Chapter 1 1 ?)±4ü6D [?eo ? 2005 MOL Chapter 1 2 D "￥D1 p ? ' [? ^ü6Dy' S 3￥<$ [ ^ B M)ù??uü6De ?v￥$  μ1 <?hn??±4ü6?8￥ü6?1Té B?si) g?ü6￥?? pb ? T1<$ [ n51 pD 3 ?1g?± 4ü6D￥'à Q?'e ?iü6  ?￥'L !?y ￥/ ? m ?μiT￥  3N$ D?¨'ü6 ? ?? Ms C L￥ü6ù5b Chapter 1 3 ? S ? ? Walter Nicholson, Microeconomic Theory: Basic Principles and Extension, 8th Edition, Southwestern, 2002 ?ó ???Sü6? 1999 M? ——? ?? ? B?Jr ? fí μ±4ü6D?Sü6? 2002 M? Chapter 1 4 ? I ? ? >ó? ú?<?IuZü6Dv  ¨?S | ?vD? ? Hal Varian, Intermediate Microeconomics: A Modern Approach, 6th Edition, Norton, 2003  ? ??ó? Z ? ó? 1994 M? ? ü? D±4ü6D E???vD? 2001 M ? James M.Henderson and R. E. Quandt, Microeconomic Theory A Mathematical Approach, McGraw-Hill Book Co. 1980b(? ? ?ó???vD? 1988 M? ) Chapter 1 5 D1 p ?  ?Q  [- -1 p4 - [? ?· ?￥  [ = ?i? H?? BQ [? ??￥T<b' [?￥ ù I k? |>  I kbKa?? ?/ 3?sF? ? ù I k] 80% ? ü HT<] 15% ? ? èD] 5%b Chapter 1 6  [ = ?# HW [?c?  [ HW ?Bc ? ? 9 6 ° ?=c h n??1 ? ? D3 p ? ? 9 8 13 15 20 ° ? ?c h n??1 ? ? 5 9 22 27 29 ° ? 1c ì  ? ? 10 11 13 18 20 ° ??c é ?? g? 10 25 27 ° Chapter 1 7 [? = ?# HW ? Bc é?? g?￥ B? (? 11 1 3 ° ? tc p ? ? 11 8 10 15 ° ??c I  11 17 22 ° ? ? c * h g? 11 24 29,12 1 ° 10? Ec 1 í g?D 1 íNì 12 6 8 13 ° ? EBc g? > 2D ±4ü6?| 12 15,20,22 ° Chapter 1 8 ±4ü6DDL ? ±4ü6DDL ^1?vD' S 3D ±4ü6D7y ?￥D?<?_b ? oμvn?¨]D V[? /μ1 [ q? è?5? = ?b ?  =ye =y31]D ì # h??^Y?￡ (www.163.com),μ 163?Q￥]D V[°¤ ¨b ? ?? :microeconomics.nease.net ? ?]D ì|???_￥y ! Chapter 1 9 Chapter One Introduction ? 2003 MOL Chapter 1 10 Chapter one includes: ? 1.1 Defining Microeconomics ? 1.2 Economic Theories and Models ü6 ? ?D ? ? 1.3 The Mathematics of Optimization ?DKa? ? 1.4 Demand-Supply Model Chapter 1 11 Outline of This Chapter ('cà ? ) ? This chapter mainly provide some background for the study of microeconomics. ? What is microeconomics ? The general approach used in microeconomics. ? How economists devise and verify simple models of economic activity. ? Positive Analysis and Normative Analysis ? The basic Supply-Demand Model ? Several Mathematical Methods that can be used to solve maximization (and minimization) problem. Chapter 1 12 Readings about this chapter ? Nicholson: Chapter 1,2, P1-61 ? Zhang: Chapter 1,P1-23 Chapter 1 13 Chapter one includes: ? 1.1 Defining Microeconomics ? 1.2 Economic Theories and Models ? 1.3 The Mathematics of Optimization ? 1.4 Demand-Supply Model Chapter 1 14 1.1 Defining Microeconomics ? Economics ? The study of the allocation of scarce resources among alternative end uses Scarcity ( ?? ) Alternative end uses Vê4￥K?¨o Choice Economics Chapter 1 15 Defining Microeconomics (1) ?Economics is composed of two branches: ?Microeconomics ?Macroeconomics Chapter 1 16 Defining Microeconomics (2) ? Microeconomics deals with: ? Behavior of individual units——Consumers ? When Consuming ? Maximizing Utility ? How we choose what to buy ? Behavior of individual units——Firms ? When Producing ? Maximizing Profit ? How we choose what to produce Chapter 1 17 Defining Microeconomics (3) ? Microeconomics deals with: ?Markets: The interaction of consumers and producers ?Output Market (Product Market)  á ? g? ?Input Market (Factor Market)  1 í g? Chapter 1 18 Defining Microeconomics(4) ? Macroeconomics deals with: ?Analysis of aggregate issues 9 ￥ù5 : ?Economic growth ü69é ?Inflation Y 3n ?Unemployment >< ?etc. Chapter 1 19 Defining Microeconomics(5) ? The Linkage(1" ) Between Micro and Macro-economics ? Microeconomics is the foundation of macroeconomic analysis Chapter 1 20 Chapter one includes: ? 1.1 Defining Microeconomics ? 1.2 Economic Theories and Models ? 1.3 The Mathematics of Optimization ? 1.4 Demand-Supply Model Chapter 1 21 1.2 Theories and Models  ? ?D ? Chapter 1 22 Theories and Models(1) ? Microeconomic Analysis——Theories ? Theories are used to explain observed phenomena in terms of a set of basic rules and assumptions. ? For example ? The Theory of the Firm ? The Theory of Consumer Behavior Chapter 1 23 Theories and Models(2) ? Microeconomic Analysis—— Models ?Models: ? Simple theoretical descriptions that capture the essentials of how the economy works Chapter 1 24 Theories and Models 3 ? General Features of Economic Model: ? 1. The Ceteris Paribus Assumption   eHq (M]L ! ? Other things equal ? 2. Optimization Assumption Ka?L ! ? Consumers——Maximizing their own well-being. ? Firms——Maximizing Profits ? Government regulator——Maximizing public welfare Chapter 1 25 Theories and Models 4 ? General Features of Economic Model: ? 3. Positive and Normative Distinction ? Positive Analysis( L￡s ) is the use of theories and models to explain how resources actually are allocated in an economy ? Try to answer: What is ^ I 1 ? For example: ? What will be the impact of an import quota é g ￥- on foreign cars? Chapter 1 26 Theories and Models 5 ? Positive and Normative Distinction ? Normative analysis ?Ss is theories that make value judgments about how the economic resource should be allocated ? Try to answer: “What ought to be?” ?? ^ I 1 ? For example: ? Whether should an import quota be used on foreign cars or not? Chapter 1 27 The Positive-Normative Distinction ? Distinction between theories that seek to explain the world as it is and theories that postulate the way the world should be ? To many economists, the correct role for theory is to explain the way the world is (positive) rather than the way it should be (normative) ? Positive economics is the primary approach of the text Chapter 1 28 How Economists Verify Theoretical Models ? Two methods are used ? Direct Approach °¤￡E : Testing Assumptions: Verifying economic models by examining validity of the assumptions on which they are based ? Indirect Approach(W¤_E ) Testing Predictions: Verifying economic models by asking if they can accurately predict real-world events Chapter 1 29 Testing Assumptions ? One approach would be to determine if the assumptions are reasonable ? The obvious problem is that people have differing opinion regarding reasonable ? Empirical evidence can also be used ? Results of such methods have had problems similar to those found in opinion polls Chapter 1 30 Testing Predictions ? Economists, such as Milton Friedman argue that all theories require unrealistic assumptions ? The theory is only useful if it can be used to predict real-world events ? Even if firms state they don’t maximize profits, if their behavior can be predicted by using this assumption, the theory is useful Chapter 1 31 Chapter one includes: ? 1.1 Defining Microeconomics ? 1.2 Economic Theories and Models ? 1.3 Basic Demand-Supply Model ? 1.4 The Mathematics of Optimization Chapter 1 32 A Popular Economic Adage “Even your parrot can become an economist—just teach it to say ‘supply and demand.’” Chapter 1 33 1.3 The Basic Supply-Demand Model ? The Supply-Demand Model ? A model describing how a good’s price is determined by the behavior of the individual’s who buy the good and the firms that sell it. ? Economists argue that market behavior can generally be explained by this model that captures the relationship between consumers’ preferences and firms’ costs. Chapter 1 34 Basic Assumptions('L ! ) about the Supply-Demand Model ? Rational Behavior Assumption ? ??1L ! ? Each person tries to choose the best alternative available to him or her ? Scarce Resource Assumption '÷  ??L ! Chapter 1 35 Marginalism H=?l and Marshall’s Model of Supply and Demand ? Ricardo’s model was unable to explain the fall in the relative prices of good during the nineteenth century so a more general model was needed ? Economists argued the willingness of people to pay for a good will decline as they have more of it Chapter 1 36 Marginalism and Marshall’s Model of Supply and Demand ? People will be willing to consume more of a good only if the price is lower ? The focus of the model was on the value of the last, or marginal, unit purchased ? Alfred Marshall (1842-1924) showed how the forces of demand and supply simultaneously ] H determined price Chapter 1 37 FIGURE 1.3: The Marshall Supply-Demand Cross Demand Supply Price 0 Quantity per week Chapter 1 38 Marginalism and Marshall’s Model of Supply and Demand ? The demand curve shows the amount people want to buy at each price and is negatively sloped reflecting the marginalism principle ? The upward sloping supply curve reflects the idea of increasing cost of making one more unit of a good as total production increases Chapter 1 39 Marginalism and Marshall’s Model of Supply and Demand ? Supply reflects increasing marginal costs and demand reflects decreasing marginal usefulness Chapter 1 40 Market Equilibrium  g? (? ? In Figure 1.3, the demand and supply curve intersect at the market equilibrium point P*, Q* ? P* is the equilibrium price: The price at which the quantity demanded by buyers of a good is equal to the quantity supplied by sellers of the good Chapter 1 41 FIGURE 1.3: The Marshall Supply-Demand Cross Price Demand Supply Equilibrium point . P* 0 Quantity per week Q* Chapter 1 42 Market Equilibrium ? Both demanders and suppliers are satisfied at this price, so there is no incentive for either to alter their behavior unless something else happens ? Marshall compared the roles of supply and demand in establishing market equilibrium to the two blades of a pair of scissors working together in order to make a cut Chapter 1 43 Nonequilibrium Outcomes ? If something causes the price to be set above P*, demanders would wish to