Consider this problem. A person s utility function depends on consumption and leisure. Of his market time (the complement to leisure), h t

Size: px
Start display at page:

Download "Consider this problem. A person s utility function depends on consumption and leisure. Of his market time (the complement to leisure), h t"

Transcription

1 VI. INEQUALITY CONSTRAINED OPTIMIZATION Application of the Kuhn-Tucker conditions to inequality constrained optimization problems is another very, very important skill to your career as an economist. If your mathematical training comes from somehere other than an economics department, it s unlikely that you ve seen much emphasis put on this. Consider this problem. A person s utility function depends on consumption and leisure. Of his market time (the complement to leisure), h t, some share i t is devoted to human capital investment (training, schooling, and such). The lifetime utility maximization problem looks like: ( ) T max! t U c t,t " h t s.t.: t & c t t ( " i t )h t + rs t " " s t ( ' t + t + $ ( i t h t ) t ( ) 0 c t, 0 h t T, 0 i t This is a version of the Heckman model of human capital investment. The first to inequality constraints are typical, that consumption must be nonnegative (try eating negative a bagel!) and that market time is also nonnegative, but also less than the total amount of time in the year. Sometimes, lazy economists ill ignore corner solutions and just assume that the solution lies in the interior. After all, e never see people consuming absolutely zero in a year you d die and have utility of negative infinity if that happened. Though sometimes corner solutions might seem silly, in other cases they are not. The interpretation of the variable i t is that it is the share of your market time (nonleisure time) devoted to making yourself a more productive orker, and the constraint is that it must be beteen 0 and 00. The case here i t is very interesting, because it is interpreted as full-time schooling. You ve probably spent some time in this situation. Retirement is another interesting corner solution. People also study hat affects the decision hether to participate in the labor market, another corner solution. In fact, in this model, the most interesting things are the corner solutions. Generally, though, e orry about solutions that suggest optimal consumption is negative. Consider a person ith the quasi-linear utility function: U(, ) + ln Solving this the usual ay, e get the demand function: Fall 2007 math class notes, page 4

2 ( )!, x(p,) (p,), (p,) " $ Here e have a potential problem. What if <? The first-order conditions tell our consumer that he should buy a negative amount of good one, but that s difficult. What makes more sense is to say these demands are optimal, provided that the person has enough ealth. If not, he just buys zero of good one and spends everything on good to: x ( p,! ) )"!, p + $ & ' if ( * +,( 0,) if < Demand functions should be ritten like this hen corner solutions are possible. We can also get unreasonable solutions hen e are dealing ith satiated preferences. Suppose that my demands for to goods (beer and ice cream) are given by the function: u(, ) 8! 2 + 8! 2 The price of each good is equal to one, and I have ealth 00. Solving the problem: & ' max8! 2 + 8! 2 s.t gives demands of 50 and 50. Hoever, let s think a bit more carefully about this. My marginal utility of consuming the n-th unit of either good is 8! 2n. That means that consuming more than four units of either good gives me disutility. If I ant to survive the evening ithout puking, I should stop after four beers and four ice creams leaving some of my money unspent. There are to ays to do inequality-constrained optimization. One is the proper ay, and the other is the simpler ay that everyone does it. We ll do the proper version first, so let s set up the general frameork for an inequality-constrained optimization problem. We ant to maximize the objective f : X! R, ith the k inequality constraints g i ( x)! b i, and the requirement that each of the n elements x i of the vector x cannot fall belo a certain fixed value, c i : max x!x f (x) s.t.: g (x) " b $! & g m (x) " b m and s.t.: ' c $ " & x n ' c n Fall 2007 math class notes, page 42

3 Typically, f ill be a utility function or a social elfare function; the function g ill be a budget constraint, or feasibility constraint; and c ill be the zero vector (for nonnegativity constraints). Once again, e define a Lagrangian: L(x,!,µ) " f (x) +!(b $ g(x)) + µ (x $ c) m n f (x) +! j ( b j $ g j (x)) + µ i x i $ c i j i ( ) The complete requirements for a maximum x * hen c 0 ) are: (called Kuhn-Tucker conditions.!l(x *,",µ)!x 0 2.!L(x *,",µ)!" 0 and! " 0, ith (!L(x *,",µ)!")" 0 3.!L(x *,",µ)!µ 0 and µ! 0, ith (!L(x *,",µ)!µ)µ 0 Which can also be ritten as:.!l!x i 0, for i,2,,n ( ) 0 2. b j! g j (x * ) " 0 and! j " 0, ith! j " b j g j (x * ) 3. x i *! c i " 0 and µ i! 0, ith µ i!(x i * " c i ) 0, for i,2,,n Though these look nasty, they have a simple interpretation. The first part of each of (2. ) and (3. ) says that each constraint must be satisfied e should certainly hope so, since that as the hole point. The second expression says that each constraint holds ith equality, or else the multiplier on the constraint is zero, hich e describe as a nonbinding constraint for instance, that either the person ends at a corner solution, or e might as ell not have orried about the constraint. This requirement that at least one of the terms equals zero is called complementary slackness. As an example of a nonbinding constraint, e could have a utility maximization problem ith the constraints the price of your consumption must be less than or equal to your ealth and the calorie values of the foods you eat must be greater than or equal to your required caloric intake. For most people the first of these constraints ould be binding, and so their budget balances exactly. On the other hand (at least in the U.S. at this time), the energy requirement ould probably be nonbinding, and they end up eating hatever they like. Okay, no it s time for an example. Hopefully, this problem ill be orked so ell (and not have any sign errors) that you can pattern your problems on this. The problem is to maximize a quasi-linear utility function ith respect to some variables x, subject to a budget constraint and some nonnegativity constraints: Fall 2007 math class notes, page 43

