Intermediate Microeconomics - Budget Constraint PDF

CHAPTER 2 BUDGET CONSTRAINT The economic theory of the consumer is very simple: economists assume that consumers choose the best bundle of goods they can aﬀord. To give content to this theory, we have to describe more precisely what we mean by “best” and what we mean by “can aﬀord.” In this chapter we will examine how to describe what a consumer can aﬀord; the next chapter will focus on the concept of how the consumer determines what is best. We will then be able to undertake a detailed study of the implications of this simple model of consumer behavior. 2.1 The Budget Constraint We begin by examining the concept of the budget constraint. Suppose that there is some set of goods from which the consumer can choose. In real life there are many goods to consume, but for our purposes it is conve- nient to consider only the case of two goods, since we can then depict the consumer’s choice behavior graphically. We will indicate the consumer’s consumption bundle by (x1 , x2 ). This is simply a list of two numbers that tells us how much the consumer is choos- ing to consume of good 1, x1 , and how much the consumer is choosing to TWO GOODS ARE OFTEN ENOUGH 21 consume of good 2, x2. Sometimes it is convenient to denote the consumer’s bundle by a single symbol like X, where X is simply an abbreviation for the list of two numbers (x1 , x2 ). We suppose that we can observe the prices of the two goods, (p1 , p2 ), and the amount of money the consumer has to spend, m. Then the budget constraint of the consumer can be written as p1 x1 + p2 x2 ≤ m. (2.1) Here p1 x1 is the amount of money the consumer is spending on good 1, and p2 x2 is the amount of money the consumer is spending on good 2. The budget constraint of the consumer requires that the amount of money spent on the two goods be no more than the total amount the consumer has to spend. The consumer’s aﬀordable consumption bundles are those that don’t cost any more than m. We call this set of aﬀordable consumption bundles at prices (p1 , p2 ) and income m the budget set of the consumer. 2.2 Two Goods Are Often Enough The two-good assumption is more general than you might think at ﬁrst, since we can often interpret one of the goods as representing everything else the consumer might want to consume. For example, if we are interested in studying a consumer’s demand for milk, we might let x1 measure his or her consumption of milk in quarts per month. We can then let x2 stand for everything else the consumer might want to consume. When we adopt this interpretation, it is convenient to think of good 2 as being the dollars that the consumer can use to spend on other goods. Under this interpretation the price of good 2 will automatically be 1, since the price of one dollar is one dollar. Thus the budget constraint will take the form p1 x1 + x2 ≤ m. (2.2) This expression simply says that the amount of money spent on good 1, p1 x1 , plus the amount of money spent on all other goods, x2 , must be no more than the total amount of money the consumer has to spend, m. We say that good 2 represents a composite good that stands for ev- erything else that the consumer might want to consume other than good 1. Such a composite good is invariably measured in dollars to be spent on goods other than good 1. As far as the algebraic form of the budget constraint is concerned, equation (2.2) is just a special case of the formula given in equation (2.1), with p2 = 1, so everything that we have to say about the budget constraint in general will hold under the composite-good interpretation. 22 BUDGET CONSTRAINT (Ch. 2) 2.3 Properties of the Budget Set The budget line is the set of bundles that cost exactly m: p1 x1 + p2 x2 = m. (2.3) These are the bundles of goods that just exhaust the consumer’s income. The budget set is depicted in Figure 2.1. The heavy line is the budget line—the bundles that cost exactly m—and the bundles below this line are those that cost strictly less than m. x2 Vertical intercept = m/p 2 Budget line; slope = – p1 /p2 Budget set Horizontal intercept = m/p1 x1 Figure The budget set. The budget set consists of all bundles that 2.1 are aﬀordable at the given prices and income. We can rearrange the budget line in equation (2.3) to give us the formula m p1 x2 = − x1. (2.4) p2 p2 This is the formula for a straight line with a vertical intercept of m/p2 and a slope of −p1 /p2. The formula tells us how many units of good 2 the consumer needs to consume in order to just satisfy the budget constraint if she is consuming x1 units of good 1. PROPERTIES OF THE BUDGET SET 23 Here is an easy way to draw a budget line given prices (p1 , p2 ) and income m. Just ask yourself how much of good 2 the consumer could buy if she spent all of her money on good 2. The answer is, of course, m/p2. Then ask how much of good 1 the consumer could buy if she spent all of her money on good 1. The answer is m/p1. Thus the horizontal and vertical intercepts measure how much the consumer could get if she spent all of her money on goods 1 and 2, respectively. In order to depict the budget line just plot these two points on the appropriate axes of the graph and connect them with a straight line. The slope of the budget line has a nice economic interpretation. It mea- sures the rate at which the market is willing to “substitute” good 1 for good 2. Suppose for example that the consumer is going to increase her consumption of good 1 by Δx1.1 How much will her consumption of good 2 have to change in order to satisfy her budget constraint? Let us use Δx2 to indicate her change in the consumption of good 2. Now note that if she satisﬁes her budget constraint before and after making the change she must satisfy p1 x1 + p2 x2 = m and p1 (x1 + Δx1 ) + p2 (x2 + Δx2 ) = m. Subtracting the ﬁrst equation from the second gives p1 Δx1 + p2 Δx2 = 0. This says that the total value of the change in her consumption must be zero. Solving for Δx2 /Δx1 , the rate at which good 2 can be substituted for good 1 while still satisfying the budget constraint, gives Δx2 p1 =−. Δx1 p2 This is just the slope of the budget line. The negative sign is there since Δx1 and Δx2 must always have opposite signs. If you consume more of good 1, you have to consume less of good 2 and vice versa if you continue to satisfy the budget constraint. Economists sometimes say that the slope of the budget line measures the opportunity cost of consuming good 1. In order to consume more of good 1 you have to give up some consumption of good 2. Giving up the opportunity to consume good 2 is the true economic cost of more good 1 consumption; and that cost is measured by the slope of the budget line. 1 The Greek letter Δ, delta, is pronounced “del-ta.” The notation Δx1 denotes the change in good 1. For more on changes and rates of changes, see the Mathematical Appendix. 24 BUDGET CONSTRAINT (Ch. 2) 2.4 How the Budget Line Changes When prices and incomes change, the set of goods that a consumer can aﬀord changes as well. How do these changes aﬀect the budget set? Let us ﬁrst consider changes in income. It is easy to see from equation (2.4) that an increase in income will increase the vertical intercept and not aﬀect the slope of the line. Thus an increase in income will result in a par- allel shift outward of the budget line as in Figure 2.2. Similarly, a decrease in income will cause a parallel shift inward. x2 m’/p2 Budget lines m/p2 Slope = –p1/p 2 m/p1 m’/p1 x1 Figure Increasing income. An increase in income causes a parallel 2.2 shift outward of the budget line. What about changes in prices? Let us ﬁrst consider increasing price 1 while holding price 2 and income ﬁxed. According to equation (2.4), increasing p1 will not change the vertical intercept, but it will make the budget line steeper since p1 /p2 will become larger. Another way to see how the budget line changes is to use the trick de- scribed earlier for drawing the budget line. If you are spending all of your money on good 2, then increasing the price of good 1 doesn’t change the maximum amount of good 2 you could buy—thus the vertical inter- cept of the budget line doesn’t change. But if you are spending all of your money on good 1, and good 1 becomes more expensive, then your HOW THE BUDGET LINE CHANGES 25 consumption of good 1 must decrease. Thus the horizontal intercept of the budget line must shift inward, resulting in the tilt depicted in Fig- ure 2.3. x2 m/p2 Budget lines Slope = –p'1 /p2 Slope = –p1 /p2 m/p'1 m/p1 x1 Increasing price. If good 1 becomes more expensive, the Figure budget line becomes steeper. 2.3 What happens to the budget line when we change the prices of good 1 and good 2 at the same time? Suppose for example that we double the prices of both goods 1 and 2. In this case both the horizontal and vertical intercepts shift inward by a factor of one-half, and therefore the budget line shifts inward by one-half as well. Multiplying both prices by two is just like dividing income by 2. We can also see this algebraically. Suppose our original budget line is p1 x1 + p2 x2 = m. Now suppose that both prices become t times as large. Multiplying both prices by t yields tp1 x1 + tp2 x2 = m. But this equation is the same as m p1 x1 + p2 x2 =. t Thus multiplying both prices by a constant amount t is just like dividing income by the same constant t. It follows that if we multiply both prices 26 BUDGET CONSTRAINT (Ch. 2) by t and we multiply income by t, then the budget line won’t change at all. We can also consider price and income changes together. What happens if both prices go up and income goes down? Think about what happens to the horizontal and vertical intercepts. If m decreases and p1 and p2 both increase, then the intercepts m/p1 and m/p2 must both decrease. This means that the budget line will shift inward. What about the slope of the budget line? If price 2 increases more than price 1, so that −p1 /p2 decreases (in absolute value), then the budget line will be ﬂatter; if price 2 increases less than price 1, the budget line will be steeper. 