# Preferences. Preferences. Budget Constraint and Line 9/10/2014

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 9/0/04 references Chapter 4 At this point, we know a lot about preferences and their representation with utilit functions references tell us how a consumer ranks a given bundle compared to another bundle, and the utilit function tells us how much utilit a consumer derives from different bundles. references However Different bundles do not alwas cost the same amount, and therefore, all of the available bundles ma not be reasonable given a consumer s budget. Therefore, in order to determine how a consumer chooses amongst different bundles, we need to take into account not onl consumer preferences but also the budget constraint. Budget Constraint and Line The budget constraint is the set of bundles that a consumer can purchase with a limited amount of income. The budget line is the set of bundles that a consumer can purchase when spending all of his or her available income. Eample.

2 9/0/04 Budget Constraint Suppose a consumer purchases onl two tpes of goods, food and clothing. Let = food = clothing = income To construct the budget line, we must mathematicall represent that the total ependiture of and coincides with the consumer s income = Eample X Y Let s assume that p = 0, p = 40, = G is unattainable at the current prices and income p + p = = \$,000 > \$ 800 Budget Line: p + p = where p total ependiture on food p total ependiture on clothing At F: p + p = illustrates the budget constraint because it states that ou can onl spend as much mone as ou have To find the slope of the budget line, we must solve for (the good in the vertical ais) p + p = p = p Notice that this equation has the linear format of =b+m, and thus, the slope of the budget line is Note that if =0, then the consumer can afford (vertical intercept), while if =0, the consumer can afford (horizontal intercept) The slope of the Budget Line tells us how much of the good on the ais we must give up to bu one more unit of good on the ais, as we move from left to right on the figure. From our eample with food and clothing 0 40 Thus, the slope indicates that our consumer must give up ½ a unit of clothing () to acquire one more unit of food (). ntuitivel this makes sense because food is half as epensive.

3 9/0/04 ncome Change But how does the budget line change when income changes? Given constant prices, the budget line will make a parallel shift outward with a raise in income, or inward with a reduction in income. The budget line remains parallel because the slope, (p /p ), remains unchanged, assuming onl income is changed Slope of BL = arallel Shift in the Budget line Slope of budget line unchanged at rice Changes What about a change in prices? A change in the price of either good changes the slope of (p /p ). For instance, a change in the price of food () from 0 to 5 changes the slope to Thus, the consumer must give up more clothing to obtain more food rice Change 0 5 Steeper budget line BL ivots nward Reduction in the set of affordable bundles

4 9/0/04 What if both income and all prices are doubled? s this good/bad news for the consumer? A proportionate change in income and prices (let s sa b doubling each) causes no change in the budget line Vertical intercept /p = /p Horizontal intercept /p = /p Slope = p /p = p /p The s cancel out and leave B.L. unchanged Hence, the consumer s purchasing power is unaffected. utting references and B.L. Together Given preferences and budget constraints, we can find a consumer s optimal choice, the optimal amount of each good to purchase. His or her optimal choice bundle will be a bundle that fulfills two conditions () Maimize Utilit (preference) () s affordable (lies on the budget line) mmediatel we can see that the optimal choice bundle must be located on the budget line. Wh? Because an bundle inside the B.L. would leave the consumer with etra mone that could be spent to further increase utilit And an bundle outside the B.L. wouldn t be affordable n this continuing eample, A is the optimal bundle. t is the onl affordable bundle that reaches a utilit level of U. 4

5 9/0/04 Wh other baskets are not optimal? D unaffordable C mone unspent could increase our utilit b spending it (generall, it could be future consumption of the good, or savings) B lower.c. than at A Notice that at the optimal bundle A, the budget line and the utilit curve U are tangent. That is, the have the same slope (This is crucial!) From chapter, we know that the slope of an indifference curve is the marginal rate of substitution, MU MU Slope of.c.= Slope of B.L. Therefore, MU MU or MU MU This is called the Tangenc Condition and it reveals nterior Optimums to us, where the optimum bundle is at a tangenc between B.L. and.c. B rearranging the terms we get MU MU (The marginal utilities per dollar of both good and at bundle A are equal, so ou get the same BANG FOR YOUR BUCK for each good) Etra utilit per dollar spent on =etra utilit per dollar spent on f this were not the case, the bundle would not be optimal because ou could take some dollars awa from the good with less marginal utilit per dollar and put it into buing more of the good with a higher marginal utilit Eample U(,) = Hence, MU =, MU = = 800, p = 0, p = 40 Tangenc condition ) p + p = = 800 ) MU 0 MU 40 MU MU p p A 0() + 40 = = 800 = 0 = = 0 = 0, 0,0. 5

