# Consumer Theory: The Mathematical Core

Size: px
Start display at page:

## Transcription

1 Consumer Theory: The Mathematical Core Dan McFadden, C13 Suppose an individual has a utility function U(x) which is a function of non-negative commodity vectors x = (x 1,x,...,x N ), and seeks to maximize this utility function subject to the budget constraint = p 1 x 1 + p x p N x N # y, where y is income and p = (p 1,p,...,p N ) is the vector of commodity prices. Define u = V(p,y) to be the value of utility attained by solving this problem; V is called the indirect utility function. Let x = X(p,y) = (X 1 (p,y),x (p,y),...,x N (p,y)) be the commodity vector that achieves the utility maximum subject to this budget constraint. The function X(p,y) is called this consumer s market or Marshallian demand function. Since this function maximizes utility subject to the budget constraint, V(p,y) / U(X(p,y)) / U(X 1 (p,y),x (p,y),...,x N (p,y)). The figure below shows for the case of two goods the budget line d-e, and the point a that maximizes utility. 1 Note that the point a is on the highest utility contour (or indifference curve) that touches the budget line, and that at a the indifference curve is tangent to the budget line, so that its slope, the marginal rate of substitution (MRS) is equal to the slope of the budget line, -p 1 /p. Note that the indirect utility function and the market demand functions all depend on income and on the price vector. For example, the demand for good 1, written out, 1 is x 1 = X 1 (p,y). When demand for good 1 is graphed against its own price p 1, then 1 changes in own price p 1 correspond to movement along the graphed demand curve, d while changes in income y or in the crossprice p correspond to shifts in the graphed a demand curve. If all income is spent, then e / y. The consumer is said to be 1 1 locally non-satiated if in each neighborhood good 1 of a commodity vector there is always another that has strictly higher utility. When the consumer is locally non-satiated, no commodity vector that costs less than y can give the maximum obtainable utility level, the utility-maximizing consumer will spend all income,, and the indirect utility function V(p,y) is strictly increasing in y. Local non-satiation rules out fat indifference curves. A sufficient condition for local non-satiation is that the utility function be continuous and monotone increasing; i.e., if xn>> xo, then U(xN) > U(xO). good 1 The graph was created using the utility function U(x) = x 1 1/3 x /3. The budget line d-e is given by 9x x = 13. The demand functions that maximize utility are X 1 (p 1,p,y) = y/3p 1 and X (p 1,p,y) = y/3p. The indirect utility function is V(p 1,p,y) = y(3p 1 ) -1/3 (3p /) -/3. 1

2 If income and all prices are scaled by a constant, the budget set and the maximum attainable utility remain the same. This implies that V(p,y) and X(p,y) are unchanged by rescaling of income and prices. Functions with this property are said to be homogeneous of degree zero in income and prices. Given utility level u, let y = M(p,u) be the minimum income needed to buy a commodity vector that gives utility u; M is called the expenditure function. Let x = H(p,u) / (H 1 (p,u),h (p,u),...,h N (p,u)) be the commodity vector that achieves the minimum expenditure subject to the constraint that a utility level u be attained. Then, M(p,u) / p"h(p,u) / p 1 H 1 (p,u) p N H N (p,u). The demand functions H(p,u) are called the Hicksian, constant utility, or compensated demand functions. If all prices are rescaled by a constant, then the commodity vector solving the minimum income problem is unchanged, so that H(p,u) is homogeneous of degree zero in p. The expenditure is scaled up by the same constant as the prices, and is said to be linear homogeneous in p. Suppose in the figure above, instead of fixing the budget line and locating the point a where the highest indifference curve touches this line, we had fixed the indifference curve and varied the level of income for given prices to find the minimum income necessary to touch this indifference curve. Provided the consumer is locally non-satiated so that all income is spent, this minimum again occurs at a. Then, solving the utility maximization problem for a fixed y and attaining a maximum utility level u, and solving the expenditure minimization problem for this u and attaining the income y, lead to the same point a. In this case, if the consumer utility level is u = V(p,y), then y is the minimum income at which utility level u can be achieved. Then, the result that the utility maximization problem and the expenditure minimization problem pick out the same point a implies that y / M(p,V(p,y)) and u / V(p,M(p,u)); i.e., V and M are inverses of each other for fixed p. Further, X(p,y) / H(p,V(p,y)) and X(p,M(p,u)) / H(p,u). The demand for a commodity is said to be normal if demand does not fall when income rises, and inferior or regressive if demand falls when income rises. A commodity is a luxury good if its budget share rises when income rises, and is otherwise a necessary good. The budget share of the first commodity is s 1 = p 1 X 1 (p,y)/y. Define the income elasticity of demand, = (y/x 1 1 (p,y)/my. This is the percentage by which demand for good 1 increases when income goes up by one percent. Show as an exercise that the income elasticity of the budget share satisfies (y/ s 1 1 /My= - 1.

