The next step is to come up with a classification of what is sensitive and what is insensitive. Consider the example below:

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1 Econ311_lecture1 8/28/2006 2:27 PM Elasticities Why? Why should we bother with elasticities? As we learn a bit more economics, one of the things we would like to do is to try to be more specific. If you think back to your first microeconomics course, we were happy with you telling us simply in what direction the price and quantity will change. As we progress, we will still want to know in what direction the price and quantity will change, but in addition, we d like to know how much the price and quantity will change. Elasticities enable us to answer these questions. Is it a big change in price or a small one? While it will be towards the middle of the semester before we get there, you will find that a firm s pricing decisions will be directly influenced by the elasticity of demand for the product it is selling. Elasticities What are they? All elasticities will be a measure of sensitivity. The (own) price elasticity of demand measures how sensitive quantity demand is to changes in (own) price. The (own) price elasticity of supply measures how sensitive quantity supplied is to changes in (own) price. The income elasticity of demand measures, you guessed it, how sensitive demand is to changes in income. The next step is to come up with a classification of what is sensitive and what is insensitive. Consider the example below: Suppose a $1 increase in the price of gas that leads to a 500 gallon decrease in gas sold. Now, suppose a $1 increase in the price of a BMW leads to a 500 fewer BMW cars to be sold. Which of these is a big quantity response? Even though both changes involve an increase of $1 and a reduction in quantity of 500, my feeling is that car change is extremely sensitive, while the gas is not so sensitive. The point here is that measuring changes is dollars and units sold might not be appropriate. $1 is a big deal for gas, but not for cars. Perhaps measuring the changes in percentages would give us a better idea? This is in fact how elasticities are measured. In general, an elasticity is the percentage change in one variable resulting from a 1% increase in another. (Own) Price Elasticity of Demand (Own) Price Elasticity of Demand is defined as the percentage change in quantity demanded of a good resulting from a 1% increase in its price. E P % Q = % P You should read this as the percentage change in quantity divided by the percentage change in price. The notation means change in. Of course, the elasticity of demand will always be a negative number as demand curves are downward sloping. An increase in price (positive denominator) will lead to a reduction in quantity demanded (negative numerator), giving us a negative value of E P. Likewise, a decrease in price (negative denominator) will lead to an increase in quantity demanded (positive numerator), and again result in a negative value of E P. Since the elasticity of demand is always negative, people often strip off the negative sign and talk about the absolute value of elasticity. If someone ever tells you the elasticity of demand is 2, what they really mean to say is that the absolute value of the elasticity of demand is 2. No matter how you slice it, we ll all know that this really means is that E P = -2. Categorization of Price Elasticity of Demand Values We ll want to classify demand curves into three categories: 1

2 Econ311_lecture1 8/28/2006 2:27 PM When you think of elasticities, think of sensitivities. The (own) price elasticity of demand is a measure of how sensitive the change in quantity demanded is to a given sized price change. If a demand curve is inelastic, this means that change in quantity demanded is relatively insensitive to changes in price. If a demand curve is elastic, this means that the change in quantity demanded is relatively sensitive to changes in price. Let s be a bit more formal: E P > 1, elastic a 1% in price will lead to a more than 1% in quantity demanded E P = 1, unit elastic a 1% in price will lead to an exactly 1% in quantity demanded E P < 1, inelastic a 1% in price will lead to an less 1% in quantity demanded Also, there are some extreme elasticity values. E P =, perfectly elastic a 1% in price will lead to an infinite reduction in quantity demanded E P = 0, perfectly inelastic a 1% in price will lead to no change in quantity demanded Are elasticities the same thing as slopes of demand curves? No. To see this, we can look at the definition of elasticities, do some rearranging, and you ll see why. First, we need to back to 6 th grade and remember how to calculate percentage changes. Remember a percentage change in just the change in the value divided by the original value. So we have: % Q = Q / Q and % P = P / P Thus, we can rewrite the elasticity formula, plug in the above definitions, and then do a little rearranging: Q % Q Q Q P EP = EP = E = P P % P P P Q Q The first term ( P ) is constant along a linear demand curve and equal to the inverse of the slope of the demand curve. The second term ( P Q ) changes as we move along a linear demand curve. On the uppermost portion of a demand curve, P will be relatively high and Q relatively low, resulting in P Q being a large number. On the lower portion of a demand curve, P will be relatively low while Q is relatively high, resulting in P Q being a low number. Therefore, as we move along a linear demand curve the elasticity of demand will change from 0 to negative infinity. In a moment, I will take you through an example, but first, let us do some background on demand curves and inverse demand functions. Aside: Algebraic Representations of Demand Curves Suppose you were told a demand curve was described by the equation Q = 8 2P. You could pull out a piece of graph paper and draw it (you should). You could just stick in various price and see what you d get for Q. You d get the right answer, but it would be tedious. You ll draw a bunch of demand curves in my class. It turns out a nice trick will be to use find the horizontal intercept and the vertical intercept of a demand curve and connect the dots. But you are always welcome to do it the long way. To find the horizontal intercept of the demand curve, find where price is equal to zero. If you stick in P = 0 into the equation for the demand curve and solve for Q, you ll find that Q = 8. That is, the horizontal intercept of the demand curve is 8. All of that is a fancy way of saying that P = 0, Q = 8 is one spot on the demand curve. Q = 8 2P P = 0 Q = 8 2(0) = 8 0 = 8 2

