Intermediate Microeconomics Theory PDF

1-1 CHAPTER 1 LECTURE – ANALYZING ECONOMIC PROBLEMS ECONOMICS can be defined as the study of how society allocates its scarce (limited) resources to satisfy unlimited wants. We look at choices and try to estimate the expected benefits and costs, and we make decisions. We also try t...

1-1 CHAPTER 1 LECTURE – ANALYZING ECONOMIC PROBLEMS ECONOMICS can be defined as the study of how society allocates its scarce (limited) resources to satisfy unlimited wants. We look at choices and try to estimate the expected benefits and costs, and we make decisions. We also try to understand the consequences of some actions, those consequences that are not so apparent in the first place. Think of some examples. MICROECONOMICS studies the economic behavior of individual economic decision makers, such as a consumer, a worker, a firm, or a manager. It also analyzes the behavior of individual households, industries, markets, labor unions, or trade associations. HOW TO STUDY MICROECONOMICS Any model, whether it is used to study chemistry, physics, or economics, must specify what variables will be taken as given in the analysis and what variables are to be determined by the model. This brings us to the important distinction between exogenous and endogenous variables. An exogenous variable is one whose value is taken as given in a model. In other words, the value of an exogenous variable is determined by some process outside the model being examined. An endogenous variable is a variable whose value is determined within the model being studied. No matter what the specific issue is—coffee prices in Qatar, or decision making by firms on the Internet— microeconomics uses the same three analytical tools: Constrained optimization Equilibrium analysis Comparative statics  1-2 CONSTRAINED OPTIMIZATION Economics is the science of constrained choice. The tool of constrained optimization is used when a decision maker seeks to make the best (optimal) choice, taking into account any possible limitations or restrictions on the choices. We can therefore think about constrained optimization problems as having two parts, an objective function and a set of constraints. An objective function is the relationship that the decision maker seeks to “optimize,” that is, either maximize or minimize. For example, a consumer may want to purchase goods to maximize her satisfaction. In this case, the objective function would be the relationship that describes how satisfied she will be when she purchases any particular set of goods. Similarly, a producer may want to plan production activities to minimize the costs of manufacturing its product. The constraints in a constrained optimization problem represent restrictions or limits that are imposed on the decision maker. Examples of Constrained Optimization (a) What is the objective function for this problem? (b) What is the constraint? (c) Which of the variables in this model (I, PA, PB. A, and B) are exogenous? Which are endogenous? Explain. Explain the Solution Marginal Reasoning and Constrained Optimization The solution to any constrained optimization problem depends on the marginal impact of the decision variables on the value of the objective function. The term marginal in microeconomics tells us how a dependent variable changes as a result of adding one unit of an independent variable. Marginal cost measures the incremental impact of the last unit of the independent variable (output) on the dependent variable (total cost).  1-3 EQUILIBRIUM ANALYSIS A second important tool in microeconomics is the analysis of equilibrium, a concept found in many branches of science. An equilibrium in a system is a state or condition that will continue indefinitely as long as exogenous factors remain unchanged—that is, as long as no outside factor upsets the equilibrium. You should be able to explain how the market moves to equilibrium if it is out of equilibrium. COMPARATIVE STATICS Our third key analytical tool, comparative statics analysis, is used to examine how a change in an exogenous variable will affect the level of an endogenous variable in an economic model. Comparative statics analysis can be applied to constrained optimization problems or to equilibrium analyses. Comparative statics allows us to do a “before and- after” analysis by comparing two snapshots of an economic model.  1-4 Looking again at the Cost-Benefit Approach - If the benefit of an activity exceeds its cost, do it. We look at the concept of Reservation Price. The reservation price of activity x: the price at which a person would be indifferent between doing x and not doing x. Another way to look at this is the price for an asset above which a buyer is not willing to pay and/or below which a seller is not will to take. This tension between the buyer wanting a low price and the seller wanting a high price helps create the market price for the asset. We will emphasize a model called the "Perfectly Competitive Market Model." This model achieves "allocative efficiency," that is, it allocates scarce resources in such a way that social welfare is maximized. We have to admit, however, that "social welfare" is very narrowly defined, and that the perfectly competitive market model does not assure equity in the distribution of goods and services. POSITIVE AND NORMATIVE ECONOMICS ONCE AGAIN Positive Economics- Deals with objective or scientific explanations of the working of the economy. Emphasis here is on EXPLANATION with OBJECTIVITY. Example: 'If a tax is imposed on a good, its price will tend to rise.' Normative Economics - Offers prescriptions or recommendations based on personal value judgements. The emphasis here is more SUBJECTIVE, or what we think OUGHT to be. Example: 'A tax SHOULD be imposed on tobacco to discourage smoking.' We are going to develop some models of the "market" system. The market system operates fundamentally via prices to solve the questions what, how, and for whom in a context of scarcity. CETERIS PARIBUS (assumption) - Translated from the Latin as all other things being equal or holding everything else constant. Ceteris paribus example - “other things being equal”. An analysis is conducted whereby one variable is changed while all other parameters are assumed stable. For example, given the following functional relationship: z = F ( x1 , x2 , x3 ,..., xn ) Then the change in the dependent variable z given a change in the independent variable x1 is: z F = = F1 So if: x1 x1 a. F1 < 0, a  x1 has a negative impact on z. b. F1 > 0, a  x1 has a positive impact on z. z is the partial derivative of variable z with respect to x1; it relates the change in z given x1 a change in x1, all other variables, i.e., x2, x3......, xn , remaining stable, i.e., ceteris paribus.  2-1 CHAPTER 2 LECTURE – DEMAND AND SUPPLY ANALYSIS In a market-oriented economy, the majority of price and output decisions are determined in the market through the forces of Demand and Supply. A. DEMAND QUANTITY DEMANDED is the amount of a good that consumers are willing to purchase at each price per unit of time. The number of bottles of soda, for example, that an individual will buy per month is Qd, or the quantity demanded of bottles of soda. The amount of a good that consumers are willing to buy will depend on factors such as price of the good itself, the income level of the consumer, tastes or preferences for the good, price of substitute good (ex. coffee versus tea) or price of complements (cameras and film), number of buyers, etc. The demand function is generally expressed as: Qd = Qd (Price, Income, Tastes or Preferences, Price of Substitutes and Complements, Number of Consumers) Demand Curve for Bottles of Soda Price d Quantity/Time The demand curve above shows the different quantities of bottles of soda that will be purchased at various prices per time period, holding the other factors that effect demand constant (ceteris paribus). Notice that as the price of the good falls, more of the good will be purchased (ceteris paribus). SUMMARIZING: LAW OF DEMAND: As the price of the good rises (P), the quantity demanded of the good falls (Qd ) and as the price of the good falls (P), the quantity demanded of the good rises(Q d ), holding all other factors constant or ceteris paribus. An easy way of writing the law of demand is: As P   Qd  and as P  Qd, ceteris paribus  2-2 We have specifically noted that various factors other than the price of the good itself will also determine the amount of a good demanded. For a given demand curve the movement along a demand results from a change in the price of the good and assumes other factor remain constant. However, if one of the other factors change, more or less of the quantity of a good will be demanded at a given price. Thus, a change in one of these factors will cause a shift of the demand curse either left or right, rather than a movement along the demand curve. CHANGE IN QUANTITY DEMANDED - The movement along the demand curve resulting from a change in the price of the good. This holds all other factors constant. CHANGE IN DEMAND - A shift in the demand curve resulting from a change in a factor other than the price of the good itself. Major factors that shift the demand curve or cause a Change in Demand include: 1). Change in Income a). Normal Good - I   D b). Inferior Good - I   D  2). Changes in Tastes or preferences T   D  and T   D  3). Substitutes - Price of substitute good rises - Demand for good rises. Price of