4 maxu(, ) + ln s.t.: +! and x i! 0. The Lagrangian function associated ith the solution to this problem is: L( x,!,µ ) " + ln +!( + ) + µ + µ 2 There are to x variables, hich give these partial derivatives: ()!L! " + µ $ 0 (2)!L! " + µ 2 $ 0 There are also three constraints, hich returns these conditions: (3)!L!" $ 0 and (!! )" 0 (4)!L!µ " 0 and: µ! 0 (5)!L!µ 2 " 0 and: µ 2! 0 I rerite the conditions () and (2) in this manner: (6) ( + µ )! (7) ( + µ 2 )! Well, no e can solve for in terms of and the multipliers and then derive to demand functions, inclusive of the multipliers: (8) ( ) ( )! ( ( + µ )! µ 2 )! ( + µ ) + µ 2 (9)! + µ! µ 2 (0) The next step is to hypothesize that each constraint is non-binding, and to see hat facts are consistent ith each hypothesis. If the hypothesis implies something illogical, e can determine that the constraint never binds. If the hypothesis implies something possible, then e have determined the conditions under hich this constraint doesn t bind. First, let s suppose that the budget constraint is non-binding, so! 0. Equation (6) ould become: () ( + µ )! 0 Fall 2007 math class notes, page 44

5 In order for the product to equal zero, either 0 or (+ µ ) 0. Provided that prices are finite, the first of these is impossible. The second ould require that µ 2!, but the Kuhn-Tucker conditions require that µ i! 0, so this is impossible as ell. Putting these together, e can determine that a non-binding budget constraint ould imply impossible things, so a non-binding budget constraint is impossible. Therefore,! > 0, and +. Next, let us ork ith the requirement that either µ 2 0 or 0. If 0, then equation (0) tells us that: (2) ( )! µ µ Multiplying both sides by (+ µ )! µ 2, e ould have that 0, hich is a very silly result this case is impossible. We can therefore conclude that > 0, so µ 2 0. This ill simplify the demand a bit. Finally, e ll examine the requirement that either µ 0 or 0. If 0 and µ > 0, then equation (9) tells us that: (3)! ( + µ )! 0 This means that: (4) ( + µ )! (5) (6) + µ! µ > 0 And so this implies that: (7) >! > There s no reason hy that couldn t occur. We have established that 0 is consistent ith >. What about the opposite case, here > 0 and µ 0? From equation (9), e have that this implies: (8)! > 0 That tells us to things: this case is consistent ith >, hich isn t impossible; and in this case, demand is. Fall 2007 math class notes, page 45

6 Putting this altogether, e can conclude that there are to possible cases. If <, then (9)! ( + µ )! 0 (0) µ! Which means that ( ( ))! (0) + µ + ( (! ))! p ( )! ( )! You spend nothing on good one, and you spend all your ealth on good to. The other possible case is that >, ith µ µ 2 0. This gives demands of: (9)! ( + µ )!! (0) + µ ( ( ))! ( ) ( )! And that s it. We might rite these demand functions as: " $ p!, if > $ 0 otherise p! p 2 if > " $ otherise There is a lot of intuition involved in dealing ith corner solutions. Sometimes e can rule certain kinds of binding constraints. Here are some general guidelines. Rule : Almost never is the number of binding constraints greater than the number of choice variables. Rule 2: The budget constraint or feasibility constraint is alays binding hen a utility or elfare function is strictly increasing in any variable:!i : "x i u x i > 0 $ > 0 Rule 3: Any variable hose marginal utility goes to infinity as that variable goes to zero ill not have a binding nonnegativity constraint: lim "u "x i $ µ i 0 x i!0 Here are some more practice problems. Exercise: max x u, Exercise: max x u c ( ) x! " x 2, s.t.: +!, x! 0 ( )! ", subject to: p! x ",! 0 Fall 2007 math class notes, page 46