2.5 The Numeraire The budget line is deﬁned by two prices and one income, but one of these variables is redundant. We could peg one of the prices, or the income, to some ﬁxed value, and adjust the other variables so as to describe exactly the same budget set. Thus the budget line p1 x1 + p2 x2 = m is exactly the same budget line as p1 m x1 + x2 = p2 p2 or p1 p2 x1 + x2 = 1, m m since the ﬁrst budget line results from dividing everything by p2 , and the second budget line results from dividing everything by m. In the ﬁrst case, we have pegged p2 = 1, and in the second case, we have pegged m = 1. Pegging the price of one of the goods or income to 1 and adjusting the other price and income appropriately doesn’t change the budget set at all. When we set one of the prices to 1, as we did above, we often refer to that price as the numeraire price. The numeraire price is the price relative to which we are measuring the other price and income. It will occasionally be convenient to think of one of the goods as being a numeraire good, since there will then be one less price to worry about. 2.6 Taxes, Subsidies, and Rationing Economic policy often uses tools that aﬀect a consumer’s budget constraint, such as taxes. For example, if the government imposes a quantity tax, this means that the consumer has to pay a certain amount to the government TAXES, SUBSIDIES, AND RATIONING 27 for each unit of the good he purchases. In the U.S., for example, we pay about 15 cents a gallon as a federal gasoline tax. How does a quantity tax aﬀect the budget line of a consumer? From the viewpoint of the consumer the tax is just like a higher price. Thus a quantity tax of t dollars per unit of good 1 simply changes the price of good 1 from p1 to p1 + t. As we’ve seen above, this implies that the budget line must get steeper. Another kind of tax is a value tax. As the name implies this is a tax on the value—the price—of a good, rather than the quantity purchased of a good. A value tax is usually expressed in percentage terms. Most states in the U.S. have sales taxes. If the sales tax is 6 percent, then a good that is priced at $1 will actually sell for $1.06. (Value taxes are also known as ad valorem taxes.) If good 1 has a price of p1 but is subject to a sales tax at rate τ , then the actual price facing the consumer is (1 + τ )p1.2 The consumer has to pay p1 to the supplier and τ p1 to the government for each unit of the good so the total cost of the good to the consumer is (1 + τ )p1. A subsidy is the opposite of a tax. In the case of a quantity subsidy, the government gives an amount to the consumer that depends on the amount of the good purchased. If, for example, the consumption of milk were subsidized, the government would pay some amount of money to each consumer of milk depending on the amount that consumer purchased. If the subsidy is s dollars per unit of consumption of good 1, then from the viewpoint of the consumer, the price of good 1 would be p1 − s. This would therefore make the budget line ﬂatter. Similarly an ad valorem subsidy is a subsidy based on the price of the good being subsidized. If the government gives you back $1 for every $2 you donate to charity, then your donations to charity are being subsidized at a rate of 50 percent. In general, if the price of good 1 is p1 and good 1 is subject to an ad valorem subsidy at rate σ, then the actual price of good 1 facing the consumer is (1 − σ)p1.3 You can see that taxes and subsidies aﬀect prices in exactly the same way except for the algebraic sign: a tax increases the price to the consumer, and a subsidy decreases it. Another kind of tax or subsidy that the government might use is a lump- sum tax or subsidy. In the case of a tax, this means that the government takes away some ﬁxed amount of money, regardless of the individual’s be- havior. Thus a lump-sum tax means that the budget line of a consumer will shift inward because his money income has been reduced. Similarly, a lump-sum subsidy means that the budget line will shift outward. Quantity taxes and value taxes tilt the budget line one way or the other depending 2 The Greek letter τ , tau, rhymes with “wow.” 3 The Greek letter σ is pronounced “sig-ma.” 28 BUDGET CONSTRAINT (Ch. 2) on which good is being taxed, but a lump-sum tax shifts the budget line inward. Governments also sometimes impose rationing constraints. This means that the level of consumption of some good is ﬁxed to be no larger than some amount. For example, during World War II the U.S. government rationed certain foods like butter and meat. Suppose, for example, that good 1 were rationed so that no more than x1 could be consumed by a given consumer. Then the budget set of the consumer would look like that depicted in Figure 2.4: it would be the old budget set with a piece lopped oﬀ. The lopped-oﬀ piece consists of all the consumption bundles that are aﬀordable but have x1 > x1. x2 Budget line Budget set x1 x1 Figure Budget set with rationing. If good 1 is rationed, the section 2.4 of the budget set beyond the rationed quantity will be lopped oﬀ. Sometimes taxes, subsidies, and rationing are combined. For example, we could consider a situation where a consumer could consume good 1 at a price of p1 up to some level x1 , and then had to pay a tax t on all consumption in excess of x1. The budget set for this consumer is depicted in Figure 2.5. Here the budget line has a slope of −p1 /p2 to the left of x1 , and a slope of −(p1 + t)/p2 to the right of x1. TAXES, SUBSIDIES, AND RATIONING 29 x2 Budget line Slope = – p1/p 2 Budget set Slope = – (p1 + t )/p 2 x1 x1 Taxing consumption greater than x1. In this budget set Figure the consumer must pay a tax only on the consumption of good 2.5 1 that is in excess of x1 , so the budget line becomes steeper to the right of x1. EXAMPLE: The Food Stamp Program Since the Food Stamp Act of 1964 the U.S. federal government has provided a subsidy on food for poor people. The details of this program have been adjusted several times. Here we will describe the economic eﬀects of one of these adjustments. Before 1979, households who met certain eligibility requirements were allowed to purchase food stamps, which could then be used to purchase food at retail outlets. In January 1975, for example, a family of four could receive a maximum monthly allotment of $153 in food coupons by participating in the program. The price of these coupons to the household depended on the household income. A family of four with an adjusted monthly income of $300 paid $83 for the full monthly allotment of food stamps. If a family of four had a monthly income of $100, the cost for the full monthly allotment would have been $25.4 The pre-1979 Food Stamp program was an ad valorem subsidy on food. The rate at which food was subsidized depended on the household income. 4 These ﬁgures are taken from Kenneth Clarkson, Food Stamps and Nutrition, Ameri- can Enterprise Institute, 1975. 30 BUDGET CONSTRAINT (Ch. 2) The family of four that was charged $83 for their allotment paid $1 to receive $1.84 worth of food (1.84 equals 153 divided by 83). Similarly, the household that paid $25 was paying $1 to receive $6.12 worth of food (6.12 equals 153 divided by 25). The way that the Food Stamp program aﬀected the budget set of a household is depicted in Figure 2.6A. Here we have measured the amount of money spent on food on the horizontal axis and expenditures on all other goods on the vertical axis. Since we are measuring each good in terms of the money spent on it, the “price” of each good is automatically 1, and the budget line will therefore have a slope of −1. If the household is allowed to buy $153 of food stamps for $25, then this represents roughly an 84 percent (= 1 − 25/153) subsidy of food purchases, so the budget line will have a slope of roughly −.16 (= 25/153) until the household has spent $153 on food. Each dollar that the household spends on food up to $153 would reduce its consumption of other goods by about 16 cents. After the household spends $153 on food, the budget line facing it would again have a slope of −1. OTHER OTHER GOODS GOODS Budget line Budget line with food with food stamps stamps Budget Budget line line without without food food stamps stamps $153 FOOD $200 FOOD A B Figure Food stamps. How the budget line is aﬀected by the Food 2.6 Stamp program. Part A shows the pre-1979 program and part B the post-1979 program. These eﬀects lead to the kind of “kink” depicted in Figure 2.6. House- holds with higher incomes had to pay more for their allotment of food stamps. Thus the slope of the budget line would become steeper as house- hold income increased. In 1979 the Food Stamp program was modiﬁed. Instead of requiring that SUMMARY 31 households purchase food stamps, they are now simply given to qualiﬁed households. Figure 2.6B shows how this aﬀects the budget set. Suppose that a household now receives a grant of $200 of food stamps a month. Then this means that the household can consume $200 more food per month, regardless of how much it is spending on other goods, which implies that the budget line will shift to the right by $200. The slope will not change: $1 less spent on food would mean $1 more to spend on other things. But since the household cannot legally sell food stamps, the maximum amount that it can spend on other goods does not change. The Food Stamp program is eﬀectively a lump-sum subsidy, except for the fact that the food stamps can’t be sold. 2.7 Budget Line Changes In the next chapter we will analyze how the consumer chooses an optimal consumption bundle from his or her budget set. But we can already state some observations here that follow from what we have learned about the movements of the budget line. First, we can observe that since the budget set doesn’t change when we multiply all prices and income by a positive number, the optimal choice of the consumer from the budget set can’t change either. Without even ana- lyzing the choice process itself, we have derived an important conclusion: a perfectly balanced inﬂation—one in which all prices and all incomes rise at the same rate—doesn’t change anybody’s budget set, and thus cannot change anybody’s optimal choice. Second, we can make some statements about how well-oﬀ the consumer can be at diﬀerent prices and incomes. Suppose that the consumer’s income increases and all prices remain the same. We know that this represents a parallel shift outward of the budget line. Thus every bundle the consumer was consuming at the lower income is also a possible choice at the higher income. But then the consumer must be at least as well-oﬀ at the higher income as at the lower income—since he or she has the same choices avail- able as before plus some more. Similarly, if one price declines and all others stay the same, the consumer must be at least as well-oﬀ. This simple ob- servation will be of considerable use later on. Summary 1. The budget set consists of all bundles of goods that the consumer can aﬀord at given prices and income. We will typically assume that there are only two goods, but this assumption is more general than it seems. 