6 9/0/04 oint B=(8,6) was not optimal because Thus, at point B MU MU MU MU MU Utilit Maimization roblem MU and (At bundle B, the marginal utilities per dollar show us that spending more mone on good and less on good will increase overall utilit) So, mathematicall we represent the constrained maimization problem as Ma U(,) subject to p + p = n words: Maimize the consumer s utilit level subject to his budget constraint We can hence use the Lagrange Multiplier Method of Constraint Maimization to solve the utilit maimization problem. (,;λ) = U(,) + λ [ p p ] B taking F.O.C.s, we come to the same two conditions as before for a bundle to be optimal (ma U subject to B.L.) Let s tr to show this using Cobb Douglas Utilit Function Eample: Cobb Douglas Utilit function:, MAX U (, ) S. T. XY (,, ) [ p p ] Taking F.O.Cs with respect to u(, ) p 0 p u(,) () And with respect to, ( ) p ( ) p ( ) u(, ) () 6

7 9/0/04 And taking FOCs with respect to 0 () Using the tangenc condition p p and From () and (), u(,) ( ) u(,) ( ) rearranging it, we obtain p =p Which is eactl the tangenc condition = since MU = p p lugging this into (), ields ( ) And MU = lugging into the tangenc condition ields Budget Shares After finding the demand for and, we can find their budget shares (dividing b the income level, ), as follows X Budget share for, Hence, demands are and Budget share for, ( ) Y Budget shares are, hence, constant in income e.g. if α = /, the budget share of good is % while that of good is 66% 7

8 9/0/04 Approach #: Using the Lagrange Method Approach #: Using the tangenc condition Approach #: Using the Lagrange method FOC FOC u (, ) (, ; ) s. t [ ] FOC We now set FOC=FOC FOC FOC Which simplifies to Further rearranging, ields =. 0 0 Use the FOCλ FOC Therefore, And the optimal consumption of good is. 8

9 9/0/04 Approach #: Using the Tangenc Condition Rearranging the tangenc condition ields MU MU p p We can now plug our result from the tangenc condition, =, in the budget line And good is ==. units. Budget Shares Let us evaluate the budget shares in each good Good, Good, p p Recall that these budget shares coincide with the eponents of the Cobb Douglas utilit function. 9

10 9/0/04 After going over this numerical eample of utilit maimization problems (UM), let us now continue with our discussion of UMs: nterior solutions. We have analzed interior solutions (positive amounts of both goods), but Corner solutions. What if the solution happens to be in a corner (impling a zero consumption of one of the two goods?) Corner Solutions What do we do in optimization cases where the B.L. is never tangent to an indifference curve? That is, if Eample MU MU Numerical Eample of Corner solutions U(,) = + 0 MU = + 0 MU = = 0, p =, p = Notice that utilit can be positive where zero units of good are consumed. This is because the.c. crosses the ais. This didn t happen in the Cobb Douglas utilit function where, if =0 or if =0, utilit was zero. Let s approach this like an interior optimum () +=0 (B.L.) X Y () MU 0 0 MU The tangenc approach gives us a negative amount of clothing which shows us that the solution is not an interior optimum. The solution will be a corner point, where =0. (+0) + = 0 4 = 0 =.5 (NO!! cannot be negative, thus we have a corner solution where = 0) B.L. 0 Since 0, then 0 Negative amounts indicate that the solution must be in a corner. 0

11 9/0/04 Then, a consumer wants to increase his consumption of as much as possible. That is, the consumer spends all his income in,, and no income in. 0 0 Bundle R is optimal At R, MU MU Y MU MU FURTHER NCREASE. Corner oint Solution with erfect Substitutes U(,) a b Let s consider the case where a consumer is perfectl willing to substitute one good for another at a constant ratio. For instance, Sara likes both chocolate and vanilla ice cream, and is alwas willing to substitute one scoop of chocolate for two scoops of vanilla. MRS choc.,vanilla = (constant MRS because of erfect substitutes) Hence, c MU MU v c v MU c MU v, MU v v and c v MU c c MU per dollar spent on v MU per dollar spent on c MRS choc.,vanilla = MU c MU v (constant MRS because of erfect substitutes) c v MU c MU v MU c MU v v c Consequentl, since the marginal utilit per dollar of vanilla ice cream is greater than that of chocolate ice cream, Sara should onl consume vanilla ice cream. B [What if price ratio was lower than MRS? Then BL becomes flatter than.c., and the consumer bus onl chocolate] Figure in net slide

