Chapter 3 Manual for Microeconomics

CHAPTER 3 Optimization: Doing the Best You Can I. Key Ideas When economic agents choose the best feasible option, they are optimizing. Optimization in levels calculates the total net benefit of different alternatives and then chooses the best alternative. Optimization in differences calculates the change in net benefits when a person switches from one alternative to another, and then uses these marginal comparisons to choose the best alternative. Optimization in levels and optimization in differences give identical answers. II. Getting Started The Big Picture Economists tend to view the world as consisting of optimizing economic actors, whether they are profit-maximizing firms setting prices or hiring workers, tax-revenue-maximizing governments, or utility-maximizing consumers. Some familiar microeconomic graphs, shown below, help us communicate the idea of optimization. For instance, a monopolist maximize profit, a government sets a per-unit tax to maximize tax revenues, employers hire workers as long as they are worth hiring, a woman drinks water until her thirst is sated (but no more—overconsumption of water eventually becomes very painful), and a consumer buys a bundle of goods X and Y depending on which good generates the most additional happiness per dollar spent. Optimization is a major theme that runs throughout the book, so this chapter gives readers a good understanding of how economists analyze decision making. For example, in later chapters on the theory of the firm, we could imagine a firm that seeks the profit-maximizing level of output could solve this optimization problem in either of two basic ways. It could compute profit for every feasible level of output and then identify the level of output that generates the maximum profit, or it could find the output level that sets marginal revenue equal to marginal cost (and then check to ensure that the firm wouldn’t prefer to shut down). Both approaches will yield the same result, so one is free to choose the optimization method that is most convenient, least expensive, or otherwise preferred. To make this accessible to students, we imagine a newly employed person who must decide how close to live to her job. Assuming that the apartments under consideration offer the same benefits (e.g., comparable amenities), we compare her total costs, which include not only the out-of-pocket rent, but also the commuting costs, which depend on the apartment’s distance from her job and on her opportunity cost of time. As we would expect, individuals with high opportunity costs of time will live closer to work— and will pay a premium for this convenience! Where We’ve Been Chapter 1 introduces the three underlying principles, including optimization. The evidence-based economics section considers the true cost of spending an hour per day on Facebook, and once students realize that the annual opportunity cost of this is about $3,650, perhaps they will re-optimize and spend more time travelling and less time Facebooking. Chapter 2 focuses mostly on empirical methods and considers a model of the returns to education. The chapter’s key optimization problem is whether a person with a given level of education should invest in an additional year of education, given the expected costs (tuition, forgone earnings) and benefits (higher future earnings, presumably based on higher human capital, and thus, value to employers). In short, readers already have some idea of what optimization is and how it relates to choices about how to spend one’s scarce resources, such as allocating an hour to either leisure or work, or if one works, doing so for either a government, business, or non-profit organization. So students should be ready for an in-depth presentation of different types of optimization and how these tools can be used to make a good decision about where to live. Where We’re Going Just as Chapter 2’s coverage of empiricism prepares students to understand the Evidence-Based Economics (EBE) vignettes found in each chapter, Chapter 3’s coverage of alternative methods of optimizing prepares students for all subsequent microeconomic chapters, each of which features one or more economic actors solving constrained optimization problems. Indeed, the authors encourage us to view the world as consisting of decision makers trying to do the best they can given limited resources and information. Here’s a quick guide to some of the numerous optimizers— and their choices— that you’ll find in the remainder of the microeconomic chapters: Chapter 4 Demand, Supply, and Equilibrium: Profit-maximizing oil companies shop around for the lowest-cost crude oil inputs. Chapter 5 Consumers and Incentives: Using marginal benefit per dollar spent (or “bang for the buck”), buyers make shopping decisions that generate demand curves; smokers decide whether it is worth $100/month to quit. Chapter 6 Sellers and Incentives: Profit-maximizing sellers decide how much to produce, thereby generating supply curves; ethanol producers build new plants if sufficiently motivated by cost-reducing government subsidies. Chapter 7 Perfect Competition and the Invisible Hand: Profit-maximizing firms determine whether to enter the paper delivery business or exit the corn hauling market. Chapter 8 Trade: Two parties with different comparative advantages decide whether to go it alone or to specialize and trade. Chapter 9 Externalities and Public