Profit Maximization and the Profit Function

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Profit Maximization and the Profit Function"

Transcription

1 Profit Maximization and the Profit Function Juan Manuel Puerta September 30, 2009

2 Profits: Difference between the revenues a firm receives and the cost it incurs Cost should include all the relevant cost (opportunity cost) Time scales: since inputs are flows, their prices are also flows (wages per hour, rental cost of machinery) We assume firms want to maximize profits. Let a be a vector of actions a firm may take and R(a) and C(a) the Revenue and Cost functions respectively. Firms seek to max a1,a 2,...,a n R(a 1, a 2,..., a n ) C(a 1, a 2,..., a n ) The optimal set of actions a are such that R(a ) a i = C(a ) a i

3 From the Marginal Revenue=Marginal Cost there are three fundamental interpretations The actions could be: output production, labor hiring and in each the principle is MR=MC Also, it is evident that in the long-run all firms should have similar profits given that they face the same cost and revenue functions In order to explore these possibilities, we have to break up the revenue (price and quantity of output) and costs (price and quantity of inputs).

4 But the prices are going to come as a result of the market interaction subject to 2 types of constraints: Technological constraints: (whatever we saw in chapter 1) Market constraints: Given by the actions of other agents (monopoly,monopsony etc) For the time being, we assume the simplest possible behavior, i.e. firms are price-takers. Price-taking firms are also referred to as competitive firms

5 The profit maximization problem Profit Function π(p) = max py, such that y is in Y Short-run Profit Function π(p,z) = max py, such that y is in Y(z) Single-Output Profit Function π(p, w)=max pf (x) wx Single-Output Cost Function c(w, y) = min wx such that x is in V(y). Single-Output restricted Cost Function c(w, y, z) = min wx such that (y,-x) is in Y(z).

6 Profit Maximizing Behavior Single output technology: p f (x ) x i = w i for i = 1,..., n Note that these are n equations. Using matrix notation pdf(x ) = w where Df(x ) = ( x x 1,..., x x n ) is the gradient of f(x), that is, the vector of first partial derivatives. Geometric interpretation of the first order conditions Intuitive/Economic interpretation of the first order condition Second-order conditions : In the two dimensional case: 2 f (x ) x 0 2 With multi-inputs: D 2 f(x ) = ( 2 f (x ) x i x j ), the Hessian matrix is negative semidefinite.

7 Factor Demand and Supply Functions For each vector of prices (p, w), profit-maximization would normally yield a set of optimal x Factor Demand Function: The function that reflects the optimal choice of inputs given the set of input and output prices (p, w). This function is denoted x(p, w). Supply Function: The function that gives the optimal choice of output given the input prices (p,w). This is simply defined as y(p, w) = f (x(p, w)) In the previous section (and typically) we will assume that these functions exist and are well-behaved. However, there are cases in which problems may arise.

8 Problems 1 and 2 The technology cannot be described by a differentiable production function (e.g. Leontieff) There might not be an interior solution for the problem. This is typically caused by non-negativity constraints (x 0) In general, the conditions to handle the boundary solutions in case of non-negativity contraints (x 0) are the following: f (x) p x i w i 0 if x i = 0 f (x) x i w i = 0 if x i > 0 p These are the so-called Kuhn-Tucker conditions (see Alpha-Chiang or appendix Big Varian)

9 Problems 1 and 2 The technology cannot be described by a differentiable production function (e.g. Leontieff) There might not be an interior solution for the problem. This is typically caused by non-negativity constraints (x 0) In general, the conditions to handle the boundary solutions in case of non-negativity contraints (x 0) are the following: f (x) p x i w i 0 if x i = 0 f (x) x i w i = 0 if x i > 0 p These are the so-called Kuhn-Tucker conditions (see Alpha-Chiang or appendix Big Varian)

10 Problems 3 and 4 There might not be a profit maximizing plan Example: If the production function is CRS and profits are positive At first sight this may seem surprising but there are some reasons why this is intuitive: 1) As a firm grows it is likely to fall into DRS, 2) price-taking behavior becomes unrealistic and 3) other firms have incentives to imitate the first one The profit maximizing plan may not be unique (conditional factor demand set).

11 Problems 3 and 4 There might not be a profit maximizing plan Example: If the production function is CRS and profits are positive At first sight this may seem surprising but there are some reasons why this is intuitive: 1) As a firm grows it is likely to fall into DRS, 2) price-taking behavior becomes unrealistic and 3) other firms have incentives to imitate the first one The profit maximizing plan may not be unique (conditional factor demand set).

12 Properties of the Demand and Supply functions The fact that these functions are the result on optimization puts some restrictions on the possible outcomes. The factor demand function is homogenous of degree 0. Fairly intuitive, if price of output and that of all inputs increase by a x%, the optimal choice of x does not change There are both theoretical and empirical reasons to consider all the restrictions derived from maximizing behavior We examine these restrictions in three ways: 1 Checking FOC 2 Checking the properties of maximizing demand and supply functions 3 Checking the properties of the associated profit and cost functions.

13 Properties of the Demand and Supply functions The fact that these functions are the result on optimization puts some restrictions on the possible outcomes. The factor demand function is homogenous of degree 0. Fairly intuitive, if price of output and that of all inputs increase by a x%, the optimal choice of x does not change There are both theoretical and empirical reasons to consider all the restrictions derived from maximizing behavior We examine these restrictions in three ways: 1 Checking FOC 2 Checking the properties of maximizing demand and supply functions 3 Checking the properties of the associated profit and cost functions.