3 Then, a luxury good has an income elasticity of its budget share greater than zero, a necessary good has an income elasticity of the budget share less than zero, and an inferior good has an income elasticity of the budget share less than -1. A graph of the demand for a good against income is called an Engle curve. The figure below shows the Engle curves for three cases. Budget Share of Good Necessary Engle Curves Regressive The income elasticities of demand at I = are -.33 for Regressive,.8 for Necessary, 1. for Luxury. Luxury 1 3 Income It is possible to trace out the locus of demand points in an indifference curve map as income changes with prices fixed; this locus is called an income-offer curve or income-expansion path. Points on an income-expansion path correspond to points on Engle curves for each of the commodities. The figure below shows an income-expansion path when good 1 is a luxury good. Locus of Demand Points Good 3 Income-Expansion Path Good 1 3

4 In an indifference curve map, it is also possible to trace out the locus of demand points as the price of a good changes; this locus is called an price offer curve. The figure below depicts a typical price offer curve. Note that the price-offer curve is the locus of tangencies between indifference curves and budget lines that pivot about one point on the vertical axis, in this case (,). As one moves out along the offer curve, one is identifying quantities demanded of good 1 as its price falls. In this diagram, falling price always results in increased demand for good 1. This is called the law of demand, but a qualification is needed to make this law hold, as described below. Locus of Demand Points Good 3 Offer Curve Good 1 If one plots the demand for good 1 against its own price, one obtains the standard picture of a demand function, given in the next figure. Remember that this is the locus of points satisfying x 1 = X 1 (p,y) as p 1 changes, with y and p,...,p N kept fixed. If y or p,...,p N were to change, this would appear as a shift in the demand function given in the picture. It is often useful to characterize demand functions in terms of their own-price elasticity, defined as, = (p 1 /X 1 1 (p)/mp 1. In this definition,, is negative in the usual case that the demand function slopes downward. When, < -1, demand is said to be elastic, and when -1 <, <, demand is said to be inelastic. Verify as an exercise that the own-price elasticity of the budget share of good 1 is 1 +,. Price of Good 1 Demand Function Quantity of Good 1

5 If one looks in more detail in an indifference curve map at the change in demand for a good resulting from a decrease in its price, one can decompose the total change into the effects of a pure income change, with relative prices fixed, and a pure price change, with income compensated to keep utility constant. The first of these effects is termed an income effect, and the second is termed a substitution effect. What we will show is that the substitution effect always operates to increase the demand for a good whose price falls. We will show that for a normal good, the income effect reinforces the substitution effect, so that falling price increases demand. However, for an inferior good, we will show that the substitution and income effects work in opposite directions. It is even possible for the income effect to be larger in magnitude than the income effect, so that a fall in price reduces demand. In these circumstances, the good is called a Giffen good. The occurrence of Giffen goods is primarily a curiosity rather than a common event, and requires that a good have a large share of the budget and be strongly inferior. An example of a Giffen good is day-old bread, which may absorb most of the income of a homeless person. If the price of day-old bread falls, this consumer needs to commit less income to subsistence on day-old bread, and may instead buy more fresh bread. The figure below shows the decomposition of a total price effect into income and substitution effects. In this figure, the good is normal, so that the income and substitution effects are reinforcing. In the diagram, start from the budget line d-e, for which utility is maximized at a. Now decrease the price of good 1 so the budget line becomes d-f. On this new budget line, utility is maximized at c. The total price effect is a-c. In this case, the total own price effect is to increase demand for good 1 (from a to c) when the price of good 1 falls. Now introduce the budget g-h which keeps prices the same as the were at d-e, but compensates income so that maximum utility (attained at b) is the same as it is at c. Then, a-b is the income effect, and b-c is the substitution effect. The substitution effect always operates in the direction of increasing demand for the good whose price is decreasing. The income effect also increases demand for good 1 in this case, since it is a normal good. However, if good 1 had been an inferior good, the income effect would partially (or completely) offset the substitution effect. good 1 g 1 d a b 1 1 h good 1 c e f

