The Curvature of Indifference Curves: Substitutes and Complements
∂ The online appendix looks at the relationship between the utility function and the shape of indifference curves. (http://glsmicro.com/appendices)
The steepness of an indifference curve tells us the rate at which a consumer is willing to trade one good for another. The curvature of an indifference curve also has a meaning. Suppose indifferences curves between two goods are almost straight, as in Figure 4.8a. In this case, a consumer (let’s call him Evan) is willing to trade about the same amount of the first good (here, decaf coffees) to get the same amount of the second good (decaf lattes), regardless of whether he has a lot of lattes relative to coffees or vice versa. Stated in terms of the marginal rates of substitution, the MRS of coffees for lattes doesn’t change much as we move along the indifference curve. In practical terms, it means that the two goods are close substitutes for each other in Evan’s utility function. That is, the relative value a consumer places on two substitute goods will typically not be very responsive to the amounts he has of one good versus the other. (It’s no coincidence that this example features two goods that many consumers would consider to be close substitutes for each other.)
Figure 4.8: Figure 4.8 Curvature of Indifference Curves
Figure 4.8: The curvature of indifference curves reflects information about the consumer’s preferences between two goods, just as its steepness does. (a) Goods that are highly substitutable (such as decaf lattes and decaf coffees) are likely to produce indifference curves that are relatively straight. This means that the MRS does not change much as the consumer moves from one point to another along the indifference curve. (b) Goods that are complementary will generally have indifference curves with more curvature. For example, if Evan has many tortilla chips and little guacamole, he will be willing to trade many tortilla chips to get some guacamole. If a consumer has a lot of guacamole and few tortilla chips, he will be less willing to trade chips for guacamole.
On the other hand, for goods such as tortilla chips and guacamole that are poor substitutes (Figure 4.8b), the relative value of one more pint of guacamole will be much greater when you have a lot of chips and not much guacamole than if you have 10 pints of guacamole but no chips. In these types of cases, indifference curves are sharply curved, as shown. The MRS of guacamole for chips is very high on the far left part of the indifference curve (where the consumer does not have much guacamole) and very low on the far right (when the consumer is awash in guacamole).
A good that a consumer can trade for another good, in fixed units, and receive the same level of utility.
A good from which the consumer receives utility dependent on its being used in a fixed proportion with another good.
Perfect Substitutes The intuition behind the meaning of the curvature of indifference curves may be easier to grasp if we focus on the most extreme cases, perfect substitutes and perfect complements. Figure 4.9 shows an example of two goods that might be perfect substitutes: 12-ounce bags of tortilla chips and 3-ounce bags of tortilla chips. If all the consumer cares about is the total amount of chips, then she is just as well off trading 4 small bags of chips for each large bag, regardless of how many of either she already has. These kinds of preferences produce linear indifference curves, and utility functions for perfect substitutes take on the general form U= aX + bY, where a indicates the marginal utility of consuming one more unit of X and b indicates the marginal utility of consuming one more unit of Y. This is precisely the situation shown in Figure 4.9. The indifference curves are straight lines with a constant slope equal to –1/4, which means that the MRSXY is also constant and equal to 1/4. We can’t actually say what values a and b take here, only that their ratio is 1 to 4—that is, a/b = 1/4. The indifference curves in the figure would be the same if a = 1 and b = 4 or if a = 40 and b = 160, for instance. This is another demonstration of the point we made above: A transformation of a utility function that does not change the order of which goods the consumer prefers implies the same preference choices.
Figure 4.9: Figure 4.9 Indifference Curves for Perfect Substitutes
Figure 4.9: Two goods that are perfect substitutes have indifference curves that are straight lines. In this case, the consumer is willing to trade one 12-ounce bag of tortilla chips for four 3-ounce bags of tortilla chips no matter how many of each she currently has, and the consumer’s preference for chips does not change along the indifference curve. The MRS is constant in this case.