14 Properties of the Demand and Supply functions The fact that these functions are the result on optimization puts some restrictions on the possible outcomes. The factor demand function is homogenous of degree 0. Fairly intuitive, if price of output and that of all inputs increase by a x%, the optimal choice of x does not change There are both theoretical and empirical reasons to consider all the restrictions derived from maximizing behavior We examine these restrictions in three ways: 1 Checking FOC 2 Checking the properties of maximizing demand and supply functions 3 Checking the properties of the associated profit and cost functions.

15 Comparative Statics (single input) Comparative statics refers to the study of how optimal choices respond to changes in the economic environment. In particular, we could answer questions like: how does the optimal choice of input x 1 changes with changes in prices of p or w 1? Using the FOC and the definition of factor demand function we have: pf (x(p, w)) w 0 (1) pf (x(p, w)) w 0

16 Comparative Statics (single input) Differentiating equation 1 with respect to w we get: pf (x(p, w)) dx(p,w) dw 1 0 And assuming f is not equal to 0, there is a regular maximum, then dx(p,w) 1 dw pf (x(p,w)) This equation tells us a couple of interesting facts about factor demands 1 Since f is negative, the factor demand slopes downward 2 If the production function is very curved then factor demands do not react much to factor prices. Geometric intuition for the 1 output and 1 input case.

17 Comparative Statics (multiple inputs) In the case of two-inputs, the factor demand functions must satisfy the following first order conditions f (x 1 (w 1, w 2 ), x 2 (w 1, w 2 )) x 1 w 1 (2) f (x 1 (w 1, w 2 ), x 2 (w 1, w 2 )) x 2 w 2 (3) Differentiating these two equations with respect to w 1 we obtain f 11 x 1 w 1 + f 12 x 2 w 1 = 1 (4) f 21 x 1 w 1 + f 22 x 2 w 1 = 0 (5)

18 and similarly, differentiating these two equations with respect to w 2 x 1 x 2 f 11 + f 12 = 0 (6) w 2 w 2 writing equations (4)-(7) in matrix form, ( ) ( ) x f11 f 1 x 1 12 w 1 w 2 = f 21 f 22 f 21 x 1 w 2 + f 22 x 2 w 2 = 1 (7) x 2 w 1 x 2 w 2 ( ) If we assume a technology that has a regular maximum, then the Hessian is negative definite and, thus, the matrix is non-singular.

19 Comparative statics (cont.) Inverting the Hessian, ( x1 w 1 x 1 w 2 x 2 w 1 x 2 w 2 ) ( f11 f = 12 f 21 f 22 ) 1 Factor demand functions (substitution matrix) under (strict) conditions of optimization implies: 1 The substitution matrix is the inverse of the Hessian and, thus, negative definite. 2 Negative Slope: xi w i 0 3 Symmetric Effects: xi w j = xj w i These derivations were done for the 2-input case, it turns out that it is straightforward to generalize it to the n-input case using matrix algebra.

20 Substitution matrix for the n-input case 1 The first order conditions are Df(x(w)) w 0 2 Differentiation with respect to w yields D 2 f(x(w))dx(w) I 0 3 Which means that Dx(w) = (D 2 f(x(w))) 1 What is the empirical meaning of a negative semi-definite matrix? 1 Changes in input demand if prices vary a bit: dx = Dx(w)dw 2 Use the Hessian formula above: dx = (D 2 f(x(w))) 1 dw 3 Premultiply by dw, dwdx = dw(d 2 f(x(w))) 1 dw 4 Negative semi-definiteness means: dwdx 0

21 Properties of the Profit Function The profit function π(p) results from the profit maximization problem. i. e. π(p) = max py, such that y is in Y. This function exhibits a number of properties: 1 Non-increasing in input prices and non-decreasing in output prices. That is, if p i p i for all the outputs and p i p i for all the inputs, π(p ) π(p) 2 Homogeneous of degree 1 in p: π(tp) = tπ(p) for t 0 3 Convex in p. If Let p = tp + (1 t)p. Then π(p ) tπ(p) + (1 t)π(p ) 4 Continuous in p. When π(p) is well-defined and p i 0 for i = 1, 2,..., n

22 Proof All these properties depend only on the profit maximization assumption and are not resulting from the monotonicity, regularity or convexity assumptions we discussed earlier. The first 2 properties are intuitive, but convexity a little bit so. Economic intuition in the convexity result This predictions are useful because they allow us to check whether a firm is maximizing profits.

