Consumer Theory: The Mathematical Core

Save this PDF as:

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

Practice Question ECON 203. Intermediate Microeconomics. Utility and Choice

Practice Question ECON 203 Intermediate Microeconomics Utility and Choice 1. As long as the principle of diminishing marginal utility is operating, any increased consumption of a good a. lowers total utility.

Two aspects of an elasticity are important: (1) whether it positive or negative and (2) whether it is greater than 1 or less than 1 in absolute value

Overview I. The Elasticity Concept - Own Price Elasticity - Elasticity and Total Revenue - Cross-Price Elasticity - Income Elasticity II. Demand Functions - Linear - Log-Linear II. Regression Analysis

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

When the price of a good rises (falls) and all other variables affecting the

ONLINE APPENDIX 1 Substitution and Income Effects of a Price Change When the price of a good rises (falls) and all other variables affecting the consumer remain the same, the law of demand tells us that

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.........................

Econ 101: Principles of Microeconomics Fall 2012

Problem 1: Use the following graph to answer the questions. a. From the graph, which good has the price change? Did the price go down or up? What is the fraction of the new price relative to the original

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

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

Economics 326: Marshallian Demand and Comparative Statics. Ethan Kaplan

Economics 326: Marshallian Demand and Comparative Statics Ethan Kaplan September 17, 2012 Outline 1. Utility Maximization: General Formulation 2. Marshallian Demand 3. Homogeneity of Degree Zero of Marshallian

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

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%

UNIT 2: CONSUMER EQUILIBRIUM AND DEMAND

KEY CONCEPTS UNIT 2: CONSUMER EQUILIBRIUM AND DEMAND 1. UTILITY A) MARGINAL UTILITY B) LAW OF DIMINISHING MARGINAL UTILITY 2. CONDITIONS OF CONSUMER S EQUILIBRIUM 3. INDIFFERENCE CURVE ANALYSIS 4. THE

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

1 st Exam. 7. Cindy's cross-price elasticity of magazine demand with respect to the price of books is

1 st Exam 1. Marginal utility measures: A) the total utility of all your consumption B) the total utility divided by the price of the good C) the increase in utility from consuming one additional unit

Consider a consumer living for two periods, and trying to decide how much money to spend today, and how much to save for tomorrow.

Consider a consumer living for two periods, and trying to decide how much money to spend today, and how much to save for tomorrow.. Suppose that the individual has income of >0 today, and will receive

Recitation #5 Week 02/08/2009 to 02/14/2009. Chapter 6 - Elasticity

Recitation #5 Week 02/08/2009 to 02/14/2009 Chapter 6 - Elasticity 1. This problem explores the midpoint method of calculating percentages and why this method is the preferred method when calculating price

Indifference Curves and the Marginal Rate of Substitution

Introduction Introduction to Microeconomics Indifference Curves and the Marginal Rate of Substitution In microeconomics we study the decisions and allocative outcomes of firms, consumers, households and

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

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

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.

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

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

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

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

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

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

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).

Econ 110: Introduction to Economic Theory. 7th Class 2/4/11. recall the consumer's solution of their optimization problem from last time

Econ 110: Introduction to Economic Theory 7th Class 2/4/11 go over problem answers from last time continue with our discussion of consumer theory recall the consumer's solution of their optimization problem

ELASTICITY AND ITS APPLICATION

5 ELASTICITY AND ITS APPLICATION CHAPTER OUTLINE: I. The Elasticity of Demand A. Definition of elasticity: a measure of the responsiveness of quantity demanded or quantity supplied to one of its determinants.

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

Substitution and Income Effect, Individual and Market Demand, Consumer Surplus. 1 Substitution Effect, Income Effect, Giffen Goods

Substitution Effect, Income Effect, Giffen Goods 4.0 Principles of Microeconomics, Fall 007 Chia-Hui Chen September 9, 007 Lecture 7 Substitution and Income Effect, Individual and Market Demand, Consumer

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.

Measuring Elasticity of Demand

C H A P T E R E I G H T P r i c e e l a s t i c i t y o f d e m a n d Measuring Elasticity of Demand Demand curves can have many different shapes and so it is important to derive a way to convey their

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.

Economics 101 Fall 2013 Answers to Homework 5 Due Tuesday, November 19, 2013

Economics 101 Fall 2013 Answers to Homework 5 Due Tuesday, November 19, 2013 Directions: The homework will be collected in a box before the lecture. Please place your name, TA name and section number on

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

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

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

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

Public Goods : (b) Efficient Provision of Public Goods. Efficiency and Private Goods

Public Goods : (b) Efficient Provision of Public Goods Efficiency and Private Goods Suppose that there are only two goods consumed in an economy, and that they are both pure private goods. Suppose as well

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

Price Elasticity of Demand & Supply

13 10 Price Elasticity of Demand & Supply A. Price elasticity of demand Elastic demand (Ed > 1) % change in quantity demanded > % change in price Inelastic demand (Ed < 1) % change in quantity demanded

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

CHAPTER 5 Applying Consumer Theory

CHAPTER 5 Applying Consumer Theory CHAPTER OUTLINE 5.1 Deriving Demand Curves 5.2 How Income Changes Shift Demand Curves A Rise in Income Shifts the Demand Curve Consumer Theory and Income Elasticities

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

10 : Theory of Demand

10 : Theory of Demand 1 Recap from last session Change in Demand Supply, Law of Supply Market Equilibrium Change in Equilibrium 2 A Shift in Both Supply and Demand Price of Ice-Cream Cone Large increase

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

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

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.