12 9/0/04 When MRS,, we have: Application : Coupons vs Cash Subsidies At oint B, MU MU MU MU so the consumer spends all his mone on good alone. Before we start with applications, one note: Composite Goods are goods that represent the composite ependitures on ever other good besides the commodit being considered. That is, we use composite goods when we wish to isolate how a consumer selects one specific good. The price of composite goods is p = so that it measures not onl units of consumed, but also total ependiture on (p =total ependiture on ) The government might consider that the amount of housing individuals select, h A, is insufficient, and seeks to promote a larger amount of housing, h B. t can achieve this goal b providing a subsid S (which shifts individuals budget line to EG), or with a coupon that can onl be used to purchase housing (which ields the new budget line KFG). n the previous case, the consumer was equall happ with either program because both achieve.c. of U. However, in the situation below the consumer is worse off with the coupon, reaching U, than the subsid, where he reaches U4. Cash subsid of S shifts budget line out b S, to B.L. EG Coupon for housing of S units onl increases the budget with respect to housing, so the new budget line with the coupon is KFG B.L. is MN SubsidBundle T, with Utilit U 4 Coupon B.L. is KRG. Bundle R with utilit U Cash: B.L. is EG. Bundle B with Utilit U Coupon: B.L. is KFG. Bundle B with Utilit u

13 9/0/04 Consider instead that the government, still seeking to induce h B units of housing, does not provide a coupon, but rather provides a voucher V, which gives an amount of cash to the individual so that his budget line shifts until the point in which he can afford to bu h B units of housing at point R. That is, budget line EG emerges from the voucher polic. However, will the individual bu bundle R? No! While R and F are both affordable, he reaches a higher utilit level, U, with bundle F than with bundle R (where he onl reaches U). Hence, the voucher polic, despite making the individual happier than the coupon polic (U>U), does not help the government achieve a minimal housing level of h B Application : Joining a Club Man consumers can join clubs to receive member discounts. Yet, the must also pa a membership fee. So is it worth joining? Let s look at an eample Before joining: = 00, p = \$0 00 5units Horizontal ntercept of B.L 0 \$ After joining: = (fee) = \$00, p = \$0 p = \$ 00 00units Vertical ntercept of B.L 00 0units 00 00units Horizontal intercept of BL Vertical intercept of BL At BL He chooses basket A Consumer chooses bundle A before joining club A similar analsis is applicable to: Cell phone service: ou pa a monthl subscription fee in order to enjo calls at lower prices per minute Gm, golf clubs, etc. After joining, consumer chooses bundle B and is better off than before because B is on.c. U

14 9/0/04 Eample: AT&T plans for its Apple ihone g (Chicago, 009) lan A: \$40 fee, 450 minutes each additional minute \$.40 lan B: \$60 fee, 900 minutes each additional minute \$.40 Drawing the B.L. for lan A: ncome: \$500 After fee, ncome = \$500 40=460 (vertical intercept) Flat for the first 450 minutes, then slope of.40 Drawing B.L. for lan B: After the \$60 fee, income is \$500 60=440 (vertical intercept) Flat for B.L. for the first 900 minutes, then slope of.40 AT&T hone Service Eample Application : Borrowing and Lending Without borrowing or lending, a consumer consumes toda and tomorrow, where is the consumer s salar toda, and is his salar tomorrow. However, introducing borrowing and lending allows a consumer to put off consuming toda so that toda s income can be lent, so tomorrow he or she can consume + (+r), with (+r) being the interest earned from lending income. e.g. 00, r 0.05 Consume 00 r 00 r.05 toda b borrowing, and not consuming anthing tomorrow. 4

15 9/0/04 How do we find the slope of the BL in this contet? ntuitivel, the price of giving up one consumption unit tomorrow ( ais) in order to gain one more unit of consumption toda ( ais) is measured b the opportunit cost of ever dollar borrowed, (+r). We can state this more formall as We know that the B.L. is = a + m = + (+r) + m a and, in addition, at = 0 we have = + /(+r), then 0 ( r) m r m ( r) ( r) ( r) ( r) ( r) r Here it is better to borrow some mone this ear to be paid back net ear, because the optimum bundle went from being on U before borrowing to U after borrowing. World Without Banks! Borrowing n this eample both the lending interest rate and borrowing interest rate are 0 percent. So far we assumed that the lending and borrowing interest rates coincide, but. What if the lending and borrowing interest rates are different? Lending, r L =.05 Net ear he will have,00 from tomorrow s income =,00 0,000 (+0.05) from savings = 0,500,700 Borrowing, r B =.0 Toda can have : 0,000 from toda s income= 0,000,00 0.0,000,000 5