6 The statement we obtained geometrically that the substitution effect always increases the demand for a good whose price is falling can also be established mathematically, with great generality. The remainder of these notes provide a simple mathematical exposition of this analysis. From the definition of the expenditure function, for any vector q, and g a small scalar, or M(p+gq,u) / (p+gq)"h(p+gq,u) # (p+gq)"h(p,u) = M(p,u) + gq"h(p,u), M(p+gq,u) - M(p,u) # gq"h(p,u). This implies lim g` (M(p+gq,u) - M(p,u))/g # q"h(p,u) # lim g_ (M(p+gq,u) - M(p,u))/g. But the left and right hand ends of this inequality both converge to q"l p M(p,u) when the derivative exists, establishing that q"h(p,u) = q"l p M(p,u). This is true for any q, so H(p,u) / L p M(p,u). This is called Shephard s identity. From this, X(p,y) / H(p,V(p,y)) / L p M(p,u). u=v(p,y). From the definition of the expenditure function, M(p+q.u) = (p+q)"h(p+q.u) = p"h(p+q,u) + q"h(p+q,u) \$ M(p,u) + M(q,u). A linear homogeneous function that satisfies this inequality is said to be concave. Mathematical implications of concavity are that M will be continuous and will almost always have first and second derivatives, the mixed second partial derivatives will be independent of the order of differentiation, and the own second partial derivatives will be non-positive. As a consequence, the Hicksian demand function exists and satisfies Shephard s identity almost everywhere in p, without any requirement that the utility function be quasi-concave. Consider budgets(pn,yn) and (po,yo), and the convex combination p* = pn+(1-)po and y* = yn+(1-)yo with < < 1. We now demonstrate that V(p*,y*) # max{v(pn,yn),v(po,yo)}. A function with this property is said to be quasi-convex. Suppose x* maximizes utility subject to the constraint # y*, so that V(p*,y*) = U(x*). Then #, implying that either # yn and hence U(x*) # V(pN,yN), or else # yo and hence U(x*) # V(pO,yO). Therefore, V(p*,y*) = U(x*) # max{v(pn,yn),v(po,yo)}, establishing the result. From the identity y / M(p,V(p,y)) and the result H(p,u) = L p M(p,u), one can take derivatives with respect to y and p to obtain 1 / L u M(p,u)). u=v(p,y) V y (p,y) and / L u M(p,u). p V(p,y) + L p M(p,u). u=v(p,y). 6

7 Then, using the first of these equations to eliminate the term L u M(p,u). u=v(p,y) in the second gives a relationship between the indirect utility function and the Marshallian demand functions that is termed Roy s identity, X 1 (p,y) / H 1 (p,v(p,y)) / MM(p,u)/Mp 1. u=v(p,y) / -(MV(p,y)/Mp 1 )/(MV(p,y)/My). From the identity X 1 (p,y) / H 1 (p,v(p,y)), by taking derivatives one obtains the relationships L y X 1 (p,y) / L u H 1 (p,u). u=v(p,y) L y V(p,y) and L p X 1 (p,y) / L u H 1 (p,u). u=v(p,y) L p V(p,y) + L p H 1 (p,u). u=v(p,y). Use the first equation to obtain L u H 1 = L y X 1 /L y V, and then substitute this into the second equation to obtain L p X 1 (p,y) / [L p V(p,y)/L y V(p,y)]L y X 1 (p,y) + L p H 1 (p,u). u=v(p,y) / - X(p,y)L I X 1 (p,y) + L p H 1 (p,u). u=v(p,y). Written out, this equation decomposes the overall effect of own price on market demand into a substitution effect MH 1 (p,u)/mp 1. u=v(p,y) = M M(p,u)/Mp 1. u=v(p,y), which corresponds to the change in demand when income is compensated to keep the consumer on the same indifference curve, and is always non-positive, and an income effect, which for the own price change is - X 1 (p,y)l y X 1 (p,y). If commodity 1 is normal, its change with income, L y X 1 (p,y), is non-negative, and the income effect reinforces the non-positive substitution effect so that demand for commodity 1 falls when its price rises, and the law of demand holds. If commodity 1 is inferior, its change with income is negative, and the income effect tends to offset the non-positive substitution effect. Usually, the income effect for an inferior good is not strong enough to completely offset the substitution effect. An interesting variant of the classical consumer demand problem occurs when instead of considering fixed income y when prices change, one considers income y = A(p,t) that is a function of prices and of policies (denoted by t) that control taxes and transfers which can alter income. For example, one might consider a consumer who has an initial endowment of goods T, and receives income A(p,t) = from selling this endowment in the market. Endowments of leisure that are sold as labor are a leading case. If one decomposes the effect on demand of a change in price into substitution and income effects, one gets as before a substitution effect that never increases demand for a good when its price rises, but now the income effect has the pattern given in the following table: Income Effect Consumer Net Demander of Good Consumer Net Supplier of Good Normal Good Reinforces substitution effect Offsets substitution effect Inferior Good Offsets substitution effect Reinforces substitution effect 7