3 Econ311_lecture1 8/28/2006 2:27 PM To find the vertical intercept of the graph, find where quantity is equal to zero. If you stick in Q = 0 into the equation for the demand curve and solve for P, you ll find that P = 4. That is, the vertical intercept of the demand curve is 4, or simply P = $4, Q = 0 is another spot on the demand curve. Q = 8 2P Q = 0 0 = 8 2( P) 2P = 8 P = 4 Q While the reason won t be clear just yet, as long as we are here, we can note in this case, ( P ) = -2. That is to say that a one unit increase in P will lead to a two unit reduction in Q. I ve drawn the demand curve below. Ignore the slope calculation for just a second if you can. P 4 3 Slope = Rise Run P 4 1 = = = Q Q For fun, you should draw a graph of a demand curve with the equation given by 1 Q = 6 3 * P. You should confirm that the vertical intercept occurs at P = 18, while the horizontal intercept is where Q = 6. In a page or so, you ll be able to confirm that the slope of the graph is -3. Aside: Inverse Demand Functions We think of the demand curves as being correctly specified above because we think that consumers choose their quantity demanded based on the market price of the good. In QBA language, quantity demanded is the dependent variable and price is the independent variable. Or quantity demand is the variable that is determined by consumers and market price is beyond their control. The weird thing here is that you have always been taught to graph the dependent variable on the y-axis and the independent variable on the x- axis. However, the powers that be in Econland have decided to do just the opposite. As such, we will often find it convenient to take the same demand function and instead of having it solved for Q (as was done above), we solve it for P. We call the result the inverse demand function. It obviously contains the same information, and will lead to the same graph (try it). In this case, the inverse demand function is P = 4 ½ Q. Again, it is found by solving the demand curve for P. 1 Q = 8 2P Q + 2P = 8 2P = 8 Q P = 4 Q 2 P 1 Now, go and calculate the slope using the graph. The slope is Q 2 (see picture above). You ll notice it s easy to grab the slope from the inverse demand function, as it is just the coefficient on Q. Aside: When should you use the demand curve and when should you use the inverse demand function? Both expressions contain the same information. Use which is ever is convenient. You ll never get the wrong answer from using one or the other. However, it will save you some time doing algebra if you pick the right one. You ll see as you get some experience. If you learn how to go from one to the other, you ll be fine. = 3

4 Econ311_lecture1 8/28/2006 2:27 PM For fun, you should figure out the inverse demand function for the demand curve 1 Q = 6 3 * P. Confirm that the correct answer is P = 18 3Q. Also, confirm a slope of -3. Are elasticities the same thing as slopes of demand curves? (Again) No. There are a range of elasticities along each demand curve. See the discussion on page 2, or perhaps a better way is to just calculate some elasticities along a linear demand curve. Suppose I gave you the demand curve Q = 8 2P, or equivalently, the inverse demand function P = 4 ½ Q. Suppose I asked you to calculate the elasticity of demand at five different spots along the demand curve, specifically where P = 4, P = 3, P = 2, P = 1, and P = 0. You d no doubt consult your notes and write down the formula for the elasticity of demand. Q P E P = P Q Q Now, hopefully your remember the first term ( ), has something to do with the slope of a demand curve P and remains constant along a demand curve. You could stare at the demand curve, Q = 8 2P, and note that a one dollar increase in price leads to a two Q 2 unit reduction in Q, and thus conclude that = = 2. Or you could stare at the inverse demand P 1 function, P = 4 1 Q, and note that a 1 unit increase in Q leads to a ½ unit decrease in P leading you again 2 Q 1 to conclude that = = 2. Or you might have calculated the slope of the demand curve as ½ and P 1/ 2 Q took the inverse. Any way you do it, you ll conclude that = 2. Now you just need P and Q. To figure out the quantity demanded for each P, it more convenient to use the demand curve. 1 BecauseQ = 8 2P, it is pretty easy to plug and chug... When P = 4, Q = 8 2(4) = 0 When P = 3, Q = 8 2(3) = 2 When P = 2. Finally, put it all together: P P = 4, Q = 0 P = 3, Q = 2 P = 2, Q =4 P = 1, Q=6 P = 0, Q = 8 E P E P E P E P E P Q P 4 = = 2 = P Q 0 Q P 3 = = 2 = 3 P Q 2 Q P 2 = = 2 = 1 P Q 4 Q P 1 = = 2 = 0.33 P Q 6 Q P 0 = = 2 = 0 P Q 8 perfectly elastic elastic unit elastic inelastic perfectly inelastic Notice that as the price increase (as you move up a demand curve), the demand curve becomes more elastic. As price decreases (as you move down a demand curve), the demand curve becomes more inelastic. 1 You, of course, will get the same answers if you use the inverse demand function, you ll just have to do more algebra. Try it. It will get old quick. 4