23 Supply and demand functions The net supply function is the net output vector that satisfy profit maximization, i.e. π(p) = py(p) It is easy to go from the supply function to the profit function. What about the other way around? Hotelling s Lemma. Let y i (p) be the firm s net supply function for good i. Then, y i (p) = π(p) p i for i=1,2,...,n 1 input/1 output case. First order conditions imply, pf (x) w = 0. By definition, the profit function is π(p, w) pf (x(p, w)) wx(p, w) Differentiation with respect to w yields, = p π(p,w) w f (x(p,w)) x(p,w) x(p,w) w [x(p, w) + w x(p,w) w ]

24 grouping by x(p,w) w f (x(p,w)) x π(p,w) w = [p w] x(p,w) w x(p, w) = x(p, w) The last equality follows from the first order conditions, which f (x(p,w)) imply p x w = 0 There are 2 effects when prices go up: 1 Direct effect: x is more expensive due to the increase in prices. 2 Indirect effect: Change in x due re-optimization. This effect is equal to 0 because we are at a profit maximizing point. Hotelling s Lemma is an application of a mathematical result: the envelope theorem

25 The envelope theorem Assume an arbitrary maximization problem M(a) = max x f (x, a) Let x(a) be the value that solves the maximization problem, M(a) = f (x(a), a). It is often interesting to check dm(a)/da The envelope theorem says that dm(a) da = f (x,a) a x=x(a) This expression means that the derivative of M with respect to a is just the derivative of the function with respect to a where x is fixed at the optimal level x(a).

26 In the 1 input/1 output case, π(p, w) = max x pf (x) wx. The a in the envelope theorem is simply (p,w), which means that π(p, w) p = f (x) x=x(p,w) = f (x(p, w)) (8) π(p, w) w = x x=x(p,w) = x(p, w) (9)

27 Comparative statics with the profit function What do the properties of the profit function mean for the net supply functions 1 π increasing in prices means net supplies positive if it is an output and negative if it is an input (this is our sign convention!). 2 π homogeneous of degree 1 (HD1) in p, means that y(p) is HD0 in prices 3 Convexity of π with respect to p means that the hessian of π ( 2 π p π p 2 p 1 2 π p 1 p 2 2 π p 2 2 ) = ( y1 y 1 p 1 p 2 y 2 y 2 p 1 p 2 is positive semi-definite and symmetric. But the Hessian of π is simply the substitution matrix. )

28 Quadratic forms and Concavity/Convexity A function is convex (concave) if and only if its Hessian is positive (negative) semi-definite A function is strictly convex (concave) if (but not only if) its Hessian is positive (negative) definite A negative (positive) definite Hessian is a sufficient second order condition for a maximum (minimum). However, a less stringent necessary SOC are often used: negative (positive) semi-definite.

Cost Minimization and the Cost Function

Cost Minimization and the Cost Function Cost Minimization and the Cost Function Juan Manuel Puerta October 5, 2009 So far we focused on profit maximization, we could look at a different problem, that is the cost minimization problem. This is

More information

Multi-variable Calculus and Optimization

Multi-variable Calculus and Optimization Multi-variable Calculus and Optimization Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 1 / 51 EC2040 Topic 3 - Multi-variable Calculus

More information

INTERMEDIATE MICROECONOMICS MATH REVIEW

INTERMEDIATE MICROECONOMICS MATH REVIEW INTERMEDIATE MICROECONOMICS MATH REVIEW August 31, 2008 OUTLINE 1. Functions Definition Inverse functions Convex and Concave functions 2. Derivative of Functions of One variable Definition Rules for finding

More information

Profit Maximization under Perfect Competition

Profit Maximization under Perfect Competition ECON 1110 Rajiv Vohra Profit Maximization under Perfect Competition Perfect Competition A firm in a perfectly competitive market knows it has no influence over the market price for its product because

More information

The Envelope Theorem 1

The Envelope Theorem 1 John Nachbar Washington University April 2, 2015 1 Introduction. The Envelope Theorem 1 The Envelope theorem is a corollary of the Karush-Kuhn-Tucker theorem (KKT) that characterizes changes in the value

More information

Profit Maximization. PowerPoint Slides prepared by: Andreea CHIRITESCU Eastern Illinois University

Profit Maximization. PowerPoint Slides prepared by: Andreea CHIRITESCU Eastern Illinois University Profit Maximization PowerPoint Slides prepared by: Andreea CHIRITESCU Eastern Illinois University 1 The Nature and Behavior of Firms A firm An association of individuals Firms Who have organized themselves

More information

Walrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.

Walrasian Demand. u(x) where B(p, w) = {x R n + : p x w}. Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian

More information

CITY AND REGIONAL PLANNING 7230. Consumer Behavior. Philip A. Viton. March 4, 2015. 1 Introduction 2

CITY AND REGIONAL PLANNING 7230. Consumer Behavior. Philip A. Viton. March 4, 2015. 1 Introduction 2 CITY AND REGIONAL PLANNING 7230 Consumer Behavior Philip A. Viton March 4, 2015 Contents 1 Introduction 2 2 Foundations 2 2.1 Consumption bundles........................ 2 2.2 Preference relations.........................

More information

Lecture Notes on Elasticity of Substitution

Lecture Notes on Elasticity of Substitution Lecture Notes on Elasticity of Substitution Ted Bergstrom, UCSB Economics 210A March 3, 2011 Today s featured guest is the elasticity of substitution. Elasticity of a function of a single variable Before

More information

Mathematical Economics - Part I - Optimization. Filomena Garcia. Fall Optimization. Filomena Garcia. Optimization. Existence of Solutions

Mathematical Economics - Part I - Optimization. Filomena Garcia. Fall Optimization. Filomena Garcia. Optimization. Existence of Solutions of Mathematical Economics - Part I - Fall 2009 of 1 in R n Problems in Problems: Some of 2 3 4 5 6 7 in 8 Continuity: The Maximum Theorem 9 Supermodularity and Monotonicity of An optimization problem in

More information

Lecture Notes on Elasticity of Substitution

Lecture Notes on Elasticity of Substitution Lecture Notes on Elasticity of Substitution Ted Bergstrom, UCSB Economics 20A October 26, 205 Today s featured guest is the elasticity of substitution. Elasticity of a function of a single variable Before

More information

or, put slightly differently, the profit maximizing condition is for marginal revenue to equal marginal cost:

or, put slightly differently, the profit maximizing condition is for marginal revenue to equal marginal cost: Chapter 9 Lecture Notes 1 Economics 35: Intermediate Microeconomics Notes and Sample Questions Chapter 9: Profit Maximization Profit Maximization The basic assumption here is that firms are profit maximizing.