### 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

### 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

### 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

### 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

### 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.........................

### 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

### Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay. Lecture - 13 Consumer Behaviour (Contd )

(Refer Slide Time: 00:28) Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay Lecture - 13 Consumer Behaviour (Contd ) We will continue our discussion

### Slutsky Equation. M. Utku Ünver Micro Theory. Boston College. M. Utku Ünver Micro Theory (BC) Slutsky Equation 1 / 15

Slutsky Equation M. Utku Ünver Micro Theory Boston College M. Utku Ünver Micro Theory (BC) Slutsky Equation 1 / 15 Effects of a Price Change: What happens when the price of a commodity decreases? 1 The

### Elasticity. Definition of the Price Elasticity of Demand: Formula for Elasticity: Types of Elasticity:

Elasticity efinition of the Elasticity of emand: The law of demand states that the quantity demanded of a good will vary inversely with the price of the good during a given time period, but it does not

### Demand. Lecture 3. August 2015. Reading: Perlo Chapter 4 1 / 58

Demand Lecture 3 Reading: Perlo Chapter 4 August 2015 1 / 58 Introduction We saw the demand curve in chapter 2. We learned about consumer decision making in chapter 3. Now we bridge the gap between the

### 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

### Theory of Demand. ECON 212 Lecture 7. Tianyi Wang. Winter 2013. Queen s Univerisity. Tianyi Wang (Queen s Univerisity) Lecture 7 Winter 2013 1 / 46

Theory of Demand ECON 212 Lecture 7 Tianyi Wang Queen s Univerisity Winter 2013 Tianyi Wang (Queen s Univerisity) Lecture 7 Winter 2013 1 / 46 Intro Note: Quiz 1 can be picked up at Distribution Center.

### Microeconomic Theory: Basic Math Concepts

Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts

### 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.

### AK 4 SLUTSKY COMPENSATION

AK 4 SLUTSKY COMPENSATION ECON 210 A. JOSEPH GUSE (1) (a) First calculate the demand at the original price p b = 2 b(p b,m) = 1000 20 5p b b 0 = b(2) = 40 In general m c = m+(p 1 b p0 b )b 0. If the price

### 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.

### MERSİN UNIVERSITY FACULTY OF ECONOMICS AND ADMINISTRATIVE SCİENCES DEPARTMENT OF ECONOMICS MICROECONOMICS MIDTERM EXAM DATE 18.11.

MERSİN UNIVERSITY FACULTY OF ECONOMICS AND ADMINISTRATIVE SCİENCES DEPARTMENT OF ECONOMICS MICROECONOMICS MIDTERM EXAM DATE 18.11.2011 TİIE 12:30 STUDENT NAME AND NUMBER MULTIPLE CHOICE. Choose the one

### 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

### Chapter 27: Taxation. 27.1: Introduction. 27.2: The Two Prices with a Tax. 27.2: The Pre-Tax Position

Chapter 27: Taxation 27.1: Introduction We consider the effect of taxation on some good on the market for that good. We ask the questions: who pays the tax? what effect does it have on the equilibrium

### Problem Set #5-Key. Economics 305-Intermediate Microeconomic Theory

Problem Set #5-Key Sonoma State University Economics 305-Intermediate Microeconomic Theory Dr Cuellar (1) Suppose that you are paying your for your own education and that your college tuition is \$200 per

### 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).

### Review of Fundamental Mathematics

Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

### REVIEW OF MICROECONOMICS

ECO 352 Spring 2010 Precepts Weeks 1, 2 Feb. 1, 8 REVIEW OF MICROECONOMICS Concepts to be reviewed Budget constraint: graphical and algebraic representation Preferences, indifference curves. Utility function

