BEE2025 Microeconomics II Lecture Notes PDF Autumn Term 2024

BEE2025: Microeconomics II Shaun Grimshaw, Max Posch, and Alejandro Robinson-Cortés University of Exeter Lecture Notes Autumn Term 2024 [Week 2] Contents 1 Introduction...

BEE2025: Microeconomics II Shaun Grimshaw, Max Posch, and Alejandro Robinson-Cortés University of Exeter Lecture Notes Autumn Term 2024 [Week 2] Contents 1 Introduction 5 1.1 Why models? Why maths?......................... 6 1.2 Optimisation and equilibrium....................... 6 1.3 What is a desirable allocation? Efficiency and fairness......... 7 1.4 How to use these notes........................... 8 2 Budget sets 10 2.1 Consumption sets, bundles, and budget sets............... 10 2.2 Changes in prices and income....................... 12 2.3 Policies affecting budget sets........................ 14 2.3.1 Per-unit and ad-valorem taxes and subsidies.......... 14 2.3.2 Lump-sum taxes and rationing.................. 14 2.3.3 Non-linear taxes and subsidies.................. 15 2.3.4 Vouchers............................... 16 3 Preferences 18 3.1 Preference relations............................. 18 3.2 Axioms of consumer theory........................ 21 3.2.1 Rational preferences: completeness and transitivity...... 21 3.2.2 Visualizing preferences when X = R2+.............. 23 3.2.3 Local non-satiation and monotonicity.............. 26 3.2.4 Convexity.............................. 28 3.2.5 Well-behaved preferences..................... 29 3.3 Goods, bads, and bliss points....................... 29 3.4 Marginal rate of substitution........................ 30 2 References 34 Glossary 35 3 Note on sources. The bulk of these lecture notes is standard undergraduate mi- croeconomic theory, which dates back to the days of Marshall (1890) and Hicks (1939). The classic reference nowadays is Varian (2010). The first draft of the notes was based on the teaching material of Grame Pearce. The notes draw mainly from Varian’s book and what we have learned in a number of classes throughout the years. We have also relied on Nicholson and Snyder (2008), and Jehle and Reny (2001) for some of the more advanced parts. The lecture notes of David Autor— publicly available online—also have been a useful resource. There are a number of well-known textbooks in microeconomics. While there is some variation across them, the core or classic theory is pretty standard. We’ve done our best to avoid typos and mistakes. Please let us know if you find any! 4  Contents 1 Introduction Economics studies how humans allocate scarce resources, individually and through institutions. In particular, microeconomics studies the actions that consumers and firms take in the presence of scarcity and how these actions interact with each other. Microeconomists give particular attention to the functioning of markets, which are some of the most common and successful institutions through which scarce re- sources are allocated in reality. In traditional markets, prices determine who gets what: you can buy it if you can pay the price, and you can sell it if someone is willing to pay for it. Two fundamental questions are: what determines prices in an economy? And, are the allocations generated by markets somehow desirable? The question of what drives prices in a market economy is one of the oldest in the discipline. Over the past century economists have developed theories that have increased greatly our understanding of markets. This has allowed economists to un- derstand better how and why markets work, as well as when they fail and how can we fix them when they do so. In this module, we will cover some of the foundations underlying classic microeconomic theory, all of which are aimed at understanding the determinants of prices. This module is focused on the theory of the consumer. Next term, you will cover the theory of the producer. Both are the foundation of what is known as general equilibrium theory. The goal of general equilibrium theory is to explain how the prices of all the goods in an economy are determined simultaneously. Near the end of the term, we will study pure exchange economies, which are the simplest setting to study general equilibrium. A key assumption behind classic general equilibrium theory is perfect compe- tition, which means that agents behave as price takers. In perfectly competitive markets, the actions of single consumers or firms do not affect market prices. While this may be a somewhat realistic assumption on the consumer side for many real- life markets, it is probably less so for firms. In real life, we see firms setting prices all the time, some of which are likely non-competitive. The study of markets in which firms are able to set prices above their competitive level, and in doing so exploit their market power, is the focus of industrial organization, one of the most active fields of study within microeconomics today. However, in this module, we will limit our- selves to study perfectly competitive markets. In other words, we will study the functioning of markets when they work correctly. 5  Contents 1.1 Why models? Why maths? Economic phenomena can be immensely complex. The objective of economic mod- els is to strip them to their most basic components in order to make them tractable and simpler to analyze. The trick lies in simplifying the analysis to make it as tractable as possible without ignoring its key components. As we go along, you will see that economic models can become increasingly complex as we add assump- tions to make them more realistic. The key trade-off then is how to add just what is necessary to prove a point that is plausibly relevant in the real world. The language of modern economics is largely based on mathematical models. Formulating arguments using maths compels us to be formal, precise, and trans- parent in our reasoning. Thanks to their universal nature, maths also allow us to communicate ideas more concisely and efficiently than when using everyday lan- guage. However, a drawback is that maths require practice and technical skill. On occasion, and at times with reason, economic models are criticized for prioritizing mathematical elegance over relevant aspects of the social reality they aim to explain. Nonetheless, despite this trade-off, many would argue that a methodology based on formal reasoning has allowed economics to become a highly specialized and professional discipline over the past half century. Nowadays, economists are less susceptible to slippery arguments and much better equipped to back their claims empirically than in the past. One of the goals of this module is to introduce you to formal modeling in economics. 1.2 Optimisation and equilibrium Standard microeconomic theory is largely based on two principles: the optimisation and the equilibrium principles. The optimisation principle states that individuals act rationally: they know what they want, and they act accordingly. This princi- ple follows from the premise that agents make choices freely and voluntarily, and, hence, choose the alternatives they deem better for themselves. The optimisation principle is not without controversy. Throughout the years, behavioural economists have documented numerous instances in which individual behavior deviates from what standard models of rational behavior predict. The critiques are typically fo- cused on how agents’ choices can be influenced by distinct aspects of an environ- ment, rather than on optimisation itself. That is, in recognizing that the optimisation 6  Contents problems agents face in reality are likely more complex than what some standard models assume. The second principle is the equilibrium principle. Loosely, this principle states that economists tend to study markets “at equilibrium,” where an equilibrium is an arrangement of individual actions that are self-enforcing or stable in some way. While