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

Exam: Principles Micro ECO 2023.U07 Fall 2009 Name

Exam: Principles Micro ECO 2023.U07 Fall 2009 Name Panther ID Instructions: 1. Please write in your name and Panther ID on the question paper. 2. Please Bubble in your name and Panther ID on the Scantron

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

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

8 POSSIBILITIES, PREFERENCES, AND CHOICES. Chapter. Consumption Possibilities

Chapter 8 POSSIBILITIES, PREFERENCES, AND CHOICES Consumption Possibilities 1) Budget lines are drawn on a diagram with the A) price of the good on the vertical axis and its quantity on the horizontal

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:

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

Economics Homework 3 Fall 2006 Stacy Dickert-Conlin

Economics 0 - Homework Fall 00 Stacy Dickert-Conlin nswer Key. ndy collects baseball and football cards. The following graph shows a few of his indifference curves. The price of a pack of baseball cards

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

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

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

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

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

UNIT 6 cont PRICING UNDER DIFFERENT MARKET STRUCTURES. Monopolistic Competition

UNIT 6 cont PRICING UNDER DIFFERENT MARKET STRUCTURES Monopolistic Competition Market Structure Perfect Competition Pure Monopoly Monopolistic Competition Oligopoly Duopoly Monopoly The further right on

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

Introduction. Agents have preferences over the two goods which are determined by a utility function. Speci cally, type 1 agents utility is given by

Introduction General equilibrium analysis looks at how multiple markets come into equilibrium simultaneously. With many markets, equilibrium analysis must take explicit account of the fact that changes

BUSINESS ECONOMICS CEC2 532-751 & 761 PRACTICE MICROECONOMICS MULTIPLE CHOICE QUESTIONS Warning: These questions have been posted to give you an opportunity to practice with the multiple choice format

10/1/2011 25% 25% 25% 25% Impact of lower price on total consumer expenditures or a firm s total revenue

Micro Chapter 7 Consumer Choice and Elasticity This chapter is an extension of the first part of Chapter 3 on demand and consumer theory Refer back to your Chapter 3 notes and mentally combine them with

Module 2 Lecture 5 Topics

Module 2 Lecture 5 Topics 2.13 Recap of Relevant Concepts 2.13.1 Social Welfare 2.13.2 Demand Curves 2.14 Elasticity of Demand 2.14.1 Perfectly Inelastic 2.14.2 Perfectly Elastic 2.15 Production & Cost

Figure 4.1 Average Hours Worked per Person in the United States

The Supply of Labor Figure 4.1 Average Hours Worked per Person in the United States 1 Table 4.1 Change in Hours Worked by Age: 1950 2000 4.1: Preferences 4.2: The Constraints 4.3: Optimal Choice I: Determination

Revealed Preference. Ichiro Obara. October 8, 2012 UCLA. Obara (UCLA) Revealed Preference October 8, / 17

Ichiro Obara UCLA October 8, 2012 Obara (UCLA) October 8, 2012 1 / 17 Obara (UCLA) October 8, 2012 2 / 17 Suppose that we obtained data of price and consumption pairs D = { (p t, x t ) R L ++ R L +, t

Problem Set- Chapter 2 Solutions

roblem Set- Chapter 2 Solutions 1. Ch 2, roblem 2.1 The demand for beer in Japan is given by the following equation: d 700 2 N + 0.1I, where is the price of beer, N is the price of nuts, and I is average

First Welfare Theorem

First Welfare Theorem Econ 2100 Fall 2015 Lecture 17, November 2 Outline 1 First Welfare Theorem 2 Preliminaries to Second Welfare Theorem Last Class Definitions A feasible allocation (x, y) is Pareto

Chapter 4: Elasticity. McTaggart, Findlay, Parkin: Microeconomics 2007 Pearson Education Australia

Chapter 4: Elasticity Objectives After studying this chapter, you will be able to: Define, calculate, and explain the factors that influence the price elasticity of demand Define, calculate, and explain

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.

Economics 165 Winter 2002 Problem Set #2

Economics 165 Winter 2002 Problem Set #2 Problem 1: Consider the monopolistic competition model. Say we are looking at sailboat producers. Each producer has fixed costs of 10 million and marginal costs

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

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

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

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

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

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

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

PROBLEM SET#3 PART I: MULTIPLE CHOICE

1 PROBLEM SET#3 PART I: MULTIPLE CHOICE 1. In general, elasticity is a measure of a. the extent to which advances in technology are adopted by producers. b. the extent to which a market is competitive.

EC130 FOUNDATIONS OF ECONOMIC ANALYSIS

EC130 FOUNDATIONS OF ECONOMIC ANALYSIS 2004-2005 DEARTMENT OF ECONOMICS UNIVERSITY OF WARWICK Topic 2 Market equilibrium Stability revisited Elasticity Tax Tax incidence Tax incidence and elasticity 1

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

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

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

Chapter 16 General Equilibrium and Economic Efficiency

Chapter 16 General Equilibrium and Economic Efficiency Questions for Review 1. Why can feedback effects make a general equilibrium analysis substantially different from a partial equilibrium analysis?

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

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

E(p,.Pr,v)=U*D. v =#6 SOLUTIONS. op, ap", Amherst College Department of Economics Economics 58 Fall2009. First Hour Test

Amherst College Department of Economics Economics 58 Fall2009 Name First Hour Test SOLUTIONS There are three questions on this 80 minute examination. Each is of equal weight in grading. 1. Suppose that

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

Chapter 4. Elasticity

Chapter 4 Elasticity -comparative static exercises in the supply and demand model give us the direction of changes in equilibrium prices and quantities -sometimes we want to know more we want to know about

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