Content
Introduction: The Micro-Filling of the Planetary Donut
One of the central themes of this course is the concept of the planetary donut from Kate Raworth, which was introduced in the previous lecture when examining sustainability indicators. This lecture goes considerably further than Raworth’s original book, which did not cover the material presented here. The contention throughout this course is that economics is genuinely valuable for a variety of reasons, and one of the most important is that it possesses a remarkably coherent theory explaining why people make the decisions they do.
Working in environmental spaces often means finding oneself in the position of an apologist for economists. This is because environmental circles frequently include people who view economists negatively, seeing them as mere representatives of the capitalist system. The constant effort involves arguing for the importance of both environment and economics simultaneously. This lecture demonstrates why economics has value and what tools an economics major can bring to environmental problems.
The concept of the “micro-filling” captures this idea: when the planetary donut is embedded in a detailed economic model, it contains tremendous depth and valuable details that enable predictions about human behavior. While AI-generated images can now produce reasonable-looking supply and demand curves, they still make errors that an undergraduate would be marked down for, such as having the minimum of the marginal revenue curve fail to intersect the average total cost curve properly.
Utility Theory: The Foundation of Welfare Economics
What Is Utility?
The specific focus of this lecture is utility. Utility is the fundamental concept that economists use to argue that the policies of the invisible hand or the free market are beneficial. Before exploring utility in detail, it is necessary to understand the tools of utility, utility curves, and how individuals who are assumed to be rational will optimize their utility. This involves the tools of constrained optimization.
This discussion falls within the broader goal of maximizing welfare. If unlimited resources existed, the question would be trivial since one could simply have unlimited consumer goods. In reality, however, choices are constrained by budgets. The lecture will tie these concepts back to environmental aspects and explain how they will be used throughout the semester.
Philosophical Origins: Utilitarianism
It is useful to consider the philosophical origins of utility theory. Modern welfare economics grows directly out of utilitarian moral philosophy. The term “welfare” in economics has a very specific meaning that differs entirely from how most people use the word. Most people think of government assistance programs when they hear “welfare.” Those programs are named for their aim of increasing welfare, but economists use the term to mean something quite specific related to utility maximization and consumer and producer surplus.
The philosophical basis comes from Jeremy Bentham and John Stuart Mill, who were the two most notable proponents of utilitarianism. This philosophy is appealing partly because it can be summarized simply. The most common phrase is “the greatest good for the greatest number of people.” To use their actual quotes, “the right action is the one that maximizes total happiness.”
Utilitarianism has been used extensively throughout policy analysis. However, it is very different from a rights-based approach. While utilitarianism sounds entirely positive, there exists a substantial philosophical disagreement about whether utilitarianism is the correct way to think about morals, or whether a rights-based approach is superior. These two frameworks are fundamentally in conflict. If one says there is a right to free speech, free expression, protest, or not being detained, that right might conflict with what utilitarianism prescribes, because a utilitarian might argue that such a right prevents doing the thing that would maximize utility.
A deep philosophical debate surrounds these issues, but this course will not dive deeply into philosophy. Instead, it will use utilitarianism because it is useful from a pragmatic standpoint. In economics, this framing proves valuable because it can be taken quite far. The approach posits that each person has a utility function, that social welfare is basically some aggregation of everybody’s utility functions, and that therefore the utilitarian or normative good policy would simply be the one that maximizes the sum of utilities.
Challenges with Utility Theory
Before unpacking utility theory in detail, it is important to discuss its challenges from the beginning rather than presenting the complete theory and only mentioning challenges at the end.
The Ordinal Nature of Utility
The first challenge is that utilities are ordinal, meaning only the order can be determined. When using utility as a concept, one can easily say that one bundle of goods gives more utility than another bundle, creating a ranking or order of different consumption choices. This is what ordinal means.
This differs from something cardinal, where an actual measurement is obtained. A cardinal measurement would be like that of a thermometer: inserting a thermometer and getting 98 degrees actually means something specific. But with utility, the situation is different. A number will be obtained when using utility functions, but it does not mean anything in the same way. There is no unit of utility like there is a unit of temperature. The first challenge, therefore, is that utility is hard to measure and is private, meaning it cannot be observed directly. Dealing with something that is unobservable directly is inherently difficult. Nevertheless, there are ways to learn a great deal about its shape.