More information

Definition and Properties of the Production Function: Lecture

Definition and Properties of the Production Function: Lecture Definition and Properties of the Production Function: Lecture II August 25, 2011 Definition and : Lecture A Brief Brush with Duality Cobb-Douglas Cost Minimization Lagrangian for the Cobb-Douglas Solution

More information

Increasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.

Increasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all. 1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.

More information

Envelope Theorem. Kevin Wainwright. Mar 22, 2004

Envelope Theorem. Kevin Wainwright. Mar 22, 2004 Envelope Theorem Kevin Wainwright Mar 22, 2004 1 Maximum Value Functions A maximum (or minimum) value function is an objective function where the choice variables have been assigned their optimal values.

More information

Readings. D Chapter 1. Lecture 2: Constrained Optimization. Cecilia Fieler. Example: Input Demand Functions. Consumer Problem

Readings. D Chapter 1. Lecture 2: Constrained Optimization. Cecilia Fieler. Example: Input Demand Functions. Consumer Problem Economics 245 January 17, 2012 : Example Readings D Chapter 1 : Example The FOCs are max p ( x 1 + x 2 ) w 1 x 1 w 2 x 2. x 1,x 2 0 p 2 x i w i = 0 for i = 1, 2. These are two equations in two unknowns,

More information

Math 2331 Linear Algebra

Math 2331 Linear Algebra 2.2 The Inverse of a Matrix Math 2331 Linear Algebra 2.2 The Inverse of a Matrix Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math2331 Jiwen He, University

More information

Monopoly and Monopsony Labor Market Behavior

Monopoly and Monopsony Labor Market Behavior Monopoly and Monopsony abor Market Behavior 1 Introduction For the purposes of this handout, let s assume that firms operate in just two markets: the market for their product where they are a seller) and

More information

Expenditure Minimisation Problem

Expenditure Minimisation Problem Expenditure Minimisation Problem This Version: September 20, 2009 First Version: October, 2008. The expenditure minimisation problem (EMP) looks at the reverse side of the utility maximisation problem

More information

Sample Exam According to Figure 6.1, A. Soup is a normal good C. Soup is a Giffen good B. Soup is an inferior good D. Bread is an inferior good

Sample Exam According to Figure 6.1, A. Soup is a normal good C. Soup is a Giffen good B. Soup is an inferior good D. Bread is an inferior good Sample Exam 2 1. Suppose the base year for a Lespeyres index is 2001. The value of the index is 1.3 in 2004 and 1.6 in 2006. By how much did the cost of the bundle increase between 2004 and 2006? A..3%

More information

Monotonic transformations: Cardinal Versus Ordinal Utility

Monotonic transformations: Cardinal Versus Ordinal Utility Natalia Lazzati Mathematics for Economics (Part I) Note 1: Quasiconcave and Pseudoconcave Functions Note 1 is based on Madden (1986, Ch. 13, 14) and Simon and Blume (1994, Ch. 21). Monotonic transformations:

More information

Firm Supply: Market Structure & Perfect Competition

Firm Supply: Market Structure & Perfect Competition Firm Supply: Market Structure & Perfect Competition Firm Supply How does a firm decide how much to supply at a given price? This depends upon the firm s goals; technology; market environment; and competitors

More information

Constrained optimization.

Constrained optimization. ams/econ 11b supplementary notes ucsc Constrained optimization. c 2010, Yonatan Katznelson 1. Constraints In many of the optimization problems that arise in economics, there are restrictions on the values

More information

The Cobb-Douglas Production Function

The Cobb-Douglas Production Function 171 10 The Cobb-Douglas Production Function This chapter describes in detail the most famous of all production functions used to represent production processes both in and out of agriculture. First used

More information

Maths for Economics. Geoff Renshaw with contributions from Norman Ireland OXPORD UNIVERSITY PRESS. Second Edition

Maths for Economics. Geoff Renshaw with contributions from Norman Ireland OXPORD UNIVERSITY PRESS. Second Edition Maths for Economics Second Edition Geoff Renshaw with contributions from Norman Ireland OXPORD UNIVERSITY PRESS Brief contents Detailed contents About the author About the book How to use the book Chapter

More information

THEORY OF SIMPLEX METHOD

THEORY OF SIMPLEX METHOD Chapter THEORY OF SIMPLEX METHOD Mathematical Programming Problems A mathematical programming problem is an optimization problem of finding the values of the unknown variables x, x,, x n that maximize

More information

ME128 Computer-Aided Mechanical Design Course Notes Introduction to Design Optimization

ME128 Computer-Aided Mechanical Design Course Notes Introduction to Design Optimization ME128 Computer-ided Mechanical Design Course Notes Introduction to Design Optimization 2. OPTIMIZTION Design optimization is rooted as a basic problem for design engineers. It is, of course, a rare situation