economic reality can be incredibly complex and subject to constant shocks, economists rely on the existence of equilibria to simplify the analysis. In our case, we will study the competitive equilibrium of a market economy. This is a situation in which prices adjust freely to clear demand and supply. At the equilibrium price, the amount of a good that consumers want to buy is equal to the amount firms want to sell. The presumption is that, if this were not the case, prices would adjust to equilibrate markets. That is, if the amount of a good demanded by consumers is greater than the amount firms are willing to supply at a given price, prices would increase to reduce demand and incentivize supply. Conversely, if consumers de- mand less quantity than what firms are willing to supply at a given price, prices would decrease to increase demand and reduce supply. For the most part, the focus of economic analysis is on the equilibrium itself, rather than on how an economy gets to it. 1.3 What is a desirable allocation? Efficiency and fairness The most common criteria economists use to assess the desirability of an allocation is efficiency. In its simplest form, efficiency is understood as a lack of unnecessary welfare loss. More formally, economists rely on the notion of Pareto efficiency. An allocation is said to be Pareto efficient if it is not possible to improve an individual without harming others. While the notion of Pareto efficiency is straightforward and intuitive, it might arguably be seen as a weak or insufficient criteria to rank al- locations. For instance, it is mute regarding how resources are distributed among individuals in an economy, neglecting aspects of fairness or inequality. Nonethe- less, for many decades economists have remained agnostic about subjective criteria that involve prioritising some individuals’ welfare over others. In this sense, one of the pillars of modern economics has been to recognize that agents differ in their tastes, and, as such, should be free to decide their own con- sumption, be it in terms of how much to spend, save, work, study, etc. While this approach has not been free of criticism, it has allowed economists to stay away of 7  Contents questions for which there appear to be no satisfactory answers. Or at least answers based on objective criteria, rather than on prioritising some individuals over others via arbitrary principles. This being said, the study of fairness and inequality have become more and more active in recent years within the discipline. While a lot of the work in this area invariably runs into some of these issues, its goal is to alleviate some of the drawbacks of focusing solely on efficiency. In this module, however, we will focus on the traditional approach based on Pareto efficiency. 1.4 How to use these notes The notes cover all the material we will see during the module. We will update them every week as we go along. While we will cover some of the material during lectures, the notes go above and beyond what we will see in class. You are expected to work through the notes on your own, even (especially) the parts that we do not cover in class. We’ve done our best to write the notes in an accessible and self- contained way. We have added a glossary at the end where we briefly introduce some of the mathematical concepts we assume you already know. If you feel like you could use a maths refresher or would like a more standard treatment of the maths used in the notes, you may find helpful to review any of the existing “math- ematics for economists” textbooks, such as Sydsæter et al. (2021), Renshaw (2021), or Jacques (2017). As you make your way through the notes, keep in mind that some parts are more difficult than others. Try your best not to let yourself down if you find the material hard the first time you read it. It might be the case that you have to go through a section more than once. We recommend you to work through the notes as you read them: solve the exercises, derive the equations on your own, try to formulate examples to test your understanding, come up with questions, and, above all, think critically. To help you navigate the notes and set expectations for the exams, we have in- dicated the difficulty of the exercises, examples, and remarks, with the following symbols on the left margin of the text with the symbols (i), (ii), (iii), and (iv). They indicate the following. (i) This means essential. You should feel absolutely comfortable solving these exer- cises by the time of the exam. 8  Contents (ii) This means important. These exercises are not the easiest, but not the hardest either. They are about the average difficulty of what you will face in the exams. (iii) This means hard, but important. These exercises are the challenging ones. Feeling comfortable solving them is a good sign that you are well prepared for the exam. (iv) This means nice to know. We use this to indicate material that is a bit more ad- vanced. While these parts will help you to grasp better the meaning and importance of the material, we recognize it might be difficult to fully dominate them during this term. If you have a hard time understanding something or get stuck, please do not hesitate to reach out and ask questions. Good luck! 9  Contents 2 Budget sets 2.1 Consumption sets, bundles, and budget sets We start by modeling the choice of a consumer. The object of choice is a consump- tion bundle, typically denoted by x ∈ X, where X is the consumer’s consumption set.1 In the standard case, we will focus in the problem a consumer faces when de- ciding how much to consume of two goods that are infinitely divisible. That is, X = R2+ , which means that a consumption bundle is a vector x = (x1 , x2 ), where x1 is the quantity of good 1 in the bundle and x2 that of good 2.2 Our first task is to characterize the set of bundles the consumer can afford to purchase. Let p = (p1 , p2 ) ∈ R2++ denote the price vector associated to X, where p1 is the price of good 1 and p2 is the price of good 2.3 Let m ≥ 0 be the amount of money or income the consumer has to spend. Then, the consumer can afford bundle x = (x1 , x2 ) if p1 x1 + p2 x2 ≤ m. The budget set of the consumer is the set containing all the bundles they can afford, that is: B = {(x1 , x2 ) ∈ R+ : p1 x1 + p2 x2 ≤ m}. The equation p1 x1 +p2 x2 ≤ m is known as the consumer’s budget constraint. When it is satisfied with equality, we refer to it as the budget line. Note that the budget constraint is satisfied with equality at bundles in which the consumer spends all their income. (i) Exercise 2.1.1. Explain what does this mean: “the budget constraint is satisfied with equality at bundles in which the consumer spends all their income.” How many bundles are on the budget line? Budget sets can be depicted graphically. To plot a budget set, rearrange the ex- pression of the budget line to express x2 as a function of x1 : m p1 x2 = − · x1. p2 p2 1. See that we are using set notation when talking about the consumption set X. If you would like a refresher on set notation, click on it to go to the Glossary. 2. When we talk about consumption bundles x = (x1 , x2 ), we use vector notation. 3. The set R2++ is the strictly positive orthant of the cartesian plane. 