5 Econ311_lecture1 8/28/2006 2:27 PM It looks like the point where demand is unit elastic is half way along the demand curve. In fact it is. You can take it on faith, and that s fine. If you like algebra, I ve written it out at the end of these notes. See the graphs next page. P E P = E P = 3 4 E P = 1 3 E P = E P = 0 In general, Q P E P = (perfectly elastic) elastic E P = 1 (unit inelastic) at the midpoint of the demand curve inelastic E P = 0 (perfectly inelastic) Q Relationship between inverse demand curves and marginal revenue You need to be familiar with how to go from an inverse demand function to a marginal revenue curve. Recall that marginal revenue is the additional revenue associated with a one unit increase in production. Inverse Demand Function: P = a bq 2 Total Revenue = P * Q = ( a bq) * Q = aq bq Marginal Revenue = ( Total Re venue) = a 2bQ Q For those of you who are calculus inclined, marginal revenue is the derivative of total revenue with respect to quantity. For those of you who are not calculus inclined, do not fret, you can use the shortcut method described below. A few things to notice... The marginal revenue curve has the same vertical intercept (a) as the demand curve does. The slope of the marginal revenue curve is twice the slope of the demand curve, or more simply the MR curve is twice as steep as the demand curve. The marginal revenue curve intersects the horizontal axis where a Q = which is half way 2 b between the origin and where the demand curve intersects the horizontal axis a Q =. b 5

6 Econ311_lecture1 8/28/2006 2:27 PM Marginal revenue can be negative. In general, the picture will look like this: P D MR Q I want everyone to be able to come up with the equation of the MR curve given the inverse demand function (or for that matter the demand function). Notice that P = a bq and that MR = a 2bQ. For those of you who aren t calculus inclined, to transform an inverse demand function to a marginal revenue function, you just have to change the coefficient on the Q term. 2 See the examples below. Example: Inverse Demand Function: P = 12 2Q 2 Total Revenue = P * Q = ( 12 2Q) * Q = 12Q 2Q Marginal Revenue = ( Total Re venue) = 12 4Q Q You should sketch both the demand function and the marginal revenue function. You should find the demand curve intersects the horizontal axis where P = $12 and the vertical axis where Q = 6. You should also find that the marginal revenue curve intersects the horizontal axis where MR = $12 and the vertical axis where Q = 3, and then continues along with negative values. If you were given an inverse demand function P = Q, the marginal revenue function would be P = 14 7Q. If you were given a demand curve of Q = 4 2P, you have to first transform the demand curve into an inverse demand function, and then use the trick. In this case, you d find an inverse demand function of P = 2 ½Q, and MR = 2 Q. If you are not sure, graph them. What is the relationship between Marginal Revenue and Total Revenue? Marginal revenue is the extra revenue associated with a one unit increase in production. The calculus inclined should realize that MR is the slope of the total revenue function. Others don t need to worry. Let s start at Q = 0 (the upper most point on the demand curve). At Q = 0, MR is positive, meaning that producing another unit will increase total revenue. However, we can see that MR is getting smaller as we increase output (though for the time being it is still positive). Thus we know that total revenue is 2 Be sure that you are using the inverse demand function when you do this trick. 6

7 Econ311_lecture1 8/28/2006 2:27 PM increasing, but increasing at decreasing rate. Eventually (at the midpoint of the linear demand curve), MR = 0. This means total revenue is neither rising nor falling. Total revenue has reached a peak. For all additional units, MR < 0, this means total revenue will fall as we produce additional units of output. We can combine this information, and the fact that we know revenue is 0 when we sell zero units (the vertical intercept of the demand curve) and the fact that we know revenue is 0 when price is zero (the horizontal intercept of the demand curve) to draw the following picture. Total revenue is maximized where MR = 0, which just so happens to be the spot where the elasticity of demand is equal to -1 (unit elastic). This is no coincidence. Those of you who like calculus can note we are setting the derivative of the total revenue function to zero to find out where total revenue is maximized. Others need not worry. P A B C (midpoint) D Total Revenue MR E Demand Q Q Given P = 18 3Q, could you calculate MR? Could you then find the quantity that will have the largest total revenue? Hint: what is the value of marginal when total revenue is at its maximum? What is the quantity where this occurs? What is the price at this quantity? The value of total revenue? The answer to the last question is $27. What is the relationship between elasticity of demand and total revenue? If a firm or manager knew what their whole demand curve looked like, it would be easy for the firm to calculate marginal revenue and elasticities. Often, though, firms will not have information on their entire demand curve, but will have a pretty good idea about the elasticity of demand for their product. What if you only know the elasticity? What could you say about what happens to total revenue if price changes? One last thing you should be able to do is tell me what happens to total revenue when there is an increase in price when demand is elastic, when demand is unit elastic and when demand is inelastic. If you combine the picture above with the picture on page two of the lecture notes to answer your have the whole story. For instance, you know on the upper half of the demand curve, demand is elastic. You also know that if you raise price (move from B to A), total revenue is decreasing. On the lower half of the demand curve, 7

