EconPathways #2: Utility Function and Budget Constraint

The major attraction of economics for me lies in its ability to rigorously and abstractly model and predict human behavior into math equations, accounting for the diverse preferences and motivations that drive decision-making. To elaborate on this concept of “diverse preferences,” economics makes the assumption that people always want to improve their own circumstances when assessing rational choices. While economics often assumes that individuals aim to make themselves “better off” in various situations, measured by a notion called “utility.” Utility is often subjective: this notion of being “better off” is far more nuanced than simply receiving monetary or tangible rewards; people derive value from intangible sources, such as the fulfillment of personal beliefs or altruistic actions. For instance, in the context of religion, serving a holy figure may not yield immediate material benefits, but for believers, such actions contribute to their sense of spiritual fulfillment, potentially leading to a better afterlife, as they perceive it. Similarly, for a kind person, helping others can provide a sense of self-esteem and personal happiness, intangible rewards that hold great value to them. By creating utility functions, which are mathematical depictions of how people value various aspects of their lives, economics takes these varied preferences into consideration. Every variable, whether real or intangible, contributes a specific quantity of “utility” according to how significant it is to the individual. By doing this, economics enables us to represent a broad variety of preferences and motives, guaranteeing that the examination of rational decision-making takes into account both concrete and intangible aspects. This framework aids in capturing the intricacy of human behavior in a manner that is both analytically sound and adaptable enough to take into account a wide range of preferences.

1. Utility Functions

Now, it will be time to formulate the notion of personal preferences into functions. To expand on the formulation of personal preferences into utility functions, let’s start by defining utility more precisely. Utility represents the satisfaction or benefit that an individual derives from consuming goods, services, or engaging in activities. In economics, we assume that individuals are rational and seek to maximize their utility based on their preferences within the constraints they face, such as income or time. Utility Functions A utility function is a mathematical representation that assigns a numerical value to each possible combination of goods, services, or actions, reflecting the level of utility an individual derives from them. These functions allow us to model and predict choices by quantifying preferences, even when the choices involve intangible rewards like altruism or religious fulfillment. The utility function is usually written as: f(x1, x2, x3,…,xn), or u(x1, x2,…, xn), where “xn”s represents the variables in the function. In order to be more intuitive, let’s first look at a simple model (my friend Evan will be used as an example): Evan is an avid foodie, ordering extra pizzas and noodles every day to satisfy his ravenous appetite outside of breakfast, lunch, and dinner. It is assumed that Evan’s level of satisfaction rises with the amount of food he eats; specifically, one pizza is worth three units of utility/satisfaction, and one bowl of ramen is worth two units. An incredibly simple and straightforward utility function for this scenario could be derived if we set the number of pizzas as x and the number of ramen as u(x,y)= 3x+2y. The graph of this function would look like the following graphed out by GeoGebra:

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Regardless of how large Evan’s stomach capacity is, his appetite will diminish as he consumes more food, leading to a decrease in the additional satisfaction (or marginal utility) gained from each extra unit. This concept of diminishing marginal utility suggests that while Evan may initially derive significant enjoyment from consuming food, each subsequent unit will provide less additional utility. Therefore, we need to adjust our original utility function 3x+2y to reflect this diminishing satisfaction from increasing consumption, ensuring the model accurately captures the decreasing utility associated with each additional unit of food. 

The trait of a natural log function would fit this requirement as we examine the graph of y=lnx:

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As we can see, the function’s slope decreases as 𝑥 increases; in other words, as the values of 𝑥 rise, the rate of growth of the function declines. Explained with calculus, the second derivative of the function lnx is -1/x^2; this expression is negative as long as x>0, meaning that as long as Evan consumes a positive amount of food (which is realistic since consuming a negative amount would imply something impossible—unless, hypothetically, Evan starts vomiting). The negative second derivative indicates that the rate of increase in Evan’s utility slows down as he consumes more food. This aligns with the concept of diminishing marginal utility: although Evan’s utility continues to increase with additional food, it increases at a slower and slower pace.

By incorporating the natural logarithm into the utility function, we can express it as u(x,y) = 3 ln⁡x + 2 ln⁡y, where x represents the amount of pizza consumed and y represents the amount of ramen consumed.

The total utility expressed by the function 𝑢(𝑥, 𝑦) = 3 ln⁡𝑥 + 2 ln𝑦 yields a value of 0 when 𝑥 = 1 and 𝑦 = 1. This might seem counterintuitive because we’d expect Evan to experience some utility even if he consumes small, non-zero amounts of food. However, despite this, the function effectively captures the relationship between total utility with the change of quantities of 𝑥  and 𝑦, and illustrates the key concept of diminishing marginal utility. The graph would look like this:

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2. Introducing Budget Constraints

Up to this point, you might be wondering: why does “optimization” even matter in this model? Wouldn’t Evan be the happiest man in the world if he could keep buying as much food as he wanted, since his utility would simply keep increasing? However, we’ve yet to address the most crucial aspect of economics—the primary reason the discipline exists in the first place: in reality, resources are limited. Optimization becomes essential because people face constraints while acquiring extra units of utilities—whether they are financial, time-related, or related to the availability of goods. Therefore, the importance of optimization arises when we consider these constraints, which restrict the variables in our utility functions and force us to make trade-offs between different options.

In the situation of Evan acquiring food, we can thus develop the following budget constraint: he has a maximum allowance of 200 dollars per month, so he can’t spend anywhere over 200 dollars on his monthly pizza/ ramen purchase. If we assume that one box of pizza costs 10 dollars and one bowl of ramen costs 3 dollars, then we would be able to develop the following budget constraint function: 10x + 3y ≤ 200. The graph of the constraint would look like such if Evan decides to spend all of his money on food.

Now, the graph would appear as follows if we combined the budget constraint and the utility function:

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As we can see, since the z-axis represents the magnitude of total utility, the optimized utility corresponds to the highest possible point on the graph. Visually, this would occur at the point where the blue budget constraint curve intersects with the purple utility curve at its greatest height. This intersection represents the optimal combination of goods that maximizes utility, given the budget constraint, where Evan gets the most satisfaction while staying within his limited resources. But how do we determine the exact coordinates of this optimal point? Or how many pizzas and bowls of ramen should Evan purchase to maximize his utility? Well, you could always ask Mr. GPT for a quick solution if you’re feeling lazy. But if you’re not much of a looser (Just kidding, of course, to keep you motivated to read my next article), I’ll suggest a couple of alternatives: you could dive into computer programming to solve optimization problems, or you could stick around for my next article, where I’ll introduce a powerful mathematical tool—Lagrange multipliers. This method, taught in college-level Calculus 3 (Multivariable Calculus), helps you solve optimization models with constraints, such as the one Evan faces. Stay tuned!

3. References

[1] “3D Calculator – GeoGebra.” Geogebra.org, 2024, www.geogebra.org/3d?lang=en

[2] “Desmos | Graphing Calculator.” Desmos, 2024, www.desmos.com/calculator.

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