7 Exercise: max x, u, Exercise: max q,x ( ) +, subject to: +!, x!! 0! ( pq! x), subject to: q! x + 4 " 2, q! 0, x! 0 Exercise: min x, ( + 2 ), subject to: +! q,! 0,! 0 ( q fixed). There is another ay to handle corner solutions, omitting the second multiplier µ. I ill use! L to denote this less-than-rigorous Lagrangian ithout µ. Instead, they allo the first order condition for!! L!x to have an inequality this is the ay that most people do it (and likely you ll end up doing, too). Except in eird cases ith, you ll get the same anser either ay, provided you re careful. The problem is the same: max f (x) s.t.: g(x)! b and to x " c Set up the Lagrangian: ( ) m!l(x,!) " f (x) +!(b $ g(x) f (x) +! i b i $ g i (x) Take the derivative of this ith respect to x i. For each x i, either the derivative of the Lagrangian ith respect to that variable equals zero (the usual first-order condition) or that variable is at a corner:! f (x * ) k!g " $ j (x * )( ' j j &!x i!x * i ) + 0 and:! f (x * ) k!g " $ j (x * )( ' j j &!x i!x * i ) xi* " c i i ( ) 0 The rules for derivatives ith respect to! j are the same as before. Pretend that the usual condition holds ith equality and solve. Look at the implied value for x i. Does it go outside the permissible range? If so, set it equal to the corner value. Unless you have some strange things going on ith a bunch of corners, that s all you have to orry about. The end. As an example, once more the problem is to maximize a quasi-linear utility function ith respect to some variables x, subject to a budget constraint and some nonnegativity constraints: maxu(, ) max + ln s.t.: x! p + " and to: x! 0. The Lagrangian function associated ith the solution to this problem is:!l ( x,! ) " + ln +!( + ) There are to x variables and one constraint, hich give these partial derivatives: Fall 2007 math class notes, page 47

8 ( ) 0 ( )!! L! " $ 0 and:! " ( ) 0 (2 )! L! " $ 0 and:! " (3 )! L!" $ 0 and: (!! )" 0 Solve these as if the inequality holds ith equality. We get the demand for good 2: (9)! Substituting into the budget constraint, the demand for good is then: (20) x`!! ( ) "! This is fine (that is, certainly nonnegative) only if!. In any other case, e ll have to set 0. So e mention this possibility hen riting don the demand function: " $ p (2)!, if > $ 0 otherise This also suggests that the predicted optimal amounts of (that is, the amount that came from solving as if the equality held) ill also be okay, but only if!. In the other case, e kno that 0, so from the budget constraint e can infer that! 0 ( ) ". So the demand function is: p! p (22) 2 if > " $ otherise These solutions both coincide ith the solutions from the other method, hich is a good thing. A fe more practice problems: Exercise: maxu c,! Exercise: maxu,, x 3 Exercise: maxu c ( ) c +! ln (!), subject to: p! c " h! and h +!! T. ( ) x! " x 2 " x $ 3, subject to: + + p 3 x 3! ( ) c! "c 2, subject to: p! c " Fall 2007 math class notes, page 48

Econ 201: Problem Set 3 Answers

Econ 201: Problem Set 3 Answers Econ 20: Problem Set 3 Ansers Instructor: Alexandre Sollaci T.A.: Ryan Hughes Winter 208 Question a) The firm s fixed cost is F C = a and variable costs are T V Cq) = 2 bq2. b) As seen in class, the optimal

More information

Business Cycles: The Classical Approach

Business Cycles: The Classical Approach San Francisco State University ECON 302 Business Cycles: The Classical Approach Introduction Michael Bar Recall from the introduction that the output per capita in the U.S. is groing steady, but there

More information

Mathematical Foundations -1- Constrained Optimization. Constrained Optimization. An intuitive approach 2. First Order Conditions (FOC) 7

Mathematical Foundations -1- Constrained Optimization. Constrained Optimization. An intuitive approach 2. First Order Conditions (FOC) 7 Mathematical Foundations -- Constrained Optimization Constrained Optimization An intuitive approach First Order Conditions (FOC) 7 Constraint qualifications 9 Formal statement of the FOC for a maximum

More information

How to Characterize Solutions to Constrained Optimization Problems

How to Characterize Solutions to Constrained Optimization Problems How to Characterize Solutions to Constrained Optimization Problems Michael Peters September 25, 2005 1 Introduction A common technique for characterizing maximum and minimum points in math is to use first

More information

The Kuhn-Tucker and Envelope Theorems

The 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 information

The Fundamental Welfare Theorems

The Fundamental Welfare Theorems The Fundamental Welfare Theorems The so-called Fundamental Welfare Theorems of Economics tell us about the relation between market equilibrium and Pareto efficiency. The First Welfare Theorem: Every Walrasian

More information

In economics, the amount of a good x demanded is a function of the price of that good. In other words,

In economics, the amount of a good x demanded is a function of the price of that good. In other words, I. UNIVARIATE CALCULUS Given two sets X and Y, a function is a rule that associates each member of X with eactly one member of Y. That is, some goes in, and some y comes out. These notations are used to

More information

CHAPTER 1-2: SHADOW PRICES

CHAPTER 1-2: SHADOW PRICES Essential Microeconomics -- CHAPTER -: SHADOW PRICES An intuitive approach: profit maimizing firm with a fied supply of an input Shadow prices 5 Concave maimization problem 7 Constraint qualifications

More information

Nonlinear Programming and the Kuhn-Tucker Conditions

Nonlinear Programming and the Kuhn-Tucker Conditions Nonlinear Programming and the Kuhn-Tucker Conditions The Kuhn-Tucker (KT) conditions are first-order conditions for constrained optimization problems, a generalization of the first-order conditions we

More information

1. Suppose preferences are represented by the Cobb-Douglas utility function, u(x1,x2) = Ax 1 a x 2

1. Suppose preferences are represented by the Cobb-Douglas utility function, u(x1,x2) = Ax 1 a x 2 Additional questions for chapter 7 1. Suppose preferences are represented by the Cobb-Douglas utility function ux1x2 = Ax 1 a x 2 1-a 0 < a < 1 &A > 0. Assuming an interior solution solve for the Marshallian

More information

Ch. 2 Math Preliminaries for Lossless Compression. Section 2.4 Coding

Ch. 2 Math Preliminaries for Lossless Compression. Section 2.4 Coding Ch. 2 Math Preliminaries for Lossless Compression Section 2.4 Coding Some General Considerations Definition: An Instantaneous Code maps each symbol into a codeord Notation: a i φ (a i ) Ex. : a 0 For Ex.