### Theoretical Tools of Public Economics. Part-2

Theoretical Tools of Public Economics Part-2 Previous Lecture Definitions and Properties Utility functions Marginal utility: positive (negative) if x is a good ( bad ) Diminishing marginal utility Indifferences

### Economics 121b: Intermediate Microeconomics Problem Set 2 1/20/10

Dirk Bergemann Department of Economics Yale University s by Olga Timoshenko Economics 121b: Intermediate Microeconomics Problem Set 2 1/20/10 This problem set is due on Wednesday, 1/27/10. Preliminary

### 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

### 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

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

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MBA 640 Survey of Microeconomics Fall 2006, Quiz 6 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A monopoly is best defined as a firm that

### Practice Questions Week 3 Day 1

Practice Questions Week 3 Day 1 Figure 4-1 Quantity Demanded \$ 2 18 3 \$ 4 14 4 \$ 6 10 5 \$ 8 6 6 \$10 2 8 Price Per Pair Quantity Supplied 1. Figure 4-1 shows the supply and demand for socks. If a price

### CONSUMER PREFERENCES THE THEORY OF THE CONSUMER

CONSUMER PREFERENCES The underlying foundation of demand, therefore, is a model of how consumers behave. The individual consumer has a set of preferences and values whose determination are outside the

### Elasticity. I. What is Elasticity?

Elasticity I. What is Elasticity? The purpose of this section is to develop some general rules about elasticity, which may them be applied to the four different specific types of elasticity discussed in

### G021 Microeconomics Lecture notes Ian Preston

G021 Microeconomics Lecture notes Ian Preston 1 Consumption set and budget set The consumption set X is the set of all conceivable consumption bundles q, usually identified with R n + The budget set B

### 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.

### Lecture notes for Choice Under Uncertainty

Lecture notes for Choice Under Uncertainty 1. Introduction In this lecture we examine the theory of decision-making under uncertainty and its application to the demand for insurance. The undergraduate

### Managerial Economics

Managerial Economics Unit 1: Demand Theory Rudolf Winter-Ebmer Johannes Kepler University Linz Winter Term 2012/13 Winter-Ebmer, Managerial Economics: Unit 1 - Demand Theory 1 / 54 OBJECTIVES Explain the

### Microeconomics Instructor Miller Practice Problems Labor Market

Microeconomics Instructor Miller Practice Problems Labor Market 1. What is a factor market? A) It is a market where financial instruments are traded. B) It is a market where stocks and bonds are traded.

### Chapter 6. Elasticity: The Responsiveness of Demand and Supply

Chapter 6. Elasticity: The Responsiveness of Demand and Supply Instructor: JINKOOK LEE Department of Economics / Texas A&M University ECON 202 504 Principles of Microeconomics Elasticity Demand curve:

### Tastes and Indifference Curves

C H A P T E R 4 Tastes and Indifference Curves This chapter begins a -chapter treatment of tastes or what we also call preferences. In the first of these chapters, we simply investigate the basic logic

### 2011 Pearson Education. Elasticities of Demand and Supply: Today add elasticity and slope, cross elasticities

2011 Pearson Education Elasticities of Demand and Supply: Today add elasticity and slope, cross elasticities What Determines Elasticity? Influences on the price elasticity of demand fall into two categories:

### PART II THEORY OF CONSUMER BEHAVIOR AND DEMAND

1 PART II THEORY OF CONSUMER BEHAVIOR AND DEMAND 2 CHAPTER 5 MARSHALL S ANALYSIS OF DEMAND Initially Alfred Marshall initially worked with objective demand curves. However by working backwards, he developed

### Intermediate Microeconomics (22014)

Intermediate Microeconomics (22014) I. Consumer Instructor: Marc Teignier-Baqué First Semester, 2011 Outline Part I. Consumer 1. umer 1.1 Budget Constraints 1.2 Preferences 1.3 Utility Function 1.4 1.5

### INTRODUCTORY MICROECONOMICS

INTRODUCTORY MICROECONOMICS UNIT-I PRODUCTION POSSIBILITIES CURVE The production possibilities (PP) curve is a graphical medium of highlighting the central problem of 'what to produce'. To decide what