Goods: A power plant determines how much electricity to produce (but it ignores the social costs of pollution); given its citizens’ demands, a social-surplus-maximizing government determines how many space missions to provide. Chapter 10 The Government in the Economy: Taxation and Regulation: To raise sufficient revenues to finance its programs, a government chooses from a set of possible taxes. Chapter 11 Markets for Factors of Production: Manufacturing firm Cheeseman weighs VMPL and the wage to determine the optimal number of workers to hire. Chapter 12 Monopoly: Patent-holding pharmaceutical firm Schering-Plough uses its monopoly power to choose the profit-maximizing price of Claritin. Chapter 13 Game Theory and Strategic Play: Each player in a prisoners’ dilemma decides whether to confess or hold out; rival surf shops decide whether to advertise; refineries decide whether to pollute in a commonly used canal; a soccer player taking a penalty kick tries to outwit the goalie; and you decide whether to trust Gary. Chapter 14 Oligopoly and Monopolistic Competition: Rival landscaping firms compete for customers by choosing prices and a monopolistically competitive Dairy Queen responds to entry by rival Baskin-Robbins. Chapter 15 Trade-offs Involving Time and Risk: A gambler decides how to bet at the roulette wheel, a TV buyer decides whether to buy an extended warranty, and a high-earning parent decides how much life insurance to purchase. Chapter 16 The Economics of Information: A used car shopper ponders how much to pay for a car of uncertain quality and Henry Ford deals with moral hazard (e.g., turnover, absenteeism) by offering an efficiency wage Chapter 17 Auctions and Bargaining: Football fans decide how much to bid for Oakland Raiders tickets and a job-hunter negotiates a starting wage with Caribou Coffee. Chapter 18 Social Economics: Players on the “Friend or Foe” TV game show decide the extent to which fairness concerns alter payoffs in a prisoners’ dilemma-style game and players in ultimatum games decide how large of a price to pay to punish others who are not being fair. Number of Lectures One could spend one-half to one full 50-minute lecture on Chapter 3, depending on students’ preparation, anticipated pace of the term, instructor faith in the ability of students to grasp nuances by reading (or even to complete the reading), and desire to connect with students early in the term through classroom discussion. Instructors with a less harried schedule might chat with students about optimization: Do students believe that they optimize? When? How? How have they optimized already today? How is optimization related to one’s commute? And to one’s length of commute? Does it seem that apartment prices fall as they get farther from the college/university? (One could collect, present, and discuss a bit of anonymous data on apartment prices and locations.) Why should total cost contain more than just rent? Who has the highest opportunity cost of time? What are the two types of optimization in the chapter and how do they work? Do students think that the same relationship between rent and proximity to the college of university holds in general? Opening Question and Evidence-Based Economics “How does location affect the rental cost of housing?” This is another great question that students will find relevant (many of them will need to do some apartment shopping soon, if they haven’t already) and timely (they’ve probably heard it’s been harder to get approved for a mortgage lately, so many living-space-hunters have looked to the apartment market … or moved home with parents). The opening question will be answered by the EBE section at the end of the chapter. To present this “Where to live?” question in class, one might replicate the chapter’s analysis of apartments around Portland, Oregon, by choosing the nearest major city and some favorite suburbs. For instance, a person with a job in Chicago’s Loop could live nearby in Streeterville, a few neighborhoods away in Ravenswood, or “far” up the Lake Michigan coast in Rogers Park. Many cities have a beltway (encircling highway), but even if it does not, we would expect higher rents in the most appealing locations. An alternative approach, which may be more appealing, is to think about apartments close to one’s college or university. We would expect the same results: Students are willing to pay more for apartments within walking distance than for apartments within a short drive. One could also cast this in terms of business location decisions: Just as students prefer to live near the center of action, firms may prefer to locate on streets with lots of foot traffic, and their rents (a capital cost) will reflect that. Organizers of college alumni functions or other social gatherings sometimes give prizes to the attendee who traveled the farthest; one could give a token prize to the student who had the longest commute, and then ask him/her a bit about the rent-convenience trade-off. (One should be sensitive to differences in student budget constraints.) III. Chapter Outline 3.1 Two Kinds of Optimization: A Matter of Focus Economists provide a useful perspective by modeling choices in terms of constrained optimization. A typical economic model features an economic actor (recall the list from Chapter 1) trying to pursue some objective (e.g., maximizing profit, utility, or social surplus) by making a choice in a complex environment