Economic Markets. The Theory of Consumer Behavior

Transcription

1 Economic Markets We want to work towards a theor of market equilibrium. We alread have a prett good idea of what that is, the prices and quantit in a market are going to be set b suppl and demand. So to reach this point we need to understand: 1) Market Equilibrium (class 7) (class 9) (class 8) 2) DEMAND (Consumer Behavior) 3) SUPPLY (Theor of the firm) We will also look at some markets that do not work the wa we think the should. These would be monopolies and duopolies. In those markets the idea of suppl will be corrupted in some wa. There will be different decisions faced b someone in a monopol as opposed to someone in a competitive market. (this will be another class) The Theor of Consumer Behavior We ll be considering these topics. Overview Utilit, how people order choices. Which choice gives more utilit, a wa to order consumption. An idea of the satisfaction I receive, the benefit. Basic Concepts in terms of utilit. Simple indifference curves. Maimization of Utilit, what level of each good, given b budget constraint, is going to give me the greatest utilit? Dependent on income level and prices of goods. Find optimum level. Demand Functions, we can find solutions given an price and income. Elasticit s, if the price changes how much more or less are we going to demand of that particular good? Given a proportional change at a particular price what is the change in the amount of good we demand. This leads to an idea of luur goods and necessar goods. Also cross elasticit. If the price changes for good 1, how much does that effect our demand for good two? Remember we onl have a fied amount of dollars to spend. FNCE317 Class 7 Page 1

2 Generalization I have used frequentl the following classic tet Microeconomic Theor, Henderson and Quandt In the following developments we will be considering a two good world. We will see how these concepts are ver general. Aioms of Rational Choice Completeness, given two choices an individual has to be able to sa If A and B are an two situations, an individual can alwas specif eactl one of these possibilities: A is preferred to B B is preferred to A A and B are equall attractive Transitivit If A is preferred to B, and B is preferred to C, then A is preferred to C Assumes that the individual s choices are internall consistent Continuit If A is preferred to B, then situations suitabl close to A must also be preferred to B (there are no jumps, the choices in between for a nice orderl line) Used to analze individuals responses to relativel small changes in income and prices To construct an idea of utilit we need some idea of what we mean b rational choice. There are different was of framing these but these aioms seem to fit with the wa we see the world (although this is not a complete list). We need a wa of comparing two situations in order to construct an idea of how much we value those two situations. Given these assumptions economist can construct Utilit Functions. All this means is people are able to rank, place in order, an given choices. Rank from most desirerable to least desirerable. FNCE317 Class 7 Page 2

3 Utilit Given these assumptions, it is possible to show that people are able to rank in order all possible situations from least desirable to most Economists call this ranking utilit If A is preferred to B, then the utilit assigned to A eceeds the utilit assigned to B U(A) > U(B) The Utilit of A or B or an choice is a number. We can place a value on the preference and if A is preferred to B then the utilit of a, U(A) is greater than the utilit of b, U(B). Bear in mind this is an ordinal ranking. We cannot sa that something is twice as preferred. It is onl a ordering sstem, the distance between the two values doesn t tell ou ver much. Given these three aioms we can rank things and construct a utilit curve for an individual. Basic Concepts Utilit Functions Two commodit case U = U(, ) is the quantit of commodit consumed, It is assumed that the function U is continuous and has continuous first and second partial derivatives (which eist) It is assumed that the first partial derivatives are strictl positive Consumer will alwas desire more of each commodit The utilit if going to be a function of the consumption of two goods. If we can sa how much of an item we have we can assign a value to that consumption choice. If I increase one of the variables then the utilit should increase. Once we get to etremes then increased consumption of a particular good doesn t necessaril increase our well being (utilit). But we will not be considering these realms. FNCE317 Class 7 Page 3

4 Basic Concepts, cont d Indifference Curves Defined b the different combinations of the two commodities that ield the same utilit Higher utilit is found the further the indifference curve is from the origin Indifference curves do not intersect The ais s represent amounts of good Y and good X. The utilit curves (indifference curves) here are the curves where the utilit remains constant for the given values of and. So if we plot the value of utilit U(,) for an values or and, on the lines above the utilit will be constant. If I can receive more of both and m utilit is going to increase (for instance from utilit curve I to II). As we get further awa from the origin (move toward the northeast) the utilit is increased. This is a result of the assumption that the first partial derivative wrt an of the variables is increasing. As I increase the amount of goods I consume I assume m utilit goes up. This means that U I () < U II () < U III (). Called indifference curves because we are indifferent, in terms of happiness and utilit, to the various points on the curve. E. Hotels: = in room services, = out of room services The hotels will construct using surves how people feel about different combinations of in room and out of room services. Then the look at there competitors and where the are on these curves the have constructed (based on the services the competitor hotels offer). Now the hotel has a certain budget to spend to improve services, either in room, out of room, or both. The would construct models of different curves based on the changes the could make. The will construct a package which fits within the budget and FNCE317 Class 7 Page 4

