required optimal value(s). At the same time, you should
More informationMultivariable Calculus and Matrix Algebra-Summer 2017
Multivariable Calculus and Matrix Algebra-Summer 017 Homework 4 Solutions Note that the solutions below are for the latest version of the problems posted. For those of you who worked on an earlier version
More informationEcon Slides from Lecture 14
Econ 205 Sobel Econ 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Theorem ( Lagrange Multipliers ) Theorem If x solves max f (x) subject to G(x) = 0 then there exists λ such that Df (x ) =
More informationECON 186 Class Notes: Derivatives and Differentials
ECON 186 Class Notes: Derivatives and Differentials Jijian Fan Jijian Fan ECON 186 1 / 27 Partial Differentiation Consider a function y = f (x 1,x 2,...,x n ) where the x i s are all independent, so each
More informationECON Answers Homework #4. = 0 6q q 2 = 0 q 3 9q = 0 q = 12. = 9q 2 108q AC(12) = 3(12) = 500
ECON 331 - Answers Homework #4 Exercise 1: (a)(i) The average cost function AC is: AC = T C q = 3q 2 54q + 500 + 2592 q (ii) In order to nd the point where the average cost is minimum, I solve the rst-order
More informationMicroeconomic Theory. Microeconomic Theory. Everyday Economics. The Course:
The Course: Microeconomic Theory This is the first rigorous course in microeconomic theory This is a course on economic methodology. The main goal is to teach analytical tools that will be useful in other
More informationECON2285: Mathematical Economics
ECON2285: Mathematical Economics Yulei Luo SEF of HKU September 9, 2017 Luo, Y. (SEF of HKU) ME September 9, 2017 1 / 81 Constrained Static Optimization So far we have focused on nding the maximum or minimum
More informationMATH 19520/51 Class 5
MATH 19520/51 Class 5 Minh-Tam Trinh University of Chicago 2017-10-04 1 Definition of partial derivatives. 2 Geometry of partial derivatives. 3 Higher derivatives. 4 Definition of a partial differential
More informationMathematics for Economics ECON MA/MSSc in Economics-2017/2018. Dr. W. M. Semasinghe Senior Lecturer Department of Economics
Mathematics for Economics ECON 53035 MA/MSSc in Economics-2017/2018 Dr. W. M. Semasinghe Senior Lecturer Department of Economics MATHEMATICS AND STATISTICS LERNING OUTCOMES: By the end of this course unit
More informationECON 255 Introduction to Mathematical Economics
Page 1 of 5 FINAL EXAMINATION Winter 2017 Introduction to Mathematical Economics April 20, 2017 TIME ALLOWED: 3 HOURS NUMBER IN THE LIST: STUDENT NUMBER: NAME: SIGNATURE: INSTRUCTIONS 1. This examination
More informationSTUDY MATERIALS. (The content of the study material is the same as that of Chapter I of Mathematics for Economic Analysis II of 2011 Admn.