10  Contents Figure 1: Generic budget set (left); coffees and sandwiches (right) This shows that the budget line takes the shape of a straight line with intercept m/p2 and slope −p1 /p2.4 Graphically, the budget set consists of all the bundles below the budget line. See the left panel of Figure 1. The intercepts of the budget line have a clear economic interpretation. If the consumer spends all their money on good 1, they can purchase m/p1 units. And if they spend it all on good 2, they can purchase m/p2 units. The slope of the budget line also has an important economic interpretation: it is the exchange rate between goods imposed by the market prices, the opportunity cost of one good in terms of the other. To see this, ask yourself: if the consumer is currently spending all their income, how many units of good 2 do they need to forego if they want to purchase an extra unit of good 1? In other words, the ratio p1 /p2 is the relative price of good 1 (i.e., the “price” of good 1 in terms of good 2). (i) Example 2.1.2. Consider a consumer who has income m = 12 and is deciding to purchase coffee and sandwiches. The price of a sandwich is p1 = 8 and the price of a coffee is p2 = 4. Further assume that sandwiches and coffee are infinitely divisible (e.g., they can buy a fraction of a coffee or a sandwich by sharing with a friend). See that if they spend all their money in sandwiches, they can afford one and a half sandwich (8 × 1.5 = 12), and if they spend it all on coffee, they can afford 3 coffees (4 × 3 = 12). They can also purchase both items, as in one sandwich and one coffee ((8 × 1) + (4 × 1) = 12), or 0.75 sandwiches and 1.5 coffees ((8 × 43 ) + (4 × 32 ) = 12). One sandwich costs the same as two coffees. The consumer could also choose not to spend all their money by purchasing, say, half sandwich and one coffee. The budget set of this consumer is depicted on the right panel of Figure 1. 4. Here we are making use of the functional form of straight lines. 11  Contents (ii) Exercise 2.1.3. Consider a consumer who has no friends, so they can only buy sand- wiches and coffees in full units. What is the budget set of this consumer? How does it look like graphically? (iv) Exercise 2.1.4. Evaluate whether the following statement is true, false, or it depends: “if a consumer has no income, i.e., if m = 0, then their budget set is empty.” 2.2 Changes in prices and income An advantage of analyzing budget sets graphically is that we can visualize how they are affected by changes in prices and income. First, note that a change in the price of either good will change the slope of the budget line, as well as its intercept on the corresponding axis. The left panel of Figure Figure 2 shows the effect of a decrease in the price of good 1, a decrease from p1 to p01 , where p01 < p1. Since the slope of the budget line decreases in absolute terms, p01 /p2 < p1 /p2 , it becomes flatter. Likewise, if the consumer spends all their money on good 1, they will be able to afford more of it, m/p01 instead of m/p1. By contrast, the amount they are able to purchase of good 2 were they to spend all their money on it remains unchanged. (i) Exercise 2.2.1. Indicate in a plot how an increase in the price of good 2 changes the budget set. (i) Exercise 2.2.2. Suppose the prices of both goods increase in the same proportion, to p01 = αp1 and p02 = αp2 , for some α > 1. How does this affect the budget set? Show your answer in a plot. A change in the consumer’s income m leaves the slope of the budget line un- changed, but it affects its intercepts. If m increases, the budget line will shift out- wards in parallel. The opportunity cost between goods remains unchanged but the consumer can afford to buy more of them; their purchasing power increases. Like- wise, if the income of the consumer decreases, the budget line shifts inward towards the origin in parallel. See the right panel of Figure 2. (ii) Exercise 2.2.3. In a first instance, suppose the prices of both goods increase in the same proportion to p01 = αp1 and p02 = αp2 , for some α > 1. Now, instead of the prices increasing, suppose that the income of the consumer changes in some quantity ∆m. That is, the new income level is m0 = m + ∆m. What is the value of ∆m such that the budget set of the consumer is the same in both situations, with an increase in prices or with a change in income? 12  Contents Figure 2: Effect of price decrease (left) and income decrease (right) (ii) Exercise 2.2.4. Suppose there is inflation, prices increase proportionally by the same factor 1 + π with π ∈ (0, 1), and the government wishes to compensate consumers for their loss in purchasing power by increasing their income.5 Who will receive a higher compensation in absolute terms, those with higher or lower incomes? Who loses more from inflation, the rich or the poor? Discuss. Note that if prices and the income of the consumer change in the same propor- tion, the budget set remains unchanged.6 The specific units with which we measure money are irrelevant for the budget set. In other words, the budget set only depends on real prices, not nominal ones. What matters for the budget set are not the abso- lute levels of prices but the relative prices between the goods, which are encoded in the price ratio. For this reason, on occasion economists normalize the price of one good to 1, and they refer to this good as the numéraire. This is equivalent to dividing the budget constraint by p1 and thinking of the price of good 2 and the consumer’s income in terms of good 1. As in, how many units of the numéraire does good 2 cost, and so on. 5. When stating π ∈ (0, 1), we are using notation for intervals. 6. To see this clearly, see that if x is affordable under prices p and income m, it is also affordable under prices αp and income αm for every α > 0. 13  Contents 2.3 Policies affecting budget sets 2.3.1 Per-unit and ad-valorem taxes and subsidies The budget set of a consumer can be affected by government policies that affect prices and incomes. For instance, suppose that the government imposes a per-unit tax on good x1 of τ > 0. For every unit purchased of good 1, consumers have to pay τ on top of its price. Then, the budget line of the consumer becomes: (p1 + τ )x1 + p2 x2 = m. Alternatively, the government can impose an ad-valorem tax on good 1 of τ ∈ (0, 1), meaning that the consumers pay a tax rate of τ on their expenditure on good 1. Their budget line then becomes: (1 + τ )p1 x1 + p2 x2 = m. (ii) Exercise 2.3.1. Show the effect of a per-unit tax and an ad-valorem tax on the budget set. Plot your answers in separate plots. How do taxes affect the purchasing power of consumers? (ii) Exercise 2.3.2. The government is considering two policies to disincentivise the con- sumption of tobacco. The first policy is a per-unit tax of £2.75 per pack of cigarettes. The second one is an ad-valorem tax of τ % on every purchase of cigarettes. Find the value of τ which makes the two policies equivalent. Does it depend on the price of cigarettes? Why would a government prefer one policy over the other one? Discuss. (ii) Exercise 2.3.3. How do per-unit and ad-valorem subsidies affect a consumer’s bud- get set? Plot your answers. Discuss how per-unit and ad-valorem subsidies are different to per-unit and ad-valorem taxes, respectively. 2.3.2 Lump-sum taxes and rationing Another common policy are lump-sum taxes which impose a direct deduction from consumers’ income. In the presence of a lump-sum tax of T > 0, the income a consumer has available to spend becomes m − T instead of m. A rationing policy imposes a cap on the quantity a consumer may purchase of a certain good. For example, consider there is rationing on good 1, and consumers 14  Contents may only buy up to x̄1 units. In this case, the budget set takes the following form: B = {(x1 , x2 ) ∈ R2+ : x1 ≤ x̄1 & p1 x1 + p2 x2 ≤ m}. (ii) Exercise 2.3.4. Plot the budget set in the presence of rationing for good 1. Assume x̄1 is the maximum amount of good 1 the consumer is allowed to purchase. Consider two cases: (a) x̄1 < m/p1 and (b) x̄1 ≥ m/p1. Discuss why these two cases are relevant. 