substitute good falls - Demand for good falls 4). Complements - Price of complementary good rises - Demand for good falls. Price of complementary falls - Demand for good rises. Factors that Shift the Demand Curve  2-3 B. SUPPLY QUANTITY SUPPLIED is the amount of a good that firms want to produce at each price per unit of time. The number of bottles of soda, for example, that the beverage manufacturer will produce per month is Qs, or the quantity supplied of bottles of soda. The amount of a good that firms will produce will depend on such factors as the price of the good itself, the price of inputs used to produce the good, the level of technology, and the number of firms. The supply function is generally expressed as: Q s = Qs (Price, Price of Inputs, Technology, Number of firms) Supply Curve for Bottles of Soda Price S Quantity/time The supply curve above shows the different quantities of bottles of soda that will be produced or made available for sale at various prices per time period, holding the other factors that effect supply constant (ceteris paribus). Notice that as the price of the good rises, more of the good will be produced (ceteris paribus). SUMMARIZING: LAW OF SUPPLY: As the price of the good rises (P), the quantity supplied of the good rises (Qs ) and as the price of the good falls (P ), the quantity supplied of the good falls (Qs ), holding all other factors constant or ceteris paribus. An easy way of writing the law of supply is: As P   Qs  and as P   Qs, ceteris paribus Just as there was a difference between a change in quantity demanded and a change in demand, we can distinguish between a change in quantity supplied and a change in supply. CHANGE IN QUANTITY SUPPLIED - The movement along the supply curve resulting from a change in the price of the good. This holds all other factors constant.  2-4 CHANGE IN SUPPLY - A shift in the supply curve resulting from a change in a factor other than the price of the good itself. Major factors that shift the supply curve or cause a Change in Supply include: 1). Price of Inputs Price of Inputs   S  Price of Inputs   S 2). Technology Tech   S Tech   S  3). Number of Firms Number of Firms   S Number of Firms   S  Factors that Shift the Supply Curve C. SYNTHESIS OF DEMAND AND SUPPLY The two forces of supply and demand determine equilibrium price and quantity. At Pe Qd = Qs If P < Pe Qd > Qs Shortage Pe If P > Pe Qs > Qd Surplus There is no guarantee that market prices will always be at equilibrium level. Actual or markets prices may differ. Once the market is in equilibrium, there is no tendency for prices to change, unless a factor affecting the demand or supply curve changes. Should such a change occur, the market moves to a new equilibrium price and quantity.  2-5 Looking at Changes in Demand and Supply Figure 1 Price S Pe‘ Pe SUMMARIZING: D D’ Increase in demand: D   Pe  and Qe  Decrease in demand: D   Pe  and Qe  Increase in supply: S   Pe  and Qe  Decrease in supply: S   Pe  and Qe  Q e Qe ‘ Q/t Figure 2 Price S S’ Pe Pe‘ D Qe Qe‘ Q/t You can give examples of changes in both Supply and Demand at the same time. Rationing function of price: the process whereby price directs existing supplies of a product to the users who value it most highly. Allocative function of price: the process whereby price acts as a signal that guides resources away from the production of goods whose prices lie below cost toward the production of goods whose prices exceed cost.  2-6 Supply and Demand - A Mathematical Approach (We can use Q or q) Supply-Demand Equilibrium Example: Q d= 1000 – 100P QS = -125 + 125P Equilibrium  Qd = QS 1000 – 100P = -125 + 125P or 225P = 1125 P* = 5 Q* = 500 Shifts in Supply-Demand Equilibrium Example What happens to the equilibrium price if either demand or supply shift? A shift in demand will lead to a new equilibrium: Original Supply and Demand Qd = 1000 – 100P QS = -125 + 125P New Demand Q’d = 1450 - 100P Q’d = 1450 - 100P = QS = -125 + 125P or 225P = 1575 P* = 7 Q* = 750 Graphing Supply Curve Sometimes we use the inverse demand and supply functions. That is, Price= f(Q). Suppose we have the demand curve Qd = 20 - P The inverse demand function is P= 20 - Qd Suppose we have the supply curve Qs = -8 + P The inverse supply function is P= 8 + Qs If we solve original equations, in equilibrium we know that Q d = Qs, we arrive at: 20 - P = -8 + P or 2P = 28 or P* = 14 Substituting this back into either the supply or demand equation gives the equilibrium quantity of Q* = 6 Solving using the inverse functions: 20 - Qd = 8 + Qs Since Qd = Qs 2Q = 12 or Q* = 6 and solving for P, P* = 14.  2-7 Graphing the Supply Curve A more general model of Demand and Supply is: Demand: qD = a – bp Supply: qS = c + dp Equilibrium  qD = qS a - bp = c + dp a−c p* = You can solve for q. d +b What happens to the equilibrium price if either demand or supply shift? An increase in demand (an increase in a) increases equilibrium price An increase in supply (an increase in c) reduces price dp* 1 dp* −1 = 0 = 0 da d + b dc d + b We Can Look at Stable and Unstable Equilibrium  2-8 More on Demand Functions A demand curve for X can be written as follows, Qd = f(Px ,Pother goods, I, T) where I is for income and T is other facts such as tastes. This simply states that the quantity demanded is a function of the variables in the brackets. So for a typical demand curve, all these other variables are constant -- only Px changes along the curve. The following equation could be a demand curve: Qd = 800 − 6 Px − 5Py + 10 I How do we interpret this? What does the “negative sign” tell us? Well, it tells us the slope and that this is indeed a demand curve because the law of demand says there is an inverse relationship between quantity demanded and price. What does the negative sign in front of Py tell us? It means that if the price of a related good goes up then the demand for X goes down. Therefore, this good must be a complement (like bread and butter). If it was a positive then the goods would be substitutes (like different kinds of bread). What does the positive sign in front of income tell us? It means that if income goes up, then demand for X goes up -- therefore it is a normal good. If it were negative, then it would be an inferior good. We can also show supply functions in as similar way. Give some examples. ELASTICITY OF DEMAND AND SUPPLY ELASTICITY can be defined as a measure of responsiveness. We want to examine how a change in one variable affects a change in another. Price elasticity of demand is defined as the percentage change in quantity demanded with respect to a percentage change in the price of the good. The symbol often used to denote price elasticity of demand is Ed even though the text uses € (epsilon). Ed is easier to write so we can us Ed. Other texts may different symbols – Economists are not consistent). We will use the following formula for price elasticity of demand: We keep in mind that Q= Qd % Q Q P Ed = = x or if we use calculus we can define elasticity as % P P Q dQ P Q P Ed = x =. We will look at using this formula a little later. dP Q P Q  2-9 Some books put a negative sign in front of the equation or take the absolute value. Thus, the value of elasticity always becomes positive. Some books do not consider the absolute value and treat elasticity as negative. BE AWARE OF HOW ELASTICITY IS DEFINED. Notice that the value of the price elasticity of demand includes the reciprocal of the slope of the demand function, P/Qd. The value of the slope of the demand function is a factor affecting the value of the elasticity. However, the values are not the same. Remember, in principles we generally looked at the arc elasticity, taking the average between to points. We defined arc elasticity as: Q P P1 + P2 Q1 + Q2 * Ed = x where P* = and Q* = P Q* 2 2 Using a little bit of algebra, the formula above reduces to: Q P1 + P2 Ed = x P Q1 + Q2 If we have a straight-line demand curve, we can use determine elasticity at a particular point. Let Q = 10 – 2P What is the value of the elasticity at P= 2? dQ dQ Solving for we obtain = -2, and at P= 2, Q = 10 – 2 (2) = 6. dP dP dQ P 2 Thus, x = ( −2) x = −0.67 dP Q 6 Special Case of a constant price elasticity demand curve a Let Q = aP − b or Q = a and b are constants Pb dQ a P = −baP − b−1 If P = P, Q = Thus, Ed = ( − baP − b−1 ) = −b dP Pb aP − b  2 - 10 If the demand function is in logarithmic form, then the coefficient of the variable is the value of the elasticity. Let Q = 2P-3 Taking the natural logs of both sides yields ln Q = ln2 – 3ln P If y = lnx, we know that the derivative of a logarithm is dy d ln x 1 dx dQ dP = = or d ln x = so d ln Q = and d ln P = dx dx x x Q P dQ Q d ln Q Elasticity is defined as Ed = = dP d ln P P d ln Q d ln 2 d ln P d −3 From before: ln Q = ln2 – 3ln P Ed = =( −3 + ln P ) = (−3) = −3 d ln P d ln P d ln P d ln P Knowing how to calculate the value of the price elasticity of demand is important. What is perhaps more important, however, is understanding what the value of elasticity means and its relationship to the concept of TOTAL REVENUE. TOTAL REVENUE is defined as price times quantity. We noted before that the value of elasticity indicates the percentage change in quantity with respect to the percentage change in price. Since elasticity is a fraction, the value calculated can be interpreted as the percentage change in quantity with respect to a one percent change in price. If, for example, Ed = -1.14, a one percent increase in price would cause quantity to fall by 1.14 percent. This can also be interpreted as saying a ten percent increase in price would cause quantity to fall by 11.4 percent. Elasticity and Total Revenue (TR) When discussing the price elasticity of demand it is useful to distinguish different ranges of elasticity values. If Ed  1 , demand is referred to a being elastic. If Ed  1 , demand is referred to a being inelastic. If Ed = 1 , demand is referred to a being unit or unitary elastic. Whether demand is elastic, inelastic or unit elastic will determine how a change in price (and the resultant change in quantity) will affect total revenue (PQ).  