Different-sized packages of the same good are just one example of why two goods might be perfect substitutes.6 Another way perfect substitutes might arise is if there are attributes of a product that a particular consumer does not care at all about. For instance, some people might not care about whether a bottle of water is branded Aquafina or Dasani. Their indifference curves when comparing Aquafina and Dasani water would therefore be straight lines. On the other hand, other consumers who do care about such features would not view the goods as perfect substitutes, and their indifference curves would be curved.
6 You could think of reasons why the different-sized bags might not be perfect substitutes—maybe there’s a convenience factor involved with the smaller ones because there’s no need to worry about storing open, partially eaten bags. But even allowing for these small differences, they’re fairly close to perfect substitutes.
It is crucial to understand that two goods being perfect substitutes does not necessarily imply that the consumer is indifferent between single items of the goods. In our tortilla chip example above, for instance, the consumer likes a big bag a lot more than a small bag. That’s why she would have to be given 4 small bags, not just 1, to be willing to trade away 1 large bag. The idea behind perfect substitutes is only that the tradeoff the consumer is willing to make between the two goods—that is, the marginal rate of substitution—doesn’t depend on how much or little she already has of each but is instead constant at every point along an indifference curve.
Perfect Complements When the utility a consumer receives from a good depends on its being used in fixed proportion with another good, the two goods are perfect complements. Figure 4.10 shows indifference curves for right and left shoes, which are an example of perfect complements (or at least something very close to it). Compare point A (2 right shoes and 2 left shoes) and point B (3 right shoes and 2 left shoes). Although the consumer has one extra shoe at point B, there is no matching shoe for the other foot, so the extra shoe is useless to her. She is therefore indifferent between these two bundles, and the bundles are on the same indifference curve. Similarly, comparing points A and C in the figure, we see that an extra left-footed shoe provides no additional utility if it isn’t paired with a right-footed shoe, so A and C must also lie on the same indifference curve. However, if you add an extra left shoe and an extra right shoe (point D compared to point A), then the consumer is better off. That’s why D is on a higher indifference curve.
Figure 4.10: Figure 4.10 Indifference Curves for Perfect Complements
Figure 4.10: When goods are perfect complements, they have L-shaped indifference curves. For example, at point A, the consumer has 2 left shoes and 2 right shoes. Adding another right shoe while keeping left shoes constant does not increase the consumer’s utility, so point B is on the same indifference curve as point A. In like manner, adding another left shoe will not increase the consumer’s utility without an additional right shoe, so point C is on the same indifference curve as points A and B. Because shoes are always consumed together, 1 right shoe and 1 left shoe, the consumer’s utility rises only when she has more of both goods (a move from point A to point D).
Perfect complements lead to distinctive L-shaped indifference curves. Mathematically, this can be represented as U = min{aX, bY}, where a and b are again numbers reflecting how consuming more units of X and Y affects utility. This mathematical structure means a consumer reaches a given utility level by consuming a minimum amount of each good X and Y. To be on the indifference curve U2, for instance, the consumer must have at least 2 left shoes and 2 right shoes. The kink in the indifference curve is the point at which she is consuming the minimum amount of each good at that utility level.7
7 The proportion in which perfect complements are consumed need not be one-for-one, as in the case of our left- and right-shoe example. Chopsticks and Chinese buffet lunches might be perfect complements for some consumers, for example, but it’s likely that they will be consumed in a proportion of 2 chopsticks to 1 buffet. It’s hard to eat with just one chopstick.
This L-shape is the most extreme case of curvature of indifference curves. It is at the other extreme from the straight-line indifference curves that arise with perfect substitutes, and its shape produces interesting results for MRSXY. At the horizontal part of the indifference curve, MRSXY equals zero, while on the vertical portion, the marginal rate of substitution is infinite. As we’ve noted, indifference curves more generally will fall somewhere in between the shapes of the indifference curves for perfect substitutes and perfect complements, with some intermediate amount of curvature.
Different Shapes for a Particular Consumer One final point to make about the curvature of indifference curves is that even for a particular consumer, indifference curves may take on a variety of shapes depending on the utility level. They don’t all have to look the same.
