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

Transcription

1 ECONS INTERMEDIATE MICROECONOMICS LECTURE NOTES Felix Munoz-Garcia 1 School of Economic Sciences Washington State University This document contains a set of partial lecture notes that are intended to serve as a starting point when coming to class, so every student can complement them with additional examples, exercises and applications discussed in class. (Do not quote) G Hulbert Hall, School of Economic Sciences, Washington State University, Pullman, WA fmunoz@wsu.edu. Tel

2 EconS 301 Intermediate Microeconomics Chapter 2 Demand and Supply Lecture notes In chapter 2 we deal with demand and supply analysis in perfectly competitive markets. Perfectly competitive markets consist of a large number of buyers and sellers. In competitive markets, the transactions of any individual buyer or seller are so small in comparison to the overall volume of the good or the service traded in the market that the buyer or seller essentially takes the price set by the market. Therefore, we consider this the perfect competition model as a model of price-taking behavior. Demand Curve: Market Demand Curve: A curve that shows us the quantity of good that consumers are willing to buy at different prices. For instance, in the graph below, the demand curve shows us the amount of corn a buyer would be willing to buy at different prices. This illustration brings us to a crucial law in economics, the Law of Demand Law of Demand: The inverse relationship between the price of a good and the quantity demanded, when all other factors that influence demand are held fixed. This law is intuitive, as a higher price will make consumers less willing to purchase that good. With almost all goods, the consumer will behave in this manner, treating price and quantity demanded as inversely related. A demand curve will take the form... Q = a bp where a = vertical intercept, and b = slope Rearranging and solving for P, we get 1

3 P = (a/b) (Q/b) This is the Inverse Demand Curve, which is simply the demand curve where P = some function of Q Example: Demand: Q = 100-2P Inverse Demand: P = 50 (Q/2) The vertical intercept is therefore 50 and represents the Choke Price, or the price at which consumers of the product will not desire any of the good. The horizontal intercept is therefore 100, and represents the amount of the good the consumer would want to purchase at a price of 0. Or more generally Demand: Q = a bp Inverse Demand: P = (a/b) (Q/b Choke Price 2

4 Supply Curve Market Supply Curve: A curve that shows us the total quantity of goods that their suppliers are willing to sell at different prices. Example: Supply Curve: Q S = P a) Supply of wheat if P = $2 Q S =.15 (2) = 2.15 P = $3 Q S =.15 (3) = 3.15 b) Sketch the supply curve Q S = P and solving for P, we get P = Q S 0.15 So the slope = 1 (coefficient of Q S ) Intercept =

5 Equilibrium As you might guess, the market equilibrium in a perfectly competitive market is the intersection of the supply and demand curves. That is, in equilibrium, a perfectly competitive market will set a price and quantity such that there is no excess supply and no excess demand, hence demand equals supply. Example: Demand Curve: Q d = 500 4P Supply Curve: Q S = P a) Let us sketch these curves on the same graph with quantity on the horizontal axis and price on the vertical axis. Inverse Demand Curve P = (500/4) (Q d /4) Inverse Supply Curve P = (Q S /2) + 50 When P=0, Q d = =500 b) At what price and quantity do you reach equilibrium? Q S = Q d 500 4P = P 600 = 6P 100 = P And then take this p=100 and plug it into either the demand or supply curve to find the equilibrium quantity Q S = 500 4(100) = 100 And so, equilibrium occurs at p=100 and Q=100 An Increase in Demand, for any given price 4

6 An increase in demand as the one depicted about can originate from an increase in income, or in consumer s preference for the good. For any given price, the quantity that consumers demand has now gone up. You can visually see that by extending a long horizontal dotted line which maintains your focus on a given (fixed price). The point where the dotted line crosses each demand curve represents the quantity demanded. A decrease in supply, for any given price 5

7 A decrease in supply might originate from an increase in production costs, which lead producers of the good to supply lower amounts of the good at any given price. [Follow similar graphical representation as above] Example: Market for Aluminum Q d = P + 10I where P=price and I=income Q S = P a) Equilibrium when I = 10 Plug in I=10 into Q d to get Q d = P + 10(10) = P Now equate Q d to Q S, P = P 1000 = 100P 10=P Q S = (10) = 100 b) Equilibrium when I = 5 Plug I = 5 into Q d to get Q d = P +10(5) = P Now equate Q d to Q S, P = P P 9.5 = P Q S = (9.5) = 75 P Summarize P=10 Supply P= Q 6

8 A change in both the demand and supply curve A decrease in demand and an increase in supply: Unambiguously produce a reduction in the equilibrium price; but the effect on the equilibrium quantity is more ambiguous. In the figure, the outward shift in the supply curve dominates the inward shift in the demand curve, producing an overall increase in the equilibrium quantity. Otherwise, the equilibrium quantity would decrease. Price Elasticity of Demand The price elasticity of demand measures the sensitivity of the quantity demanded to price changes. Price Elasticity of Demand: A measure of the rate of percentage change of quantity demanded with respect to price, holding all other determinants of demand constant. Or, it is the percentage change in quantity demanded (Q) brought about by a 1% change in the price. When P=10, quantity is Q=50, When P=12, quantity is Q=45 (So Q = -5) % % Example:, Remember, (P/Q) represents the initial price and quantity 7

9 Why is the elasticity of demand always negative? Because ( Q/ P) is the slope of the demand curve, which is negative by the law of demand and (P/Q) is always positive because neither P nor Q can ever be negative. Example Let s find the elasticities for different prices along the demand curve Q = 100 2(P), first when o A) P=40, so Q= 20 o B) P=25, so Q= 50 o C) P=10, so Q= 80 Remember, the demand curve is in the form Q = a bp, so the elasticity equation is є d = -b (P/Q) A) Є d = -2 (40/20) = -4 B) Є d = -2 (25/50) = -1 C) Є d = -2 (10/80) = -(1/4) Vertical Intercept: є d = -2 (50/0) = - Horizontal Intercept: Є d = -2 (0/100) = 0 Elasticity of demand in the linear demand curve, Q = a-bp 8