2. The budget line is written as p1 x1 + p2 x2 = m. It has a slope of −p1 /p2 , a vertical intercept of m/p2 , and a horizontal intercept of m/p1. 32 BUDGET CONSTRAINT (Ch. 2) 3. Increasing income shifts the budget line outward. Increasing the price of good 1 makes the budget line steeper. Increasing the price of good 2 makes the budget line ﬂatter. 4. Taxes, subsidies, and rationing change the slope and position of the budget line by changing the prices paid by the consumer. REVIEW QUESTIONS 1. Originally the consumer faces the budget line p1 x1 + p2 x2 = m. Then the price of good 1 doubles, the price of good 2 becomes 8 times larger, and income becomes 4 times larger. Write down an equation for the new budget line in terms of the original prices and income. 2. What happens to the budget line if the price of good 2 increases, but the price of good 1 and income remain constant? 3. If the price of good 1 doubles and the price of good 2 triples, does the budget line become ﬂatter or steeper? 4. What is the deﬁnition of a numeraire good? 5. Suppose that the government puts a tax of 15 cents a gallon on gasoline and then later decides to put a subsidy on gasoline at a rate of 7 cents a gallon. What net tax is this combination equivalent to? 6. Suppose that a budget equation is given by p1 x1 + p2 x2 = m. The government decides to impose a lump-sum tax of u, a quantity tax on good 1 of t, and a quantity subsidy on good 2 of s. What is the formula for the new budget line? 7. If the income of the consumer increases and one of the prices decreases at the same time, will the consumer necessarily be at least as well-oﬀ? CHAPTER 3 PREFERENCES We saw in Chapter 2 that the economic model of consumer behavior is very simple: people choose the best things they can aﬀord. The last chapter was devoted to clarifying the meaning of “can aﬀord,” and this chapter will be devoted to clarifying the economic concept of “best things.” We call the objects of consumer choice consumption bundles. This is a complete list of the goods and services that are involved in the choice problem that we are investigating. The word “complete” deserves empha- sis: when you analyze a consumer’s choice problem, make sure that you include all of the appropriate goods in the deﬁnition of the consumption bundle. If we are analyzing consumer choice at the broadest level, we would want not only a complete list of the goods that a consumer might consume, but also a description of when, where, and under what circumstances they would become available. After all, people care about how much food they will have tomorrow as well as how much food they have today. A raft in the middle of the Atlantic Ocean is very diﬀerent from a raft in the middle of the Sahara Desert. And an umbrella when it is raining is quite a diﬀerent good from an umbrella on a sunny day. It is often useful to think of the 34 PREFERENCES (Ch. 3) “same” good available in diﬀerent locations or circumstances as a diﬀerent good, since the consumer may value the good diﬀerently in those situations. However, when we limit our attention to a simple choice problem, the relevant goods are usually pretty obvious. We’ll often adopt the idea de- scribed earlier of using just two goods and calling one of them “all other goods” so that we can focus on the tradeoﬀ between one good and ev- erything else. In this way we can consider consumption choices involving many goods and still use two-dimensional diagrams. So let us take our consumption bundle to consist of two goods, and let x1 denote the amount of one good and x2 the amount of the other. The complete consumption bundle is therefore denoted by (x1 , x2 ). As noted before, we will occasionally abbreviate this consumption bundle by X. 3.1 Consumer Preferences We will suppose that given any two consumption bundles, (x1 , x2 ) and (y1 , y2 ), the consumer can rank them as to their desirability. That is, the consumer can determine that one of the consumption bundles is strictly better than the other, or decide that she is indiﬀerent between the two bundles. We will use the symbol ≻ to mean that one bundle is strictly preferred to another, so that (x1 , x2 ) ≻ (y1 , y2 ) should be interpreted as saying that the consumer strictly prefers (x1 , x2 ) to (y1 , y2 ), in the sense that she deﬁnitely wants the x-bundle rather than the y-bundle. This preference relation is meant to be an operational notion. If the consumer prefers one bundle to another, it means that he or she would choose one over the other, given the opportunity. Thus the idea of preference is based on the consumer’s behavior. In order to tell whether one bundle is preferred to another, we see how the consumer behaves in choice situations involving the two bundles. If she always chooses (x1 , x2 ) when (y1 , y2 ) is available, then it is natural to say that this consumer prefers (x1 , x2 ) to (y1 , y2 ). If the consumer is indiﬀerent between two bundles of goods, we use the symbol ∼ and write (x1 , x2 ) ∼ (y1 , y2 ). Indiﬀerence means that the consumer would be just as satisﬁed, according to her own preferences, consuming the bundle (x1 , x2 ) as she would be consuming the other bundle, (y1 , y2 ). If the consumer prefers or is indiﬀerent between the two bundles we say that she weakly prefers (x1 , x2 ) to (y1 , y2 ) and write (x1 , x2 ) (y1 , y2 ). These relations of strict preference, weak preference, and indiﬀerence are not independent concepts; the relations are themselves related! For example, if (x1 , x2 ) (y1 , y2 ) and (y1 , y2 ) (x1 , x2 ) we can conclude that (x1 , x2 ) ∼ (y1 , y2 ). That is, if the consumer thinks that (x1 , x2 ) is at least as good as (y1 , y2 ) and that (y1 , y2 ) is at least as good as (x1 , x2 ), then the consumer must be indiﬀerent between the two bundles of goods. ASSUMPTIONS ABOUT PREFERENCES 35 Similarly, if (x1 , x2 ) (y1 , y2 ) but we know that it is not the case that (x1 , x2 ) ∼ (y1 , y2 ), we can conclude that we must have (x1 , x2 ) ≻ (y1 , y2 ). This just says that if the consumer thinks that (x1 , x2 ) is at least as good as (y1 , y2 ), and she is not indiﬀerent between the two bundles, then it must be that she thinks that (x1 , x2 ) is strictly better than (y1 , y2 ). 3.2 Assumptions about Preferences Economists usually make some assumptions about the “consistency” of consumers’ preferences. For example, it seems unreasonable—not to say contradictory—to have a situation where (x1 , x2 ) ≻ (y1 , y2 ) and, at the same time, (y1 , y2 ) ≻ (x1 , x2 ). For this would mean that the consumer strictly prefers the x-bundle to the y-bundle... and vice versa. So we usually make some assumptions about how the preference relations work. Some of the assumptions about preferences are so fundamental that we can refer to them as “axioms” of consumer theory. Here are three such axioms about consumer preference. Complete. We assume that any two bundles can be compared. That is, given any x-bundle and any y-bundle, we assume that (x1 , x2 ) (y1 , y2 ), or (y1 , y2 ) (x1 , x2 ), or both, in which case the consumer is indiﬀerent between the two bundles. Reﬂexive. We assume that any bundle is at least as good as itself: (x1 , x2 ) (x1 , x2 ). Transitive. If (x1 , x2 ) (y1 , y2 ) and (y1 , y2 ) (z1 , z2 ), then we assume that (x1 , x2 ) (z1 , z2 ). In other words, if the consumer thinks that X is at least as good as Y and that Y is at least as good as Z, then the consumer thinks that X is at least as good as Z. The ﬁrst axiom, completeness, is hardly objectionable, at least for the kinds of choices economists generally examine. To say that any two bundles can be compared is simply to say that the consumer is able to make a choice between any two given bundles. One might imagine extreme situations involving life or death choices where ranking the alternatives might be diﬃcult, or even impossible, but these choices are, for the most part, outside the domain of economic analysis. The second axiom, reﬂexivity, is trivial. Any bundle is certainly at least as good as an identical bundle. Parents of small children may occasionally observe behavior that violates this assumption, but it seems plausible for most adult behavior. The third axiom, transitivity, is more problematic. It isn’t clear that transitivity of preferences is necessarily a property that preferences would have to have. The assumption that preferences are transitive doesn’t seem 36 PREFERENCES (Ch. 3) compelling on grounds of pure logic alone. In fact it’s not. Transitivity is a hypothesis about people’s choice behavior, not a statement of pure logic. Whether it is a basic fact of logic or not isn’t the point: it is whether or not it is a reasonably accurate description of how people behave that matters. What would you think about a person who said that he preferred a bundle X to Y , and preferred Y to Z, but then also said that he preferred Z to X? This would certainly be taken as evidence of peculiar behavior. More importantly, how would this consumer behave if faced with choices among the three bundles X, Y , and Z? If we asked him to choose his most preferred bundle, he would have quite a problem, for whatever bundle he chose, there would always be one that was preferred to it. If we are to have a theory where people are making “best” choices, preferences must satisfy the transitivity axiom or something very much like it. If preferences were not transitive there could well be a set of bundles for which there is no best choice. 