2 - 11 a) If Ed  1 , or demand is elastic, a decrease in price will increase the value of total revenue. An increase in price will decrease the value of total revenue. b) If Ed  1 , or demand is inelastic, a decrease in price will decrease the value of total revenue. An increase in price will increase the value of total revenue. c) If Ed = 1 , demand is unit elastic, and any change in price will have no effect on total revenue. Thus, total revenue is constant. At the price and quantity in which demand is unit elastic, total revenue is also maximized. Elasticity and Marginal Revenue (MR) From above, it was shown that the value of elasticity determines the effect of a price (quantity) change on total revenue. A change in total revenue with respect to a change in quantity is called Marginal Revenue (MR). TR dTR TR dTR Marginal Revenue = = (We will let Q = Qd so = MR = = Qd dQd Q dQ Deriving the relationship between MR and TR. We know TR = PQ TR Q P P Look at changes: TR = PQ + QP MR = =P +Q = P+Q Q Q Q Q Q P P PQP Look at the term. Multiplying by yields Q P PQ Q P 1 1 Remembering that Ed = MR = P + P ( ) MR = P(1 + ) P Q Ed Ed This can also be solved be taking the derivative of TR with respect to Q. dTR dQ dP dP Q dP 1 1 =P +Q = MR = P + Q = P+ P MR = P(1 + ) or MR = P (1 − ) dQ dQ dQ dQ P dQ Ed Ed NOW REMEMBER THAT Ed will be negative. Sometimes you will see the use of the absolute values of Ed or Ed Substituting into the MR function yields: 1 1 If Ed  1 , { (1 + ) } > 0 or (1 − ) >0., and MR > 0. As P rises (Q falls), TR falls. Ed Ed 1 1 If Ed  1, { (1 + ) } < 0, or (1 − ) 1, then dTR/dPx < 0. In plain English, this says that when demand is price elastic, the relationship between price and total revenue is negative. That is, an increase in price will decrease total revenue and a decrease in price will have the opposite effect on total revenue. b) If Ed < 1, then dTR/dPx > 0. Again, in plain English, this says that when demand is price inelastic, the relationship between price and total revenue is positive. c) If Ed = 1.0, then dTR/dPx = 0. Thus, a change in price will have no effect on total revenue. Linear demand curve If we have a linear demand curve of the form Q = a -bP, then when we can solve for Q and P, we find Elasticity and Total Revenue (TR) Diagram  2 - 13 Special Cases of Elasticity (You can show the shape of the demand function) If Ed = - , demand is referred to a being perfectly elastic. If Ed = 0, demand is referred to a being perfectly inelastic. NOTE: Although we know that along a (straight line) demand curve the value of elasticity will change, a steep demand curve is often referred to as being relatively inelastic. A flat demand curve is referred to as being relatively elastic. Factors that affect the price elasticity of demand. Substitution possibilities: the substitution effect of a price change tends to be small for goods with no close substitutes. Budget share: the larger the share of total expenditures accounted for by the prod- uct, the more important will be the income effect of a price change. Direction of income effect: a normal good will have a higher price elasticity than an inferior good. Time: demand for a good will be more responsive to price in the long-run than in the short-run. Price Elasticity Is Greater in the Long Run than in the Short Run  2 - 14 FITTING LINEAR DEMAND CURVES USING QUANTITY, PRICE, AND ELASTICITY INFORMATION Suppose demand is linear: 𝑄𝑑 = 𝑎 − 𝑏𝑃 Hence, elasticity is 𝜀𝑄,𝑃 = −𝑏𝑃⁄𝑄 If we have data on ε, Q, and P, we can calculate b from elasticity equation and then calculate a by substituting into demand. Example: Per capita consumption is 70 pounds – price 70¢ per pound 𝜀𝑄,𝑃 = −0.55 𝜀 = − 𝑏𝑃⁄𝑄  𝑏 = −𝜀 (𝑄⁄𝑃) = − (−.55(70⁄0.7)) = 55 𝑎 = 𝑄𝑑 + 𝑏𝑃 = 70+. (55 ∗ 70) = 108.5 𝑄𝑑 = 108.5 − 55𝑃 Other Types of Elasticity 1. Cross price elasticity of demand is defined as the percentage change in quantity demanded of good two with respect to a percentage change in the price of good one. %Q2 Q2 P1 dQ2 P1 Q2 P1 E1,2 = = x or x or x %P1 P1 Q2 dP1 Q2 P1 Q2 If E1,2  0 , goods 1 and 2 are substitutes. If E1,2  0 , goods 1 and 2 are complements. If E1,2 = 0 , goods 1 and 2 are not related. 2. Income elasticity of demand is defined as the percentage change in the quantity of a good with respect to a percentage change in income. The symbol commonly used to denote income elasticity of demand is µ, the Greek symbol mu. (Although η (eta) is used by the book). The book also uses Y to represent Income, but M or I are also used. %Q Q I dQ I Q I η= = x or x or x %I I Q dI Q I Q If 0 < η ≤ 1, the good is considered a normal good. If η < 0, the good is considered an inferior good. If η > 1, the good is considered superior  2 - 15 3. Elasticity of Supply is defined as the percentage change in quantity supplied with respect to a percentage change in the price of the good. The symbol commonly used to denote the price elasticity of supply is the letter e. The formula for elasticity of supply is: %  Qs Qs P dQs P Qs P e = = x or x or x % P P Qs dP Qs P Qs Supply curves which you may often encounter in economics are: P e=0 e = , Q/t An example of a good that has a perfectly inelastic supply curve is land. Land is a resource that is in fixed supply and no matter how high price rises output cannot increase. An example of a resource that has a perfectly elastic supply curve is unskilled labor. Firms can hire all the labor necessary at the going wage rate. In this case, wage is the price of labor and quantity is the amount of labor available. Range of the Coefficient of Elasticity of Supply: 0  e   If e > 1 then, elastic price elasticity of supply. If e = 1 then, unitary elastic price elasticity of supply. If e < 1 then, inelastic price elasticity of supply. The elasticity of supply tends to be greater in the long run, when all adjustments to the higher or lower relative price have been made by producers (than in shorter periods of time).  2 - 16 Alternative Market Equilibrium Definitions based on the Price Elasticity of Supply The definitions are based on the work of Alfred Marshal, who emphasized the time element in competitive price equilibrium. 1. Momentary Equilibrium (Market Period) Producers are totally unresponsive to a price change. Why? There is no time for producers to adjust output levels in response to the change in price! Supply is perfectly inelastic; therefore, demand determines price. 2. Short-Run Equilibrium Firms can respond to the change in market price for the good by increasing the variable input in production. That is to say, produces produce more by using their equipment and/or plants more intensively. Short-run production function: Q = F (K, L), where K (capital) is the fixed input and L is the variable input. Hence, firms are able to increase output in the short-run if they increase their variable (labor) input. 3. Long-Run Equilibrium (or “Normal Price”) All inputs are variable; hence, firms can increase their capital stock, e.g., build new plants, and new firms can enter the industry or old ones leave. Q = F (K, L). 4. Very-Long Run Equilibrium Technology is improving, so that for a given amount of capital and labor input more output is forthcoming. The supply curve is becoming more elastic! Q = T{Q (K, L)}.  3-1 CHAPTER 3 LECTURE - CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY Our objective is to construct a simple model of consumer behavior that will permit us to predict consumers' reactions to changes in their opportunities and constraints. We will take tastes and preferences as given, but we will represent them with a very general analytical model. Consumer Preferences Utility: theoretical concept that represents the level of satisfaction or enjoyment that a consumer receives from consumption of a good. We do not measure utility. Consumers do not measure their utility in any units of measure, but they can rank their utilities from different consumption bundles. Ordinal Theory – places market basket in the order of most preferred to least preferred, but doesn’t indicate by how much one market basket is preferred to another. Cardinal Utility Theory - quantitatively measuring a consumer’s satisfaction, but is mostly a theoretical concept. We define Total Utility as the utility that a consumer receives from all of the units of a particular good that she consumes. Define Marginal Utility as the increase in Total Utility that corresponds to a one-unit increase in consumption of a good. Diminishing Marginal Utility: plays a very important role in our analysis of consumer behavior. As the quantity of a good consumed increases (ceteris paribus), the marginal utility attached to consuming additional units of the good eventually diminishes.  