10 Usual mistake: say that the price-elasticity of demand is equal to the slope of the demand curve. NO! We just saw that a demand curve with a constant slope (-b) can have different price-elasticities of demand, depending on the price at which the elasticity is evaluated. Why do we make things so difficult? Wouldn t it be easier to simply talk about the slope of the demand curve, rather than the price-elasticity? The reason we use price elasticity of demand (and not simply the slope of the demand curve) is because by using the former we can produce a unit-free measure of how sensitive is the demand curve to changes in prices. Indeed, note that units cancel out when you use the formula of price-elasticity of demand, but they wouldn t if you were simply using the slope of the demand curve. Different Types of Elasticities of Demand As we have seen, elasticities show us how one component of a demand equation changes with another component. So far we have only compared how quantity demanded varies with price, in order to see how price sensitive certain commodities are. This analysis, however, can be extended to compare more than just quantity with price. As we will show, we can compare changes in quantity with changes in income, or even with the price of other goods. Constant Elasticity Demand Curve: A demand curve of the form Q = ap -b where a and b are positive constants. The term b is the price elasticity of demand along this curve. If we take the derivative of Q with respect to P, we can see that the elasticity is simply the b exponent Income Elasticity of Demand: The ratio of the percentage change in quantity demanded to the percentage change in income, holding price and all other determinants of demand constant.,, Cross-Price Elasticity of Demand: The ratio of the percentage change of the quantity of one good demanded with respect to the percentage change in the price of another good. For example, research shows that a 1% increase in the price of the Nissan Seutra cause a 0.454% change in the quantity demanded of the Ford Escort Coke vs. Pepsi 9

11 Coke Pepsi Price Elasticity of Demand є Q,P Cross-Price Elasticity of Demand Income Elasticity of Demand SO a 1% in the price of Coke causes a 1.47% drop in the Q d of coke, but a 0.52% increase in the Q d of pepsi. This shows us that they are substitutes, so as the price of one rises consumers will demand less of that product and more of its substitute. Price Elasticity of Supply This is a very similar analysis to price elasticity of demand, except we are seeing how supply reacts to a 1% increase in the price. Simply, this elasticity will tell us how sensitive quantity supplied is to the price of that good., Fitting Linear Demand Curves Now that we know how a demand curve is constructed and how to use its various parts to analysis the relationship between quantity demand, quantity supplied, and price, we can work backwards to actually construct a demand curve. More simply, if we know the prevailing market Q and P and є Q,P, we can have all the components we need to construct the actual demand curve equation of Q d = a - bp 1) Є Q,P = -b(p/q) So b = -є Q,P (Q/P) 2) Q = a bp So a = Q + bp a = Q + (-є Q,P (Q/P))P a = Q + (-є Q,P (Q)) a = (1 -є Q,P ) Q 10

12 CHAPTER 3 - Consumer Preferences and Utility Bundle (food, clothing)- each bundle/basket is represented by a point on the graph below. There are three assumptions that must hold true in consumer preference theory. Consumer preferences: These three preferences must hold true and 1. Complete ability to compare, either A >B or B >A, or A~B (in other words the consumer has to state their preferences. Whether they prefer A over B, or B over A or if they Rationality both make them equally happy.) 2. Transitive if A >B and B >C, then A >C (the choices that the consumer makes must be consistent with each other. Explaining the symbols above, the consumer prefers A over B, and B over C, such that the consumer should prefer A over C) Example of non-transitive preferences: I would rather have two cookies than a glass of milk. And I would rather have a glass of milk than a slice of pizza. However (puzzlingly) I would rather have a slice of pizza than two cookies. 3. Nonsatiated (more is better)- a fairly easy concept to understand because a consumer should prefer two hot dogs to one, and three hot dogs to two. As consumers we want more. Ranking Systems: There are two type of ranking systems that the book references. 1

13 1) Ordinal Ranking: allows us to gather information about the order in which a consumer ranks a bundle. We can know that a consumer prefers basket A to B but not how much more they prefer it. 2) Cardinal Ranking: allows us to answer the intensity at which a consumer prefers one bundle to another. For example, I like basket A twice as much as basket B. It is usually hard for consumers to articulate their intensity but both ranking systems are important to recognize. Utility Function: measures the level of satisfaction from consuming different bundles. It represents the consumer s preferences. The unit of measurement is Utils, which does not have a real life translation. Instead of considering the absolute or numerical value for the level of satisfaction we compare relative values. For example, the consumer receives more utility from consuming an orange than from consuming an apple. Different consumers may have different utility functions. Thus knowing that John receives 5 Utils from consuming an apple and Jill receives 6 Utils does not give us any useful information. If we knew John received 7 Utils from consuming an orange we could say that John would prefer to consume an orange more than an apple. But again we can only compare relative values. We can think of Utils as a consumers level of happiness in consuming one good. Let s begin with just one good, u(y)=(y) 1/2. This utility function represents consumer preferences that satisfy: 1) Completeness 2) Transitive For any A, B we have u(a) or u(b) that indicates either A >B, A <B, or A ~B. For example what if: A=1, B=4, C=5, then u(a) = (1) 1/2 = 1 u(b) = (4) 1/2 = 2 u(c) = (3) 1/2 > 2 Thus if u(a) > u(b) and u(b) > u(c), then we have that u(a) > u(c). Therefore the preferences are transitive because if A >B and B >C then A >C 3) Non-satiation (more is better) Indeed is satisfied since u(c) > u(b) > u(a) 2

14 Marginal utility: rate at which utility changes as consumption increases. Often thought of as, how much better off we would be if we received one more of something. In mathematical terms as you will see below, the marginal utility is the first derivative of the utility function. MU y = du/dy = change in u/change in y Each representation is equivalent. MU y = u/ y = U(y)/ y MU y (at y=4)=slope of U at y=4 Example: If u(y)=(y) 1/2, then u/ y = ½ y 1/2-1 = ½ y -1/2 = ½ (1/y 1/2 ) = 1/(2 y 1/2 ) 3