3.3 Indifference Curves It turns out that the whole theory of consumer choice can be formulated in terms of preferences that satisfy the three axioms described above, plus a few more technical assumptions. However, we will ﬁnd it convenient to describe preferences graphically by using a construction known as indif- ference curves. Consider Figure 3.1 where we have illustrated two axes representing a consumer’s consumption of goods 1 and 2. Let us pick a certain consump- tion bundle (x1 , x2 ) and shade in all of the consumption bundles that are weakly preferred to (x1 , x2 ). This is called the weakly preferred set. The bundles on the boundary of this set—the bundles for which the consumer is just indiﬀerent to (x1 , x2 )—form the indiﬀerence curve. We can draw an indiﬀerence curve through any consumption bundle we want. The indiﬀerence curve through a consumption bundle consists of all bundles of goods that leave the consumer indiﬀerent to the given bundle. One problem with using indiﬀerence curves to describe preferences is that they only show you the bundles that the consumer perceives as being indiﬀerent to each other—they don’t show you which bundles are better and which bundles are worse. It is sometimes useful to draw small arrows on the indiﬀerence curves to indicate the direction of the preferred bundles. We won’t do this in every case, but we will do it in a few of the examples where confusion might arise. If we make no further assumptions about preferences, indiﬀerence curves can take very peculiar shapes indeed. But even at this level of generality, we can state an important principle about indiﬀerence curves: indiﬀerence curves representing distinct levels of preference cannot cross. That is, the situation depicted in Figure 3.2 cannot occur. EXAMPLES OF PREFERENCES 37 x2 Weakly preferred set: bundles weakly preferred to (x1, x2 ) x2 Indifference curve: bundles indifferent to (x1, x2 ) x1 x1 Weakly preferred set. The shaded area consists of all bun- Figure dles that are at least as good as the bundle (x1 , x2 ). 3.1 In order to prove this, let us choose three bundles of goods, X, Y , and Z, such that X lies only on one indiﬀerence curve, Y lies only on the other indiﬀerence curve, and Z lies at the intersection of the indiﬀerence curves. By assumption the indiﬀerence curves represent distinct levels of prefer- ence, so one of the bundles, say X, is strictly preferred to the other bundle, Y. We know that X ∼ Z and Z ∼ Y , and the axiom of transitivity there- fore implies that X ∼ Y. But this contradicts the assumption that X ≻ Y. This contradiction establishes the result—indiﬀerence curves representing distinct levels of preference cannot cross. What other properties do indiﬀerence curves have? In the abstract, the answer is: not many. Indiﬀerence curves are a way to describe preferences. Nearly any “reasonable” preferences that you can think of can be depicted by indiﬀerence curves. The trick is to learn what kinds of preferences give rise to what shapes of indiﬀerence curves. 3.4 Examples of Preferences Let us try to relate preferences to indiﬀerence curves through some exam- ples. We’ll describe some preferences and then see what the indiﬀerence curves that represent them look like. 38 PREFERENCES (Ch. 3) x2 Alleged indifference curves X Z Y x1 Figure Indiﬀerence curves cannot cross. If they did, X, Y , and 3.2 Z would all have to be indiﬀerent to each other and thus could not lie on distinct indiﬀerence curves. There is a general procedure for constructing indiﬀerence curves given a “verbal” description of the preferences. First plop your pencil down on the graph at some consumption bundle (x1 , x2 ). Now think about giving a little more of good 1, Δx1 , to the consumer, moving him to (x1 + Δx1 , x2 ). Now ask yourself how would you have to change the consumption of x2 to make the consumer indiﬀerent to the original consumption point? Call this change Δx2. Ask yourself the question “For a given change in good 1, how does good 2 have to change to make the consumer just indiﬀerent between (x1 + Δx1 , x2 + Δx2 ) and (x1 , x2 )?” Once you have determined this movement at one consumption bundle you have drawn a piece of the indiﬀerence curve. Now try it at another bundle, and so on, until you develop a clear picture of the overall shape of the indiﬀerence curves. Perfect Substitutes Two goods are perfect substitutes if the consumer is willing to substitute one good for the other at a constant rate. The simplest case of perfect substitutes occurs when the consumer is willing to substitute the goods on a one-to-one basis. Suppose, for example, that we are considering a choice between red pen- cils and blue pencils, and the consumer involved likes pencils, but doesn’t care about color at all. Pick a consumption bundle, say (10, 10). Then for this consumer, any other consumption bundle that has 20 pencils in it is EXAMPLES OF PREFERENCES 39 just as good as (10, 10). Mathematically speaking, any consumption bun- dle (x1 , x2 ) such that x1 + x2 = 20 will be on this consumer’s indiﬀerence curve through (10, 10). Thus the indiﬀerence curves for this consumer are all parallel straight lines with a slope of −1, as depicted in Figure 3.3. Bundles with more total pencils are preferred to bundles with fewer total pencils, so the direction of increasing preference is up and to the right, as illustrated in Figure 3.3. How does this work in terms of general procedure for drawing indiﬀerence curves? If we are at (10, 10), and we increase the amount of the ﬁrst good by one unit to 11, how much do we have to change the second good to get back to the original indiﬀerence curve? The answer is clearly that we have to decrease the second good by 1 unit. Thus the indiﬀerence curve through (10, 10) has a slope of −1. The same procedure can be carried out at any bundle of goods with the same results—in this case all the indiﬀerence curves have a constant slope of −1. x2 Indifference curves x1 Perfect substitutes. The consumer only cares about the total Figure number of pencils, not about their colors. Thus the indiﬀerence 3.3 curves are straight lines with a slope of −1. The important fact about perfect substitutes is that the indiﬀerence curves have a constant slope. Suppose, for example, that we graphed blue pencils on the vertical axis and pairs of red pencils on the horizontal axis. The indiﬀerence curves for these two goods would have a slope of −2, since the consumer would be willing to give up two blue pencils to get one more pair of red pencils. 40 PREFERENCES (Ch. 3) In the textbook we’ll primarily consider the case where goods are perfect substitutes on a one-for-one basis, and leave the treatment of the general case for the workbook. Perfect Complements Perfect complements are goods that are always consumed together in ﬁxed proportions. In some sense the goods “complement” each other. A nice example is that of right shoes and left shoes. The consumer likes shoes, but always wears right and left shoes together. Having only one out of a pair of shoes doesn’t do the consumer a bit of good. Let us draw the indiﬀerence curves for perfect complements. Suppose we pick the consumption bundle (10, 10). Now add 1 more right shoe, so we have (11, 10). By assumption this leaves the consumer indiﬀerent to the original position: the extra shoe doesn’t do him any good. The same thing happens if we add one more left shoe: the consumer is also indiﬀerent between (10, 11) and (10, 10). Thus the indiﬀerence curves are L-shaped, with the vertex of the L oc- curring where the number of left shoes equals the number of right shoes as in Figure 3.4. LEFT SHOES Indifference curves RIGHT SHOES Figure Perfect complements. The consumer always wants to con- 3.4 sume the goods in ﬁxed proportions to each other. Thus the indiﬀerence curves are L-shaped. EXAMPLES OF PREFERENCES 41 Increasing both the number of left shoes and the number of right shoes at the same time will move the consumer to a more preferred position, so the direction of increasing preference is again up and to the right, as illustrated in the diagram. The important thing about perfect complements is that the consumer prefers to consume the goods in ﬁxed proportions, not necessarily that the proportion is one-to-one. If a consumer always uses two teaspoons of sugar in her cup of tea, and doesn’t use sugar for anything else, then the indiﬀerence curves will still be L-shaped. In this case the corners of the L will occur at (2 teaspoons sugar, 1 cup tea), (4 teaspoons sugar, 2 cups tea) and so on, rather than at (1 right shoe, 1 left shoe), (2 right shoes, 2 left shoes), and so on. In the textbook we’ll primarily consider the case where the goods are consumed in proportions of one-for-one and leave the treatment of the general case for the workbook. Bads A bad is a commodity that the consumer doesn’t like. For example, sup- pose that the commodities in question are now pepperoni and anchovies— and the consumer loves pepperoni but dislikes anchovies. But let us suppose there is some possible tradeoﬀ between pepperoni and anchovies. That is, there would be some amount of pepperoni on a pizza that would compen- sate the consumer for having to consume a given amount of anchovies. How could we represent these preferences using indiﬀerence curves? Pick a bundle (x1 , x2 ) consisting of some pepperoni and some anchovies. If we give the consumer more anchovies, what do we have to do with the pepperoni to keep him on the same indiﬀerence curve? Clearly, we have to give him some extra pepperoni to compensate him for having to put up with the anchovies. Thus this consumer must have indiﬀerence curves that slope up and to the right as depicted in Figure 3.5. The direction of increasing preference is down and to the right—that is, toward the direction of decreased anchovy consumption and increased pepperoni consumption, just as the arrows in the diagram illustrate. Neutrals A good is a neutral good if the consumer doesn’t care about it one way or the other. What if a consumer is just neutral about anchovies?1 In this case his indiﬀerence curves will be vertical lines as depicted in Figure 3.6. 1 Is anybody neutral about anchovies? 