Problem 26 - Applied Calculus

Question

In economics, total utility refers to the total satisfaction from consuming some commodity. According to the economist Samuelson: \(^{25}\) As you consume more of the same good, the total (psychological) utility increases. However, \(\ldots\) with successive new units of the good, your total utility will grow at a slower and slower rate because of a fundamental tendency for your psychological ability to appreciate more of the good to become less keen. (a) Sketch the total utility as a function of the number of units consumed. (b) In terms of derivatives, what is Samuelson saying?

Step-by-Step Solution

Answer

(a) Sketch a concave downward curve. (b) Marginal utility diminishes; \( U'(x) > 0 \), \( U''(x) < 0 \).

Step 1: Understand Total Utility

Total utility is the overall satisfaction or happiness received from consuming a certain quantity of goods or services. According to Samuelson, as more units of a good are consumed, the total utility continues to increase, but the rate of increase slows down with each additional unit.

Step 2: Identify Relationship for Part (a)

For part (a), you need to sketch a graph of total utility as a function of the number of units consumed. Imagine a curve that starts from the origin, increases as quantity increases, but eventually flattens out, showing slower growth in utility as more units are consumed. This is like a curve that concaves downward.

Step 3: Sketch the Graph

Draw the x-axis as the number of units consumed and the y-axis as total utility. Start the curve from the origin. As you increase the number of units on the x-axis, plot points reflecting an upward slope that gradually becomes less steep. This shows increasing utility at a decreasing rate.

Step 4: Understand Samuelson's Statement for Part (b)

Samuelson is discussing the concept of diminishing marginal utility. In terms of derivatives, he is suggesting that while the first derivative of the utility function (marginal utility) is positive (as total utility is increasing), the second derivative is negative. This negative second derivative indicates the decreasing rate of increase in total utility.

Step 5: Interpret Derivatives (Part b)

Mathematically, if we let \( U(x) \) be the total utility as a function of units \( x \), then \( U'(x) > 0 \) for all \( x \) because total utility is increasing. However, \( U''(x) < 0 \) indicating that as more units are consumed, the rate of increase of total utility is decreasing.

Key Concepts

Total Utility

Total utility represents the complete satisfaction or happiness an individual derives from consuming a certain amount of goods or services. Samuelson explains that as you consume more of a specific good, your total utility will continue to rise. This means you feel more satisfied overall. However, the speed at which your happiness increases starts to slow down with each additional unit you consume. Imagine enjoying an ice cream cone on a hot day. The first few scoops are delightful! But as you keep eating, the joy starts to taper off, even though you still enjoy the treat.
So, when we draw a graph of total utility against the number of units consumed, we see the curve rise quickly at first but then begin to flatten out. This illustrates the core concept of diminishing marginal utility, where each additional unit contributes less to your overall satisfaction.

Derivative Analysis

The concept of derivative analysis helps us understand Samuelson's point in terms of mathematical functions and curves. Imagine your total utility from consuming a good is represented by a function, let's call it \( U(x) \), where \( x \) is the number of units consumed. Derivatives tell us how the utility function changes, very much like determining the slope of a hill.
Firstly, the first derivative \( U'(x) \) shows the rate at which total utility changes with additional units. A positive \( U'(x) \) means utility is increasing, which is exactly what happens here—consuming more makes you happier. However, Samuelson points that this increase slows down over time. This is explained by the second derivative, \( U''(x) \), which is negative. A negative second derivative indicates that even though you're becoming happier, you're doing so at a decreasing rate. The hill's slope isn't as steep as it was initially.

Economic Concepts

Understanding diminishing marginal utility and derivative analysis involves grasping essential economic concepts. In economics, these ideas help explain consumer behavior and choices. As people consume more, they weigh the benefit of consuming an additional unit against its cost, since the extra satisfaction gained from each additional unit wanes over time.
This is why businesses strategize pricing and quantity distribution, aiming to balance consumer satisfaction with their product prices. Moreover, understanding these concepts assists policymakers in predicting how changes in the economy or prices might influence consumer habits. Through evaluating utility, derivatives, and marginal outcomes, economists can craft models and strategies that reflect real-world scenarios and guide economic growth and policy decisions.