3-2 Looking at the mathematics: U dU U We define Marginal utility (y for example) as simply: MUY = = =. Y dY Y 1 Suppose we have a U=f(Y) = U (Y ) = Y = Y 2 dU 1 − 12 1 Solving for MUy MU Y = = Y = dY 2 2 Y Is this subject to diminishing marginal utility? Yes. HOW WOULD YOU PROVE? S.O.C. dMUY d 2U 1 − 32 = = − Y 0 S.O.C for maximization satisfied. dY dY 2 4 Is More Always Better? PREFERENCES WITH MULTIPLE GOODS: MARGINAL UTILITY, INDIFFERENCE CURVES, AND THE MARGINAL RATE OF SUBSTITUTION 1 1 Suppose the Utility Function is U = X Y. The level of utility is shown on the vertical axis, and 2 2 the amounts of food (x) and clothing (y) are shown, respectively, on the right and left axes. Contours representing lines of constant utility are also shown. For example, the consumer is indifferent between baskets A, B, and C because they all yield the same level of utility (U = 4) We previously defined Marginal utility in the one good case (y for example) as: U dU U MUY = = =. Y dY Y In the multiple good case, for example where U =U(X,Y) U dU U U dU U MU X Y held constant = = = MUY X held constant = = = X dX X Y dY Y  3-3 1 1 U 1 − 12 12 U 1 12 − 12 If U = X 2 Y 2 MU X Y held constant = = X Y MUY X held constant = = X Y X 2 Y 2 You can show that both X and Y are subject to diminishing marginal utility. INDIFFERENCE CURVES The First Steps in Understanding Utility – Make some assumptions about people/individuals. Premises of the model: 1. Individual tastes or preferences determine the amount of pleasure people derive from the goods and services they consume. 2. Consumers face constraints, or limits, on their choices. 3. Consumers maximize their well-being or pleasure from consumption subject to the budget and other constraints they face. Assumption One: Completeness. We assume that an individual has preferences over any two bundles of goods (a bundle can be a single good or a bunch of goods). In other words, they can choose between them or decide they are indifferent. Prefer A to B or Prefer B to A or be Indifferent Assumption Two: More is better. This is simply that if a person considers something to be a good (i.e. they value it) then more of it is preferred to less. From these two assumptions alone, we can construct an indifference curve. An Indifference Curve is a line (curve) that shows all the possible combinations of two goods between which a person is indifferent. In other words, it shows the consumption of different combinations of two goods that will give the same utility (satisfaction) to the person. Now this curve is a locus of bundles we are indifferent between. Let’s graph this. Y B 11 A 6 4 6 X  3-4 So what do we know? Well, that tells us that an indifference curve must be negatively sloped; in other words, must go from the Northwest to the southeast. But it doesn’t tell us much about the shape of the curve. Is it concave (bowed out), convex (bowed in), or straight? What do these curves show? Y Y Y X X X For that we need our third postulate. Assumption Three: Diminishing Marginal Rate of Substitution What that means in English is that as I acquire more and more of a good I am willing to give up less and less of other goods to obtain it. That implies an indifference curve will be convex (or bowed in) – the middle graph above. We’ll will come back to that to discuss why that is the case. There is one more thing we need to know about indifference curves and that comes from the fourth assumption. Assumption Four: Transitivity What does that mean? Well, it means my preferences or choices are consistent. In other words, if I prefer A to B and B to C, then I also prefer A to C. Or equivalently, I am indifferent between A and B…and indifferent between B and C, then I am also indifferent between A and C. What this implies is that indifference curves can never touch or cross each other. X  3-5 Interpreting the Indifference Curves Interpreting/reading indifference curves takes us back to that third assumption of diminishing marginal rate of substitution. What is the marginal rate of substitution? This is the size of the reduction in the variable on the vertical axis that leaves the individual indifferent following an increase of one unit of the variable on the horizontal axis. This is much clearer if we do it on a graph. Y A 40 30 B C 20 22 D 8 9 12 13 X Another way of saying it is that the MRS is the maximum amount a person will give up to obtain one more unit. The maximum amount is the amount that leaves him just indifferent. This should ring a bell in regard to demand: remember how we interpreted the height of a demand curve. It was the maximum amount a person would pay – in effect, how much he valued that unit. Well, we’ll link all this up. So this is what we mean by Marginal Rate of Substitution – but what about this diminishing part? The diminishing part comes from what happens as this individual consumes more and more of X. As you can see from the slope, it is getting flatter. What this means is that as he has more X, the maximum amount of Y he is willing to give up to get more X gets less and less. Up at the top (steep slope), he is willing to give up 10Y (go from 40Y to 30Y) to get one more X (from bundle A to bundle B) but down here, he is only willing to give up 2Y to get one more X (from bundle C to bundle D). The rate I am willing to substitute in X for Y is diminishing as I consume more X. A good example of an indifference map  3-6 Other Examples - How would you interpret indifference curves that looked like this? A B Look at A. In this case you must consume these two goods in fixed ratios – like one left and one right shoe, or one set of eye glass frames and one set of lenses – things like that. When looking at the graph you can see that having one right shoe and two left shoes that the additional left shoe doesn’t make you any better off. Look at B - These are perfect substitutes The Slope of the Indifference Curve Along an indifference curve, we say that utility is constant -- in other words, if I am indifferent between two bundles that means it gives me the same level of utility. Therefore, we have the following: TU = TUX + TUY You can show that MUX∆X + MUY∆Y =0 The first terms is the addition in utility resulting from additional X (so ∆X>0); the second term is the decrease in utility resulting from the decrease in Y (so ∆Y 𝟏, the marginal utility of 𝒙 increases as 𝒙 increases.  AX  −1Y  Y c) What is the MRSx, y? MRS X ,Y = =  AX  Y  −1  X d) Is MRSx, y diminishing, constant, or increasing as the consumer substitutes x for y along an indifference curve? As the consumer substitutes 𝒙 for 𝒚, the 𝑴𝑹𝑺𝒙,𝒚 will diminish. 2) Linear Utility Function In general form it may be written as U = aX + bY The linear nature of this utility function implies that the goods are perfect substitutes. Using calculus to find the MU’s of X and Y U U 𝑎 MU X = =a MU Y = =b We can see the MRSx, y = X Y 𝑏  3-9 Can you answer this problem? Consider the utility function U(x, y) = 3x + y, with MUx = 3 and MUy = 1. a) Is the assumption that more is better satisfied for both goods? Yes, the “more is better” assumption is satisfied for both goods since both marginal utilities are always positive. b) Does the marginal utility of x diminish, remain constant, or increase as the consumer buys more x? Explain. The marginal utility of 𝒙 remains constant at 3 for all values of x. c) What is MRSx, y? 𝑀𝑅𝑺𝒙,𝒚 = 𝟑 d) Is MRSx, y diminishing, constant, or increasing as the consumer substitutes x for y along an indifference curve? The 𝑴𝑹𝑺𝒙,𝒚 remains constant moving along the indifference curve. Can you answer this problem? Consider the utility function U = x2 + y2. The marginal utilities are MUx = 2x and MUy = 2y. a) Is the assumption that more is better satisfied for both goods? Yes, the “more is better” assumption is satisfied for both goods since both marginal utilities are always positive. b) Does the marginal utility of x diminish, remain constant, or increase as the consumer buys more x? Explain. The marginal utility of 𝒙 increases as the consumer buys more 𝒙. 2x x c) What is MRSx, y? MRS X ,Y = = 2y y d) Is MRSx, y diminishing, constant, or increasing as the consumer substitutes x for y along an indifference curve? As the consumer substitutes 𝒙 for 𝒚, the 𝑴𝑹𝑺𝒙,𝒚 will increase. 3 ) Constant Elasticity of Substitution (CES) Utility Function. 1 U =  X  + Y    The value ρ is a number between -∞ and 1. This is called a constant elasticity of substitution (CES) utility function. CES functions are also used in production theory. You should be able to show that the marginal utilities for this utility function are given by 1 1  -1  -1 MU x = [X  + Y  ] X  −1 MU y = [X  + Y  ] Y  −1  3 - 10 Does this utility function exhibit the property of diminishing MRSx,y? 𝑀𝑈 Recall that 𝑀𝑅𝑆𝑥,𝑦 = 𝑀𝑈𝑥. Substituting in the marginal utilities given above yields 𝑦  −1 x MRS X ,Y = y  −1 Now, because  < 1, x  - 1 decreases as x increases. By the same logic, y  - 1 increases as y decreases. As we “slide down” an indifference curve, x increases and y decreases, so it follows that MRSx,y decreases. Thus, this utility function exhibits diminishing marginal rate of substitution of x for y. – For  = 1: U(x,y) corresponds to perfect substitutes case – For  approaching 0: U(x,y) approaches similar to Cobb-Douglas – For  approaching -: U(x,y) approaches the case of perfect complements 4) Quasilinear utility function - A utility function that is linear in at least one of the goods consumed, but may be a nonlinear function of the other good(s) The equation for a quasilinear utility function is U(x, y) = v(x) + by, where b is a positive constant and v(x) is a function that increases in x—the value of v(x) increases as x increases [e.g., v(x) = X2 or v(x)= √𝑋 ]. This utility function is linear in y, but generally not linear in x. That is why it is called quasilinear. Below are indifference curves for a quasilinear utility function. The distinguishing characteristic of a quasilinear utility function is that, as we move due north on the indifference map, the marginal rate of substitution of x for y remains the same. That is, at any value of x, the slopes of all of the indifference curves will be the same, so the indifference curves are parallel to each other.  