Aggregation Challenges
The second set of challenges involves aggregation. The idea of summing together individual utility functions was mentioned briefly, but this turns out to be extremely challenging to do in a value-free way. However utilities are summed together, values are fundamentally involved. If utilities are summed equally, that expresses a view that equality matters. Alternatively, they might be summed according to some distribution rule, such as ensuring nobody is extremely poor. These considerations are especially important because how society’s welfare is aggregated into a utility function representing everybody is fundamentally challenging in terms of debates between equality versus efficiency.
This tension will appear extensively in upcoming classes, particularly with respect to present value versus future generations. How much to care about someone born 100 years from now who may be struggling with an unstable climate is a genuinely difficult question. The discussion will return to this extensively when examining the discount rate.
Revealed Preferences: From Behavior to Utility
Inferring Utility from Observed Behavior
Although utility cannot be observed directly, substantial thinking and theory exists about what can be observed and used in computational settings. While utility itself cannot be observed, things about it can be inferred from observed behavior through revealed choices or revealed preferences. This involves going from unambiguous observed data to a utility function. The exact number of units of utility cannot be known, but preference information can be determined, such as “I prefer A over B.” This can be observed through behavior like purchasing A instead of B. This information constitutes observable data about a person, whether observed in a store or obtained by asking them directly.
The Structure of Preferences
Consider an example where the options to choose from are A, B, and C. Option A might be a bundle of goods and services called “Go to the Movies.” Option B might be “Go to a Museum,” and Option C might be “Watch Trees Grow.” Any given person, or at least according to the theory of revealed preferences, can make a mapping of how they prefer one option to another. This involves statements like “I prefer going to the movies over the museum, and I prefer the museum over watching a tree grow.” The specific order does not matter; all revealed preferences requires is that these preference relationships are known and can be expressed.
From Preferences to Utility Functions
From preferences, economists argue that a utility function can be constructed that is compatible with all observed preferences. This looks quite similar to preference ranking, where the utility from choice A is greater than the utility of choice B, and the utility of B is greater than the utility of C. The similarity to preference ranking is apparent, but the claim is that these preferences can be plugged into a numerical function that produces output numbers. What those numbers mean or how they can be interpreted remains unknown. There is no such thing as an observable unit of utility, which people sometimes jokingly call a “util.” Rather, the argument is simply that functions can be created that preserve the ordering of preferences.
The Role of Preferences in Economic Theory
In intermediate-level microeconomics courses, there is debate about whether preferences are even worth discussing. Some instructors include them, others do not. However, at the PhD level in economics, considerable time is spent on preferences because they constitute the fundamental reason why economics claims to be good, correct, or powerful.
A criticism sometimes leveled at economics is what might be called “physics envy.” Many economists want to see the world and the theories they create as something akin to physics, as if fundamental truths about human behavior can be obtained. The film A Beautiful Mind about John Nash illustrates this mentality: the pursuit of fundamental mathematics revealing beautiful relationships, like Newtonian physics describing a ball flying through the air, but applied to the complexity of humans.
This perspective can be dangerous because humans are so much more complex than a ball flying through the air. A PhD in economics spends considerable time on preferences and mathematical theorems applicable to preferences or utility functions. This is worth emphasizing because unlike a standard intermediate or advanced undergraduate microeconomics course, a course focusing on the environment deals fundamentally with hard-to-value things, like the value of watching a tree grow or the value of a species’ existence. Therefore, it is worth spending more time building out the utilitarianism, revealed preferences, and utility maximization chain of logic that will be used throughout, since this is embedded in environmental decision-making through tools like cost-benefit analysis.
The key point is that assuming a specific number can be placed on all different things represents a huge leap. How many units of utility does contemplating a painting in a museum provide? This is difficult to determine. But the approach will be used anyway because it is useful and predictive.
The Utility Function and Budget Constraint
Defining the Utility Function
The discussion now turns to defining consumer preferences more precisely. The utility function will be denoted throughout as capital U. Like any function, it takes inputs. For the moment, minimal detail will be specified other than having two different goods that might be consumed: good X and good Y. This should be review from intermediate courses. X is simply the number of units of good X that can be purchased, and Y is the number of units of good Y that can be purchased. When these two quantities are plugged into the utility function, a utility metric emerges.