15 Diminishing marginal utility: additional units add less utility ( U smaller and smaller) but we will assume that U > 0 always. That is, more is always better, although it is not as good as the last unit consumed. There are three important points in mind when drawing total and marginal utility curves. 1) Total Utility and marginal utility cannot be plotted on the same graph, because the Y-axis variable differs on each graph. 2) The marginal utility function is the slope of the total utility function. 3) The relationship between total and marginal utility holds for other measures in economics that will be discussed later on. For example: copies of the same DVD movie. When you get the first one you re very excited and value the DVD movie greatly. When you get a second one you re not as happy because you already own a copy of the DVD. The property that you receive less utility from the 2 nd copy of the DVD than the 1 st copy and less utility from the 3 rd copy than from the 2 nd copy is called diminishing marginal utility. Is more always better? More than one good Example Utility function: U(x,y) = (x y) 1/2 = x 1/2 y 1/2 If x = 2, y = 8 then we plug into the utility function to find (2 8) 1/2 = (16) 1/2 = 4 utils. In the 3D graph below points A, B, and C all represent the same level of utility x = 4, y = 4 4 utils x = 8, y = 2 4 utils x = 2, y = 8 4 utils 4

16 MU x = U/ x y is held constant = U(x,y)/ x y is held constant MU y = U/ y x is held constant = U(x,y)/ y x is held constant This notation represents partial derivatives. When we take a derivative with respect to x we must hold y constant. For example if x is apples and y is oranges. We would take a partial derivative with respect to x in order to find out how much utility increases from consuming one more apple. We must hold the number of oranges consumed constant because consuming more oranges would also cause utility to increase. Calculus example: Then MU x = U(x,y)/ x = ½ x -1/2 y 1/2 = 1/(2x 1/2 ) y 1/2 MU y = U(x,y)/ y = ½ x 1/2 y -1/2 = 1/(2y 1/2 ) x 1/2 Learning-by-doing 3.1 U(x,y) = (x y) 1/2 1. Satisfies more is better? 2. Satisfies diminishing marginal utility? 1. More is better can be checked in two different ways: a) Increase in x increase in U and increase in y increase in U b) MU x > 0 and MU y > 0 for any x,y > 0 2. Diminishing marginal utility just tells us that the additional utility we get from consuming amount of goods is smaller and smaller. The additional utility we get from consuming additional goods is MU x and MU y. So we notice that: 5

17 MU x is decreasing in x (x is in the denominator) MU y is decreasing in y (y is in the denominator) Mathematically we may take a derivative twice to check for diminishing marginal utility. a) MU xx < 0 and MU yy < 0 for any x,y > 0 Indifference curves: curves connecting consumption bundles that yield the same level of utility. Properties: 1) When the consumer likes both goods (MU x >0, MU y >0), indifference curves are negatively sloped (Figure) 2) Indifference curves cannot intercept. They cannot intersect because if they did, then a bundle could creat two different levels of utility which would destroy rational thinking. In the Figure the ICs increase to the Northeast (upper right) 3) Every consumption bundle lies on one and only one IC (Figure) 4) ICs are not thick They are not thick because then it would violate the more is better assumption if a bundle with more of one good and a constant amount of the other would create the same utils. 6

18 MRS (marginal rate of substitution) The rate at which a consumer will give up one good to get one additional unit of another good, keeping his or her utility level constant. In our earlier example the consumer gets the same utility from 4x and 4y as she does from 8x and 2y. This in the second situation she d be exactly indifferent between a trade of 4x for 2y. In this situation we would say the MRS of x for y is 4 for 2, or 2 for 1, or just 2. Graph depicting MRS Moving along the curve we see if the consumer has a lot of good x she will trade a lot of good x for a single unit of good y. In the same way if the consumer has a lot of good y she will trade a lot of good y for a single unit of good x. This is known as diminishing MRS. y/ x is the rate at which the consumer is giving up y in order to get an additional unit of x. MRS x,y = - slope of I.C. = MU x /MU y Totally differentiating Diminishing MRS Initially you are willing to give up many glasses of lemonade (y) for m additional hamburgers (x). When you have few glasses of lemonade and many hamburgers you will not be willing to give up as many (or none) glasses of lemonade in order to get an additional hamburger. 7

19 Totally differentially x and y (since I am lowering y in order to get more of x), we obtain Δu = MU x Δx + MU y Δy MU x Δx = MU y Δy MU x MU y = Δy Δx -MRS=slope of Ind. Curve MRS= -slope of Ind. Curve ICs are bowed in toward the origin. Application 3.2. Demand for attributes in cars U(x,y) = (x y) 1/2 where x is horsepower y is gas mileage MRS x,y represents how a typical consumer will be willing to forgo horsepower x in order to get an additional mile per gallon (y). In 1969, MRS x,y =

20 In 1986, MRS x,y = 0.71 The decrease means that people became more willing to give up horsepower in order to get an additional mile per gallon (y). Learning-by-doing 3.3 U(x,y) = x y MU x = U(x,y)/ x = y MU y = U(x,y)/ y = x a) Draw the IC correspondingly to U 1 = 128 x y = 128 in many of its combinations For example, G: x = 8, y = 16 H: x = 16, y = 8 and connect them I: x = 32, y = 4 Does I.C. intersect either axis? No, if IC intersects x-axis, then y = 0, U 1 = y-axis, then x = 0, U 1 = Does IC indicate that MRSx,y is diminishing? Yes, since IC is bowed in toward the origin. 9