5 brings them to a higher indifference curve than their competitors. In this wa the know that a particular combination of improvements is going to be the most beneficial. Tells them the can charge more for the rooms because the have the services customers will pa for. Eample Shell (gals) Gasoline Stations Notice that these are perfectl straight lines. This is because of tank capacit for one thing but also because we are indifference between gas from Shell and gas from Eon. As long as the price is relativel close these are substitutes. Eon (gals) Right Shoe Eample Right and left shoe. The indifference curves indicate that we cannot reach a new level of satisfaction b having more than one right shoe and onl one left shoe, there is no increase in utilit. To get to a greater level of satisfaction (utilit) I need more of each, another right and another left shoe Left Shoe This is an eample complements. I need both to increase m utilit. Here we are seeing that if I do not get the eact mi that I need then increasing one or the other does not increase m utilit. Will sta at a given sweet spot unless I can improve on both. Hotdogs and buns, hamburgers and ketchup would be other eamples. FNCE317 Class 7 Page 5

6 We usuall assume that the curves are conve to the origin. An two points on the curve joined b a straight line, conve to the origin means that the straight line is never an closer to the origin than the indifference curve. This implies that a well balanced bundle of goods is alwas preferred to having an etreme of one good and onl a little of the other. A bundle is a combination of the two goods. The graph of ½ the Y range and ½ the X range is showing that utilit of the balanced bundle is greater than the indifference curve. Both of these points lead to the conclusion: IF I JOIN TWO POINTS ON THE INDIFFERENCE CURVE THE RESULT WILL ALWAYS GIVE ME HIGHER INDIFFERENCE. Joining two points will never give less utilit (given the framework of these assumptions). WE ARE SEEING HERE THAT PEOPLE DO NOT LIKE EXTREMES, GIVE A CHOICE OF TWO PARTICULAR GOODS THEY WILL ALWAYS PREFER A BALANCED MIXTURE. FNCE317 Class 7 Page 6

7 NOTATION The following notation will be use and all mean the same. U U du d d U d 1 2 U d U d U d Basic Concepts, cont d Rate of Commodit Substitution The total differential of the utilit function is: du = U d + U d Where U and U are the partial derivative of U w.r.t. and Suppose we fi an indifference curve and decrease b a small amount and compensate b increasing b a small amount Rate of Commodit refers to the idea that if I travel along the indifference curve at an particular point how much am I giving up of one good to receive another good? If I fi m utilit, move down that curve, what rate am I echanging one good for another in order to remain at the same utilit? Tet books will call this Marginal Rate Commodit Substitution (MRCS). Rate of Commodit Substitution, cont d Notice that U d is approimatel the loss in utilit b decreasing b a small amount and U d is approimatel the gain in utilit b increasing b a small amount Setting du = 0 ields (how much to change in order to compensate for a change in ) U d + U d = 0, so U d = -U d, and d d This is wh it is called Rate Commodit Substitution. If I want to maintain the same utilit, if I give up some Y, how much etra X do I need (and visa-versa). We are moving down one of the indifference curves. We set the small change in utilit to 0 in order to sta on the same indifference curve. U U Notice U d is approimatel the loss in utilit b decreasing b a small amount. U d is approimatel the gain in utilit b increasing b a small amount. I ve set the change in utilit to 0 (du = 0) so I m moving along an indifference curve. FNCE317 Class 7 Page 7

8 Y Tangent to the utilit curve. If I want to change one good for another how much do I need to change it in order to maintain the same level of utilit? d d X d/d gives the slope of the curve. If I am moving down an indifference curve, the slope tells me how much I have to compensate for a loss in b increasing to sta on the same indifference curve. Eample: Utilit and the RCS Suppose an individual s preferences for beer () and pizzas () can be represented b This form of the utilit utilit 20 function is called Cobb- Dobles (?) Suppose we are interested in the indifference curve where m utilit is 20. Utilit is a number, it is some measure of m world view. What I am interested in is the miture of and that give me a utilit of 20. A simple wa or epressing the Rate of Commodit Substitution there are a number of was but probabl the simplest is to epress as a function of and then take the derivative. Solving for, we get = 400/ Solving for RCS = -d/d: RCS = -d/d = 400/ 2 RCS is the derivative Started b setting our utilit function to the value we are interested in, 20, solving for in terms of, and taking the derivative which we call RCS. So now we know that we need to divide 400 b X in order to maintain a utilit of 20. This describes the utilit curve where m utilit is 20. Given an amount of pizza () I can tell ou how much beer () I need to generate that level of utilit (20). FNCE317 Class 7 Page 8

