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.