More information

More is Better. an investigation of monotonicity assumption in economics. Joohyun Shon. August 2008

More is Better. an investigation of monotonicity assumption in economics. Joohyun Shon. August 2008 More is Better an investigation of monotonicity assumption in economics Joohyun Shon August 2008 Abstract Monotonicity of preference is assumed in the conventional economic theory to arrive at important

More information

NAME: INTERMEDIATE MICROECONOMIC THEORY SPRING 2008 ECONOMICS 300/010 & 011 Midterm II April 30, 2008

NAME: INTERMEDIATE MICROECONOMIC THEORY SPRING 2008 ECONOMICS 300/010 & 011 Midterm II April 30, 2008 NAME: INTERMEDIATE MICROECONOMIC THEORY SPRING 2008 ECONOMICS 300/010 & 011 Section I: Multiple Choice (4 points each) Identify the choice that best completes the statement or answers the question. 1.

More information

General Equilibrium Theory: Examples

General Equilibrium Theory: Examples General Equilibrium Theory: Examples 3 examples of GE: pure exchange (Edgeworth box) 1 producer - 1 consumer several producers and an example illustrating the limits of the partial equilibrium approach

More information

SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA

SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA This handout presents the second derivative test for a local extrema of a Lagrange multiplier problem. The Section 1 presents a geometric motivation for the

More information

Consumer Theory. The consumer s problem

Consumer Theory. The consumer s problem Consumer Theory The consumer s problem 1 The Marginal Rate of Substitution (MRS) We define the MRS(x,y) as the absolute value of the slope of the line tangent to the indifference curve at point point (x,y).

More information

Lecture 5 Principal Minors and the Hessian

Lecture 5 Principal Minors and the Hessian Lecture 5 Principal Minors and the Hessian Eivind Eriksen BI Norwegian School of Management Department of Economics October 01, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and

More information

By W.E. Diewert. July, Linear programming problems are important for a number of reasons:

By W.E. Diewert. July, Linear programming problems are important for a number of reasons: APPLIED ECONOMICS By W.E. Diewert. July, 3. Chapter : Linear Programming. Introduction The theory of linear programming provides a good introduction to the study of constrained maximization (and minimization)

More information

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a real-valued

More information

MATH MODULE 11. Maximizing Total Net Benefit. 1. Discussion M11-1

MATH MODULE 11. Maximizing Total Net Benefit. 1. Discussion M11-1 MATH MODULE 11 Maximizing Total Net Benefit 1. Discussion In one sense, this Module is the culminating module of this Basic Mathematics Review. In another sense, it is the starting point for all of the

More information

Convex analysis and profit/cost/support functions

Convex analysis and profit/cost/support functions CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m

More information

Name. Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions.

Name. Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions. Name Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions. Question 1. A firm has a production function F (x 1, x 2 ) = ( x 1 + x 2 ) 2. It is a price

More information

Advanced Topics in Machine Learning (Part II)

Advanced Topics in Machine Learning (Part II) Advanced Topics in Machine Learning (Part II) 3. Convexity and Optimisation February 6, 2009 Andreas Argyriou 1 Today s Plan Convex sets and functions Types of convex programs Algorithms Convex learning

More information

Lecture 4: Equality Constrained Optimization. Tianxi Wang

Lecture 4: Equality Constrained Optimization. Tianxi Wang Lecture 4: Equality Constrained Optimization Tianxi Wang wangt@essex.ac.uk 2.1 Lagrange Multiplier Technique (a) Classical Programming max f(x 1, x 2,..., x n ) objective function where x 1, x 2,..., x

More information

Tangent and normal lines to conics

Tangent and normal lines to conics 4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

More information

POTENTIAL OUTPUT and LONG RUN AGGREGATE SUPPLY

POTENTIAL OUTPUT and LONG RUN AGGREGATE SUPPLY POTENTIAL OUTPUT and LONG RUN AGGREGATE SUPPLY Aggregate Supply represents the ability of an economy to produce goods and services. In the Long-run this ability to produce is based on the level of production

More information

1 Homogenous and Homothetic Functions

1 Homogenous and Homothetic Functions 1 Homogenous and Homothetic Functions Reading: [Simon], Chapter 20, p. 483-504. 1.1 Homogenous Functions Definition 1 A real valued function f(x 1,..., x n ) is homogenous of degree k if for all t > 0

More information

Lecture 3: Convex Sets and Functions

Lecture 3: Convex Sets and Functions EE 227A: Convex Optimization and Applications January 24, 2012 Lecture 3: Convex Sets and Functions Lecturer: Laurent El Ghaoui Reading assignment: Chapters 2 (except 2.6) and sections 3.1, 3.2, 3.3 of

More information

Economics 401. Sample questions 1

Economics 401. Sample questions 1 Economics 40 Sample questions. Do each of the following choice structures satisfy WARP? (a) X = {a, b, c},b = {B, B 2, B 3 }, B = {a, b}, B 2 = {b, c}, B 3 = {c, a}, C (B ) = {a}, C (B 2 ) = {b}, C (B

More information

Chapter 11 Perfect Competition

Chapter 11 Perfect Competition Chapter 11 Perfect Competition Perfect Competition Conditions for Perfectly competitive markets Product firms are perfect substitutes (homogeneous product) Firms are price takers Reasonable with many firms,