9 Utilit and the RCS RCS = -d/d = 400/ 2 Note that as rises, RCS falls when = 5, RCS = 16 when = 20, RCS = 1 -d Rate of Commodit Substitution = RCS = d In our eample = 400/ so RCS = -d/d = 400/ 2 This tells me how much beer I need to give up for pizza in order to maintain the utilit of 20. Notice that as increases the rate of commodit substitution falls. What does this mean? This is saing as I get more and more pizza compared to beer, I m less willing to echange more to add to m pizza and give up beer. If I m sitting at 50 pizzas and I onl have 3 beers, I m not going to go to 51 pizzas to give up one beer. It s not going to give Beer me an benefit. It s a function of the indifference curve being sloped in this wa. As I move up this curve I get less and less benefit from adding more of at the epense of. I have to get far more in echange. At the other etreme, if I ve got 400 beers and onl 2 pizzas I m not going to give up ver much pizza for one more beer. It s not going to give me much more value. Pizza When = 5, RCS = 16, when I have 5 pizzas I m willing to echange beer for pizza at a rate of 16. When = 20, RCS = 1, at 20 pizzas the rate of commodit substitution has fallen to 1. I m not willing to give up m beer in order to add pizza, I m alread saturated with pizzas. FNCE317 Class 7 Page 9

10 Beer Pizza 5 if I give up a beer I don t epect much pizza back. Consider = 5, then = 400/5 = 80 in order to keep our satisfaction at 20. Suppose I give up one beer, how much more pizza do I need to sta at U=20? If I give up one beer m goes from 80 to 79 and m goes from 5 to 400/79 = So I ve given up one beer and not gotten ver much pizza. But we are not surprised because we are in a high slope area of the curve, in this region Previousl we had calculated that at =5 the RCS is 16, steep slope. This is the graphical representation. The small change in is 1 beer. The small change in = 1/.06 = On a fied indifference curve, at a fied level of utilit, how much do I have to echange of one good for another to remain at the same level of utilit. BE AWARE THAT THE RATE OF COMMODITY SUBSTITUTION AS I GO DOWN THE SAME INDIFFERENCE CURVE. THIS IS JUST THE SLOPE CHANGING AT DIFFERENT POINTS IN THE FUNCTION. We can see that in this eample, b the time I get to 20 pizzas I m giving them up 1 for 1 (from =1 gives RCS = 400/20 2 = 1). As a result of our function being conve wrt to origin, the rate of commodit substitution is diminishing in. As I come down the curve from high levels of to low levels of the slope becomes shallower. -d Verif RCS = d U U U U 1 U 2 U U 400 substitute = to get 2 U FNCE317 Class 7 Page 10

11 Maimization of Utilit To maimize utilit, given a fied amount of income to spend, an individual will bu the goods and services: That ehaust her total income For which the rate of trade-off between an goods (the RCS) is equal to the rate at which goods can be traded for one another in the marketplace Assume that the individual s RCS = 1 Willing to trade one unit of for one unit of Suppose the price of = $2 and the price of = $1 The individual can be made better off trade 1 unit of for 2 units of in the marketplace If m ependiture is not constrained then I can never maimize m utilit. If someone just continues to give me funds m utilit is just going to increase. Therefore to maimize m utilit I will have to make the assumption that I have a certain budget over a certain period of time. Given that amount of mone what miture of the two goods will I choose that maimizes m well being? In our model there is no carr over of mone from one period to another, all income is spent. Now how do we find the best miture given our monetar budget? It s going to be when the trade off of the two goods on our indifference curve is equal to the difference in the price of those two goods. FNCE317 Class 7 Page 11

12 We could spend all of our mone on one of the goods, these are the line end points where income I is over price of that commodit (p or p ). Or I could spend m mone on an combination of goods between the two end points. How will we maimize our utilit? We introduce an indifference curve FNCE317 Class 7 Page 12