16 9/0/04 Here the interest rates for borrowing and lending are different. Borrowing ( rl ( rb ) ) Here the.c. of a saver would reach tangenc on segment AE, while the.c. of a borrower would reach tangenc on segment AG Application V: Quantit Discounts Sometimes consumers are offered quantit discounts, so that the price of a good is not constant but changes after some defined quantit (i.e., a discount on electricit after number of units). Notice this will mean the budget line has a kink where the slope changes because of the new price ratio. Let s consider the case of electricit where the prices are as follows p = \$ p = \$ for the first 9 units (e.g., kw/h) \$5.5 for all additional units How to find the vertical and horizontal intercepts in this case? Vertical ntercept = \$440 \$ 440 Horizontal intercept of BL (no discounts) = Horizontal intercept of BL (with discounts) = Slope is on segment on MA otential Util. gain from the quantit discount if preferences are like these Slope is 5.5 on AR where we subtract the amount of mone alread spent on good before eperiencing the price discount, 9 units \$ per unit, to the consumer s initial income At point A, we can use the BL in order to find its height: +(9*)=440, which ields a value of =440 99=4 6

17 9/0/04 BL for Frequent Fler rograms Another Eample: Frequent fler programs, e.g, after 00k miles the price of another ticket decreases. For this eample, we can actuall have man quantit discounts, as the net figure illustrates. Utilit maimization problems and Ependiture minimization problems (Appendi to Chapter 4) Eample : Utilit Maimization roblem (UM) with the Cobb Douglas utilit function. Let us first consider an eample of a utilit maimization problem. Take the Cobb Douglas function epressed b u,, The budget constraint : 00 So we know that: price of good is \$, that of good is \$, and income is \$00. 7

18 9/0/04 Setting up the Lagrangian: L(, ; ) (00 ) We can take first order conditions: L 0 L 0 L 00 0 Taking the ratio of the first two terms shows that: or rearranging, (Note that this eactl coincides with the tangenc condition between the.c. and the budget line, that s wh the tangenc condition and the Lagrangian method ield the same results.) u ndeed, the MU is u while the MU is Hence, the MRS is MRS, Substituting the tangenc condition budget set 00 gives 00 5 units into the Therefore, the utilit level that the consumer reaches is given b plugging 5 and 50 into the utilit function u, And about good, we have 5 50 units This utilit level from selecting utilit maimizing bundles is often referred to as indirect utilit 8

19 9/0/04 Eample : Ependiture minimization problem (EM) with a Cobb Douglas utilit function. Another wa to think about optimum bundles is as Ependiture Minimization roblems: the bundle that will give ou a given level of utilit at the lowest ependiture. Min Ependiture p + p Subject to U(,) = (reaching a given level of utilit) u Eample : Ependiture minimization problem (EM) with a Cobb Douglas utilit function. Consider a consumer with Cobb Douglas utilit function, who tries to reach a utilit level of u=5.5. That is, u (, ) and p, p, u5.5. To find the compensated demands for goods and, let us solve the ependiture minimization problem (EM) min ( ) s.t. 5.5 u Setting up the Lagrangian in this case, ields, ; 5.5 L where μ denotes the Lagrange multiplier of this ependiture minimization problem (we use μ rather than λ, which is the Greek letter we used in the UM). n the case of interior solutions, the above first order conditions become L 0 L 0 L

20 9/0/04 Let s solve the previous sstem of equations, 4 4 From, 4 From, 4 Since 4 From, lugging 4 into Using 4, 5 50 Our old friend!! The same eact tangenc condition we found when finding the optimal consumption bundle using the UM approach. 4 For the Cobb Douglas utilit function u (, ) considered above, we just found that the compensated demands for goods and are ( p, u) 5, ( p, u) 50 c c which coincide with the optimal consumption bundles that we found when solving the UM!! Therefore, the total ependiture required to purchase these compensated demands is e( p, u) p ( p, u) p ( p, u) c c which coincides with the budget constraint of the consumer in the UM (where we started saing the consumer had an income of \$00 to spend). 0

22 9/0/04 Here is an eample where a consumer fails to maimize utilit ncome = \$4 ( p, p ) = (\$4, \$) (, ) = (5, ) A ~ ~, = (\$, \$) (, ) = (, 6) B Let us see wh he fails to maimize utilit. BL : () when facing BL, the consumer chose A instead of C, in spite of being available (affordable), A C AC BA B () Since C is to the northeast of B, C>B A >B BL : when facing BL, he chose B in spite of D being available, B D Since D is to the northeast of A, D>A BD AB A B >A Contradiction! Bundle A cannot be strongl preferred to B, and B strongl preferred to A in a case where both are affordable. Alternative Approach, using total ependiture Let us tr a few more BL : Consumer prefers basket A. Cost of basket A, \$4 5 + \$ = \$4 Cost of basket B, \$4 + \$ 6 = \$0 X Y X Y BL : Consumer prefers basket B. Thus B >A ~ ~ Cost of basket B, \$ + \$ 6 = \$4 X Y ~ ~ Cost of basket A, \$ 5 + \$ = \$ X Y We need X Y X Y (\$4 > \$0), and ~ ~ X Y X Y ~ ~ Hence, he is NOT maimizing utilit (but in this case we obtain (\$4\$). /