characterized by various real-world constraints. Optimization refers to the objective-pursuing choice, and there are two basic approaches: Either choosing the alternative with the highest total net benefit (“optimization in levels”) or using marginal comparisons to find the alternative(s) for which no switch can increase net benefits (“optimization in differences”). The example of choosing between two Halloween bags of candy clearly illustrates the difference between the two methods of optimization. Teaching Idea: It’s helpful to remind students (now and then) that there is a difference between reality and economic models, and economists tend to believe that people do their best to optimize, but people make mistakes, face incomplete information, have different levels of risk tolerance and patience, and are otherwise imperfect at making decisions. Math-phobic students might be especially skeptical about modeling choice in a quantitative way or the argument that economics is more compelling than other social sciences. One way around this is to present some real world scenarios in which people seem to optimize. For example, a man at a salad bar considers the set of available salad components, the size of his empty plate, the difficulty of transporting a heaping plate, his friends’ expected comments on his choices, dietary restrictions, the quality of the items, and whether it would be a good idea to take the last tomato, and then assembles his ideal salad, given the circumstances. Alternatively, a woman with an evening flight who needs to get to the airport may choose between taking a taxi, train, bus, bicycle, walking, getting a ride from a friend, or hitchhiking; depending on the distance, weather, flight departure time and sense of adventure, any of these modes of transportation could be optimal. Teaching Idea: Some students are motivated by the possibility of making money or solving social problems. They might be interested in how many smartphone applications help users optimize by providing information or streamlining market functions. Consider an app that helps drivers find nearby gas stations and compare gas prices; reducing the time (and gas) one spends searching for inexpensive gasoline frees up time to run other errands and get more done, so this app may be highly valued by users who might be willing to pay for it, thereby rewarding the designer or programmer. “People’s behavior is approximated by optimization.” This means that an optimization model can provide a reasonable explanation of current and past behavior and should by useful in helping us predicting future behavior. Alternative Teaching Examples: The example of comparing bags of Halloween candy is great, but if one wanted to reinterpret it in an academic setting, one could compare two four-course schedules and replace one course (or one instructor). Choice & Consequence: Do People Really Optimize?—When people face unfamiliar decisions, they may struggle through a trial-and-error phase, but once they gain experience, they make better decisions and look more like optimizers. One could tell students that rookie mistakes are part of life and today’s experts were once newbies. If a goal of economics is efficiency (prudent use of limited resources), it’s a good idea to seek wise counsel, perhaps in office hours. Then again, there is value in early struggles—the answer to a tough question may make more sense and stick longer if a student has made a real effort to answer it before seeking assistance. During the first few times playing a card game, one may try to optimize, but without understanding the nuances of the game, may end up with disappointing results. It is important to note that poor results do not imply that the player was not optimizing, but the results may have been due to misunderstanding, bad luck, or actions by opportunistic veteran players. 3.2 Optimization in Levels This means identifying all of the costs and benefits associated with each feasible option, converting all costs and benefits into dollars (so we can make direct comparisons across alternatives), computing the total net benefit of each option, and picking the option with the maximum net benefit. “To keep things simple, we will omit other factors for now, even though they are important in practice.” As we teach the next generation how to do economics, an important lesson is how to boil down a complex scenario into its essential elements, and why it is necessary to make simplifying assumptions if one wants to make headway when trying to tackle a difficult problem. Some students get hung up on what they perceive to be unrealistic assumptions. The authors introduce the phrase “all else held constant” in Chapter 4, but one may choose to introduce the phrase ceteris paribus (Latin for “all else held constant”) here. Students who live near campus will find it easy and fun to generate a list of apartment benefits that have been assumed away in this section. Exhibit 3.1 illustrates four apartments, which differ only on location and price (rent). All four rest on one highway (going from the city to the southwest suburbs), but this is only because this looks nice and corresponds to the table entries. Reality is often messier, and they needn’t all rest on the same highway; the key question is how far is each from the city center, regardless of the direction. Just as the two bags of Halloween candy only differed in whether there was a Milky