More information

GRA6035 Mathematics. Eivind Eriksen and Trond S. Gustavsen. Department of Economics

GRA6035 Mathematics. Eivind Eriksen and Trond S. Gustavsen. Department of Economics GRA635 Mathematics Eivind Eriksen and Trond S. Gustavsen Department of Economics c Eivind Eriksen, Trond S. Gustavsen. Edition. Edition Students enrolled in the course GRA635 Mathematics for the academic

More information

5. Economic costs are synonymous with A) Accounting costs B) Sunk costs C) Opportunity costs D) Implicit costs

5. Economic costs are synonymous with A) Accounting costs B) Sunk costs C) Opportunity costs D) Implicit costs Ch. 7 1. Opportunity cost for a firm is A) Costs that involve a direct monetary outlay B) The sum of the firm's implicit costs C) The total of explicit costs that have been incurred in the past D) The

More information

Partial Equilibrium: Positive Analysis

Partial Equilibrium: Positive Analysis Partial Equilibrium: Positive Analysis This Version: November 28, 2009 First Version: December 1, 2008. In this Chapter we consider consider the interaction between different agents and firms, and solve

More information

constraint. Let us penalize ourselves for making the constraint too big. We end up with a

constraint. Let us penalize ourselves for making the constraint too big. We end up with a Chapter 4 Constrained Optimization 4.1 Equality Constraints (Lagrangians) Suppose we have a problem: Maximize 5, (x 1, 2) 2, 2(x 2, 1) 2 subject to x 1 +4x 2 =3 If we ignore the constraint, we get the

More information

Section 12.6: Directional Derivatives and the Gradient Vector

Section 12.6: Directional Derivatives and the Gradient Vector Section 26: Directional Derivatives and the Gradient Vector Recall that if f is a differentiable function of x and y and z = f(x, y), then the partial derivatives f x (x, y) and f y (x, y) give the rate

More information

Game Theory: Supermodular Games 1

Game Theory: Supermodular Games 1 Game Theory: Supermodular Games 1 Christoph Schottmüller 1 License: CC Attribution ShareAlike 4.0 1 / 22 Outline 1 Introduction 2 Model 3 Revision questions and exercises 2 / 22 Motivation I several solution

More information

Shiqian Ma, SEEM 5121, Dept. of SEEM, CUHK 1. Chapter 2. Convex Optimization

Shiqian Ma, SEEM 5121, Dept. of SEEM, CUHK 1. Chapter 2. Convex Optimization Shiqian Ma, SEEM 5121, Dept. of SEEM, CUHK 1 Chapter 2 Convex Optimization Shiqian Ma, SEEM 5121, Dept. of SEEM, CUHK 2 2.1. Convex Optimization General optimization problem: min f 0 (x) s.t., f i (x)

More information

Microeconomics: CH 9-10 Take home quiz.

Microeconomics: CH 9-10 Take home quiz. Microeconomics: CH 9-10 Take home quiz. Mark your answers on a Scantron BEFORE class. Bring your Scantron to Class On Monday, November 26. Be sure to be on time, late Scantron forms will be penalized.

More information

Math 313 Lecture #10 2.2: The Inverse of a Matrix

Math 313 Lecture #10 2.2: The Inverse of a Matrix Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is

More information

Lecture 6 Part I. Markets without market power: Perfect competition

Lecture 6 Part I. Markets without market power: Perfect competition Lecture 6 Part I Markets without market power: Perfect competition Market power Market power: Ability to control, or at least affect, the terms and conditions of the exchanges in which one participates

More information

Chapter 9 Profit Maximization

Chapter 9 Profit Maximization Chater 9 Profit Maximization Economic theory normally uses the rofit maximization assumtion in studying the firm just as it uses the utility maximization assumtion for the individual consumer. This aroach

More information

c 2009 Je rey A. Miron

c 2009 Je rey A. Miron Lecture 22: Factor Markets c 2009 Je rey A. Miron Outline 1. Introduction 2. Monopoly in the Output Market 3. Monopsony 4. Upstream and Downstream Monopolies 1 Introduction The analysis in earlier lectures

More information

Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions

Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions

More information

Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions

Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions D. R. Wilkins Copyright c David R. Wilkins 2016 Contents 3 Functions 43 3.1 Functions between Sets...................... 43 3.2 Injective

More information

Sample Midterm Solutions

Sample Midterm Solutions Sample Midterm Solutions Instructions: Please answer both questions. You should show your working and calculations for each applicable problem. Correct answers without working will get you relatively few

More information

A Detailed Price Discrimination Example

A Detailed Price Discrimination Example A Detailed Price Discrimination Example Suppose that there are two different types of customers for a monopolist s product. Customers of type 1 have demand curves as follows. These demand curves include

More information

An increase in the number of students attending college. shifts to the left. An increase in the wage rate of refinery workers.

An increase in the number of students attending college. shifts to the left. An increase in the wage rate of refinery workers. 1. Which of the following would shift the demand curve for new textbooks to the right? a. A fall in the price of paper used in publishing texts. b. A fall in the price of equivalent used text books. c.