STUDY MATERIALS MATHEMATICAL TOOLS FOR ECONOMICS III (The content of the study material is the same as that of Chapter I of Mathematics for Economic Analysis II of 2011 Admn.) & MATHEMATICAL TOOLS FOR
More informationLet f(x) = x, but the domain of f is the interval 0 x 1. Note
I.g Maximum and Minimum. Lagrange Multipliers Recall: Suppose we are given y = f(x). We recall that the maximum/minimum points occur at the following points: (1) where f = 0; (2) where f does not exist;
More information1 Objective. 2 Constrained optimization. 2.1 Utility maximization. Dieter Balkenborg Department of Economics
BEE020 { Basic Mathematical Economics Week 2, Lecture Thursday 2.0.0 Constrained optimization Dieter Balkenborg Department of Economics University of Exeter Objective We give the \ rst order conditions"
More informationx +3y 2t = 1 2x +y +z +t = 2 3x y +z t = 7 2x +6y +z +t = a
UCM Final Exam, 05/8/014 Solutions 1 Given the parameter a R, consider the following linear system x +y t = 1 x +y +z +t = x y +z t = 7 x +6y +z +t = a (a (6 points Discuss the system depending on the
More informationEC /11. Math for Microeconomics September Course, Part II Problem Set 1 with Solutions. a11 a 12. x 2
LONDON SCHOOL OF ECONOMICS Professor Leonardo Felli Department of Economics S.478; x7525 EC400 2010/11 Math for Microeconomics September Course, Part II Problem Set 1 with Solutions 1. Show that the general
More informationMATH529 Fundamentals of Optimization Constrained Optimization I
MATH529 Fundamentals of Optimization Constrained Optimization I Marco A. Montes de Oca Mathematical Sciences, University of Delaware, USA 1 / 26 Motivating Example 2 / 26 Motivating Example min cost(b)
More informationBi-Variate Functions - ACTIVITES
Bi-Variate Functions - ACTIVITES LO1. Students to consolidate basic meaning of bi-variate functions LO2. Students to learn how to confidently use bi-variate functions in economics Students are given the
More informationECONOMICS 207 SPRING 2008 PROBLEM SET 13
ECONOMICS 207 SPRING 2008 PROBLEM SET 13 Problem 1. The cost function for a firm is a rule or mapping that tells the minimum total cost of production of any output level produced by the firm for a fixed
More informationINTRODUCTORY MATHEMATICS FOR ECONOMICS MSCS. LECTURE 3: MULTIVARIABLE FUNCTIONS AND CONSTRAINED OPTIMIZATION. HUW DAVID DIXON CARDIFF BUSINESS SCHOOL.
INTRODUCTORY MATHEMATICS FOR ECONOMICS MSCS. LECTURE 3: MULTIVARIABLE FUNCTIONS AND CONSTRAINED OPTIMIZATION. HUW DAVID DIXON CARDIFF BUSINESS SCHOOL. SEPTEMBER 2009. 3.1 Functions of more than one variable.
More informationMTAEA Implicit Functions
School of Economics, Australian National University February 12, 2010 Implicit Functions and Their Derivatives Up till now we have only worked with functions in which the endogenous variables are explicit
More informationProblem Set 2 Solutions
EC 720 - Math for Economists Samson Alva Department of Economics Boston College October 4 2011 1. Profit Maximization Problem Set 2 Solutions (a) The Lagrangian for this problem is L(y k l λ) = py rk wl
More informationEconomics 101. Lecture 2 - The Walrasian Model and Consumer Choice
Economics 101 Lecture 2 - The Walrasian Model and Consumer Choice 1 Uncle Léon The canonical model of exchange in economics is sometimes referred to as the Walrasian Model, after the early economist Léon
More informationSeptember Math Course: First Order Derivative
September Math Course: First Order Derivative Arina Nikandrova Functions Function y = f (x), where x is either be a scalar or a vector of several variables (x,..., x n ), can be thought of as a rule which
More informationAdvanced Microeconomic Analysis, Lecture 6
Advanced Microeconomic Analysis, Lecture 6 Prof. Ronaldo CARPIO April 10, 017 Administrative Stuff Homework # is due at the end of class. I will post the solutions on the website later today. The midterm
More informationA DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any
Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y $ Z % Y Y x x } / % «] «] # z» & Y X»
More informationThe Kuhn-Tucker and Envelope Theorems
The Kuhn-Tucker and Envelope Theorems Peter Ireland EC720.01 - Math for Economists Boston College, Department of Economics Fall 2010 The Kuhn-Tucker and envelope theorems can be used to characterize the
More informationLecture Notes: Math Refresher 1
Lecture Notes: Math Refresher 1 Math Facts The following results from calculus will be used over and over throughout the course. A more complete list of useful results from calculus is posted on the course
More informationTvestlanka Karagyozova University of Connecticut
September, 005 CALCULUS REVIEW Tvestlanka Karagyozova University of Connecticut. FUNCTIONS.. Definition: A function f is a rule that associates each value of one variable with one and only one value of
More informationLecture Notes for January 8, 2009; part 2
Economics 200B Prof. R. Starr UCSD Winter 2009 Lecture Notes for January 8, 2009; part 2 Integrating Production and Multiple Consumption Decisions: A2 2 2 Model Competitive Equilibrium: Production and
More informationEC611--Managerial Economics
EC611--Managerial Economics Optimization Techniques and New Management Tools Dr. Savvas C Savvides, European University Cyprus Models and Data Model a framework based on simplifying assumptions it helps
More informationHigher order derivative
2 Î 3 á Higher order derivative Î 1 Å Iterated partial derivative Iterated partial derivative Suppose f has f/ x, f/ y and ( f ) = 2 f x x x 2 ( f ) = 2 f x y x y ( f ) = 2 f y x y x ( f ) = 2 f y y y
More informationChapter 4 Differentiation
Chapter 4 Differentiation 08 Section 4. The derivative of a function Practice Problems (a) (b) (c) 3 8 3 ( ) 4 3 5 4 ( ) 5 3 3 0 0 49 ( ) 50 Using a calculator, the values of the cube function, correct
More informationLecture Notes October 18, Reading assignment for this lecture: Syllabus, section I.