buy less than Q* while suppliers would produce more than Q* ? If something causes the price to be set below P*, demanders would wish to buy more than Q* while suppliers would produce less than Q* Chapter 1 44 Change in Market Equilibrium: Increased Demand ? Figure 1.4 shows the case where people’s demand for the good increases as represented by the shift of the demand curve from D to D’ ? A new equilibrium is established where the equilibrium price has increased to P** Chapter 1 45 FIGURE 1.4: An increase in Demand Alters Equilibrium Price and Quantity Price D D’ S P* P** Q* Q** Quantity per week 0 Chapter 1 46 Change in Market Equilibrium: decrease in Supply ? In Figure 1.5 the supply curve has shifted leftward reflecting a decrease in supply brought about because of an increase in supplier costs (say an increase in wages) ? At the new equilibrium price P** consumers respond by reducing quantity demanded along the Demand curve D Chapter 1 47 FIGURE 1.5: A shift in Supply Alters Equilibrium Price and Quantity Price D S P* Quantity per week 0 Q* Chapter 1 48 FIGURE 1.5: A shift in Supply Alters Equilibrium Price and Quantity Price D S’ S P* P** Quantity per week 0 Q** Q* Chapter 1 49 How We Verify the Supply- Demand Models ? Two methods are used ? Testing Assumptions: Verifying economic models by examining validity of the assumptions on which they are based ? Testing Predictions: Verifying economic models by asking if they can accurately predict real-world events Chapter 1 50 ?An Experimental Economics Example Chapter 1 51 Double Auction ?Z ? ? ? Buyers attributed marginal values ? Sellers attributed marginal costs ? Buyers only know own values ? Sellers only know own costs highest bids lowest asks trading prices ….made public Chapter 1 52 Double Auction: Example Buyers’ values: Buyer unit1 unit2 B1 5.2 4.8 B2 5.0 --- B3 4.6 4.4 B4 4.2 4.0 Chapter 1 53 Double Auction: Example Sellers’ costs: Seller unit1 unit2 S1 3.7 4.4 S2 3.8 4.2 S3 3.9 4.0 S4 4.1 --- Chapter 1 54 Double Auction: Example Aggregate Demand and Supply:(marginal values give demand function &Marginal costs give supply function) 3.7 3.9 4.1 4.3 4.5 4.7 4.9 5.1 5.3 3.5 price 12 43 5678 unit Chapter 1 55 Double Auction: Example 3.7 3.9 4.1 4.3 4.5 4.7 4.9 5.1 5.3 3.5 price B2 B1 B3 B3 B4 B4 B1 B1 5.2 4.8 B2 5.0 --- B3 4.6 4.4 B4 4.2 4.0 12 43 5678 unit Chapter 1 56 Double Auction: Example 3.7 3.9 4.1 4.3 4.5 4.7 4.9 5.1 5.3 3.5 price B2 B1 B3 B3 B4 B4 S1 3.7 4.4 S2 3.8 4.2 S3 3.9 4.0 S4 4.1 --- S1 S2 S3 S3 S4 S2 S1 B1 12 43 5678 unit Chapter 1 57 Double Auction: Example 3.7 3.9 4.1 4.3 4.5 4.7 4.9 5.1 5.3 3.5 price B1 B2 B1 B3 B3 B4 B4 S1 S2 S3 S3 S4 S2 S1 Q*=5 or 6 Competitive predictions P*=4.2 12 43 5678 unit Chapter 1 58 Chapter one includes: ? 1.1 Defining Microeconomics ? 1.2 Economic Theories and Models ? 1.3 Basic Demand-Supply Model ? 1.4 The Mathematics of Optimization Chapter 1 59 1.4 The Mathematics of Optimization  ?D￥Ka?ù5 ? Optimization of Economic Behavior ? Maximization Kv? ? Minimization Kl? ? Type of Optimization ? Unconstrained Optimization (í? ?￥Ka? ) ? Unconstrained Optimization  μ? ?￥Ka? Chapter 1 60 Unconstrained Optimization 1 ? Function of One Varibale(Bíf ? ) ? For example Y=f(X) ? First-Order Condition B¨Hq for Maximization ? Second-Order Condition (=¨Hq ) for Maximization 0= ? =xx dx df 0 2 2 < ? =xx dx fd Chapter 1 61 Unconstrained Optimization ? 2. Function of Several Variables í f ? ? Y=f(X 1 ,X 2 ……X n ) ? F.O.C for Max.(Min.) f 1 =f 2 =…=f n =0 ? S.E.C For Max(Min) Hessian Matrix(Z × ? ) for this function is negative (positive) semidefinite(μ?? ) ?? ? Chapter 1 62 Constrained Optimaztion ? MAX. Y=f(X 1 ,X 2 ……X n ) ? S.t. g(X 1 ,X 2 ……X n )=0 ? Construct Lagrange Function Chapter 1 63 The End Chapter 1 64 Last Revised: September. 4, 2005 Chapter 2 Consumer Behavior and Demand Theory (hn??1D3 p ? ? ) ? 2005 MOL 2 This Chapter will Include: ? 2.1 Preference and Utility ? 2.2 Utility Maximization and Choice ? 2.3 Income and Substitution Effects  l ?r?D9}r? ? 2.4 Market Demand and Elasticity 3 Readings about this chapter ? Nicholson: Chapter 2-5,7 ? Zhang: Chapter 3, 4 Outline of Today’s Class ? 2.1 Preference and Utility ? Assumption of Consumer’s Preference ? From Preference to Utility Function ? Indifference Curve íμs wL ? Characteristic of Indifference Curve ? Diminishing MRS (H=9} q?hE5 ) 5 Consumer Preferences  hn? êz ? Consumer’s Rationality in Economics ? Basic behavioral postulate: Maximizing his well- being ? Consumer’s optimization is constrained by his income or time ? Decision-maker chooses best available alternative ? So, to model consumer choice must model consumer preferences 6 Defining Preference When an individual reports that “X is preferred to Y,” it is taken to mean that all things considered, he or she feel better off under situation X than under situation Y. Note: X, or Y is called a bundle of consumption goods (hn ? ?F? ) 7 How to Express Preference Relations(1) ? Bundles of consumption goods: ),( 21 xxX = ),( 21 yyY = ),(),( 2121 yyxx f ),(),( 2121 xxyy f ),(~),( 2121 xxyy If bundle X is preferred to Y If bundle Y is preferred to X If consumer indifferent between X and Y For example, x 1 =5 bananas, x 2 =6 oranges, y 1 =3 coconuts and y 2 =2 packets of biscuits 8 How to Express Preference Relations(2) ? Consumers can compare two different consumption bundles, X and y: ? strict preference ?ì êz : ? weak preference D êz : X >Y ? indifference: X ~ Y ? Preference Relations ? If X > Y and Y > X, then X~Y ? If X > Y but not Y> X, then X > Y YX f 9 How to Express Preference Relations(3) ? These are ordinal relations ? ?1" ? Only rank alternative bundles by order (e.g. first,second,…) ? Do not specify magnitude ?  of preference differences 10 Assumptions of Consumer’s Preference(1) ? Assumption 1:Completeness ?!?L ! If X and Y are any situations, the individual can always specify exactly one of the following three possibilities: ? “X is preferred to Y”, i.e. ? “Y is preferred to X”, i.e. ? “X and Y are equally attractive”, i.e. ),(),( 2121 yyxx f ),(),( 2121 yyxx p Complete preferences means consumer can always make a clear choice over any two bundles ),(~),( 2121 yyxx 11 Assumptions of Consumer’s Preference(2) ? Assumption 2:Transitivity .??L ! if an individual reports that “X is preferred to Y” and that “Y is preferred to Z,” then he or she must also report that “X is preferred to Z” ),(),( 2121 yyxx f ),(),( 2121 zzxx f ),(),( 2121 zzyy f , and If then Transitivity means a consumers choices must be logically consistent 12 Assumptions of Consumer’s Preference(3) ? Assumption 3: Reflexive. 1Q?L ! ? Assumption 4:Non Satiation or More is good. d??L ! An economic good is one that yields positive benefits to people. Thus, more of a good is, by definition, better.(m ? ) xx≥ 13 Assumptions of Consumer’s Preference(4) ? Assumption 5: Continuity  ?? ? if an individual reports that “X is preferred to Y” then situations suitably “close to” X must also be preferred to Y” ? Continuity: small bundle changes cause only small changes to the preference level 14 From Preference to Utility Functions ? If consumer preferences are ? Complete ? Reflexive ? Transitive ? Non-satiation ? Continuous (no jumps) ? Then, can be represented by a continuous utility function. 15 Utility Functions r¨f ? ? A utility function U(x) represents a preference relation if and only if(? O?? ): x’ > x” U(x’) > U(x”) x’ < x” U(x’) < U(x”) x’ ～ x” U(x’) = U(x”). ? The utility function must preserve the preference ordering 16 Utility Functions ? Utility is an ordinal (ordering) concept ? Example ?If U(x) = 6 and U(y) = 2, then x > y ?But x not preferred three times as much as y 17 General Expression for Utility Function ? we assume that a person receives utility from the consumption of n goods X 1, X 2,…. X n, which we can show in functional notation by ? The other things that appear after the semicolon are assumed to be held constant. gs).other thin ;,( 21 n XXXUUtility L= 18 Defining Marginal Utility(H=r¨ ) Marginal utility MU : Change in utility from small change in one commodity holding other commodities fixed 1 21211 1 1 ),,(),,( x xxxuxxxxu x u MU nn ? ??+ = ? ? = LL 1 21 1 ),,( x xxxu MU n ? ? = L …… 2 21 2 ),,( x xxxu MU n ? ? = L 2 21 ),,( x xxxu MU n n ? ? = L 19 Non-uniqueness of Utility Measures(1) r¨ ￥d·B? ? U(A) = 5 and U(B) = 4 or U(A) = 1,000,000 and U(B) = 0.5. ? In either case the numbers imply that A is preferred to B. It show that we can express the same preference relations by the different utility functions. 20 Non-uniqueness of Utility Measures(2) Bundle U 1 U 2 U 3 X67967-2 Y74-4 Z 1 -6 -8 Example Each of the three utility functions ranks the bundles in same way The preference ordering unaffected by a monotonic transformation(??MD ) of the utility function 21 Non-uniqueness of Utility Functions  3 ? There is no unique utility function representation of a preference relation ? Suppose U(x 1 ,x 2 ) = x 1 x 2 ? Again consider bundles (4,1), (2,3), (2,2) ? U(2,3) = 6 > U(4,1) = U(2,2) = 4 ? Or, suppose V(x 1 ,x 2 ) =U 2 = x 1 2 x 2 2 ? Then V(2,3) = 36 > U(4,1) = U(2,2) = 16 ? V preserves same order, reflects same preferences 22 Non-uniqueness of Utility Functions 4 ? Or W(x 1 ,x 2 ) = 2U + 10 = 2 x 1 x 2 +10 ? For same bundles (4,1), (2,3), (2,2) ? W(2,3) = 22 > U(4,1) = U(2,2) = 18 ? Again, preserves same ordering, represents same preferences ? True for any monotonic (strictly increasing) transformation ?