2.3.3 Non-linear taxes and subsidies When deriving the budget line in the presence of policies, it is useful to have in mind what a budget constraint represents more generally. In general, the consumer’s bud- get constraint can be seen as: resources needed to resources available ≤ consume bundle to spend This is especially helpful when analysing the effect of non-linear policies. For ex- ample, consider that the government imposes an ad-valorem tax on good 1 for the quantities purchased above x̄1 > 0. That is, the consumer can purchase up to x̄1 units of good 1 tax-free, but has to pay an ad-valorem tax of τ for units consumed beyond the first x̄1 units. To derive the effect of this policy on the budget set, notice that the consumption set can be split into two: first, the set of bundles in which the consumer pays no tax (those in which x1 ≤ x̄1 ), and second, those in which the consumer will pay some tax (those in which x1 > x̄1 ). In the first case, if x1 ≤ x̄1 , the budget constraint remains unchanged. However, in the second case, if x1 > x̄1 , the expenditure of the consumer will be: (1) p1 x̄1 + (1 + τ )p1 (x1 − x̄1 ) + p2 x2. The first term is the expenditure on the first x̄1 units of good 1, p1 x̄1 , which the consumer gets tax-free. The second term, (1 + τ )p1 (x1 − x̄1 ), is the expenditure on the units beyond the first x̄1 units, of which there are x1 − x̄1. For these units, the consumer pays the price plus the ad-valorem tax.7 And the third term is the 7. If, say, x̄1 = 5 and x1 = 7.5, then the consumer gets the first five units tax-free, but pays tax on the remaining 2.5 units. So the expenditure in good 1 is 5p1 + 2.5(1 + τ )p1. 15  Contents expenditure on good 2, which pays no tax. Hence, we can write the budget set in this case as the union of two sets. Namely, B = B1 ∪ B2 , where B1 = {(x1 , x2 ) ∈ R2+ : x1 ≤ x̄1 & p1 x1 + p2 x2 ≤ m} B2 = {(x1 , x2 ) ∈ R2+ : x1 > x̄1 & p1 x̄1 + (1 + τ )p1 (x − x̄1 ) + p2 x2 ≤ m}. To plot the budget set, consider the two cases, when x1 ≤ x̄1 , which corresponds to B1 , and when x1 > x̄1 , which corresponds to B2. In the first case, the budget line takes the usual shape, with slope −p1 /p2 and intercept m/p2. In the second one, by setting the expenditure in (1) to be less than or equal to m, and rearranging, we obtain: m + τ p1 x̄1 (1 + τ )p1 x2 ≤ − · x1 p2 p2 From the above, we see that the budget line has a slope −(1 + τ )p1 /p2. This means that the budget line is steeper when x1 > x̄1. Likewise, see that when the consumer only consumes good 1, they can afford to consume (m + τ p1 x̄1 )/(1 + τ )p1 units of it. Putting the two cases together, one obtains the budget set plotted in Figure 3, which has a kink outwards. This is intuitive since we know that an increase in the price of good 1 causes the budget line to be steeper and moves its intercept with the x-axis closer to the origin. In this case, the same occurs, but only on the area to the right of x̄1 , where the tax kicks in. (ii) Exercise 2.3.5. Consider the government imposes a per-unit subsidy of s > 0 for the first x̄2 units bought of good 2, where x̄2 ∈ (0, m/p2 ). Plot the effect of this policy in the consumer’s budget set. Contrast your answer with the case of a tax for the units bought above x̄2. (ii) Exercise 2.3.6. Consider the government imposes two tax thresholds for good 1, whereby any consumption of good 1 between x̄1 and x̄2 is taxed at a per-unit tax of τ1 and consumption of good 1 above x̄2 is taxed at a per-unit tax of τ2. Assume 0 < x̄1 < x̄2 < m/p1 , and 0 < τ1 < τ2. Plot the budget line of this consumer. 2.3.4 Vouchers A voucher is a specific amount of money the government gives to a consumer which may only be spent on one good. For example, consider a voucher worth v > 0 for good 1. To plot the effect of vouchers in the budget set, we follow a similar reasoning 16  Contents Figure 3: Ad-valorem tax of τ for quantities of x1 above x̄1 as with non-linear taxes. See that the consumer will spend none of their income on good 1 if they purchase an amount that is worth below v. And if they spend above this amount on good 1, they will only pay for the expenditure above v. In other words, if p1 x1 ≤ v, the consumer’s actual expenditure will only consist on what they spend on good 2, so their budget set will be: B1 = {(x1 , x2 ) ∈ R2+ : p1 x1 ≤ v & p2 x2 ≤ m}. Alternatively, if p1 x1 > v the consumer will have to pay what they spend on good 1 above v, so their budget set will be: B2 = {(x1 , x2 ) ∈ R2+ : p1 x1 > v & (p1 x1 − v) + p2 x2 ≤ m}. Taking the union of the sets B1 and B2 results in the consumer’s budget set. (ii) Exercise 2.3.7. Plot the budget line of a consumer who has a voucher worth v > 0 for good 1. Compare your answer with a budget set without any vouchers. 17  Contents 3 Preferences In the previous section, we analyzed the set of consumption bundles consumers can afford to purchase given the market prices and their income. In this section and the next, we turn to the question of how consumers rank consumption bundles. 3.1 Preference relations We model consumer preferences for goods using preference relations. A preference relation % is defined over the consumption set X. For any pair of consumption bundles x, x0 ∈ X, we write: x % x0 if x is at least as preferred as x0. That is, we assume that a consumer’s preferences over goods can be summed up by a preference relation %, which specifies which bundles are “at least as preferred” as others. Importantly, for any pair of bundles x and x0 , one and just one of the following must be true: either the consumer prefers x at least as x0 (and we write x % x0 ), or they do not prefer x at least as x0 (and we write x % / x0 ). Preference relations denoted by % are also known as weak preference relations. If x % x0 , we also say that the consumer weakly prefers x to x0. If a consumer prefers x at least as x0 , but they do not prefer x0 at least as x, we say that the consumer strictly prefers x to x0. Formally, given a weak preference relation %, we define a strict preference relation as follows: x  x0 if x % x0 and x0 % / x. In other words, we say that x is strictly preferred to x0 if it is at least as preferred to it, and x0 is not at least as preferred to x. See that if a consumer strictly prefers x to x0 , then they also weakly prefer it. However, the converse may not be true. If a consumer prefers x at least as x0 , and they also prefer x0 at least as x, we say that the consumer is indifferent between x and x0. Formally, given %, we define the indifference relation as: x ∼ x0 if x % x0 and x0 % x. 18  Contents In other words, we say that a consumer is indifferent over bundles x and x0 if both of them are at least as preferred as the other one. (i) Example 3.1.1. Consider two consumers who are choosing among a finite number of alter- natives, for example, what color of car to buy. Assume that the options are Red, Blue, and Green, so X = {Red, Blue, Green}. A preference relation % encodes the preferences of a consumer for all the alternatives in the consumption set.8 Call the two consumers 1 and 2. Suppose that consumer 1 likes the red car more than the blue one, and they like the green car the least, less than the red and less than the blue. Consumer 2 also likes the green car the least, but they are indifferent between the red and the blue cars. Then, we may describe consumer 1’s preferences by %1 , where %1 is such that: Red %1 Blue, Blue % / 1 Red, Red %1 Green, Green % / 1 Red, Blue %1 Green, and Green % / 1 Blue. See that this is equivalent to: Red 1 Blue, Red 1 Green, and Blue 1 Green, where 1 is the strict preference relation of consumer 1. More succinctly, we may also write this as: Red 1 Blue 1 Green. Since consumer 2 is indifferent between the red and the blue cars, their preferences will be described by a different preference relation, call it %2. Then, we have Red %2 Blue, / 2 Red, Blue %2 Green, and Green % Blue %2 Red, Red %2 Green, Green % / 2 Blue. Using the indifference relation of consumer 2, ∼2 , and their strict preference relation 2 , we can describe their preference as: Red ∼2 Blue, Red 2 Green, and Blue 2 Green. Or, more succinctly, as: Red ∼2 Blue 2 Green. 