3 - 11 Additional Notes 1) Cobb-Douglas (CD) General Form: U = X  Y  or U = X  Y 1− (special case), where α and β are constants. Shortcut for finding the MRS: For Cobb-Douglass utility functions, it is always true that Y MRS =. The indifference curves are always strictly convex (that is, the MRS diminishes). X 2) Perfect complements General form: U = min{ax,by}, where a and b are constants. Examples: U = min{x,y} U = min{2x,y} U = min{2x,4y} U = min{(1/3)x,(1/4)y} Finding the MRS: The utility function is not differentiable, so we cannot use calculus techniques to find the MRS. However we know that the ICs are “L”-shaped, so all we need to know in order to draw them is the point at which they are kinked. It is always the case that these kinked points of the IC lie on a line whose equation you can derive by setting ax = by. 3) Perfect substitutes General form: U = ax + by, where a and b are constants. Examples: U = x + y U = 2x + y U = 2x + 3y U = (1/2)x + (3/4)y Finding the MRS: The MRS is always equal to a/b, a constant. That is, ICs are straight lines with slope (negative) a/b. 4) Quasi-linear General form: U = ax + f(y) or U = f(x) + ay, where a is a constant. Examples: U = 2x + ln y U = 0.5x + y0.5 U = 6 ln x + y U = 2x0.5 + (3/4)y Finding the MRS: No shortcut for finding the MRS: you will need to take the ratio of partial derivatives in every case.  4-1 CHAPTER 4 LECTURE - CONSUMER CHOICE Our objective is to construct a simple model of consumer behavior that will permit us to predict consumers' reactions to changes in their opportunities and constraints. We will take tastes and preferences as given, but we will represent them with a very general analytical model. We will Now Look at the Consumer’s Opportunity Set or Budget Constraint General Formulation: Assume for the moment there are only two goods in the world -- this is a simplification we will relax later -- a person therefore will spend all his income on these two goods. This can be depicted as I = PXX + PYY, where the first term is the person's total expenditure on X and the second term the total expenditure on Y. Some texts use M for Income. Let's graph this using some numbers, suppose I = $200, PX = $5, and PY = $10 -- note, as is usually the case, this person is a price taker. From this information we can graph a budget constraint. What is the maximum amount of X (Movies) this person can buy? 40 units. How does one know that? From the general formula, (Income or I)/PX. a) What is the maximum amount of Y this person can buy? 20 units. b) Suppose she wants one unit of X -- now what is the maximum amount of Y she could purchase? It would be 18 -- so when you buy ONE X, you give up TWO Y -- this is the REAL or RELATIVE price of X. Of course, we can put this into a general formula, as follows: I = PX X + PY Y PY Y = I − PX X I  PX  or Y= − X PY  PY   4-2 The first term is the intercept of the Y-axis; the second term is the slope. Note the slope is the PX price of X over the price of Y -- the relative price or −. PY We noted that anything within (below and to the left) of the budget line is obtainable; anything beyond the budget line is not obtainable. In this world where we are spending all our income, we will always be on the budget line. Changes in the Budget Constraint. There are three givens when we construct the budget line, income, price of X, and price of Y - - if any of these things change, the budget line changes. What happens if: a) Income increases? b) Price of X goes up? c) Price of Y goes down? d) Given a gift certificate of $20 for X? What has happened here?  4-3 Consumer Equilibrium Now it's time to bring these two concepts together and build a model of human behavior. Look at the following graph: We want to Maximize Utility, U(x,y), subject to our constraint PXX+PYY ≤I U2 Y U1 "A E Y0 B X0 X At point A, what is the slope of the indifference curve? Remember the slope is the marginal rate of substitution that tells us what the maximum amount of Y the person is willing to give up to obtain one more unit of X -- let's say the slope is 10. At point A, what is the slope of the budget constraint? Remember, the slope is the relative price of X -- in other words, what do I have to pay? Let's say it is 2 -- which means that to obtain one more unit of X, I give up two units of Y. Now, if I am willing to give up 10 but only have to give up 2, what will happen to my utility -- clearly I am better off and therefore, by definition, I will move up a higher indifference curve. So if we move up a higher indifference curve, we go through the same decision process -- is what I'm willing to pay greater than what I have to pay? -- if so, I buy it and am better off as a result -- this is identical to what we did with a demand curve -- in fact, later this week we will derive a demand curve from this graph. Now look at point E -- at this point, the indifference curve is tangent to the budget line. At the point of tangency, what can you say about the slopes? By definition, they are equal. So what does that say about this person? Well the maximum amount I am willing to pay is equal to the amount I have to pay -- i.e. MRS is to the price ratio. If I go beyond that, then the maximum amount I'm willing to pay is less than what I have to pay and therefore, if I purchase more, I make myself worse off. So point E gives us is one point on the demand curve. For a given income, price of X, and price of Y, and given my preferences this is how much X (and Y) I will consume. So E is the equilibrium point -- that is where we will always be (or get to very quickly).  4-4 Remember we are doing comparative statics -- which is just the comparison of different equilibrium points. Notice that At point E, the slopes are equal. We know the slope of the indifference curve is the marginal rate of substitution: MU X PX MRS = − The slope of the budget constraint is the price ratio: −. MU Y PY MU X PX MU X MU Y Therefore, at point E the following is true: = or = MU Y PY PX PY The economic interpretation of this is that at equilibrium the utility per dollar spent must be equal across all goods. In other words, if I can take a dollar away from Y and spend it on X and get more utility, I will do that (and vice versa). Another Example Explain why Point B in the above diagram is NOT OPTIMAL We can also look at this as an Expenditure Minimization Problem. Minimize (x,y) expenditure = PXX+PYY subject to constraint that U(x,y)=U*  4-5 Some Special Cases (Corner Solution) (Equilibrium with Perfect Substitutes) Consumer Choice with Composite Goods Housing with Subsidy versus Voucher Quantity Discounts  4-6 REVEALED PREFERENCE Suppose we do not know the consumer’s indifference map, but we do have observations about consumer choice with two different budget lines. When the budget line is BL 1, the consumer chooses basket A. When the budget line is BL2, the consumer chooses basket B. What does the consumer’s behavior reveal about his preferences? As shown by the analysis in the text, the consumer’s indifference curve through A must pass somewhere through the yellow area, perhaps including other baskets on EF. First, the consumer chooses basket A when he could afford any other basket on or inside BL 1, such as basket B. Therefore, A is at least as preferred as B (A preferred to B). But he has revealed even more about how he ranks A and B. Consider basket C. Since the consumer chooses A when he can afford C, we know that A is preferred to C. And since C lies to the northeast of B, C must be strongly preferred to B (C ≻ B). Then, by transitivity, A must be strongly preferred to B (if A is preferred to C and C ≻ B, then A ≻ B) MATHEMATICAL ANALYSIS OF CONSUMER BEHAVIOR 1. The Indifference Curve The total utility function is given as: U(X, Y). We have an indifference curve that is given by: U(X, Y) = C, where “C” is a constant level of utility along the stable indifference curve. Take the total differential of this indifference curve and set it equal to zero (Why?): We call this the F.O.C. or first order condition. U U U U U = U(X, Y) dU = dX + dY = 0 or dX + dY = 0 X Y X Y (Note: By taking the total differential of an indifference curve and setting it equal to zero, both X and Y are allowed to change simultaneously, but there is no change in total utility.)  4-7 We note that some texts define the MRS as the negative of the slope. If that is the case, solving for the negative of the slope of the indifference curve, -dY/dX: U MU X − dY = MRS X ,Y = X or MRS X ,Y = dX U MUY Y The slope of an indifference curve is the MRSX,Y; which is equal to the ratio of the marginal utility of the two goods consumed. MRSX,Y should be positive if both MUx and MUy are positive. 