21 Note that MRS x,y = MU x /MU y = y/x so we more from left to the right ( x), we get smaller ratios, smaller MRS x,y (diminishing). b) Similar, but for U 2 = 200 Not graded: Learning-by-doing 3.4. Increasing MRS x,y U = Ax 2 + By 2 MRS x,y = MU x MU y MU x = 2Ax MU y = 2By = 2Ax 2Ay = Ax, which increases in x By MRS = 2Ax/2By = Ax/By, which increases in x Special Utility Functions-There are certain goods that will create unique utility functions. These functions are important to recognize because they will not always have diminishing MRS, and it is important to check what type of utility function your working with. Perfect Substitutes: Aquafina versus Dasani bottled water or Kingston versus Scandisk memory sticks. Close substitutes: butter versus margarine or coffee versus black tea. MRS B,M = MRS M,B = 1 but it can be any constant Example 10

22 U = ab + am, then MU B = a and MU M = a Hence, MRS B,M = MU B /MU M = a/a = 1 Then, slope of IC = -1 Example: Pancakes and Waffles U = P + 2W, MUp = 1 and MU W = 2 The consumer is always willing to trade 2 pancakes for 1 waffle. MRS P,W = MUP/MU W = ½ slope of IC = -1/2 Constant slope of IC- perfect substitutes will always have a linear indifference curve and are the simplest to sketch. Constant MRS (not diminishing as in previous examples) Similarly, U = ax + by, MRS x,y = a/b Perfect compliments left shoe and right shoe (consumer wants them in fixed proportions), or two scoops of peanut butter to one scoop of jelly for the perfect PB&J sandwhich. Example: U = 10 min(r,l) 11

23 G R = 2 and L = 2, then U 2 = 10 2 = 20 H R = 3 and L = 2, then U 2 = 10 2 = 20 (consumption choices should be at the kink) Slope of IC (MRS) is: Zero at the flat segment of the IC Infinity at the vertical segment of the IC Undefined at the kink of the IC (infinitely many slopes Cobb-Douglas U = (x y) 1/2 and U = x y are examples of Cobb-Douglas utility functions Generally, U = Ax α y β Properties: if α = β = ½, then U = Ax 1/2 y 1/2 = A(xy) 1/2 if α = β = 1 and A =1, then U = xy 1) MU x > 0 and MU y > 0. That is, more is better is satisfied MU x = U(x,y)/ x = Aαx α-1 y β > 0 for any x MU y = U(x,y)/ y = Aβx α y β-1 > 0 for any y 2) Since MU x > 0 and MU y > 0, then IC is downward slopping (recall property 1 of ICs) [what about the case in which the consumer dislikes some goods, MU x < 0? Review Session] 12

24 3) Diminishing MRS x,y : MRS x,y = Aαx α-1 y β / Aβx α y β-1 = α/β x α-1-α y β-(β-1) = α/β 1/x y = α/β y/x The movements from left to right ( x) will shrink this ratio, and reduce MRS x,y Diminishing MRS x,y Quasi-Linear preferences Used in economic applications for situations in which the amount of a commodity (such as toothpaste or garlic) doesn t change very much to income. ICs are parallel displacements to each other Same slope, same MRS between goods even if income increases General form U(x,y) = ν(x) + by (linear in y, but not generally linear in x) where ν(x) is a function that increases in x, such as ν(x) = (x) 1/2 or ν(x) = x 2. where b > 0. For example you make $400 dollars and spend $5 on toothpaste and $395 on pizza. With quasilinear preferences if got a better job and made $500 dollars you would still spend $5 on toothpaste and increase pizza purchases to $

25 Application 3.4 Pet Rock Fad Pet Rock Fad is not completely over For more practice on derivatives see: APP. A3 and A4, pp Another way to find MRS U = MU x x + MU y y 0 = MU x x + MU y y -MU x x = MU y y -MU x / MU y = y/ x -MRS = slope of IC or MRS = - slope of IC 14

26 CHAPTER 4 Consumer Choice Chapter 4 introduces us to consumer choice. That is, we are introduced to how consumers choose between different bundles of goods in order to maximize their utility. As we will see, this requires that we know not only consumer preferences but also the constraints that any consumer is bound by, the budget constraint. Using both preferences and constraints we can deduct which baskets a utility-maximizing will choose in a given situation between various goods. Preferences Preferences tell us how a consumer ranks some bundle compared to another bundle, and the utility function representation tells us how much utility a consumer derives from different bundles. Remember also that from a utility function we can create our indifference map, which will later be crucial in choosing an optimal bundle. However, the bundles are not equally costly, and some may be unavailable at the current prices and income. Hence, in order to understand how a consumer chooses among different goods we need to take into account both consumer preferences and budget constraint. Simply put, given that a consumer faces a certain budget constraint (i.e. he or she only has so many resources by which to attain goods), what bundle is the affordable and utility maximizing? Budget constraint The budget constraint defines the set of baskets that a consumer can purchase with a limited amount of income. Graphically, the budget constraint will be represented by a specific region on the graph where all bundles in that region are affordable. To find the limit (outermost edge) of our budget constraint we must construct a budget line (B.L.). The B.L. indicates all of the combinations of a given set of goods that a consumer can afford to consume if she spends all of her money. x food, p x y clothing, p y I income Budget Line: p x x + p y y = I where p x x total expenditure on food p y y total expenditure on clothing Combination of x and y that the consumer can buy when spending her entire income Example: Let s use the example from the book where Eric is faced with the decision of purchasing some combination of food and clothing. Let p x =price of food, p y =price of clothing, and I=Eric s income over the period. 1

27 p x = 20, p y = 40, I = 800 Budget line Unattainable at the current prices and income p x x + p y y = 20μ μ15 = $1,000 > $ 800 p x x + p y y I meaning the you can only spend as much money as you have this is called the budget constraint (B.C.). Notice that both the x and y intercepts are simply Eric s income divided by the price of the good on that respective axis. Each intercept therefore represents how much food or clothing Eric could consume if he were to spend his entire income on that one good. Slope of Budget Line (B.L.) p x x + p y y = I p y y = I - p x x y = I/p y (p x /p y ) x Notice that this equation resembles the general equation of a straight line y = b + m x, where b = vertical intercept, m = slope Hence, the slope of the B.L. = (p x /p y ) and indicates, when moving from left to right along the budget line, how many units of y the consumer must give up to increase his consumption of x in one unit. An easy way to remember the slope is simply as the negative of the price ratio. But why is the slope negative? Because to get more of good x while staying on the same budget line you must give up some of good y. If the slope were positive, you would be getting more of good x by also getting more of good y, which intuitively doesn t make sense when we are constrained by a budget. Example (p x /p y ) = - 20/40 = -1/2, and the consumer must give up ½ units of clothing (y) in order to obtain an additional unit of food (x). This ratio is constant because prices are constant. Income is the symbol for change in 2