Lecture Notes October 18, 2012 Reading assignment for this lecture: Syllabus, section I. Economic General Equilibrium Partial and General Economic Equilibrium PARTIAL EQUILIBRIUM S k (p o ) = D k k (po
More informationg(t) = f(x 1 (t),..., x n (t)).
Reading: [Simon] p. 313-333, 833-836. 0.1 The Chain Rule Partial derivatives describe how a function changes in directions parallel to the coordinate axes. Now we shall demonstrate how the partial derivatives
More informationFunctions. A function is a rule that gives exactly one output number to each input number.
Functions A function is a rule that gives exactly one output number to each input number. Why it is important to us? The set of all input numbers to which the rule applies is called the domain of the function.
More informationproblem. max Both k (0) and h (0) are given at time 0. (a) Write down the Hamilton-Jacobi-Bellman (HJB) Equation in the dynamic programming
1. Endogenous Growth with Human Capital Consider the following endogenous growth model with both physical capital (k (t)) and human capital (h (t)) in continuous time. The representative household solves
More informationIntroductory Microeconomics
Prof. Wolfram Elsner Faculty of Business Studies and Economics iino Institute of Institutional and Innovation Economics Introductory Microeconomics The Ideal Neoclassical Market and General Equilibrium
More informationEconomics 203: Intermediate Microeconomics. Calculus Review. A function f, is a rule assigning a value y for each value x.
Economics 203: Intermediate Microeconomics Calculus Review Functions, Graphs and Coordinates Econ 203 Calculus Review p. 1 Functions: A function f, is a rule assigning a value y for each value x. The following
More informationMonetary Economics: Solutions Problem Set 1
Monetary Economics: Solutions Problem Set 1 December 14, 2006 Exercise 1 A Households Households maximise their intertemporal utility function by optimally choosing consumption, savings, and the mix of
More informationDEPARTMENT OF MANAGEMENT AND ECONOMICS Royal Military College of Canada. ECE Modelling in Economics Instructor: Lenin Arango-Castillo
Page 1 of 5 DEPARTMENT OF MANAGEMENT AND ECONOMICS Royal Military College of Canada ECE 256 - Modelling in Economics Instructor: Lenin Arango-Castillo Final Examination 13:00-16:00, December 11, 2017 INSTRUCTIONS
More informationMaximum Theorem, Implicit Function Theorem and Envelope Theorem
Maximum Theorem, Implicit Function Theorem and Envelope Theorem Ping Yu Department of Economics University of Hong Kong Ping Yu (HKU) MIFE 1 / 25 1 The Maximum Theorem 2 The Implicit Function Theorem 3
More informationSchool of Business. Blank Page
Maxima and Minima 9 This unit is designed to introduce the learners to the basic concepts associated with Optimization. The readers will learn about different types of functions that are closely related
More informationConstrained optimization BUSINESS MATHEMATICS
Constrained optimization BUSINESS MATHEMATICS 1 CONTENTS Unconstrained optimization Constrained optimization Lagrange method Old exam question Further study 2 UNCONSTRAINED OPTIMIZATION Recall extreme
More informationStudy Skills in Mathematics. Edited by D. Burkhardt and D. Rutherford. Nottingham: Shell Centre for Mathematical Education (Revised edn 1981).