42 PREFERENCES (Ch. 3) ANCHOVIES Indifference curves PEPPERONI Figure Bads. Here anchovies are a “bad,” and pepperoni is a “good” 3.5 for this consumer. Thus the indiﬀerence curves have a positive slope. ANCHOVIES Indifference curves PEPPERONI Figure A neutral good. The consumer likes pepperoni but is neutral 3.6 about anchovies, so the indiﬀerence curves are vertical lines. He only cares about the amount of pepperoni he has and doesn’t care at all about how many anchovies he has. The more pepperoni the better, but adding more anchovies doesn’t aﬀect him one way or the other. EXAMPLES OF PREFERENCES 43 Satiation We sometimes want to consider a situation involving satiation, where there is some overall best bundle for the consumer, and the “closer” he is to that best bundle, the better oﬀ he is in terms of his own preferences. For example, suppose that the consumer has some most preferred bundle of goods (x1 , x2 ), and the farther away he is from that bundle, the worse oﬀ he is. In this case we say that (x1 , x2 ) is a satiation point, or a bliss point. The indiﬀerence curves for the consumer look like those depicted in Figure 3.7. The best point is (x1 , x2 ) and points farther away from this bliss point lie on “lower” indiﬀerence curves. x2 Indifference curves x2 Satiation point x1 x1 Satiated preferences. The bundle (x1 , x2 ) is the satiation Figure point or bliss point, and the indiﬀerence curves surround this 3.7 point. In this case the indiﬀerence curves have a negative slope when the con- sumer has “too little” or “too much” of both goods, and a positive slope when he has “too much” of one of the goods. When he has too much of one of the goods, it becomes a bad—reducing the consumption of the bad good moves him closer to his “bliss point.” If he has too much of both goods, they both are bads, so reducing the consumption of each moves him closer to the bliss point. Suppose, for example, that the two goods are chocolate cake and ice cream. There might well be some optimal amount of chocolate cake and 44 PREFERENCES (Ch. 3) ice cream that you would want to eat per week. Any less than that amount would make you worse oﬀ, but any more than that amount would also make you worse oﬀ. If you think about it, most goods are like chocolate cake and ice cream in this respect—you can have too much of nearly anything. But people would generally not voluntarily choose to have too much of the goods they consume. Why would you choose to have more than you want of something? Thus the interesting region from the viewpoint of economic choice is where you have less than you want of most goods. The choices that people actually care about are choices of this sort, and these are the choices with which we will be concerned. Discrete Goods Usually we think of measuring goods in units where fractional amounts make sense—you might on average consume 12.43 gallons of milk a month even though you buy it a quart at a time. But sometimes we want to examine preferences over goods that naturally come in discrete units. For example, consider a consumer’s demand for automobiles. We could deﬁne the demand for automobiles in terms of the time spent using an automobile, so that we would have a continuous variable, but for many purposes it is the actual number of cars demanded that is of interest. There is no diﬃculty in using preferences to describe choice behavior for this kind of discrete good. Suppose that x2 is money to be spent on other goods and x1 is a discrete good that is only available in integer amounts. We have illustrated the appearance of indiﬀerence “curves” and a weakly preferred set for this kind of good in Figure 3.8. In this case the bundles indiﬀerent to a given bundle will be a set of discrete points. The set of bundles at least as good as a particular bundle will be a set of line segments. The choice of whether to emphasize the discrete nature of a good or not will depend on our application. If the consumer chooses only one or two units of the good during the time period of our analysis, recognizing the discrete nature of the choice may be important. But if the consumer is choosing 30 or 40 units of the good, then it will probably be convenient to think of this as a continuous good. 3.5 Well-Behaved Preferences We’ve now seen some examples of indiﬀerence curves. As we’ve seen, many kinds of preferences, reasonable or unreasonable, can be described by these simple diagrams. But if we want to describe preferences in general, it will be convenient to focus on a few general shapes of indiﬀerence curves. In WELL-BEHAVED PREFERENCES 45 GOOD GOOD 2 2 Bundles weakly preferred to (1, x 2) x2 x2 1 2 3 GOOD GOOD 1 1 2 3 1 A Indifference "curves" B Weakly preferrred set A discrete good. Here good 1 is only available in integer Figure amounts. In panel A the dashed lines connect together the 3.8 bundles that are indiﬀerent, and in panel B the vertical lines represent bundles that are at least as good as the indicated bundle. this section we will describe some more general assumptions that we will typically make about preferences and the implications of these assumptions for the shapes of the associated indiﬀerence curves. These assumptions are not the only possible ones; in some situations you might want to use diﬀerent assumptions. But we will take them as the deﬁning features for well-behaved indiﬀerence curves. First we will typically assume that more is better, that is, that we are talking about goods, not bads. More precisely, if (x1 , x2 ) is a bundle of goods and (y1 , y2 ) is a bundle of goods with at least as much of both goods and more of one, then (y1 , y2 ) ≻ (x1 , x2 ). This assumption is sometimes called monotonicity of preferences. As we suggested in our discussion of satiation, more is better would probably only hold up to a point. Thus the assumption of monotonicity is saying only that we are going to ex- amine situations before that point is reached—before any satiation sets in—while more still is better. Economics would not be a very interesting subject in a world where everyone was satiated in their consumption of every good. What does monotonicity imply about the shape of indiﬀerence curves? It implies that they have a negative slope. Consider Figure 3.9. If we start at a bundle (x1 , x2 ) and move anywhere up and to the right, we must be moving to a preferred position. If we move down and to the left we must be moving to a worse position. So if we are moving to an indiﬀerent position, we must be moving either left and up or right and down: the indiﬀerence curve must have a negative slope. 46 PREFERENCES (Ch. 3) Second, we are going to assume that averages are preferred to extremes. That is, if we take two bundles of goods (x1 , x2 ) and (y1 , y2 ) on the same indiﬀerence curve and take a weighted average of the two bundles such as 1 1 1 1 x1 + y1 , x2 + y2 , 2 2 2 2 then the average bundle will be at least as good as or strictly preferred to each of the two extreme bundles. This weighted-average bundle has the average amount of good 1 and the average amount of good 2 that is present in the two bundles. It therefore lies halfway along the straight line connecting the x–bundle and the y–bundle. x2 Better bundles (x1, x2 ) Worse bundles x1 Figure Monotonic preferences. More of both goods is a better 3.9 bundle for this consumer; less of both goods represents a worse bundle. Actually, we’re going to assume this for any weight t between 0 and 1, not just 1/2. Thus we are assuming that if (x1 , x2 ) ∼ (y1 , y2 ), then (tx1 + (1 − t)y1 , tx2 + (1 − t)y2 ) (x1 , x2 ) for any t such that 0 ≤ t ≤ 1. This weighted average of the two bundles gives a weight of t to the x-bundle and a weight of 1 − t to the y-bundle. Therefore, the distance from the x-bundle to the average bundle is just a fraction t of the distance from the x-bundle to the y-bundle, along the straight line connecting the two bundles. WELL-BEHAVED PREFERENCES 47 What does this assumption about preferences mean geometrically? It means that the set of bundles weakly preferred to (x1 , x2 ) is a convex set. For suppose that (y1 , y2 ) and (x1 , x2 ) are indiﬀerent bundles. Then, if aver- ages are preferred to extremes, all of the weighted averages of (x1 , x2 ) and (y1 , y2 ) are weakly preferred to (x1 , x2 ) and (y1 , y2 ). A convex set has the property that if you take any two points in the set and draw the line seg- ment connecting those two points, that line segment lies entirely in the set. Figure 3.10A depicts an example of convex preferences, while Figures 3.10B and 3.10C show two examples of nonconvex preferences. Figure 3.10C presents preferences that are so nonconvex that we might want to call them “concave preferences.” x2 x2 x2 (y1, y2) (y1, y2) Averaged (y1, y2) bundle Averaged bundle Averaged (x1, x2) bundle (x1, x2) (x1, x2) x1 x1 x1 A Convex B Nonconvex C Concave preferences preferences preferences Various kinds of preferences. Panel A depicts convex pref- Figure erences, panel B depicts nonconvex preferences, and panel C 3.10 depicts “concave” preferences. Can you think of preferences that are not convex? One possibility might be something like my preferences for ice cream and olives. I like ice cream and I like olives... but I don’t like to have them together! In considering my consumption in the next hour, I might be indiﬀerent between consuming 8 ounces of ice cream and 2 ounces of olives, or 8 ounces of olives and 2 ounces of ice cream. But either one of these bundles would be better than consuming 5 ounces of each! These are the kind of preferences depicted in Figure 3.10C. Why do we want to assume that well-behaved preferences are convex? Because, for the most part, goods are consumed together. The kinds of preferences depicted in Figures 3.10B and 3.10C imply that the con- 48 PREFERENCES (Ch. 3) sumer would prefer to specialize, at least to some degree, and to consume only one of the goods. However, the normal case is where the consumer would want to trade some of one good for the other and end up consuming some of each, rather than specializing in consuming only one of the two goods. In fact, if we look at my preferences for monthly consumption of ice cream and olives, rather than at my immediate consumption, they would tend to look much more like Figure 3.10A than Figure 3.10C. Each month I would prefer having some ice cream and some olives—albeit at diﬀerent times—to specializing in consuming either one for the entire month. Finally, one extension of the assumption of convexity is the assumption of strict convexity. This means that the weighted average of two in- diﬀerent bundles is strictly preferred to the two extreme bundles. Convex preferences may have ﬂat spots, while strictly convex preferences must have indiﬀerences curves that are “rounded.” The preferences for two goods that are perfect substitutes are convex, but not strictly convex. 