### Demand and Consumer Behavior emand is a model of consumer behavior. It attempts to identify the factors

R. Larry Reynolds Demand and Consumer Behavior R. Larry Reynolds (005) Demand and Consumer Behavior emand is a model of consumer behavior. It attempts to identify the factors D that influence the choices

### Problem Set #3 Answer Key

Problem Set #3 Answer Key Economics 305: Macroeconomic Theory Spring 2007 1 Chapter 4, Problem #2 a) To specify an indifference curve, we hold utility constant at ū. Next, rearrange in the form: C = ū

### Figure 4-1 Price Quantity Quantity Per Pair Demanded Supplied \$ 2 18 3 \$ 4 14 4 \$ 6 10 5 \$ 8 6 6 \$10 2 8

Econ 101 Summer 2005 In-class Assignment 2 & HW3 MULTIPLE CHOICE 1. A government-imposed price ceiling set below the market's equilibrium price for a good will produce an excess supply of the good. a.

### Market Structure: Perfect Competition and Monopoly

WSG8 7/7/03 4:34 PM Page 113 8 Market Structure: Perfect Competition and Monopoly OVERVIEW One of the most important decisions made by a manager is how to price the firm s product. If the firm is a profit

### The Point-Slope Form

7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

### Insurance. Michael Peters. December 27, 2013

Insurance Michael Peters December 27, 2013 1 Introduction In this chapter, we study a very simple model of insurance using the ideas and concepts developed in the chapter on risk aversion. You may recall

### SUPPLY AND DEMAND : HOW MARKETS WORK

SUPPLY AND DEMAND : HOW MARKETS WORK Chapter 4 : The Market Forces of and and demand are the two words that economists use most often. and demand are the forces that make market economies work. Modern

### Understanding the Slutsky Decomposition: Substitution & Income Effect

Understanding the Slutsky Decomposition: Substitution & Income Effect age 1 lacement of the Final Bundle when p : Substitute or Complement Goods? egion A egion B egion C BC 2 S When p, BC rotates inwards

### a. Meaning: The amount (as a percentage of total) that quantity demanded changes as price changes. b. Factors that make demand more price elastic

Things to know about elasticity. 1. Price elasticity of demand a. Meaning: The amount (as a percentage of total) that quantity demanded changes as price changes. b. Factors that make demand more price

### CHAPTER 10 MARKET POWER: MONOPOLY AND MONOPSONY

CHAPTER 10 MARKET POWER: MONOPOLY AND MONOPSONY EXERCISES 3. A monopolist firm faces a demand with constant elasticity of -.0. It has a constant marginal cost of \$0 per unit and sets a price to maximize

### U = x 1 2. 1 x 1 4. 2 x 1 4. What are the equilibrium relative prices of the three goods? traders has members who are best off?

Chapter 7 General Equilibrium Exercise 7. Suppose there are 00 traders in a market all of whom behave as price takers. Suppose there are three goods and the traders own initially the following quantities:

### 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

### CHAPTER 3 CONSUMER BEHAVIOR

CHAPTER 3 CONSUMER BEHAVIOR EXERCISES 2. Draw the indifference curves for the following individuals preferences for two goods: hamburgers and beer. a. Al likes beer but hates hamburgers. He always prefers

### Chapter 4 Online Appendix: The Mathematics of Utility Functions

Chapter 4 Online Appendix: The Mathematics of Utility Functions We saw in the text that utility functions and indifference curves are different ways to represent a consumer s preferences. Calculus can

### 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

economicsentrance.weebly.com Basic Exercises Micro Economics AKG 09 Table of Contents MICRO ECONOMICS Budget Constraint... 4 Practice problems... 4 Answers... 4 Supply and Demand... 7 Practice Problems...

### Elasticity and Its Application

Elasticity and Its Application Chapter 5 All rights reserved. Copyright 2001 by Harcourt, Inc. Requests for permission to make copies of any part of the work should be mailed to: Permissions Department,

### Examples on Monopoly and Third Degree Price Discrimination

1 Examples on Monopoly and Third Degree Price Discrimination This hand out contains two different parts. In the first, there are examples concerning the profit maximizing strategy for a firm with market

### 2013 MBA Jump Start Program

2013 MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Algebra Review Calculus Permutations and Combinations [Online Appendix: Basic Mathematical Concepts] 2 1 Equation of