13 First-Order Conditions for a Maimum We can add the individual s utilit map to show the utilitmaimization process P = price of 1 unit of good. P = price of 1 unit of good. Consider point C on U 3. This is beond the range our budget can achieve, we cannot reach this level. There is no miture of and that I can afford which will get me to that level. We can reach a number of points on curve U 1 (A for instance) but these are sub-optimal. Point B represents the tangential intersection of the budget constraint and the utilit curve U 2. This is the best utilit that I can achieve given that miture of prices and that level of income. All the maimization problem does if find the indifference curve. That just touches the budget constraint. Can t go an further because not enough mone. Don t want to drop down because I could consume more and be better off. -d At point B the slope of the curve is RCS = d p At this point or an point on the straight line the slope is the ratio of the two prices,. p When I maimize m utilit the rate at which I am willing to give up the goods is equal to the price ratio between the two goods. THE RATE OF COMMODITY SUBSTITUTION IS EQUAL TO NEGATIVE THE RATIO OF THE PRICES TRADE OFF BETWEEN ANY TWO GOODS, THE RCS IS EQUAL TO THE RATE AT WHICH THE GOODS CAN BE TRADED FOR ONE ANOTHER. THAT IS THE RATIO OF THE PRICES. FNCE317 Class 7 Page 13

14 Suppose I have an individual at a particular mi of and (the rate of commodit substitution changes as ou move along an indifference curve) and at this particular point the RCS is 1 to 1. Remember, commodit substitution is measured in change in the number of goods, no dollar value associated. Giving up one good for one good, RCS = 1. Now suppose the price of is $2 and the price of is $1. I onl have a certain amount to spend. Therefore if I trade one unit of (sell one unit of or not purchase one unit of ) I m better off b $2. I can take that two dollars and bu two units of. That is going to make me strictl better off. Because at this point I m willing to trade 1 for 1, but because of the market place and the price I can actuall give up one and receive two. It all comes down to the fact that at this point the rate of substitution is equal to the ratio of the prices. If I can echange better then I can receive more and reinvest it in the other commodit. The shaded area represents combinations that I can possibl consume. In the sense of efficient markets we can be in the shaded area if we wish. Because I cannot carr forward mone I will not be in the shaded area, I will be on the boundar, because there is no point in throwing awa mone (since I cannot carr it forward). Awa from the boundar is in effect just burning ecess cash. Since indifference curves increase as ou move northeast (awa from the origin) and the are conve to the origin and we are alwas on the boundar of our spending line, we will alwas seek point B where spending is optimal to utilit. (the curves cannot be wiggl like below). This curve implies that if there is no budget constraint we would just keep increasing our utilit level. FNCE317 Class 7 Page 14

15 Second-Order Conditions for a Maimum The tangenc rule is onl a necessar condition we need RCS to be diminishing Would be better off at point B because higher utilit level. This is wh we must assume our utilit curves are conve to the origin. It can be proven that the are. Notice that as long as our rate of commodit substitution is diminishing then we are alwas safe (?). 400/ 2 is certainl diminishing. The Lagrangian The individual s objective is to maimize utilit = U(, ) subject to the budget constraint: I = p + p How do we find the optimal point? There is a wa of maimizing a function given a constraint. There are several requirements to using this method but basicall as long as our RCS is diminishing were basicall safe. (skipping some possible solutions). What we will do is maimize U(,) subject to the constraint that the answer has to lie on the budget line. We will define this line as Income equals the price of good times the number of units of good plus the price of good times the number of units of good. When is zero When RCS = Budget Constraint: I = p + p I I p and when is zero. The slope of this line is. p p p p p the Lagrangian method we are optimal. To solve this tpe of constraint optimization we use Set up the Lagrangian: L = U(, ) + (I p p ) FNCE317 Class 7 Page 15

16 Works because the budgetar term is alwas set to zero and we solve for the best value of utilit we can find along that line. The Lagrangian, cont d First-order conditions for an interior maimum: L/ = U/ p = 0 L/ = U/ p = 0 L/ = I p p = 0 Now we take the partial derivatives wrt,, and lambda setting each equal to zero. Notice that when taking the derivative wrt lambda the budget epression is treated as a constant. Implications of First-Order Conditions For the two goods, (using the above equations) U / p U / p This is telling us that the Rate of Commodit Substitution (RCS) is equal to the slope of the budget constraint line. The rate at which I m willing to give up for is equal to the ratio of the price of to the price of. I m giving up units of goods for units of goods. The optimal point is when the rate of echange is equal to the ratio of the market prices. If I can give up two slices of pizza for one glass of beer and be indifferent (the price of each is one-to-one) then I m going to do it because I can earn $2 and bu more beer. The solution will be found where this is the case. This implies that at the optimal allocation of income RCS ( for ) Interpreting the Lagrangian Multiplier p p Lambda is described as the shadow price of the constraint. U / U / Lambda represents how much m utilit improves if I can achieve another dollar of income. Tells me how much p p better off I am if I receive additional income to spend. Lambda is telling me if I can shift m budget constraint b a small amount how much m utilit improves. is the marginal utilit of an etra dollar of consumption ependiture. WHAT HAPPENED TO THE NEGATIVE SIGN? FNCE317 Class 7 Page 16