ì?9￥ ??MD of U(x 1 ,x 2 ) 23 Monotonic Transformation of the Utility Monotonic Transformation Non-Monotonic Transformation 24 Example: The below are examples of the Cobb-Douglas function 2121 ),( xxxxu = 2121 ),( xtxxxu = 2 21 2 2 2 121 )(),( xxxxxxu == dc xxxxu 2121 ),( = c and d are constants reflecting the importance of that commodity Note for later: We can always find a monotonic transformation such that the above becomes aa xxxxu ? = 1 2121 ),( a+(1-a)=1 25 Utility Functions and Indifference Curve(íμs wL ) ? Utility is an ordinal (ordering) concept ? Example ? If U(x) = 6 and U(y) = 2, then x > y ? But x not preferred three times as much as y ? Indifference curve (set): all equally preferred bundles ? Equal preference ? same utility level ? Therefore, all bundles on an indifference curve have same utility level 26 Utility Functions and Indifference Curves ? Example: ? Suppose bundles (4,1) and (2,2) are on the indifference curve with utility level U ≡4 ? Bundle (2,3) is in the indifference curve with utility level U ≡ 6. ? Implies indifference curve: 27 Utility Functions and Indifference Curves U ≡ 6 U ≡ 4 (2,3) (2,2) ～ (4,1) x 1 x 2 p 28 Utility Functions and Indifference Curves ? Or could visualize in three dimensions: ? Plot consumption bundles in horizontal plane ? Plot the utility level on a vertical axis ? Start with several points ? Add indifference curves through points ? Add more indifference curves to represent all preferences over all consumption bundles 29 Utility Functions and Indifference Curves U(2,3) = 6 U(2,2) = 4 U(4,1) = 4 3-D plot ??m of consumption, utility levels for 3 bundles x 1 x 2 Utility 30 Utility Functions and Indifference Curves U ≡4 U ≡6 Higher indifference curves contain more preferred bundles. Utility x 2 x 1 31 Utility Functions and Indifference Curves U ≡ 6 U ≡ 5 U ≡ 4 U ≡ 3 U ≡ 2 U ≡ 1 x 1 x 2 Utility 32 Utility Functions and Indifference Curves ? Let orange to green represent increasing utility levels ? Can completely represents the consumer’s preferences in 3-D ? This is an “indifference map” that fully represents a utility function 33 Utility Functions and Indifference Curves x 1 x 2 34 Utility Functions and Indifference Curves x 1 x 2 35 Utility Functions and Indifference Curves x 1 x 2 36 Utility Functions and Indifference Curves x 1 x 2 37 Utility Functions and Indifference Curves x 1 x 2 38 Utility Functions and Indifference Curves x 1 x 2 39 Utility Functions and Indifference Curves x 1 40 Utility Functions and Indifference Curves x 1 41 Utility Functions and Indifference Curves x 1 42 Utility Functions and Indifference Curves x 1 43 Utility Functions and Indifference Curves x 1 44 Utility Functions and Indifference Curves x 1 45 Utility Functions and Indifference Curves x 1 46 Utility Functions and Indifference Curves x 1 47 Utility Functions and Indifference Curves x 1 48 Utility Functions and Indifference Curves x 1 49 The Characteristics of the Indifference Curve ? 1.There are many (or countless) indifference curves in the X-Y plane, in which each point must have an indifference curve passing through it (by assumption 5 ) ? 2.An indifference curve above and to the right of another represents preferred combination of commodities. (by assumption 4 ). 50 The Characteristic of the Indifference Curve ? 3.An indifference curve has negative slope (by assumption 4). ? 4.Indifference curve can never intersect (by assumption 3 and 4) ? 5.The Shape of indifference curve is convex to the Origin  j_e? . by Diminishing marginal rates of substitution. ? The slope of an difference curve diminishes as we move from left to right along its length. 51 03.02 Transitivity and Non-satiation of preferences implies indifference curves cannot cross . X Figure for Two Indifference Curves Cannot Intersect 52 the Marginal Rate of Substitution(H=9} q ) ? Marginal Rate of Substitution (MRS): The rate at which an individual is willing to reduce consumption of one good when he or she gets one more unit of another good in order to hold his or her utility constant. ? MRS is also the negative of the slope of an indifference curve. ? The MRS between points A and B on U 1 in Figure of next page is (approximately) 2. 53 FIGURE : Indifference Curve and MRS X 2 6 A B C E F D U 1 4 3 2 X 1 234560 2 1 1 2 / / xU xU xd xd MRS ?? ?? =?= 54 Marginal Utilities and the MRS ? The equation of an indifference curve is U(x 1 ,x 2 ) ≡ k, a constant ? Totally differentiating this identity gives MRS )MU(x )MU(x xU/ xU/ dx dx or,dx x U dx x U == ?? ?? =? = ? ? + ? ? 2 1 2 1 1 2 2 2 1 1 0 55 Example of Marginal Utilities and MRS ? Suppose U(x 1 ,x 2 ) = x 1 x 2 . Then ? ? ? ? U x xx U x xx 1 22 2 11 1 1 == == ()( ) ()() 1 2 2 1 1 2 / / x x xU xU xd xd MRS ==?= ?? ?? so 56 Example of Marginal Utilities and MRS 1 2 x x MRS = MRS(1,8) = 8/1 = 8 MRS(6,6) = 6/6 = 1. x 1 x 2 8 6 16 U = 8 U = 36 U(x 1 ,x 2 ) = x 1 x 2 ; 57 MRS for Quasi-linear Utility Function( EL?r¨f ? ) ? Quasi-linear utility function: ? U(x 1 ,x 2 ) = f(x 1 ) + x 2 ? ? U x fx 1 1 = ′() ? ? U x 2 1= ).( / / 1 2 1 1 2 xf xU xU xd xd MRS ′ ==?= ?? ?? 58 MRS for Quasi-linear Utility Functions x 2 x 1 Each curve is a vertically shifted copy of the others. MRS is a constant along any line for which x 1 is constant. MRS = f(x 1 ’) MRS = f(x 1 ”) x 1 ’ x 1 ” 59 Monotonic Transformations and MRS ? Monotonic transformation of U does not change underlying preference structure ? Example: U(x 1 ,x 2 ) = x 1 x 2 with MRS = x 2 /x 1 ? Consider V = U 2, or V(x 1 ,x 2 ) = x 1 2 x 2 2 which is the same as the MRS for U ? Monotonic transformation does not change MRS 1 2 2 2 1 2 21 2 1 2 2 / / x x xx xx xV xV MRS === ?? ?? 60 ?Special Utility Functions and Their Indifference Curve 61 Diminishing MRS Along an Indifference Curve 62 Example: Perfect Substitutes  ? ?9} ? ? Suppose goods are perfect substitutes for example, Coke and Pepsi; red pencil and blue pencil, etc. ? Consumer cares only about total of both goods ? So, preferences can be represented by utility function of the form V(x 1 ,x 2 ) = x 1 + x 2 . ? Along an IC, utility is constant (k), so x 2 = k – x 1 ? Linear with slope of -1 63 Perfect Substitutes Indifference Curves 5 5 9 9 13 13 x 1 x 2 x 1 + x 2 = 5 x 1 + x 2 = 9 x 1 + x 2 = 13 V(x 1 ,x 2 ) = x 1 + x 2 64 Perfect substitutes (Linear utility) 2121 ),( bxaxxxu += Generally Marginal Utilities aMU = 1 bMU = 2 MRS=a/b 65 Example: Perfect Complements  ? ?o? ? ? Suppose goods are perfect complements, e.g. Left Shoe and Right Shoe ? Consumer can use goods only in pairs ? Goods consumed in fixed proportions ? Can be represented by utility function W(x 1 ,x 2 ) = min{x 1 ,x 2 }. ? Gives “right angle” °? IC’s with vertices on ray through the origin 66 Perfect Complements Indifference Curves x 2 x 1 45 o min{x 1 ,x 2 } = 8 3 5 8 3 5 8 min{x 1 ,x 2 } = 5 min{x 1 ,x 2 } = 3 W(x 1 ,x 2 ) = min{x 1 ,x 2 } 67 indifference curves slope upwards | q_ ￥íμs w L 03.05 Smoking a bad Drinking a good Indifference curves positive slope Need more Smoking to compensate for accepting more Drinking Drinking Smoking 68 A Neutral Good ??t ? 03.06 My Wages Li’s Wags 69 Example: Cobb-Douglas Utility Function( S??ì ? ?r¨f ? ) ? Cobb-Douglas utility function has form U(x 1 ,x 2 ) = x 1 a x 2 b , a > 0, b > 0 ? Examples: ? U(x 1 ,x 2 ) = x 1 1/2 x 2 1/2 (a = b = 1/2) ? V(x 1 ,x 2 ) = x 1 x 2 3 (a = 1, b = 3) 70 Cobb-Douglas Indifference Curves x 2 x 1 All curves are hyperbolic  ?à ? wL , asymptoting to, but never touching any axis 71 Example: Cobb-Douglas utility aa xxxxu ? = 1 2121 ),( 1 1 2 1 11 x aU xaxMU aa == ?? 2 212 )1( )1( x Ua xxaMU aa ? =?= ? 1 2 2 1 1 x x a a MU MU MRS ? == Slope of the indifference curve (Marginal rate of substitution) Marginal Utilities a+b=1 72 Quasilinear Preferences EL? êz 2121 )(),( xxvxxu += Linear in one of the commodities The End Last Revised: September. 9, 2005 1 T<#5 [? ? ? T<5|'? ?? b?  h]D ì? H? ?b ? ?QT<?/?B (17 ° )  [ Hμò?D?? l ?ó5 [ o =b ? > o =?5 [ HW? 1 1-2?1? ?? 213b ? f o =?5 [ HW?? 7-8?1?è? 112b Chapter 2 Consumer Behavior and Demand Theory 3 Chapter 2 includes: ? 2.1 Preference and Utility ? 2.2 Utility Maximization and Choice ? 2.3 Income and Substitution Effects ? 2.4 Market Demand and Elasticity 4 Overview of Last Class ? Assumptions of Consumer’s Preference ? Defining Utility ? How to Get Indifference Curve ? Characteristic of Indifference Curve ? Diminishing MRS H=9} q?hE 5 5 Outline of Today’s Class ? Budget Constraint(? ?? ? ) ? Application of Budget Constraint ? Utility Maximization (Consumer’s Optimal Choice) ? Types of Optimal Solution 6 Utility Maximization: An Initial Survey ? Economists assume that when a person is faced with a choice among several possible options, he or she will choose the one that yields the highest utility- utility maximization. ? Economists assume that people know their own minds and make choices consistent with their preferences. 7 Choices are Constrained s? ?￥ ? People are constrained in their choices by the size of their incomes and other resources. ? Of the choices the individual can afford, the person will choose the one that yields the most utility. 