### 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

### Problems: Table 1: Quilt Dress Quilts Dresses Helen 50 10 1.8 9 Carolyn 90 45 1 2

Problems: Table 1: Labor Hours needed to make one Amount produced in 90 hours: Quilt Dress Quilts Dresses Helen 50 10 1.8 9 Carolyn 90 45 1 2 1. Refer to Table 1. For Carolyn, the opportunity cost of 1

### Chapter 9: Perfect Competition

Chapter 9: Perfect Competition Perfect Competition Law of One Price Short-Run Equilibrium Long-Run Equilibrium Maximize Profit Market Equilibrium Constant- Cost Industry Increasing- Cost Industry Decreasing-

### Price Elasticity of Supply; Consumer Preferences

1 Price Elasticity of Supply 1 14.01 Principles of Microeconomics, Fall 2007 Chia-Hui Chen September 12, 2007 Lecture 4 Price Elasticity of Supply; Consumer Preferences Outline 1. Chap 2: Elasticity -

### Notes on indifference curve analysis of the choice between leisure and labor, and the deadweight loss of taxation. Jon Bakija

Notes on indifference curve analysis of the choice between leisure and labor, and the deadweight loss of taxation Jon Bakija This example shows how to use a budget constraint and indifference curve diagram

### Pre-Test Chapter 18 ed17

Pre-Test Chapter 18 ed17 Multiple Choice Questions 1. (Consider This) Elastic demand is analogous to a and inelastic demand to a. A. normal wrench; socket wrench B. Ace bandage; firm rubber tie-down C.

### Constrained Optimisation

CHAPTER 9 Constrained Optimisation Rational economic agents are assumed to make choices that maximise their utility or profit But their choices are usually constrained for example the consumer s choice

### LIST OF MEMBERS WHO PREPARED QUESTION BANK FOR ECONOMICS FOR CLASS XII TEAM MEMBERS. Sl. No. Name Designation

LIST OF MEMBERS WHO PREPARED QUESTION BANK FOR ECONOMICS FOR CLASS XII TEAM MEMBERS Sl. No. Name Designation 1. Mrs. Neelam Vinayak V. Principal (Team Leader) G.G.S.S. Deputy Ganj, Sadar Bazar Delhi-110006

### Market for cream: P 1 P 2 D 1 D 2 Q 2 Q 1. Individual firm: W Market for labor: W, S MRP w 1 w 2 D 1 D 1 D 2 D 2

Factor Markets Problem 1 (APT 93, P2) Two goods, coffee and cream, are complements. Due to a natural disaster in Brazil that drastically reduces the supply of coffee in the world market the price of coffee

### Chapter 5 Elasticity of Demand and Supply. These slides supplement the textbook, but should not replace reading the textbook

Chapter 5 Elasticity of Demand and Supply These slides supplement the textbook, but should not replace reading the textbook 1 What is total revenue? Price multiplied by the quantity sold at that price

### AP Microeconomics Chapter 12 Outline

I. Learning Objectives In this chapter students will learn: A. The significance of resource pricing. B. How the marginal revenue productivity of a resource relates to a firm s demand for that resource.

### 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

### Elasticity: The Responsiveness of Demand and Supply

Chapter 6 Elasticity: The Responsiveness of Demand and Supply Chapter Outline 61 LEARNING OBJECTIVE 61 The Price Elasticity of Demand and Its Measurement Learning Objective 1 Define the price elasticity

### Chapter 3 Consumer Behavior

Chapter 3 Consumer Behavior Read Pindyck and Rubinfeld (2013), Chapter 3 Microeconomics, 8 h Edition by R.S. Pindyck and D.L. Rubinfeld Adapted by Chairat Aemkulwat for Econ I: 2900111 1/29/2015 CHAPTER

### Preferences. M. Utku Ünver Micro Theory. Boston College. M. Utku Ünver Micro Theory (BC) Preferences 1 / 20

Preferences M. Utku Ünver Micro Theory Boston College M. Utku Ünver Micro Theory (BC) Preferences 1 / 20 Preference Relations Given any two consumption bundles x = (x 1, x 2 ) and y = (y 1, y 2 ), the

### Deriving Demand Functions - Examples 1

Deriving Demand Functions - Examples 1 What follows are some examples of different preference relations and their respective demand functions. In all the following examples, assume we have two goods x

### Practice Problem Set 2 (ANSWERS)

Economics 370 Professor H.J. Schuetze Practice Problem Set 2 (ANSWERS) 1. See the figure below, where the initial budget constraint is given by ACE. After the new legislation is passed, the budget constraint

### Productioin OVERVIEW. WSG5 7/7/03 4:35 PM Page 63. Copyright 2003 by Academic Press. All rights of reproduction in any form reserved.