8 Econ311_lecture1 8/28/2006 2:27 PM demand is inelastic. You see from the picture above if you raise price (moving from E to D), total revenue is increasing. And finally, if you are right in the middle of the demand curve, demand is unit elastic. You know if you make a very small change in price at point C, there is no change in revenue. You might try and figure out what is going on for price decreases for each of inelastic, elastic, and unit elastic. Confirm your answers in the chart below. Price Decrease Price Increase Demand Elastic TR TR Demand Unit Elastic no TR no TR Demand Inelastic TR TR Solving for Equilibrium given a demand curve and a supply Curve Suppose you were told that the demand curve is given by: Q D = 160 8P And were told that the supply curve is given by Q S = P. Can we determine exactly what will be the equilibrium price and quantity in this case? We have to be a little more careful about the notation here because we have to sets of prices and quantities. Q D of course will represent quantity demanded, while Q S will represent quantity supplied. You know that in equilibrium, quantity demanded equals quantity supplied. One way you could answer this question is to graph the demand and supply curve on a piece of graph paper, and simply see where they intersect. Fortunately, we can save time by figuring out the answer algebraically. Simply set Q S = Q D, and solve for the price. Math jocks will realize we simply have two equations and two unknowns. Q = Q P = 160 8P P = P = 90 P = 6 S D Now, plug the price back into the supply curve or the demand curve to find out the equilibrium quantity. In fact, it is a pretty good idea to plug it into both to double check that you get the same quantity (to confirm that quantity demanded does indeed equal quantity supplied). Q D Q S = 160 8P = 160 8*6 = = 112 = P = *6 = = 112 So indeed, P = $6 and Q = 112 is the equilibrium price and quantity. Try this one out: Q D = 40 ½ P Q S = 20 + P Check to see if Q D = Q S at the equilibrium price you find. If it is, you will know you have it correct. 8

9 Econ311_lecture1 8/28/2006 2:27 PM What should I read? For deep background on supply and demand, see For even more review, see my old Econ 211 notes, lectures 3 5. For elasticity of demand, see 2.4. While not assigned, if you are interested in arc elasticities, they are not difficult. See 2.4 if you are interested, and/or the appendix attached to these notes. For a proof of the fact that the elasticity of demand is equal to -1 at the midpoint of a linear demand curve, see the appendix attached to these notes. This is not something you have to do. If you are feeling very adventurous, and really like algebra, for more algebra on supply, demand, and equilibrium, see 2.6. I think your text goes overboard here, but you might like it? Appendix: Algebra that proves the elasticity of demand is equal to -1 at the midpoint of the demand curve Let s start with an inverse demand function where P = a bq. First, let s find the vertical intercept of the demand curve. This would occur where P = a. I m interested in the point half way along the vertical axis, so I am interested in a P =. Now let s figure out the quantity that corresponds to this price along the 2 demand curve: P= a bq a a a a P= = a bq = bq Q = b The 2 nd a term in our elasticity formula is P. From above, P 2 a 2b = = = b a Q Q 2b 2 a The first term in my elasticity formula is Q. To figure it out, first solve for the demand function: P a 1 P= a bq bq = a P = Q P b b Then it is pretty clear that Q = 1 P b % Q Q P 1 = = b % P P Q b Finally, = ( ) = 1 E P The own price elasticity of demand calculated at the midpoint of a linear demand curve is -1. 9

10 Econ311_lecture1 8/28/2006 2:27 PM Appendix: Info on Arc Elasticity Thus far we have talked only about elasticities on a particular spot on the demand curve. What if we wanted to calculate the elasticity of demand along a range of the demand curve? Say for instance we know that originally, P = $5 and Q = 8. Then, something changed and now the new situation is that P = $8 and Q = 6. The kicker is that the direction we are going (from original to new or from new to original ) when we calculate the elasticity makes a difference in the answer. Let s see how. Suppose we start from the original price and go to the new price. It s clear that P = $3 and Q = -2. Using the original figures (P = $5, Q = $8), we d calculate: E P % Q Q = = % P P P Q = = = Suppose we went from the new price to the original price. It s clear that P =- $3 and Q = 2, so that part doesn t make any difference. But using the new figures (P = $8, Q = $6), we d calculate: E P % Q Q = = % P P P Q = = = The order matters. Which is right? Neither is more right or wrong. So the solution is to calculate the arc elasticity of demand. Basically, rather than using the original P and Q, or the new P and Q, we average the price and average the quantities. Using the averaged prices and quantities gives us an average of the two calculations above (though it isn t necessarily half way between the values above). Let P be the average of the two prices and Q be the average of the two quantities. Then, we simply calculate as we did before, only substituting P and Q in the 2 nd term in the elasticity formula. E P % Q Q P = = where % P P Q P + P2 P = 2 Q1 + Q = 2 1 Q2 In our example, E P % Q Q = = % P P P Q = = = Notice you would not get the right answer if you simply average the two elasticities we calculated previously. 10