8 Budget Constraint Set ? ?? ?" ? An individual’s budget constraint Set (Consumption Choice set) is the set that income places on the combinations of goods and services that a person can buy. 9 Consumption Choice (Budget) Sets ? Consumption choice (or budget) set ? All consumption choices available to consumer ? “Available” – subject to constraints 10 Budget Sets and Constraints ? Consumption bundle: (x 1 , x 2 , … , x n ) ? Commodity prices: p 1 , p 2 , … , p n ? Budget set defined by income constraint: ? x 1 ≥ 0, …, x n ≥0 ? p 1 x 1 + … + p n x n ≤ I ? I = consumer’s (disposable) income ? Budget constraint assumes all income is spent ? p 1 x 1 + … + p n x n = I ? Is thus upper boundary of budget set ? ?? ?"￥  H? 11 Budget Set, Constraint for Two Goods x 2 x 1 Budget constraint is p 1 x 1 + p 2 x 2 = I Intercepts ?  are I/p 1 and I/p 2 I/p 1 I/p 2 12 Budget Set, Constraint for Two Goods x 2 x 1 Budget constraint is p 1 x 1 + p 2 x 2 = I I/p 2 I/p 1 13 Budget Set, Constraint for Two Goods x 2 x 1 Budget constraint is p 1 x 1 + p 2 x 2 = I. I/p 1 Just affordable I/p 2 14 Budget Set, Constraint for Two Goods x 2 x 1 Budget constraint is p 1 x 1 + p 2 x 2 = I. I/p 1 Just affordable Not affordable I/p 2 15 Budget Set, Constraint for Two Goods x 2 x 1 Budget constraint is p 1 x 1 + p 2 x 2 = I. I/p 1 Affordable with cash left Just affordable Not affordable I/p 2 16 Budget Set, Constraint for Two Goods x 2 x 1 Budget constraint is p 1 x 1 + p 2 x 2 = I I/p 1 Budget Set Collection of all affordable bundles I/p 2 17 Budget Set, Constraint for Two Goods x 2 x 1 Solve p 1 x 1 + p 2 x 2 = I for x 2 : x 2 = I/p 2 -(p 1 /p 2 )x 1 I/p 1 Budget Set I/p 2 18 Slope of Budget Constraint (2 Goods) ? x 2 = I/p 2 -(p 1 /p 2 )x 1 ? Slope of Budget Line is - (p 1 /p 2 ) 19 Budget Constraint (2 Goods) x 1 Slope is - p 1 /p 2 +1 -p 1 /p 2 x 2 20 Effects of increase in income I Original budget set x 2 x 1 21 Effects of increase in income I Original budget set New affordable consumption choices – budget set expands x 1 Original and new budget constraints are parallel (same slope). x 2 22 Effects of decrease in income I x 2 Original budget set x 1 23 Effects of decrease in income m x 2 x 1 New, smaller budget set Consumption bundles that are no longer affordable. Old and new constraints are parallel. 24 Effects of p 1 Decrease from p 1 ’ to p 1 ” Original budget set x 2 x 1 I/p 2 I/p 1 ’ I/p 1 ” -p 1 ’/p 2 25 Effects of p 1 decrease from p 1 ’ to p 1 ” Original budget set x 2 x 1 I/p 2 I/p 1 ’ I/p 1 ” New affordable choices Set expands – welfare up -p 1 ’/p 2 26 Effects of p 1 decrease from p 1 ’ to p 1 ” Original budget set x 2 x 1 I/p 2 I/p 1 ’ I/p 1 ” New affordable choices Budget constraint pivots; slope flattens from -p 1 ’/p 2 to -p 1 ”/p 2 -p 1 ’/p 2 -p 1 ”/p 2 27 Applications of Budget Constraint Set 28 Example of Price Changes: Sales Taxes ? Proportional or ad valorem sales tax VN ￥ ? Levied at proportional rate t ? Increases price from p to (1+t)p ? Changes budget constraint to (1+t)p 1 x 1 + (1+t)p 2 x 2 = I p 1 x 1 + p 2 x 2 = I/(1+t). 29 Uniform Ad Valorem Sales Taxes x 2 x 1 1 p I p 1 x 1 + p 2 x 2 = I 2 p I 30 Uniform Ad Valorem Sales Taxes x 2 x 1 1 p I p 1 x 1 + p 2 x 2 = I p 1 x 1 + p 2 x 2 = I/(1+t) 1 )1( pt I + 2 p I 2 )1( pt I + 31 Uniform Ad Valorem Sales Taxes x 2 x 1 1 )1( pt I + 1 p I Equivalent income loss I t t t I I + = + ? 11 2 p I 2 )1( pt I + 32 An Example of A Budget Set With Rationing Lt ￥ó 02.04 Ixpxp ≤+ 2211 11 xx ≤ Suppose good 1 were rationed so that no more than could be consumed by a given consumer. 1 x 33 Taxing Consumption Greater Than 1 x 02.05 In this budget line the consumer must pay a tax only on the consumption of good 1 that is in excess of so the budget line becomes steeper to the right of 1 x 1 x 34 The Food Stamp Program J ? ? ? Food stamps ? Popular income support program ? Coupons a? ? given to poor (used to be sold) ? Can be legally exchanged only for food ? Popular with some donors (“in kind” transfers) ? Popular with agricultural interests 35 The Food Stamp Program ? What is effect on budget constraint? ? Suppose I = $100 ? p F = $1 ? Price of “all other goods” is p G = $1 ? Budget constraint is F + G =100 ? Key factor: Income available for “other goods” does not change with receipt of food stamps ? Suppose receive food stamps good for 40 units of food 36 The Food Stamp Program G F 100 100 F + G = 100, before food stamps 37 The Food Stamp Program J ? ?9? G F 100 100 F + G = 100: before stamps Budget set after 40 food stamps issued 14040 38 The Food Stamp Program G F 100 100 140 Welfare up since budget set is enlarged 40 39 The Food Stamp Program ? If food stamp program is generous, families may be at “kink” |?) of budget set ? What if food stamps can be traded on a black market for $0.50 each? 40 The Food Stamp Program G F 100 100 140 120 Budget constraint with black market trading, $0.50 on dollar 40 41 The Food Stamp Program G F 100 100 140 120 Black market trading expands budget set further 40 42 Quantity-Based Prices [ ? 1$ ￥Nì ? Price may be a function of quantity ? Discounts for large buyers, or, ? Penalties for buying “too much” ? Budget constraints “kinked” where p changes ? Suppose quantity discount: ? p 2 constant at $1 and I=100 ? p 1 =$2 for 0 ≤ x 1 ≤ 20; -p 1 /p 2 = -2 ? p 1 =$1 for x 1 >20; -p 1 /p 2 = -1 43 Budget Constraints / Quantity Discount I = $100 20 units @ p=2 60 units @ p=1 50 100 20 Slope = - 2 Slope = - 1 80 x 2 x 1 44 Budget Constraints / Quantity Discount I = $100 50 100 20 80 x 2 x 1 Budget Set Budget Constraint 45 Budget Constraints / Quantity Penalty x 2 x 1 Budget Set Budget Constraint 46 Utility Maximization r¨Kv? ? An individual can afford all bundles of X and Y that fall within the budget constraint represented by the shaded area in Figure 2.7. ? Point A is affordable but not all of the consumer’s income would be spent. ? Point B is affordable but is not on the highest indifference curve that can be reached by the consumer. 47 FIGURE 2.7: Graphic Demonstration of Utility Maximization X 2 X 2 * B A C D Income U 2 X 1 0 X 1 * U 3 U 1 48 Utility Maximization ? Point D is on a higher indifference curve than C, but is not affordable given the budget constraint. ? Point C, where the consumer chooses X 1 * , X 2 * is the point that is affordable that lies on the highest indifference curve, so it represents utility maximization. 49 Utility Maximization——Optimal Choice ? At point C all income is spent. ? At point C indifference curve U 2 is tangent to the budget line so that the ? or curve ceindifferen of Slope constraintbudget of Slope = . 2 1 MRS P P = 50 Optimal choice: Typically where slope of indifference curve equals slope budget line 2 1 2 1 2 21 1 21 12 ),( ),( p p MU MU x xxu x xxu MRS == ? ? ? ? = Rewrite this condition as: 2 2 1 1 p MU p MU = The marginal utility of the last penny spent on each good is the same 51 An example with well behaved preferences 5.0 2 5.0 1 xxu += 5.0 1 1 5.0 x MU = 5.0 2 2 5.0 x MU = 2 1 5.0 1 2 )( p p x x MRS == 2 1 2 2 11 x p p xp = Ixpxp =+ 2211 Max. S.T. Use this in the budget constraint to solve for the demand function for x 1 ,x 2 : I ppp p x ][ 211 2 1 + = I ppp p x ][ 212 1 2 + = 52 The Lagrangean approach is an alternative approach for Constrained optimization problem ][),(),,( 22112121 IxpxpxxuxxL ?+?= λλ Utility is to be maximised The lagrange multiplier The budget constraint Take first order necessary conditions for maximum 0 ),( 1 1 21 =? ? ? p x xxu λ 0 ),( 2 2 21 =? ? ? p x xxu λ 0 2211 =?+ Ixpxp These conditions give the same three equations as in previous slide 53 Types of the Optimal Solution For Maximization Utility 54 Optimal choice with well behaved preferences——interior solution Optimal choice with well behaved preferences—— interior solution 55 Optimal choice with not well-behaved preferences——Corner Solution(?3 ) The tangency condition will not hold between two goods if one is not consumed 2 2 1 1 p MU p MU > Even when all income spent on good 1 56 Optimal Choice with Perfect Substitutes 05.05 57 21 bxaxu += Perfect substitutes: aMU = 1 bMU = 2 2 1 p p b a MRS == Since MRS is constant, the tangency condition only holds if both slopes the same 2 1 p p b a > 0, 2 1 1 == x p I x Consume all x 1 If 2 1 p p b a < Consume all x 2 x 0, 1 2 2 == x p I If 58 Optimal Choice with Perfect complements 05.06 59 Optimal Choice for Perfect complements ),min( 21 bxaxu = Tangency condition will not hold since indifference curves not continuous. Slope is either zero, infinity or not defined a b x x = 2 1 Consumer will always be on the corner of the indifference curve consuming in fixed proportions Use this condition in the budget constraint to solve for x1 or x2 2211 xpxpI += bpap bI x 21 1 + = bpap aI x 21 2 + = 60 More than one Solution Non-convex preferences may lead to more than one optimal outcome 61 Optimal Choice with Concave(?? ) Preferences 05.08 Concave preferences consume at a corner to maximise utility 62 Using the Model of Choice ? The utility maximization model can be used to explain many common observations. ? Figure in next page shows people with the same income still consume different bundles of goods. 63 Differences in Preferences Result in Differing Choices Hamburgers per week 8 04 Income 2 16 U 0 U 1 U 2 Soft drinks per week 20 Soft drinks per week Soft drinks per week Hamburgers per week Hamburgers per week U 0 U 0 U 1 U 1 U 2 U 2 Income Income (a) Zhang San (c) Wang Wu(b) LI Si 64 Using the Model of Choice ? Figure in next page shows the four indifference curve maps with a budget constraint and the utility maximizing choice labeled E. ? Panel (a) shows that people will not buy useless goods and (b) shows they will not buy bads. ? Panel (c) shows that people will buy the least expensive of the two perfect substitutes while (d) shows that perfect complements will be purchased together. 