17 is the marginal utilit of an etra dollar of consumption ependiture the marginal utilit of income Lambda does not measure a dollar increase in m utilit. It tells me how much m well being increases for a given dollar change in m income. Utilit is not measured in dollars. Utilit is just a sense of well being, an ordering. Eample Suppose we have a utilit function of the form: U(, ) = Suppose that p = $2 and p = $5 The consumer s income is $100 Therefore the budget constraint is given b: = 0 zero means we are going to spend everthing Price of good is $2 and price of good is $5. Income for period is $100. Set up the Lagrangian: L = U(, ) + (I p p ) = + ( ) First-order conditions for an interior maimum: L/ = U/ p = 2 = 0 L/ = U/ p = 5 = 0 L/ = I p p = = 0 Solve these three equations with three variables for the value of,, and lambda > = > = > -5-5= > =10 Y = 10 = 25 = 5 This is how much of each good (at these prices and this budget constraint) I want to consume. is telling us how much the utilit increases if our income is increased b $1, this is the shadow price of income. FNCE317 Class 7 Page 17

18 What is our level of utilit at this point? U(25, 10) = (25)(10) = 250. Now suppose we increase our budget b $1 to $101. Now completel redo the solution using the new budget of $101. Will find that: = 10.1 = = (10.1)(25.25) = The new utilit is approimatel $5 more than the original utilit, this confirms solution above (because is onl approimate). Eample, cont d Solve to get = 25, = 10 and = 5 Thus, if the budget constraint can be increased b $1, then utilit will increase b approimatel = 5 Check to see: U with budget set at $100 is = 25 * 10 = 250 Solve again with budget set at $101 gives = 101/4, = 101/10 and = BACK TO ORIGINAL SOLUTION: Y = 10 = 25 = 5 = 250 d 250 Epress as = 250 / then plug in the solution for to get 2 d d RCS = d P = P This sas I m willing to trade for at a 2/5 ratio which also ties up with the value in the market place. This is saing that I can be no better off moving down the indifference curve. The ratio I m willing to give up goods to the market place equals the ratio of the market place price for the goods. This leads into demand functions We want to construct the market places suppl versus demand. We ve built an idea of to decide how much the consumer is willing to consume of a bundle of goods and where the utilit is maimized. We will take the solutions to that and turn the process around. We will construct demand functions. FNCE317 Class 7 Page 18

19 Demand Functions Ordinar Demand Functions An individual s demand for depends on preferences, all prices, and income: * = (p, p, I) It ma be convenient to graph the individual s demand for assuming that income and the price of (p ) are held constant The demand function for a particular good is how much of that good we want to consume given the price of that good. We can see b the arguments we are using in the utilit maimization that the amount that we actuall consume of the good will be dependent on our income, the price of other goods in the market place, and on the price of our particular good that we are looking to consume. How does the demand for a good var as it s price changes assuming everthing else remains constant? We often assume when crafting these tpes of things that as we look at the demand for given the price of holding the price of and the income that we have constant. See how our demand changes. Create a demand curve for good against it s price holding everthing else constant. (This is our goal). Continuing on with our eample Demand Function Eample Suppose we have a utilit function of the form: U(, ) = Budget Constraint: I p p Once again set up the Lagrangian: L = U(, ) + (I p p ) = + (I p p ) Notice we are not assuming an value of p, p or I in this eample First-order conditions for an interior maimum: L/ = U/ p = p = 0 L/ = U/ p = p = 0 L/ = I p p = 0 Solving for and gives the demand functions Solve for and the solutions are the demand curves. FNCE317 Class 7 Page 19