More information

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

More information

Economics 2020a / HBS 4010 / HKS API-111 FALL 2010 Solutions to Practice Problems for Lectures 1 to 4

Economics 2020a / HBS 4010 / HKS API-111 FALL 2010 Solutions to Practice Problems for Lectures 1 to 4 Economics 00a / HBS 4010 / HKS API-111 FALL 010 Solutions to Practice Problems for Lectures 1 to 4 1.1. Quantity Discounts and the Budget Constraint (a) The only distinction between the budget line with

More information

Managerial Economics & Business Strategy Chapter 8. Managing in Competitive, Monopolistic, and Monopolistically Competitive Markets

Managerial Economics & Business Strategy Chapter 8. Managing in Competitive, Monopolistic, and Monopolistically Competitive Markets Managerial Economics & Business Strategy Chapter 8 Managing in Competitive, Monopolistic, and Monopolistically Competitive Markets I. Perfect Competition Overview Characteristics and profit outlook. Effect

More information

Lecture 8: Market Structure and Competitive Strategy. Managerial Economics September 11, 2014

Lecture 8: Market Structure and Competitive Strategy. Managerial Economics September 11, 2014 Lecture 8: Market Structure and Competitive Strategy Managerial Economics September 11, 2014 Focus of This Lecture Examine optimal price and output decisions of managers operating in environments with

More information

Profit Maximization. 2. product homogeneity

Profit Maximization. 2. product homogeneity Perfectly Competitive Markets It is essentially a market in which there is enough competition that it doesn t make sense to identify your rivals. There are so many competitors that you cannot single out

More information

2.3 Convex Constrained Optimization Problems

2.3 Convex Constrained Optimization Problems 42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

More information

Review of Likelihood Theory

Review of Likelihood Theory Appendix A Review of Likelihood Theory This is a brief summary of some of the key results we need from likelihood theory. A.1 Maximum Likelihood Estimation Let Y 1,..., Y n be n independent random variables

More information

Advanced Microeconomics

Advanced Microeconomics Advanced Microeconomics Ordinal preference theory Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 68 Part A. Basic decision and preference theory 1 Decisions

More information

Economics 326: Duality and the Slutsky Decomposition. Ethan Kaplan

Economics 326: Duality and the Slutsky Decomposition. Ethan Kaplan Economics 326: Duality and the Slutsky Decomposition Ethan Kaplan September 19, 2011 Outline 1. Convexity and Declining MRS 2. Duality and Hicksian Demand 3. Slutsky Decomposition 4. Net and Gross Substitutes

More information

Describe the characteristics of different market structures: perfect competition, monopolistic competition, oligopoly, and pure monopoly

Describe the characteristics of different market structures: perfect competition, monopolistic competition, oligopoly, and pure monopoly www.edupristine.com Describe the characteristics of different market structures: perfect competition, monopolistic competition, oligopoly, and pure monopoly Prerequisite Characteristics of different market

More information

Lecture 2: Consumer Theory

Lecture 2: Consumer Theory Lecture 2: Consumer Theory Preferences and Utility Utility Maximization (the primal problem) Expenditure Minimization (the dual) First we explore how consumers preferences give rise to a utility fct which

More information

Short-Run Production and Costs

Short-Run Production and Costs Short-Run Production and Costs The purpose of this section is to discuss the underlying work of firms in the short-run the production of goods and services. Why is understanding production important to

More information

Chapter 5 The Production Process and Costs

Chapter 5 The Production Process and Costs Managerial Economics & Business Strategy Chapter 5 The Production Process and Costs McGraw-Hill/Irwin Copyright 2010 by the McGraw-Hill Companies, Inc. All rights reserved. Overview I. Production Analysis

More information

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

In economics, the amount of a good x demanded is a function of a person s wealth and the price of that good. In other words, LABOR NOTES, PART TWO: REVIEW OF MATH 2.1 Univariate calculus Given two sets X andy, a function is a rule that associates each member of X with exactly one member ofy. Intuitively, y is a function of x

More information

TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

More information

HOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba

HOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba HOMEWORK SOLUTIONS All questions are from Vector Calculus, by Marsden and Tromba Question :..6 Let w = f(x, y) be a function of two variables, and let x = u + v, y = u v. Show that Solution. By the chain

More information

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1. MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

More information

Price Competition Under Product Differentiation

Price Competition Under Product Differentiation Competition Under Product Differentiation Chapter 10: Competition 1 Introduction In a wide variety of markets firms compete in prices Internet access Restaurants Consultants Financial services Without

More information

Mathematics for Business Economics. Herbert Hamers, Bob Kaper, John Kleppe

Mathematics for Business Economics. Herbert Hamers, Bob Kaper, John Kleppe Mathematics for Business Economics Herbert Hamers, Bob Kaper, John Kleppe Mathematics for Business Economics Herbert Hamers Bob Kaper John Kleppe You can obtain further information about this and other

More information

Gauss-Markov Theorem. The Gauss-Markov Theorem is given in the following regression model and assumptions:

Gauss-Markov Theorem. The Gauss-Markov Theorem is given in the following regression model and assumptions: Gauss-Markov Theorem The Gauss-Markov Theorem is given in the following regression model and assumptions: The regression model y i = β 1 + β x i + u i, i = 1,, n (1) Assumptions (A) or Assumptions (B):

More information

Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand

Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such

More information

Cost of Living Indexes and Exact Index Numbers Revised, August 6, 2009.

Cost of Living Indexes and Exact Index Numbers Revised, August 6, 2009. 1 Cost of Living Indexes and Exact Index Numbers Revised, August 6, 2009. W. Erwin Diewert, 1 Discussion Paper 09-06, Department of Economics, University of British Columbia, Vancouver, Canada, V6T 1Z1.