65 Utility-Maximizing Choices for Special Types of Goods E Income Houseflies per week U 1 U 2 U 3 Food per week010 U 1 E U 2 U 3 Food per week Income 010 E E Income Income (c) Perfect substitute U 1 U 2 U 3 Gallons of Mobil per week 0 (d) Perfect complements Right shoes per week U 3 U 1 U 2 Left shoes per week 0 2 2 Smoke grinders per week (b) An economic bad(a) A useless good Gallons of Exxon per week 66 APPLICATION : Quantity Discounts and Frequent- Flier Programs ? When consumers receive quantity discounts or have to pay excessive use fees, the budget line is no longer straight. ? In Figure 1, the consumer pays regular price for good X up to X D but receive a quantity discount beyond that as shown by the flatter budget line after consuming X D . 67 FIGURE 1: Kinked Budget Constraint Resulting from a Quantity Discount Quantity of X per period 0 X 0 B U 1 A Quantity of Y per period 68 APPLICATION : Quantity Discounts and Frequent- Flier Programs ? Since the consumer is indifferent between points A and B, a slightly larger discount would cause the consumer to reach a higher indifference curve by using the discount. ? All major airlines use frequent-flier programs that provide such quantity discounts and enable the airlines to gain revenues on seats that otherwise would remain empty. The End Last Revised: September 12, 2005 ?vDü6Dy 1 Chapter 2 Consumer Behavior and Demand Theory ?vDü6Dy 2 Chapter 2 includes: ? 2.1 Preference and Utility ? 2.2 Utility Maximization and Choice ? 2.3 Income and Substitution Effects ? 2.4 Market Demand and Elasticity ?vDü6Dy 3 Overview of Last Class ? Budget Constraint(? ?? ? ) ? Application of Budget Constraint ? Utility Maximization (Consumer’s Optimal Choice) ? Types of Optimal Solution ?vDü6Dy 4 Outline of Today Class ? How to Get Individual Demand Function (Algebra and Graph) ? Marshallian ?u: demand function ? Price-Consumption Curve NìhnL and Demand Curve ? Income-Consumption Curve l ?hnL and Engel Curve 6ì: wL ? Examples of relative static equilibrium ?vDü6Dy 5 A Survey of This Section(2.3) ? This section studies how people change their choices when conditions such as income or changes in the prices of goods affect the amount that people choose to consume. ? This section then compares the new choices with those that were made before conditions changed ? The main result of this approach is to construct an individual’s demand curve ?vDü6Dy 6 Defining Individual Demand Function ? Individual demand Function: An individual demand Function shows the relationship between the price of a good and the quantity of that good purchased by an individual assuming that all other determinants of demand are held constant. ?vDü6Dy 7 Representation of General Individual Demand Function );,,(d x spreferenceIPPd YXx = ? The three elements that determine the quantity demanded are the prices of X and Y, the person’s income (I), and the person’s preferences for X and Y. ? Preferences appear to the right of the semicolon because we assume that preferences do not change during the analysis. ?vDü6Dy 8 How to Get Individual Demand Function by Algebra Approach(} ?ZE ) ? With 2 goods, the individual’s objective is to maximize utility from these 2 goods: U=U(x 1 ,x 2 ) ? Subject to the budget constraint: I=x 1 P 1 +x 2 P 2 ? For maximizing a utility function subject to a constraint, we set up the Lagrangian expression: ?vDü6Dy 9 How to Get Individual Demand Function by Algebra Approach ][),(),,( 22112121 IxpxpxxuxxL ?+?= λλ Take first order necessary conditions for maximum 0 ),( 1 1 21 =? ? ? p x xxu λ 0 ),( 2 2 21 =? ? ? p x xxu λ 0 2211 =?+ Ixpxp ?vDü6Dy 10 How to Get Individual Demand Function by Algebra Approach ? From 3 first-order conditions, we can get three equations, and then solve them to get individual demand functions for two goods. x 1 * =x 1 (p 1 ,p 2 ,I) x 2 * =x 2 (p 1 ,p 2 ,I) ? x i *= x i (p 1 ,p 2 I) ( i=1,2) is called ordinary demand function( ?Y3 pf ? ), is also called Marshallian Demand Function ?u :3 pf ? ?vDü6Dy 11 Ordinary Demands: Cobb-Douglas Example ? Given p 1 , p 2 and I, how to calculate optimal ordinary demands (x 1 *,x 2 *)? ? Assume Cobb-Douglas preferences: 1 21 2 2 2 1 1 1 1 2121 ),( ? ? == == = ba ba ba xbx x U MU xax x U MU xxxxU ? ? ? ? ?vDü6Dy 12 Ordinary Demands: Cobb-Douglas Example ? So the MRS is ? At (x 1 *,x 2 *), MRS = -p 1 /p 2 so . / / 1 2 1 21 2 1 1 2 1 1 2 bx ax xbx xax xU xU dx dx MRS ba ba ?=?=?== ? ? ?? ?? )1( * 1 2 1 * 2 2 1 * 1 * 2 x ap bp x p p bx ax =??=? ? Also, at (x 1 *,x 2 *), the budget is exhausted, so )2( * 22 * 11 Ixpxp =+ ?vDü6Dy 13 Ordinary Demands: Cobb-Douglas Example ? Substitute for x 2 * from (1) into (2) to get Ix ap bp pxp =+ * 1 2 1 2 * 11 ? Solve for x 1 * to get 1 * 1 )( pba aI x + = ? Substitute x 1 * into (2) and solve for x 2 * to get 2 * 2 )( pba bI x + = ?vDü6Dy 14 Ordinary Demands: Cobb-Douglas Example ba xxxxU 2121 ),( = 1 * 1 )( pba Ia x + = x 1 x 2 2 * 2 )( pba Ib x + = ?vDü6Dy 15 Ordinary Demands: Cobb-Douglas Example ? Note that for Cobb-Douglas utility function ? Demands are linear in income ? Expenditure shares are constant ? Expenditure shares sum to one . )( )( * 22 * 11 ba bI xp ba aI xp + = + = ?vDü6Dy 16 How to Get Demand Curve by Graphic Approaches m^E p¤3 p wL ? By the way of comparative static analysis, if we let the price of a good change holding other factors constant, we get ordinary demand curve ? Example: the price of a good increase ? Suppose p 1 increases, from p 1 ’to p 1 ’’, and to p 1 ’’’ ? Hold p 2 and I constant ?vDü6Dy 17 the price of a good Changes x 1 x 2 p 1 =p 1 ’ p 1 x 1 + p 2 x 2 = I ?vDü6Dy 18 the price of a good Increase p 1 = p 1 ’’ x 1 x 2 p 1 = p 1 ’ p 1 x 1 + p 2 x 2 = I ?vDü6Dy 19 the price of a good Increases x 2 x 1 p 1 = p 1 ’’ p 1 = p 1 ’’’ p 1 = p 1 ’ p 1 x 1 + p 2 x 2 = I ?vDü6Dy 20 the price of a good Changes x 1 x 1 *(p 1 ’) p 1 = p 1 ’ x 2 ?vDü6Dy 21 the price of a good Changes x 2 x 1 x 1 *(p 1 ’) p 1 x 1 *(p 1 ’) p 1 ’ x 1 * p 1 = p 1 ’ ?vDü6Dy 22 the price of a good Changes p 1 x 1 *(p 1 ’)x 1 *(p 1 ’’) x 1 *(p 1 ’) p 1 ’ p 1 = p 1 ’’ x 1 * 2 x 1 x ?vDü6Dy 23 the price of a good Changes x 2 x 1 x 1 *(p 1 ’)x 1 *(p 1 ’’) p 1 x 1 *(p 1 ’) x 1 *(p 1 ’’) p 1 ’ p 1 ’’ x 1 * ?vDü6Dy 24 the price of a good Changes x 2 x 1x 1 *(p 1 ’’’) x 1 *(p 1 ’) x 1 *(p 1 ’’) p 1 x 1 *(p 1 ’) x 1 *(p 1 ’’) p 1 ’ p 1 ’’ p 1 = p 1 ’’’ x 1 * 2 1 ?vDü6Dy 25 the price of a good Changes x 2 x 1 x 1 *(p 1 ’’’) x 1 *(p 1 ’) x 1 *(p 1 ’’) p 1 x 1 *(p 1 ’)x 1 *(p 1 ’’’) x 1 *(p 1 ’’) p 1 ’ p 1 ’’ p 1 ’’’ x 1 * x 1 x 2 ?vDü6Dy 26 the price of a good Changes x 2 x 1 x 1 *(p 1 ’’’) x 1 *(p 1 ’) x 1 *(p 1 ’’) p 1 x 1 *(p 1 ’)x 1 *(p 1 ’’’) x 1 *(p 1 ’’) p 1 ’ p 1 ’’ p 1 ’’’ x 1 * demand curve x 1 x 2 ?vDü6Dy 27 the price of a good Changes p 1 x 1 *(p 1 ’)x 1 *(p 1 ’’’) x 1 *(p 1 ’’) p 1 ’ p 1 ’’ p 1 ’’’ x 1 * demand curve x 1 x 2 x 1 *(p 1 ’’’) x 1 *(p 1 ’) x 1 *(p 1 ’’) ?vDü6Dy 28 the price of a good Changes and Price consumption curve(NìhnL ) p 1 x 1 *(p 1 ’)x 1 *(p 1 ’’’) x 1 *(p 1 ’’) p 1 ’ p 1 ’’ p 1 ’’’ x 1 * demand curve P.C.C x 1 x 2 x 1 *(p 1 ’’’) x 1 *(p 1 ’) x 1 *(p 1 ’’) ?vDü6Dy 29 Summary: the price of a good Changes ? Price consumption curve P.C.C : ? contains all utility-maximizing bundles traced out as p 1 changes ? p 2 and I constant ? Ordinary demand curve for commodity 1: ? Plot of x 1 -coordinate of p 1 - price offer curve for each value of p 1 ? Reflects optimal consumption of x 1 at each p 1 ?vDü6Dy 30 Example: Cobb-Douglas Utility ? Assume Cobb-Douglas Utility function ? Ordinary demand functions are ? x 2 * is constant (flat) – not f(p 1 ) ? x 1 * demand is rectangular hyperbola(°? ? wL ) (px=k) 2 21 * 2 1 21 * 1 ),,(,),,( p I ba b Ippx p I ba a Ippx + = + = ba xxxxU 2121 ),( = ?vDü6Dy 31 Example: Cobb-Douglas Utility x 1 *(p 1 ’’’) x 1 *(p 1 ’) x 1 *(p 1 ’’) x 2 x 1 2 * 2 1 * 1 )( )( pba bI x pba aI x + = + = x 1 x 2 P.C.C ?vDü6Dy 32 Example: Cobb-Douglas Utility p 1 x 1 * demand curve 1 * 1 )( pba aI x + = x 2 x x 1 x 2 x 1 *(p 1 ’’’) x 1 *(p 1 ’) x 1 *(p 1 ’’) P.C.C 1 ?vDü6Dy 33 Inverse Demand Functions(Q3 pf ? ) Inverse Demand Functions: At what price are x 1 ’ units demanded? p 1 x 1 * x 1 ’ ?vDü6Dy 34 Inverse Demand Functions Inverse Demand Functions: At what price are x 1 ’ units demanded? p 1 ’. x 1 ’ p 1 x 1 * p 1 ’ ?vDü6Dy 35 Inverse Demand Functions ? Cobb-Douglas ordinary demand function: ? Cobb-Douglas inverse demand function: x aI abp 1 1 * () = + p aI abx 1 1 = + () * ?vDü6Dy 36 Income Changes, Holding Prices Constant ? Change income I, holding prices constant ? Income consumption curve(I.C.C) l ?h nL : resulting consumer equilibria for both goods ? Engle curve: graph of quantity demanded and income ?vDü6Dy 37 Income Changes 1 I’ < I’’ < I’’’ x 2 x x 1 x 2 Income changes, holding prices constant ?vDü6Dy 38 Income Changes x 2 x 1 x 1 x 2 ?vDü6Dy 39 Income Changes x 2 x 1 x 2 ’’’ x 2 ’’ x 2 ’ x 1 x 2 x 1 ’’’ x 1 ’’ x 1 ’ ?vDü6Dy 40 Income Changes Income consumption curve x 1 x 2 x 1 ’’’ x 1 ’’ x 1 ’ x 2 ’’’ x 2 ’’ x 2 ’ ?vDü6Dy 41 Income Changes x 2 x 1 x 1 ’’’ x 1 ’’ x 1 ’ x 2 ’’’ x 2 ’’ x 2 ’ I.C.C x 1 * I x 1 ’’’ x 1 ’’ x 1 ’ I’ I’’ I’’’ x 1 x 2 Engel Curve ?vDü6Dy 42 Income Changes x x 2 x 1 x 1 ’’’ x 1 ’’ x 1 ’ x 2 ’’’ x 2 ’’ x 2 ’ I.C.C x 1 x 2 x 1 ’’’ x 1 ’’ x 1 ’ I’ I’’ I’’’ x 1 Engel Curve x 2 ’’’ x 2 ’’ x 2 ’ I’ I’’ I’’’ x 2 Engel Curve ?vDü6Dy 43 Example: Cobb-Douglas Utility ? Cobb-Douglas Engel curves: ? Ordinary demands: U(x 1 , x 2 ) = x 1 a x 2 b x bI abp 2 2 () . = + x aI abp 1 1 ** () ; = + ?vDü6Dy 44 Example: Cobb-Douglas Utility ? Rearranging to solve for I: ? Engel curve for good 1: ? Engel curve for good 2: I abp a x = + () * 1 1 + I abp b x = () * 2 2 ?vDü6Dy 45 Example: Cobb-Douglas Utility x 1 * I abp a x = + () * 1 1 x 2 * + I abp b x = () * 2 2 I I ?vDü6Dy 46 Linear Engel Curves and Homothetic(]ê? ) (x 1 ,x 2 ) < (y 1 ,y 2 ) (kx 1 ,kx 2 ) < (ky 1 ,ky 2 ) ? Engel curves are straight lines if consumer’s preferences are homothetic (all examples thus far) ? A consumer’s preferences are homothetic if and only if, for every k>0, ? That is, the consumer’s MRS is the same anywhere on a straight line drawn from the origin ? Often assumed in dynamic optimization problems ? ?vDü6Dy 47 Income Effects: Nonhomothetic Example ? Quasilinear preferences are not homothetic U(x 1 , x 2 ) = f(x 1 ) + x 2 ? For example, Ux x x x(, ) . 