Question

Show that the function \(V=A e^{r t}\) (with \(A, f>0\) ) and the function \(A=V e^{-r t}\) (with \(V, r>0)\) are both strictly monotonic, but in opposite directions, and that they are both strictly convex in shape (cf. Exercise \(10.2-5\) ).

Short answer

The function \( V = A e^{rt} \) is strictly increasing and convex, while \( A = V e^{-rt} \) is strictly decreasing and convex.

Step-by-step solution

Understanding the Functions

The first function given is \( V = A e^{rt} \). Here, \( A \) is a constant, and \( e^{rt} \) is the exponential growth component because the base \( e \) is greater than 1 when \( r > 0 \). The second function is \( A = V e^{-rt} \), where \( V \) is a constant and the function includes \( e^{-rt} \), which represents exponential decay due to the negative exponent.

Determine Monotonicity for \( V = A e^{rt} \)

Compute the derivative with respect to \( t \): \( \frac{dV}{dt} = A re^{rt} \). Since \( A > 0 \) and \( r > 0 \), the derivative \( A re^{rt} > 0 \) for all \( t \). Therefore, \( V \) is a strictly increasing function.

Determine Monotonicity for \( A = V e^{-rt} \)

Compute the derivative with respect to \( t \): \( \frac{dA}{dt} = -V re^{-rt} \). Since \( V > 0 \) and \( r > 0 \), the derivative \( -V re^{-rt} < 0 \) for all \( t \). Therefore, \( A \) is a strictly decreasing function.

Determine Convexity for \( V = A e^{rt} \)

Compute the second derivative: \( \frac{d^2V}{dt^2} = A r^2 e^{rt} \). Since \( A > 0 \) and \( r > 0 \), \( A r^2 e^{rt} > 0 \) for all \( t \). Therefore, \( V \) is strictly convex.

Determine Convexity for \( A = V e^{-rt} \)

Compute the second derivative: \( \frac{d^2A}{dt^2} = V r^2 e^{-rt} \). Since \( V > 0 \) and \( r > 0 \), \( V r^2 e^{-rt} > 0 \) for all \( t \). Therefore, \( A \) is strictly convex.

Key concepts

Monotonic Functions

Monotonic functions are mathematical functions that are either entirely non-increasing or non-decreasing over an interval. This means that as the input increases, the output can only go in one direction—either up or down. In economics, understanding monotonicity helps to predict whether a variable, like revenue or cost, increases or decreases in response to changes in other variables such as time or quantity.
To determine if a function is monotonic, we look at its first derivative. If the derivative is always positive, the function increases with time, showing strictly increasing behavior. If the derivative is always negative, the function decreases, showing strictly decreasing behavior.
In our example, the function \( V = A e^{rt} \) is strictly increasing because its derivative \( \frac{dV}{dt} = A re^{rt} \) is positive. Conversely, the function \( A = V e^{-rt} \) is strictly decreasing since its derivative \( \frac{dA}{dt} = -V re^{-rt} \) is negative.
Understanding these behaviors helps economists and analysts in forecasting future trends and making informed decisions.

Convexity

Convexity in functions refers to the shape of the graph and its bending towards one direction. A function is convex if a line segment joining any two points on its graph lies above or on the graph. In simpler terms, a convex function curves "upward."
Determining whether a function is convex involves analyzing the second derivative. If the second derivative is positive at all points within the interval, the function is convex. This property indicates that the rate of increase of the function itself is increasing, making it "bend" upwards more and more.
In our case, both functions \( V = A e^{rt} \) and \( A = V e^{-rt} \) are strictly convex. The second derivative of \( V = A e^{rt} \) is \( \frac{d^2V}{dt^2} = A r^2 e^{rt} \), which is positive, affirming its convexity. Similarly, \( A = V e^{-rt} \) has its second derivative \( \frac{d^2A}{dt^2} = V r^2 e^{-rt} \) positive as well, ensuring both functions have an upward bending curve.
In economics, convexity can describe scenarios like increasing returns to scale, where each additional unit produced contributes more to total output than the previous one.

Exponential Growth and Decay

Exponential growth and decay describe processes that increase or decrease at rates proportional to their current value. This behavior is modeled using exponential functions, where the base of the exponent dictates whether the function exhibits growth or decay.
An exponential growth model is characterized by a positive rate of growth. The function \( V = A e^{rt} \), embodies this as it grows over time since the base \( e \), in combination with a positive exponent \( r \), multiplies itself, causing \( V \) to increase rapidly.
Alternatively, exponential decay involves a negative rate, where the function decreases over time. The function \( A = V e^{-rt} \) represents decay with its negative exponent, gradually reducing \( A \) as time progresses.
In economics, recognizing exponential growth helps to understand phenomena like compounding interest or inflation over time, while exponential decay might describe decreasing depreciation values of assets. Understanding these models is crucial for making sound financial forecasts and decisions.