More information

Recitation #2 (August 31st, 2018)

Recitation #2 (August 31st, 2018) Recitation #2 (August 1st, 2018) 1. [Checking properties of the Cobb-Douglas utility function.] Consider the utility function u(x) = n i=1 xα i i, where x denotes a vector of n different goods x R n +,

More information

Dynamic Macroeconomic Theory Notes. David L. Kelly. Department of Economics University of Miami Box Coral Gables, FL

Dynamic Macroeconomic Theory Notes. David L. Kelly. Department of Economics University of Miami Box Coral Gables, FL Dynamic Macroeconomic Theory Notes David L. Kelly Department of Economics University of Miami Box 248126 Coral Gables, FL 33134 dkelly@miami.edu Current Version: Fall 2013/Spring 2013 I Introduction A

More information

Chapter 3. Systems of Linear Equations: Geometry

Chapter 3. Systems of Linear Equations: Geometry Chapter 3 Systems of Linear Equations: Geometry Motiation We ant to think about the algebra in linear algebra (systems of equations and their solution sets) in terms of geometry (points, lines, planes,

More information

E 600 Chapter 4: Optimization

E 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 information

The Fundamental Welfare Theorems

The Fundamental Welfare Theorems The Fundamental Welfare Theorems The so-called Fundamental Welfare Theorems of Economics tell us about the relation between market equilibrium and Pareto efficiency. The First Welfare Theorem: Every Walrasian

More information

Seminars on Mathematics for Economics and Finance Topic 5: Optimization Kuhn-Tucker conditions for problems with inequality constraints 1

Seminars on Mathematics for Economics and Finance Topic 5: Optimization Kuhn-Tucker conditions for problems with inequality constraints 1 Seminars on Mathematics for Economics and Finance Topic 5: Optimization Kuhn-Tucker conditions for problems with inequality constraints 1 Session: 15 Aug 2015 (Mon), 10:00am 1:00pm I. Optimization with

More information

Minimize Cost of Materials

Minimize Cost of Materials Question 1: Ho do you find the optimal dimensions of a product? The size and shape of a product influences its functionality as ell as the cost to construct the product. If the dimensions of a product

More information

Recitation 2-09/01/2017 (Solution)

Recitation 2-09/01/2017 (Solution) Recitation 2-09/01/2017 (Solution) 1. Checking properties of the Cobb-Douglas utility function. Consider the utility function u(x) Y n i1 x i i ; where x denotes a vector of n di erent goods x 2 R n +,

More information

Announcements Wednesday, September 06

Announcements Wednesday, September 06 Announcements Wednesday, September 06 WeBWorK due on Wednesday at 11:59pm. The quiz on Friday coers through 1.2 (last eek s material). My office is Skiles 244 and my office hours are Monday, 1 3pm and

More information

Sometimes the domains X and Z will be the same, so this might be written:

Sometimes 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 information

f( x) f( y). Functions which are not one-to-one are often called many-to-one. Take the domain and the range to both be all the real numbers:

f( x) f( y). Functions which are not one-to-one are often called many-to-one. Take the domain and the range to both be all the real numbers: I. UNIVARIATE CALCULUS Given two sets X and Y, a function is a rule that associates each member of X with exactly one member of Y. That is, some x goes in, and some y comes out. These notations are used

More information

COS 511: Theoretical Machine Learning. Lecturer: Rob Schapire Lecture # 12 Scribe: Indraneel Mukherjee March 12, 2008

COS 511: Theoretical Machine Learning. Lecturer: Rob Schapire Lecture # 12 Scribe: Indraneel Mukherjee March 12, 2008 COS 511: Theoretical Machine Learning Lecturer: Rob Schapire Lecture # 12 Scribe: Indraneel Mukherjee March 12, 2008 In the previous lecture, e ere introduced to the SVM algorithm and its basic motivation

More information

The Dot Product

The Dot Product The Dot Product 1-9-017 If = ( 1,, 3 ) and = ( 1,, 3 ) are ectors, the dot product of and is defined algebraically as = 1 1 + + 3 3. Example. (a) Compute the dot product (,3, 7) ( 3,,0). (b) Compute the

More information

Economics 201b Spring 2010 Solutions to Problem Set 1 John Zhu

Economics 201b Spring 2010 Solutions to Problem Set 1 John Zhu Economics 201b Spring 2010 Solutions to Problem Set 1 John Zhu 1a The following is a Edgeworth box characterization of the Pareto optimal, and the individually rational Pareto optimal, along with some

More information

Economics 101. Lecture 2 - The Walrasian Model and Consumer Choice

Economics 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 information

Long run input use-input price relations and the cost function Hessian. Ian Steedman Manchester Metropolitan University

Long run input use-input price relations and the cost function Hessian. Ian Steedman Manchester Metropolitan University Long run input use-input price relations and the cost function Hessian Ian Steedman Manchester Metropolitan University Abstract By definition, to compare alternative long run equilibria is to compare alternative

More information

Microeconomic Theory. Microeconomic Theory. Everyday Economics. The Course:

Microeconomic 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 information

Adding Production to the Theory

Adding Production to the Theory Adding Production to the Theory We begin by considering the simplest situation that includes production: two goods, both of which have consumption value, but one of which can be transformed into the other.