For instance, indifference curve UA in Figure 4.11 is almost a straight line. This means that at low levels of utility, this consumer considers bananas and strawberries almost perfect substitutes. Her marginal rate of substitution barely changes whether she starts with a relatively high number of bananas to strawberries, or a relatively small number. If all she is worried about is getting enough calories to survive (which might be the case at really low utility levels like that represented by UA), how something tastes won’t matter much to her. She’s not going to be picky about the mix of fruit she eats. This leads the indifference curve to be fairly straight, like UA.
Figure 4.11: Figure 4.11 The Same Consumer Can Have Indifference Curves with Different Shapes
Figure 4.11: Indifference curves for a consumer can take on a variety of shapes, depending on the utility level. For example, at low levels of utility, bananas and strawberries may be substitutes and the consumer may just want to buy something to give her calories, not caring whether it is a banana or a strawberry. This means that the indifference curve will be close to linear, as is the case of UA. But, at higher levels of utility, the consumer may prefer a variety of fruit. This means that she will be willing to give up many bananas for another strawberry when she has a lot of bananas, but is not willing to do so when she only has a few bananas. Here, the consumer’s indifference curve will have more curvature, such as UB.
Indifference curve UB, on the other hand, is very sharply curved. This means that at higher utility levels, the two goods are closer to perfect complements. When this consumer has plenty of fruit, she is more concerned with enjoying variety when she eats. This leads her to prefer some of each fruit rather than a lot of one or the other. If she already has a lot of one good, she will be willing to give up quite a lot of that good in order to receive a unit of the good she has less of. This leads to the more curved indifference curve. Remember, however, that even as the shapes of a consumer’s indifference curves vary with her utility levels, the indifference curves will never intersect.
figure it out 4.2
Jasmine can watch hours of baseball (B) or hours of reality shows (R) on TV. Watching more baseball makes Jasmine happier, but she really doesn’t care about reality shows—good or bad. Draw a diagram showing a set of Jasmine’s indifference curves for hours of baseball and hours of reality shows. (Put reality shows on the horizontal axis.) What is Jasmine’s MRSRB when she is consuming 1 unit of each good?
The easiest way to diagram Jasmine’s preferences is to consider various bundles of reality shows and baseball, and determine whether they lie on the same or different indifference curves. For example, suppose she watches 1 hour of reality TV and 1 hour of baseball. Plot this in Figure A as point A. Now, suppose she watches 1 hour of reality TV and 2 hours of baseball. Plot this as point B. Because watching more hours of baseball makes Jasmine happier, point B must lie on a higher indifference curve than point A.
Now, try another point with 2 hours of reality TV and 1 hour of baseball. Call this point C. Compare point A with point C. Point C has the same number of hours of baseball as point A, but provides Jasmine with more reality TV. Jasmine neither likes nor dislikes reality TV, however, so her utility is unchanged by having more reality TV. Points A and C must therefore lie on the same indifference curve. This would also be true of points D and E. Economists often refer to a good that has no impact on utility as a “neutral good.”
Looking at Figure A, we see that there will be an indifference curve that is a horizontal line going through points A, C, D, and E. Will all of the indifference curves be horizontal lines? Let’s consider another bundle to make sure. Suppose that Jasmine watches 3 hours of reality TV and 2 hours of baseball, as at point F. It is clear that Jasmine will prefer point F to point D because she gets more baseball. It should also be clear that Jasmine will be equally happy between points B and F; she has the same hours of baseball, and reality shows have no effect on her utility. As shown in Figure B, points B and F lie on the same indifference curve (U2) and provide a greater level of utility than the bundles on the indifference curve below (U1).
To calculate the marginal rate of substitution when Jasmine is consuming 1 unit of each good, we need to calculate the slope of U1 at point A. Because the indifference curve is a horizontal line, the slope is zero. Therefore, MRSRB is zero. This makes sense; Jasmine is not willing to give up any baseball to watch more reality TV because reality TV has no impact on her utility. Remember that MRSRB equals MUR/MUB. Because MUR is zero, MRSRB will also equal zero.
