# Utility Functions

Have you ever been to a fast food chain and frozen while you are ordering your food? Then you probably know the struggle of selection and never-ending questions about your menu that you ask yourself. What type of main dish do you want? Do you want nuggets or fries? Or both? What type of drink do you want with your food? Soda or milkshake?

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It is practically normal to be bombarded by these questions. Any agent in an economy -- whether an individual, a firm, or even the government -- faces similar questions in their decision-making processes. If your decision while ordering food from the menu is so hard to make, how can firms and governments make decisions? Economics has a stylish answer to these questions. Keep reading to learn about 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?

The utility is an abstract value that an agent gets from a preference. It may also be defined as the satisfaction gained from a selection.

Let’s assume that you would prefer grapes over lemons at any time. Economists argue that your utility gained from grapes is higher than your utility gained from lemons. Maybe you like sweet fruits over a bitter lemon, or you prefer something with smaller seeds. Your satisfaction gained from grapes is higher than your satisfaction gained from lemons. Thus, your utility is higher if you would select grapes over lemons.

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 are a special type of functions that connect or map the amount of utility gained from preferences or bundles of goods to a ranking system or a set of numbers.

Let’s say that your utility from consuming a bottle of ginger soda is denoted with $$u(\hbox{ginger})$$, and your utility gained from consuming a cup of lemonade is denoted with $$u(\hbox{lemonade})$$. Now let’s assume that you would prefer a cup of lemonade over a bottle of ginger soda. We can denote the relationship between your utility functions as $$u(\hbox{lemonade}) > u(\hbox{ginger})$$.

### 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, $$y_1$$.

The graph above is similar to a specific logarithmic graph. For example, if we plot the graph of $$log (x^2)$$ or $$log({x}^{10})$$, we will get a similar curve. It is highly common to denote utility curves in the form of logarithmic functions.

If you have questions about the shape of the graph, please be sure to check our explanation: Diminishing Marginal Returns!

### 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.

 Peach / Slices of Watermelon 2 Slices of Watermelon 4 Slices of Watermelon 7 Slices of Watermelon 2 Peaches $$u(2,2) = 2^2 + 2 = 6$$ $$u(2,4) = 2^2 + 4 = 8$$ $$u(2,7) = 2^2 + 7 = 11$$ 3 Peaches $$u(3,2) = 3^2 + 2 = 11$$ $$u(3,4) = 3^2 + 4 = 13$$ $$u(3,7) = 3^2 + 7 = 16$$

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.

The indifference curves show the combinations of preferences that provide the same amount of utility.

We have covered the Indifference Curves in detail. Don't hesitate to check it out!

## 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.

Marginal utility is the change in the total utility with respect to the increase or decrease in consumption by one unit.

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?

A bundle of goods is perfect complements if the goods in the bundle are consumed together with the same proportion.

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?

A bundle of goods contains perfect substitutes if and only if the goods in the bundle can be used for each other’s places in precisely the same way.

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.

#### Cobb-Douglas Utility Function

Cobb-Douglas utility functions are another common sub-type of utility functions since they are extremely flexible and open to alteration. This function is generally used for explaining the marginal rate of substitution between two goods. The basic Cobb-Douglas utility functions take this form:

$$u(x,y) = x^a y^b | a,b \in R, a+b = 1$$

To find the marginal rate of substitution, we focus on the partial derivative of x and y and their relationship. First, we will calculate the partial derivative with respect to x:

$$\dfrac{\partial u(x,y)}{\partial x} = ax^{a-1}y^b$$

Then, we can take the partial derivative with respect to y:

$$\dfrac{\partial u(x,y)}{\partial y} = x^aby^{b-1}$$

Now if we take their ratio, we can find their marginal rate of substitution:

$$\dfrac{\dfrac{\partial u(x,y)}{\partial x}}{\dfrac{\partial u(x,y)}{\partial y}} = \dfrac{ax^{a-1}y^b}{x^aby^{b-1}} = \dfrac {ay}{bx}$$

If you think you are missing some parts in Cobb-Douglas utility functions, please be sure to check the relevant part in our explanation: Marginal Rate of Substitution!