28 Unchanged slope since (p x /p y ) = - 20/40 = -1/2 Hence, new B.L. is parallel to old B.L., but shifted outward. The set of available choices increases with the increases in Income. Why is the B.L. still parallel after an increase or decrease in income? Remember that the slope of the B.L. is the negative of the price ratio between food and clothing. A shift in income has no effect on the prices of the two goods and thus leaves the slope of the line unchanged. p x What if the price of good x increases from 20 to 25? The consumer must give up 5/8 units of y to obtain one more unit of x. New slope = (p x /p y ) = -25/40 = -5/8 since -5/8 > -1/2, BL 2 is steeper than BL 1. In this case where prices are changed, the slope of the B.L. is inevitably changed as well, as shown in the above graph. Now, to stay on the budget line as Eric moves from left to right, his tradeoff between clothing and food is now occurring in a different ration than before the change in the price of food. 3

29 Good news or Bad News? What if both income and all prices are doubled? Intuitively it would seem that if both prices and income double, the two effects would cancel each other out and Eric would be left no better or worse off than before the change. Mathematically, this proves to be the case Vertical intercept 2I/2p y = I/p y Horizontal intercept 2I/2p x = I/p x Slope = - 2p x /2p y = - p x /p y The 2s cancel out, thus no element of B.L. changes PUTTING PREFERENCES AND B.L. TOGETHER: Having now identified what a consumer is constrained by when making a choice between bundles, we can put together both his preferences and constraint to find the optimal bundle. This process is central to intermediate micro and will come up time and time again throughout the course, so let s learn it well Optimal choice Consumer choice of a basket of goods that 1) maximizes utility 2) is affordable (in the B.C.) No points inside B.L. (utility could be increased) No points outside B.L. (unaffordable) Intuitively, an optimal consumption bundle will not only maximize a consumer s utility, but it will also be affordable. These very simple conditions are what we translate into mathematical condition by which to discover the optimal bundle. Also note that a consumer would not choose any bundle inside or outside of the budget line as his optimal bundle. Any point outside the line would be unaffordable for the consumer, and any point inside the line will not maximize utility because it would leave income to be spent which would further increase his utility (remember that this consumer spends all of his income to maximize his utility). Here is our mathematical expression of the two conditions for optimization Max U(x,y) reaches the highest I.C. max U(x,y) means x and y are the choice variables x,y x,y subject to p x x + p y y = I that still belongs to the set of affordable baskets(on or inside B.L.) 4

30 Looking at the above graph, why are other baskets not optimal? D unaffordable (outside of B.L.) C money unspent could increase our utility by spending it (generally, it could be future consumption of the good, or savings) B lower I.C. than at A (less utility at same cost) Tangency condition at the optimum, A We refer to the condition for the optimal bundle as a tangency condition. This is because at the optimum bundle A the I.C. and the B.L. are tangent, that is, they have the same slope. Knowing this allows us to simply express that optimum condition as Slope of B.L. = slope of I.C. at A (p x /p y ) = - MU x /MU y = MRS x,y (remember MRS is the slope of I.C.) p x /p y = MU x /MU y MU y / p y = MU x / p x Extra utility per dollar spent on y = Extra utility per dollar spent on x Same BANG FOR THE BUCK spent across different goods. The expression MU y / p y = MU x / p x tells us that at the optimum bundle A both the marginal utility per dollar of good x and the marginal utility of good y are equal. So, at this bundle each dollar spent on each good result in the same level of utility received from consuming that good. If these marginal utilities per dollar were different this bundle would not be optimal. Why is this the case? Because if one good had a higher MU per dollar than it wouldn t make sense to consume that bundle. It would make sense to consume more of the good with the higher MU per dollar, thus increasing your utility, and thus changing the bundle. 5

31 Example U(x,y) = xy MU x = y, MU y = x I = 800, p x = 20, p y = 40 B.L. 1) p x x + p y y = I 20x + 40y = 800 Tangency Condition 2) MU x /MU y = p x /p y y/x = 20/40 2y = x Plug (2y=x) back into the B.L. equation so that we only have one variable 20(2y) + 40y = y = 800 y = 10 x = 2y = 2μ10 = 20 NOTE: In an optimization problem where the MU s are both positive, the optimal bundle we find at the point of tangency is referred to as an interior optimum. An interior optimum is simply an optimal basket at which the consumer will be purchasing positive amounts of all commodities. Point B in the graph was not optimal since: MU y / p y = MU x / p x y/20 x/40 16/20 8/ In particular, MU y / p y > MU x / p x x Thus, spending less Money on y and more Money on x increases total utility (different marginal utilities per dollar). USING THE LAGRANGE MULTIPLIER METHOD OF CONSTRAINT MAXIMIZATION IN THE CONSUMER CHOICE PROBLEM Using the Lagrange method, we can mathematically show this tangency condition as the optimal bundle. Here we again see that the optimal bundle is where MU x /P x =MU y /P y. Max U(x,y) x,y subject to p x x + p y y = I (x,y;l) = U(x,y) + l [I - p x x - p y y] F.O.C.s: / x = U(x,y)/ x - lp x = 0 MU x / p x = l (1) MU x 6