12 1 2 = + ?vDü6Dy 48 Quasi-linear Indifference Curves x 2 x 1 Each curve is vertically shifted copy of others Each curve intersects both axes. ?vDü6Dy 49 Income Changes: Quasi-linear Utility x 1 ~ x 1 x 2 ?vDü6Dy 50 Income Changes: Quasi-linear Utility x 1 * x 1 ~ Engel Curve I x 1 ~ x 1 x 2 ?vDü6Dy 51 Income Changes: Quasi-linear Utility x 2 * Engel Curve I x 1 ~ x 1 x 2 ?vDü6Dy 52 Income Changes: Quasi-linear Utility x 1 * x 1 ~ x 1 Engel Curve I x 1 ~ x 1 x 2 I x 2 * x 2 Engel Curve ?vDü6Dy 53 Normal and Inferior Goods ? Normal good(?è ? ): ? quantity demanded rises with income ? Engel curve is positively sloped ? Inferior good ?? ? : ? quantity demanded falls with income ? Engel curve is negatively sloped ?vDü6Dy 54 x 2 x 1 ’’’ x 1 ’’ x 1 ’ x 2 ’’’ x 2 ’’ x 2 ’ I.C.C x 1 x 2 Income Changes: x 1 , x 2 Normal Goods 1 x 1 ’’’ x 1 ’’ x 1 ’ I’ I’’ I’’’ x 1 Engel Curve x 2 ’’’ x 2 ’’ x 2 ’ I’ I’’ I’’’ x 2 Engel Curve ?vDü6Dy 55 Income Changes: x 2 Normal, x 1 Inferior x 2 x 1 ?vDü6Dy 56 Income Changes: x 2 Normal, x 1 Inferior x 2 x 1 ?vDü6Dy 57 Income Changes: x 2 Normal, x 1 Inferior x 2 x 1 ?vDü6Dy 58 Income Changes: x 2 Normal, x 1 Inferior x 2 x 1 ?vDü6Dy 59 Income Changes: x 2 Normal, x 1 Inferior x 2 x 1 ?vDü6Dy 60 Income Changes: x 2 Normal, x 1 Inferior x 2 x 1 I.C.C ?vDü6Dy 61 Income Changes: x 2 Normal, x 1 Inferior x 2 x 1 I Engel Curve x 1 * ?vDü6Dy 62 Income Changes: x 2 Normal, x 1 Inferior x 2 x 1 x 1 * x 2 * I x 1 Engel Curve x 2 Engel Curve I ?vDü6Dy 63 Ordinary Goods x 2 Ordinary Good: Q demanded down as own price increases x 1 ?vDü6Dy 64 Ordinary Goods x 2 P.C.C x 1 ?vDü6Dy 65 Ordinary Goods x 1 * Downward sloping demand curve Good 1 is ordinary ? p 1 x 2 P.C.C x 1 ?vDü6Dy 66 Ordinary Goods x 2 Giffen Good: Quantity demanded rises as price increases (for some prices) x 1 ?vDü6Dy 67 Ordinary Goods x 2 P.C.C x 1 ?vDü6Dy 68 Ordinary Goods x 1 * D-curve has positively sloped portion Good 1 is Giffen ? p 1 x 2 P.C.C x 1 ?vDü6Dy 69 The End ?vDü6Dy 70 Last Revised: September 19, 2005 Slide 1 Chapter 2 Consumer Behavior and Demand Theory Slide 2 Chapter 2 includes: ? 2.1 Preference and Utility ? 2.2 Utility Maximization and Choice ? 2.3 Income and Substitution Effects ? 2.4 Market Demand and Elasticity Slide 3 Overview of Last Class ? How to Get Demand Function (Algebra and Graph) ? Marshallian ?u: Demand Function ? Price-Consumption Curve NìhnL and Demand Curve ? Income-Consumption Curve l ?hnL and Engel Curve 6ì: wL ? Examples of relative static equilibrium Slide 4 Outline of Today’s Class ? Substitution and Income Effect ? Hicks Approach  X ?E ) ? Slutsky Approach ? ?GE ? SE and IE for Normal, Inferior and Giffen Good Slide 5 Review: Construction of Individual Demand Curves ? An individual demand curve ? |3 p w L is a graphic representation between the price of a good and the quantity of that good purchased by a person holding all other factors (preferences, the prices of other goods, and income) constant. ? Demand curves limit the study to the relationship between the quantity demanded and changes in the own price of the good. Slide 6 Review: Construction of Individual Demand Curves ? In Panel of Figure in next page an individual’s indifference curve map is drawn using three different budget constraints in which the price of X decreases. ? The decreasing prices are P’ X , P” X , and P’’’ X respectively. ? The individual’s utility maximizing choices of X are X’, X’, and X’’’ respectively. Slide 7 FIGURE: Construction of an Individual’s Demand Curve Quantity of Y per week U 1 U 2 U 3 X Quantity of X per week X’ X” X’” ’ (a) Individual's indifference curve map 0 Price P 9 X P 0 X P - dx Quantity of XX’ X” X’” (b) Demand curve 0 Budget constraint for P’ X Budget constraint for P’’ X Budget constraint for P’’’ X Slide 8 Construction of Individual Demand Curves ? These three choices show that the quantity demanded of X increases as the price of X falls. ? Panel b shows how the three price and quantity choices can be used to construct the demand curve. Slide 9 Construction of Individual Demand Curves ? The demand curve (d X ) is downward sloping showing that when the price of X falls, the quantity demanded of X increases.——Law of Demand(3 pE5 ) ? this result follows from the substitution(9 }r? ) and income effects l ?r? . Slide 10 y x 0 P X lowers, Demand rise from OF 1 to OF 2 A E2 B1 F1 B2F2 Slide 11 Income and Substitution Effects A change in the price of a good on the demand has two effects: Substitution & Income Effect Substitution Effect(9}r? ) the extra purchase of a good now that it is, after the price of this good fall, relatively cheaper than other substitutes in consumption. Income Effect( l ?r? ) the extra purchase of a good that is caused by an increase in real purchasing power L= ? ? of the consumer when the price of one good falls. Slide 12 Two Definitions of holding real purchasing power constant y x 0 A B1 B2 E1 F1 E2 F2 Slutsky Compensated Budget Line ? ?G?ê? ?L — —Holding original bundle of goods constant Hicks Compensated Budget Line( X ??ê? ?L )——Holding Utility (Satisfaction) Constantb Slide 13 Two Methods of Substitute and Income Effect Analysis ? According two definitions of real purchasing power, there are two approaches that can be used to analyze Substitute and Income Effect z Hicks approach( X ?sE ) z Slutsky approach ? ?GsE Slide 14 Hicks Approach: Income Effect and Substitution Effect Slide 15 Hicks Approach: Income and Substitution Effects: A Normal Good Case The income effect, EF 2 , ( from D to B) keeps relative prices constant but increases purchasing power. Food (units per month) O Clothing (units per month) R F 1 S C 1 A U 1 Income Effect C 2 F 2 T U 2 B When the price of food falls, consumption increases by F 1 F 2 as the consumer moves from A to B. E Total Effect Substitution Effect D The substitution effect,F 1 E, (from point A to D), changes the relative prices but keeps real income (satisfaction) constant. Slide 16 Hicks Approach: Income and Substitution Effects ? Substitution Effect z The substitution effect is the change in a good’s consumption associated with a change in the price of the good, with the level of utility held constant. z When the price of a good declines, the substitution effect always leads to an increase in the quantity of the good demanded. Slide 17 Hicks Approach: Income and Substitution Effects ? Income Effect z The income effect is the change in a good’s consumption brought about by the increase in purchasing power, with the price of the good held constant. z When a person’s income increases, the quantity demanded for the product may increase or decrease. Slide 18 When the price of a good increases, the quantity of the good demanded will declines. Slide 19 Hicks Approach: Income and Substitution Effects: Inferior Good Food (units per month) O R Clothing (units per month) F 1 S F 2 T A U 1 E Substitution Effect D Total Effect Since food is an inferior good, the income effect is negative. However, the substitution effect is larger than the income effect. B Income Effect U 2 Slide 20 Giffen’s Paradox(oc ? ) ? When the price of potatoes rose in Ireland the consumption of potatoes also increased. ? Potatoes were not only an inferior good but constituted the source of a large portion of Irish people’s income. ? The situation in which an increase in a good’s price leads people to consume more of the good is called Giffen’s paradox. Slide 21 Giffen’s Paradox ? If the income effect of a price change is strong enough with an inferior good, it is possible for the quantity demanded to change in the same direction as the price change. ? this phenomenon was observed by English economist Robert Giffen. Slide 22 Hicks Approach: Income and Substitution Effects_ Giffen Good ? The Giffen Good o ? z The income effect may theoretically be large enough to cause the demand curve for a good to slope upward. z This rarely occurs and is of little practical interest. Slide 23 C If the income effect swamps the substitution effect, the total effect is positive: A GIFFEN GOOD. Suppose F is a giffen good, and is also a inferior good, so the income effect is negative. At the same time, the income effect is larger than the substitution effect. F 0 BC 0 F 1 F 2 E BC 1 B ’ income effect: substitution effect A B D Hicks Approach: Income and Substitution Effects: Giffen’s Good Total Effect Slide 24 Slutsky Approach: Income Effect and Substitution Effect Slide 25 Effects of a Price Change x 2 x 1 Original choice Consumer’s budget is $I. 2 p I Slide 26 Effects of a Price Change x 1 Lowered price for commodity 1 pivots outwards the budget constraint. Consumer’s budget is $I. x 2 2 p I Slide 27 Effects of a Price Change x 1 Lowered price for commodity 1 pivots outwards the budget constraint. Consumer’s budget is $I. x 2 2 p I 2 p I Now only $I’ are needed to buy the original bundle at the new prices. It is as if the consumer’s income has increased by $I - $I’. Slide 28 Effects of a Price Change ? The change to quantities demanded due to this ‘extra’ income is the income effect of the price change. Slide 29 Effects of a Price Change ? Slutsky’s insight was that the effects on quantities demanded of any price change can always be decomposed into a substitution effect and an income effect. ? The overall change in quantities demanded due to a price change is the sum of the substitution and income effects. Slide 30 Substitution Effect ? Slutsky isolated the change in quantities demanded due only to the change in relative prices by asking “What is the change in quantities demanded when the consumer’s income is adjusted so that, at the new prices, she can only just buy the original bundle?” Slide 31 Substitution Effect Only x 2 x 1 x 1 ’ x 1 ’’ Lowering p 1 makes good 1 relatively cheaper and causes a substitution from good 2 to good 1. The change from point A to point C is the substitution effect. Slusky Substitution Effect A C Slide 32 And Now The Income Effect x 2 x 1 x 1 ’ x 1 ’’ The income effect is the change from point C to point B. Slutsky Income Effect Slutsky Substitution Effect A B C Slide 33 The Overall Change in Demand x 2 x 1 x 1 ’ x 1 ’’ B The overall effect on demands of the change in p 1 is the sum of the income and substitution effects. This is the change from point A to point B. A C Slide 34 Slutsky’s Effects for Normal Goods x 2 x 1 x 1 ’ x 1 ’’ B Good 1 is normal because an increase to income causes demand to rise. So the income and substitution effects Reinforce each other. A C Slide 35 Slutsky’s Effects for Normal Goods ? Since both the substitution and income effects increase demand when own-price decreases, the ordinary demand curve for a normal good must slope downwards. ? The Law of Downward-Sloping Demand therefore always applies to normal goods. Slide 36 Slutsky’s Effects for Income-Inferior Goods ? Some goods are income-inferior; that is, demand is reduced by an increase in income. ? Slutsky showed that, for income- inferior goods, the substitution and income effects oppose each other when a good’s own price changes. Slide 37 Slutsky’s Effects for Income-Inferior Goods x 2 x 1 x 1 ’ x 1 ’’ B The overall changes to demand are the sums of the substitution and income effects.A C Slide 38 Giffen Goods ? In the rare case of extreme income- inferiority, the income effect may be larger in size than the substitution effect, causing quantity demanded to decrease as own-price decreases. ? Such goods are called Giffen goods. Slide 39 Slutsky’s Effects for Giffen Goods x 2 x 1 x 1 ’ x 1 ’’x 1 ’’’ Substitution effect Income effect A decrease in p 1 causes a decrease in the quantity demanded of good 1. A C B Slide 40 Summary of income and substitution effects of a price change Types of Goods Change of the price S.E. I.E. |SE| and |IE| - + |SE|>|IE| |SE|<|IE| - + - + T.E. ++ +Normal good -- - +- + -+ - +- - -+ + Giffen good Inferior good Slide 41 The overall effect (9r? ) ? The overall effect is the sum of the substitution effect and the income effect. ? Slutsky equation: Slide 42 Application Slide 43 °¤?L?W¤?L ???1  ? ?ùhn ?t ?B??? | ?Z T z°¤óCá ?hn?°¤ ? ?ê zW¤óaò ? V[u|￥Z T ? ?ê  ? h1?? ??|￥a ?? Slide 44 Slide 45 The End Sept. 22 2005 ?vDü6Dy 1 Chapter 2 Consumer Behavior and Demand Theory(5) ?vDü6Dy 2 Chapter 2 includes: ? 2.1 Preference and Utility ? 2.2 Utility Maximization and Choice ? 2.3 Income and Substitution Effects ? 2.4 Market Demand and Elasticity ?vDü6Dy 3 Overview of Last Class ? Substitution and Income Effect 9}r?? l ?r? ? Hicks Approach  X ?E ) ? Slutsky Approach ? ?GE ? SE and IE for Normal, Inferior and Giffen Good ?vDü6Dy 4 Outline of Today’s Class ? From Individual demand to Market Demand (Algebra and Graph) ? Own-price elasticity of demand  3 pNì?? ? Relationships between Revenue (Expenditure) and Own-Price Elasticity of Demand ? Cross-price elasticity of demand  3 p￥?-? ? ? Income elasticity of demand  3 p￥ l ??? ? Consumer’s Surplus (CS) (Marshallian CS) ?vDü6Dy 5 From Individual to Market Demand: Introduction ? Suppose know individual consumer demands ? Need to aggregate to total demand ? Suppose economy contains n consumers, i = 1, … ,n ? and two goods, x and Y ? Consumer i’s ordinary demand function for good X is ),,( i yx i xi IppdX = ?vDü6Dy 6 From Individual to Market Demand Functions ? Suppose all consumers are price-takers ? Then market demand for good j is ? If all consumers are identical ]B￥ ]é￥ , then ),,(),,( IppdnMppX yxxyx ×= where M = nI = aggregate income ),,,,,( ),,( 21 11 n yxx i yx n i i x n i i IIIppD IppdXX L= == ∑∑ == ?vDü6Dy 7 Market Demand Functions g?3 pf ? ? In general, individual demands differ ? Market demand curve is “horizontal sum” ￡ ü F9 of individual consumers’ curves ? At each price, add up quantities demanded ? A graphical example ? Assume two consumers: i = A,B ?vDü6Dy 8 Market Demand: Adding D-Curve Graphs p 1 x A 1 * 20 p 1 ’ p 1 ” The “horizontal sum” of D-curves of A & B p 1 B x 1 15 p 1 ’ p 1 ” xx A B 11 * + p 1 35 p 1 ’ p 1 ” + = ?vDü6Dy 9 From Individual Demand to Market Demand: An algebra example ? Suppose there are two consumers in a market, their inverse individual demand functions are: ? PA=100-qA ? PB=180-2qB ? What is this market demand? ?vDü6Dy 10 Elasticities: Introduction ? Often want to measure responsiveness of one variable in response to changes in another ? Elasticity measures this “sensitivity”( ù?? ) ? Elasticity of x=f(p) is defined as pp xx e px / / , ? ? = ?vDü6Dy 11 Types of Demand Elasticity(3 p??￥ ?? ) ? Own-price elasticity of demand   ?￥1 &N ì3 p?? -- quantity demanded of good i and price of good i ? Cross-price elasticity of demand  3 p￥?-? ? -- demand for good i and price of good j ? Income elasticity of demand  3 p￥ l ?? ? -- demand for good i and income ?vDü6Dy 12 Defining of Own-Price Elasticity of Demand ? Own-Price elasticity of demand (e Q,p ) ? This elasticity shows how Q changes (in percentage terms) in response to a percentage change in P. ? Because of law of demand, dQ/dP is usually negative. ? We make e Q,P positive by the way of the absolute value of dQ/dP. Q P dP dQ or Q P P Q P P Q Q e PQ ??= ? ? ? ?= ? ? ?= , ?vDü6Dy 13 Measures of Elasticities: Arc and Point Elasticities ? Suppose demand function is X=X(p) ? Arc-elasticity(??? ) measures finite response ? “average” elasticity ? computed by a mid-point formula ?? T )( )( )( 2 1 )( 2 1 / / 21 21 12 12 21 21 12 12 , XX pp pp XX XX pp pp XX X p p X pp XX e arc pX + + ? ? ? ?= + + ? ? ? ?= ? ? ? ?= ? ? ?= ?vDü6Dy 14 Point Elasticity ? Point elasticity ??? measures response to infinitesimal í kl change in p at specific value of p X p dp dX pdp XdX e po pX ??=?= / / int , ?vDü6Dy 15 Own-Price Demand Point Elasticity p dp dX X p e pX ?= , Own-price demand elasticity at point (X, p) is p X X ?vDü6Dy 16 Example: Linear Demand Curve L?3 p wL ? Suppose inverse demand Q3 pf ? is linear p i = a - bX i X i = a/b - p i /b dX i /dp i = -1/b ? Own price elasticity of demand is e X p i i i i ii p ap b b p ap , ()/ = ? ? ? ? ? ? ? ? = ? 1 ?vDü6Dy 17 Own-Price Demand (Point) Elasticity p i p i = a - bX i i i pX pa p e ii ? = , a a/b X i ?vDü6Dy 18 Own-Price Demand (Point) Elasticity p i X i 00 =?= ep 0=e a a/b i i pX pa p e ii ? = , ?vDü6Dy 19 Own-Price Demand (Point) Elasticity p i X i a a/b 1 2/ 2/ 2 = ? =?= aa a e a p 1=e 0=e a/2 a/2b i i pX pa p e ii ? = ?vDü6Dy 20 Own-Price Demand (Point) Elasticity p i X i a a/b ∞= ? =?= aa a eap 1=e 0=e a/2 a/2b ∞=e i i pX pa p e ii ? = , ?vDü6Dy 21 The Geometrical Meanings of Price Elasticity of Demand(3 p wL￥+?il ) A BF OB AF AE OC CE CE AC OC AC SlopeX P dX dP X P X P dP dX e A PX === ?=?= ? ?=??= 11 1 , p F B EOC X ?vDü6Dy 22 Own-Price (Point) Elasticity: Summary p i X i a a/b 1=e 0=e a/2 a/2b ∞=e own-price elastic(?μ?? ) own-price inelastic ?C?? own-price unit elastic ?ê?? ?vDü6Dy 23 ? Suppose demand is ? Own price elasticity of demand is Xkp ii -a = 1?? ?= α i i i pak dp dX e Xp i i -a i -a ii p kp (-ka p , =? = a ?1 ? (a>0) ) Example: Constant Elasticity Demand  %???3 p ?vDü6Dy 24 Example: Constant Elasticity Demand (a=2) 2 2 i i a ii p k kpkpX === ?? 2=e everywhere along the demand curve p i X i ?vDü6Dy 25 Revenue( lm ) and Own-Price Elasticity of Demand ? Inelastic demand ? P increase raises revenue if Q response small ? Occurs if demand own-price inelastic ? Elastic demand ? P increase reduces revenue if Q response large ? Occurs if demand own-price elastic ?vDü6Dy 26 Revenue and Own-Price Elasticity of Demand ).()( pXppR =Sellers’ revenue is dp dX ppX dp dR += )( ? ? ? ? ? ? += dp dX pX p pX )( 1)( []()1 =? Xp e ?vDü6Dy 27 Revenue and Own-Price Elasticity of Demand []epX dp dR ?= 1)( 1=eIf dR dp = 0 then And a change to price doesn’t alter revenue ?vDü6Dy 28 Revenue and Own-Price Elasticity of Demand []epX dp dR ?