11 Econ311_lecture2 8/30/2006 2:44 PM Consumer Behavior Off we go. If we want to understand markets, we have to understand something about consumers. Our first task will be to develop a theory of consumer behavior - of how consumers make decisions. I think your text does a good job of organizing this material. By the way, while the last set of notes was all around the textbook, now would be a good time to begin reading the book. Consider these notes a complement to the textbook, not a substitute. The first thing we will focus on is consumer preferences. We will describe these preferences primarily graphically, though on occasion we will do some math here. The big tool here will be an indifference curve. We will learn what an indifference curve is, how it expresses preferences, and how different indifference curves express different preferences. Next, as you know, consumer preferences are only a portion of the story. Consumers face constraints if not we would consume ourselves into a happy nirvana. The tool here will be a budget constraint. The budget constraint will not only be an indication of the consumer s income, but will also reflect information on prices. This section will concentrate on what is feasible. Finally, we will examine how consumers make their choices. We will combine the information on preferences (indifference curves) and on constraints (budget constraints), and assume that consumers (optimally) choose their maximum level of satisfaction. Some almost useless jargon Your textbook will talk about market baskets or sometimes market bundles. All that is meant by these terms is some combination of goods. One package of laxatives, three scratch off lottery tickets, and a carton of cigarettes is one market basket. A bedpan, 2 packages of Nicoderm, some codeine and a box of animal crackers is another market basket. A word on units is in order - an apple is not quite the same as an orange. Should we measure cigarettes in term of the number of cigarettes or the number of packs? Animal crackers in boxes or pounds? Who knows? As such, we will always talk about the generic term units and not worry about this. In the end, we are not going to need to know exactly how many units of satisfaction each good provides. We will only need to know which market bundle provides more satisfaction - which bundle is preferred. Thus, the units in which we measures these goods will not be of any concern. Assumptions about Consumer Preferences We will make three assumptions about preferences. These are technical and will not rock your world, but you should know a bit about them. 1. Completeness We will assume preferences are complete. Suppose you have two market bundles, A and B. By assuming preferences are complete, we are assuming that consumers can look at two bundles and rank them. They will either tell you: they prefer A to B, they prefer B to A, or they are indifferent between A and B. 1 The only think that completeness rules out is somewhere looking at A and B and saying something like it is impossible for me to compare these two bundles. Basically, either: A f B A is preferred to B 1 Saying that they are indifferent A and B means that A and B provide the exact same level of satisfaction. Folks sometimes get confused about the difference between indifferent and incomplete. Indifferent means (someone has compared them and) the alternatives provide the same level of satisfaction. Incomplete means that you are incapable of comparing them. 1

12 Econ311_lecture2 8/30/2006 2:44 PM A p B B is preferred to A A ~ B Indifferent between A and B 2. Transitivity We will assume that preferences are transitive. This means if a consumer prefers A to B and prefers B to C, you can then it must be that the consumer prefers A to C. Likewise if a consumer is indifferent between A and B and indifferent between B and C, then they are indifferent between A and C. If If A f B and B f C, then A f C A ~ B and B ~ C, then A ~ C 3. More is preferred to less This means that goods are good. Cookies are good, but more cookies are even better. Consumers are never satiated. I am sure you can think of examples where this is not true. Trash is not good, and more trash is even worse. Or cockroaches in your house, toxic waste, weeds in your lawn. These types of items often are called economic bads or just bads, instead of goods. A quick fix is that if trash is a bad, then trash removal is a good. If weeds are good, weed killers are a good. However, even without this trick, we can still handle bads. For now, unless you hear explicitly otherwise, assume that goods are good. 4. Diminishing marginal rate of substitution we will hold off on this until later in the notes. Indifference Curves I think I may belabor the indifference curves a bit, but I do not want you think these came from right field. An indifference curve shows the various combinations of market bundles that will yield some specific level of utility or satisfaction to the individual. 2 If two bundles are on the same indifference curve, they yield the same level of satisfaction. Likewise, if two bundles yield the same level of satisfaction, they will be on the same indifference curve. People invariably get shook up by the word utility. If you do not like this word, you will be fine if you think satisfaction, welfare, or enjoyment. Ultimately, we will be interested in what these indifference curves look like. With a little introspection we can learn quite a bit about their shape and slope and how they might differ across types of people. Are indifference curves positively or negatively sloped? Look at the following diagram and focus your attention on point A. The first thing to ask is what other point(s) could possibly be on the same indifference curve as point A, bearing in mind that two points on the same indifference curve indicate the same level of satisfaction. 2 We could draw indifference curves for any two goods. If our two goods where apples and pears, this indifference curve would show combinations of apples and pears that would yield the same level of utility. For those of you who have seen indifference curves before, changing the names of the goods (to leisure and real income or food and clothing) will not change the results in any meaningful fashion. What you learned before is still pertinent. 2

13 Econ311_lecture2 8/30/2006 2:44 PM Clothing (units per week) Y E W A Z G X D First, compare point A and point E. Can they represent the same level of satisfaction and therefore be on the same indifference curve? Clearly not, as point E has more food and more clothing than point A. Since more is preferred to less, people would prefer point E to point A, and thus point E would yield a higher satisfaction level. Since A and E cannot be on the same indifference curve, we know indifference curves will not be upward sloping. Compare point A and point Y. These cannot be on the same indifference curve either, as both A and Y have the same amount of food, but point Y has more clothing. People would prefer Y to A. Since A and Y cannot be on the same indifference curve, we know indifference curves cannot be vertical lines. Likewise, people will prefer point Z to point A. They both have the same amount of clothing, but point Z has more food and thus Z is preferred to A. Since A and Z cannot be on the same indifference curve, we know indifference curves cannot be horizontal lines. By using the same arguments, we know that none of W, G, and X can be on the same indifference curve as A. Therefore, the indifference curve through point A cannot be in any of the shaded parts of the graph, so it must be downward sloping. Another way to see this is to compare points A and D. Is it possible that these two points lie on the same indifference curve? Point D has more food but less clothing than point A. Is it possible that larger amount of food is just enough to offset the smaller amount of clothing? It is. So it is possible that A and D are on the same indifference curve. If you add some food, and want to stay on the same indifference curve (same level of satisfaction), you have to take away some clothing. What have we learned after all this? We have learned that indifference curves are downward sloping. See your textbook for a very similar discussion. Are Indifference Curves Convex, Concave, or Straight? Food (units per week) Knowing they are downward sloping, there are still three possibilities. Indifference curves could be straight lines, they could be convex, or they could be concave. The straight-line indifference curve is in black, the convex indifference curve is blue (bowed inward farthest from the origin), and the concave indifference curve in red (bowed outward closest to the origin). 3