23 9/0/04 Case : () C>A since it is to the northeast of A () B C since B was chosen when C was affordable Case : Affordable () BL : From above, we know that B >A since A is inside BL Under BL () BL : Additionall, A B since both were affordable when facing BL Contradiction, not utilit maimizing behavior Case : () BL : the consumer chose A when both A and B were affordable, A B () BL : the consumer chose B when A wasn t affordable Case 4: () BL : chose A but B wasn t affordable () BL : chose B but A wasn t affordable B Transitivit B C A B A B >A Ranking A B is all we can sa We can t infer anthing from these choices because we don t know which bundle the consumer would choose if both were affordable

### 19. At a consumer's interior optimum solution, which of the following will not necessarily hold true? MU = MU

Ch. 4 1. The budget line A) represents the set of all baskets the consumer can afford. B) represents the set of all baskets the consumer can afford while spending all available income. C) represents the

### Problem Set Chapter 3 & 4 Solutions

Problem Set Chapter 3 & 4 Solutions. Graph a tpical indifference curve for the following utilit functions and determine whether the obe the assumption of diminishing MRS: a. (, ) 3 + 3 Slope -3 3 Since

### A Utility Maximization Example

A Utilit Maximization Example Charlie Gibbons Universit of California, Berkele September 17, 2007 Since we couldn t finish the utilit maximization problem in section, here it is solved from the beginning.

### Introduction. Introduction

Introduction Solving Sstems of Equations Let s start with an eample. Recall the application of sales forecasting from the Working with Linear Equations module. We used historical data to derive the equation

### Price Theory Lecture 3: Theory of the Consumer

Price Theor Lecture 3: Theor of the Consumer I. Introduction The purpose of this section is to delve deeper into the roots of the demand curve, to see eactl how it results from people s tastes, income,

### 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

### 2.7 Applications of Derivatives to Business

80 CHAPTER 2 Applications of the Derivative 2.7 Applications of Derivatives to Business and Economics Cost = C() In recent ears, economic decision making has become more and more mathematicall oriented.

### The Graph of a Linear Equation

4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

### Chapter 3: Utility Maximization

Chapter 3: Consumer Preferences Chapter 3: Utility Maximization Axiom Based Revealed Preference Preference Ordering Budget Constraint Utility Maximization Outline and Conceptual Inquiries Consider a Household

### Felix Munoz-Garcia 1. School of Economic Sciences. Washington State University

ECONS 301 - INTERMEDIATE MICROECONOMICS LECTURE NOTES Felix Munoz-Garcia 1 School of Economic Sciences Washington State University This document contains a set of partial lecture notes that are intended

### A graphical introduction to the budget constraint and utility maximization

EC 35: ntermediate Microeconomics, Lecture 4 Economics 35: ntermediate Microeconomics Notes and Assignment Chater 4: tilit Maimization and Choice This chater discusses how consumers make consumtion decisions

### 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

### Problem Problem 3.21(b) Does the marginal utility of x diminish, remain constant, or increase as the consumer buys more of x? Explain.

EconS 301 Problem 3.1 Suppose a consumer s preferences for two goods can be represented by the Cobb Douglas utility function U = Ax α y β where A, α, and β are positive constants. The marginal utilities

### Deriving MRS from Utility Function, Budget Constraints, and Interior Solution of Optimization

Utilit Function, Deriving MRS. Principles of Microeconomics, Fall Chia-Hui Chen September, Lecture Deriving MRS from Utilit Function, Budget Constraints, and Interior Solution of Optimization Outline.

### Review of Essential Skills and Knowledge

Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope

### Alex and Morgan were asked to graph the equation y = 2x + 1

Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and -intercept wa First, I made a table. I chose some -values, then plugged

### 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

### LINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0

LINEAR FUNCTIONS As previousl described, a linear equation can be defined as an equation in which the highest eponent of the equation variable is one. A linear function is a function of the form f ( )

### Florida Algebra I EOC Online Practice Test

Florida Algebra I EOC Online Practice Test 1 Directions: This practice test contains 65 multiple-choice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end

### 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

### Econ306 Intermediate Microeconomics Fall 2007 Midterm exam Solutions

Econ306 Intermediate Microeconomics Fall 007 Midterm exam Solutions Question ( points) Perry lives on avocado and beans. The price of avocados is \$0, the price of beans is \$, and his income is \$0. (i)

### Quiz #2 Week 03/08/2009 to 03/14/2009

Quiz #2 Week 03/08/2009 to 03/14/2009 You have 25 minutes to answer the following 14 multiple choice questions. Record your answers in the bubble sheet. Your grade in this quiz will count for 1% of your

### 2.2 Absolute Value Functions

. Absolute Value Functions 7. Absolute Value Functions There are a few was to describe what is meant b the absolute value of a real number. You ma have been taught that is the distance from the real number

### 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

### Section 7.2 Linear Programming: The Graphical Method

Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function

### Exponential and Logarithmic Functions

Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,

### Utility Maximization

Utility Maimization Given the consumer's income, M, and prices, p and p y, the consumer's problem is to choose the a ordable bundle that maimizes her utility. The feasible set (budget set): total ependiture

### 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

### Q (x 1, y 1 ) m = y 1 y 0

. Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that two distinct points in the plane determine

### servings of fries, income is exhausted and MU per dollar spent is the same for both goods, so this is the equilibrium.