2. The Budget Line Budget Line (or Income Constraint) The locus of budgets (total expenditures on alternative combinations of goods) that can be purchased if all money income is spent. Algebraically: I = Px X + Py Y, Where I is total money income, PX is the nominal price of good X, PY is the nominal price of good Y, and X and Y are the quantities of both goods purchased. I P Solving for Y: Y = − X X (Note: The slope of the budget line is the ratios of the PY PY PX price of X to Y.) Slope of Budget Line = − PY If I = $200, PX is $20, and PY is $10, then the budget line or income constraint is: $200 $20 PX 20 Y = − X or Y = 20 − 2 X Thus: − =− = −2 $10 $10 PY 10 We often ignore the negative sign. A consumer maximizes his/her total utility from consumption subject to the budget constraint. Consumer equilibrium implies: 1. The consumer’s highest indifference curve is tangent to the budget line. (Why?) 2. Equality between the MRSX,Y and the relative price ratio of good X. (Why?) 3. The ratios of the MU to the price are equal for all goods. (Why?) (Consumer allocates expenditures so that the utility of the last dollar spent on each good is equal.)  4-8 PX MU X P The point of consumer equilibrium is where: MRS XforY = or = X PY MU Y PY MU X MU Y Cross-multiplying: = PX PY Formal Mathematical Proof of Consumer Equilibrium Maximize: U = U(X, Y) Subject to: Px X + Py Y = I, (Note: A consumer maximizes his/her total utility given the budget constraint.) This is a constrained maximization problem, hence, the Lagrangean function is: £ = U ( X , Y ) −  ( PX X + PY Y − I ) Taking the derivative of £ with respect to each good (i.e., X and Y) and setting it equal to zero:  £ U = −  PX = 0 X X  £ U = −  PY = 0 Y Y £ = PX X + PY Y − I = 0  MU X MU Y Solving for : = and = PX PY MU X MU Y Equating these two equations: = PX Py Consumer equilibrium requires that the ratio of the marginal utility and the money price of each good be equalized for all goods consumed. MU X PX Alternatively stated: = MU Y Py Consumer equilibrium requires that the ratio of the marginal utilities of the goods equal the relative price ratio. NOTE: The negative sign in front of λ is arbitrary; a positive sign works equally well. In the case we have £ = U ( X , Y ) +  ( I − PX X − PY Y )  4-9 Examples Example 1 - Julie’s preferences for food (measured by F) and clothing (measured by C) are described by the utility function U(F,C) = FC. Suppose food costs $1 per unit and clothing costs $2 per unit. She has an income of $12 to spend on food and clothing. For this problem the Lagrangian function is: £ = 𝐹𝐶 − 𝜆(𝑃𝐹 𝐹 + 𝑃𝐶 𝐶 − 𝐼) = 𝐹𝐶 − 𝜆(𝐹 + 2𝐶 − 𝐼) Solving: 𝑀𝑈𝐹 − 𝜆𝑃𝐹 = 0 => 𝐶 − 𝜆 = 0 𝑀𝑈𝐶 − 𝜆𝑃𝐶 = 0 => 𝐹 − 2𝜆 = 0 𝑃𝐹 𝐹 + 𝑃𝐶 𝐶 − 𝐼 = 0 > 𝐹 + 2𝐶 − 12 = 0 Her marginal utilities are MUF = C and MUC = F. Thus, both F and C are subject to constant marginal utility. We have three equations and three unknowns. When we combine the first two equations, we find that Julie should equate the marginal utility per dollar spent on each good. 𝑀𝑈𝐹 𝑀𝑈𝐶 𝐶 𝐹 𝜆= = => 𝜆 = = 𝑎𝑛𝑑 𝐹 = 2𝐶. 𝑃𝐹 𝑃𝐶 1 2 Substituting 𝐹 = 2𝐶 into the budget constraint, we find that 𝐹 + 2𝐶 − 12 = 0 => 2𝐶 ∓ 2𝐶 − 12 = 0 4𝐶 = 12 > 𝐶 = 3 and 𝐹 = 6. 𝐶 𝐹 3 6 The meaning of 𝝀. We found that 𝜆= = => 𝜆 = = = 3. This is the rate 1 2 1 2 ∆𝑈 of change when Julie’s income is exactly 12. If her income were to rise by a dollar, we would ∆𝐼 expect to see her utility increase by about 3. When Julie’s income is 12, she chooses C = 3 and F = 6. Her utility is U = FC = (6)(3) = 18. When her income is 13, she chooses F = 6.5 and C = 3.25. Her utility is U = FC = (6.5)(4.25) = 21.25. Julie’s utility increased by 3.25 (from 18 to 21.25) when her income increased by 1 (from 12 to 13). This is close to the values of 𝜆 we found in parts (c) and (d), as we expected.  4 - 10 Example 2 - The utility that Ann receives from consuming food (measured by F) and clothing (measured by C) is described by the utility function U(F,C) = FC + F. Suppose food costs $1 per unit and clothing costs $2 per unit. She has an income of $22 to spend on food and clothing. We can see that Ann has a diminishing marginal rate of substitution of food for clothing. The marginal utilities for the two goods are MUF = C+1 and MUC = F. Since both marginal utilities are positive, we know that the indifference curves are negatively sloped. When we increase F along an indifference curve, the level of C must therefore fall. We know 𝑀𝑈𝐹 𝐶+1 that 𝑀𝑅𝑆𝐹,𝐶 = =. As we increase F along an indifference curve (and C falls), the 𝑀𝑈 𝐶 𝐹 value of 𝑀𝑅𝑆𝐹,𝐶 falls. Therefore, we do have diminishing 𝑀𝑅𝑆𝐹,𝐶. This is important because it guarantees that the solution, we find using the method of Lagrange will maximize utility while satisfying the budget constraint. For this problem the Lagrangian function is: £ = (𝐹𝐶 + 𝐹) + 𝜆(𝐼 − 𝑃𝐹 𝐹 − 𝑃𝐶 𝐶) = (𝐹𝐶 + 𝐹) + 𝜆(22 − 𝐹 − 2𝐶) We set it up with a positive in front of 𝜆 just for fun. 𝑀𝑈𝐹 − 𝜆𝑃𝐹 = 0 => 𝐶 + 1 − 𝜆 = 0 𝑀𝑈𝐶 − 𝜆𝑃𝐶 = 0 => 𝐹 − 2𝜆 = 0 𝐼 − 𝑃𝐹 𝐹 − 𝑃𝐶 𝐶 = 0 => 22 − 𝐹 − 2𝐶 = 0 We have three equations and three unknowns. When we combine the first two equations, we find that Ann should equate the marginal utility per dollar spent on each good. 𝑀𝑈𝐹 𝑀𝑈𝐶 𝐶+1 𝐹 𝜆= = => 𝜆 = = 𝑎𝑛𝑑 𝐹 = 2𝐶 + 2. 𝑃𝐹 𝑃𝐶 1 2 Substituting 𝐹 = 2𝐶 + 2 into the budget constraint, we find that 22 − 𝐹 − 2𝐶 = 0 => 22 − 2𝐶 − 2 − 2𝐶 = 0 => 4𝐶 = 20 𝑜𝑟 𝐶 = 5 and 𝐹 = 12. Example 3 - Omar consumes only two goods, whose quantities are measured by x and y. His preferences are described by the utility function U(x,y) = xy + 10(x + y). The prices of the goods are PX = $9 and Py = $3. He has a daily income of $30. The marginal utilities for the two goods are MUx = y+10 and MUy = x+10. Since both marginal utilities are positive, we know that the indifference curves are negatively sloped. When we increase x along an indifference curve, the level of y must therefore fall.  4 - 11 𝑀𝑈𝑥 𝑦+10 We know that 𝑀𝑅𝑆𝑥,𝑦 = = 𝑥+10. As we increase x along an indifference curve (and y 𝑀𝑈𝑦 falls), the value of 𝑀𝑅𝑆𝑥,𝑦 falls. Therefore, we do have diminishing 𝑀𝑅𝑆𝑥,𝑦. For this problem the Lagrangian function is: £ = 𝑥𝑦 + 10(𝑥 + 𝑦) − 𝜆(𝑃𝑥 𝑥 + 𝑃𝑦 𝑦 − 𝐼) = 𝑥𝑦 + 10(𝑥 + 𝑦) − 𝜆(9𝑥 + 3𝑦 − 30) 𝑀𝑈𝑥 − 𝜆𝑃𝑥 = 0 => 𝑦 + 10 − 9 𝜆 = 0 𝑥>0 𝑀𝑈𝑦 − 𝜆𝑃𝑦 = 0 => 𝑥 + 10 − 3 𝜆 = 0 𝑦>0 𝑃𝑥 𝑥 + 𝑃𝑦 𝑦 − 𝐼 = 0 => 9𝑥 + 3𝑦 − 30 = 0 𝜆>0 We now have three equations and three unknowns. If Omar can do so, he should equate the marginal utility per dollar spent on each good. Combining the first two equations, we see that equating the marginal utility per dollar spent would require that: 𝑀𝑈𝑥 𝑀𝑈𝑦 𝑦 + 10 𝑥 + 10 𝜆= = => 𝜆 = = 𝑎𝑛𝑑 𝑦 = 3𝑥 + 20. 𝑃𝑥 𝑃𝑦 9 3 Substituting 𝑦 = 3𝑥 + 20 into the budget constraint, we find that 5 30 − 9𝑥 − 3𝑦 = 0 => 30 − 9𝑥 − 3(3𝑥 + 20) = 0 => 𝑥 = − 3 and 𝑦 = 15. But x cannot be negative so the utility-maximizing basket must be at a corner point, with either x = 0 or y = 0. Homothetic Preferences - If the MRS depends only on the ratio of the amounts of the two goods, not on the quantities of the goods, the utility function is homothetic U = X  Y  We can show using the Cobb – Douglas function. U = X Y  U  X  −1Y   Y MRS = X = =  U X Y  −1  X yY  4 - 12 Practice Problems 1. An individual consumes products X and Y and spends $24 per time period. The prices of the two goods are $3 per unit for X and $2 per unit for Y. The consumer in this case has a utility function expressed as: 1 1 U = X 2Y 2 a. Determine the values of X and Y that will maximize utility in the consumption of X and Y. b. Determine the total utility that will be generated per unit of time for this individual.  5-1 CHAPTER 5 LECTURE – THEORY OF DEMAND Deriving a Demand Curve - Using the graph below we can derive a demand curve. Remember along a demand curve, what is constant? Income, preferences, price of other goods, etc. -- the only things that change are price and quantity. The following graph depicts that situation where the price of X has fallen and total expenditure has risen. The Price Consumption Curve Price-Consumption Curve (PCC): for a good X is the set of optimal bundles traced on an indifference map as the price of X varies (holding income and the price of Y constant).  5-2 Effect of Changes in Income: Income Consumption Curve (Also called ‘offer curve’) Income-Consumption Curve (ICC): for a good X is the set of optimal bundles traced on an indifference map as income varies (holding the prices of X and Y constant). Engel Curve: curve that plots the relationship between the quantity of X consumed and income.  5-3 Normal goods: demand increases with income Inferior goods: demand falls with income  5-4 Mathematical Analysis Looking at deriving demand functions Maximize: U = U(X,Y) Subject to: PyX + PyY = I 1 1 Example: Suppose U = X 2Y 2 Find utility maximizing choice with prices, Px and Py income I 1 −1 1 1 1 −1 Px/Py = MU /MU = Y/X (since MU = X 2 Y 2 and MU = X 2Y 2 x y x 2 Y 2 so PxX = PyY Looking at the Budget Constraint I = 2PxX or I = 2PYY Demand curve for x is X = I/2Px and Demand curve for Y = I/2Py From these demand curves, can you determine the price elasticity of demand for good X and Y? X PX 1 PX Demand for Good X = I/2Px or X = (IPX-1)/2 X = = − IPX −2 = −1 PX X 2 1 IPX −1 2 Y PY 1 PY Demand curve for Y = I/2Py Y = = − IPY −2 = −1 PY Y 2 1 −1 IPY 2 From these demand curves, can you determine the Income elasticity of demand for good X and Y? X I 1 I Demand for Good X = I/2Px or X = (IPX-1)/2  I ,X = = =1 I X 2 PX I 2 PX Y I 1 I Demand curve for Y = I/2Py  I ,X = = =1 I Y 2 PY I 2 PY You could also take the natural log of the demand functions and then solve. For example. I 1 X= becomes ln X = ln + ln I − ln PX 2 PX 2 You can see that the values for the cross-price elasticities are equal to 0.  