14 Econ311_lecture2 8/30/2006 2:44 PM Clothing (per week) A Convex Concave Food (per week) We will start with the concave indifference curve, see what it implies about people s tradeoff between food and clothing, and then decide if it makes sense. Concave Indifference Curves? Clothing (per week) A B C 1 C D C 2 E F Food (per week) Imagine staring at point A. The person is consuming all clothing and no food. Consider how much clothing a person would be willing to give up to get an additional unit of food. The indifference curve suggests that the person would be willing to give up a tiny amount of clothing for that first unit of food. It is too small to label. Weird, eh? Now, continue with the same logic at point C. Now, the consumer has a bit less clothing than before and more food than previously. Consider again how much clothing the person would be willing to give up for an additional unit of food. The indifference curve suggests that the consumer would be willing to give up a fair chuck of clothing ( C 1 ) for an extra unit of food. Finally, continue on to point E. Here the consumer has a quite a bit of food, and little clothing. Again, ask how much clothing the consumer would be willing to give up for an additional unit of food. The indifference curve suggests that answer would be ( C 2 ). 4

15 Econ311_lecture2 8/30/2006 2:44 PM Does it make sense that as a consumer has more and more food that the consumer would be willing to give up larger amounts of clothing to get an additional unit of food? It does not make sense to me in fact, it seems entirely backwards. This suggests that indifference curves are not, in fact, concave. Convex Indifference Curves! Clothing (per week) C 1 A B C 2 C D E F Food (per week) Now, try that same train of thought, only this time, with the convex indifference curve. Imagine starting at point A. The person has relatively large amount of clothing, but little food. Consider giving up some clothing to increase food by one unit. The indifference curve suggests that it would a person would give up a sizable amount of clothing ( C 1 ) in order to be able to consume one additional unit of food. Now consider completing this process again, only this time starting at point C. The person already has less clothing and more food than they did at point A. Again, consider giving up some clothing to increase food by another unit. The indifference curve suggests it the person would be willing to give up a smaller amount of clothing ( C 2 ) to get another unit of food. Repeat this one last time starting at point E. As clothing becomes less plentiful and food more plentiful, the shape of this indifference curve suggests that the person would be willing to only extremely small amounts of clothing for an additional unit of food (or looked a bit differently, the consumer would be willing to give up a lot of food for a small amount of clothing). I think that makes more sense than the concave case. In fact, indifference curves are convex. What have we learned? Indifference curves are convex. 3 3 Hey, what happened to linear? A linear indifference curve would imply a constant tradeoff between food and clothing. A person who would be willing to give up their first unit of food for, say, a unit of clothing, would also be willing to give up their last unit of food for a unit of clothing. We reject linear indifference curves by the same argument with which we rejected concave indifference curves. Also, see the discussion of perfect substitutes at the end of these notes 5

16 Econ311_lecture2 8/30/2006 2:44 PM Marginal Rate of Substitution The marginal rate of substitution of food for clothing (MRS) is the maximum amount of clothing that a person is willing to give up to obtain an additional unit of food. A way you can remember this is that you are giving up food for clothing. To be consistent, your textbook and I will always define the MRS as the amount of the good on the vertical axis that the consumer is willing to give up in order to obtain one extra unit of the good on the horizontal axis. You can think of the MRS as the value the individual places on an extra unit of a good (in terms of another good). In our case, the MRS is the value of a unit of food, measured in clothing. The MRS is also equal to the absolute value of the slope of the indifference curve. Here s a picture ripped right out of your book (Figure 3.5) Clothing (per week) 16 A 10 B 6 D 4 3 E G Food (per week) Slope Slope MRS Steepness C F 6 1 From A to B = = 6 C F 4 1 From B to D = = 4 C F 2 = = 1 From D to E 2 C F 1 1 From E to G = = 1 C F C F C F C F 6 = 1 4 = 1 2 = 1 1 = 1 = 6 = 6 = 4 = 4 = 2 = 2 = 1 = 1 6 steepest 4 steep 2 flat 1 flattest One important thing to see is that as we move to the right along an indifference curve, the absolute value of the slope decreases (the indifference curve becomes flatter), so the MRS decreases. Alternatively, as 6

