Problem 59 - Brief Applied Calculus

Question

59-60. ECONOMICS: Marginal Utility Generally, the more you have of something, the less valuable each additional unit becomes. For example, a dollar is less valuable to a millionaire than to a beggar. Economists define a person's "utility function" \(U(x)\) for a product as the "perceived value" of having \(x\) units of that product. The derivative of \(U(x)\) is called the marginal utility function, \(M U(x)=U^{\prime}(x)\). Suppose that a person's utility function for money is given by the function below. That is, \(U(x)\) is the utility (perceived value) of \(x\) dollars. a. Find the marginal utility function \(M U(x)\). $$ U(x)=100 \sqrt{x} $$ b. Find \(M U(1)\), the marginal utility of the first dollar. c. Find \(M U(1,000,000)\), the marginal utility of the millionth dollar.

Step-by-Step Solution

Answer

a. \( MU(x) = \frac{50}{\sqrt{x}} \); b. 50; c. 0.05.

Step 1: Understand the Utility Function

The person's utility function is given as \( U(x) = 100 \sqrt{x} \). This represents the perceived value of having \( x \) dollars.

Step 2: Differentiate to Find Marginal Utility Function

To find the marginal utility function \( MU(x) \), or \( U'(x) \), we need to differentiate \( U(x) \) with respect to \( x \). Using the power rule, the derivative of \( \sqrt{x} = x^{\frac{1}{2}} \) is \( \frac{1}{2}x^{-\frac{1}{2}} \). Thus:\[ MU(x) = U'(x) = 100 \cdot \frac{1}{2}x^{-\frac{1}{2}} = 50 x^{-\frac{1}{2}} = \frac{50}{\sqrt{x}} \]

Step 3: Calculate Marginal Utility of the First Dollar

Substitute \( x = 1 \) into the marginal utility function.\[ MU(1) = \frac{50}{\sqrt{1}} = 50 \]

Step 4: Calculate Marginal Utility of the Millionth Dollar

Substitute \( x = 1,000,000 \) into the marginal utility function.\[ MU(1,000,000) = \frac{50}{\sqrt{1,000,000}} = \frac{50}{1000} = 0.05 \]

Key Concepts

Utility Function

The utility function, expressed as \( U(x) \), is a key concept in economics. It measures how much satisfaction or value a person derives from possessing \( x \) units of a particular product or good. In simple terms, it's like a happiness score for owning a certain amount of something, whether it's money, chocolate bars, or any other item. Economists use these functions to determine how different quantities of goods contribute to an individual's overall well-being.

The utility function can differ from person to person, depending on their preferences and needs. For example:

• A millionaire might have a utility function where each additional dollar offers little extra happiness, illustrating diminishing returns.
• A person living on a tight budget might gain significant utility from each extra dollar they acquire.

This concept helps to detail why resources like money can bring different levels of satisfaction to different people. Understanding utility functions is crucial for analyzing consumer behavior and their purchasing decisions.

Derivative

In mathematics, the derivative is a powerful tool used to understand how functions change. Essentially, it shows how one quantity reacts or changes in response to changes in another.

When we talk about the derivative of a utility function, what we are interested in is how the perceived value (or utility) changes as the quantity of the product changes. The derivative gives us the marginal utility function, which measures the increase in utility gained from an additional unit.

For example, let’s look at the utility function\( U(x) = 100 \sqrt{x} \), the derivative \( U'(x) \) is computed by using the power rule:

• Convert \( \sqrt{x} \) to \( x^{1/2} \).
• Apply the power rule: the derivative of \( x^{n} \) is \( n \cdot x^{n-1} \).
• The function becomes \( U'(x) = 50 x^{-1/2} = \frac{50}{\sqrt{x}} \).

This shows us how utility increases with additional units, giving insights into consumer behavior and decision-making.

Calculus

Calculus is a branch of mathematics that studies continuous change and is quite essential in economics for understanding dynamic systems. Within calculus, differentiation allows economists to determine marginal concepts such as marginal utility and marginal cost.

By differentiating a utility function like \( U(x) = 100 \sqrt{x} \), economists can derive insights such as how the utility (satisfaction) alters with a change in quantity. This basic derivative approach provides us with the marginal utility, a concept showing the rate of change of utility with respect to changing amounts of goods.

Consider the marginal utility function \( MU(x) = \frac{50}{\sqrt{x}} \):

• This function confirms that utility grows at a decreasing rate as more units are added, embodying the law of diminishing marginal utility.
• With each additional unit of money, the added satisfaction decreases.

Thus, calculus facilitates a deeper understanding of consumer satisfaction and allows businesses and economists to make informed decisions.

Economics

Economics is the social science that examines how individuals and societies allocate limited resources to satisfy their needs and desires. Within this field, the concept of marginal utility is vital, as it describes the additional utility gained from consuming one more unit of a good. This principle is rooted in the law of diminishing marginal utility, which states that as a person consumes more of a good, the incremental gain in utility typically decreases.

Understanding marginal utility helps economists and businesses:

• Determine the optimal allocation of resources to maximize satisfaction or profit.
• Set prices based on the perceived value of goods to consumers.

Consider how this applies to a millionaire buying another dollar's worth of a product:

- For them, the marginal utility might be low, suggesting they perceive less value in the extra dollar compared to someone with fewer financial resources.
This illustrates that economics values each unit's utility to form decisions on pricing, production, and resource distribution.