5-5 General Form for a Cobb-Douglas Utility Function where α+β=1. U(x,y) = xy, where α+β=1 Setting up the Lagrangian: ℒ = xy + (I - pxx - pyy) First-order conditions for a maximum: ℒ/x = x-1y - px = 0 ℒ/y = xy-1 - py = 0 ℒ/ = I - pxx - pyy = 0 We can see that consumer’s choice is y/x = px/py Since  +  = 1: pyy = (/)pxx = [(1- )/]pxx Substituting into the budget constraint: I = pxx + [(1- )/]pxx = (1/)pxx I I (1 −  ) I Solving: X* = You can show that Y* = or Y * = PX PY PY You can also see that an individual will allocate  percent of his income to good x and  percent of his income to good y. PX X =  I PY Y =  I = (1 −  )Y I I If  +  ≠ 1 x* = and y* = ( +  ) Px ( +  ) Py The weights are as shown above.  5-6 Example with a CES Utility Function MU X (Y ) ( MU Y ) We define the elasticity of substitution as  = [ X x ] MU X (Y ( ) X ) MU Y "The shape of the indifference curve indicates the degree of substitutability of the goods…" General form of CES Utility Function U(x,y) = [x + y]1/ , where   1,   0 Assume that  = 0.5 U(x,y) = [x0.5 + y0.5]2 We can take the square root so our utility function becomes U(x,y) = [x0.5 + y0.5]. Setting up the Lagrangian: ℒ = x0.5 + y0.5 - (pxx + pyy- I) First-order conditions for a maximum: ℒ/x = 0.5x -0.5 - px = 0 ℒ/y = 0.5y -0.5 - py = 0 ℒ/ = I - pxx - pyy = 0 This means that: (y/x)0.5 = px/py Substituting into the budget constraint, we can solve for the demand functions I I py I I px x* = or y* = or px [1 + ( px / p y )] px [ px + p y )] p y [1 + ( p y / px )] p y [ p y + px )] The share of income spent on either x or y is not a constant. Depends on the ratio of the two prices. The higher is the relative price of x, the smaller will be the share of income spent on x.  5-7 Finding a Demand Curve with a Corner Point Solution A consumer purchases two goods, food and clothing. She has the utility function U(x,y) = xy + 10x where x denotes the amount of food consumed and y the amount of clothing. You can see the marginal utilities are MUx = y + 10 and MUy = x. The consumer’s income is $100, and the price of food is $1. The price of clothing is Py. We can show that the equation for the consumer’s demand curve for clothing is 100 - 10PY Y= , when Py < 10 and y = 0, when Py ≥ 10 2PY Consider a consumer with the utility function U(x, y) = min(3x, 5y), that is, the two goods are perfect complements in the ratio 3:5. The prices of the two goods are Px = $5 and Py = $10, and the consumer’s income is $220. Determine the optimum consumption basket. This question cannot be solved using the usual tangency condition. However, you can see from the graph below that the optimum basket will necessarily lie on the “elbow” of some indifference curve, such as (5, 3), (10, 6) etc. If the consumer were at some other point, he could always move to such a point, keeping utility constant and decreasing his expenditure. The equation of all these “elbow” points is 3x = 5y, or y = 0.6x. Therefore, the optimum point must be such that 3x = 5y. The usual budget constraint must hold of course. That is, 5𝑥 + 10𝑦 = 220. Combining these two conditions, we get (x, y) = (20, 12).  5-8 Income and Substitution Effects Substitution Effect: that component of the total effect of a price change that results from the associated change in the relative attractiveness of other goods. Income Effect: that component of the total effect of a price change that results from the associated change in real purchasing power. Total Effect: the sum of the substitution and income effects. Total Effect of a Price Change is the change in quantity demanded as the consumer moves from one equilibrium consumption bundle to another. Total Effect = Substitution Effect + Income Effect = Qd caused by a Qd caused by a  relative price, +  real income, holding real income holding relative price stable. stable Substitution Effect - The change in quantity demanded given a change in the relative price after compensating for the change in real income (which is caused by the change in relative prices). 1. In general, the substitution effect is negative. a. If relative price falls for a good, then quantity demanded increases. b. If relative price increases for a good, then quantity demanded falls. 2. Graphically it is shown by keeping the consumer on the same indifference curve, but with the new relative price ratio! Income Effect - The change in quantity demanded of a good resulting solely from a change in real income (which is caused by the change in relative prices). 1. In general, the income effect is positive. a. If real income increases, consumers buy more of the good. b. If real income falls, consumers buy less of the good. 2. Normal and superior goods have a positive income effect. 3. Inferior goods have a negative income effect. 4. Graphically it is shown by moving the consumer to a higher or lower indifference curve. In general, a positive income effect reinforces a negative substitution effect.  5-9 We will do this by graphically breaking down a price decrease. HICKS METHOD Consider the following graph: Two things have happened with the price decrease; X has become cheaper relative to other goods (this is the substitution effect); plus, this person can buy more of all goods, including X (this is the income effect). AB is substitution effect and BC is Income effect.  5 - 10 Note: the two effects are in the same direction -- when price decreases, the substitution effect is positive (buys more); the income effect is also positive -- but does it have to be? No, it doesn't. It is in this case because we have assumed the good is normal. Let's depict what would happen if the good were inferior; this we show in the following graph: Everything is the same as last time except we're going to assume the good is inferior. This doesn't affect the substitution effect; it is exactly the same as before, from XA to XB. But as we decrease the income (in effect, shifting the budget constraint to BLd, we decrease our consumption of X (from XB to XC). Do we still have a downward sloping demand curve? Yes, we do. The total effect, from X1 to X3 is still positive showing that our consumption of X increases as the price decreases. The reason it is positive is that the substitution effect is greater than the income effect. Conditions for an Upward Sloping Demand Curve –Giffen Good First, it must be an inferior good. Second, for that inferior good the income effect must be greater than the substitution effect. Not only does a good have to be inferior -- which is relatively rare -- but also the good has to take up a 'large' portion of a person's total budget -- which is even rarer. This is why we do not observe upward sloping demand curves even though they are theoretically possible.  5 - 11 Finding Income and Substitution Effects Algebraically (Hicks Method) The consumer’s utility function is U(x, y) = xy, where x denotes the amount of food consumed and y the amount of clothing. You can see that MUx = y and MUy = x. Now suppose that he has an income of $72 per week and that the price of clothing is Py = $1 per unit and the price of food is initially Px = $9. Problem: Find the numerical values of the income and substitution effects on food consumption. Solution: Step 1. You can show the consumer will maximize utility such that Y/X = 9/1 or Y = 9X. Plugging in values in the budget line give us 72 = 9X + 9X Solving we find X = 4 and Y = 36. So, at basket A, the consumer purchases 4 units of food and 36 units of clothing each week. Step 2. Find the final consumption basket C when the price of food is $4. We repeat step 1, but now with the price of a unit of food of $4, which again yields two equations with two unknowns: 4X+Y =72 (coming from the budget line) Y=4X (coming from the tangency condition). When we solve these two equations, we find that X = 9 and Y = 36. So, at basket C, the consumer purchases 9 units of food and 36 units of clothing each week. Step 3. Find the decomposition basket B. The decomposition basket must satisfy two conditions. First, it must lie on the original indifference curve U1 along with basket A. Recall that this consumer’s utility function is U(x, y) = xy, so at basket A, utility U 1 = 4(36) = 144. At basket B, the amounts of food and clothing must also satisfy U = xy = 144. Second, the decomposition basket must be at the point where the decomposition budget line is tangent to the indifference curve. Remember that the price of food P X on the decomposition budget line is the final price of $4. The tangency occurs when MUX/MUY = PX/PY, that is, when Y/X = 4/1, or Y = 4X. When we solve the two equations XY = 144 and Y = 4X, we find that, at the decomposition basket, X = 6 units of food and Y = 24 units of clothing. Now we can find the income and substitution effects. The substitution effect is the increase in food purchased as the consumer moves along initial indifference curve U 1 from basket A (at which he purchases 4 units of food) to basket B (at which he purchases 6 units of food). The substitution effect is therefore 6 − 4 = 2 units of food. The income effect is the increase in food purchased as he moves from basket B (at which he purchases 6 units of food) to basket C (at which he purchases 9 units of food). The income effect is therefore 9 − 6 = 3 units of food.  5 - 12 Income and Substitution Effects The above method of showing the income and substitution effect is based on the Hicks Method. That is, utility is held