Application: Indifference Curves for “Bads”
A good or service that provides a consumer with negative utility.
Let’s look at some real-life data on house prices. A research paper by Steven Levitt and Chad Syverson looked at how houses’ sales prices change with their characteristics.8 Their analysis, like Nevo’s work on breakfast cereals, uses hedonic regressions. While the extent to which these types of analyses directly measure utility functions is controversial among economists, the details of that argument are not important to understanding our basic point here: Some things are the opposite of “goods.” We call them “bads.”
8 Steven Levitt and Chad Syverson, “Market Distortions When Agents Are Better Informed: The Value of Information in Real Estate Transactions,” Review of Economics and Statistics 90, no. 4 (2008): 599–611.
Using regression techniques to compare otherwise similar houses (e.g., those having the same architectural style, the same type of siding material, and even located on the same city block), Levitt and Syverson found that an extra bedroom adds about 5% to the sales price of a house. Central air-conditioning adds about 7% to a house’s price, a “state-of-the-art kitchen” adds almost 8%, and a fireplace brings in another 4%. The correlation between each of these amenities and the sales price of the house suggests that homebuyers are willing to pay for them. In other words, bedrooms, central air, state-of-the-art kitchens, and fireplaces are all “goods” (they just happen to be sold as premade bundles in the form of houses).
But Levitt and Syverson also uncovered a characteristic that had a negative effect on a house’s price, at least up to a point: its age. Again, comparing otherwise similar houses, they found that a house between 6–10 years old sold for 9% less than one that was built less than 6 years ago. Houses 11–25 years old, in turn, had sales prices that were 6% lower than those recorded for 6- to 10-year-old houses. There was another 3% drop in price for houses that were between 11–25 and 26–50 years old. (For houses older than that, prices stopped dropping and even rose slightly, suggesting that eventually “age” may be regarded as “charm.”) Therefore, at least for houses less than 50 years old, age is a “bad”; all else equal, buyers receive lower utility from an older house than they would from an otherwise comparable but newer one.
The indifference curves for bundles of age (a bad) and bedrooms (a good) are shown in Figure 4.12. We see that the indifference curves slope upward, not downward. Why? Let’s first consider bundles A and B, which lie on the same indifference curve, U1. A homebuyer who does not like old houses needs more bedrooms for the utility of bundle B (greater age, more bedrooms) to equal that of bundle A (younger age, fewer bedrooms). Bundle C has more space than bundle A but is the same age, so C must be preferred to A. Indifference curves that lie higher (more space) and to the left (not as old) indicate greater levels of utility.
Figure 4.12: Figure 4.12 Indifference Curves for a “Bad”
Figure 4.12: An economic “bad” is a product that reduces a consumer’s utility. This homebuyer’s utility from a house falls as the age of the house increases. Therefore, to keep her utility constant, we must provide the homebuyer with more bedrooms if we increase the house’s age. This leads to upward-sloping indifference curves. Indifference curve U2 provides more utility than U1 because (holding the number of bedrooms constant) bundle B contains an older house than bundle C, making the homebuyer worse off. Alternatively, bundles A and C contain houses of the same age, but the house in bundle C has more bedrooms. Thus, the homebuyer is better off at point C (on U2) than at point A (on U1).
Do bads violate our assumption that more is better? Not if we redefine the age factor as “newness,” the absence of age. Newness then becomes a good, and our indifference curves for newness versus bedrooms then slope downward. Graphing our homebuyer’s indifference curves in terms of how new a house is—the opposite of age—produces the standard, downward-sloping indifference curves we’ve been working with, as in Figure 4.13. 
Figure 4.13: Figure 4.13 Indifference Curves for the Absence of a “Bad”
Figure 4.13: An economic “bad” can be converted into a “good.” By changing the economic bad of a house’s “increased age” into the economic good of a house’s “newness,” we can have two goods that our homeowner desires and produce typical downward-sloping, convex indifference curves. The homebuyer’s utility increases with either an increase in the number of bedrooms in a house or an increase in the house’s newness.