# Preferences and Utility Theory

Preferences and utility theory are the essential building blocks of explaining the behavioral decision-making of households in economic theory. Therefore, understanding preferences and utility theory is crucial for all studying economics and business administration. This article will explain the elements of preferences and utility theory that help determine households’ rational decision-making under certainty as economic subjects.

## Introduction

Why do we make the decisions we choose? Behavior is an act of an individual’s preference or choice and means to decide to act. Any economics textbook will teach you that households make consumption and income decisions or choices based on their preferences and will depict those preferences using certain utility functions. Let us ask the following critical questions: how do preferences and utility relate to each other?

### What are Preferences? Introduction to Preference Theory

Preferences explain how decision-makers make a rational decision when confronted with the task of comparing one bundle of goods with another. In this case, we concentrate on household decision-making, where households must decide which consumer goods, income generation, and savings bundles they prefer to satisfy their needs. First, households must choose their labor supply and provide production factors to the firms (production decision-makers) to receive income. Afterward, they can decide how to spend their income by consuming goods and saving (creating wealth). How such decisions are made is explained in economics using the preferences of rational economic agents. Later we shall discuss the properties of rational choices.

An Example of preferences can be viewed as follows: First, assume that you must decide the amounts of two goods. Let us say you choose the number of apples $X_1$ and bananas $X_2$ in your shopping cart, representing a bundle of goods $X (x_{1}, x_{2})$. Which bundle would you prefer over another if you had to choose among the following amounts of goods found in a bundle? Could you explain why bundle $E(2, 3)$ with two Apples and three Bananas will be preferred over all other bundles? Using these five bundles, we can discuss some fundamentals of preference theory.

• Bundle $A(0, 0)$
• Bundle $B(1, 0)$
• Bundle $C(0, 1)$
• Bundle $D(1, 1)$
• Bundle $E(2, 3)$

### What is Utility? Introduction to Utility Theory

Utility and preferences go hand in hand in economic theory. First, utility theory uses utility functions $U(X_{1}, X_{2})$ to virtualize and formalize the preference theory. The goal of utility theory is to explain how rational decision-making households order the choices they face between available bundles of goods and to qualify the level of satisfaction of households making decisions. Broadly, utility theory develops mathematical concepts that help explain basic preferences, such as those of (perfect and imperfect) complements, substitutes, and quasi-linear choices. Later we will learn how utility theory helps quantify the opportunity costs of decision-making in household optimum.

Following the example in the previous subsection, we can view the utility function as a means of ordering goods bundles. Comparing Bundle A, B, and C, we can conclude that, where rational preferences apply, the Utility of Bundle A will be least the Utility of B and C. Still, we cannot compare the Utility of B and C without further information about the preferences of the individual. We can identify that something is better than nothing regarding preferences over the bundles. Therefore any utility function should show that $U(0, 0) < U(1, 0)$ and $U(0, 0) < U(0, 1)$, while comparing bundle A with B and C.

Regarding the behavior of households, utility theory helps economists to utilize mathematical concepts while developing models that explain economic behavior. Later, we will learn how utility theory helps reveal the optimal household demand for individual goods in an economy.

## Characteristics of Rational Preferences

Some preference assumptions must be fulfilled for economists to eliminate irrational behavior in their economic models. Choices are said to be rational if and only if the following conditions are fulfilled:

• Completeness: Decisionmakers know the choices the would make, when come across potential bundles of goods.
• Transitiveness: Households are clear while comparing different consecutive bundles, and there are no vicious circles in the choices.
• Continuity: Households do not ignore slight changes in their bundles, order the slightest change in their fortune accordingly, and an increase, Ceteris Paribus, leads to the higher ordering of the new bundle.
• Convexity: extremes are not preferred.
• Monotonous: More is better than less whenever the household is confronted with goods. “Bads” are excluded.

Later in economic theory, irrational behavior is reintroduced using the ideas of bounded rationality by lifting some of the abovementioned assumptions.

## Type of Preferences and Utility in Theory

In economics, the behavior of households is analyzed based on how consumers react to bundles of goods presented to them and are classified into general categories of preferences as follows:

• Preferences for substitutes, e.g., the amounts of food and water for lunch consumed can be compensated against each other.
• Preferences for compliments, e.g., you probably prefer a pair of shoes. Another example is that an office chair is consumed together with a desk.

### Preferences and Utility of perfect and imperfect Substitutes

Amongst preferences for substitutes, we can differentiate between goods that individual views as perfect substitutes from those that the individual deems to be imperfect substitutes. While perfect substitutes allow for the complete elimination of one good from the consumption bundle keeping the utility level constant (along an indifference curve), imperfect substitutes do not enable the complete elimination of one good from the Bundle, e.g., Cobb-Douglas-Preferences.

Preferences of perfect substitutes are presented using the following form of utility function:

U(x_1,x_2)=u(x_1)+u(x_2) \\ \text {for a bundle with two goods or a bundle with n goods} \\ U(x_1,x_3,..., x_n)=\sum_{i=1}^{n}{U(x_i)}

On the other hand, preference of imperfect substitutes can be presented using the following form of utility function:

U(x_1,x_2)=u(x_1) \cdot u(x_2) \\ \text {for a bundle with two goods or a bundle with n goods} \\ U(x_1,x_3,..., x_n)=\prod_{i=1}^{n}{U(x_i)}

### Preferences and Utility of perfect Compliments

Preference for a perfect compliment reflects the necessity of goods to be consumed in a specific combination of amounts, e.g., most people prefer to wear a pair of shoes. They would only consider owning them in pairs compared to owning one shoe. Such preferences of perfect compliments can be presented using the following form of utility function:

U(x_1,x_2)= \min {(u(x_1);u(x_2))} \\ \text {for a bundle with two goods or a bundle with n goods} \\ U(x_1,x_3,..., x_n)=\min_{i=1}^{n}{U(x_i)}