32 / y = U(x,y)/ y - lp y = 0 MU y / p y = l (2) MU y / l = I - p x x - p y y = 0 (3) From (1) and (2), we have MU y / p y = MU x / p x (same bang for the buck ), and from (3) we have the B.L. These are the same two conditions we use to find optimal consumption choice for the consumer. More on maximization problems: APP.A5 pp constrained optimization: APP.A7 pp and Lagrange multipliers: APP.A.8 pp Example: Cobb-Douglas Utility Here we use the Lagrange method with a Cobb-Douglas Utility Function. U(x,y) = x a y 1-a subject to p x x + p y y = I (x,y;l) = x a y 1-a + l [I - p x x - p y y] F.O.C.s: / x = ax a-1 y 1-a MU x - lp x = 0 ax a-1 y 1-a (1/x) = lp x U(x,y) au(x,y) / xp x = l (1) / y = (1-a)x a y 1-a-1 - lp y = 0 (1-a)x a y -a (1/y) = lp y MU y U(x,y) (1-a)U(x,y) / yp y = l (2) / l = I - p x x - p y y = 0 p x x + p y y = I (3) From (1) and (2) we have au(x,y) / xp x = (1-a)U(x,y) / yp y xp x / yp y = a/1-a Or alternatively, p x x = (a/1-a ) p y y Substituting this result into (3), 7

33 (a/1-a ) p y y + p y y = I (1+ a/1-a ) p y y = I (1 - a + a/1-a ) p y y = I y = (1 - a) I/p y And similarly for x. Starting with p y y = (1-a/a ) p x x, and plugging it into (3), (1-a/a ) p x x + p x x = I (1+ 1-a/a ) p x x = I (a+ 1-a/a ) p x x = I x = (I/p x ) a Budget shares The proportion of total income spent on each good. Budget share of x: p x x/i = p x (I/ p x a) / I = I a / I = a Budget share of y: p y y/i = p y (I/ p y (1-a)) / I = I (1-a) / I = (1-a) Notice from an empirical point of view: if you observe a consumer spending a % of his wealth on x and (1-a) % on y, you know his utility preferences function from consuming these two goods. Expenditure minimization problem (EMP) alternative way to think about optimality This is simply another way to look at the optimization conditions. Whereas before we were maximizing utility given a certain budget constraint, here we approach optimization by minimizing expenditure subject to a certain utility level. This shows that the answer you reach through the tangency condition is the cheapest bundle at a certain utility level. As you can see with bundle A above, it is the least expensive and in fact the only affordable bundle that reaches the utility level of U 2. This approach is mathematically expressed as Min Expenditure = p x x + p y y (x,y) Subject to U(x,y) = U (x,y) (given level of utility) U(x,y) = x a y 1-a Budget p x x + p y y 8

34 Min p x x + p y y x,y subject to U(x,y) = x a y 1-a Ū (x,y;l) = p x x + p y y + l [Ū - x a y 1-a ] F.O.C.s: / x = p x - l ax a-1 y 1-a = 0 p x = lax a-1 y 1-a (1/x) = U(x,y) p x x / au(x,y) = l (1) / y = p y - l (1-a)x a y 1-a-1 = 0 p y = l (1-a)x a y -a (1/y) U(x,y) p y y/ (1-a)U(x,y) = l (2) / l = Ū - x a y 1-a = 0 (3) From (1) and (2) we have p x x / au(x,y) = p y y/ (1-a)U(x,y) (1-a)p x x = a p y y xp x / yp y = a/1-a Hence, p y y= p x x (1-a)/a, or y = (p x x/ p y ) ((1-a)/a) Substituting this result into (3), we obtain, Ū = x a [(p x x/ p y ) ((1-a)/a)] 1-a Ū = x a x 1-a [(p x / p y ) ((1-a)/a)] 1-a Solving for x, x = Ū/ [(p x / p y ) ((1-a)/a)] 1-a = Ū [(p x / p y ) ((1-a)/a)] a-1 And similarly for y, we start from x = (a/1-a) ( p y / p x ) y. Substituting it into (3). We have Ū = [(a/1-a) ( p y / p x )y] a y 1-a = [(a/1-a) ( p y / p x )] a y a y 1-a Solving for y, y = Ū/ [(a/1-a) ( p y / p x )] a = Ū [(a/1-a) ( p y / p x )] -a 9

35 p y y = (1-a/a ) p x x, and plugging it into (3), (1-a/a ) p x x + p x x = I (1+ 1-a/a ) p x x = I (a+ 1-a/a ) p x x = I x = (I/p x ) a Corner Points Solutions So far we have been dealing with optimal bundles where the consumer purchases positive amounts of all commodities (interior optimum). However, in the real world this is not always the case. Many times a consumer will not consume a good at all, and thus, we need a way to find optimal bundles in these situations. This search will take us to what we refer to as corner point solutions: a solution to the consumer s optimal choice problem at which some good is not being consumed at all, in which case the optimal basket lies on an axis. Example: U(x,y) = xy + 10x MU x = y + 10 MU y = x I = 10, p x = 1, p y = 2 1) x + 2y = 10 2) MU x / MU y = p x / p y (y+10)/x = ½ 2y + 20 = x (2y+20) + 2y = 10 4y = -10 y = -2.5 (NO!! y cannot be negative, thus we have a corner solution where y = 0) x = 2y + 20 = = 5 Attempting to find the optimal bundle via the tangency condition gives us a negative amount consumed of good clothing. Since we can obviously not consume negative clothing, this simply informs us that our optimal bundle is going to be where we consume 0 units of clothing. This is a corner point. Then, consumer wants to increase his consumption of x as much as possible. That is, the consumer spends all his income in x, I/ p x = 10/1 = 10, and no income in y. 10