More information

The Kuhn-Tucker and Envelope Theorems

The Kuhn-Tucker and Envelope Theorems The Kuhn-Tucker and Envelope Theorems Peter Ireland ECON 77200 - Math for Economists Boston College, Department of Economics Fall 207 The Kuhn-Tucker and envelope theorems can be used to characterize the

More information

Final Examination with Answers: Economics 210A

Final Examination with Answers: Economics 210A Final Examination with Answers: Economics 210A December, 2016, Ted Bergstrom, UCSB I asked students to try to answer any 7 of the 8 questions. I intended the exam to have some relatively easy parts and

More information

Chapter 2. Mathematical Reasoning. 2.1 Mathematical Models

Chapter 2. Mathematical Reasoning. 2.1 Mathematical Models Contents Mathematical Reasoning 3.1 Mathematical Models........................... 3. Mathematical Proof............................ 4..1 Structure of Proofs........................ 4.. Direct Method..........................

More information

Fundamentals of Operations Research. Prof. G. Srinivasan. Indian Institute of Technology Madras. Lecture No. # 15

Fundamentals of Operations Research. Prof. G. Srinivasan. Indian Institute of Technology Madras. Lecture No. # 15 Fundamentals of Operations Research Prof. G. Srinivasan Indian Institute of Technology Madras Lecture No. # 15 Transportation Problem - Other Issues Assignment Problem - Introduction In the last lecture

More information

Solving with Absolute Value

Solving with Absolute Value Solving with Absolute Value Who knew two little lines could cause so much trouble? Ask someone to solve the equation 3x 2 = 7 and they ll say No problem! Add just two little lines, and ask them to solve

More information

September Math Course: First Order Derivative

September 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 information

Constrained Optimization

Constrained 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 information

The Envelope Theorem

The 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 information

Elements of Economic Analysis II Lecture VII: Equilibrium in a Competitive Market

Elements of Economic Analysis II Lecture VII: Equilibrium in a Competitive Market Elements of Economic Analysis II Lecture VII: Equilibrium in a Competitive Market Kai Hao Yang 10/31/2017 1 Partial Equilibrium in a Competitive Market In the previous lecture, e derived the aggregate

More information

Optimization. A first course on mathematics for economists

Optimization. A first course on mathematics for economists Optimization. A first course on mathematics for economists Xavier Martinez-Giralt Universitat Autònoma de Barcelona xavier.martinez.giralt@uab.eu II.3 Static optimization - Non-Linear programming OPT p.1/45

More information

Microeconomic Theory -1- Introduction

Microeconomic Theory -1- Introduction Microeconomic Theory -- Introduction. Introduction. Profit maximizing firm with monopoly power 6 3. General results on maximizing with two variables 8 4. Model of a private ownership economy 5. Consumer

More information

( )! ±" and g( x)! ±" ], or ( )! 0 ] as x! c, x! c, x! c, or x! ±". If f!(x) g!(x) "!,

( )! ± and g( x)! ± ], or ( )! 0 ] as x! c, x! c, x! c, or x! ±. If f!(x) g!(x) !, IV. MORE CALCULUS There are some miscellaneous calculus topics to cover today. Though limits have come up a couple of times, I assumed prior knowledge, or at least that the idea makes sense. Limits are

More information

Midterm Exam - Solutions

Midterm Exam - Solutions EC 70 - Math for Economists Samson Alva Department of Economics, Boston College October 13, 011 Midterm Exam - Solutions 1 Quasilinear Preferences (a) There are a number of ways to define the Lagrangian

More information

( )( b + c) = ab + ac, but it can also be ( )( a) = ba + ca. Let s use the distributive property on a couple of

( )( b + c) = ab + ac, but it can also be ( )( a) = ba + ca. Let s use the distributive property on a couple of Factoring Review for Algebra II The saddest thing about not doing well in Algebra II is that almost any math teacher can tell you going into it what s going to trip you up. One of the first things they

More information

Econ Slides from Lecture 14

Econ 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 information

Optimization Over Time

Optimization Over Time Optimization Over Time Joshua Wilde, revised by Isabel Tecu and Takeshi Suzuki August 26, 21 Up to this point, we have only considered constrained optimization problems at a single point in time. However,

More information

FIN 550 Exam answers. A. Every unconstrained problem has at least one interior solution.