8. Note that the alternatives in this example are colors, and that we have a finite number of them. This example is somewhat simpler than the standard consumption set, where the alternatives are bundles of two infinitely divisible goods (i.e., when X = R2+ ). Dealing with finite consumption sets that only have few alternatives is useful to gain intuition when first encountering preference rela- tions. Nonetheless, the consumption set we are ultimately interested is R2+ , which contains an infinite number of alternatives. 19  Contents (i) Exercise 3.1.2. Consider the set of universities you applied to back when you grad- uated high school. What is the weak preference relation that describes your pref- erences over these universities? Can you describe the same preferences using the strict preference and indifference relations? (i) Example 3.1.3. Consider consumers 3 and 4 who are also thinking about what color of car to buy. However, these two consumers have some “strange” preferences. Consumer 3 is indecisive. They know that they prefer red to blue, but nothing else. They cannot make up their mind about whether the green car is better than the others, the same, or worse. Consumer 4 has made up their mind about all cars, but cannot decide which one is their favorite: they like red better than blue, blue better than green, and green better than red! Can we describe the preferences of these two consumer using preference relations? The answer is “yes!” Consumer 3’s preference can be described by %3 such that Red %3 / 3 Red, which is equivalent to Red 3 Blue. Since we have no further Blue and Blue % information about consumer 3’s preferences, we have Red % / 3 Green, Blue % / 3 Green, / 3 Red, and Green % Green % / 3 Blue. Consumer 4’s preferences can be described by %4 such that: Red %4 Blue, Blue % / 4 Red, Blue %4 Green, Green % / 4 Blue, Green %4 Red, and Red % / 4 Green, which is equivalent to: Red 4 Blue, Blue 4 Green, and Green 4 Red. While these preferences might seem odd, there’s nothing in the theory so far that precludes a consumer from having them. (i) Exercise 3.1.4. Can you think of examples in which you have preferences like con- sumers 3 and 4. That is, cases in which you are unable to rank all alternatives and have “incomplete” preferences, or others in which your preferences are somehow “inconsistent” and contradict each other? (iv) Remark 3.1.5. Formally, a weak preference relation % is a subset of ordered-pairs of bun- dles; that is, % ⊆ X × X. For every ordered-pair of bundles (x, x0 ), we have either (x, x0 ) ∈ %, or (x, x0 ) ∈ / %. In the first case, instead of writing (x, x0 ) ∈ %, we typically write x % x0 and say that x is at least as preferred as x0. In the second case, instead of writing (x, x0 ) ∈ / %, 20  Contents / x0 and say that x is not at least as preferred as x0. The same holds for we typically write x % the strict preference relation  and the indifference relation ∼. An ordered-pair (x, x0 ) is in , in which case we write x  x0 , if (x, x0 ) is in %, but not (x0 , x). And it is in ∼, in which case we write x ∼ x0 , if both (x, x0 ) and (x0 , x) are in %. For example, two “extreme” preference relations are (i) the empty set % = ∅, which rep- resents the preferences of a consumer who cannot rank anything (i.e., we have no information about their preferences), and (ii) the “full indifference,” given by % = X × X, which rep- resents a consumer who is indifferent between every pair of alternatives. See that, from the viewpoint of consumer theory, “having no preference” is not the same as being “indifferent.” In microeconomic parlance, being “indifferent” does imply that a preference exists. 3.2 Axioms of consumer theory 3.2.1 Rational preferences: completeness and transitivity As the example above with consumers 3 and 4 shows, we might run into problems when defining preference relations freely. While the examples might be descriptive of some real behavior, people are typically able to make up their minds when choos- ing what to consume (e.g., what color of car to choose). In order to impose some structure to preference relations, we need some properties, also known as axioms. Let X be a consumption set, the preference relation % over X is: 1. complete if x % x0 , x0 % x, or both, for every x, x0 ∈ X; 2. transitive if x % x0 and x0 % x00 imply x % x00 for every x, x0 , x00 ∈ X; rational if it is complete and transitive. Completeness means that we can compare any pair of bundles with %, so that pref- erences are “defined everywhere.” Having complete preferences rules out prefer- ences as the ones of consumer 3 above. Transitivity means that if we prefer x at least as much as x0 , and x0 at least as much as x00 , then we must also prefer x at least as much as x00. Transitivity rules out preferences as the ones of consumer 4 above. In simple words, rationality means that consumers know what they like, and their preferences do not contradict each other. Indeed, as we will see below, rationality actually implies something stronger, namely, that consumers can rank all the al- 21  Contents ternatives and determine their favorite(s). In other words, rationality implies that consumers can rank alternatives and choose at least one. (i) Remark 3.2.1. A preference relation % over a consumption set X is reflexive if x % x for every x ∈ X. Note that every preference relation that is complete is also reflexive (one can take x0 = x in the definition of completeness), but not the other way around. For example, if X = {x, x0 }, and % only specifies x % x and x0 % x0 , then % is reflexive but not complete. (ii) Exercise 3.2.2. Explain why is it the case that if a preference relation is complete, then for every pair of bundles x and x0 , one and only one of the following must be true: (a) x  x0 , (b) x0  x, or (c) x ∼ x0. (iv) Remark 3.2.3. The axioms can also be applied to the strict preference relation and to the indifference relation. Indeed, if % is rational, we can show that both  and ∼ are transitive but, in general, not complete. To see why neither is complete, consider a consumption set with two alternatives X = {x, x0 }. If x  x0 then ∼ is not complete. And, if x ∼ x0 , then  is not complete. Formally showing that  and ∼ are transitive if % is transitive goes beyond the scope of this module. However, you are free to give it a go and try to prove it! (ii) Exercise 3.2.4. Explain in your own words what does it mean for  and ∼ to be transitive but not complete. (iv) Remark 3.2.5. Rationality is a key assumption in consumer theory because it implies that consumers can rank all alternatives consistently, and thus determine which is the best one (or the best ones if there are more than one). This point can be appreciated with a finite consumption set. For the sake of the argument, suppose the consumer is choosing a bundle from a finite number of alternatives. Define the set of “best” bundles as follows: X ∗ = x ∈ X : x % x0 for all x0 ∈ X.  The best bundles are the ones that are at least as preferred as every other bundle in X. See that the bundles in X ∗ are the ones for which there is no other bundle that is strictly preferred to them, i.e., x ∈ X ∗ if there does not exist x0 such that x0  x. The set X ∗ might have multiple bundles, in which case all the bundles in X ∗ will be indifferent among themselves.9 A key result in consumer theory is that X ∗ will always be non-empty if % is rational; there will always be at least one “best” bundle. To see why this is the case, start by picking any 9. For example, if X = {x, x0 , x00 }, and x ∼ x0  x00 , then X ∗ = {x, x0 }. 