3.6 The Marginal Rate of Substitution We will often ﬁnd it useful to refer to the slope of an indiﬀerence curve at a particular point. This idea is so useful that it even has a name: the slope of an indiﬀerence curve is known as the marginal rate of substitution (MRS). The name comes from the fact that the MRS measures the rate at which the consumer is just willing to substitute one good for the other. Suppose that we take a little of good 1, Δx1 , away from the consumer. Then we give him Δx2 , an amount that is just suﬃcient to put him back on his indiﬀerence curve, so that he is just as well oﬀ after this substitution of x2 for x1 as he was before. We think of the ratio Δx2 /Δx1 as being the rate at which the consumer is willing to substitute good 2 for good 1. Now think of Δx1 as being a very small change—a marginal change. Then the rate Δx2 /Δx1 measures the marginal rate of substitution of good 2 for good 1. As Δx1 gets smaller, Δx2 /Δx1 approaches the slope of the indiﬀerence curve, as can be seen in Figure 3.11. When we write the ratio Δx2 /Δx1 , we will always think of both the numerator and the denominator as being small numbers—as describing marginal changes from the original consumption bundle. Thus the ratio deﬁning the MRS will always describe the slope of the indiﬀerence curve: the rate at which the consumer is just willing to substitute a little more consumption of good 2 for a little less consumption of good 1. One slightly confusing thing about the MRS is that it is typically a negative number. We’ve already seen that monotonic preferences imply that indiﬀerence curves must have a negative slope. Since the MRS is the numerical measure of the slope of an indiﬀerence curve, it will naturally be a negative number. THE MARGINAL RATE OF SUBSTITUTION 49 x2 Indifference curve ∆x2 Slope = = marginal rate ∆x1 of substitution ∆x2 ∆x1 x1 The marginal rate of substitution (MRS). The marginal Figure rate of substitution measures the slope of the indiﬀerence curve. 3.11 The marginal rate of substitution measures an interesting aspect of the consumer’s behavior. Suppose that the consumer has well-behaved prefer- ences, that is, preferences that are monotonic and convex, and that he is currently consuming some bundle (x1 , x2 ). We now will oﬀer him a trade: he can exchange good 1 for 2, or good 2 for 1, in any amount at a “rate of exchange” of E. That is, if the consumer gives up Δx1 units of good 1, he can get EΔx1 units of good 2 in exchange. Or, conversely, if he gives up Δx2 units of good 2, he can get Δx2 /E units of good 1. Geometrically, we are oﬀering the consumer an opportunity to move to any point along a line with slope −E that passes through (x1 , x2 ), as depicted in Figure 3.12. Moving up and to the left from (x1 , x2 ) involves exchanging good 1 for good 2, and moving down and to the right involves exchanging good 2 for good 1. In either movement, the exchange rate is E. Since exchange always involves giving up one good in exchange for another, the exchange rate E corresponds to a slope of −E. We can now ask what would the rate of exchange have to be in order for the consumer to want to stay put at (x1 , x2 )? To answer this question, we simply note that any time the exchange line crosses the indiﬀerence curve, there will be some points on that line that are preferred to (x1 , x2 )—that lie above the indiﬀerence curve. Thus, if there is to be no movement from 50 PREFERENCES (Ch. 3) (x1 , x2 ), the exchange line must be tangent to the indiﬀerence curve. That is, the slope of the exchange line, −E, must be the slope of the indiﬀerence curve at (x1 , x2 ). At any other rate of exchange, the exchange line would cut the indiﬀerence curve and thus allow the consumer to move to a more preferred point. x2 Indifference curves Slope = – E x2 x1 x1 Figure Trading at an exchange rate. Here we are allowing the con- 3.12 sumer to trade the goods at an exchange rate E, which implies that the consumer can move along a line with slope −E. Thus the slope of the indiﬀerence curve, the marginal rate of substitution, measures the rate at which the consumer is just on the margin of trading or not trading. At any rate of exchange other than the MRS, the consumer would want to trade one good for the other. But if the rate of exchange equals the MRS, the consumer wants to stay put. 3.7 Other Interpretations of the MRS We have said that the MRS measures the rate at which the consumer is just on the margin of being willing to substitute good 1 for good 2. We could also say that the consumer is just on the margin of being willing to “pay” some of good 1 in order to buy some more of good 2. So sometimes BEHAVIOR OF THE MRS 51 you hear people say that the slope of the indiﬀerence curve measures the marginal willingness to pay. If good 2 represents the consumption of “all other goods,” and it is measured in dollars that you can spend on other goods, then the marginal- willingness-to-pay interpretation is very natural. The marginal rate of sub- stitution of good 2 for good 1 is how many dollars you would just be willing to give up spending on other goods in order to consume a little bit more of good 1. Thus the MRS measures the marginal willingness to give up dollars in order to consume a small amount more of good 1. But giving up those dollars is just like paying dollars in order to consume a little more of good 1. If you use the marginal-willingness-to-pay interpretation of the MRS, you should be careful to emphasize both the “marginal” and the “willingness” aspects. The MRS measures the amount of good 2 that one is willing to pay for a marginal amount of extra consumption of good 1. How much you actually have to pay for some given amount of extra consumption may be diﬀerent than the amount you are willing to pay. How much you have to pay will depend on the price of the good in question. How much you are willing to pay doesn’t depend on the price—it is determined by your preferences. Similarly, how much you may be willing to pay for a large change in consumption may be diﬀerent from how much you are willing to pay for a marginal change. How much you actually end up buying of a good will depend on your preferences for that good and the prices that you face. How much you would be willing to pay for a small amount extra of the good is a feature only of your preferences. 3.8 Behavior of the MRS It is sometimes useful to describe the shapes of indiﬀerence curves by de- scribing the behavior of the marginal rate of substitution. For example, the “perfect substitutes” indiﬀerence curves are characterized by the fact that the MRS is constant at −1. The “neutrals” case is characterized by the fact that the MRS is everywhere inﬁnite. The preferences for “perfect complements” are characterized by the fact that the MRS is either zero or inﬁnity, and nothing in between. We’ve already pointed out that the assumption of monotonicity implies that indiﬀerence curves must have a negative slope, so the MRS always involves reducing the consumption of one good in order to get more of another for monotonic preferences. The case of convex indiﬀerence curves exhibits yet another kind of be- havior for the MRS. For strictly convex indiﬀerence curves, the MRS—the slope of the indiﬀerence curve—decreases (in absolute value) as we increase x1. Thus the indiﬀerence curves exhibit a diminishing marginal rate of 52 PREFERENCES (Ch. 3) substitution. This means that the amount of good 1 that the person is willing to give up for an additional amount of good 2 increases the amount of good 1 increases. Stated in this way, convexity of indiﬀerence curves seems very natural: it says that the more you have of one good, the more willing you are to give some of it up in exchange for the other good. (But remember the ice cream and olives example—for some pairs of goods this assumption might not hold!) Summary 1. Economists assume that a consumer can rank various consumption pos- sibilities. The way in which the consumer ranks the consumption bundles describes the consumer’s preferences. 2. Indiﬀerence curves can be used to depict diﬀerent kinds of preferences. 3. Well-behaved preferences are monotonic (meaning more is better) and convex (meaning averages are preferred to extremes). 4. The marginal rate of substitution (MRS) measures the slope of the in- diﬀerence curve. This can be interpreted as how much the consumer is willing to give up of good 2 to acquire more of good 1. REVIEW QUESTIONS 1. If we observe a consumer choosing (x1 , x2 ) when (y1 , y2 ) is available one time, are we justiﬁed in concluding that (x1 , x2 ) ≻ (y1 , y2 )? 2. Consider a group of people A, B, C and the relation “at least as tall as,” as in “A is at least as tall as B.” Is this relation transitive? Is it complete? 3. Take the same group of people and consider the relation “strictly taller than.” Is this relation transitive? Is it reﬂexive? Is it complete? 4. A college football coach says that given any two linemen A and B, he always prefers the one who is bigger and faster. Is this preference relation transitive? Is it complete? 5. Can an indiﬀerence curve cross itself? For example, could Figure 3.2 depict a single indiﬀerence curve? 6. Could Figure 3.2 be a single indiﬀerence curve if preferences are mono- tonic? REVIEW QUESTIONS 53 7. If both pepperoni and anchovies are bads, will the indiﬀerence curve have a positive or a negative slope? 8. Explain why convex preferences means that “averages are preferred to extremes.” 9. What is your marginal rate of substitution of $1 bills for $5 bills? 10. If good 1 is a “neutral,” what is its marginal rate of substitution for good 2? 11. Think of some other goods for which your preferences might be concave. CHAPTER 4 UTILITY In Victorian days, philosophers and economists talked blithely of “utility” as an indicator of a person’s overall well-being. Utility was thought of as a numeric measure of a person’s happiness. Given this idea, it was natural to think of consumers making choices so as to maximize their utility, that is, to make themselves as happy as possible. The trouble is that these classical economists never really described how we were to measure utility. How are we supposed to quantify the “amount” of utility associated with diﬀerent choices? Is one person’s utility the same as another’s? What would it mean to say that an extra candy bar would give me twice as much utility as an extra carrot? Does the concept of utility have any independent meaning other than its being what people maximize? Because of these conceptual problems, economists have abandoned the old-fashioned view of utility as being a measure of happiness. Instead, the theory of consumer behavior has been reformulated entirely in terms of consumer preferences, and utility is seen only as a way to describe preferences. Economists gradually came to recognize that all that mattered about utility as far as choice behavior was concerned was whether one bundle had a higher utility than another—how much higher didn’t really matter. UTILITY 55 Originally, preferences were deﬁned in terms of utility: to say a bundle (x1 , x2 ) was preferred to a bundle (y1 , y2 ) meant that the x-bundle had a higher utility than the y-bundle. But now we tend to think of things the other way around. The preferences of the consumer are the fundamen- tal description useful for analyzing choice, and utility is simply a way of describing preferences. A utility function is a way of assigning a number to every possible consumption bundle such that