In principle, a large number of different goods could go into this function. A real consumer considers not just two things but thousands of choices per day. The mathematics is elegant because it scales up well; utility maximization can be performed with many goods. For most drawings, however, two goods will suffice.
The Budget Constraint
The utility function by itself is not particularly interesting because more utility can always be obtained by simply plugging in more and more. But economics is the study of choices under scarcity. For utility theory to be useful, something must be scarce, and that will be the budget, as introduced in introductory economics.
Information on income and prices can be combined graphically. With the price of good Y at some value and the price of good X at another value, a budget line can be determined. The budget line represents the outer bound of what can be afforded. Everything inside the line is affordable, everything outside is unaffordable, and everything right on the line is affordable but represents spending all available money. This will be used because the goal is to maximize utility subject to feasibility.
The Constrained Optimization Problem
The approach involves applying utility maximization. The notation indicates maximizing by choosing different levels of X and Y for the utility function, subject to the budget constraint. The budget constraint states that the price of X times the number of units of X purchased, plus the price of Y times the number of units of Y purchased, equals total expenditures. For this to be feasible, that total must be less than or equal to income I.
The problem is: maximize utility by choosing X and Y such that the combination is actually affordable, given prices and income. This departs from introductory economics, which presents the budget line without a framework applying a decision rule to optimize within that setting.
When discussing rational agents and all the assumptions of Homo economicus or perfectly rational decision makers, this is actually what they are assumed to be doing: somehow examining all possible combinations of X and Y, or thousands of goods, and rationally and correctly optimizing this mathematical equation.
Applying Utility Theory: The Beach Exercise
The Challenge of Valuation
To unpack constrained optimization, consider a brief exercise. Suppose there are $1,000 to spend on fun activities, with the options being: A, go to the beach; B, go to an indoor water park; or C, sit by a backyard kiddie pool. The task is to allocate that $1,000 across these three options, spending it all. The allocation could be $333 for each, or all $1,000 to the beach with nothing to the others.
This exercise is genuinely difficult. Information about prices is missing. Also, sitting by a backyard kiddie pool might be fun once, but $1,000 could cover doing that all summer long. Even with only partial information, without knowing the prices, and even if prices were provided, it would not be enough to do the exercise quantitatively because the number of units of utility from each activity is unknown. The point of rational utility maximization is assuming that all sorts of different bundles, including unusual ones, can be accurately analyzed within the utility framework.
Environmental Goods and Utility
Putting a numerical measure on going to an indoor water park is challenging. But saying how much utility comes from simply knowing that a kangaroo exists is even harder. Or consider a species that one knows will never be seen. There would likely still be some positive value assigned to that existence, even knowing the species will never be personally encountered.
This sets up a recurring theme: utility theory makes many assumptions that work reasonably well for items purchased at Target, but the assumptions become harder to apply when analyzing anything about the environment. Nonetheless, utility theory assumes people are perfectly rational, can perfectly rank different options no matter how different they are, that resources are scarce, and that people are well-informed with preferences over all possible combinations.
Solving the Optimization Problem Graphically
The Lagrangian Approach (Briefly)
The mathematical method for solving constrained optimization problems is the Lagrangian, which is covered extensively in intermediate and advanced microeconomics. It is elegant mathematics and beautiful, serving as the workhorse behind solving what a rational agent would do when facing such decisions. Given goods and prices, one maximizes over all choices subject to the budget constraint. The Lagrangian is a mathematical way of solving for optimal choices.
Graphical Analysis with Indifference Curves
Instead of the Lagrangian, this lecture uses graphical analysis, which is mathematically identical. Anything done graphically can be computed mathematically. Where this approach departs from introductory economics is spending more time on indifference curves, which are typically not discussed in introductory courses because those courses do not address utility maximization and revealed preferences.
Dynamic Indifference Curve
What Is an Indifference Curve?
An indifference curve represents a relationship between two different goods, with good X on the x-axis and good Y on the y-axis. An indifference curve is a line showing all combinations of Y and X that leave the consumer indifferent. This sounds somewhat odd, so consider it more carefully.
The curve typically looks like a downward-sloping arc where utility is the same for every point on that indifference curve. This means equal happiness with a lot of X and only a little Y compared to a lot of Y and only a little X. All points on the curve generate the exact same number when plugged into the utility function.