Study Skills in Mathematics. Edited by D. Burkhardt and D. Rutherford. Nottingham: Shell Centre for Mathematical Education (Revised edn 1981). (Copies are available from the Shell Centre for Mathematical
More information(a) Write down the Hamilton-Jacobi-Bellman (HJB) Equation in the dynamic programming
1. Government Purchases and Endogenous Growth Consider the following endogenous growth model with government purchases (G) in continuous time. Government purchases enhance production, and the production
More informationMath 10C - Fall Final Exam
Math 1C - Fall 217 - Final Exam Problem 1. Consider the function f(x, y) = 1 x 2 (y 1) 2. (i) Draw the level curve through the point P (1, 2). Find the gradient of f at the point P and draw the gradient
More informationFACULTY OF ARTS AND SCIENCE University of Toronto FINAL EXAMINATIONS, APRIL 2016 MAT 133Y1Y Calculus and Linear Algebra for Commerce
FACULTY OF ARTS AND SCIENCE University of Toronto FINAL EXAMINATIONS, APRIL 206 MAT YY Calculus and Linear Algebra for Commerce Duration: Examiners: hours N. Hoell A. Igelfeld D. Reiss L. Shorser J. Tate
More informationVolume 30, Issue 3. Ramsey Fiscal Policy and Endogenous Growth: A Comment. Jenn-hong Tang Department of Economics, National Tsing-Hua University
Volume 30, Issue 3 Ramsey Fiscal Policy and Endogenous Growth: A Comment Jenn-hong Tang Department of Economics, National Tsing-Hua University Abstract Recently, Park (2009, Economic Theory 39, 377--398)
More informationReview of Optimization Methods
Review of Optimization Methods Prof. Manuela Pedio 20550 Quantitative Methods for Finance August 2018 Outline of the Course Lectures 1 and 2 (3 hours, in class): Linear and non-linear functions on Limits,
More informationHomework 1 Solutions
Homework Solutions Econ 50 - Stanford University - Winter Quarter 204/5 January 6, 205 Exercise : Constrained Optimization with One Variable (a) For each function, write the optimal value(s) of x on the
More informationwhere u is the decision-maker s payoff function over her actions and S is the set of her feasible actions.
Seminars on Mathematics for Economics and Finance Topic 3: Optimization - interior optima 1 Session: 11-12 Aug 2015 (Thu/Fri) 10:00am 1:00pm I. Optimization: introduction Decision-makers (e.g. consumers,
More informationEconS 301. Math Review. Math Concepts
EconS 301 Math Review Math Concepts Functions: Functions describe the relationship between input variables and outputs y f x where x is some input and y is some output. Example: x could number of Bananas
More informationII. An Application of Derivatives: Optimization
Anne Sibert Autumn 2013 II. An Application of Derivatives: Optimization In this section we consider an important application of derivatives: finding the minimum and maximum points. This has important applications
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Study Questions for OPMT 5701 Most Questions come from Chapter 17. The Answer Key has a section code for each. Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers
More informationUnit #24 - Lagrange Multipliers Section 15.3
Unit #24 - Lagrange Multipliers Section 1.3 Some material from Calculus, Single and MultiVariable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 200 by John Wiley & Sons, Inc. This material is
More informationLagrange Multipliers
Optimization with Constraints As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each
More information3 Applications of partial differentiation
Advanced Calculus Chapter 3 Applications of partial differentiation 37 3 Applications of partial differentiation 3.1 Stationary points Higher derivatives Let U R 2 and f : U R. The partial derivatives
More informationModelling Production