17 Econ311_lecture2 8/30/2006 2:44 PM people have more and more food, they are willing to give up less and less clothing for an additional unit of food. 4 If you go back to our assumptions about preferences, item 4 was that we assumed that preferences displayed a diminishing marginal rate of substitution. Since a diminishing marginal rate of substitution occurs along a convex indifference curve, this is the same as assuming that indifference curves are convex. And in fact, if you wanted to know the MRS at any one point on an indifference curve, all you would have to do is draw a tangent line at that point and figure out the (absolute value) of that tangent line s slope. See the review of tangent lines at the end of these notes if you need a refresher. Indifference Map We have only drawn one indifference curve so far. If you go back to the first picture in these notes, the only one that was drawn went through point A. We noted that point A and Point E were not on the same indifference curve. However, point E is on some indifference curve. Every combination of food and clothing is on some indifference curve. There are hundreds, thousands, in fact an infinite number of indifference curves. Thank goodness, we do not want to take the time to draw them all. Often a series of indifference curves is called a family of indifference curves or an indifference map. Usually we will draw in one or two to illustrate the general shape. From time to time in this class, you will want to add in an indifference curve. Go ahead! I have drawn a family of indifference curves below. It is important to remember that as we move to the northeast, the level of satisfaction associated with an indifference curve increases. Do you know why? On which indifference curve would you rather be? Remember, goods are good. Clothing (per week) U 1 U 2 U 3 Satisfaction Level Increases Food (per week) Can Indifference Curves cross? No. It would contradict our assumptions about preferences. Suppose for a second indifference curve did cross. Then, because A and B are on the same indifference curve, we could conclude that A~B. Because A and C are on same indifference curve, A~C. From transitivity, we can conclude B~C. However, clearly, this is false. Goods are goods, and therefore C is preferred to B. Therefore, indifference curves cannot cross. 4 If you are having a deja-vu moment back to the discussion on convex indifference curves, you are right. We chose convex indifference curves because they are consistent with a MRS that decreases as we move to the right along indifference curve. A linear indifference curve would have a constant MRS, while a concave indifference curve would produce a MRS that increase as we move to the right along an indifference curve. 7

18 Econ311_lecture2 8/30/2006 2:44 PM Clothing (per week) U 1 U 2 B C A Food (per week) Different Preferences Hey, wait a minute, I though indifference curves were about people s preferences. All this talk about trading off food clothing but aren t we supposed to be discussing preferences? Surely, it is the case that different people have different preferences. Can we show different preferences with differently shaped indifference curves? We sure can. Think of the first part as flat vs. steep. Food Lovers Below is a picture with relatively steep indifference curves. As you know, steep indifference curves mean large values of the MRS. A large value of the MRS means the consumer is willing to give up a lot of clothing for an additional unit of food. This means that a unit of food is very valuable. Clothing (per week) U 1 U 2 U 3 Clothing lovers 0 Food (per week) Below is a picture with relatively flat indifference curves. Again, flat indifference curves mean low values of the MRS. This means a person would be willing to give up very little clothing for an additional unit of food. Looked a bit differently, this person would be willing to give up a large amount of food for an additional unit of clothing. 8

19 Econ311_lecture2 8/30/2006 2:44 PM Clothing (per week) U 3 U 2 U 1 Food (per week) We have to be careful about comparing these two pictures maybe I am being a bit uptight but I am cheating a bit. Since the slope of the indifference curves (the steepness of the indifference curves) varies along each individual indifference curve, what we can just say that the second set of indifference curves are flatter than the first it really depends on where we measure the slope or steepness. To be super precise, we would really have to compare the slope of the two different sets of indifference curves at point on the graph. In the picture below, at point C, it is clear that person A (the food lover) has a steeper indifference curve (a higher MRS) than person B (the clothing lover). Since person has a migher MRS than person B, person A is displays a stronger preference for food than person B person A is willing to give up more clothing for an additional unit of food. Clothing (per week) C U B U A 0 Food (per week) Weird preferences perfect substitutes and perfect complements See if you can t figure this one out yourselves. Say I asked you to draw two sets of indifference curves. One of them represents goods that are perfect substitutes. Think of perhaps red jelly beans and green jelly beans. Consumers don t care whether they have red or green they are perfect substitutes. Can you draw a family of indifference curves? The other set of preferences represents perfect complements. Think of one good being right shoe and the other good a left shoe. What would the indifferences curves look like here? Do no cheat and look at the pictures just yet. First, think about jelly beans. Suppose we considered a couple of bundles. Will they all be on the same indifference curve? What does this imply about the shape of the indifference curves? 9

20 Econ311_lecture2 8/30/2006 2:44 PM Bundle A 2 green jelly beans, 2 red jelly beans Bundle B 1 green jelly bean, 3 red jelly beans Bundle C 3 green jelly beans, 1 red jelly bean Bunlde D 2 green jelly beans, 3 red jelly beans The answers are below in the footnote. 5 Now, think about shoes. Consider the following bundles. Will they all be on the same indifference curve? The answers are below in the footnote. 6 Bundle Z 2 right shoes, 2 left shoes Bundle Y 1 left shoe, 3 right shoes Bundle X 3 left shoes, 1 right shoe Bundle W 2 right shoes, 3 left shoes Bundle V 3 right shoes, 2 left shoes Bundle U 4 right shoes, 2 left shoes Take a look at the pictures: Green Jelly Beans Perfect Substitutes Left Shoes Perfect Complements U 3 U 2 U 1 U 1 U 2 U 3 Red Jelly Beans Right Shoes Indifference curves contain a lot of information Indifference curves do truly contain a lot of information. The steepness of the indifference curves tells you something about relative preferences for the goods on the x axis and y-axis. The curvature of the indifference curve tells you about whether the consumers view the goods as substitutes (straighter) or complements (curvier). The convexity of the indifference curves tell you that people have diminishing marginal rates of substitution. Next up, we will talk about the constraints. Next, we will talk about the constraints What should I read? Believe it or not, it is all in Chapter Bundles A, B, and C are all on the same indifference curves. You will thus find that if the goods are perfect substitutes, indifference curves will be a straight line. Bundle D will be on a higher indifference curve. 6 Bundles Y and X are on the same indifference curve, while bundles U, V, W, and Z are all on the same indifference curve (which is higher level of satisfaction than the indifference curve than contains Y and X). With perfect complementarity, a right shoe (if not paired with a left show) provides no satisfaction. 10

