Question

Let \(T=\phi(x)\) be a total function (e.g., total product or total cost): (a) Write out the expressions for the marginal function \(M\) and the average function \(A\). (b) Show that, when \(A\) reaches a relative extremum, \(M\) and \(A\) must have the same value. (c) What general principle does this suggest for the drawing of a marginal curve and an average curve in the same diagram? (d) What can you conclude about the elasticity of the total function \(T\) at the point where \(A\) reaches an extreme value?

Short answer

(a) Marginal: \( M(x) = \frac{dT}{dx} \), Average: \( A(x) = \frac{T(x)}{x} \). (b) At extremum, \( M(x) = A(x) \). (c) Marginal curve intersects average curve at extremum. (d) Elasticity is 1 where \( A \) is extreme.

Step-by-step solution

Understanding Marginal and Average Functions

The marginal function, denoted as \( M(x) \), describes the rate of change of the total function \( T \) with respect to \( x \). Mathematically, this is the derivative of \( T \) with respect to \( x \): \( M(x) = \frac{dT}{dx} \). The average function, denoted as \( A(x) \), is the total function divided by \( x \): \( A(x) = \frac{T(x)}{x} \).

Finding Extremum Condition for Average Function

To find a relative extremum of the average function \( A(x) \), we need to set the derivative of \( A(x) \) with respect to \( x \) to zero: \( \frac{d}{dx} \left( \frac{T(x)}{x} \right) = 0 \). Using the quotient rule, the derivative of \( A(x) \) becomes: \[ \frac{d}{dx} \left( \frac{T(x)}{x} \right) = \frac{x\frac{dT}{dx} - T(x)}{x^2} = 0 \]. This simplifies to \( xM(x) - T(x) = 0 \), which implies \( M(x) = A(x) \).

Principle for Drawing Marginal and Average Curves

The extremum condition \( M(x) = A(x) \) suggests that whenever the average function \( A(x) \) reaches an extremum, the marginal function \( M(x) \) crosses it at the same point. So, when drawing their curves, the marginal curve should intersect the average curve at points of extremum for the average curve.

Elasticity of Total Function at Extremum

Elasticity of the total function \( T \) with respect to \( x \) at any point is given by \( E = \frac{x\frac{dT}{dx}}{T(x)} \). At the point where \( A(x) \) is at an extremum, since \( M(x) = A(x) \), it follows that \( E = 1 \). This means the elasticity is 1 at the point where \( A \) reaches an extreme value.

Key concepts

average functions

An average function is a concept that helps us understand how a quantity behaves as it's distributed over a certain number of units. Think of it like calculating the average score on a test or the average speed during a trip. For functions, the average function, typically denoted as \(A(x)\), represents the total quantity, \(T(x)\), divided by the number of units, \(x\). In a mathematical form, it is expressed as:
\[A(x) = \frac{T(x)}{x}\]
This formula gives us a measure of the per-unit value of the total function. Whether this is average cost, average product, or any other average measure, it helps in understanding trends and analyzing the efficiency or performance in economic or mathematical terms.

• It provides a simplified view by leveling out fluctuations.
• Useful for making comparisons between different scenarios or over different quantities.
• Also aids in identifying consistent weaknesses or strengths at a per-unit level.

The study of the average function is essential because it often interacts with other functions, like the marginal function, to reveal deeper insights about the behavior of the total function.

extremum

Finding an extremum (either a maximum or minimum) of a function is key in calculus, both in mathematical theory and practical applications. When considering the average function \(A(x) = \frac{T(x)}{x}\), its extremum can be identified by setting its derivative equal to zero. This derivative uses the quotient rule:
\[ \frac{d}{dx} \left( \frac{T(x)}{x} \right) = \frac{x\frac{dT}{dx} - T(x)}{x^2} \]
When set to zero, this simplifies to \( xM(x) - T(x) = 0 \), showing that the marginal function \(M(x)\) and the average function \(A(x)\) are equal at the extremum.

• This means that the rate of change of the total function per unit is equal to the average amount per unit.
• The point where this occurs is significant because it usually indicates efficient resource allocation or other economic interpretations.

Understanding postextremum behavior helps predict patterns that might occur after reaching these extremal points, such as declining efficiency or increasing costs.

total functions

Total functions act as the foundation in many analytical works, describing the complete sum of a quantity, such as total cost or total production. Represented as \(T(x)\), these functions encapsulate the entire outcome produced or expended over a range \(x\).

The elasticity of a total function is a measure of how sensitive the function is to changes in \(x\). It is calculated as:
\[ E = \frac{x\frac{dT}{dx}}{T(x)} \]
At points where the average function reaches an extremum, the elasticity is exactly 1, indicating unit elasticity.

• This is crucial in identifying linear proportionality where percentage change in total is equal to that in \(x\).
• Provides insights into the behavior of the function with respect to scale.
• Helps in strategic decision-making like pricing, production planning, and resource allocation when total function hits crucial points.

Understanding total functions and their derivatives, such as marginal functions, is vital for optimizing economic processes and mathematical analysis.