FIN 550 Exam answers. A. Every unconstrained problem has at least one interior solution. FIN 0 Exam answers Phil Dybvig December 3, 0. True-False points A. Every unconstrained problem has at least one interior solution. False. An unconstrained problem may not have any solution at all. For

More information

What is proof? Lesson 1

What is proof? Lesson 1 What is proof? Lesson The topic for this Math Explorer Club is mathematical proof. In this post we will go over what was covered in the first session. The word proof is a normal English word that you might

More information

Pareto Efficiency (also called Pareto Optimality)

Pareto Efficiency (also called Pareto Optimality) Pareto Efficiency (also called Pareto Optimality) 1 Definitions and notation Recall some of our definitions and notation for preference orderings. Let X be a set (the set of alternatives); we have the

More information

A Generalization of a result of Catlin: 2-factors in line graphs

A Generalization of a result of Catlin: 2-factors in line graphs AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 72(2) (2018), Pages 164 184 A Generalization of a result of Catlin: 2-factors in line graphs Ronald J. Gould Emory University Atlanta, Georgia U.S.A. rg@mathcs.emory.edu

More information

Study skills for mathematicians

Study skills for mathematicians PART I Study skills for mathematicians CHAPTER 1 Sets and functions Everything starts somewhere, although many physicists disagree. Terry Pratchett, Hogfather, 1996 To think like a mathematician requires

More information

Stat 8112 Lecture Notes Weak Convergence in Metric Spaces Charles J. Geyer January 23, Metric Spaces

Stat 8112 Lecture Notes Weak Convergence in Metric Spaces Charles J. Geyer January 23, Metric Spaces Stat 8112 Lecture Notes Weak Convergence in Metric Spaces Charles J. Geyer January 23, 2013 1 Metric Spaces Let X be an arbitrary set. A function d : X X R is called a metric if it satisfies the folloing

More information

Lecture 3a: The Origin of Variational Bayes

Lecture 3a: The Origin of Variational Bayes CSC535: 013 Advanced Machine Learning Lecture 3a: The Origin of Variational Bayes Geoffrey Hinton The origin of variational Bayes In variational Bayes, e approximate the true posterior across parameters

More information

Dynamic Problem Set 1 Solutions

Dynamic Problem Set 1 Solutions Dynamic Problem Set 1 Solutions Jonathan Kreamer July 15, 2011 Question 1 Consider the following multi-period optimal storage problem: An economic agent imizes: c t} T β t u(c t ) (1) subject to the period-by-period

More information

Lecture Notes: Math Refresher 1

Lecture 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 information

Race car Damping Some food for thought

Race car Damping Some food for thought Race car Damping Some food for thought The first article I ever rote for Racecar Engineering as ho to specify dampers using a dual rate damper model. This as an approach that my colleagues and I had applied

More information

Vector Spaces. Addition : R n R n R n Scalar multiplication : R R n R n.

Vector Spaces. Addition : R n R n R n Scalar multiplication : R R n R n. Vector Spaces Definition: The usual addition and scalar multiplication of n-tuples x = (x 1,..., x n ) R n (also called vectors) are the addition and scalar multiplication operations defined component-wise:

More information

where u is the decision-maker s payoff function over her actions and S is the set of her feasible actions.

where 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 information

Tutorial 3: Optimisation

Tutorial 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 information

Increasingly, economists are asked not just to study or explain or interpret markets, but to design them.

Increasingly, economists are asked not just to study or explain or interpret markets, but to design them. What is market design? Increasingly, economists are asked not just to study or explain or interpret markets, but to design them. This requires different tools and ideas than neoclassical economics, which

More information

Constrained optimization.

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 information

Differentiable Welfare Theorems Existence of a Competitive Equilibrium: Preliminaries

Differentiable Welfare Theorems Existence of a Competitive Equilibrium: Preliminaries Differentiable Welfare Theorems Existence of a Competitive Equilibrium: Preliminaries Econ 2100 Fall 2017 Lecture 19, November 7 Outline 1 Welfare Theorems in the differentiable case. 2 Aggregate excess

More information

Logic Effort Revisited

Logic Effort Revisited Logic Effort Revisited Mark This note ill take another look at logical effort, first revieing the asic idea ehind logical effort, and then looking at some of the more sutle issues in sizing transistors..0

More information

Problem Set 2 Solutions

Problem 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 information

y(x) = x w + ε(x), (1)

y(x) = x w + ε(x), (1) Linear regression We are ready to consider our first machine-learning problem: linear regression. Suppose that e are interested in the values of a function y(x): R d R, here x is a d-dimensional vector-valued

More information

Sequences and infinite series

Sequences and infinite series Sequences and infinite series D. DeTurck University of Pennsylvania March 29, 208 D. DeTurck Math 04 002 208A: Sequence and series / 54 Sequences The lists of numbers you generate using a numerical method

More information

Homework Set 2 Solutions

Homework Set 2 Solutions MATH 667-010 Introduction to Mathematical Finance Prof. D. A. Edards Due: Feb. 28, 2018 Homeork Set 2 Solutions 1. Consider the ruin problem. Suppose that a gambler starts ith ealth, and plays a game here

More information

1 Slutsky Matrix og Negative deniteness

1 Slutsky Matrix og Negative deniteness Slutsky Matrix og Negative deniteness This is exercise 2.F. from the book. Given the demand function x(p,) from the book page 23, here β = and =, e shall :. Calculate the Slutsky matrix S = D p x(p, )

More information

EconS Cost Structures

EconS Cost Structures EconS 425 - Cost Structures Eric Dunaway Washington State University eric.dunaway@wsu.edu Industrial Organization Eric Dunaway (WSU) EconS 425 Industrial Organization 1 / 34 Introduction Today, we ll review

More information

Suggested solutions to the 6 th seminar, ECON4260

Suggested solutions to the 6 th seminar, ECON4260 1 Suggested solutions to the 6 th seminar, ECON4260 Problem 1 a) What is a public good game? See, for example, Camerer (2003), Fehr and Schmidt (1999) p.836, and/or lecture notes, lecture 1 of Topic 3.