21 Econ311_lecture2 8/30/2006 2:44 PM Appendix: Review of Tangent Lines / Slopes Surely, you remember that the equation of the slope of a line from fifth grade: slope = rise run = y x 1 1 y x 0 0 This is super easy when the function is a straight line, as the slope of a straight line is constant. That is, it does not change depending on where we are on the line. In this class, we will be look at some relations ships where the slope changes as we move along our function or curve. To determine the slope of a function at a particular spot we can draw what is called a tangent line. To draw a tangent line you simply find a line that just barely touches the function at that point. See the picture below, where I have drawn four tangent lines one each for points A, B, C and D. To determine the slope of the function at each point, we would just figure out the slope of the tangent line. Pull out a piece of paper and a straight edge, draw something like this function, and draw a couple tangent lines. It is not hard. C D B A What you will be able to need to recognize for this course is how the slope changes. At point A, the slope of the function is relative low. As we move from point A to point B, the slope of the function is increasing. Notice at point B, the function is steeper than it was at point A. In addition, notice as we move from B to C to D, the slope of the function is decreasing. The tangent lines are becoming flatter. Flat tangent lines are associated with lower slopes while steep tangent lines indicate larger slopes. This gets a bit trickier when it comes to a function that has a negative slope like the one pictured below. It is still the same process to draw the tangent lines, and we will still figure out the slope by looking at the slope of the tangent lines. What is trickier is how to talk about comparing these slopes. 11

22 Econ311_lecture2 8/30/2006 2:44 PM A B C Perhaps some made up numbers are in order here. At point A, the slope might be something like -4, while at point B the slope might be -3, and at point C The confusion comes about when we talk about which has the larger slope. Technically, point C has the largest slope because -1.5 is the largest of those three numbers (it is the least negative). Now the confusion becomes that steep lines are associated with lower slopes and flat lines are associated with larger slopes. Unfortunately, this is the exact opposite as we had in the situation above. We are not going to like this confusion. So instead of talking about where the slope is the largest, what we will do is talk about where the slope is the steepest. I think everyone would agree that the function is the steepest at point A and become less steep at we scoot down to point C. I think everyone would agree that the function becomes more flat as we move from point A to B to C. If you look at the picture on the previous page, point B is the steepest. If you like math and are not scared away by absolute values, we could say that slopes that are large in absolute value are steeper, while slopes that are small in absolute value are flatter. We could also say that the absolute value of the slope falls as we move from point A to B to C. If you can handle this, this is the way to go. If you cannot, just think in terms of "steeper" and "flatter" and you will be fine. 12

23 Econ311_lecture3 9/11/ :04 AM Budget Constraints and Consumer Choice The last topic discussed consumer s preferences. We next move on to the consumer s constraints. While we have learned how consumers might rank bundles, some market bundles are not affordable or attainable. The tool we will use to depict those attainable bundles will be the budget constraint or budget line. I think you will find budget constraints are easy. When we figure these out, we ll finally let the consumer choose the optimal bundle. This will be called consumer choice. Budget Constraints Your text s definition of a budget line is that it indicates all combinations of food and clothing for which the total amount of money spending is equal to income. I think it is slightly better to consider the budget line as indicating those combinations of food and clothing that are attainable for a given level of income. Points inside the budget line are attainable (affordable), but there would still be money leftover to spend. Points outside the budget line are unattainable the consumer does not have enough income to purchase them. Points on the budget line are attainable and result in the consumer s income being exhausted there is no income leftover. The budget line is expressed as: PF F + PC C = I Where P F and P C are the prices of food and clothing, respectively, F and C are the amounts of food and clothing, respectively, and I is the consumer s income. In this format, it is clear to the see that the first term (P F * F) represents the amount spent on food and the second term (P C * C) represents the amount spent on clothing. If all money is spent, then these two terms will add to the total income level. 1 We, of course, would like to graph this budget line. As we put C on the vertical axis, if we were to solve this expression for C, it will be easier to graph the budget line. PC C = I PF F I PF C = F P P C C This expression, I believe, can lead to a few insights. It turns out that the first term (I / P C ) is the vertical intercept of the budget constraint. It represents the amount of clothing that could be purchased, if all income were spent on clothing. The slope of the line is clearly given by (-P F /P C ). This tells us that the slope or steepness of the budget line depends on the relative prices of food and clothing. One more tip for drawing these budget lines, the horizontal intercept will be given by I / P F, and would represent the amount of food that could be purchased if all income were spend on food. 1 What about saving and/or borrowing? For simplicity, we will ignore the possibility of saving or borrowing for the time being. Essentially, we are assuming that consumer must decide to spend their income on either food (today) or clothing (today). 1