36 The graph shows our corner solution where all income is spent on good x. In this situation, the MU per dollar of food is always higher than the MU per dollar of clothing, so you will this consumer will continually substitute clothing for more food until he reaches 0 units of clothing (i.e. all his income is spent on food at bundle R). Corner Points with Perfect Substitutes What if a consumer is always willing to substitute clothing for food, but in a constant ratio? This is a case of perfect substitutes. Instead of a curved I.C., the I.C. s will be straight line, since the slope is constant. MRS choc.,vanilla = MU c /MU v = 2 (constant MRS because of Perfect substitutes) p c /p v > MU c /MU v MU v /p v > MU c /p c and as a consequence the consumer should only buy vanilla ice cream. Once again, the MU per dollar for both goods is never equal, which will in turn lead the optimal bundle to a corner point where 0 units of the less preferred good is consumed. In other words, the consumer only likes chocolate ice cream twice as much as vanilla ice cream, but vanilla ice cream is three times cheaper than chocolate ice cream. p c /p v = 3 Don t confuse this Indifference curve with B.L. [What if price ratio was lower than MRS? Then BL becomes flatter than I.C., and the consumer buys only chocolate] 11

37 Coupons vs cash subsidies Occasionally governments and other institutions want to increase the consumption of a particular good. They can do this by issuing coupons, which can only be spent on the good of interest, or through cash subsidies, which are lump sum payments that can be spent on any good (but which the issuer hopes will be spent on the good of interest). To isolate the good of interest, say housing, we lump all other goods into a variable we refer to as a composite good. A composite good is simple a good which represents the collective expenditures on every other good except the commodity being considered. Cash subsidies: B.L. moves from KJ to EG Vouchers (coupon): B.L. moves from KJ to KFG In the above graph, both methods of either issuing a coupon or a subsidy push the consumer from I.C. U 1 to U 2. Thus, both achieve the desired effect of raising housing consumption to h B. However, this is not always the case. Notice that if the U 2 I.C. was tangent to the segment EF, the coupon would not allow the consumer to move to a higher I.C. while the subsidy would. Joining a Club You pay a lower price if you are a club member, however you must pay to join the club. Is this trade off worth joining the club? Let s look at an example I = 300 p x = $20 p y = $1 12

38 I = (fee) = $200 p x = $10 p y = $1 Before joining: I= 300, p x = $20 I/ p x = 300/20 = 15 units p y = $1 I/ p y = 300/1 = 300 units The consumer chooses basket A. After joining: I = (fee) = $200, p x = $10 I/ p x = 200/10 = 20 units p y = $1 I/ p y = 200/1 = 200 units Flatter BL, and the consumer selects basket B. Basket B is on a higher I.C. than basket A, and therefore this consumer is better off after joining the club. A similar analysis is applicable for 1) cell phone service (higher monthly subscription charge in order to have lower price per minute in calls) 2) gym, golf club Example: Sprint Cellular in 2006 Plan A: $ 30 monthly charge, 200 minutes, and $0.10 beyond these minutes. Plan B: $60 monthly charge, 1,000 minutes, and $0.10 beyond these minutes. Drawing Plan A: income is $120, but the consumer pays $30 on monthly charge 200 first minutes are free, then the slope of BL is 0.1 If he didn t get free minutes, the highest amount of minutes he could consume is 90/0.1 = 900, but he gets 200 min.for free = 1,100 MRT is the BL A 13

39 Drawing Plan B: income is $120, but the consumer pays $60 on monthly charge 1,000 first minutes are free, then the slope of BL is 0.1 If he didn t get free minutes, the highest amount of minutes he could consume is 60/0.1 = 600, but he gets 1,000 min.for free = 1,600 MRT is the BL B If the consumer were to spend all income on cell phone minutes he would choose Plan B to receive more minutes. Borrowing and Lending So far we have considered the consumer s income as fixed within a time period. However, this is not always the case in the real world. As we well know as college students, consumers can both borrow and lend money. Borrowing and lending is really a tradeoff in consumption. In borrowing you consuming more today by consuming less tomorrow. In lending you are consuming less today so that you can consume more tomorrow. But how does this affect our optimization problem? Without borrowing and lending consume I 1 today and I 2 tomorrow (as we have considered thus far) With borrow and lending (allowing for) If you don t consume anything today, then you have I 2 + I 1 (1+r) tomorrow because you lent some consumption today to be paid back with interest (1+r) tomorrow, where r=interest rate earned by loaning the money If you don t consume anything tomorrow, then you can consume: I 1 + I 2 /(1+r) today by borrowing How to find a slope? Our x-axis here will be consumption today while the y-axis will represent consumption tomorrow Intuitively, the price of giving up one unit of consumption tomorrow (y axis) in order to gain one more unit of consumption today (x axis) is measured by the opportunity costs of every dollar, (1+r). How to prove it more formally? We know that the B.L. is y = a + m x y = I 2 + I 1 (1+r) + m x a and, in addition, y = 0 and x = I 1 + I 2 /(1+r), Then, 0 = I 2 + I 1 (1+r)+ m ( I 1 + I 2 /(1+r)) m = [I 2 + I 1 (1+r)]/-[ I 1 + I 2 /(1+r)] = [I 2 + I 1 (1+r)]/-[ (I 1 (1+r)+ I 2 )/(1+r)] = - (1+r)[I 2 + I 1 (1+r)]/ [I 2 + I 1 (1+r)] = - (1+r) 14

40 A consumer with these preferences chooses to : o Borrow today if C1B > I1 o Repay the loan tomorrow if C2B<I2 How can we draw the ind. curve of an individual that prefers to: o Save today? o Get the returns of his savings tomorrow? By drawing ind.curve where the tangency point (optimal consumption) is above the point of no-borrowing or lending (A) WHAT IF THE INTEREST RATE FOR BORROWING AND LENDING DO NOT COINCIDE? Assume I 1 = 10,000 I 2 = 13,200 15

41 Example: Lending, r L = 0.05 Next year he will have 13,200 from tomorrow s income = 13,200 10,000 (1+0.05) from savings = 10, ,700 (Vertical intercept, all of consumer s income was consumed next year) Borrowing, r B = 0.10 Today I can have : 10,000 from today s income = 10,000 13,200/ = 12,000 = 12, ,000 Where is the I.C. of a saver? Tangency point at segment AE Where is the I.C. of a borrower? Tangency point at segment AG Quantity Discounts Firms can also incentivize the consumption of a product by offering quantity discounts (i.e. discounts after you purchase over a given number of units). In this case, the budget line for a consumer will have a kink where the discounted price kicks in. Let s look at an example p y = $1 p x = $11 for the first 9 units $5.5 for all additional units Slope = - p x / p y = - 5.5/1 = -5.5 Slope = - p x / p y = - 11/1 = -11 How to find the vertical and horizontal intercepts in this case? Vertical Intercept = I = $440 P y $1 = 440 I = 440 P x 11 = 40 16