## 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. We have covered the Marginal Rate of Substitution in detail. Don't hesitate to check it out! 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.  $$Q_y = 0$$ $$Q_y = 1$$ $$Q_y = 2$$ $$Q_y = 3$$ $$Q_y = 4$$ $$Q_y = 5$$ $$Q_x = 0$$$0, $$u = 0$$ $2, $$u = 2$$$4, $$u =4$$ $6, $$u =6$$$8, $$u =8$$ $10, $$u =10$$ $$Q_x= 2$$$2, $$u = 2$$ $4, $$u =4$$$6, $$u =6$$ $8, $$u = 8$$$10, $$u =10$$ $12, $$u = 12$$ $$Q_x= 4$$$4, $$u = 4$$ $6, $$u =6$$$8, $$u =8$$ $10, $$u = 10$$$12, $$u = 12$$ $14, $$u = 14$$ $$Q_x= 6$$$6, $$u = 6$$ $8, $$u = 8$$$10, $$u =10$$ $12, $$u = 12$$$14, $$u = 14$$ $16, $$u =16$$ $$Q_x= 8$$$8, $$u = 8$$ $10, $$u =10$$$12, $$u = 12$$ $14, $$u =14$$$16, $$u =16$$ $18, $$u = 18$$ $$Q_x= 10$$$10, $$u = 10$$ $12, $$u = 12$$$14, $$u = 14$$ $16, $$u =16$$$18, $$u =18$$ $20, $$u = 20$$ 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.  $$Q_y = 0$$ $$Q_y = 1$$ $$Q_y = 2$$ $$Q_y = 3$$ $$Q_y = 4$$ $$Q_y = 5$$ $$Q_x = 0$$$0, $$u = 0$$ $2, $$u = 2$$$4, $$u =4$$ $6, $$u =6$$$8, $$u =8$$ $10, $$u =10$$ $$Q_x= 2$$$2, $$u = 2$$ $4, $$u =4$$$6, $$u =6$$ $8, $$u = 8$$$10, $$u =10$$ $12, $$u = 12$$ $$Q_x= 4$$$4, $$u = 4$$ $6, $$u =6$$$8, $$u =8$$ $10, $$u = 10$$$12, $$u = 12$$ $14, $$u = 14$$ $$Q_x= 6$$$6, $$u = 6$$ $8, $$u = 8$$$10, $$u =10$$ $12, $$u = 12$$$14, $$u = 14$$ $16, $$u =16$$ $$Q_x= 8$$$8, $$u = 8$$ $10, $$u =10$$$12, $$u = 12$$ $14, $$u =14$$$16, $$u =16$$ $18, $$u = 18$$ $$Q_x= 10$$$10, $$u = 10$$ $12, $$u = 12$$$14, $$u = 14$$ $16, $$u =16$$$18, $$u =18$$ \$20, $$u = 20$$

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.

## Utility Functions - Key takeaways

• The utility is an abstract value that an agent gets from a preference. It may also be defined as the satisfaction gained from a selection.
• Utility functions are a special type of functions that connect or map the amount of utility gained from preferences or bundles of goods.
• There are four common types of utility functions: linear, perfect substitutes, perfect complements, and Cobb-Douglas. (Nonetheless, it is better to keep in mind that utility functions can take many shapes, these are just the most common ones.)
• A bundle of goods is perfect complements if the goods in the bundle are consumed together with the same proportion.
• A bundle of goods contains perfect substitutes if and only if the goods in the bundle can be used for each other’s places precisely the same way.
• If the results of a utility function are the same between different bundle combinations, we can say that the consumer is indifferent between these two bundles. Thus, if we draw a curve between these points, we will get an indifference curve.
• Marginal utility is the change in the total utility with respect to the change in consumption by one unit.

#### Flashcards in Utility Functions 15

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What do you mean by utility function?

Utility is an abstract value that an agent gets from a preference. It may also be defined as satisfaction gained from a selection.

Utility functions are a special type of function that connects or maps the amount of utility gained from preferences or bundles of goods to a ranking system or a set of numbers.

What are the types of utility functions in economics?

Utility functions have a wide range of structures. The most commonly used utility functions are linear utility functions, Cobb-Douglas utility functions, utility functions of perfect complements, utility functions of perfect substitutes.

How do you solve a utility function problem?

You may find the maximum utility by solving for when the marginal utility equals 0.

What is an example of a utility in microeconomics?

The utility can be anything that an agent gets satisfaction with. If you are getting satisfaction from a good or a service, you are gaining utility from it. For example, eating a burger gives you utility. This may be subjective, like its taste, or objective like the ingredients that you get from a burger.

What are the properties of utility functions?

Utility functions are a special type of function that connects or maps the amount of utility gained from preferences or bundles of goods to a ranking system or a set of numbers.

## Test your knowledge with multiple choice flashcards

Following function is a linear utility function.$$u(x) = ax + x^2 | x ,a \in R^{+}$$

Following function is a linear utility function.$$u(x) = mx + c | x,m,c \in R^{+}$$

The following function can be given as the general structure of the Cobb-Douglas utility functions.$$u(x,y) = x^a y^b | a,b \in R, a+b = 1$$

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