Problem Set # 5 Unless told otherwise, assume that individuals think that more of any good is better (that is, marginal utility is positive). Also assume that indifference curves have their normal shape,

### INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

### Graphing Linear Equations

6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are

### 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

### Higher. Functions and Graphs. Functions and Graphs 18

hsn.uk.net Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 8 Sets 8 Functions 9 Composite Functions 4 Inverse Functions 5 Eponential Functions 4 6 Introduction to Logarithms

### 1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model

. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses

### Constrained Optimization: The Method of Lagrange Multipliers:

Constrained Optimization: The Method of Lagrange Multipliers: Suppose the equation p(x,) x 60x 7 00 models profit when x represents the number of handmade chairs and is the number of handmade rockers produced

### 2.4 Inequalities with Absolute Value and Quadratic Functions

08 Linear and Quadratic Functions. Inequalities with Absolute Value and Quadratic Functions In this section, not onl do we develop techniques for solving various classes of inequalities analticall, we

### The fundamental question in economics is 2. Consumer Preferences

A Theory of Consumer Behavior Preliminaries 1. Introduction The fundamental question in economics is 2. Consumer Preferences Given limited resources, how are goods and service allocated? 1 3. Indifference

### 1.2 GRAPHS OF EQUATIONS

000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the - and -intercepts of graphs of equations. Write the standard forms of equations of

### Direct Variation. 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship

6.5 Direct Variation 6.5 OBJECTIVES 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship Pedro makes \$25 an hour as an electrician. If he works

### Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

### 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

### Anytime plan TalkMore plan

CONDENSED L E S S O N 6.1 Solving Sstems of Equations In this lesson ou will represent situations with sstems of equations use tables and graphs to solve sstems of linear equations A sstem of equations

### Choices. Preferences. Indifference Curves. Preference Relations. ECON 370: Microeconomic Theory Summer 2004 Rice University Stanley Gilbert

Choices Preferences ECON 370: Microeconomic Theor Summer 2004 Rice Universit Stanle Gilbert The theor of consumer preferences is based fundamentall on choices The steak dinner or the salad bar Major in

### 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

### THE POWER RULES. Raising an Exponential Expression to a Power

8 (5-) Chapter 5 Eponents and Polnomials 5. THE POWER RULES In this section Raising an Eponential Epression to a Power Raising a Product to a Power Raising a Quotient to a Power Variable Eponents Summar

### 2.2 Absolute Value Functions

. Absolute Value Functions 7. Absolute Value Functions There are a few was to describe what is meant b the absolute value of a real number. You ma have been taught that is the distance from the real number

### LINEAR FUNCTIONS OF 2 VARIABLES

CHAPTER 4: LINEAR FUNCTIONS OF 2 VARIABLES 4.1 RATES OF CHANGES IN DIFFERENT DIRECTIONS From Precalculus, we know that is a linear function if the rate of change of the function is constant. I.e., for

### Quadratic Functions. MathsStart. Topic 3

MathsStart (NOTE Feb 2013: This is the old version of MathsStart. New books will be created during 2013 and 2014) Topic 3 Quadratic Functions 8 = 3 2 6 8 ( 2)( 4) ( 3) 2 1 2 4 0 (3, 1) MATHS LEARNING CENTRE

### Course 2 Answer Key. 1.1 Rational & Irrational Numbers. Defining Real Numbers Student Logbook. The Square Root Function Student Logbook

Course Answer Ke. Rational & Irrational Numbers Defining Real Numbers. integers; 0. terminates; repeats 3. two; number 4. ratio; integers 5. terminating; repeating 6. rational; irrational 7. real 8. root

### Economics 301 Problem Set 4 5 October 2007

Economics 301 Name Problem Set 4 5 October 2007 Budget Lines and Indifference Curves and the Consumer Optimum 1. Parvez, a pharmacology student, has allocated \$120 per month to spend on paperback novels

. Quadratic Functions 9. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions: the

### Quadratic Functions and Models. The Graph of a Quadratic Function. These functions are examples of polynomial functions. Why you should learn it