Modelling Production David N. DeJong University of Pittsburgh Econ. 1540, Spring 2010 DND () Production Econ. 1540, Spring 2010 1 / 23 Introduction The production function is the foundation upon which
More informationComparative Statics. Autumn 2018
Comparative Statics Autumn 2018 What is comparative statics? Contents 1 What is comparative statics? 2 One variable functions Multiple variable functions Vector valued functions Differential and total
More informationE 600 Chapter 4: Optimization
E 600 Chapter 4: Optimization Simona Helmsmueller August 8, 2018 Goals of this lecture: Every theorem in these slides is important! You should understand, remember and be able to apply each and every one
More informationMathematical Economics: Lecture 16
Mathematical Economics: Lecture 16 Yu Ren WISE, Xiamen University November 26, 2012 Outline 1 Chapter 21: Concave and Quasiconcave Functions New Section Chapter 21: Concave and Quasiconcave Functions Concave
More informationFACULTY OF ARTS AND SCIENCE University of Toronto FINAL EXAMINATIONS, APRIL 2012 MAT 133Y1Y Calculus and Linear Algebra for Commerce
FACULTY OF ARTS AND SCIENCE University of Toronto FINAL EXAMINATIONS, APRIL 0 MAT 33YY Calculus and Linear Algebra for Commerce Duration: Examiners: 3 hours N. Francetic A. Igelfeld P. Kergin J. Tate LEAVE
More informationThe Envelope Theorem
The Envelope Theorem In an optimization problem we often want to know how the value of the objective function will change if one or more of the parameter values changes. Let s consider a simple example:
More informationConstrained Optimization
Constrained Optimization Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 13, 2013 1 General Problem Consider the following general constrained optimization problem:
More informationEquilibrium in Factors Market: Properties
Equilibrium in Factors Market: Properties Ram Singh Microeconomic Theory Lecture 12 Ram Singh: (DSE) Factor Prices Lecture 12 1 / 17 Questions What is the relationship between output prices and the wage
More information1 Recursive Competitive Equilibrium
Feb 5th, 2007 Let s write the SPP problem in sequence representation: max {c t,k t+1 } t=0 β t u(f(k t ) k t+1 ) t=0 k 0 given Because of the INADA conditions we know that the solution is interior. So
More informationOptimization. Sherif Khalifa. Sherif Khalifa () Optimization 1 / 50
Sherif Khalifa Sherif Khalifa () Optimization 1 / 50 Y f(x 0 ) Y=f(X) X 0 X Sherif Khalifa () Optimization 2 / 50 Y Y=f(X) f(x 0 ) X 0 X Sherif Khalifa () Optimization 3 / 50 A necessary condition for
More informationEcon 110: Introduction to Economic Theory. 8th Class 2/7/11
Econ 110: Introduction to Economic Theory 8th Class 2/7/11 go over problem answers from last time; no new problems today given you have your problem set to work on; we'll do some problems for these concepts
More informationProblem Set #06- Answer key First CompStat Problem Set
Fall 2015 Problem Set #06- Answer key First CompStat Problem Set ARE211 (1) Inproblem 3 on your last problem set, you foundthemaximum and minimumdistance from the origin to the ellipse x 2 1 +x 1x 2 +x
More informationPartial Differentiation
CHAPTER 7 Partial Differentiation From the previous two chapters we know how to differentiate functions of one variable But many functions in economics depend on several variables: output depends on both
More informationMacroeconomic Theory and Analysis Suggested Solution for Midterm 1
Macroeconomic Theory and Analysis Suggested Solution for Midterm February 25, 2007 Problem : Pareto Optimality The planner solves the following problem: u(c ) + u(c 2 ) + v(l ) + v(l 2 ) () {c,c 2,l,l
More informationTopic 2. Consumption/Saving and Productivity shocks
14.452. Topic 2. Consumption/Saving and Productivity shocks Olivier Blanchard April 2006 Nr. 1 1. What starting point? Want to start with a model with at least two ingredients: Shocks, so uncertainty.