42 Horizontal intercept of BL 1 (no discounts) = Horizontal intercept of BL 2 (with discounts) = = 62 Notice that this consumer is better off with the quantity discount, and that she is consuming 7 more units than before. Frequent flyer programs are another example of quantity discounts. Revealed Preferences We have learned how to find optimal baskets for a consumer if we know her preferences and budget line. But, what if we don t know the consumer s preferences? Her choices in different settings will help us infer her preferences. That is, by looking at how she chooses bundles in different situation, we can infer which bundles she prefers to others. Note that the dark shaded region represents that the area to the northeast of any bundle will always be preferred to that bundle. Through revealed preferences we can establish if a consumer weakly prefers a bundle to another (i.e. prefers it at least as much as the other bundle when both choices are affordable), or if he strongly prefers one bundle to another (i.e. prefers the bundle more when both choices are affordable). 1) When facing BL 1, consumer chose A instead of any point inside BL 1, such as B. Then A B 2) A and C were equally costly, but he chose A. Hence A C A >B 3) Since C lies to the northwest of B, then C >B Choices: 17

43 24/2=12 Failing to max utility I= $24 ( p x, p y ) = ($4, $2) (x 1, y 1 ) = (5, 2) A ( p x, p y ) = ($3, $3) (x 2, y 2 ) = (2, 6) B Here the consumer fails to maximize utility. Let s see why 24/3=8 24/4=6 24/3=8 BL 1 : 1) when facing BL 1, the consumer chose A instead of C, in spite of being available (affordable) A C 2) Since C is to the northwest of B, C>B A >B BL 2 : 1) when facing B in spite of D was available B D 2) Since D is to the northwest of A, D>A B >A Contradiction occurs because under both budget lines A and B were affordable. If A were chosen over B under BL 1 then it should always be chosen over B whenever both are affordable consumer wasn t maximizing utility when 18

44 Choosing A within BL 1 Choosing B within BL 2 Alternative approach BL 1 : Consumer prefers basket A. Thus A >B Cost of basket A, $4 5 + $2 2 = $24 Cost of basket B, $4 2 + $2 6 = $20 BL 2 : Consumer prefers basket B. Thus B >A Cost of basket B, $3 2 + $3 6 = $24 Cost of basket A, $3 5 + $3 2 = $21 Case 1. Any point on B.L. is strongly preferred to points inside B.L. BL 2 Case 2 C>A since it is to the northwest of A B C since B was chosen when C was affordable B >A An example of choices that are inconsistent with utility-maximizing behavior BL 2 : From above, we know that B >A since A is inside BL 2 BL 1 : Additionally, A B since both were affordable when facing BL 1 Contradiction Hence, consumer is NOT maximizing utility Case 3: BL 1 : the consumer chose A when both A and B were affordable, A B BL 2 : the consumer chose B when A wasn t affordable Case 4 BL 1 : chose A but B wasn t affordable We can t infer anything from these choices because we don t know which bundle the consumer would choose if both were affordable 19

45 BL 2 : chose B but A wasn t affordable The theory of revealed preference: 1) Allows us to infer how a consumer ranks one basket of goods over others her preferences 2) We can discover when a consumer is not maximizing his utility 20

46 EconS 301 Intermediate Microeconomics Chapter 5 Optimal Choice and Demand It is important to realize that given the consumers preferences, income, and price of all goods we could determine how much they will buy. That is simply one point on their demand curve for that good. We can find more point by repeating the process by changing the price of that one good for example ice cream. Figure 5.1 Does a good job of explaining how to use indifference curves and budget lines to find the demand curve. 1

47 The curve used to find the demand curve is known as the price consumption curve (the set of utility maximizing baskets as the price of one good varies while holding constant income and the prices of the other goods). Effects of a change in Income: We can also change the income of the consumer and observe how their demand shifts based on income. Figure 5.2 displays how we use an income consumption curve to see shifting demands. From the picture above we can see the Income Consumption Curve is the set of utility maximizing baskets as income varies while prices are held constant! There is one last way of showing how a consumer s choice of a particular good varies with income and that is to draw an Engle curve (a curve that relates the amount of commodity purchased to the level of income, holding constant the prices of all goods). It is important to remember there are more than one type of good, and an Engle curve can help you distinguish which type of good the consumer is dealing with. A normal good is a good that a consumer purchases more of as income rises and an inferior good is a good that a consumer purchases less of as income rises. 2

48 The next page contains algebraically how to find the demand curve as well as the engle curve with visual representation. Finding a demand curve (Learning by doing 5.2): A consumer purchase two goods X and Y where the utility function is represented by U = xy MUx = y and MUy = x 1) px x + py y = I MU x px y px pxx 2) MRS = = = y = plug into 1 MU y py x py py px x * I px x + py = I 2px x = I x = plug into y p y 2 px I px * 2 px I y = = p 2 p y y * * The last equations ( x and y ) represent the demand curve. Given any value for Income and for prices of goods, we can find the quantity of good the consumer will purchase. Demand with corner point solution: Using the same goods from previous example but here we change the utility function such that U = xy+ 10x MU = y+ 10 and MU = x Suppose that I = 100 and p = 1 p = unknow then ( ) 1) p = 1 x+ p y = 100 x+ p y = 100 MU x px y ) MRS = = = x = ( y + 10 ) py plug into 1 MU p x p p y+ 10 py + py y = py y + 10 py = 100 y= 2 p ( ) p 2 p x y y y y x y y y x y 0 only if p p 10 p py if py <10 y = 2 py 0 otherwise Only the consumer demand positive quantities of normal goods. y y y y y y 3