More information

STATC141 Spring 2005 The materials are from Pairwise Sequence Alignment by Robert Giegerich and David Wheeler

STATC141 Spring 2005 The materials are from Pairwise Sequence Alignment by Robert Giegerich and David Wheeler STATC141 Spring 2005 The materials are from Pairise Sequence Alignment by Robert Giegerich and David Wheeler Lecture 6, 02/08/05 The analysis of multiple DNA or protein sequences (I) Sequence similarity

More information

Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 1 / Reny, 32

Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 1 / Reny, 32 Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer Theory (Jehle and Reny, Chapter 1) Tsun-Feng Chiang* *School of Economics, Henan University, Kaifeng, China November 1, 2015 Week 7: The Consumer

More information

- a value calculated or derived from the data.

- a value calculated or derived from the data. Descriptive statistics: Note: I'm assuming you know some basics. If you don't, please read chapter 1 on your own. It's pretty easy material, and it gives you a good background as to why we need statistics.

More information

IE 5531 Midterm #2 Solutions

IE 5531 Midterm #2 Solutions IE 5531 Midterm #2 s Prof. John Gunnar Carlsson November 9, 2011 Before you begin: This exam has 9 pages and a total of 5 problems. Make sure that all pages are present. To obtain credit for a problem,

More information

Bucket handles and Solenoids Notes by Carl Eberhart, March 2004

Bucket handles and Solenoids Notes by Carl Eberhart, March 2004 Bucket handles and Solenoids Notes by Carl Eberhart, March 004 1. Introduction A continuum is a nonempty, compact, connected metric space. A nonempty compact connected subspace of a continuum X is called

More information

Mathematics Review Revised: January 9, 2008

Mathematics Review Revised: January 9, 2008 Global Economy Chris Edmond Mathematics Review Revised: January 9, 2008 Mathematics is a precise and efficient language for expressing quantitative ideas, including many that come up in business. What

More information

Second Welfare Theorem

Second Welfare Theorem Second Welfare Theorem Econ 2100 Fall 2015 Lecture 18, November 2 Outline 1 Second Welfare Theorem From Last Class We want to state a prove a theorem that says that any Pareto optimal allocation is (part

More information

The Necessity of the Transversality Condition at Infinity: A (Very) Special Case

The Necessity of the Transversality Condition at Infinity: A (Very) Special Case The Necessity of the Transversality Condition at Infinity: A (Very) Special Case Peter Ireland ECON 772001 - Math for Economists Boston College, Department of Economics Fall 2017 Consider a discrete-time,

More information

It is convenient to introduce some notation for this type of problems. I will write this as. max u (x 1 ; x 2 ) subj. to. p 1 x 1 + p 2 x 2 m ;

It is convenient to introduce some notation for this type of problems. I will write this as. max u (x 1 ; x 2 ) subj. to. p 1 x 1 + p 2 x 2 m ; 4 Calculus Review 4.1 The Utility Maimization Problem As a motivating eample, consider the problem facing a consumer that needs to allocate a given budget over two commodities sold at (linear) prices p

More information

CS 124 Math Review Section January 29, 2018

CS 124 Math Review Section January 29, 2018 CS 124 Math Review Section CS 124 is more math intensive than most of the introductory courses in the department. You re going to need to be able to do two things: 1. Perform some clever calculations to

More information

Operations and Supply Chain Management Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras

Operations and Supply Chain Management Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Operations and Supply Chain Management Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Lecture - 12 Marginal Quantity Discount, Multiple Item Inventory Constraint

More information

Bivariate Uniqueness in the Logistic Recursive Distributional Equation

Bivariate Uniqueness in the Logistic Recursive Distributional Equation Bivariate Uniqueness in the Logistic Recursive Distributional Equation Antar Bandyopadhyay Technical Report # 629 University of California Department of Statistics 367 Evans Hall # 3860 Berkeley CA 94720-3860

More information

Complex Numbers and the Complex Exponential

Complex Numbers and the Complex Exponential Complex Numbers and the Complex Exponential φ (2+i) i 2 θ φ 2+i θ 1 2 1. Complex numbers The equation x 2 + 1 0 has no solutions, because for any real number x the square x 2 is nonnegative, and so x 2

More information

1 Numbers, Sets, Algebraic Expressions

1 Numbers, Sets, Algebraic Expressions AAU - Business Mathematics I Lecture #1, February 27, 2010 1 Numbers, Sets, Algebraic Expressions 1.1 Constants, Variables, and Sets A constant is something that does not change, over time or otherwise:

More information

The Probability of Pathogenicity in Clinical Genetic Testing: A Solution for the Variant of Uncertain Significance

The Probability of Pathogenicity in Clinical Genetic Testing: A Solution for the Variant of Uncertain Significance International Journal of Statistics and Probability; Vol. 5, No. 4; July 2016 ISSN 1927-7032 E-ISSN 1927-7040 Published by Canadian Center of Science and Education The Probability of Pathogenicity in Clinical