more-preferred bundles get assigned larger numbers than less-preferred bundles. That is, a bundle (x1 , x2 ) is preferred to a bundle (y1 , y2 ) if and only if the utility of (x1 , x2 ) is larger than the utility of (y1 , y2 ): in symbols, (x1 , x2 ) ≻ (y1 , y2 ) if and only if u(x1 , x2 ) > u(y1 , y2 ). The only property of a utility assignment that is important is how it orders the bundles of goods. The magnitude of the utility function is only important insofar as it ranks the diﬀerent consumption bundles; the size of the utility diﬀerence between any two consumption bundles doesn’t matter. Because of this emphasis on ordering bundles of goods, this kind of utility is referred to as ordinal utility. Consider for example Table 4.1, where we have illustrated several dif- ferent ways of assigning utilities to three bundles of goods, all of which order the bundles in the same way. In this example, the consumer prefers A to B and B to C. All of the ways indicated are valid utility functions that describe the same preferences because they all have the property that A is assigned a higher number than B, which in turn is assigned a higher number than C. Diﬀerent ways to assign utilities. Table 4.1 Bundle U1 U2 U3 A 3 17 −1 B 2 10 −2 C 1.002 −3 Since only the ranking of the bundles matters, there can be no unique way to assign utilities to bundles of goods. If we can ﬁnd one way to assign utility numbers to bundles of goods, we can ﬁnd an inﬁnite number of ways to do it. If u(x1 , x2 ) represents a way to assign utility numbers to the bundles (x1 , x2 ), then multiplying u(x1 , x2 ) by 2 (or any other positive number) is just as good a way to assign utilities. Multiplication by 2 is an example of a monotonic transformation. A 56 UTILITY (Ch. 4) monotonic transformation is a way of transforming one set of numbers into another set of numbers in a way that preserves the order of the numbers. We typically represent a monotonic transformation by a function f (u) that transforms each number u into some other number f (u), in a way that preserves the order of the numbers in the sense that u1 > u2 implies f (u1 ) > f (u2 ). A monotonic transformation and a monotonic function are essentially the same thing. Examples of monotonic transformations are multiplication by a positive number (e.g., f (u) = 3u), adding any number (e.g., f (u) = u + 17), raising u to an odd power (e.g., f (u) = u3 ), and so on.1 The rate of change of f (u) as u changes can be measured by looking at the change in f between two values of u, divided by the change in u: Δf f (u2 ) − f (u1 ) =. Δu u2 − u1 For a monotonic transformation, f (u2 ) − f (u1 ) always has the same sign as u2 − u1. Thus a monotonic function always has a positive rate of change. This means that the graph of a monotonic function will always have a positive slope, as depicted in Figure 4.1A. v v v = f (u ) v = f (u ) u u A B Figure A positive monotonic transformation. Panel A illustrates 4.1 a monotonic function—one that is always increasing. Panel B illustrates a function that is not monotonic, since it sometimes increases and sometimes decreases. 1 What we are calling a “monotonic transformation” is, strictly speaking, called a “posi- tive monotonic transformation,” in order to distinguish it from a “negative monotonic transformation,” which is one that reverses the order of the numbers. Monotonic transformations are sometimes called “monotonous transformations,” which seems unfair, since they can actually be quite interesting. CARDINAL UTILITY 57 If f (u) is any monotonic transformation of a utility function that repre- sents some particular preferences, then f (u(x1 , x2 )) is also a utility function that represents those same preferences. Why? The argument is given in the following three statements: 1. To say that u(x1 , x2 ) represents some particular preferences means that u(x1 , x2 ) > u(y1 , y2 ) if and only if (x1 , x2 ) ≻ (y1 , y2 ). 2. But if f (u) is a monotonic transformation, then u(x1 , x2 ) > u(y1 , y2 ) if and only if f (u(x1 , x2 )) > f (u(y1 , y2 )). 3. Therefore, f (u(x1 , x2 )) > f (u(y1 , y2 )) if and only if (x1 , x2 ) ≻ (y1 , y2 ), so the function f (u) represents the preferences in the same way as the original utility function u(x1 , x2 ). We summarize this discussion by stating the following principle: a mono- tonic transformation of a utility function is a utility function that represents the same preferences as the original utility function. Geometrically, a utility function is a way to label indiﬀerence curves. Since every bundle on an indiﬀerence curve must have the same utility, a utility function is a way of assigning numbers to the diﬀerent indiﬀerence curves in a way that higher indiﬀerence curves get assigned larger num- bers. Seen from this point of view a monotonic transformation is just a relabeling of indiﬀerence curves. As long as indiﬀerence curves containing more-preferred bundles get a larger label than indiﬀerence curves contain- ing less-preferred bundles, the labeling will represent the same preferences. 4.1 Cardinal Utility There are some theories of utility that attach a signiﬁcance to the magni- tude of utility. These are known as cardinal utility theories. In a theory of cardinal utility, the size of the utility diﬀerence between two bundles of goods is supposed to have some sort of signiﬁcance. We know how to tell whether a given person prefers one bundle of goods to another: we simply oﬀer him or her a choice between the two bundles and see which one is chosen. Thus we know how to assign an ordinal utility to the two bundles of goods: we just assign a higher utility to the chosen bundle than to the rejected bundle. Any assignment that does this will be a utility function. Thus we have an operational criterion for determining whether one bundle has a higher utility than another bundle for some individual. But how do we tell if a person likes one bundle twice as much as another? How could you even tell if you like one bundle twice as much as another? One could propose various deﬁnitions for this kind of assignment: I like one bundle twice as much as another if I am willing to pay twice as much for it. Or, I like one bundle twice as much as another if I am willing to run 58 UTILITY (Ch. 4) twice as far to get it, or to wait twice as long, or to gamble for it at twice the odds. There is nothing wrong with any of these deﬁnitions; each one would give rise to a way of assigning utility levels in which the magnitude of the numbers assigned had some operational signiﬁcance. But there isn’t much right about them either. Although each of them is a possible interpretation of what it means to want one thing twice as much as another, none of them appears to be an especially compelling interpretation of that statement. Even if we did ﬁnd a way of assigning utility magnitudes that seemed to be especially compelling, what good would it do us in describing choice behavior? To tell whether one bundle or another will be chosen, we only have to know which is preferred—which has the larger utility. Knowing how much larger doesn’t add anything to our description of choice. Since cardinal utility isn’t needed to describe choice behavior and there is no compelling way to assign cardinal utilities anyway, we will stick with a purely ordinal utility framework. 4.2 Constructing a Utility Function But are we assured that there is any way to assign ordinal utilities? Given a preference ordering can we always ﬁnd a utility function that will order bundles of goods in the same way as those preferences? Is there a utility function that describes any reasonable preference ordering? Not all kinds of preferences can be represented by a utility function. For example, suppose that someone had intransitive preferences so that A ≻ B ≻ C ≻ A. Then a utility function for these preferences would have to consist of numbers u(A), u(B), and u(C) such that u(A) > u(B) > u(C) > u(A). But this is impossible. However, if we rule out perverse cases like intransitive preferences, it turns out that we will typically be able to ﬁnd a utility function to represent preferences. We will illustrate one construction here, and another one in Chapter 14. Suppose that we are given an indiﬀerence map as in Figure 4.2. We know that a utility function is a way to label the indiﬀerence curves such that higher indiﬀerence curves get larger numbers. How can we do this? One easy way is to draw the diagonal line illustrated and label each indiﬀerence curve with its distance from the origin measured along the line. How do we know that this is a utility function? It is not hard to see that if preferences are monotonic then the line through the origin must intersect every indiﬀerence curve exactly once. Thus every bundle is getting a label, and those bundles on higher indiﬀerence curves are getting larger labels— and that’s all it takes to be a utility function. SOME EXAMPLES OF UTILITY FUNCTIONS 59 x2 Measures distance from origin 4 3 2 1 Indifference curves 0 x1 Constructing a utility function from indiﬀerence curves. Figure Draw a diagonal line and label each indiﬀerence curve with how 4.2 far it is from the origin measured along the line. This gives us one way to ﬁnd a labeling of indiﬀerence curves, at least as long as preferences are monotonic. This won’t always be the most natural way in any given case, but at least it shows that the idea of an ordinal utility function is pretty general: nearly any kind of “reasonable” preferences can be represented by a utility function. 