20 = p = p use I p p = 0 to get the solution = I/2p and = I/2p Notice these demand functions are the special case in which the demand for each commodit depends onl upon its own price and income It is eas to show that for an utilit if all prices and income change in the same proportion, the quantities demanded remain unchanged Given income and price we can tell what demand is. This is a ver special case. The demand for is onl dependent on the price of and our income level, same is true for, onl dependent on income level and price of good. Generall this will not be the case, onl have this because we used the simple utilit function U(, ) =. If we add a little compleit to the utilit function the and demand will be dependent on the price of the other good as well. But keep in mind that when we are plotting demand curves we are assuming that those other prices are constant. Another point, it s actuall ver eas to show, and this is true for an utilit, if I increase m income and I increase m prices b the same ratio I will still get the same solutions. Still will consume the same amount of goods. FNCE317 Class 7 Page 20

21 The Individual s Demand Curve I P Demand Curve I P Given the budget constraint and the shape of the indifference curve and utilit level above we can get a ver general answer. The amount of goods I will consume at a certain level. Given a certain price I will consume so much of good. Now we will change the price of, we will decrease the price of. If we do so what will happen to the budget constraint? Consider the endpoints of the budget constraint lines, I/P and I/P. If I decrease m price the budget constraint line becomes shallower. This sas that if I onl consume I can consume more of it (because the price has come down). If m price comes down from 10 to 5, if all I consume is I can consume more for a given level of income. (So the graph on the left is showing the price of coming down?) So the line becomes shallower. Now I do the maimization problem again. I receive another utilit and I m better off because the price of good has fallen, I don t have to spend so much on good. I can afford to spend more on good or consume more on good or both. But what phsicall happens? As the price falls I m almost certainl going to consume more of. Now we reduce the price of even further. Budget constraint line becomes shallower, utilit comes to some new level, the amount of I consume will increase. Therefore the amount I consume at the lowest price is even higher. As the price of a good falls ou end up consuming more and more of the good. So we ve seen that as the price of falls the number of units that I consume increases. FNCE317 Class 7 Page 21

22 The Individual s Demand Curve An individual demand curve shows the relationship between the price of a good and the quantit of that good purchased b an individual assuming that all other determinants of demand are held constant The individuals demand curve will show the relationship between the price of the good and how much good I m going to purchase if I hold everthing else constant. I solve m capitalization problem, I lock in on that particular good, I solve for that particular good, the amount that I consume is going to be a function of m income and the price of all other goods. The demand curve describes that relationship if I hold the income and the other prices constant. Shifts in the Demand Curve Three factors are held constant when a demand curve is derived income prices of other goods (p ) the individual s preferences If an of these factors change, the demand curve will shift to a new position A movement along a given demand curve is caused b a change in the price of the good a change in quantit demanded A shift in the demand curve is caused b changes in income, prices of other goods, or preferences a change in demand curve Demand curves will shift, the will change. If m income changes m demand curve will change. Remember, I m holding m income constant. If I increase m income it s going to shift m demand curve. Prices of the other goods, if I don t hold that constant I m going to end up with new demand curves. Individuals preferences, if I change m utilit function then m demand curve is going to change. FNCE317 Class 7 Page 22

23 An of these factors will shift the demand curve for a particular good for a particular consumer. Moving along the demand curve is caused b a change in the price of the good. Price goes up, quantit falls. Price goes down, quantit will go up. Shift in the demand curve, actual change in the entire curve, will occur because of income, prices of the other goods, or preferences. Holding all things constant and I get changes in price/good relationships, if I var then m curves actuall change. Demand Elasticities The own price elasticit of demand for (which we will call ) is defined as the proportionate rate of change amount consumed, divided b the proportionate rate of change of its own price, holding income and the other price constant p p All we are considering here is how much the demand for a good changes given a change in prices. The simplest one to think of is the own price elasticit. Given a change in the price of a good, how much does the demand for that good change? How much do I move along the demand curve? We define this as eplison sub. What this means is the change in the amount of a good that I demand given a change in the price of that good. change in the amount of a good that I demand, given a change in the price of that good I change the price of P b a small amount, a tin little piece of. E., the price is $5 I change it to $5.01. The proportionate change is the small amount I m going to change divided b current price. If the price is a dollar and I increase the price b.01, then m proportionate change is a 1% increase. That s going to lead to a change in the goods that I want to consume. How much I want to consume of. What s the proportionate change? This will lead to a change in good (in the literature, the good is the capital letter, the amount consumed is the small letter). So a good currentl at is changed b a small amount. Price goes up b 1%, what is the effect on the amount demanded? Proportionate Change = FNCE317 Class 7 Page 23