0_00.qd 8 /7/05 Chapter. 9:0 AM Page 8 Polnomial and Rational Functions Quadratic Functions and Models What ou should learn Analze graphs of quadratic functions. Write quadratic functions in standard form

### 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

### Essential Question How can you describe the graph of the equation Ax + By = C? Number of adult tickets. adult

3. Graphing Linear Equations in Standard Form Essential Question How can ou describe the graph of the equation A + B = C? Using a Table to Plot Points Work with a partner. You sold a total of \$16 worth

### P1. Plot the following points on the real. P2. Determine which of the following are solutions

Section 1.5 Rectangular Coordinates and Graphs of Equations 9 PART II: LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES 1.5 Rectangular Coordinates and Graphs of Equations OBJECTIVES 1 Plot Points in

### 4.4 Logarithmic Functions

SECTION 4.4 Logarithmic Functions 87 4.4 Logarithmic Functions PREPARING FOR THIS SECTION Before getting started, review the following: Solving Inequalities (Appendi, Section A.8, pp. 04 05) Polnomial

### C1: Coordinate geometry of straight lines

B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the

### To Be or Not To Be a Linear Equation: That Is the Question

To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not

### 7.3 Graphing Rational Functions

Section 7.3 Graphing Rational Functions 639 7.3 Graphing Rational Functions We ve seen that the denominator of a rational function is never allowed to equal zero; division b zero is not defined. So, with

### (b) Can Charlie afford any bundles that give him a utility of 150? (c) Can Charlie afford any bundles that give him a utility of 300?

Micro PS2 - Choice, Demand and Consumer Surplus 1. We begin again with Charlie of the apples and bananas. Recall that Charlie s utility function is U(x A, x B ) = x A x B. Suppose that the price of apples

### 4 Writing Linear Functions

Writing Linear Functions.1 Writing Equations in Slope-Intercept Form. Writing Equations in Point-Slope Form.3 Writing Equations in Standard Form. Writing Equations of Parallel and Perpendicular Lines.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

### Exponential Functions

CHAPTER Eponential Functions 010 Carnegie Learning, Inc. Georgia has two nuclear power plants: the Hatch plant in Appling Count, and the Vogtle plant in Burke Count. Together, these plants suppl about

### Systems of linear equations (simultaneous equations)

Before starting this topic ou should review how to graph equations of lines. The link below will take ou to the appropriate location on the Academic Skills site. http://www.scu.edu.au/academicskills/numerac/inde.php/1

### The Fisher Model. and the. Foundations of the Net Present Value Rule

The Fisher Model and the Foundations of the Net Present Value Rule Dr. Richard MacMinn mailto:macminn@mail.utexas.edu Finance 374c Introduction Financial markets perform of role of allowing individuals

### CHAPTER 4 INDIVIDUAL AND MARKET DEMAND

CHAPTER 4 INDIVIDUAL AND MARKET DEMAND EXERCISES 1. The ACME corporation determines that at current prices the demand for its computer chips has a price elasticity of -2 in the short run, while the price

### 3.4 The Point-Slope Form of a Line

Section 3.4 The Point-Slope Form of a Line 293 3.4 The Point-Slope Form of a Line In the last section, we developed the slope-intercept form of a line ( = m + b). The slope-intercept form of a line is

### Higher. Polynomials and Quadratics 64

hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining

### 5.1. A Formula for Slope. Investigation: Points and Slope CONDENSED

CONDENSED L E S S O N 5.1 A Formula for Slope In this lesson ou will learn how to calculate the slope of a line given two points on the line determine whether a point lies on the same line as two given

### Problem Set #3-Key. (1) What is the largest possible number of pizza slices you could consume in a month given your budget?

Problem Set #3-Key Sonoma State University Economics 305-Intermediate Microeconomic Theory Dr. Cuellar Suppose that you consume only two goods pizza (X) and beer (Y). Your budget on pizza and beer is \$120

### ECON 300 Homework 2 Due Wednesday, October 20 th

ECON 300 Homework 2 Due Wednesday, October 20 th 1. Jack s preferences for jeans (J) and shirts (S) are given by the utility function: a. Derive a formula for the marginal utility of jeans b. Derive a

### 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

### Economic Principles Solutions to Problem Set 1

Economic Principles Solutions to Problem Set 1 Question 1. Let < be represented b u : R n +! R. Prove that u (x) is strictl quasiconcave if and onl if < is strictl convex. If part: ( strict convexit of

### 9.2 LINEAR PROGRAMMING INVOLVING TWO VARIABLES

86 CHAPTER 9 LINEAR PROGRAMMING 9. LINEAR PROGRAMMING INVOLVING TWO VARIABLES Figure 9.0 Feasible solutions Man applications in business and economics involve a process called optimization, in which we