Way bar or a 3 Musketeers bar, the four apartments differ only on location and rent. “An extra hour of time has value to you whatever you would do with that time, including napping, socializing, watching videos, taking longer showers, or working.” Once we have a plausible economic model, it may be useful to a wide variety of people. Economists appreciate general analyses that can be applied to a wide variety of similar scenarios. The dollar cost of the commute is , and we notice that the “hours” terms cancel. Some students will find such math procedures very easy, while other students may find such math procedures intimidating, though they may not admit it. It only takes a few seconds to talk through such an example, and some will quietly appreciate it. Back in Chapter 1’s discussion of opportunity cost, a common example is the choice of whether to stay in college for a fifth year; one could show how to crudely estimate annual earnings given an hourly wage: . “We first need to decide on a common unit of account. Let’s pick dollars per month for now.” Some students are put off by the omnipresence of dollars in an economics course, as if economists only care about money. One response is that we all understand the value of $1, so in America, this is probably the clearest way to express value, though we could always use Hershey Bars, which also have value, even for people who don’t like chocolate (because they can be traded). In Chapter 1, we found that the cost of an hour of Facebooking includes the often-overlooked opportunity cost of that hour. Here, it is easy to recognize the direct cost of rent— because one writes a check due on the first of the month— but it may be easy to overlook the indirect cost of commuting time. The authors provide the three steps of optimization in levels. One might add that it is important to identify all of the relevant (incremental) costs and benefits before translating them into dollars. Comparative Statics—We solve an economic model for the status quo, change some assumption, resolve the model in the new environment, and then compare the two answers. This approach helps us understand both exogenous shocks and changes based on willfully implemented interventionist policies. To make the concept of comparative statics palatable for students, one might present a list of policies and try to get the class to predict the likely responses. For example, ask how many students get flu shots now, and how many more students would get flu shots if they were free? What if a student received two free tickets to a big basketball game when getting a flu shot? Or, at the state or national level, what will happen to employment if the government adopts a higher minimum wage? If one wants to go through the “Where to live” optimization problem, Exhibit 3.6 is the key diagram to show because it’s clear that total costs bottom out at the Close apartment for employees with the high ($15/hour) opportunity cost of time, whereas they are minimize at the Far apartment for employees with low ($10/hour) opportunity cost of time. Textbooks inevitably become dated, so it is nice when one can weave in some timely real-world anecdotes. This section’s comparative statics analysis is based on increasing the opportunity cost of time from $10/hour to $15/hour. Fortuitously, the minimum wage has been in the news a lot lately, particularly Seattle’s decision to increase its minimum wage to $15/hour. Comparing Exhibits 3.2 and 3.4, we see that the total-cost-minimizing apartment changes from Apartment Far to Apartment Close when we hold rent fixed and vary the opportunity cost of time. One might point out that another comparative statics problem not covered here would be to vary something else. For example, perhaps a major, multi-year, highway construction project changes the commuting times or events (e.g., the bankruptcy and decline of Detroit, a city revitalization project, or the construction of a new stadium or other major attraction) change the relative apartment rents as people migrate to the city center or out to the suburbs. In short, comparative statics involves measuring the impact of a change in one variable, holding the other variables constant. 3.3 Optimization in Differences: Marginal Analysis This approach to optimization involves identifying all relevant costs and benefits of each option, translating all of these costs and benefits into dollars, computing the change in net benefits due to moving from one option to another, and continuing to move better options until one identifies the option that has the highest net benefit, and also the property that one’s well-being improves by moving to it, and one’s well-being falls (or stays the same) by moving away from it. Better students will ask what happens if there are multiple optima, i.e., alternatives that generate the same total net benefits. Some might be scared by talk of strong preferences and weak preferences, but they may appreciate analogies to physical structures such as mountain peaks (one optimum), ski hill moguls (many separate optima), and buttes (many connected or adjacent optima). Common Mistakes or Misunderstandings: Careless application of optimization in differences can lead one to identify a local optimum, but miss the global optimum. For example, if one were standing at the top of the shortest of the three Great Pyramids of Giza on a moonless night, one’s altitude would temporarily fall with