22  Contents bundle from X, say x. Then, pick all the bundles in X that are indifferent to x, and put them all together in a box [x]. Since ∼ is transitive, all the bundles in [x] will be indifferent to x and each other. Then, pick another bundle from X that is not in [x], say x0. By completeness, x0 will be either strictly preferred to all bundles in [x] or strictly less preferred to them. Without loss of generality, say it is strictly preferred. Then, take all the bundles indifferent to x0 and put them in another box [x0 ]. By transitivity, all the bundles in [x0 ] will be indifferent to each other and more preferred than any bundle in [x]. Abusing notation, we can write [x0 ]  [x], meaning that all the bundles in [x0 ] are strictly preferred to all bundles in [x]. Then, continue this process, picking bundles which haven’t been assigned to any box, seeing where they fall in the order defined by , and grouping them with all the bundles they are indifferent to in a new box. When we finish (which we will since X is finite), we will end up with something that looks like this: [x1 ]  [x2 ]  · · ·  [xn ], where n is be the number of boxes we are able to form, and the numbering 1, 2,... ,n indicates the boxes that are more preferred to others. Then, we obtain X ∗ = [x1 ], meaning that we will always find a box that is better than all the other boxes, and that box will have at least one bundle. To see why X ∗ might be empty when we do not have a rational preference relation, con- sider X = {x, x0 , x00 } and % such that x  x0 , x0  x00 , and x00  x. In this case, it is impossible to determine the “best” bundle. What would this consumer choose?! Finally, see that X ∗ may also be non-empty even if preferences are not rational, though it is not guaranteed to be so. In other words, rationality implies that we can always find at least one alternative that is as good as all others. However, being able to find at least one alternative that is as good as all others does not mean someone is rational. Can you think in an example to prove this claim? 3.2.2 Visualizing preferences when X = R2+ In the section above we introduced the notion of a preference relation. We dis- cussed weak preference relations, strict ones, and indifference relations. We also introduced the first axioms of the theory of the consumer: completeness and tran- sitivity, which together form the basis of what economists refer to when talking about rationality. We discussed the importance of these axioms by noting that they 23  Contents allow to rank all the alternatives in the consumption set, and, in particular, guar- antee that the consumer will have at least one alternative that is more preferred to all others. However, during the discussion we assumed that the consumption set X was finite. In this section we return to the standard case, in which the consumption set is X = R2+ , and an alternative is a consumption bundle, x = (x1 , x2 ), that speci- fies the amount consumed of goods 1 and 2, both of which are infinitely divisible. When X = R2+ , the number of alternatives in the consumption set is infinite. This complicates things a bit since it becomes unfeasible to describe the content of preference relations by listing them, as we have done above. Indeed, this is only feasible, or at least practical, when the number of alternatives is relatively small. Nonetheless, if you go back to the definition of a preference relation and the axioms in the previous sections, you will notice that we defined them for a general con- sumption set X. That is, we can still define a preference relation %, and derive its strict part  and the corresponding indifference relation ∼, when X = R2+ , even if we have an infinite number of alternatives. However, if we cannot list the content of a preference relation in this setting, since it is effectively an infinite list, the question becomes how to visualize it. To visualize a preference relation in R2+ , it is useful to define the following sets. For a bundle x ∈ R2+ , the “at least as good” set of x: %(x) = x0 ∈ R2+ : x0 % x ;  the “not better than” set of x: -(x) = x0 ∈ R2+ : x % x0 ;  the “preferred to” set of x: (x) = x0 ∈ R2+ : x0  x ;  the “worse than” set of x: ≺(x) = x0 ∈ R2+ : x  x0 ;  the “indifference” set of x: ∼(x) = x0 ∈ R2+ : x0 ∼ x.  (ii) Exercise 3.2.6. Explain in your own words what bundles are in each of the sets above. The sets above allow to better appreciate how the strict preference relation and the indifference relation are a partition of the information encoded in the weak pref- erence relation. Indeed, for any x ∈ R2+ , we can show that the at least as good set %(x) is equal to the union of the preferred to set (x) and the indifference set ∼(x). And, moreover, that the sets (x) and ∼(x) are disjoint. To see this, take a bundle from %(x) , say x0 ; that is, x0 % x. See that one and only one of the two must hold, 24  Contents Figure 4: Example of rational preferences in R2+ either (i) x % / x0 , in which case x0  x and x0 ∈ (x) , or (ii) x % x0 , in which case x0 ∼ x and x0 ∈ ∼(x). Similarly, we can show, in the other direction, that if we start with x0 in either (x) or ∼(x) , it will be in %(x). In other words, we have shown that %(x) = (x) ∪ ∼(x) , and that there is no bundle that is in both (x) and ∼(x). (iii) Exercise 3.2.7. Explain why (i) ≺(x) ∪ ∼(x) equals -(x) , and (ii) the sets ≺(x) and ∼(x) are disjoint. Completeness implies that, for every x ∈ R2+ , every consumption bundle in R2+ must lie in one of these three sets: (x) , ≺(x) , or ∼(x) , all of which are disjoint (mutually exclusive). That is, for any x we pick, every bundle is either preferred to x, worse than it, or indifferent to it. Therefore, if we fix an x, we can split the consumption set R2+ into these three sets. Transitivity implies that the indifference sets of distinct bundles are either the same (if the bundles are indifferent to each other), or do not overlap (if one is strictly preferred to the other). This, in turn, implies that the “preferred to” and “worse than” sets will be the same for bundles that are indifferent among themselves. In other words, if x and x0 are indifferent to each other, then their “at least as good,” “not better than,” “preferred to,” “worse than,” and “indifference” sets will be the same. And, if they are not indifferent to each other, then their indifference sets will be disjoint. The sets (x) , ≺(x) , and ∼(x) can be depicted graphically as in Figure 4. See that rational preference relations still allow for preferences that seem rather odd. They have thick indifference sets with irregular curvature. For example, in the prefer- 25  Contents ences depicted in the figure, if the consumer is consuming x and we increase a little bit the consumption of good 1, they are better off (moving to the right from x gets us to (x) ). However, if we keep increasing the consumption of good 1, suddenly we are worse off (cross onto ≺(x) ), but if we then increase the consumption of good 1 the consumer ends up indifferent. Nonetheless, at least in principle, these pref- erences look rational: every bundle is in either (x) , ≺(x) , or ∼(x) , and, as far as we can tell, the indifference sets of bundles that are not indifferent to each other do not cross. In other words, there are rational preferences that look like those in the fig- ure.10 In what follows, we consider additional axioms that impose more structure to preferences and make them both more plausible and tractable. (iv) Remark 3.2.8. If you think about it, the sets in Figure 4 could actually look even more odd. For instance, in the figure, the indifference set (depicted by the two thick gray sets and the black line) lies “between” the preferred to and worse than sets. In principle, this could not be the case. Nothing precludes the preferred to and the worse than set to be “side by side” and “touch” each other. This would imply that if we have a bundle that is, say, less preferred than x, and we make an infinitesimal change to it, suddenly it becomes more preferred to x. To rule out these type of preferences that have “jumps” for infinitesimal changes, there is a technical axiom called continuity. While important from a technical perspective, it is a bit more advanced than what we will cover in the module, so we will not discuss it in detail. Nevertheless, it is important to bare in mind that we will be making this additional technical assumption that allows us to “separate” the preferred to and the worse than sets with the indifference set. 3.2.3 Local non-satiation and monotonicity The next three axioms remove the thick portions from the indifference sets and turn them into indifference curves. Let X = R2+ , the preference relation % over X is: 4a. locally non-satiated if for every x ∈ X, among the bundles that are close to x, no matter how close we look at, there is always at least one bundle that is strictly preferred to x;11 10. A keen observer will note that the plot in Figure 4 does not actually encode all the information in a preference relation. For example, from the plot, we cannot tell how the indifference sets of bundles that are not indifferent to x look like. This is another undesirable feature of working with preferences that are only rational in R2+ ; they are not very tractable. 