24 The own price elasticit is the percentage change divided b the percentage change. (?) Percentage change in This is approimatel equal to. Percentage change in P If the prices go up b 1% what happens to the amount of good that I receive? It s going to be negative. Demand is going to fall. Elasticit tells us what that ratio is. For loanable funds interest is the price of loanable funds. This is saing that interest is inelastic if the amount that the government wishes to barrow doesn t change ver much given a change in interest rates. If interest rates go up b 100%, sa from 5% to 10%, the government is not going to see an significant change in the amount of funds it demands. The epsilon ratio will be close to zero. The government demand for loanable funds is inelastic. Phsicall what is going on is that is the demand function for the amount of goods that we are going to consume, ou calculate the partial derivative of wrt it s price. (eample below) Two stage process, find the demand functions and then calculate the partial derivatives. FNCE317 Class 7 Page 24

25 Own Price Elasticit of Demand The own price elasticit of demand is almost alwas negative The size of the elasticit is important < 1, demand is elastic (luuries) > 1, demand is inelastic (necessities) = 1, demand is unit elastic If the size of the elasticit is <-1, demand is called elastic. Steep Slope. Given a change in prices there is a larger change in the goods consumed. A 1% change in prices is going to lead to a > 1% decrease in the goods I consume. Slope is steep, less than -1. These items are called luuries. As the price of luuries goes up the demand falls sharpl. If the size of the elasticit is > -1, demand is called inelastic. Remember, ZERO is greater than -1. So at the etreme there is no change in demand given a change in price. These things are called necessities. Will have to pa whatever the price, inelastic. Right at -1 the demand is called unit elastic. Back to our eample FNCE317 Class 7 Page 25

26 Price Elasticit and Total Spending Total spending on is equal to total spending = p Using elasticit, we can determine how total spending changes when the price of changes p p p p p 1 1 p I U = we have Want to calculate it s Own Price Elasticit so what I need to 2P δ -I calculate is first the rate of change of wrt it s own price =. This is one of the δp 2P factors in Own Price Elasticit. Therefore the Own Price Elasticit for will be P P I 1 in this case the Own Price Elasticit is UNIT ELASTIC. 2 P I 2P 2P What does this phsicall tell me? If the price falls b 1%, demand increases b 1%. NOTE: THIS IS SPECIFIC TO U(X,Y) = XY. Other utilit functions will have other results! Because the utilit function is completel smmetrical I also know that Own Price Elasticit of will also be -1. Look back to the demand functions for, it was just =I/2P. Redo the calculation wrt, will get -1. Nothing special about or in this utilit function. EXAM If given a smmetrical utilit function and we are asked to calculate two Own Price Elasticit s, we onl reall have to do one, the other is smmetrical! FNCE317 Class 7 Page 26

27 Price Elasticit and Total Spending The sign of this derivative depends on whether is greater or less than 1 > 1, demand is inelastic and price and total spending move in the same direction < 1, demand is elastic and price and total spending move in opposite directions If I change the price of a good, I m not now interested in how the demand for that good changes, I want to know how much I spend on that good changes. We have said that as the price goes up the number of goods consumed declines. But if the price doesn t go up fast enough, the price times the number of goods, the actual dollar amount I spend, could go up and could go down. How much I actuall phsicall spend on that particular good ma go up or ma go down depending on how quickl the price moving. Is the price increasing quicker than the demand is falling? This is where we will have to eamine to find the tipping point. It s ver eas. If I lock in on the total spending, whatever the price and whatever the demand, remember these are solutions from our utilit maimization, given m utilit and given m income, this is the amount that I spend, and this is the number of goods I consume. Lock in on this, if I change the price how much does m total spending on good change? That s what we want, the dollar amount we spent on. If the price changes how am I moving on m demand curve? P Two stage derivative, P P P P P P P P P (1 P ) (1 ) P P P If I decrease the price is m dollar ependiture going to go up or go down? If I decrease the price then the amount of goods I demand goes up, is going to go up. If P goes down, goes up. Change in dollar value P, is that going to go up or down? There si no obvious wa of knowing because ones going up and ones going down. This gives us the answer. If the elasticit < -1, then (1 + ) is negative (?). Is the old price elasticit greater or less than -1? If it s greater than (1-1), demand is inelastic, the are necessities. If the price goes up, total spending goes up. Think about it like water, if the price of water goes up I still basicall consume the same amount (the demand comes down a little bit, I might consume a little bit less but I will end up spending more in dollar terms). So therefore I end up spending more. Whereas for a luur good, if the (err?) of elasticit is less than -1, if the price goes up I consume so much less that I end up spending less in dollar terms. FNCE317 Class 7 Page 27