WSG5 7/7/03 4:35 PM Page 63 5 Productioin OVERVIEW This chapter reviews the general problem of transforming productive resources in goods and services for sale in the market. A production function is the

### Demand, Supply and Elasticity

Demand, Supply and Elasticity CHAPTER 2 OUTLINE 2.1 Demand and Supply Definitions, Determinants and Disturbances 2.2 The Market Mechanism 2.3 Changes in Market Equilibrium 2.4 Elasticities of Supply and

### Principles of Economics: Micro: Exam #2: Chapters 1-10 Page 1 of 9

Principles of Economics: Micro: Exam #2: Chapters 1-10 Page 1 of 9 print name on the line above as your signature INSTRUCTIONS: 1. This Exam #2 must be completed within the allocated time (i.e., between

### 1. Briefly explain what an indifference curve is and how it can be graphically derived.

Chapter 2: Consumer Choice Short Answer Questions 1. Briefly explain what an indifference curve is and how it can be graphically derived. Answer: An indifference curve shows the set of consumption bundles

### Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS

DUSP 11.203 Frank Levy Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS These notes have three purposes: 1) To explain why some simple calculus formulae are useful in understanding

### 1 Mathematical Models of Cost, Revenue and Profit

Section 1.: Mathematical Modeling Math 14 Business Mathematics II Minh Kha Goals: to understand what a mathematical model is, and some of its examples in business. Definition 0.1. Mathematical Modeling

### Week 1: Functions and Equations

Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.1-2.2, and Chapter

### 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

### Econ 100A: Intermediate Microeconomics Notes on Consumer Theory

Econ 100A: Interediate Microeconoics Notes on Consuer Theory Linh Bun Winter 2012 (UCSC 1. Consuer Theory Utility Functions 1.1. Types of Utility Functions The following are soe of the type of the utility

### Long-Run Average Cost. Econ 410: Micro Theory. Long-Run Average Cost. Long-Run Average Cost. Economies of Scale & Scope Minimizing Cost Mathematically

Slide 1 Slide 3 Econ 410: Micro Theory & Scope Minimizing Cost Mathematically Friday, November 9 th, 2007 Cost But, at some point, average costs for a firm will tend to increase. Why? Factory space and

### ANSWER KEY 3 UTILITY FUNCTIONS, THE CONSUMER S PROBLEM, DEMAND CURVES

ANSWER KEY 3 UTILITY FUNCTIONS, THE CONSUMER S PROBLEM, DEMAND CURVES ECON 210 (1) Perfect Substitutes. Suppose that Jack s utility is entirely based on number of hours spent camping (c) and skiing (s).

### Elements of a graph. Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section Elements of a graph Linear equations and their graphs What is slope? Slope and y-intercept in the equation of a line Comparing lines on

### 17. Suppose demand is given by Q d = 400 15P + I, where Q d is quantity demanded, P is. I = 100, equilibrium quantity is A) 15 B) 20 C) 25 D) 30

Ch. 2 1. A relationship that shows the quantity of goods that consumers are willing to buy at different prices is the A) elasticity B) market demand curve C) market supply curve D) market equilibrium 2.

### Elasticity. Ratio of Percentage Changes. Elasticity and Its Application. Price Elasticity of Demand. Price Elasticity of Demand. Elasticity...

Elasticity and Its Application Chapter 5 All rights reserved. Copyright 21 by Harcourt, Inc. Requests for permission to make copies of any part of the work should be mailed to: Permissions Department,

### In this chapter, you will learn to use cost-volume-profit analysis.

2.0 Chapter Introduction In this chapter, you will learn to use cost-volume-profit analysis. Assumptions. When you acquire supplies or services, you normally expect to pay a smaller price per unit as the

### Problem Set II: budget set, convexity

Problem Set II: budget set, convexity Paolo Crosetto paolo.crosetto@unimi.it Exercises will be solved in class on January 25th, 2010 Recap: Walrasian Budget set, definition Definition (Walrasian budget

### Employment and Pricing of Inputs

Employment and Pricing of Inputs Previously we studied the factors that determine the output and price of goods. In chapters 16 and 17, we will focus on the factors that determine the employment level