Utility Functions

Utility Functions Meaning

Utility functions are mathematical relationships that map the preferences to the amount of utility gained from that preference. But what is utility in economics?

Utility functions are a special type of function that measures the utility gained from a preference. In economics, we generally denote utility with \(u\), and we denote utility gained from preference \(x\) with \(u(x)\).

Utility Functions with a Single Argument

Single-argument utility functions are a common way to show the utility gained from a single good or preference. Let’s assume a utility function, \(u\), which shows the amount of utility gained from peach consumption. We can show this function with the following curve.

Fig. 1 - Utility curve of peach consumption

If an agent with the utility curve \(u\) consumes \(x_1\) amount of peaches, she will gain utility that is equal to \(y_1\). This is a utility function with a single argument. It takes its only one argument, \(x_1\), and matches it with a value, \(y_1\).

Utility Functions with Multiple Arguments

Another common usage of utility functions is their application over a bundle of goods. Let’s assume the previous scenario but now, instead of comparing two different utilities, let’s measure the combined utility. Let’s assume that our utility function \(u\) takes two arguments, \(x_1\) and \(x_2\). It maps the results to \(x_1\) and \(x_2\) according to the following rule:

\(u(x_1,x_2) = x_1^2 + x_2\)

Now, assume that you ate two peaches which are denoted as \(x_1\), and one slice of watermelon, which is denoted as \(x_2\). We can calculate the total utility with the following approach.

\(u(\hbox{peaches, watermelon}) = \hbox{peaches}^2 + \hbox{watermelon}\)

Let’s elaborate on this to a greater extent. For a better understanding, we can demonstrate our equations on a Cartesian coordinate system. This will unveil the relationship between indifference curves and utility functions.

Utility Functions vs Indifference Curves

Utility functions and indifference curves are related in a close manner. We can explain their relationship with an example. Let's assume that our utility function of the fruits is still the same, which can be denoted as:

\(u(\hbox{peaches, watermelon}) = \hbox{peaches}^2 + \hbox{watermelon}\)

Now, we can create a table for different amounts of consumption behaviors.

Table. 1 - Different consumption bundles of watermelons and peaches

Did you notice anything between the first and the second row? The third column of the first row equals 11, and similarly, the first column of the second row equals 11. This points out the fact that the utility of eating two peaches and seven slices of watermelon is the same as the utility of eating 3 peaches and 2 slices of watermelon. Obviously, your value gained from eating a single peach is much more than eating a slice of watermelon. We can represent these values on a Cartesian coordinate system as in Figure 2 below.

Fig. 2 - Showing the values on a cartesian coordinate system

Note that with respect to different combinations, our utility will change. If we follow the line shown, our utility will increase continuously. Now let’s assume that there exist combination points that give the same amount of utility. In the previous example, the combination of three peaches and two slices of watermelon gave the same amount of utility as the combination of two peaches and seven slices of watermelon. Since the utility gained from these combinations is the same, we can denote them on the same indifference curve as in Figure 3 below.

Fig. 3 - Different utility curves with respect to different bundles of the goods

Did you notice something similar between utility functions and indifference curves? We can say that a utility function maps the utility gained from a bundle of goods to an indifference curve. Here, \(u_3\) is an indifference curve. The agent is indifferent between any combinations on \(u_3\) because it gives them the same amount of utility.

Long story short, we can say that utility functions measure the utility from a combination of preferences or a bundle of goods. We can denote the amount of utility gained from a specific bundle with an indifference curve. That’s why utility functions and indifference curves are closely related.

Utility Functions Types

Although utility functions can appear in a variety of forms, there are some common types of utility functions used for economic modeling, policy-making, and understanding general individual behaviors. In this section, we will go over these common types of utility functions and try to grasp their structure.

Utility Functions Formula

Since utility functions come in different flavors, it is impossible to state them with one general formula. On the other hand, there are some common utility function structures that are widely used in the economics literature. We can list them as follows:

• Linear utility functions
• Utility functions of perfect complements
• Utility functions of perfect substitutes
• Cobb - Douglas utility function

Linear Utility Functions

The most well-known and basic utility functions are linear utility functions. Linear utility functions are called linear due to their structure. Let’s denote a utility function, \(u\), with the following conditions.

\(u(x_1,x_2,x_3,...,x_n) = m_1x_1 + m_2x_2 + m_3x_3+...+m_nx_n\),

\(m_1, m_2,...,m_n \in R^{+}\)

In these types of utility functions, utility from the consumption increases in a linear manner. If we are trying to find the utility gained from consuming one unit of the good, we can take the partial derivative of the function with respect to that good.

With regard to linear utility functions, we can find the marginal utility by taking the partial derivative: \(\hbox{Marginal utility} = \dfrac{\partial \hbox{Total utility}}{\partial \hbox{Total consumption}}\).

Let’s assume that we have a linear utility function as above and we want to find the amount of utility gained from the consumption of one unit of \(x_1\).

We know that our utility function is:

\(u(x_1,x_2,x_3,...,x_n) = m_1x_1 + m_2x_2 + m_3x_3+...+m_nx_n\),

and if we want to find the marginal utility of \(x_1\), we should take the partial derivative with respect to \(x_1\). Thus,

\(\dfrac{\partial u}{\partial x_1} = m_1\)

Thus, if we consume one unit of \(x_1\), our utility will increase by \(m_1\).

Utility Functions of Perfect Complements

What are perfect complements?

Utility functions for perfect complements goods contain the minimum arrangement since the goods can only be consumed together with respect to the same proportion. Thus, we can represent the utility function of perfect complements with \(u(x,y) = min(x,y) \).

Utility Functions of Perfect Substitutes

What are perfect substitutes?

Utility functions for these types of goods can be denoted as \(u(x,y) = x + y\). Their marginal rate of substitution is equal to one.

Examples of Utility Functions

Since we have covered the general aspects of utility functions, now, it is better to give an example of utility functions and bind our knowledge with it.

Let’s denote a utility function \(u\) that takes two commodities as arguments, \(x\) and \(y\), and map them as below:

\(u(x,y) = x + 2y | u(x,y) \geq 0 \land x,y \in R\)

Now, let's denote a budget line, \(w\), which is equal to $10, while a unit of x costs $1 and a unit of y costs $2:

\(w = \$10, x = \$1, y = \$2 \)

Can you find the bundle of goods that one can get with respect to the budget? What is the maximum amount of utility a consumer can get from this bundle of goods?

In such questions, we have two options. One is to list all possible combinations, and the other is to find the marginal rate of substitution between two goods. We are going to use the possible combinations approach here.

Since we are going to list some combinations, it is better to create a table with different possible values. In the table below, we are going to show the cost of the combination and the utility gained from the combination with respect to previously given equations.

Table. 2 - Table with different combinations of the two goods

We should keep in mind that some of the combinations here are not feasible. Since our budget is $10, we can eliminate the options that are above $10. Therefore let's denote the unfeasible cells with a different color.

Table. 3 - Table with the infeasible combinations marked in red

Thus, the maximum amount of utility one can get is \(u = 10\). There exist many combinations for reaching this utility level. We can clearly see the connection between indifference curves and utility functions.