11. Formally, for every ε > 0, there always exists a bundle within an ε-radius of x that is strictly preferred to it. 26  Contents Figure 5: Local non-satiation and strict monotonicity 4b. monotonic (a.k.a. weakly monotonic) if bundles that have more of both goods are strictly preferred; that is, if x > x0 imply x  x0 for every x, x0 ∈ X;12 4c. strictly monotonic if bundles that have more of at least one good, and not less of the other, are strictly preferred; that is, if x ≥ x0 and x 6= x0 imply x  x0 for every x, x0 ∈ X.13 Local non-satiation is a weaker axiom than weak monotonicity, which in turn is weaker than strict monotonicity. Preferences that are strictly monotonic are weakly monotonic, and those that are weakly monotonic are locally non-satiated. But the opposite do not hold. The left panel of Figure 5 shows graphically the effect of assuming local non-satiation. Local non-satiation removes the thick portions from the indifference sets, so that all bundles that are indifferent to x lie in the same curve. Weak and strict monotonicity further restrict the portions of the plane through which indifference curves can pass. Namely, the bundles that have more of either or both goods (depending on whether the monotonicity is weak or strict) must lie in the “preferred to” set. Similarly, the bundles that have less of either or both goods must lie in the “worse than” set. The right panel of Figure 5 shows an indifference curve of a preference relation satisfying strict monotonicity. (i) Exercise 3.2.9. Plot the indifference curves of a preference relation that is weakly monotonic but not strictly monotonic. 12. For x = (x1 , x2 ) and x0 = (x01 , x02 ) in R2+ , x > x0 means x1 > x01 and x2 > x02. 13. For x = (x1 , x2 ) and x0 = (x01 , x02 ) in R2+ , x ≥ x0 means x1 ≥ x01 and x2 ≥ x02. 27  Contents Figure 6: Strictly convex preferences 3.2.4 Convexity The final two axioms impose structure over the curvature of the indifference curves. Let X = R2+ , the preference relation % over X is: 5a. convex (a.k.a. weakly convex) if combinations of bundles are weakly pre- ferred; that is, for every x, x0 ∈ X such that x ∼ x0 , αx + (1 − α)x0 % x for every α ∈ (0, 1); 5b. strictly convex if combinations of bundles are strictly preferred; that is, for every x, x0 ∈ X such that x ∼ x0 and x 6= x0 , αx + (1 − α)x0  x for every α ∈ (0, 1). Intuitively, consumers with convex preferences prefer bundles which mix goods. Averages are preferred to extremes. To see this clearly, note that the bundle αx + (1 − α)x0 is the bundle that has αx1 + (1 − α)x01 of good 1, and αx2 + (1 − α)x02 of good 2. The indifference curves of convex preferences are convex, as shown in the two panels of Figure 6. Strict convexity is a stronger axiom than convexity since every preference relation that is strictly convex is also convex, but not the other way around. (i) Exercise 3.2.10. Plot the indifference curves of a preference relation that is weakly convex but not strictly convex. 28  Contents 3.2.5 Well-behaved preferences Together, completeness, transitivity, monotonicity, and convexity, including their different variations, are the axioms of consumer theory. Indeed, a preference rela- tion is said to be: well-behaved if it is rational, monotonic, and convex. Typically, though not always, we will assume that preferences are well-behaved. While usually assumed together, the motivation for each axiom is slightly different. Completeness and transitivity are necessary in order to work with preferences that are defined everywhere and are internally consistent. Intuitively, a consumer who has rational preferences knows what they prefer and is capable of ranking alter- natives consistently. Monotonicity and convexity impose structure over consumers’ tastes for goods. Monotonicity implies that consuming more is preferred to consum- ing less, and convexity that consuming goods combined is preferred to consuming them separately. Importantly, the axioms of consumer theory are the basis of the theory of the consumer. All the conclusions we will draw in the next sections are derived solely from these axioms. (ii) Exercise 3.2.11. A consumer has preferences that are complete, transitive, and mono- tonic. Their preferences over all bundles x = (x1 , x2 ) and x0 = (x01 , x02 ) in the con- sumption set are such that, if x ∼ x0 and x 6= x0 , x % αx + (1 − α)x0 for α ∈ (0, 1). Draw the indifference curves of this consumer. Clearly label the “at least as good” set. 3.3 Goods, bads, and bliss points While we will often assume well-behaved preferences, there are cases worth analyz- ing in which preferences are not well-behaved. A typical example is when we are considering bads instead of goods. That is, when the consumption set is composed, at least in part, by goods the consumer may wish to avoid consuming. A leading example of a bad is labour. While certain people might actually enjoy working for the sake of it, the majority of people probably work because it is a means of acquir- ing income, which then they spend on goods they actually enjoy consuming. Other examples of bads are pollution, noise, and, in some cases, even pain. 29  Contents Preferences over bads are clearly not monotonic, more is not better. One can see whether preferences are monotonic by looking at the slope of their indifference curves. The indifference curves of monotonic preferences slope downwards. If the consumer increases consumption of one good, the consumption of the other good must be reduced for them to remain indifferent. When one of the goods is a bad, the indifference curves have a positive slope. If the consumption of a good increases, the consumption of the bad must also increase for the consumer to remain indifferent. See the left two panels in Figure 7. In the top panel, good 1 is a bad and good 2 is a good. The “preferred to” set lies above and to the left of the indifference curve. In the plot, we indicate the direction in which the bundles are more preferred with arrows pointing in that direction. The bottom-left panel shows the case in which good 1 is a good and good 2 is a bad. In this case, the “preferred to” set lies below and to the right of the indifference curve. When both goods are bads, as in the top-right panel of Figure 7, the indifference curves have negative slope, as in the case with monotonic preferences. However, the “preferred to” set lies closer to the origin, below and to the left of the indiffer- ence curve. The consumer prefers less of both goods rather than more. Yet another example is when a consumer has a bliss point, as in the bottom-right panel of Fig- ure 7. See that in this case both goods behave as goods initially, up to the bliss point (x̄1 , x̄2 ). Past this point, the goods behave as bads. These preferences not only violate monotonicity, but also local non-satiation. This plot includes multiple indifference curves, not just the one where x lies. (ii) Exercise 3.3.1. State which axioms of the consumer theory each of the indifference curves shown in Figure 7 satisfies. Provide examples of goods which preferences might look like them. 3.4 Marginal rate of substitution A useful way of analyzing consumer preferences is by looking at the slope of their indifference curves. The slope of the indifference curve is called the marginal rate of substitution (MRS). This rate measures the willingness of a consumer to substitute both goods and remain indifferent. Say the consumer increases their consumption of good 1, the marginal rate of substitution measures how much of good 2 would they need to forego to remain indifferent. Importantly, the marginal rate of substi- tution might differ along the same indifference curve. 