More information

The Generalized Roy Model and Treatment Effects

The Generalized Roy Model and Treatment Effects The Generalized Roy Model and Treatment Effects Christopher Taber University of Wisconsin November 10, 2016 Introduction From Imbens and Angrist we showed that if one runs IV, we get estimates of the Local

More information

5.2 Infinite Series Brian E. Veitch

5.2 Infinite Series Brian E. Veitch 5. Infinite Series Since many quantities show up that cannot be computed exactly, we need some way of representing it (or approximating it). One way is to sum an infinite series. Recall that a n is the

More information

Is there a political support for the double burden on prolonged activity? 1

Is there a political support for the double burden on prolonged activity? 1 Is there a political support for the double burden on prolonged activity? 1 Georges Casamatta 2, Helmuth Cremer 3 and Pierre Pestieau 4 February 2004, revised November 2004 1 The paper has been presented

More information

FINANCIAL OPTIMIZATION

FINANCIAL OPTIMIZATION FINANCIAL OPTIMIZATION Lecture 1: General Principles and Analytic Optimization Philip H. Dybvig Washington University Saint Louis, Missouri Copyright c Philip H. Dybvig 2008 Choose x R N to minimize f(x)

More information

Equilibrium in a Production Economy

Equilibrium in a Production Economy Equilibrium in a Production Economy Prof. Eric Sims University of Notre Dame Fall 2012 Sims (ND) Equilibrium in a Production Economy Fall 2012 1 / 23 Production Economy Last time: studied equilibrium in

More information

x 1 + x 2 2 x 1 x 2 1 x 2 2 min 3x 1 + 2x 2

x 1 + x 2 2 x 1 x 2 1 x 2 2 min 3x 1 + 2x 2 Lecture 1 LPs: Algebraic View 1.1 Introduction to Linear Programming Linear programs began to get a lot of attention in 1940 s, when people were interested in minimizing costs of various systems while

More information

5 Quantum Wells. 1. Use a Multimeter to test the resistance of your laser; Record the resistance for both polarities.

5 Quantum Wells. 1. Use a Multimeter to test the resistance of your laser; Record the resistance for both polarities. Measurement Lab 0: Resistance The Diode laser is basically a diode junction. Same as all the other semiconductor diode junctions, e should be able to see difference in resistance for different polarities.

More information

Day 1: Over + Over Again

Day 1: Over + Over Again Welcome to Morning Math! The current time is... huh, that s not right. Day 1: Over + Over Again Welcome to PCMI! We know you ll learn a great deal of mathematics here maybe some new tricks, maybe some

More information

Quadratic Equations Part I

Quadratic Equations Part I Quadratic Equations Part I Before proceeding with this section we should note that the topic of solving quadratic equations will be covered in two sections. This is done for the benefit of those viewing

More information

Algebra Exam. Solutions and Grading Guide

Algebra Exam. Solutions and Grading Guide Algebra Exam Solutions and Grading Guide You should use this grading guide to carefully grade your own exam, trying to be as objective as possible about what score the TAs would give your responses. Full

More information

Econ Spring Review Set 1 - Answers ELEMENTS OF LOGIC. NECESSARY AND SUFFICIENT. SET THE-

Econ Spring Review Set 1 - Answers ELEMENTS OF LOGIC. NECESSARY AND SUFFICIENT. SET THE- Econ 4808 - Spring 2008 Review Set 1 - Answers ORY ELEMENTS OF LOGIC. NECESSARY AND SUFFICIENT. SET THE- 1. De ne a thing or action in words. Refer to this thing or action as A. Then de ne a condition

More information

Competitive Equilibrium

Competitive Equilibrium Competitive Equilibrium Econ 2100 Fall 2017 Lecture 16, October 26 Outline 1 Pareto Effi ciency 2 The Core 3 Planner s Problem(s) 4 Competitive (Walrasian) Equilibrium Decentralized vs. Centralized Economic

More information

8. TRANSFORMING TOOL #1 (the Addition Property of Equality)

8. TRANSFORMING TOOL #1 (the Addition Property of Equality) 8 TRANSFORMING TOOL #1 (the Addition Property of Equality) sentences that look different, but always have the same truth values What can you DO to a sentence that will make it LOOK different, but not change

More information

+ τ t R t 1B t 1 + M t 1. = R t 1B t 1 + M t 1. = λ t (1 + γ f t + γ f t v t )

+ τ t R t 1B t 1 + M t 1. = R t 1B t 1 + M t 1. = λ t (1 + γ f t + γ f t v t ) Eco504, Part II Spring 2006 C. Sims FTPL WITH MONEY 1. FTPL WITH MONEY This model is that of Sims (1994). Agent: [ ] max E β t log C t {C t,m t,b t } t=0 s.t. C t (1 + γ f (v t )) + M t + B t + τ t R t

More information

Lecture 4: Labour Economics and Wage-Setting Theory

Lecture 4: Labour Economics and Wage-Setting Theory ecture 4: abour Economics and Wage-Setting Theory Spring 203 ars Calmfors iterature: Chapter 5 Cahuc-Zylberberg (pp 257-26) Chapter 7 Cahuc-Zylberberg (pp 369-390 and 393-397) Topics The monopsony model

More information

Econ Macroeconomics I Optimizing Consumption with Uncertainty

Econ 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 information