constant. NOTE: SOME TEXTS LOOK AT THE SLUTSKY SUBSTITUTION EFFECT. IN THIS CASE, WE HOLD REAL INCOME CONSTANT. THIS IS SHOWN IN THE GRAPH BELOW. In this case we take away enough nominal income so real income has not changed. This is at point X0. Here the substitution effect is X0 to Xc and income effect is XC to Xf.  5 - 13 Slutsky method with a increase in Price of X (adapted from Varian). SLUTSKY METHOD  5 - 14 Example with decrease in Price of X: The consumer’s utility function is U(x, y) = xy, where x denotes the amount of food consumed and y the amount of clothing. You can see that MUx = y and MUy = x. Now suppose that he has an income of $72 per week and that the price of clothing is Py = $1 per unit and the price of food is initially Px = $9. Problem: Find the numerical values of the income and substitution effects on food consumption. Solution: Step 1. We know the demand function at PX= 9 is X = I/2Px or X = 72/2(9) = 4 What happens if Px fall to 4? We know the demand function at PX’= 4 is X = I/2Px or X = 72/2(4) =9 So, at basket A (original point), the consumer purchases 4 units of food and at point C (total effect), 9 units of food. Step 2. I′ = I+ (I′−I) = I + (Px’ – Px)[X(Px, PY, I)] = 72 + (4 – 9)4 = 52 Step 3. X’ = I’/2Px’ or X = 52/2(4) =6.5 Substitution effect = 6.5 – 4 = 2.5 Income effect = 9 - 6.5 = 2.5 COMPENSATING VARIATION AND EQUIVALENT VARIATION We have shown how a price change affects the level of utility for a consumer. However, there is no natural measure for the units of utility. Economists therefore often measure the impact of a price change on a consumer’s well-being in monetary terms. How can we estimate the monetary value that a consumer would assign to a change in the price of a good? We can look at how much income the consumer would be willing to give up after a price reduction, or how much additional income the consumer would need after a price increase, to maintain the level of utility she had before the price change. We call this change in income the compensating variation (because it is the change in income that would exactly compensate the consumer for the impact of the price change). The compensating variation for a price reduction is positive because the price reduction makes the consumer better off.  5 - 15 For a price increase, the compensating variation is negative because the price increase makes the consumer worse off. Second, we see how much additional income the consumer would need before a price reduction, or how much less income the consumer would need before a price increase, to give the consumer the level of utility she would have after the price change. We call this change in income the equivalent variation (because it is the change in income that would be equivalent to the price change in its impact on the consumer). The equivalent variation for a price reduction is positive because the price reduction makes the consumer better off. For a price increase, the equivalent variation is negative because the price increase makes the consumer worse off. Compensating variation: A measure of how much money a consumer would be willing to give up after a reduction in the price of a good to be just as well off as before the price decrease. Equivalent variation: A measure of how much additional money a consumer would need before a price reduction to be as well off as after the price decrease. The price change from Px1 to Px2 has a positive income effect, so the compensating variation (the length of the segment KL) and the equivalent variation (the length of the segment JK) are not equal. In this case, JK > KL.  5 - 16 Duality in Consumer Theory We can look at behavior in two ways: 1. Maximize utility given budget constraint 2. Minimize expenditure necessary to achieve a given level of Utility (U*) Result is the same no matter how we state problem. Let E denote expenditure, so E = Px X + Py Y Utility function is U = U(X, Y) Minimize expenditure subject to constraint that U* = U(X, Y) This is a constrained minimization problem, hence, the Lagrangean function is: £ = PX X + PY Y −  (U ( X , Y ) − U *) Taking the derivative of L with respect to each good (i.e., X and Y) and setting it equal to zero: L U L U Solving for : MU X and = PX −  =0 = PY −  =0 = = MU Y X X Y Y PX PY Equating these two equations yields the same result as the before: MU X MU Y = PX Py Looking once again at deriving demand functions Expenditure Function: E=Px X+ Py Y Subject to Constraint: U(X,Y) = XY = U* Find Expenditure minimizing choice with prices, Px and Py , income M and U=XY Short cut: Px/Py = MU /MU = Y/X (since MU = Y and MU = X) so P X = P Y x y x Y x y Looking at the Budget Constraint E (or M) = 2PxX or M = 2PYY Demand curve for X is X = M/2P and Demand curve for Y = M/2Py Same result as before. x Ordinary vs. Income-Compensated Demand Curves for a Normal Good Compensated (HICKSIAN) Demand Curves and Functions Actual level of utility varies along the demand curve As the price of x falls, the individual moves to higher indifference curves Assumption: nominal income is held constant as the demand curve is derived “Real” income rises as the price of x falls o Alternative approach o Hold real income (or utility) constant while examining reactions to changes in Px o The effects of the price change are “compensated” so as to force the individual to remain on the same indifference curve Reactions to price changes include only substitution effects  5 - 17 Aggregating Individual Demand Curves  5 - 18 CONSUMER PRICE INDICES Substitution Bias in the Consumer Price Index In year 1, the consumer has an income of $480, the price of food is $3, and the price of clothing is $8. The consumer chooses basket A. In year 2, the price of food rises to $6, and the price of clothing rises to $9. The consumer could maintain his initial level of utility U1 at the new prices by purchasing basket B, costing $720. An ideal cost of living index would be 1.5 (=$720/$480), telling us that the cost of living has increased by 50 percent. However, the actual CPI assumes the consumer does not substitute clothing for food as relative prices change, but continues to buy basket A at the new prices, for which he would need an income of $750. The CPI ($750/$480 = 1.56) suggests that the consumer’s cost of living has increased by about 56 percent, which overstates the actual increase in the cost of living. In fact, if the consumer’s income in year 2 were $750, he could choose a basket such as E on BL3 and achieve a higher level of utility than U1.  6-1 CHAPTER 6 LECTURE – INPUTS AND PRODUCTION FUNCTIONS Production is the entire process of making goods and services. In theoretical terms production involves the transformation of inputs into outputs, and in practice involves input sourcing, capital equipment acquisition, industrial engineering, adaptation of new technology, personnel training and management, and a great deal of managerial coordination. Production is a firm's response not only to the consumer market signals of 'what' type of good or service is demanded, but also to the input market signals of 'how' to produce, given the costs associated with various labor and capital/technological inputs. Production Functions A production function is a formal mathematical relation that describes the efficient process of transforming inputs into outputs. The word efficient is in the definition because embedded in the concept of the production function is the notion that firms will want to squeeze the maximum output from a given set of inputs. A simple production function describes the process of transforming a set of inputs K, L, etc. into a quantity Q of output (goods or services): Q = Q(K, L etc.) Conceptually, a specific production function, Q(K, L), defines a technology. This is extremely similar to a recipe in so many ounces of flour, liquid, and sweetener, properly combined then baked, creates a cake or so many ounces of steel, fasteners, rubber, and glass, properly combined, creates an automobile. Different recipes to make cakes and automobiles represent different technologies.  6-2 Production Decisions in the Short – Run The short run is defined as the time frame in which there are fixed factors of production Variable and fixed factors of production – variable factors are the inputs a manager can adjust to alter production. Fixed factors are the inputs the manager cannot adjust for a period of time Usually capital (K) is fixed (denoted by K ) in the short run and labor (L) is variable, so we can rewrite the production function as, Q = f (L) = F ( K , L) Total, Marginal, and Average Product Total product (TP) is the entire output of the production process, and often denoted as the Q, or quantity of output. Marginal product (MP) of a particular factor of production, for example labor, is defined to be the change in output (Q) resulting from a one-unit change in the input (ex: one more hour worked, or one more worker employed). If again we use the symbol Δ to mean 'change in', then we have: Q MPX = = where X is a particular input, such as labor, and Q is output of the final good or X service produced. Marginal product is the slope of total product. If we have a nice continuous total product curve, then marginal product at a particular point on the total product curve is simply the slope of the tangent line at that point on the total product curve. Q Using calculus MPX = X Average product (AP) is the average amount of output produced with each unit of input: Q APX = X