= 1)( 10 ≤< e If demand inelastic dR dp > 0 then And a price increase raises sellers’ revenue ?vDü6Dy 29 Revenue and Own-Price Elasticity of Demand []epX dp dR ?= 1)( 1>eAnd if demand elastic dR dp < 0 then and a price increase reduces sellers’ revenue ?vDü6Dy 30 Revenue and Own-Price Elasticity: Summary ? Own-price inelastic demand: 0< e ≤ 1 ? Price rise causes rise in seller’s revenue ? Own-price unit elastic demand: e = 1 ? Price rise causes no change in seller’s revenue ? Own-price elastic demand: e >1 ? Price rise causes fall in seller’s revenue ?vDü6Dy 31 Judging ) x,p from P.C.C ? Relationships between revenue and e x,p is also called relationships between consumer’s expenditure and e x,p ? If 0< e ≤ 1,Price rise causes rise in consumer’s expenditure; ? if e = 1,Price rise causes no change in consumer’s expenditure; ? if e >1,Price rise causes fall in consumer’s expenditure ?vDü6Dy 32 Judging ) x,p from P.C.C ? Budget line and consumer’s expenditure. ?vDü6Dy 33 ) x,p and P.C.C e x,p <1 e x,p >1 ?vDü6Dy 34 e x,p =1 e x,p <0 ?vDü6Dy 35 Marginal Revenue(H= lm ) and Own-Price Elasticity ? Marginal revenue = change in revenue as number of units sold changes ? p(q) is seller’s inverse demand function (price at which can sell q units) ? So, R(q)=p(q) q dR q dq () MR() = q ?vDü6Dy 36 MR and Own-Price Elasticity of Demand R(q)=p(q) q MR q dR q dq dp q dq qpq() () () ()== + ? ? ? q pq dp q dq() () =+ ? ? pq()1 ? MR q p q() () =? ? ? ? ? ? ? 1 1 e ?vDü6Dy 37 MR and Own-Price Elasticity of Demand ? ? ? ? ? ? ?= e qpqMR 1 1)()( ? If e = 1 then MR(q) = 0 ? If 0< e ≤ 1 (inelastic demand) then MR(q) < 0 ? If e >1 (elastic demand) then MR(q) > 0 ?vDü6Dy 38 Example: Linear Demand Curve p = a - bq R(q)=p(q)q=(a-bq)q=aq-bq 2 MR(q)=dR(q)/dq=a-2bq ?vDü6Dy 39 MR and Own-Price Elasticity $ pq a bq()= ? MR q a bq()= ? 2 a a/b p q a/2b q a/ba/2b R(q) ?vDü6Dy 40 Some Real-World Price Elasticities of Demand Good or Service Elasticity Elastic Demand Metals 1.52 Electrical engineering products 1.30 Mechanical engineering products 1.30 Furniture 1.26 Motor vehicles 1.14 Instrument engineering products 1.10 Professional services 1.09 Transportation services 1.03 Inelastic Demand Gas, electricity, and water 0.92 Oil 0.91 Chemicals 0.89 Beverages (all types) 0.78 Clothing 0.64 Tobacco 0.61 Banking and insurance services 0.56 Housing services 0.55 Agricultural and fish products 0.42 Books, magazines, and newspapers 0.34 Food 0.12 ?vDü6Dy 41 Factors That Influence the Elasticity of Demand Closeness of Substitutes. The closer the substitutes, the more elastic the demand. ?vDü6Dy 42 Factors That Influence Elasticity the Elasticity of Demand Proportion of Income Spent on the Good The greater the proportion of income spent on a good, the more elastic the demand. Time Elapsed Hì Since Price Change The longer the time, the more elastic the demand. ? Short-run demand ? Long-run demand ?vDü6Dy 43 Price Elasticities in 20 Countries ?vDü6Dy 44 More Elasticities of Demand Cross elasticity of demand  3 p￥?-?? Measures the responsiveness of the demand for a good to a change in the price of a substitute or complement good. Cross elasticity of demand = Percentage change in quantity demanded Percentage change in price of a substitute or complement i j j i c X P P X e ? ? ? = ?vDü6Dy 45 Cross Elasticity of Demand Price of pizzas Price of soda, a complement, falls. Negative cross elasticity. D 2 Price of a burger, a substitute, falls. Positive cross elasticity. D 0 D 1 e c <0 e c >0 0 Quantity of pizzas ?vDü6Dy 46 Income Elasticity of Demand Income elasticity  3 p￥ l ??? Measures the responsiveness of the demand to a change in income. Income elasticity of demand = Percentage change in quantity demanded Percentage change in income i i I X I I X e ? ? ? = ?vDü6Dy 47 Income Elasticity of Demand Income elasticity can be: 1 ) Greater than 1 (normal good, income elastic) Luxury Good(  ? ) 2 ) Between zero and 1 (normal good, income inelastic) necessity 3 ) Less than zero (inferior good) ?vDü6Dy 48 Income Elasticity of Demand Normal Good - demand increases by a greater percentage than the change in income, ex. luxury goods Inferior Good - demand increases by a smaller percentage than the change in income, ex. food and necessities ?vDü6Dy 49 I ?vDü6Dy 50 Income Elasticity of Demand Elasticity greater than 1 Elasticity between zero and 1 Elasticity less than 1 and becomes negative Income Income Income Quantity demanded Quantity demanded Quantity demanded m Income inelastic Income elastic Positive income elasticity Negative income elasticity ?vDü6Dy 51 Some Real-World Income Elasticities of Demand Elastic Demand - Normal Goods Airline Travel 5.82 Movies 3.41 Foreign Travel 3.08 Electricity 1.94 Restaurant meals 1.61 Local buses and trains 1.38 Haircutting 1.36 Cars 1.07 Inelastic Demand - Inferior Goods Tobacco 0.86 Alcoholic beverages 0.62 Furniture 0.53 Clothing 0.51 Newspapers and magazines 0.38 Telephone 0.32 Food 0.14 ?vDü6Dy 52 Income Elasticities for Food in 15 Countries ?vDü6Dy 53 Relationships among Elasticities ? Sum of Income elasticities for all goods ? That is : I YP s I XP s eses Y Y X X IYYIXX == =+ , 1 ,, ?vDü6Dy 54 Slutsky Equation in Elasticities ? Slutsky Equation: I X X P X P X UU XX ? ? ? ? ? = ? ? = 0 X I I X I XP X P P X X P P X X UU X X X X ? ? ? ? ? +? ? ? ?=? ? ? ? = 0 IXX S PXPX esee XX ,,, += Slutsky Equation in Elasticities ?vDü6Dy 55 Homogeneity(]Q? ) ? Demand Function X=X(P X ,P Y ,I) is homogeneous of degree zero in all prices and income. ? So, according to Euler’s theorem x ? ? ? for homogeneous function, we can get: 0=? ? ? +? ? ? +? ? ? I I X P P X P P X Y Y X X 0=? ? ? +? ? ? +? ? ? X I I X X P P X X P P X Y X X X 0 ,,, =++? IXPXPX eee YX ?vDü6Dy 56 Consumer Surplus: introduction ? Consumers gain (or welfare) from trading at single price or policy change. ? Need a measure of these gains from trade ? Want total dollar value of entering market ? Calculated for market quantity purchased ? Reflects maximum willingness to pay to enter market ? Ideally, want utility-based measure ? But typically use observed market quantities ?vDü6Dy 57 Dollar Equivalent Utility Gains ? Measure marginal utility gain as willingness to pay for marginal unit ? Suppose gasoline bought only in gallons ? Reservation price ( =Nì )(r 1 ) of 1 st gallon = most consumer would pay for 1st gallon = dollar equivalent of MU 1 st gallon ? Same for r 2 thru r n ?vDü6Dy 58 Dollar Equivalent Utility Gains ? r 1 + … + r n = dollar equivalent of total change in utility from n gallons at zero price ? r 1 + … + r n -p G n = dollar equivalent of total change in utility from n gallons of gasoline at unit price of $p G ? Reservation-price curve ? plot of r 1 , r 2 , … , r n , … against n ?vDü6Dy 59 Dollar Equivalent Utility Gains Reservation Price Curve for Gasoline ($) Res. Values 0 2 4 6 8 10 Gasoline (gallons) 1 23456 r 1 r 2 r 3 r 4 r 5 r 6 ?vDü6Dy 60 Dollar Equivalent Utility Gains ? Dollar equivalent net utility gain from trading in gas market at $p G ? For 1 st gal = $(r 1 -p G ) ? For 2 nd gal = $(r 2 -p G ), etc. ? Total dollar value of gain from trade = $(r 1 -p G ) + $(r 2 -p G ) + … for all n such that r n -p G > 0 ? Consumer “surplus” - excess of marginal valuation over price paid ?vDü6Dy 61 Dollar Equivalent Utility Gains 0 2 4 6 8 10 ($) Res. Values 1 23456 p G r 1 r 2 r 3 r 4 r 5 r 6 Gasoline (gallons) ?vDü6Dy 62 Dollar Equivalent Utility Gains dollar value of net utility gains from trade 0 2 4 6 8 10 ($) Res. Values 1 23456 p G r 1 r 2 r 3 r 4 r 5 r 6 ?vDü6Dy 63 Reservation Price Curve for Half Gallons Reservation Price Curve for Gasoline 0 2 4 6 8 10 Gasoline (half gallons) ($) Res. Values 123456 r 1 r 3 r 5 r 7 r 9 r 11 7891011 ?vDü6Dy 64 Reservation Price Curve for Half Gallons dollar value of net utility gains from trade Gasoline (half gallons) ($) Res. Values 1234567891011 p G 0 2 4 6 8 10 r 1 r 3 r 5 r 7 r 9 r 11 ?vDü6Dy 65 Infinitely-Divisible Quantities of Gas Reservation Price Curve for Gasoline Gasoline ($) Res. Prices ?vDü6Dy 66 Infinitely Divisible Quantities of Gas $ value of net utility gains from trade Gasoline ($) Res. Prices p G ?vDü6Dy 67 Consumer’s Surplus: Quasi-Linear Utility ? Assume no income effects on x 1 demand ? That is, utility is quasi-linear in x 2 ? Assume p 2 = 1 ? Then CS is exact measure of dollar equivalent of utility gain from trade ?vDü6Dy 68 Consumer’s Surplus: Quasi-Linear Utility( EL?r¨ ) ? Consumer’s optimization problem is max. U(x 1 ,x 2 ) = v(x 1 ) + x 2 subject to p 1 x 1 + x 2 = I ? Demand for x 1 (MRS=v’(x 1 )/1=p 1 /1) can be solved from v’(x 1 )=p 1 ?vDü6Dy 69 Consumer’s Surplus: Quasi-Linear Utility Ordinary Inverse D-curve p 1 pvx 11 = '( ) x 1 * x 1 ' CS p 1 ' ?vDü6Dy 70 Consumer’s Surplus: Quasi-Linear Utility p 1 x 1 * x 1 ' CS CS = ∫ vx dx px ? '( ) '' 1111 0 x ' 1 1 p ' ?vDü6Dy 71 Consumer’s Surplus: Quasi-Linear Utility p 1 x 1 * CS =?? vx v px() () ''' 111 0 CS = ∫ vx dx px ? '( ) '' 1111 0 x ' 1 x 1 ' p ' Exact measure of dollar equiv. of utility gain from consuming x 1 ?vDü6Dy 72 Changes in Consumer’s Surplus p 1 CS before * x 1 x 1 ' p ' ?vDü6Dy 73 Changes in Consumer’s Surplus p 1 CS after " p 1 " x 1 p 1 (x 1 ) increases ' p 1 * x 1 ' x 1 ?vDü6Dy 74 Changes in Consumer’s Surplus p 1 Lost CS " p 1 " x 1 ' p 1 * x 1 ' x 1 ?vDü6Dy 75 The End ?vDü6Dy 76 Last Revised: Sept. 25, 2005