More informationBasics of Calculus and Algebra
Monika Department of Economics ISCTE-IUL September 2012 Basics of linear algebra Real valued Functions Differential Calculus Integral Calculus Optimization Introduction I A matrix is a rectangular array
More informationMATH Midterm 1 Sample 4
1. (15 marks) (a) (4 marks) Given the function: f(x, y) = arcsin x 2 + y, find its first order partial derivatives at the point (0, 3). Simplify your answers. Solution: Compute the first order partial
More informationEcon Macroeconomics I Optimizing Consumption with Uncertainty
Econ 07 - Macroeconomics I Optimizing Consumption with Uncertainty Prof. David Weil September 16, 005 Overall, this is one of the most important sections of this course. The key to not getting lost is
More informationPermanent Income Hypothesis Intro to the Ramsey Model
Consumption and Savings Permanent Income Hypothesis Intro to the Ramsey Model Lecture 10 Topics in Macroeconomics November 6, 2007 Lecture 10 1/18 Topics in Macroeconomics Consumption and Savings Outline
More informationLinear Algebra. Chapter 8: Eigenvalues: Further Applications and Computations Section 8.2. Applications to Geometry Proofs of Theorems.
Linear Algebra Chapter 8: Eigenvalues: Further Applications and Computations Section 8.2. Applications to Geometry Proofs of Theorems May 1, 2018 () Linear Algebra May 1, 2018 1 / 8 Table of contents 1
More informationThe Inconsistent Neoclassical Theory of the Firm and Its Remedy
The Inconsistent Neoclassical Theory of the Firm and Its Remedy Consider the following neoclassical theory of the firm, as developed by Walras (1874/1954, Ch.21), Marshall (1907/1948, Ch.13), Samuelson
More informationLESSON 7.1 FACTORING POLYNOMIALS I
LESSON 7.1 FACTORING POLYNOMIALS I LESSON 7.1 FACTORING POLYNOMIALS I 293 OVERVIEW Here s what you ll learn in this lesson: Greatest Common Factor a. Finding the greatest common factor (GCF) of a set of
More informationLecture 4: Optimization. Maximizing a function of a single variable
Lecture 4: Optimization Maximizing or Minimizing a Function of a Single Variable Maximizing or Minimizing a Function of Many Variables Constrained Optimization Maximizing a function of a single variable
More informationSometimes the domains X and Z will be the same, so this might be written:
II. MULTIVARIATE CALCULUS The first lecture covered functions where a single input goes in, and a single output comes out. Most economic applications aren t so simple. In most cases, a number of variables
More informationAMC Lecture Notes. Justin Stevens. Cybermath Academy
AMC Lecture Notes Justin Stevens Cybermath Academy Contents 1 Systems of Equations 1 1 Systems of Equations 1.1 Substitution Example 1.1. Solve the system below for x and y: x y = 4, 2x + y = 29. Solution.
More informationBusiness Mathematics. Lecture Note #13 Chapter 7-(1)
1 Business Mathematics Lecture Note #13 Chapter 7-(1) Applications of Partial Differentiation 1. Differentials and Incremental Changes 2. Production functions: Cobb-Douglas production function, MP L, MP
More informationFinal Exam 2011 Winter Term 2 Solutions
. (a Find the radius of convergence of the series: ( k k+ x k. Solution: Using the Ratio Test, we get: L = lim a k+ a k = lim ( k+ k+ x k+ ( k k+ x k = lim x = x. Note that the series converges for L
More informationQueen s University. Department of Economics. Instructor: Kevin Andrew
Figure 1: 1b) GDP Queen s University Department of Economics Instructor: Kevin Andrew Econ 320: Math Assignment Solutions 1. (10 Marks) On the course website is provided an Excel file containing quarterly
More informationMath 10C Practice Final Solutions
Math 1C Practice Final Solutions March 9, 216 1. (6 points) Let f(x, y) x 3 y + 12x 2 8y. (a) Find all critical points of f. SOLUTION: f x 3x 2 y + 24x 3x(xy + 8) x,xy 8 y 8 x f y x 3 8 x 3 8 x 3 8 2 So
More informationCES functions and Dixit-Stiglitz Formulation
CES functions and Dixit-Stiglitz Formulation Weijie Chen Department of Political and Economic Studies University of Helsinki September, 9 4 8 3 7 Labour 6 5 4 5 Labour 5 Capital 3 4 6 8 Capital Any suggestion
More informationHomework 3 Suggested Answers
Homework 3 Suggested Answers Answers from Simon and Blume are on the back of the book. Answers to questions from Dixit s book: 2.1. We are to solve the following budget problem, where α, β, p, q, I are
More information