Question

Which of the following are monotonic transformations? \((1) u=2 v-13\) (2) \(u=-1 / v^{2} ;\) (3) \(u=1 / v^{2}\) \((4) u=\ln v\) \((5) u=-e^{-v}\) \((6) u=v^{2}\) \((7) u=v^{2}\) for \(v>0 ;(8) u=v^{2}\) for \(v<0\)

Short answer

Functions 1, 2, 3, 4, 5, 7, and 8 are monotonic.

Step-by-step solution

Understand Monotonic Transformations

A function \( u = f(v) \) is called a monotonic transformation if it is either entirely non-increasing or non-decreasing on its domain. This means that as \( v \) increases, \( u \) can either consistently increase or decrease, but not switch between the two.

Analyze Function 1: \( u = 2v - 13 \)

This function is linear with a positive slope. As \( v \) increases, \( u \) increases as well. Thus, it is a monotonic transformation.

Analyze Function 2: \( u = -1 / v^2 \)

This function is a rational function that decreases as \( v \) increases for all non-zero \( v \). Therefore, it is a monotonic transformation.

Analyze Function 3: \( u = 1 / v^2 \)

This function is similar to function 2, but it increases as \( v \) moves away from zero. This function is also monotonic in its domain of \( v eq 0 \).

Analyze Function 4: \( u = \ln v \)

The natural logarithm is defined only for \( v > 0 \), and \( \ln v \) is an increasing function over this domain. Hence, it is a monotonic transformation.

Analyze Function 5: \( u = -e^{-v} \)

The function \( -e^{-v} \) decreases as \( v \) increases, so it is a monotonic transformation.

Analyze Function 6: \( u = v^2 \)

The function \( u = v^2 \) is not monotonic over the entire real line since it decreases for \( v < 0 \) and increases for \( v > 0 \).

Analyze Function 7: \( u = v^2 \) for \( v > 0 \)

When restricted to \( v > 0 \), \( u = v^2 \) is an increasing function, thus it is monotonic on this domain.

Analyze Function 8: \( u = v^2 \) for \( v < 0 \)

Within this domain, \( v < 0 \), the function is decreasing as \( v \) becomes less negative. Hence, this is a monotonic transformation on this domain.

Key concepts

Linear Functions

Linear functions are fundamental in mathematics and are often used to model real-world phenomena. A linear function can be expressed in the form \( u = av + b \). Here, \( a \) is the slope of the line, and \( b \) is the y-intercept.

• The slope \( a \) determines the steepness and direction of the line. If \( a > 0 \), the function is increasing; if \( a < 0 \), it is decreasing.
• The y-intercept \( b \) is the point where the line crosses the y-axis.

The simplicity of linear functions makes them tools to identify monotonic transformations. For example, the linear function \( u = 2v - 13 \) has a positive slope, indicating that it is a monotonically increasing function. This means as \( v \) increases, \( u \) also increases consistently with no reversal in direction.

Rational Functions

Rational functions consist of ratios of polynomials. They are written in the form \( u = \frac{P(v)}{Q(v)} \), where both \( P(v) \) and \( Q(v) \) are polynomials.

• The domain of a rational function is determined by where the polynomial in the denominator \( Q(v) \) does not equal zero.
• Behavior can vary greatly within their domain due to asymptotes and intercepts.

For instance, consider \( u = -\frac{1}{v^2} \), a rational function where the numerator is constant and the denominator is a squared variable.

• As \( v \) moves away from zero, the function consistently decreases, making it a monotonic transformation.
• Similarly, \( u = \frac{1}{v^2} \) behaves monotonically but increases as \( v \) moves further from zero.

Rational functions can have interesting domain-specific monotonic behaviors depending on the polynomial powers and signs.

Natural Logarithm

The natural logarithm is a crucial function in calculus and economics. Written as \( \ln v \), it’s only defined for \( v > 0 \).

• The natural logarithm function represents the time needed to reach a certain level of continuous growth starting from an initial amount of 1.
• Its main attribute is that it increases gradually as \( v \) increases.

As a monotonic function, \( \ln v \) offers particular advantages in economics:

• It linearizes exponential growth models for easier analysis.
• In utility functions, it can describe diminishing returns, indicating that each slice of additional resource input results in a smaller increase in output.

Incorporated into models, the logarithm's monotonically increasing nature facilitates understanding and predicting trends.

Monotonicity in Economics

Monotonicity is a valuable concept in economics, helping to describe functions that are either non-decreasing or non-increasing.

• Understanding monotonic transformations enables economists to transform functions while preserving order relations.
• This is essential in utility functions, where economists want to transform utility representations without changing the preference order.

Two common examples related to economics are:

• The function \( u = -e^{-v} \) represents a decaying exponential that always decreases as \( v \) increases. This might be used to model decaying variables such as depreciation value.
• Squaring functions like \( u = v^2 \) behave differently over domains. While not monotonic over all real numbers, parting the domain, such as \( v > 0 \), results in monotonic behavior relevant in specific models.

By applying monotonicity, economists can manage the complexity of economic models, optimizing decision processes and improving predictive accuracy.