More information

Fixed Point Theorems

Fixed Point Theorems Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation

More information

Cost OVERVIEW. WSG6 7/7/03 4:36 PM Page 79. Copyright 2003 by Academic Press. All rights of reproduction in any form reserved.

Cost OVERVIEW. WSG6 7/7/03 4:36 PM Page 79. Copyright 2003 by Academic Press. All rights of reproduction in any form reserved. WSG6 7/7/03 4:36 PM Page 79 6 Cost OVERVIEW The previous chapter reviewed the theoretical implications of the technological process whereby factors of production are efficiently transformed into goods

More information

A FIRST COURSE IN OPTIMIZATION THEORY

A FIRST COURSE IN OPTIMIZATION THEORY A FIRST COURSE IN OPTIMIZATION THEORY RANGARAJAN K. SUNDARAM New York University CAMBRIDGE UNIVERSITY PRESS Contents Preface Acknowledgements page xiii xvii 1 Mathematical Preliminaries 1 1.1 Notation

More information

In this section, we will consider techniques for solving problems of this type.

In this section, we will consider techniques for solving problems of this type. Constrained optimisation roblems in economics typically involve maximising some quantity, such as utility or profit, subject to a constraint for example income. We shall therefore need techniques for solving

More information

Kuhn-Tucker theorem foundations and its application in mathematical economics

Kuhn-Tucker theorem foundations and its application in mathematical economics MPRA Munich Personal RePEc Archive Kuhn-Tucker theorem foundations and its application in mathematical economics Dushko Josheski and Elena Gelova University Goce Delcev-Stip 12. October 2013 Online at

More information

ECONOMIC THEORY AND OPERATIONS ANALYSIS

ECONOMIC THEORY AND OPERATIONS ANALYSIS WILLIAM J. BAUMOL Professor of Economics Princeton University ECONOMIC THEORY AND OPERATIONS ANALYSIS Second Edition Prentice-Hall, I Inc. Engkwood Cliffs, New Jersey CONTENTS PART 7 ANALYTIC TOOLS OF

More information

Intermediate Microeconomics Video Handbook, IM6 Faculty, 2014

Intermediate Microeconomics Video Handbook, IM6 Faculty, 2014 A. MATH TOPICS 1. Calculus Review Machina 2. Elasticity Machina 3. Level Curves Machina a. Level Curves of a Function Machina b. Slopes of Level Curves Machina 4. Scale Properties of Functions a. Scale

More information

Producer Theory. Chapter 5

Producer Theory. Chapter 5 Chapter 5 Producer Theory Markets have two sides: consumers and producers. Up until now we have been studying the consumer side of the market. We now begin our study of the producer side of the market.

More information

LINE INTEGRALS OF VECTOR FUNCTIONS: GREEN S THEOREM. Contents. 2. Green s Theorem 3

LINE INTEGRALS OF VECTOR FUNCTIONS: GREEN S THEOREM. Contents. 2. Green s Theorem 3 LINE INTEGRALS OF VETOR FUNTIONS: GREEN S THEOREM ontents 1. A differential criterion for conservative vector fields 1 2. Green s Theorem 3 1. A differential criterion for conservative vector fields We

More information

Choose the single best answer for each question. Do all of your scratch-work in the side and bottom margins of pages.

Choose the single best answer for each question. Do all of your scratch-work in the side and bottom margins of pages. Econ 0, Sections 3 and 4, S, Schroeter Exam #4, Special code = 000 Choose the single best answer for each question. Do all of your scratch-work in the side and bottom margins of pages.. Gordon is the owner

More information

Inducing Teamwork in Rank-Order Tournaments

Inducing Teamwork in Rank-Order Tournaments Inducing Teamwork in Rank-Order Tournaments Vikram Ahuja Dmitry Lubensky January 10, 2013 Abstract We analyze the design of an optimal contract in a two-person partnership in which all the firm s proceeds

More information

ECONOMICS 101 WINTER 2004 TA LECTURE 5: GENERAL EQUILIBRIUM

ECONOMICS 101 WINTER 2004 TA LECTURE 5: GENERAL EQUILIBRIUM ECONOMICS 0 WINTER 2004 TA LECTURE 5: GENERAL EQUILIBRIUM KENNETH R. AHERN. Production Possibility Frontier We now move from partial equilibrium, where we examined only one market at a time, to general

More information

EconS 301 Review Session #8 Chapter 11: Monopoly and Monopsony

EconS 301 Review Session #8 Chapter 11: Monopoly and Monopsony EconS 301 Review Session #8 Chapter 11: Monopoly and Monopsony 1. Which of the following describes a correct relation between price elasticity of demand and a monopolist s marginal revenue when inverse

More information

The Classical Linear Regression Model

The Classical Linear Regression Model The Classical Linear Regression Model 1 September 2004 A. A brief review of some basic concepts associated with vector random variables Let y denote an n x l vector of random variables, i.e., y = (y 1,

More information

Bertrand with complements

Bertrand with complements Microeconomics, 2 nd Edition David Besanko and Ron Braeutigam Chapter 13: Market Structure and Competition Prepared by Katharine Rockett Dieter Balkenborg Todd Kaplan Miguel Fonseca Bertrand with complements

More information