This explains why it is called an indifference curve: there is indifference between different points on the curve. For any two pairs of goods, like pens and notebooks, consider whether it is better to have 100 pens and one notebook or 100 notebooks and one pen for school success. This is difficult to answer, but some relationship exists, some combination where utility is roughly equal between different options. It simply expresses different bundles between which there is indifference.
Properties of Indifference Curves
Multiple indifference curves can be imagined. As consumption increases, having more X and more Y with more budget to spend allows for improvement by getting more of both. Indifference curves, moving away from the origin up and to the right, represent new curves. They are still indifference curves where utility is the same at any point on that curve, but they continuously move outward.
The typical shape drawn relates to the law of diminishing marginal returns from introductory economics. The classic example is: if starving, how much utility would come from successive slices of pizza? The first slice provides tremendous utility due to hunger. The second, third, and fourth provide considerable utility. By the fifth slice, satiation begins. By the sixth, fullness sets in, and at some point eating stops because the additional utility or happiness from eating reaches zero.
This is described as a law because many real-world phenomena, both in consumption and production, exhibit diminishing returns. In the consumption case, with diminishing returns, as more of one good is accumulated, the relative value of the other good increases. Shapes like the typical indifference curve indicate that a mix of X and Y will often be optimal.
Specific Examples with Goods
Consider specific units: quantity of pens and quantity of mugs. In the normal case, few people would want to live with lots of pens and no mugs, with nothing to drink coffee from. Or live with only mugs and no pens to write notes on. The curve bows outward, indicating partial substitutability. The more bowed it is, the less substitutable the goods are.
The normal case is that more is better, but a mix is best. This is the key takeaway from indifference curves and their typical shape.
Perfect Substitutes
The analysis can be extended to other extremes. A curve could be a perfectly straight line. This is called perfect substitutes, where the value of an X is exactly the same as the value of a Y, making a one-for-one exchange comfortable with no real value from having a mixture. An example might be blue pens versus black pens. One might argue that black pens are superior to blue pens, which would make the curve slightly bowed. But more or less, a blue pen is a pretty good substitute for a black pen for note-taking, so the curve would be closer to flat.
Perfect Complements
Perfect complements represent the opposite extreme. Instead of being straight, the curve bows inward so far that it reaches an extreme case where it is bowed as much as possible, creating an L-shape. The classic example is left shoes compared to right shoes. For collectors, having just left shoes might be acceptable. But for walking, the value of a left shoe is only obtainable with a right shoe. There is no substitutability. Having two right shoes provides far less happiness than having one right and one left shoe.
Utility Maximization with the Budget Constraint
Combining Indifference Curves and Budget Constraints
These indifference curves will be used to solve the optimization problem. Instead of the Lagrangian, the budget constraint will be added to the indifference curve analysis. The indifference curve describes the utility function; the budget constraint is what gets added.
With mugs and pens as goods X and Y, plugging them into the budget constraint function shows the constraint. If all money were spent on mugs (good Y), the amount that could be purchased depends on income and price. With $800 of income and a price of $1 per unit of mugs, 800 mugs and 0 pens could be purchased. That is one point. Conversely, buying all X means spending zero on the other good. The budget constraint is simply the line connecting these endpoints.
This takes a step beyond introductory courses by expressing it as a functional constraint, but the idea is the same. The purchasable set lies inside the budget line, and the things on the line are purchasable with all money spent.
Finding the Optimal Point
Drawing indifference curves on the budget constraint reveals the solution. Consider a utility level U0. The consumer is at U0 units of happiness, but could things be better since the goal is to maximize utility?
If a particular combination of pens and mugs is chosen on the budget line, that combination provides a certain amount of utility. But improvement is possible by sliding to a higher indifference curve, toward the upper right. The process continues getting happier until reaching the optimal indifference curve, marked U1*. The star indicates optimality. What is interesting is that there is now a single point: the point of exact tangency. That specific amount of pens and that specific amount of mugs is optimal.
This contrasts with wishes for more. More utility would come from reaching a higher indifference curve, but that is not possible given the budget. Not enough money exists to purchase any combination on higher curves. Constrained optimization, the Lagrangian, or any other method, simply gets to that utility-maximized point where the two curves just barely touch.