30  Contents Figure 7: Examples of not well-behaved preferences For example, consider the indifference curve in Figure 8. At bundles with rela- tively low quantities of good 1, such as x0 in the figure (depicted in blue), the MRS is high, meaning that the consumer is willing to forego a relatively large amount of good 2 in order to increase the amount of good 1. By contrast, when good 1 is plentiful relative to good 2, as in bundle x00 (depicted in red in the figure), the con- sumer is willing to forego a smaller amount of good 2 to increase consumption of good 1. In other words, the marginal rate of substitution is decreasing in good 1. Intuitively, good 1 is more “valuable” to the consumer when it is scarce relative to good 2. As the consumer has more of good 1, their willingness to substitute good 2 for it decreases. This is what is known as the principle of decreasing marginal rate of substitution. Note that a decreasing marginal rate of substitution is a fea- ture of convex preferences. Hence, an alternative way of interpreting or justifying a 31  Contents Figure 8: Decreasing marginal rate of substitution consumer having convex preferences is by them having a decreasing marginal rate of substitution between goods. (ii) Exercise 3.4.1. State which axioms of the consumer theory each of the indifference curves shown in Figure 9 satisfies. For each of them, also discuss whether the marginal rate of substitution is decreasing, increasing, or constant. Finally, provide examples of goods which preferences might look like them. 32 Figure 9: Examples of indifference curves 33  Contents References Hicks, John R. 1939. Value and Capital: An Inquiry Into Some Fundamental Principles of Economic Theory. Clarendon Press Oxford, Oxfordshire. (Cited on page 4). Jacques, Ian. 2017. Mathematics for Economics and Business. Pearson. (Cited on page 8). Jehle, Geoffrey A., and Philip J. Reny. 2001. Advanced Microeconomic Theory. Boston: Addison-Wesley. (Cited on page 4). Marshall, Alfred. 1890. The Principles of Economics. McMaster University Archive for the History of Economic Thought. (Cited on page 4). Nicholson, Walter, and Mark C. Snyder. 2008. Microeconomic Theory: Basic Principles and Extensions. Thomson/South-Western. (Cited on page 4). Renshaw, Geoff. 2021. Maths for Economics. Fifth edition. Oxford University Press. (Cited on page 8). Sydsæter, Knut, Peter J. Hammond, Arne Strøm, and Andrés Carvajal. 2021. Es- sential mathematics for economic analysis. Sixth edition. Hoboken , NJ: Pearson. (Cited on page 8). Varian, Hal R. 2010. Intermediate Microeconomics: A Modern Approach. (Cited on page 4). 34  Contents Glossary disjoint sets Two sets A and B are disjoint if they have no elements in common. The intersection of two disjoint sets is the empty set. 24 empty set The empty set, denoted ∅ or {}, is the set that contains no elements. 12 intervals An interval (a, b) in the real line, also known as an open interval, is the set of all numbers that are strictly greater than a and strictly lower than b, i.e., / (a, b).14 A closed interval [a, b] (a, b) = {x ∈ R : a < x, x < b}. See that a, b ∈ is the same but includes a and b, i.e., [a, b] = {x ∈ R : a ≤ x, x ≤ b}. One can also define semiclosed intervals, as in (a, b] or [a, b). 13 ordered-pairs The set X × X is the set containing all the ordered-pairs formed by elements in X; that is, X × X = {(x, x0 ) : x ∈ X, x0 ∈ X}. For example, R2+ = R+ × R+ , where R+ is the set of non-negative numbers in the real line. 20 orthant of the cartesian plane The cartesian plane, denoted by R2 , is the set of two-dimensional real vectors, that is, that take the shape (x1 , x2 ), where x1 and x2 are real numbers: x1 , x2 ∈ R. The positive orthant of the cartesian plane is the set defined by: R2+ = {(x1 , x2 ) : x1 , x2 ∈ R, x1 , x2 ≥ 0}, which is a fancy way of writing that R2+ is the set of vectors (x1 , x2 ) such that both x1 and x2 are non-negative real numbers. (When defining sets, colons “:” stand for “such that” and commas “,” stand for “and”.) The strictly positive orthant is then defined by: R2++ = {(x1 , x2 ) : x1 , x2 ∈ R, x1 , x2 > 0}. 10 14. If you are thinking that the notation for open intervals and vectors is confusingly similar, we hear you. Perhaps surprisingly, it is not uncommon in mathematics to use similar notation for different concepts. This is why it is important to always introduce and define clearly the concepts one uses. 35  Contents set notation A set is an unordered collection of elements, typically denoted X = {x, y, z}, where X is the name of the set, and x, y, and z are its elements. It is unordered because the sets {x, y, z} and {z, x, y} are the same set. To write that an element x belongs in a set X, we write x ∈ X. To write that it does not belong to the set, we write x ∈ / X. 10 straight lines A straight line in two dimensions, i.e., in R2 , is characterized by the equation y = mx + b, where y is the variable we typically plot on the y-axis and x the one we plot on the x-axis (hence the names of the axes). The con- stants m and b indicate the slope of the line and its intercept with the y-axis, respectively. We can obtain the intercept of the line with the x-axis by asking “what value does x take when y = 0?” That is, what is the value of x that solves mx + b = 0, which yields x = −b/m. When talking about slopes, re- member that m > 0 characterizes lines with positive slopes, m < 0 lines with negative slopes, m = 0 horizontally flat lines, and m = +/ − ∞ fully vertical lines. Steeper lines have slopes that are larger in absolute value (that are far- ther away from zero). This can get tricky when comparing slopes, especially negative ones. Say we have m0 < m < 0, then m0 is lower than m but it indi- cates a steeper line (since it is higher in absolute value). If slopes are positive, the opposite holds: say we have m0 > m > 0, then m0 is larger both in sign and absolute value, so the line with slope m0 is steeper. 11 subset A set A is a subset of another set B if all the elements in A are also in B, in which case we write A ⊆ B. For example, (0, 1) ⊆ [0, 1], and [0, 100] ⊆ R. The set A \ B contains all the elements that are in A and not in B. Given a set A and a subset of it, B, we can always partition A into B and A \ B. That is, every element in A will be either in B or in A \ B. It has to be in at least one of them, and cannot be in both. For example, the set of natural numbers N = {1, 2, 3, 4,... } can be partitioned into the set of odd numbers, given by O = {1, 3, 5, 7,... }, and the set of even number, given by E = {2, 4, 6, 8,... }. 20 union of two sets The union of two sets, A and B, denoted A ∪ B, is the set that contains all the elements that are in A, in B, or in both. 16, 24 vector notation A vector is an ordered pair of numbers, typically denoted x = (x1 , x2 ), where x is the name of the vector, x1 is its value on its first dimen- 36  Contents sion and x2 in its second dimension. See that when talking about vectors, the order does matter: the vector (0, 1) is not the same as the vector (1, 0). 10 37