### Solving Systems of Linear Equations

5 Solving Sstems of Linear Equations 5. Solving Sstems of Linear Equations b Graphing 5. Solving Sstems of Linear Equations b Substitution 5.3 Solving Sstems of Linear Equations b Elimination 5. Solving

### Simplification of Rational Expressions and Functions

7.1 Simplification of Rational Epressions and Functions 7.1 OBJECTIVES 1. Simplif a rational epression 2. Identif a rational function 3. Simplif a rational function 4. Graph a rational function Our work

### Economics 100A. Final Exam

Name form number 1 Economics 100A Final Exam Fill in the bubbles on your scantron with your id number (starting from the left side of the box), your name, and the form type. Students who do this successfully

### Solving Systems of Linear Equations

5 Solving Sstems of Linear Equations 5. Solving Sstems of Linear Equations b Graphing 5. Solving Sstems of Linear Equations b Substitution 5.3 Solving Sstems of Linear Equations b Elimination 5. Solving

### Graphing Linear Equations in Slope-Intercept Form

4.4. Graphing Linear Equations in Slope-Intercept Form equation = m + b? How can ou describe the graph of the ACTIVITY: Analzing Graphs of Lines Work with a partner. Graph each equation. Find the slope

### 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

### 4 Non-Linear relationships

NUMBER AND ALGEBRA Non-Linear relationships A Solving quadratic equations B Plotting quadratic relationships C Parabolas and transformations D Sketching parabolas using transformations E Sketching parabolas

### Rational Functions. 7.1 A Rational Existence. 7.2 A Rational Shift in Behavior. 7.3 A Rational Approach. 7.4 There s a Hole In My Function, Dear Liza

Rational Functions 7 The ozone laer protects Earth from harmful ultraviolet radiation. Each ear, this laer thins dramaticall over the poles, creating ozone holes which have stretched as far as Australia

### 6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:

Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph

### CLAS. Utility Functions Handout

Utility Functions Handout A big chunk of this class revolves around utility functions (AKA preferences). Bottom line, utility functions tell us how we prefer to consume goods so that we maximize our utility

### 15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors

SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential

### Math 1400/1650 (Cherry): Quadratic Functions

Math 100/1650 (Cherr): Quadratic Functions A quadratic function is a function Q() of the form Q() = a + b + c with a 0. For eample, Q() = 3 7 + 5 is a quadratic function, and here a = 3, b = 7 and c =

### Systems of Equations. from Campus to Careers Fashion Designer

Sstems of Equations from Campus to Careers Fashion Designer Radius Images/Alam. Solving Sstems of Equations b Graphing. Solving Sstems of Equations Algebraicall. Problem Solving Using Sstems of Two Equations.

### 4.1 Piecewise-Defined Functions

Section 4.1 Piecewise-Defined Functions 335 4.1 Piecewise-Defined Functions In preparation for the definition of the absolute value function, it is etremel important to have a good grasp of the concept

### 4 Constrained Optimization: The Method of Lagrange Multipliers. Chapter 7 Section 4 Constrained Optimization: The Method of Lagrange Multipliers 551

Chapter 7 Section 4 Constrained Optimization: The Method of Lagrange Multipliers 551 LEVEL CURVES 2 7 2 45. f(, ) ln 46. f(, ) 6 2 12 4 16 3 47. f(, ) 2 4 4 2 (11 18) 48. Sometimes ou can classif the critical

### Systems of Linear Equations: Solving by Substitution

8.3 Sstems of Linear Equations: Solving b Substitution 8.3 OBJECTIVES 1. Solve sstems using the substitution method 2. Solve applications of sstems of equations In Sections 8.1 and 8.2, we looked at graphing

### 4.1 Ordinal versus cardinal utility

Microeconomics I. Antonio Zabalza. Universit of Valencia 1 Micro I. Lesson 4. Utilit In the previous lesson we have developed a method to rank consistentl all bundles in the (,) space and we have introduced

### Solving Special Systems of Linear Equations

5. Solving Special Sstems of Linear Equations Essential Question Can a sstem of linear equations have no solution or infinitel man solutions? Using a Table to Solve a Sstem Work with a partner. You invest

### SOLVING SYSTEMS OF EQUATIONS

SOLVING SYSTEMS OF EQUATIONS 4.. 4..4 Students have been solving equations even before Algebra. Now the focus on what a solution means, both algebraicall and graphicall. B understanding the nature of solutions,

### LESSON EIII.E EXPONENTS AND LOGARITHMS

LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential

### Chapter 7. Costs. C = FC + VC Marginal cost MC = C/ q Note that FC will not change, so marginal cost also means marginal variable cost.

Chapter 7. Costs Short-run costs Long-run costs Lowering costs in the long-run 0. Economic cost and accounting cost Opportunity cost : the highest value of other alternative activities forgone. To determine