Utility Maximization Game
Why This Matters
This is interesting for two reasons. First, it provides a way of solving the optimization problem. Second, it offers a powerful way to predict how human behavior might react to changes like price changes. Prices constantly change, and those changes affect purchasing decisions.
If the price of pens increased, the budget curve would rotate inward. The utility maximizing approach still works, but there would be a loss of utility. The previous indifference curve is no longer attainable, requiring a fall back to a lower curve.
This directly illustrates, from a utilitarian philosophy perspective, why price increases are bad. Applying this to all of society, jumping ahead conceptually, society itself would be worse off, so whatever caused that price increase is a bad thing, ceteris paribus. This ties back explicitly to philosophy and morality.
Summary of Consumer Theory
Tying it all together, the graphical solution is the same as the mathematical solution. Consumers choose whatever bundle of X and Y maximizes their utility from the set of affordable bundles. Mathematically: maximize U(X,Y) subject to PxX + PyY ≤ I.
This represents the combination of basic consumer theory. The concepts should be review, but having them fresh in memory is valuable for what follows.
The Cobb-Douglas Utility Function
Specifying a Functional Form
Usually the utility function is not specified. It is more useful to think about it as some unspecified function satisfying the preference relationship, without caring about the specific numbers. However, to actually mathematically optimize, a specific shape must be given. This course will use Cobb-Douglas utility functions.
Instead of the general form, the Cobb-Douglas provides a specific computable form: the number of units of X raised to the alpha power times the number of units of Y raised to the (1-alpha) power. Based on alpha, different shapes of indifference curves result, but all have the standard convex bowed-in shape. This will be used throughout.
Interactive Tools and Visualization
Web Applications for Exploring Utility Theory
To develop an intuitive sense of indifference curves, web-based tools are available on the course website. Under the Games tab, also found under Micro Foundations (the micro-filling), there is a list of games to play.
A web app has been programmed that illustrates what happens when different variables change. As discussed, if the price of Y increases, the budget line rotates. The app allows playing around with all the different variables and trying out different levels of alpha to see what a Cobb-Douglas looks like, observing how it bends the relative importance of one good versus the other.
The Utility Maximization Game
An interactive game allows playing with these concepts at home. Different combinations of X and Y can be tested to see which yields the most utility. The current utility is displayed as a function of the quantities of good X and good Y. It changes as combinations vary, but going past the budget constraint is not possible since those combinations are unaffordable.
Extensions to Higher Dimensions
The claim that utility theory works for any number of goods was made earlier. So far, only two dimensions have been considered. But in three dimensions, the Cobb-Douglas takes a similar form with X, Y, and Z, and all coefficients summing to 1. The same idea holds: as more of X is produced, curves shift. The optimum occurs when the indifference surface is just tangent to the budget constraint, which, with three goods, becomes a plane instead of a line.
The logic can be extended further: four dimensions using color as the fourth dimension, five dimensions, and even a user interface allowing sliders to optimize a utility function subject to a budget constraint with 15 goods and 15 prices. This was accomplished through “vibe coding,” using AI to make things very rapidly. While vibe coding is often considered problematic, it enabled creating these applications in minutes rather than the long time it would have taken previously.
Connecting to Environmental Economics
The Central Challenge
The fundamental goal is developing an intuitive sense of this beautiful mathematics. It forms the fundamental core of the moral basis of utilitarianism, but also the fundamental challenge for environmental economics: how to value things that may not have a clear quantification method or price.
The next assignment will involve playing around with one of the interactive tools. The goal is building intuition for utility maximization while recognizing its limitations for environmental goods. This foundation will be used throughout the semester when addressing cost-benefit analysis and other environmental decision-making tools.
Looking Ahead
This lecture intentionally built out the full chain of logic from utilitarianism through revealed preferences to utility maximization, because this chain is embedded throughout environmental economics. The challenges identified at the beginning, especially the ordinal nature of utility and the aggregation problems, become particularly acute when dealing with environmental goods. How to value the existence of a species, the aesthetic value of watching trees grow, or the well-being of future generations facing climate change all require grappling with these foundational concepts. Future lectures will address specific tools for dealing with these challenges while keeping these philosophical foundations in mind.