movement in any compass direction, leading one to believe that s/he was at a global maximum. Later, when discussing firms, we’ll see that setting marginal revenue equal to marginal cost generates the maximum profit from producing, but it could be the case that shutting down allows one to break even rather than incur a loss. Usually in economics marginal means “due to an additional unit,” and we study marginal changes in profit, revenue, cost, utility, etc. Some may object to using the term marginal cost to apply to cost changes between different options; incremental cost might be a palatable alternative. Marginal Cost—The additional cost incurred when one moves between adjacent feasible options. Later, we will revisit this in the context of profit-maximizing firms. For now it makes sense to hold off on the formal definition, . Alternative Teaching Example: Intuitively, if one plans to make pancakes, the marginal cost of an upgrade to blueberry pancakes is simply the additional cost of the blueberries because the other ingredients are all the same. Optimization by levels would consider the cost of all of the other ingredients, whereas optimization by differences focuses only on the changed costs and/or benefits. Evidence-Based Economics: How does location affect the rental cost of housing?— This EBE case shows that apartment rent tends to fall as distance to the city center rises, largely because people are willing to pay higher rent to shorten their commute. Essentially, people value convenience (e.g., the less time spent sitting in traffic or on a bus the better), but as many potential tenants compete for a scarce amount of apartments in preferred locations, they drive up the price, so there is a trade-off between rental cost and commuting cost—as one goes up, the other goes down. If comparable apartments offer similar benefits, then the maximum net benefit will occur when total costs are minimized. Because people have heterogeneous opportunity costs of time, they are solving different optimization problems, and we expect to see a range of solutions; the chief executive officer who values an hour at $1,000 may live a block away from work while a fast-food worker who values an hour at $20 may live many miles away, at least until he becomes CEO, at which point he moves much closer to work. This example gives us a chance to compare optimization techniques, revisit the concept of opportunity cost, introduce marginal analysis in general (and marginal cost in particular), use comparative statics, and talk about other choices in the housing market. As shown in Exhibit 3.9, the flattening of rents around 12 miles from the city center due to a “ring road” (or beltway) is a neat result that students may find interesting. The last page of the EBE case inspires several excellent discussion questions. For example, what are some of the other ways of allocating scarce apartments besides market prices? [A: first come, first served; random/lottery; nepotism/connections/favoritism.] Also, is it fair that those with the highest opportunity cost of time tend to pay the most and live close to the city center? VI. Active Learning Exercises 1. (Optimization in Levels) Suppose there are three possible airports near your home that you use for a flight to Boston. At the closest airport you find an airfare of $400, the middle airport (the second closest to your home) has an airfare of $325, and you would pay $275 to fly from the farthest airport. Assume you want to take a taxi to the airport and the roundtrip cost to the close airport is $20 and will take 15 minutes. The taxi fare to the second airport is $30 and will take 30 minutes. The taxi fare to the farthest airport is $60 and will take one hour. You have a part-time job where you earn $12 per hour. Create a table similar to Exhibit 3.2 and determine which airport is the optimum for you to use. Solution: The optimum is the airport farthest from your house. Airport Time to Airport Cost of Time Traveling to Airport Taxi Fare Airfare Total Cost Closest 15 minutes $3 $20 $400 $423 Middle 30 minutes $6 $30 $325 $361 Furthest 60 minutes $12 $60 $275 $347 2. (Optimization in Levels; Comparative Statics) Use all the data from exercise 1 for this problem. Assume you have a next door neighbor who is a lawyer who earns $200 per hour. Create a table similar to Exhibit 3.2 and determine which airport is the optimum for your neighbor to use. Solution: The optimum for your neighbor is the airport that is in the middle (or second closest to your house). Airport Time to Airport Cost of Time Traveling to Airport Taxi Fare Airfare Total Cost Closest 15 minutes $50 $20 $400 $470 Middle 30 minutes $100 $30 $325 $455 Furthest 60 minutes $200 $60 $275 $535 3. (Optimization in Differences: Marginal Analysis) Use the data from exercise 1 for this problem. Create a table using marginal analysis to confirm that the answer for exercise 1 is the same when marginal analysis is used instead of optimization in levels. Solution: When using marginal analysis you need to add a column to compute the change in the total cost as you move the closest airport to the middle airport. The optimum for you remains the airport that is farthest from your house. Airport Time to Airport Cost of Time Traveling to Airport Taxi Fare Airfare Total Cost Marginal Cost Closest 15 minutes $3 $20 $400 $423 X Middle 30 minutes $6 $30 $325 $361 –$62 Furthest 60 minutes $12 $60 $275 $347 –$14