4.3 Some Examples of Utility Functions In Chapter 3 we described some examples of preferences and the indiﬀer- ence curves that represented them. We can also represent these preferences by utility functions. If you are given a utility function, u(x1 , x2 ), it is rel- atively easy to draw the indiﬀerence curves: you just plot all the points (x1 , x2 ) such that u(x1 , x2 ) equals a constant. In mathematics, the set of all (x1 , x2 ) such that u(x1 , x2 ) equals a constant is called a level set. For each diﬀerent value of the constant, you get a diﬀerent indiﬀerence curve. EXAMPLE: Indifference Curves from Utility Suppose that the utility function is given by: u(x1 , x2 ) = x1 x2. What do the indiﬀerence curves look like? 60 UTILITY (Ch. 4) We know that a typical indiﬀerence curve is just the set of all x1 and x2 such that k = x1 x2 for some constant k. Solving for x2 as a function of x1 , we see that a typical indiﬀerence curve has the formula: k x2 =. x1 This curve is depicted in Figure 4.3 for k = 1, 2, 3 · · ·. x2 Indifference curves k=3 k=2 k=1 x1 Figure Indiﬀerence curves. The indiﬀerence curves k = x1 x2 for 4.3 diﬀerent values of k. Let’s consider another example. Suppose that we were given a utility function v(x1 , x2 ) = x21 x22. What do its indiﬀerence curves look like? By the standard rules of algebra we know that: v(x1 , x2 ) = x21 x22 = (x1 x2 )2 = u(x1 , x2 )2. Thus the utility function v(x1 , x2 ) is just the square of the utility func- tion u(x1 , x2 ). Since u(x1 , x2 ) cannot be negative, it follows that v(x1 , x2 ) is a monotonic transformation of the previous utility function, u(x1 , x2 ). This means that the utility function v(x1 , x2 ) = x21 x22 has to have exactly the same shaped indiﬀerence curves as those depicted in Figure 4.3. The labeling of the indiﬀerence curves will be diﬀerent—the labels that were 1, 2, 3, · · · will now be 1, 4, 9, · · ·—but the set of bundles that has v(x1 , x2 ) = SOME EXAMPLES OF UTILITY FUNCTIONS 61 9 is exactly the same as the set of bundles that has u(x1 , x2 ) = 3. Thus v(x1 , x2 ) describes exactly the same preferences as u(x1 , x2 ) since it orders all of the bundles in the same way. Going the other direction—ﬁnding a utility function that represents some indiﬀerence curves—is somewhat more diﬃcult. There are two ways to proceed. The ﬁrst way is mathematical. Given the indiﬀerence curves, we want to ﬁnd a function that is constant along each indiﬀerence curve and that assigns higher values to higher indiﬀerence curves. The second way is a bit more intuitive. Given a description of the pref- erences, we try to think about what the consumer is trying to maximize— what combination of the goods describes the choice behavior of the con- sumer. This may seem a little vague at the moment, but it will be more meaningful after we discuss a few examples. Perfect Substitutes Remember the red pencil and blue pencil example? All that mattered to the consumer was the total number of pencils. Thus it is natural to measure utility by the total number of pencils. Therefore we provisionally pick the utility function u(x1 , x2 ) = x1 +x2. Does this work? Just ask two things: is this utility function constant along the indiﬀerence curves? Does it assign a higher label to more-preferred bundles? The answer to both questions is yes, so we have a utility function. Of course, this isn’t the only utility function that we could use. We could also use the square of the number of pencils. Thus the utility function v(x1 , x2 ) = (x1 + x2 )2 = x21 + 2x1 x2 + x22 will also represent the perfect- substitutes preferences, as would any other monotonic transformation of u(x1 , x2 ). What if the consumer is willing to substitute good 1 for good 2 at a rate that is diﬀerent from one-to-one? Suppose, for example, that the consumer would require two units of good 2 to compensate him for giving up one unit of good 1. This means that good 1 is twice as valuable to the consumer as good 2. The utility function therefore takes the form u(x1 , x2 ) = 2x1 + x2. Note that this utility yields indiﬀerence curves with a slope of −2. In general, preferences for perfect substitutes can be represented by a utility function of the form u(x1 , x2 ) = ax1 + bx2. Here a and b are some positive numbers that measure the “value” of goods 1 and 2 to the consumer. Note that the slope of a typical indiﬀerence curve is given by −a/b. 62 UTILITY (Ch. 4) Perfect Complements This is the left shoe–right shoe case. In these preferences the consumer only cares about the number of pairs of shoes he has, so it is natural to choose the number of pairs of shoes as the utility function. The number of complete pairs of shoes that you have is the minimum of the number of right shoes you have, x1 , and the number of left shoes you have, x2. Thus the utility function for perfect complements takes the form u(x1 , x2 ) = min{x1 , x2 }. To verify that this utility function actually works, pick a bundle of goods such as (10, 10). If we add one more unit of good 1 we get (11, 10), which should leave us on the same indiﬀerence curve. Does it? Yes, since min{10, 10} = min{11, 10} = 10. So u(x1 , x2 ) = min{x1 , x2 } is a possible utility function to describe per- fect complements. As usual, any monotonic transformation would be suit- able as well. What about the case where the consumer wants to consume the goods in some proportion other than one-to-one? For example, what about the consumer who always uses 2 teaspoons of sugar with each cup of tea? If x1 is the number of cups of tea available and x2 is the number of teaspoons of sugar available, then the number of correctly sweetened cups of tea will be min{x1 , 12 x2 }. This is a little tricky so we should stop to think about it. If the number of cups of tea is greater than half the number of teaspoons of sugar, then we know that we won’t be able to put 2 teaspoons of sugar in each cup. In this case, we will only end up with 12 x2 correctly sweetened cups of tea. (Substitute some numbers in for x1 and x2 to convince yourself.) Of course, any monotonic transformation of this utility function will describe the same preferences. For example, we might want to multiply by 2 to get rid of the fraction. This gives us the utility function u(x1 , x2 ) = min{2x1 , x2 }. In general, a utility function that describes perfect-complement prefer- ences is given by u(x1 , x2 ) = min{ax1 , bx2 }, where a and b are positive numbers that indicate the proportions in which the goods are consumed. Quasilinear Preferences Here’s a shape of indiﬀerence curves that we haven’t seen before. Suppose that a consumer has indiﬀerence curves that are vertical translates of one another, as in Figure 4.4. This means that all of the indiﬀerence curves are just vertically “shifted” versions of one indiﬀerence curve. It follows that SOME EXAMPLES OF UTILITY FUNCTIONS 63 the equation for an indiﬀerence curve takes the form x2 = k − v(x1 ), where k is a diﬀerent constant for each indiﬀerence curve. This equation says that the height of each indiﬀerence curve is some function of x1 , −v(x1 ), plus a constant k. Higher values of k give higher indiﬀerence curves. (The minus sign is only a convention; we’ll see why it is convenient below.) x2 Indifference curves x1 Quasilinear preferences. Each indiﬀerence curve is a verti- Figure cally shifted version of a single indiﬀerence curve. 4.4 The natural way to label indiﬀerence curves here is with k—roughly speaking, the height of the indiﬀerence curve along the vertical axis. Solv- ing for k and setting it equal to utility, we have u(x1 , x2 ) = k = v(x1 ) + x2. In this case the utility function is linear in good 2, but (possibly) non- linear in good 1; hence the name quasilinear utility, meaning “partly linear” utility. Speciﬁc examples of quasilinear utility would be u(x1 , x2 ) = √ x1 + x2 , or u(x1 , x2 ) = ln x1 + x2. Quasilinear utility functions are not particularly realistic, but they are very easy to work with, as we’ll see in several examples later on in the book. Cobb-Douglas Preferences Another commonly used utility function is the Cobb-Douglas utility func- tion u(x1 , x2 ) = xc1 xd2 , 64 UTILITY (Ch. 4) where c and d are positive numbers that describe the preferences of the consumer.2 The Cobb-Douglas utility function will be useful in several examples. The preferences represented by the Cobb-Douglas utility function have the general shape depicted in Figure 4.5. In Figure 4.5A, we have illustrated the indiﬀerence curves for c = 1/2, d = 1/2. In Figure 4.5B, we have illustrated the indiﬀerence curves for c = 1/5, d = 4/5. Note how diﬀerent values of the parameters c and d lead to diﬀerent shapes of the indiﬀerence curves. x2 x2 x1 x1 A c = 1/2 d =1/2 B c = 1/5 d =4/5 Figure Cobb-Douglas indiﬀerence curves. Panel A shows the case 4.5 where c = 1/2, d = 1/2 and panel B shows the case where c = 1/5, d = 4/5. Cobb-Douglas indiﬀerence curves look just like the nice convex mono- tonic indiﬀerence curves that we referred to as “well-behaved indiﬀerence curves” in Chapter 3. Cobb-Douglas preferences are the standard exam- ple of indiﬀerence curves that look well-behaved, and in fact the formula describing them is about the simplest algebraic expression that generates well-behaved preferences. We’ll ﬁnd Cobb-Douglas preferences quite useful to present algebraic examples of the economic ideas we’ll study later. Of course a monotonic transformation of the Cobb-Douglas utility func- tion will represent exactly the same preferences, and it is useful to see a couple of examples of these transformations. 2 Paul Douglas was a twentieth-century economist at the University of Chicago who later became a U.S. senator. Charles Cobb was a mathematician at Amherst College. The Cobb-Douglas functional form was originally used to study production behavior. MARGINAL UTILITY 65 First, if we take the natural log of utility, the product of the terms will become a sum so that we have v(x1 , x2 ) = ln(xc1 xd2 ) = c ln x1 + d ln x2. The indiﬀerence curves for this utility function will look just like the ones for the ﬁrst Cobb-Douglas function, since the logarithm is a monotonic transformation. (For a brief review of natural logarithms, see the Mathe- matical Appendix at the end of the book.) For the second example, suppose that we start with the Cobb-Douglas form v(x1 , x2 ) = xc1 xd2. Then raising utility to the 1/(c + d) power, we have c d x1c+d x2c+d. Now deﬁne a new number c a=. c+d We can now write our utility function as v(x1 , x2 ) = xa1 x1−a 2. This means that we can always take a monotonic transformation of the Cobb-Douglas utility function that make the exponents sum to 1. This will turn out to have a useful interpretation later on. The Cobb-Douglas utility function can be expressed in a variety of ways; you should learn to recognize them, as this family of preferences is very useful for examples. 4.4 Marginal Utility Consider a consumer who is consuming some bundle of goods, (x1 , x2 ). How does this consumer’s utility change as we give him or her a little more of good 1? This rate of change is called the marginal utility with respect to good 1. We write it as M U1 and think of it as being a ratio, ΔU u(x1 + Δx1 , x2 ) − u(x1 , x2 ) M U1 = = , Δx1 Δx1 that measures the rate of change in utility (ΔU ) associated with a small change in the amount of good 1 (Δx1 ). Note that the amount of good 2 is held ﬁxed in this calculation.3 3 See the appendix to this chapter for a calculus treatment of marginal utility. 66 UTILITY (

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