28 The whole ke is weather (1 + ) is greater than or less than -1. In the case of our eample the Own Price Elasticit is -1. What that told me is a 1% increase in price represents a 1% decrease in value to the consumer. So in our eample the dollar amount we spend on does not change. I just consume less to compensate for having to spend more. But I sta at the same dollar value. Cross-Price Elasticit of Demand The cross-price elasticit of demand for with respect to price p (which we will call ) is defined as the proportionate rate of change amount consumed, divided b the proportionate rate of change in the price of, holding income and the other price constant p p Cross-price elasticities ma be positive or negative Substitutes have positive cross price elasticities: Butter & Margarine Complements have negative cross price elasticities: DVD machines and the rental price of DVDs Same idea, it s how much the demand for changes if the price of changes. Wh would it, the price of? Well we onl have so much to spend, if the price of one good changes then it ma have an impact on how much I spend on another good. What we are looking at is how much does the amount of, in percentage terms, change given the change in the price of? if the price of goes up 1%, what happens to the amount of that I consume? Whereas Own Price Elasticities are negative (price up means the number of goods I consume goes down), with Cross Price Elasticit who knows! I ma consume more and I ma consume less. The connection comes from weather or not the are substitutes or are the compliments. Under these circumstances if the price of hamburgers goes up I am likel to consume less rolls, if the price of CDs goes up I might start using analog tape. The can move in opposite directions dependent on weather the are compliments or substitutes. FNCE317 Class 7 Page 28

29 For our particular case, the amount consumed of = I/2P and the amount consumed of = I/2P. Therefore if I take d/dp it will be the same as d/dp (these are partical derivatives) and the are both zero. If the price in our model, where the utilit function has the form, if the price of one good changes it has no impact on the other good. We can take the analsis a step further. When we calculated this thing (?) we came to the conclusion that the dollar amount that I was going to spend on would remain constant. Because the price elasticit was minus 1. It set this thing (?) equal to zero. So the dollar value that I spent on remain constant. That is wh this is zero. Sending the same dollar amount on so therefore I still have the same dollar amount to spend on. So therefore changing the price of, in this eample, does not change the value of that I consume. Cournot Aggregation Consider the budget constraint: I = p + p Differentiating both sides b p ields: I p p p p p p p Since we are holding income and the price of p constant b assumption then: I p 0 p p Therefore p p 0 p p FNCE317 Class 7 Page 29

30 Cournot Aggregation, cont d Multipling all terms b p /I gives: p p p p p 0 p I I p I Or Where = p /I and = p /I The proportion of income spent on each good Cournot Aggregation, cont d The epression: Is described as the Cournot aggregation condition Useful since it allows cross price elasticities to be calculated without construction demand functions If = 1 then = 0 If < 1 then > 0 If > 1 then < 0 If the Own Price Elasticit is -1 then this relationship tells us I have minus a constant and the same minus constant on the other side. Therefore the cross price elasticit is zero. If the price elasticit is < -1, the some term is less than minus alpha, therefore we solve and end up with a positive value. Which implies that this thing is greater than 0. Comes back down to this, if m dollar amount increases for a particular good, then the amount that I consume for the other good has to decrease to compensate. When does the dollar amount increase? When this is less than or greater than -1. Understand wh this is the case, wh this is tied to this particular interpretation. Remember, we are on the same budget constraint line, if m dollar amount of 1 changes, goes down, the dollar amount of the other goes up. If the price is the same then the amount I consume goes up. FNCE317 Class 7 Page 30

31 Income Elasticit of Demand Defined as the proportionate change in the purchases of a commodit relative to the proportionate change in income with prices held constant Can be positive or zero, but usuall found to be negative 1 Generalization The optimal levels of 1, 2,, n can be epressed as functions of all prices and income These can be epressed as n demand functions of the form: 1 * = 1 (p 1, p 2,, p n, I) 2 * = 2 (p 1, p 2,, p n, I) n * = n (p 1, p 2,, p n, I) Conclusion I I I can alwas do this for two variables if I consider one good and everthing else. If one good is food and the other good is everthing else, based on this relationship it tells ou what happens when the price of food goes up. Conclusion We understand utilit functions, how to maimize utilit given a budget constraint, from that maimization process we can find demand functions, those demand functions can give us interesting analsis of goods, which allows us to solve elasticit questions. This allows us to solve interesting things such as if the price goes up what happens to how much I consume in dollar terms, and what impact does it have on the other goods that I consume. FNCE317 Class 7 Page 31